summaryrefslogtreecommitdiff
path: root/macros/latex/contrib/aomart/aomsample.tex
blob: f8279246704a8ed33f2c14a9d0209021d632829e (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
%%% ====================================================================
%%% @LaTeX-file{
%%%   filename  = "aomsample.tex",
%%%   copyright = "Copyright 1995, 1999 American Mathematical Society,
%%%                2005 Hebrew University Magnes Press,
%%%                all rights reserved.  Copying of this file is
%%%                authorized only if either:
%%%                (1) you make absolutely no changes to your copy,
%%%                including name; OR
%%%                (2) if you do make changes, you first rename it
%%%                to some other name.",
%%% }
%%% ====================================================================
\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX)
[1994/12/01]% LaTeX date must December 1994 or later
\documentclass[manuscript, printscheme]{aomart}
%\usepackage[english]{babel}
%\documentclass[screen]{aomart}
%\documentclass{aomart}
%    Some definitions useful in producing this sort of documentation:
\chardef\bslash=`\\ % p. 424, TeXbook
%    Normalized (nonbold, nonitalic) tt font, to avoid font
%    substitution warning messages if tt is used inside section
%    headings and other places where odd font combinations might
%    result.
\newcommand{\ntt}{\normalfont\ttfamily}
%    command name
\newcommand{\cn}[1]{{\protect\ntt\bslash#1}}
%    LaTeX package name
\newcommand{\pkg}[1]{{\protect\ntt#1}}
%    File name
\newcommand{\fn}[1]{{\protect\ntt#1}}
%    environment name
\newcommand{\env}[1]{{\protect\ntt#1}}
\hfuzz1pc % Don't bother to report overfull boxes if overage is < 1pc

%       Theorem environments

%% \theoremstyle{plain} %% This is the default
\newtheorem[{}\it]{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{ax}{Axiom}

\theoremstyle{definition}
\newtheorem{defn}{Definition}[section]
\newtheorem{rem}{Remark}[section]
\newtheorem*[{}\it]{notation}{Notation}
\newtheorem{step}{Step}

%\numberwithin{equation}{section}

\newcommand{\thmref}[1]{Theorem~\ref{#1}}
\newcommand{\secref}[1]{\S\ref{#1}}
\newcommand{\lemref}[1]{Lemma~\ref{#1}}


%       Math definitions

\newcommand{\A}{\mathcal{A}}
\newcommand{\B}{\mathcal{B}}
\newcommand{\st}{\sigma}
\newcommand{\XcY}{{(X,Y)}}
\newcommand{\SX}{{S_X}}
\newcommand{\SY}{{S_Y}}
\newcommand{\SXY}{{S_{X,Y}}}
\newcommand{\SXgYy}{{S_{X|Y}(y)}}
\newcommand{\Cw}[1]{{\hat C_#1(X|Y)}}
\newcommand{\G}{{G(X|Y)}}
\newcommand{\PY}{{P_{\mathcal{Y}}}}
\newcommand{\X}{\mathcal{X}}
\newcommand{\wt}{\widetilde}
\newcommand{\wh}{\widehat}

\DeclareMathOperator{\per}{per}
\DeclareMathOperator{\cov}{cov}
\DeclareMathOperator{\non}{non}
\DeclareMathOperator{\cf}{cf}
\DeclareMathOperator{\add}{add}
\DeclareMathOperator{\Cham}{Cham}
\DeclareMathOperator{\IM}{Im}
\DeclareMathOperator{\esssup}{ess\,sup}
\DeclareMathOperator{\meas}{meas}
\DeclareMathOperator{\seg}{seg}

%    \interval is used to provide better spacing after a [ that
%    is used as a closing delimiter.
\newcommand{\interval}[1]{\mathinner{#1}}

%    Notation for an expression evaluated at a particular condition. The
%    optional argument can be used to override automatic sizing of the
%    right vert bar, e.g. \eval[\biggr]{...}_{...}
\newcommand{\eval}[2][\right]{\relax
  \ifx#1\right\relax \left.\fi#2#1\rvert}

%    Enclose the argument in vert-bar delimiters:
\newcommand{\envert}[1]{\left\lvert#1\right\rvert}
\let\abs=\envert

%    Enclose the argument in double-vert-bar delimiters:
\newcommand{\enVert}[1]{\left\lVert#1\right\rVert}
\let\norm=\enVert

%\setcounter{tocdepth}{5}

\title[Sample Paper]{Sample Paper for the \texttt{aomart}
  Class}
\author[AMS]{American Mathematical Society}
\address{AMS\\
Providence, Rhode Island}
\email{tech-support@ams.org} 
\author{Boris Veytsman}
\address{George Mason University\\
Fairfax, Virginia}
\fulladdress{School of Systems Biology\\
George Mason University\\
Fairfax, VA 22030}
\email{borisv@lk.net}
\urladdr{http://borisv.lk.net}
\givenname{Boris}
\surname{Veytsman}
\copyrightyear{2008--2020}
\copyrightnote{\textcopyright~2008--2020 Boris Veytsman}
\contrib[with appendix by]{Frodo Baggins}
\givenname{Frodo}
\surname{Baggins}
\contrib{Bilbo Baggins}
\givenname{Bilbo}
\surname{Baggins}
\contrib[with afterword by]{Bilbo Baggins}
\givenname{Bilbo}
\surname{Baggins}
\address{The Unseen Univeresity}

\thanks{The class was commissioned by Annals of Mathematics}

\keyword{Hamiltonian paths}
\keyword{Typesetting}
\subject{primary}{matsc2020}{1AB5}
\subject{primary}{matsc2020}{2FD5}
\subject{secondary}{matsc2020}{FFFF}
\subject{secondary}{matsc2020}{G25}

\received{\formatdate{2004-12-24}}
\revised{\formatdate{2006-04-12}}
\accepted{\formatdate{2007-10-02}}
\published{\formatdate{2008-04-02}}
\publishedonline{\formatdate{2007-11-12}}

\volumenumber{160}
\issuenumber{1}
\publicationyear{2008}
\papernumber{12}
\startpage{17}
\endpage{}

\proposed{E. D. Itor}
\seconded{A. S. Sociate}
\corresponding{A. U. Thor}
\version{2.1}

%\doinumber{10.1215/S0012-7094-92-06520-3}
\mrnumber{MR1154181}
\zblnumber{0774.14039}
\arxivnumber{1234.567890}
%\oldsubsections


\begin{document}

\begin{abstract}
  This is a test file for \textsf{aomart} class based on the
  \texttt{testmath.tex} file from the \textsf{amsmath} distribution.  

