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<h2 class="titleHead">Sample Paper for the amsmath Package<br />
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File name: testmath.tex</h2>
20
<div class="author" align="center"><span
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class="cmr-12">American Mathematical Society</span></div>
23
<div class="date" align="center"><span
24
class="cmr-12">Version 2.0, 1999/11/15</span></div>
26
<h3 class="sectionHead"><span class="titlemark">1</span> <a
27
name="x1-10001"></a>Introduction</h3>
28
<!--l. 147--><p class="noindent">This paper contains examples of various features from <span
29
class="cmsy-10"><!--span
30
class="htf-calligraphy"-->A<!--/span--></span><span
31
class="cmsy-10"><!--span
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class="htf-calligraphy"-->M<!--/span--></span> <span
33
class="cmsy-10"><!--span
34
class="htf-calligraphy"-->S<!--/span--></span>-<span class="LATEX">L<span class="A">A</span><span class="TEX">T<span
35
class="E">E</span>X</span></span>.
37
<h3 class="sectionHead"><span class="titlemark">2</span> <a
38
name="x1-20002"></a>Enumeration of Hamiltonian paths in a graph</h3>
39
<!--l. 151--><p class="noindent">Let <!--l. 151--><math
40
xmlns="http://www.w3.org/1998/Math/MathML"
41
mode="inline"> <mi class="mathbf">A</mi> <mo
42
class="MathClass-rel">=</mo> <mrow><mo
43
class="MathClass-open">(</mo><msub
45
class="MathClass-ord">a</mi><mrow
47
class="MathClass-ord">i</mi><mi
48
class="MathClass-ord">j</mi></mrow></msub
50
class="MathClass-close">)</mo></mrow></math> be the adjacency matrix
51
of graph <!--l. 151--><math
52
xmlns="http://www.w3.org/1998/Math/MathML"
54
class="MathClass-ord">G</mi></math>. The corresponding
55
Kirchhoff matrix <!--l. 152--><math
56
xmlns="http://www.w3.org/1998/Math/MathML"
57
mode="inline"> <mi class="mathbf">K</mi> <mo
58
class="MathClass-rel">=</mo> <mrow><mo
59
class="MathClass-open">(</mo><msub
61
class="MathClass-ord">k</mi><mrow
63
class="MathClass-ord">i</mi><mi
64
class="MathClass-ord">j</mi></mrow></msub
66
class="MathClass-close">)</mo></mrow></math>
67
is obtained from <!--l. 153--><math
68
xmlns="http://www.w3.org/1998/Math/MathML"
69
mode="inline"> <mi class="mathbf">A</mi></math>
70
by replacing in <!--l. 153--><math
71
xmlns="http://www.w3.org/1998/Math/MathML"
73
class="MathClass-bin">−</mo><mi class="mathbf">A</mi></math>
74
each diagonal entry by the degree of its corresponding vertex; i.e., the <!--l. 154--><math
75
xmlns="http://www.w3.org/1998/Math/MathML"
78
class="MathClass-ord">i</mi></math>th
79
diagonal entry is identified with the degree of the <!--l. 155--><math
80
xmlns="http://www.w3.org/1998/Math/MathML"
83
class="MathClass-ord">i</mi></math>th
84
vertex. It is well known that </p><table class="equation"><tr><td>
87
xmlns="http://www.w3.org/1998/Math/MathML"
90
class="equation"><mtr><mtd>
92
> det</mo> <mi class="mathbf">K</mi><mrow><mo
93
class="MathClass-open">(</mo><mi
94
class="MathClass-ord">i</mi><mo
95
class="MathClass-rel">∣</mo><mi
96
class="MathClass-ord">i</mi><mo
97
class="MathClass-close">)</mo></mrow> <mo
98
class="MathClass-rel">=</mo> <mrow
99
class="text"><mtext > the number of spanning trees of </mtext><mrow
101
class="MathClass-ord">G</mi></mrow><mtext ></mtext></mrow><mo
102
class="MathClass-punc">,</mo> <mspace width="1em" class="quad"/><mi
103
class="MathClass-ord">i</mi> <mo
104
class="MathClass-rel">=</mo> <mn
105
class="MathClass-ord">1</mn><mo
106
class="MathClass-punc">,</mo> <mo>…</mo><mo
107
class="MathClass-punc">,</mo> <mi
108
class="MathClass-ord">n</mi></mtd><mtd><mspace
109
id="x1-2001r1" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
111
<!--l. 159--><p class="nopar"></p></td><td width="5%">(1)</td></tr></table>
112
where <!--l. 160--><math
113
xmlns="http://www.w3.org/1998/Math/MathML"
114
mode="inline"> <mi class="mathbf">K</mi><mrow><mo
115
class="MathClass-open">(</mo><mi
116
class="MathClass-ord">i</mi><mo
117
class="MathClass-rel">∣</mo><mi
118
class="MathClass-ord">i</mi><mo
119
class="MathClass-close">)</mo></mrow></math> is the <!--l. 160--><math
120
xmlns="http://www.w3.org/1998/Math/MathML"
123
class="MathClass-ord">i</mi></math>th principal
124
submatrix of <!--l. 161--><math
125
xmlns="http://www.w3.org/1998/Math/MathML"
126
mode="inline"> <mi class="mathbf">K</mi></math>.
129
class="verbatim"><tr class="verbatim"><td
130
class="verbatim"><pre class="verbatim">
131
 \det\mathbf{K}(i|i)=\text{ the number of spanning trees of $G$},
134
<!--l. 166--><p class="indent"> Let <!--l. 166--><math
135
xmlns="http://www.w3.org/1998/Math/MathML"
138
class="MathClass-ord">C</mi><mrow
140
class="MathClass-ord">i</mi><mrow><mo
141
class="MathClass-open">(</mo><mi
142
class="MathClass-ord">j</mi><mo
143
class="MathClass-close">)</mo></mrow></mrow></msub
144
></math> be the set of
145
graphs obtained from <!--l. 166--><math
146
xmlns="http://www.w3.org/1998/Math/MathML"
148
class="MathClass-ord">G</mi></math>
149
by attaching edge <!--l. 167--><math
150
xmlns="http://www.w3.org/1998/Math/MathML"
151
mode="inline"> <mrow><mo
152
class="MathClass-open">(</mo><msub
154
class="MathClass-ord">v</mi><mrow
156
class="MathClass-ord">i</mi></mrow></msub
159
class="MathClass-ord">v</mi><mrow
161
class="MathClass-ord">j</mi></mrow></msub
163
class="MathClass-close">)</mo></mrow></math> to
164
each spanning tree of <!--l. 167--><math
165
xmlns="http://www.w3.org/1998/Math/MathML"
167
class="MathClass-ord">G</mi></math>.
168
Denote by <!--l. 167--><math
169
xmlns="http://www.w3.org/1998/Math/MathML"
172
class="MathClass-ord">C</mi><mrow
174
class="MathClass-ord">i</mi></mrow></msub
176
class="MathClass-rel">=</mo><msub
178
class="MathClass-op">⋃</mo>
181
class="MathClass-ord">j</mi></mrow></msub
184
class="MathClass-ord">C</mi><mrow
186
class="MathClass-ord">i</mi><mrow><mo
187
class="MathClass-open">(</mo><mi
188
class="MathClass-ord">j</mi><mo
189
class="MathClass-close">)</mo></mrow></mrow></msub
191
It is obvious that the collection of Hamiltonian cycles is a subset of <!--l. 169--><math
192
xmlns="http://www.w3.org/1998/Math/MathML"
196
class="MathClass-ord">C</mi><mrow
198
class="MathClass-ord">i</mi></mrow></msub
199
></math>. Note that the
200
cardinality of <!--l. 169--><math
201
xmlns="http://www.w3.org/1998/Math/MathML"
204
class="MathClass-ord">C</mi><mrow
206
class="MathClass-ord">i</mi></mrow></msub
208
is <!--l. 169--><math
209
xmlns="http://www.w3.org/1998/Math/MathML"
212
class="MathClass-ord">k</mi><mrow
214
class="MathClass-ord">i</mi><mi
215
class="MathClass-ord">i</mi></mrow></msub
217
> det</mo> <mi class="mathbf">K</mi><mrow><mo
218
class="MathClass-open">(</mo><mi
219
class="MathClass-ord">i</mi><mo
220
class="MathClass-rel">∣</mo><mi
221
class="MathClass-ord">i</mi><mo
222
class="MathClass-close">)</mo></mrow></math>.
223
Let <!--l. 170--><math
224
xmlns="http://www.w3.org/1998/Math/MathML"
225
mode="inline"> <munderover
228
class="MathClass-ord">X</mi></mrow><mrow
230
></mrow></munderover> <mo
231
class="MathClass-rel">=</mo> <mrow><mo
232
class="MathClass-open">{</mo><msub
236
class="MathClass-ord">x</mi></mrow><mrow
239
class="MathClass-ord">̂</mi></mrow></munderover><mrow
241
class="MathClass-ord">1</mn></mrow></msub
243
class="MathClass-punc">,</mo> <mo
244
class="MathClass-op">…</mo><mo
245
class="MathClass-punc">,</mo><msub
249
class="MathClass-ord">x</mi></mrow><mrow
252
class="MathClass-ord">̂</mi></mrow></munderover><mrow
254
class="MathClass-ord">n</mi></mrow></msub
256
class="MathClass-close">}</mo></mrow></math>.
260
class="verbatim"><tr class="verbatim"><td
261
class="verbatim"><pre class="verbatim">
262
 $\wh X=\{\hat x_1,\dots,\hat x_n\}$
265
<!--l. 174--><p class="indent"> Define multiplication for the elements of <!--l. 174--><math
266
xmlns="http://www.w3.org/1998/Math/MathML"
271
class="MathClass-ord">X</mi></mrow><mrow
273
></mrow></munderover></math> by
274
</p><table class="equation"><tr><td>
276
xmlns="http://www.w3.org/1998/Math/MathML"
279
class="equation"><mtr><mtd>
284
class="MathClass-ord">x</mi></mrow><mrow
287
class="MathClass-ord">̂</mi></mrow></munderover><mrow
289
class="MathClass-ord">i</mi></mrow></msub
294
class="MathClass-ord">x</mi></mrow><mrow
297
class="MathClass-ord">̂</mi></mrow></munderover><mrow
299
class="MathClass-ord">j</mi></mrow></msub
301
class="MathClass-rel">=</mo><msub
305
class="MathClass-ord">x</mi></mrow><mrow
308
class="MathClass-ord">̂</mi></mrow></munderover><mrow
310
class="MathClass-ord">j</mi></mrow></msub
315
class="MathClass-ord">x</mi></mrow><mrow
318
class="MathClass-ord">̂</mi></mrow></munderover><mrow
320
class="MathClass-ord">i</mi></mrow></msub
322
class="MathClass-punc">,</mo> <mspace width="1em" class="quad"/><msubsup
326
class="MathClass-ord">x</mi></mrow><mrow
329
class="MathClass-ord">̂</mi></mrow></munderover><mrow
331
class="MathClass-ord">i</mi></mrow><mrow
333
class="MathClass-ord">2</mn></mrow></msubsup
335
class="MathClass-rel">=</mo> <mn
336
class="MathClass-ord">0</mn><mo
337
class="MathClass-punc">,</mo> <mspace width="1em" class="quad"/><mi
338
class="MathClass-ord">i</mi><mo
339
class="MathClass-punc">,</mo> <mi
340
class="MathClass-ord">j</mi> <mo
341
class="MathClass-rel">=</mo> <mn
342
class="MathClass-ord">1</mn><mo
343
class="MathClass-punc">,</mo> <mo
344
class="MathClass-op">…</mo><mo
345
class="MathClass-punc">,</mo> <mi
346
class="MathClass-ord">n</mi><mo
347
class="MathClass-punc">.</mo></mtd><mtd><mspace
348
id="x1-2002r2" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
350
<!--l. 178--><p class="nopar"></p></td><td width="5%">(2)</td></tr></table>
351
Let <!--l. 179--><math
352
xmlns="http://www.w3.org/1998/Math/MathML"
357
class="MathClass-ord">k</mi></mrow><mrow
360
class="MathClass-ord">̂</mi></mrow></munderover><mrow
362
class="MathClass-ord">i</mi><mi
363
class="MathClass-ord">j</mi></mrow></msub
365
class="MathClass-rel">=</mo> <msub
367
class="MathClass-ord">k</mi><mrow
369
class="MathClass-ord">i</mi><mi
370
class="MathClass-ord">j</mi></mrow></msub
375
class="MathClass-ord">x</mi></mrow><mrow
378
class="MathClass-ord">̂</mi></mrow></munderover><mrow
380
class="MathClass-ord">j</mi></mrow></msub
381
></math> and <!--l. 179--><math
382
xmlns="http://www.w3.org/1998/Math/MathML"
388
class="MathClass-ord">k</mi></mrow><mrow
391
class="MathClass-ord">̂</mi></mrow></munderover><mrow
394
class="MathClass-ord">i</mi><mi
395
class="MathClass-ord">j</mi></mrow></msub
397
class="MathClass-rel">=</mo> <mo
398
class="MathClass-bin">−</mo><msub
400
class="MathClass-op">∑</mo>
403
class="MathClass-ord">j</mi><mo
404
class="MathClass-rel">/</mo><mo
405
class="MathClass-rel">=</mo><mi
406
class="MathClass-ord">i</mi></mrow></msub
411
class="MathClass-ord">k</mi></mrow><mrow
414
class="MathClass-ord">̂</mi></mrow></munderover><mrow
416
class="MathClass-ord">i</mi><mi
417
class="MathClass-ord">j</mi></mrow></msub
418
></math>. Then the number of
419
Hamiltonian cycles <!--l. 180--><math
420
xmlns="http://www.w3.org/1998/Math/MathML"
423
class="MathClass-ord">H</mi><mrow
425
class="MathClass-ord">c</mi></mrow></msub
427
is given by the relation <span class="cite">[<a
428
href="#Xliuchow:formalsum">8</a>]</span> <table class="equation"><tr><td>
431
xmlns="http://www.w3.org/1998/Math/MathML"
434
class="equation"><mtr><mtd>
436
class="MathClass-open">(
439
class="MathClass-op">∏</mo>
442
class="MathClass-ord">j</mi> <mo
443
class="MathClass-rel">=</mo> <mn
444
class="MathClass-ord">1</mn></mrow><mrow
446
class="MathClass-ord">n</mi></mrow></msubsup
451
class="MathClass-ord">x</mi></mrow><mrow
454
class="MathClass-ord">̂</mi></mrow></munderover><mrow
456
class="MathClass-ord">j</mi></mrow></msub
458
class="MathClass-close">)
461
class="MathClass-ord">H</mi><mrow
463
class="MathClass-ord">c</mi></mrow></msub
465
class="MathClass-rel">=</mo> <mfrac><mrow
467
class="MathClass-ord">1</mn></mrow>
470
class="MathClass-ord">2</mn></mrow></mfrac><msub
474
class="MathClass-ord">k</mi></mrow><mrow
477
class="MathClass-ord">̂</mi></mrow></munderover><mrow
479
class="MathClass-ord">i</mi><mi
480
class="MathClass-ord">j</mi></mrow></msub
482
> det</mo> <munderover
484
><mi class="mathbf">K</mi></mrow><mrow
486
></mrow></munderover><mrow><mo
487
class="MathClass-open">(</mo><mi
488
class="MathClass-ord">i</mi><mo
489
class="MathClass-rel">∣</mo><mi
490
class="MathClass-ord">i</mi><mo
491
class="MathClass-close">)</mo></mrow><mo
492
class="MathClass-punc">,</mo> <mspace width="2em" class="qquad"/><mi
493
class="MathClass-ord">i</mi> <mo
494
class="MathClass-rel">=</mo> <mn
495
class="MathClass-ord">1</mn><mo
496
class="MathClass-punc">,</mo> <mo>…</mo><mo
497
class="MathClass-punc">,</mo> <mi
498
class="MathClass-ord">n</mi><mo
499
class="MathClass-punc">.</mo></mtd><mtd><mspace
500
id="x1-2003r3" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
502
<!--l. 185--><p class="nopar"></p></td><td width="5%">(3)</td></tr></table>
503
The task here is to express (<a
504
href="#x1-2003r3">3<!--tex4ht:ref: H-cycles--></a>) in a form free of any <!--l. 187--><math
505
xmlns="http://www.w3.org/1998/Math/MathML"
511
class="MathClass-ord">x</mi></mrow><mrow
514
class="MathClass-ord">̂</mi></mrow></munderover><mrow
517
class="MathClass-ord">i</mi></mrow></msub
518
></math>, <!--l. 188--><math
519
xmlns="http://www.w3.org/1998/Math/MathML"
522
class="MathClass-ord">i</mi> <mo
523
class="MathClass-rel">=</mo> <mn
524
class="MathClass-ord">1</mn><mo
525
class="MathClass-punc">,</mo> <mo
526
class="MathClass-op">…</mo><mo
527
class="MathClass-punc">,</mo> <mi
528
class="MathClass-ord">n</mi></math>. The
529
result also leads to the resolution of enumeration of Hamiltonian paths in a
531
<!--l. 191--><p class="indent"> It is well known that the enumeration of Hamiltonian cycles and paths in a complete graph <!--l. 192--><math
532
xmlns="http://www.w3.org/1998/Math/MathML"
536
class="MathClass-ord">K</mi><mrow
538
class="MathClass-ord">n</mi></mrow></msub
539
></math> and in a complete
540
bipartite graph <!--l. 192--><math
541
xmlns="http://www.w3.org/1998/Math/MathML"
544
class="MathClass-ord">K</mi><mrow
547
class="MathClass-ord">n</mi><mrow
549
class="MathClass-ord">1</mn></mrow></msub
552
class="MathClass-ord">n</mi><mrow
554
class="MathClass-ord">2</mn></mrow></msub
557
can only be found from <span
558
class="cmti-10">first combinatorial principles </span><span class="cite">[<a
559
href="#Xhapa:graphenum">4</a>]</span>. One wonders if
560
there exists a formula which can be used very efficiently to produce <!--l. 195--><math
561
xmlns="http://www.w3.org/1998/Math/MathML"
565
class="MathClass-ord">K</mi><mrow
567
class="MathClass-ord">n</mi></mrow></msub
568
></math> and <!--l. 195--><math
569
xmlns="http://www.w3.org/1998/Math/MathML"
573
class="MathClass-ord">K</mi><mrow
576
class="MathClass-ord">n</mi><mrow
578
class="MathClass-ord">1</mn></mrow></msub
581
class="MathClass-ord">n</mi><mrow
583
class="MathClass-ord">2</mn></mrow></msub
586
Recently, using Lagrangian methods, Goulden and Jackson have shown that <!--l. 196--><math
587
xmlns="http://www.w3.org/1998/Math/MathML"
591
class="MathClass-ord">H</mi><mrow
593
class="MathClass-ord">c</mi></mrow></msub
595
be expressed in terms of the determinant and permanent of the adjacency
596
matrix <span class="cite">[<a
597
href="#Xgouja:lagrmeth">3</a>]</span>. However, the formula of Goulden and Jackson determines neither <!--l. 199--><math
598
xmlns="http://www.w3.org/1998/Math/MathML"
602
class="MathClass-ord">K</mi><mrow
604
class="MathClass-ord">n</mi></mrow></msub
605
></math> nor <!--l. 199--><math
606
xmlns="http://www.w3.org/1998/Math/MathML"
610
class="MathClass-ord">K</mi><mrow
613
class="MathClass-ord">n</mi><mrow
615
class="MathClass-ord">1</mn></mrow></msub
618
class="MathClass-ord">n</mi><mrow
620
class="MathClass-ord">2</mn></mrow></msub
622
></math> effectively. In
623
this paper, using an algebraic method, we parametrize the adjacency matrix. The resulting
624
formula also involves the determinant and permanent, but it can easily be applied to <!--l. 202--><math
625
xmlns="http://www.w3.org/1998/Math/MathML"
629
class="MathClass-ord">K</mi><mrow
631
class="MathClass-ord">n</mi></mrow></msub
632
></math> and <!--l. 202--><math
633
xmlns="http://www.w3.org/1998/Math/MathML"
637
class="MathClass-ord">K</mi><mrow
640
class="MathClass-ord">n</mi><mrow
642
class="MathClass-ord">1</mn></mrow></msub
645
class="MathClass-ord">n</mi><mrow
647
class="MathClass-ord">2</mn></mrow></msub
649
></math>. In addition, we eliminate
650
the permanent from <!--l. 203--><math
651
xmlns="http://www.w3.org/1998/Math/MathML"
654
class="MathClass-ord">H</mi><mrow
656
class="MathClass-ord">c</mi></mrow></msub
658
and show that <!--l. 203--><math
659
xmlns="http://www.w3.org/1998/Math/MathML"
662
class="MathClass-ord">H</mi><mrow
664
class="MathClass-ord">c</mi></mrow></msub
666
can be represented by a determinantal function of multivariables, each variable with domain <!--l. 205--><math
667
xmlns="http://www.w3.org/1998/Math/MathML"
670
class="MathClass-open">{</mo><mn
671
class="MathClass-ord">0</mn><mo
672
class="MathClass-punc">,</mo> <mn
673
class="MathClass-ord">1</mn><mo
674
class="MathClass-close">}</mo></mrow></math>. Furthermore, we
675
show that <!--l. 205--><math
676
xmlns="http://www.w3.org/1998/Math/MathML"
679
class="MathClass-ord">H</mi><mrow
681
class="MathClass-ord">c</mi></mrow></msub
683
can be written by number of spanning trees of subgraphs.
684
Finally, we apply the formulas to a complete multigraph <!--l. 207--><math
685
xmlns="http://www.w3.org/1998/Math/MathML"
689
class="MathClass-ord">K</mi><mrow
692
class="MathClass-ord">n</mi><mrow
694
class="MathClass-ord">1</mn></mrow></msub
696
class="MathClass-op">…</mo><msub
698
class="MathClass-ord">n</mi><mrow
700
class="MathClass-ord">p</mi></mrow></msub
703
</p><!--l. 209--><p class="indent"> The conditions <!--l. 209--><math
704
xmlns="http://www.w3.org/1998/Math/MathML"
707
class="MathClass-ord">a</mi><mrow
709
class="MathClass-ord">i</mi><mi
710
class="MathClass-ord">j</mi></mrow></msub
712
class="MathClass-rel">=</mo> <msub
714
class="MathClass-ord">a</mi><mrow
716
class="MathClass-ord">j</mi><mi
717
class="MathClass-ord">i</mi></mrow></msub
721
xmlns="http://www.w3.org/1998/Math/MathML"
723
class="MathClass-ord">i</mi><mo
724
class="MathClass-punc">,</mo> <mi
725
class="MathClass-ord">j</mi> <mo
726
class="MathClass-rel">=</mo> <mn
727
class="MathClass-ord">1</mn><mo
728
class="MathClass-punc">,</mo> <mo
729
class="MathClass-op">…</mo><mo
730
class="MathClass-punc">,</mo> <mi
731
class="MathClass-ord">n</mi></math>, are
732
not required in this paper. All formulas can be extended to a digraph simply by multiplying <!--l. 211--><math
733
xmlns="http://www.w3.org/1998/Math/MathML"
737
class="MathClass-ord">H</mi><mrow
739
class="MathClass-ord">c</mi></mrow></msub
743
<h3 class="sectionHead"><span class="titlemark">3</span> <a
744
name="x1-30003"></a>Main Theorem</h3>
745
<div class="newtheorem">
746
<!--l. 216--><p class="noindent"><span class="head">
748
class="cmti-10">Notation.</span> </span>For <!--l. 216--><math
749
xmlns="http://www.w3.org/1998/Math/MathML"
751
class="MathClass-ord">p</mi><mo
752
class="MathClass-punc">,</mo> <mi
753
class="MathClass-ord">q</mi> <mo
754
class="MathClass-rel">∈</mo> <mi
755
class="MathClass-ord">P</mi></math>
756
and <!--l. 216--><math
757
xmlns="http://www.w3.org/1998/Math/MathML"
759
class="MathClass-ord">n</mi> <mo
760
class="MathClass-rel">∈</mo> <mi
761
class="MathClass-ord">ω</mi></math> we
762
write <!--l. 217--><math
763
xmlns="http://www.w3.org/1998/Math/MathML"
764
mode="inline"> <mrow><mo
765
class="MathClass-open">(</mo><mi
766
class="MathClass-ord">q</mi><mo
767
class="MathClass-punc">,</mo> <mi
768
class="MathClass-ord">n</mi><mo
769
class="MathClass-close">)</mo></mrow> <mo
770
class="MathClass-rel">≤</mo> <mrow><mo
771
class="MathClass-open">(</mo><mi
772
class="MathClass-ord">p</mi><mo
773
class="MathClass-punc">,</mo> <mi
774
class="MathClass-ord">n</mi><mo
775
class="MathClass-close">)</mo></mrow></math>
776
if <!--l. 217--><math
777
xmlns="http://www.w3.org/1998/Math/MathML"
779
class="MathClass-ord">q</mi> <mo
780
class="MathClass-rel">≤</mo> <mi
781
class="MathClass-ord">p</mi></math>
782
and <!--l. 217--><math
783
xmlns="http://www.w3.org/1998/Math/MathML"
786
class="MathClass-ord">A</mi><mrow
788
class="MathClass-ord">q</mi><mo
789
class="MathClass-punc">,</mo><mi
790
class="MathClass-ord">n</mi></mrow></msub
792
class="MathClass-rel">=</mo> <msub
794
class="MathClass-ord">A</mi><mrow
796
class="MathClass-ord">p</mi><mo
797
class="MathClass-punc">,</mo><mi
798
class="MathClass-ord">n</mi></mrow></msub
803
class="verbatim"><tr class="verbatim"><td
804
class="verbatim"><pre class="verbatim">
805
 \begin{notation} For $p,q\in P$ and $n\in\omega$
807
 \end{notation}
811
<!--l. 225--><p class="indent"> Let <!--l. 225--><math
812
xmlns="http://www.w3.org/1998/Math/MathML"
813
mode="inline"> <mi class="mathbf">B</mi> <mo
814
class="MathClass-rel">=</mo> <mrow><mo
815
class="MathClass-open">(</mo><msub
817
class="MathClass-ord">b</mi><mrow
819
class="MathClass-ord">i</mi><mi
820
class="MathClass-ord">j</mi></mrow></msub
822
class="MathClass-close">)</mo></mrow></math> be
823
an <!--l. 225--><math
824
xmlns="http://www.w3.org/1998/Math/MathML"
826
class="MathClass-ord">n</mi> <mo
827
class="MathClass-bin">×</mo> <mi
828
class="MathClass-ord">n</mi></math> matrix.
829
Let <!--l. 225--><math
830
xmlns="http://www.w3.org/1998/Math/MathML"
831
mode="inline"> <mi class="mathbf">n</mi> <mo
832
class="MathClass-rel">=</mo> <mrow><mo
833
class="MathClass-open">{</mo><mn
834
class="MathClass-ord">1</mn><mo
835
class="MathClass-punc">,</mo> <mo
836
class="MathClass-op">…</mo><mo
837
class="MathClass-punc">,</mo> <mi
838
class="MathClass-ord">n</mi><mo
839
class="MathClass-close">}</mo></mrow></math>.
840
Using the properties of (<a
841
href="#x1-2002r2">2<!--tex4ht:ref: multdef--></a>), it is readily seen that
843
<div class="newtheorem">
844
<!--l. 229--><p class="noindent"><span class="head">
846
name="x1-3001r1"></a>
848
class="cmbx-10">Lemma 3.1.</span> </span> </p><table class="equation"><tr><td>
850
xmlns="http://www.w3.org/1998/Math/MathML"
853
class="equation"><mtr><mtd>
856
class="MathClass-op">∏</mo>
859
class="MathClass-ord">i</mi><mo
860
class="MathClass-rel">∈</mo><mi class="mathbf">n</mi></mrow></msub
862
class="MathClass-open">(
865
class="MathClass-op">∑</mo>
868
class="MathClass-ord">j</mi><mo
869
class="MathClass-rel">∈</mo><mi class="mathbf">n</mi></mrow></msub
872
class="MathClass-ord">b</mi><mrow
874
class="MathClass-ord">i</mi><mi
875
class="MathClass-ord">j</mi></mrow></msub
880
class="MathClass-ord">x</mi></mrow><mrow
883
class="MathClass-ord">̂</mi></mrow></munderover><mrow
885
class="MathClass-ord">i</mi></mrow></msub
887
class="MathClass-close">)
889
class="MathClass-rel">=</mo><mrow><mo
890
class="MathClass-open">(
893
class="MathClass-op">∏</mo>
896
class="MathClass-ord">i</mi><mo
897
class="MathClass-rel">∈</mo><mi class="mathbf">n</mi></mrow></msub
902
class="MathClass-ord">x</mi></mrow><mrow
905
class="MathClass-ord">̂</mi></mrow></munderover><mrow
907
class="MathClass-ord">i</mi></mrow></msub
909
class="MathClass-close">)
911
class="MathClass-op"> per</mo><!--nolimits--> <mi class="mathbf">B</mi></mtd><mtd><mspace
912
id="x1-3002r4" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
914
<!--l. 234--><p class="nopar"></p></td><td width="5%">(4)</td></tr></table>
916
class="cmti-10">where </span><!--l. 235--><math
917
xmlns="http://www.w3.org/1998/Math/MathML"
919
class="MathClass-op">per</mo><!--nolimits--> <mi class="mathbf">B</mi></math> <span
920
class="cmti-10">is the</span>
922
class="cmti-10">permanent of </span><!--l. 235--><math
923
xmlns="http://www.w3.org/1998/Math/MathML"
924
mode="inline"> <mi class="mathbf">B</mi></math><span
925
class="cmti-10">.</span>
928
<!--l. 238--><p class="indent"> Let <!--l. 238--><math
929
xmlns="http://www.w3.org/1998/Math/MathML"
930
mode="inline"> <munderover
933
class="MathClass-ord">Y</mi> </mrow><mrow
935
></mrow></munderover> <mo
936
class="MathClass-rel">=</mo> <mrow><mo
937
class="MathClass-open">{</mo><msub
939
class="MathClass-ord">ŷ</mi><mrow
941
class="MathClass-ord">1</mn></mrow></msub
943
class="MathClass-punc">,</mo> <mo
944
class="MathClass-op">…</mo><mo
945
class="MathClass-punc">,</mo><msub
947
class="MathClass-ord">ŷ</mi><mrow
949
class="MathClass-ord">n</mi></mrow></msub
951
class="MathClass-close">}</mo></mrow></math>. Define multiplication
952
for the elements of <!--l. 239--><math
953
xmlns="http://www.w3.org/1998/Math/MathML"
954
mode="inline"> <munderover
957
class="MathClass-ord">Y</mi> </mrow><mrow
959
></mrow></munderover></math>
960
by </p><table class="equation"><tr><td>
962
xmlns="http://www.w3.org/1998/Math/MathML"
965
class="equation"><mtr><mtd>
968
class="MathClass-ord">ŷ</mi><mrow
970
class="MathClass-ord">i</mi></mrow></msub
973
class="MathClass-ord">ŷ</mi><mrow
975
class="MathClass-ord">j</mi></mrow></msub
977
class="MathClass-bin">+</mo><msub
979
class="MathClass-ord">ŷ</mi><mrow
981
class="MathClass-ord">j</mi></mrow></msub
984
class="MathClass-ord">ŷ</mi><mrow
986
class="MathClass-ord">i</mi></mrow></msub
988
class="MathClass-rel">=</mo> <mn
989
class="MathClass-ord">0</mn><mo
990
class="MathClass-punc">,</mo> <mspace width="1em" class="quad"/><mi
991
class="MathClass-ord">i</mi><mo
992
class="MathClass-punc">,</mo> <mi
993
class="MathClass-ord">j</mi> <mo
994
class="MathClass-rel">=</mo> <mn
995
class="MathClass-ord">1</mn><mo
996
class="MathClass-punc">,</mo> <mo
997
class="MathClass-op">…</mo><mo
998
class="MathClass-punc">,</mo> <mi
999
class="MathClass-ord">n</mi><mo
1000
class="MathClass-punc">.</mo></mtd><mtd><mspace
1001
id="x1-3003r5" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
1003
<!--l. 242--><p class="nopar"></p></td><td width="5%">(5)</td></tr></table>
1004
Then, it follows that <div class="newtheorem">
1005
<!--l. 244--><p class="noindent"><span class="head">
1007
name="x1-3004r2"></a>
1009
class="cmbx-10">Lemma 3.2.</span> </span> </p><table class="equation"><tr><td>
1012
xmlns="http://www.w3.org/1998/Math/MathML"
1015
class="equation"><mtr><mtd>
1018
class="MathClass-op">∏</mo>
1021
class="MathClass-ord">i</mi><mo
1022
class="MathClass-rel">∈</mo><mi class="mathbf">n</mi></mrow></msub
1024
class="MathClass-open">(
1027
class="MathClass-op">∑</mo>
1030
class="MathClass-ord">j</mi><mo
1031
class="MathClass-rel">∈</mo><mi class="mathbf">n</mi></mrow></msub
1034
class="MathClass-ord">b</mi><mrow
1036
class="MathClass-ord">i</mi><mi
1037
class="MathClass-ord">j</mi></mrow></msub
1040
class="MathClass-ord">ŷ</mi><mrow
1042
class="MathClass-ord">j</mi></mrow></msub
1044
class="MathClass-close">)
1046
class="MathClass-rel">=</mo><mrow><mo
1047
class="MathClass-open">(
1050
class="MathClass-op">∏</mo>
1053
class="MathClass-ord">i</mi><mo
1054
class="MathClass-rel">∈</mo><mi class="mathbf">n</mi></mrow></msub
1057
class="MathClass-ord">ŷ</mi><mrow
1059
class="MathClass-ord">i</mi></mrow></msub
1061
class="MathClass-close">)
1063
> det</mo> <mi class="mathbf">B</mi><mo
1064
class="MathClass-punc">.</mo></mtd><mtd><mspace
1065
id="x1-3005r6" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
1067
<!--l. 249--><p class="nopar"></p></td><td width="5%">(6)</td></tr></table>
1069
<!--l. 252--><p class="indent"> Note that all basic properties of determinants are direct consequences of Lemma
1071
href="#x1-3004r2">3.2<!--tex4ht:ref: lem-det--></a>. Write </p><table class="equation"><tr><td>
1073
xmlns="http://www.w3.org/1998/Math/MathML"
1076
class="equation"><mtr><mtd>
1079
class="MathClass-op">∑</mo>
1082
class="MathClass-ord">j</mi><mo
1083
class="MathClass-rel">∈</mo><mi class="mathbf">n</mi></mrow></msub
1086
class="MathClass-ord">b</mi><mrow
1088
class="MathClass-ord">i</mi><mi
1089
class="MathClass-ord">j</mi></mrow></msub
1092
class="MathClass-ord">ŷ</mi><mrow
1094
class="MathClass-ord">j</mi></mrow></msub
1096
class="MathClass-rel">=</mo><msub
1098
class="MathClass-op">∑</mo>
1101
class="MathClass-ord">j</mi><mo
1102
class="MathClass-rel">∈</mo><mi class="mathbf">n</mi></mrow></msub
1105
class="MathClass-ord">b</mi><mrow
1107
class="MathClass-ord">i</mi><mi
1108
class="MathClass-ord">j</mi></mrow><mrow
1110
class="MathClass-open">(</mo><mi
1111
class="MathClass-ord">λ</mi><mo
1112
class="MathClass-close">)</mo></mrow></mrow></msubsup
1115
class="MathClass-ord">ŷ</mi><mrow
1117
class="MathClass-ord">j</mi></mrow></msub
1119
class="MathClass-bin">+</mo> <mrow><mo
1120
class="MathClass-open">(</mo><msub
1122
class="MathClass-ord">b</mi><mrow
1124
class="MathClass-ord">i</mi><mi
1125
class="MathClass-ord">i</mi></mrow></msub
1127
class="MathClass-bin">−</mo> <msub
1129
class="MathClass-ord">λ</mi><mrow
1131
class="MathClass-ord">i</mi></mrow></msub
1133
class="MathClass-close">)</mo></mrow><msub
1135
class="MathClass-ord">ŷ</mi><mrow
1137
class="MathClass-ord">i</mi></mrow></msub
1139
class="MathClass-ord">ŷ</mi></mtd><mtd><mspace
1140
id="x1-3006r7" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
1142
<!--l. 257--><p class="nopar"></p></td><td width="5%">(7)</td></tr></table>
1143
where <table class="equation"><tr><td>
1146
xmlns="http://www.w3.org/1998/Math/MathML"
1149
class="equation"><mtr><mtd>
1152
class="MathClass-ord">b</mi><mrow
1154
class="MathClass-ord">i</mi><mi
1155
class="MathClass-ord">i</mi></mrow><mrow
1157
class="MathClass-open">(</mo><mi
1158
class="MathClass-ord">λ</mi><mo
1159
class="MathClass-close">)</mo></mrow></mrow></msubsup
1161
class="MathClass-rel">=</mo> <msub
1163
class="MathClass-ord">λ</mi><mrow
1165
class="MathClass-ord">i</mi></mrow></msub
1167
class="MathClass-punc">,</mo> <mspace width="1em" class="quad"/><msubsup
1169
class="MathClass-ord">b</mi><mrow
1171
class="MathClass-ord">i</mi><mi
1172
class="MathClass-ord">j</mi></mrow><mrow
1174
class="MathClass-open">(</mo><mi
1175
class="MathClass-ord">λ</mi><mo
1176
class="MathClass-close">)</mo></mrow></mrow></msubsup
1178
class="MathClass-rel">=</mo> <msub
1180
class="MathClass-ord">b</mi><mrow
1182
class="MathClass-ord">i</mi><mi
1183
class="MathClass-ord">j</mi></mrow></msub
1185
class="MathClass-punc">,</mo> <mspace width="1em" class="quad"/><mi
1186
class="MathClass-ord">i</mi> <mo
1187
class="MathClass-rel">/</mo><mo
1188
class="MathClass-rel">=</mo> <mi
1189
class="MathClass-ord">j</mi><mo
1190
class="MathClass-punc">.</mo></mtd><mtd><mspace
1191
id="x1-3007r8" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
1193
<!--l. 262--><p class="nopar"></p></td><td width="5%">(8)</td></tr></table>
1194
Let <!--l. 263--><math
1195
xmlns="http://www.w3.org/1998/Math/MathML"
1196
mode="inline"> <msup
1197
><mi class="mathbf">B</mi><mrow
1199
class="MathClass-open">(</mo><mi
1200
class="MathClass-ord">λ</mi><mo
1201
class="MathClass-close">)</mo></mrow></mrow></msup
1203
class="MathClass-rel">=</mo> <mrow><mo
1204
class="MathClass-open">(</mo><msubsup
1206
class="MathClass-ord">b</mi><mrow
1208
class="MathClass-ord">i</mi><mi
1209
class="MathClass-ord">j</mi></mrow><mrow
1211
class="MathClass-open">(</mo><mi
1212
class="MathClass-ord">λ</mi><mo
1213
class="MathClass-close">)</mo></mrow></mrow></msubsup
1215
class="MathClass-close">)</mo></mrow></math>.
1217
href="#x1-3005r6">6<!--tex4ht:ref: detprod--></a>) and (<a
1218
href="#x1-3006r7">7<!--tex4ht:ref: sum-bij--></a>), it is straightforward to show the following result: <div class="newtheorem">
1219
<!--l. 267--><p class="noindent"><span class="head">
1221
name="x1-3008r3"></a>
1223
class="cmbx-10">Theorem 3.3.</span> </span> </p><table class="equation"><tr><td>
1225
xmlns="http://www.w3.org/1998/Math/MathML"
1228
class="equation"><mtr><mtd>
1230
> det</mo> <mi class="mathbf">B</mi> <mo
1231
class="MathClass-rel">=</mo><msubsup
1233
class="MathClass-op">∑</mo>
1236
class="MathClass-ord">l</mi> <mo
1237
class="MathClass-rel">=</mo> <mn
1238
class="MathClass-ord">0</mn></mrow><mrow
1240
class="MathClass-ord">n</mi></mrow></msubsup
1243
class="MathClass-op">∑</mo>
1247
class="MathClass-ord">I</mi><mrow
1249
class="MathClass-ord">l</mi></mrow></msub
1251
class="MathClass-rel">⊆</mo><mi
1252
class="MathClass-ord">n</mi></mrow></msub
1255
class="MathClass-op">∏</mo>
1258
class="MathClass-ord">i</mi><mo
1259
class="MathClass-rel">∈</mo><msub
1261
class="MathClass-ord">I</mi><mrow
1263
class="MathClass-ord">l</mi></mrow></msub
1266
class="MathClass-open">(</mo><msub
1268
class="MathClass-ord">b</mi><mrow
1270
class="MathClass-ord">i</mi><mi
1271
class="MathClass-ord">i</mi></mrow></msub
1273
class="MathClass-bin">−</mo> <msub
1275
class="MathClass-ord">λ</mi><mrow
1277
class="MathClass-ord">i</mi></mrow></msub
1279
class="MathClass-close">)</mo></mrow><mo
1281
> <mi class="mathbf">B</mi><mrow
1283
class="MathClass-open">(</mo><mi
1284
class="MathClass-ord">λ</mi><mo
1285
class="MathClass-close">)</mo></mrow></mrow></msup
1287
class="MathClass-open">(</mo><msub
1289
class="MathClass-ord">I</mi><mrow
1292
class="MathClass-ord">l</mi></mrow></msub
1294
class="MathClass-rel">∣</mo><msub
1296
class="MathClass-ord">I</mi><mrow
1298
class="MathClass-ord">l</mi></mrow></msub
1300
class="MathClass-close">)</mo></mrow><mo
1301
class="MathClass-punc">,</mo> </mtd><mtd><mspace
1302
id="x1-3009r9" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
1304
<!--l. 273--><p class="nopar"></p></td><td width="5%">(9)</td></tr></table>
1306
class="cmti-10">where </span><!--l. 274--><math
1307
xmlns="http://www.w3.org/1998/Math/MathML"
1308
mode="inline"> <msub
1310
class="MathClass-ord">I</mi><mrow
1312
class="MathClass-ord">l</mi></mrow></msub
1314
class="MathClass-rel">=</mo> <mrow><mo
1315
class="MathClass-open">{</mo><msub
1317
class="MathClass-ord">i</mi><mrow
1319
class="MathClass-ord">1</mn></mrow></msub
1321
class="MathClass-punc">,</mo> <mo
1322
class="MathClass-op">…</mo><mo
1323
class="MathClass-punc">,</mo> <msub
1325
class="MathClass-ord">i</mi><mrow
1327
class="MathClass-ord">l</mi></mrow></msub
1329
class="MathClass-close">}</mo></mrow></math> <span
1330
class="cmti-10">and </span><!--l. 274--><math
1331
xmlns="http://www.w3.org/1998/Math/MathML"
1334
><mi class="mathbf">B</mi><mrow
1336
class="MathClass-open">(</mo><mi
1337
class="MathClass-ord">λ</mi><mo
1338
class="MathClass-close">)</mo></mrow></mrow></msup
1340
class="MathClass-open">(</mo><msub
1342
class="MathClass-ord">I</mi><mrow
1344
class="MathClass-ord">l</mi></mrow></msub
1346
class="MathClass-rel">∣</mo><msub
1348
class="MathClass-ord">I</mi><mrow
1350
class="MathClass-ord">l</mi></mrow></msub
1352
class="MathClass-close">)</mo></mrow></math> <span
1353
class="cmti-10">is the principal submatrix</span>
1355
class="cmti-10">obtained from </span><!--l. 275--><math
1356
xmlns="http://www.w3.org/1998/Math/MathML"
1357
mode="inline"> <msup
1358
><mi class="mathbf">B</mi><mrow
1360
class="MathClass-open">(</mo><mi
1361
class="MathClass-ord">λ</mi><mo
1362
class="MathClass-close">)</mo></mrow></mrow></msup
1365
class="cmti-10">by deleting its </span><!--l. 276--><math
1366
xmlns="http://www.w3.org/1998/Math/MathML"
1367
mode="inline"> <msub
1369
class="MathClass-ord">i</mi><mrow
1371
class="MathClass-ord">1</mn></mrow></msub
1373
class="MathClass-punc">,</mo> <mo
1374
class="MathClass-op">…</mo><mo
1375
class="MathClass-punc">,</mo> <msub
1377
class="MathClass-ord">i</mi><mrow
1379
class="MathClass-ord">l</mi></mrow></msub
1382
class="cmti-10">rows and columns.</span>
1385
<div class="newtheorem">
1386
<!--l. 279--><p class="noindent"><span class="head">
1388
name="x1-3010r1"></a>
1390
class="cmti-10">Remark 3.1.</span> </span>Let <!--l. 280--><math
1391
xmlns="http://www.w3.org/1998/Math/MathML"
1393
<mi class="mathbf">M</mi></math>
1394
be an <!--l. 280--><math
1395
xmlns="http://www.w3.org/1998/Math/MathML"
1398
class="MathClass-ord">n</mi> <mo
1399
class="MathClass-bin">×</mo> <mi
1400
class="MathClass-ord">n</mi></math>
1401
matrix. The convention <!--l. 281--><math
1402
xmlns="http://www.w3.org/1998/Math/MathML"
1404
<mi class="mathbf">M</mi><mrow><mo
1405
class="MathClass-open">(</mo><mi class="mathbf">n</mi><mo
1406
class="MathClass-rel">∣</mo><mi class="mathbf">n</mi><mo
1407
class="MathClass-close">)</mo></mrow> <mo
1408
class="MathClass-rel">=</mo> <mn
1409
class="MathClass-ord">1</mn></math>
1410
has been used in (<a
1411
href="#x1-3009r9">9<!--tex4ht:ref: detB--></a>) and hereafter.
1414
<!--l. 285--><p class="indent"> Before proceeding with our discussion, we pause to note that Theorem <a
1415
href="#x1-3008r3">3.3<!--tex4ht:ref: thm-main--></a> yields
1416
immediately a fundamental formula which can be used to compute the coefficients of
1417
a characteristic polynomial <span class="cite">[<a
1418
href="#Xmami:matrixth">9</a>]</span>: </p><div class="newtheorem">
1419
<!--l. 289--><p class="noindent"><span class="head">
1421
name="x1-3011r4"></a>
1423
class="cmbx-10">Corollary 3.4.</span> </span> <span
1424
class="cmti-10">Write </span><!--l. 290--><math
1425
xmlns="http://www.w3.org/1998/Math/MathML"
1428
class="MathClass-open">(</mo><mi class="mathbf">B</mi> <mo
1429
class="MathClass-bin">−</mo> <mi
1430
class="MathClass-ord">x</mi><mi class="mathbf">I</mi><mo
1431
class="MathClass-close">)</mo></mrow> <mo
1432
class="MathClass-rel">=</mo><msubsup
1434
class="MathClass-op">∑</mo>
1437
class="MathClass-ord">l</mi> <mo
1438
class="MathClass-rel">=</mo> <mn
1439
class="MathClass-ord">0</mn></mrow><mrow
1441
class="MathClass-ord">n</mi></mrow></msubsup
1444
class="MathClass-open">(</mo><mo
1445
class="MathClass-bin">−</mo><mn
1446
class="MathClass-ord">1</mn><mo
1447
class="MathClass-close">)</mo></mrow><mrow
1449
class="MathClass-ord">l</mi></mrow></msup
1452
class="MathClass-ord">b</mi><mrow
1454
class="MathClass-ord">l</mi></mrow></msub
1457
class="MathClass-ord">x</mi><mrow
1459
class="MathClass-ord">l</mi></mrow></msup
1461
class="cmti-10">.</span>
1463
class="cmti-10">Then</span> </p><table class="equation"><tr><td>
1465
xmlns="http://www.w3.org/1998/Math/MathML"
1468
class="equation"><mtr><mtd>
1471
class="MathClass-ord">b</mi><mrow
1473
class="MathClass-ord">l</mi></mrow></msub
1475
class="MathClass-rel">=</mo><msub
1477
class="MathClass-op">∑</mo>
1481
class="MathClass-ord">I</mi><mrow
1483
class="MathClass-ord">l</mi></mrow></msub
1485
class="MathClass-rel">⊆</mo><mi class="mathbf">n</mi></mrow></msub
1487
> det</mo> <mi class="mathbf">B</mi><mrow><mo
1488
class="MathClass-open">(</mo><msub
1490
class="MathClass-ord">I</mi><mrow
1492
class="MathClass-ord">l</mi></mrow></msub
1494
class="MathClass-rel">∣</mo><msub
1496
class="MathClass-ord">I</mi><mrow
1498
class="MathClass-ord">l</mi></mrow></msub
1500
class="MathClass-close">)</mo></mrow><mo
1501
class="MathClass-punc">.</mo></mtd><mtd><mspace
1502
id="x1-3012r10" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
1504
<!--l. 294--><p class="nopar"></p></td><td width="5%">(10)</td></tr></table>
1506
<!--l. 296--><p class="indent"> Let </p><table class="equation"><tr><td>
1509
xmlns="http://www.w3.org/1998/Math/MathML"
1512
class="equation"><mtr><mtd>
1513
<mi class="mathbf">K</mi><mrow><mo
1514
class="MathClass-open">(</mo><mi
1515
class="MathClass-ord">t</mi><mo
1516
class="MathClass-punc">,</mo> <msub
1518
class="MathClass-ord">t</mi><mrow
1520
class="MathClass-ord">1</mn></mrow></msub
1522
class="MathClass-punc">,</mo> <mo
1523
class="MathClass-op">…</mo><mo
1524
class="MathClass-punc">,</mo> <msub
1526
class="MathClass-ord">t</mi><mrow
1528
class="MathClass-ord">n</mi></mrow></msub
1530
class="MathClass-close">)</mo></mrow> <mo
1531
class="MathClass-rel">=</mo> <mfenced
1532
open="(" close=")" ><mtable
1533
equalrows="false" equalcolumns="false" class="array"><mtr><mtd
1534
class="array" > <msub
1536
class="MathClass-ord">D</mi><mrow
1538
class="MathClass-ord">1</mn></mrow></msub
1540
class="MathClass-ord">t</mi> </mtd> <mtd
1542
class="MathClass-bin">−</mo><msub
1544
class="MathClass-ord">a</mi><mrow
1546
class="MathClass-ord">1</mn><mn
1547
class="MathClass-ord">2</mn></mrow></msub
1550
class="MathClass-ord">t</mi><mrow
1552
class="MathClass-ord">2</mn></mrow></msub
1555
class="MathClass-op">…</mo> </mtd><mtd
1557
class="MathClass-bin">−</mo><msub
1559
class="MathClass-ord">a</mi><mrow
1561
class="MathClass-ord">1</mn><mi
1562
class="MathClass-ord">n</mi></mrow></msub
1565
class="MathClass-ord">t</mi><mrow
1567
class="MathClass-ord">n</mi></mrow></msub
1571
class="MathClass-bin">−</mo><msub
1573
class="MathClass-ord">a</mi><mrow
1575
class="MathClass-ord">2</mn><mn
1576
class="MathClass-ord">1</mn></mrow></msub
1579
class="MathClass-ord">t</mi><mrow
1581
class="MathClass-ord">1</mn></mrow></msub
1583
class="array" > <msub
1585
class="MathClass-ord">D</mi><mrow
1587
class="MathClass-ord">2</mn></mrow></msub
1589
class="MathClass-ord">t</mi> </mtd> <mtd
1591
class="MathClass-op">…</mo> </mtd><mtd
1593
class="MathClass-bin">−</mo><msub
1595
class="MathClass-ord">a</mi><mrow
1597
class="MathClass-ord">2</mn><mi
1598
class="MathClass-ord">n</mi></mrow></msub
1601
class="MathClass-ord">t</mi><mrow
1603
class="MathClass-ord">n</mi></mrow></msub
1606
class="array" columnspan="4" columnalign="center"><mrow
1607
class="multicolumn-columnalign-center"> <mo
1608
class="MathClass-punc">.</mo> <mo
1609
class="MathClass-punc">.</mo> <mo
1610
class="MathClass-punc">.</mo> <mo
1611
class="MathClass-punc">.</mo> <mo
1612
class="MathClass-punc">.</mo> <mo
1613
class="MathClass-punc">.</mo> <mo
1614
class="MathClass-punc">.</mo> <mo
1615
class="MathClass-punc">.</mo> <mo
1616
class="MathClass-punc">.</mo> <mo
1617
class="MathClass-punc">.</mo> <mo
1618
class="MathClass-punc">.</mo> <mo
1619
class="MathClass-punc">.</mo> <mo
1620
class="MathClass-punc">.</mo> <mo
1621
class="MathClass-punc">.</mo> <mo
1622
class="MathClass-punc">.</mo></mrow>
1623
</mtd></mtr><mtr><mtd
1625
class="MathClass-bin">−</mo><msub
1627
class="MathClass-ord">a</mi><mrow
1629
class="MathClass-ord">n</mi><mn
1630
class="MathClass-ord">1</mn></mrow></msub
1633
class="MathClass-ord">t</mi><mrow
1635
class="MathClass-ord">1</mn></mrow></msub
1638
class="MathClass-bin">−</mo><msub
1640
class="MathClass-ord">a</mi><mrow
1642
class="MathClass-ord">n</mi><mn
1643
class="MathClass-ord">2</mn></mrow></msub
1646
class="MathClass-ord">t</mi><mrow
1648
class="MathClass-ord">2</mn></mrow></msub
1651
class="MathClass-op">…</mo> </mtd><mtd
1652
class="array" > <msub
1654
class="MathClass-ord">D</mi><mrow
1656
class="MathClass-ord">n</mi></mrow></msub
1658
class="MathClass-ord">t</mi> </mtd></mtr> <!--*\c@MaxMatrixCols c--></mtable> </mfenced> <mo
1659
class="MathClass-punc">,</mo> </mtd><mtd><mspace
1660
id="x1-3013r11" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /></mtd></mtr></mtable>
1662
<!--l. 303--><p class="nopar"></p></td><td width="5%">(11)</td></tr></table>
1665
class="verbatim"><tr class="verbatim"><td
1666
class="verbatim"><pre class="verbatim">
1667
 \begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\
1668
 -a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\
1669
 \hdotsfor[2]{4}\\
1670
 -a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix}
1673
<!--l. 310--><p class="indent"> where </p><table class="equation"><tr><td>
1675
xmlns="http://www.w3.org/1998/Math/MathML"
1678
class="equation"><mtr><mtd>
1681
class="MathClass-ord">D</mi><mrow
1683
class="MathClass-ord">i</mi></mrow></msub
1685
class="MathClass-rel">=</mo><msub
1687
class="MathClass-op">∑</mo>
1690
class="MathClass-ord">j</mi><mo
1691
class="MathClass-rel">∈</mo><mi class="mathbf">n</mi></mrow></msub
1694
class="MathClass-ord">a</mi><mrow
1696
class="MathClass-ord">i</mi><mi
1697
class="MathClass-ord">j</mi></mrow></msub
1700
class="MathClass-ord">t</mi><mrow
1702
class="MathClass-ord">j</mi></mrow></msub
1704
class="MathClass-punc">,</mo> <mspace width="1em" class="quad"/><mi
1705
class="MathClass-ord">i</mi> <mo
1706
class="MathClass-rel">=</mo> <mn
1707
class="MathClass-ord">1</mn><mo
1708
class="MathClass-punc">,</mo> <mo
1709
class="MathClass-op">…</mo><mo
1710
class="MathClass-punc">,</mo> <mi
1711
class="MathClass-ord">n</mi><mo
1712
class="MathClass-punc">.</mo></mtd><mtd><mspace
1713
id="x1-3014r12" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
1715
<!--l. 313--><p class="nopar"></p></td><td width="5%">(12)</td></tr></table>
1716
<!--l. 315--><p class="indent"> Set </p><table class="equation"><tr><td>
1719
xmlns="http://www.w3.org/1998/Math/MathML"
1722
class="equation"><mtr><mtd>
1724
class="MathClass-ord">D</mi><mrow><mo
1725
class="MathClass-open">(</mo><msub
1727
class="MathClass-ord">t</mi><mrow
1729
class="MathClass-ord">1</mn></mrow></msub
1731
class="MathClass-punc">,</mo> <mo
1732
class="MathClass-op">…</mo><mo
1733
class="MathClass-punc">,</mo> <msub
1735
class="MathClass-ord">t</mi><mrow
1737
class="MathClass-ord">n</mi></mrow></msub
1739
class="MathClass-close">)</mo></mrow> <mo
1740
class="MathClass-rel">=</mo> <mfrac><mrow
1742
class="MathClass-ord">δ</mi></mrow>
1745
class="MathClass-ord">δ</mi><mi
1746
class="MathClass-ord">t</mi></mrow></mfrac><msub
1748
open="" close="|" ><mo
1749
>det</mo> <mi class="mathbf">K</mi><mrow><mo
1750
class="MathClass-open">(</mo><mi
1751
class="MathClass-ord">t</mi><mo
1752
class="MathClass-punc">,</mo> <msub
1754
class="MathClass-ord">t</mi><mrow
1756
class="MathClass-ord">1</mn></mrow></msub
1758
class="MathClass-punc">,</mo> <mo>…</mo><mo
1759
class="MathClass-punc">,</mo> <msub
1761
class="MathClass-ord">t</mi><mrow
1763
class="MathClass-ord">n</mi></mrow></msub
1765
class="MathClass-close">)</mo></mrow></mfenced> <mrow
1767
class="MathClass-ord">t</mi><mo
1768
class="MathClass-rel">=</mo><mn
1769
class="MathClass-ord">1</mn></mrow></msub
1771
class="MathClass-punc">.</mo></mtd><mtd> </mtd></mtr></mtable>
1773
<!--l. 319--><p class="nopar"></p></td></tr></table>
1774
Then <table class="equation"><tr><td>
1776
xmlns="http://www.w3.org/1998/Math/MathML"
1779
class="equation"><mtr><mtd>
1781
class="MathClass-ord">D</mi><mrow><mo
1782
class="MathClass-open">(</mo><msub
1784
class="MathClass-ord">t</mi><mrow
1786
class="MathClass-ord">1</mn></mrow></msub
1788
class="MathClass-punc">,</mo> <mo
1789
class="MathClass-op">…</mo><mo
1790
class="MathClass-punc">,</mo> <msub
1792
class="MathClass-ord">t</mi><mrow
1794
class="MathClass-ord">n</mi></mrow></msub
1796
class="MathClass-close">)</mo></mrow> <mo
1797
class="MathClass-rel">=</mo><msub
1799
class="MathClass-op">∑</mo>
1802
class="MathClass-ord">i</mi><mo
1803
class="MathClass-rel">∈</mo><mi class="mathbf">n</mi></mrow></msub
1806
class="MathClass-ord">D</mi><mrow
1808
class="MathClass-ord">i</mi></mrow></msub
1810
> det</mo> <mi class="mathbf">K</mi><mrow><mo
1811
class="MathClass-open">(</mo><mi
1812
class="MathClass-ord">t</mi> <mo
1813
class="MathClass-rel">=</mo> <mn
1814
class="MathClass-ord">1</mn><mo
1815
class="MathClass-punc">,</mo> <msub
1817
class="MathClass-ord">t</mi><mrow
1819
class="MathClass-ord">1</mn></mrow></msub
1821
class="MathClass-punc">,</mo> <mo>…</mo><mo
1822
class="MathClass-punc">,</mo> <msub
1824
class="MathClass-ord">t</mi><mrow
1826
class="MathClass-ord">n</mi></mrow></msub
1828
class="MathClass-punc">;</mo> <mi
1829
class="MathClass-ord">i</mi><mo
1830
class="MathClass-rel">∣</mo><mi
1831
class="MathClass-ord">i</mi><mo
1832
class="MathClass-close">)</mo></mrow><mo
1833
class="MathClass-punc">,</mo> </mtd><mtd><mspace
1834
id="x1-3015r13" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
1836
<!--l. 324--><p class="nopar"></p></td><td width="5%">(13)</td></tr></table>
1837
where <!--l. 325--><math
1838
xmlns="http://www.w3.org/1998/Math/MathML"
1839
mode="inline"> <mi class="mathbf">K</mi><mrow><mo
1840
class="MathClass-open">(</mo><mi
1841
class="MathClass-ord">t</mi> <mo
1842
class="MathClass-rel">=</mo> <mn
1843
class="MathClass-ord">1</mn><mo
1844
class="MathClass-punc">,</mo> <msub
1846
class="MathClass-ord">t</mi><mrow
1848
class="MathClass-ord">1</mn></mrow></msub
1850
class="MathClass-punc">,</mo> <mo
1851
class="MathClass-op">…</mo><mo
1852
class="MathClass-punc">,</mo> <msub
1854
class="MathClass-ord">t</mi><mrow
1856
class="MathClass-ord">n</mi></mrow></msub
1858
class="MathClass-punc">;</mo> <mi
1859
class="MathClass-ord">i</mi><mo
1860
class="MathClass-rel">∣</mo><mi
1861
class="MathClass-ord">i</mi><mo
1862
class="MathClass-close">)</mo></mrow></math> is the <!--l. 325--><math
1863
xmlns="http://www.w3.org/1998/Math/MathML"
1866
class="MathClass-ord">i</mi></math>th principal
1867
submatrix of <!--l. 326--><math
1868
xmlns="http://www.w3.org/1998/Math/MathML"
1869
mode="inline"> <mi class="mathbf">K</mi><mrow><mo
1870
class="MathClass-open">(</mo><mi
1871
class="MathClass-ord">t</mi> <mo
1872
class="MathClass-rel">=</mo> <mn
1873
class="MathClass-ord">1</mn><mo
1874
class="MathClass-punc">,</mo> <msub
1876
class="MathClass-ord">t</mi><mrow
1878
class="MathClass-ord">1</mn></mrow></msub
1880
class="MathClass-punc">,</mo> <mo
1881
class="MathClass-op">…</mo><mo
1882
class="MathClass-punc">,</mo> <msub
1884
class="MathClass-ord">t</mi><mrow
1886
class="MathClass-ord">n</mi></mrow></msub
1888
class="MathClass-close">)</mo></mrow></math>.
1889
<!--l. 328--><p class="indent"> Theorem  <a
1890
href="#x1-3008r3">3.3<!--tex4ht:ref: thm-main--></a> leads to </p><table class="equation"><tr><td>
1893
xmlns="http://www.w3.org/1998/Math/MathML"
1896
class="equation"><mtr><mtd>
1898
> det</mo> <mi class="mathbf">K</mi><mrow><mo
1899
class="MathClass-open">(</mo><msub
1901
class="MathClass-ord">t</mi><mrow
1903
class="MathClass-ord">1</mn></mrow></msub
1905
class="MathClass-punc">,</mo> <msub
1907
class="MathClass-ord">t</mi><mrow
1909
class="MathClass-ord">1</mn></mrow></msub
1911
class="MathClass-punc">,</mo> <mo>…</mo><mo
1912
class="MathClass-punc">,</mo> <msub
1914
class="MathClass-ord">t</mi><mrow
1916
class="MathClass-ord">n</mi></mrow></msub
1918
class="MathClass-close">)</mo></mrow> <mo
1919
class="MathClass-rel">=</mo><msub
1921
class="MathClass-op">∑</mo>
1924
class="MathClass-ord">I</mi><mo
1925
class="MathClass-rel">∈</mo><mi class="mathbf">n</mi></mrow></msub
1928
class="MathClass-open">(</mo><mo
1929
class="MathClass-bin">−</mo><mn
1930
class="MathClass-ord">1</mn><mo
1931
class="MathClass-close">)</mo></mrow><mrow
1933
open="|" close="|" ><mi
1934
class="MathClass-ord">I</mi> </mfenced></mrow></msup
1937
class="MathClass-ord">t</mi><mrow
1939
class="MathClass-ord">n</mi><mo
1940
class="MathClass-bin">−</mo> <mfenced
1941
open="|" close="|" ><mi
1942
class="MathClass-ord">I</mi> </mfenced></mrow></msup
1945
class="MathClass-op">∏</mo>
1948
class="MathClass-ord">i</mi><mo
1949
class="MathClass-rel">∈</mo><mi
1950
class="MathClass-ord">I</mi></mrow></msub
1953
class="MathClass-ord">t</mi><mrow
1955
class="MathClass-ord">i</mi></mrow></msub
1958
class="MathClass-op">∏</mo>
1961
class="MathClass-ord">j</mi><mo
1962
class="MathClass-rel">∈</mo><mi
1963
class="MathClass-ord">I</mi></mrow></msub
1965
class="MathClass-open">(</mo><msub
1967
class="MathClass-ord">D</mi><mrow
1969
class="MathClass-ord">j</mi></mrow></msub
1971
class="MathClass-bin">+</mo> <msub
1973
class="MathClass-ord">λ</mi><mrow
1975
class="MathClass-ord">j</mi></mrow></msub
1978
class="MathClass-ord">t</mi><mrow
1980
class="MathClass-ord">j</mi></mrow></msub
1982
class="MathClass-close">)</mo></mrow><mo
1984
> <mi class="mathbf">A</mi><mrow
1986
class="MathClass-open">(</mo><mi
1987
class="MathClass-ord">λ</mi><mi
1988
class="MathClass-ord">t</mi><mo
1989
class="MathClass-close">)</mo></mrow></mrow></msup
1991
class="MathClass-open">(</mo><mover
1992
class="mml-overline"><mrow><mi
1993
class="MathClass-ord">I</mi></mrow><mo
1994
accent="true">‾</mo></mover><mo
1995
class="MathClass-rel">∣</mo><mover
1996
class="mml-overline"><mrow><mi
1997
class="MathClass-ord">I</mi></mrow><mo
1998
accent="true">‾</mo></mover><mo
1999
class="MathClass-close">)</mo></mrow><mo
2000
class="MathClass-punc">.</mo></mtd><mtd><mspace
2001
id="x1-3016r14" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /></mtd></mtr></mtable>
2003
<!--l. 334--><p class="nopar"></p></td><td width="5%">(14)</td></tr></table>
2004
Note that <table class="equation"><tr><td>
2006
xmlns="http://www.w3.org/1998/Math/MathML"
2009
class="equation"><mtr><mtd>
2011
> det</mo> <mi class="mathbf">K</mi><mrow><mo
2012
class="MathClass-open">(</mo><mi
2013
class="MathClass-ord">t</mi> <mo
2014
class="MathClass-rel">=</mo> <mn
2015
class="MathClass-ord">1</mn><mo
2016
class="MathClass-punc">,</mo> <msub
2018
class="MathClass-ord">t</mi><mrow
2020
class="MathClass-ord">1</mn></mrow></msub
2022
class="MathClass-punc">,</mo> <mo>…</mo><mo
2023
class="MathClass-punc">,</mo> <msub
2025
class="MathClass-ord">t</mi><mrow
2027
class="MathClass-ord">n</mi></mrow></msub
2029
class="MathClass-close">)</mo></mrow> <mo
2030
class="MathClass-rel">=</mo><msub
2032
class="MathClass-op">∑</mo>
2035
class="MathClass-ord">I</mi><mo
2036
class="MathClass-rel">∈</mo><mi class="mathbf">n</mi></mrow></msub
2039
class="MathClass-open">(</mo><mo
2040
class="MathClass-bin">−</mo><mn
2041
class="MathClass-ord">1</mn><mo
2042
class="MathClass-close">)</mo></mrow><mrow
2044
open="|" close="|" ><mi
2045
class="MathClass-ord">I</mi> </mfenced></mrow></msup
2048
class="MathClass-op">∏</mo>
2051
class="MathClass-ord">i</mi><mo
2052
class="MathClass-rel">∈</mo><mi
2053
class="MathClass-ord">I</mi></mrow></msub
2056
class="MathClass-ord">t</mi><mrow
2058
class="MathClass-ord">i</mi></mrow></msub
2061
class="MathClass-op">∏</mo>
2064
class="MathClass-ord">j</mi><mo
2065
class="MathClass-rel">∈</mo><mi
2066
class="MathClass-ord">I</mi></mrow></msub
2068
class="MathClass-open">(</mo><msub
2070
class="MathClass-ord">D</mi><mrow
2072
class="MathClass-ord">j</mi></mrow></msub
2074
class="MathClass-bin">+</mo> <msub
2076
class="MathClass-ord">λ</mi><mrow
2078
class="MathClass-ord">j</mi></mrow></msub
2081
class="MathClass-ord">t</mi><mrow
2083
class="MathClass-ord">j</mi></mrow></msub
2085
class="MathClass-close">)</mo></mrow><mo
2087
> <mi class="mathbf">A</mi><mrow
2089
class="MathClass-open">(</mo><mi
2090
class="MathClass-ord">λ</mi><mo
2091
class="MathClass-close">)</mo></mrow></mrow></msup
2093
class="MathClass-open">(</mo><mover
2094
class="mml-overline"><mrow><mi
2095
class="MathClass-ord">I</mi></mrow><mo
2096
accent="true">‾</mo></mover><mo
2097
class="MathClass-rel">∣</mo><mover
2098
class="mml-overline"><mrow><mi
2099
class="MathClass-ord">I</mi></mrow><mo
2100
accent="true">‾</mo></mover><mo
2101
class="MathClass-close">)</mo></mrow> <mo
2102
class="MathClass-rel">=</mo> <mn
2103
class="MathClass-ord">0</mn><mo
2104
class="MathClass-punc">.</mo></mtd><mtd><mspace
2105
id="x1-3017r15" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /></mtd></mtr></mtable>
2107
<!--l. 340--><p class="nopar"></p></td><td width="5%">(15)</td></tr></table>
2108
<!--l. 342--><p class="indent"> Let <!--l. 342--><math
2109
xmlns="http://www.w3.org/1998/Math/MathML"
2110
mode="inline"> <msub
2112
class="MathClass-ord">t</mi><mrow
2114
class="MathClass-ord">i</mi></mrow></msub
2116
class="MathClass-rel">=</mo><msub
2120
class="MathClass-ord">x</mi></mrow><mrow
2123
class="MathClass-ord">̂</mi></mrow></munderover><mrow
2125
class="MathClass-ord">i</mi></mrow></msub
2127
class="MathClass-punc">,</mo> <mi
2128
class="MathClass-ord">i</mi> <mo
2129
class="MathClass-rel">=</mo> <mn
2130
class="MathClass-ord">1</mn><mo
2131
class="MathClass-punc">,</mo> <mo
2132
class="MathClass-op">…</mo><mo
2133
class="MathClass-punc">,</mo> <mi
2134
class="MathClass-ord">n</mi></math>.
2136
href="#x1-3001r1">3.1<!--tex4ht:ref: lem-per--></a> yields
2138
</p><!--l. 343--><math
2139
xmlns="http://www.w3.org/1998/Math/MathML"
2144
class="multline"></mtd><mtd><mrow><mo
2145
class="MathClass-open">(
2148
class="MathClass-op">∑</mo>
2151
class="MathClass-ord">i</mi><mo
2152
class="MathClass-rel">∈</mo><mi class="mathbf">n</mi></mrow></msub
2155
class="MathClass-ord">a</mi><mrow
2158
class="MathClass-ord">l</mi><mrow
2160
class="MathClass-ord">i</mi></mrow></msub
2164
class="MathClass-ord">x</mi><mrow
2166
class="MathClass-ord">i</mi></mrow></msub
2168
class="MathClass-close">)
2170
> det</mo> <mi class="mathbf">K</mi><mrow><mo
2171
class="MathClass-open">(</mo><mi
2172
class="MathClass-ord">t</mi> <mo
2173
class="MathClass-rel">=</mo> <mn
2174
class="MathClass-ord">1</mn><mo
2175
class="MathClass-punc">,</mo> <msub
2177
class="MathClass-ord">x</mi><mrow
2179
class="MathClass-ord">1</mn></mrow></msub
2181
class="MathClass-punc">,</mo> <mo>…</mo><mo
2182
class="MathClass-punc">,</mo> <msub
2184
class="MathClass-ord">x</mi><mrow
2186
class="MathClass-ord">n</mi></mrow></msub
2188
class="MathClass-punc">;</mo> <mi
2189
class="MathClass-ord">l</mi><mo
2190
class="MathClass-rel">∣</mo><mi
2191
class="MathClass-ord">l</mi><mo
2192
class="MathClass-close">)</mo></mrow>
2193
</mtd></mtr><mtr><mtd
2194
class="multline"></mtd><mtd> <mo
2195
class="MathClass-rel">=</mo><mrow><mo
2196
class="MathClass-open">(
2199
class="MathClass-op">∏</mo>
2202
class="MathClass-ord">i</mi><mo
2203
class="MathClass-rel">∈</mo><mi class="mathbf">n</mi></mrow></msub
2208
class="MathClass-ord">x</mi></mrow><mrow
2211
class="MathClass-ord">̂</mi></mrow></munderover><mrow
2213
class="MathClass-ord">i</mi></mrow></msub
2215
class="MathClass-close">)
2218
class="MathClass-op">∑</mo>
2221
class="MathClass-ord">I</mi><mo
2222
class="MathClass-rel">⊆</mo><mi class="mathbf">n</mi><mo
2223
class="MathClass-bin">−</mo><mrow><mo
2224
class="MathClass-open">{</mo><mi
2225
class="MathClass-ord">l</mi><mo
2226
class="MathClass-close">}</mo></mrow></mrow></msub
2229
class="MathClass-open">(</mo><mo
2230
class="MathClass-bin">−</mo><mn
2231
class="MathClass-ord">1</mn><mo
2232
class="MathClass-close">)</mo></mrow><mrow
2234
open="|" close="|" ><mi
2235
class="MathClass-ord">I</mi> </mfenced></mrow></msup
2237
class="MathClass-op"> per</mo><!--nolimits--><msup
2238
> <mi class="mathbf">A</mi><mrow
2240
class="MathClass-open">(</mo><mi
2241
class="MathClass-ord">λ</mi><mo
2242
class="MathClass-close">)</mo></mrow></mrow></msup
2244
class="MathClass-open">(</mo><mi
2245
class="MathClass-ord">I</mi><mo
2246
class="MathClass-rel">∣</mo><mi
2247
class="MathClass-ord">I</mi><mo
2248
class="MathClass-close">)</mo></mrow><mo
2250
> <mi class="mathbf">A</mi><mrow
2252
class="MathClass-open">(</mo><mi
2253
class="MathClass-ord">λ</mi><mo
2254
class="MathClass-close">)</mo></mrow></mrow></msup
2256
class="MathClass-open">(</mo><mover
2257
class="mml-overline"><mrow><mi
2258
class="MathClass-ord">I</mi></mrow><mo
2259
accent="true">‾</mo></mover> <mo
2260
class="MathClass-bin">∪</mo> <mrow><mo
2261
class="MathClass-open">{</mo><mi
2262
class="MathClass-ord">l</mi><mo
2263
class="MathClass-close">}</mo></mrow><mo
2264
class="MathClass-rel">∣</mo><mover
2265
class="mml-overline"><mrow><mi
2266
class="MathClass-ord">I</mi></mrow><mo
2267
accent="true">‾</mo></mover> <mo
2268
class="MathClass-bin">∪</mo> <mrow><mo
2269
class="MathClass-open">{</mo><mi
2270
class="MathClass-ord">l</mi><mo
2271
class="MathClass-close">}</mo></mrow><mo
2272
class="MathClass-close">)</mo></mrow><mo
2273
class="MathClass-punc">.</mo><mspace
2274
id="x1-3018r16" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /></mtd><mtd><mrow><mo
2275
class="MathClass-open">(</mo><mn
2276
class="MathClass-ord">1</mn><mn
2277
class="MathClass-ord">6</mn><mo
2278
class="MathClass-close">)</mo></mrow> </mtd></mtr></mtable>
2280
<!--l. 352--><p class="nopar">
2284
class="verbatim"><tr class="verbatim"><td
2285
class="verbatim"><pre class="verbatim">
2286
 \begin{multline}
2287
 \biggl(\sum_{\,i\in\mathbf{n}}a_{l _i}x_i\biggr)
2288
 \det\mathbf{K}(t=1,x_1,\dots,x_n;l |l )\\
2289
 =\biggl(\prod_{\,i\in\mathbf{n}}\hat x_i\biggr)
2290
 \sum_{I\subseteq\mathbf{n}-\{l \}}
2291
 (-1)ˆ{\envert{I}}\per\mathbf{A}ˆ{(\lambda)}(I|I)
2292
 \det\mathbf{A}ˆ{(\lambda)}
2293
 (\overline I\cup\{l \}|\overline I\cup\{l \}).