  It was changed to test the features of the Annals of Mathematics
  class. 
\end{abstract}

\maketitle
\tableofcontents

\section{Introduction}


This paper demonstrates the use of \texttt{aomart} class.  It is based
on \texttt{testmath.tex} from  \AmS-\LaTeX{} distribution.  The text
is (slightly) reformatted according to the requirements of the
\texttt{aomart} style.  See also
\cite{Lenstra74,Zarhin92,Minasyan08,Arnold89:MathMethodsMechanics,%
Michal48,Michal38,Zarhin:AG,Zarhin:AC,degroot1992stochastic}.
\EditorialComment{Are these 
  quotations necessary?}


It is always a pleasure to cite Knuth~\cite{Knuth94:TheTeXbook}.  

\section{Enumeration of Hamiltonian paths in a graph}

Let $\mathbf{A}=(a_{ij})$ be the adjacency matrix of graph $G$. The
corresponding Kirchhoff matrix $\mathbf{K}=(k_{ij})$ is obtained from
$\mathbf{A}$ by replacing in $-\mathbf{A}$ each diagonal entry by the
degree of its corresponding vertex; i.e., the $i$th diagonal entry is
identified with the degree of the $i$th vertex. It is well known that
\begin{equation}
\det\mathbf{K}(i|i)=\text{ the number of spanning trees of $G$},
\quad i=1,\dots,n
\end{equation}
where $\mathbf{K}(i|i)$ is the $i$th principal submatrix of
$\mathbf{K}$.
\begin{verbatim}
\det\mathbf{K}(i|i)=\text{ the number of spanning trees of $G$},
\end{verbatim}

Let $C_{i(j)}$ be the set of graphs obtained from $G$ by attaching edge
$(v_iv_j)$ to each spanning tree of $G$. Denote by $C_i=\bigcup_j
C_{i(j)}$. It is obvious that the collection of Hamiltonian cycles is a
subset of $C_i$. Note that the cardinality of $C_i$ is $k_{ii}\det
\mathbf{K}(i|i)$. Let $\wh X=\{\hat x_1,\dots,\hat x_n\}$.
\begin{verbatim}
$\wh X=\{\hat x_1,\dots,\hat x_n\}$
\end{verbatim}
Define multiplication for the elements of $\wh X$ by
\begin{equation}\label{multdef}
\hat x_i\hat x_j=\hat x_j\hat x_i,\quad \hat x^2_i=0,\quad
i,j=1,\dots,n.
\end{equation}
Let $\hat k_{ij}=k_{ij}\hat x_j$ and $\hat k_{ij}=-\sum_{j\not=i} \hat
k_{ij}$. Then the number of Hamiltonian cycles $H_c$ is given by the
relation \cite{liuchow:formalsum}
\begin{equation}\label{H-cycles}
\biggl(\prod^n_{\,j=1}\hat x_j\biggr)H_c=\frac{1}{2}\hat k_{ij}\det
\wh{\mathbf{K}}(i|i),\qquad i=1,\dots,n.
\end{equation}
The task here is to express \eqref{H-cycles}
in a form free of any $\hat x_i$,
$i=1,\dots,n$. The result also leads to the resolution of enumeration of
Hamiltonian paths in a graph.

It is well known that the enumeration of Hamiltonian cycles and paths in
a complete graph $K_n$ and in a complete bipartite graph $K_{n_1n_2}$
can only be found from \textit{first combinatorial principles}
\cite{hapa:graphenum}. One wonders if there exists a formula which can
be used very efficiently to produce $K_n$ and $K_{n_1n_2}$. Recently,
using Lagrangian methods, Goulden and Jackson have shown that $H_c$ can
be expressed in terms of the determinant and permanent of the adjacency
matrix \cite{gouja:lagrmeth}. However, the formula of Goulden and
Jackson determines neither $K_n$ nor $K_{n_1n_2}$ effectively. In this
paper, using an algebraic method, we parametrize the adjacency matrix.
The resulting formula also involves the determinant and permanent, but
it can easily be applied to $K_n$ and $K_{n_1n_2}$. In addition, we
eliminate the permanent from $H_c$ and show that $H_c$ can be
represented by a determinantal function of multivariables, each variable
with domain $\{0,1\}$. Furthermore, we show that $H_c$ can be written by
number of spanning trees of subgraphs. Finally, we apply the formulas to
a complete multigraph $K_{n_1\dots n_p}$.

The conditions $a_{ij}=a_{ji}$, $i,j=1,\dots,n$, are not required in
this paper. All formulas can be extended to a digraph simply by
multiplying $H_c$ by 2.  Some other discussion can be found
in~\cite{fre:riesz,fre:cichon}.


\section{Main theorem}
\label{s:mt}

\begin{notation} For $p,q\in P$ and $n\in\omega$ we write
$(q,n)\le(p,n)$ if $q\le p$ and $A_{q,n}=A_{p,n}$.
\begin{verbatim}
\begin{notation} For $p,q\in P$ and $n\in\omega$
...
\end{notation}
\end{verbatim}
\end{notation}

Let $\mathbf{B}=(b_{ij})$ be an $n\times n$ matrix. Let $\mathbf{n}=\{1,
\dots,n\}$. Using the properties of \eqref{multdef}, it is readily seen
that

\begin{lem}\label{lem-per}
\begin{equation}
\prod_{i\in\mathbf{n}}
\biggl(\sum_{\,j\in\mathbf{n}}b_{ij}\hat x_i\biggr)
=\biggl(\prod_{\,i\in\mathbf{n}}\hat x_i\biggr)\per \mathbf{B}
\end{equation}
where $\per \mathbf{B}$ is the permanent of $\mathbf{B}$.
\end{lem}

Let $\wh Y=\{\hat y_1,\dots,\hat y_n\}$. Define multiplication
for the elements of $\wh Y$ by
\begin{equation}
\hat y_i\hat y_j+\hat y_j\hat y_i=0,\quad i,j=1,\dots,n.
\end{equation}
Then, it follows that
\begin{lem}\label{lem-det}
\begin{equation}\label{detprod}
\prod_{i\in\mathbf{n}}
\biggl(\sum_{\,j\in\mathbf{n}}b_{ij}\hat y_j\biggr)
=\biggl(\prod_{\,i\in\mathbf{n}}\hat y_i\biggr)\det\mathbf{B}.
\end{equation}
\end{lem}

Note that all basic properties of determinants are direct consequences
of Lemma~\ref{lem-det}. Write
\begin{equation}\label{sum-bij}
\sum_{j\in\mathbf{n}}b_{ij}\hat y_j=\sum_{j\in\mathbf{n}}b^{(\lambda)}
_{ij}\hat y_j+(b_{ii}-\lambda_i)\hat y_i\hat y
\end{equation}
where
\begin{equation}
b^{(\lambda)}_{ii}=\lambda_i,\quad b^{(\lambda)}_{ij}=b_{ij},
\quad i\not=j.
\end{equation}
Let $\mathbf{B}^{(\lambda)}=(b^{(\lambda)}_{ij})$. By \eqref{detprod}
and \eqref{sum-bij}, it is
straightforward to show the following
result:
\begin{thm}\label{thm-main}
\begin{equation}\label{detB}
\det\mathbf{B}=
\sum^n_{l =0}\sum_{I_l \subseteq n}
\prod_{i\in I_l}(b_{ii}-\lambda_i)
\det\mathbf{B}^{(\lambda)}(I_l |I_l ),
\end{equation}
where $I_l =\{i_1,\dots,i_l \}$ and $\mathbf{B}^{(\lambda)}(I_l |I_l )$
is the principal submatrix (obtained from $\mathbf{B}^{(\lambda)}$
by deleting its $i_1,\dots,i_l $ rows and columns).
\end{thm}

\begin{rem}[convention]
Let $\mathbf{M}$ be an $n\times n$ matrix. The convention
$\mathbf{M}(\mathbf{n}|\mathbf{n})=1$ has been used in \eqref{detB} and
hereafter.
\end{rem}

Before proceeding with our discussion, we pause to note that
\thmref{thm-main} yields immediately a fundamental formula which can be
used to compute the coefficients of a characteristic polynomial
\cite{mami:matrixth}:
\begin{cor}\label{BI}
Write $\det(\mathbf{B}-x\mathbf{I})=\sum^n_{l =0}(-1)
^l b_l x^l $. Then
\begin{equation}\label{bl-sum}
b_l =\sum_{I_l \subseteq\mathbf{n}}\det\mathbf{B}(I_l |I_l ).
\end{equation}
\end{cor}
Let
\begin{equation}
\mathbf{K}(t,t_1,\dots,t_n)
=\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\
-a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\
\hdotsfor[2]{4}\\
-a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix},
\end{equation}
\begin{verbatim}
\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\
-a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\
\hdotsfor[2]{4}\\
-a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix}
\end{verbatim}
where
\begin{equation}
D_i=\sum_{j\in\mathbf{n}}a_{ij}t_j,\quad i=1,\dots,n.
\end{equation}

Set
\begin{equation*}
D(t_1,\dots,t_n)=\frac{\delta}{\delta t}\eval{\det\mathbf{K}(t,t_1,\dots,t_n)
}_{t=1}.
\end{equation*}
Then
\begin{equation}\label{sum-Di}
D(t_1,\dots,t_n)
=\sum_{i\in\mathbf{n}}D_i\det\mathbf{K}(t=1,t_1,\dots,t_n; i|i),
\end{equation}
where $\mathbf{K}(t=1,t_1,\dots,t_n; i|i)$ is the $i$th principal
submatrix of $\mathbf{K}(t=1,t_1,\dots,t_n)$.

Theorem~\ref{thm-main} leads to
\begin{equation}\label{detK1}
\det\mathbf{K}(t_1,t_1,\dots,t_n)
=\sum_{I\in\mathbf{n}}(-1)^{\envert{I}}t^{n-\envert{I}}
\prod_{i\in I}t_i\prod_{j\in I}(D_j+\lambda_jt_j)\det\mathbf{A}
^{(\lambda t)}(\overline{I}|\overline I).
\end{equation}
Note that
\begin{equation}\label{detK2}
\det\mathbf{K}(t=1,t_1,\dots,t_n)=\sum_{I\in\mathbf{n}}(-1)^{\envert{I}}
\prod_{i\in I}t_i\prod_{j\in I}(D_j+\lambda_jt_j)\det\mathbf{A}
^{(\lambda)}(\overline{I}|\overline{I})=0.
\end{equation}

Let $t_i=\hat x_i,i=1,\dots,n$. Lemma~\ref{lem-per} yields
\begin{multline}
\biggl(\sum_{\,i\in\mathbf{n}}a_{l _i}x_i\biggr)
\det\mathbf{K}(t=1,x_1,\dots,x_n;l |l )\\
=\biggl(\prod_{\,i\in\mathbf{n}}\hat x_i\biggr)
\sum_{I\subseteq\mathbf{n}-\{l \}}
(-1)^{\envert{I}}\per\mathbf{A}^{(\lambda)}(I|I)
\det\mathbf{A}^{(\lambda)}
(\overline I\cup\{l \}|\overline I\cup\{l \}).
\label{sum-ali}
\end{multline}
\begin{verbatim}
\begin{multline}
\biggl(\sum_{\,i\in\mathbf{n}}a_{l _i}x_i\biggr)
\det\mathbf{K}(t=1,x_1,\dots,x_n;l |l )\\
=\biggl(\prod_{\,i\in\mathbf{n}}\hat x_i\biggr)
\sum_{I\subseteq\mathbf{n}-\{l \}}
(-1)^{\envert{I}}\per\mathbf{A}^{(\lambda)}(I|I)
\det\mathbf{A}^{(\lambda)}
(\overline I\cup\{l \}|\overline I\cup\{l \}).
\label{sum-ali}
\end{multline}
\end{verbatim}

By \eqref{H-cycles}, \eqref{detprod}, and \eqref{sum-bij}, we have
\begin{prop}\label{prop:eg}
\begin{equation}
H_c=\frac1{2n}\sum^n_{l =0}(-1)^{l}
D_{l},
\end{equation}
where
\begin{equation}\label{delta-l}
D_{l}=\eval[2]{\sum_{I_{l}\subseteq \mathbf{n}}
D(t_1,\dots,t_n)}_{t_i=\left\{\begin{smallmatrix}
0,& \text{if }i\in I_{l}\quad\\% \quad added for centering
1,& \text{otherwise}\end{smallmatrix}\right.\;,\;\; i=1,\dots,n}.
\end{equation}
\end{prop}

\section{Application}
\label{lincomp}

We consider here the applications of Theorems~\ref{th-info-ow-ow}
and~\ref{th-weak-ske-owf} on page~\pageref{th-weak-ske-owf} to a complete
multipartite graph $K_{n_1\dots n_p}$. It can be shown that the
number of spanning trees of $K_{n_1\dots n_p}$
may be written
\begin{equation}\label{e:st}
T=n^{p-2}\prod^p_{i=1}
(n-n_i)^{n_i-1}
\end{equation}
where
\begin{equation}
n=n_1+\dots+n_p.
\end{equation}