2294
 \label{sum-ali}
2295
 \end{multline}
2298
<!--l. 366--><p class="indent"> By (<a
2299
href="#x1-2003r3">3<!--tex4ht:ref: H-cycles--></a>), (<a
2300
href="#x1-3005r6">6<!--tex4ht:ref: detprod--></a>), and (<a
2301
href="#x1-3006r7">7<!--tex4ht:ref: sum-bij--></a>), we have </p><div class="newtheorem">
2302
<!--l. 367--><p class="noindent"><span class="head">
2304
name="x1-3019r5"></a>
2306
class="cmbx-10">Proposition 3.5.</span> </span> </p><table class="equation"><tr><td>
2308
xmlns="http://www.w3.org/1998/Math/MathML"
2311
class="equation"><mtr><mtd>
2314
class="MathClass-ord">H</mi><mrow
2316
class="MathClass-ord">c</mi></mrow></msub
2318
class="MathClass-rel">=</mo> <mfrac><mrow
2320
class="MathClass-ord">1</mn></mrow>
2323
class="MathClass-ord">2</mn><mi
2324
class="MathClass-ord">n</mi></mrow></mfrac><msubsup
2326
class="MathClass-op">∑</mo>
2329
class="MathClass-ord">l</mi> <mo
2330
class="MathClass-rel">=</mo> <mn
2331
class="MathClass-ord">0</mn></mrow><mrow
2333
class="MathClass-ord">n</mi></mrow></msubsup
2336
class="MathClass-open">(</mo><mo
2337
class="MathClass-bin">−</mo><mn
2338
class="MathClass-ord">1</mn><mo
2339
class="MathClass-close">)</mo></mrow><mrow
2341
class="MathClass-ord">l</mi></mrow></msup
2344
class="MathClass-ord">D</mi><mrow
2347
class="MathClass-ord">l</mi></mrow></msub
2349
class="MathClass-punc">,</mo> </mtd><mtd><mspace
2350
id="x1-3020r17" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
2352
<!--l. 371--><p class="nopar"></p></td><td width="5%">(17)</td></tr></table>
2354
class="cmti-10">where </span><table class="equation"><tr><td>
2357
xmlns="http://www.w3.org/1998/Math/MathML"
2360
class="equation"><mtr><mtd>
2363
class="MathClass-ord">D</mi><mrow
2365
class="MathClass-ord">l</mi></mrow></msub
2367
class="MathClass-rel">=</mo><msub
2369
class="MathClass-op">∑</mo>
2373
class="MathClass-ord">I</mi><mrow
2375
class="MathClass-ord">l</mi></mrow></msub
2377
class="MathClass-rel">⊆</mo><mi class="mathbf">n</mi></mrow></msub
2379
class="MathClass-ord">D</mi><mrow><mo
2380
class="MathClass-open">(</mo><msub
2382
class="MathClass-ord">t</mi><mrow
2384
class="MathClass-ord">1</mn></mrow></msub
2386
class="MathClass-punc">,</mo> <mo
2387
class="MathClass-op">…</mo><mo
2388
class="MathClass-punc">,</mo> <msub
2390
class="MathClass-ord">t</mi><mrow
2392
class="MathClass-ord">n</mi></mrow></msub
2394
class="MathClass-close">)</mo></mrow><mn
2395
class="MathClass-ord">2</mn><msub
2397
class="MathClass-rel">∣</mo><mrow
2401
class="MathClass-ord">t</mi><mrow
2403
class="MathClass-ord">i</mi></mrow></msub
2405
class="MathClass-rel">=</mo><mfenced
2406
open="{" close="" ><mtable><mtr
2407
class="smallmatrix">
2409
class="MathClass-ord">0</mn><mo
2410
class="MathClass-punc">,</mo></mtd>
2412
class="text"><mtext >if </mtext></mrow><mi
2413
class="MathClass-ord">i</mi><mo
2414
class="MathClass-rel">∈</mo><msub
2416
class="MathClass-ord">I</mi><mrow
2418
class="MathClass-ord">l</mi></mrow></msub
2419
><mspace width="1em" class="quad"/></mtd>
2421
class="smallmatrix">
2423
class="MathClass-ord">1</mn><mo
2424
class="MathClass-punc">,</mo></mtd>
2426
class="text"><mtext >otherwise</mtext></mrow></mtd>
2427
</mtr> </mtable></mfenced> <mo
2428
class="MathClass-punc">,</mo> <mi
2429
class="MathClass-ord">i</mi><mo
2430
class="MathClass-rel">=</mo><mn
2431
class="MathClass-ord">1</mn><mo
2432
class="MathClass-punc">,</mo><mo
2433
class="MathClass-op">…</mo><mo
2434
class="MathClass-punc">,</mo><mi
2435
class="MathClass-ord">n</mi></mrow></msub
2437
class="MathClass-punc">.</mo></mtd><mtd><mspace
2438
id="x1-3021r18" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /></mtd></mtr></mtable>
2440
<!--l. 378--><p class="nopar"></p></td><td width="5%">(18)</td></tr></table>
2442
<h3 class="sectionHead"><span class="titlemark">4</span> <a
2443
name="x1-40004"></a>Application</h3>
2444
<!--l. 384--><p class="noindent">We consider here the applications of Theorems <a
2445
href="#x1-5002r1">5.1<!--tex4ht:ref: th-info-ow-ow--></a> and  <a
2446
href="#x1-5003r2">5.2<!--tex4ht:ref: th-weak-ske-owf--></a> to a complete multipartite graph <!--l. 386--><math
2447
xmlns="http://www.w3.org/1998/Math/MathML"
2451
class="MathClass-ord">K</mi><mrow
2454
class="MathClass-ord">n</mi><mrow
2456
class="MathClass-ord">1</mn></mrow></msub
2458
class="MathClass-op">…</mo><msub
2460
class="MathClass-ord">n</mi><mrow
2462
class="MathClass-ord">p</mi></mrow></msub
2465
It can be shown that the number of spanning trees of <!--l. 387--><math
2466
xmlns="http://www.w3.org/1998/Math/MathML"
2470
class="MathClass-ord">K</mi><mrow
2473
class="MathClass-ord">n</mi><mrow
2475
class="MathClass-ord">1</mn></mrow></msub
2477
class="MathClass-op">…</mo><msub
2479
class="MathClass-ord">n</mi><mrow
2481
class="MathClass-ord">p</mi></mrow></msub
2484
be written </p><table class="equation"><tr><td>
2487
xmlns="http://www.w3.org/1998/Math/MathML"
2490
class="equation"><mtr><mtd>
2492
class="MathClass-ord">T</mi> <mo
2493
class="MathClass-rel">=</mo> <msup
2495
class="MathClass-ord">n</mi><mrow
2497
class="MathClass-ord">p</mi><mo
2498
class="MathClass-bin">−</mo><mn
2499
class="MathClass-ord">2</mn></mrow></msup
2502
class="MathClass-op">∏</mo>
2505
class="MathClass-ord">i</mi> <mo
2506
class="MathClass-rel">=</mo> <mn
2507
class="MathClass-ord">1</mn></mrow><mrow
2509
class="MathClass-ord">p</mi></mrow></msubsup
2512
class="MathClass-open">(</mo><mi
2513
class="MathClass-ord">n</mi> <mo
2514
class="MathClass-bin">−</mo> <msub
2516
class="MathClass-ord">n</mi><mrow
2518
class="MathClass-ord">i</mi></mrow></msub
2520
class="MathClass-close">)</mo></mrow><mrow
2523
class="MathClass-ord">n</mi><mrow
2525
class="MathClass-ord">i</mi></mrow></msub
2527
class="MathClass-bin">−</mo><mn
2528
class="MathClass-ord">1</mn></mrow></msup
2530
id="x1-4001r19" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
2532
<!--l. 392--><p class="nopar"></p></td><td width="5%">(19)</td></tr></table>
2533
where <table class="equation"><tr><td>
2535
xmlns="http://www.w3.org/1998/Math/MathML"
2538
class="equation"><mtr><mtd>
2540
class="MathClass-ord">n</mi> <mo
2541
class="MathClass-rel">=</mo> <msub
2543
class="MathClass-ord">n</mi><mrow
2545
class="MathClass-ord">1</mn></mrow></msub
2547
class="MathClass-bin">+</mo> <mo
2548
class="MathClass-rel">⋯</mo> <mo
2549
class="MathClass-bin">+</mo> <msub
2551
class="MathClass-ord">n</mi><mrow
2553
class="MathClass-ord">p</mi></mrow></msub
2555
class="MathClass-punc">.</mo></mtd><mtd><mspace
2556
id="x1-4002r20" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
2558
<!--l. 396--><p class="nopar"></p></td><td width="5%">(20)</td></tr></table>
2559
<!--l. 398--><p class="indent"> It follows from Theorems <a
2560
href="#x1-5002r1">5.1<!--tex4ht:ref: th-info-ow-ow--></a> and  <a
2561
href="#x1-5003r2">5.2<!--tex4ht:ref: th-weak-ske-owf--></a> that </p><table class="equation"><tr><td>
2564
xmlns="http://www.w3.org/1998/Math/MathML"
2567
class="equation"><mtr><mtd>
2568
<mtable class="split"><mtr><mtd>
2570
class="split-mtr"></mrow><mrow
2571
class="split-mtd"></mrow> <msub
2573
class="MathClass-ord">H</mi><mrow
2575
class="MathClass-ord">c</mi></mrow></msub
2577
class="split-mtd"></mrow> <mo
2578
class="MathClass-rel">=</mo> <mfrac><mrow
2580
class="MathClass-ord">1</mn></mrow>
2583
class="MathClass-ord">2</mn><mi
2584
class="MathClass-ord">n</mi></mrow></mfrac><msubsup
2586
class="MathClass-op">∑</mo>
2589
class="MathClass-ord">l</mi> <mo
2590
class="MathClass-rel">=</mo> <mn
2591
class="MathClass-ord">0</mn></mrow><mrow
2593
class="MathClass-ord">n</mi></mrow></msubsup
2596
class="MathClass-open">(</mo><mo
2597
class="MathClass-bin">−</mo><mn
2598
class="MathClass-ord">1</mn><mo
2599
class="MathClass-close">)</mo></mrow><mrow
2601
class="MathClass-ord">l</mi></mrow></msup
2604
class="MathClass-open">(</mo><mi
2605
class="MathClass-ord">n</mi> <mo
2606
class="MathClass-bin">−</mo> <mi
2607
class="MathClass-ord">l</mi><mo
2608
class="MathClass-close">)</mo></mrow><mrow
2610
class="MathClass-ord">p</mi><mo
2611
class="MathClass-bin">−</mo><mn
2612
class="MathClass-ord">2</mn></mrow></msup
2615
class="MathClass-op">∑</mo>
2619
class="MathClass-ord">l</mi><mrow
2621
class="MathClass-ord">1</mn></mrow></msub
2623
class="MathClass-bin">+</mo><mo
2624
class="MathClass-rel">⋯</mo><mo
2625
class="MathClass-bin">+</mo><msub
2627
class="MathClass-ord">l</mi><mrow
2629
class="MathClass-ord">p</mi></mrow></msub
2631
class="MathClass-rel">=</mo><mi
2632
class="MathClass-ord">l</mi></mrow></msub
2635
class="MathClass-op">∏</mo>
2638
class="MathClass-ord">i</mi> <mo
2639
class="MathClass-rel">=</mo> <mn
2640
class="MathClass-ord">1</mn></mrow><mrow
2642
class="MathClass-ord">p</mi></mrow></msubsup
2643
><mfenced open="(" close=")" class="binom"><mover
2644
class="binom"><mrow><msub
2646
class="MathClass-ord">n</mi><mrow
2648
class="MathClass-ord">i</mi></mrow></msub
2651
class="MathClass-ord">l</mi><mrow
2653
class="MathClass-ord">i</mi></mrow></msub
2654
></mrow></mover></mfenced>
2656
class="split-mtr"></mrow><mrow
2657
class="split-mtd"></mrow> <mrow
2658
class="split-mtd"></mrow><mspace width="1em" class="quad"/> <mo
2659
class="MathClass-punc">·</mo> <msup
2661
class="MathClass-open">[</mo><mrow><mo
2662
class="MathClass-open">(</mo><mi
2663
class="MathClass-ord">n</mi> <mo
2664
class="MathClass-bin">−</mo> <mi
2665
class="MathClass-ord">l</mi><mo
2666
class="MathClass-close">)</mo></mrow> <mo
2667
class="MathClass-bin">−</mo> <mrow><mo
2668
class="MathClass-open">(</mo><msub
2670
class="MathClass-ord">n</mi><mrow
2672
class="MathClass-ord">i</mi></mrow></msub
2674
class="MathClass-bin">−</mo> <msub
2676
class="MathClass-ord">l</mi><mrow
2678
class="MathClass-ord">i</mi></mrow></msub
2680
class="MathClass-close">)</mo></mrow><mo
2681
class="MathClass-close">]</mo></mrow><mrow
2684
class="MathClass-ord">n</mi><mrow
2686
class="MathClass-ord">i</mi></mrow></msub
2688
class="MathClass-bin">−</mo><msub
2690
class="MathClass-ord">l</mi><mrow
2692
class="MathClass-ord">i</mi></mrow></msub
2696
class="MathClass-punc">·</mo><mrow><mo
2697
class="MathClass-open">[
2700
class="MathClass-open">(</mo><mi
2701
class="MathClass-ord">n</mi> <mo
2702
class="MathClass-bin">−</mo> <mi
2703
class="MathClass-ord">l</mi><mo
2704
class="MathClass-close">)</mo></mrow><mrow
2706
class="MathClass-ord">2</mn></mrow></msup
2708
class="MathClass-bin">−</mo><msubsup
2710
class="MathClass-op">∑</mo>
2713
class="MathClass-ord">j</mi> <mo
2714
class="MathClass-rel">=</mo> <mn
2715
class="MathClass-ord">1</mn></mrow><mrow
2717
class="MathClass-ord">p</mi></mrow></msubsup
2720
class="MathClass-open">(</mo><msub
2722
class="MathClass-ord">n</mi><mrow
2724
class="MathClass-ord">i</mi></mrow></msub
2726
class="MathClass-bin">−</mo> <msub
2728
class="MathClass-ord">l</mi><mrow
2730
class="MathClass-ord">i</mi></mrow></msub
2732
class="MathClass-close">)</mo></mrow><mrow
2734
class="MathClass-ord">2</mn></mrow></msup
2736
class="MathClass-close">]
2738
class="MathClass-punc">.</mo>
2739
</mtd></mtr></mtable> </mtd><mtd><mspace
2740
id="x1-4003r21" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /></mtd></mtr></mtable>
2742
<!--l. 408--><p class="nopar"></p></td><td width="5%">(21)</td></tr></table>
2745
class="verbatim"><tr class="verbatim"><td
2746
class="verbatim"><pre class="verbatim">
2747
 ... \binom{n_i}{l _i}\\
2750
<!--l. 412--><p class="indent"> and </p><table class="equation"><tr><td>
2752
xmlns="http://www.w3.org/1998/Math/MathML"
2755
class="equation"><mtr><mtd>
2756
<mtable class="split"><mtr><mtd>
2758
class="split-mtr"></mrow><mrow
2759
class="split-mtd"></mrow> <msub
2761
class="MathClass-ord">H</mi><mrow
2763
class="MathClass-ord">c</mi></mrow></msub
2765
class="split-mtd"></mrow> <mo
2766
class="MathClass-rel">=</mo> <mfrac><mrow
2768
class="MathClass-ord">1</mn></mrow>
2771
class="MathClass-ord">2</mn></mrow></mfrac><msubsup
2773
class="MathClass-op">∑</mo>
2776
class="MathClass-ord">l</mi> <mo
2777
class="MathClass-rel">=</mo> <mn
2778
class="MathClass-ord">0</mn></mrow><mrow
2780
class="MathClass-ord">n</mi> <mo
2781
class="MathClass-bin">−</mo> <mn
2782
class="MathClass-ord">1</mn></mrow></msubsup
2785
class="MathClass-open">(</mo><mo
2786
class="MathClass-bin">−</mo><mn
2787
class="MathClass-ord">1</mn><mo
2788
class="MathClass-close">)</mo></mrow><mrow
2790
class="MathClass-ord">l</mi></mrow></msup
2793
class="MathClass-open">(</mo><mi
2794
class="MathClass-ord">n</mi> <mo
2795
class="MathClass-bin">−</mo> <mi
2796
class="MathClass-ord">l</mi><mo
2797
class="MathClass-close">)</mo></mrow><mrow
2799
class="MathClass-ord">p</mi><mo
2800
class="MathClass-bin">−</mo><mn
2801
class="MathClass-ord">2</mn></mrow></msup
2804
class="MathClass-op">∑</mo>
2808
class="MathClass-ord">l</mi><mrow
2810
class="MathClass-ord">1</mn></mrow></msub
2812
class="MathClass-bin">+</mo><mo
2813
class="MathClass-rel">⋯</mo><mo
2814
class="MathClass-bin">+</mo><msub
2816
class="MathClass-ord">l</mi><mrow
2818
class="MathClass-ord">p</mi></mrow></msub
2820
class="MathClass-rel">=</mo><mi
2821
class="MathClass-ord">l</mi></mrow></msub
2824
class="MathClass-op">∏</mo>
2827
class="MathClass-ord">i</mi> <mo
2828
class="MathClass-rel">=</mo> <mn
2829
class="MathClass-ord">1</mn></mrow><mrow
2831
class="MathClass-ord">p</mi></mrow></msubsup
2832
><mfenced open="(" close=")" class="binom"><mover
2833
class="binom"><mrow><msub
2835
class="MathClass-ord">n</mi><mrow
2837
class="MathClass-ord">i</mi></mrow></msub
2840
class="MathClass-ord">l</mi><mrow
2842
class="MathClass-ord">i</mi></mrow></msub
2843
></mrow></mover></mfenced>
2845
class="split-mtr"></mrow><mrow
2846
class="split-mtd"></mrow> <mrow
2847
class="split-mtd"></mrow><mspace width="1em" class="quad"/> <mo
2848
class="MathClass-punc">·</mo> <msup
2850
class="MathClass-open">[</mo><mrow><mo
2851
class="MathClass-open">(</mo><mi
2852
class="MathClass-ord">n</mi> <mo
2853
class="MathClass-bin">−</mo> <mi
2854
class="MathClass-ord">l</mi><mo
2855
class="MathClass-close">)</mo></mrow> <mo
2856
class="MathClass-bin">−</mo> <mrow><mo
2857
class="MathClass-open">(</mo><msub
2859
class="MathClass-ord">n</mi><mrow
2861
class="MathClass-ord">i</mi></mrow></msub
2863
class="MathClass-bin">−</mo> <msub
2865
class="MathClass-ord">l</mi><mrow
2867
class="MathClass-ord">i</mi></mrow></msub
2869
class="MathClass-close">)</mo></mrow><mo
2870
class="MathClass-close">]</mo></mrow><mrow
2873
class="MathClass-ord">n</mi><mrow
2875
class="MathClass-ord">i</mi></mrow></msub
2877
class="MathClass-bin">−</mo><msub
2879
class="MathClass-ord">l</mi><mrow
2881
class="MathClass-ord">i</mi></mrow></msub
2885
open="(" close=")" ><mn
2886
class="MathClass-ord">1</mn> <mo
2887
class="MathClass-bin">−</mo> <mfrac><mrow
2890
class="MathClass-ord">l</mi><mrow
2892
class="MathClass-ord">p</mi></mrow></msub
2897
class="MathClass-ord">n</mi><mrow
2899
class="MathClass-ord">p</mi></mrow></msub
2900
></mrow></mfrac> </mfenced> <mrow><mo
2901
class="MathClass-open">[</mo><mrow><mo
2902
class="MathClass-open">(</mo><mi
2903
class="MathClass-ord">n</mi> <mo
2904
class="MathClass-bin">−</mo> <mi
2905
class="MathClass-ord">l</mi><mo
2906
class="MathClass-close">)</mo></mrow> <mo
2907
class="MathClass-bin">−</mo> <mrow><mo
2908
class="MathClass-open">(</mo><msub
2910
class="MathClass-ord">n</mi><mrow
2912
class="MathClass-ord">p</mi></mrow></msub
2914
class="MathClass-bin">−</mo> <msub
2916
class="MathClass-ord">l</mi><mrow
2918
class="MathClass-ord">p</mi></mrow></msub
2920
class="MathClass-close">)</mo></mrow><mo
2921
class="MathClass-close">]</mo></mrow><mo
2922
class="MathClass-punc">.</mo>
2923
</mtd></mtr></mtable> </mtd><mtd><mspace
2924
id="x1-4004r22" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /></mtd></mtr></mtable>
2926
<!--l. 423--><p class="nopar"></p></td><td width="5%">(22)</td></tr></table>
2927
<!--l. 425--><p class="indent"> The enumeration of <!--l. 425--><math
2928
xmlns="http://www.w3.org/1998/Math/MathML"
2929
mode="inline"> <msub
2931
class="MathClass-ord">H</mi><mrow
2933
class="MathClass-ord">c</mi></mrow></msub
2935
in a <!--l. 425--><math
2936
xmlns="http://www.w3.org/1998/Math/MathML"
2937
mode="inline"> <msub
2939
class="MathClass-ord">K</mi><mrow
2942
class="MathClass-ord">n</mi><mrow
2944
class="MathClass-ord">1</mn></mrow></msub
2946
class="MathClass-rel">⋯</mo><msub
2948
class="MathClass-ord">n</mi><mrow
2950
class="MathClass-ord">p</mi></mrow></msub
2953
graph can also be carried out by Theorem  <a
2954
href="#x1-9005r2">7.2<!--tex4ht:ref: thm-H-param--></a> or  <a
2955
href="#x1-9012r3">7.3<!--tex4ht:ref: thm-asym--></a> together with the algebraic
2957
href="#x1-2002r2">2<!--tex4ht:ref: multdef--></a>). Some elegant representations may be obtained. For example, <!--l. 428--><math
2958
xmlns="http://www.w3.org/1998/Math/MathML"
2962
class="MathClass-ord">H</mi><mrow
2964
class="MathClass-ord">c</mi></mrow></msub
2965
></math> in a <!--l. 429--><math
2966
xmlns="http://www.w3.org/1998/Math/MathML"
2970
class="MathClass-ord">K</mi><mrow
2973
class="MathClass-ord">n</mi><mrow
2975
class="MathClass-ord">1</mn></mrow></msub
2978
class="MathClass-ord">n</mi><mrow
2980
class="MathClass-ord">2</mn></mrow></msub
2983
class="MathClass-ord">n</mi><mrow
2985
class="MathClass-ord">3</mn></mrow></msub
2988
may be written </p><table class="equation"><tr><td>
2991
xmlns="http://www.w3.org/1998/Math/MathML"
2994
class="equation"><mtr><mtd>
2995
<mtable class="split"><mtr><mtd>
2997
class="split-mtr"></mrow><mrow
2998
class="split-mtd"></mrow> <msub
3000
class="MathClass-ord">H</mi><mrow
3002
class="MathClass-ord">c</mi></mrow></msub
3004
class="MathClass-rel">=</mo><mrow
3005
class="split-mtd"></mrow> <mfrac><mrow
3008
class="MathClass-ord">n</mi><mrow
3010
class="MathClass-ord">1</mn></mrow></msub
3012
class="MathClass-ord">!</mi> <msub
3014
class="MathClass-ord">n</mi><mrow
3016
class="MathClass-ord">2</mn></mrow></msub
3018
class="MathClass-ord">!</mi> <msub
3020
class="MathClass-ord">n</mi><mrow
3022
class="MathClass-ord">3</mn></mrow></msub
3024
class="MathClass-ord">!</mi></mrow>
3028
class="MathClass-ord">n</mi><mrow
3030
class="MathClass-ord">1</mn></mrow></msub
3032
class="MathClass-bin">+</mo> <msub
3034
class="MathClass-ord">n</mi><mrow
3036
class="MathClass-ord">2</mn></mrow></msub
3038
class="MathClass-bin">+</mo> <msub
3040
class="MathClass-ord">n</mi><mrow
3042
class="MathClass-ord">3</mn></mrow></msub
3043
></mrow></mfrac><msub
3045
class="MathClass-op">∑</mo>
3048
class="MathClass-ord">i</mi></mrow></msub
3050
open="[" close="" ><mfenced open="(" close=")" class="binom"><mover
3051
class="binom"><mrow><msub
3053
class="MathClass-ord">n</mi><mrow
3055
class="MathClass-ord">1</mn></mrow></msub
3057
class="MathClass-ord">i</mi></mrow></mover></mfenced><mfenced open="(" close=")" class="binom"><mover
3058
class="binom"><mrow><msub
3060
class="MathClass-ord">n</mi><mrow
3062
class="MathClass-ord">2</mn></mrow></msub
3065
class="MathClass-ord">n</mi><mrow
3067
class="MathClass-ord">3</mn></mrow></msub
3069
class="MathClass-bin">−</mo> <msub
3071
class="MathClass-ord">n</mi><mrow
3073
class="MathClass-ord">1</mn></mrow></msub
3075
class="MathClass-bin">+</mo> <mi
3076
class="MathClass-ord">i</mi></mrow></mover></mfenced><mfenced open="(" close=")" class="binom"><mover
3077
class="binom"><mrow><msub
3079
class="MathClass-ord">n</mi><mrow
3081
class="MathClass-ord">3</mn></mrow></msub
3084
class="MathClass-ord">n</mi><mrow
3086
class="MathClass-ord">3</mn></mrow></msub
3088
class="MathClass-bin">−</mo> <msub
3090
class="MathClass-ord">n</mi><mrow
3092
class="MathClass-ord">2</mn></mrow></msub
3094
class="MathClass-bin">+</mo> <mi
3095
class="MathClass-ord">i</mi></mrow></mover></mfenced></mfenced>
3097
class="split-mtr"></mrow><mrow
3098
class="split-mtd"></mrow> <mrow
3099
class="split-mtd"></mrow> <mo
3100
class="MathClass-bin">+</mo> <mfenced
3101
open="" close="]" ><mfenced open="(" close=")" class="binom"><mover
3102
class="binom"><mrow><msub
3104
class="MathClass-ord">n</mi><mrow
3106
class="MathClass-ord">1</mn></mrow></msub
3108
class="MathClass-bin">−</mo> <mn
3109
class="MathClass-ord">1</mn></mrow><mrow><mi
3110
class="MathClass-ord">i</mi></mrow></mover></mfenced><mfenced open="(" close=")" class="binom"><mover
3111
class="binom"><mrow><msub
3113
class="MathClass-ord">n</mi><mrow
3115
class="MathClass-ord">2</mn></mrow></msub
3117
class="MathClass-bin">−</mo> <mn
3118
class="MathClass-ord">1</mn></mrow><mrow><msub
3120
class="MathClass-ord">n</mi><mrow
3122
class="MathClass-ord">3</mn></mrow></msub
3124
class="MathClass-bin">−</mo> <msub
3126
class="MathClass-ord">n</mi><mrow
3128
class="MathClass-ord">1</mn></mrow></msub
3130
class="MathClass-bin">+</mo> <mi
3131
class="MathClass-ord">i</mi></mrow></mover></mfenced><mfenced open="(" close=")" class="binom"><mover
3132
class="binom"><mrow><msub
3134
class="MathClass-ord">n</mi><mrow
3136
class="MathClass-ord">3</mn></mrow></msub
3138
class="MathClass-bin">−</mo> <mn
3139
class="MathClass-ord">1</mn></mrow><mrow><msub
3141
class="MathClass-ord">n</mi><mrow
3143
class="MathClass-ord">3</mn></mrow></msub
3145
class="MathClass-bin">−</mo> <msub
3147
class="MathClass-ord">n</mi><mrow
3149
class="MathClass-ord">2</mn></mrow></msub
3151
class="MathClass-bin">+</mo> <mi
3152
class="MathClass-ord">i</mi></mrow></mover></mfenced></mfenced> <mo
3153
class="MathClass-punc">.</mo>
3154
</mtd></mtr></mtable> </mtd><mtd><mspace
3155
id="x1-4005r23" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /></mtd></mtr></mtable>
3157
<!--l. 439--><p class="nopar"></p></td><td width="5%">(23)</td></tr></table>
3158
<h3 class="sectionHead"><span class="titlemark">5</span> <a
3159
name="x1-50005"></a>Secret Key Exchanges</h3>
3160
<!--l. 444--><p class="noindent">Modern cryptography is fundamentally concerned with the problem of secure private
3161
communication. A Secret Key Exchange is a protocol where Alice and Bob, having no
3162
secret information in common to start, are able to agree on a common secret key,
3163
conversing over a public channel. The notion of a Secret Key Exchange protocol was
3164
first introduced in the seminal paper of Diffie and Hellman <span class="cite">[<a
3165
href="#Xdihe:newdir">1</a>]</span>. <span class="cite">[<a
3166
href="#Xdihe:newdir">1</a>]</span> presented a
3167
concrete implementation of a Secret Key Exchange protocol, dependent on a specific
3168
assumption (a variant on the discrete log), specially tailored to yield Secret Key
3169
Exchange. Secret Key Exchange is of course trivial if trapdoor permutations
3170
exist. However, there is no known implementation based on a weaker general
3172
</p><!--l. 457--><p class="indent"> The concept of an informationally one-way function was introduced in <span class="cite">[<a
3173
href="#Ximlelu:oneway">5</a>]</span>. We
3174
give only an informal definition here:
3176
<div class="newtheorem">
3177
<!--l. 460--><p class="noindent"><span class="head">
3179
name="x1-5001r1"></a>
3181
class="cmbx-10">Definition 5.1.</span> </span>A polynomial time computable function <!--l. 461--><math
3182
xmlns="http://www.w3.org/1998/Math/MathML"
3184
class="MathClass-ord">f</mi> <mo
3185
class="MathClass-rel">=</mo> <mrow><mo
3186
class="MathClass-open">{</mo><msub
3188
class="MathClass-ord">f</mi><mrow
3190
class="MathClass-ord">k</mi></mrow></msub
3192
class="MathClass-close">}</mo></mrow></math>
3193
is informationally one-way if there is no probabilistic polynomial time algorithm
3194
which (with probability of the form <!--l. 463--><math
3195
xmlns="http://www.w3.org/1998/Math/MathML"
3197
class="MathClass-ord">1</mn> <mo
3198
class="MathClass-bin">−</mo> <msup
3200
class="MathClass-ord">k</mi><mrow
3202
class="MathClass-bin">−</mo><mi
3203
class="MathClass-ord">e</mi></mrow></msup
3205
for some <!--l. 463--><math
3206
xmlns="http://www.w3.org/1998/Math/MathML"
3208
class="MathClass-ord">e</mi> <mo
3209
class="MathClass-rel">></mo> <mn
3210
class="MathClass-ord">0</mn></math>)
3211
returns on input <!--l. 464--><math
3212
xmlns="http://www.w3.org/1998/Math/MathML"
3214
class="MathClass-ord">y</mi> <mo
3215
class="MathClass-rel">∈</mo> <msup
3217
class="MathClass-open">{</mo><mn
3218
class="MathClass-ord">0</mn><mo
3219
class="MathClass-punc">,</mo> <mn
3220
class="MathClass-ord">1</mn><mo
3221
class="MathClass-close">}</mo></mrow><mrow
3223
class="MathClass-ord">k</mi></mrow></msup
3225
a random element of <!--l. 464--><math
3226
xmlns="http://www.w3.org/1998/Math/MathML"
3227
mode="inline"> <msup
3229
class="MathClass-ord">f</mi><mrow
3231
class="MathClass-bin">−</mo><mn
3232
class="MathClass-ord">1</mn></mrow></msup
3234
class="MathClass-open">(</mo><mi
3235
class="MathClass-ord">y</mi><mo
3236
class="MathClass-close">)</mo></mrow></math>.
3240
<!--l. 466--><p class="indent"> In the non-uniform setting <span class="cite">[<a
3241
href="#Ximlelu:oneway">5</a>]</span> show that these are not weaker than one-way
3242
functions: </p><div class="newtheorem">
3243
<!--l. 468--><p class="noindent"><span class="head">
3245
name="x1-5002r1"></a>
3247
class="cmbx-10">Theorem 5.1 (</span><span class="cite"><span
3248
class="cmbx-10">[</span><a
3249
href="#Ximlelu:oneway"><span
3250
class="cmbx-10">5</span></a><span
3251
class="cmbx-10">]</span></span> <span
3252
class="cmbx-10">(non-uniform)).</span> </span> <span
3253
class="cmti-10">The existence of informationally</span>
3255
class="cmti-10">one-way functions implies the existence of one-way functions.</span>
3258
<!--l. 473--><p class="indent"> We will stick to the convention introduced above of saying “non-uniform” before
3259
the theorem statement when the theorem makes use of non-uniformity. It should be
3260
understood that if nothing is said then the result holds for both the uniform and the
3262
</p><!--l. 479--><p class="indent"> It now follows from Theorem <a
3263
href="#x1-5002r1">5.1<!--tex4ht:ref: th-info-ow-ow--></a> that
3265
<div class="newtheorem">
3266
<!--l. 481--><p class="noindent"><span class="head">
3268
name="x1-5003r2"></a>
3270
class="cmbx-10">Theorem 5.2 (non-uniform).</span> </span> <span
3271
class="cmti-10">Weak SKE implies the existence of a one-way</span>
3273
class="cmti-10">function.</span>
3276
<!--l. 485--><p class="indent"> More recently, the polynomial-time, interior point algorithms for linear
3277
programming have been extended to the case of convex quadratic programs
3278
<span class="cite">[<a
3279
href="#Xmoad:quadpro">11</a>, <a
3280
href="#Xye:intalg">13</a>]</span>, certain linear complementarity problems <span class="cite">[<a
3281
href="#Xkomiyo:lincomp">7</a>, <a
3282
href="#Xmiyoki:lincomp">10</a>]</span>, and the nonlinear
3283
complementarity problem <span class="cite">[<a
3284
href="#Xkomiyo:unipfunc">6</a>]</span>. The connection between these algorithms and
3285
the classical Newton method for nonlinear equations is well explained in
3286
<span class="cite">[<a
3287
href="#Xkomiyo:lincomp">7</a>]</span>.
3289
<h3 class="sectionHead"><span class="titlemark">6</span> <a
3290
name="x1-60006"></a>Review</h3>
3291
<!--l. 496--><p class="noindent">We begin our discussion with the following definition:
3293
<div class="newtheorem">
3294
<!--l. 498--><p class="noindent"><span class="head">
3296
name="x1-6001r1"></a>
3298
class="cmbx-10">Definition 6.1.</span> </span>
3299
</p><!--l. 500--><p class="indent"> A function <!--l. 500--><math
3300
xmlns="http://www.w3.org/1998/Math/MathML"
3302
class="MathClass-ord">H</mi> <mo
3303
class="MathClass-punc">:</mo> <msup
3305
class="MathClass-ord">ℜ</mi><mrow
3307
class="MathClass-ord">n</mi></mrow></msup
3309
class="MathClass-rel">→</mo> <msup
3311
class="MathClass-ord">ℜ</mi><mrow
3313
class="MathClass-ord">n</mi></mrow></msup
3316
class="cmti-10">B-differentiable </span>at the point <!--l. 501--><math
3317
xmlns="http://www.w3.org/1998/Math/MathML"
3319
class="MathClass-ord">z</mi></math>
3320
if (i) <!--l. 501--><math
3321
xmlns="http://www.w3.org/1998/Math/MathML"
3323
class="MathClass-ord">H</mi></math>
3325
is Lipschitz continuous in a neighborhood of <!--l. 502--><math
3326
xmlns="http://www.w3.org/1998/Math/MathML"
3328
class="MathClass-ord">z</mi></math>,
3329
and (ii)  there exists a positive homogeneous function <!--l. 503--><math
3330
xmlns="http://www.w3.org/1998/Math/MathML"
3332
class="MathClass-ord">B</mi><mi
3333
class="MathClass-ord">H</mi><mrow><mo
3334
class="MathClass-open">(</mo><mi
3335
class="MathClass-ord">z</mi><mo
3336
class="MathClass-close">)</mo></mrow><mo
3337
class="MathClass-punc">:</mo> <msup
3339
class="MathClass-ord">ℜ</mi><mrow
3341
class="MathClass-ord">n</mi></mrow></msup
3343
class="MathClass-rel">→</mo> <msup
3345
class="MathClass-ord">ℜ</mi><mrow
3347
class="MathClass-ord">n</mi></mrow></msup
3350
class="cmti-10">B-derivative </span>of <!--l. 504--><math
3351
xmlns="http://www.w3.org/1998/Math/MathML"
3353
class="MathClass-ord">H</mi></math>
3354
at <!--l. 504--><math
3355
xmlns="http://www.w3.org/1998/Math/MathML"
3357
class="MathClass-ord">z</mi></math>,
3358
such that <!--l. 505--><math
3359
xmlns="http://www.w3.org/1998/Math/MathML"
3360
mode="display"> <mrow
3366
class="MathClass-ord">v</mi><mo
3367
class="MathClass-rel">→</mo><mn
3368
class="MathClass-ord">0</mn></mrow></msub
3371
class="MathClass-ord">H</mi><mrow><mo
3372
class="MathClass-open">(</mo><mi
3373
class="MathClass-ord">z</mi> <mo
3374
class="MathClass-bin">+</mo> <mi
3375
class="MathClass-ord">v</mi><mo
3376
class="MathClass-close">)</mo></mrow> <mo
3377
class="MathClass-bin">−</mo> <mi
3378
class="MathClass-ord">H</mi><mrow><mo
3379
class="MathClass-open">(</mo><mi
3380
class="MathClass-ord">z</mi><mo
3381
class="MathClass-close">)</mo></mrow> <mo
3382
class="MathClass-bin">−</mo> <mi
3383
class="MathClass-ord">B</mi><mi
3384
class="MathClass-ord">H</mi><mrow><mo
3385
class="MathClass-open">(</mo><mi
3386
class="MathClass-ord">z</mi><mo
3387
class="MathClass-close">)</mo></mrow><mi
3388
class="MathClass-ord">v</mi></mrow>
3391
open="|" close="|" ><mi
3392
class="MathClass-ord">v</mi></mfenced> </mrow></mfrac> <mo
3393
class="MathClass-rel">=</mo> <mn
3394
class="MathClass-ord">0</mn><mo
3395
class="MathClass-punc">.</mo>
3397
The function <!--l. 506--><math
3398
xmlns="http://www.w3.org/1998/Math/MathML"
3400
class="MathClass-ord">H</mi></math>
3402
class="cmti-10">B-differentiable in set </span><!--l. 506--><math
3403
xmlns="http://www.w3.org/1998/Math/MathML"
3405
class="MathClass-ord">S</mi></math>
3406
if it is B-differentiable at every point in <!--l. 507--><math
3407
xmlns="http://www.w3.org/1998/Math/MathML"
3409
class="MathClass-ord">S</mi></math>.
3410
The B-derivative <!--l. 507--><math
3411
xmlns="http://www.w3.org/1998/Math/MathML"
3413
class="MathClass-ord">B</mi><mi
3414
class="MathClass-ord">H</mi><mrow><mo
3415
class="MathClass-open">(</mo><mi
3416
class="MathClass-ord">z</mi><mo
3417
class="MathClass-close">)</mo></mrow></math>
3419
class="cmti-10">strong </span>if <!--l. 509--><math
3420
xmlns="http://www.w3.org/1998/Math/MathML"
3421
mode="display"> <mrow
3427
class="MathClass-open">(</mo><mi
3428
class="MathClass-ord">v</mi><mo
3429
class="MathClass-punc">,</mo><mi
3430
class="MathClass-ord">v</mi><mi
3431
class="MathClass-ord">′</mi><mo
3432
class="MathClass-close">)</mo></mrow><mo
3433
class="MathClass-rel">→</mo><mrow><mo
3434
class="MathClass-open">(</mo><mn
3435
class="MathClass-ord">0</mn><mo
3436
class="MathClass-punc">,</mo><mn
3437
class="MathClass-ord">0</mn><mo
3438
class="MathClass-close">)</mo></mrow></mrow></msub
3441
class="MathClass-ord">H</mi><mrow><mo
3442
class="MathClass-open">(</mo><mi
3443
class="MathClass-ord">z</mi> <mo
3444
class="MathClass-bin">+</mo> <mi
3445
class="MathClass-ord">v</mi><mo
3446
class="MathClass-close">)</mo></mrow> <mo
3447
class="MathClass-bin">−</mo> <mi
3448
class="MathClass-ord">H</mi><mrow><mo
3449
class="MathClass-open">(</mo><mi
3450
class="MathClass-ord">z</mi> <mo
3451
class="MathClass-bin">+</mo> <mi
3452
class="MathClass-ord">v</mi><mi
3453
class="MathClass-ord">′</mi><mo
3454
class="MathClass-close">)</mo></mrow> <mo
3455
class="MathClass-bin">−</mo> <mi
3456
class="MathClass-ord">B</mi><mi
3457
class="MathClass-ord">H</mi><mrow><mo
3458
class="MathClass-open">(</mo><mi
3459
class="MathClass-ord">z</mi><mo
3460
class="MathClass-close">)</mo></mrow><mrow><mo
3461
class="MathClass-open">(</mo><mi
3462
class="MathClass-ord">v</mi> <mo
3463
class="MathClass-bin">−</mo> <mi
3464
class="MathClass-ord">v</mi><mi
3465
class="MathClass-ord">′</mi><mo
3466
class="MathClass-close">)</mo></mrow></mrow>
3469
open="|" close="|" ><mi
3470
class="MathClass-ord">v</mi> <mo
3471
class="MathClass-bin">−</mo> <mi
3472
class="MathClass-ord">v</mi><mi
3473
class="MathClass-ord">′</mi> </mfenced> </mrow></mfrac> <mo
3474
class="MathClass-rel">=</mo> <mn
3475
class="MathClass-ord">0</mn><mo
3476
class="MathClass-punc">.</mo>
3480
<div class="newtheorem">
3481
<!--l. 514--><p class="noindent"><span class="head">
3483
name="x1-6002r1"></a>
3485
class="cmbx-10">Lemma 6.1.</span> </span> <span
3486
class="cmti-10">There exists a smooth function </span><!--l. 514--><math
3487
xmlns="http://www.w3.org/1998/Math/MathML"
3491
class="MathClass-ord">ψ</mi><mrow
3493
class="MathClass-ord">0</mn></mrow></msub
3495
class="MathClass-open">(</mo><mi
3496
class="MathClass-ord">z</mi><mo
3497
class="MathClass-close">)</mo></mrow></math> <span
3498
class="cmti-10">defined for</span>
3500
xmlns="http://www.w3.org/1998/Math/MathML"
3501
mode="inline"> <mfenced
3502
open="|" close="|" ><mi
3503
class="MathClass-ord">z</mi></mfenced> <mo
3504
class="MathClass-rel">></mo> <mn
3505
class="MathClass-ord">1</mn> <mo
3506
class="MathClass-bin">−</mo> <mn
3507
class="MathClass-ord">2</mn><mi
3508
class="MathClass-ord">a</mi></math>
3510
class="cmti-10">satisfying the following properties</span>:
3512
</p><ol type="1" class="enumerate1"
3514
<li class="enumerate"><a
3515
name="x1-6004x1"></a><!--l. 518--><math
3516
xmlns="http://www.w3.org/1998/Math/MathML"
3517
mode="inline"> <msub
3519
class="MathClass-ord">ψ</mi><mrow
3521
class="MathClass-ord">0</mn></mrow></msub
3523
class="MathClass-open">(</mo><mi
3524
class="MathClass-ord">z</mi><mo
3525
class="MathClass-close">)</mo></mrow></math>
3527
class="cmti-10">is bounded above and below by positive constants </span><!--l. 519--><math
3528
xmlns="http://www.w3.org/1998/Math/MathML"
3529
mode="inline"> <msub
3531
class="MathClass-ord">c</mi><mrow
3533
class="MathClass-ord">1</mn></mrow></msub
3535
class="MathClass-rel">≤</mo> <msub
3537
class="MathClass-ord">ψ</mi><mrow
3539
class="MathClass-ord">0</mn></mrow></msub
3541
class="MathClass-open">(</mo><mi
3542
class="MathClass-ord">z</mi><mo
3543
class="MathClass-close">)</mo></mrow> <mo
3544
class="MathClass-rel">≤</mo> <msub
3546
class="MathClass-ord">c</mi><mrow
3548
class="MathClass-ord">2</mn></mrow></msub
3550
class="cmti-10">.</span>
3552
<li class="enumerate"><a
3553
name="x1-6006x2"></a><span
3554
class="cmti-10">If </span><!--l. 520--><math
3555
xmlns="http://www.w3.org/1998/Math/MathML"
3556
mode="inline"> <mfenced
3557
open="|" close="|" ><mi
3558
class="MathClass-ord">z</mi></mfenced> <mo
3559
class="MathClass-rel">></mo> <mn
3560
class="MathClass-ord">1</mn></math><span
3561
class="cmti-10">,</span>
3563
class="cmti-10">then </span><!--l. 520--><math
3564
xmlns="http://www.w3.org/1998/Math/MathML"
3565
mode="inline"> <msub
3567
class="MathClass-ord">ψ</mi><mrow
3569
class="MathClass-ord">0</mn></mrow></msub
3571
class="MathClass-open">(</mo><mi
3572
class="MathClass-ord">z</mi><mo
3573
class="MathClass-close">)</mo></mrow> <mo
3574
class="MathClass-rel">=</mo> <mn
3575
class="MathClass-ord">1</mn></math><span
3576
class="cmti-10">.</span>
3578
<li class="enumerate"><a
3579
name="x1-6008x3"></a><span
3580
class="cmti-10">For all </span><!--l. 521--><math
3581
xmlns="http://www.w3.org/1998/Math/MathML"
3583
class="MathClass-ord">z</mi></math>
3585
class="cmti-10">in the domain of </span><!--l. 521--><math
3586
xmlns="http://www.w3.org/1998/Math/MathML"
3587
mode="inline"> <msub
3589
class="MathClass-ord">ψ</mi><mrow
3591
class="MathClass-ord">0</mn></mrow></msub
3593
class="cmti-10">,</span>
3595
xmlns="http://www.w3.org/1998/Math/MathML"
3596
mode="inline"> <msub
3598
class="MathClass-ord">Δ</mi><mrow
3600
class="MathClass-ord">0</mn></mrow></msub
3602
> ln</mo><!--nolimits--> <msub
3604
class="MathClass-ord">ψ</mi><mrow
3606
class="MathClass-ord">0</mn></mrow></msub
3608
class="MathClass-rel">≥</mo> <mn
3609
class="MathClass-ord">0</mn></math><span
3610
class="cmti-10">.</span>
3612
<li class="enumerate"><a
3613
name="x1-6010x4"></a><span
3614
class="cmti-10">If </span><!--l. 522--><math
3615
xmlns="http://www.w3.org/1998/Math/MathML"
3617
class="MathClass-ord">1</mn> <mo
3618
class="MathClass-bin">−</mo> <mn
3619
class="MathClass-ord">2</mn><mi
3620
class="MathClass-ord">a</mi> <mo
3621
class="MathClass-rel"><</mo> <mfenced
3622
open="|" close="|" ><mi
3623
class="MathClass-ord">z</mi></mfenced> <mo
3624
class="MathClass-rel"><</mo> <mn
3625
class="MathClass-ord">1</mn> <mo
3626
class="MathClass-bin">−</mo> <mi
3627
class="MathClass-ord">a</mi></math><span
3628
class="cmti-10">,</span>
3630
class="cmti-10">then </span><!--l. 522--><math
3631
xmlns="http://www.w3.org/1998/Math/MathML"
3632
mode="inline"> <msub
3634
class="MathClass-ord">Δ</mi><mrow
3636
class="MathClass-ord">0</mn></mrow></msub
3638
> ln</mo><!--nolimits--> <msub
3640
class="MathClass-ord">ψ</mi><mrow
3642
class="MathClass-ord">0</mn></mrow></msub
3644
class="MathClass-rel">≥</mo> <msub
3646
class="MathClass-ord">c</mi><mrow
3648
class="MathClass-ord">3</mn></mrow></msub
3650
class="MathClass-rel">></mo> <mn
3651
class="MathClass-ord">0</mn></math><span
3652
class="cmti-10">.</span></li></ol>
3654
<!--l. 527--><p class="indent">
3656
<div class="proof"><span class="head">
3658
class="cmti-10">Proof.</span> </span>We choose <!--l. 528--><math
3659
xmlns="http://www.w3.org/1998/Math/MathML"
3660
mode="inline"> <msub
3662
class="MathClass-ord">ψ</mi><mrow
3664
class="MathClass-ord">0</mn></mrow></msub
3666
class="MathClass-open">(</mo><mi
3667
class="MathClass-ord">z</mi><mo
3668
class="MathClass-close">)</mo></mrow></math>
3669
to be a radial function depending only on <!--l. 528--><math
3670
xmlns="http://www.w3.org/1998/Math/MathML"
3672
class="MathClass-ord">r</mi> <mo
3673
class="MathClass-rel">=</mo> <mfenced
3674
open="|" close="|" ><mi
3675
class="MathClass-ord">z</mi></mfenced></math>.
3676
Let <!--l. 529--><math
3677
xmlns="http://www.w3.org/1998/Math/MathML"
3679
class="MathClass-ord">h</mi><mrow><mo
3680
class="MathClass-open">(</mo><mi
3681
class="MathClass-ord">r</mi><mo
3682
class="MathClass-close">)</mo></mrow> <mo
3683
class="MathClass-rel">≥</mo> <mn
3684
class="MathClass-ord">0</mn></math>
3685
be a suitable smooth function satisfying <!--l. 529--><math
3686
xmlns="http://www.w3.org/1998/Math/MathML"
3688
class="MathClass-ord">h</mi><mrow><mo
3689
class="MathClass-open">(</mo><mi
3690
class="MathClass-ord">r</mi><mo
3691
class="MathClass-close">)</mo></mrow> <mo
3692
class="MathClass-rel">≥</mo> <msub
3694
class="MathClass-ord">c</mi><mrow
3696
class="MathClass-ord">3</mn></mrow></msub
3698
for <!--l. 530--><math
3699
xmlns="http://www.w3.org/1998/Math/MathML"
3701
class="MathClass-ord">1</mn> <mo
3702
class="MathClass-bin">−</mo> <mn
3703
class="MathClass-ord">2</mn><mi
3704
class="MathClass-ord">a</mi> <mo
3705
class="MathClass-rel"><</mo> <mfenced
3706
open="|" close="|" ><mi
3707
class="MathClass-ord">z</mi></mfenced> <mo
3708
class="MathClass-rel"><</mo> <mn
3709
class="MathClass-ord">1</mn> <mo
3710
class="MathClass-bin">−</mo> <mi
3711
class="MathClass-ord">a</mi></math>,
3712
and <!--l. 530--><math
3713
xmlns="http://www.w3.org/1998/Math/MathML"
3715
class="MathClass-ord">h</mi><mrow><mo
3716
class="MathClass-open">(</mo><mi
3717
class="MathClass-ord">r</mi><mo
3718
class="MathClass-close">)</mo></mrow> <mo
3719
class="MathClass-rel">=</mo> <mn
3720
class="MathClass-ord">0</mn></math>
3721
for <!--l. 530--><math
3722
xmlns="http://www.w3.org/1998/Math/MathML"
3723
mode="inline"> <mfenced
3724
open="|" close="|" ><mi
3725
class="MathClass-ord">z</mi></mfenced> <mo
3726
class="MathClass-rel">></mo> <mn
3727
class="MathClass-ord">1</mn> <mo
3728
class="MathClass-bin">−</mo> <mfrac class="tfrac"><mrow><mi
3729
class="MathClass-ord">a</mi></mrow><mrow><mn
3730
class="MathClass-ord">2</mn></mrow></mfrac></math>.
3731
The radial Laplacian <!--l. 532--><math
3732
xmlns="http://www.w3.org/1998/Math/MathML"
3733
mode="display"> <mrow
3737
class="MathClass-ord">Δ</mi><mrow
3739
class="MathClass-ord">0</mn></mrow></msub
3741
> ln</mo><!--nolimits--> <msub
3743
class="MathClass-ord">ψ</mi><mrow
3745
class="MathClass-ord">0</mn></mrow></msub
3747
class="MathClass-open">(</mo><mi
3748
class="MathClass-ord">r</mi><mo
3749
class="MathClass-close">)</mo></mrow> <mo
3750
class="MathClass-rel">=</mo> <mfenced
3751
open="(" close=")" > <mfrac><mrow
3754
class="MathClass-ord">d</mi><mrow
3756
class="MathClass-ord">2</mn></mrow></msup
3760
class="MathClass-ord">d</mi><msup
3762
class="MathClass-ord">r</mi><mrow
3764
class="MathClass-ord">2</mn></mrow></msup
3765
></mrow></mfrac> <mo
3766
class="MathClass-bin">+</mo> <mfrac><mrow
3768
class="MathClass-ord">1</mn></mrow>
3771
class="MathClass-ord">r</mi></mrow></mfrac> <mfrac><mrow
3773
class="MathClass-ord">d</mi></mrow>
3776
class="MathClass-ord">d</mi><mi
3777
class="MathClass-ord">r</mi></mrow></mfrac></mfenced><mo
3778
> ln</mo><!--nolimits--> <msub
3780
class="MathClass-ord">ψ</mi><mrow
3782
class="MathClass-ord">0</mn></mrow></msub
3784
class="MathClass-open">(</mo><mi
3785
class="MathClass-ord">r</mi><mo
3786
class="MathClass-close">)</mo></mrow>
3788
has smooth coefficients for <!--l. 534--><math
3789
xmlns="http://www.w3.org/1998/Math/MathML"
3791
class="MathClass-ord">r</mi> <mo
3792
class="MathClass-rel">></mo> <mn
3793
class="MathClass-ord">1</mn> <mo
3794
class="MathClass-bin">−</mo> <mn
3795
class="MathClass-ord">2</mn><mi
3796
class="MathClass-ord">a</mi></math>.
3797
Therefore, we may apply the existence and uniqueness theory for ordinary
3798
differential equations. Simply let <!--l. 536--><math
3799
xmlns="http://www.w3.org/1998/Math/MathML"
3801
>ln</mo><!--nolimits--> <msub
3803
class="MathClass-ord">ψ</mi><mrow
3805
class="MathClass-ord">0</mn></mrow></msub
3807
class="MathClass-open">(</mo><mi
3808
class="MathClass-ord">r</mi><mo
3809
class="MathClass-close">)</mo></mrow></math>
3811
be the solution of the differential equation <!--l. 538--><math
3812
xmlns="http://www.w3.org/1998/Math/MathML"
3813
mode="display"> <mrow
3816
open="(" close=")" > <mfrac><mrow
3819
class="MathClass-ord">d</mi><mrow
3821
class="MathClass-ord">2</mn></mrow></msup
3825
class="MathClass-ord">d</mi><msup
3827
class="MathClass-ord">r</mi><mrow
3829
class="MathClass-ord">2</mn></mrow></msup
3830
></mrow></mfrac> <mo
3831
class="MathClass-bin">+</mo> <mfrac><mrow
3833
class="MathClass-ord">1</mn></mrow>
3836
class="MathClass-ord">r</mi></mrow></mfrac> <mfrac><mrow
3838
class="MathClass-ord">d</mi></mrow>
3841
class="MathClass-ord">d</mi><mi
3842
class="MathClass-ord">r</mi></mrow></mfrac></mfenced><mo
3843
> ln</mo><!--nolimits--> <msub
3845
class="MathClass-ord">ψ</mi><mrow
3847
class="MathClass-ord">0</mn></mrow></msub
3849
class="MathClass-open">(</mo><mi
3850
class="MathClass-ord">r</mi><mo
3851
class="MathClass-close">)</mo></mrow> <mo
3852
class="MathClass-rel">=</mo> <mi
3853
class="MathClass-ord">h</mi><mrow><mo
3854
class="MathClass-open">(</mo><mi
3855
class="MathClass-ord">r</mi><mo
3856
class="MathClass-close">)</mo></mrow>
3858
with initial conditions given by <!--l. 539--><math
3859
xmlns="http://www.w3.org/1998/Math/MathML"
3861
>ln</mo><!--nolimits--> <msub
3863
class="MathClass-ord">ψ</mi><mrow
3865
class="MathClass-ord">0</mn></mrow></msub
3867
class="MathClass-open">(</mo><mn
3868
class="MathClass-ord">1</mn><mo
3869
class="MathClass-close">)</mo></mrow> <mo
3870
class="MathClass-rel">=</mo> <mn
3871
class="MathClass-ord">0</mn></math>
3872
and <!--l. 540--><math
3873
xmlns="http://www.w3.org/1998/Math/MathML"
3875
>ln</mo><!--nolimits--> <msub
3877
class="MathClass-ord">ψ</mi><mrow
3879
class="MathClass-ord">0</mn></mrow></msub
3881
class="MathClass-ord">′</mi><mrow><mo
3882
class="MathClass-open">(</mo><mn
3883
class="MathClass-ord">1</mn><mo
3884
class="MathClass-close">)</mo></mrow> <mo
3885
class="MathClass-rel">=</mo> <mn
3886
class="MathClass-ord">0</mn></math>.
3887
<!--l. 542--><p class="indent"> Next, let <!--l. 542--><math
3888
xmlns="http://www.w3.org/1998/Math/MathML"
3889
mode="inline"> <msub
3891
class="MathClass-ord">D</mi><mrow
3893
class="MathClass-ord">ν</mi></mrow></msub
3895
be a finite collection of pairwise disjoint disks, all of which
3896
are contained in the unit disk centered at the origin in <!--l. 544--><math
3897
xmlns="http://www.w3.org/1998/Math/MathML"
3900
class="MathClass-ord">C</mi></math>. We assume that <!--l. 544--><math
3901
xmlns="http://www.w3.org/1998/Math/MathML"
3905
class="MathClass-ord">D</mi><mrow
3907
class="MathClass-ord">ν</mi></mrow></msub
3909
class="MathClass-rel">=</mo> <mrow><mo
3910
class="MathClass-open">{</mo><mi
3911
class="MathClass-ord">z</mi> <mo
3912
class="MathClass-rel">∣</mo> <mfenced
3913
open="|" close="|" ><mi
3914
class="MathClass-ord">z</mi> <mo
3915
class="MathClass-bin">−</mo> <msub
3917
class="MathClass-ord">z</mi><mrow
3919
class="MathClass-ord">ν</mi></mrow></msub
3921
class="MathClass-rel"><</mo> <mi
3922
class="MathClass-ord">δ</mi><mo
3923
class="MathClass-close">}</mo></mrow></math>. Suppose that <!--l. 545--><math
3924
xmlns="http://www.w3.org/1998/Math/MathML"
3928
class="MathClass-ord">D</mi><mrow
3930
class="MathClass-ord">ν</mi></mrow></msub
3932
class="MathClass-open">(</mo><mi
3933
class="MathClass-ord">a</mi><mo
3934
class="MathClass-close">)</mo></mrow></math> denotes the smaller concentric
3935
disk <!--l. 545--><math
3936
xmlns="http://www.w3.org/1998/Math/MathML"
3937
mode="inline"> <msub
3939
class="MathClass-ord">D</mi><mrow
3941
class="MathClass-ord">ν</mi></mrow></msub
3943
class="MathClass-open">(</mo><mi
3944
class="MathClass-ord">a</mi><mo
3945
class="MathClass-close">)</mo></mrow> <mo
3946
class="MathClass-rel">=</mo> <mrow><mo
3947
class="MathClass-open">{</mo><mi
3948
class="MathClass-ord">z</mi> <mo
3949
class="MathClass-rel">∣</mo> <mfenced
3950
open="|" close="|" ><mi
3951
class="MathClass-ord">z</mi> <mo
3952
class="MathClass-bin">−</mo> <msub
3954
class="MathClass-ord">z</mi><mrow
3956
class="MathClass-ord">ν</mi></mrow></msub
3958
class="MathClass-rel">≤</mo> <mrow><mo
3959
class="MathClass-open">(</mo><mn
3960
class="MathClass-ord">1</mn> <mo
3961
class="MathClass-bin">−</mo> <mn
3962
class="MathClass-ord">2</mn><mi
3963
class="MathClass-ord">a</mi><mo
3964
class="MathClass-close">)</mo></mrow><mi
3965
class="MathClass-ord">δ</mi><mo
3966
class="MathClass-close">}</mo></mrow></math>. We define a
3967
smooth weight function <!--l. 547--><math
3968
xmlns="http://www.w3.org/1998/Math/MathML"
3969
mode="inline"> <msub
3971
class="MathClass-ord">Φ</mi><mrow
3973
class="MathClass-ord">0</mn></mrow></msub
3975
class="MathClass-open">(</mo><mi
3976
class="MathClass-ord">z</mi><mo
3977
class="MathClass-close">)</mo></mrow></math>
3978
for <!--l. 547--><math
3979
xmlns="http://www.w3.org/1998/Math/MathML"
3981
class="MathClass-ord">z</mi> <mo
3982
class="MathClass-rel">∈</mo> <mi
3983
class="MathClass-ord">C</mi> <mo
3984
class="MathClass-bin">−</mo><msub
3986
class="MathClass-op">⋃</mo>
3989
class="MathClass-ord">ν</mi></mrow></msub
3992
class="MathClass-ord">D</mi><mrow
3994
class="MathClass-ord">ν</mi></mrow></msub
3996
class="MathClass-open">(</mo><mi
3997
class="MathClass-ord">a</mi><mo
3998
class="MathClass-close">)</mo></mrow></math> by
3999
setting <!--l. 547--><math
4000
xmlns="http://www.w3.org/1998/Math/MathML"
4001
mode="inline"> <msub
4003
class="MathClass-ord">Φ</mi><mrow
4005
class="MathClass-ord">0</mn></mrow></msub
4007
class="MathClass-open">(</mo><mi
4008
class="MathClass-ord">z</mi><mo
4009
class="MathClass-close">)</mo></mrow> <mo
4010
class="MathClass-rel">=</mo> <mn
4011
class="MathClass-ord">1</mn></math>
4012
when <!--l. 548--><math
4013
xmlns="http://www.w3.org/1998/Math/MathML"
4015
class="MathClass-ord">z</mi><mo
4016
class="MathClass-rel">∉</mo><msub
4018
class="MathClass-op">⋃</mo>
4021
class="MathClass-ord">ν</mi></mrow></msub
4024
class="MathClass-ord">D</mi><mrow
4026
class="MathClass-ord">ν</mi></mrow></msub
4029
xmlns="http://www.w3.org/1998/Math/MathML"
4030
mode="inline"> <msub
4032
class="MathClass-ord">Φ</mi><mrow
4034
class="MathClass-ord">0</mn></mrow></msub
4036
class="MathClass-open">(</mo><mi
4037
class="MathClass-ord">z</mi><mo
4038
class="MathClass-close">)</mo></mrow> <mo
4039
class="MathClass-rel">=</mo> <msub
4041
class="MathClass-ord">ψ</mi><mrow
4043
class="MathClass-ord">0</mn></mrow></msub
4045
class="MathClass-open">(</mo><mrow><mo
4046
class="MathClass-open">(</mo><mi
4047
class="MathClass-ord">z</mi> <mo
4048
class="MathClass-bin">−</mo> <msub
4050
class="MathClass-ord">z</mi><mrow
4052
class="MathClass-ord">ν</mi></mrow></msub
4054
class="MathClass-close">)</mo></mrow><mo
4055
class="MathClass-bin">/</mo><mi
4056
class="MathClass-ord">δ</mi><mo
4057
class="MathClass-close">)</mo></mrow></math> when <!--l. 549--><math
4058
xmlns="http://www.w3.org/1998/Math/MathML"
4061
class="MathClass-ord">z</mi></math> is an element of <!--l. 549--><math
4062
xmlns="http://www.w3.org/1998/Math/MathML"
4066
class="MathClass-ord">D</mi><mrow
4068
class="MathClass-ord">ν</mi></mrow></msub
4069
></math>. It follows from
4071
href="#x1-6002r1">6.1<!--tex4ht:ref: limbog--></a> that <!--l. 550--><math
4072
xmlns="http://www.w3.org/1998/Math/MathML"
4073
mode="inline"> <msub
4075
class="MathClass-ord">Φ</mi><mrow
4077
class="MathClass-ord">0</mn></mrow></msub
4079
satisfies the properties:
4080
</p><ol type="1" class="enumerate1"
4082
<li class="enumerate"><a
4083
name="x1-6012x1"></a><!--l. 553--><math
4084
xmlns="http://www.w3.org/1998/Math/MathML"
4085
mode="inline"> <msub
4087
class="MathClass-ord">Φ</mi><mrow
4089
class="MathClass-ord">0</mn></mrow></msub
4091
class="MathClass-open">(</mo><mi
4092
class="MathClass-ord">z</mi><mo
4093
class="MathClass-close">)</mo></mrow></math>
4094
is bounded above and below by positive constants <!--l. 554--><math
4095
xmlns="http://www.w3.org/1998/Math/MathML"
4096
mode="inline"> <msub
4098
class="MathClass-ord">c</mi><mrow
4100
class="MathClass-ord">1</mn></mrow></msub
4102
class="MathClass-rel">≤</mo> <msub
4104
class="MathClass-ord">Φ</mi><mrow
4106
class="MathClass-ord">0</mn></mrow></msub
4108
class="MathClass-open">(</mo><mi
4109
class="MathClass-ord">z</mi><mo
4110
class="MathClass-close">)</mo></mrow> <mo
4111
class="MathClass-rel">≤</mo> <msub
4113
class="MathClass-ord">c</mi><mrow
4115
class="MathClass-ord">2</mn></mrow></msub
4118
<li class="enumerate"><a
4119
name="x1-6014x2"></a><!--l. 555--><math
4120
xmlns="http://www.w3.org/1998/Math/MathML"
4121
mode="inline"> <msub
4123
class="MathClass-ord">Δ</mi><mrow
4125
class="MathClass-ord">0</mn></mrow></msub
4127
> ln</mo><!--nolimits--> <msub
4129
class="MathClass-ord">Φ</mi><mrow
4131
class="MathClass-ord">0</mn></mrow></msub
4133
class="MathClass-rel">≥</mo> <mn
4134
class="MathClass-ord">0</mn></math>
4135
for all <!--l. 556--><math
4136
xmlns="http://www.w3.org/1998/Math/MathML"
4138
class="MathClass-ord">z</mi> <mo
4139
class="MathClass-rel">∈</mo> <mi
4140
class="MathClass-ord">C</mi> <mo
4141
class="MathClass-bin">−</mo><msub
4143
class="MathClass-op">⋃</mo>
4146
class="MathClass-ord">ν</mi></mrow></msub
4149
class="MathClass-ord">D</mi><mrow
4151
class="MathClass-ord">ν</mi></mrow></msub
4153
class="MathClass-open">(</mo><mi
4154
class="MathClass-ord">a</mi><mo
4155
class="MathClass-close">)</mo></mrow></math>,
4156
the domain where the function <!--l. 557--><math
4157
xmlns="http://www.w3.org/1998/Math/MathML"
4158
mode="inline"> <msub
4160
class="MathClass-ord">Φ</mi><mrow
4162
class="MathClass-ord">0</mn></mrow></msub
4166
<li class="enumerate"><a
4167
name="x1-6016x3"></a><!--l. 558--><math
4168
xmlns="http://www.w3.org/1998/Math/MathML"
4169
mode="inline"> <msub
4171
class="MathClass-ord">Δ</mi><mrow
4173
class="MathClass-ord">0</mn></mrow></msub
4175
> ln</mo><!--nolimits--> <msub
4177
class="MathClass-ord">Φ</mi><mrow
4179
class="MathClass-ord">0</mn></mrow></msub
4181
class="MathClass-rel">≥</mo> <msub
4183
class="MathClass-ord">c</mi><mrow
4185
class="MathClass-ord">3</mn></mrow></msub
4188
class="MathClass-ord">δ</mi><mrow
4190
class="MathClass-bin">−</mo><mn
4191
class="MathClass-ord">2</mn></mrow></msup
4193
when <!--l. 559--><math
4194
xmlns="http://www.w3.org/1998/Math/MathML"
4195
mode="inline"> <mrow><mo
4196
class="MathClass-open">(</mo><mn
4197
class="MathClass-ord">1</mn> <mo
4198
class="MathClass-bin">−</mo> <mn
4199
class="MathClass-ord">2</mn><mi
4200
class="MathClass-ord">a</mi><mo
4201
class="MathClass-close">)</mo></mrow><mi
4202
class="MathClass-ord">δ</mi> <mo
4203
class="MathClass-rel"><</mo> <mfenced
4204
open="|" close="|" ><mi
4205
class="MathClass-ord">z</mi> <mo
4206
class="MathClass-bin">−</mo> <msub
4208
class="MathClass-ord">z</mi><mrow
4210
class="MathClass-ord">ν</mi></mrow></msub
4212
class="MathClass-rel"><</mo> <mrow><mo
4213
class="MathClass-open">(</mo><mn
4214
class="MathClass-ord">1</mn> <mo
4215
class="MathClass-bin">−</mo> <mi
4216
class="MathClass-ord">a</mi><mo
4217
class="MathClass-close">)</mo></mrow><mi
4218
class="MathClass-ord">δ</mi></math>.</li></ol>
4219
<!--l. 561--><p class="noindent">Let <!--l. 561--><math
4220
xmlns="http://www.w3.org/1998/Math/MathML"
4221
mode="inline"> <msub
4223
class="MathClass-ord">A</mi><mrow
4225
class="MathClass-ord">ν</mi></mrow></msub
4228
annulus <!--l. 561--><math
4229
xmlns="http://www.w3.org/1998/Math/MathML"
4230
mode="inline"> <msub
4232
class="MathClass-ord">A</mi><mrow
4234
class="MathClass-ord">ν</mi></mrow></msub
4236
class="MathClass-rel">=</mo> <mrow><mo
4237
class="MathClass-open">{</mo><mrow><mo
4238
class="MathClass-open">(</mo><mn
4239
class="MathClass-ord">1</mn> <mo
4240
class="MathClass-bin">−</mo> <mn
4241
class="MathClass-ord">2</mn><mi
4242
class="MathClass-ord">a</mi><mo
4243
class="MathClass-close">)</mo></mrow><mi
4244
class="MathClass-ord">δ</mi> <mo
4245
class="MathClass-rel"><</mo> <mfenced
4246
open="|" close="|" ><mi
4247
class="MathClass-ord">z</mi> <mo
4248
class="MathClass-bin">−</mo> <msub
4250
class="MathClass-ord">z</mi><mrow
4252
class="MathClass-ord">ν</mi></mrow></msub
4254
class="MathClass-rel"><</mo> <mrow><mo
4255
class="MathClass-open">(</mo><mn
4256
class="MathClass-ord">1</mn> <mo
4257
class="MathClass-bin">−</mo> <mi
4258
class="MathClass-ord">a</mi><mo
4259
class="MathClass-close">)</mo></mrow><mi
4260
class="MathClass-ord">δ</mi><mo
4261
class="MathClass-close">}</mo></mrow></math>, and set <!--l. 562--><math
4262
xmlns="http://www.w3.org/1998/Math/MathML"
4265
class="MathClass-ord">A</mi> <mo
4266
class="MathClass-rel">=</mo><msub
4268
class="MathClass-op">⋃</mo>
4271
class="MathClass-ord">ν</mi></mrow></msub
4274
class="MathClass-ord">A</mi><mrow
4276
class="MathClass-ord">ν</mi></mrow></msub
4277
></math>. The properties (<a
4278
href="#x1-6014x2">2<!--tex4ht:ref: d:over--></a>)
4280
href="#x1-6016x3">3<!--tex4ht:ref: d:ad--></a>) of <!--l. 563--><math
4281
xmlns="http://www.w3.org/1998/Math/MathML"
4282
mode="inline"> <msub
4284
class="MathClass-ord">Φ</mi><mrow
4286
class="MathClass-ord">0</mn></mrow></msub
4288
summarized as <!--l. 564--><math
4289
xmlns="http://www.w3.org/1998/Math/MathML"
4290
mode="inline"> <msub
4292
class="MathClass-ord">Δ</mi><mrow
4294
class="MathClass-ord">0</mn></mrow></msub
4296
> ln</mo><!--nolimits--> <msub
4298
class="MathClass-ord">Φ</mi><mrow
4300
class="MathClass-ord">0</mn></mrow></msub
4302
class="MathClass-rel">≥</mo> <msub
4304
class="MathClass-ord">c</mi><mrow
4306
class="MathClass-ord">3</mn></mrow></msub
4309
class="MathClass-ord">δ</mi><mrow
4311
class="MathClass-bin">−</mo><mn
4312
class="MathClass-ord">2</mn></mrow></msup
4315
class="MathClass-ord">χ</mi><mrow
4317
class="MathClass-ord">A</mi></mrow></msub
4318
></math>, where <!--l. 565--><math
4319
xmlns="http://www.w3.org/1998/Math/MathML"
4323
class="MathClass-ord">χ</mi><mrow
4325
class="MathClass-ord">A</mi></mrow></msub
4326
></math> is the characteristic
4327
function of <!--l. 565--><math
4328
xmlns="http://www.w3.org/1998/Math/MathML"
4330
class="MathClass-ord">A</mi></math>.
4332
<!--l. 568--><p class="noindent">Suppose that <!--l. 568--><math
4333
xmlns="http://www.w3.org/1998/Math/MathML"
4335
class="MathClass-ord">α</mi></math>
4336
is a nonnegative real constant. We apply Proposition <a
4337
href="#x1-3019r5">3.5<!--tex4ht:ref: prop:eg--></a> with <!--l. 569--><math
4338
xmlns="http://www.w3.org/1998/Math/MathML"
4341
class="MathClass-ord">Φ</mi><mrow><mo
4342
class="MathClass-open">(</mo><mi
4343
class="MathClass-ord">z</mi><mo
4344
class="MathClass-close">)</mo></mrow> <mo
4345
class="MathClass-rel">=</mo> <msub
4347
class="MathClass-ord">Φ</mi><mrow
4349
class="MathClass-ord">0</mn></mrow></msub
4351
class="MathClass-open">(</mo><mi
4352
class="MathClass-ord">z</mi><mo
4353
class="MathClass-close">)</mo></mrow><msup
4355
class="MathClass-ord">e</mi><mrow
4357
class="MathClass-ord">α</mi><msup
4359
open="|" close="|" ><mi
4360
class="MathClass-ord">z</mi> </mfenced><mrow
4362
class="MathClass-ord">2</mn></mrow></msup
4365
></math>. If <!--l. 570--><math
4366
xmlns="http://www.w3.org/1998/Math/MathML"
4369
class="MathClass-ord">u</mi> <mo
4370
class="MathClass-rel">∈</mo> <msubsup
4372
class="MathClass-ord">C</mi><mrow
4374
class="MathClass-ord">0</mn></mrow><mrow
4376
class="MathClass-ord">∞</mi></mrow></msubsup
4378
class="MathClass-open">(</mo><msup
4380
class="MathClass-ord">R</mi><mrow
4382
class="MathClass-ord">2</mn></mrow></msup
4384
class="MathClass-bin">−</mo><msub
4386
class="MathClass-op">⋃</mo>
4389
class="MathClass-ord">ν</mi></mrow></msub
4392
class="MathClass-ord">D</mi><mrow
4394
class="MathClass-ord">ν</mi></mrow></msub
4396
class="MathClass-open">(</mo><mi
4397
class="MathClass-ord">a</mi><mo
4398
class="MathClass-close">)</mo></mrow><mo
4399
class="MathClass-close">)</mo></mrow></math>, assume
4400
that <!--l. 570--><math
4401
xmlns="http://www.w3.org/1998/Math/MathML"
4403
class="MathClass-ord"><!--span
4404
class="htf-calligraphy"-->D<!--/span--></mi></math>
4405
is a bounded domain containing the support of <!--l. 571--><math
4406
xmlns="http://www.w3.org/1998/Math/MathML"
4409
class="MathClass-ord">u</mi></math> and <!--l. 571--><math
4410
xmlns="http://www.w3.org/1998/Math/MathML"
4413
class="MathClass-ord">A</mi> <mo
4414
class="MathClass-rel">⊂</mo> <mi
4415
class="MathClass-ord"><!--span
4416
class="htf-calligraphy"-->D<!--/span--></mi> <mo
4417
class="MathClass-rel">⊂</mo> <msup
4419
class="MathClass-ord">R</mi><mrow
4421
class="MathClass-ord">2</mn></mrow></msup
4423
class="MathClass-bin">−</mo><msub
4425
class="MathClass-op">⋃</mo>
4428
class="MathClass-ord">ν</mi></mrow></msub
4431
class="MathClass-ord">D</mi><mrow
4433
class="MathClass-ord">ν</mi></mrow></msub
4435
class="MathClass-open">(</mo><mi
4436
class="MathClass-ord">a</mi><mo
4437
class="MathClass-close">)</mo></mrow></math>. A calculation
4438
gives <!--l. 573--><math
4439
xmlns="http://www.w3.org/1998/Math/MathML"
4440
mode="display"> <mrow
4444
class="MathClass-op">∫</mo>
4447
class="MathClass-ord"><!--span
4448
class="htf-calligraphy"-->D<!--/span--></mi></mrow></msub
4451
open="|" close="|" ><mover
4452
class="mml-overline"><mrow><mi
4453
class="MathClass-ord">∂</mi></mrow><mo
4454
accent="true">‾</mo></mover><mi
4455
class="MathClass-ord">u</mi></mfenced> <mrow
4457
class="MathClass-ord">2</mn></mrow></msup
4460
class="MathClass-ord">Φ</mi><mrow
4463
class="MathClass-ord">0</mn></mrow></msub
4465
class="MathClass-open">(</mo><mi
4466
class="MathClass-ord">z</mi><mo
4467
class="MathClass-close">)</mo></mrow><msup
4469
class="MathClass-ord">e</mi><mrow
4471
class="MathClass-ord">α</mi><msup
4473
open="|" close="|" ><mi
4474
class="MathClass-ord">z</mi> </mfenced><mrow
4476
class="MathClass-ord">2</mn></mrow></msup
4480
class="MathClass-rel">≥</mo> <msub
4482
class="MathClass-ord">c</mi><mrow
4484
class="MathClass-ord">4</mn></mrow></msub
4486
class="MathClass-ord">α</mi><msub
4488
class="MathClass-op">∫</mo>
4491
class="MathClass-ord"><!--span
4492
class="htf-calligraphy"-->D<!--/span--></mi></mrow></msub
4495
open="|" close="|" ><mi
4496
class="MathClass-ord">u</mi></mfenced> <mrow
4498
class="MathClass-ord">2</mn></mrow></msup
4501
class="MathClass-ord">Φ</mi><mrow
4504
class="MathClass-ord">0</mn></mrow></msub
4507
class="MathClass-ord">e</mi><mrow
4509
class="MathClass-ord">α</mi><msup
4511
open="|" close="|" ><mi
4512
class="MathClass-ord">z</mi> </mfenced><mrow
4514
class="MathClass-ord">2</mn></mrow></msup
4518
class="MathClass-bin">+</mo> <msub
4520
class="MathClass-ord">c</mi><mrow
4522
class="MathClass-ord">5</mn></mrow></msub
4525
class="MathClass-ord">δ</mi><mrow
4527
class="MathClass-bin">−</mo><mn
4528
class="MathClass-ord">2</mn></mrow></msup
4531
class="MathClass-op">∫</mo>
4534
class="MathClass-ord">A</mi></mrow></msub
4537
open="|" close="|" ><mi
4538
class="MathClass-ord">u</mi></mfenced> <mrow
4540
class="MathClass-ord">2</mn></mrow></msup
4543
class="MathClass-ord">Φ</mi><mrow
4546
class="MathClass-ord">0</mn></mrow></msub
4549
class="MathClass-ord">e</mi><mrow
4551
class="MathClass-ord">α</mi><msup
4553
open="|" close="|" ><mi
4554
class="MathClass-ord">z</mi> </mfenced><mrow
4556
class="MathClass-ord">2</mn></mrow></msup
4560
class="MathClass-punc">.</mo>
4562
</p><!--l. 577--><p class="indent"> The boundedness, property (<a
4563
href="#x1-6012x1">1<!--tex4ht:ref: boundab--></a>) of <!--l. 577--><math
4564
xmlns="http://www.w3.org/1998/Math/MathML"
4568
class="MathClass-ord">Φ</mi><mrow
4570
class="MathClass-ord">0</mn></mrow></msub
4571
></math>, then yields <!--l. 578--><math
4572
xmlns="http://www.w3.org/1998/Math/MathML"
4578
class="MathClass-op">∫</mo>
4581
class="MathClass-ord"><!--span
4582
class="htf-calligraphy"-->D<!--/span--></mi></mrow></msub
4585
open="|" close="|" ><mover
4586
class="mml-overline"><mrow><mi
4587
class="MathClass-ord">∂</mi></mrow><mo
4588
accent="true">‾</mo></mover><mi
4589
class="MathClass-ord">u</mi></mfenced> <mrow
4591
class="MathClass-ord">2</mn></mrow></msup
4594
class="MathClass-ord">e</mi><mrow
4596
class="MathClass-ord">α</mi><msup
4598
open="|" close="|" ><mi
4599
class="MathClass-ord">z</mi> </mfenced><mrow
4601
class="MathClass-ord">2</mn></mrow></msup
4605
class="MathClass-rel">≥</mo> <msub
4607
class="MathClass-ord">c</mi><mrow
4609
class="MathClass-ord">6</mn></mrow></msub
4611
class="MathClass-ord">α</mi><msub
4613
class="MathClass-op">∫</mo>
4616
class="MathClass-ord"><!--span
4617
class="htf-calligraphy"-->D<!--/span--></mi></mrow></msub
4620
open="|" close="|" ><mi
4621
class="MathClass-ord">u</mi></mfenced> <mrow
4623
class="MathClass-ord">2</mn></mrow></msup
4626
class="MathClass-ord">e</mi><mrow
4628
class="MathClass-ord">α</mi><msup
4630
open="|" close="|" ><mi
4631
class="MathClass-ord">z</mi> </mfenced><mrow
4633
class="MathClass-ord">2</mn></mrow></msup
4637
class="MathClass-bin">+</mo> <msub
4639
class="MathClass-ord">c</mi><mrow
4641
class="MathClass-ord">7</mn></mrow></msub
4644
class="MathClass-ord">δ</mi><mrow
4646
class="MathClass-bin">−</mo><mn
4647
class="MathClass-ord">2</mn></mrow></msup
4650
class="MathClass-op">∫</mo>
4653
class="MathClass-ord">A</mi></mrow></msub
4656
open="|" close="|" ><mi
4657
class="MathClass-ord">u</mi></mfenced> <mrow
4659
class="MathClass-ord">2</mn></mrow></msup
4662
class="MathClass-ord">e</mi><mrow
4664
class="MathClass-ord">α</mi><msup
4666
open="|" close="|" ><mi
4667
class="MathClass-ord">z</mi> </mfenced><mrow
4669
class="MathClass-ord">2</mn></mrow></msup
4673
class="MathClass-punc">.</mo>
4675
</p><!--l. 582--><p class="indent"> Let <!--l. 582--><math
4676
xmlns="http://www.w3.org/1998/Math/MathML"
4678
class="MathClass-ord">B</mi><mrow><mo
4679
class="MathClass-open">(</mo><mi
4680
class="MathClass-ord">X</mi><mo
4681
class="MathClass-close">)</mo></mrow></math> be the
4683
set of blocks of <!--l. 582--><math
4684
xmlns="http://www.w3.org/1998/Math/MathML"
4685
mode="inline"> <msub
4687
class="MathClass-ord">Λ</mi><mrow
4689
class="MathClass-ord">X</mi></mrow></msub
4691
and let <!--l. 583--><math
4692
xmlns="http://www.w3.org/1998/Math/MathML"
4694
class="MathClass-ord">b</mi><mrow><mo
4695
class="MathClass-open">(</mo><mi
4696
class="MathClass-ord">X</mi><mo
4697
class="MathClass-close">)</mo></mrow> <mo
4698
class="MathClass-rel">=</mo> <mfenced
4699
open="|" close="|" ><mi
4700
class="MathClass-ord">B</mi><mrow><mo
4701
class="MathClass-open">(</mo><mi
4702
class="MathClass-ord">X</mi><mo
4703
class="MathClass-close">)</mo></mrow></mfenced></math>.