It follows from Theorems~\ref{th-info-ow-ow} and
~\ref{th-weak-ske-owf} that
\begin{equation}\label{e:barwq}
\begin{split}
H_c&=\frac1{2n}
\sum^n_{{l}=0}(-1)^{l}(n-{l})^{p-2}
\sum_{l _1+\dots+l _p=l}\prod^p_{i=1}
\binom{n_i}{l _i}\\
&\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i}\cdot
\biggl[(n-l )^2-\sum^p_{j=1}(n_i-l _i)^2\biggr].\end{split}
\end{equation}
\begin{verbatim}
... \binom{n_i}{l _i}\\
\end{verbatim}
and
\begin{equation}\label{joe}
\begin{split}
H_c&=\frac12\sum^{n-1}_{l =0}
(-1)^{l}(n-l )^{p-2}
\sum_{l _1+\dots+l _p=l}
\prod^p_{i=1}\binom{n_i}{l _i}\\
&\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i}
\left(1-\frac{l _p}{n_p}\right)
[(n-l )-(n_p-l _p)].
\end{split}
\end{equation}

The enumeration of $H_c$ in a $K_{n_1\dotsm n_p}$ graph can also be
carried out by Theorem~\ref{thm-H-param} or~\ref{thm-asym}
together with the algebraic method of \eqref{multdef}.
Some elegant representations may be obtained. For example, $H_c$ in
a $K_{n_1n_2n_3}$ graph may be written
\begin{equation}\label{j:mark}
\begin{split}
H_c=&
\frac{n_1!\,n_2!\,n_3!}
{n_1+n_2+n_3}\sum_i\left[\binom{n_1}{i}
\binom{n_2}{n_3-n_1+i}\binom{n_3}{n_3-n_2+i}\right.\\
&+\left.\binom{n_1-1}{i}
\binom{n_2-1}{n_3-n_1+i}
\binom{n_3-1}{n_3-n_2+i}\right].\end{split}
\end{equation}

\section{Secret key exchanges}
\label{SKE}

Modern cryptography is fundamentally concerned with the problem of
secure private communication.  A Secret Key Exchange is a protocol
where Alice and Bob, having no secret information in common to start,
are able to agree on a common secret key, conversing over a public
channel.  The notion of a Secret Key Exchange protocol was first
introduced in the seminal paper of Diffie and Hellman
\cite{dihe:newdir}. \cite{dihe:newdir} presented a concrete
implementation of a Secret Key Exchange protocol, dependent on a
specific assumption (a variant on the discrete log), specially
tailored to yield Secret Key Exchange. Secret Key Exchange is of
course trivial if trapdoor permutations exist. However, there is no
known implementation based on a weaker general assumption.

The concept of an informationally one-way function was introduced
in \cite{imlelu:oneway}. We give only an informal definition here:

\begin{defn}[one way] A polynomial time
computable function $f = \{f_k\}$ is informationally
one-way if there is no probabilistic polynomial time algorithm which
(with probability of the form $1 - k^{-e}$ for some $e > 0$)
returns on input $y \in \{0,1\}^{k}$ a random element of $f^{-1}(y)$.
\end{defn}
In the non-uniform setting \cite{imlelu:oneway} show that these are not
weaker than one-way functions:
\begin{thm}[\cite{imlelu:oneway} (non-uniform)]
\label{th-info-ow-ow}
The existence of informationally one-way functions
implies the existence of one-way functions.
\end{thm}
We will stick to the convention introduced above of saying
``non-uniform'' before the theorem statement when the theorem
makes use of non-uniformity. It should be understood that
if nothing is said then the result holds for both the uniform and
the non-uniform models.

It now follows from \thmref{th-info-ow-ow} that

\begin{thm}[non-uniform]\label{th-weak-ske-owf} Weak SKE
implies the existence of a one-way function.
\end{thm}

More recently, the polynomial-time, interior point algorithms for linear
programming have been extended to the case of convex quadratic programs
\cite{moad:quadpro,ye:intalg}, certain linear complementarity problems
\cite{komiyo:lincomp,miyoki:lincomp}, and the nonlinear complementarity
problem \cite{komiyo:unipfunc}. The connection between these algorithms
and the classical Newton method for nonlinear equations is well
explained in \cite{komiyo:lincomp}.

\section{Review}
\label{computation}

We begin our discussion with the following definition:

\begin{defn}

A function $H\colon \Re^n \to \Re^n$ is said to be
\emph{B-differentiable} at the point $z$ if (i)~$H$ is Lipschitz
continuous in a neighborhood of $z$, and (ii)~ there exists a positive
homogeneous function $BH(z)\colon \Re^n \to \Re^n$, called the
\emph{B-derivative} of $H$ at $z$, such that
\[ \lim_{v \to 0} \frac{H(z+v) - H(z) - BH(z)v}{\enVert{v}} = 0. \]
The function $H$ is \textit{B-differentiable in set $S$} if it is
B-differentiable at every point in $S$. The B-derivative $BH(z)$ is said
to be \textit{strong} if
\[ \lim_{(v,v') \to (0,0)} \frac{H(z+v) - H(z+v') - BH(z)(v
 -v')}{\enVert{v - v'}} = 0. \]
\end{defn}


\begin{lem}\label{limbog} There exists a smooth function $\psi_0(z)$
defined for $\abs{z}>1-2a$ satisfying the following properties\textup{:}
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $\psi_0(z)$ is bounded above and below by positive constants
$c_1\leq \psi_0(z)\leq c_2$.
\item If $\abs{z}>1$, then $\psi_0(z)=1$.
\item For all $z$ in the domain of $\psi_0$, $\Delta_0\ln \psi_0\geq 0$.
\item If $1-2a<\abs{z}<1-a$, then $\Delta_0\ln \psi_0\geq
c_3>0$.
\end{enumerate}
\end{lem}

\begin{proof}
We choose $\psi_0(z)$ to be a radial function depending only on $r=\abs{z}$.
Let $h(r)\geq 0$ be a suitable smooth function satisfying $h(r)\geq c_3$
for $1-2a<\abs{z}<1-a$, and $h(r)=0$ for $\abs{z}>1-\tfrac a2$. The radial
Laplacian
\[\Delta_0\ln\psi_0(r)=\left(\frac {d^2}{dr^2}+\frac
1r\frac d{dr}\right)\ln\psi_0(r)\]
has smooth coefficients for $r>1-2a$. Therefore, we may
apply the existence and uniqueness theory for ordinary differential
equations. Simply let $\ln \psi_0(r)$ be the solution of the differential
equation
\[\left(\frac{d^2}{dr^2}+\frac 1r\frac d{dr}\right)\ln \psi_0(r)=h(r)\]
with initial conditions given by $\ln \psi_0(1)=0$ and
$\ln\psi_0'(1)=0$.

Next, let $D_\nu$ be a finite collection of pairwise disjoint disks,
all of which are contained in the unit disk centered at the origin in
$C$. We assume that $D_\nu=\{z\mid \abs{z-z_\nu}<\delta\}$. Suppose that
$D_\nu(a)$ denotes the smaller concentric disk $D_\nu(a)=\{z\mid
\abs{z-z_\nu}\leq (1-2a)\delta\}$. We define a smooth weight function
$\Phi_0(z)$ for $z\in C-\bigcup_\nu D_\nu(a)$ by setting $\Phi_
0(z)=1$ when $z\notin \bigcup_\nu D_\nu$ and $\Phi_
0(z)=\psi_0((z-z_\nu)/\delta)$ when $z$ is an element of $D_\nu$. It
follows from \lemref{limbog} that $\Phi_ 0$ satisfies the properties:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item \label{boundab}$\Phi_ 0(z)$ is bounded above and below by
positive constants $c_1\leq \Phi_ 0(z)\leq c_2$.
\item \label{d:over}$\Delta_0\ln\Phi_ 0\geq 0$ for all
$z\in C-\bigcup_\nu D_\nu(a)$,
the domain where the function $\Phi_ 0$ is defined.
\item \label{d:ad}$\Delta_0\ln\Phi_ 0\geq c_3\delta^{-2}$
when $(1-2a)\delta<\abs{z-z_\nu}<(1-a)\delta$.
\end{enumerate}
Let $A_\nu$ denote the annulus $A_\nu=\{(1-2a)\delta<\abs{z-z_\nu}<(1-a)
\delta \}$, and set $A=\bigcup_\nu A_\nu$. The
properties (\ref{d:over}) and (\ref{d:ad}) of $\Phi_ 0$
may be summarized as $\Delta_0\ln \Phi_ 0\geq c_3\delta^{-2}\chi_A$,
where $\chi _A$ is the characteristic function of $A$.
\end{proof}

Suppose that $\alpha$ is a nonnegative real constant. We apply
Proposition~\ref{prop:eg} with $\Phi(z)=\Phi_ 0(z) e^{\alpha\abs{z}^2}$. If
$u\in C^\infty_0(R^2-\bigcup_\nu D_\nu(a))$, assume that $\mathcal{D}$
is a bounded domain containing the support of $u$ and $A\subset
\mathcal{D}\subset R^2-\bigcup_\nu D_\nu(a)$. A calculation gives
\[\int_{\mathcal{D}}\abs{\overline\partial u}^2\Phi_ 0(z) e^{\alpha\abs{z}^2}
\geq c_4\alpha\int_{\mathcal{D}}\abs{u}^2\Phi_ 0e^{\alpha\abs{z}^2}
+c_5\delta^{-2}\int_ A\abs{u}^2\Phi_ 0e^{\alpha\abs{z}^2}.\]

The boundedness, property (\ref{boundab}) of $\Phi_ 0$, then yields
\[\int_{\mathcal{D}}\abs{\overline\partial u}^2e^{\alpha\abs{z}^2}\geq c_6\alpha
\int_{\mathcal{D}}\abs{u}^2e^{\alpha\abs{z}^2}
+c_7\delta^{-2}\int_ A\abs{u}^2e^{\alpha\abs{z}^2}.\]

Let $B(X)$ be the set of blocks of $\Lambda_{X}$
and let $b(X) = \abs{B(X)}$. If $\phi \in Q_{X}$ then
$\phi$ is constant on the blocks of $\Lambda_{X}$.
\begin{equation}\label{far-d}
 P_{X} = \{ \phi \in M \mid \Lambda_{\phi} = \Lambda_{X} \},
\qquad
Q_{X} = \{\phi \in M \mid \Lambda_{\phi} \geq \Lambda_{X} \}.
\end{equation}
If $\Lambda_{\phi} \geq \Lambda_{X}$ then
$\Lambda_{\phi} = \Lambda_{Y}$ for some $Y \geq X$ so that
\[ Q_{X} = \bigcup_{Y \geq X} P_{Y}. \]
Thus by M\"obius inversion
\[ \abs{P_{Y}}= \sum_{X\geq Y} \mu (Y,X)\abs{Q_{X}}.\]
Thus there is a bijection from $Q_{X}$ to $W^{B(X)}$.
In particular $\abs{Q_{X}} = w^{b(X)}$.

Next note that $b(X)=\dim X$. We see this by choosing a
basis for $X$ consisting of vectors $v^{k}$ defined by
\[v^{k}_{i}=
\begin{cases} 1 & \text{if $i \in \Lambda_{k}$},\\
0 &\text{otherwise.} \end{cases}
\]
\begin{verbatim}
\[v^{k}_{i}=
\begin{cases} 1 & \text{if $i \in \Lambda_{k}$},\\
0 &\text{otherwise.} \end{cases}
\]
\end{verbatim}

\begin{lem}\label{p0201}
Let $\A$ be an arrangement. Then
\[ \chi (\A,t) = \sum_{\B \subseteq \A}
(-1)^{\abs{\B}} t^{\dim T(\B)}. \]
\end{lem}

In order to compute $R''$ recall the definition
of $S(X,Y)$ from \lemref{lem-per}. Since $H \in \B$,
$\A_{H} \subseteq \B$. Thus if $T(\B) = Y$ then
$\B \in S(H,Y)$. Let $L'' = L(\A'')$. Then
\begin{equation}\label{E_SXgYy}
\begin{split}
R''&= \sum_{H\in \B \subseteq \A} (-1)^{\abs{\B}}
t^{\dim T(\B)}\\
&= \sum_{Y \in L''} \sum_{\B \in S(H,Y)}
(-1)^{\abs{\B}}t^{\dim Y} \\
&= -\sum_{Y \in L''} \sum_{\B \in S(H,Y)} (-1)^
{\abs{\B - \A_{H}}} t^{\dim Y} \\
&= -\sum_{Y \in L''} \mu (H,Y)t^{\dim Y} \\
&= -\chi (\A '',t).
\end{split}
\end{equation}

\begin{cor}\label{tripleA}
Let $(\A,\A',\A'')$ be a triple of arrangements. Then
\[ \pi (\A,t) = \pi (\A',t) + t \pi (\A'',t). \]
\end{cor}

\begin{defn}
Let $(\A,\A',\A'')$ be a triple with respect to
the hyperplane $H \in \A$. Call $H$ a \textit{separator}
if $T(\A) \not\in L(\A')$.
\end{defn}

\begin{cor}\label{nsep}
Let $(\A,\A',\A'')$ be a triple with respect to $H \in \A$.
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item
If $H$ is a separator then
\[ \mu (\A) = - \mu (\A'') \]
and hence
\[ \abs{\mu (\A)} = \abs{ \mu (\A'')}. \]