4704
If <!--l. 583--><math
4705
xmlns="http://www.w3.org/1998/Math/MathML"
4707
class="MathClass-ord">φ</mi> <mo
4708
class="MathClass-rel">∈</mo> <msub
4710
class="MathClass-ord">Q</mi><mrow
4712
class="MathClass-ord">X</mi></mrow></msub
4713
></math> then <!--l. 584--><math
4714
xmlns="http://www.w3.org/1998/Math/MathML"
4717
class="MathClass-ord">φ</mi></math> is constant on the
4718
blocks of <!--l. 584--><math
4719
xmlns="http://www.w3.org/1998/Math/MathML"
4720
mode="inline"> <msub
4722
class="MathClass-ord">Λ</mi><mrow
4724
class="MathClass-ord">X</mi></mrow></msub
4726
</p><table class="equation"><tr><td>
4728
xmlns="http://www.w3.org/1998/Math/MathML"
4731
class="equation"><mtr><mtd>
4734
class="MathClass-ord">P</mi><mrow
4736
class="MathClass-ord">X</mi></mrow></msub
4738
class="MathClass-rel">=</mo> <mrow><mo
4739
class="MathClass-open">{</mo><mi
4740
class="MathClass-ord">φ</mi> <mo
4741
class="MathClass-rel">∈</mo> <mi
4742
class="MathClass-ord">M</mi> <mo
4743
class="MathClass-rel">∣</mo> <msub
4745
class="MathClass-ord">Λ</mi><mrow
4747
class="MathClass-ord">φ</mi></mrow></msub
4749
class="MathClass-rel">=</mo> <msub
4751
class="MathClass-ord">Λ</mi><mrow
4753
class="MathClass-ord">X</mi></mrow></msub
4755
class="MathClass-close">}</mo></mrow><mo
4756
class="MathClass-punc">,</mo> <mspace width="2em" class="qquad"/><msub
4758
class="MathClass-ord">Q</mi><mrow
4760
class="MathClass-ord">X</mi></mrow></msub
4762
class="MathClass-rel">=</mo> <mrow><mo
4763
class="MathClass-open">{</mo><mi
4764
class="MathClass-ord">φ</mi> <mo
4765
class="MathClass-rel">∈</mo> <mi
4766
class="MathClass-ord">M</mi> <mo
4767
class="MathClass-rel">∣</mo> <msub
4769
class="MathClass-ord">Λ</mi><mrow
4771
class="MathClass-ord">φ</mi></mrow></msub
4773
class="MathClass-rel">≥</mo> <msub
4775
class="MathClass-ord">Λ</mi><mrow
4777
class="MathClass-ord">X</mi></mrow></msub
4779
class="MathClass-close">}</mo></mrow><mo
4780
class="MathClass-punc">.</mo></mtd><mtd><mspace
4781
id="x1-6017r24" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
4783
<!--l. 589--><p class="nopar"></p></td><td width="5%">(24)</td></tr></table>
4784
If <!--l. 590--><math
4785
xmlns="http://www.w3.org/1998/Math/MathML"
4786
mode="inline"> <msub
4788
class="MathClass-ord">Λ</mi><mrow
4790
class="MathClass-ord">φ</mi></mrow></msub
4792
class="MathClass-rel">≥</mo> <msub
4794
class="MathClass-ord">Λ</mi><mrow
4796
class="MathClass-ord">X</mi></mrow></msub
4797
></math> then <!--l. 591--><math
4798
xmlns="http://www.w3.org/1998/Math/MathML"
4802
class="MathClass-ord">Λ</mi><mrow
4804
class="MathClass-ord">φ</mi></mrow></msub
4806
class="MathClass-rel">=</mo> <msub
4808
class="MathClass-ord">Λ</mi><mrow
4810
class="MathClass-ord">Y</mi> </mrow></msub
4811
></math> for some <!--l. 591--><math
4812
xmlns="http://www.w3.org/1998/Math/MathML"
4815
class="MathClass-ord">Y</mi> <mo
4816
class="MathClass-rel">≥</mo> <mi
4817
class="MathClass-ord">X</mi></math> so that <!--l. 592--><math
4818
xmlns="http://www.w3.org/1998/Math/MathML"
4824
class="MathClass-ord">Q</mi><mrow
4826
class="MathClass-ord">X</mi></mrow></msub
4828
class="MathClass-rel">=</mo><msub
4830
class="MathClass-op">⋃</mo>
4833
class="MathClass-ord">Y</mi> <mo
4834
class="MathClass-rel">≥</mo><mi
4835
class="MathClass-ord">X</mi></mrow></msub
4838
class="MathClass-ord">P</mi><mrow
4840
class="MathClass-ord">Y</mi> </mrow></msub
4842
class="MathClass-punc">.</mo>
4843
</mrow></math> Thus by Möbius
4845
inversion <!--l. 594--><math
4846
xmlns="http://www.w3.org/1998/Math/MathML"
4847
mode="display"> <mrow
4850
open="|" close="|" ><msub
4852
class="MathClass-ord">P</mi><mrow
4854
class="MathClass-ord">Y</mi> </mrow></msub
4856
class="MathClass-rel">=</mo><msub
4858
class="MathClass-op">∑</mo>
4861
class="MathClass-ord">X</mi><mo
4862
class="MathClass-rel">≥</mo><mi
4863
class="MathClass-ord">Y</mi> </mrow></msub
4865
class="MathClass-ord">μ</mi><mrow><mo
4866
class="MathClass-open">(</mo><mi
4867
class="MathClass-ord">Y</mi><mo
4868
class="MathClass-punc">,</mo> <mi
4869
class="MathClass-ord">X</mi><mo
4870
class="MathClass-close">)</mo></mrow> <mfenced
4871
open="|" close="|" ><msub
4873
class="MathClass-ord">Q</mi><mrow
4875
class="MathClass-ord">X</mi></mrow></msub
4877
class="MathClass-punc">.</mo>
4878
</mrow></math> Thus there
4879
is a bijection from <!--l. 595--><math
4880
xmlns="http://www.w3.org/1998/Math/MathML"
4881
mode="inline"> <msub
4883
class="MathClass-ord">Q</mi><mrow
4885
class="MathClass-ord">X</mi></mrow></msub
4887
to <!--l. 595--><math
4888
xmlns="http://www.w3.org/1998/Math/MathML"
4889
mode="inline"> <msup
4891
class="MathClass-ord">W</mi><mrow
4893
class="MathClass-ord">B</mi><mrow><mo
4894
class="MathClass-open">(</mo><mi
4895
class="MathClass-ord">X</mi><mo
4896
class="MathClass-close">)</mo></mrow></mrow></msup
4898
particular <!--l. 596--><math
4899
xmlns="http://www.w3.org/1998/Math/MathML"
4900
mode="inline"> <mfenced
4901
open="|" close="|" ><msub
4903
class="MathClass-ord">Q</mi><mrow
4905
class="MathClass-ord">X</mi></mrow></msub
4907
class="MathClass-rel">=</mo> <msup
4909
class="MathClass-ord">w</mi><mrow
4911
class="MathClass-ord">b</mi><mrow><mo
4912
class="MathClass-open">(</mo><mi
4913
class="MathClass-ord">X</mi><mo
4914
class="MathClass-close">)</mo></mrow></mrow></msup
4916
<!--l. 598--><p class="indent"> Next note that <!--l. 598--><math
4917
xmlns="http://www.w3.org/1998/Math/MathML"
4919
class="MathClass-ord">b</mi><mrow><mo
4920
class="MathClass-open">(</mo><mi
4921
class="MathClass-ord">X</mi><mo
4922
class="MathClass-close">)</mo></mrow> <mo
4923
class="MathClass-rel">=</mo><mo
4924
> dim</mo><!--nolimits--> <mi
4925
class="MathClass-ord">X</mi></math>. We see
4926
this by choosing a basis for <!--l. 599--><math
4927
xmlns="http://www.w3.org/1998/Math/MathML"
4929
class="MathClass-ord">X</mi></math>
4930
consisting of vectors <!--l. 599--><math
4931
xmlns="http://www.w3.org/1998/Math/MathML"
4932
mode="inline"> <msup
4934
class="MathClass-ord">v</mi><mrow
4936
class="MathClass-ord">k</mi></mrow></msup
4938
defined by <!--l. 600--><math
4939
xmlns="http://www.w3.org/1998/Math/MathML"
4940
mode="display"> <mrow
4944
class="MathClass-ord">v</mi><mrow
4946
class="MathClass-ord">i</mi></mrow><mrow
4948
class="MathClass-ord">k</mi></mrow></msubsup
4950
class="MathClass-rel">=</mo> <mfenced
4951
open="{" close="" ><mtable
4952
equalrows="false" equalcolumns="false" class="array"><mtr><mtd
4954
class="MathClass-ord">1</mn><mspace width="1em" class="quad"/></mtd><mtd
4955
class="array" ><mrow
4956
class="text"><mtext >if </mtext><mrow
4958
class="MathClass-ord">i</mi> <mo
4959
class="MathClass-rel">∈</mo> <msub
4961
class="MathClass-ord">Λ</mi><mrow
4963
class="MathClass-ord">k</mi></mrow></msub
4964
></mrow><mtext ></mtext></mrow><mo
4965
class="MathClass-punc">,</mo> </mtd>
4968
class="MathClass-ord">0</mn><mspace width="1em" class="quad"/></mtd><mtd
4969
class="array" ><mrow
4970
class="text"><mtext >otherwise.</mtext></mrow></mtd></mtr> <!--@{}l@{\quad }l@{}--></mtable> </mfenced>
4975
class="verbatim"><tr class="verbatim"><td
4976
class="verbatim"><pre class="verbatim">
4977
 \[vˆ{k}_{i}=
4978
 \begin{cases} 1 & \text{if $i \in \Lambda_{k}$},\\
4979
 0 &\text{otherwise.} \end{cases}
4983
<div class="newtheorem">
4984
<!--l. 611--><p class="noindent"><span class="head">
4986
name="x1-6018r2"></a>
4988
class="cmbx-10">Lemma 6.2.</span> </span> <span
4989
class="cmti-10">Let </span><!--l. 612--><math
4990
xmlns="http://www.w3.org/1998/Math/MathML"
4993
class="MathClass-ord"><!--span
4994
class="htf-calligraphy"-->A<!--/span--></mi></math>
4996
class="cmti-10">be an arrangement. Then </span><!--l. 613--><math
4997
xmlns="http://www.w3.org/1998/Math/MathML"
5002
class="MathClass-ord">χ</mi><mrow><mo
5003
class="MathClass-open">(</mo><mi
5004
class="MathClass-ord"><!--span
5005
class="htf-calligraphy"-->A<!--/span--></mi><mo
5006
class="MathClass-punc">,</mo> <mi
5007
class="MathClass-ord">t</mi><mo
5008
class="MathClass-close">)</mo></mrow> <mo
5009
class="MathClass-rel">=</mo><msub
5011
class="MathClass-op">∑</mo>
5014
class="MathClass-ord"><!--span
5015
class="htf-calligraphy"-->B<!--/span--></mi><mo
5016
class="MathClass-rel">⊆</mo><mi
5017
class="MathClass-ord"><!--span
5018
class="htf-calligraphy"-->A<!--/span--></mi></mrow></msub
5021
class="MathClass-open">(</mo><mo
5022
class="MathClass-bin">−</mo><mn
5023
class="MathClass-ord">1</mn><mo
5024
class="MathClass-close">)</mo></mrow><mrow
5026
open="|" close="|" ><mi
5027
class="MathClass-ord"><!--span
5028
class="htf-calligraphy"-->B<!--/span--></mi> </mfenced></mrow></msup
5031
class="MathClass-ord">t</mi><mrow
5033
>dim</mo><!--nolimits--> <mi
5034
class="MathClass-ord">T</mi><mrow><mo
5035
class="MathClass-open">(</mo><mi
5036
class="MathClass-ord"><!--span
5037
class="htf-calligraphy"-->B<!--/span--></mi><mo
5038
class="MathClass-close">)</mo></mrow></mrow></msup
5040
class="MathClass-punc">.</mo>
5044
<!--l. 617--><p class="indent"> In order to compute <!--l. 617--><math
5045
xmlns="http://www.w3.org/1998/Math/MathML"
5047
class="MathClass-ord">R</mi><mi
5048
class="MathClass-ord">′</mi><mi
5049
class="MathClass-ord">′</mi></math>
5050
recall the definition of <!--l. 618--><math
5051
xmlns="http://www.w3.org/1998/Math/MathML"
5053
class="MathClass-ord">S</mi><mrow><mo
5054
class="MathClass-open">(</mo><mi
5055
class="MathClass-ord">X</mi><mo
5056
class="MathClass-punc">,</mo> <mi
5057
class="MathClass-ord">Y</mi> <mo
5058
class="MathClass-close">)</mo></mrow></math>
5059
from Lemma <a
5060
href="#x1-3001r1">3.1<!--tex4ht:ref: lem-per--></a>. Since <!--l. 618--><math
5061
xmlns="http://www.w3.org/1998/Math/MathML"
5063
class="MathClass-ord">H</mi> <mo
5064
class="MathClass-rel">∈</mo> <mi
5065
class="MathClass-ord"><!--span
5066
class="htf-calligraphy"-->B<!--/span--></mi></math>,
5068
xmlns="http://www.w3.org/1998/Math/MathML"
5069
mode="inline"> <msub
5071
class="MathClass-ord"><!--span
5072
class="htf-calligraphy"-->A<!--/span--></mi><mrow
5074
class="MathClass-ord">H</mi></mrow></msub
5076
class="MathClass-rel">⊆</mo> <mi
5077
class="MathClass-ord"><!--span
5078
class="htf-calligraphy"-->B<!--/span--></mi></math>. Thus
5079
if <!--l. 619--><math
5080
xmlns="http://www.w3.org/1998/Math/MathML"
5082
class="MathClass-ord">T</mi><mrow><mo
5083
class="MathClass-open">(</mo><mi
5084
class="MathClass-ord"><!--span
5085
class="htf-calligraphy"-->B<!--/span--></mi><mo
5086
class="MathClass-close">)</mo></mrow> <mo
5087
class="MathClass-rel">=</mo> <mi
5088
class="MathClass-ord">Y</mi> </math>
5089
then <!--l. 620--><math
5090
xmlns="http://www.w3.org/1998/Math/MathML"
5092
class="MathClass-ord"><!--span
5093
class="htf-calligraphy"-->B<!--/span--></mi> <mo
5094
class="MathClass-rel">∈</mo> <mi
5095
class="MathClass-ord">S</mi><mrow><mo
5096
class="MathClass-open">(</mo><mi
5097
class="MathClass-ord">H</mi><mo
5098
class="MathClass-punc">,</mo> <mi
5099
class="MathClass-ord">Y</mi> <mo
5100
class="MathClass-close">)</mo></mrow></math>.
5101
Let <!--l. 620--><math
5102
xmlns="http://www.w3.org/1998/Math/MathML"
5104
class="MathClass-ord">L</mi><mi
5105
class="MathClass-ord">′</mi><mi
5106
class="MathClass-ord">′</mi> <mo
5107
class="MathClass-rel">=</mo> <mi
5108
class="MathClass-ord">L</mi><mrow><mo
5109
class="MathClass-open">(</mo><mi
5110
class="MathClass-ord"><!--span
5111
class="htf-calligraphy"-->A<!--/span--></mi><mi
5112
class="MathClass-ord">′</mi><mi
5113
class="MathClass-ord">′</mi><mo
5114
class="MathClass-close">)</mo></mrow></math>.
5115
Then </p><table class="equation"><tr><td>
5118
xmlns="http://www.w3.org/1998/Math/MathML"
5121
class="equation"><mtr><mtd>
5122
<mtable class="split"><mtr><mtd>
5124
class="split-mtr"></mrow><mrow
5125
class="split-mtd"></mrow> <mi
5126
class="MathClass-ord">R</mi><mi
5127
class="MathClass-ord">′</mi><mi
5128
class="MathClass-ord">′</mi><mrow
5129
class="split-mtd"></mrow> <mo
5130
class="MathClass-rel">=</mo><msub
5132
class="MathClass-op">∑</mo>
5135
class="MathClass-ord">H</mi><mo
5136
class="MathClass-rel">∈</mo><mi
5137
class="MathClass-ord"><!--span
5138
class="htf-calligraphy"-->B<!--/span--></mi><mo
5139
class="MathClass-rel">⊆</mo><mi
5140
class="MathClass-ord"><!--span
5141
class="htf-calligraphy"-->A<!--/span--></mi></mrow></msub
5144
class="MathClass-open">(</mo><mo
5145
class="MathClass-bin">−</mo><mn
5146
class="MathClass-ord">1</mn><mo
5147
class="MathClass-close">)</mo></mrow><mrow
5149
open="|" close="|" ><mi
5150
class="MathClass-ord"><!--span
5151
class="htf-calligraphy"-->B<!--/span--></mi> </mfenced></mrow></msup
5154
class="MathClass-ord">t</mi><mrow
5156
>dim</mo><!--nolimits--> <mi
5157
class="MathClass-ord">T</mi><mrow><mo
5158
class="MathClass-open">(</mo><mi
5159
class="MathClass-ord"><!--span
5160
class="htf-calligraphy"-->B<!--/span--></mi><mo
5161
class="MathClass-close">)</mo></mrow></mrow></msup
5164
class="split-mtr"></mrow><mrow
5165
class="split-mtd"></mrow> <mrow
5166
class="split-mtd"></mrow> <mo
5167
class="MathClass-rel">=</mo><msub
5169
class="MathClass-op">∑</mo>
5172
class="MathClass-ord">Y</mi> <mo
5173
class="MathClass-rel">∈</mo><mi
5174
class="MathClass-ord">L</mi><mi
5175
class="MathClass-ord">′</mi><mi
5176
class="MathClass-ord">′</mi> </mrow></msub
5179
class="MathClass-op">∑</mo>
5182
class="MathClass-ord"><!--span
5183
class="htf-calligraphy"-->B<!--/span--></mi><mo
5184
class="MathClass-rel">∈</mo><mi
5185
class="MathClass-ord">S</mi><mrow><mo
5186
class="MathClass-open">(</mo><mi
5187
class="MathClass-ord">H</mi><mo
5188
class="MathClass-punc">,</mo><mi
5189
class="MathClass-ord">Y</mi> <mo
5190
class="MathClass-close">)</mo></mrow></mrow></msub
5193
class="MathClass-open">(</mo><mo
5194
class="MathClass-bin">−</mo><mn
5195
class="MathClass-ord">1</mn><mo
5196
class="MathClass-close">)</mo></mrow><mrow
5198
open="|" close="|" ><mi
5199
class="MathClass-ord"><!--span
5200
class="htf-calligraphy"-->B<!--/span--></mi> </mfenced></mrow></msup
5203
class="MathClass-ord">t</mi><mrow
5205
>dim</mo><!--nolimits--> <mi
5206
class="MathClass-ord">Y</mi> </mrow></msup
5209
class="split-mtr"></mrow><mrow
5210
class="split-mtd"></mrow> <mrow
5211
class="split-mtd"></mrow> <mo
5212
class="MathClass-rel">=</mo> <mo
5213
class="MathClass-bin">−</mo><msub
5215
class="MathClass-op">∑</mo>
5218
class="MathClass-ord">Y</mi> <mo
5219
class="MathClass-rel">∈</mo><mi
5220
class="MathClass-ord">L</mi><mi
5221
class="MathClass-ord">′</mi><mi
5222
class="MathClass-ord">′</mi> </mrow></msub
5225
class="MathClass-op">∑</mo>
5228
class="MathClass-ord"><!--span
5229
class="htf-calligraphy"-->B<!--/span--></mi><mo
5230
class="MathClass-rel">∈</mo><mi
5231
class="MathClass-ord">S</mi><mrow><mo
5232
class="MathClass-open">(</mo><mi
5233
class="MathClass-ord">H</mi><mo
5234
class="MathClass-punc">,</mo><mi
5235
class="MathClass-ord">Y</mi> <mo
5236
class="MathClass-close">)</mo></mrow></mrow></msub
5239
class="MathClass-open">(</mo><mo
5240
class="MathClass-bin">−</mo><mn
5241
class="MathClass-ord">1</mn><mo
5242
class="MathClass-close">)</mo></mrow><mrow
5244
open="|" close="|" ><mi
5245
class="MathClass-ord"><!--span
5246
class="htf-calligraphy"-->B<!--/span--></mi><mo
5247
class="MathClass-bin">−</mo><msub
5249
class="MathClass-ord"><!--span
5250
class="htf-calligraphy"-->A<!--/span--></mi><mrow
5252
class="MathClass-ord">H</mi> </mrow></msub
5258
class="MathClass-ord">t</mi><mrow
5260
>dim</mo><!--nolimits--> <mi
5261
class="MathClass-ord">Y</mi> </mrow></msup
5264
class="split-mtr"></mrow><mrow
5265
class="split-mtd"></mrow> <mrow
5266
class="split-mtd"></mrow> <mo
5267
class="MathClass-rel">=</mo> <mo
5268
class="MathClass-bin">−</mo><msub
5270
class="MathClass-op">∑</mo>
5273
class="MathClass-ord">Y</mi> <mo
5274
class="MathClass-rel">∈</mo><mi
5275
class="MathClass-ord">L</mi><mi
5276
class="MathClass-ord">′</mi><mi
5277
class="MathClass-ord">′</mi> </mrow></msub
5279
class="MathClass-ord">μ</mi><mrow><mo
5280
class="MathClass-open">(</mo><mi
5281
class="MathClass-ord">H</mi><mo
5282
class="MathClass-punc">,</mo> <mi
5283
class="MathClass-ord">Y</mi> <mo
5284
class="MathClass-close">)</mo></mrow><msup
5286
class="MathClass-ord">t</mi><mrow
5288
>dim</mo><!--nolimits--> <mi
5289
class="MathClass-ord">Y</mi> </mrow></msup
5292
class="split-mtr"></mrow><mrow
5293
class="split-mtd"></mrow> <mrow
5294
class="split-mtd"></mrow> <mo
5295
class="MathClass-rel">=</mo> <mo
5296
class="MathClass-bin">−</mo><mi
5297
class="MathClass-ord">χ</mi><mrow><mo
5298
class="MathClass-open">(</mo><mi
5299
class="MathClass-ord"><!--span
5300
class="htf-calligraphy"-->A<!--/span--></mi><mi
5301
class="MathClass-ord">′</mi><mi
5302
class="MathClass-ord">′</mi><mo
5303
class="MathClass-punc">,</mo> <mi
5304
class="MathClass-ord">t</mi><mo
5305
class="MathClass-close">)</mo></mrow><mo
5306
class="MathClass-punc">.</mo>
5307
</mtd></mtr></mtable> </mtd><mtd><mspace
5308
id="x1-6019r25" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /></mtd></mtr></mtable>
5310
<!--l. 632--><p class="nopar"></p></td><td width="5%">(25)</td></tr></table>
5311
<div class="newtheorem">
5312
<!--l. 634--><p class="noindent"><span class="head">
5314
name="x1-6020r3"></a>
5316
class="cmbx-10">Corollary 6.3.</span> </span> <span
5317
class="cmti-10">Let </span><!--l. 635--><math
5318
xmlns="http://www.w3.org/1998/Math/MathML"
5321
class="MathClass-open">(</mo><mi
5322
class="MathClass-ord"><!--span
5323
class="htf-calligraphy"-->A<!--/span--></mi><mo
5324
class="MathClass-punc">,</mo> <mi
5325
class="MathClass-ord"><!--span
5326
class="htf-calligraphy"-->A<!--/span--></mi><mi
5327
class="MathClass-ord">′</mi><mo
5328
class="MathClass-punc">,</mo> <mi
5329
class="MathClass-ord"><!--span
5330
class="htf-calligraphy"-->A<!--/span--></mi><mi
5331
class="MathClass-ord">′</mi><mi
5332
class="MathClass-ord">′</mi><mo
5333
class="MathClass-close">)</mo></mrow></math>
5335
class="cmti-10">be a triple of arrangements. Then </span><!--l. 636--><math
5336
xmlns="http://www.w3.org/1998/Math/MathML"
5341
class="MathClass-ord">π</mi><mrow><mo
5342
class="MathClass-open">(</mo><mi
5343
class="MathClass-ord"><!--span
5344
class="htf-calligraphy"-->A<!--/span--></mi><mo
5345
class="MathClass-punc">,</mo> <mi
5346
class="MathClass-ord">t</mi><mo
5347
class="MathClass-close">)</mo></mrow> <mo
5348
class="MathClass-rel">=</mo> <mi
5349
class="MathClass-ord">π</mi><mrow><mo
5350
class="MathClass-open">(</mo><mi
5351
class="MathClass-ord"><!--span
5352
class="htf-calligraphy"-->A<!--/span--></mi><mi
5353
class="MathClass-ord">′</mi><mo
5354
class="MathClass-punc">,</mo> <mi
5355
class="MathClass-ord">t</mi><mo
5356
class="MathClass-close">)</mo></mrow> <mo
5357
class="MathClass-bin">+</mo> <mi
5358
class="MathClass-ord">t</mi><mi
5359
class="MathClass-ord">π</mi><mrow><mo
5360
class="MathClass-open">(</mo><mi
5361
class="MathClass-ord"><!--span
5362
class="htf-calligraphy"-->A<!--/span--></mi><mi
5363
class="MathClass-ord">′</mi><mi
5364
class="MathClass-ord">′</mi><mo
5365
class="MathClass-punc">,</mo> <mi
5366
class="MathClass-ord">t</mi><mo
5367
class="MathClass-close">)</mo></mrow><mo
5368
class="MathClass-punc">.</mo>
5373
<div class="newtheorem">
5374
<!--l. 639--><p class="noindent"><span class="head">
5376
name="x1-6021r2"></a>
5378
class="cmbx-10">Definition 6.2.</span> </span>Let <!--l. 640--><math
5379
xmlns="http://www.w3.org/1998/Math/MathML"
5380
mode="inline"> <mrow><mo
5381
class="MathClass-open">(</mo><mi
5382
class="MathClass-ord"><!--span
5383
class="htf-calligraphy"-->A<!--/span--></mi><mo
5384
class="MathClass-punc">,</mo> <mi
5385
class="MathClass-ord"><!--span
5386
class="htf-calligraphy"-->A<!--/span--></mi><mi
5387
class="MathClass-ord">′</mi><mo
5388
class="MathClass-punc">,</mo> <mi
5389
class="MathClass-ord"><!--span
5390
class="htf-calligraphy"-->A<!--/span--></mi><mi
5391
class="MathClass-ord">′</mi><mi
5392
class="MathClass-ord">′</mi><mo
5393
class="MathClass-close">)</mo></mrow></math>
5394
be a triple with respect to the hyperplane <!--l. 641--><math
5395
xmlns="http://www.w3.org/1998/Math/MathML"
5397
class="MathClass-ord">H</mi> <mo
5398
class="MathClass-rel">∈</mo> <mi
5399
class="MathClass-ord"><!--span
5400
class="htf-calligraphy"-->A<!--/span--></mi></math>.
5401
Call <!--l. 641--><math
5402
xmlns="http://www.w3.org/1998/Math/MathML"
5404
class="MathClass-ord">H</mi></math>
5406
class="cmti-10">separator </span>if <!--l. 642--><math
5407
xmlns="http://www.w3.org/1998/Math/MathML"
5409
class="MathClass-ord">T</mi><mrow><mo
5410
class="MathClass-open">(</mo><mi
5411
class="MathClass-ord"><!--span
5412
class="htf-calligraphy"-->A<!--/span--></mi><mo
5413
class="MathClass-close">)</mo></mrow> <mo
5414
class="MathClass-rel">/</mo><mo
5415
class="MathClass-rel">∈</mo> <mi
5416
class="MathClass-ord">L</mi><mrow><mo
5417
class="MathClass-open">(</mo><mi
5418
class="MathClass-ord"><!--span
5419
class="htf-calligraphy"-->A<!--/span--></mi><mi
5420
class="MathClass-ord">′</mi><mo
5421
class="MathClass-close">)</mo></mrow></math>.
5424
<div class="newtheorem">
5425
<!--l. 645--><p class="noindent"><span class="head">
5427
name="x1-6022r4"></a>
5429
class="cmbx-10">Corollary 6.4.</span> </span> <span
5430
class="cmti-10">Let </span><!--l. 646--><math
5431
xmlns="http://www.w3.org/1998/Math/MathML"
5432
mode="inline"> <mrow><mo
5433
class="MathClass-open">(</mo><mi
5434
class="MathClass-ord"><!--span
5435
class="htf-calligraphy"-->A<!--/span--></mi><mo
5436
class="MathClass-punc">,</mo> <mi
5437
class="MathClass-ord"><!--span
5438
class="htf-calligraphy"-->A<!--/span--></mi><mi
5439
class="MathClass-ord">′</mi><mo
5440
class="MathClass-punc">,</mo> <mi
5441
class="MathClass-ord"><!--span
5442
class="htf-calligraphy"-->A<!--/span--></mi><mi
5443
class="MathClass-ord">′</mi><mi
5444
class="MathClass-ord">′</mi><mo
5445
class="MathClass-close">)</mo></mrow></math> <span
5446
class="cmti-10">be</span>
5448
class="cmti-10">a triple with respect to </span><!--l. 646--><math
5449
xmlns="http://www.w3.org/1998/Math/MathML"
5451
class="MathClass-ord">H</mi> <mo
5452
class="MathClass-rel">∈</mo> <mi
5453
class="MathClass-ord"><!--span
5454
class="htf-calligraphy"-->A<!--/span--></mi></math><span
5455
class="cmti-10">.</span>
5456
</p><ol type="1" class="enumerate1"
5458
<li class="enumerate"><a
5459
name="x1-6024x1"></a><span
5460
class="cmti-10">If </span><!--l. 650--><math
5461
xmlns="http://www.w3.org/1998/Math/MathML"
5463
class="MathClass-ord">H</mi></math>
5465
class="cmti-10">is a separator then </span><!--l. 651--><math
5466
xmlns="http://www.w3.org/1998/Math/MathML"
5467
mode="display"> <mrow
5470
class="MathClass-ord">μ</mi><mrow><mo
5471
class="MathClass-open">(</mo><mi
5472
class="MathClass-ord"><!--span
5473
class="htf-calligraphy"-->A<!--/span--></mi><mo
5474
class="MathClass-close">)</mo></mrow> <mo
5475
class="MathClass-rel">=</mo> <mo
5476
class="MathClass-bin">−</mo><mi
5477
class="MathClass-ord">μ</mi><mrow><mo
5478
class="MathClass-open">(</mo><mi
5479
class="MathClass-ord"><!--span
5480
class="htf-calligraphy"-->A<!--/span--></mi><mi
5481
class="MathClass-ord">′</mi><mi
5482
class="MathClass-ord">′</mi><mo
5483
class="MathClass-close">)</mo></mrow>
5486
class="cmti-10">and hence </span><!--l. 653--><math
5487
xmlns="http://www.w3.org/1998/Math/MathML"
5488
mode="display"> <mrow
5491
open="|" close="|" ><mi
5492
class="MathClass-ord">μ</mi><mrow><mo
5493
class="MathClass-open">(</mo><mi
5494
class="MathClass-ord"><!--span
5495
class="htf-calligraphy"-->A<!--/span--></mi><mo
5496
class="MathClass-close">)</mo></mrow></mfenced> <mo
5497
class="MathClass-rel">=</mo> <mfenced
5498
open="|" close="|" ><mi
5499
class="MathClass-ord">μ</mi><mrow><mo
5500
class="MathClass-open">(</mo><mi
5501
class="MathClass-ord"><!--span
5502
class="htf-calligraphy"-->A<!--/span--></mi><mi
5503
class="MathClass-ord">′</mi><mi
5504
class="MathClass-ord">′</mi><mo
5505
class="MathClass-close">)</mo></mrow></mfenced> <mo
5506
class="MathClass-punc">.</mo>
5510
<li class="enumerate"><a
5511
name="x1-6026x2"></a><span
5512
class="cmti-10">If </span><!--l. 655--><math
5513
xmlns="http://www.w3.org/1998/Math/MathML"
5515
class="MathClass-ord">H</mi></math>
5517
class="cmti-10">is not a separator then </span><!--l. 656--><math
5518
xmlns="http://www.w3.org/1998/Math/MathML"
5519
mode="display"> <mrow
5522
class="MathClass-ord">μ</mi><mrow><mo
5523
class="MathClass-open">(</mo><mi
5524
class="MathClass-ord"><!--span
5525
class="htf-calligraphy"-->A<!--/span--></mi><mo
5526
class="MathClass-close">)</mo></mrow> <mo
5527
class="MathClass-rel">=</mo> <mi
5528
class="MathClass-ord">μ</mi><mrow><mo
5529
class="MathClass-open">(</mo><mi
5530
class="MathClass-ord"><!--span
5531
class="htf-calligraphy"-->A<!--/span--></mi><mi
5532
class="MathClass-ord">′</mi><mo
5533
class="MathClass-close">)</mo></mrow> <mo
5534
class="MathClass-bin">−</mo> <mi
5535
class="MathClass-ord">μ</mi><mrow><mo
5536
class="MathClass-open">(</mo><mi
5537
class="MathClass-ord"><!--span
5538
class="htf-calligraphy"-->A<!--/span--></mi><mi
5539
class="MathClass-ord">′</mi><mi
5540
class="MathClass-ord">′</mi><mo
5541
class="MathClass-close">)</mo></mrow>
5544
class="cmti-10">and </span><!--l. 658--><math
5545
xmlns="http://www.w3.org/1998/Math/MathML"
5546
mode="display"> <mrow
5549
open="|" close="|" ><mi
5550
class="MathClass-ord">μ</mi><mrow><mo
5551
class="MathClass-open">(</mo><mi
5552
class="MathClass-ord"><!--span
5553
class="htf-calligraphy"-->A<!--/span--></mi><mo
5554
class="MathClass-close">)</mo></mrow></mfenced> <mo
5555
class="MathClass-rel">=</mo> <mfenced
5556
open="|" close="|" ><mi
5557
class="MathClass-ord">μ</mi><mrow><mo
5558
class="MathClass-open">(</mo><mi
5559
class="MathClass-ord"><!--span
5560
class="htf-calligraphy"-->A<!--/span--></mi><mi
5561
class="MathClass-ord">′</mi><mo
5562
class="MathClass-close">)</mo></mrow></mfenced> <mo
5563
class="MathClass-bin">+</mo> <mfenced
5564
open="|" close="|" ><mi
5565
class="MathClass-ord">μ</mi><mrow><mo
5566
class="MathClass-open">(</mo><mi
5567
class="MathClass-ord"><!--span
5568
class="htf-calligraphy"-->A<!--/span--></mi><mi
5569
class="MathClass-ord">′</mi><mi
5570
class="MathClass-ord">′</mi><mo
5571
class="MathClass-close">)</mo></mrow></mfenced> <mo
5572
class="MathClass-punc">.</mo>
5573
</mrow></math></li></ol>
5575
<!--l. 662--><p class="indent">
5577
<div class="proof"><span class="head">
5579
class="cmti-10">Proof.</span> </span>It follows from Theorem <a
5580
href="#x1-5002r1">5.1<!--tex4ht:ref: th-info-ow-ow--></a> that <!--l. 663--><math
5581
xmlns="http://www.w3.org/1998/Math/MathML"
5583
class="MathClass-ord">π</mi><mrow><mo
5584
class="MathClass-open">(</mo><mi
5585
class="MathClass-ord"><!--span
5586
class="htf-calligraphy"-->A<!--/span--></mi><mo
5587
class="MathClass-punc">,</mo> <mi
5588
class="MathClass-ord">t</mi><mo
5589
class="MathClass-close">)</mo></mrow></math>
5591
has leading term <!--l. 665--><math
5592
xmlns="http://www.w3.org/1998/Math/MathML"
5593
mode="display"> <mrow
5597
class="MathClass-open">(</mo><mo
5598
class="MathClass-bin">−</mo><mn
5599
class="MathClass-ord">1</mn><mo
5600
class="MathClass-close">)</mo></mrow><mrow
5602
class="MathClass-ord">r</mi><mrow><mo
5603
class="MathClass-open">(</mo><mi
5604
class="MathClass-ord"><!--span
5605
class="htf-calligraphy"-->A<!--/span--></mi><mo
5606
class="MathClass-close">)</mo></mrow></mrow></msup
5608
class="MathClass-ord">μ</mi><mrow><mo
5609
class="MathClass-open">(</mo><mi
5610
class="MathClass-ord"><!--span
5611
class="htf-calligraphy"-->A<!--/span--></mi><mo
5612
class="MathClass-close">)</mo></mrow><msup
5614
class="MathClass-ord">t</mi><mrow
5616
class="MathClass-ord">r</mi><mrow><mo
5617
class="MathClass-open">(</mo><mi
5618
class="MathClass-ord"><!--span
5619
class="htf-calligraphy"-->A<!--/span--></mi><mo
5620
class="MathClass-close">)</mo></mrow></mrow></msup
5622
class="MathClass-punc">.</mo>
5624
The conclusion follows by comparing coefficients of the leading terms on both
5625
sides of the equation in Corollary <a
5626
href="#x1-6020r3">6.3<!--tex4ht:ref: tripleA--></a>. If <!--l. 669--><math
5627
xmlns="http://www.w3.org/1998/Math/MathML"
5629
class="MathClass-ord">H</mi></math>
5630
is a separator then <!--l. 670--><math
5631
xmlns="http://www.w3.org/1998/Math/MathML"
5633
class="MathClass-ord">r</mi><mrow><mo
5634
class="MathClass-open">(</mo><mi
5635
class="MathClass-ord"><!--span
5636
class="htf-calligraphy"-->A<!--/span--></mi><mi
5637
class="MathClass-ord">′</mi><mo
5638
class="MathClass-close">)</mo></mrow> <mo
5639
class="MathClass-rel"><</mo> <mi
5640
class="MathClass-ord">r</mi><mrow><mo
5641
class="MathClass-open">(</mo><mi
5642
class="MathClass-ord"><!--span
5643
class="htf-calligraphy"-->A<!--/span--></mi><mo
5644
class="MathClass-close">)</mo></mrow></math>
5645
and there is no contribution from <!--l. 671--><math
5646
xmlns="http://www.w3.org/1998/Math/MathML"
5648
class="MathClass-ord">π</mi><mrow><mo
5649
class="MathClass-open">(</mo><mi
5650
class="MathClass-ord"><!--span
5651
class="htf-calligraphy"-->A<!--/span--></mi><mi
5652
class="MathClass-ord">′</mi><mo
5653
class="MathClass-punc">,</mo> <mi
5654
class="MathClass-ord">t</mi><mo
5655
class="MathClass-close">)</mo></mrow></math>.
5657
<!--l. 674--><p class="noindent">The Poincaré polynomial of an arrangement will appear repeatedly
5658
in these notes. It will be shown to equal the Poincaré polynomial
5659
of the graded algebras which we are going to associate with <!--l. 679--><math
5660
xmlns="http://www.w3.org/1998/Math/MathML"
5663
class="MathClass-ord"><!--span
5664
class="htf-calligraphy"-->A<!--/span--></mi></math>.
5665
It is also the Poincaré polynomial of the complement <!--l. 680--><math
5666
xmlns="http://www.w3.org/1998/Math/MathML"
5669
class="MathClass-ord">M</mi><mrow><mo
5670
class="MathClass-open">(</mo><mi
5671
class="MathClass-ord"><!--span
5672
class="htf-calligraphy"-->A<!--/span--></mi><mo
5673
class="MathClass-close">)</mo></mrow></math> for
5674
a complex arrangement. Here we prove that the Poincaré polynomial is
5675
the chamber counting function for a real arrangement. The complement <!--l. 684--><math
5676
xmlns="http://www.w3.org/1998/Math/MathML"
5679
class="MathClass-ord">M</mi><mrow><mo
5680
class="MathClass-open">(</mo><mi
5681
class="MathClass-ord"><!--span
5682
class="htf-calligraphy"-->A<!--/span--></mi><mo
5683
class="MathClass-close">)</mo></mrow></math> is a disjoint union
5684
of chambers <!--l. 685--><math
5685
xmlns="http://www.w3.org/1998/Math/MathML"
5686
mode="display"> <mrow
5689
class="MathClass-ord">M</mi><mrow><mo
5690
class="MathClass-open">(</mo><mi
5691
class="MathClass-ord"><!--span
5692
class="htf-calligraphy"-->A<!--/span--></mi><mo
5693
class="MathClass-close">)</mo></mrow> <mo
5694
class="MathClass-rel">=</mo><msub
5696
class="MathClass-op">⋃</mo>
5699
class="MathClass-ord">C</mi><mo
5700
class="MathClass-rel">∈</mo><mo
5701
class="MathClass-op">Cham</mo><!--nolimits--><mrow><mo
5702
class="MathClass-open">(</mo><mi
5703
class="MathClass-ord"><!--span
5704
class="htf-calligraphy"-->A<!--/span--></mi><mo
5705
class="MathClass-close">)</mo></mrow></mrow></msub
5707
class="MathClass-ord">C</mi><mo
5708
class="MathClass-punc">.</mo>
5710
The number of chambers is determined by the Poincaré polynomial as
5713
<div class="newtheorem">
5715
<!--l. 690--><p class="noindent"><span class="head">
5717
name="x1-6027r5"></a>
5719
class="cmbx-10">Theorem 6.5.</span> </span> <span
5720
class="cmti-10">Let </span><!--l. 691--><math
5721
xmlns="http://www.w3.org/1998/Math/MathML"
5725
class="MathClass-ord"><!--span
5726
class="htf-calligraphy"-->A<!--/span--></mi><mrow
5727
><mi class="mathbf">R</mi></mrow></msub
5730
class="cmti-10">be a real arrangement. Then </span><!--l. 692--><math
5731
xmlns="http://www.w3.org/1998/Math/MathML"
5736
open="|" close="|" ><mo
5737
class="MathClass-op">Cham</mo><!--nolimits--><mrow><mo
5738
class="MathClass-open">(</mo><msub
5740
class="MathClass-ord"><!--span
5741
class="htf-calligraphy"-->A<!--/span--></mi><mrow
5742
><mi class="mathbf">R</mi></mrow></msub
5744
class="MathClass-close">)</mo></mrow></mfenced> <mo
5745
class="MathClass-rel">=</mo> <mi
5746
class="MathClass-ord">π</mi><mrow><mo
5747
class="MathClass-open">(</mo><msub
5749
class="MathClass-ord"><!--span
5750
class="htf-calligraphy"-->A<!--/span--></mi><mrow
5751
><mi class="mathbf">R</mi></mrow></msub
5753
class="MathClass-punc">,</mo> <mn
5754
class="MathClass-ord">1</mn><mo
5755
class="MathClass-close">)</mo></mrow><mo
5756
class="MathClass-punc">.</mo>
5760
<!--l. 695--><p class="indent">
5762
<div class="proof"><span class="head">
5764
class="cmti-10">Proof.</span> </span>We check the properties required in Corollary <a
5765
href="#x1-6022r4">6.4<!--tex4ht:ref: nsep--></a>: (i) follows from <!--l. 697--><math
5766
xmlns="http://www.w3.org/1998/Math/MathML"
5768
class="MathClass-ord">π</mi><mrow><mo
5769
class="MathClass-open">(</mo><msub
5771
class="MathClass-ord">Φ</mi><mrow
5773
class="MathClass-ord">l</mi></mrow></msub
5775
class="MathClass-punc">,</mo> <mi
5776
class="MathClass-ord">t</mi><mo
5777
class="MathClass-close">)</mo></mrow> <mo
5778
class="MathClass-rel">=</mo> <mn
5779
class="MathClass-ord">1</mn></math>,
5780
and (ii) is a consequence of Corollary <a
5781
href="#x1-3011r4">3.4<!--tex4ht:ref: BI--></a>. </div> □
5782
<hr class="figure" /><div align="center" class="figure"
5783
><table class="figure"><tr class="figure"><td class="figure"
5787
name="x1-60281"></a>
5789
<br /> <div align="center" class="caption"><table class="caption"
5790
><tr valign="baseline" class="caption"><td class="id">Figure 1: </td><td
5791
class="content"><!--l. 703--><math
5792
xmlns="http://www.w3.org/1998/Math/MathML"
5795
class="MathClass-ord">Q</mi><mrow><mo
5796
class="MathClass-open">(</mo><msub
5798
class="MathClass-ord"><!--span
5799
class="htf-calligraphy"-->A<!--/span--></mi><mrow
5801
class="MathClass-ord">1</mn></mrow></msub
5803
class="MathClass-close">)</mo></mrow> <mo
5804
class="MathClass-rel">=</mo> <mi
5805
class="MathClass-ord">x</mi><mi
5806
class="MathClass-ord">y</mi><mi
5807
class="MathClass-ord">z</mi><mrow><mo
5808
class="MathClass-open">(</mo><mi
5809
class="MathClass-ord">x</mi> <mo
5810
class="MathClass-bin">−</mo> <mi
5811
class="MathClass-ord">z</mi><mo
5812
class="MathClass-close">)</mo></mrow><mrow><mo
5813
class="MathClass-open">(</mo><mi
5814
class="MathClass-ord">x</mi> <mo
5815
class="MathClass-bin">+</mo> <mi
5816
class="MathClass-ord">z</mi><mo
5817
class="MathClass-close">)</mo></mrow><mrow><mo
5818
class="MathClass-open">(</mo><mi
5819
class="MathClass-ord">y</mi> <mo
5820
class="MathClass-bin">−</mo> <mi
5821
class="MathClass-ord">z</mi><mo
5822
class="MathClass-close">)</mo></mrow><mrow><mo
5823
class="MathClass-open">(</mo><mi
5824
class="MathClass-ord">y</mi> <mo
5825
class="MathClass-bin">+</mo> <mi
5826
class="MathClass-ord">z</mi><mo
5827
class="MathClass-close">)</mo></mrow></math></td></tr></table></div><!--tex4ht:label?: x1-60281-->
5829
</td></tr></table></div><hr class="endfigure" />
5830
<hr class="figure" /><div align="center" class="figure"
5831
><table class="figure"><tr class="figure"><td class="figure"
5835
name="x1-60292"></a>
5837
<br /> <div align="center" class="caption"><table class="caption"
5838
><tr valign="baseline" class="caption"><td class="id">Figure 2: </td><td
5839
class="content"><!--l. 708--><math
5840
xmlns="http://www.w3.org/1998/Math/MathML"
5843
class="MathClass-ord">Q</mi><mrow><mo
5844
class="MathClass-open">(</mo><msub
5846
class="MathClass-ord"><!--span
5847
class="htf-calligraphy"-->A<!--/span--></mi><mrow
5849
class="MathClass-ord">2</mn></mrow></msub
5851
class="MathClass-close">)</mo></mrow> <mo
5852
class="MathClass-rel">=</mo> <mi
5853
class="MathClass-ord">x</mi><mi
5854
class="MathClass-ord">y</mi><mi
5855
class="MathClass-ord">z</mi><mrow><mo
5856
class="MathClass-open">(</mo><mi
5857
class="MathClass-ord">x</mi> <mo
5858
class="MathClass-bin">+</mo> <mi
5859
class="MathClass-ord">y</mi> <mo
5860
class="MathClass-bin">+</mo> <mi
5861
class="MathClass-ord">z</mi><mo
5862
class="MathClass-close">)</mo></mrow><mrow><mo
5863
class="MathClass-open">(</mo><mi
5864
class="MathClass-ord">x</mi> <mo
5865
class="MathClass-bin">+</mo> <mi
5866
class="MathClass-ord">y</mi> <mo
5867
class="MathClass-bin">−</mo> <mi
5868
class="MathClass-ord">z</mi><mo
5869
class="MathClass-close">)</mo></mrow><mrow><mo
5870
class="MathClass-open">(</mo><mi
5871
class="MathClass-ord">x</mi> <mo
5872
class="MathClass-bin">−</mo> <mi
5873
class="MathClass-ord">y</mi> <mo
5874
class="MathClass-bin">+</mo> <mi
5875
class="MathClass-ord">z</mi><mo
5876
class="MathClass-close">)</mo></mrow><mrow><mo
5877
class="MathClass-open">(</mo><mi
5878
class="MathClass-ord">x</mi> <mo
5879
class="MathClass-bin">−</mo> <mi
5880
class="MathClass-ord">y</mi> <mo
5881
class="MathClass-bin">−</mo> <mi
5882
class="MathClass-ord">z</mi><mo
5883
class="MathClass-close">)</mo></mrow></math></td></tr></table></div><!--tex4ht:label?: x1-60292-->
5885
</td></tr></table></div><hr class="endfigure" />
5886
<div class="newtheorem">
5887
<!--l. 712--><p class="noindent"><span class="head">
5889
name="x1-6030r6"></a>
5891
class="cmbx-10">Theorem 6.6.</span> </span> <span
5892
class="cmti-10">Let </span><!--l. 714--><math
5893
xmlns="http://www.w3.org/1998/Math/MathML"
5895
class="MathClass-ord">φ</mi></math>
5897
class="cmti-10">be a protocol for a random pair </span><!--l. 714--><math
5898
xmlns="http://www.w3.org/1998/Math/MathML"
5899
mode="inline"> <mrow><mo
5900
class="MathClass-open">(</mo><mi
5901
class="MathClass-ord">X</mi><mo
5902
class="MathClass-punc">,</mo> <mi
5903
class="MathClass-ord">Y</mi> <mo
5904
class="MathClass-close">)</mo></mrow></math><span
5905
class="cmti-10">.</span>
5907
class="cmti-10">If one of </span><!--l. 715--><math
5908
xmlns="http://www.w3.org/1998/Math/MathML"
5909
mode="inline"> <msub
5911
class="MathClass-ord">σ</mi><mrow
5913
class="MathClass-ord">φ</mi></mrow></msub
5915
class="MathClass-open">(</mo><mi
5916
class="MathClass-ord">x</mi><mi
5917
class="MathClass-ord">′</mi><mo
5918
class="MathClass-punc">,</mo> <mi
5919
class="MathClass-ord">y</mi><mo
5920
class="MathClass-close">)</mo></mrow></math>
5922
class="cmti-10">and </span><!--l. 715--><math
5923
xmlns="http://www.w3.org/1998/Math/MathML"
5924
mode="inline"> <msub
5926
class="MathClass-ord">σ</mi><mrow
5928
class="MathClass-ord">φ</mi></mrow></msub
5930
class="MathClass-open">(</mo><mi
5931
class="MathClass-ord">x</mi><mo
5932
class="MathClass-punc">,</mo> <mi
5933
class="MathClass-ord">y</mi><mi
5934
class="MathClass-ord">′</mi><mo
5935
class="MathClass-close">)</mo></mrow></math>
5937
class="cmti-10">is a prefix of the other and </span><!--l. 716--><math
5938
xmlns="http://www.w3.org/1998/Math/MathML"
5939
mode="inline"> <mrow><mo
5940
class="MathClass-open">(</mo><mi
5941
class="MathClass-ord">x</mi><mo
5942
class="MathClass-punc">,</mo> <mi
5943
class="MathClass-ord">y</mi><mo
5944
class="MathClass-close">)</mo></mrow> <mo
5945
class="MathClass-rel">∈</mo> <msub
5947
class="MathClass-ord">S</mi><mrow
5949
class="MathClass-ord">X</mi><mo
5950
class="MathClass-punc">,</mo><mi
5951
class="MathClass-ord">Y</mi> </mrow></msub
5953
class="cmti-10">,</span>
5955
class="cmti-10">then </span><!--l. 717--><math
5956
xmlns="http://www.w3.org/1998/Math/MathML"
5957
mode="display"> <mrow
5961
class="MathClass-open">〈</mo><msub
5963
class="MathClass-ord">σ</mi><mrow
5965
class="MathClass-ord">j</mi></mrow></msub
5967
class="MathClass-open">(</mo><mi
5968
class="MathClass-ord">x</mi><mi
5969
class="MathClass-ord">′</mi><mo
5970
class="MathClass-punc">,</mo> <mi
5971
class="MathClass-ord">y</mi><mo
5972
class="MathClass-close">)</mo></mrow><mo
5973
class="MathClass-close">〉</mo></mrow><mrow
5975
class="MathClass-ord">j</mi> <mo
5976
class="MathClass-rel">=</mo> <mn
5977
class="MathClass-ord">1</mn></mrow><mrow
5979
class="MathClass-ord">∞</mi></mrow></msubsup
5981
class="MathClass-rel">=</mo> <msubsup
5983
class="MathClass-open">〈</mo><msub
5985
class="MathClass-ord">σ</mi><mrow
5988
class="MathClass-ord">j</mi></mrow></msub
5990
class="MathClass-open">(</mo><mi
5991
class="MathClass-ord">x</mi><mo
5992
class="MathClass-punc">,</mo> <mi
5993
class="MathClass-ord">y</mi><mo
5994
class="MathClass-close">)</mo></mrow><mo
5995
class="MathClass-close">〉</mo></mrow><mrow
5997
class="MathClass-ord">j</mi> <mo
5998
class="MathClass-rel">=</mo> <mn
5999
class="MathClass-ord">1</mn></mrow><mrow
6001
class="MathClass-ord">∞</mi></mrow></msubsup
6003
class="MathClass-rel">=</mo> <msubsup
6005
class="MathClass-open">〈</mo><msub
6007
class="MathClass-ord">σ</mi><mrow
6009
class="MathClass-ord">j</mi></mrow></msub
6011
class="MathClass-open">(</mo><mi
6012
class="MathClass-ord">x</mi><mo
6013
class="MathClass-punc">,</mo> <mi
6014
class="MathClass-ord">y</mi><mi
6015
class="MathClass-ord">′</mi><mo
6016
class="MathClass-close">)</mo></mrow><mo
6017
class="MathClass-close">〉</mo></mrow><mrow
6019
class="MathClass-ord">j</mi> <mo
6020
class="MathClass-rel">=</mo> <mn
6021
class="MathClass-ord">1</mn></mrow><mrow
6023
class="MathClass-ord">∞</mi></mrow></msubsup
6025
class="MathClass-punc">.</mo>
6029
<!--l. 723--><p class="indent">
6031
<div class="proof"><span class="head">
6033
class="cmti-10">Proof.</span> </span>We show by induction on <!--l. 724--><math
6034
xmlns="http://www.w3.org/1998/Math/MathML"
6036
class="MathClass-ord">i</mi></math>
6037
that <!--l. 725--><math
6038
xmlns="http://www.w3.org/1998/Math/MathML"
6039
mode="display"> <mrow
6043
class="MathClass-open">〈</mo><msub
6045
class="MathClass-ord">σ</mi><mrow
6047
class="MathClass-ord">j</mi></mrow></msub
6049
class="MathClass-open">(</mo><mi
6050
class="MathClass-ord">x</mi><mi
6051
class="MathClass-ord">′</mi><mo
6052
class="MathClass-punc">,</mo> <mi
6053
class="MathClass-ord">y</mi><mo
6054
class="MathClass-close">)</mo></mrow><mo
6055
class="MathClass-close">〉</mo></mrow><mrow
6057
class="MathClass-ord">j</mi> <mo
6058
class="MathClass-rel">=</mo> <mn
6059
class="MathClass-ord">1</mn></mrow><mrow
6061
class="MathClass-ord">i</mi></mrow></msubsup
6063
class="MathClass-rel">=</mo> <msubsup
6065
class="MathClass-open">〈</mo><msub
6067
class="MathClass-ord">σ</mi><mrow
6070
class="MathClass-ord">j</mi></mrow></msub
6072
class="MathClass-open">(</mo><mi
6073
class="MathClass-ord">x</mi><mo
6074
class="MathClass-punc">,</mo> <mi
6075
class="MathClass-ord">y</mi><mo
6076
class="MathClass-close">)</mo></mrow><mo
6077
class="MathClass-close">〉</mo></mrow><mrow
6079
class="MathClass-ord">j</mi> <mo
6080
class="MathClass-rel">=</mo> <mn
6081
class="MathClass-ord">1</mn></mrow><mrow
6083
class="MathClass-ord">i</mi></mrow></msubsup
6085
class="MathClass-rel">=</mo> <msubsup
6087
class="MathClass-open">〈</mo><msub
6089
class="MathClass-ord">σ</mi><mrow
6091
class="MathClass-ord">j</mi></mrow></msub
6093
class="MathClass-open">(</mo><mi
6094
class="MathClass-ord">x</mi><mo
6095
class="MathClass-punc">,</mo> <mi
6096
class="MathClass-ord">y</mi><mi
6097
class="MathClass-ord">′</mi><mo
6098
class="MathClass-close">)</mo></mrow><mo
6099
class="MathClass-close">〉</mo></mrow><mrow
6101
class="MathClass-ord">j</mi> <mo
6102
class="MathClass-rel">=</mo> <mn
6103
class="MathClass-ord">1</mn></mrow><mrow
6105
class="MathClass-ord">i</mi></mrow></msubsup
6107
class="MathClass-punc">.</mo>
6110
The induction hypothesis holds vacuously for <!--l. 730--><math
6111
xmlns="http://www.w3.org/1998/Math/MathML"
6113
class="MathClass-ord">i</mi> <mo
6114
class="MathClass-rel">=</mo> <mn
6115
class="MathClass-ord">0</mn></math>.
6116
Assume it holds for <!--l. 731--><math
6117
xmlns="http://www.w3.org/1998/Math/MathML"
6119
class="MathClass-ord">i</mi> <mo
6120
class="MathClass-bin">−</mo> <mn
6121
class="MathClass-ord">1</mn></math>,
6122
in particular <!--l. 732--><math
6123
xmlns="http://www.w3.org/1998/Math/MathML"
6124
mode="inline"> <msubsup
6126
class="MathClass-open">[</mo><msub
6128
class="MathClass-ord">σ</mi><mrow
6130
class="MathClass-ord">j</mi></mrow></msub
6132
class="MathClass-open">(</mo><mi
6133
class="MathClass-ord">x</mi><mi
6134
class="MathClass-ord">′</mi><mo
6135
class="MathClass-punc">,</mo> <mi
6136
class="MathClass-ord">y</mi><mo
6137
class="MathClass-close">)</mo></mrow><mo
6138
class="MathClass-close">]</mo></mrow><mrow
6140
class="MathClass-ord">j</mi> <mo
6141
class="MathClass-rel">=</mo> <mn
6142
class="MathClass-ord">1</mn></mrow><mrow
6144
class="MathClass-ord">i</mi> <mo
6145
class="MathClass-bin">−</mo> <mn
6146
class="MathClass-ord">1</mn></mrow></msubsup
6148
class="MathClass-rel">=</mo> <msubsup
6150
class="MathClass-open">[</mo><msub
6152
class="MathClass-ord">σ</mi><mrow
6154
class="MathClass-ord">j</mi></mrow></msub
6156
class="MathClass-open">(</mo><mi
6157
class="MathClass-ord">x</mi><mo
6158
class="MathClass-punc">,</mo> <mi
6159
class="MathClass-ord">y</mi><mi
6160
class="MathClass-ord">′</mi><mo
6161
class="MathClass-close">)</mo></mrow><mo
6162
class="MathClass-close">]</mo></mrow><mrow
6164
class="MathClass-ord">j</mi> <mo
6165
class="MathClass-rel">=</mo> <mn
6166
class="MathClass-ord">1</mn></mrow><mrow
6168
class="MathClass-ord">i</mi> <mo
6169
class="MathClass-bin">−</mo> <mn
6170
class="MathClass-ord">1</mn></mrow></msubsup
6172
Then one of <!--l. 733--><math
6173
xmlns="http://www.w3.org/1998/Math/MathML"
6174
mode="inline"> <msubsup
6176
class="MathClass-open">[</mo><msub
6178
class="MathClass-ord">σ</mi><mrow
6180
class="MathClass-ord">j</mi></mrow></msub
6182
class="MathClass-open">(</mo><mi
6183
class="MathClass-ord">x</mi><mi
6184
class="MathClass-ord">′</mi><mo
6185
class="MathClass-punc">,</mo> <mi
6186
class="MathClass-ord">y</mi><mo
6187
class="MathClass-close">)</mo></mrow><mo
6188
class="MathClass-close">]</mo></mrow><mrow
6190
class="MathClass-ord">j</mi> <mo
6191
class="MathClass-rel">=</mo> <mi
6192
class="MathClass-ord">i</mi></mrow><mrow
6194
class="MathClass-ord">∞</mi></mrow></msubsup
6196
and <!--l. 733--><math
6197
xmlns="http://www.w3.org/1998/Math/MathML"
6198
mode="inline"> <msubsup
6200
class="MathClass-open">[</mo><msub
6202
class="MathClass-ord">σ</mi><mrow
6204
class="MathClass-ord">j</mi></mrow></msub
6206
class="MathClass-open">(</mo><mi
6207
class="MathClass-ord">x</mi><mo
6208
class="MathClass-punc">,</mo> <mi
6209
class="MathClass-ord">y</mi><mi
6210
class="MathClass-ord">′</mi><mo
6211
class="MathClass-close">)</mo></mrow><mo
6212
class="MathClass-close">]</mo></mrow><mrow
6214
class="MathClass-ord">j</mi> <mo
6215
class="MathClass-rel">=</mo> <mi
6216
class="MathClass-ord">i</mi></mrow><mrow
6218
class="MathClass-ord">∞</mi></mrow></msubsup
6220
is a prefix of the other which implies that one of <!--l. 734--><math
6221
xmlns="http://www.w3.org/1998/Math/MathML"
6222
mode="inline"> <msub
6224
class="MathClass-ord">σ</mi><mrow
6226
class="MathClass-ord">i</mi></mrow></msub
6228
class="MathClass-open">(</mo><mi
6229
class="MathClass-ord">x</mi><mi
6230
class="MathClass-ord">′</mi><mo
6231
class="MathClass-punc">,</mo> <mi
6232
class="MathClass-ord">y</mi><mo
6233
class="MathClass-close">)</mo></mrow></math>
6234
and <!--l. 735--><math
6235
xmlns="http://www.w3.org/1998/Math/MathML"
6236
mode="inline"> <msub
6238
class="MathClass-ord">σ</mi><mrow
6240
class="MathClass-ord">i</mi></mrow></msub
6242
class="MathClass-open">(</mo><mi
6243
class="MathClass-ord">x</mi><mo
6244
class="MathClass-punc">,</mo> <mi
6245
class="MathClass-ord">y</mi><mi
6246
class="MathClass-ord">′</mi><mo
6247
class="MathClass-close">)</mo></mrow></math>
6248
is a prefix of the other. If the <!--l. 735--><math
6249
xmlns="http://www.w3.org/1998/Math/MathML"
6251
class="MathClass-ord">i</mi></math>th
6252
message is transmitted by <!--l. 736--><math
6253
xmlns="http://www.w3.org/1998/Math/MathML"
6254
mode="inline"> <msub
6256
class="MathClass-ord">P</mi><mrow
6258
class="MathClass-ord"><!--span
6259
class="htf-calligraphy"-->X<!--/span--></mi> </mrow></msub
6261
then, by the separate-transmissions property and the induction hypothesis, <!--l. 737--><math
6262
xmlns="http://www.w3.org/1998/Math/MathML"
6266
class="MathClass-ord">σ</mi><mrow
6268
class="MathClass-ord">i</mi></mrow></msub
6270
class="MathClass-open">(</mo><mi
6271
class="MathClass-ord">x</mi><mo
6272
class="MathClass-punc">,</mo> <mi
6273
class="MathClass-ord">y</mi><mo
6274
class="MathClass-close">)</mo></mrow> <mo
6275
class="MathClass-rel">=</mo> <msub
6277
class="MathClass-ord">σ</mi><mrow
6279
class="MathClass-ord">i</mi></mrow></msub
6281
class="MathClass-open">(</mo><mi
6282
class="MathClass-ord">x</mi><mo
6283
class="MathClass-punc">,</mo> <mi
6284
class="MathClass-ord">y</mi><mi
6285
class="MathClass-ord">′</mi><mo
6286
class="MathClass-close">)</mo></mrow></math>,
6287
hence one of <!--l. 738--><math
6288
xmlns="http://www.w3.org/1998/Math/MathML"
6289
mode="inline"> <msub
6291
class="MathClass-ord">σ</mi><mrow
6293
class="MathClass-ord">i</mi></mrow></msub
6295
class="MathClass-open">(</mo><mi
6296
class="MathClass-ord">x</mi><mo
6297
class="MathClass-punc">,</mo> <mi
6298
class="MathClass-ord">y</mi><mo
6299
class="MathClass-close">)</mo></mrow></math>
6300
and <!--l. 738--><math
6301
xmlns="http://www.w3.org/1998/Math/MathML"
6302
mode="inline"> <msub
6304
class="MathClass-ord">σ</mi><mrow
6306
class="MathClass-ord">i</mi></mrow></msub
6308
class="MathClass-open">(</mo><mi
6309
class="MathClass-ord">x</mi><mi
6310
class="MathClass-ord">′</mi><mo
6311
class="MathClass-punc">,</mo> <mi
6312
class="MathClass-ord">y</mi><mo
6313
class="MathClass-close">)</mo></mrow></math>
6314
is a prefix of the other. By the implicit-termination property, neither <!--l. 739--><math
6315
xmlns="http://www.w3.org/1998/Math/MathML"
6316
mode="inline"> <msub
6318
class="MathClass-ord">σ</mi><mrow
6320
class="MathClass-ord">i</mi></mrow></msub
6322
class="MathClass-open">(</mo><mi
6323
class="MathClass-ord">x</mi><mo
6324
class="MathClass-punc">,</mo> <mi
6325
class="MathClass-ord">y</mi><mo
6326
class="MathClass-close">)</mo></mrow></math>
6327
nor <!--l. 739--><math
6328
xmlns="http://www.w3.org/1998/Math/MathML"
6329
mode="inline"> <msub
6331
class="MathClass-ord">σ</mi><mrow
6333
class="MathClass-ord">i</mi></mrow></msub
6335
class="MathClass-open">(</mo><mi
6336
class="MathClass-ord">x</mi><mi
6337
class="MathClass-ord">′</mi><mo
6338
class="MathClass-punc">,</mo> <mi
6339
class="MathClass-ord">y</mi><mo
6340
class="MathClass-close">)</mo></mrow></math>
6341
can be a proper prefix of the other, hence they must be the same and <!--l. 741--><math
6342
xmlns="http://www.w3.org/1998/Math/MathML"
6343
mode="inline"> <msub
6345
class="MathClass-ord">σ</mi><mrow
6347
class="MathClass-ord">i</mi></mrow></msub
6349
class="MathClass-open">(</mo><mi
6350
class="MathClass-ord">x</mi><mi
6351
class="MathClass-ord">′</mi><mo
6352
class="MathClass-punc">,</mo> <mi
6353
class="MathClass-ord">y</mi><mo
6354
class="MathClass-close">)</mo></mrow> <mo
6355
class="MathClass-rel">=</mo> <msub
6357
class="MathClass-ord">σ</mi><mrow
6359
class="MathClass-ord">i</mi></mrow></msub
6361
class="MathClass-open">(</mo><mi
6362
class="MathClass-ord">x</mi><mo
6363
class="MathClass-punc">,</mo> <mi
6364
class="MathClass-ord">y</mi><mo
6365
class="MathClass-close">)</mo></mrow> <mo
6366
class="MathClass-rel">=</mo> <msub
6368
class="MathClass-ord">σ</mi><mrow
6370
class="MathClass-ord">i</mi></mrow></msub
6372
class="MathClass-open">(</mo><mi
6373
class="MathClass-ord">x</mi><mo
6374
class="MathClass-punc">,</mo> <mi
6375
class="MathClass-ord">y</mi><mi
6376
class="MathClass-ord">′</mi><mo
6377
class="MathClass-close">)</mo></mrow></math>.
6378
If the <!--l. 741--><math
6379
xmlns="http://www.w3.org/1998/Math/MathML"
6381
class="MathClass-ord">i</mi></math>th
6382
message is transmitted by <!--l. 742--><math
6383
xmlns="http://www.w3.org/1998/Math/MathML"
6384
mode="inline"> <msub
6386
class="MathClass-ord">P</mi><mrow
6388
class="MathClass-ord"><!--span
6389
class="htf-calligraphy"-->Y<!--/span--></mi></mrow></msub
6391
then, symmetrically, <!--l. 742--><math
6392
xmlns="http://www.w3.org/1998/Math/MathML"
6393
mode="inline"> <msub
6395
class="MathClass-ord">σ</mi><mrow
6397
class="MathClass-ord">i</mi></mrow></msub
6399
class="MathClass-open">(</mo><mi
6400
class="MathClass-ord">x</mi><mo
6401
class="MathClass-punc">,</mo> <mi
6402
class="MathClass-ord">y</mi><mo
6403
class="MathClass-close">)</mo></mrow> <mo
6404
class="MathClass-rel">=</mo> <msub
6406
class="MathClass-ord">σ</mi><mrow
6408
class="MathClass-ord">i</mi></mrow></msub
6410
class="MathClass-open">(</mo><mi
6411
class="MathClass-ord">x</mi><mi
6412
class="MathClass-ord">′</mi><mo
6413
class="MathClass-punc">,</mo> <mi
6414
class="MathClass-ord">y</mi><mo
6415
class="MathClass-close">)</mo></mrow></math>
6416
by the induction hypothesis and the separate-transmissions property, and, then, <!--l. 744--><math
6417
xmlns="http://www.w3.org/1998/Math/MathML"
6421
class="MathClass-ord">σ</mi><mrow
6423
class="MathClass-ord">i</mi></mrow></msub
6425
class="MathClass-open">(</mo><mi
6426
class="MathClass-ord">x</mi><mo
6427
class="MathClass-punc">,</mo> <mi
6428
class="MathClass-ord">y</mi><mo
6429
class="MathClass-close">)</mo></mrow> <mo
6430
class="MathClass-rel">=</mo> <msub
6432
class="MathClass-ord">σ</mi><mrow
6434
class="MathClass-ord">i</mi></mrow></msub
6436
class="MathClass-open">(</mo><mi
6437
class="MathClass-ord">x</mi><mo
6438
class="MathClass-punc">,</mo> <mi
6439
class="MathClass-ord">y</mi><mi
6440
class="MathClass-ord">′</mi><mo
6441
class="MathClass-close">)</mo></mrow></math>
6442
by the implicit-termination property, proving the induction step. </div> □
6443
<!--l. 748--><p class="noindent">If <!--l. 748--><math
6444
xmlns="http://www.w3.org/1998/Math/MathML"
6446
class="MathClass-ord">φ</mi></math> is a
6447
protocol for <!--l. 748--><math
6448
xmlns="http://www.w3.org/1998/Math/MathML"
6449
mode="inline"> <mrow><mo
6450
class="MathClass-open">(</mo><mi
6451
class="MathClass-ord">X</mi><mo
6452
class="MathClass-punc">,</mo> <mi
6453
class="MathClass-ord">Y</mi> <mo
6454
class="MathClass-close">)</mo></mrow></math>,
6455
and <!--l. 748--><math
6456
xmlns="http://www.w3.org/1998/Math/MathML"
6457
mode="inline"> <mrow><mo
6458
class="MathClass-open">(</mo><mi
6459
class="MathClass-ord">x</mi><mo
6460
class="MathClass-punc">,</mo> <mi
6461
class="MathClass-ord">y</mi><mo
6462
class="MathClass-close">)</mo></mrow></math>, <!--l. 748--><math
6463
xmlns="http://www.w3.org/1998/Math/MathML"
6466
class="MathClass-open">(</mo><mi
6467
class="MathClass-ord">x</mi><mi
6468
class="MathClass-ord">′</mi><mo
6469
class="MathClass-punc">,</mo> <mi
6470
class="MathClass-ord">y</mi><mo
6471
class="MathClass-close">)</mo></mrow></math> are distinct inputs in <!--l. 749--><math
6472
xmlns="http://www.w3.org/1998/Math/MathML"
6476
class="MathClass-ord">S</mi><mrow
6478
class="MathClass-ord">X</mi><mo
6479
class="MathClass-punc">,</mo><mi
6480
class="MathClass-ord">Y</mi> </mrow></msub
6481
></math>, then, by the correct-decision
6482
property, <!--l. 750--><math
6483
xmlns="http://www.w3.org/1998/Math/MathML"
6484
mode="inline"> <msubsup
6486
class="MathClass-open">〈</mo><msub
6488
class="MathClass-ord">σ</mi><mrow
6490
class="MathClass-ord">j</mi></mrow></msub
6492
class="MathClass-open">(</mo><mi
6493
class="MathClass-ord">x</mi><mo
6494
class="MathClass-punc">,</mo> <mi
6495
class="MathClass-ord">y</mi><mo
6496
class="MathClass-close">)</mo></mrow><mo
6497
class="MathClass-close">〉</mo></mrow><mrow
6499
class="MathClass-ord">j</mi> <mo
6500
class="MathClass-rel">=</mo> <mn
6501
class="MathClass-ord">1</mn></mrow><mrow
6503
class="MathClass-ord">∞</mi></mrow></msubsup
6505
class="MathClass-rel">≠</mo><msubsup
6507
class="MathClass-open">〈</mo><msub
6509
class="MathClass-ord">σ</mi><mrow
6511
class="MathClass-ord">j</mi></mrow></msub
6513
class="MathClass-open">(</mo><mi
6514
class="MathClass-ord">x</mi><mi
6515
class="MathClass-ord">′</mi><mo
6516
class="MathClass-punc">,</mo> <mi
6517
class="MathClass-ord">y</mi><mo
6518
class="MathClass-close">)</mo></mrow><mo
6519
class="MathClass-close">〉</mo></mrow><mrow
6521
class="MathClass-ord">j</mi> <mo
6522
class="MathClass-rel">=</mo> <mn
6523
class="MathClass-ord">1</mn></mrow><mrow
6525
class="MathClass-ord">∞</mi></mrow></msubsup
6527
</p><!--l. 753--><p class="indent"> Equation (<a
6528
href="#x1-6019r25">25<!--tex4ht:ref: E_SXgYy--></a>) defined <!--l. 753--><math
6529
xmlns="http://www.w3.org/1998/Math/MathML"
6530
mode="inline"> <msub
6532
class="MathClass-ord">P</mi><mrow
6534
class="MathClass-ord"><!--span
6535
class="htf-calligraphy"-->Y<!--/span--></mi></mrow></msub
6537
ambiguity set <!--l. 753--><math
6538
xmlns="http://www.w3.org/1998/Math/MathML"
6539
mode="inline"> <msub
6541
class="MathClass-ord">S</mi><mrow
6543
class="MathClass-ord">X</mi><mo
6544
class="MathClass-rel">∣</mo><mi
6545
class="MathClass-ord">Y</mi> </mrow></msub
6547
class="MathClass-open">(</mo><mi
6548
class="MathClass-ord">y</mi><mo
6549
class="MathClass-close">)</mo></mrow></math> to be the
6550
set of possible <!--l. 754--><math
6551
xmlns="http://www.w3.org/1998/Math/MathML"
6553
class="MathClass-ord">X</mi></math> values
6554
when <!--l. 754--><math
6555
xmlns="http://www.w3.org/1998/Math/MathML"
6557
class="MathClass-ord">Y</mi> <mo
6558
class="MathClass-rel">=</mo> <mi
6559
class="MathClass-ord">y</mi></math>. The last corollary
6560
implies that for all <!--l. 755--><math
6561
xmlns="http://www.w3.org/1998/Math/MathML"
6563
class="MathClass-ord">y</mi> <mo
6564
class="MathClass-rel">∈</mo> <msub
6566
class="MathClass-ord">S</mi><mrow
6568
class="MathClass-ord">Y</mi> </mrow></msub
6571
href="testmath2.xml" name="testmath2.xml" ><sup>1</sup></a> of
6572
codewords <!--l. 759--><math
6573
xmlns="http://www.w3.org/1998/Math/MathML"
6574
mode="inline"> <mrow><mo
6575
class="MathClass-open">{</mo><msub
6577
class="MathClass-ord">σ</mi><mrow
6579
class="MathClass-ord">φ</mi></mrow></msub
6581
class="MathClass-open">(</mo><mi
6582
class="MathClass-ord">x</mi><mo
6583
class="MathClass-punc">,</mo> <mi
6584
class="MathClass-ord">y</mi><mo
6585
class="MathClass-close">)</mo></mrow> <mo
6586
class="MathClass-punc">:</mo> <mi
6587
class="MathClass-ord">x</mi> <mo
6588
class="MathClass-rel">∈</mo> <msub
6590
class="MathClass-ord">S</mi><mrow
6592
class="MathClass-ord">X</mi><mo
6593
class="MathClass-rel">∣</mo><mi
6594
class="MathClass-ord">Y</mi> </mrow></msub
6596
class="MathClass-open">(</mo><mi
6597
class="MathClass-ord">y</mi><mo
6598
class="MathClass-close">)</mo></mrow><mo
6599
class="MathClass-close">}</mo></mrow></math>
6603
<h3 class="sectionHead"><span class="titlemark">7</span> <a
6604
name="x1-70007"></a>One-Way Complexity</h3>
6605
<!--l. 764--><p class="noindent"><!--l. 764--><math
6606
xmlns="http://www.w3.org/1998/Math/MathML"
6607
mode="inline"> <msub
6609
class="MathClass-ord">Ĉ</mi><mrow
6611
class="MathClass-ord">1</mn></mrow></msub
6613
class="MathClass-open">(</mo><mi
6614
class="MathClass-ord">X</mi><mo
6615
class="MathClass-rel">∣</mo><mi
6616
class="MathClass-ord">Y</mi> <mo
6617
class="MathClass-close">)</mo></mrow></math>,
6618
the one-way complexity of a random pair <!--l. 764--><math
6619
xmlns="http://www.w3.org/1998/Math/MathML"
6622
class="MathClass-open">(</mo><mi
6623
class="MathClass-ord">X</mi><mo
6624
class="MathClass-punc">,</mo> <mi
6625
class="MathClass-ord">Y</mi> <mo
6626
class="MathClass-close">)</mo></mrow></math>, is the number of bits <!--l. 765--><math
6627
xmlns="http://www.w3.org/1998/Math/MathML"
6631
class="MathClass-ord">P</mi><mrow
6633
class="MathClass-ord"><!--span
6634
class="htf-calligraphy"-->X<!--/span--></mi> </mrow></msub
6635
></math> must transmit in the
6636
worst case when <!--l. 766--><math
6637
xmlns="http://www.w3.org/1998/Math/MathML"
6638
mode="inline"> <msub
6640
class="MathClass-ord">P</mi><mrow
6642
class="MathClass-ord"><!--span
6643
class="htf-calligraphy"-->Y<!--/span--></mi></mrow></msub
6645
is not permitted to transmit any feedback messages. Starting with <!--l. 767--><math
6646
xmlns="http://www.w3.org/1998/Math/MathML"
6650
class="MathClass-ord">S</mi><mrow
6652
class="MathClass-ord">X</mi><mo
6653
class="MathClass-punc">,</mo><mi
6654
class="MathClass-ord">Y</mi> </mrow></msub
6655
></math>, the support set of <!--l. 767--><math
6656
xmlns="http://www.w3.org/1998/Math/MathML"
6659
class="MathClass-open">(</mo><mi
6660
class="MathClass-ord">X</mi><mo
6661
class="MathClass-punc">,</mo> <mi
6662
class="MathClass-ord">Y</mi> <mo
6663
class="MathClass-close">)</mo></mrow></math>, we define <!--l. 767--><math
6664
xmlns="http://www.w3.org/1998/Math/MathML"
6667
class="MathClass-ord">G</mi><mrow><mo
6668
class="MathClass-open">(</mo><mi
6669
class="MathClass-ord">X</mi><mo
6670
class="MathClass-rel">∣</mo><mi
6671
class="MathClass-ord">Y</mi> <mo
6672
class="MathClass-close">)</mo></mrow></math>, the <span
6673
class="cmti-10">characteristic</span>
6675
class="cmti-10">hypergraph </span>of <!--l. 768--><math
6676
xmlns="http://www.w3.org/1998/Math/MathML"
6677
mode="inline"> <mrow><mo
6678
class="MathClass-open">(</mo><mi
6679
class="MathClass-ord">X</mi><mo
6680
class="MathClass-punc">,</mo> <mi
6681
class="MathClass-ord">Y</mi> <mo
6682
class="MathClass-close">)</mo></mrow></math>,
6683
and show that <!--l. 769--><math
6684
xmlns="http://www.w3.org/1998/Math/MathML"
6685
mode="display"> <mrow
6689
class="MathClass-ord">Ĉ</mi><mrow
6691
class="MathClass-ord">1</mn></mrow></msub
6693
class="MathClass-open">(</mo><mi
6694
class="MathClass-ord">X</mi><mo
6695
class="MathClass-rel">∣</mo><mi
6696
class="MathClass-ord">Y</mi> <mo
6697
class="MathClass-close">)</mo></mrow> <mo
6698
class="MathClass-rel">=</mo> <mrow><mo
6699
class="MathClass-open">⌈</mo> <mo
6700
>log</mo><!--nolimits--> <mi
6701
class="MathClass-ord">χ</mi><mrow><mo
6702
class="MathClass-open">(</mo><mi
6703
class="MathClass-ord">G</mi><mrow><mo
6704
class="MathClass-open">(</mo><mi
6705
class="MathClass-ord">X</mi><mo
6706
class="MathClass-rel">∣</mo><mi
6707
class="MathClass-ord">Y</mi> <mo
6708
class="MathClass-close">)</mo></mrow><mo
6709
class="MathClass-close">)</mo></mrow><mo
6710
class="MathClass-close">⌉</mo></mrow><mspace class="nbsp" /><mo
6711
class="MathClass-punc">.</mo>
6713
</p><!--l. 773--><p class="indent"> Let <!--l. 773--><math
6714
xmlns="http://www.w3.org/1998/Math/MathML"
6715
mode="inline"> <mrow><mo
6716
class="MathClass-open">(</mo><mi
6717
class="MathClass-ord">X</mi><mo
6718
class="MathClass-punc">,</mo> <mi
6719
class="MathClass-ord">Y</mi> <mo
6720
class="MathClass-close">)</mo></mrow></math> be a random
6721
pair. For each <!--l. 773--><math
6722
xmlns="http://www.w3.org/1998/Math/MathML"
6724
class="MathClass-ord">y</mi></math> in <!--l. 773--><math
6725
xmlns="http://www.w3.org/1998/Math/MathML"
6729
class="MathClass-ord">S</mi><mrow
6731
class="MathClass-ord">Y</mi> </mrow></msub
6732
></math>, the support set of <!--l. 774--><math
6733
xmlns="http://www.w3.org/1998/Math/MathML"
6736
class="MathClass-ord">Y</mi> </math>, Equation (<a
6737
href="#x1-6019r25">25<!--tex4ht:ref: E_SXgYy--></a>) defined <!--l. 774--><math
6738
xmlns="http://www.w3.org/1998/Math/MathML"
6742
class="MathClass-ord">S</mi><mrow
6744
class="MathClass-ord">X</mi><mo
6745
class="MathClass-rel">∣</mo><mi
6746
class="MathClass-ord">Y</mi> </mrow></msub
6748
class="MathClass-open">(</mo><mi
6749
class="MathClass-ord">y</mi><mo
6750
class="MathClass-close">)</mo></mrow></math> to be the set of possible <!--l. 775--><math
6751
xmlns="http://www.w3.org/1998/Math/MathML"
6754
class="MathClass-ord">x</mi></math> values when <!--l. 775--><math
6755
xmlns="http://www.w3.org/1998/Math/MathML"
6758
class="MathClass-ord">Y</mi> <mo
6759
class="MathClass-rel">=</mo> <mi
6760
class="MathClass-ord">y</mi></math>. The <span
6761
class="cmti-10">characteristic</span>
6763
class="cmti-10">hypergraph </span><!--l. 775--><math
6764
xmlns="http://www.w3.org/1998/Math/MathML"
6766
class="MathClass-ord">G</mi><mrow><mo
6767
class="MathClass-open">(</mo><mi
6768
class="MathClass-ord">X</mi><mo
6769
class="MathClass-rel">∣</mo><mi
6770
class="MathClass-ord">Y</mi> <mo
6771
class="MathClass-close">)</mo></mrow></math>
6772
of <!--l. 776--><math
6773
xmlns="http://www.w3.org/1998/Math/MathML"
6774
mode="inline"> <mrow><mo
6775
class="MathClass-open">(</mo><mi
6776
class="MathClass-ord">X</mi><mo
6777
class="MathClass-punc">,</mo> <mi
6778
class="MathClass-ord">Y</mi> <mo
6779
class="MathClass-close">)</mo></mrow></math> has <!--l. 776--><math
6780
xmlns="http://www.w3.org/1998/Math/MathML"
6784
class="MathClass-ord">S</mi><mrow
6786
class="MathClass-ord">X</mi></mrow></msub
6787
></math> as its vertex set and
6788
the hyperedge <!--l. 776--><math
6789
xmlns="http://www.w3.org/1998/Math/MathML"
6790
mode="inline"> <msub
6792
class="MathClass-ord">S</mi><mrow
6794
class="MathClass-ord">X</mi><mo
6795
class="MathClass-rel">∣</mo><mi
6796
class="MathClass-ord">Y</mi> </mrow></msub
6798
class="MathClass-open">(</mo><mi
6799
class="MathClass-ord">y</mi><mo
6800
class="MathClass-close">)</mo></mrow></math>
6801
for each <!--l. 777--><math
6802
xmlns="http://www.w3.org/1998/Math/MathML"
6804
class="MathClass-ord">y</mi> <mo
6805
class="MathClass-rel">∈</mo> <msub
6807
class="MathClass-ord">S</mi><mrow
6809
class="MathClass-ord">Y</mi> </mrow></msub
6811
</p><!--l. 780--><p class="indent"> We can now prove a continuity theorem. </p><div class="newtheorem">
6812
<!--l. 781--><p class="noindent"><span class="head">
6814
name="x1-7001r1"></a>
6816
class="cmbx-10">Theorem 7.1.</span> </span> <span
6817
class="cmti-10">Let </span><!--l. 782--><math
6818
xmlns="http://www.w3.org/1998/Math/MathML"
6820
class="MathClass-ord">Ω</mi> <mo
6821
class="MathClass-rel">⊂</mo><msup
6822
> <mi class="mathbf">R</mi><mrow
6824
class="MathClass-ord">n</mi></mrow></msup
6827
class="cmti-10">be an open set, let </span><!--l. 783--><math
6828
xmlns="http://www.w3.org/1998/Math/MathML"
6830
class="MathClass-ord">u</mi> <mo
6831
class="MathClass-rel">∈</mo> <mi
6832
class="MathClass-ord">B</mi><mi
6833
class="MathClass-ord">V</mi> <mrow><mo
6834
class="MathClass-open">(</mo><mi
6835
class="MathClass-ord">Ω</mi><mo
6836
class="MathClass-punc">;</mo><msup
6837
> <mi class="mathbf">R</mi><mrow
6839
class="MathClass-ord">m</mi></mrow></msup
6841
class="MathClass-close">)</mo></mrow></math><span
6842
class="cmti-10">,</span>
6845
class="cmti-10">and let</span> </p><table class="equation"><tr><td>
6847
xmlns="http://www.w3.org/1998/Math/MathML"
6850
class="equation"><mtr><mtd>
6853
class="MathClass-ord">T</mi><mrow
6855
class="MathClass-ord">x</mi></mrow><mrow
6857
class="MathClass-ord">u</mi></mrow></msubsup
6859
class="MathClass-rel">=</mo> <mfenced
6860
open="{" close="}" ><mi
6861
class="MathClass-ord">y</mi> <mo
6862
class="MathClass-rel">∈</mo><msup
6863
> <mi class="mathbf">R</mi><mrow
6865
class="MathClass-ord">m</mi></mrow></msup
6867
class="MathClass-punc">:</mo> <mi
6868
class="MathClass-ord">y</mi> <mo
6869
class="MathClass-rel">=</mo> <mi
6870
class="MathClass-ord">ũ</mi><mrow><mo
6871
class="MathClass-open">(</mo><mi
6872
class="MathClass-ord">x</mi><mo
6873
class="MathClass-close">)</mo></mrow> <mo
6874
class="MathClass-bin">+</mo> <mfenced
6875
open="〈" close="〉" > <mfrac><mrow
6877
class="MathClass-ord">D</mi><mi
6878
class="MathClass-ord">u</mi></mrow>
6881
open="|" close="|" ><mi
6882
class="MathClass-ord">D</mi><mi
6883
class="MathClass-ord">u</mi></mfenced> </mrow></mfrac><mrow><mo
6884
class="MathClass-open">(</mo><mi
6885
class="MathClass-ord">x</mi><mo
6886
class="MathClass-close">)</mo></mrow><mo
6887
class="MathClass-punc">,</mo> <mi
6888
class="MathClass-ord">z</mi></mfenced> <mrow
6889
class="text"><mtext > for some </mtext></mrow><mi
6890
class="MathClass-ord">z</mi> <mo
6891
class="MathClass-rel">∈</mo><msup
6892
> <mi class="mathbf">R</mi><mrow
6894
class="MathClass-ord">n</mi></mrow></msup
6895
> </mfenced> </mtd><mtd><mspace
6896
id="x1-7002r26" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
6898
<!--l. 788--><p class="nopar"></p></td><td width="5%">(26)</td></tr></table>
6900
class="cmti-10">for every </span><!--l. 789--><math
6901
xmlns="http://www.w3.org/1998/Math/MathML"
6903
class="MathClass-ord">x</mi> <mo
6904
class="MathClass-rel">∈</mo> <mi
6905
class="MathClass-ord">Ω</mi><mo
6906
class="MathClass-bin">∖</mo><msub
6908
class="MathClass-ord">S</mi><mrow
6910
class="MathClass-ord">u</mi></mrow></msub
6912
class="cmti-10">. Let </span><!--l. 789--><math
6913
xmlns="http://www.w3.org/1998/Math/MathML"
6916
class="MathClass-ord">f</mi> <mo
6917
class="MathClass-punc">:</mo><msup
6918
> <mi class="mathbf">R</mi><mrow
6920
class="MathClass-ord">m</mi></mrow></msup
6922
class="MathClass-rel">→</mo><msup
6923
> <mi class="mathbf">R</mi><mrow
6925
class="MathClass-ord">k</mi></mrow></msup
6927
class="cmti-10">be a Lipschitz continuous</span>
6929
class="cmti-10">function such that </span><!--l. 790--><math
6930
xmlns="http://www.w3.org/1998/Math/MathML"
6932
class="MathClass-ord">f</mi><mrow><mo
6933
class="MathClass-open">(</mo><mn
6934
class="MathClass-ord">0</mn><mo
6935
class="MathClass-close">)</mo></mrow> <mo
6936
class="MathClass-rel">=</mo> <mn
6937
class="MathClass-ord">0</mn></math><span
6938
class="cmti-10">,</span>
6940
class="cmti-10">and let </span><!--l. 791--><math
6941
xmlns="http://www.w3.org/1998/Math/MathML"
6943
class="MathClass-ord">v</mi> <mo
6944
class="MathClass-rel">=</mo> <mi
6945
class="MathClass-ord">f</mi><mrow><mo
6946
class="MathClass-open">(</mo><mi
6947
class="MathClass-ord">u</mi><mo
6948
class="MathClass-close">)</mo></mrow><mo
6949
class="MathClass-punc">:</mo> <mi
6950
class="MathClass-ord">Ω</mi> <mo
6951
class="MathClass-rel">→</mo><msup
6952
> <mi class="mathbf">R</mi><mrow
6954
class="MathClass-ord">k</mi></mrow></msup
6956
class="cmti-10">.</span>
6958
class="cmti-10">Then </span><!--l. 791--><math
6959
xmlns="http://www.w3.org/1998/Math/MathML"
6961
class="MathClass-ord">v</mi> <mo
6962
class="MathClass-rel">∈</mo> <mi
6963
class="MathClass-ord">B</mi><mi
6964
class="MathClass-ord">V</mi> <mrow><mo
6965
class="MathClass-open">(</mo><mi
6966
class="MathClass-ord">Ω</mi><mo
6967
class="MathClass-punc">;</mo><msup
6968
> <mi class="mathbf">R</mi><mrow
6970
class="MathClass-ord">k</mi></mrow></msup
6972
class="MathClass-close">)</mo></mrow></math>
6974
class="cmti-10">and </span><table class="equation"><tr><td>
6976
xmlns="http://www.w3.org/1998/Math/MathML"
6979
class="equation"><mtr><mtd>
6981
class="MathClass-ord">J</mi><mi
6982
class="MathClass-ord">v</mi> <mo
6983
class="MathClass-rel">=</mo><msub
6985
open="" close="|" ><mrow><mo
6986
class="MathClass-open">(</mo><mi
6987
class="MathClass-ord">f</mi><mrow><mo
6988
class="MathClass-open">(</mo><msup
6990
class="MathClass-ord">u</mi><mrow
6992
class="MathClass-bin">+</mo></mrow></msup
6994
class="MathClass-close">)</mo></mrow> <mo
6995
class="MathClass-bin">−</mo> <mi
6996
class="MathClass-ord">f</mi><mrow><mo
6997
class="MathClass-open">(</mo><msup
6999
class="MathClass-ord">u</mi><mrow
7001
class="MathClass-bin">−</mo></mrow></msup
7003
class="MathClass-close">)</mo></mrow><mo
7004
class="MathClass-close">)</mo></mrow> <mo
7005
class="MathClass-bin">⊗</mo> <msub
7007
class="MathClass-ord">ν</mi><mrow
7010
class="MathClass-ord">u</mi></mrow></msub
7012
class="MathClass-punc">·</mo> <msub
7014
class="MathClass-ord"><!--span
7015
class="htf-calligraphy"-->H<!--/span--></mi><mrow
7017
class="MathClass-ord">n</mi><mo
7018
class="MathClass-bin">−</mo><mn
7019
class="MathClass-ord">1</mn></mrow></msub
7023
class="MathClass-ord">S</mi><mrow
7025
class="MathClass-ord">u</mi></mrow></msub
7028
class="MathClass-punc">.</mo></mtd><mtd><mspace
7029
id="x1-7003r27" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
7032
<!--l. 796--><p class="nopar"></p></td><td width="5%">(27)</td></tr></table>
7034
class="cmti-10">In addition, for </span><!--l. 797--><math
7035
xmlns="http://www.w3.org/1998/Math/MathML"
7036
mode="inline"> <mfenced
7037
open="|" close="|" ><munderover
7040
class="MathClass-ord">D</mi></mrow><mrow
7042
></mrow></munderover><mi
7043
class="MathClass-ord">u</mi></mfenced></math><span
7044
class="cmti-10">-almost</span>
7046
class="cmti-10">every </span><!--l. 797--><math
7047
xmlns="http://www.w3.org/1998/Math/MathML"
7049
class="MathClass-ord">x</mi> <mo
7050
class="MathClass-rel">∈</mo> <mi
7051
class="MathClass-ord">Ω</mi></math> <span
7052
class="cmti-10">the restriction</span>
7054
class="cmti-10">of the function </span><!--l. 798--><math
7055
xmlns="http://www.w3.org/1998/Math/MathML"
7057
class="MathClass-ord">f</mi></math> <span
7058
class="cmti-10">to </span><!--l. 798--><math
7059
xmlns="http://www.w3.org/1998/Math/MathML"
7063
class="MathClass-ord">T</mi><mrow
7065
class="MathClass-ord">x</mi></mrow><mrow
7067
class="MathClass-ord">u</mi></mrow></msubsup
7069
class="cmti-10">is differentiable</span>
7071
class="cmti-10">at </span><!--l. 798--><math
7072
xmlns="http://www.w3.org/1998/Math/MathML"
7074
class="MathClass-ord">ũ</mi><mrow><mo
7075
class="MathClass-open">(</mo><mi
7076
class="MathClass-ord">x</mi><mo
7077
class="MathClass-close">)</mo></mrow></math>
7079
class="cmti-10">and </span><table class="equation"><tr><td>
7081
xmlns="http://www.w3.org/1998/Math/MathML"
7084
class="equation"><mtr><mtd>
7088
class="MathClass-ord">D</mi></mrow><mrow
7090
></mrow></munderover><mi
7091
class="MathClass-ord">v</mi> <mo
7092
class="MathClass-rel">=</mo> <mi
7093
class="MathClass-ord">∇</mi><mrow><mo
7094
class="MathClass-open">(</mo><msub
7096
open="" close="|" ><mi
7097
class="MathClass-ord">f</mi></mfenced> <mrow
7100
class="MathClass-ord">T</mi><mrow
7102
class="MathClass-ord">x</mi></mrow><mrow
7104
class="MathClass-ord">u</mi></mrow></msubsup
7107
class="MathClass-close">)</mo></mrow><mrow><mo
7108
class="MathClass-open">(</mo><mi
7109
class="MathClass-ord">ũ</mi><mo
7110
class="MathClass-close">)</mo></mrow> <mfrac><mrow
7114
class="MathClass-ord">D</mi></mrow><mrow
7116
></mrow></munderover><mi
7117
class="MathClass-ord">u</mi></mrow>
7120
open="|" close="|" ><munderover
7123
class="MathClass-ord">D</mi></mrow><mrow
7125
></mrow></munderover><mi
7126
class="MathClass-ord">u</mi></mfenced> </mrow></mfrac> <mo
7127
class="MathClass-punc">·</mo> <mfenced
7128
open="|" close="|" ><munderover
7131
class="MathClass-ord">D</mi></mrow><mrow
7133
></mrow></munderover><mi
7134
class="MathClass-ord">u</mi></mfenced> <mo
7135
class="MathClass-punc">.</mo></mtd><mtd><mspace
7136
id="x1-7004r28" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
7138
<!--l. 802--><p class="nopar"></p></td><td width="5%">(28)</td></tr></table>
7140
<!--l. 805--><p class="indent"> Before proving the theorem, we state without proof three elementary remarks
7141
which will be useful in the sequel. </p><div class="newtheorem">
7142
<!--l. 807--><p class="noindent"><span class="head">
7144
name="x1-7005r1"></a>
7146
class="cmti-10">Remark 7.1.</span> </span> Let <!--l. 808--><math
7147
xmlns="http://www.w3.org/1998/Math/MathML"
7149
class="MathClass-ord">ω</mi><mo
7150
class="MathClass-punc">:</mo> <mfenced
7151
open="]" close="[" ><mn
7152
class="MathClass-ord">0</mn><mo
7153
class="MathClass-punc">,</mo> <mo
7154
class="MathClass-bin">+</mo><mi
7155
class="MathClass-ord">∞</mi></mfenced> <mo
7156
class="MathClass-rel">→</mo> <mfenced
7157
open="]" close="[" ><mn
7158
class="MathClass-ord">0</mn><mo
7159
class="MathClass-punc">,</mo> <mo
7160
class="MathClass-bin">+</mo><mi
7161
class="MathClass-ord">∞</mi></mfenced></math>
7162
be a continuous function such that <!--l. 809--><math
7163
xmlns="http://www.w3.org/1998/Math/MathML"
7165
class="MathClass-ord">ω</mi><mrow><mo
7166
class="MathClass-open">(</mo><mi
7167
class="MathClass-ord">t</mi><mo
7168
class="MathClass-close">)</mo></mrow> <mo
7169
class="MathClass-rel">→</mo> <mn
7170
class="MathClass-ord">0</mn></math>
7171
as <!--l. 809--><math
7172
xmlns="http://www.w3.org/1998/Math/MathML"
7174
class="MathClass-ord">t</mi> <mo
7175
class="MathClass-rel">→</mo> <mn
7176
class="MathClass-ord">0</mn></math>.