\item If $H$ is not a separator then
\[\mu (\A) = \mu (\A') - \mu (\A'') \]
and
\[ \abs{\mu (\A)} = \abs{\mu (\A')} + \abs{\mu (\A'')}. \]
\end{enumerate}
\end{cor}

\begin{proof}
It follows from \thmref{th-info-ow-ow} that $\pi(\A,t)$
has leading term
\[(-1)^{r(\A)}\mu (\A)t^{r(\A)}.\]
The conclusion
follows by comparing coefficients of the leading
terms on both sides of the equation in
Corollary~\ref{tripleA}. If $H$ is a separator then
$r(\A') < r(\A)$ and there is no contribution
from $\pi (\A',t)$.
\end{proof}

The Poincar\'e polynomial of an arrangement
will appear repeatedly
in these notes. It will be shown to equal the
Poincar\'e polynomial
of the graded algebras which we are going to
associate with $\A$. It is also the Poincar\'e
polynomial of the complement $M(\A)$ for a
complex arrangement. Here we prove
that the Poincar\'e polynomial is the chamber
counting function for a real arrangement. The
complement $M(\A)$ is a disjoint union of chambers
\[M(\A) = \bigcup_{C \in \Cham(\A)} C.\]
The number
of chambers is determined by the Poincar\'e
polynomial as follows.

\begin{thm}\label{th-realarr}
Let $\A_{\mathbf{R}}$ be a real arrangement. Then
\[ \abs{\Cham(\A_{\mathbf{R}})} = \pi (\A_{\mathbf{R}},1). \]
\end{thm}

\begin{proof}
We check the properties required in Corollary~\ref{nsep}:
(i) follows from $\pi (\Phi_{ l},t) = 1$, and (ii) is a
consequence of Corollary~\ref{BI}.
\end{proof}

\begin{figure}
\vspace{5cm}
\caption[]{$Q(\A_{1}) = xyz(x-z)(x+z)(y-z)(y+z)$}
\end{figure}

\begin{figure}
\vspace{5cm}
\caption[]{$Q(\A_{2})= xyz(x+y+z)(x+y-z)(x-y+z)(x-y-z)$}
\end{figure}


\begin{thm}
\label{T_first_the_int}
Let $\phi$ be a protocol for a random pair $\XcY$.
If one of $\st_\phi(x',y)$ and $\st_\phi(x,y')$ is a prefix of the other
and $(x,y)\in\SXY$, then
\[
\langle \st_j(x',y)\rangle_{j=1}^\infty
=\langle \st_j(x,y)\rangle_{j=1}^\infty
=\langle \st_j(x,y')\rangle_{j=1}^\infty .
\]
\end{thm}
\begin{proof}
We show by induction on $i$ that
\[
\langle \st_j(x',y)\rangle_{j=1}^i
=\langle \st_j(x,y)\rangle_{j=1}^i
=\langle \st_j(x,y')\rangle_{j=1}^i.
\]
The induction hypothesis holds vacuously for $i=0$. Assume it holds for
$i-1$, in particular
$[\st_j(x',y)]_{j=1}^{i-1}=[\st_j(x,y')]_{j=1}^{i-1}$. Then one of
$[\st_j(x',y)]_{j=i}^{\infty}$ and $[\st_j(x,y')]_{j=i}^{\infty}$ is a
prefix of the other which implies that one of $\st_i(x',y)$ and
$\st_i(x,y')$ is a prefix of the other. If the $i$th message is
transmitted by $P_\X$ then, by the separate-transmissions property and
the induction hypothesis, $\st_i(x,y)=\st_i(x,y')$, hence one of
$\st_i(x,y)$ and $\st_i(x',y)$ is a prefix of the other. By the
implicit-termination property, neither $\st_i(x,y)$ nor $\st_i(x',y)$
can be a proper prefix of the other, hence they must be the same and
$\st_i(x',y)=\st_i(x,y)=\st_i(x,y')$. If the $i$th message is
transmitted by $\PY$ then, symmetrically, $\st_i(x,y)=\st_i(x',y)$ by
the induction hypothesis and the separate-transmissions property, and,
then, $\st_i(x,y)=\st_i(x,y')$ by the implicit-termination property,
proving the induction step.
\end{proof}

If $\phi$ is a protocol for $(X,Y)$, and $(x,y)$, $(x',y)$ are distinct
inputs in $\SXY$, then, by the correct-decision property,
$\langle\st_j(x,y)\rangle_{j=1}^\infty\ne\langle
\st_j(x',y)\rangle_{j=1}^\infty$.

Equation~(\ref{E_SXgYy}) defined $\PY$'s ambiguity set $\SXgYy$
to be the set of possible $X$ values when $Y=y$.
The last corollary implies that for all $y\in\SY$,
the multiset%
\footnote{A multiset allows multiplicity of elements.
Hence, $\{0,01,01\}$ is prefix free as a set, but not as a multiset.}
of codewords $\{\st_\phi(x,y):x\in\SXgYy\}$ is prefix free.

\section{One-way complexity}
\label{S_Cp1}

$\Cw1$, the one-way complexity of a random pair $\XcY$,
is the number of bits $P_\X$ must transmit in the worst case
when $\PY$ is not permitted to transmit any feedback messages.
Starting with $\SXY$, the support set of $\XcY$, we define $\G$,
the \textit{characteristic hypergraph} of $\XcY$, and show that
\[
\Cw1=\lceil\,\log\chi(\G)\rceil\ .
\]

Let $\XcY$ be a random pair. For each $y$ in $\SY$, the support set of
$Y$, equation~(\ref{E_SXgYy}) defined $\SXgYy$ to be the set of possible
$x$ values when $Y=y$. The \textit{characteristic hypergraph} $\G$ of
$\XcY$ has $\SX$ as its vertex set and the hyperedge $\SXgYy$ for each
$y\in\SY$.