7178
Then <!--l. 811--><math
7179
xmlns="http://www.w3.org/1998/Math/MathML"
7180
mode="display"> <mrow
7186
class="MathClass-ord">h</mi><mo
7187
class="MathClass-rel">→</mo><msup
7189
class="MathClass-ord">0</mn><mrow
7191
class="MathClass-bin">+</mo></mrow></msup
7194
class="MathClass-ord">g</mi><mrow><mo
7195
class="MathClass-open">(</mo><mi
7196
class="MathClass-ord">ω</mi><mrow><mo
7197
class="MathClass-open">(</mo><mi
7198
class="MathClass-ord">h</mi><mo
7199
class="MathClass-close">)</mo></mrow><mo
7200
class="MathClass-close">)</mo></mrow> <mo
7201
class="MathClass-rel">=</mo> <mi
7202
class="MathClass-ord">L</mi> <mo
7203
class="MathClass-rel">⇔</mo><msub
7207
class="MathClass-ord">h</mi><mo
7208
class="MathClass-rel">→</mo><msup
7210
class="MathClass-ord">0</mn><mrow
7212
class="MathClass-bin">+</mo></mrow></msup
7215
class="MathClass-ord">g</mi><mrow><mo
7216
class="MathClass-open">(</mo><mi
7217
class="MathClass-ord">h</mi><mo
7218
class="MathClass-close">)</mo></mrow> <mo
7219
class="MathClass-rel">=</mo> <mi
7220
class="MathClass-ord">L</mi>
7222
for any function <!--l. 813--><math
7223
xmlns="http://www.w3.org/1998/Math/MathML"
7225
class="MathClass-ord">g</mi><mo
7226
class="MathClass-punc">:</mo> <mfenced
7227
open="]" close="[" ><mn
7228
class="MathClass-ord">0</mn><mo
7229
class="MathClass-punc">,</mo> <mo
7230
class="MathClass-bin">+</mo><mi
7231
class="MathClass-ord">∞</mi></mfenced> <mo
7232
class="MathClass-rel">→</mo> <mi class="mathbf">R</mi></math>.
7235
<div class="newtheorem">
7236
<!--l. 815--><p class="noindent"><span class="head">
7238
name="x1-7006r2"></a>
7240
class="cmti-10">Remark 7.2.</span> </span> Let <!--l. 816--><math
7241
xmlns="http://www.w3.org/1998/Math/MathML"
7243
class="MathClass-ord">g</mi><mo
7244
class="MathClass-punc">:</mo><msup
7245
> <mi class="mathbf">R</mi><mrow
7247
class="MathClass-ord">n</mi></mrow></msup
7249
class="MathClass-rel">→</mo> <mi class="mathbf">R</mi></math>
7250
be a Lipschitz continuous function and assume that <!--l. 818--><math
7251
xmlns="http://www.w3.org/1998/Math/MathML"
7252
mode="display"> <mrow
7255
class="MathClass-ord">L</mi><mrow><mo
7256
class="MathClass-open">(</mo><mi
7257
class="MathClass-ord">z</mi><mo
7258
class="MathClass-close">)</mo></mrow> <mo
7259
class="MathClass-rel">=</mo><msub
7263
class="MathClass-ord">h</mi><mo
7264
class="MathClass-rel">→</mo><msup
7266
class="MathClass-ord">0</mn><mrow
7268
class="MathClass-bin">+</mo></mrow></msup
7272
class="MathClass-ord">g</mi><mrow><mo
7273
class="MathClass-open">(</mo><mi
7274
class="MathClass-ord">h</mi><mi
7275
class="MathClass-ord">z</mi><mo
7276
class="MathClass-close">)</mo></mrow> <mo
7277
class="MathClass-bin">−</mo> <mi
7278
class="MathClass-ord">g</mi><mrow><mo
7279
class="MathClass-open">(</mo><mn
7280
class="MathClass-ord">0</mn><mo
7281
class="MathClass-close">)</mo></mrow></mrow>
7284
class="MathClass-ord">h</mi></mrow></mfrac>
7286
exists for every <!--l. 819--><math
7287
xmlns="http://www.w3.org/1998/Math/MathML"
7289
class="MathClass-ord">z</mi> <mo
7290
class="MathClass-rel">∈</mo><msup
7291
> <mi class="mathbf">Q</mi><mrow
7293
class="MathClass-ord">n</mi></mrow></msup
7295
and that <!--l. 819--><math
7296
xmlns="http://www.w3.org/1998/Math/MathML"
7298
class="MathClass-ord">L</mi></math>
7299
is a linear function of <!--l. 820--><math
7300
xmlns="http://www.w3.org/1998/Math/MathML"
7302
class="MathClass-ord">z</mi></math>.
7303
Then <!--l. 820--><math
7304
xmlns="http://www.w3.org/1998/Math/MathML"
7306
class="MathClass-ord">g</mi></math>
7307
is differentiable at 0.
7310
<div class="newtheorem">
7311
<!--l. 822--><p class="noindent"><span class="head">
7313
name="x1-7007r3"></a>
7316
class="cmti-10">Remark 7.3.</span> </span> Let <!--l. 823--><math
7317
xmlns="http://www.w3.org/1998/Math/MathML"
7320
class="MathClass-ord">A</mi><mo
7321
class="MathClass-punc">:</mo><msup
7322
> <mi class="mathbf">R</mi><mrow
7324
class="MathClass-ord">n</mi></mrow></msup
7326
class="MathClass-rel">→</mo><msup
7327
> <mi class="mathbf">R</mi><mrow
7329
class="MathClass-ord">m</mi></mrow></msup
7331
be a linear function, and let <!--l. 824--><math
7332
xmlns="http://www.w3.org/1998/Math/MathML"
7335
class="MathClass-ord">f</mi> <mo
7336
class="MathClass-punc">:</mo><msup
7337
> <mi class="mathbf">R</mi><mrow
7339
class="MathClass-ord">m</mi></mrow></msup
7341
class="MathClass-rel">→</mo> <mi class="mathbf">R</mi></math>
7342
be a function. Then the restriction of <!--l. 825--><math
7343
xmlns="http://www.w3.org/1998/Math/MathML"
7346
class="MathClass-ord">f</mi></math>
7347
to the range of <!--l. 825--><math
7348
xmlns="http://www.w3.org/1998/Math/MathML"
7351
class="MathClass-ord">A</mi></math>
7352
is differentiable at 0 if and only if <!--l. 826--><math
7353
xmlns="http://www.w3.org/1998/Math/MathML"
7356
class="MathClass-ord">f</mi><mrow><mo
7357
class="MathClass-open">(</mo><mi
7358
class="MathClass-ord">A</mi><mo
7359
class="MathClass-close">)</mo></mrow><mo
7360
class="MathClass-punc">:</mo><msup
7361
> <mi class="mathbf">R</mi><mrow
7363
class="MathClass-ord">n</mi></mrow></msup
7365
class="MathClass-rel">→</mo> <mi class="mathbf">R</mi></math>
7366
is differentiable at 0 and <!--l. 828--><math
7367
xmlns="http://www.w3.org/1998/Math/MathML"
7372
class="MathClass-ord">∇</mi><mrow><mo
7373
class="MathClass-open">(</mo><msub
7375
open="" close="|" ><mi
7376
class="MathClass-ord">f</mi></mfenced> <mrow
7378
class="MathClass-op">Im</mo><!--nolimits--><mrow><mo
7379
class="MathClass-open">(</mo><mi
7380
class="MathClass-ord">A</mi><mo
7381
class="MathClass-close">)</mo></mrow></mrow></msub
7383
class="MathClass-close">)</mo></mrow><mrow><mo
7384
class="MathClass-open">(</mo><mn
7385
class="MathClass-ord">0</mn><mo
7386
class="MathClass-close">)</mo></mrow><mi
7387
class="MathClass-ord">A</mi> <mo
7388
class="MathClass-rel">=</mo> <mi
7389
class="MathClass-ord">∇</mi><mrow><mo
7390
class="MathClass-open">(</mo><mi
7391
class="MathClass-ord">f</mi><mrow><mo
7392
class="MathClass-open">(</mo><mi
7393
class="MathClass-ord">A</mi><mo
7394
class="MathClass-close">)</mo></mrow><mo
7395
class="MathClass-close">)</mo></mrow><mrow><mo
7396
class="MathClass-open">(</mo><mn
7397
class="MathClass-ord">0</mn><mo
7398
class="MathClass-close">)</mo></mrow><mo
7399
class="MathClass-punc">.</mo>
7403
<!--l. 831--><p class="indent">
7405
<div class="proof"><span class="head">
7407
class="cmti-10">Proof.</span> </span>We begin by showing that <!--l. 832--><math
7408
xmlns="http://www.w3.org/1998/Math/MathML"
7410
class="MathClass-ord">v</mi> <mo
7411
class="MathClass-rel">∈</mo> <mi
7412
class="MathClass-ord">B</mi><mi
7413
class="MathClass-ord">V</mi> <mrow><mo
7414
class="MathClass-open">(</mo><mi
7415
class="MathClass-ord">Ω</mi><mo
7416
class="MathClass-punc">;</mo><msup
7417
> <mi class="mathbf">R</mi><mrow
7419
class="MathClass-ord">k</mi></mrow></msup
7421
class="MathClass-close">)</mo></mrow></math>
7422
and <table class="equation"><tr><td>
7425
xmlns="http://www.w3.org/1998/Math/MathML"
7428
class="equation"><mtr><mtd>
7430
open="|" close="|" ><mi
7431
class="MathClass-ord">D</mi><mi
7432
class="MathClass-ord">v</mi></mfenced> <mrow><mo
7433
class="MathClass-open">(</mo><mi
7434
class="MathClass-ord">B</mi><mo
7435
class="MathClass-close">)</mo></mrow> <mo
7436
class="MathClass-rel">≤</mo> <mi
7437
class="MathClass-ord">K</mi> <mfenced
7438
open="|" close="|" ><mi
7439
class="MathClass-ord">D</mi><mi
7440
class="MathClass-ord">u</mi></mfenced> <mrow><mo
7441
class="MathClass-open">(</mo><mi
7442
class="MathClass-ord">B</mi><mo
7443
class="MathClass-close">)</mo></mrow><mspace width="2em" class="qquad"/><mi
7444
class="MathClass-ord">∀</mi><mi
7445
class="MathClass-ord">B</mi> <mo
7446
class="MathClass-rel">∈</mo> <mi class="mathbf">B</mi><mrow><mo
7447
class="MathClass-open">(</mo><mi
7448
class="MathClass-ord">Ω</mi><mo
7449
class="MathClass-close">)</mo></mrow><mo
7450
class="MathClass-punc">,</mo> </mtd><mtd><mspace
7451
id="x1-7008r29" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
7453
<!--l. 835--><p class="nopar"></p></td><td width="5%">(29)</td></tr></table>
7454
where <!--l. 836--><math
7455
xmlns="http://www.w3.org/1998/Math/MathML"
7457
class="MathClass-ord">K</mi> <mo
7458
class="MathClass-rel">></mo> <mn
7459
class="MathClass-ord">0</mn></math> is the Lipschitz
7460
constant of <!--l. 836--><math
7461
xmlns="http://www.w3.org/1998/Math/MathML"
7463
class="MathClass-ord">f</mi></math>.
7465
href="#x1-3015r13">13<!--tex4ht:ref: sum-Di--></a>) and by the approximation result quoted in <span
7466
class="cmsy-10">§</span><a
7467
href="#x1-30003">3<!--tex4ht:ref: s:mt--></a>, it is possible to find a sequence <!--l. 838--><math
7468
xmlns="http://www.w3.org/1998/Math/MathML"
7471
class="MathClass-open">(</mo><msub
7473
class="MathClass-ord">u</mi><mrow
7475
class="MathClass-ord">h</mi></mrow></msub
7477
class="MathClass-close">)</mo></mrow> <mo
7478
class="MathClass-rel">⊂</mo> <msup
7480
class="MathClass-ord">C</mi><mrow
7482
class="MathClass-ord">1</mn></mrow></msup
7484
class="MathClass-open">(</mo><mi
7485
class="MathClass-ord">Ω</mi><mo
7486
class="MathClass-punc">;</mo><msup
7487
> <mi class="mathbf">R</mi><mrow
7489
class="MathClass-ord">m</mi></mrow></msup
7491
class="MathClass-close">)</mo></mrow></math> converging
7492
to <!--l. 838--><math
7493
xmlns="http://www.w3.org/1998/Math/MathML"
7495
class="MathClass-ord">u</mi></math> in <!--l. 839--><math
7496
xmlns="http://www.w3.org/1998/Math/MathML"
7500
class="MathClass-ord">L</mi><mrow
7502
class="MathClass-ord">1</mn></mrow></msup
7504
class="MathClass-open">(</mo><mi
7505
class="MathClass-ord">Ω</mi><mo
7506
class="MathClass-punc">;</mo><msup
7507
> <mi class="mathbf">R</mi><mrow
7509
class="MathClass-ord">m</mi></mrow></msup
7511
class="MathClass-close">)</mo></mrow></math> and such that <!--l. 840--><math
7512
xmlns="http://www.w3.org/1998/Math/MathML"
7520
class="MathClass-ord">h</mi><mo
7521
class="MathClass-rel">→</mo><mo
7522
class="MathClass-bin">+</mo><mi
7523
class="MathClass-ord">∞</mi></mrow></msub
7526
class="MathClass-op">∫</mo>
7529
class="MathClass-ord">Ω</mi></mrow></msub
7531
open="|" close="|" ><mi
7532
class="MathClass-ord">∇</mi><msub
7534
class="MathClass-ord">u</mi><mrow
7536
class="MathClass-ord">h</mi></mrow></msub
7538
class="MathClass-ord">d</mi><mi
7539
class="MathClass-ord">x</mi> <mo
7540
class="MathClass-rel">=</mo> <mfenced
7541
open="|" close="|" ><mi
7542
class="MathClass-ord">D</mi><mi
7543
class="MathClass-ord">u</mi></mfenced> <mrow><mo
7544
class="MathClass-open">(</mo><mi
7545
class="MathClass-ord">Ω</mi><mo
7546
class="MathClass-close">)</mo></mrow><mo
7547
class="MathClass-punc">.</mo>
7548
</mrow></math> The functions <!--l. 841--><math
7549
xmlns="http://www.w3.org/1998/Math/MathML"
7553
class="MathClass-ord">v</mi><mrow
7555
class="MathClass-ord">h</mi></mrow></msub
7557
class="MathClass-rel">=</mo> <mi
7558
class="MathClass-ord">f</mi><mrow><mo
7559
class="MathClass-open">(</mo><msub
7561
class="MathClass-ord">u</mi><mrow
7563
class="MathClass-ord">h</mi></mrow></msub
7565
class="MathClass-close">)</mo></mrow></math> are locally Lipschitz
7566
continuous in <!--l. 841--><math
7567
xmlns="http://www.w3.org/1998/Math/MathML"
7569
class="MathClass-ord">Ω</mi></math>,
7570
and the definition of differential implies that <!--l. 842--><math
7571
xmlns="http://www.w3.org/1998/Math/MathML"
7574
open="|" close="|" ><mi
7575
class="MathClass-ord">∇</mi><msub
7577
class="MathClass-ord">v</mi><mrow
7579
class="MathClass-ord">h</mi></mrow></msub
7581
class="MathClass-rel">≤</mo> <mi
7582
class="MathClass-ord">K</mi> <mfenced
7583
open="|" close="|" ><mi
7584
class="MathClass-ord">∇</mi><msub
7586
class="MathClass-ord">u</mi><mrow
7588
class="MathClass-ord">h</mi></mrow></msub
7589
> </mfenced></math> almost everywhere
7590
in <!--l. 843--><math
7591
xmlns="http://www.w3.org/1998/Math/MathML"
7593
class="MathClass-ord">Ω</mi></math>.
7594
The lower semicontinuity of the total variation and (<a
7595
href="#x1-3015r13">13<!--tex4ht:ref: sum-Di--></a>) yield <table class="equation"><tr><td>
7598
xmlns="http://www.w3.org/1998/Math/MathML"
7601
class="equation"><mtr><mtd>
7602
<mtable class="split"><mtr><mtd>
7604
class="split-mtr"></mrow><mrow
7605
class="split-mtd"></mrow> <mfenced
7606
open="|" close="|" ><mi
7607
class="MathClass-ord">D</mi><mi
7608
class="MathClass-ord">v</mi></mfenced> <mrow><mo
7609
class="MathClass-open">(</mo><mi
7610
class="MathClass-ord">Ω</mi><mo
7611
class="MathClass-close">)</mo></mrow> <mo
7612
class="MathClass-rel">≤</mo><msub
7614
> lim inf</mo> <mrow
7616
class="MathClass-ord">h</mi><mo
7617
class="MathClass-rel">→</mo><mo
7618
class="MathClass-bin">+</mo><mi
7619
class="MathClass-ord">∞</mi></mrow></msub
7621
open="|" close="|" ><mi
7622
class="MathClass-ord">D</mi><msub
7624
class="MathClass-ord">v</mi><mrow
7626
class="MathClass-ord">h</mi></mrow></msub
7627
> </mfenced> <mrow><mo
7628
class="MathClass-open">(</mo><mi
7629
class="MathClass-ord">Ω</mi><mo
7630
class="MathClass-close">)</mo></mrow><mrow
7631
class="split-mtd"></mrow> <mo
7632
class="MathClass-rel">=</mo><msub
7634
> lim inf</mo> <mrow
7636
class="MathClass-ord">h</mi><mo
7637
class="MathClass-rel">→</mo><mo
7638
class="MathClass-bin">+</mo><mi
7639
class="MathClass-ord">∞</mi></mrow></msub
7642
class="MathClass-op">∫</mo>
7645
class="MathClass-ord">Ω</mi></mrow></msub
7647
open="|" close="|" ><mi
7648
class="MathClass-ord">∇</mi><msub
7650
class="MathClass-ord">v</mi><mrow
7652
class="MathClass-ord">h</mi></mrow></msub
7654
class="MathClass-ord">d</mi><mi
7655
class="MathClass-ord">x</mi>
7657
class="split-mtr"></mrow><mrow
7658
class="split-mtd"></mrow> <mrow
7659
class="split-mtd"></mrow> <mo
7660
class="MathClass-rel">≤</mo> <mi
7661
class="MathClass-ord">K</mi><msub
7663
> lim inf</mo> <mrow
7665
class="MathClass-ord">h</mi><mo
7666
class="MathClass-rel">→</mo><mo
7667
class="MathClass-bin">+</mo><mi
7668
class="MathClass-ord">∞</mi></mrow></msub
7671
class="MathClass-op">∫</mo>
7674
class="MathClass-ord">Ω</mi></mrow></msub
7676
open="|" close="|" ><mi
7677
class="MathClass-ord">∇</mi><msub
7679
class="MathClass-ord">u</mi><mrow
7681
class="MathClass-ord">h</mi></mrow></msub
7683
class="MathClass-ord">d</mi><mi
7684
class="MathClass-ord">x</mi> <mo
7685
class="MathClass-rel">=</mo> <mi
7686
class="MathClass-ord">K</mi> <mfenced
7687
open="|" close="|" ><mi
7688
class="MathClass-ord">D</mi><mi
7689
class="MathClass-ord">u</mi></mfenced> <mrow><mo
7690
class="MathClass-open">(</mo><mi
7691
class="MathClass-ord">Ω</mi><mo
7692
class="MathClass-close">)</mo></mrow><mo
7693
class="MathClass-punc">.</mo>
7694
</mtd></mtr></mtable> </mtd><mtd><mspace
7695
id="x1-7009r30" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /></mtd></mtr></mtable>
7697
<!--l. 851--><p class="nopar"></p></td><td width="5%">(30)</td></tr></table>
7698
Since <!--l. 852--><math
7699
xmlns="http://www.w3.org/1998/Math/MathML"
7701
class="MathClass-ord">f</mi><mrow><mo
7702
class="MathClass-open">(</mo><mn
7703
class="MathClass-ord">0</mn><mo
7704
class="MathClass-close">)</mo></mrow> <mo
7705
class="MathClass-rel">=</mo> <mn
7706
class="MathClass-ord">0</mn></math>, we
7707
have also <!--l. 853--><math
7708
xmlns="http://www.w3.org/1998/Math/MathML"
7709
mode="display"> <mrow
7713
class="MathClass-op">∫</mo>
7716
class="MathClass-ord">Ω</mi></mrow></msub
7718
open="|" close="|" ><mi
7719
class="MathClass-ord">v</mi></mfenced> <mi
7720
class="MathClass-ord">d</mi><mi
7721
class="MathClass-ord">x</mi> <mo
7722
class="MathClass-rel">≤</mo> <mi
7723
class="MathClass-ord">K</mi><msub
7725
class="MathClass-op">∫</mo>
7728
class="MathClass-ord">Ω</mi></mrow></msub
7730
open="|" close="|" ><mi
7731
class="MathClass-ord">u</mi></mfenced> <mi
7732
class="MathClass-ord">d</mi><mi
7733
class="MathClass-ord">x</mi><mo
7734
class="MathClass-punc">;</mo>
7736
therefore <!--l. 854--><math
7737
xmlns="http://www.w3.org/1998/Math/MathML"
7739
class="MathClass-ord">u</mi> <mo
7740
class="MathClass-rel">∈</mo> <mi
7741
class="MathClass-ord">B</mi><mi
7742
class="MathClass-ord">V</mi> <mrow><mo
7743
class="MathClass-open">(</mo><mi
7744
class="MathClass-ord">Ω</mi><mo
7745
class="MathClass-punc">;</mo><msup
7746
> <mi class="mathbf">R</mi><mrow
7748
class="MathClass-ord">k</mi></mrow></msup
7750
class="MathClass-close">)</mo></mrow></math>.
7751
Repeating the same argument for every open set <!--l. 855--><math
7752
xmlns="http://www.w3.org/1998/Math/MathML"
7755
class="MathClass-ord">A</mi> <mo
7756
class="MathClass-rel">⊂</mo> <mi
7757
class="MathClass-ord">Ω</mi></math>, we get (<a
7758
href="#x1-7008r29">29<!--tex4ht:ref: e:bomb--></a>) for
7759
every <!--l. 856--><math
7760
xmlns="http://www.w3.org/1998/Math/MathML"
7762
class="MathClass-ord">B</mi> <mo
7763
class="MathClass-rel">∈</mo> <mi class="mathbf">B</mi><mrow><mo
7764
class="MathClass-open">(</mo><mi
7765
class="MathClass-ord">Ω</mi><mo
7766
class="MathClass-close">)</mo></mrow></math>,
7767
because <!--l. 856--><math
7768
xmlns="http://www.w3.org/1998/Math/MathML"
7769
mode="inline"> <mfenced
7770
open="|" close="|" ><mi
7771
class="MathClass-ord">D</mi><mi
7772
class="MathClass-ord">v</mi></mfenced></math>,
7774
xmlns="http://www.w3.org/1998/Math/MathML"
7775
mode="inline"> <mfenced
7776
open="|" close="|" ><mi
7777
class="MathClass-ord">D</mi><mi
7778
class="MathClass-ord">u</mi></mfenced></math>
7779
are Radon measures. To prove Lemma <a
7780
href="#x1-6002r1">6.1<!--tex4ht:ref: limbog--></a>, first we observe that <table class="equation"><tr><td>
7783
xmlns="http://www.w3.org/1998/Math/MathML"
7786
class="equation"><mtr><mtd>
7789
class="MathClass-ord">S</mi><mrow
7791
class="MathClass-ord">v</mi></mrow></msub
7793
class="MathClass-rel">⊂</mo> <msub
7795
class="MathClass-ord">S</mi><mrow
7797
class="MathClass-ord">u</mi></mrow></msub
7799
class="MathClass-punc">,</mo> <mspace width="2em" class="qquad"/><mi
7800
class="MathClass-ord">ṽ</mi><mrow><mo
7801
class="MathClass-open">(</mo><mi
7802
class="MathClass-ord">x</mi><mo
7803
class="MathClass-close">)</mo></mrow> <mo
7804
class="MathClass-rel">=</mo> <mi
7805
class="MathClass-ord">f</mi><mrow><mo
7806
class="MathClass-open">(</mo><mi
7807
class="MathClass-ord">ũ</mi><mrow><mo
7808
class="MathClass-open">(</mo><mi
7809
class="MathClass-ord">x</mi><mo
7810
class="MathClass-close">)</mo></mrow><mo
7811
class="MathClass-close">)</mo></mrow><mspace width="2em" class="qquad"/><mi
7812
class="MathClass-ord">∀</mi><mi
7813
class="MathClass-ord">x</mi> <mo
7814
class="MathClass-rel">∈</mo> <mi
7815
class="MathClass-ord">Ω</mi><mo
7816
class="MathClass-bin">∖</mo><msub
7818
class="MathClass-ord">S</mi><mrow
7820
class="MathClass-ord">u</mi></mrow></msub
7822
class="MathClass-punc">.</mo></mtd><mtd><mspace
7823
id="x1-7010r31" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
7825
<!--l. 860--><p class="nopar"></p></td><td width="5%">(31)</td></tr></table>
7826
In fact, for every <!--l. 861--><math
7827
xmlns="http://www.w3.org/1998/Math/MathML"
7829
class="MathClass-ord">ɛ</mi> <mo
7830
class="MathClass-rel">></mo> <mn
7831
class="MathClass-ord">0</mn></math>
7832
we have <!--l. 862--><math
7833
xmlns="http://www.w3.org/1998/Math/MathML"
7834
mode="display"> <mrow
7837
class="MathClass-open">{</mo><mi
7838
class="MathClass-ord">y</mi> <mo
7839
class="MathClass-rel">∈</mo> <msub
7841
class="MathClass-ord">B</mi><mrow
7843
class="MathClass-ord">ρ</mi></mrow></msub
7845
class="MathClass-open">(</mo><mi
7846
class="MathClass-ord">x</mi><mo
7847
class="MathClass-close">)</mo></mrow> <mo
7848
class="MathClass-punc">:</mo> <mfenced
7849
open="|" close="|" ><mi
7850
class="MathClass-ord">v</mi><mrow><mo
7851
class="MathClass-open">(</mo><mi
7852
class="MathClass-ord">y</mi><mo
7853
class="MathClass-close">)</mo></mrow> <mo
7854
class="MathClass-bin">−</mo> <mi
7855
class="MathClass-ord">f</mi><mrow><mo
7856
class="MathClass-open">(</mo><mi
7857
class="MathClass-ord">ũ</mi><mrow><mo
7858
class="MathClass-open">(</mo><mi
7859
class="MathClass-ord">x</mi><mo
7860
class="MathClass-close">)</mo></mrow><mo
7861
class="MathClass-close">)</mo></mrow></mfenced> <mo
7862
class="MathClass-rel">></mo> <mi
7863
class="MathClass-ord">ɛ</mi><mo
7864
class="MathClass-close">}</mo></mrow> <mo
7865
class="MathClass-rel">⊂</mo> <mrow><mo
7866
class="MathClass-open">{</mo><mi
7867
class="MathClass-ord">y</mi> <mo
7868
class="MathClass-rel">∈</mo> <msub
7870
class="MathClass-ord">B</mi><mrow
7872
class="MathClass-ord">ρ</mi></mrow></msub
7874
class="MathClass-open">(</mo><mi
7875
class="MathClass-ord">x</mi><mo
7876
class="MathClass-close">)</mo></mrow> <mo
7877
class="MathClass-punc">:</mo> <mfenced
7878
open="|" close="|" ><mi
7879
class="MathClass-ord">u</mi><mrow><mo
7880
class="MathClass-open">(</mo><mi
7881
class="MathClass-ord">y</mi><mo
7882
class="MathClass-close">)</mo></mrow> <mo
7883
class="MathClass-bin">−</mo> <mi
7884
class="MathClass-ord">ũ</mi><mrow><mo
7885
class="MathClass-open">(</mo><mi
7886
class="MathClass-ord">x</mi><mo
7887
class="MathClass-close">)</mo></mrow></mfenced> <mo
7888
class="MathClass-rel">></mo> <mi
7889
class="MathClass-ord">ɛ</mi><mo
7890
class="MathClass-bin">/</mo><mi
7891
class="MathClass-ord">K</mi><mo
7892
class="MathClass-close">}</mo></mrow><mo
7893
class="MathClass-punc">,</mo>
7894
</mrow></math> hence
7896
xmlns="http://www.w3.org/1998/Math/MathML"
7897
mode="display"> <mrow
7903
class="MathClass-ord">ρ</mi><mo
7904
class="MathClass-rel">→</mo><msup
7906
class="MathClass-ord">0</mn><mrow
7908
class="MathClass-bin">+</mo></mrow></msup
7912
open="|" close="|" ><mrow><mo
7913
class="MathClass-open">{</mo><mi
7914
class="MathClass-ord">y</mi> <mo
7915
class="MathClass-rel">∈</mo> <msub
7917
class="MathClass-ord">B</mi><mrow
7919
class="MathClass-ord">ρ</mi></mrow></msub
7921
class="MathClass-open">(</mo><mi
7922
class="MathClass-ord">x</mi><mo
7923
class="MathClass-close">)</mo></mrow> <mo
7924
class="MathClass-punc">:</mo> <mfenced
7925
open="|" close="|" ><mi
7926
class="MathClass-ord">v</mi><mrow><mo
7927
class="MathClass-open">(</mo><mi
7928
class="MathClass-ord">y</mi><mo
7929
class="MathClass-close">)</mo></mrow> <mo
7930
class="MathClass-bin">−</mo> <mi
7931
class="MathClass-ord">f</mi><mrow><mo
7932
class="MathClass-open">(</mo><mi
7933
class="MathClass-ord">ũ</mi><mrow><mo
7934
class="MathClass-open">(</mo><mi
7935
class="MathClass-ord">x</mi><mo
7936
class="MathClass-close">)</mo></mrow><mo
7937
class="MathClass-close">)</mo></mrow></mfenced> <mo
7938
class="MathClass-rel">></mo> <mi
7939
class="MathClass-ord">ɛ</mi><mo
7940
class="MathClass-close">}</mo></mrow></mfenced> </mrow>
7944
class="MathClass-ord">ρ</mi><mrow
7946
class="MathClass-ord">n</mi></mrow></msup
7947
></mrow></mfrac> <mo
7948
class="MathClass-rel">=</mo> <mn
7949
class="MathClass-ord">0</mn>
7950
</mrow></math> whenever <!--l. 867--><math
7951
xmlns="http://www.w3.org/1998/Math/MathML"
7954
class="MathClass-ord">x</mi> <mo
7955
class="MathClass-rel">∈</mo> <mi
7956
class="MathClass-ord">Ω</mi><mo
7957
class="MathClass-bin">∖</mo><msub
7959
class="MathClass-ord">S</mi><mrow
7961
class="MathClass-ord">u</mi></mrow></msub
7962
></math>. By a similar argument, if <!--l. 867--><math
7963
xmlns="http://www.w3.org/1998/Math/MathML"
7966
class="MathClass-ord">x</mi> <mo
7967
class="MathClass-rel">∈</mo> <msub
7969
class="MathClass-ord">S</mi><mrow
7971
class="MathClass-ord">u</mi></mrow></msub
7972
></math> is a point such that there
7973
exists a triplet <!--l. 868--><math
7974
xmlns="http://www.w3.org/1998/Math/MathML"
7975
mode="inline"> <mrow><mo
7976
class="MathClass-open">(</mo><msup
7978
class="MathClass-ord">u</mi><mrow
7980
class="MathClass-bin">+</mo></mrow></msup
7982
class="MathClass-punc">,</mo> <msup
7984
class="MathClass-ord">u</mi><mrow
7986
class="MathClass-bin">−</mo></mrow></msup
7988
class="MathClass-punc">,</mo> <msub
7990
class="MathClass-ord">ν</mi><mrow
7992
class="MathClass-ord">u</mi></mrow></msub
7994
class="MathClass-close">)</mo></mrow></math> satisfying
7997
href="#x1-3016r14">14<!--tex4ht:ref: detK1--></a>), (<a
7998
href="#x1-3017r15">15<!--tex4ht:ref: detK2--></a>), then <!--l. 870--><math
7999
xmlns="http://www.w3.org/1998/Math/MathML"
8000
mode="display"> <mrow
8003
class="MathClass-open">(</mo><msup
8005
class="MathClass-ord">v</mi><mrow
8007
class="MathClass-bin">+</mo></mrow></msup
8009
class="MathClass-open">(</mo><mi
8010
class="MathClass-ord">x</mi><mo
8011
class="MathClass-close">)</mo></mrow> <mo
8012
class="MathClass-bin">−</mo> <msup
8014
class="MathClass-ord">v</mi><mrow
8016
class="MathClass-bin">−</mo></mrow></msup
8018
class="MathClass-open">(</mo><mi
8019
class="MathClass-ord">x</mi><mo
8020
class="MathClass-close">)</mo></mrow><mo
8021
class="MathClass-close">)</mo></mrow> <mo
8022
class="MathClass-bin">⊗</mo> <msub
8024
class="MathClass-ord">ν</mi><mrow
8027
class="MathClass-ord">v</mi></mrow></msub
8029
class="MathClass-rel">=</mo> <mrow><mo
8030
class="MathClass-open">(</mo><mi
8031
class="MathClass-ord">f</mi><mrow><mo
8032
class="MathClass-open">(</mo><msup
8034
class="MathClass-ord">u</mi><mrow
8036
class="MathClass-bin">+</mo></mrow></msup
8038
class="MathClass-open">(</mo><mi
8039
class="MathClass-ord">x</mi><mo
8040
class="MathClass-close">)</mo></mrow><mo
8041
class="MathClass-close">)</mo></mrow> <mo
8042
class="MathClass-bin">−</mo> <mi
8043
class="MathClass-ord">f</mi><mrow><mo
8044
class="MathClass-open">(</mo><msup
8046
class="MathClass-ord">u</mi><mrow
8048
class="MathClass-bin">−</mo></mrow></msup
8050
class="MathClass-open">(</mo><mi
8051
class="MathClass-ord">x</mi><mo
8052
class="MathClass-close">)</mo></mrow><mo
8053
class="MathClass-close">)</mo></mrow><mo
8054
class="MathClass-close">)</mo></mrow> <mo
8055
class="MathClass-bin">⊗</mo> <msub
8057
class="MathClass-ord">ν</mi><mrow
8060
class="MathClass-ord">u</mi></mrow></msub
8061
><mspace width="1em" class="quad"/><mrow
8062
class="text"><mtext >if </mtext></mrow><mi
8063
class="MathClass-ord">x</mi> <mo
8064
class="MathClass-rel">∈</mo> <msub
8066
class="MathClass-ord">S</mi><mrow
8068
class="MathClass-ord">v</mi></mrow></msub
8071
and <!--l. 874--><math
8072
xmlns="http://www.w3.org/1998/Math/MathML"
8074
class="MathClass-ord">f</mi><mrow><mo
8075
class="MathClass-open">(</mo><msup
8077
class="MathClass-ord">u</mi><mrow
8079
class="MathClass-bin">−</mo></mrow></msup
8081
class="MathClass-open">(</mo><mi
8082
class="MathClass-ord">x</mi><mo
8083
class="MathClass-close">)</mo></mrow><mo
8084
class="MathClass-close">)</mo></mrow> <mo
8085
class="MathClass-rel">=</mo> <mi
8086
class="MathClass-ord">f</mi><mrow><mo
8087
class="MathClass-open">(</mo><msup
8089
class="MathClass-ord">u</mi><mrow
8091
class="MathClass-bin">+</mo></mrow></msup
8093
class="MathClass-open">(</mo><mi
8094
class="MathClass-ord">x</mi><mo
8095
class="MathClass-close">)</mo></mrow><mo
8096
class="MathClass-close">)</mo></mrow></math>
8097
if <!--l. 874--><math
8098
xmlns="http://www.w3.org/1998/Math/MathML"
8100
class="MathClass-ord">x</mi> <mo
8101
class="MathClass-rel">∈</mo> <msub
8103
class="MathClass-ord">S</mi><mrow
8105
class="MathClass-ord">u</mi></mrow></msub
8107
class="MathClass-bin">∖</mo><msub
8109
class="MathClass-ord">S</mi><mrow
8111
class="MathClass-ord">v</mi></mrow></msub
8113
Hence, by (1.8) we get <table class="equation"><tr><td>
8115
xmlns="http://www.w3.org/1998/Math/MathML"
8118
class="equation"><mtr><mtd>
8119
<mtable class="split"><mtr><mtd>
8121
class="split-mtr"></mrow><mrow
8122
class="split-mtd"></mrow> <mi
8123
class="MathClass-ord">J</mi><mi
8124
class="MathClass-ord">v</mi><mrow><mo
8125
class="MathClass-open">(</mo><mi
8126
class="MathClass-ord">B</mi><mo
8127
class="MathClass-close">)</mo></mrow> <mo
8128
class="MathClass-rel">=</mo><msub
8130
class="MathClass-op">∫</mo>
8133
class="MathClass-ord">B</mi><mo
8134
class="MathClass-bin">∩</mo><msub
8136
class="MathClass-ord">S</mi><mrow
8138
class="MathClass-ord">v</mi></mrow></msub
8141
class="MathClass-open">(</mo><msup
8143
class="MathClass-ord">v</mi><mrow
8145
class="MathClass-bin">+</mo></mrow></msup
8147
class="MathClass-bin">−</mo> <msup
8149
class="MathClass-ord">v</mi><mrow
8151
class="MathClass-bin">−</mo></mrow></msup
8153
class="MathClass-close">)</mo></mrow> <mo
8154
class="MathClass-bin">⊗</mo> <msub
8156
class="MathClass-ord">ν</mi><mrow
8159
class="MathClass-ord">v</mi></mrow></msub
8161
class="MathClass-ord">d</mi><msub
8163
class="MathClass-ord"><!--span
8164
class="htf-calligraphy"-->H<!--/span--></mi><mrow
8166
class="MathClass-ord">n</mi><mo
8167
class="MathClass-bin">−</mo><mn
8168
class="MathClass-ord">1</mn></mrow></msub
8170
class="split-mtd"></mrow> <mo
8171
class="MathClass-rel">=</mo><msub
8173
class="MathClass-op">∫</mo>
8176
class="MathClass-ord">B</mi><mo
8177
class="MathClass-bin">∩</mo><msub
8179
class="MathClass-ord">S</mi><mrow
8181
class="MathClass-ord">v</mi></mrow></msub
8184
class="MathClass-open">(</mo><mi
8185
class="MathClass-ord">f</mi><mrow><mo
8186
class="MathClass-open">(</mo><msup
8188
class="MathClass-ord">u</mi><mrow
8190
class="MathClass-bin">+</mo></mrow></msup
8192
class="MathClass-close">)</mo></mrow> <mo
8193
class="MathClass-bin">−</mo> <mi
8194
class="MathClass-ord">f</mi><mrow><mo
8195
class="MathClass-open">(</mo><msup
8197
class="MathClass-ord">u</mi><mrow
8199
class="MathClass-bin">−</mo></mrow></msup
8201
class="MathClass-close">)</mo></mrow><mo
8202
class="MathClass-close">)</mo></mrow> <mo
8203
class="MathClass-bin">⊗</mo> <msub
8205
class="MathClass-ord">ν</mi><mrow
8208
class="MathClass-ord">u</mi></mrow></msub
8210
class="MathClass-ord">d</mi><msub
8212
class="MathClass-ord"><!--span
8213
class="htf-calligraphy"-->H<!--/span--></mi><mrow
8215
class="MathClass-ord">n</mi><mo
8216
class="MathClass-bin">−</mo><mn
8217
class="MathClass-ord">1</mn></mrow></msub
8220
class="split-mtr"></mrow><mrow
8221
class="split-mtd"></mrow> <mrow
8222
class="split-mtd"></mrow> <mo
8223
class="MathClass-rel">=</mo><msub
8225
class="MathClass-op">∫</mo>
8228
class="MathClass-ord">B</mi><mo
8229
class="MathClass-bin">∩</mo><msub
8231
class="MathClass-ord">S</mi><mrow
8233
class="MathClass-ord">u</mi></mrow></msub
8236
class="MathClass-open">(</mo><mi
8237
class="MathClass-ord">f</mi><mrow><mo
8238
class="MathClass-open">(</mo><msup
8240
class="MathClass-ord">u</mi><mrow
8242
class="MathClass-bin">+</mo></mrow></msup
8244
class="MathClass-close">)</mo></mrow> <mo
8245
class="MathClass-bin">−</mo> <mi
8246
class="MathClass-ord">f</mi><mrow><mo
8247
class="MathClass-open">(</mo><msup
8249
class="MathClass-ord">u</mi><mrow
8251
class="MathClass-bin">−</mo></mrow></msup
8253
class="MathClass-close">)</mo></mrow><mo
8254
class="MathClass-close">)</mo></mrow> <mo
8255
class="MathClass-bin">⊗</mo> <msub
8257
class="MathClass-ord">ν</mi><mrow
8260
class="MathClass-ord">u</mi></mrow></msub
8262
class="MathClass-ord">d</mi><msub
8264
class="MathClass-ord"><!--span
8265
class="htf-calligraphy"-->H<!--/span--></mi><mrow
8267
class="MathClass-ord">n</mi><mo
8268
class="MathClass-bin">−</mo><mn
8269
class="MathClass-ord">1</mn></mrow></msub
8271
</mtd></mtr></mtable> </mtd><mtd></mtd></mtr></mtable>
8273
<!--l. 880--><p class="nopar"></p></td></tr></table>
8275
href="#x1-6002r1">6.1<!--tex4ht:ref: limbog--></a> is proved. </div> □
8276
<!--l. 884--><p class="noindent">To prove (<a
8277
href="#x1-7010r31">31<!--tex4ht:ref: e:SS--></a>), it is not restrictive to assume that <!--l. 884--><math
8278
xmlns="http://www.w3.org/1998/Math/MathML"
8281
class="MathClass-ord">k</mi> <mo
8282
class="MathClass-rel">=</mo> <mn
8283
class="MathClass-ord">1</mn></math>.
8284
Moreover, to simplify our notation, from now on we shall assume that <!--l. 886--><math
8285
xmlns="http://www.w3.org/1998/Math/MathML"
8288
class="MathClass-ord">Ω</mi> <mo
8289
class="MathClass-rel">=</mo><msup
8290
> <mi class="mathbf">R</mi><mrow
8292
class="MathClass-ord">n</mi></mrow></msup
8293
></math>. The proof of
8296
href="#x1-7010r31">31<!--tex4ht:ref: e:SS--></a>) is divided into two steps. In the first step we prove the statement in the one-dimensional
8297
case <!--l. 888--><math
8298
xmlns="http://www.w3.org/1998/Math/MathML"
8299
mode="inline"> <mrow><mo
8300
class="MathClass-open">(</mo><mi
8301
class="MathClass-ord">n</mi> <mo
8302
class="MathClass-rel">=</mo> <mn
8303
class="MathClass-ord">1</mn><mo
8304
class="MathClass-close">)</mo></mrow></math>,
8305
using Theorem <a
8306
href="#x1-5003r2">5.2<!--tex4ht:ref: th-weak-ske-owf--></a>. In the second step we achieve the general result using
8308
href="#x1-7001r1">7.1<!--tex4ht:ref: t:conl--></a>.
8310
<h4 class="likesubsectionHead"><a
8311
name="x1-80007"></a>Step 1</h4> Assume that <!--l. 892--><math
8312
xmlns="http://www.w3.org/1998/Math/MathML"
8314
class="MathClass-ord">n</mi> <mo
8315
class="MathClass-rel">=</mo> <mn
8316
class="MathClass-ord">1</mn></math>.
8317
Since <!--l. 892--><math
8318
xmlns="http://www.w3.org/1998/Math/MathML"
8319
mode="inline"> <msub
8321
class="MathClass-ord">S</mi><mrow
8323
class="MathClass-ord">u</mi></mrow></msub
8324
></math> is at most countable,
8326
href="#x1-3006r7">7<!--tex4ht:ref: sum-bij--></a>) yields that <!--l. 893--><math
8327
xmlns="http://www.w3.org/1998/Math/MathML"
8328
mode="inline"> <mfenced
8329
open="|" close="|" ><munderover
8332
class="MathClass-ord">D</mi></mrow><mrow
8334
></mrow></munderover><mi
8335
class="MathClass-ord">v</mi></mfenced> <mrow><mo
8336
class="MathClass-open">(</mo><msub
8338
class="MathClass-ord">S</mi><mrow
8340
class="MathClass-ord">u</mi></mrow></msub
8342
class="MathClass-bin">∖</mo><msub
8344
class="MathClass-ord">S</mi><mrow
8346
class="MathClass-ord">v</mi></mrow></msub
8348
class="MathClass-close">)</mo></mrow> <mo
8349
class="MathClass-rel">=</mo> <mn
8350
class="MathClass-ord">0</mn></math>, so that
8352
href="#x1-4001r19">19<!--tex4ht:ref: e:st--></a>) and (<a
8353
href="#x1-4003r21">21<!--tex4ht:ref: e:barwq--></a>) imply that <!--l. 894--><math
8354
xmlns="http://www.w3.org/1998/Math/MathML"
8356
class="MathClass-ord">D</mi><mi
8357
class="MathClass-ord">v</mi> <mo
8358
class="MathClass-rel">=</mo> <munderover
8361
class="MathClass-ord">D</mi></mrow><mrow
8363
></mrow></munderover><mi
8364
class="MathClass-ord">v</mi> <mo
8365
class="MathClass-bin">+</mo> <mi
8366
class="MathClass-ord">J</mi><mi
8367
class="MathClass-ord">v</mi></math>
8368
is the Radon-Nikodým decomposition of <!--l. 895--><math
8369
xmlns="http://www.w3.org/1998/Math/MathML"
8372
class="MathClass-ord">D</mi><mi
8373
class="MathClass-ord">v</mi></math>
8374
in absolutely continuous and singular part with respect to <!--l. 896--><math
8375
xmlns="http://www.w3.org/1998/Math/MathML"
8378
open="|" close="|" ><munderover
8381
class="MathClass-ord">D</mi></mrow><mrow
8383
></mrow></munderover><mi
8384
class="MathClass-ord">u</mi></mfenced></math>. By
8386
href="#x1-5003r2">5.2<!--tex4ht:ref: th-weak-ske-owf--></a>, we have <table class="equation"><tr><td>
8388
xmlns="http://www.w3.org/1998/Math/MathML"
8391
class="equation"><mtr><mtd>
8396
class="MathClass-ord">D</mi></mrow><mrow
8398
></mrow></munderover><mi
8399
class="MathClass-ord">v</mi></mrow>
8402
open="|" close="|" ><munderover
8405
class="MathClass-ord">D</mi></mrow><mrow
8407
></mrow></munderover><mi
8408
class="MathClass-ord">u</mi></mfenced> </mrow></mfrac><mrow><mo
8409
class="MathClass-open">(</mo><mi
8410
class="MathClass-ord">t</mi><mo
8411
class="MathClass-close">)</mo></mrow> <mo
8412
class="MathClass-rel">=</mo><msub
8416
class="MathClass-ord">s</mi><mo
8417
class="MathClass-rel">→</mo><msup
8419
class="MathClass-ord">t</mi><mrow
8421
class="MathClass-bin">+</mo></mrow></msup
8425
class="MathClass-ord">D</mi><mi
8426
class="MathClass-ord">v</mi><mrow><mo
8427
class="MathClass-open">(</mo><mrow class="mathinner"> <mfenced
8428
open="[" close="[" ><mi
8429
class="MathClass-ord">t</mi><mo
8430
class="MathClass-punc">,</mo> <mi
8431
class="MathClass-ord">s</mi></mfenced> </mrow><mo
8432
class="MathClass-close">)</mo></mrow></mrow>
8435
open="|" close="|" ><munderover
8438
class="MathClass-ord">D</mi></mrow><mrow
8440
></mrow></munderover><mi
8441
class="MathClass-ord">u</mi></mfenced> <mrow><mo
8442
class="MathClass-open">(</mo><mrow class="mathinner"> <mfenced
8443
open="[" close="[" ><mi
8444
class="MathClass-ord">t</mi><mo
8445
class="MathClass-punc">,</mo> <mi
8446
class="MathClass-ord">s</mi></mfenced> </mrow><mo
8447
class="MathClass-close">)</mo></mrow></mrow></mfrac><mo
8448
class="MathClass-punc">,</mo> <mspace width="2em" class="qquad"/> <mfrac><mrow
8452
class="MathClass-ord">D</mi></mrow><mrow
8454
></mrow></munderover><mi
8455
class="MathClass-ord">u</mi></mrow>
8458
open="|" close="|" ><munderover
8461
class="MathClass-ord">D</mi></mrow><mrow
8463
></mrow></munderover><mi
8464
class="MathClass-ord">u</mi></mfenced> </mrow></mfrac><mrow><mo
8465
class="MathClass-open">(</mo><mi
8466
class="MathClass-ord">t</mi><mo
8467
class="MathClass-close">)</mo></mrow> <mo
8468
class="MathClass-rel">=</mo><msub
8472
class="MathClass-ord">s</mi><mo
8473
class="MathClass-rel">→</mo><msup
8475
class="MathClass-ord">t</mi><mrow
8477
class="MathClass-bin">+</mo></mrow></msup
8481
class="MathClass-ord">D</mi><mi
8482
class="MathClass-ord">u</mi><mrow><mo
8483
class="MathClass-open">(</mo><mrow class="mathinner"> <mfenced
8484
open="[" close="[" ><mi
8485
class="MathClass-ord">t</mi><mo
8486
class="MathClass-punc">,</mo> <mi
8487
class="MathClass-ord">s</mi></mfenced> </mrow><mo
8488
class="MathClass-close">)</mo></mrow></mrow>
8491
open="|" close="|" ><munderover
8494
class="MathClass-ord">D</mi></mrow><mrow
8496
></mrow></munderover><mi
8497
class="MathClass-ord">u</mi></mfenced> <mrow><mo
8498
class="MathClass-open">(</mo><mrow class="mathinner"> <mfenced
8499
open="[" close="[" ><mi
8500
class="MathClass-ord">t</mi><mo
8501
class="MathClass-punc">,</mo> <mi
8502
class="MathClass-ord">s</mi></mfenced> </mrow><mo
8503
class="MathClass-close">)</mo></mrow></mrow></mfrac></mtd><mtd> </mtd></mtr></mtable>
8505
<!--l. 905--><p class="nopar"></p></td></tr></table>
8507
xmlns="http://www.w3.org/1998/Math/MathML"
8508
mode="inline"> <mfenced
8509
open="|" close="|" ><munderover
8512
class="MathClass-ord">D</mi></mrow><mrow
8514
></mrow></munderover><mi
8515
class="MathClass-ord">u</mi></mfenced></math>-almost
8516
everywhere in <!--l. 906--><math
8517
xmlns="http://www.w3.org/1998/Math/MathML"
8518
mode="inline"> <mi class="mathbf">R</mi></math>.
8519
It is well known (see, for instance, <span class="cite">[<a
8520
href="#Xste:sint">12</a>, 2.5.16]</span>) that every one-dimensional function of bounded
8521
variation <!--l. 908--><math
8522
xmlns="http://www.w3.org/1998/Math/MathML"
8524
class="MathClass-ord">w</mi></math>
8525
has a unique left continuous representative, i.e., a function <!--l. 909--><math
8526
xmlns="http://www.w3.org/1998/Math/MathML"
8529
class="MathClass-ord">ŵ</mi></math> such that <!--l. 909--><math
8530
xmlns="http://www.w3.org/1998/Math/MathML"
8533
class="MathClass-ord">ŵ</mi> <mo
8534
class="MathClass-rel">=</mo> <mi
8535
class="MathClass-ord">w</mi></math> almost everywhere
8536
and <!--l. 910--><math
8537
xmlns="http://www.w3.org/1998/Math/MathML"
8538
mode="inline"> <msub
8542
class="MathClass-ord">s</mi><mo
8543
class="MathClass-rel">→</mo><msup
8545
class="MathClass-ord">t</mi><mrow
8547
class="MathClass-bin">−</mo></mrow></msup
8550
class="MathClass-ord">ŵ</mi><mrow><mo
8551
class="MathClass-open">(</mo><mi
8552
class="MathClass-ord">s</mi><mo
8553
class="MathClass-close">)</mo></mrow> <mo
8554
class="MathClass-rel">=</mo> <mi
8555
class="MathClass-ord">ŵ</mi><mrow><mo
8556
class="MathClass-open">(</mo><mi
8557
class="MathClass-ord">t</mi><mo
8558
class="MathClass-close">)</mo></mrow></math> for
8559
every <!--l. 910--><math
8560
xmlns="http://www.w3.org/1998/Math/MathML"
8562
class="MathClass-ord">t</mi> <mo
8563
class="MathClass-rel">∈</mo> <mi class="mathbf">R</mi></math>.
8565
These conditions imply <table class="equation"><tr><td>
8567
xmlns="http://www.w3.org/1998/Math/MathML"
8570
class="equation"><mtr><mtd>
8572
class="MathClass-ord">û</mi><mrow><mo
8573
class="MathClass-open">(</mo><mi
8574
class="MathClass-ord">t</mi><mo
8575
class="MathClass-close">)</mo></mrow> <mo
8576
class="MathClass-rel">=</mo> <mi
8577
class="MathClass-ord">D</mi><mi
8578
class="MathClass-ord">u</mi><mrow><mo
8579
class="MathClass-open">(</mo><mrow class="mathinner"> <mfenced
8580
open="]" close="[" ><mo
8581
class="MathClass-bin">−</mo><mi
8582
class="MathClass-ord">∞</mi><mo
8583
class="MathClass-punc">,</mo> <mi
8584
class="MathClass-ord">t</mi></mfenced> </mrow><mo
8585
class="MathClass-close">)</mo></mrow><mo
8586
class="MathClass-punc">,</mo> <mspace width="2em" class="qquad"/><munderover
8589
class="MathClass-ord">v</mi></mrow><mrow
8592
class="MathClass-ord">̂</mi></mrow></munderover><mrow><mo
8593
class="MathClass-open">(</mo><mi
8594
class="MathClass-ord">t</mi><mo
8595
class="MathClass-close">)</mo></mrow> <mo
8596
class="MathClass-rel">=</mo> <mi
8597
class="MathClass-ord">D</mi><mi
8598
class="MathClass-ord">v</mi><mrow><mo
8599
class="MathClass-open">(</mo><mrow class="mathinner"> <mfenced
8600
open="]" close="[" ><mo
8601
class="MathClass-bin">−</mo><mi
8602
class="MathClass-ord">∞</mi><mo
8603
class="MathClass-punc">,</mo> <mi
8604
class="MathClass-ord">t</mi></mfenced> </mrow><mo
8605
class="MathClass-close">)</mo></mrow><mspace width="2em" class="qquad"/><mi
8606
class="MathClass-ord">∀</mi><mi
8607
class="MathClass-ord">t</mi> <mo
8608
class="MathClass-rel">∈</mo> <mi class="mathbf">R</mi></mtd><mtd><mspace
8609
id="x1-8001r32" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
8611
<!--l. 916--><p class="nopar"></p></td><td width="5%">(32)</td></tr></table>
8612
and <table class="equation"><tr><td>
8614
xmlns="http://www.w3.org/1998/Math/MathML"
8617
class="equation"><mtr><mtd>
8621
class="MathClass-ord">v</mi></mrow><mrow
8624
class="MathClass-ord">̂</mi></mrow></munderover><mrow><mo
8625
class="MathClass-open">(</mo><mi
8626
class="MathClass-ord">t</mi><mo
8627
class="MathClass-close">)</mo></mrow> <mo
8628
class="MathClass-rel">=</mo> <mi
8629
class="MathClass-ord">f</mi><mrow><mo
8630
class="MathClass-open">(</mo><mi
8631
class="MathClass-ord">û</mi><mrow><mo
8632
class="MathClass-open">(</mo><mi
8633
class="MathClass-ord">t</mi><mo
8634
class="MathClass-close">)</mo></mrow><mo
8635
class="MathClass-close">)</mo></mrow><mspace width="2em" class="qquad"/><mi
8636
class="MathClass-ord">∀</mi><mi
8637
class="MathClass-ord">t</mi> <mo
8638
class="MathClass-rel">∈</mo> <mi class="mathbf">R</mi><mo
8639
class="MathClass-punc">.</mo></mtd><mtd><mspace
8640
id="x1-8002r33" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
8642
<!--l. 919--><p class="nopar"></p></td><td width="5%">(33)</td></tr></table>
8643
Let <!--l. 920--><math
8644
xmlns="http://www.w3.org/1998/Math/MathML"
8646
class="MathClass-ord">t</mi> <mo
8647
class="MathClass-rel">∈</mo> <mi class="mathbf">R</mi></math> be
8648
such that <!--l. 921--><math
8649
xmlns="http://www.w3.org/1998/Math/MathML"
8650
mode="inline"> <mfenced
8651
open="|" close="|" ><munderover
8654
class="MathClass-ord">D</mi></mrow><mrow
8656
></mrow></munderover><mi
8657
class="MathClass-ord">u</mi></mfenced> <mrow><mo
8658
class="MathClass-open">(</mo><mrow class="mathinner"> <mfenced
8659
open="[" close="[" ><mi
8660
class="MathClass-ord">t</mi><mo
8661
class="MathClass-punc">,</mo> <mi
8662
class="MathClass-ord">s</mi></mfenced> </mrow><mo
8663
class="MathClass-close">)</mo></mrow> <mo
8664
class="MathClass-rel">></mo> <mn
8665
class="MathClass-ord">0</mn></math>
8666
for every <!--l. 921--><math
8667
xmlns="http://www.w3.org/1998/Math/MathML"
8669
class="MathClass-ord">s</mi> <mo
8670
class="MathClass-rel">></mo> <mi
8671
class="MathClass-ord">t</mi></math>
8672
and assume that the limits in (<a
8673
href="#x1-4004r22">22<!--tex4ht:ref: joe--></a>) exist. By (<a
8674
href="#x1-4005r23">23<!--tex4ht:ref: j:mark--></a>) and (<a
8675
href="#x1-6017r24">24<!--tex4ht:ref: far-d--></a>) we get <table class="equation"><tr><td>
8678
xmlns="http://www.w3.org/1998/Math/MathML"
8681
class="equation"><mtr><mtd>
8682
<mtable class="split"><mtr><mtd>
8684
class="split-mtr"></mrow><mrow
8685
class="split-mtd"></mrow> <mfrac><mrow
8689
class="MathClass-ord">v</mi></mrow><mrow
8692
class="MathClass-ord">̂</mi></mrow></munderover><mrow><mo
8693
class="MathClass-open">(</mo><mi
8694
class="MathClass-ord">s</mi><mo
8695
class="MathClass-close">)</mo></mrow> <mo
8696
class="MathClass-bin">−</mo> <munderover
8699
class="MathClass-ord">v</mi></mrow><mrow
8702
class="MathClass-ord">̂</mi></mrow></munderover><mrow><mo
8703
class="MathClass-open">(</mo><mi
8704
class="MathClass-ord">t</mi><mo
8705
class="MathClass-close">)</mo></mrow></mrow>
8708
open="|" close="|" ><munderover
8711
class="MathClass-ord">D</mi></mrow><mrow
8713
></mrow></munderover><mi
8714
class="MathClass-ord">u</mi></mfenced> <mrow><mo
8715
class="MathClass-open">(</mo><mrow class="mathinner"> <mfenced
8716
open="[" close="[" ><mi
8717
class="MathClass-ord">t</mi><mo
8718
class="MathClass-punc">,</mo> <mi
8719
class="MathClass-ord">s</mi></mfenced> </mrow><mo
8720
class="MathClass-close">)</mo></mrow></mrow></mfrac><mrow
8721
class="split-mtd"></mrow> <mo
8722
class="MathClass-rel">=</mo> <mfrac><mrow
8724
class="MathClass-ord">f</mi><mrow><mo
8725
class="MathClass-open">(</mo><mi
8726
class="MathClass-ord">û</mi><mrow><mo
8727
class="MathClass-open">(</mo><mi
8728
class="MathClass-ord">s</mi><mo
8729
class="MathClass-close">)</mo></mrow><mo
8730
class="MathClass-close">)</mo></mrow> <mo
8731
class="MathClass-bin">−</mo> <mi
8732
class="MathClass-ord">f</mi><mrow><mo
8733
class="MathClass-open">(</mo><mi
8734
class="MathClass-ord">û</mi><mrow><mo
8735
class="MathClass-open">(</mo><mi
8736
class="MathClass-ord">t</mi><mo
8737
class="MathClass-close">)</mo></mrow><mo
8738
class="MathClass-close">)</mo></mrow></mrow>
8741
open="|" close="|" ><munderover
8744
class="MathClass-ord">D</mi></mrow><mrow
8746
></mrow></munderover><mi
8747
class="MathClass-ord">u</mi></mfenced> <mrow><mo
8748
class="MathClass-open">(</mo><mrow class="mathinner"> <mfenced
8749
open="[" close="[" ><mi
8750
class="MathClass-ord">t</mi><mo
8751
class="MathClass-punc">,</mo> <mi
8752
class="MathClass-ord">s</mi></mfenced> </mrow><mo
8753
class="MathClass-close">)</mo></mrow></mrow></mfrac>
8755
class="split-mtr"></mrow><mrow
8756
class="split-mtd"></mrow> <mrow
8757
class="split-mtd"></mrow> <mo
8758
class="MathClass-rel">=</mo> <mfrac><mrow
8760
class="MathClass-ord">f</mi><mrow><mo
8761
class="MathClass-open">(</mo><mi
8762
class="MathClass-ord">û</mi><mrow><mo
8763
class="MathClass-open">(</mo><mi
8764
class="MathClass-ord">s</mi><mo
8765
class="MathClass-close">)</mo></mrow><mo
8766
class="MathClass-close">)</mo></mrow> <mo
8767
class="MathClass-bin">−</mo> <mi
8768
class="MathClass-ord">f</mi><mrow><mo
8769
class="MathClass-open">(</mo><mi
8770
class="MathClass-ord">û</mi><mrow><mo
8771
class="MathClass-open">(</mo><mi
8772
class="MathClass-ord">t</mi><mo
8773
class="MathClass-close">)</mo></mrow> <mo
8774
class="MathClass-bin">+</mo> <mfrac class="dfrac"><mrow><munderover
8777
class="MathClass-ord">D</mi></mrow><mrow
8779
></mrow></munderover><mi
8780
class="MathClass-ord">u</mi></mrow><mrow><mfenced
8781
open="|" close="|" ><munderover
8784
class="MathClass-ord">D</mi></mrow><mrow
8786
></mrow></munderover><mi
8787
class="MathClass-ord">u</mi></mfenced></mrow></mfrac><mrow><mo
8788
class="MathClass-open">(</mo><mi
8789
class="MathClass-ord">t</mi><mo
8790
class="MathClass-close">)</mo></mrow> <mfenced
8791
open="|" close="|" ><munderover
8794
class="MathClass-ord">D</mi></mrow><mrow
8796
></mrow></munderover><mi
8797
class="MathClass-ord">u</mi></mfenced> <mrow><mo
8798
class="MathClass-open">(</mo><mrow class="mathinner"> <mfenced
8799
open="[" close="[" ><mi
8800
class="MathClass-ord">t</mi><mo
8801
class="MathClass-punc">,</mo> <mi
8802
class="MathClass-ord">s</mi></mfenced> </mrow><mo
8803
class="MathClass-close">)</mo></mrow><mo
8804
class="MathClass-close">)</mo></mrow></mrow>
8807
open="|" close="|" ><munderover
8810
class="MathClass-ord">D</mi></mrow><mrow
8812
></mrow></munderover><mi
8813
class="MathClass-ord">u</mi></mfenced> <mrow><mo
8814
class="MathClass-open">(</mo><mrow class="mathinner"> <mfenced
8815
open="[" close="[" ><mi
8816
class="MathClass-ord">t</mi><mo
8817
class="MathClass-punc">,</mo> <mi
8818
class="MathClass-ord">s</mi></mfenced> </mrow><mo
8819
class="MathClass-close">)</mo></mrow></mrow></mfrac>
8821
class="split-mtr"></mrow><mrow
8822
class="split-mtd"></mrow> <mrow
8823
class="split-mtd"></mrow> <mo
8824
class="MathClass-bin">+</mo> <mfrac><mrow
8826
class="MathClass-ord">f</mi><mrow><mo
8827
class="MathClass-open">(</mo><mi
8828
class="MathClass-ord">û</mi><mrow><mo
8829
class="MathClass-open">(</mo><mi
8830
class="MathClass-ord">t</mi><mo
8831
class="MathClass-close">)</mo></mrow> <mo
8832
class="MathClass-bin">+</mo> <mfrac class="dfrac"><mrow><munderover
8835
class="MathClass-ord">D</mi></mrow><mrow
8837
></mrow></munderover><mi
8838
class="MathClass-ord">u</mi></mrow><mrow><mfenced
8839
open="|" close="|" ><munderover
8842
class="MathClass-ord">D</mi></mrow><mrow
8844
></mrow></munderover><mi
8845
class="MathClass-ord">u</mi></mfenced></mrow></mfrac><mrow><mo
8846
class="MathClass-open">(</mo><mi
8847
class="MathClass-ord">t</mi><mo
8848
class="MathClass-close">)</mo></mrow> <mfenced
8849
open="|" close="|" ><munderover
8852
class="MathClass-ord">D</mi></mrow><mrow
8854
></mrow></munderover><mi
8855
class="MathClass-ord">u</mi></mfenced> <mrow><mo
8856
class="MathClass-open">(</mo><mrow class="mathinner"> <mfenced
8857
open="[" close="[" ><mi
8858
class="MathClass-ord">t</mi><mo
8859
class="MathClass-punc">,</mo> <mi
8860
class="MathClass-ord">s</mi></mfenced> </mrow><mo
8861
class="MathClass-close">)</mo></mrow><mo
8862
class="MathClass-close">)</mo></mrow> <mo
8863
class="MathClass-bin">−</mo> <mi
8864
class="MathClass-ord">f</mi><mrow><mo
8865
class="MathClass-open">(</mo><mi
8866
class="MathClass-ord">û</mi><mrow><mo
8867
class="MathClass-open">(</mo><mi
8868
class="MathClass-ord">t</mi><mo
8869
class="MathClass-close">)</mo></mrow><mo
8870
class="MathClass-close">)</mo></mrow></mrow>
8873
open="|" close="|" ><munderover
8876
class="MathClass-ord">D</mi></mrow><mrow
8878
></mrow></munderover><mi
8879
class="MathClass-ord">u</mi></mfenced> <mrow><mo
8880
class="MathClass-open">(</mo><mrow class="mathinner"> <mfenced
8881
open="[" close="[" ><mi
8882
class="MathClass-ord">t</mi><mo
8883
class="MathClass-punc">,</mo> <mi
8884
class="MathClass-ord">s</mi></mfenced> </mrow><mo
8885
class="MathClass-close">)</mo></mrow></mrow></mfrac>
8886
</mtd></mtr></mtable> </mtd><mtd></mtd></mtr></mtable>
8888
<!--l. 936--><p class="nopar"></p></td></tr></table>
8889
for every <!--l. 937--><math
8890
xmlns="http://www.w3.org/1998/Math/MathML"
8892
class="MathClass-ord">s</mi> <mo
8893
class="MathClass-rel">></mo> <mi
8894
class="MathClass-ord">t</mi></math>. Using the
8895
Lipschitz condition on <!--l. 937--><math
8896
xmlns="http://www.w3.org/1998/Math/MathML"
8898
class="MathClass-ord">f</mi></math>
8902
xmlns="http://www.w3.org/1998/Math/MathML"
8905
class="multline-star">
8907
class="multline-star"> <mfenced
8908
open="|" close="|" ><mfrac><mrow
8912
class="MathClass-ord">v</mi></mrow><mrow
8915
class="MathClass-ord">̂</mi></mrow></munderover><mrow><mo
8916
class="MathClass-open">(</mo><mi
8917
class="MathClass-ord">s</mi><mo
8918
class="MathClass-close">)</mo></mrow> <mo
8919
class="MathClass-bin">−</mo> <munderover
8922
class="MathClass-ord">v</mi></mrow><mrow
8925
class="MathClass-ord">̂</mi></mrow></munderover><mrow><mo
8926
class="MathClass-open">(</mo><mi
8927
class="MathClass-ord">t</mi><mo
8928
class="MathClass-close">)</mo></mrow></mrow>
8931
open="|" close="|" ><munderover
8934
class="MathClass-ord">D</mi></mrow><mrow
8936
></mrow></munderover><mi
8937
class="MathClass-ord">u</mi></mfenced> <mrow><mo
8938
class="MathClass-open">(</mo><mrow class="mathinner"> <mfenced
8939
open="[" close="[" ><mi
8940
class="MathClass-ord">t</mi><mo
8941
class="MathClass-punc">,</mo> <mi
8942
class="MathClass-ord">s</mi></mfenced> </mrow><mo
8943
class="MathClass-close">)</mo></mrow></mrow></mfrac> <mo
8944
class="MathClass-bin">−</mo> <mfrac><mrow
8946
class="MathClass-ord">f</mi><mrow><mo
8947
class="MathClass-open">(</mo><mi
8948
class="MathClass-ord">û</mi><mrow><mo
8949
class="MathClass-open">(</mo><mi
8950
class="MathClass-ord">t</mi><mo
8951
class="MathClass-close">)</mo></mrow> <mo
8952
class="MathClass-bin">+</mo> <mfrac class="dfrac"><mrow><munderover
8955
class="MathClass-ord">D</mi></mrow><mrow
8957
></mrow></munderover><mi
8958
class="MathClass-ord">u</mi></mrow><mrow><mfenced
8959
open="|" close="|" ><munderover
8962
class="MathClass-ord">D</mi></mrow><mrow
8964
></mrow></munderover><mi
8965
class="MathClass-ord">u</mi></mfenced></mrow></mfrac><mrow><mo
8966
class="MathClass-open">(</mo><mi
8967
class="MathClass-ord">t</mi><mo
8968
class="MathClass-close">)</mo></mrow> <mfenced
8969
open="|" close="|" ><munderover
8972
class="MathClass-ord">D</mi></mrow><mrow
8974
></mrow></munderover><mi
8975
class="MathClass-ord">u</mi></mfenced> <mrow><mo
8976
class="MathClass-open">(</mo><mrow class="mathinner"> <mfenced
8977
open="[" close="[" ><mi
8978
class="MathClass-ord">t</mi><mo
8979
class="MathClass-punc">,</mo> <mi
8980
class="MathClass-ord">s</mi></mfenced> </mrow><mo
8981
class="MathClass-close">)</mo></mrow><mo
8982
class="MathClass-close">)</mo></mrow> <mo
8983
class="MathClass-bin">−</mo> <mi
8984
class="MathClass-ord">f</mi><mrow><mo
8985
class="MathClass-open">(</mo><mi
8986
class="MathClass-ord">û</mi><mrow><mo
8987
class="MathClass-open">(</mo><mi
8988
class="MathClass-ord">t</mi><mo
8989
class="MathClass-close">)</mo></mrow><mo
8990
class="MathClass-close">)</mo></mrow></mrow>
8993
open="|" close="|" ><munderover
8996
class="MathClass-ord">D</mi></mrow><mrow
8998
></mrow></munderover><mi
8999
class="MathClass-ord">u</mi></mfenced> <mrow><mo
9000
class="MathClass-open">(</mo><mrow class="mathinner"> <mfenced
9001
open="[" close="[" ><mi
9002
class="MathClass-ord">t</mi><mo
9003
class="MathClass-punc">,</mo> <mi
9004
class="MathClass-ord">s</mi></mfenced> </mrow><mo
9005
class="MathClass-close">)</mo></mrow></mrow></mfrac> </mfenced>
9006
</mtd></mtr><mtr><mtd
9007
class="multline-star"> <mo
9008
class="MathClass-rel">≤</mo> <mi
9009
class="MathClass-ord">K</mi> <mfenced
9010
open="|" close="|" ><mfrac><mrow
9012
class="MathClass-ord">û</mi><mrow><mo
9013
class="MathClass-open">(</mo><mi
9014
class="MathClass-ord">s</mi><mo
9015
class="MathClass-close">)</mo></mrow> <mo
9016
class="MathClass-bin">−</mo> <mi
9017
class="MathClass-ord">û</mi><mrow><mo
9018
class="MathClass-open">(</mo><mi
9019
class="MathClass-ord">t</mi><mo
9020
class="MathClass-close">)</mo></mrow></mrow>
9023
open="|" close="|" ><munderover
9026
class="MathClass-ord">D</mi></mrow><mrow
9028
></mrow></munderover><mi
9029
class="MathClass-ord">u</mi></mfenced> <mrow><mo
9030
class="MathClass-open">(</mo><mrow class="mathinner"> <mfenced
9031
open="[" close="[" ><mi
9032
class="MathClass-ord">t</mi><mo
9033
class="MathClass-punc">,</mo> <mi
9034
class="MathClass-ord">s</mi></mfenced> </mrow><mo
9035
class="MathClass-close">)</mo></mrow></mrow></mfrac> <mo
9036
class="MathClass-bin">−</mo> <mfrac><mrow
9040
class="MathClass-ord">D</mi></mrow><mrow
9042
></mrow></munderover><mi
9043
class="MathClass-ord">u</mi></mrow>
9046
open="|" close="|" ><munderover
9049
class="MathClass-ord">D</mi></mrow><mrow
9051
></mrow></munderover><mi
9052
class="MathClass-ord">u</mi></mfenced> </mrow></mfrac><mrow><mo
9053
class="MathClass-open">(</mo><mi
9054
class="MathClass-ord">t</mi><mo
9055
class="MathClass-close">)</mo></mrow></mfenced> <mo
9056
class="MathClass-punc">.</mo> </mtd></mtr></mtable>
9058
<!--l. 949--><p class="nopar">
9060
href="#x1-7008r29">29<!--tex4ht:ref: e:bomb--></a>), the function <!--l. 951--><math
9061
xmlns="http://www.w3.org/1998/Math/MathML"
9063
class="MathClass-ord">s</mi> <mo
9064
class="MathClass-rel">→</mo> <mfenced
9065
open="|" close="|" ><munderover
9068
class="MathClass-ord">D</mi></mrow><mrow
9070
></mrow></munderover><mi
9071
class="MathClass-ord">u</mi></mfenced> <mrow><mo
9072
class="MathClass-open">(</mo><mrow class="mathinner"> <mfenced
9073
open="[" close="[" ><mi
9074
class="MathClass-ord">t</mi><mo
9075
class="MathClass-punc">,</mo> <mi
9076
class="MathClass-ord">s</mi></mfenced> </mrow><mo
9077
class="MathClass-close">)</mo></mrow></math> is
9078
continuous and converges to 0 as <!--l. 953--><math
9079
xmlns="http://www.w3.org/1998/Math/MathML"
9081
class="MathClass-ord">s</mi> <mi
9082
class="MathClass-ord">↓</mi> <mi
9083
class="MathClass-ord">t</mi></math>.