We can now prove a continuity theorem.
\begin{thm}\label{t:conl}
Let $\Omega \subset\mathbf{R}^n$ be an open set, let
$u\in BV(\Omega ;\mathbf{R}^m)$, and let
\begin{equation}\label{quts}
T^u_x=\left\{y\in\mathbf{R}^m:
 y=\tilde u(x)+\left\langle \frac{Du}{\abs{Du}}(x),z
\right\rangle \text{ for some }z\in\mathbf{R}^n\right\}
\end{equation}
for every $x\in\Omega \backslash S_u$. Let $f\colon \mathbf{R}^m\to
\mathbf{R}^k$ be a Lipschitz continuous function such that $f(0)=0$, and
let $v=f(u)\colon \Omega \to \mathbf{R}^k$. Then $v\in BV(\Omega
;\mathbf{R}^k)$ and
\begin{equation}
Jv=\eval{(f(u^+)-f(u^-))\otimes \nu_u\cdot\,
\mathcal{H}_{n-1}}_{S_u}.
\end{equation}
In addition, for $\abs{\wt{D}u}$-almost every $x\in\Omega $ the
restriction of the function $f$ to $T^u_x$ is differentiable at $\tilde
u(x)$ and
\begin{equation}
\wt{D}v=\nabla (\eval{f}_{T^u_x})(\tilde u)
\frac{\wt{D}u}{\abs{\wt{D}u}}\cdot\abs{\wt{D}u}.\end{equation}
\end{thm}

Before proving the theorem, we state without proof three elementary
remarks which will be useful in the sequel.
\begin{rem}\label{r:omb}
Let $\omega\colon \left]0,+\infty\right[\to \left]0,+\infty\right[$
be a continuous function such that $\omega (t)\to 0$ as $t\to
0$. Then
\[\lim_{h\to 0^+}g(\omega(h))=L\Leftrightarrow\lim_{h\to
0^+}g(h)=L\]
for any function $g\colon \left]0,+\infty\right[\to \mathbf{R}$.
\end{rem}
\begin{rem}\label{r:dif}
Let $g \colon  \mathbf{R}^n\to \mathbf{R}$ be a Lipschitz
continuous function and assume that
\[L(z)=\lim_{h\to 0^+}\frac{g(hz)-g(0)}h\]
exists for every $z\in\mathbf{Q}^n$ and that $L$ is a linear function of
$z$. Then $g$ is differentiable at 0.
\end{rem}
\begin{rem}\label{r:dif0}
Let $A \colon \mathbf{R}^n\to \mathbf{R}^m$ be a linear function, and
let $f \colon \mathbf{R}^m\to \mathbf{R}$ be a function. Then the
restriction of $f$ to the range of $A$ is differentiable at 0 if and
only if $f(A)\colon \mathbf{R}^n\to \mathbf{R}$ is differentiable at 0
and
\[\nabla(\eval{f}_{\IM(A)})(0)A=\nabla (f(A))(0).\]
\end{rem}

\begin{proof}
 We begin by showing that $v\in BV(\Omega;\mathbf{R}^k)$ and
\begin{equation}\label{e:bomb}
\abs{Dv}(B)\le K\abs{Du}(B)\qquad\forall B\in\mathbf{B}(\Omega ),
\end{equation}
where $K>0$ is the Lipschitz constant of $f$. By \eqref{sum-Di} and by
the approximation result quoted in \secref{s:mt}, it is possible to find
a sequence $(u_h)\subset C^1(\Omega ;\mathbf{R}^m)$ converging to $u$ in
$L^1(\Omega ;\mathbf{R}^m)$ and such that
\[\lim_{h\to +\infty}\int_\Omega \abs{\nabla u_h}\,dx=\abs{Du}(\Omega ).\]
The functions $v_h=f(u_h)$ are locally Lipschitz continuous in $\Omega
$, and the definition of differential implies that $\abs{\nabla v_h}\le
K\abs{\nabla u_h}$ almost everywhere in $\Omega $. The lower semicontinuity
of the total variation and \eqref{sum-Di} yield
\begin{equation}
\begin{split}
\abs{Dv}(\Omega )\le\liminf_{h\to +\infty}\abs{Dv_h}(\Omega) &
=\liminf_{h\to +\infty}\int_\Omega \abs{\nabla v_h}\,dx\\
&\le K\liminf_{h\to +\infty}\int_\Omega
\abs{\nabla u_h}\,dx=K\abs{Du}(\Omega).
\end{split}\end{equation}
Since $f(0)=0$, we have also
\[\int_\Omega \abs{v}\,dx\le K\int_\Omega \abs{u}\,dx;\]
therefore $u\in BV(\Omega ;\mathbf{R}^k)$. Repeating the same argument
for every open set $A\subset\Omega $, we get \eqref{e:bomb} for every
$B\in\mathbf{B}(\Omega)$, because $\abs{Dv}$, $\abs{Du}$ are Radon measures. To
prove \lemref{limbog}, first we observe that
\begin{equation}\label{e:SS}
S_v\subset S_u,\qquad\tilde v(x)=f(\tilde u(x))\qquad \forall x\in\Omega
\backslash S_u.\end{equation}
In fact, for every $\varepsilon >0$ we have
\[\{y\in B_\rho(x): \abs{v(y)-f(\tilde u(x))}>\varepsilon \}\subset \{y\in
B_\rho(x): \abs{u(y)-\tilde u(x)}>\varepsilon /K\},\]
hence
\[\lim_{\rho\to 0^+}\frac{\abs{\{y\in B_\rho(x): \abs{v(y)-f(\tilde u(x))}>
\varepsilon \}}}{\rho^n}=0\]
whenever $x\in\Omega \backslash S_u$. By a similar argument, if $x\in
S_u$ is a point such that there exists a triplet $(u^+,u^-,\nu_u)$
satisfying \eqref{detK1}, \eqref{detK2}, then
\[
(v^+(x)-v^-(x))\otimes \nu_v=(f(u^+(x))-f(u^-(x)))\otimes\nu_u\quad
\text{if }x\in S_v
\]
and $f(u^-(x))=f(u^+(x))$ if $x\in S_u\backslash S_v$. Hence, by (1.8)
we get
\begin{equation*}\begin{split}
Jv(B)=\int_{B\cap S_v}(v^+-v^-)\otimes \nu_v\,d\mathcal{H}_{n-1}&=
\int_{B\cap S_v}(f(u^+)-f(u^-))\otimes \nu_u\,d\mathcal{H}_{n-1}\\
&=\int_{B\cap S_u}(f(u^+)-f(u^-))\otimes \nu_u\,d\mathcal{H}_{n-1}
\end{split}\end{equation*}
and \lemref{limbog} is proved.
\end{proof}