9084
Therefore Remark <a
9085
href="#x1-7005r1">7.1<!--tex4ht:ref: r:omb--></a> and the previous inequality imply <!--l. 955--><math
9086
xmlns="http://www.w3.org/1998/Math/MathML"
9094
class="MathClass-ord">D</mi></mrow><mrow
9096
></mrow></munderover><mi
9097
class="MathClass-ord">v</mi></mrow>
9100
open="|" close="|" ><munderover
9103
class="MathClass-ord">D</mi></mrow><mrow
9105
></mrow></munderover><mi
9106
class="MathClass-ord">u</mi></mfenced> </mrow></mfrac><mrow><mo
9107
class="MathClass-open">(</mo><mi
9108
class="MathClass-ord">t</mi><mo
9109
class="MathClass-close">)</mo></mrow> <mo
9110
class="MathClass-rel">=</mo><msub
9114
class="MathClass-ord">h</mi><mo
9115
class="MathClass-rel">→</mo><msup
9117
class="MathClass-ord">0</mn><mrow
9119
class="MathClass-bin">+</mo></mrow></msup
9123
class="MathClass-ord">f</mi><mrow><mo
9124
class="MathClass-open">(</mo><mi
9125
class="MathClass-ord">û</mi><mrow><mo
9126
class="MathClass-open">(</mo><mi
9127
class="MathClass-ord">t</mi><mo
9128
class="MathClass-close">)</mo></mrow> <mo
9129
class="MathClass-bin">+</mo> <mi
9130
class="MathClass-ord">h</mi><mfrac class="dfrac"><mrow><munderover
9133
class="MathClass-ord">D</mi></mrow><mrow
9135
></mrow></munderover><mi
9136
class="MathClass-ord">u</mi></mrow><mrow><mfenced
9137
open="|" close="|" ><munderover
9140
class="MathClass-ord">D</mi></mrow><mrow
9142
></mrow></munderover><mi
9143
class="MathClass-ord">u</mi></mfenced></mrow></mfrac><mrow><mo
9144
class="MathClass-open">(</mo><mi
9145
class="MathClass-ord">t</mi><mo
9146
class="MathClass-close">)</mo></mrow><mo
9147
class="MathClass-close">)</mo></mrow> <mo
9148
class="MathClass-bin">−</mo> <mi
9149
class="MathClass-ord">f</mi><mrow><mo
9150
class="MathClass-open">(</mo><mi
9151
class="MathClass-ord">û</mi><mrow><mo
9152
class="MathClass-open">(</mo><mi
9153
class="MathClass-ord">t</mi><mo
9154
class="MathClass-close">)</mo></mrow><mo
9155
class="MathClass-close">)</mo></mrow></mrow>
9158
class="MathClass-ord">h</mi></mrow></mfrac> <mspace width="1em" class="quad"/> <mfenced
9159
open="|" close="|" ><munderover
9162
class="MathClass-ord">D</mi></mrow><mrow
9164
></mrow></munderover><mi
9165
class="MathClass-ord">u</mi></mfenced> <mrow
9166
class="text"><mtext >-a.e. in </mtext></mrow><mi class="mathbf">R</mi><mo
9167
class="MathClass-punc">.</mo>
9168
</mrow></math> By (<a
9169
href="#x1-4004r22">22<!--tex4ht:ref: joe--></a>), <!--l. 958--><math
9170
xmlns="http://www.w3.org/1998/Math/MathML"
9173
class="MathClass-ord">û</mi><mrow><mo
9174
class="MathClass-open">(</mo><mi
9175
class="MathClass-ord">x</mi><mo
9176
class="MathClass-close">)</mo></mrow> <mo
9177
class="MathClass-rel">=</mo> <mi
9178
class="MathClass-ord">ũ</mi><mrow><mo
9179
class="MathClass-open">(</mo><mi
9180
class="MathClass-ord">x</mi><mo
9181
class="MathClass-close">)</mo></mrow></math> for every
9183
xmlns="http://www.w3.org/1998/Math/MathML"
9185
class="MathClass-ord">x</mi> <mo
9186
class="MathClass-rel">∈</mo> <mi class="mathbf">R</mi><mo
9187
class="MathClass-bin">∖</mo><msub
9189
class="MathClass-ord">S</mi><mrow
9191
class="MathClass-ord">u</mi></mrow></msub
9193
moreover, applying the same argument to the functions <!--l. 960--><math
9194
xmlns="http://www.w3.org/1998/Math/MathML"
9197
class="MathClass-ord">u</mi><mi
9198
class="MathClass-ord">′</mi><mrow><mo
9199
class="MathClass-open">(</mo><mi
9200
class="MathClass-ord">t</mi><mo
9201
class="MathClass-close">)</mo></mrow> <mo
9202
class="MathClass-rel">=</mo> <mi
9203
class="MathClass-ord">u</mi><mrow><mo
9204
class="MathClass-open">(</mo><mo
9205
class="MathClass-bin">−</mo><mi
9206
class="MathClass-ord">t</mi><mo
9207
class="MathClass-close">)</mo></mrow></math>, <!--l. 960--><math
9208
xmlns="http://www.w3.org/1998/Math/MathML"
9211
class="MathClass-ord">v</mi><mi
9212
class="MathClass-ord">′</mi><mrow><mo
9213
class="MathClass-open">(</mo><mi
9214
class="MathClass-ord">t</mi><mo
9215
class="MathClass-close">)</mo></mrow> <mo
9216
class="MathClass-rel">=</mo> <mi
9217
class="MathClass-ord">f</mi><mrow><mo
9218
class="MathClass-open">(</mo><mi
9219
class="MathClass-ord">u</mi><mi
9220
class="MathClass-ord">′</mi><mrow><mo
9221
class="MathClass-open">(</mo><mi
9222
class="MathClass-ord">t</mi><mo
9223
class="MathClass-close">)</mo></mrow><mo
9224
class="MathClass-close">)</mo></mrow> <mo
9225
class="MathClass-rel">=</mo> <mi
9226
class="MathClass-ord">v</mi><mrow><mo
9227
class="MathClass-open">(</mo><mo
9228
class="MathClass-bin">−</mo><mi
9229
class="MathClass-ord">t</mi><mo
9230
class="MathClass-close">)</mo></mrow></math>, we get <!--l. 961--><math
9231
xmlns="http://www.w3.org/1998/Math/MathML"
9240
class="MathClass-ord">D</mi></mrow><mrow
9242
></mrow></munderover><mi
9243
class="MathClass-ord">v</mi></mrow>
9246
open="|" close="|" ><munderover
9249
class="MathClass-ord">D</mi></mrow><mrow
9251
></mrow></munderover><mi
9252
class="MathClass-ord">u</mi></mfenced> </mrow></mfrac><mrow><mo
9253
class="MathClass-open">(</mo><mi
9254
class="MathClass-ord">t</mi><mo
9255
class="MathClass-close">)</mo></mrow> <mo
9256
class="MathClass-rel">=</mo><msub
9260
class="MathClass-ord">h</mi><mo
9261
class="MathClass-rel">→</mo><mn
9262
class="MathClass-ord">0</mn></mrow></msub
9265
class="MathClass-ord">f</mi><mrow><mo
9266
class="MathClass-open">(</mo><mi
9267
class="MathClass-ord">ũ</mi><mrow><mo
9268
class="MathClass-open">(</mo><mi
9269
class="MathClass-ord">t</mi><mo
9270
class="MathClass-close">)</mo></mrow> <mo
9271
class="MathClass-bin">+</mo> <mi
9272
class="MathClass-ord">h</mi><mfrac class="dfrac"><mrow><munderover
9275
class="MathClass-ord">D</mi></mrow><mrow
9277
></mrow></munderover><mi
9278
class="MathClass-ord">u</mi></mrow><mrow><mfenced
9279
open="|" close="|" ><munderover
9282
class="MathClass-ord">D</mi></mrow><mrow
9284
></mrow></munderover><mi
9285
class="MathClass-ord">u</mi></mfenced></mrow></mfrac><mrow><mo
9286
class="MathClass-open">(</mo><mi
9287
class="MathClass-ord">t</mi><mo
9288
class="MathClass-close">)</mo></mrow><mo
9289
class="MathClass-close">)</mo></mrow> <mo
9290
class="MathClass-bin">−</mo> <mi
9291
class="MathClass-ord">f</mi><mrow><mo
9292
class="MathClass-open">(</mo><mi
9293
class="MathClass-ord">ũ</mi><mrow><mo
9294
class="MathClass-open">(</mo><mi
9295
class="MathClass-ord">t</mi><mo
9296
class="MathClass-close">)</mo></mrow><mo
9297
class="MathClass-close">)</mo></mrow></mrow>
9300
class="MathClass-ord">h</mi></mrow></mfrac> <mspace width="2em" class="qquad"/> <mfenced
9301
open="|" close="|" ><munderover
9304
class="MathClass-ord">D</mi></mrow><mrow
9306
></mrow></munderover><mi
9307
class="MathClass-ord">u</mi></mfenced> <mrow
9308
class="text"><mtext >-a.e. in </mtext></mrow><mi class="mathbf">R</mi>
9310
our statement is proved.
9312
<h4 class="likesubsectionHead"><a
9313
name="x1-90007"></a>Step 2</h4>
9314
<!--l. 969--><p class="noindent">Let us consider now the general case <!--l. 969--><math
9315
xmlns="http://www.w3.org/1998/Math/MathML"
9318
class="MathClass-ord">n</mi> <mo
9319
class="MathClass-rel">></mo> <mn
9320
class="MathClass-ord">1</mn></math>. Let <!--l. 969--><math
9321
xmlns="http://www.w3.org/1998/Math/MathML"
9324
class="MathClass-ord">ν</mi> <mo
9325
class="MathClass-rel">∈</mo><msup
9326
> <mi class="mathbf">R</mi><mrow
9328
class="MathClass-ord">n</mi></mrow></msup
9329
></math> be such that <!--l. 970--><math
9330
xmlns="http://www.w3.org/1998/Math/MathML"
9333
open="|" close="|" ><mi
9334
class="MathClass-ord">ν</mi></mfenced> <mo
9335
class="MathClass-rel">=</mo> <mn
9336
class="MathClass-ord">1</mn></math>, and let <!--l. 970--><math
9337
xmlns="http://www.w3.org/1998/Math/MathML"
9341
class="MathClass-ord">π</mi><mrow
9343
class="MathClass-ord">ν</mi></mrow></msub
9345
class="MathClass-rel">=</mo> <mrow><mo
9346
class="MathClass-open">{</mo><mi
9347
class="MathClass-ord">y</mi> <mo
9348
class="MathClass-rel">∈</mo><msup
9349
> <mi class="mathbf">R</mi><mrow
9351
class="MathClass-ord">n</mi></mrow></msup
9353
class="MathClass-punc">:</mo> <mrow><mo
9354
class="MathClass-open">〈</mo><mi
9355
class="MathClass-ord">y</mi><mo
9356
class="MathClass-punc">,</mo> <mi
9357
class="MathClass-ord">ν</mi><mo
9358
class="MathClass-close">〉</mo></mrow> <mo
9359
class="MathClass-rel">=</mo> <mn
9360
class="MathClass-ord">0</mn><mo
9361
class="MathClass-close">}</mo></mrow></math>. In the following, we
9362
shall identify <!--l. 971--><math
9363
xmlns="http://www.w3.org/1998/Math/MathML"
9364
mode="inline"> <msup
9365
><mi class="mathbf">R</mi><mrow
9367
class="MathClass-ord">n</mi></mrow></msup
9368
></math> with <!--l. 972--><math
9369
xmlns="http://www.w3.org/1998/Math/MathML"
9373
class="MathClass-ord">π</mi><mrow
9375
class="MathClass-ord">ν</mi></mrow></msub
9377
class="MathClass-bin">×</mo> <mi class="mathbf">R</mi></math>, and we shall denote by <!--l. 972--><math
9378
xmlns="http://www.w3.org/1998/Math/MathML"
9381
class="MathClass-ord">y</mi></math> the variable ranging
9382
in <!--l. 973--><math
9383
xmlns="http://www.w3.org/1998/Math/MathML"
9384
mode="inline"> <msub
9386
class="MathClass-ord">π</mi><mrow
9388
class="MathClass-ord">ν</mi></mrow></msub
9389
></math> and by <!--l. 973--><math
9390
xmlns="http://www.w3.org/1998/Math/MathML"
9393
class="MathClass-ord">t</mi></math> the variable
9394
ranging in <!--l. 973--><math
9395
xmlns="http://www.w3.org/1998/Math/MathML"
9396
mode="inline"> <mi class="mathbf">R</mi></math>.
9397
By the just proven one-dimensional result, and by Theorem <a
9398
href="#x1-3008r3">3.3<!--tex4ht:ref: thm-main--></a>, we get <!--l. 975--><math
9399
xmlns="http://www.w3.org/1998/Math/MathML"
9407
class="MathClass-ord">h</mi><mo
9408
class="MathClass-rel">→</mo><mn
9409
class="MathClass-ord">0</mn></mrow></msub
9412
class="MathClass-ord">f</mi><mrow><mo
9413
class="MathClass-open">(</mo><mi
9414
class="MathClass-ord">ũ</mi><mrow><mo
9415
class="MathClass-open">(</mo><mi
9416
class="MathClass-ord">y</mi> <mo
9417
class="MathClass-bin">+</mo> <mi
9418
class="MathClass-ord">t</mi><mi
9419
class="MathClass-ord">ν</mi><mo
9420
class="MathClass-close">)</mo></mrow> <mo
9421
class="MathClass-bin">+</mo> <mi
9422
class="MathClass-ord">h</mi><mfrac class="dfrac"><mrow><munderover
9425
class="MathClass-ord">D</mi></mrow><mrow
9427
></mrow></munderover><msub
9429
class="MathClass-ord">u</mi><mrow
9431
class="MathClass-ord">y</mi></mrow></msub
9432
></mrow><mrow> <mfenced
9433
open="|" close="|" ><munderover
9436
class="MathClass-ord">D</mi></mrow><mrow
9438
></mrow></munderover><msub
9440
class="MathClass-ord">u</mi><mrow
9442
class="MathClass-ord">y</mi></mrow></msub
9443
> </mfenced></mrow></mfrac><mrow><mo
9444
class="MathClass-open">(</mo><mi
9445
class="MathClass-ord">t</mi><mo
9446
class="MathClass-close">)</mo></mrow><mo
9447
class="MathClass-close">)</mo></mrow> <mo
9448
class="MathClass-bin">−</mo> <mi
9449
class="MathClass-ord">f</mi><mrow><mo
9450
class="MathClass-open">(</mo><mi
9451
class="MathClass-ord">ũ</mi><mrow><mo
9452
class="MathClass-open">(</mo><mi
9453
class="MathClass-ord">y</mi> <mo
9454
class="MathClass-bin">+</mo> <mi
9455
class="MathClass-ord">t</mi><mi
9456
class="MathClass-ord">ν</mi><mo
9457
class="MathClass-close">)</mo></mrow><mo
9458
class="MathClass-close">)</mo></mrow></mrow>
9461
class="MathClass-ord">h</mi></mrow></mfrac> <mo
9462
class="MathClass-rel">=</mo> <mfrac><mrow
9466
class="MathClass-ord">D</mi></mrow><mrow
9468
></mrow></munderover><msub
9470
class="MathClass-ord">v</mi><mrow
9472
class="MathClass-ord">y</mi></mrow></msub
9476
open="|" close="|" ><munderover
9479
class="MathClass-ord">D</mi></mrow><mrow
9481
></mrow></munderover><msub
9483
class="MathClass-ord">u</mi><mrow
9485
class="MathClass-ord">y</mi></mrow></msub
9486
> </mfenced> </mrow></mfrac> <mrow><mo
9487
class="MathClass-open">(</mo><mi
9488
class="MathClass-ord">t</mi><mo
9489
class="MathClass-close">)</mo></mrow><mspace width="2em" class="qquad"/> <mfenced
9490
open="|" close="|" ><munderover
9493
class="MathClass-ord">D</mi></mrow><mrow
9495
></mrow></munderover><msub
9497
class="MathClass-ord">u</mi><mrow
9499
class="MathClass-ord">y</mi></mrow></msub
9501
class="text"><mtext >-a.e. in </mtext></mrow><mi class="mathbf">R</mi>
9502
</mrow></math> for <!--l. 978--><math
9503
xmlns="http://www.w3.org/1998/Math/MathML"
9507
class="MathClass-ord"><!--span
9508
class="htf-calligraphy"-->H<!--/span--></mi><mrow
9510
class="MathClass-ord">n</mi><mo
9511
class="MathClass-bin">−</mo><mn
9512
class="MathClass-ord">1</mn></mrow></msub
9514
every <!--l. 978--><math
9515
xmlns="http://www.w3.org/1998/Math/MathML"
9517
class="MathClass-ord">y</mi> <mo
9518
class="MathClass-rel">∈</mo> <msub
9520
class="MathClass-ord">π</mi><mrow
9522
class="MathClass-ord">ν</mi></mrow></msub
9525
We claim that </p><table class="equation"><tr><td>
9527
xmlns="http://www.w3.org/1998/Math/MathML"
9530
class="equation"><mtr><mtd>
9533
class="MathClass-open">〈</mo><munderover
9536
class="MathClass-ord">D</mi></mrow><mrow
9538
></mrow></munderover><mi
9539
class="MathClass-ord">u</mi><mo
9540
class="MathClass-punc">,</mo> <mi
9541
class="MathClass-ord">ν</mi><mo
9542
class="MathClass-close">〉</mo></mrow></mrow>
9545
open="|" close="|" ><mrow><mo
9546
class="MathClass-open">〈</mo><munderover
9549
class="MathClass-ord">D</mi></mrow><mrow
9551
></mrow></munderover><mi
9552
class="MathClass-ord">u</mi><mo
9553
class="MathClass-punc">,</mo> <mi
9554
class="MathClass-ord">ν</mi><mo
9555
class="MathClass-close">〉</mo></mrow></mfenced> </mrow></mfrac><mrow><mo
9556
class="MathClass-open">(</mo><mi
9557
class="MathClass-ord">y</mi> <mo
9558
class="MathClass-bin">+</mo> <mi
9559
class="MathClass-ord">t</mi><mi
9560
class="MathClass-ord">ν</mi><mo
9561
class="MathClass-close">)</mo></mrow> <mo
9562
class="MathClass-rel">=</mo> <mfrac><mrow
9566
class="MathClass-ord">D</mi></mrow><mrow
9568
></mrow></munderover><msub
9570
class="MathClass-ord">u</mi><mrow
9572
class="MathClass-ord">y</mi></mrow></msub
9576
open="|" close="|" ><munderover
9579
class="MathClass-ord">D</mi></mrow><mrow
9581
></mrow></munderover><msub
9583
class="MathClass-ord">u</mi><mrow
9585
class="MathClass-ord">y</mi></mrow></msub
9586
> </mfenced> </mrow></mfrac> <mrow><mo
9587
class="MathClass-open">(</mo><mi
9588
class="MathClass-ord">t</mi><mo
9589
class="MathClass-close">)</mo></mrow><mspace width="2em" class="qquad"/> <mfenced
9590
open="|" close="|" ><munderover
9593
class="MathClass-ord">D</mi></mrow><mrow
9595
></mrow></munderover><msub
9597
class="MathClass-ord">u</mi><mrow
9599
class="MathClass-ord">y</mi></mrow></msub
9601
class="text"><mtext >-a.e. in </mtext></mrow><mi class="mathbf">R</mi></mtd><mtd><mspace
9602
id="x1-9001r34" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
9604
<!--l. 983--><p class="nopar"></p></td><td width="5%">(34)</td></tr></table>
9605
for <!--l. 984--><math
9606
xmlns="http://www.w3.org/1998/Math/MathML"
9607
mode="inline"> <msub
9609
class="MathClass-ord"><!--span
9610
class="htf-calligraphy"-->H<!--/span--></mi><mrow
9612
class="MathClass-ord">n</mi><mo
9613
class="MathClass-bin">−</mo><mn
9614
class="MathClass-ord">1</mn></mrow></msub
9616
every <!--l. 984--><math
9617
xmlns="http://www.w3.org/1998/Math/MathML"
9619
class="MathClass-ord">y</mi> <mo
9620
class="MathClass-rel">∈</mo> <msub
9622
class="MathClass-ord">π</mi><mrow
9624
class="MathClass-ord">ν</mi></mrow></msub
9627
href="#x1-3018r16">16<!--tex4ht:ref: sum-ali--></a>) and (<a
9628
href="#x1-3021r18">18<!--tex4ht:ref: delta-l--></a>) we get
9631
xmlns="http://www.w3.org/1998/Math/MathML"
9634
class="multline-star">
9636
class="multline-star"><msub
9638
class="MathClass-op">∫</mo>
9642
class="MathClass-ord">π</mi><mrow
9644
class="MathClass-ord">ν</mi> </mrow></msub
9650
class="MathClass-ord">D</mi></mrow><mrow
9652
></mrow></munderover><msub
9654
class="MathClass-ord">u</mi><mrow
9656
class="MathClass-ord">y</mi></mrow></msub
9660
open="|" close="|" ><munderover
9663
class="MathClass-ord">D</mi></mrow><mrow
9665
></mrow></munderover><msub
9667
class="MathClass-ord">u</mi><mrow
9669
class="MathClass-ord">y</mi></mrow></msub
9670
> </mfenced> </mrow></mfrac> <mo
9671
class="MathClass-punc">·</mo> <mfenced
9672
open="|" close="|" ><munderover
9675
class="MathClass-ord">D</mi></mrow><mrow
9677
></mrow></munderover><msub
9679
class="MathClass-ord">u</mi><mrow
9681
class="MathClass-ord">y</mi></mrow></msub
9683
class="MathClass-ord">d</mi><msub
9685
class="MathClass-ord"><!--span
9686
class="htf-calligraphy"-->H<!--/span--></mi><mrow
9688
class="MathClass-ord">n</mi><mo
9689
class="MathClass-bin">−</mo><mn
9690
class="MathClass-ord">1</mn></mrow></msub
9692
class="MathClass-open">(</mo><mi
9693
class="MathClass-ord">y</mi><mo
9694
class="MathClass-close">)</mo></mrow> <mo
9695
class="MathClass-rel">=</mo><msub
9697
class="MathClass-op">∫</mo>
9701
class="MathClass-ord">π</mi><mrow
9703
class="MathClass-ord">ν</mi> </mrow></msub
9708
class="MathClass-ord">D</mi></mrow><mrow
9710
></mrow></munderover><msub
9712
class="MathClass-ord">u</mi><mrow
9714
class="MathClass-ord">y</mi></mrow></msub
9716
class="MathClass-ord">d</mi><msub
9718
class="MathClass-ord"><!--span
9719
class="htf-calligraphy"-->H<!--/span--></mi><mrow
9721
class="MathClass-ord">n</mi><mo
9722
class="MathClass-bin">−</mo><mn
9723
class="MathClass-ord">1</mn></mrow></msub
9725
class="MathClass-open">(</mo><mi
9726
class="MathClass-ord">y</mi><mo
9727
class="MathClass-close">)</mo></mrow>
9728
</mtd></mtr><mtr><mtd
9729
class="multline-star"> <mo
9730
class="MathClass-rel">=</mo> <mrow><mo
9731
class="MathClass-open">〈</mo><munderover
9734
class="MathClass-ord">D</mi></mrow><mrow
9736
></mrow></munderover><mi
9737
class="MathClass-ord">u</mi><mo
9738
class="MathClass-punc">,</mo> <mi
9739
class="MathClass-ord">ν</mi><mo
9740
class="MathClass-close">〉</mo></mrow> <mo
9741
class="MathClass-rel">=</mo> <mfrac><mrow
9743
class="MathClass-open">〈</mo><munderover
9746
class="MathClass-ord">D</mi></mrow><mrow
9748
></mrow></munderover><mi
9749
class="MathClass-ord">u</mi><mo
9750
class="MathClass-punc">,</mo> <mi
9751
class="MathClass-ord">ν</mi><mo
9752
class="MathClass-close">〉</mo></mrow></mrow>
9755
open="|" close="|" ><mrow><mo
9756
class="MathClass-open">〈</mo><munderover
9759
class="MathClass-ord">D</mi></mrow><mrow
9761
></mrow></munderover><mi
9762
class="MathClass-ord">u</mi><mo
9763
class="MathClass-punc">,</mo> <mi
9764
class="MathClass-ord">ν</mi><mo
9765
class="MathClass-close">〉</mo></mrow></mfenced> </mrow></mfrac> <mo
9766
class="MathClass-punc">·</mo> <mfenced
9767
open="|" close="|" ><mrow><mo
9768
class="MathClass-open">〈</mo><munderover
9771
class="MathClass-ord">D</mi></mrow><mrow
9773
></mrow></munderover><mi
9774
class="MathClass-ord">u</mi><mo
9775
class="MathClass-punc">,</mo> <mi
9776
class="MathClass-ord">ν</mi><mo
9777
class="MathClass-close">〉</mo></mrow></mfenced> <mo
9778
class="MathClass-rel">=</mo><msub
9780
class="MathClass-op">∫</mo>
9784
class="MathClass-ord">π</mi><mrow
9786
class="MathClass-ord">ν</mi> </mrow></msub
9790
class="MathClass-open">〈</mo><munderover
9793
class="MathClass-ord">D</mi></mrow><mrow
9795
></mrow></munderover><mi
9796
class="MathClass-ord">u</mi><mo
9797
class="MathClass-punc">,</mo> <mi
9798
class="MathClass-ord">ν</mi><mo
9799
class="MathClass-close">〉</mo></mrow></mrow>
9802
open="|" close="|" ><mrow><mo
9803
class="MathClass-open">〈</mo><munderover
9806
class="MathClass-ord">D</mi></mrow><mrow
9808
></mrow></munderover><mi
9809
class="MathClass-ord">u</mi><mo
9810
class="MathClass-punc">,</mo> <mi
9811
class="MathClass-ord">ν</mi><mo
9812
class="MathClass-close">〉</mo></mrow></mfenced> </mrow></mfrac><mrow><mo
9813
class="MathClass-open">(</mo><mi
9814
class="MathClass-ord">y</mi> <mo
9815
class="MathClass-bin">+</mo> <mo
9816
class="MathClass-punc">·</mo><mi
9817
class="MathClass-ord">ν</mi><mo
9818
class="MathClass-close">)</mo></mrow> <mo
9819
class="MathClass-punc">·</mo> <mfenced
9820
open="|" close="|" ><munderover
9823
class="MathClass-ord">D</mi></mrow><mrow
9825
></mrow></munderover><msub
9827
class="MathClass-ord">u</mi><mrow
9829
class="MathClass-ord">y</mi></mrow></msub
9831
class="MathClass-ord">d</mi><msub
9833
class="MathClass-ord"><!--span
9834
class="htf-calligraphy"-->H<!--/span--></mi><mrow
9836
class="MathClass-ord">n</mi><mo
9837
class="MathClass-bin">−</mo><mn
9838
class="MathClass-ord">1</mn></mrow></msub
9840
class="MathClass-open">(</mo><mi
9841
class="MathClass-ord">y</mi><mo
9842
class="MathClass-close">)</mo></mrow> </mtd></mtr></mtable>
9844
<!--l. 994--><p class="nopar">
9846
href="#x1-6017r24">24<!--tex4ht:ref: far-d--></a>) follows from (<a
9847
href="#x1-3015r13">13<!--tex4ht:ref: sum-Di--></a>). By the same argument it is possible to prove that
9848
</p><table class="equation"><tr><td>
9850
xmlns="http://www.w3.org/1998/Math/MathML"
9853
class="equation"><mtr><mtd>
9856
class="MathClass-open">〈</mo><munderover
9859
class="MathClass-ord">D</mi></mrow><mrow
9861
></mrow></munderover><mi
9862
class="MathClass-ord">v</mi><mo
9863
class="MathClass-punc">,</mo> <mi
9864
class="MathClass-ord">ν</mi><mo
9865
class="MathClass-close">〉</mo></mrow></mrow>
9868
open="|" close="|" ><mrow><mo
9869
class="MathClass-open">〈</mo><munderover
9872
class="MathClass-ord">D</mi></mrow><mrow
9874
></mrow></munderover><mi
9875
class="MathClass-ord">u</mi><mo
9876
class="MathClass-punc">,</mo> <mi
9877
class="MathClass-ord">ν</mi><mo
9878
class="MathClass-close">〉</mo></mrow></mfenced> </mrow></mfrac><mrow><mo
9879
class="MathClass-open">(</mo><mi
9880
class="MathClass-ord">y</mi> <mo
9881
class="MathClass-bin">+</mo> <mi
9882
class="MathClass-ord">t</mi><mi
9883
class="MathClass-ord">ν</mi><mo
9884
class="MathClass-close">)</mo></mrow> <mo
9885
class="MathClass-rel">=</mo> <mfrac><mrow
9889
class="MathClass-ord">D</mi></mrow><mrow
9891
></mrow></munderover><msub
9893
class="MathClass-ord">v</mi><mrow
9895
class="MathClass-ord">y</mi></mrow></msub
9899
open="|" close="|" ><munderover
9902
class="MathClass-ord">D</mi></mrow><mrow
9904
></mrow></munderover><msub
9906
class="MathClass-ord">u</mi><mrow
9908
class="MathClass-ord">y</mi></mrow></msub
9909
> </mfenced> </mrow></mfrac> <mrow><mo
9910
class="MathClass-open">(</mo><mi
9911
class="MathClass-ord">t</mi><mo
9912
class="MathClass-close">)</mo></mrow><mspace width="2em" class="qquad"/> <mfenced
9913
open="|" close="|" ><munderover
9916
class="MathClass-ord">D</mi></mrow><mrow
9918
></mrow></munderover><msub
9920
class="MathClass-ord">u</mi><mrow
9922
class="MathClass-ord">y</mi></mrow></msub
9924
class="text"><mtext >-a.e. in </mtext></mrow><mi class="mathbf">R</mi></mtd><mtd><mspace
9925
id="x1-9002r35" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
9927
<!--l. 1000--><p class="nopar"></p></td><td width="5%">(35)</td></tr></table>
9928
for <!--l. 1001--><math
9929
xmlns="http://www.w3.org/1998/Math/MathML"
9930
mode="inline"> <msub
9932
class="MathClass-ord"><!--span
9933
class="htf-calligraphy"-->H<!--/span--></mi><mrow
9935
class="MathClass-ord">n</mi><mo
9936
class="MathClass-bin">−</mo><mn
9937
class="MathClass-ord">1</mn></mrow></msub
9938
></math>-almost every <!--l. 1001--><math
9939
xmlns="http://www.w3.org/1998/Math/MathML"
9942
class="MathClass-ord">y</mi> <mo
9943
class="MathClass-rel">∈</mo> <msub
9945
class="MathClass-ord">π</mi><mrow
9947
class="MathClass-ord">ν</mi></mrow></msub
9949
href="#x1-6017r24">24<!--tex4ht:ref: far-d--></a>) and (<a
9950
href="#x1-6019r25">25<!--tex4ht:ref: E_SXgYy--></a>)
9952
we get <!--l. 1003--><math
9953
xmlns="http://www.w3.org/1998/Math/MathML"
9954
mode="display"> <mrow
9960
class="MathClass-ord">h</mi><mo
9961
class="MathClass-rel">→</mo><mn
9962
class="MathClass-ord">0</mn></mrow></msub
9965
class="MathClass-ord">f</mi><mrow><mo
9966
class="MathClass-open">(</mo><mi
9967
class="MathClass-ord">ũ</mi><mrow><mo
9968
class="MathClass-open">(</mo><mi
9969
class="MathClass-ord">y</mi> <mo
9970
class="MathClass-bin">+</mo> <mi
9971
class="MathClass-ord">t</mi><mi
9972
class="MathClass-ord">ν</mi><mo
9973
class="MathClass-close">)</mo></mrow> <mo
9974
class="MathClass-bin">+</mo> <mi
9975
class="MathClass-ord">h</mi><mfrac class="dfrac"><mrow><mrow><mo
9976
class="MathClass-open">〈</mo><munderover
9979
class="MathClass-ord">D</mi></mrow><mrow
9981
></mrow></munderover><mi
9982
class="MathClass-ord">u</mi><mo
9983
class="MathClass-punc">,</mo> <mi
9984
class="MathClass-ord">ν</mi><mo
9985
class="MathClass-close">〉</mo></mrow></mrow><mrow><mfenced
9986
open="|" close="|" ><mrow><mo
9987
class="MathClass-open">〈</mo><munderover
9990
class="MathClass-ord">D</mi></mrow><mrow
9992
></mrow></munderover><mi
9993
class="MathClass-ord">u</mi><mo
9994
class="MathClass-punc">,</mo> <mi
9995
class="MathClass-ord">ν</mi><mo
9996
class="MathClass-close">〉</mo></mrow></mfenced></mrow></mfrac><mrow><mo
9997
class="MathClass-open">(</mo><mi
9998
class="MathClass-ord">y</mi> <mo
9999
class="MathClass-bin">+</mo> <mi
10000
class="MathClass-ord">t</mi><mi
10001
class="MathClass-ord">ν</mi><mo
10002
class="MathClass-close">)</mo></mrow><mo
10003
class="MathClass-close">)</mo></mrow> <mo
10004
class="MathClass-bin">−</mo> <mi
10005
class="MathClass-ord">f</mi><mrow><mo
10006
class="MathClass-open">(</mo><mi
10007
class="MathClass-ord">ũ</mi><mrow><mo
10008
class="MathClass-open">(</mo><mi
10009
class="MathClass-ord">y</mi> <mo
10010
class="MathClass-bin">+</mo> <mi
10011
class="MathClass-ord">t</mi><mi
10012
class="MathClass-ord">ν</mi><mo
10013
class="MathClass-close">)</mo></mrow><mo
10014
class="MathClass-close">)</mo></mrow></mrow>
10017
class="MathClass-ord">h</mi></mrow></mfrac> <mo
10018
class="MathClass-rel">=</mo> <mfrac><mrow
10020
class="MathClass-open">〈</mo><munderover
10021
accent="true"><mrow
10023
class="MathClass-ord">D</mi></mrow><mrow
10025
></mrow></munderover><mi
10026
class="MathClass-ord">v</mi><mo
10027
class="MathClass-punc">,</mo> <mi
10028
class="MathClass-ord">ν</mi><mo
10029
class="MathClass-close">〉</mo></mrow></mrow>
10032
open="|" close="|" ><mrow><mo
10033
class="MathClass-open">〈</mo><munderover
10034
accent="true"><mrow
10036
class="MathClass-ord">D</mi></mrow><mrow
10038
></mrow></munderover><mi
10039
class="MathClass-ord">u</mi><mo
10040
class="MathClass-punc">,</mo> <mi
10041
class="MathClass-ord">ν</mi><mo
10042
class="MathClass-close">〉</mo></mrow></mfenced> </mrow></mfrac><mrow><mo
10043
class="MathClass-open">(</mo><mi
10044
class="MathClass-ord">y</mi><mo
10045
class="MathClass-bin">+</mo><mi
10046
class="MathClass-ord">t</mi><mi
10047
class="MathClass-ord">ν</mi><mo
10048
class="MathClass-close">)</mo></mrow>
10049
</mrow></math> for <!--l. 1009--><math
10050
xmlns="http://www.w3.org/1998/Math/MathML"
10054
class="MathClass-ord"><!--span
10055
class="htf-calligraphy"-->H<!--/span--></mi><mrow
10057
class="MathClass-ord">n</mi><mo
10058
class="MathClass-bin">−</mo><mn
10059
class="MathClass-ord">1</mn></mrow></msub
10060
></math>-almost every <!--l. 1009--><math
10061
xmlns="http://www.w3.org/1998/Math/MathML"
10064
class="MathClass-ord">y</mi> <mo
10065
class="MathClass-rel">∈</mo> <msub
10067
class="MathClass-ord">π</mi><mrow
10069
class="MathClass-ord">ν</mi></mrow></msub
10070
></math>, and using again (<a
10071
href="#x1-3016r14">14<!--tex4ht:ref: detK1--></a>),
10073
href="#x1-3017r15">15<!--tex4ht:ref: detK2--></a>) we get <!--l. 1011--><math
10074
xmlns="http://www.w3.org/1998/Math/MathML"
10075
mode="display"> <mrow
10081
class="MathClass-ord">h</mi><mo
10082
class="MathClass-rel">→</mo><mn
10083
class="MathClass-ord">0</mn></mrow></msub
10086
class="MathClass-ord">f</mi><mrow><mo
10087
class="MathClass-open">(</mo><mi
10088
class="MathClass-ord">ũ</mi><mrow><mo
10089
class="MathClass-open">(</mo><mi
10090
class="MathClass-ord">x</mi><mo
10091
class="MathClass-close">)</mo></mrow> <mo
10092
class="MathClass-bin">+</mo> <mi
10093
class="MathClass-ord">h</mi><mfrac class="dfrac"><mrow><mrow><mo
10094
class="MathClass-open">〈</mo><munderover
10095
accent="true"><mrow
10097
class="MathClass-ord">D</mi></mrow><mrow
10099
></mrow></munderover><mi
10100
class="MathClass-ord">u</mi><mo
10101
class="MathClass-punc">,</mo> <mi
10102
class="MathClass-ord">ν</mi><mo
10103
class="MathClass-close">〉</mo></mrow></mrow><mrow><mfenced
10104
open="|" close="|" ><mrow><mo
10105
class="MathClass-open">〈</mo><munderover
10106
accent="true"><mrow
10108
class="MathClass-ord">D</mi></mrow><mrow
10110
></mrow></munderover><mi
10111
class="MathClass-ord">u</mi><mo
10112
class="MathClass-punc">,</mo> <mi
10113
class="MathClass-ord">ν</mi><mo
10114
class="MathClass-close">〉</mo></mrow></mfenced></mrow></mfrac><mrow><mo
10115
class="MathClass-open">(</mo><mi
10116
class="MathClass-ord">x</mi><mo
10117
class="MathClass-close">)</mo></mrow><mo
10118
class="MathClass-close">)</mo></mrow> <mo
10119
class="MathClass-bin">−</mo> <mi
10120
class="MathClass-ord">f</mi><mrow><mo
10121
class="MathClass-open">(</mo><mi
10122
class="MathClass-ord">ũ</mi><mrow><mo
10123
class="MathClass-open">(</mo><mi
10124
class="MathClass-ord">x</mi><mo
10125
class="MathClass-close">)</mo></mrow><mo
10126
class="MathClass-close">)</mo></mrow></mrow>
10129
class="MathClass-ord">h</mi></mrow></mfrac> <mo
10130
class="MathClass-rel">=</mo> <mfrac><mrow
10132
class="MathClass-open">〈</mo><munderover
10133
accent="true"><mrow
10135
class="MathClass-ord">D</mi></mrow><mrow
10137
></mrow></munderover><mi
10138
class="MathClass-ord">v</mi><mo
10139
class="MathClass-punc">,</mo> <mi
10140
class="MathClass-ord">ν</mi><mo
10141
class="MathClass-close">〉</mo></mrow></mrow>
10144
open="|" close="|" ><mrow><mo
10145
class="MathClass-open">〈</mo><munderover
10146
accent="true"><mrow
10148
class="MathClass-ord">D</mi></mrow><mrow
10150
></mrow></munderover><mi
10151
class="MathClass-ord">u</mi><mo
10152
class="MathClass-punc">,</mo> <mi
10153
class="MathClass-ord">ν</mi><mo
10154
class="MathClass-close">〉</mo></mrow></mfenced> </mrow></mfrac><mrow><mo
10155
class="MathClass-open">(</mo><mi
10156
class="MathClass-ord">x</mi><mo
10157
class="MathClass-close">)</mo></mrow>
10159
<!--l. 1017--><math
10160
xmlns="http://www.w3.org/1998/Math/MathML"
10161
mode="inline"> <mfenced
10162
open="|" close="|" ><mrow><mo
10163
class="MathClass-open">〈</mo><munderover
10164
accent="true"><mrow
10166
class="MathClass-ord">D</mi></mrow><mrow
10168
></mrow></munderover><mi
10169
class="MathClass-ord">u</mi><mo
10170
class="MathClass-punc">,</mo> <mi
10171
class="MathClass-ord">ν</mi><mo
10172
class="MathClass-close">〉</mo></mrow></mfenced></math>-a.e.
10173
in <!--l. 1017--><math
10174
xmlns="http://www.w3.org/1998/Math/MathML"
10175
mode="inline"> <msup
10176
><mi class="mathbf">R</mi><mrow
10178
class="MathClass-ord">n</mi></mrow></msup
10180
<!--l. 1019--><p class="indent"> Since the function <!--l. 1019--><math
10181
xmlns="http://www.w3.org/1998/Math/MathML"
10182
mode="inline"> <mfenced
10183
open="|" close="|" ><mrow><mo
10184
class="MathClass-open">〈</mo><munderover
10185
accent="true"><mrow
10187
class="MathClass-ord">D</mi></mrow><mrow
10189
></mrow></munderover><mi
10190
class="MathClass-ord">u</mi><mo
10191
class="MathClass-punc">,</mo> <mi
10192
class="MathClass-ord">ν</mi><mo
10193
class="MathClass-close">〉</mo></mrow></mfenced> <mo
10194
class="MathClass-bin">/</mo> <mfenced
10195
open="|" close="|" ><munderover
10196
accent="true"><mrow
10198
class="MathClass-ord">D</mi></mrow><mrow
10200
></mrow></munderover><mi
10201
class="MathClass-ord">u</mi></mfenced></math> is
10202
strictly positive <!--l. 1020--><math
10203
xmlns="http://www.w3.org/1998/Math/MathML"
10204
mode="inline"> <mfenced
10205
open="|" close="|" ><mrow><mo
10206
class="MathClass-open">〈</mo><munderover
10207
accent="true"><mrow
10209
class="MathClass-ord">D</mi></mrow><mrow
10211
></mrow></munderover><mi
10212
class="MathClass-ord">u</mi><mo
10213
class="MathClass-punc">,</mo> <mi
10214
class="MathClass-ord">ν</mi><mo
10215
class="MathClass-close">〉</mo></mrow></mfenced></math>-almost
10216
everywhere, we obtain also
10218
</p><!--l. 1022--><math
10219
xmlns="http://www.w3.org/1998/Math/MathML"
10222
class="multline-star">
10224
class="multline-star"><msub
10228
class="MathClass-ord">h</mi><mo
10229
class="MathClass-rel">→</mo><mn
10230
class="MathClass-ord">0</mn></mrow></msub
10233
class="MathClass-ord">f</mi><mrow><mo
10234
class="MathClass-open">(</mo><mi
10235
class="MathClass-ord">ũ</mi><mrow><mo
10236
class="MathClass-open">(</mo><mi
10237
class="MathClass-ord">x</mi><mo
10238
class="MathClass-close">)</mo></mrow> <mo
10239
class="MathClass-bin">+</mo> <mi
10240
class="MathClass-ord">h</mi><mfrac class="dfrac"><mrow><mfenced
10241
open="|" close="|" ><mrow><mo
10242
class="MathClass-open">〈</mo><munderover
10243
accent="true"><mrow
10245
class="MathClass-ord">D</mi></mrow><mrow
10247
></mrow></munderover><mi
10248
class="MathClass-ord">u</mi><mo
10249
class="MathClass-punc">,</mo> <mi
10250
class="MathClass-ord">ν</mi><mo
10251
class="MathClass-close">〉</mo></mrow></mfenced></mrow><mrow><mfenced
10252
open="|" close="|" ><munderover
10253
accent="true"><mrow
10255
class="MathClass-ord">D</mi></mrow><mrow
10257
></mrow></munderover><mi
10258
class="MathClass-ord">u</mi></mfenced></mrow></mfrac><mrow><mo
10259
class="MathClass-open">(</mo><mi
10260
class="MathClass-ord">x</mi><mo
10261
class="MathClass-close">)</mo></mrow><mfrac class="dfrac"><mrow><mrow><mo
10262
class="MathClass-open">〈</mo><munderover
10263
accent="true"><mrow
10265
class="MathClass-ord">D</mi></mrow><mrow
10267
></mrow></munderover><mi
10268
class="MathClass-ord">u</mi><mo
10269
class="MathClass-punc">,</mo> <mi
10270
class="MathClass-ord">ν</mi><mo
10271
class="MathClass-close">〉</mo></mrow></mrow><mrow><mfenced
10272
open="|" close="|" ><mrow><mo
10273
class="MathClass-open">〈</mo><munderover
10274
accent="true"><mrow
10276
class="MathClass-ord">D</mi></mrow><mrow
10278
></mrow></munderover><mi
10279
class="MathClass-ord">u</mi><mo
10280
class="MathClass-punc">,</mo> <mi
10281
class="MathClass-ord">ν</mi><mo
10282
class="MathClass-close">〉</mo></mrow></mfenced></mrow></mfrac><mrow><mo
10283
class="MathClass-open">(</mo><mi
10284
class="MathClass-ord">x</mi><mo
10285
class="MathClass-close">)</mo></mrow><mo
10286
class="MathClass-close">)</mo></mrow> <mo
10287
class="MathClass-bin">−</mo> <mi
10288
class="MathClass-ord">f</mi><mrow><mo
10289
class="MathClass-open">(</mo><mi
10290
class="MathClass-ord">ũ</mi><mrow><mo
10291
class="MathClass-open">(</mo><mi
10292
class="MathClass-ord">x</mi><mo
10293
class="MathClass-close">)</mo></mrow><mo
10294
class="MathClass-close">)</mo></mrow></mrow>
10297
class="MathClass-ord">h</mi></mrow></mfrac>
10298
</mtd></mtr><mtr><mtd
10299
class="multline-star"> <mo
10300
class="MathClass-rel">=</mo> <mfrac><mrow
10302
open="|" close="|" ><mrow><mo
10303
class="MathClass-open">〈</mo><munderover
10304
accent="true"><mrow
10306
class="MathClass-ord">D</mi></mrow><mrow
10308
></mrow></munderover><mi
10309
class="MathClass-ord">u</mi><mo
10310
class="MathClass-punc">,</mo> <mi
10311
class="MathClass-ord">ν</mi><mo
10312
class="MathClass-close">〉</mo></mrow></mfenced> </mrow>
10315
open="|" close="|" ><munderover
10316
accent="true"><mrow
10318
class="MathClass-ord">D</mi></mrow><mrow
10320
></mrow></munderover><mi
10321
class="MathClass-ord">u</mi></mfenced> </mrow></mfrac> <mrow><mo
10322
class="MathClass-open">(</mo><mi
10323
class="MathClass-ord">x</mi><mo
10324
class="MathClass-close">)</mo></mrow> <mfrac><mrow
10326
class="MathClass-open">〈</mo><munderover
10327
accent="true"><mrow
10329
class="MathClass-ord">D</mi></mrow><mrow
10331
></mrow></munderover><mi
10332
class="MathClass-ord">v</mi><mo
10333
class="MathClass-punc">,</mo> <mi
10334
class="MathClass-ord">ν</mi><mo
10335
class="MathClass-close">〉</mo></mrow></mrow>
10338
open="|" close="|" ><mrow><mo
10339
class="MathClass-open">〈</mo><munderover
10340
accent="true"><mrow
10342
class="MathClass-ord">D</mi></mrow><mrow
10344
></mrow></munderover><mi
10345
class="MathClass-ord">u</mi><mo
10346
class="MathClass-punc">,</mo> <mi
10347
class="MathClass-ord">ν</mi><mo
10348
class="MathClass-close">〉</mo></mrow></mfenced> </mrow></mfrac><mrow><mo
10349
class="MathClass-open">(</mo><mi
10350
class="MathClass-ord">x</mi><mo
10351
class="MathClass-close">)</mo></mrow> </mtd></mtr></mtable>
10353
<!--l. 1029--><p class="nopar">
10354
<!--l. 1030--><math
10355
xmlns="http://www.w3.org/1998/Math/MathML"
10356
mode="inline"> <mfenced
10357
open="|" close="|" ><mrow><mo
10358
class="MathClass-open">〈</mo><munderover
10359
accent="true"><mrow
10361
class="MathClass-ord">D</mi></mrow><mrow
10363
></mrow></munderover><mi
10364
class="MathClass-ord">u</mi><mo
10365
class="MathClass-punc">,</mo> <mi
10366
class="MathClass-ord">ν</mi><mo
10367
class="MathClass-close">〉</mo></mrow></mfenced></math>-almost
10368
everywhere in <!--l. 1030--><math
10369
xmlns="http://www.w3.org/1998/Math/MathML"
10370
mode="inline"> <msup
10371
><mi class="mathbf">R</mi><mrow
10373
class="MathClass-ord">n</mi></mrow></msup
10375
</p><!--l. 1032--><p class="indent"> Finally, since
10376
<!--tex4ht:inline--></p><!--l. 1044--><math
10377
xmlns="http://www.w3.org/1998/Math/MathML"
10378
display="block"><mtable
10379
class="align-star">
10381
class="align-odd"></mtd> <mtd
10382
class="align-even"><mfrac><mrow
10384
open="|" close="|" ><mrow><mo
10385
class="MathClass-open">〈</mo><munderover
10386
accent="true"><mrow
10388
class="MathClass-ord">D</mi></mrow><mrow
10390
></mrow></munderover><mi
10391
class="MathClass-ord">u</mi><mo
10392
class="MathClass-punc">,</mo> <mi
10393
class="MathClass-ord">ν</mi><mo
10394
class="MathClass-close">〉</mo></mrow></mfenced> </mrow>
10397
open="|" close="|" ><munderover
10398
accent="true"><mrow
10400
class="MathClass-ord">D</mi></mrow><mrow
10402
></mrow></munderover><mi
10403
class="MathClass-ord">u</mi></mfenced> </mrow></mfrac> <mfrac><mrow
10405
class="MathClass-open">〈</mo><munderover
10406
accent="true"><mrow
10408
class="MathClass-ord">D</mi></mrow><mrow
10410
></mrow></munderover><mi
10411
class="MathClass-ord">u</mi><mo
10412
class="MathClass-punc">,</mo> <mi
10413
class="MathClass-ord">ν</mi><mo
10414
class="MathClass-close">〉</mo></mrow></mrow>
10417
open="|" close="|" ><mrow><mo
10418
class="MathClass-open">〈</mo><munderover
10419
accent="true"><mrow
10421
class="MathClass-ord">D</mi></mrow><mrow
10423
></mrow></munderover><mi
10424
class="MathClass-ord">u</mi><mo
10425
class="MathClass-punc">,</mo> <mi
10426
class="MathClass-ord">ν</mi><mo
10427
class="MathClass-close">〉</mo></mrow></mfenced> </mrow></mfrac> <mo
10428
class="MathClass-rel">=</mo> <mfrac><mrow
10430
class="MathClass-open">〈</mo><munderover
10431
accent="true"><mrow
10433
class="MathClass-ord">D</mi></mrow><mrow
10435
></mrow></munderover><mi
10436
class="MathClass-ord">u</mi><mo
10437
class="MathClass-punc">,</mo> <mi
10438
class="MathClass-ord">ν</mi><mo
10439
class="MathClass-close">〉</mo></mrow></mrow>
10442
open="|" close="|" ><munderover
10443
accent="true"><mrow
10445
class="MathClass-ord">D</mi></mrow><mrow
10447
></mrow></munderover><mi
10448
class="MathClass-ord">u</mi></mfenced> </mrow></mfrac> <mo
10449
class="MathClass-rel">=</mo> <mfenced
10450
open="〈" close="〉" > <mfrac><mrow
10452
accent="true"><mrow
10454
class="MathClass-ord">D</mi></mrow><mrow
10456
></mrow></munderover><mi
10457
class="MathClass-ord">u</mi></mrow>
10460
open="|" close="|" ><munderover
10461
accent="true"><mrow
10463
class="MathClass-ord">D</mi></mrow><mrow
10465
></mrow></munderover><mi
10466
class="MathClass-ord">u</mi></mfenced> </mrow></mfrac><mo
10467
class="MathClass-punc">,</mo> <mi
10468
class="MathClass-ord">ν</mi></mfenced> <mspace width="2em" class="qquad"/> <mfenced
10469
open="|" close="|" ><munderover
10470
accent="true"><mrow
10472
class="MathClass-ord">D</mi></mrow><mrow
10474
></mrow></munderover><mi
10475
class="MathClass-ord">u</mi></mfenced> <mrow
10476
class="text"><mtext >-a.e. in </mtext></mrow><msup
10477
><mi class="mathbf">R</mi><mrow
10479
class="MathClass-ord">n</mi></mrow></msup
10482
class="align-odd"></mtd> <mtd
10483
class="align-even"> <mfrac><mrow
10485
open="|" close="|" ><mrow><mo
10486
class="MathClass-open">〈</mo><munderover
10487
accent="true"><mrow
10489
class="MathClass-ord">D</mi></mrow><mrow
10491
></mrow></munderover><mi
10492
class="MathClass-ord">u</mi><mo
10493
class="MathClass-punc">,</mo> <mi
10494
class="MathClass-ord">ν</mi><mo
10495
class="MathClass-close">〉</mo></mrow></mfenced> </mrow>
10498
open="|" close="|" ><munderover
10499
accent="true"><mrow
10501
class="MathClass-ord">D</mi></mrow><mrow
10503
></mrow></munderover><mi
10504
class="MathClass-ord">u</mi></mfenced> </mrow></mfrac> <mfrac><mrow
10506
class="MathClass-open">〈</mo><munderover
10507
accent="true"><mrow
10509
class="MathClass-ord">D</mi></mrow><mrow
10511
></mrow></munderover><mi
10512
class="MathClass-ord">v</mi><mo
10513
class="MathClass-punc">,</mo> <mi
10514
class="MathClass-ord">ν</mi><mo
10515
class="MathClass-close">〉</mo></mrow></mrow>
10518
open="|" close="|" ><mrow><mo
10519
class="MathClass-open">〈</mo><munderover
10520
accent="true"><mrow
10522
class="MathClass-ord">D</mi></mrow><mrow
10524
></mrow></munderover><mi
10525
class="MathClass-ord">u</mi><mo
10526
class="MathClass-punc">,</mo> <mi
10527
class="MathClass-ord">ν</mi><mo
10528
class="MathClass-close">〉</mo></mrow></mfenced> </mrow></mfrac> <mo
10529
class="MathClass-rel">=</mo> <mfrac><mrow
10531
class="MathClass-open">〈</mo><munderover
10532
accent="true"><mrow
10534
class="MathClass-ord">D</mi></mrow><mrow
10536
></mrow></munderover><mi
10537
class="MathClass-ord">v</mi><mo
10538
class="MathClass-punc">,</mo> <mi
10539
class="MathClass-ord">ν</mi><mo
10540
class="MathClass-close">〉</mo></mrow></mrow>
10543
open="|" close="|" ><munderover
10544
accent="true"><mrow
10546
class="MathClass-ord">D</mi></mrow><mrow
10548
></mrow></munderover><mi
10549
class="MathClass-ord">u</mi></mfenced> </mrow></mfrac> <mo
10550
class="MathClass-rel">=</mo> <mfenced
10551
open="〈" close="〉" > <mfrac><mrow
10553
accent="true"><mrow
10555
class="MathClass-ord">D</mi></mrow><mrow
10557
></mrow></munderover><mi
10558
class="MathClass-ord">v</mi></mrow>
10561
open="|" close="|" ><munderover
10562
accent="true"><mrow
10564
class="MathClass-ord">D</mi></mrow><mrow
10566
></mrow></munderover><mi
10567
class="MathClass-ord">u</mi></mfenced> </mrow></mfrac><mo
10568
class="MathClass-punc">,</mo> <mi
10569
class="MathClass-ord">ν</mi></mfenced> <mspace width="2em" class="qquad"/> <mfenced
10570
open="|" close="|" ><munderover
10571
accent="true"><mrow
10573
class="MathClass-ord">D</mi></mrow><mrow
10575
></mrow></munderover><mi
10576
class="MathClass-ord">u</mi></mfenced> <mrow
10577
class="text"><mtext >-a.e. in </mtext></mrow><msup
10578
><mi class="mathbf">R</mi><mrow
10580
class="MathClass-ord">n</mi></mrow></msup
10582
</mtr></mtable></math>
10583
and since both sides of (<a
10584
href="#x1-8002r33">33<!--tex4ht:ref: alimo--></a>) are zero <!--l. 1046--><math
10585
xmlns="http://www.w3.org/1998/Math/MathML"
10588
open="|" close="|" ><munderover
10589
accent="true"><mrow
10591
class="MathClass-ord">D</mi></mrow><mrow
10593
></mrow></munderover><mi
10594
class="MathClass-ord">u</mi></mfenced></math>-almost everywhere on <!--l. 1047--><math
10595
xmlns="http://www.w3.org/1998/Math/MathML"
10598
open="|" close="|" ><mrow><mo
10599
class="MathClass-open">〈</mo><munderover
10600
accent="true"><mrow
10602
class="MathClass-ord">D</mi></mrow><mrow
10604
></mrow></munderover><mi
10605
class="MathClass-ord">u</mi><mo
10606
class="MathClass-punc">,</mo> <mi
10607
class="MathClass-ord">ν</mi><mo
10608
class="MathClass-close">〉</mo></mrow></mfenced></math>-negligible sets, we
10610
conclude that <!--l. 1048--><math
10611
xmlns="http://www.w3.org/1998/Math/MathML"
10612
mode="display"> <mrow
10618
class="MathClass-ord">h</mi><mo
10619
class="MathClass-rel">→</mo><mn
10620
class="MathClass-ord">0</mn></mrow></msub
10623
class="MathClass-ord">f</mi> <mfenced
10624
open="(" close=")" ><mi
10625
class="MathClass-ord">ũ</mi><mrow><mo
10626
class="MathClass-open">(</mo><mi
10627
class="MathClass-ord">x</mi><mo
10628
class="MathClass-close">)</mo></mrow> <mo
10629
class="MathClass-bin">+</mo> <mi
10630
class="MathClass-ord">h</mi> <mfenced
10631
open="〈" close="〉" ><mfrac class="dfrac"><mrow><munderover
10632
accent="true"><mrow
10634
class="MathClass-ord">D</mi></mrow><mrow
10636
></mrow></munderover><mi
10637
class="MathClass-ord">u</mi></mrow><mrow><mfenced
10638
open="|" close="|" ><munderover
10639
accent="true"><mrow
10641
class="MathClass-ord">D</mi></mrow><mrow
10643
></mrow></munderover><mi
10644
class="MathClass-ord">u</mi></mfenced></mrow></mfrac><mrow><mo
10645
class="MathClass-open">(</mo><mi
10646
class="MathClass-ord">x</mi><mo
10647
class="MathClass-close">)</mo></mrow><mo
10648
class="MathClass-punc">,</mo> <mi
10649
class="MathClass-ord">ν</mi></mfenced></mfenced> <mo
10650
class="MathClass-bin">−</mo> <mi
10651
class="MathClass-ord">f</mi><mrow><mo
10652
class="MathClass-open">(</mo><mi
10653
class="MathClass-ord">ũ</mi><mrow><mo
10654
class="MathClass-open">(</mo><mi
10655
class="MathClass-ord">x</mi><mo
10656
class="MathClass-close">)</mo></mrow><mo
10657
class="MathClass-close">)</mo></mrow></mrow>
10660
class="MathClass-ord">h</mi></mrow></mfrac> <mo
10661
class="MathClass-rel">=</mo> <mfenced
10662
open="〈" close="〉" > <mfrac><mrow
10664
accent="true"><mrow
10666
class="MathClass-ord">D</mi></mrow><mrow
10668
></mrow></munderover><mi
10669
class="MathClass-ord">v</mi></mrow>
10672
open="|" close="|" ><munderover
10673
accent="true"><mrow
10675
class="MathClass-ord">D</mi></mrow><mrow
10677
></mrow></munderover><mi
10678
class="MathClass-ord">u</mi></mfenced> </mrow></mfrac><mrow><mo
10679
class="MathClass-open">(</mo><mi
10680
class="MathClass-ord">x</mi><mo
10681
class="MathClass-close">)</mo></mrow><mo
10682
class="MathClass-punc">,</mo> <mi
10683
class="MathClass-ord">ν</mi></mfenced> <mo
10684
class="MathClass-punc">,</mo>
10686
<!--l. 1054--><math
10687
xmlns="http://www.w3.org/1998/Math/MathML"
10688
mode="inline"> <mfenced
10689
open="|" close="|" ><munderover
10690
accent="true"><mrow
10692
class="MathClass-ord">D</mi></mrow><mrow
10694
></mrow></munderover><mi
10695
class="MathClass-ord">u</mi></mfenced></math>-a.e.
10696
in <!--l. 1054--><math
10697
xmlns="http://www.w3.org/1998/Math/MathML"
10698
mode="inline"> <msup
10699
><mi class="mathbf">R</mi><mrow
10701
class="MathClass-ord">n</mi></mrow></msup
10703
Since <!--l. 1055--><math
10704
xmlns="http://www.w3.org/1998/Math/MathML"
10706
class="MathClass-ord">ν</mi></math>
10707
is arbitrary, by Remarks <a
10708
href="#x1-7006r2">7.2<!--tex4ht:ref: r:dif--></a> and <a
10709
href="#x1-7007r3">7.3<!--tex4ht:ref: r:dif0--></a> the restriction of <!--l. 1056--><math
10710
xmlns="http://www.w3.org/1998/Math/MathML"
10713
class="MathClass-ord">f</mi></math> to the affine space <!--l. 1057--><math
10714
xmlns="http://www.w3.org/1998/Math/MathML"
10718
class="MathClass-ord">T</mi><mrow
10720
class="MathClass-ord">x</mi></mrow><mrow
10722
class="MathClass-ord">u</mi></mrow></msubsup
10723
></math> is differentiable
10724
at <!--l. 1057--><math
10725
xmlns="http://www.w3.org/1998/Math/MathML"
10727
class="MathClass-ord">ũ</mi><mrow><mo
10728
class="MathClass-open">(</mo><mi
10729
class="MathClass-ord">x</mi><mo
10730
class="MathClass-close">)</mo></mrow></math> for <!--l. 1057--><math
10731
xmlns="http://www.w3.org/1998/Math/MathML"
10734
open="|" close="|" ><munderover
10735
accent="true"><mrow
10737
class="MathClass-ord">D</mi></mrow><mrow
10739
></mrow></munderover><mi
10740
class="MathClass-ord">u</mi></mfenced></math>-almost
10741
every <!--l. 1058--><math
10742
xmlns="http://www.w3.org/1998/Math/MathML"
10744
class="MathClass-ord">x</mi> <mo
10745
class="MathClass-rel">∈</mo><msup
10746
> <mi class="mathbf">R</mi><mrow
10748
class="MathClass-ord">n</mi></mrow></msup
10751
href="#x1-7002r26">26<!--tex4ht:ref: quts--></a>) holds. □
10752
It follows from (<a
10753
href="#x1-3015r13">13<!--tex4ht:ref: sum-Di--></a>), (<a
10754
href="#x1-3016r14">14<!--tex4ht:ref: detK1--></a>), and (<a
10755
href="#x1-3017r15">15<!--tex4ht:ref: detK2--></a>) that <table class="equation"><tr><td>
10756
<!--l. 1061--><math
10757
xmlns="http://www.w3.org/1998/Math/MathML"
10760
class="equation"><mtr><mtd>
10762
class="MathClass-ord">D</mi><mrow><mo
10763
class="MathClass-open">(</mo><msub
10765
class="MathClass-ord">t</mi><mrow
10767
class="MathClass-ord">1</mn></mrow></msub
10769
class="MathClass-punc">,</mo> <mo
10770
class="MathClass-op">…</mo><mo
10771
class="MathClass-punc">,</mo> <msub
10773
class="MathClass-ord">t</mi><mrow
10775
class="MathClass-ord">n</mi></mrow></msub
10777
class="MathClass-close">)</mo></mrow> <mo
10778
class="MathClass-rel">=</mo><msub
10780
class="MathClass-op">∑</mo>
10783
class="MathClass-ord">I</mi><mo
10784
class="MathClass-rel">∈</mo><mi class="mathbf">n</mi></mrow></msub
10787
class="MathClass-open">(</mo><mo
10788
class="MathClass-bin">−</mo><mn
10789
class="MathClass-ord">1</mn><mo
10790
class="MathClass-close">)</mo></mrow><mrow
10792
open="|" close="|" ><mi
10793
class="MathClass-ord">I</mi> </mfenced><mo
10794
class="MathClass-bin">−</mo><mn
10795
class="MathClass-ord">1</mn></mrow></msup
10797
open="|" close="|" ><mi
10798
class="MathClass-ord">I</mi></mfenced><msub
10800
class="MathClass-op">∏</mo>
10803
class="MathClass-ord">i</mi><mo
10804
class="MathClass-rel">∈</mo><mi
10805
class="MathClass-ord">I</mi></mrow></msub
10808
class="MathClass-ord">t</mi><mrow
10810
class="MathClass-ord">i</mi></mrow></msub
10813
class="MathClass-op">∏</mo>
10816
class="MathClass-ord">j</mi><mo
10817
class="MathClass-rel">∈</mo><mi
10818
class="MathClass-ord">I</mi></mrow></msub
10820
class="MathClass-open">(</mo><msub
10822
class="MathClass-ord">D</mi><mrow
10824
class="MathClass-ord">j</mi></mrow></msub
10826
class="MathClass-bin">+</mo> <msub
10828
class="MathClass-ord">λ</mi><mrow
10830
class="MathClass-ord">j</mi></mrow></msub
10833
class="MathClass-ord">t</mi><mrow
10835
class="MathClass-ord">j</mi></mrow></msub
10837
class="MathClass-close">)</mo></mrow><mo
10839
> <mi class="mathbf">A</mi><mrow
10841
class="MathClass-open">(</mo><mi
10842
class="MathClass-ord">λ</mi><mo
10843
class="MathClass-close">)</mo></mrow></mrow></msup
10845
class="MathClass-open">(</mo><mover
10846
class="mml-overline"><mrow><mi
10847
class="MathClass-ord">I</mi></mrow><mo
10848
accent="true">‾</mo></mover><mo
10849
class="MathClass-rel">∣</mo><mover
10850
class="mml-overline"><mrow><mi
10851
class="MathClass-ord">I</mi></mrow><mo
10852
accent="true">‾</mo></mover><mo
10853
class="MathClass-close">)</mo></mrow><mo
10854
class="MathClass-punc">.</mo></mtd><mtd><mspace
10855
id="x1-9003r36" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
10857
<!--l. 1065--><p class="nopar"></p></td><td width="5%">(36)</td></tr></table>
10858
Let <!--l. 1066--><math
10859
xmlns="http://www.w3.org/1998/Math/MathML"
10860
mode="inline"> <msub
10862
class="MathClass-ord">t</mi><mrow
10864
class="MathClass-ord">i</mi></mrow></msub
10866
class="MathClass-rel">=</mo><msub
10868
accent="true"><mrow
10870
class="MathClass-ord">x</mi></mrow><mrow
10873
class="MathClass-ord">̂</mi></mrow></munderover><mrow
10875
class="MathClass-ord">i</mi></mrow></msub
10876
></math>, <!--l. 1066--><math
10877
xmlns="http://www.w3.org/1998/Math/MathML"
10881
class="MathClass-ord">i</mi> <mo
10882
class="MathClass-rel">=</mo> <mn
10883
class="MathClass-ord">1</mn><mo
10884
class="MathClass-punc">,</mo> <mo
10885
class="MathClass-op">…</mo><mo
10886
class="MathClass-punc">,</mo> <mi
10887
class="MathClass-ord">n</mi></math>.
10888
Lemma 1 leads to <table class="equation"><tr><td>
10889
<!--l. 1067--><math
10890
xmlns="http://www.w3.org/1998/Math/MathML"
10893
class="equation"><mtr><mtd>
10895
class="MathClass-ord">D</mi><mrow><mo
10896
class="MathClass-open">(</mo><msub
10898
accent="true"><mrow
10900
class="MathClass-ord">x</mi></mrow><mrow
10903
class="MathClass-ord">̂</mi></mrow></munderover><mrow
10905
class="MathClass-ord">1</mn></mrow></msub
10907
class="MathClass-punc">,</mo> <mo
10908
class="MathClass-op">…</mo><mo
10909
class="MathClass-punc">,</mo><msub
10911
accent="true"><mrow
10913
class="MathClass-ord">x</mi></mrow><mrow
10916
class="MathClass-ord">̂</mi></mrow></munderover><mrow
10918
class="MathClass-ord">n</mi></mrow></msub
10920
class="MathClass-close">)</mo></mrow> <mo
10921
class="MathClass-rel">=</mo><msub
10923
class="MathClass-op">∏</mo>
10926
class="MathClass-ord">i</mi><mo
10927
class="MathClass-rel">∈</mo><mi class="mathbf">n</mi></mrow></msub
10930
accent="true"><mrow
10932
class="MathClass-ord">x</mi></mrow><mrow
10935
class="MathClass-ord">̂</mi></mrow></munderover><mrow
10937
class="MathClass-ord">i</mi></mrow></msub
10940
class="MathClass-op">∑</mo>
10943
class="MathClass-ord">I</mi><mo
10944
class="MathClass-rel">∈</mo><mi class="mathbf">n</mi></mrow></msub
10947
class="MathClass-open">(</mo><mo
10948
class="MathClass-bin">−</mo><mn
10949
class="MathClass-ord">1</mn><mo
10950
class="MathClass-close">)</mo></mrow><mrow
10952
open="|" close="|" ><mi
10953
class="MathClass-ord">I</mi> </mfenced><mo
10954
class="MathClass-bin">−</mo><mn
10955
class="MathClass-ord">1</mn></mrow></msup
10957
open="|" close="|" ><mi
10958
class="MathClass-ord">I</mi></mfenced><mo
10959
class="MathClass-op"> per</mo><!--nolimits--><msup
10960
> <mi class="mathbf">A</mi><mrow
10962
class="MathClass-open">(</mo><mi
10963
class="MathClass-ord">λ</mi><mo
10964
class="MathClass-close">)</mo></mrow></mrow></msup
10966
class="MathClass-open">(</mo><mi
10967
class="MathClass-ord">I</mi><mo
10968
class="MathClass-rel">∣</mo><mi
10969
class="MathClass-ord">I</mi><mo
10970
class="MathClass-close">)</mo></mrow><mo
10972
> <mi class="mathbf">A</mi><mrow
10974
class="MathClass-open">(</mo><mi
10975
class="MathClass-ord">λ</mi><mo
10976
class="MathClass-close">)</mo></mrow></mrow></msup
10978
class="MathClass-open">(</mo><mover
10979
class="mml-overline"><mrow><mi
10980
class="MathClass-ord">I</mi></mrow><mo
10981
accent="true">‾</mo></mover><mo
10982
class="MathClass-rel">∣</mo><mover
10983
class="mml-overline"><mrow><mi
10984
class="MathClass-ord">I</mi></mrow><mo
10985
accent="true">‾</mo></mover><mo
10986
class="MathClass-close">)</mo></mrow><mo
10987
class="MathClass-punc">.</mo></mtd><mtd><mspace
10988
id="x1-9004r37" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
10990
<!--l. 1071--><p class="nopar"></p></td><td width="5%">(37)</td></tr></table>
10992
href="#x1-2003r3">3<!--tex4ht:ref: H-cycles--></a>), (<a
10993
href="#x1-3015r13">13<!--tex4ht:ref: sum-Di--></a>), and (<a
10994
href="#x1-9004r37">37<!--tex4ht:ref: Dx--></a>), we have the following result: <div class="newtheorem">
10995
<!--l. 1074--><p class="noindent"><span class="head">
10997
name="x1-9005r2"></a>
10999
class="cmbx-10">Theorem 7.2.</span> </span> </p><table class="equation"><tr><td>
11001
<!--l. 1075--><math
11002
xmlns="http://www.w3.org/1998/Math/MathML"
11005
class="equation"><mtr><mtd>
11008
class="MathClass-ord">H</mi><mrow
11010
class="MathClass-ord">c</mi></mrow></msub
11012
class="MathClass-rel">=</mo> <mfrac><mrow
11014
class="MathClass-ord">1</mn></mrow>
11017
class="MathClass-ord">2</mn><mi
11018
class="MathClass-ord">n</mi></mrow></mfrac><msubsup
11020
class="MathClass-op">∑</mo>
11023
class="MathClass-ord">l</mi> <mo
11024
class="MathClass-rel">=</mo> <mn
11025
class="MathClass-ord">1</mn></mrow><mrow
11027
class="MathClass-ord">n</mi></mrow></msubsup
11029
class="MathClass-ord">l</mi><msup
11031
class="MathClass-open">(</mo><mo
11032
class="MathClass-bin">−</mo><mn
11033
class="MathClass-ord">1</mn><mo
11034
class="MathClass-close">)</mo></mrow><mrow
11036
class="MathClass-ord">l</mi><mo
11037
class="MathClass-bin">−</mo><mn
11038
class="MathClass-ord">1</mn></mrow></msup
11041
class="MathClass-ord">A</mi><mrow
11043
class="MathClass-ord">l</mi></mrow><mrow
11045
class="MathClass-open">(</mo><mi
11046
class="MathClass-ord">λ</mi><mo
11047
class="MathClass-close">)</mo></mrow></mrow></msubsup
11049
class="MathClass-punc">,</mo> </mtd><mtd><mspace
11050
id="x1-9006r38" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
11052
<!--l. 1078--><p class="nopar"></p></td><td width="5%">(38)</td></tr></table>
11054
class="cmti-10">where </span><table class="equation"><tr><td>
11055
<!--l. 1080--><math
11056
xmlns="http://www.w3.org/1998/Math/MathML"
11059
class="equation"><mtr><mtd>
11062
class="MathClass-ord">A</mi><mrow
11064
class="MathClass-ord">l</mi></mrow><mrow
11066
class="MathClass-open">(</mo><mi
11067
class="MathClass-ord">λ</mi><mo
11068
class="MathClass-close">)</mo></mrow></mrow></msubsup
11070
class="MathClass-rel">=</mo><msub
11072
class="MathClass-op">∑</mo>
11076
class="MathClass-ord">I</mi><mrow
11078
class="MathClass-ord">l</mi></mrow></msub
11080
class="MathClass-rel">⊆</mo><mi class="mathbf">n</mi></mrow></msub
11082
class="MathClass-op"> per</mo><!--nolimits--><msup
11083
> <mi class="mathbf">A</mi><mrow
11085
class="MathClass-open">(</mo><mi
11086
class="MathClass-ord">λ</mi><mo
11087
class="MathClass-close">)</mo></mrow></mrow></msup
11089
class="MathClass-open">(</mo><msub
11091
class="MathClass-ord">I</mi><mrow
11094
class="MathClass-ord">l</mi></mrow></msub
11096
class="MathClass-rel">∣</mo><msub
11098
class="MathClass-ord">I</mi><mrow
11100
class="MathClass-ord">l</mi></mrow></msub
11102
class="MathClass-close">)</mo></mrow><mo
11104
> <mi class="mathbf">A</mi><mrow
11106
class="MathClass-open">(</mo><mi
11107
class="MathClass-ord">λ</mi><mo
11108
class="MathClass-close">)</mo></mrow></mrow></msup
11110
class="MathClass-open">(</mo><msub
11112
class="mml-overline"><mrow><mi
11113
class="MathClass-ord">I</mi></mrow><mo
11114
accent="true">‾</mo></mover><mrow
11117
class="MathClass-ord">l</mi></mrow></msub
11119
class="MathClass-rel">∣</mo><msub
11121
class="mml-overline"><mrow><mi
11122
class="MathClass-ord">I</mi></mrow><mo
11123
accent="true">‾</mo></mover><mrow
11125
class="MathClass-ord">l</mi></mrow></msub
11127
class="MathClass-close">)</mo></mrow><mo
11128
class="MathClass-punc">,</mo> <mfenced
11129
open="|" close="|" ><msub
11131
class="MathClass-ord">I</mi><mrow
11133
class="MathClass-ord">l</mi></mrow></msub
11135
class="MathClass-rel">=</mo> <mi
11136
class="MathClass-ord">l</mi><mo
11137
class="MathClass-punc">.</mo></mtd><mtd><mspace
11138
id="x1-9007r39" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
11140
<!--l. 1084--><p class="nopar"></p></td><td width="5%">(39)</td></tr></table>
11142
<!--l. 1087--><p class="indent"> It is worth noting that <!--l. 1087--><math
11143
xmlns="http://www.w3.org/1998/Math/MathML"
11144
mode="inline"> <msubsup
11146
class="MathClass-ord">A</mi><mrow
11148
class="MathClass-ord">l</mi></mrow><mrow
11150
class="MathClass-open">(</mo><mi
11151
class="MathClass-ord">λ</mi><mo
11152
class="MathClass-close">)</mo></mrow></mrow></msubsup
11155
href="#x1-9007r39">39<!--tex4ht:ref: A-l-lambda--></a>) is similar to the coefficients <!--l. 1088--><math
11156
xmlns="http://www.w3.org/1998/Math/MathML"
11157
mode="inline"> <msub
11159
class="MathClass-ord">b</mi><mrow
11161
class="MathClass-ord">l</mi></mrow></msub
11163
of the characteristic polynomial of (<a
11164
href="#x1-3012r10">10<!--tex4ht:ref: bl-sum--></a>). It is well known in graph theory that the coefficients <!--l. 1090--><math
11165
xmlns="http://www.w3.org/1998/Math/MathML"
11169
class="MathClass-ord">b</mi><mrow
11171
class="MathClass-ord">l</mi></mrow></msub
11173
be expressed as a sum over certain subgraphs. It is interesting to see whether <!--l. 1091--><math
11174
xmlns="http://www.w3.org/1998/Math/MathML"
11178
class="MathClass-ord">A</mi><mrow
11180
class="MathClass-ord">l</mi></mrow></msub
11181
></math>, <!--l. 1091--><math
11182
xmlns="http://www.w3.org/1998/Math/MathML"
11185
class="MathClass-ord">λ</mi> <mo
11186
class="MathClass-rel">=</mo> <mn
11187
class="MathClass-ord">0</mn></math>,
11188
structural properties of a graph.
11190
</p><!--l. 1094--><p class="indent"> We may call (<a
11191
href="#x1-9006r38">38<!--tex4ht:ref: H-param--></a>) a parametric representation of <!--l. 1094--><math
11192
xmlns="http://www.w3.org/1998/Math/MathML"
11196
class="MathClass-ord">H</mi><mrow
11198
class="MathClass-ord">c</mi></mrow></msub
11199
></math>. In computation,
11200
the parameter <!--l. 1095--><math
11201
xmlns="http://www.w3.org/1998/Math/MathML"
11202
mode="inline"> <msub
11204
class="MathClass-ord">λ</mi><mrow
11206
class="MathClass-ord">i</mi></mrow></msub
11208
plays very important roles. The choice of the parameter usually
11209
depends on the properties of the given graph. For a complete graph <!--l. 1097--><math
11210
xmlns="http://www.w3.org/1998/Math/MathML"
11214
class="MathClass-ord">K</mi><mrow
11216
class="MathClass-ord">n</mi></mrow></msub
11217
></math>, let <!--l. 1097--><math
11218
xmlns="http://www.w3.org/1998/Math/MathML"
11222
class="MathClass-ord">λ</mi><mrow
11224
class="MathClass-ord">i</mi></mrow></msub
11226
class="MathClass-rel">=</mo> <mn
11227
class="MathClass-ord">1</mn></math>, <!--l. 1097--><math
11228
xmlns="http://www.w3.org/1998/Math/MathML"
11231
class="MathClass-ord">i</mi> <mo
11232
class="MathClass-rel">=</mo> <mn
11233
class="MathClass-ord">1</mn><mo
11234
class="MathClass-punc">,</mo> <mo
11235
class="MathClass-op">…</mo><mo
11236
class="MathClass-punc">,</mo> <mi
11237
class="MathClass-ord">n</mi></math>. It
11239
href="#x1-9007r39">39<!--tex4ht:ref: A-l-lambda--></a>) that </p><table class="equation"><tr><td>
11240
<!--l. 1099--><math
11241
xmlns="http://www.w3.org/1998/Math/MathML"
11244
class="equation"><mtr><mtd>
11247
class="MathClass-ord">A</mi><mrow
11249
class="MathClass-ord">l</mi></mrow><mrow
11251
class="MathClass-open">(</mo><mn
11252
class="MathClass-ord">1</mn><mo
11253
class="MathClass-close">)</mo></mrow></mrow></msubsup
11255
class="MathClass-rel">=</mo> <mfenced
11256
open="{" close="" ><mtable
11257
equalrows="false" equalcolumns="false" class="array"><mtr><mtd
11259
class="MathClass-ord">n</mi><mi
11260
class="MathClass-ord">!</mi><mo
11261
class="MathClass-punc">,</mo><mspace width="1em" class="quad"/></mtd><mtd
11262
class="array" ><mrow
11263
class="text"><mtext >if </mtext></mrow><mi
11264
class="MathClass-ord">l</mi> <mo
11265
class="MathClass-rel">=</mo> <mn
11266
class="MathClass-ord">1</mn> </mtd>
11269
class="MathClass-ord">0</mn><mo
11270
class="MathClass-punc">,</mo> <mspace width="1em" class="quad"/></mtd><mtd
11271
class="array" ><mrow
11272
class="text"><mtext >otherwise</mtext></mrow><mo
11273
class="MathClass-punc">.</mo></mtd></mtr> <!--@{}l@{\quad }l@{}--></mtable> </mfenced> </mtd><mtd><mspace
11274
id="x1-9008r40" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /></mtd></mtr></mtable>
11276
<!--l. 1102--><p class="nopar"></p></td><td width="5%">(40)</td></tr></table>
11278
href="#x1-9006r38">38<!--tex4ht:ref: H-param--></a>) <table class="equation"><tr><td>
11280
<!--l. 1104--><math
11281
xmlns="http://www.w3.org/1998/Math/MathML"
11284
class="equation"><mtr><mtd>
11287
class="MathClass-ord">H</mi><mrow
11289
class="MathClass-ord">c</mi></mrow></msub
11291
class="MathClass-rel">=</mo> <mfrac><mrow
11293
class="MathClass-ord">1</mn></mrow>
11296
class="MathClass-ord">2</mn></mrow></mfrac><mrow><mo
11297
class="MathClass-open">(</mo><mi
11298
class="MathClass-ord">n</mi> <mo
11299
class="MathClass-bin">−</mo> <mn
11300
class="MathClass-ord">1</mn><mo
11301
class="MathClass-close">)</mo></mrow><mi
11302
class="MathClass-ord">!</mi><mo
11303
class="MathClass-punc">.</mo></mtd><mtd><mspace
11304
id="x1-9009r41" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
11306
<!--l. 1106--><p class="nopar"></p></td><td width="5%">(41)</td></tr></table>
11307
For a complete bipartite graph <!--l. 1107--><math
11308
xmlns="http://www.w3.org/1998/Math/MathML"
11309
mode="inline"> <msub
11311
class="MathClass-ord">K</mi><mrow
11314
class="MathClass-ord">n</mi><mrow
11316
class="MathClass-ord">1</mn></mrow></msub
11319
class="MathClass-ord">n</mi><mrow
11321
class="MathClass-ord">2</mn></mrow></msub
11324
let <!--l. 1107--><math
11325
xmlns="http://www.w3.org/1998/Math/MathML"
11326
mode="inline"> <msub
11328
class="MathClass-ord">λ</mi><mrow
11330
class="MathClass-ord">i</mi></mrow></msub
11332
class="MathClass-rel">=</mo> <mn
11333
class="MathClass-ord">0</mn></math>, <!--l. 1107--><math
11334
xmlns="http://www.w3.org/1998/Math/MathML"
11337
class="MathClass-ord">i</mi> <mo
11338
class="MathClass-rel">=</mo> <mn
11339
class="MathClass-ord">1</mn><mo
11340
class="MathClass-punc">,</mo> <mo
11341
class="MathClass-op">…</mo><mo
11342
class="MathClass-punc">,</mo> <mi
11343
class="MathClass-ord">n</mi></math>. By
11345
href="#x1-9007r39">39<!--tex4ht:ref: A-l-lambda--></a>), <table class="equation"><tr><td>
11346
<!--l. 1109--><math
11347
xmlns="http://www.w3.org/1998/Math/MathML"
11350
class="equation"><mtr><mtd>
11353
class="MathClass-ord">A</mi><mrow
11355
class="MathClass-ord">l</mi></mrow></msub
11357
class="MathClass-rel">=</mo> <mfenced
11358
open="{" close="" ><mtable
11359
equalrows="false" equalcolumns="false" class="array"><mtr><mtd
11361
class="MathClass-bin">−</mo><msub
11363
class="MathClass-ord">n</mi><mrow
11365
class="MathClass-ord">1</mn></mrow></msub
11367
class="MathClass-ord">!</mi><msub
11369
class="MathClass-ord">n</mi><mrow
11371
class="MathClass-ord">2</mn></mrow></msub
11373
class="MathClass-ord">!</mi><msub
11375
class="MathClass-ord">δ</mi><mrow
11378
class="MathClass-ord">n</mi><mrow
11380
class="MathClass-ord">1</mn></mrow></msub
11383
class="MathClass-ord">n</mi><mrow
11385
class="MathClass-ord">2</mn></mrow></msub
11388
class="MathClass-punc">,</mo><mspace width="1em" class="quad"/></mtd><mtd
11389
class="array" ><mrow
11390
class="text"><mtext >if </mtext></mrow><mi
11391
class="MathClass-ord">l</mi> <mo
11392
class="MathClass-rel">=</mo> <mn
11393
class="MathClass-ord">2</mn> </mtd>
11396
class="MathClass-ord">0</mn><mo
11397
class="MathClass-punc">,</mo> <mspace width="1em" class="quad"/></mtd><mtd
11398
class="array" ><mrow
11399
class="text"><mtext >otherwise </mtext></mrow><mo
11400
class="MathClass-punc">.</mo></mtd></mtr> <!--@{}l@{\quad }l@{}--></mtable> </mfenced> </mtd><mtd><mspace
11401
id="x1-9010r42" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /></mtd></mtr></mtable>
11403
<!--l. 1114--><p class="nopar"></p></td><td width="5%">(42)</td></tr></table>
11405
href="#x1-9005r2">7.2<!--tex4ht:ref: thm-H-param--></a> leads to <table class="equation"><tr><td>
11407
<!--l. 1117--><math
11408
xmlns="http://www.w3.org/1998/Math/MathML"
11411
class="equation"><mtr><mtd>
11414
class="MathClass-ord">H</mi><mrow
11416
class="MathClass-ord">c</mi></mrow></msub
11418
class="MathClass-rel">=</mo> <mfrac><mrow
11420
class="MathClass-ord">1</mn></mrow>
11424
class="MathClass-ord">n</mi><mrow
11426
class="MathClass-ord">1</mn></mrow></msub
11428
class="MathClass-bin">+</mo> <msub
11430
class="MathClass-ord">n</mi><mrow
11432
class="MathClass-ord">2</mn></mrow></msub
11433
></mrow></mfrac><msub
11435
class="MathClass-ord">n</mi><mrow
11437
class="MathClass-ord">1</mn></mrow></msub
11439
class="MathClass-ord">!</mi><msub
11441
class="MathClass-ord">n</mi><mrow
11443
class="MathClass-ord">2</mn></mrow></msub
11445
class="MathClass-ord">!</mi><msub
11447
class="MathClass-ord">δ</mi><mrow
11450
class="MathClass-ord">n</mi><mrow
11452
class="MathClass-ord">1</mn></mrow></msub
11455
class="MathClass-ord">n</mi><mrow
11457
class="MathClass-ord">2</mn></mrow></msub
11460
class="MathClass-punc">.</mo></mtd><mtd><mspace
11461
id="x1-9011r43" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
11463
<!--l. 1119--><p class="nopar"></p></td><td width="5%">(43)</td></tr></table>
11464
<!--l. 1121--><p class="indent"> Now, we consider an asymmetrical approach. Theorem <a
11465
href="#x1-3008r3">3.3<!--tex4ht:ref: thm-main--></a> leads to
11466
</p><!--l. 1122--><math
11467
xmlns="http://www.w3.org/1998/Math/MathML"
11472
class="multline"></mtd><mtd><mo
11473
> det</mo> <mi class="mathbf">K</mi><mrow><mo
11474
class="MathClass-open">(</mo><mi
11475
class="MathClass-ord">t</mi> <mo
11476
class="MathClass-rel">=</mo> <mn
11477
class="MathClass-ord">1</mn><mo
11478
class="MathClass-punc">,</mo> <msub
11480
class="MathClass-ord">t</mi><mrow
11482
class="MathClass-ord">1</mn></mrow></msub
11484
class="MathClass-punc">,</mo> <mo>…</mo><mo
11485
class="MathClass-punc">,</mo> <msub
11487
class="MathClass-ord">t</mi><mrow
11489
class="MathClass-ord">n</mi></mrow></msub
11491
class="MathClass-punc">;</mo> <mi
11492
class="MathClass-ord">l</mi><mo
11493
class="MathClass-rel">∣</mo><mi
11494
class="MathClass-ord">l</mi><mo
11495
class="MathClass-close">)</mo></mrow>
11496
</mtd></mtr><mtr><mtd
11497
class="multline"></mtd><mtd> <mo
11498
class="MathClass-rel">=</mo><msub
11500
class="MathClass-op">∑</mo>
11503
class="MathClass-ord">I</mi><mo
11504
class="MathClass-rel">⊆</mo><mi class="mathbf">n</mi><mo
11505
class="MathClass-bin">−</mo><mrow><mo
11506
class="MathClass-open">{</mo><mi
11507
class="MathClass-ord">l</mi><mo
11508
class="MathClass-close">}</mo></mrow></mrow></msub
11511
class="MathClass-open">(</mo><mo
11512
class="MathClass-bin">−</mo><mn
11513
class="MathClass-ord">1</mn><mo
11514
class="MathClass-close">)</mo></mrow><mrow
11516
open="|" close="|" ><mi
11517
class="MathClass-ord">I</mi> </mfenced></mrow></msup
11520
class="MathClass-op">∏</mo>
11523
class="MathClass-ord">i</mi><mo
11524
class="MathClass-rel">∈</mo><mi
11525
class="MathClass-ord">I</mi></mrow></msub
11528
class="MathClass-ord">t</mi><mrow
11530
class="MathClass-ord">i</mi></mrow></msub
11533
class="MathClass-op">∏</mo>
11536
class="MathClass-ord">j</mi><mo
11537
class="MathClass-rel">∈</mo><mi
11538
class="MathClass-ord">I</mi></mrow></msub
11540
class="MathClass-open">(</mo><msub
11542
class="MathClass-ord">D</mi><mrow
11544
class="MathClass-ord">j</mi></mrow></msub
11546
class="MathClass-bin">+</mo> <msub
11548
class="MathClass-ord">λ</mi><mrow
11550
class="MathClass-ord">j</mi></mrow></msub
11553
class="MathClass-ord">t</mi><mrow
11555
class="MathClass-ord">j</mi></mrow></msub
11557
class="MathClass-close">)</mo></mrow><mo
11559
> <mi class="mathbf">A</mi><mrow
11561
class="MathClass-open">(</mo><mi
11562
class="MathClass-ord">λ</mi><mo
11563
class="MathClass-close">)</mo></mrow></mrow></msup
11565
class="MathClass-open">(</mo><mover
11566
class="mml-overline"><mrow><mi
11567
class="MathClass-ord">I</mi></mrow><mo
11568
accent="true">‾</mo></mover> <mo
11569
class="MathClass-bin">∪</mo> <mrow><mo
11570
class="MathClass-open">{</mo><mi
11571
class="MathClass-ord">l</mi><mo
11572
class="MathClass-close">}</mo></mrow><mo
11573
class="MathClass-rel">∣</mo><mover
11574
class="mml-overline"><mrow><mi
11575
class="MathClass-ord">I</mi></mrow><mo
11576
accent="true">‾</mo></mover> <mo
11577
class="MathClass-bin">∪</mo> <mrow><mo
11578
class="MathClass-open">{</mo><mi
11579
class="MathClass-ord">l</mi><mo
11580
class="MathClass-close">}</mo></mrow><mo
11581
class="MathClass-close">)</mo></mrow><mo
11582
class="MathClass-punc">.</mo></mtd><mtd><mrow><mo
11583
class="MathClass-open">(</mo><mn
11584
class="MathClass-ord">4</mn><mn
11585
class="MathClass-ord">4</mn><mo
11586
class="MathClass-close">)</mo></mrow> </mtd></mtr></mtable>
11588
<!--l. 1128--><p class="nopar">
11589
</p><!--l. 1130--><p class="indent"> By (<a
11590
href="#x1-2003r3">3<!--tex4ht:ref: H-cycles--></a>) and (<a
11591
href="#x1-3018r16">16<!--tex4ht:ref: sum-ali--></a>) we have the following asymmetrical result: </p><div class="newtheorem">
11592
<!--l. 1132--><p class="noindent"><span class="head">
11594
name="x1-9012r3"></a>
11596
class="cmbx-10">Theorem 7.3.</span> </span> </p><table class="equation"><tr><td>
11598
<!--l. 1133--><math
11599
xmlns="http://www.w3.org/1998/Math/MathML"
11602
class="equation"><mtr><mtd>
11605
class="MathClass-ord">H</mi><mrow
11607
class="MathClass-ord">c</mi></mrow></msub
11609
class="MathClass-rel">=</mo> <mfrac><mrow
11611
class="MathClass-ord">1</mn></mrow>
11614
class="MathClass-ord">2</mn></mrow></mfrac><msub
11616
class="MathClass-op">∑</mo>
11619
class="MathClass-ord">I</mi><mo
11620
class="MathClass-rel">⊆</mo><mi class="mathbf">n</mi><mo
11621
class="MathClass-bin">−</mo><mrow><mo
11622
class="MathClass-open">{</mo><mi
11623
class="MathClass-ord">l</mi><mo
11624
class="MathClass-close">}</mo></mrow></mrow></msub
11627
class="MathClass-open">(</mo><mo
11628
class="MathClass-bin">−</mo><mn
11629
class="MathClass-ord">1</mn><mo
11630
class="MathClass-close">)</mo></mrow><mrow
11632
open="|" close="|" ><mi
11633
class="MathClass-ord">I</mi> </mfenced></mrow></msup
11635
class="MathClass-op"> per</mo><!--nolimits--><msup
11636
> <mi class="mathbf">A</mi><mrow
11638
class="MathClass-open">(</mo><mi
11639
class="MathClass-ord">λ</mi><mo
11640
class="MathClass-close">)</mo></mrow></mrow></msup
11642
class="MathClass-open">(</mo><mi
11643
class="MathClass-ord">I</mi><mo
11644
class="MathClass-rel">∣</mo><mi
11645
class="MathClass-ord">I</mi><mo
11646
class="MathClass-close">)</mo></mrow><mo
11648
> <mi class="mathbf">A</mi><mrow
11650
class="MathClass-open">(</mo><mi
11651
class="MathClass-ord">λ</mi><mo
11652
class="MathClass-close">)</mo></mrow></mrow></msup
11654
class="MathClass-open">(</mo><mover
11655
class="mml-overline"><mrow><mi
11656
class="MathClass-ord">I</mi></mrow><mo
11657
accent="true">‾</mo></mover> <mo
11658
class="MathClass-bin">∪</mo> <mrow><mo
11659
class="MathClass-open">{</mo><mi
11660
class="MathClass-ord">l</mi><mo
11661
class="MathClass-close">}</mo></mrow><mo
11662
class="MathClass-rel">∣</mo><mover
11663
class="mml-overline"><mrow><mi
11664
class="MathClass-ord">I</mi></mrow><mo
11665
accent="true">‾</mo></mover> <mo
11666
class="MathClass-bin">∪</mo> <mrow><mo
11667
class="MathClass-open">{</mo><mi
11668
class="MathClass-ord">l</mi><mo
11669
class="MathClass-close">}</mo></mrow><mo
11670
class="MathClass-close">)</mo></mrow></mtd><mtd><mspace
11671
id="x1-9013r45" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
11673
<!--l. 1138--><p class="nopar"></p></td><td width="5%">(45)</td></tr></table>
11675
class="cmti-10">which reduces to Goulden Jackson’s formula when </span><!--l. 1139--><math
11676
xmlns="http://www.w3.org/1998/Math/MathML"
11680
class="MathClass-ord">λ</mi><mrow
11682
class="MathClass-ord">i</mi></mrow></msub
11684
class="MathClass-rel">=</mo> <mn
11685
class="MathClass-ord">0</mn><mo
11686
class="MathClass-punc">,</mo> <mi
11687
class="MathClass-ord">i</mi> <mo
11688
class="MathClass-rel">=</mo> <mn
11689
class="MathClass-ord">1</mn><mo
11690
class="MathClass-punc">,</mo> <mo
11691
class="MathClass-op">…</mo><mo
11692
class="MathClass-punc">,</mo> <mi
11693
class="MathClass-ord">n</mi></math>
11694
<span class="cite"><span
11695
class="cmti-10">[</span><a
11696
href="#Xmami:matrixth"><span
11697
class="cmti-10">9</span></a><span
11698
class="cmti-10">]</span></span><span
11699
class="cmti-10">.</span>
11701
<h3 class="sectionHead"><span class="titlemark">8</span> <a
11702
name="x1-100008"></a>Various font features of the <span
11703
class="cmtt-10">amsmath </span>package</h3>
11704
<h4 class="subsectionHead"><span class="titlemark">8.1</span> <a
11705
name="x1-110008.1"></a>Bold versions of special symbols</h4>
11706
<!--l. 1147--><p class="noindent">In the <span
11707
class="cmtt-10">amsmath </span>package <span
11708
class="cmtt-10">\boldsymbol </span>is used for getting individual bold math
11709
symbols and bold Greek letters everything in math except for letters of the Latin
11710
alphabet, where you’d use <span
11711
class="cmtt-10">\mathbf</span>. For example,
11714
<table width="100%"
11715
class="verbatim"><tr class="verbatim"><td
11716
class="verbatim"><pre class="verbatim">
11717
 A_\infty + \pi A_0 \sim
11718
 \mathbf{A}_{\boldsymbol{\infty}} \boldsymbol{+}
11719
 \boldsymbol{\pi} \mathbf{A}_{\boldsymbol{0}}
11722
<!--l. 1156--><p class="indent"> looks like this: <!--l. 1157--><math
11723
xmlns="http://www.w3.org/1998/Math/MathML"
11724
mode="display"> <mrow
11728
class="MathClass-ord">A</mi><mrow
11730
class="MathClass-ord">∞</mi></mrow></msub
11732
class="MathClass-bin">+</mo> <mi
11733
class="MathClass-ord">π</mi><msub
11735
class="MathClass-ord">A</mi><mrow
11737
class="MathClass-ord">0</mn></mrow></msub
11739
class="MathClass-rel">∼</mo><msub
11740
> <mi class="mathbf">A</mi><mrow
11742
class="MathClass-ord">∞</mi></mrow></msub
11744
class="MathClass-bin"> +</mo> <mi
11745
class="MathClass-ord">π</mi><msub
11746
><mi class="mathbf">A</mi><mrow
11748
class="MathClass-ord">0</mn></mrow></msub
11752
<h4 class="subsectionHead"><span class="titlemark">8.2</span> <a
11753
name="x1-120008.2"></a>“Poor man’s bold”</h4> If a bold version of a particular symbol doesn’t
11754
exist in the available fonts, then <span
11755
class="cmtt-10">\boldsymbol </span>can’t be used to make that
11756
symbol bold. At the present time, this means that <span
11757
class="cmtt-10">\boldsymbol </span>can’t be
11758
used with symbols from the <span
11759
class="cmtt-10">msam </span>and <span
11760
class="cmtt-10">msbm </span>fonts, among others. In some
11761
cases, poor man’s bold (<span
11762
class="cmtt-10">\pmb</span>) can be used instead of <span
11763
class="cmtt-10">\boldsymbol</span>: <!--l. 1172--><math
11764
xmlns="http://www.w3.org/1998/Math/MathML"
11770
class="MathClass-ord">∂</mi><mi
11771
class="MathClass-ord">x</mi></mrow>
11774
class="MathClass-ord">∂</mi><mi
11775
class="MathClass-ord">y</mi></mrow></mfrac><mfenced
11776
open="|" close="" ></mfenced><mfenced
11777
open="|" close="" ></mfenced><mfenced
11778
open="|" close="" ></mfenced><mfrac><mrow
11780
class="MathClass-ord">∂</mi><mi
11781
class="MathClass-ord">y</mi></mrow>
11784
class="MathClass-ord">∂</mi><mi
11785
class="MathClass-ord">z</mi></mrow></mfrac>
11788
<table width="100%"
11789
class="verbatim"><tr class="verbatim"><td
11790
class="verbatim"><pre class="verbatim">
11791
 \[\frac{\partial x}{\partial y}
11792
 \pmb{\bigg\vert}
11793
 \frac{\partial y}{\partial z}\]
11796
<!--l. 1180--><p class="indent"> So-called “large operator” symbols such as <!--l. 1180--><math
11797
xmlns="http://www.w3.org/1998/Math/MathML"
11800
class="MathClass-op">∑</mo>
11801
</math> and <!--l. 1180--><math
11802
xmlns="http://www.w3.org/1998/Math/MathML"
11805
class="MathClass-op">∏</mo>
11807
require an additional command, <span
11808
class="cmtt-10">\mathop</span>, to produce proper spacing
11809
and limits when <span
11810
class="cmtt-10">\pmb </span>is used. For further details see <span
11811
class="cmti-10">The </span><span class="TEX"><span
11812
class="cmti-10">T</span><span
11814
class="cmti-10">E</span></span><span
11815
class="cmti-10">X</span></span><span
11816
class="cmti-10">book</span>. <!--l. 1184--><math
11817
xmlns="http://www.w3.org/1998/Math/MathML"
11823
class="MathClass-op">∑</mo>
11826
<mtable class="subarray-c"><mtr><mtd><mi
11827
class="MathClass-ord">i</mi><mo
11828
class="MathClass-rel"><</mo><mi
11829
class="MathClass-ord">B</mi>
11830
</mtd></mtr><mtr><mtd>
11831
</mtd></mtr><mtr><mtd>
11832
</mtd></mtr><mtr><mtd><mrow
11833
class="text"><mtext ></mtext><mrow
11835
class="MathClass-ord">i</mi></mrow><mtext > odd</mtext></mrow> </mtd></mtr> </mtable></mrow></msub
11838
class="MathClass-op">∏</mo>
11841
class="MathClass-ord">κ</mi></mrow></msub
11843
class="MathClass-ord">κ</mi><mi
11844
class="MathClass-ord">F</mi><mrow><mo
11845
class="MathClass-open">(</mo><msub
11847
class="MathClass-ord">r</mi><mrow
11849
class="MathClass-ord">i</mi></mrow></msub
11851
class="MathClass-close">)</mo></mrow><mspace width="2em" class="qquad"/><msub
11853
class="MathClass-op">∑
11858
<mtable class="subarray-c"><mtr><mtd><mi
11859
class="MathClass-ord">i</mi><mo
11860
class="MathClass-rel"><</mo><mi
11861
class="MathClass-ord">B</mi>
11862
</mtd></mtr><mtr><mtd>
11863
</mtd></mtr><mtr><mtd>
11864
</mtd></mtr><mtr><mtd><mrow
11865
class="text"><mtext ></mtext><mrow
11867
class="MathClass-ord">i</mi></mrow><mtext > odd</mtext></mrow> </mtd></mtr> </mtable></mrow></msub
11870
class="MathClass-op">∏
11875
class="MathClass-ord">κ</mi></mrow></msub
11877
class="MathClass-ord">κ</mi><mrow><mo
11878
class="MathClass-open">(</mo><msub
11880
class="MathClass-ord">r</mi><mrow
11882
class="MathClass-ord">i</mi></mrow></msub
11884
class="MathClass-close">)</mo></mrow>
11888
<table width="100%"
11889
class="verbatim"><tr class="verbatim"><td
11890
class="verbatim"><pre class="verbatim">
11891
 \[\sum_{\substack{i<B\\\text{$i$ odd}}}
11892
 \prod_\kappa \kappa F(r_i)\qquad
11893
 \mathop{\pmb{\sum}}_{\substack{i<B\\\text{$i$ odd}}}
11894
 \mathop{\pmb{\prod}}_\kappa \kappa(r_i)
11898
<h3 class="sectionHead"><span class="titlemark">9</span> <a
11899
name="x1-130009"></a>Compound symbols and other features</h3>
11900
<h4 class="subsectionHead"><span class="titlemark">9.1</span> <a
11901
name="x1-140009.1"></a>Multiple integral signs</h4>
11902
<!--l. 1201--><p class="noindent"><span
11903
class="cmtt-10">\iint</span>, <span
11904
class="cmtt-10">\iiint</span>, and <span
11905
class="cmtt-10">\iiiint </span>give multiple integral signs with the spacing between
11906
them nicely adjusted, in both text and display style. <span
11907
class="cmtt-10">\idotsint </span>gives two integral
11908
signs with dots between them.
11909
</p><!--l. 1205--><math
11910
xmlns="http://www.w3.org/1998/Math/MathML"
11917
class="MathClass-op"> ∬</mo> <mrow
11919
class="MathClass-ord">A</mi></mrow></msub
11921
class="MathClass-ord">f</mi><mrow><mo
11922
class="MathClass-open">(</mo><mi
11923
class="MathClass-ord">x</mi><mo
11924
class="MathClass-punc">,</mo> <mi
11925
class="MathClass-ord">y</mi><mo
11926
class="MathClass-close">)</mo></mrow> <mi
11927
class="MathClass-ord">d</mi><mi
11928
class="MathClass-ord">x</mi> <mi
11929
class="MathClass-ord">d</mi><mi
11930
class="MathClass-ord">y</mi><mspace width="2em" class="qquad"/><msub
11932
class="MathClass-op"> ∭</mo> <mrow
11934
class="MathClass-ord">A</mi></mrow></msub
11936
class="MathClass-ord">f</mi><mrow><mo
11937
class="MathClass-open">(</mo><mi
11938
class="MathClass-ord">x</mi><mo
11939
class="MathClass-punc">,</mo> <mi
11940
class="MathClass-ord">y</mi><mo
11941
class="MathClass-punc">,</mo> <mi
11942
class="MathClass-ord">z</mi><mo
11943
class="MathClass-close">)</mo></mrow> <mi
11944
class="MathClass-ord">d</mi><mi
11945
class="MathClass-ord">x</mi> <mi
11946
class="MathClass-ord">d</mi><mi
11947
class="MathClass-ord">y</mi> <mi
11948
class="MathClass-ord">d</mi><mi
11949
class="MathClass-ord">z</mi></mtd>
11951
id="x1-14001r46" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
11952
class="MathClass-open">(</mo><mn
11953
class="MathClass-ord">4</mn><mn
11954
class="MathClass-ord">6</mn><mo
11955
class="MathClass-close">)</mo></mrow></mtd>
11959
class="MathClass-op"> ∬<!--nolimits--> ∬<!--nolimits--></mo> <mrow
11961
class="MathClass-ord">A</mi></mrow></msub
11963
class="MathClass-ord">f</mi><mrow><mo
11964
class="MathClass-open">(</mo><mi
11965
class="MathClass-ord">w</mi><mo
11966
class="MathClass-punc">,</mo> <mi
11967
class="MathClass-ord">x</mi><mo
11968
class="MathClass-punc">,</mo> <mi
11969
class="MathClass-ord">y</mi><mo
11970
class="MathClass-punc">,</mo> <mi
11971
class="MathClass-ord">z</mi><mo
11972
class="MathClass-close">)</mo></mrow> <mi
11973
class="MathClass-ord">d</mi><mi
11974
class="MathClass-ord">w</mi> <mi
11975
class="MathClass-ord">d</mi><mi
11976
class="MathClass-ord">x</mi> <mi
11977
class="MathClass-ord">d</mi><mi
11978
class="MathClass-ord">y</mi> <mi
11979
class="MathClass-ord">d</mi><mi
11980
class="MathClass-ord">z</mi><mspace width="2em" class="qquad"/><msub
11982
class="MathClass-op"> ∫
11983
· · · ∫</mo>
11986
class="MathClass-ord">A</mi></mrow></msub
11988
class="MathClass-ord">f</mi><mrow><mo
11989
class="MathClass-open">(</mo><msub
11991
class="MathClass-ord">x</mi><mrow
11993
class="MathClass-ord">1</mn></mrow></msub
11995
class="MathClass-punc">,</mo> <mo
11996
class="MathClass-op">…</mo><mo
11997
class="MathClass-punc">,</mo> <msub
11999
class="MathClass-ord">x</mi><mrow
12001
class="MathClass-ord">k</mi></mrow></msub
12003
class="MathClass-close">)</mo></mrow></mtd>
12005
id="x1-14002r47" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
12006
class="MathClass-open">(</mo><mn
12007
class="MathClass-ord">4</mn><mn
12008
class="MathClass-ord">7</mn><mo
12009
class="MathClass-close">)</mo></mrow></mtd> </mtr></mtable>
12011
<!--l. 1210--><p class="nopar">
12014
<h4 class="subsectionHead"><span class="titlemark">9.2</span> <a
12015
name="x1-150009.2"></a>Over and under arrows</h4>
12016
<!--l. 1214--><p class="noindent">Some extra over and under arrow operations are provided in the <span
12017
class="cmtt-10">amsmath </span>package.
12018
(Basic <span class="LATEX">L<span class="A">A</span><span class="TEX">T<span
12019
class="E">E</span>X</span></span> provides <span
12020
class="cmtt-10">\overrightarrow </span>and <span
12021
class="cmtt-10">\overleftarrow</span>).
12022
<!--tex4ht:inline--></p><!--l. 1224--><math
12023
xmlns="http://www.w3.org/1998/Math/MathML"
12024
display="block"><mtable
12025
class="align-star">
12027
class="align-odd"><mover><mrow class="fill"> <mo
12028
class="MathClass-rel">→</mo> </mrow><mrow
12031
class="MathClass-ord">ψ</mi><mrow
12033
class="MathClass-ord">δ</mi></mrow></msub
12035
class="MathClass-open">(</mo><mi
12036
class="MathClass-ord">t</mi><mo
12037
class="MathClass-close">)</mo></mrow><msub
12039
class="MathClass-ord">E</mi><mrow
12041
class="MathClass-ord">t</mi></mrow></msub
12043
class="MathClass-ord">h</mi></mrow></mover></mtd> <mtd
12044
class="align-even"> <mo
12045
class="MathClass-rel">=</mo> <munder><mrow
12047
class="MathClass-rel">→</mo> </mrow><mrow
12048
></mrow></munder></mtd>
12050
class="align-odd"><mover><mrow class="fill"> <mo
12051
class="MathClass-rel">←</mo> </mrow><mrow
12054
class="MathClass-ord">ψ</mi><mrow
12056
class="MathClass-ord">δ</mi></mrow></msub
12058
class="MathClass-open">(</mo><mi
12059
class="MathClass-ord">t</mi><mo
12060
class="MathClass-close">)</mo></mrow><msub
12062
class="MathClass-ord">E</mi><mrow
12064
class="MathClass-ord">t</mi></mrow></msub
12066
class="MathClass-ord">h</mi></mrow></mover></mtd> <mtd
12067
class="align-even"> <mo
12068
class="MathClass-rel">=</mo> <munder><mrow
12070
class="MathClass-rel">←</mo> </mrow><mrow
12071
></mrow></munder></mtd>
12073
class="align-odd"><mover><mrow class="fill"> <mo
12074
class="MathClass-rel">↔</mo> </mrow><mrow
12077
class="MathClass-ord">ψ</mi><mrow
12079
class="MathClass-ord">δ</mi></mrow></msub
12081
class="MathClass-open">(</mo><mi
12082
class="MathClass-ord">t</mi><mo
12083
class="MathClass-close">)</mo></mrow><msub
12085
class="MathClass-ord">E</mi><mrow
12087
class="MathClass-ord">t</mi></mrow></msub
12089
class="MathClass-ord">h</mi></mrow></mover></mtd> <mtd
12090
class="align-even"> <mo
12091
class="MathClass-rel">=</mo> <munder><mrow
12093
class="MathClass-rel">↔</mo> </mrow><mrow
12094
></mrow></munder></mtd>
12095
</mtr></mtable></math>
12097
<table width="100%"
12098
class="verbatim"><tr class="verbatim"><td
12099
class="verbatim"><pre class="verbatim">
12100
 \begin{align*}
12101
 \overrightarrow{\psi_\delta(t) E_t h}&
12102
 =\underrightarrow{\psi_\delta(t) E_t h}\\
12103
 \overleftarrow{\psi_\delta(t) E_t h}&
12104
 =\underleftarrow{\psi_\delta(t) E_t h}\\
12105
 \overleftrightarrow{\psi_\delta(t) E_t h}&
12106
 =\underleftrightarrow{\psi_\delta(t) E_t h}
12107
 \end{align*}
12110
<!--l. 1235--><p class="indent"> These all scale properly in subscript sizes: <!--l. 1236--><math
12111
xmlns="http://www.w3.org/1998/Math/MathML"
12117
class="MathClass-op">∫</mo>
12119
><mover><mrow class="fill"><mo
12120
class="MathClass-rel">→</mo></mrow><mrow
12122
class="MathClass-ord">A</mi><mi
12123
class="MathClass-ord">B</mi></mrow></mover></mrow></msub
12125
class="MathClass-ord">a</mi><mi
12126
class="MathClass-ord">x</mi> <mi
12127
class="MathClass-ord">d</mi><mi
12128
class="MathClass-ord">x</mi>
12132
<table width="100%"
12133
class="verbatim"><tr class="verbatim"><td
12134
class="verbatim"><pre class="verbatim">
12135
 \[\int_{\overrightarrow{AB}} ax\,dx\]
12138
<h4 class="subsectionHead"><span class="titlemark">9.3</span> <a
12139
name="x1-160009.3"></a>Dots</h4>
12140
<!--l. 1243--><p class="noindent">Normally you need only type <span
12141
class="cmtt-10">\dots </span>for ellipsis dots in a math formula. The main
12142
exception is when the dots fall at the end of the formula; then you need to specify
12144
class="cmtt-10">\dotsc </span>(series dots, after a comma), <span
12145
class="cmtt-10">\dotsb </span>(binary dots, for binary relations
12146
or operators), <span
12147
class="cmtt-10">\dotsm </span>(multiplication dots), or <span
12148
class="cmtt-10">\dotsi </span>(dots after an integral). For
12152
<table width="100%"
12153
class="verbatim"><tr class="verbatim"><td
12154
class="verbatim"><pre class="verbatim">
12155
 Then we have the series $A_1,A_2,\dotsc$,
12156
 the regional sum $A_1+A_2+\dotsb$,
12157
 the orthogonal product $A_1A_2\dotsm$,
12158
 and the infinite integral
12159
 \[\int_{A_1}\int_{A_2}\dotsi\].
12162
<!--l. 1257--><p class="indent"> produces
12163
</p><!--l. 1259--><p class="indent"> Then we have the series <!--l. 1259--><math
12164
xmlns="http://www.w3.org/1998/Math/MathML"
12165
mode="inline"> <msub
12167
class="MathClass-ord">A</mi><mrow
12169
class="MathClass-ord">1</mn></mrow></msub
12171
class="MathClass-punc">,</mo> <msub
12173
class="MathClass-ord">A</mi><mrow
12175
class="MathClass-ord">2</mn></mrow></msub
12177
class="MathClass-punc">,</mo> <mo
12178
class="MathClass-op">…</mo></math>,
12179
the regional sum <!--l. 1260--><math
12180
xmlns="http://www.w3.org/1998/Math/MathML"
12181
mode="inline"> <msub
12183
class="MathClass-ord">A</mi><mrow
12185
class="MathClass-ord">1</mn></mrow></msub
12187
class="MathClass-bin">+</mo> <msub
12189
class="MathClass-ord">A</mi><mrow
12191
class="MathClass-ord">2</mn></mrow></msub
12193
class="MathClass-bin">+</mo> <mo
12194
class="MathClass-rel">⋯</mo></math>,
12195
the orthogonal product <!--l. 1261--><math
12196
xmlns="http://www.w3.org/1998/Math/MathML"
12197
mode="inline"> <msub
12199
class="MathClass-ord">A</mi><mrow
12201
class="MathClass-ord">1</mn></mrow></msub
12204
class="MathClass-ord">A</mi><mrow
12206
class="MathClass-ord">2</mn></mrow></msub
12208
class="MathClass-rel">⋯</mo></math>,
12209
and the infinite integral <!--l. 1263--><math
12210
xmlns="http://www.w3.org/1998/Math/MathML"
12211
mode="display"> <mrow
12215
class="MathClass-op">∫</mo>
12219
class="MathClass-ord">A</mi><mrow
12221
class="MathClass-ord">1</mn></mrow></msub
12225
class="MathClass-op">∫</mo>
12229
class="MathClass-ord">A</mi><mrow
12231
class="MathClass-ord">2</mn></mrow></msub
12234
class="MathClass-rel">⋯</mo>
12237
<h4 class="subsectionHead"><span class="titlemark">9.4</span> <a
12238
name="x1-170009.4"></a>Accents in math</h4>
12240
<!--l. 1268--><p class="noindent">Double accents: <!--l. 1269--><math
12241
xmlns="http://www.w3.org/1998/Math/MathML"
12242
mode="display"> <mrow
12245
accent="true"><mrow
12247
class="MathClass-ord">Ĥ</mi></mrow><mrow
12250
class="MathClass-ord">̂</mi></mrow></munderover><mspace width="1em" class="quad"/><munderover
12251
accent="true"><mrow
12253
class="MathClass-ord">Č</mi></mrow><mrow
12256
class="MathClass-ord">̌</mi></mrow></munderover><mspace width="1em" class="quad"/><munderover
12257
accent="true"><mrow
12259
accent="true"><mrow
12261
class="MathClass-ord">T</mi></mrow><mrow
12264
class="MathClass-ord">̃</mi></mrow></munderover></mrow><mrow
12267
class="MathClass-ord">̃</mi></mrow></munderover><mspace width="1em" class="quad"/><munderover
12268
accent="true"><mrow
12270
class="MathClass-ord">Á</mi></mrow><mrow
12273
class="MathClass-ord">́</mi></mrow></munderover><mspace width="1em" class="quad"/><munderover
12274
accent="true"><mrow
12276
accent="true"><mrow
12278
class="MathClass-ord">G</mi></mrow><mrow
12281
class="MathClass-ord">̀</mi></mrow></munderover></mrow><mrow
12284
class="MathClass-ord">̀</mi></mrow></munderover><mspace width="1em" class="quad"/><munderover
12285
accent="true"><mrow
12287
class="MathClass-ord">Ḋ</mi></mrow><mrow
12290
class="MathClass-ord">.</mi></mrow></munderover><mspace width="1em" class="quad"/><munderover
12291
accent="true"><mrow
12293
accent="true"><mrow
12295
class="MathClass-ord">D</mi></mrow><mrow
12298
class="MathClass-ord">̈</mi></mrow></munderover></mrow><mrow
12301
class="MathClass-ord">̈</mi></mrow></munderover><mspace width="1em" class="quad"/><munderover
12302
accent="true"><mrow
12304
accent="true"><mrow
12306
class="MathClass-ord">B</mi></mrow><mrow
12309
class="MathClass-ord">˘</mi></mrow></munderover></mrow><mrow
12312
class="MathClass-ord">˘</mi></mrow></munderover><mspace width="1em" class="quad"/><munderover
12313
accent="true"><mrow
12315
accent="true"><mrow
12317
class="MathClass-ord">B</mi></mrow><mrow
12320
class="MathClass-ord">̄</mi></mrow></munderover></mrow><mrow
12323
class="MathClass-ord">̄</mi></mrow></munderover><mspace width="1em" class="quad"/><munderover
12324
accent="true"><mrow
12326
accent="true"><mrow
12328
class="MathClass-ord">V</mi> </mrow><mrow
12331
class="MathClass-ord">→</mi></mrow></munderover></mrow><mrow
12334
class="MathClass-ord">→</mi></mrow></munderover>
12338
<table width="100%"
12339
class="verbatim"><tr class="verbatim"><td
12340
class="verbatim"><pre class="verbatim">
12341
 \[\Hat{\Hat{H}}\quad\Check{\Check{C}}\quad
12342
 \Tilde{\Tilde{T}}\quad\Acute{\Acute{A}}\quad
12343
 \Grave{\Grave{G}}\quad\Dot{\Dot{D}}\quad
12344
 \Ddot{\Ddot{D}}\quad\Breve{\Breve{B}}\quad
12345
 \Bar{\Bar{B}}\quad\Vec{\Vec{V}}\]
12348
<!--l. 1281--><p class="indent"> This double accent operation is complicated and tends to slow down the
12349
processing of a <span class="LATEX">L<span class="A">A</span><span class="TEX">T<span
12350
class="E">E</span>X</span></span> file.
12352
<h4 class="subsectionHead"><span class="titlemark">9.5</span> <a
12353
name="x1-180009.5"></a>Dot accents</h4> <span
12354
class="cmtt-10">\dddot </span>and <span
12355
class="cmtt-10">\ddddot </span>are available to produce triple and quadruple
12356
dot accents in addition to the <span
12357
class="cmtt-10">\dot </span>and <span
12358
class="cmtt-10">\ddot </span>accents already available in <span class="LATEX">L<span class="A">A</span><span class="TEX">T<span
12359
class="E">E</span>X</span></span>: <!--l. 1290--><math
12360
xmlns="http://www.w3.org/1998/Math/MathML"
12365
accent="true"><mrow
12367
class="MathClass-ord">Q</mi></mrow><mrow
12370
class="MathClass-ord">...</mi></mrow></munderover><mspace width="2em" class="qquad"/><munderover
12371
accent="true"><mrow
12373
class="MathClass-ord">R</mi></mrow><mrow
12376
class="MathClass-ord">....</mi></mrow></munderover>
12379
<table width="100%"
12380
class="verbatim"><tr class="verbatim"><td
12381
class="verbatim"><pre class="verbatim">
12382
 \[\dddot{Q}\qquad\ddddot{R}\]
12385
<h4 class="subsectionHead"><span class="titlemark">9.6</span> <a
12386
name="x1-190009.6"></a>Roots</h4>
12387
<!--l. 1297--><p class="noindent">In the <span
12388
class="cmtt-10">amsmath </span>package <span
12389
class="cmtt-10">\leftroot </span>and <span
12390
class="cmtt-10">\uproot </span>allow you to adjust the position of
12391
the root index of a radical:
12394
<table width="100%"
12395
class="verbatim"><tr class="verbatim"><td
12396
class="verbatim"><pre class="verbatim">
12397
 \sqrt[\leftroot{-2}\uproot{2}\beta]{k}
12400
<!--l. 1302--><p class="indent"> gives good positioning of the <!--l. 1302--><math
12401
xmlns="http://www.w3.org/1998/Math/MathML"
12403
class="MathClass-ord">β</mi></math>:
12404
<!--l. 1303--><math
12405
xmlns="http://www.w3.org/1998/Math/MathML"
12406
mode="display"> <mrow
12409
class="MathClass-ord">β</mi><mrow
12411
class="MathClass-ord">k</mi></mrow></msqrt>
12414
<h4 class="subsectionHead"><span class="titlemark">9.7</span> <a
12415
name="x1-200009.7"></a>Boxed formulas</h4> The command <span
12416
class="cmtt-10">\boxed </span>puts a box around its argument, like
12418
class="cmtt-10">\fbox </span>except that the contents are in math mode:
12420
<table width="100%"
12421
class="verbatim"><tr class="verbatim"><td
12422
class="verbatim"><pre class="verbatim">
12423
 \boxed{W_t-F\subseteq V(P_i)\subseteq W_t}
12426
<!--l. 1310--><p class="indent"> <!--l. 1310--><math
12427
xmlns="http://www.w3.org/1998/Math/MathML"
12428
mode="display"> <mrow
12432
class="MathClass-ord">W</mi><mrow
12434
class="MathClass-ord">t</mi></mrow></msub
12436
class="MathClass-bin">−</mo> <mi
12437
class="MathClass-ord">F</mi> <mo
12438
class="MathClass-rel">⊆</mo> <mi
12439
class="MathClass-ord">V</mi> <mrow><mo
12440
class="MathClass-open">(</mo><msub
12442
class="MathClass-ord">P</mi><mrow
12444
class="MathClass-ord">i</mi></mrow></msub
12446
class="MathClass-close">)</mo></mrow> <mo
12447
class="MathClass-rel">⊆</mo> <msub
12449
class="MathClass-ord">W</mi><mrow
12451
class="MathClass-ord">t</mi></mrow></msub
12453
class="MathClass-punc">.</mo>
12456
<h4 class="subsectionHead"><span class="titlemark">9.8</span> <a
12457
name="x1-210009.8"></a>Extensible arrows</h4> <span
12458
class="cmtt-10">\xleftarrow </span>and <span
12459
class="cmtt-10">\xrightarrow </span>produce arrows that extend
12460
automatically to accommodate unusually wide subscripts or superscripts. The text of the
12461
subscript or superscript are given as an optional resp. mandatory argument: Example: <!--l. 1318--><math
12462
xmlns="http://www.w3.org/1998/Math/MathML"
12467
class="MathClass-ord">0</mn><munderover><mrow
12469
class="MathClass-ord">ζ</mi></mrow> <mo
12470
class="MathClass-rel">←</mo> <mrow
12472
class="MathClass-ord">α</mi></mrow></munderover><mi
12473
class="MathClass-ord">F</mi> <mo
12474
class="MathClass-bin">×</mo> <mo
12475
class="MathClass-bin">△</mo><mrow><mo
12476
class="MathClass-open">[</mo><mi
12477
class="MathClass-ord">n</mi> <mo
12478
class="MathClass-bin">−</mo> <mn
12479
class="MathClass-ord">1</mn><mo
12480
class="MathClass-close">]</mo></mrow><munderover><mrow
12482
class="MathClass-rel">→</mo> <mrow
12485
class="MathClass-ord">∂</mi><mrow
12487
class="MathClass-ord">0</mn></mrow></msub
12489
class="MathClass-ord">α</mi><mrow><mo
12490
class="MathClass-open">(</mo><mi
12491
class="MathClass-ord">b</mi><mo
12492
class="MathClass-close">)</mo></mrow></mrow></munderover><msup
12494
class="MathClass-ord">E</mi><mrow
12497
class="MathClass-ord">∂</mi><mrow
12499
class="MathClass-ord">0</mn></mrow></msub
12501
class="MathClass-ord">b</mi></mrow></msup
12505
<table width="100%"
12506
class="verbatim"><tr class="verbatim"><td
12507
class="verbatim"><pre class="verbatim">
12508
 \[0 \xleftarrow[\zeta]{\alpha} F\times\triangle[n-1]
12509
   \xrightarrow{\partial_0\alpha(b)} Eˆ{\partial_0b}\]
12512
<h4 class="subsectionHead"><span class="titlemark">9.9</span> <a
12513
name="x1-220009.9"></a><span
12514
class="cmtt-10">\overset</span>, <span
12515
class="cmtt-10">\underset</span>, and <span
12516
class="cmtt-10">\sideset</span></h4> Examples: <!--l. 1327--><math
12517
xmlns="http://www.w3.org/1998/Math/MathML"
12523
class="MathClass-bin">∗</mo> </mrow><mrow
12525
class="MathClass-ord">X</mi></mrow></mover><mspace width="2em" class="qquad"/><munder><mrow
12527
class="MathClass-bin">∗</mo> </mrow><mrow
12529
class="MathClass-ord">X</mi></mrow></munder><mspace width="2em" class="qquad"/><mover><mrow
12531
class="MathClass-ord">a</mi></mrow><mrow
12534
class="MathClass-ord">b</mi></mrow><mrow
12536
class="MathClass-ord">X</mi></mrow></munder></mrow></mover>
12539
<table width="100%"
12540
class="verbatim"><tr class="verbatim"><td
12541
class="verbatim"><pre class="verbatim">
12542
 \[\overset{*}{X}\qquad\underset{*}{X}\qquad
12543
 \overset{a}{\underset{b}{X}}\]
12546
<!--l. 1334--><p class="indent"> The command <span
12547
class="cmtt-10">\sideset </span>is for a rather special purpose: putting symbols at
12548
the subscript and superscript corners of a large operator symbol such as <!--l. 1336--><math
12549
xmlns="http://www.w3.org/1998/Math/MathML"
12552
class="MathClass-op">∑</mo>
12553
</math> or <!--l. 1336--><math
12554
xmlns="http://www.w3.org/1998/Math/MathML"
12557
class="MathClass-op">∏</mo>
12559
without affecting the placement of limits. Examples: <!--l. 1339--><math
12560
xmlns="http://www.w3.org/1998/Math/MathML"
12568
class="MathClass-bin">∗</mo> </mrow><mrow
12570
class="MathClass-bin">∗</mo> </mrow></msubsup
12573
class="MathClass-op">∏</mo>
12576
class="MathClass-bin">∗</mo> </mrow><mrow
12578
class="MathClass-bin">∗</mo> </mrow></msubsup
12581
class="MathClass-ord">k</mi></mrow></msub
12582
><mspace width="2em" class="qquad"/><msub
12584
class="MathClass-op">∑</mo>
12586
class="MathClass-ord">′</mi>
12589
class="MathClass-ord">0</mn><mo
12590
class="MathClass-rel">≤</mo><mi
12591
class="MathClass-ord">i</mi><mo
12592
class="MathClass-rel">≤</mo><mi
12593
class="MathClass-ord">m</mi></mrow></msub
12596
class="MathClass-ord">E</mi><mrow
12598
class="MathClass-ord">i</mi></mrow></msub
12600
class="MathClass-ord">β</mi><mi
12601
class="MathClass-ord">x</mi>
12605
<table width="100%"
12606
class="verbatim"><tr class="verbatim"><td
12607
class="verbatim"><pre class="verbatim">
12608
 \[\sideset{_*ˆ*}{_*ˆ*}\prod_k\qquad
12609
 \sideset{}{’}\sum_{0\le i\le m} E_i\beta x
12613
<h4 class="subsectionHead"><span class="titlemark">9.10</span> <a
12614
name="x1-230009.10"></a>The <span
12615
class="cmtt-10">\text </span>command</h4> The main use of the command <span
12616
class="cmtt-10">\text </span>is for words or phrases in a
12617
display: <!--l. 1351--><math
12618
xmlns="http://www.w3.org/1998/Math/MathML"
12619
mode="display"> <mrow
12621
<mi class="mathbf">y</mi> <mo
12622
class="MathClass-rel">=</mo> <mi class="mathbf">y</mi><mi
12623
class="MathClass-ord">′</mi><mspace width="1em" class="quad"/><mrow
12624
class="text"><mtext >if and only if</mtext></mrow><mspace width="1em" class="quad"/><mi
12625
class="MathClass-ord">y</mi><msub
12627
class="MathClass-ord">′</mi><mrow
12630
class="MathClass-ord">k</mi></mrow></msub
12632
class="MathClass-rel">=</mo> <msub
12634
class="MathClass-ord">δ</mi><mrow
12636
class="MathClass-ord">k</mi></mrow></msub
12639
class="MathClass-ord">y</mi><mrow
12641
class="MathClass-ord">τ</mi><mrow><mo
12642
class="MathClass-open">(</mo><mi
12643
class="MathClass-ord">k</mi><mo
12644
class="MathClass-close">)</mo></mrow></mrow></msub
12648
<table width="100%"
12649
class="verbatim"><tr class="verbatim"><td
12650
class="verbatim"><pre class="verbatim">
12651
 \[\mathbf{y}=\mathbf{y}’\quad\text{if and only if}\quad
12652
 y’_k=\delta_k y_{\tau(k)}\]
12655
<h4 class="subsectionHead"><span class="titlemark">9.11</span> <a
12656
name="x1-240009.11"></a>Operator names</h4> The more common math functions such as <!--l. 1359--><math
12657
xmlns="http://www.w3.org/1998/Math/MathML"
12660
>log</mo><!--nolimits--></math>, <!--l. 1359--><math
12661
xmlns="http://www.w3.org/1998/Math/MathML"
12664
>sin</mo><!--nolimits--></math>, and <!--l. 1359--><math
12665
xmlns="http://www.w3.org/1998/Math/MathML"
12668
>lim</mo></math> have
12669
predefined control sequences: <span class="obeylines-h"><span
12670
class="cmtt-10">\log</span></span>, <span class="obeylines-h"><span
12671
class="cmtt-10">\sin</span></span>, <span class="obeylines-h"><span
12672
class="cmtt-10">\lim</span></span>. The <span
12673
class="cmtt-10">amsmath </span>package provides
12675
class="cmtt-10">\DeclareMathOperator </span>and <span
12676
class="cmtt-10">\DeclareMathOperator* </span>for producing new
12677
function names that will have the same typographical treatment. Examples: <!--l. 1367--><math
12678
xmlns="http://www.w3.org/1998/Math/MathML"
12684
open="|" close="|" ><mi
12685
class="MathClass-ord">f</mi></mfenced> <mrow
12687
class="MathClass-ord">∞</mi></mrow></msub
12689
class="MathClass-rel">=</mo><msub
12691
class="MathClass-op"> ess sup</mo><!--nolimits--> <mrow
12693
class="MathClass-ord">x</mi><mo
12694
class="MathClass-rel">∈</mo><msup
12696
class="MathClass-ord">R</mi><mrow
12698
class="MathClass-ord">n</mi></mrow></msup
12701
open="|" close="|" ><mi
12702
class="MathClass-ord">f</mi><mrow><mo
12703
class="MathClass-open">(</mo><mi
12704
class="MathClass-ord">x</mi><mo
12705
class="MathClass-close">)</mo></mrow></mfenced>
12708
<table width="100%"
12709
class="verbatim"><tr class="verbatim"><td
12710
class="verbatim"><pre class="verbatim">
12711
 \[\norm{f}_\infty=
12712
 \esssup_{x\in Rˆn}\abs{f(x)}\]
12715
<!--l. 1373--><p class="indent"> <!--l. 1373--><math
12716
xmlns="http://www.w3.org/1998/Math/MathML"
12717
mode="display"> <mrow
12721
class="MathClass-op">meas</mo><!--nolimits--> <mrow
12723
class="MathClass-ord">1</mn></mrow></msub
12725
class="MathClass-open">{</mo><mi
12726
class="MathClass-ord">u</mi> <mo
12727
class="MathClass-rel">∈</mo> <msubsup
12729
class="MathClass-ord">R</mi><mrow
12731
class="MathClass-bin">+</mo> </mrow><mrow
12733
class="MathClass-ord">1</mn></mrow></msubsup
12735
class="MathClass-punc">:</mo> <msup
12737
class="MathClass-ord">f</mi><mrow
12739
class="MathClass-bin">∗</mo></mrow></msup
12741
class="MathClass-open">(</mo><mi
12742
class="MathClass-ord">u</mi><mo
12743
class="MathClass-close">)</mo></mrow> <mo
12744
class="MathClass-rel">></mo> <mi
12745
class="MathClass-ord">α</mi><mo
12746
class="MathClass-close">}</mo></mrow> <mo
12747
class="MathClass-rel">=</mo><msub
12749
class="MathClass-op"> meas</mo><!--nolimits--> <mrow
12752
class="MathClass-ord">n</mi></mrow></msub
12754
class="MathClass-open">{</mo><mi
12755
class="MathClass-ord">x</mi> <mo
12756
class="MathClass-rel">∈</mo> <msup
12758
class="MathClass-ord">R</mi><mrow
12760
class="MathClass-ord">n</mi></mrow></msup
12762
class="MathClass-punc">:</mo> <mfenced
12763
open="|" close="|" ><mi
12764
class="MathClass-ord">f</mi><mrow><mo
12765
class="MathClass-open">(</mo><mi
12766
class="MathClass-ord">x</mi><mo
12767
class="MathClass-close">)</mo></mrow></mfenced> <mo
12768
class="MathClass-rel">≥</mo> <mi
12769
class="MathClass-ord">α</mi><mo
12770
class="MathClass-close">}</mo></mrow><mspace width="1em" class="quad"/><mi
12771
class="MathClass-ord">∀</mi><mi
12772
class="MathClass-ord">α</mi> <mo
12773
class="MathClass-rel">></mo> <mn
12774
class="MathClass-ord">0</mn><mo
12775
class="MathClass-punc">.</mo>
12779
<table width="100%"
12780
class="verbatim"><tr class="verbatim"><td
12781
class="verbatim"><pre class="verbatim">
12782
 \[\meas_1\{u\in R_+ˆ1\colon fˆ*(u)>\alpha\}
12783
 =\meas_n\{x\in Rˆn\colon \abs{f(x)}\geq\alpha\}
12784
 \quad \forall\alpha>0.\]
12787
<!--l. 1381--><p class="indent"> <span
12788
class="cmtt-10">\esssup </span>and <span
12789
class="cmtt-10">\meas </span>would be defined in the document preamble as
12792
<table width="100%"
12793
class="verbatim"><tr class="verbatim"><td
12794
class="verbatim"><pre class="verbatim">
12795
 \DeclareMathOperator*{\esssup}{ess\,sup}
12796
 \DeclareMathOperator{\meas}{meas}
12799
<!--l. 1387--><p class="indent"> The following special operator names are predefined in the <span
12800
class="cmtt-10">amsmath </span>package:
12802
class="cmtt-10">\varlimsup</span>, <span
12803
class="cmtt-10">\varliminf</span>, <span
12804
class="cmtt-10">\varinjlim</span>, and <span
12805
class="cmtt-10">\varprojlim</span>. Here’s what they look like
12807
<!--tex4ht:inline--></p><!--l. 1397--><math
12808
xmlns="http://www.w3.org/1998/Math/MathML"
12809
display="block"><mtable
12812
class="align-odd"></mtd> <mtd
12813
class="align-even"><msub
12815
class="mml-overline"><mrow><mo
12816
>lim</mo></mrow><mo
12817
accent="true">‾</mo></mover><mrow
12819
class="MathClass-ord">n</mi><mo
12820
class="MathClass-rel">→</mo><mi
12821
class="MathClass-ord">∞</mi></mrow></msub
12823
class="MathClass-ord"><!--span
12824
class="htf-calligraphy"-->Q<!--/span--></mi><mrow><mo
12825
class="MathClass-open">(</mo><msub
12827
class="MathClass-ord">u</mi><mrow
12829
class="MathClass-ord">n</mi></mrow></msub
12831
class="MathClass-punc">,</mo> <msub
12833
class="MathClass-ord">u</mi><mrow
12835
class="MathClass-ord">n</mi></mrow></msub
12837
class="MathClass-bin">−</mo> <msup
12839
class="MathClass-ord">u</mi><mrow
12841
class="MathClass-ord">#</mi></mrow></msup
12843
class="MathClass-close">)</mo></mrow> <mo
12844
class="MathClass-rel">≤</mo> <mn
12845
class="MathClass-ord">0</mn> </mtd> <mtd
12846
class="align-label"> <mspace
12847
id="x1-24001r48" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
12848
class="MathClass-open">(</mo><mn
12849
class="MathClass-ord">4</mn><mn
12850
class="MathClass-ord">8</mn><mo
12851
class="MathClass-close">)</mo></mrow>
12852
</mtd></mtr><mtr><mtd
12853
class="align-odd"></mtd> <mtd
12854
class="align-even"><msub
12855
><mover class="mml-underline"><mo
12856
accent="true">‾</mo><mrow><mo
12857
>lim</mo></mrow></mover><mrow
12859
class="MathClass-ord">n</mi><mo
12860
class="MathClass-rel">→</mo><mi
12861
class="MathClass-ord">∞</mi></mrow></msub
12863
open="|" close="|" ><msub
12865
class="MathClass-ord">a</mi><mrow
12867
class="MathClass-ord">n</mi><mo
12868
class="MathClass-bin">+</mo><mn
12869
class="MathClass-ord">1</mn></mrow></msub
12871
class="MathClass-bin">/</mo> <mfenced
12872
open="|" close="|" ><msub
12874
class="MathClass-ord">a</mi><mrow
12876
class="MathClass-ord">n</mi></mrow></msub
12878
class="MathClass-rel">=</mo> <mn
12879
class="MathClass-ord">0</mn></mtd> <mtd
12880
class="align-label"> <mspace
12881
id="x1-24002r49" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
12882
class="MathClass-open">(</mo><mn
12883
class="MathClass-ord">4</mn><mn
12884
class="MathClass-ord">9</mn><mo
12885
class="MathClass-close">)</mo></mrow>
12886
</mtd></mtr><mtr><mtd
12887
class="align-odd"></mtd> <mtd
12888
class="align-even"><munder><mrow
12890
class="MathClass-rel">→</mo> </mrow><mrow
12891
></mrow></munder><msup
12893
class="MathClass-open">(</mo><msubsup
12895
class="MathClass-ord">m</mi><mrow
12897
class="MathClass-ord">i</mi></mrow><mrow
12899
class="MathClass-ord">λ</mi></mrow></msubsup
12901
class="MathClass-punc">·</mo><mo
12902
class="MathClass-close">)</mo></mrow><mrow
12904
class="MathClass-bin">∗</mo></mrow></msup
12906
class="MathClass-rel">≤</mo> <mn
12907
class="MathClass-ord">0</mn></mtd> <mtd
12908
class="align-label"> <mspace
12909
id="x1-24003r50" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
12910
class="MathClass-open">(</mo><mn
12911
class="MathClass-ord">5</mn><mn
12912
class="MathClass-ord">0</mn><mo
12913
class="MathClass-close">)</mo></mrow>
12914
</mtd></mtr><mtr><mtd
12915
class="align-odd"></mtd> <mtd
12916
class="align-even"><msub
12919
class="MathClass-rel">←</mo> </mrow><mrow
12920
></mrow></munder><mrow
12922
class="MathClass-ord">p</mi><mo
12923
class="MathClass-rel">∈</mo><mi
12924
class="MathClass-ord">S</mi><mrow><mo
12925
class="MathClass-open">(</mo><mi
12926
class="MathClass-ord">A</mi><mo
12927
class="MathClass-close">)</mo></mrow></mrow></msub
12930
class="MathClass-ord">A</mi><mrow
12932
class="MathClass-ord">p</mi></mrow></msub
12934
class="MathClass-rel">≤</mo> <mn
12935
class="MathClass-ord">0</mn></mtd> <mtd
12936
class="align-label"> <mspace
12937
id="x1-24004r51" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
12938
class="MathClass-open">(</mo><mn
12939
class="MathClass-ord">5</mn><mn
12940
class="MathClass-ord">1</mn><mo
12941
class="MathClass-close">)</mo></mrow>
12942
</mtd></mtr></mtable></math>
12944
<table width="100%"
12945
class="verbatim"><tr class="verbatim"><td
12946
class="verbatim"><pre class="verbatim">
12947
 \begin{align}
12948
 &\varlimsup_{n\rightarrow\infty}
12949
        \mathcal{Q}(u_n,u_n-uˆ{\#})\le0\\
12950
 &\varliminf_{n\rightarrow\infty}
12951
   \left\lvert a_{n+1}\right\rvert/\left\lvert a_n\right\rvert=0\\
12952
 &\varinjlim (m_iˆ\lambda\cdot)ˆ*\le0\\
12953
 &\varprojlim_{p\in S(A)}A_p\le0
12954
 \end{align}
12957
<h4 class="subsectionHead"><span class="titlemark">9.12</span> <a
12958
name="x1-250009.12"></a><span
12959
class="cmtt-10">\mod </span>and its relatives</h4> The commands <span
12960
class="cmtt-10">\mod </span>and <span
12961
class="cmtt-10">\pod </span>are variants of <span
12962
class="cmtt-10">\pmod</span>
12963
preferred by some authors; <span
12964
class="cmtt-10">\mod </span>omits the parentheses, whereas <span
12965
class="cmtt-10">\pod </span>omits the
12966
‘mod’ and retains the parentheses. Examples:
12967
<!--tex4ht:inline--><!--l. 1418--><math
12968
xmlns="http://www.w3.org/1998/Math/MathML"
12969
display="block"><mtable
12972
class="align-odd"><mi
12973
class="MathClass-ord">x</mi></mtd> <mtd
12974
class="align-even"> <mo
12975
class="MathClass-rel">≡</mo> <mi
12976
class="MathClass-ord">y</mi> <mo
12977
class="MathClass-bin">+</mo> <mn
12978
class="MathClass-ord">1</mn> <mrow><mo
12979
class="MathClass-open">(</mo><mi
12980
class="MathClass-ord">m</mi><mi
12981
class="MathClass-ord">o</mi><mi
12982
class="MathClass-ord">d</mi> <msup
12984
class="MathClass-ord">m</mi><mrow
12986
class="MathClass-ord">2</mn></mrow></msup
12988
class="MathClass-close">)</mo></mrow></mtd> <mtd
12989
class="align-label"> <mspace
12990
id="x1-25001r52" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
12991
class="MathClass-open">(</mo><mn
12992
class="MathClass-ord">5</mn><mn
12993
class="MathClass-ord">2</mn><mo
12994
class="MathClass-close">)</mo></mrow>
12995
</mtd></mtr><mtr><mtd
12996
class="align-odd"><mi
12997
class="MathClass-ord">x</mi></mtd> <mtd
12998
class="align-even"> <mo
12999
class="MathClass-rel">≡</mo> <mi
13000
class="MathClass-ord">y</mi> <mo
13001
class="MathClass-bin">+</mo> <mn
13002
class="MathClass-ord">1</mn> <mi
13003
class="MathClass-ord">m</mi><mi
13004
class="MathClass-ord">o</mi><mi
13005
class="MathClass-ord">d</mi> <msup
13007
class="MathClass-ord">m</mi><mrow
13009
class="MathClass-ord">2</mn></mrow></msup
13011
class="align-label"> <mspace
13012
id="x1-25002r53" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
13013
class="MathClass-open">(</mo><mn
13014
class="MathClass-ord">5</mn><mn
13015
class="MathClass-ord">3</mn><mo
13016
class="MathClass-close">)</mo></mrow>
13017
</mtd></mtr><mtr><mtd
13018
class="align-odd"><mi
13019
class="MathClass-ord">x</mi></mtd> <mtd
13020
class="align-even"> <mo
13021
class="MathClass-rel">≡</mo> <mi
13022
class="MathClass-ord">y</mi> <mo
13023
class="MathClass-bin">+</mo> <mn
13024
class="MathClass-ord">1</mn> <mrow><mo
13025
class="MathClass-open">(</mo><msup
13027
class="MathClass-ord">m</mi><mrow
13029
class="MathClass-ord">2</mn></mrow></msup
13031
class="MathClass-close">)</mo></mrow></mtd> <mtd
13032
class="align-label"> <mspace
13033
id="x1-25003r54" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
13034
class="MathClass-open">(</mo><mn
13035
class="MathClass-ord">5</mn><mn
13036
class="MathClass-ord">4</mn><mo
13037
class="MathClass-close">)</mo></mrow>
13038
</mtd></mtr></mtable></math>
13040
<table width="100%"
13041
class="verbatim"><tr class="verbatim"><td
13042
class="verbatim"><pre class="verbatim">
13043
 \begin{align}
13044
 x&\equiv y+1\pmod{mˆ2}\\
13045
 x&\equiv y+1\mod{mˆ2}\\
13046
 x&\equiv y+1\pod{mˆ2}
13047
 \end{align}
13050
<h4 class="subsectionHead"><span class="titlemark">9.13</span> <a
13051
name="x1-260009.13"></a>Fractions and related constructions</h4>
13052
<!--l. 1430--><p class="noindent">The usual notation for binomials is similar to the fraction concept, so it has a similar
13054
class="cmtt-10">\binom </span>with two arguments. Example: </p><table class="equation"><tr><td>
13055
<!--l. 1432--><math
13056
xmlns="http://www.w3.org/1998/Math/MathML"
13059
class="equation"><mtr><mtd>
13060
<mtable class="split"><mtr><mtd>
13062
class="split-mtr"></mrow><mrow
13063
class="split-mtd"></mrow> <msub
13065
class="MathClass-op">∑</mo>
13068
class="MathClass-ord">γ</mi><mo
13069
class="MathClass-rel">∈</mo><msub
13071
class="MathClass-ord">Γ</mi><mrow
13073
class="MathClass-ord">C</mi> </mrow></msub
13077
class="MathClass-ord">I</mi><mrow
13079
class="MathClass-ord">γ</mi></mrow></msub
13081
class="split-mtd"></mrow> <mo
13082
class="MathClass-rel">=</mo> <msup
13084
class="MathClass-ord">2</mn><mrow
13086
class="MathClass-ord">k</mi></mrow></msup
13088
class="MathClass-bin">−</mo> <mfenced open="(" close=")" class="binom"><mover
13089
class="binom"><mrow><mi
13090
class="MathClass-ord">k</mi></mrow><mrow><mn
13091
class="MathClass-ord">1</mn></mrow></mover></mfenced><msup
13093
class="MathClass-ord">2</mn><mrow
13095
class="MathClass-ord">k</mi><mo
13096
class="MathClass-bin">−</mo><mn
13097
class="MathClass-ord">1</mn></mrow></msup
13099
class="MathClass-bin">+</mo> <mfenced open="(" close=")" class="binom"><mover
13100
class="binom"><mrow><mi
13101
class="MathClass-ord">k</mi></mrow><mrow><mn
13102
class="MathClass-ord">2</mn></mrow></mover></mfenced><msup
13104
class="MathClass-ord">2</mn><mrow
13106
class="MathClass-ord">k</mi><mo
13107
class="MathClass-bin">−</mo><mn
13108
class="MathClass-ord">2</mn></mrow></msup
13111
class="split-mtr"></mrow><mrow
13112
class="split-mtd"></mrow> <mrow
13113
class="split-mtd"></mrow><mspace width="1em" class="quad"/> <mo
13114
class="MathClass-bin">+</mo> <mo
13115
class="MathClass-rel">⋯</mo> <mo
13116
class="MathClass-bin">+</mo> <msup
13118
class="MathClass-open">(</mo><mo
13119
class="MathClass-bin">−</mo><mn
13120
class="MathClass-ord">1</mn><mo
13121
class="MathClass-close">)</mo></mrow><mrow
13123
class="MathClass-ord">l</mi></mrow></msup
13124
><mfenced open="(" close=")" class="binom"><mover
13125
class="binom"><mrow><mi
13126
class="MathClass-ord">k</mi></mrow><mrow><mi
13127
class="MathClass-ord">l</mi></mrow></mover></mfenced><msup
13129
class="MathClass-ord">2</mn><mrow
13131
class="MathClass-ord">k</mi><mo
13132
class="MathClass-bin">−</mo><mi
13133
class="MathClass-ord">l</mi></mrow></msup
13135
class="MathClass-bin">+</mo> <mo
13136
class="MathClass-rel">⋯</mo> <mo
13137
class="MathClass-bin">+</mo> <msup
13139
class="MathClass-open">(</mo><mo
13140
class="MathClass-bin">−</mo><mn
13141
class="MathClass-ord">1</mn><mo
13142
class="MathClass-close">)</mo></mrow><mrow
13144
class="MathClass-ord">k</mi></mrow></msup
13147
class="split-mtr"></mrow><mrow
13148
class="split-mtd"></mrow> <mrow
13149
class="split-mtd"></mrow> <mo
13150
class="MathClass-rel">=</mo> <msup
13152
class="MathClass-open">(</mo><mn
13153
class="MathClass-ord">2</mn> <mo
13154
class="MathClass-bin">−</mo> <mn
13155
class="MathClass-ord">1</mn><mo
13156
class="MathClass-close">)</mo></mrow><mrow
13158
class="MathClass-ord">k</mi></mrow></msup
13160
class="MathClass-rel">=</mo> <mn
13161
class="MathClass-ord">1</mn>
13162
</mtd></mtr></mtable> </mtd><mtd><mspace
13163
id="x1-26001r55" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /></mtd></mtr></mtable>
13165
<!--l. 1440--><p class="nopar"></p></td><td width="5%">(55)</td></tr></table>
13167
<table width="100%"
13168
class="verbatim"><tr class="verbatim"><td
13169
class="verbatim"><pre class="verbatim">
13170
 \begin{equation}
13171
 \begin{split}
13172
 [\sum_{\gamma\in\Gamma_C} I_\gamma&
13173
 =2ˆk-\binom{k}{1}2ˆ{k-1}+\binom{k}{2}2ˆ{k-2}\\
13174
 &\quad+\dots+(-1)ˆl\binom{k}{l}2ˆ{k-l}
13175
 +\dots+(-1)ˆk\\
13176
 &=(2-1)ˆk=1
13177
 \end{split}
13178
 \end{equation}
13181
<!--l. 1452--><p class="indent"> There are also abbreviations
13184
<table width="100%"
13185
class="verbatim"><tr class="verbatim"><td
13186
class="verbatim"><pre class="verbatim">
13187
 \dfrac        \dbinom
13188
 \tfrac        \tbinom
13191
<!--l. 1457--><p class="indent"> for the commonly needed constructions
13194
<table width="100%"
13195
class="verbatim"><tr class="verbatim"><td
13196
class="verbatim"><pre class="verbatim">
13197
 {\displaystyle\frac ... }   {\displaystyle\binom ... }
13198
 {\textstyle\frac ... }      {\textstyle\binom ... }
13201
<!--l. 1463--><p class="indent"> The generalized fraction command <span
13202
class="cmtt-10">\genfrac </span>provides full access to the six <span class="TEX">T<span
13203
class="E">E</span>X</span>
13204
fraction primitives:
13205
<!--tex4ht:inline--></p><!--l. 1475--><math
13206
xmlns="http://www.w3.org/1998/Math/MathML"
13207
display="block"><mtable
13210
class="align-odd"><mrow
13211
class="text"><mtext >\over: </mtext></mrow></mtd> <mtd
13212
class="align-even"><mi
13213
class="MathClass-ord">n</mi> <mo
13214
class="MathClass-bin">+</mo> <mn
13215
class="MathClass-ord">1</mn><mn
13216
class="MathClass-ord">2</mn></mtd> <mtd
13217
class="align-odd"><mrow
13218
class="text"><mtext >\overwithdelims: </mtext></mrow></mtd> <mtd
13219
class="align-even"><mi
13220
class="MathClass-ord">n</mi> <mo
13221
class="MathClass-bin">+</mo> <mn
13222
class="MathClass-ord">1</mn><mn
13223
class="MathClass-ord">2</mn></mtd> <mtd
13224
class="align-label"> <mspace
13225
id="x1-26002r56" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
13226
class="MathClass-open">(</mo><mn
13227
class="MathClass-ord">5</mn><mn
13228
class="MathClass-ord">6</mn><mo
13229
class="MathClass-close">)</mo></mrow>
13230
</mtd></mtr><mtr><mtd
13231
class="align-odd"><mrow
13232
class="text"><mtext >\atop: </mtext></mrow></mtd> <mtd
13233
class="align-even"><mi
13234
class="MathClass-ord">n</mi> <mo
13235
class="MathClass-bin">+</mo> <mn
13236
class="MathClass-ord">1</mn><mn
13237
class="MathClass-ord">2</mn></mtd> <mtd
13238
class="align-odd"><mrow
13239
class="text"><mtext >\atopwithdelims: </mtext></mrow></mtd> <mtd
13240
class="align-even"><mi
13241
class="MathClass-ord">n</mi> <mo
13242
class="MathClass-bin">+</mo> <mn
13243
class="MathClass-ord">1</mn><mn
13244
class="MathClass-ord">2</mn></mtd> <mtd
13245
class="align-label"> <mspace
13246
id="x1-26003r57" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
13247
class="MathClass-open">(</mo><mn
13248
class="MathClass-ord">5</mn><mn
13249
class="MathClass-ord">7</mn><mo
13250
class="MathClass-close">)</mo></mrow>
13251
</mtd></mtr><mtr><mtd
13252
class="align-odd"><mrow
13253
class="text"><mtext >\above: </mtext></mrow></mtd> <mtd
13254
class="align-even"><mi
13255
class="MathClass-ord">n</mi> <mo
13256
class="MathClass-bin">+</mo> <mn
13257
class="MathClass-ord">1</mn><mn
13258
class="MathClass-ord">2</mn></mtd> <mtd
13259
class="align-odd"><mrow
13260
class="text"><mtext >\abovewithdelims: </mtext></mrow></mtd> <mtd
13261
class="align-even"><mi
13262
class="MathClass-ord">n</mi> <mo
13263
class="MathClass-bin">+</mo> <mn
13264
class="MathClass-ord">1</mn><mn
13265
class="MathClass-ord">2</mn></mtd> <mtd
13266
class="align-label"> <mspace
13267
id="x1-26004r58" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
13268
class="MathClass-open">(</mo><mn
13269
class="MathClass-ord">5</mn><mn
13270
class="MathClass-ord">8</mn><mo
13271
class="MathClass-close">)</mo></mrow>
13272
</mtd></mtr></mtable></math>
13274
<table width="100%"
13275
class="verbatim"><tr class="verbatim"><td
13276
class="verbatim"><pre class="verbatim">
13277
 \text{\cn{over}: }&\genfrac{}{}{}{}{n+1}{2}&
13278
 \text{\cn{overwithdelims}: }&
13279
   \genfrac{\langle}{\rangle}{}{}{n+1}{2}\\
13280
 \text{\cn{atop}: }&\genfrac{}{}{0pt}{}{n+1}{2}&
13281
 \text{\cn{atopwithdelims}: }&
13282
   \genfrac{(}{)}{0pt}{}{n+1}{2}\\
13283
 \text{\cn{above}: }&\genfrac{}{}{1pt}{}{n+1}{2}&
13284
 \text{\cn{abovewithdelims}: }&
13285
   \genfrac{[}{]}{1pt}{}{n+1}{2}
13288
<h4 class="subsectionHead"><span class="titlemark">9.14</span> <a
13289
name="x1-270009.14"></a>Continued fractions</h4> The continued fraction <table class="equation"><tr><td>
13290
<!--l. 1490--><math
13291
xmlns="http://www.w3.org/1998/Math/MathML"
13294
class="equation"><mtr><mtd>
13297
class="MathClass-ord">1</mn></mrow>
13300
class="MathClass-ord">√</mi>
13302
class="MathClass-ord">2</mn></msqrt> <mo
13303
class="MathClass-bin">+</mo> <mfrac><mrow
13305
class="MathClass-ord">1</mn></mrow>
13308
class="MathClass-ord">√</mi>
13310
class="MathClass-ord">2</mn></msqrt> <mo
13311
class="MathClass-bin">+</mo> <mfrac><mrow
13313
class="MathClass-ord">1</mn></mrow>
13316
class="MathClass-ord">√</mi>
13318
class="MathClass-ord">2</mn></msqrt> <mo
13319
class="MathClass-bin">+</mo> <mfrac><mrow
13321
class="MathClass-ord">1</mn></mrow>
13324
class="MathClass-ord">√</mi>
13326
class="MathClass-ord">2</mn></msqrt> <mo
13327
class="MathClass-bin">+</mo> <mfrac><mrow
13329
class="MathClass-ord">1</mn></mrow>
13332
class="MathClass-ord">√</mi>
13334
class="MathClass-ord">2</mn></msqrt> <mo
13335
class="MathClass-bin">+</mo> <mo
13336
class="MathClass-rel">⋯</mo></mrow></mfrac></mrow></mfrac></mrow></mfrac></mrow></mfrac></mrow></mfrac></mtd><mtd><mspace
13337
id="x1-27001r59" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /> </mtd></mtr></mtable>
13339
<!--l. 1497--><p class="nopar"></p></td><td width="5%">(59)</td></tr></table>
13340
can be obtained by typing
13342
<table width="100%"
13343
class="verbatim"><tr class="verbatim"><td
13344
class="verbatim"><pre class="verbatim">
13345
 \cfrac{1}{\sqrt{2}+
13346
  \cfrac{1}{\sqrt{2}+
13347
   \cfrac{1}{\sqrt{2}+
13348
    \cfrac{1}{\sqrt{2}+
13349
     \cfrac{1}{\sqrt{2}+\dotsb
13353
<!--l. 1507--><p class="indent"> Left or right placement of any of the numerators is accomplished by using
13355
class="cmtt-10">\cfrac[l] </span>or <span
13356
class="cmtt-10">\cfrac[r] </span>instead of <span
13357
class="cmtt-10">\cfrac</span>.
13359
<h4 class="subsectionHead"><span class="titlemark">9.15</span> <a
13360
name="x1-280009.15"></a>Smash</h4>
13361
<!--l. 1512--><p class="noindent">In <span
13362
class="cmtt-10">amsmath </span>there are optional arguments <span class="obeylines-h"><span
13363
class="cmtt-10">t</span></span> and <span class="obeylines-h"><span
13364
class="cmtt-10">b</span></span> for the plain <span class="TEX">T<span
13365
class="E">E</span>X</span> command <span
13366
class="cmtt-10">\smash</span>,
13367
because sometimes it is advantageous to be able to ‘smash’ only the top or only the
13368
bottom of something while retaining the natural depth or height. In the formula <!--l. 1516--><math
13369
xmlns="http://www.w3.org/1998/Math/MathML"
13373
class="MathClass-ord">X</mi><mrow
13375
class="MathClass-ord">j</mi></mrow></msub
13377
class="MathClass-rel">=</mo> <mrow><mo
13378
class="MathClass-open">(</mo><mn
13379
class="MathClass-ord">1</mn><mo
13380
class="MathClass-bin">/</mo><msqrt><!--<mi
13381
class="MathClass-ord">√</mi>
13384
class="MathClass-ord">λ</mi><mrow
13386
class="MathClass-ord">j</mi></mrow></msub
13388
class="MathClass-close">)</mo></mrow><msub
13390
class="MathClass-ord">X</mi><mrow
13392
class="MathClass-ord">j</mi></mrow></msub
13394
class="MathClass-ord">′</mi></math>
13396
class="cmtt-10">\smash</span><span class="obeylines-h"><span
13397
class="cmtt-10">[b]</span></span> has been used to limit the size of the radical symbol.
13400
<table width="100%"
13401
class="verbatim"><tr class="verbatim"><td
13402
class="verbatim"><pre class="verbatim">
13403
 $X_j=(1/\sqrt{\smash[b]{\lambda_j}})X_j’$
13406
<!--l. 1521--><p class="indent"> Without the use of <span
13407
class="cmtt-10">\smash</span><span class="obeylines-h"><span
13408
class="cmtt-10">[b]</span></span> the formula would have appeared thus: <!--l. 1522--><math
13409
xmlns="http://www.w3.org/1998/Math/MathML"
13413
class="MathClass-ord">X</mi><mrow
13415
class="MathClass-ord">j</mi></mrow></msub
13417
class="MathClass-rel">=</mo> <mrow><mo
13418
class="MathClass-open">(</mo><mn
13419
class="MathClass-ord">1</mn><mo
13420
class="MathClass-bin">/</mo><msqrt><!--<mi
13421
class="MathClass-ord">∘</mi>
13424
class="MathClass-ord">λ</mi><mrow
13426
class="MathClass-ord">j</mi></mrow></msub
13428
class="MathClass-close">)</mo></mrow><msub
13430
class="MathClass-ord">X</mi><mrow
13432
class="MathClass-ord">j</mi></mrow></msub
13434
class="MathClass-ord">′</mi></math>,
13435
with the radical extending to encompass the depth of the subscript <!--l. 1523--><math
13436
xmlns="http://www.w3.org/1998/Math/MathML"
13439
class="MathClass-ord">j</mi></math>.
13441
<h4 class="subsectionHead"><span class="titlemark">9.16</span> <a
13442
name="x1-290009.16"></a>The ‘cases’ environment</h4> ‘Cases’ constructions like the following can be
13443
produced using the <span
13444
class="cmtt-10">cases </span>environment. <table class="equation"><tr><td>
13445
<!--l. 1528--><math
13446
xmlns="http://www.w3.org/1998/Math/MathML"
13449
class="equation"><mtr><mtd>
13452
class="MathClass-ord">P</mi><mrow
13454
class="MathClass-ord">r</mi><mo
13455
class="MathClass-bin">−</mo><mi
13456
class="MathClass-ord">j</mi></mrow></msub
13458
class="MathClass-rel">=</mo> <mfenced
13459
open="{" close="" ><mtable
13460
equalrows="false" equalcolumns="false" class="array"><mtr><mtd
13462
class="MathClass-ord">0</mn> <mspace width="1em" class="quad"/></mtd><mtd
13463
class="array" ><mrow
13464
class="text"><mtext >if </mtext><mrow
13466
class="MathClass-ord">r</mi> <mo
13467
class="MathClass-bin">−</mo> <mi
13468
class="MathClass-ord">j</mi></mrow><mtext > is odd</mtext></mrow><mo
13469
class="MathClass-punc">,</mo> </mtd>
13472
class="MathClass-ord">r</mi><mi
13473
class="MathClass-ord">!</mi> <msup
13475
class="MathClass-open">(</mo><mo
13476
class="MathClass-bin">−</mo><mn
13477
class="MathClass-ord">1</mn><mo
13478
class="MathClass-close">)</mo></mrow><mrow
13480
class="MathClass-open">(</mo><mi
13481
class="MathClass-ord">r</mi><mo
13482
class="MathClass-bin">−</mo><mi
13483
class="MathClass-ord">j</mi><mo
13484
class="MathClass-close">)</mo></mrow><mo
13485
class="MathClass-bin">/</mo><mn
13486
class="MathClass-ord">2</mn></mrow></msup
13487
><mspace width="1em" class="quad"/></mtd><mtd
13488
class="array" ><mrow
13489
class="text"><mtext >if </mtext><mrow
13491
class="MathClass-ord">r</mi> <mo
13492
class="MathClass-bin">−</mo> <mi
13493
class="MathClass-ord">j</mi></mrow><mtext > is even</mtext></mrow><mo
13494
class="MathClass-punc">.</mo></mtd></mtr> <!--@{}l@{\quad }l@{}--></mtable> </mfenced> </mtd><mtd><mspace
13495
id="x1-29001r60" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /></mtd></mtr></mtable>
13497
<!--l. 1534--><p class="nopar"></p></td><td width="5%">(60)</td></tr></table>
13499
<table width="100%"
13500
class="verbatim"><tr class="verbatim"><td
13501
class="verbatim"><pre class="verbatim">
13502
 \begin{equation} P_{r-j}=
13503
   \begin{cases}
13504
     0&  \text{if $r-j$ is odd},\\
13505
     r!\,(-1)ˆ{(r-j)/2}&  \text{if $r-j$ is even}.
13506
   \end{cases}
13507
 \end{equation}
13510
<!--l. 1543--><p class="indent"> Notice the use of <span
13511
class="cmtt-10">\text </span>and the embedded math.
13513
<h4 class="subsectionHead"><span class="titlemark">9.17</span> <a
13514
name="x1-300009.17"></a>Matrix</h4>
13515
<!--l. 1547--><p class="noindent">Here are samples of the matrix environments, <span
13516
class="cmtt-10">\matrix</span>, <span
13517
class="cmtt-10">\pmatrix</span>, <span
13518
class="cmtt-10">\bmatrix</span>,
13520
class="cmtt-10">\Bmatrix</span>, <span
13521
class="cmtt-10">\vmatrix </span>and <span
13522
class="cmtt-10">\Vmatrix</span>: </p><table class="equation"><tr><td>
13523
<!--l. 1550--><math
13524
xmlns="http://www.w3.org/1998/Math/MathML"
13527
class="equation"><mtr><mtd>
13529
equalrows="false" equalcolumns="false" class="array"><mtr><mtd
13531
class="MathClass-ord">ϑ</mi></mtd> <mtd
13532
class="array" > <mi
13533
class="MathClass-ord">ϱ</mi> </mtd>
13536
class="MathClass-ord">ϕ</mi></mtd><mtd
13538
class="MathClass-ord">ϖ</mi></mtd></mtr> <!--*\c@MaxMatrixCols c--></mtable> <mspace width="1em" class="quad"/> <mfenced
13539
open="(" close=")" ><mtable
13540
equalrows="false" equalcolumns="false" class="array"><mtr><mtd
13542
class="MathClass-ord">ϑ</mi></mtd> <mtd
13543
class="array" > <mi
13544
class="MathClass-ord">ϱ</mi> </mtd>
13547
class="MathClass-ord">ϕ</mi></mtd><mtd
13549
class="MathClass-ord">ϖ</mi></mtd></mtr> <!--*\c@MaxMatrixCols c--></mtable> </mfenced> <mspace width="1em" class="quad"/> <mfenced
13550
open="[" close="]" ><mtable
13551
equalrows="false" equalcolumns="false" class="array"><mtr><mtd
13553
class="MathClass-ord">ϑ</mi></mtd> <mtd
13554
class="array" > <mi
13555
class="MathClass-ord">ϱ</mi> </mtd>
13558
class="MathClass-ord">ϕ</mi></mtd><mtd
13560
class="MathClass-ord">ϖ</mi></mtd></mtr> <!--*\c@MaxMatrixCols c--></mtable> </mfenced> <mspace width="1em" class="quad"/> <mfenced
13561
open="{" close="}" ><mtable
13562
equalrows="false" equalcolumns="false" class="array"><mtr><mtd
13564
class="MathClass-ord">ϑ</mi></mtd> <mtd
13565
class="array" > <mi
13566
class="MathClass-ord">ϱ</mi> </mtd>
13569
class="MathClass-ord">ϕ</mi></mtd><mtd
13571
class="MathClass-ord">ϖ</mi></mtd></mtr> <!--*\c@MaxMatrixCols c--></mtable> </mfenced> <mspace width="1em" class="quad"/> <mfenced
13572
open="|" close="|" ><mtable
13573
equalrows="false" equalcolumns="false" class="array"><mtr><mtd
13575
class="MathClass-ord">ϑ</mi></mtd> <mtd
13576
class="array" > <mi
13577
class="MathClass-ord">ϱ</mi> </mtd>
13580
class="MathClass-ord">ϕ</mi></mtd><mtd
13582
class="MathClass-ord">ϖ</mi></mtd></mtr> <!--*\c@MaxMatrixCols c--></mtable> </mfenced> <mspace width="1em" class="quad"/> <mfenced
13583
open="|" close="|" ><mtable
13584
equalrows="false" equalcolumns="false" class="array"><mtr><mtd
13586
class="MathClass-ord">ϑ</mi></mtd> <mtd
13587
class="array" > <mi
13588
class="MathClass-ord">ϱ</mi> </mtd>
13591
class="MathClass-ord">ϕ</mi></mtd><mtd
13593
class="MathClass-ord">ϖ</mi></mtd></mtr> <!--*\c@MaxMatrixCols c--></mtable> </mfenced> </mtd><mtd><mspace
13594
id="x1-30001r61" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /></mtd></mtr></mtable>
13596
<!--l. 1569--><p class="nopar"></p></td><td width="5%">(61)</td></tr></table>
13598
<table width="100%"
13599
class="verbatim"><tr class="verbatim"><td
13600
class="verbatim"><pre class="verbatim">
13601
 \begin{matrix}
13602
 \vartheta& \varrho\\\varphi& \varpi
13603
 \end{matrix}\quad
13604
 \begin{pmatrix}
13605
 \vartheta& \varrho\\\varphi& \varpi
13606
 \end{pmatrix}\quad
13607
 \begin{bmatrix}
13608
 \vartheta& \varrho\\\varphi& \varpi
13609
 \end{bmatrix}\quad
13610
 \begin{Bmatrix}
13611
 \vartheta& \varrho\\\varphi& \varpi
13612
 \end{Bmatrix}\quad
13613
 \begin{vmatrix}
13614
 \vartheta& \varrho\\\varphi& \varpi
13615
 \end{vmatrix}\quad
13616
 \begin{Vmatrix}
13617
 \vartheta& \varrho\\\varphi& \varpi
13618
 \end{Vmatrix}
13621
<!--l. 1592--><p class="indent"> To produce a small matrix suitable for use in text, use the <span
13622
class="cmtt-10">smallmatrix</span>
13626
<table width="100%"
13627
class="verbatim"><tr class="verbatim"><td
13628
class="verbatim"><pre class="verbatim">
13629
 \begin{math}
13630
   \bigl( \begin{smallmatrix}
13631
       a&b\\ c&d
13632
     \end{smallmatrix} \bigr)
13633
 \end{math}
13636
<!--l. 1601--><p class="indent"> To show the effect of the matrix on the surrounding lines of a paragraph, we put it here: <!--l. 1603--><math
13637
xmlns="http://www.w3.org/1998/Math/MathML"
13640
class="MathClass-open">(
13642
class="smallmatrix">
13644
class="MathClass-ord">a</mi></mtd>
13646
class="MathClass-ord">b</mi></mtd>
13648
class="smallmatrix">
13651
class="smallmatrix">
13654
class="smallmatrix">
13656
class="MathClass-ord">c</mi></mtd>
13658
class="MathClass-ord">d</mi></mtd>
13659
</mtr> </mtable><mo
13660
class="MathClass-close">)
13661
</mo></mrow></math>
13662
and follow it with enough text to ensure that there will be at least one full line below
13664
</p><!--l. 1611--><p class="noindent"><span
13665
class="cmtt-10">\hdotsfor</span><span class="obeylines-h"><span
13666
class="cmtt-10">{</span></span><span
13667
class="cmti-10">number</span><span class="obeylines-h"><span
13668
class="cmtt-10">}</span></span> produces a row of dots in a matrix spanning the given number of
13669
columns: <!--l. 1613--><math
13670
xmlns="http://www.w3.org/1998/Math/MathML"
13671
mode="display"> <mrow
13674
class="MathClass-ord">W</mi><mrow><mo
13675
class="MathClass-open">(</mo><mi
13676
class="MathClass-ord">Φ</mi><mo
13677
class="MathClass-close">)</mo></mrow> <mo
13678
class="MathClass-rel">=</mo> <mfenced
13679
open="|" close="|" ><mtable
13680
equalrows="false" equalcolumns="false" class="array"><mtr><mtd
13681
class="array" > <mfrac class="dfrac"><mrow><mi
13682
class="MathClass-ord">ϕ</mi></mrow><mrow><mrow><mo
13683
class="MathClass-open">(</mo><msub
13685
class="MathClass-ord">ϕ</mi><mrow
13687
class="MathClass-ord">1</mn></mrow></msub
13689
class="MathClass-punc">,</mo> <msub
13691
class="MathClass-ord">ɛ</mi><mrow
13693
class="MathClass-ord">1</mn></mrow></msub
13695
class="MathClass-close">)</mo></mrow></mrow></mfrac> </mtd><mtd
13696
class="array" > <mn
13697
class="MathClass-ord">0</mn> </mtd><mtd
13699
class="MathClass-op">…</mo> </mtd><mtd
13700
class="array" > <mn
13701
class="MathClass-ord">0</mn> </mtd>
13703
class="array" ><mfrac class="dfrac"><mrow><mi
13704
class="MathClass-ord">ϕ</mi><msub
13706
class="MathClass-ord">k</mi><mrow
13708
class="MathClass-ord">n</mi><mn
13709
class="MathClass-ord">2</mn></mrow></msub
13710
></mrow><mrow><mrow><mo
13711
class="MathClass-open">(</mo><msub
13713
class="MathClass-ord">ϕ</mi><mrow
13715
class="MathClass-ord">2</mn></mrow></msub
13717
class="MathClass-punc">,</mo> <msub
13719
class="MathClass-ord">ɛ</mi><mrow
13721
class="MathClass-ord">1</mn></mrow></msub
13723
class="MathClass-close">)</mo></mrow></mrow></mfrac></mtd> <mtd
13724
class="array" > <mfrac class="dfrac"><mrow><mi
13725
class="MathClass-ord">ϕ</mi></mrow><mrow><mrow><mo
13726
class="MathClass-open">(</mo><msub
13728
class="MathClass-ord">ϕ</mi><mrow
13730
class="MathClass-ord">2</mn></mrow></msub
13732
class="MathClass-punc">,</mo> <msub
13734
class="MathClass-ord">ɛ</mi><mrow
13736
class="MathClass-ord">2</mn></mrow></msub
13738
class="MathClass-close">)</mo></mrow></mrow></mfrac> </mtd><mtd
13740
class="MathClass-op">…</mo> </mtd><mtd
13741
class="array" > <mn
13742
class="MathClass-ord">0</mn> </mtd>
13744
class="array" columnspan="5" columnalign="center"><mrow
13745
class="multicolumn-columnalign-center"><mo
13746
class="MathClass-punc">.</mo> <mo
13747
class="MathClass-punc">.</mo><mo
13748
class="MathClass-punc">.</mo><mo
13749
class="MathClass-punc">.</mo><mo
13750
class="MathClass-punc">.</mo><mo
13751
class="MathClass-punc">.</mo><mo
13752
class="MathClass-punc">.</mo><mo
13753
class="MathClass-punc">.</mo><mo
13754
class="MathClass-punc">.</mo><mo
13755
class="MathClass-punc">.</mo><mo
13756
class="MathClass-punc">.</mo><mo
13757
class="MathClass-punc">.</mo><mo
13758
class="MathClass-punc">.</mo><mo
13759
class="MathClass-punc">.</mo><mo
13760
class="MathClass-punc">.</mo><mo
13761
class="MathClass-punc">.</mo><mo
13762
class="MathClass-punc">.</mo><mo
13763
class="MathClass-punc">.</mo><mo
13764
class="MathClass-punc">.</mo><mo
13765
class="MathClass-punc">.</mo><mo
13766
class="MathClass-punc">.</mo><mo
13767
class="MathClass-punc">.</mo><mo
13768
class="MathClass-punc">.</mo><mo
13769
class="MathClass-punc">.</mo><mo
13770
class="MathClass-punc">.</mo><mo
13771
class="MathClass-punc">.</mo><mo
13772
class="MathClass-punc">.</mo><mo
13773
class="MathClass-punc">.</mo><mo
13774
class="MathClass-punc">.</mo><mo
13775
class="MathClass-punc">.</mo><mo
13776
class="MathClass-punc">.</mo><mo
13777
class="MathClass-punc">.</mo><mo
13778
class="MathClass-punc">.</mo><mo
13779
class="MathClass-punc">.</mo><mo
13780
class="MathClass-punc">.</mo><mo
13781
class="MathClass-punc">.</mo><mo
13782
class="MathClass-punc">.</mo><mo
13783
class="MathClass-punc">.</mo><mo
13784
class="MathClass-punc">.</mo><mo
13785
class="MathClass-punc">.</mo><mo
13786
class="MathClass-punc">.</mo><mo
13787
class="MathClass-punc">.</mo><mo
13788
class="MathClass-punc">.</mo><mo
13789
class="MathClass-punc">.</mo><mo
13790
class="MathClass-punc">.</mo><mo
13791
class="MathClass-punc">.</mo><mo
13792
class="MathClass-punc">.</mo><mo
13793
class="MathClass-punc">.</mo><mo
13794
class="MathClass-punc">.</mo><mo
13795
class="MathClass-punc">.</mo><mo
13796
class="MathClass-punc">.</mo><mo
13797
class="MathClass-punc">.</mo></mrow>
13798
</mtd></mtr><mtr><mtd
13799
class="array" ><mfrac class="dfrac"><mrow><mi
13800
class="MathClass-ord">ϕ</mi><msub
13802
class="MathClass-ord">k</mi><mrow
13804
class="MathClass-ord">n</mi><mn
13805
class="MathClass-ord">1</mn></mrow></msub
13806
></mrow><mrow><mrow><mo
13807
class="MathClass-open">(</mo><msub
13809
class="MathClass-ord">ϕ</mi><mrow
13811
class="MathClass-ord">n</mi></mrow></msub
13813
class="MathClass-punc">,</mo> <msub
13815
class="MathClass-ord">ɛ</mi><mrow
13817
class="MathClass-ord">1</mn></mrow></msub
13819
class="MathClass-close">)</mo></mrow></mrow></mfrac></mtd><mtd
13820
class="array" ><mfrac class="dfrac"><mrow><mi
13821
class="MathClass-ord">ϕ</mi><msub
13823
class="MathClass-ord">k</mi><mrow
13825
class="MathClass-ord">n</mi><mn
13826
class="MathClass-ord">2</mn></mrow></msub
13827
></mrow><mrow><mrow><mo
13828
class="MathClass-open">(</mo><msub
13830
class="MathClass-ord">ϕ</mi><mrow
13832
class="MathClass-ord">n</mi></mrow></msub
13834
class="MathClass-punc">,</mo> <msub
13836
class="MathClass-ord">ɛ</mi><mrow
13838
class="MathClass-ord">2</mn></mrow></msub
13840
class="MathClass-close">)</mo></mrow></mrow></mfrac></mtd><mtd
13842
class="MathClass-op">…</mo> </mtd><mtd
13843
class="array" ><mfrac class="dfrac"><mrow><mi
13844
class="MathClass-ord">ϕ</mi><msub
13846
class="MathClass-ord">k</mi><mrow
13848
class="MathClass-ord">n</mi> <mi
13849
class="MathClass-ord">n</mi><mo
13850
class="MathClass-bin">−</mo><mn
13851
class="MathClass-ord">1</mn></mrow></msub
13852
></mrow><mrow><mrow><mo
13853
class="MathClass-open">(</mo><msub
13855
class="MathClass-ord">ϕ</mi><mrow
13857
class="MathClass-ord">n</mi></mrow></msub
13859
class="MathClass-punc">,</mo> <msub
13861
class="MathClass-ord">ɛ</mi><mrow
13863
class="MathClass-ord">n</mi><mo
13864
class="MathClass-bin">−</mo><mn
13865
class="MathClass-ord">1</mn></mrow></msub
13867
class="MathClass-close">)</mo></mrow></mrow></mfrac></mtd><mtd
13868
class="array" ><mfrac class="dfrac"><mrow><mi
13869
class="MathClass-ord">ϕ</mi></mrow><mrow><mrow><mo
13870
class="MathClass-open">(</mo><msub
13872
class="MathClass-ord">ϕ</mi><mrow
13874
class="MathClass-ord">n</mi></mrow></msub
13876
class="MathClass-punc">,</mo> <msub
13878
class="MathClass-ord">ɛ</mi><mrow
13880
class="MathClass-ord">n</mi></mrow></msub
13882
class="MathClass-close">)</mo></mrow></mrow></mfrac></mtd></mtr> <!--*\c@MaxMatrixCols c--></mtable> </mfenced>
13886
<table width="100%"
13887
class="verbatim"><tr class="verbatim"><td
13888
class="verbatim"><pre class="verbatim">
13889
 \[W(\Phi)= \begin{Vmatrix}
13890
 \dfrac\varphi{(\varphi_1,\varepsilon_1)}&0&\dots&0\\
13891
 \dfrac{\varphi k_{n2}}{(\varphi_2,\varepsilon_1)}&
13892
 \dfrac\varphi{(\varphi_2,\varepsilon_2)}&\dots&0\\
13893
 \hdotsfor{5}\\
13894
 \dfrac{\varphi k_{n1}}{(\varphi_n,\varepsilon_1)}&
13895
 \dfrac{\varphi k_{n2}}{(\varphi_n,\varepsilon_2)}&\dots&
13896
 \dfrac{\varphi k_{n\,n-1}}{(\varphi_n,\varepsilon_{n-1})}&
13897
 \dfrac{\varphi}{(\varphi_n,\varepsilon_n)}
13898
 \end{Vmatrix}\]
13901
<!--l. 1635--><p class="indent"> The spacing of the dots can be varied through use of a square-bracket option, for
13902
example, <span class="obeylines-h"><span
13903
class="cmtt-10">\hdotsfor[1.5]{3}</span></span>. The number in square brackets will be used as a
13904
multiplier; the normal value is 1.
13906
<h4 class="subsectionHead"><span class="titlemark">9.18</span> <a
13907
name="x1-310009.18"></a>The <span
13908
class="cmtt-10">\substack </span>command</h4>
13909
<!--l. 1641--><p class="noindent">The <span
13910
class="cmtt-10">\substack </span>command can be used to produce a multiline subscript or
13911
superscript: for example
13914
<table width="100%"
13915
class="verbatim"><tr class="verbatim"><td
13916
class="verbatim"><pre class="verbatim">
13917
 \sum_{\substack{0\le i\le m\\ 0<j<n}} P(i,j)
13920
<!--l. 1647--><p class="indent"> produces a two-line subscript underneath the sum: </p><table class="equation"><tr><td>
13921
<!--l. 1648--><math
13922
xmlns="http://www.w3.org/1998/Math/MathML"
13925
class="equation"><mtr><mtd>
13928
class="MathClass-op">∑</mo>
13930
><mtable class="subarray-c"><mtr><mtd><mn
13931
class="MathClass-ord">0</mn><mo
13932
class="MathClass-rel">≤</mo><mi
13933
class="MathClass-ord">i</mi><mo
13934
class="MathClass-rel">≤</mo><mi
13935
class="MathClass-ord">m</mi>
13936
</mtd></mtr><mtr><mtd><mn
13937
class="MathClass-ord">0</mn><mo
13938
class="MathClass-rel"><</mo><mi
13939
class="MathClass-ord">j</mi><mo
13940
class="MathClass-rel"><</mo><mi
13941
class="MathClass-ord">n</mi>
13942
</mtd></mtr> </mtable></mrow></msub
13944
class="MathClass-ord">P</mi><mrow><mo
13945
class="MathClass-open">(</mo><mi
13946
class="MathClass-ord">i</mi><mo
13947
class="MathClass-punc">,</mo> <mi
13948
class="MathClass-ord">j</mi><mo
13949
class="MathClass-close">)</mo></mrow></mtd><mtd><mspace
13950
id="x1-31001r62" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /></mtd></mtr></mtable>
13952
<!--l. 1650--><p class="nopar"></p></td><td width="5%">(62)</td></tr></table>
13953
A slightly more generalized form is the <span
13954
class="cmtt-10">subarray </span>environment which allows you to
13955
specify that each line should be left-aligned instead of centered, as here:
13956
<table class="equation"><tr><td>
13958
<!--l. 1654--><math
13959
xmlns="http://www.w3.org/1998/Math/MathML"
13962
class="equation"><mtr><mtd>
13965
class="MathClass-op">∑</mo>
13967
><mtable class="subarray"><mtr><mtd><mn
13968
class="MathClass-ord">0</mn><mo
13969
class="MathClass-rel">≤</mo><mi
13970
class="MathClass-ord">i</mi><mo
13971
class="MathClass-rel">≤</mo><mi
13972
class="MathClass-ord">m</mi>
13973
</mtd></mtr><mtr><mtd><mn
13974
class="MathClass-ord">0</mn><mo
13975
class="MathClass-rel"><</mo><mi
13976
class="MathClass-ord">j</mi><mo
13977
class="MathClass-rel"><</mo><mi
13978
class="MathClass-ord">n</mi>
13979
</mtd></mtr> </mtable></mrow></msub
13981
class="MathClass-ord">P</mi><mrow><mo
13982
class="MathClass-open">(</mo><mi
13983
class="MathClass-ord">i</mi><mo
13984
class="MathClass-punc">,</mo> <mi
13985
class="MathClass-ord">j</mi><mo
13986
class="MathClass-close">)</mo></mrow></mtd><mtd><mspace
13987
id="x1-31002r63" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /></mtd></mtr></mtable>
13989
<!--l. 1659--><p class="nopar"></p></td><td width="5%">(63)</td></tr></table>
13991
<table width="100%"
13992
class="verbatim"><tr class="verbatim"><td
13993
class="verbatim"><pre class="verbatim">
13994
 \sum_{\begin{subarray}{l}
13995
         0\le i\le m\\ 0<j<n
13996
       \end{subarray}}
13997
  P(i,j)
14000
<h4 class="subsectionHead"><span class="titlemark">9.19</span> <a
14001
name="x1-320009.19"></a>Big-g-g delimiters</h4> Here are some big delimiters, first in <span
14002
class="cmtt-10">\normalsize</span>: <!--l. 1670--><math
14003
xmlns="http://www.w3.org/1998/Math/MathML"
14008
class="MathClass-open">(
14010
><mi class="mathbf">E</mi><mrow
14012
class="MathClass-ord">y</mi></mrow></msub
14015
class="MathClass-op">∫</mo>
14018
class="MathClass-ord">0</mn></mrow><mrow
14021
class="MathClass-ord">t</mi><mrow
14023
class="MathClass-ord">ɛ</mi></mrow></msub
14027
class="MathClass-ord">L</mi><mrow
14029
class="MathClass-ord">x</mi><mo
14030
class="MathClass-punc">,</mo><msup
14032
class="MathClass-ord">y</mi><mrow
14034
class="MathClass-ord">x</mi></mrow></msup
14036
class="MathClass-open">(</mo><mi
14037
class="MathClass-ord">s</mi><mo
14038
class="MathClass-close">)</mo></mrow></mrow></msub
14040
class="MathClass-ord">ϕ</mi><mrow><mo
14041
class="MathClass-open">(</mo><mi
14042
class="MathClass-ord">x</mi><mo
14043
class="MathClass-close">)</mo></mrow> <mi
14044
class="MathClass-ord">d</mi><mi
14045
class="MathClass-ord">s</mi><mo
14046
class="MathClass-close">)
14050
<table width="100%"
14051
class="verbatim"><tr class="verbatim"><td
14052
class="verbatim"><pre class="verbatim">
14053
 \[\biggl(\mathbf{E}_{y}
14054
   \int_0ˆ{t_\varepsilon}L_{x,yˆx(s)}\varphi(x)\,ds
14055
   \biggr)
14059
<!--l. 1680--><p class="indent"> and now in <span
14060
class="cmtt-10">\Large </span>size: <!--l. 1682--><math
14061
xmlns="http://www.w3.org/1998/Math/MathML"
14062
mode="display"> <mrow
14065
class="MathClass-open">(
14067
><mi class="mathbf">E</mi><mrow
14069
class="MathClass-ord">y</mi></mrow></msub
14072
class="MathClass-op">∫</mo>
14075
class="MathClass-ord">0</mn></mrow><mrow
14078
class="MathClass-ord">t</mi><mrow
14080
class="MathClass-ord">ɛ</mi></mrow></msub
14084
class="MathClass-ord">L</mi><mrow
14086
class="MathClass-ord">x</mi><mo
14087
class="MathClass-punc">,</mo><msup
14089
class="MathClass-ord">y</mi><mrow
14091
class="MathClass-ord">x</mi></mrow></msup
14093
class="MathClass-open">(</mo><mi
14094
class="MathClass-ord">s</mi><mo
14095
class="MathClass-close">)</mo></mrow></mrow></msub
14097
class="MathClass-ord">ϕ</mi><mrow><mo
14098
class="MathClass-open">(</mo><mi
14099
class="MathClass-ord">x</mi><mo
14100
class="MathClass-close">)</mo></mrow> <mi
14101
class="MathClass-ord">d</mi><mi
14102
class="MathClass-ord">s</mi><mo
14103
class="MathClass-close">)
14108
<table width="100%"
14109
class="verbatim"><tr class="verbatim"><td
14110
class="verbatim"><pre class="verbatim">
14111
 {\Large
14112
 \[\biggl(\mathbf{E}_{y}
14113
   \int_0ˆ{t_\varepsilon}L_{x,yˆx(s)}\varphi(x)\,ds
14114
   \biggr)
14119
<h3 class="sectionHead"><span class="titlemark">A</span> <a
14120
name="x1-33000A"></a>Examples of multiple-line equation structures</h3>
14121
<!--l. 1724--><p class="noindent"><span
14122
class="cmbx-12">Note: Starting on this page, vertical rules are added at</span>
14124
class="cmbx-12">the margins so that the positioning of various display</span>
14126
class="cmbx-12">elements with respect to the margins can be seen more</span>
14128
class="cmbx-12">clearly.</span>
14130
<h4 class="subsectionHead"><span class="titlemark">A.1</span> <a
14131
name="x1-34000A.1"></a>Split</h4> The <span
14132
class="cmtt-10">split </span>environment is not an independent environment but should
14133
be used inside something else such as <span
14134
class="cmtt-10">equation </span>or <span
14135
class="cmtt-10">align</span>.
14136
<!--l. 1731--><p class="noindent">If there is not enough room for it, the equation number for a <span
14137
class="cmtt-10">split </span>will be shifted to
14138
the previous line, when equation numbers are on the left; the number shifts down to
14139
the next line when numbers are on the right. </p><table class="equation"><tr><td>
14140
<!--l. 1735--><math
14141
xmlns="http://www.w3.org/1998/Math/MathML"
14144
class="equation"><mtr><mtd>
14145
<mtable class="split"><mtr><mtd>
14147
class="split-mtr"></mrow><mrow
14148
class="split-mtd"></mrow> <msub
14150
class="MathClass-ord">f</mi><mrow
14152
class="MathClass-ord">h</mi><mo
14153
class="MathClass-punc">,</mo><mi
14154
class="MathClass-ord">ɛ</mi></mrow></msub
14156
class="MathClass-open">(</mo><mi
14157
class="MathClass-ord">x</mi><mo
14158
class="MathClass-punc">,</mo> <mi
14159
class="MathClass-ord">y</mi><mo
14160
class="MathClass-close">)</mo></mrow><mrow
14161
class="split-mtd"></mrow> <mo
14162
class="MathClass-rel">=</mo> <mi
14163
class="MathClass-ord">ɛ</mi><msub
14164
><mi class="mathbf">E</mi><mrow
14166
class="MathClass-ord">x</mi><mo
14167
class="MathClass-punc">,</mo><mi
14168
class="MathClass-ord">y</mi></mrow></msub
14171
class="MathClass-op">∫</mo>
14174
class="MathClass-ord">0</mn></mrow><mrow
14177
class="MathClass-ord">t</mi><mrow
14179
class="MathClass-ord">ɛ</mi></mrow></msub
14183
class="MathClass-ord">L</mi><mrow
14185
class="MathClass-ord">x</mi><mo
14186
class="MathClass-punc">,</mo><msub
14188
class="MathClass-ord">y</mi><mrow
14190
class="MathClass-ord">ɛ</mi></mrow></msub
14192
class="MathClass-open">(</mo><mi
14193
class="MathClass-ord">ɛ</mi><mi
14194
class="MathClass-ord">u</mi><mo
14195
class="MathClass-close">)</mo></mrow></mrow></msub
14197
class="MathClass-ord">ϕ</mi><mrow><mo
14198
class="MathClass-open">(</mo><mi
14199
class="MathClass-ord">x</mi><mo
14200
class="MathClass-close">)</mo></mrow> <mi
14201
class="MathClass-ord">d</mi><mi
14202
class="MathClass-ord">u</mi>
14204
class="split-mtr"></mrow><mrow
14205
class="split-mtd"></mrow> <mrow
14206
class="split-mtd"></mrow> <mo
14207
class="MathClass-rel">=</mo> <mi
14208
class="MathClass-ord">h</mi> <mo
14209
class="MathClass-op">∫</mo>
14212
class="MathClass-ord">L</mi><mrow
14214
class="MathClass-ord">x</mi><mo
14215
class="MathClass-punc">,</mo><mi
14216
class="MathClass-ord">z</mi></mrow></msub
14218
class="MathClass-ord">ϕ</mi><mrow><mo
14219
class="MathClass-open">(</mo><mi
14220
class="MathClass-ord">x</mi><mo
14221
class="MathClass-close">)</mo></mrow><msub
14223
class="MathClass-ord">ρ</mi><mrow
14225
class="MathClass-ord">x</mi></mrow></msub
14227
class="MathClass-open">(</mo><mi
14228
class="MathClass-ord">d</mi><mi
14229
class="MathClass-ord">z</mi><mo
14230
class="MathClass-close">)</mo></mrow>
14232
class="split-mtr"></mrow><mrow
14233
class="split-mtd"></mrow> <mrow
14234
class="split-mtd"></mrow><mspace width="1em" class="quad"/> <mo
14235
class="MathClass-bin">+</mo> <mi
14236
class="MathClass-ord">h</mi><mrow><mo
14237
class="MathClass-open">[
14240
class="MathClass-ord">1</mn></mrow>
14244
class="MathClass-ord">t</mi><mrow
14246
class="MathClass-ord">ɛ</mi></mrow></msub
14247
></mrow></mfrac> <mrow><mo
14248
class="MathClass-open">(
14250
><mi class="mathbf">E</mi><mrow
14252
class="MathClass-ord">y</mi></mrow></msub
14255
class="MathClass-op">∫</mo>
14258
class="MathClass-ord">0</mn></mrow><mrow
14261
class="MathClass-ord">t</mi><mrow
14263
class="MathClass-ord">ɛ</mi></mrow></msub
14267
class="MathClass-ord">L</mi><mrow
14269
class="MathClass-ord">x</mi><mo
14270
class="MathClass-punc">,</mo><msup
14272
class="MathClass-ord">y</mi><mrow
14274
class="MathClass-ord">x</mi></mrow></msup
14276
class="MathClass-open">(</mo><mi
14277
class="MathClass-ord">s</mi><mo
14278
class="MathClass-close">)</mo></mrow></mrow></msub
14280
class="MathClass-ord">ϕ</mi><mrow><mo
14281
class="MathClass-open">(</mo><mi
14282
class="MathClass-ord">x</mi><mo
14283
class="MathClass-close">)</mo></mrow> <mi
14284
class="MathClass-ord">d</mi><mi
14285
class="MathClass-ord">s</mi> <mo
14286
class="MathClass-bin">−</mo> <msub
14288
class="MathClass-ord">t</mi><mrow
14290
class="MathClass-ord">ɛ</mi></mrow></msub
14292
class="MathClass-op">∫</mo>
14295
class="MathClass-ord">L</mi><mrow
14297
class="MathClass-ord">x</mi><mo
14298
class="MathClass-punc">,</mo><mi
14299
class="MathClass-ord">z</mi></mrow></msub
14301
class="MathClass-ord">ϕ</mi><mrow><mo
14302
class="MathClass-open">(</mo><mi
14303
class="MathClass-ord">x</mi><mo
14304
class="MathClass-close">)</mo></mrow><msub
14306
class="MathClass-ord">ρ</mi><mrow
14308
class="MathClass-ord">x</mi></mrow></msub
14310
class="MathClass-open">(</mo><mi
14311
class="MathClass-ord">d</mi><mi
14312
class="MathClass-ord">z</mi><mo
14313
class="MathClass-close">)</mo></mrow><mo
14314
class="MathClass-close">)
14317
class="split-mtr"></mrow><mrow
14318
class="split-mtd"></mrow> <mrow
14319
class="split-mtd"></mrow> <mo
14320
class="MathClass-bin">+</mo> <mfrac><mrow
14322
class="MathClass-ord">1</mn></mrow>
14326
class="MathClass-ord">t</mi><mrow
14328
class="MathClass-ord">ɛ</mi></mrow></msub
14329
></mrow></mfrac> <mrow><mo
14330
class="MathClass-open">(
14332
><mi class="mathbf">E</mi><mrow
14334
class="MathClass-ord">y</mi></mrow></msub
14337
class="MathClass-op">∫</mo>
14340
class="MathClass-ord">0</mn></mrow><mrow
14343
class="MathClass-ord">t</mi><mrow
14345
class="MathClass-ord">ɛ</mi></mrow></msub
14349
class="MathClass-ord">L</mi><mrow
14351
class="MathClass-ord">x</mi><mo
14352
class="MathClass-punc">,</mo><msup
14354
class="MathClass-ord">y</mi><mrow
14356
class="MathClass-ord">x</mi></mrow></msup
14358
class="MathClass-open">(</mo><mi
14359
class="MathClass-ord">s</mi><mo
14360
class="MathClass-close">)</mo></mrow></mrow></msub
14362
class="MathClass-ord">ϕ</mi><mrow><mo
14363
class="MathClass-open">(</mo><mi
14364
class="MathClass-ord">x</mi><mo
14365
class="MathClass-close">)</mo></mrow> <mi
14366
class="MathClass-ord">d</mi><mi
14367
class="MathClass-ord">s</mi> <mo
14368
class="MathClass-bin">−</mo><msub
14369
> <mi class="mathbf">E</mi><mrow
14371
class="MathClass-ord">x</mi><mo
14372
class="MathClass-punc">,</mo><mi
14373
class="MathClass-ord">y</mi></mrow></msub
14376
class="MathClass-op">∫</mo>
14379
class="MathClass-ord">0</mn></mrow><mrow
14382
class="MathClass-ord">t</mi><mrow
14384
class="MathClass-ord">ɛ</mi></mrow></msub
14388
class="MathClass-ord">L</mi><mrow
14390
class="MathClass-ord">x</mi><mo
14391
class="MathClass-punc">,</mo><msub
14393
class="MathClass-ord">y</mi><mrow
14395
class="MathClass-ord">ɛ</mi></mrow></msub
14397
class="MathClass-open">(</mo><mi
14398
class="MathClass-ord">ɛ</mi><mi
14399
class="MathClass-ord">s</mi><mo
14400
class="MathClass-close">)</mo></mrow></mrow></msub
14402
class="MathClass-ord">ϕ</mi><mrow><mo
14403
class="MathClass-open">(</mo><mi
14404
class="MathClass-ord">x</mi><mo
14405
class="MathClass-close">)</mo></mrow> <mi
14406
class="MathClass-ord">d</mi><mi
14407
class="MathClass-ord">s</mi><mo
14408
class="MathClass-close">)
14410
class="MathClass-close">]
14413
class="split-mtr"></mrow><mrow
14414
class="split-mtd"></mrow> <mrow
14415
class="split-mtd"></mrow> <mo
14416
class="MathClass-rel">=</mo> <mi
14417
class="MathClass-ord">h</mi><msub
14419
accent="true"><mrow
14421
class="MathClass-ord">L</mi></mrow><mrow
14423
></mrow></munderover><mrow
14425
class="MathClass-ord">x</mi></mrow></msub
14427
class="MathClass-ord">ϕ</mi><mrow><mo
14428
class="MathClass-open">(</mo><mi
14429
class="MathClass-ord">x</mi><mo
14430
class="MathClass-close">)</mo></mrow> <mo
14431
class="MathClass-bin">+</mo> <mi
14432
class="MathClass-ord">h</mi><msub
14434
class="MathClass-ord">θ</mi><mrow
14436
class="MathClass-ord">ɛ</mi></mrow></msub
14438
class="MathClass-open">(</mo><mi
14439
class="MathClass-ord">x</mi><mo
14440
class="MathClass-punc">,</mo> <mi
14441
class="MathClass-ord">y</mi><mo
14442
class="MathClass-close">)</mo></mrow><mo
14443
class="MathClass-punc">,</mo>
14444
</mtd></mtr></mtable> </mtd><mtd><mspace
14445
id="x1-34001r64" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /></mtd></mtr></mtable>
14447
<!--l. 1751--><p class="nopar"></p></td><td width="5%">(64)</td></tr></table>
14448
Some text after to test the below-display spacing.
14451
<table width="100%"
14452
class="verbatim"><tr class="verbatim"><td
14453
class="verbatim"><pre class="verbatim">
14454
 \begin{equation}
14455
 \begin{split}
14456
 f_{h,\varepsilon}(x,y)
14457
 &=\varepsilon\mathbf{E}_{x,y}\int_0ˆ{t_\varepsilon}
14458
 L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\
14459
 &= h\int L_{x,z}\varphi(x)\rho_x(dz)\\
14460
 &\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\mathbf{E}_{y}
14461
   \int_0ˆ{t_\varepsilon}L_{x,yˆx(s)}\varphi(x)\,ds
14462
   -t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\
14463
 &\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon}
14464
   \biggl(\mathbf{E}_{y}\int_0ˆ{t_\varepsilon}L_{x,yˆx(s)}
14465
     \varphi(x)\,ds -\mathbf{E}_{x,y}\int_0ˆ{t_\varepsilon}
14466
    L_{x,y_\varepsilon(\varepsilon s)}
14467
    \varphi(x)\,ds\biggr)\biggr]\\
14468
 &=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y),
14469
 \end{split}
14470
 \end{equation}
14475
<!--l. 1776--><p class="indent"> Unnumbered version: </p><table class="equation"><tr><td>
14476
<!--l. 1777--><math
14477
xmlns="http://www.w3.org/1998/Math/MathML"
14480
class="equation"><mtr><mtd>
14481
<mtable class="split"><mtr><mtd>
14483
class="split-mtr"></mrow><mrow
14484
class="split-mtd"></mrow> <msub
14486
class="MathClass-ord">f</mi><mrow
14488
class="MathClass-ord">h</mi><mo
14489
class="MathClass-punc">,</mo><mi
14490
class="MathClass-ord">ɛ</mi></mrow></msub
14492
class="MathClass-open">(</mo><mi
14493
class="MathClass-ord">x</mi><mo
14494
class="MathClass-punc">,</mo> <mi
14495
class="MathClass-ord">y</mi><mo
14496
class="MathClass-close">)</mo></mrow><mrow
14497
class="split-mtd"></mrow> <mo
14498
class="MathClass-rel">=</mo> <mi
14499
class="MathClass-ord">ɛ</mi><msub
14500
><mi class="mathbf">E</mi><mrow
14502
class="MathClass-ord">x</mi><mo
14503
class="MathClass-punc">,</mo><mi
14504
class="MathClass-ord">y</mi></mrow></msub
14507
class="MathClass-op">∫</mo>
14510
class="MathClass-ord">0</mn></mrow><mrow
14513
class="MathClass-ord">t</mi><mrow
14515
class="MathClass-ord">ɛ</mi></mrow></msub
14519
class="MathClass-ord">L</mi><mrow
14521
class="MathClass-ord">x</mi><mo
14522
class="MathClass-punc">,</mo><msub
14524
class="MathClass-ord">y</mi><mrow
14526
class="MathClass-ord">ɛ</mi></mrow></msub
14528
class="MathClass-open">(</mo><mi
14529
class="MathClass-ord">ɛ</mi><mi
14530
class="MathClass-ord">u</mi><mo
14531
class="MathClass-close">)</mo></mrow></mrow></msub
14533
class="MathClass-ord">ϕ</mi><mrow><mo
14534
class="MathClass-open">(</mo><mi
14535
class="MathClass-ord">x</mi><mo
14536
class="MathClass-close">)</mo></mrow> <mi
14537
class="MathClass-ord">d</mi><mi
14538
class="MathClass-ord">u</mi>
14540
class="split-mtr"></mrow><mrow
14541
class="split-mtd"></mrow> <mrow
14542
class="split-mtd"></mrow> <mo
14543
class="MathClass-rel">=</mo> <mi
14544
class="MathClass-ord">h</mi> <mo
14545
class="MathClass-op">∫</mo>
14548
class="MathClass-ord">L</mi><mrow
14550
class="MathClass-ord">x</mi><mo
14551
class="MathClass-punc">,</mo><mi
14552
class="MathClass-ord">z</mi></mrow></msub
14554
class="MathClass-ord">ϕ</mi><mrow><mo
14555
class="MathClass-open">(</mo><mi
14556
class="MathClass-ord">x</mi><mo
14557
class="MathClass-close">)</mo></mrow><msub
14559
class="MathClass-ord">ρ</mi><mrow
14561
class="MathClass-ord">x</mi></mrow></msub
14563
class="MathClass-open">(</mo><mi
14564
class="MathClass-ord">d</mi><mi
14565
class="MathClass-ord">z</mi><mo
14566
class="MathClass-close">)</mo></mrow>
14568
class="split-mtr"></mrow><mrow
14569
class="split-mtd"></mrow> <mrow
14570
class="split-mtd"></mrow><mspace width="1em" class="quad"/> <mo
14571
class="MathClass-bin">+</mo> <mi
14572
class="MathClass-ord">h</mi><mrow><mo
14573
class="MathClass-open">[
14576
class="MathClass-ord">1</mn></mrow>
14580
class="MathClass-ord">t</mi><mrow
14582
class="MathClass-ord">ɛ</mi></mrow></msub
14583
></mrow></mfrac> <mrow><mo
14584
class="MathClass-open">(
14586
><mi class="mathbf">E</mi><mrow
14588
class="MathClass-ord">y</mi></mrow></msub
14591
class="MathClass-op">∫</mo>
14594
class="MathClass-ord">0</mn></mrow><mrow
14597
class="MathClass-ord">t</mi><mrow
14599
class="MathClass-ord">ɛ</mi></mrow></msub
14603
class="MathClass-ord">L</mi><mrow
14605
class="MathClass-ord">x</mi><mo
14606
class="MathClass-punc">,</mo><msup
14608
class="MathClass-ord">y</mi><mrow
14610
class="MathClass-ord">x</mi></mrow></msup
14612
class="MathClass-open">(</mo><mi
14613
class="MathClass-ord">s</mi><mo
14614
class="MathClass-close">)</mo></mrow></mrow></msub
14616
class="MathClass-ord">ϕ</mi><mrow><mo
14617
class="MathClass-open">(</mo><mi
14618
class="MathClass-ord">x</mi><mo
14619
class="MathClass-close">)</mo></mrow> <mi
14620
class="MathClass-ord">d</mi><mi
14621
class="MathClass-ord">s</mi> <mo
14622
class="MathClass-bin">−</mo> <msub
14624
class="MathClass-ord">t</mi><mrow
14626
class="MathClass-ord">ɛ</mi></mrow></msub
14628
class="MathClass-op">∫</mo>
14631
class="MathClass-ord">L</mi><mrow
14633
class="MathClass-ord">x</mi><mo
14634
class="MathClass-punc">,</mo><mi
14635
class="MathClass-ord">z</mi></mrow></msub
14637
class="MathClass-ord">ϕ</mi><mrow><mo
14638
class="MathClass-open">(</mo><mi
14639
class="MathClass-ord">x</mi><mo
14640
class="MathClass-close">)</mo></mrow><msub
14642
class="MathClass-ord">ρ</mi><mrow
14644
class="MathClass-ord">x</mi></mrow></msub
14646
class="MathClass-open">(</mo><mi
14647
class="MathClass-ord">d</mi><mi
14648
class="MathClass-ord">z</mi><mo
14649
class="MathClass-close">)</mo></mrow><mo
14650
class="MathClass-close">)
14653
class="split-mtr"></mrow><mrow
14654
class="split-mtd"></mrow> <mrow
14655
class="split-mtd"></mrow> <mo
14656
class="MathClass-bin">+</mo> <mfrac><mrow
14658
class="MathClass-ord">1</mn></mrow>
14662
class="MathClass-ord">t</mi><mrow
14664
class="MathClass-ord">ɛ</mi></mrow></msub
14665
></mrow></mfrac> <mrow><mo
14666
class="MathClass-open">(
14668
><mi class="mathbf">E</mi><mrow
14670
class="MathClass-ord">y</mi></mrow></msub
14673
class="MathClass-op">∫</mo>
14676
class="MathClass-ord">0</mn></mrow><mrow
14679
class="MathClass-ord">t</mi><mrow
14681
class="MathClass-ord">ɛ</mi></mrow></msub
14685
class="MathClass-ord">L</mi><mrow
14687
class="MathClass-ord">x</mi><mo
14688
class="MathClass-punc">,</mo><msup
14690
class="MathClass-ord">y</mi><mrow
14692
class="MathClass-ord">x</mi></mrow></msup
14694
class="MathClass-open">(</mo><mi
14695
class="MathClass-ord">s</mi><mo
14696
class="MathClass-close">)</mo></mrow></mrow></msub
14698
class="MathClass-ord">ϕ</mi><mrow><mo
14699
class="MathClass-open">(</mo><mi
14700
class="MathClass-ord">x</mi><mo
14701
class="MathClass-close">)</mo></mrow> <mi
14702
class="MathClass-ord">d</mi><mi
14703
class="MathClass-ord">s</mi> <mo
14704
class="MathClass-bin">−</mo><msub
14705
> <mi class="mathbf">E</mi><mrow
14707
class="MathClass-ord">x</mi><mo
14708
class="MathClass-punc">,</mo><mi
14709
class="MathClass-ord">y</mi></mrow></msub
14712
class="MathClass-op">∫</mo>
14715
class="MathClass-ord">0</mn></mrow><mrow
14718
class="MathClass-ord">t</mi><mrow
14720
class="MathClass-ord">ɛ</mi></mrow></msub
14724
class="MathClass-ord">L</mi><mrow
14726
class="MathClass-ord">x</mi><mo
14727
class="MathClass-punc">,</mo><msub
14729
class="MathClass-ord">y</mi><mrow
14731
class="MathClass-ord">ɛ</mi></mrow></msub
14733
class="MathClass-open">(</mo><mi
14734
class="MathClass-ord">ɛ</mi><mi
14735
class="MathClass-ord">s</mi><mo
14736
class="MathClass-close">)</mo></mrow></mrow></msub
14738
class="MathClass-ord">ϕ</mi><mrow><mo
14739
class="MathClass-open">(</mo><mi
14740
class="MathClass-ord">x</mi><mo
14741
class="MathClass-close">)</mo></mrow> <mi
14742
class="MathClass-ord">d</mi><mi
14743
class="MathClass-ord">s</mi><mo
14744
class="MathClass-close">)
14746
class="MathClass-close">]
14749
class="split-mtr"></mrow><mrow
14750
class="split-mtd"></mrow> <mrow
14751
class="split-mtd"></mrow> <mo
14752
class="MathClass-rel">=</mo> <mi
14753
class="MathClass-ord">h</mi><msub
14755
accent="true"><mrow
14757
class="MathClass-ord">L</mi></mrow><mrow
14759
></mrow></munderover><mrow
14761
class="MathClass-ord">x</mi></mrow></msub
14763
class="MathClass-ord">ϕ</mi><mrow><mo
14764
class="MathClass-open">(</mo><mi
14765
class="MathClass-ord">x</mi><mo
14766
class="MathClass-close">)</mo></mrow> <mo
14767
class="MathClass-bin">+</mo> <mi
14768
class="MathClass-ord">h</mi><msub
14770
class="MathClass-ord">θ</mi><mrow
14772
class="MathClass-ord">ɛ</mi></mrow></msub
14774
class="MathClass-open">(</mo><mi
14775
class="MathClass-ord">x</mi><mo
14776
class="MathClass-punc">,</mo> <mi
14777
class="MathClass-ord">y</mi><mo
14778
class="MathClass-close">)</mo></mrow><mo
14779
class="MathClass-punc">,</mo>
14780
</mtd></mtr></mtable> </mtd><mtd></mtd></mtr></mtable>
14782
<!--l. 1793--><p class="nopar"></p></td></tr></table>
14783
Some text after to test the below-display spacing.
14786
<table width="100%"
14787
class="verbatim"><tr class="verbatim"><td
14788
class="verbatim"><pre class="verbatim">
14789
 \begin{equation*}
14790
 \begin{split}
14791
 f_{h,\varepsilon}(x,y)
14792
 &=\varepsilon\mathbf{E}_{x,y}\int_0ˆ{t_\varepsilon}
14793
 L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\
14794
 &= h\int L_{x,z}\varphi(x)\rho_x(dz)\\
14795
 &\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\mathbf{E}_{y}
14796
   \int_0ˆ{t_\varepsilon}L_{x,yˆx(s)}\varphi(x)\,ds
14797
   -t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\
14798
 &\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon}
14799
   \biggl(\mathbf{E}_{y}\int_0ˆ{t_\varepsilon}L_{x,yˆx(s)}
14800
     \varphi(x)\,ds -\mathbf{E}_{x,y}\int_0ˆ{t_\varepsilon}
14801
    L_{x,y_\varepsilon(\varepsilon s)}
14802
    \varphi(x)\,ds\biggr)\biggr]\\
14803
 &=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y),
14804
 \end{split}
14805
 \end{equation*}
14810
<!--l. 1818--><p class="indent"> If the option <span
14811
class="cmtt-10">centertags </span>is included in the options list of the <span
14812
class="cmtt-10">amsmath </span>package,
14813
the equation numbers for <span
14814
class="cmtt-10">split </span>environments will be centered vertically on the
14815
height of the <span
14816
class="cmtt-10">split</span>: </p><table class="equation"><tr><td>
14817
<!--l. 1824--><math
14818
xmlns="http://www.w3.org/1998/Math/MathML"
14821
class="equation"><mtr><mtd>
14822
<mtable class="split"><mtr><mtd>
14824
class="split-mtr"></mrow><mrow
14825
class="split-mtd"></mrow> <mfenced
14826
open="|" close="|" ><msub
14828
class="MathClass-ord">I</mi><mrow
14830
class="MathClass-ord">2</mn></mrow></msub
14832
class="split-mtd"></mrow> <mo
14833
class="MathClass-rel">=</mo> <mfenced
14834
open="|" close="|" ><msubsup
14836
class="MathClass-op">∫</mo>
14839
class="MathClass-ord">0</mn></mrow><mrow
14841
class="MathClass-ord">T</mi></mrow></msubsup
14843
class="MathClass-ord">ψ</mi><mrow><mo
14844
class="MathClass-open">(</mo><mi
14845
class="MathClass-ord">t</mi><mo
14846
class="MathClass-close">)</mo></mrow> <mfenced
14847
open="{" close="}" ><mi
14848
class="MathClass-ord">u</mi><mrow><mo
14849
class="MathClass-open">(</mo><mi
14850
class="MathClass-ord">a</mi><mo
14851
class="MathClass-punc">,</mo> <mi
14852
class="MathClass-ord">t</mi><mo
14853
class="MathClass-close">)</mo></mrow> <mo
14854
class="MathClass-bin">−</mo><msubsup
14856
class="MathClass-op">∫</mo>
14859
class="MathClass-ord">γ</mi><mrow><mo
14860
class="MathClass-open">(</mo><mi
14861
class="MathClass-ord">t</mi><mo
14862
class="MathClass-close">)</mo></mrow></mrow><mrow
14864
class="MathClass-ord">a</mi></mrow></msubsup
14867
class="MathClass-ord">d</mi><mi
14868
class="MathClass-ord">θ</mi></mrow>
14871
class="MathClass-ord">k</mi><mrow><mo
14872
class="MathClass-open">(</mo><mi
14873
class="MathClass-ord">θ</mi><mo
14874
class="MathClass-punc">,</mo> <mi
14875
class="MathClass-ord">t</mi><mo
14876
class="MathClass-close">)</mo></mrow></mrow></mfrac><msubsup
14878
class="MathClass-op">∫</mo>
14881
class="MathClass-ord">a</mi></mrow><mrow
14883
class="MathClass-ord">θ</mi></mrow></msubsup
14885
class="MathClass-ord">c</mi><mrow><mo
14886
class="MathClass-open">(</mo><mi
14887
class="MathClass-ord">ξ</mi><mo
14888
class="MathClass-close">)</mo></mrow><msub
14890
class="MathClass-ord">u</mi><mrow
14892
class="MathClass-ord">t</mi></mrow></msub
14894
class="MathClass-open">(</mo><mi
14895
class="MathClass-ord">ξ</mi><mo
14896
class="MathClass-punc">,</mo> <mi
14897
class="MathClass-ord">t</mi><mo
14898
class="MathClass-close">)</mo></mrow> <mi
14899
class="MathClass-ord">d</mi><mi
14900
class="MathClass-ord">ξ</mi></mfenced> <mi
14901
class="MathClass-ord">d</mi><mi
14902
class="MathClass-ord">t</mi></mfenced>
14904
class="split-mtr"></mrow><mrow
14905
class="split-mtd"></mrow> <mrow
14906
class="split-mtd"></mrow> <mo
14907
class="MathClass-rel">≤</mo> <msub
14909
class="MathClass-ord">C</mi><mrow
14911
class="MathClass-ord">6</mn></mrow></msub
14913
open="|" close="|" ><mfenced
14914
open="|" close="|" ><mi
14915
class="MathClass-ord">f</mi><msub
14917
class="MathClass-op">∫</mo>
14920
class="MathClass-ord">Ω</mi></mrow></msub
14922
open="|" close="|" ><msubsup
14924
accent="true"><mrow
14926
class="MathClass-ord">S</mi></mrow><mrow
14928
></mrow></munderover><mrow
14930
class="MathClass-ord">a</mi><mo
14931
class="MathClass-punc">,</mo> <mo
14932
class="MathClass-bin">−</mo></mrow><mrow
14934
class="MathClass-bin">−</mo> <mn
14935
class="MathClass-ord">1</mn><mo
14936
class="MathClass-punc">,</mo> <mn
14937
class="MathClass-ord">0</mn></mrow></msubsup
14940
class="MathClass-ord">W</mi><mrow
14942
class="MathClass-ord">2</mn></mrow></msub
14944
class="MathClass-open">(</mo><mi
14945
class="MathClass-ord">Ω</mi><mo
14946
class="MathClass-punc">,</mo> <msub
14948
class="MathClass-ord">Γ</mi><mrow
14950
class="MathClass-ord">l</mi></mrow></msub
14952
class="MathClass-close">)</mo></mrow></mfenced></mfenced> <mfenced
14953
open="|" close="|" ><mfenced
14954
open="|" close="|" ><mi
14955
class="MathClass-ord">u</mi></mfenced> <mover><mrow
14957
class="MathClass-bin">◦</mo> </mrow><mrow
14959
class="MathClass-rel">→</mo> </mrow></mover><msubsup
14961
class="MathClass-ord">W</mi><mrow
14963
class="MathClass-ord">2</mn></mrow><mrow
14965
accent="true"><mrow
14967
class="MathClass-ord">A</mi></mrow><mrow
14969
></mrow></munderover></mrow></msubsup
14971
class="MathClass-open">(</mo><mi
14972
class="MathClass-ord">Ω</mi><mo
14973
class="MathClass-punc">;</mo> <msub
14975
class="MathClass-ord">Γ</mi><mrow
14977
class="MathClass-ord">r</mi></mrow></msub
14979
class="MathClass-punc">,</mo> <mi
14980
class="MathClass-ord">T</mi><mo
14981
class="MathClass-close">)</mo></mrow></mfenced></mfenced> <mo
14982
class="MathClass-punc">.</mo>
14983
</mtd></mtr></mtable> </mtd><mtd><mspace
14984
id="x1-34002r65" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /></mtd></mtr></mtable>
14986
<!--l. 1834--><p class="nopar"></p></td><td width="5%">(65)</td></tr></table>
14987
Some text after to test the below-display spacing.
14990
<!--l. 1840--><p class="indent"> Use of <span
14991
class="cmtt-10">split </span>within <span
14992
class="cmtt-10">align</span>:
14993
<!--tex4ht:inline--></p><!--l. 1864--><math
14994
xmlns="http://www.w3.org/1998/Math/MathML"
14995
display="block"><mtable
14998
class="align-odd"> <mtable class="split"><mtr><mtd>
15000
class="split-mtr"></mrow><mrow
15001
class="split-mtd"></mrow> <mfenced
15002
open="|" close="|" ><msub
15004
class="MathClass-ord">I</mi><mrow
15006
class="MathClass-ord">1</mn></mrow></msub
15008
class="split-mtd"></mrow> <mo
15009
class="MathClass-rel">=</mo> <mfenced
15010
open="|" close="|" ><msub
15012
class="MathClass-op">∫</mo>
15015
class="MathClass-ord">Ω</mi></mrow></msub
15017
class="MathClass-ord">g</mi><mi
15018
class="MathClass-ord">R</mi><mi
15019
class="MathClass-ord">u</mi> <mi
15020
class="MathClass-ord">d</mi><mi
15021
class="MathClass-ord">Ω</mi></mfenced>
15023
class="split-mtr"></mrow><mrow
15024
class="split-mtd"></mrow> <mrow
15025
class="split-mtd"></mrow> <mo
15026
class="MathClass-rel">≤</mo> <msub
15028
class="MathClass-ord">C</mi><mrow
15030
class="MathClass-ord">3</mn></mrow></msub
15033
open="[" close="]" ><msub
15035
class="MathClass-op">∫</mo>
15038
class="MathClass-ord">Ω</mi></mrow></msub
15041
open="(" close=")" ><msubsup
15043
class="MathClass-op">∫</mo>
15046
class="MathClass-ord">a</mi></mrow><mrow
15048
class="MathClass-ord">x</mi></mrow></msubsup
15050
class="MathClass-ord">g</mi><mrow><mo
15051
class="MathClass-open">(</mo><mi
15052
class="MathClass-ord">ξ</mi><mo
15053
class="MathClass-punc">,</mo> <mi
15054
class="MathClass-ord">t</mi><mo
15055
class="MathClass-close">)</mo></mrow> <mi
15056
class="MathClass-ord">d</mi><mi
15057
class="MathClass-ord">ξ</mi></mfenced> <mrow
15059
class="MathClass-ord">2</mn></mrow></msup
15061
class="MathClass-ord">d</mi><mi
15062
class="MathClass-ord">Ω</mi></mfenced> <mrow
15064
class="MathClass-ord">1</mn><mo
15065
class="MathClass-bin">/</mo><mn
15066
class="MathClass-ord">2</mn></mrow></msup
15069
class="split-mtr"></mrow><mrow
15070
class="split-mtd"></mrow> <mrow
15071
class="split-mtd"></mrow><mspace width="1em" class="quad"/> <mo
15072
class="MathClass-bin">×</mo><msup
15074
open="[" close="]" ><msub
15076
class="MathClass-op">∫</mo>
15079
class="MathClass-ord">Ω</mi></mrow></msub
15081
open="{" close="}" ><msubsup
15083
class="MathClass-ord">u</mi><mrow
15085
class="MathClass-ord">x</mi></mrow><mrow
15087
class="MathClass-ord">2</mn></mrow></msubsup
15089
class="MathClass-bin">+</mo> <mfrac><mrow
15091
class="MathClass-ord">1</mn></mrow>
15094
class="MathClass-ord">k</mi></mrow></mfrac><msup
15096
open="(" close=")" ><msubsup
15098
class="MathClass-op">∫</mo>
15101
class="MathClass-ord">a</mi></mrow><mrow
15103
class="MathClass-ord">x</mi></mrow></msubsup
15105
class="MathClass-ord">c</mi><msub
15107
class="MathClass-ord">u</mi><mrow
15109
class="MathClass-ord">t</mi></mrow></msub
15111
class="MathClass-ord">d</mi><mi
15112
class="MathClass-ord">ξ</mi></mfenced> <mrow
15114
class="MathClass-ord">2</mn></mrow></msup
15116
class="MathClass-ord">c</mi><mi
15117
class="MathClass-ord">Ω</mi></mfenced> <mrow
15119
class="MathClass-ord">1</mn><mo
15120
class="MathClass-bin">/</mo><mn
15121
class="MathClass-ord">2</mn></mrow></msup
15124
class="split-mtr"></mrow><mrow
15125
class="split-mtd"></mrow> <mrow
15126
class="split-mtd"></mrow> <mo
15127
class="MathClass-rel">≤</mo> <msub
15129
class="MathClass-ord">C</mi><mrow
15131
class="MathClass-ord">4</mn></mrow></msub
15133
open="|" close="|" ><mfenced
15134
open="|" close="|" ><mi
15135
class="MathClass-ord">f</mi> <mfenced
15136
open="|" close="|" ><msubsup
15138
accent="true"><mrow
15140
class="MathClass-ord">S</mi></mrow><mrow
15142
></mrow></munderover><mrow
15144
class="MathClass-ord">a</mi><mo
15145
class="MathClass-punc">,</mo> <mo
15146
class="MathClass-bin">−</mo></mrow><mrow
15148
class="MathClass-bin">−</mo> <mn
15149
class="MathClass-ord">1</mn><mo
15150
class="MathClass-punc">,</mo> <mn
15151
class="MathClass-ord">0</mn></mrow></msubsup
15154
class="MathClass-ord">W</mi><mrow
15156
class="MathClass-ord">2</mn></mrow></msub
15158
class="MathClass-open">(</mo><mi
15159
class="MathClass-ord">Ω</mi><mo
15160
class="MathClass-punc">,</mo> <msub
15162
class="MathClass-ord">Γ</mi><mrow
15164
class="MathClass-ord">l</mi></mrow></msub
15166
class="MathClass-close">)</mo></mrow></mfenced></mfenced> <mfenced
15167
open="|" close="|" ><mfenced
15168
open="|" close="|" ><mi
15169
class="MathClass-ord">u</mi></mfenced> <mover><mrow
15171
class="MathClass-bin">◦</mo> </mrow><mrow
15173
class="MathClass-rel">→</mo> </mrow></mover><msubsup
15175
class="MathClass-ord">W</mi><mrow
15177
class="MathClass-ord">2</mn></mrow><mrow
15179
accent="true"><mrow
15181
class="MathClass-ord">A</mi></mrow><mrow
15183
></mrow></munderover></mrow></msubsup
15185
class="MathClass-open">(</mo><mi
15186
class="MathClass-ord">Ω</mi><mo
15187
class="MathClass-punc">;</mo> <msub
15189
class="MathClass-ord">Γ</mi><mrow
15191
class="MathClass-ord">r</mi></mrow></msub
15193
class="MathClass-punc">,</mo> <mi
15194
class="MathClass-ord">T</mi><mo
15195
class="MathClass-close">)</mo></mrow></mfenced></mfenced> <mo
15196
class="MathClass-punc">.</mo>
15197
</mtd></mtr></mtable> </mtd> <mtd
15198
class="align-even"></mtd> <mtd
15199
class="align-label"> <mspace
15200
id="x1-34003r66" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
15201
class="MathClass-open">(</mo><mn
15202
class="MathClass-ord">6</mn><mn
15203
class="MathClass-ord">6</mn><mo
15204
class="MathClass-close">)</mo></mrow>
15205
</mtd></mtr><mtr><mtd
15206
class="align-odd"> <mtable class="split"><mtr><mtd>
15208
class="split-mtr"></mrow><mrow
15209
class="split-mtd"></mrow> <mfenced
15210
open="|" close="|" ><msub
15212
class="MathClass-ord">I</mi><mrow
15214
class="MathClass-ord">2</mn></mrow></msub
15216
class="split-mtd"></mrow> <mo
15217
class="MathClass-rel">=</mo> <mfenced
15218
open="|" close="|" ><msubsup
15220
class="MathClass-op">∫</mo>
15223
class="MathClass-ord">0</mn></mrow><mrow
15225
class="MathClass-ord">T</mi></mrow></msubsup
15227
class="MathClass-ord">ψ</mi><mrow><mo
15228
class="MathClass-open">(</mo><mi
15229
class="MathClass-ord">t</mi><mo
15230
class="MathClass-close">)</mo></mrow> <mfenced
15231
open="{" close="}" ><mi
15232
class="MathClass-ord">u</mi><mrow><mo
15233
class="MathClass-open">(</mo><mi
15234
class="MathClass-ord">a</mi><mo
15235
class="MathClass-punc">,</mo> <mi
15236
class="MathClass-ord">t</mi><mo
15237
class="MathClass-close">)</mo></mrow> <mo
15238
class="MathClass-bin">−</mo><msubsup
15240
class="MathClass-op">∫</mo>
15243
class="MathClass-ord">γ</mi><mrow><mo
15244
class="MathClass-open">(</mo><mi
15245
class="MathClass-ord">t</mi><mo
15246
class="MathClass-close">)</mo></mrow></mrow><mrow
15248
class="MathClass-ord">a</mi></mrow></msubsup
15251
class="MathClass-ord">d</mi><mi
15252
class="MathClass-ord">θ</mi></mrow>
15255
class="MathClass-ord">k</mi><mrow><mo
15256
class="MathClass-open">(</mo><mi
15257
class="MathClass-ord">θ</mi><mo
15258
class="MathClass-punc">,</mo> <mi
15259
class="MathClass-ord">t</mi><mo
15260
class="MathClass-close">)</mo></mrow></mrow></mfrac><msubsup
15262
class="MathClass-op">∫</mo>
15265
class="MathClass-ord">a</mi></mrow><mrow
15267
class="MathClass-ord">θ</mi></mrow></msubsup
15269
class="MathClass-ord">c</mi><mrow><mo
15270
class="MathClass-open">(</mo><mi
15271
class="MathClass-ord">ξ</mi><mo
15272
class="MathClass-close">)</mo></mrow><msub
15274
class="MathClass-ord">u</mi><mrow
15276
class="MathClass-ord">t</mi></mrow></msub
15278
class="MathClass-open">(</mo><mi
15279
class="MathClass-ord">ξ</mi><mo
15280
class="MathClass-punc">,</mo> <mi
15281
class="MathClass-ord">t</mi><mo
15282
class="MathClass-close">)</mo></mrow> <mi
15283
class="MathClass-ord">d</mi><mi
15284
class="MathClass-ord">ξ</mi></mfenced> <mi
15285
class="MathClass-ord">d</mi><mi
15286
class="MathClass-ord">t</mi></mfenced>
15288
class="split-mtr"></mrow><mrow
15289
class="split-mtd"></mrow> <mrow
15290
class="split-mtd"></mrow> <mo
15291
class="MathClass-rel">≤</mo> <msub
15293
class="MathClass-ord">C</mi><mrow
15295
class="MathClass-ord">6</mn></mrow></msub
15297
open="|" close="|" ><mfenced
15298
open="|" close="|" ><mi
15299
class="MathClass-ord">f</mi><msub
15301
class="MathClass-op">∫</mo>
15304
class="MathClass-ord">Ω</mi></mrow></msub
15306
open="|" close="|" ><msubsup
15308
accent="true"><mrow
15310
class="MathClass-ord">S</mi></mrow><mrow
15312
></mrow></munderover><mrow
15314
class="MathClass-ord">a</mi><mo
15315
class="MathClass-punc">,</mo> <mo
15316
class="MathClass-bin">−</mo></mrow><mrow
15318
class="MathClass-bin">−</mo> <mn
15319
class="MathClass-ord">1</mn><mo
15320
class="MathClass-punc">,</mo> <mn
15321
class="MathClass-ord">0</mn></mrow></msubsup
15324
class="MathClass-ord">W</mi><mrow
15326
class="MathClass-ord">2</mn></mrow></msub
15328
class="MathClass-open">(</mo><mi
15329
class="MathClass-ord">Ω</mi><mo
15330
class="MathClass-punc">,</mo> <msub
15332
class="MathClass-ord">Γ</mi><mrow
15334
class="MathClass-ord">l</mi></mrow></msub
15336
class="MathClass-close">)</mo></mrow></mfenced></mfenced> <mfenced
15337
open="|" close="|" ><mfenced
15338
open="|" close="|" ><mi
15339
class="MathClass-ord">u</mi></mfenced> <mover><mrow
15341
class="MathClass-bin">◦</mo> </mrow><mrow
15343
class="MathClass-rel">→</mo> </mrow></mover><msubsup
15345
class="MathClass-ord">W</mi><mrow
15347
class="MathClass-ord">2</mn></mrow><mrow
15349
accent="true"><mrow
15351
class="MathClass-ord">A</mi></mrow><mrow
15353
></mrow></munderover></mrow></msubsup
15355
class="MathClass-open">(</mo><mi
15356
class="MathClass-ord">Ω</mi><mo
15357
class="MathClass-punc">;</mo> <msub
15359
class="MathClass-ord">Γ</mi><mrow
15361
class="MathClass-ord">r</mi></mrow></msub
15363
class="MathClass-punc">,</mo> <mi
15364
class="MathClass-ord">T</mi><mo
15365
class="MathClass-close">)</mo></mrow></mfenced></mfenced> <mo
15366
class="MathClass-punc">.</mo>
15367
</mtd></mtr></mtable> </mtd> <mtd
15368
class="align-even"></mtd> <mtd
15369
class="align-label"> <mspace
15370
id="x1-34004r67" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
15371
class="MathClass-open">(</mo><mn
15372
class="MathClass-ord">6</mn><mn
15373
class="MathClass-ord">7</mn><mo
15374
class="MathClass-close">)</mo></mrow>
15375
</mtd></mtr></mtable></math>
15376
Some text after to test the below-display spacing.
15379
<table width="100%"
15380
class="verbatim"><tr class="verbatim"><td
15381
class="verbatim"><pre class="verbatim">
15382
 \begin{align}
15383
 \begin{split}\abs{I_1}
15384
   &=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\
15385
 &\le C_3\left[\int_\Omega\left(\int_{a}ˆx
15386
   g(\xi,t)\,d\xi\right)ˆ2d\Omega\right]ˆ{1/2}\\
15387
 &\quad\times \left[\int_\Omega\left\{uˆ2_x+\frac{1}{k}
15388
   \left(\int_{a}ˆx cu_t\,d\xi\right)ˆ2\right\}
15389
   c\Omega\right]ˆ{1/2}\\
15390
 &\le C_4\left\lvert \left\lvert f\left\lvert \wt{S}ˆ{-1,0}_{a,-}
15391
   W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
15392
   \left\lvert \abs{u}\overset{\circ}\to W_2ˆ{\wt{A}}
15393
   (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
15394
 \end{split}\label{eq:A}\\
15395
 \begin{split}\abs{I_2}&=\left\lvert \int_{0}ˆT \psi(t)\left\{u(a,t)
15396
   -\int_{\gamma(t)}ˆa\frac{d\theta}{k(\theta,t)}
15397
   \int_{a}ˆ\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\
15398
 &\le C_6\left\lvert \left\lvert f\int_\Omega
15399
   \left\lvert \wt{S}ˆ{-1,0}_{a,-}
15400
   W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
15401
   \left\lvert \abs{u}\overset{\circ}\to W_2ˆ{\wt{A}}
15402
   (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
15403
 \end{split}
15404
 \end{align}
15409
<!--l. 1896--><p class="indent"> Unnumbered <span
15410
class="cmtt-10">align</span>, with a number on the second <span
15411
class="cmtt-10">split</span>:
15412
<!--tex4ht:inline--></p><!--l. 1918--><math
15413
xmlns="http://www.w3.org/1998/Math/MathML"
15414
display="block"><mtable
15415
class="align-star">
15417
class="align-odd"> <mtable class="split"><mtr><mtd>
15419
class="split-mtr"></mrow><mrow
15420
class="split-mtd"></mrow> <mfenced
15421
open="|" close="|" ><msub
15423
class="MathClass-ord">I</mi><mrow
15425
class="MathClass-ord">1</mn></mrow></msub
15427
class="split-mtd"></mrow> <mo
15428
class="MathClass-rel">=</mo> <mfenced
15429
open="|" close="|" ><msub
15431
class="MathClass-op">∫</mo>
15434
class="MathClass-ord">Ω</mi></mrow></msub
15436
class="MathClass-ord">g</mi><mi
15437
class="MathClass-ord">R</mi><mi
15438
class="MathClass-ord">u</mi> <mi
15439
class="MathClass-ord">d</mi><mi
15440
class="MathClass-ord">Ω</mi></mfenced>
15442
class="split-mtr"></mrow><mrow
15443
class="split-mtd"></mrow> <mrow
15444
class="split-mtd"></mrow> <mo
15445
class="MathClass-rel">≤</mo> <msub
15447
class="MathClass-ord">C</mi><mrow
15449
class="MathClass-ord">3</mn></mrow></msub
15452
open="[" close="]" ><msub
15454
class="MathClass-op">∫</mo>
15457
class="MathClass-ord">Ω</mi></mrow></msub
15460
open="(" close=")" ><msubsup
15462
class="MathClass-op">∫</mo>
15465
class="MathClass-ord">a</mi></mrow><mrow
15467
class="MathClass-ord">x</mi></mrow></msubsup
15469
class="MathClass-ord">g</mi><mrow><mo
15470
class="MathClass-open">(</mo><mi
15471
class="MathClass-ord">ξ</mi><mo
15472
class="MathClass-punc">,</mo> <mi
15473
class="MathClass-ord">t</mi><mo
15474
class="MathClass-close">)</mo></mrow> <mi
15475
class="MathClass-ord">d</mi><mi
15476
class="MathClass-ord">ξ</mi></mfenced> <mrow
15478
class="MathClass-ord">2</mn></mrow></msup
15480
class="MathClass-ord">d</mi><mi
15481
class="MathClass-ord">Ω</mi></mfenced> <mrow
15483
class="MathClass-ord">1</mn><mo
15484
class="MathClass-bin">/</mo><mn
15485
class="MathClass-ord">2</mn></mrow></msup
15488
class="split-mtr"></mrow><mrow
15489
class="split-mtd"></mrow> <mrow
15490
class="split-mtd"></mrow> <mo
15491
class="MathClass-bin">×</mo><msup
15493
open="[" close="]" ><msub
15495
class="MathClass-op">∫</mo>
15498
class="MathClass-ord">Ω</mi></mrow></msub
15500
open="{" close="}" ><msubsup
15502
class="MathClass-ord">u</mi><mrow
15504
class="MathClass-ord">x</mi></mrow><mrow
15506
class="MathClass-ord">2</mn></mrow></msubsup
15508
class="MathClass-bin">+</mo> <mfrac><mrow
15510
class="MathClass-ord">1</mn></mrow>
15513
class="MathClass-ord">k</mi></mrow></mfrac><msup
15515
open="(" close=")" ><msubsup
15517
class="MathClass-op">∫</mo>
15520
class="MathClass-ord">a</mi></mrow><mrow
15522
class="MathClass-ord">x</mi></mrow></msubsup
15524
class="MathClass-ord">c</mi><msub
15526
class="MathClass-ord">u</mi><mrow
15528
class="MathClass-ord">t</mi></mrow></msub
15530
class="MathClass-ord">d</mi><mi
15531
class="MathClass-ord">ξ</mi></mfenced> <mrow
15533
class="MathClass-ord">2</mn></mrow></msup
15535
class="MathClass-ord">c</mi><mi
15536
class="MathClass-ord">Ω</mi></mfenced> <mrow
15538
class="MathClass-ord">1</mn><mo
15539
class="MathClass-bin">/</mo><mn
15540
class="MathClass-ord">2</mn></mrow></msup
15543
class="split-mtr"></mrow><mrow
15544
class="split-mtd"></mrow> <mrow
15545
class="split-mtd"></mrow> <mo
15546
class="MathClass-rel">≤</mo> <msub
15548
class="MathClass-ord">C</mi><mrow
15550
class="MathClass-ord">4</mn></mrow></msub
15552
open="|" close="|" ><mfenced
15553
open="|" close="|" ><mi
15554
class="MathClass-ord">f</mi> <mfenced
15555
open="|" close="|" ><msubsup
15557
accent="true"><mrow
15559
class="MathClass-ord">S</mi></mrow><mrow
15561
></mrow></munderover><mrow
15563
class="MathClass-ord">a</mi><mo
15564
class="MathClass-punc">,</mo> <mo
15565
class="MathClass-bin">−</mo></mrow><mrow
15567
class="MathClass-bin">−</mo> <mn
15568
class="MathClass-ord">1</mn><mo
15569
class="MathClass-punc">,</mo> <mn
15570
class="MathClass-ord">0</mn></mrow></msubsup
15573
class="MathClass-ord">W</mi><mrow
15575
class="MathClass-ord">2</mn></mrow></msub
15577
class="MathClass-open">(</mo><mi
15578
class="MathClass-ord">Ω</mi><mo
15579
class="MathClass-punc">,</mo> <msub
15581
class="MathClass-ord">Γ</mi><mrow
15583
class="MathClass-ord">l</mi></mrow></msub
15585
class="MathClass-close">)</mo></mrow></mfenced></mfenced> <mfenced
15586
open="|" close="|" ><mfenced
15587
open="|" close="|" ><mi
15588
class="MathClass-ord">u</mi></mfenced> <mover><mrow
15590
class="MathClass-bin">◦</mo> </mrow><mrow
15592
class="MathClass-rel">→</mo> </mrow></mover><msubsup
15594
class="MathClass-ord">W</mi><mrow
15596
class="MathClass-ord">2</mn></mrow><mrow
15598
accent="true"><mrow
15600
class="MathClass-ord">A</mi></mrow><mrow
15602
></mrow></munderover></mrow></msubsup
15604
class="MathClass-open">(</mo><mi
15605
class="MathClass-ord">Ω</mi><mo
15606
class="MathClass-punc">;</mo> <msub
15608
class="MathClass-ord">Γ</mi><mrow
15610
class="MathClass-ord">r</mi></mrow></msub
15612
class="MathClass-punc">,</mo> <mi
15613
class="MathClass-ord">T</mi><mo
15614
class="MathClass-close">)</mo></mrow></mfenced></mfenced> <mo
15615
class="MathClass-punc">.</mo>
15616
</mtd></mtr></mtable> </mtd>
15618
class="align-odd"> <mtable class="split"><mtr><mtd>
15620
class="split-mtr"></mrow><mrow
15621
class="split-mtd"></mrow> <mfenced
15622
open="|" close="|" ><msub
15624
class="MathClass-ord">I</mi><mrow
15626
class="MathClass-ord">2</mn></mrow></msub
15628
class="split-mtd"></mrow> <mo
15629
class="MathClass-rel">=</mo> <mfenced
15630
open="|" close="|" ><msubsup
15632
class="MathClass-op">∫</mo>
15635
class="MathClass-ord">0</mn></mrow><mrow
15637
class="MathClass-ord">T</mi></mrow></msubsup
15639
class="MathClass-ord">ψ</mi><mrow><mo
15640
class="MathClass-open">(</mo><mi
15641
class="MathClass-ord">t</mi><mo
15642
class="MathClass-close">)</mo></mrow> <mfenced
15643
open="{" close="}" ><mi
15644
class="MathClass-ord">u</mi><mrow><mo
15645
class="MathClass-open">(</mo><mi
15646
class="MathClass-ord">a</mi><mo
15647
class="MathClass-punc">,</mo> <mi
15648
class="MathClass-ord">t</mi><mo
15649
class="MathClass-close">)</mo></mrow> <mo
15650
class="MathClass-bin">−</mo><msubsup
15652
class="MathClass-op">∫</mo>
15655
class="MathClass-ord">γ</mi><mrow><mo
15656
class="MathClass-open">(</mo><mi
15657
class="MathClass-ord">t</mi><mo
15658
class="MathClass-close">)</mo></mrow></mrow><mrow
15660
class="MathClass-ord">a</mi></mrow></msubsup
15663
class="MathClass-ord">d</mi><mi
15664
class="MathClass-ord">θ</mi></mrow>
15667
class="MathClass-ord">k</mi><mrow><mo
15668
class="MathClass-open">(</mo><mi
15669
class="MathClass-ord">θ</mi><mo
15670
class="MathClass-punc">,</mo> <mi
15671
class="MathClass-ord">t</mi><mo
15672
class="MathClass-close">)</mo></mrow></mrow></mfrac><msubsup
15674
class="MathClass-op">∫</mo>
15677
class="MathClass-ord">a</mi></mrow><mrow
15679
class="MathClass-ord">θ</mi></mrow></msubsup
15681
class="MathClass-ord">c</mi><mrow><mo
15682
class="MathClass-open">(</mo><mi
15683
class="MathClass-ord">ξ</mi><mo
15684
class="MathClass-close">)</mo></mrow><msub
15686
class="MathClass-ord">u</mi><mrow
15688
class="MathClass-ord">t</mi></mrow></msub
15690
class="MathClass-open">(</mo><mi
15691
class="MathClass-ord">ξ</mi><mo
15692
class="MathClass-punc">,</mo> <mi
15693
class="MathClass-ord">t</mi><mo
15694
class="MathClass-close">)</mo></mrow> <mi
15695
class="MathClass-ord">d</mi><mi
15696
class="MathClass-ord">ξ</mi></mfenced> <mi
15697
class="MathClass-ord">d</mi><mi
15698
class="MathClass-ord">t</mi></mfenced>
15700
class="split-mtr"></mrow><mrow
15701
class="split-mtd"></mrow> <mrow
15702
class="split-mtd"></mrow> <mo
15703
class="MathClass-rel">≤</mo> <msub
15705
class="MathClass-ord">C</mi><mrow
15707
class="MathClass-ord">6</mn></mrow></msub
15709
open="|" close="|" ><mfenced
15710
open="|" close="|" ><mi
15711
class="MathClass-ord">f</mi><msub
15713
class="MathClass-op">∫</mo>
15716
class="MathClass-ord">Ω</mi></mrow></msub
15718
open="|" close="|" ><msubsup
15720
accent="true"><mrow
15722
class="MathClass-ord">S</mi></mrow><mrow
15724
></mrow></munderover><mrow
15726
class="MathClass-ord">a</mi><mo
15727
class="MathClass-punc">,</mo> <mo
15728
class="MathClass-bin">−</mo></mrow><mrow
15730
class="MathClass-bin">−</mo> <mn
15731
class="MathClass-ord">1</mn><mo
15732
class="MathClass-punc">,</mo> <mn
15733
class="MathClass-ord">0</mn></mrow></msubsup
15736
class="MathClass-ord">W</mi><mrow
15738
class="MathClass-ord">2</mn></mrow></msub
15740
class="MathClass-open">(</mo><mi
15741
class="MathClass-ord">Ω</mi><mo
15742
class="MathClass-punc">,</mo> <msub
15744
class="MathClass-ord">Γ</mi><mrow
15746
class="MathClass-ord">l</mi></mrow></msub
15748
class="MathClass-close">)</mo></mrow></mfenced></mfenced> <mfenced
15749
open="|" close="|" ><mfenced
15750
open="|" close="|" ><mi
15751
class="MathClass-ord">u</mi></mfenced> <mover><mrow
15753
class="MathClass-bin">◦</mo> </mrow><mrow
15755
class="MathClass-rel">→</mo> </mrow></mover><msubsup
15757
class="MathClass-ord">W</mi><mrow
15759
class="MathClass-ord">2</mn></mrow><mrow
15761
accent="true"><mrow
15763
class="MathClass-ord">A</mi></mrow><mrow
15765
></mrow></munderover></mrow></msubsup
15767
class="MathClass-open">(</mo><mi
15768
class="MathClass-ord">Ω</mi><mo
15769
class="MathClass-punc">;</mo> <msub
15771
class="MathClass-ord">Γ</mi><mrow
15773
class="MathClass-ord">r</mi></mrow></msub
15775
class="MathClass-punc">,</mo> <mi
15776
class="MathClass-ord">T</mi><mo
15777
class="MathClass-close">)</mo></mrow></mfenced></mfenced> <mo
15778
class="MathClass-punc">.</mo>
15779
</mtd></mtr></mtable> </mtd> <mtd
15780
class="align-even"></mtd> <mtd
15781
class="align-label"> <mrow><mo
15782
class="MathClass-open">(</mo><mn
15783
class="MathClass-ord">6</mn><mn
15784
class="MathClass-ord">7</mn><mi
15785
class="MathClass-ord">′</mi><mo
15786
class="MathClass-close">)</mo></mrow><mspace
15787
id="x1-34006r67" class="label" width="0pt" /><mspace class="endlabel" width="0pt" />
15788
</mtd></mtr></mtable></math>
15789
Some text after to test the below-display spacing.
15792
<table width="100%"
15793
class="verbatim"><tr class="verbatim"><td
15794
class="verbatim"><pre class="verbatim">
15795
 \begin{align*}
15796
 \begin{split}\abs{I_1}&=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\
15797
   &\le C_3\left[\int_\Omega\left(\int_{a}ˆx
15798
   g(\xi,t)\,d\xi\right)ˆ2d\Omega\right]ˆ{1/2}\\
15799
 &\phantom{=}\times \left[\int_\Omega\left\{uˆ2_x+\frac{1}{k}
15800
   \left(\int_{a}ˆx cu_t\,d\xi\right)ˆ2\right\}
15801
   c\Omega\right]ˆ{1/2}\\
15802
 &\le C_4\left\lvert \left\lvert f\left\lvert \wt{S}ˆ{-1,0}_{a,-}
15803
   W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
15804
   \left\lvert \abs{u}\overset{\circ}\to W_2ˆ{\wt{A}}
15805
   (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
15806
 \end{split}\\
15807
 \begin{split}\abs{I_2}&=\left\lvert \int_{0}ˆT \psi(t)\left\{u(a,t)
15808
   -\int_{\gamma(t)}ˆa\frac{d\theta}{k(\theta,t)}
15809
   \int_{a}ˆ\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\
15810
 &\le C_6\left\lvert \left\lvert f\int_\Omega
15811
   \left\lvert \wt{S}ˆ{-1,0}_{a,-}
15812
   W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
15813
   \left\lvert \abs{u}\overset{\circ}\to W_2ˆ{\wt{A}}
15814
   (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
15815
 \end{split}\tag{\theequation$’$}
15816
 \end{align*}
15821
<h4 class="subsectionHead"><span class="titlemark">A.2</span> <a
15822
name="x1-35000A.2"></a>Multline</h4> Numbered version:
15823
<!--l. 1950--><math
15824
xmlns="http://www.w3.org/1998/Math/MathML"
15829
class="multline"></mtd><mtd><mspace
15830
id="x1-35001r68" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><msubsup
15832
class="MathClass-op">∫</mo>
15835
class="MathClass-ord">a</mi></mrow><mrow
15837
class="MathClass-ord">b</mi></mrow></msubsup
15839
class="MathClass-open">{
15842
class="MathClass-op">∫</mo>
15845
class="MathClass-ord">a</mi></mrow><mrow
15847
class="MathClass-ord">b</mi></mrow></msubsup
15849
class="MathClass-open">[</mo><mi
15850
class="MathClass-ord">f</mi><msup
15852
class="MathClass-open">(</mo><mi
15853
class="MathClass-ord">x</mi><mo
15854
class="MathClass-close">)</mo></mrow><mrow
15856
class="MathClass-ord">2</mn></mrow></msup
15858
class="MathClass-ord">g</mi><msup
15860
class="MathClass-open">(</mo><mi
15861
class="MathClass-ord">y</mi><mo
15862
class="MathClass-close">)</mo></mrow><mrow
15864
class="MathClass-ord">2</mn></mrow></msup
15866
class="MathClass-bin">+</mo> <mi
15867
class="MathClass-ord">f</mi><msup
15869
class="MathClass-open">(</mo><mi
15870
class="MathClass-ord">y</mi><mo
15871
class="MathClass-close">)</mo></mrow><mrow
15873
class="MathClass-ord">2</mn></mrow></msup
15875
class="MathClass-ord">g</mi><msup
15877
class="MathClass-open">(</mo><mi
15878
class="MathClass-ord">x</mi><mo
15879
class="MathClass-close">)</mo></mrow><mrow
15881
class="MathClass-ord">2</mn></mrow></msup
15883
class="MathClass-close">]</mo></mrow> <mo
15884
class="MathClass-bin">−</mo> <mn
15885
class="MathClass-ord">2</mn><mi
15886
class="MathClass-ord">f</mi><mrow><mo
15887
class="MathClass-open">(</mo><mi
15888
class="MathClass-ord">x</mi><mo
15889
class="MathClass-close">)</mo></mrow><mi
15890
class="MathClass-ord">g</mi><mrow><mo
15891
class="MathClass-open">(</mo><mi
15892
class="MathClass-ord">x</mi><mo
15893
class="MathClass-close">)</mo></mrow><mi
15894
class="MathClass-ord">f</mi><mrow><mo
15895
class="MathClass-open">(</mo><mi
15896
class="MathClass-ord">y</mi><mo
15897
class="MathClass-close">)</mo></mrow><mi
15898
class="MathClass-ord">g</mi><mrow><mo
15899
class="MathClass-open">(</mo><mi
15900
class="MathClass-ord">y</mi><mo
15901
class="MathClass-close">)</mo></mrow> <mi
15902
class="MathClass-ord">d</mi><mi
15903
class="MathClass-ord">x</mi><mo
15904
class="MathClass-close">}
15906
class="MathClass-ord">d</mi><mi
15907
class="MathClass-ord">y</mi>
15908
</mtd></mtr><mtr><mtd
15909
class="multline"></mtd><mtd> <mo
15910
class="MathClass-rel">=</mo><msubsup
15912
class="MathClass-op">∫</mo>
15915
class="MathClass-ord">a</mi></mrow><mrow
15917
class="MathClass-ord">b</mi></mrow></msubsup
15919
class="MathClass-open">{
15921
class="MathClass-ord">g</mi><msup
15923
class="MathClass-open">(</mo><mi
15924
class="MathClass-ord">y</mi><mo
15925
class="MathClass-close">)</mo></mrow><mrow
15927
class="MathClass-ord">2</mn></mrow></msup
15930
class="MathClass-op">∫</mo>
15933
class="MathClass-ord">a</mi></mrow><mrow
15935
class="MathClass-ord">b</mi></mrow></msubsup
15938
class="MathClass-ord">f</mi><mrow
15940
class="MathClass-ord">2</mn></mrow></msup
15942
class="MathClass-bin">+</mo> <mi
15943
class="MathClass-ord">f</mi><msup
15945
class="MathClass-open">(</mo><mi
15946
class="MathClass-ord">y</mi><mo
15947
class="MathClass-close">)</mo></mrow><mrow
15949
class="MathClass-ord">2</mn></mrow></msup
15952
class="MathClass-op">∫</mo>
15955
class="MathClass-ord">a</mi></mrow><mrow
15957
class="MathClass-ord">b</mi></mrow></msubsup
15960
class="MathClass-ord">g</mi><mrow
15962
class="MathClass-ord">2</mn></mrow></msup
15964
class="MathClass-bin">−</mo> <mn
15965
class="MathClass-ord">2</mn><mi
15966
class="MathClass-ord">f</mi><mrow><mo
15967
class="MathClass-open">(</mo><mi
15968
class="MathClass-ord">y</mi><mo
15969
class="MathClass-close">)</mo></mrow><mi
15970
class="MathClass-ord">g</mi><mrow><mo
15971
class="MathClass-open">(</mo><mi
15972
class="MathClass-ord">y</mi><mo
15973
class="MathClass-close">)</mo></mrow><msubsup
15975
class="MathClass-op">∫</mo>
15978
class="MathClass-ord">a</mi></mrow><mrow
15980
class="MathClass-ord">b</mi></mrow></msubsup
15982
class="MathClass-ord">f</mi><mi
15983
class="MathClass-ord">g</mi><mo
15984
class="MathClass-close">}
15986
class="MathClass-ord">d</mi><mi
15987
class="MathClass-ord">y</mi></mtd><mtd><mrow><mo
15988
class="MathClass-open">(</mo><mn
15989
class="MathClass-ord">6</mn><mn
15990
class="MathClass-ord">8</mn><mo
15991
class="MathClass-close">)</mo></mrow> </mtd></mtr></mtable>
15993
<!--l. 1955--><p class="nopar">
15994
To test the use of <span class="obeylines-h"><span
15995
class="cmtt-10">\label</span></span> and <span class="obeylines-h"><span
15996
class="cmtt-10">\ref</span></span>, we refer to the number of this equation here:
15998
href="#x1-35001r68">68<!--tex4ht:ref: eq:E--></a>).
16002
<table width="100%"
16003
class="verbatim"><tr class="verbatim"><td
16004
class="verbatim"><pre class="verbatim">
16005
 \begin{multline}\label{eq:E}
16006
 \int_aˆb\biggl\{\int_aˆb[f(x)ˆ2g(y)ˆ2+f(y)ˆ2g(x)ˆ2]
16007
  -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\
16008
  =\int_aˆb\biggl\{g(y)ˆ2\int_aˆbfˆ2+f(y)ˆ2
16009
   \int_aˆb gˆ2-2f(y)g(y)\int_aˆb fg\biggr\}\,dy
16010
 \end{multline}
16013
<!--l. 1970--><p class="indent"> Unnumbered version:
16014
</p><!--l. 1971--><math
16015
xmlns="http://www.w3.org/1998/Math/MathML"
16018
class="multline-star">
16020
class="multline-star"><msubsup
16022
class="MathClass-op">∫</mo>
16025
class="MathClass-ord">a</mi></mrow><mrow
16027
class="MathClass-ord">b</mi></mrow></msubsup
16029
class="MathClass-open">{
16032
class="MathClass-op">∫</mo>
16035
class="MathClass-ord">a</mi></mrow><mrow
16037
class="MathClass-ord">b</mi></mrow></msubsup
16039
class="MathClass-open">[</mo><mi
16040
class="MathClass-ord">f</mi><msup
16042
class="MathClass-open">(</mo><mi
16043
class="MathClass-ord">x</mi><mo
16044
class="MathClass-close">)</mo></mrow><mrow
16046
class="MathClass-ord">2</mn></mrow></msup
16048
class="MathClass-ord">g</mi><msup
16050
class="MathClass-open">(</mo><mi
16051
class="MathClass-ord">y</mi><mo
16052
class="MathClass-close">)</mo></mrow><mrow
16054
class="MathClass-ord">2</mn></mrow></msup
16056
class="MathClass-bin">+</mo> <mi
16057
class="MathClass-ord">f</mi><msup
16059
class="MathClass-open">(</mo><mi
16060
class="MathClass-ord">y</mi><mo
16061
class="MathClass-close">)</mo></mrow><mrow
16063
class="MathClass-ord">2</mn></mrow></msup
16065
class="MathClass-ord">g</mi><msup
16067
class="MathClass-open">(</mo><mi
16068
class="MathClass-ord">x</mi><mo
16069
class="MathClass-close">)</mo></mrow><mrow
16071
class="MathClass-ord">2</mn></mrow></msup
16073
class="MathClass-close">]</mo></mrow> <mo
16074
class="MathClass-bin">−</mo> <mn
16075
class="MathClass-ord">2</mn><mi
16076
class="MathClass-ord">f</mi><mrow><mo
16077
class="MathClass-open">(</mo><mi
16078
class="MathClass-ord">x</mi><mo
16079
class="MathClass-close">)</mo></mrow><mi
16080
class="MathClass-ord">g</mi><mrow><mo
16081
class="MathClass-open">(</mo><mi
16082
class="MathClass-ord">x</mi><mo
16083
class="MathClass-close">)</mo></mrow><mi
16084
class="MathClass-ord">f</mi><mrow><mo
16085
class="MathClass-open">(</mo><mi
16086
class="MathClass-ord">y</mi><mo
16087
class="MathClass-close">)</mo></mrow><mi
16088
class="MathClass-ord">g</mi><mrow><mo
16089
class="MathClass-open">(</mo><mi
16090
class="MathClass-ord">y</mi><mo
16091
class="MathClass-close">)</mo></mrow> <mi
16092
class="MathClass-ord">d</mi><mi
16093
class="MathClass-ord">x</mi><mo
16094
class="MathClass-close">}
16096
class="MathClass-ord">d</mi><mi
16097
class="MathClass-ord">y</mi>
16098
</mtd></mtr><mtr><mtd
16099
class="multline-star"> <mo
16100
class="MathClass-rel">=</mo><msubsup
16102
class="MathClass-op">∫</mo>
16105
class="MathClass-ord">a</mi></mrow><mrow
16107
class="MathClass-ord">b</mi></mrow></msubsup
16109
class="MathClass-open">{
16111
class="MathClass-ord">g</mi><msup
16113
class="MathClass-open">(</mo><mi
16114
class="MathClass-ord">y</mi><mo
16115
class="MathClass-close">)</mo></mrow><mrow
16117
class="MathClass-ord">2</mn></mrow></msup
16120
class="MathClass-op">∫</mo>
16123
class="MathClass-ord">a</mi></mrow><mrow
16125
class="MathClass-ord">b</mi></mrow></msubsup
16128
class="MathClass-ord">f</mi><mrow
16130
class="MathClass-ord">2</mn></mrow></msup
16132
class="MathClass-bin">+</mo> <mi
16133
class="MathClass-ord">f</mi><msup
16135
class="MathClass-open">(</mo><mi
16136
class="MathClass-ord">y</mi><mo
16137
class="MathClass-close">)</mo></mrow><mrow
16139
class="MathClass-ord">2</mn></mrow></msup
16142
class="MathClass-op">∫</mo>
16145
class="MathClass-ord">a</mi></mrow><mrow
16147
class="MathClass-ord">b</mi></mrow></msubsup
16150
class="MathClass-ord">g</mi><mrow
16152
class="MathClass-ord">2</mn></mrow></msup
16154
class="MathClass-bin">−</mo> <mn
16155
class="MathClass-ord">2</mn><mi
16156
class="MathClass-ord">f</mi><mrow><mo
16157
class="MathClass-open">(</mo><mi
16158
class="MathClass-ord">y</mi><mo
16159
class="MathClass-close">)</mo></mrow><mi
16160
class="MathClass-ord">g</mi><mrow><mo
16161
class="MathClass-open">(</mo><mi
16162
class="MathClass-ord">y</mi><mo
16163
class="MathClass-close">)</mo></mrow><msubsup
16165
class="MathClass-op">∫</mo>
16168
class="MathClass-ord">a</mi></mrow><mrow
16170
class="MathClass-ord">b</mi></mrow></msubsup
16172
class="MathClass-ord">f</mi><mi
16173
class="MathClass-ord">g</mi><mo
16174
class="MathClass-close">}
16176
class="MathClass-ord">d</mi><mi
16177
class="MathClass-ord">y</mi> </mtd></mtr></mtable>
16179
<!--l. 1976--><p class="nopar">
16180
Some text after to test the below-display spacing.
16184
<table width="100%"
16185
class="verbatim"><tr class="verbatim"><td
16186
class="verbatim"><pre class="verbatim">
16187
 \begin{multline*}
16188
 \int_aˆb\biggl\{\int_aˆb[f(x)ˆ2g(y)ˆ2+f(y)ˆ2g(x)ˆ2]
16189
  -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\
16190
  =\int_aˆb\biggl\{g(y)ˆ2\int_aˆbfˆ2+f(y)ˆ2
16191
   \int_aˆb gˆ2-2f(y)g(y)\int_aˆb fg\biggr\}\,dy
16192
 \end{multline*}
16197
<h4 class="subsectionHead"><span class="titlemark">A.3</span> <a
16198
name="x1-36000A.3"></a>Gather</h4> Numbered version with <span class="obeylines-h"><span
16199
class="cmtt-10">\notag</span></span> on the second line:
16200
<!--l. 2042--><math
16201
xmlns="http://www.w3.org/1998/Math/MathML"
16207
class="MathClass-ord">D</mi><mrow><mo
16208
class="MathClass-open">(</mo><mi
16209
class="MathClass-ord">a</mi><mo
16210
class="MathClass-punc">,</mo> <mi
16211
class="MathClass-ord">r</mi><mo
16212
class="MathClass-close">)</mo></mrow> <mo
16213
class="MathClass-rel">≡</mo> <mrow><mo
16214
class="MathClass-open">{</mo><mi
16215
class="MathClass-ord">z</mi> <mo
16216
class="MathClass-rel">∈</mo> <mi class="mathbf">C</mi><mo
16217
class="MathClass-punc">:</mo> <mfenced
16218
open="|" close="|" ><mi
16219
class="MathClass-ord">z</mi> <mo
16220
class="MathClass-bin">−</mo> <mi
16221
class="MathClass-ord">a</mi></mfenced> <mo
16222
class="MathClass-rel"><</mo> <mi
16223
class="MathClass-ord">r</mi><mo
16224
class="MathClass-close">}</mo></mrow><mo
16225
class="MathClass-punc">,</mo> </mtd>
16227
id="x1-36001r69" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
16228
class="MathClass-open">(</mo><mn
16229
class="MathClass-ord">6</mn><mn
16230
class="MathClass-ord">9</mn><mo
16231
class="MathClass-close">)</mo></mrow></mtd>
16234
class="MathClass-op"> seg</mo><!--nolimits--><mrow><mo
16235
class="MathClass-open">(</mo><mi
16236
class="MathClass-ord">a</mi><mo
16237
class="MathClass-punc">,</mo> <mi
16238
class="MathClass-ord">r</mi><mo
16239
class="MathClass-close">)</mo></mrow> <mo
16240
class="MathClass-rel">≡</mo> <mrow><mo
16241
class="MathClass-open">{</mo><mi
16242
class="MathClass-ord">z</mi> <mo
16243
class="MathClass-rel">∈</mo> <mi class="mathbf">C</mi><mo
16244
class="MathClass-punc">:</mo> <mi
16245
class="MathClass-ord">ℑ</mi><mi
16246
class="MathClass-ord">z</mi> <mo
16247
class="MathClass-rel">=</mo> <mi
16248
class="MathClass-ord">ℑ</mi><mi
16249
class="MathClass-ord">a</mi><mo
16250
class="MathClass-punc">,</mo> <mspace class="nbsp" /> <mfenced
16251
open="|" close="|" ><mi
16252
class="MathClass-ord">z</mi> <mo
16253
class="MathClass-bin">−</mo> <mi
16254
class="MathClass-ord">a</mi></mfenced> <mo
16255
class="MathClass-rel"><</mo> <mi
16256
class="MathClass-ord">r</mi><mo
16257
class="MathClass-close">}</mo></mrow><mo
16258
class="MathClass-punc">,</mo> </mtd>
16262
class="MathClass-ord">c</mi><mrow><mo
16263
class="MathClass-open">(</mo><mi
16264
class="MathClass-ord">e</mi><mo
16265
class="MathClass-punc">,</mo> <mi
16266
class="MathClass-ord">θ</mi><mo
16267
class="MathClass-punc">,</mo> <mi
16268
class="MathClass-ord">r</mi><mo
16269
class="MathClass-close">)</mo></mrow> <mo
16270
class="MathClass-rel">≡</mo> <mrow><mo
16271
class="MathClass-open">{</mo><mrow><mo
16272
class="MathClass-open">(</mo><mi
16273
class="MathClass-ord">x</mi><mo
16274
class="MathClass-punc">,</mo> <mi
16275
class="MathClass-ord">y</mi><mo
16276
class="MathClass-close">)</mo></mrow> <mo
16277
class="MathClass-rel">∈</mo> <mi class="mathbf">C</mi><mo
16278
class="MathClass-punc">:</mo> <mfenced
16279
open="|" close="|" ><mi
16280
class="MathClass-ord">x</mi> <mo
16281
class="MathClass-bin">−</mo> <mi
16282
class="MathClass-ord">e</mi></mfenced> <mo
16283
class="MathClass-rel"><</mo> <mi
16284
class="MathClass-ord">y</mi><mo
16285
> tan</mo><!--nolimits--> <mi
16286
class="MathClass-ord">θ</mi><mo
16287
class="MathClass-punc">,</mo> <mspace class="nbsp" /><mn
16288
class="MathClass-ord">0</mn> <mo
16289
class="MathClass-rel"><</mo> <mi
16290
class="MathClass-ord">y</mi> <mo
16291
class="MathClass-rel"><</mo> <mi
16292
class="MathClass-ord">r</mi><mo
16293
class="MathClass-close">}</mo></mrow><mo
16294
class="MathClass-punc">,</mo> </mtd>
16296
id="x1-36002r70" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
16297
class="MathClass-open">(</mo><mn
16298
class="MathClass-ord">7</mn><mn
16299
class="MathClass-ord">0</mn><mo
16300
class="MathClass-close">)</mo></mrow></mtd>
16303
class="MathClass-ord">C</mi><mrow><mo
16304
class="MathClass-open">(</mo><mi
16305
class="MathClass-ord">E</mi><mo
16306
class="MathClass-punc">,</mo> <mi
16307
class="MathClass-ord">θ</mi><mo
16308
class="MathClass-punc">,</mo> <mi
16309
class="MathClass-ord">r</mi><mo
16310
class="MathClass-close">)</mo></mrow> <mo
16311
class="MathClass-rel">≡</mo><msub
16313
class="MathClass-op">⋃</mo>
16316
class="MathClass-ord">e</mi><mo
16317
class="MathClass-rel">∈</mo><mi
16318
class="MathClass-ord">E</mi></mrow></msub
16320
class="MathClass-ord">c</mi><mrow><mo
16321
class="MathClass-open">(</mo><mi
16322
class="MathClass-ord">e</mi><mo
16323
class="MathClass-punc">,</mo> <mi
16324
class="MathClass-ord">θ</mi><mo
16325
class="MathClass-punc">,</mo> <mi
16326
class="MathClass-ord">r</mi><mo
16327
class="MathClass-close">)</mo></mrow><mo
16328
class="MathClass-punc">.</mo></mtd>
16330
id="x1-36003r71" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
16331
class="MathClass-open">(</mo><mn
16332
class="MathClass-ord">7</mn><mn
16333
class="MathClass-ord">1</mn><mo
16334
class="MathClass-close">)</mo></mrow></mtd> </mtr></mtable>
16336
<!--l. 2049--><p class="nopar">
16340
<table width="100%"
16341
class="verbatim"><tr class="verbatim"><td
16342
class="verbatim"><pre class="verbatim">
16343
 \begin{gather}
16344
 D(a,r)\equiv\{z\in\mathbf{C}\colon \abs{z-a}<r\},\\
16345
 \seg(a,r)\equiv\{z\in\mathbf{C}\colon
16346
 \Im z= \Im a,\ \abs{z-a}<r\},\notag\\
16347
 c(e,\theta,r)\equiv\{(x,y)\in\mathbf{C}
16348
 \colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\
16349
 C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r).
16350
 \end{gather}
16353
<!--l. 2062--><p class="indent"> Unnumbered version.
16354
</p><!--l. 2063--><math
16355
xmlns="http://www.w3.org/1998/Math/MathML"
16358
class="gather-star">
16361
class="MathClass-ord">D</mi><mrow><mo
16362
class="MathClass-open">(</mo><mi
16363
class="MathClass-ord">a</mi><mo
16364
class="MathClass-punc">,</mo> <mi
16365
class="MathClass-ord">r</mi><mo
16366
class="MathClass-close">)</mo></mrow> <mo
16367
class="MathClass-rel">≡</mo> <mrow><mo
16368
class="MathClass-open">{</mo><mi
16369
class="MathClass-ord">z</mi> <mo
16370
class="MathClass-rel">∈</mo> <mi class="mathbf">C</mi><mo
16371
class="MathClass-punc">:</mo> <mfenced
16372
open="|" close="|" ><mi
16373
class="MathClass-ord">z</mi> <mo
16374
class="MathClass-bin">−</mo> <mi
16375
class="MathClass-ord">a</mi></mfenced> <mo
16376
class="MathClass-rel"><</mo> <mi
16377
class="MathClass-ord">r</mi><mo
16378
class="MathClass-close">}</mo></mrow><mo
16379
class="MathClass-punc">,</mo> </mtd>
16383
class="MathClass-op"> seg</mo><!--nolimits--><mrow><mo
16384
class="MathClass-open">(</mo><mi
16385
class="MathClass-ord">a</mi><mo
16386
class="MathClass-punc">,</mo> <mi
16387
class="MathClass-ord">r</mi><mo
16388
class="MathClass-close">)</mo></mrow> <mo
16389
class="MathClass-rel">≡</mo> <mrow><mo
16390
class="MathClass-open">{</mo><mi
16391
class="MathClass-ord">z</mi> <mo
16392
class="MathClass-rel">∈</mo> <mi class="mathbf">C</mi><mo
16393
class="MathClass-punc">:</mo> <mi
16394
class="MathClass-ord">ℑ</mi><mi
16395
class="MathClass-ord">z</mi> <mo
16396
class="MathClass-rel">=</mo> <mi
16397
class="MathClass-ord">ℑ</mi><mi
16398
class="MathClass-ord">a</mi><mo
16399
class="MathClass-punc">,</mo> <mspace class="nbsp" /> <mfenced
16400
open="|" close="|" ><mi
16401
class="MathClass-ord">z</mi> <mo
16402
class="MathClass-bin">−</mo> <mi
16403
class="MathClass-ord">a</mi></mfenced> <mo
16404
class="MathClass-rel"><</mo> <mi
16405
class="MathClass-ord">r</mi><mo
16406
class="MathClass-close">}</mo></mrow><mo
16407
class="MathClass-punc">,</mo> </mtd>
16411
class="MathClass-ord">c</mi><mrow><mo
16412
class="MathClass-open">(</mo><mi
16413
class="MathClass-ord">e</mi><mo
16414
class="MathClass-punc">,</mo> <mi
16415
class="MathClass-ord">θ</mi><mo
16416
class="MathClass-punc">,</mo> <mi
16417
class="MathClass-ord">r</mi><mo
16418
class="MathClass-close">)</mo></mrow> <mo
16419
class="MathClass-rel">≡</mo> <mrow><mo
16420
class="MathClass-open">{</mo><mrow><mo
16421
class="MathClass-open">(</mo><mi
16422
class="MathClass-ord">x</mi><mo
16423
class="MathClass-punc">,</mo> <mi
16424
class="MathClass-ord">y</mi><mo
16425
class="MathClass-close">)</mo></mrow> <mo
16426
class="MathClass-rel">∈</mo> <mi class="mathbf">C</mi><mo
16427
class="MathClass-punc">:</mo> <mfenced
16428
open="|" close="|" ><mi
16429
class="MathClass-ord">x</mi> <mo
16430
class="MathClass-bin">−</mo> <mi
16431
class="MathClass-ord">e</mi></mfenced> <mo
16432
class="MathClass-rel"><</mo> <mi
16433
class="MathClass-ord">y</mi><mo
16434
> tan</mo><!--nolimits--> <mi
16435
class="MathClass-ord">θ</mi><mo
16436
class="MathClass-punc">,</mo> <mspace class="nbsp" /><mn
16437
class="MathClass-ord">0</mn> <mo
16438
class="MathClass-rel"><</mo> <mi
16439
class="MathClass-ord">y</mi> <mo
16440
class="MathClass-rel"><</mo> <mi
16441
class="MathClass-ord">r</mi><mo
16442
class="MathClass-close">}</mo></mrow><mo
16443
class="MathClass-punc">,</mo> </mtd>
16447
class="MathClass-ord">C</mi><mrow><mo
16448
class="MathClass-open">(</mo><mi
16449
class="MathClass-ord">E</mi><mo
16450
class="MathClass-punc">,</mo> <mi
16451
class="MathClass-ord">θ</mi><mo
16452
class="MathClass-punc">,</mo> <mi
16453
class="MathClass-ord">r</mi><mo
16454
class="MathClass-close">)</mo></mrow> <mo
16455
class="MathClass-rel">≡</mo><msub
16457
class="MathClass-op">⋃</mo>
16460
class="MathClass-ord">e</mi><mo
16461
class="MathClass-rel">∈</mo><mi
16462
class="MathClass-ord">E</mi></mrow></msub
16464
class="MathClass-ord">c</mi><mrow><mo
16465
class="MathClass-open">(</mo><mi
16466
class="MathClass-ord">e</mi><mo
16467
class="MathClass-punc">,</mo> <mi
16468
class="MathClass-ord">θ</mi><mo
16469
class="MathClass-punc">,</mo> <mi
16470
class="MathClass-ord">r</mi><mo
16471
class="MathClass-close">)</mo></mrow><mo
16472
class="MathClass-punc">.</mo></mtd>
16473
<mtd></mtd> </mtr></mtable>
16475
<!--l. 2070--><p class="nopar">
16476
Some text after to test the below-display spacing.
16480
<table width="100%"
16481
class="verbatim"><tr class="verbatim"><td
16482
class="verbatim"><pre class="verbatim">
16483
 \begin{gather*}
16484
 D(a,r)\equiv\{z\in\mathbf{C}\colon \abs{z-a}<r\},\\
16485
 \seg (a,r)\equiv\{z\in\mathbf{C}\colon
16486
 \Im z= \Im a,\ \abs{z-a}<r\},\\
16487
 c(e,\theta,r)\equiv\{(x,y)\in\mathbf{C}
16488
  \colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\
16489
 C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r).
16490
 \end{gather*}
16495
<h4 class="subsectionHead"><span class="titlemark">A.4</span> <a
16496
name="x1-37000A.4"></a>Align</h4> Numbered version:
16497
<!--tex4ht:inline--><!--l. 2092--><math
16498
xmlns="http://www.w3.org/1998/Math/MathML"
16499
display="block"><mtable
16502
class="align-odd"><msub
16504
class="MathClass-ord">γ</mi><mrow
16506
class="MathClass-ord">x</mi></mrow></msub
16508
class="MathClass-open">(</mo><mi
16509
class="MathClass-ord">t</mi><mo
16510
class="MathClass-close">)</mo></mrow></mtd> <mtd
16511
class="align-even"> <mo
16512
class="MathClass-rel">=</mo> <mrow><mo
16513
class="MathClass-open">(</mo><mo
16514
>cos</mo><!--nolimits--> <mi
16515
class="MathClass-ord">t</mi><mi
16516
class="MathClass-ord">u</mi> <mo
16517
class="MathClass-bin">+</mo><mo
16518
> sin</mo><!--nolimits--> <mi
16519
class="MathClass-ord">t</mi><mi
16520
class="MathClass-ord">x</mi><mo
16521
class="MathClass-punc">,</mo> <mi
16522
class="MathClass-ord">v</mi><mo
16523
class="MathClass-close">)</mo></mrow><mo
16524
class="MathClass-punc">,</mo> </mtd> <mtd
16525
class="align-label"> <mspace
16526
id="x1-37001r72" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
16527
class="MathClass-open">(</mo><mn
16528
class="MathClass-ord">7</mn><mn
16529
class="MathClass-ord">2</mn><mo
16530
class="MathClass-close">)</mo></mrow>
16531
</mtd></mtr><mtr><mtd
16532
class="align-odd"><msub
16534
class="MathClass-ord">γ</mi><mrow
16536
class="MathClass-ord">y</mi></mrow></msub
16538
class="MathClass-open">(</mo><mi
16539
class="MathClass-ord">t</mi><mo
16540
class="MathClass-close">)</mo></mrow></mtd> <mtd
16541
class="align-even"> <mo
16542
class="MathClass-rel">=</mo> <mrow><mo
16543
class="MathClass-open">(</mo><mi
16544
class="MathClass-ord">u</mi><mo
16545
class="MathClass-punc">,</mo><mo
16546
> cos</mo><!--nolimits--> <mi
16547
class="MathClass-ord">t</mi><mi
16548
class="MathClass-ord">v</mi> <mo
16549
class="MathClass-bin">+</mo><mo
16550
> sin</mo><!--nolimits--> <mi
16551
class="MathClass-ord">t</mi><mi
16552
class="MathClass-ord">y</mi><mo
16553
class="MathClass-close">)</mo></mrow><mo
16554
class="MathClass-punc">,</mo> </mtd> <mtd
16555
class="align-label"> <mspace
16556
id="x1-37002r73" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
16557
class="MathClass-open">(</mo><mn
16558
class="MathClass-ord">7</mn><mn
16559
class="MathClass-ord">3</mn><mo
16560
class="MathClass-close">)</mo></mrow>
16561
</mtd></mtr><mtr><mtd
16562
class="align-odd"><msub
16564
class="MathClass-ord">γ</mi><mrow
16566
class="MathClass-ord">z</mi></mrow></msub
16568
class="MathClass-open">(</mo><mi
16569
class="MathClass-ord">t</mi><mo
16570
class="MathClass-close">)</mo></mrow></mtd> <mtd
16571
class="align-even"> <mo
16572
class="MathClass-rel">=</mo> <mfenced
16573
open="(" close=")" ><mo
16574
>cos</mo><!--nolimits--> <mi
16575
class="MathClass-ord">t</mi><mi
16576
class="MathClass-ord">u</mi> <mo
16577
class="MathClass-bin">+</mo> <mfrac><mrow
16579
class="MathClass-ord">α</mi></mrow>
16582
class="MathClass-ord">β</mi></mrow></mfrac><mo
16583
> sin</mo><!--nolimits--> <mi
16584
class="MathClass-ord">t</mi><mi
16585
class="MathClass-ord">v</mi><mo
16586
class="MathClass-punc">,</mo> <mo
16587
class="MathClass-bin">−</mo><mfrac><mrow
16589
class="MathClass-ord">β</mi></mrow>
16592
class="MathClass-ord">α</mi></mrow></mfrac><mo
16593
> sin</mo><!--nolimits--> <mi
16594
class="MathClass-ord">t</mi><mi
16595
class="MathClass-ord">u</mi> <mo
16596
class="MathClass-bin">+</mo><mo
16597
> cos</mo><!--nolimits--> <mi
16598
class="MathClass-ord">t</mi><mi
16599
class="MathClass-ord">v</mi></mfenced> <mo
16600
class="MathClass-punc">.</mo> </mtd> <mtd
16601
class="align-label"> <mspace
16602
id="x1-37003r74" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
16603
class="MathClass-open">(</mo><mn
16604
class="MathClass-ord">7</mn><mn
16605
class="MathClass-ord">4</mn><mo
16606
class="MathClass-close">)</mo></mrow>
16607
</mtd></mtr></mtable></math>
16608
Some text after to test the below-display spacing.
16611
<table width="100%"
16612
class="verbatim"><tr class="verbatim"><td
16613
class="verbatim"><pre class="verbatim">
16614
 \begin{align}
16615
 \gamma_x(t)&=(\cos tu+\sin tx,v),\\
16616
 \gamma_y(t)&=(u,\cos tv+\sin ty),\\
16617
 \gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv,
16618
   -\frac\beta\alpha\sin tu+\cos tv\right).
16619
 \end{align}
16622
<!--l. 2105--><p class="indent"> Unnumbered version:
16623
<!--tex4ht:inline--></p><!--l. 2111--><math
16624
xmlns="http://www.w3.org/1998/Math/MathML"
16625
display="block"><mtable
16626
class="align-star">
16628
class="align-odd"><msub
16630
class="MathClass-ord">γ</mi><mrow
16632
class="MathClass-ord">x</mi></mrow></msub
16634
class="MathClass-open">(</mo><mi
16635
class="MathClass-ord">t</mi><mo
16636
class="MathClass-close">)</mo></mrow></mtd> <mtd
16637
class="align-even"> <mo
16638
class="MathClass-rel">=</mo> <mrow><mo
16639
class="MathClass-open">(</mo><mo
16640
>cos</mo><!--nolimits--> <mi
16641
class="MathClass-ord">t</mi><mi
16642
class="MathClass-ord">u</mi> <mo
16643
class="MathClass-bin">+</mo><mo
16644
> sin</mo><!--nolimits--> <mi
16645
class="MathClass-ord">t</mi><mi
16646
class="MathClass-ord">x</mi><mo
16647
class="MathClass-punc">,</mo> <mi
16648
class="MathClass-ord">v</mi><mo
16649
class="MathClass-close">)</mo></mrow><mo
16650
class="MathClass-punc">,</mo> </mtd>
16652
class="align-odd"><msub
16654
class="MathClass-ord">γ</mi><mrow
16656
class="MathClass-ord">y</mi></mrow></msub
16658
class="MathClass-open">(</mo><mi
16659
class="MathClass-ord">t</mi><mo
16660
class="MathClass-close">)</mo></mrow></mtd> <mtd
16661
class="align-even"> <mo
16662
class="MathClass-rel">=</mo> <mrow><mo
16663
class="MathClass-open">(</mo><mi
16664
class="MathClass-ord">u</mi><mo
16665
class="MathClass-punc">,</mo><mo
16666
> cos</mo><!--nolimits--> <mi
16667
class="MathClass-ord">t</mi><mi
16668
class="MathClass-ord">v</mi> <mo
16669
class="MathClass-bin">+</mo><mo
16670
> sin</mo><!--nolimits--> <mi
16671
class="MathClass-ord">t</mi><mi
16672
class="MathClass-ord">y</mi><mo
16673
class="MathClass-close">)</mo></mrow><mo
16674
class="MathClass-punc">,</mo> </mtd>
16676
class="align-odd"><msub
16678
class="MathClass-ord">γ</mi><mrow
16680
class="MathClass-ord">z</mi></mrow></msub
16682
class="MathClass-open">(</mo><mi
16683
class="MathClass-ord">t</mi><mo
16684
class="MathClass-close">)</mo></mrow></mtd> <mtd
16685
class="align-even"> <mo
16686
class="MathClass-rel">=</mo> <mfenced
16687
open="(" close=")" ><mo
16688
>cos</mo><!--nolimits--> <mi
16689
class="MathClass-ord">t</mi><mi
16690
class="MathClass-ord">u</mi> <mo
16691
class="MathClass-bin">+</mo> <mfrac><mrow
16693
class="MathClass-ord">α</mi></mrow>
16696
class="MathClass-ord">β</mi></mrow></mfrac><mo
16697
> sin</mo><!--nolimits--> <mi
16698
class="MathClass-ord">t</mi><mi
16699
class="MathClass-ord">v</mi><mo
16700
class="MathClass-punc">,</mo> <mo
16701
class="MathClass-bin">−</mo><mfrac><mrow
16703
class="MathClass-ord">β</mi></mrow>
16706
class="MathClass-ord">α</mi></mrow></mfrac><mo
16707
> sin</mo><!--nolimits--> <mi
16708
class="MathClass-ord">t</mi><mi
16709
class="MathClass-ord">u</mi> <mo
16710
class="MathClass-bin">+</mo><mo
16711
> cos</mo><!--nolimits--> <mi
16712
class="MathClass-ord">t</mi><mi
16713
class="MathClass-ord">v</mi></mfenced> <mo
16714
class="MathClass-punc">.</mo> </mtd>
16715
</mtr></mtable></math>
16716
Some text after to test the below-display spacing.
16719
<table width="100%"
16720
class="verbatim"><tr class="verbatim"><td
16721
class="verbatim"><pre class="verbatim">
16722
 \begin{align*}
16723
 \gamma_x(t)&=(\cos tu+\sin tx,v),\\
16724
 \gamma_y(t)&=(u,\cos tv+\sin ty),\\
16725
 \gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv,
16726
   -\frac\beta\alpha\sin tu+\cos tv\right).
16727
 \end{align*}
16730
<!--l. 2124--><p class="indent"> A variation:
16731
<!--tex4ht:inline--></p><!--l. 2129--><math
16732
xmlns="http://www.w3.org/1998/Math/MathML"
16733
display="block"><mtable
16736
class="align-odd"><mi
16737
class="MathClass-ord">x</mi></mtd> <mtd
16738
class="align-even"> <mo
16739
class="MathClass-rel">=</mo> <mi
16740
class="MathClass-ord">y</mi></mtd> <mtd
16741
class="align-odd"></mtd> <mtd
16742
class="align-even"><mrow
16743
class="text"><mtext >by (</mtext><mtext
16744
name="x1-39003r84" class="label" >84<!--tex4ht:ref: eq:C--></mtext><mtext class="endlabel">)</mtext></mrow></mtd> <mtd
16745
class="align-label"> <mspace
16746
id="x1-37004r75" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
16747
class="MathClass-open">(</mo><mn
16748
class="MathClass-ord">7</mn><mn
16749
class="MathClass-ord">5</mn><mo
16750
class="MathClass-close">)</mo></mrow>
16751
</mtd></mtr><mtr><mtd
16752
class="align-odd"><mi
16753
class="MathClass-ord">x</mi><mi
16754
class="MathClass-ord">′</mi></mtd> <mtd
16755
class="align-even"> <mo
16756
class="MathClass-rel">=</mo> <mi
16757
class="MathClass-ord">y</mi><mi
16758
class="MathClass-ord">′</mi></mtd> <mtd
16759
class="align-odd"></mtd> <mtd
16760
class="align-even"><mrow
16761
class="text"><mtext >by (</mtext><mtext
16762
name="x1-39004r85" class="label" >85<!--tex4ht:ref: eq:D--></mtext><mtext class="endlabel">)</mtext></mrow></mtd> <mtd
16763
class="align-label"> <mspace
16764
id="x1-37005r76" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
16765
class="MathClass-open">(</mo><mn
16766
class="MathClass-ord">7</mn><mn
16767
class="MathClass-ord">6</mn><mo
16768
class="MathClass-close">)</mo></mrow>
16769
</mtd></mtr><mtr><mtd
16770
class="align-odd"><mi
16771
class="MathClass-ord">x</mi> <mo
16772
class="MathClass-bin">+</mo> <mi
16773
class="MathClass-ord">x</mi><mi
16774
class="MathClass-ord">′</mi></mtd> <mtd
16775
class="align-even"> <mo
16776
class="MathClass-rel">=</mo> <mi
16777
class="MathClass-ord">y</mi> <mo
16778
class="MathClass-bin">+</mo> <mi
16779
class="MathClass-ord">y</mi><mi
16780
class="MathClass-ord">′</mi></mtd> <mtd
16781
class="align-odd"></mtd> <mtd
16782
class="align-even"><mrow
16783
class="text"><mtext >by Axiom 1.</mtext></mrow></mtd> <mtd
16784
class="align-label"> <mspace
16785
id="x1-37006r77" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
16786
class="MathClass-open">(</mo><mn
16787
class="MathClass-ord">7</mn><mn
16788
class="MathClass-ord">7</mn><mo
16789
class="MathClass-close">)</mo></mrow>
16790
</mtd></mtr></mtable></math>
16791
Some text after to test the below-display spacing.
16794
<table width="100%"
16795
class="verbatim"><tr class="verbatim"><td
16796
class="verbatim"><pre class="verbatim">
16797
 \begin{align}
16798
 x& =y && \text {by (\ref{eq:C})}\\
16799
 x’& = y’ && \text {by (\ref{eq:D})}\\
16800
 x+x’ & = y+y’ && \text {by Axiom 1.}
16801
 \end{align}
16806
<h4 class="subsectionHead"><span class="titlemark">A.5</span> <a
16807
name="x1-38000A.5"></a>Align and split within gather</h4> When using the <span
16808
class="cmtt-10">align </span>environment within the
16810
class="cmtt-10">gather </span>environment, one or the other, or both, should be unnumbered (using the
16811
<span class="obeylines-h"><span
16812
class="cmtt-10">*</span></span> form); numbering both the outer and inner environment would cause a
16814
<!--l. 2148--><p class="noindent">Automatically numbered <span
16815
class="cmtt-10">gather </span>with <span
16816
class="cmtt-10">split </span>and <span
16817
class="cmtt-10">align*</span>:
16818
</p><!--l. 2149--><math
16819
xmlns="http://www.w3.org/1998/Math/MathML"
16824
<mtd> <mtable class="split"><mtr><mtd>
16826
class="split-mtr"></mrow><mrow
16827
class="split-mtd"></mrow> <mi
16828
class="MathClass-ord">ϕ</mi><mrow><mo
16829
class="MathClass-open">(</mo><mi
16830
class="MathClass-ord">x</mi><mo
16831
class="MathClass-punc">,</mo> <mi
16832
class="MathClass-ord">z</mi><mo
16833
class="MathClass-close">)</mo></mrow><mrow
16834
class="split-mtd"></mrow> <mo
16835
class="MathClass-rel">=</mo> <mi
16836
class="MathClass-ord">z</mi> <mo
16837
class="MathClass-bin">−</mo> <msub
16839
class="MathClass-ord">γ</mi><mrow
16841
class="MathClass-ord">1</mn><mn
16842
class="MathClass-ord">0</mn></mrow></msub
16844
class="MathClass-ord">x</mi> <mo
16845
class="MathClass-bin">−</mo> <msub
16847
class="MathClass-ord">γ</mi><mrow
16849
class="MathClass-ord">m</mi><mi
16850
class="MathClass-ord">n</mi></mrow></msub
16853
class="MathClass-ord">x</mi><mrow
16855
class="MathClass-ord">m</mi></mrow></msup
16858
class="MathClass-ord">z</mi><mrow
16860
class="MathClass-ord">n</mi></mrow></msup
16863
class="split-mtr"></mrow><mrow
16864
class="split-mtd"></mrow> <mrow
16865
class="split-mtd"></mrow> <mo
16866
class="MathClass-rel">=</mo> <mi
16867
class="MathClass-ord">z</mi> <mo
16868
class="MathClass-bin">−</mo> <mi
16869
class="MathClass-ord">M</mi><msup
16871
class="MathClass-ord">r</mi><mrow
16873
class="MathClass-bin">−</mo><mn
16874
class="MathClass-ord">1</mn></mrow></msup
16876
class="MathClass-ord">x</mi> <mo
16877
class="MathClass-bin">−</mo> <mi
16878
class="MathClass-ord">M</mi><msup
16880
class="MathClass-ord">r</mi><mrow
16882
class="MathClass-bin">−</mo><mrow><mo
16883
class="MathClass-open">(</mo><mi
16884
class="MathClass-ord">m</mi><mo
16885
class="MathClass-bin">+</mo><mi
16886
class="MathClass-ord">n</mi><mo
16887
class="MathClass-close">)</mo></mrow></mrow></msup
16890
class="MathClass-ord">x</mi><mrow
16892
class="MathClass-ord">m</mi></mrow></msup
16895
class="MathClass-ord">z</mi><mrow
16897
class="MathClass-ord">n</mi></mrow></msup
16899
</mtd></mtr></mtable> </mtd>
16901
id="x1-38001r78" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
16902
class="MathClass-open">(</mo><mn
16903
class="MathClass-ord">7</mn><mn
16904
class="MathClass-ord">8</mn><mo
16905
class="MathClass-close">)</mo></mrow></mtd>
16909
class="align-star">
16911
class="align-odd"><msup
16913
class="MathClass-ord">ζ</mi><mrow
16915
class="MathClass-ord">0</mn></mrow></msup
16917
class="align-even"> <mo
16918
class="MathClass-rel">=</mo> <msup
16920
class="MathClass-open">(</mo><msup
16922
class="MathClass-ord">ξ</mi><mrow
16924
class="MathClass-ord">0</mn></mrow></msup
16926
class="MathClass-close">)</mo></mrow><mrow
16928
class="MathClass-ord">2</mn></mrow></msup
16930
class="MathClass-punc">,</mo></mtd>
16932
class="align-odd"><msup
16934
class="MathClass-ord">ζ</mi><mrow
16936
class="MathClass-ord">1</mn></mrow></msup
16938
class="align-even"> <mo
16939
class="MathClass-rel">=</mo> <msup
16941
class="MathClass-ord">ξ</mi><mrow
16943
class="MathClass-ord">0</mn></mrow></msup
16946
class="MathClass-ord">ξ</mi><mrow
16948
class="MathClass-ord">1</mn></mrow></msup
16950
class="MathClass-punc">,</mo></mtd>
16952
class="align-odd"><msup
16954
class="MathClass-ord">ζ</mi><mrow
16956
class="MathClass-ord">2</mn></mrow></msup
16958
class="align-even"> <mo
16959
class="MathClass-rel">=</mo> <msup
16961
class="MathClass-open">(</mo><msup
16963
class="MathClass-ord">ξ</mi><mrow
16965
class="MathClass-ord">1</mn></mrow></msup
16967
class="MathClass-close">)</mo></mrow><mrow
16969
class="MathClass-ord">2</mn></mrow></msup
16971
class="MathClass-punc">,</mo></mtd>
16972
</mtr></mtable></mrow> </mtd>
16973
<mtd></mtd> </mtr></mtable>
16975
<!--l. 2159--><p class="nopar">
16977
class="cmtt-10">split </span>environment gets a number from the outer <span
16978
class="cmtt-10">gather </span>environment;
16979
numbers for individual lines of the <span
16980
class="cmtt-10">align* </span>are suppressed because of the
16985
<table width="100%"
16986
class="verbatim"><tr class="verbatim"><td
16987
class="verbatim"><pre class="verbatim">
16988
 \begin{gather}
16989
 \begin{split} \varphi(x,z)
16990
 &=z-\gamma_{10}x-\gamma_{mn}xˆmzˆn\\
16991
 &=z-Mrˆ{-1}x-Mrˆ{-(m+n)}xˆmzˆn
16992
 \end{split}\\[6pt]
16993
 \begin{align*}
16994
 \zetaˆ0 &=(\xiˆ0)ˆ2,\\
16995
 \zetaˆ1 &=\xiˆ0\xiˆ1,\\
16996
 \zetaˆ2 &=(\xiˆ1)ˆ2,
16997
 \end{align*}
16998
 \end{gather}
17001
<!--l. 2179--><p class="indent"> The <span class="obeylines-h"><span
17002
class="cmtt-10">*</span></span>-ed form of <span
17003
class="cmtt-10">gather </span>with the non-<span class="obeylines-h"><span
17004
class="cmtt-10">*</span></span>-ed form of <span
17005
class="cmtt-10">align</span>.
17006
</p><!--l. 2181--><math
17007
xmlns="http://www.w3.org/1998/Math/MathML"
17010
class="gather-star">
17012
<mtd> <mtable class="split"><mtr><mtd>
17014
class="split-mtr"></mrow><mrow
17015
class="split-mtd"></mrow> <mi
17016
class="MathClass-ord">ϕ</mi><mrow><mo
17017
class="MathClass-open">(</mo><mi
17018
class="MathClass-ord">x</mi><mo
17019
class="MathClass-punc">,</mo> <mi
17020
class="MathClass-ord">z</mi><mo
17021
class="MathClass-close">)</mo></mrow><mrow
17022
class="split-mtd"></mrow> <mo
17023
class="MathClass-rel">=</mo> <mi
17024
class="MathClass-ord">z</mi> <mo
17025
class="MathClass-bin">−</mo> <msub
17027
class="MathClass-ord">γ</mi><mrow
17029
class="MathClass-ord">1</mn><mn
17030
class="MathClass-ord">0</mn></mrow></msub
17032
class="MathClass-ord">x</mi> <mo
17033
class="MathClass-bin">−</mo> <msub
17035
class="MathClass-ord">γ</mi><mrow
17037
class="MathClass-ord">m</mi><mi
17038
class="MathClass-ord">n</mi></mrow></msub
17041
class="MathClass-ord">x</mi><mrow
17043
class="MathClass-ord">m</mi></mrow></msup
17046
class="MathClass-ord">z</mi><mrow
17048
class="MathClass-ord">n</mi></mrow></msup
17051
class="split-mtr"></mrow><mrow
17052
class="split-mtd"></mrow> <mrow
17053
class="split-mtd"></mrow> <mo
17054
class="MathClass-rel">=</mo> <mi
17055
class="MathClass-ord">z</mi> <mo
17056
class="MathClass-bin">−</mo> <mi
17057
class="MathClass-ord">M</mi><msup
17059
class="MathClass-ord">r</mi><mrow
17061
class="MathClass-bin">−</mo><mn
17062
class="MathClass-ord">1</mn></mrow></msup
17064
class="MathClass-ord">x</mi> <mo
17065
class="MathClass-bin">−</mo> <mi
17066
class="MathClass-ord">M</mi><msup
17068
class="MathClass-ord">r</mi><mrow
17070
class="MathClass-bin">−</mo><mrow><mo
17071
class="MathClass-open">(</mo><mi
17072
class="MathClass-ord">m</mi><mo
17073
class="MathClass-bin">+</mo><mi
17074
class="MathClass-ord">n</mi><mo
17075
class="MathClass-close">)</mo></mrow></mrow></msup
17078
class="MathClass-ord">x</mi><mrow
17080
class="MathClass-ord">m</mi></mrow></msup
17083
class="MathClass-ord">z</mi><mrow
17085
class="MathClass-ord">n</mi></mrow></msup
17087
</mtd></mtr></mtable> </mtd>
17094
class="align-odd"><msup
17096
class="MathClass-ord">ζ</mi><mrow
17098
class="MathClass-ord">0</mn></mrow></msup
17100
class="align-even"> <mo
17101
class="MathClass-rel">=</mo> <msup
17103
class="MathClass-open">(</mo><msup
17105
class="MathClass-ord">ξ</mi><mrow
17107
class="MathClass-ord">0</mn></mrow></msup
17109
class="MathClass-close">)</mo></mrow><mrow
17111
class="MathClass-ord">2</mn></mrow></msup
17113
class="MathClass-punc">,</mo></mtd> <mtd
17114
class="align-label"> <mspace
17115
id="x1-38002r79" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
17116
class="MathClass-open">(</mo><mn
17117
class="MathClass-ord">7</mn><mn
17118
class="MathClass-ord">9</mn><mo
17119
class="MathClass-close">)</mo></mrow>
17120
</mtd></mtr><mtr><mtd
17121
class="align-odd"><msup
17123
class="MathClass-ord">ζ</mi><mrow
17125
class="MathClass-ord">1</mn></mrow></msup
17127
class="align-even"> <mo
17128
class="MathClass-rel">=</mo> <msup
17130
class="MathClass-ord">ξ</mi><mrow
17132
class="MathClass-ord">0</mn></mrow></msup
17135
class="MathClass-ord">ξ</mi><mrow
17137
class="MathClass-ord">1</mn></mrow></msup
17139
class="MathClass-punc">,</mo></mtd> <mtd
17140
class="align-label"> <mspace
17141
id="x1-38003r80" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
17142
class="MathClass-open">(</mo><mn
17143
class="MathClass-ord">8</mn><mn
17144
class="MathClass-ord">0</mn><mo
17145
class="MathClass-close">)</mo></mrow>
17146
</mtd></mtr><mtr><mtd
17147
class="align-odd"><msup
17149
class="MathClass-ord">ζ</mi><mrow
17151
class="MathClass-ord">2</mn></mrow></msup
17153
class="align-even"> <mo
17154
class="MathClass-rel">=</mo> <msup
17156
class="MathClass-open">(</mo><msup
17158
class="MathClass-ord">ξ</mi><mrow
17160
class="MathClass-ord">1</mn></mrow></msup
17162
class="MathClass-close">)</mo></mrow><mrow
17164
class="MathClass-ord">2</mn></mrow></msup
17166
class="MathClass-punc">,</mo></mtd> <mtd
17167
class="align-label"> <mspace
17168
id="x1-38004r81" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
17169
class="MathClass-open">(</mo><mn
17170
class="MathClass-ord">8</mn><mn
17171
class="MathClass-ord">1</mn><mo
17172
class="MathClass-close">)</mo></mrow>
17173
</mtd></mtr></mtable></mrow> </mtd>
17174
<mtd></mtd></mtr></mtable>
17177
<!--l. 2190--><p class="nopar">
17178
Some text after to test the below-display spacing.
17182
<table width="100%"
17183
class="verbatim"><tr class="verbatim"><td
17184
class="verbatim"><pre class="verbatim">
17185
 \begin{gather*}
17186
 \begin{split} \varphi(x,z)
17187
 &=z-\gamma_{10}x-\gamma_{mn}xˆmzˆn\\
17188
 &=z-Mrˆ{-1}x-Mrˆ{-(m+n)}xˆmzˆn
17189
 \end{split}\\[6pt]
17190
 \begin{align} \zetaˆ0&=(\xiˆ0)ˆ2,\\
17191
 \zetaˆ1 &=\xiˆ0\xiˆ1,\\
17192
 \zetaˆ2 &=(\xiˆ1)ˆ2,
17193
 \end{align}
17194
 \end{gather*}
17199
<h4 class="subsectionHead"><span class="titlemark">A.6</span> <a
17200
name="x1-39000A.6"></a>Alignat</h4> Numbered version:
17201
<!--tex4ht:inline--><!--l. 2216--><math
17202
xmlns="http://www.w3.org/1998/Math/MathML"
17203
display="block"><mtable
17206
class="align-odd"><msub
17208
class="MathClass-ord">V</mi> <mrow
17210
class="MathClass-ord">i</mi></mrow></msub
17212
class="align-even"> <mo
17213
class="MathClass-rel">=</mo> <msub
17215
class="MathClass-ord">v</mi><mrow
17217
class="MathClass-ord">i</mi></mrow></msub
17219
class="MathClass-bin">−</mo> <msub
17221
class="MathClass-ord">q</mi><mrow
17223
class="MathClass-ord">i</mi></mrow></msub
17226
class="MathClass-ord">v</mi><mrow
17228
class="MathClass-ord">j</mi></mrow></msub
17230
class="MathClass-punc">,</mo></mtd> <mtd
17231
class="align-odd"><mspace width="2em" class="qquad"/><msub
17233
class="MathClass-ord">X</mi><mrow
17235
class="MathClass-ord">i</mi></mrow></msub
17237
class="align-even"> <mo
17238
class="MathClass-rel">=</mo> <msub
17240
class="MathClass-ord">x</mi><mrow
17242
class="MathClass-ord">i</mi></mrow></msub
17244
class="MathClass-bin">−</mo> <msub
17246
class="MathClass-ord">q</mi><mrow
17248
class="MathClass-ord">i</mi></mrow></msub
17251
class="MathClass-ord">x</mi><mrow
17253
class="MathClass-ord">j</mi></mrow></msub
17255
class="MathClass-punc">,</mo></mtd> <mtd
17256
class="align-odd"><mspace width="2em" class="qquad"/><msub
17258
class="MathClass-ord">U</mi><mrow
17260
class="MathClass-ord">i</mi></mrow></msub
17262
class="align-even"> <mo
17263
class="MathClass-rel">=</mo> <msub
17265
class="MathClass-ord">u</mi><mrow
17267
class="MathClass-ord">i</mi></mrow></msub
17269
class="MathClass-punc">,</mo> <mspace width="2em" class="qquad"/><mrow
17270
class="text"><mtext >for </mtext><mrow
17272
class="MathClass-ord">i</mi><mo
17273
class="MathClass-rel">≠</mo><mi
17274
class="MathClass-ord">j</mi></mrow><mtext >;</mtext></mrow></mtd> <mtd
17275
class="align-label"> <mspace
17276
id="x1-39001r82" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
17277
class="MathClass-open">(</mo><mn
17278
class="MathClass-ord">8</mn><mn
17279
class="MathClass-ord">2</mn><mo
17280
class="MathClass-close">)</mo></mrow>
17281
</mtd></mtr><mtr><mtd
17282
class="align-odd"><msub
17284
class="MathClass-ord">V</mi> <mrow
17286
class="MathClass-ord">j</mi></mrow></msub
17288
class="align-even"> <mo
17289
class="MathClass-rel">=</mo> <msub
17291
class="MathClass-ord">v</mi><mrow
17293
class="MathClass-ord">j</mi></mrow></msub
17295
class="MathClass-punc">,</mo></mtd> <mtd
17296
class="align-odd"><mspace width="2em" class="qquad"/><msub
17298
class="MathClass-ord">X</mi><mrow
17300
class="MathClass-ord">j</mi></mrow></msub
17302
class="align-even"> <mo
17303
class="MathClass-rel">=</mo> <msub
17305
class="MathClass-ord">x</mi><mrow
17307
class="MathClass-ord">j</mi></mrow></msub
17309
class="MathClass-punc">,</mo></mtd> <mtd
17310
class="align-odd"><mspace width="2em" class="qquad"/><msub
17312
class="MathClass-ord">U</mi><mrow
17314
class="MathClass-ord">j</mi></mrow></msub
17316
class="align-even"><msub
17318
class="MathClass-ord">u</mi><mrow
17320
class="MathClass-ord">j</mi></mrow></msub
17322
class="MathClass-bin">+</mo><msub
17324
class="MathClass-op">∑</mo>
17327
class="MathClass-ord">i</mi><mo
17328
class="MathClass-rel">≠</mo><mi
17329
class="MathClass-ord">j</mi></mrow></msub
17332
class="MathClass-ord">q</mi><mrow
17334
class="MathClass-ord">i</mi></mrow></msub
17337
class="MathClass-ord">u</mi><mrow
17339
class="MathClass-ord">i</mi></mrow></msub
17341
class="MathClass-punc">.</mo></mtd> <mtd
17342
class="align-label"> <mspace
17343
id="x1-39002r83" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
17344
class="MathClass-open">(</mo><mn
17345
class="MathClass-ord">8</mn><mn
17346
class="MathClass-ord">3</mn><mo
17347
class="MathClass-close">)</mo></mrow>
17348
</mtd></mtr></mtable></math>
17349
Some text after to test the below-display spacing.
17352
<table width="100%"
17353
class="verbatim"><tr class="verbatim"><td
17354
class="verbatim"><pre class="verbatim">
17355
 \begin{alignat}{3}
17356
 V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j,
17357
  & \qquad U_i & = u_i,
17358
  \qquad \text{for $i\ne j$;}\label{eq:B}\\
17359
 V_j & = v_j, & \qquad X_j & = x_j,
17360
   & \qquad U_j & u_j + \sum_{i\ne j} q_i u_i.
17361
 \end{alignat}
17364
<!--l. 2230--><p class="indent"> Unnumbered version:
17365
<!--tex4ht:inline--></p><!--l. 2237--><math
17366
xmlns="http://www.w3.org/1998/Math/MathML"
17367
display="block"><mtable
17368
class="alignat-star">
17370
class="align-odd"><msub
17372
class="MathClass-ord">V</mi> <mrow
17374
class="MathClass-ord">i</mi></mrow></msub
17376
class="align-even"> <mo
17377
class="MathClass-rel">=</mo> <msub
17379
class="MathClass-ord">v</mi><mrow
17381
class="MathClass-ord">i</mi></mrow></msub
17383
class="MathClass-bin">−</mo> <msub
17385
class="MathClass-ord">q</mi><mrow
17387
class="MathClass-ord">i</mi></mrow></msub
17390
class="MathClass-ord">v</mi><mrow
17392
class="MathClass-ord">j</mi></mrow></msub
17394
class="MathClass-punc">,</mo></mtd> <mtd
17395
class="align-odd"><mspace width="2em" class="qquad"/><msub
17397
class="MathClass-ord">X</mi><mrow
17399
class="MathClass-ord">i</mi></mrow></msub
17401
class="align-even"> <mo
17402
class="MathClass-rel">=</mo> <msub
17404
class="MathClass-ord">x</mi><mrow
17406
class="MathClass-ord">i</mi></mrow></msub
17408
class="MathClass-bin">−</mo> <msub
17410
class="MathClass-ord">q</mi><mrow
17412
class="MathClass-ord">i</mi></mrow></msub
17415
class="MathClass-ord">x</mi><mrow
17417
class="MathClass-ord">j</mi></mrow></msub
17419
class="MathClass-punc">,</mo></mtd> <mtd
17420
class="align-odd"><mspace width="2em" class="qquad"/><msub
17422
class="MathClass-ord">U</mi><mrow
17424
class="MathClass-ord">i</mi></mrow></msub
17426
class="align-even"> <mo
17427
class="MathClass-rel">=</mo> <msub
17429
class="MathClass-ord">u</mi><mrow
17431
class="MathClass-ord">i</mi></mrow></msub
17433
class="MathClass-punc">,</mo> <mspace width="2em" class="qquad"/><mrow
17434
class="text"><mtext >for </mtext><mrow
17436
class="MathClass-ord">i</mi><mo
17437
class="MathClass-rel">≠</mo><mi
17438
class="MathClass-ord">j</mi></mrow><mtext >;</mtext></mrow></mtd>
17440
class="align-odd"><msub
17442
class="MathClass-ord">V</mi> <mrow
17444
class="MathClass-ord">j</mi></mrow></msub
17446
class="align-even"> <mo
17447
class="MathClass-rel">=</mo> <msub
17449
class="MathClass-ord">v</mi><mrow
17451
class="MathClass-ord">j</mi></mrow></msub
17453
class="MathClass-punc">,</mo></mtd> <mtd
17454
class="align-odd"><mspace width="2em" class="qquad"/><msub
17456
class="MathClass-ord">X</mi><mrow
17458
class="MathClass-ord">j</mi></mrow></msub
17460
class="align-even"> <mo
17461
class="MathClass-rel">=</mo> <msub
17463
class="MathClass-ord">x</mi><mrow
17465
class="MathClass-ord">j</mi></mrow></msub
17467
class="MathClass-punc">,</mo></mtd> <mtd
17468
class="align-odd"><mspace width="2em" class="qquad"/><msub
17470
class="MathClass-ord">U</mi><mrow
17472
class="MathClass-ord">j</mi></mrow></msub
17474
class="align-even"><msub
17476
class="MathClass-ord">u</mi><mrow
17478
class="MathClass-ord">j</mi></mrow></msub
17480
class="MathClass-bin">+</mo><msub
17482
class="MathClass-op">∑</mo>
17485
class="MathClass-ord">i</mi><mo
17486
class="MathClass-rel">≠</mo><mi
17487
class="MathClass-ord">j</mi></mrow></msub
17490
class="MathClass-ord">q</mi><mrow
17492
class="MathClass-ord">i</mi></mrow></msub
17495
class="MathClass-ord">u</mi><mrow
17497
class="MathClass-ord">i</mi></mrow></msub
17499
class="MathClass-punc">.</mo></mtd>
17500
</mtr></mtable></math>
17501
Some text after to test the below-display spacing.
17504
<table width="100%"
17505
class="verbatim"><tr class="verbatim"><td
17506
class="verbatim"><pre class="verbatim">
17507
 \begin{alignat*}3
17508
 V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j,
17509
  & \qquad U_i & = u_i,
17510
  \qquad \text{for $i\ne j$;} \\
17511
 V_j & = v_j, & \qquad X_j & = x_j,
17512
   & \qquad U_j & u_j + \sum_{i\ne j} q_i u_i.
17513
 \end{alignat*}
17518
<!--l. 2252--><p class="indent"> The most common use for <span
17519
class="cmtt-10">alignat </span>is for things like
17520
<!--tex4ht:inline--></p><!--l. 2257--><math
17521
xmlns="http://www.w3.org/1998/Math/MathML"
17522
display="block"><mtable
17525
class="align-odd"><mi
17526
class="MathClass-ord">x</mi></mtd> <mtd
17527
class="align-even"> <mo
17528
class="MathClass-rel">=</mo> <mi
17529
class="MathClass-ord">y</mi></mtd> <mtd
17530
class="align-odd"></mtd> <mtd
17531
class="align-even"><mspace width="2em" class="qquad"/><mrow
17532
class="text"><mtext >by (</mtext><mtext
17533
name="x1-34003r66" class="label" >66<!--tex4ht:ref: eq:A--></mtext><mtext class="endlabel">)</mtext></mrow></mtd> <mtd
17534
class="align-label"> <mspace
17535
id="x1-39003r84" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
17536
class="MathClass-open">(</mo><mn
17537
class="MathClass-ord">8</mn><mn
17538
class="MathClass-ord">4</mn><mo
17539
class="MathClass-close">)</mo></mrow>
17540
</mtd></mtr><mtr><mtd
17541
class="align-odd"><mi
17542
class="MathClass-ord">x</mi><mi
17543
class="MathClass-ord">′</mi></mtd> <mtd
17544
class="align-even"> <mo
17545
class="MathClass-rel">=</mo> <mi
17546
class="MathClass-ord">y</mi><mi
17547
class="MathClass-ord">′</mi></mtd> <mtd
17548
class="align-odd"></mtd> <mtd
17549
class="align-even"><mspace width="2em" class="qquad"/><mrow
17550
class="text"><mtext >by (</mtext><mtext
17551
name="x1-39001r82" class="label" >82<!--tex4ht:ref: eq:B--></mtext><mtext class="endlabel">)</mtext></mrow></mtd> <mtd
17552
class="align-label"> <mspace
17553
id="x1-39004r85" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
17554
class="MathClass-open">(</mo><mn
17555
class="MathClass-ord">8</mn><mn
17556
class="MathClass-ord">5</mn><mo
17557
class="MathClass-close">)</mo></mrow>
17558
</mtd></mtr><mtr><mtd
17559
class="align-odd"><mi
17560
class="MathClass-ord">x</mi> <mo
17561
class="MathClass-bin">+</mo> <mi
17562
class="MathClass-ord">x</mi><mi
17563
class="MathClass-ord">′</mi></mtd> <mtd
17564
class="align-even"> <mo
17565
class="MathClass-rel">=</mo> <mi
17566
class="MathClass-ord">y</mi> <mo
17567
class="MathClass-bin">+</mo> <mi
17568
class="MathClass-ord">y</mi><mi
17569
class="MathClass-ord">′</mi></mtd> <mtd
17570
class="align-odd"></mtd> <mtd
17571
class="align-even"><mspace width="2em" class="qquad"/><mrow
17572
class="text"><mtext >by Axiom 1.</mtext></mrow></mtd> <mtd
17573
class="align-label"> <mspace
17574
id="x1-39005r86" class="label" width="0pt" /><mspace class="endlabel" width="0pt" /><mrow><mo
17575
class="MathClass-open">(</mo><mn
17576
class="MathClass-ord">8</mn><mn
17577
class="MathClass-ord">6</mn><mo
17578
class="MathClass-close">)</mo></mrow>
17579
</mtd></mtr></mtable></math>
17580
Some text after to test the below-display spacing.
17583
<table width="100%"
17584
class="verbatim"><tr class="verbatim"><td
17585
class="verbatim"><pre class="verbatim">
17586
 \begin{alignat}{2}
17587
 x& =y && \qquad \text {by (\ref{eq:A})}\label{eq:C}\\
17588
 x’& = y’ && \qquad \text {by (\ref{eq:B})}\label{eq:D}\\
17589
 x+x’ & = y+y’ && \qquad \text {by Axiom 1.}
17590
 \end{alignat}
17595
<h3 class="likesectionHead"><a
17596
name="x1-40000A.6"></a>References</h3>
17597
<div class="thebibliography"><p class="bibitem"><span class="biblabel">
17598
[1]<span class="bibsp">   </span></span><a
17599
name="Xdihe:newdir"></a>W. Diffie and E. Hellman, <span
17600
class="cmti-10">New directions in cryptography</span>, IEEE
17601
Transactions on Information Theory <span
17602
class="cmbx-10">22 </span>(1976), no. 5, 644 654.
17603
</p><p class="bibitem"><span class="biblabel">
17604
[2]<span class="bibsp">   </span></span><a
17605
name="Xfre:cichon"></a>D. H. Fremlin, <span
17606
class="cmti-10">Cichon’s diagram</span>, 1983/1984, presented at the Séminaire
17607
Initiation à l’Analyse, G. Choquet, M. Rogalski, J. Saint Raymond, at the
17608
Université Pierre et Marie Curie, Paris, 23e année.
17609
</p><p class="bibitem"><span class="biblabel">
17610
[3]<span class="bibsp">   </span></span><a
17611
name="Xgouja:lagrmeth"></a>I. P. Goulden and D. M. Jackson, <span
17612
class="cmti-10">The enumeration of directed closed Euler</span>
17614
class="cmti-10">trails and directed Hamiltonian circuits by Langrangian methods</span>, European
17616
class="cmbx-10">2 </span>(1981), 131 212.
17617
</p><p class="bibitem"><span class="biblabel">
17618
[4]<span class="bibsp">   </span></span><a
17619
name="Xhapa:graphenum"></a>F. Harary and E. M. Palmer, <span
17620
class="cmti-10">Graphical enumeration</span>, Academic Press, 1973.
17621
</p><p class="bibitem"><span class="biblabel">
17622
[5]<span class="bibsp">   </span></span><a
17623
name="Ximlelu:oneway"></a>R. Impagliazzo, L. Levin, and M. Luby, <span
17624
class="cmti-10">Pseudo-random generation from</span>
17626
class="cmti-10">one-way functions</span>, Proc. 21st STOC (1989), ACM, New York, pp. 12 24.
17627
</p><p class="bibitem"><span class="biblabel">
17628
[6]<span class="bibsp">   </span></span><a
17629
name="Xkomiyo:unipfunc"></a>M. Kojima, S. Mizuno, and A. Yoshise, <span
17630
class="cmti-10">A new continuation method for</span>
17632
class="cmti-10">complementarity problems with uniform p-functions</span>, Tech. Report B-194,
17633
Tokyo Inst. of Technology, Tokyo, 1987, Dept. of Information Sciences.
17634
</p><p class="bibitem"><span class="biblabel">
17635
[7]<span class="bibsp">   </span></span><a
17636
name="Xkomiyo:lincomp"></a>______ , <span
17637
class="cmti-10">A polynomial-time algorithm for a class of linear complementarity</span>
17639
class="cmti-10">problems</span>, Tech. Report B-193, Tokyo Inst. of Technology, Tokyo, 1987,
17640
Dept. of Information Sciences.
17641
</p><p class="bibitem"><span class="biblabel">
17642
[8]<span class="bibsp">   </span></span><a
17643
name="Xliuchow:formalsum"></a>C. J. Liu and Yutze Chow, <span
17644
class="cmti-10">On operator and formal sum methods for graph</span>
17646
class="cmti-10">enumeration problems</span>, SIAM J. Algorithms Discrete Methods <span
17647
class="cmbx-10">5 </span>(1984),
17649
</p><p class="bibitem"><span class="biblabel">
17650
[9]<span class="bibsp">   </span></span><a
17651
name="Xmami:matrixth"></a>M. Marcus and H. Minc, <span
17652
class="cmti-10">A survey of matrix theory and matrix inequalities</span>,
17653
Complementary Series in Math. <span
17654
class="cmbx-10">14 </span>(1964), 21 48.
17655
</p><p class="bibitem"><span class="biblabel">
17656
[10]<span class="bibsp">   </span></span><a
17657
name="Xmiyoki:lincomp"></a>S. Mizuno, A. Yoshise, and T. Kikuchi, <span
17658
class="cmti-10">Practical polynomial time algorithms</span>
17660
class="cmti-10">for linear complementarity problems</span>, Tech. Report 13, Tokyo Inst. of
17661
Technology, Tokyo, April 1988, Dept. of Industrial Engineering and
17664
</p><p class="bibitem"><span class="biblabel">
17665
[11]<span class="bibsp">   </span></span><a
17666
name="Xmoad:quadpro"></a>R. D. Monteiro and I. Adler, <span
17667
class="cmti-10">Interior path following primal-dual algorithms,</span>
17669
class="cmti-10">part II: Quadratic programming</span>, August 1987, Working paper, Dept. of
17670
Industrial Engineering and Operations Research.
17671
</p><p class="bibitem"><span class="biblabel">
17672
[12]<span class="bibsp">   </span></span><a
17673
name="Xste:sint"></a>E. M. Stein, <span
17674
class="cmti-10">Singular integrals and differentiability properties of functions</span>,
17675
Princeton Univ. Press, Princeton, N.J., 1970.
17676
</p><p class="bibitem"><span class="biblabel">
17677
[13]<span class="bibsp">   </span></span><a
17678
name="Xye:intalg"></a>Y. Ye, <span
17679
class="cmti-10">Interior algorithms for linear, quadratic and linearly constrained</span>
17681
class="cmti-10">convex programming</span>, Ph.D. thesis, Stanford Univ., Palo Alto, Calif., July
17682
1987, Dept. of Engineering Economic Systems, unpublished.
added
removed
doc/mathml/testmath.xml