To prove \eqref{e:SS}, it is not restrictive to assume that $k=1$.
Moreover, to simplify our notation, from now on we shall assume that
$\Omega = \mathbf{R}^n$. The proof of \eqref{e:SS} is divided into two
steps. In the first step we prove the statement in the one-dimensional
case $(n=1)$, using \thmref{th-weak-ske-owf}. In the second step we
achieve the general result using \thmref{t:conl}.

\begin{step}
Assume that $n=1$. Since $S_u$ is at most countable, \eqref{sum-bij}
yields that $\abs{\wt{D}v}(S_u\backslash S_v)=0$, so that~\eqref{e:st}
and \eqref{e:barwq} imply that $Dv=\wt{D}v+Jv$ is 
the Radon-Nikod\'ym decomposition of $Dv$ in absolutely continuous and
singular part with respect to $\abs{\wt{D} u}$. By
\thmref{th-weak-ske-owf}, we have
\begin{equation*}
\frac{\wt{D}v}{\abs{\wt{D}u}}(t)=\lim_{s\to t^+}
\frac{Dv(\interval{\left[t,s\right[})}
{\abs{\wt{D}u}(\interval{\left[t,s\right[})},\qquad
\frac{\wt{D}u}{\abs{\wt{D}u}}(t)=\lim_{s\to t^+}
\frac{Du(\interval{\left[t,s\right[})}
{\abs{\wt{D}u}(\interval{\left[t,s\right[})}
\end{equation*}
$\abs{\wt{D}u}$-almost everywhere in $\mathbf{R}$. It is well known
(see, for instance, \cite[2.5.16]{ste:sint}) that every one-dimensional
function of bounded variation $w$ has a unique left continuous
representative, i.e., a function $\hat w$ such that $\hat w=w$ almost
everywhere and $\lim_{s\to t^-}\hat w(s)=\hat w(t)$ for every $t\in
\mathbf{R}$. These conditions imply
\begin{equation}
\hat u(t)=Du(\interval{\left]-\infty,t\right[}),
\qquad \hat v(t)=Dv(\interval{\left]-\infty,t\right[})\qquad
\forall t\in\mathbf{R}
\end{equation}
and
\begin{equation}\label{alimo}
\hat v(t)=f(\hat u(t))\qquad\forall t\in\mathbf{R}.\end{equation}
Let $t\in\mathbf{R}$ be such that
$\abs{\wt{D}u}(\interval{\left[t,s\right[})>0$ for every $s>t$ and
assume that the limits in \eqref{joe} exist. By \eqref{j:mark}
and~\eqref{far-d} we get 
\begin{equation*}\begin{split}
\frac{\hat v(s)-\hat
v(t)}{\abs{\wt{D}u}(\interval{\left[t,s\right[})}&=\frac {f(\hat
u(s))-f(\hat u(t))}{\abs{\wt{D}u}(\interval{\left[t,s\right[})}\\
&=\frac{f(\hat u(s))-f(\hat
u(t)+\dfrac{\wt{D}u}{\abs{\wt{D}u}}(t)\abs{\wt{D}u
}(\interval{\left[t,s\right[}))}%
{\abs{\wt{D}u}(\interval{\left[t,s\right[})}\\
&+\frac
{f(\hat u(t)+\dfrac{\wt{D}u}{\abs{\wt{D}u}}(t)\abs{\wt{D}
u}(\interval{\left[t,s\right[}))-f(\hat
u(t))}{\abs{\wt{D}u}(\interval{\left[t,s\right[})}
\end{split}\end{equation*}
for every $s>t$. Using the Lipschitz condition on $f$ we find
{\setlength{\multlinegap}{0pt}
\begin{multline*}
\left\lvert\frac{\hat v(s)-\hat
v(t)}{\abs{\wt{D}u}(\interval{\left[t,s\right[})} -\frac{f(\hat
u(t)+\dfrac{\wt{D}u}{\abs{\wt{D}u}}(t)
\abs{\wt{D}u}(\interval{\left[t,s\right[}))-f(\hat
u(t))}{\abs{\wt{D}u}(\interval{\left[t,s\right[})}\right\rvert\\
\le K\left\lvert
\frac{\hat u(s)-\hat u(t)}
  {\abs{\wt{D}u}(\interval{\left[t,s\right[})}
-\frac{\wt{D}u}{\abs{
\wt{D}u}}(t)\right\rvert.\end{multline*}
}% end of group with \multlinegap=0pt
By \eqref{e:bomb}, the function $s\to
\abs{\wt{D}u}(\interval{\left[t,s\right[})$ is continuous and
converges to 0 as $s\downarrow t$. Therefore Remark~\ref{r:omb} and the
previous inequality imply
\[\frac{\wt{D}v}{\abs{\wt{D}u}}(t)=\lim_{h\to 0^+}
\frac{f(\hat u(t)+h\dfrac{\wt{D}u}{\abs{\wt{D}u}}
(t))-f(\hat u(t))}h\quad\abs{\wt{D}u}\text{-a.e. in }\mathbf{R}.\]
By \eqref{joe}, $\hat u(x)=\tilde u(x)$ for every
$x\in\mathbf{R}\backslash S_u$; moreover, applying the same argument to
the functions $u'(t)=u(-t)$, $v'(t)=f(u'(t))=v(-t)$, we get
\[\frac{\wt{D}v}{\abs{\wt{D}u}}(t)=\lim_{h\to 0}
\frac{f(\tilde u(t)
+h\dfrac{\wt{D}u}{\abs{\wt{D}u}}(t))-f(\tilde u(t))}{h}
\qquad\abs{\wt{D}u}\text{-a.e. in }\mathbf{R}\]
and our statement is proved.
\end{step}

\begin{step}
Let us consider now the general case $n>1$. Let $\nu\in \mathbf{R}^n$ be
such that $\abs{\nu}=1$, and let $\pi_\nu=\{y\in\mathbf{R}^n: \langle
y,\nu\rangle =0\}$. In the following, we shall identify $\mathbf{R}^n$
with $\pi_\nu\times\mathbf{R}$, and we shall denote by $y$ the variable
ranging in $\pi_\nu$ and by $t$ the variable ranging in $\mathbf{R}$. By
the just proven one-dimensional result, and by \thmref{thm-main}, we get
\[\lim_{h\to 0}\frac{f(\tilde u(y+t\nu)+h\dfrac{\wt{D}u_y}{\abs{
\wt{D}u_y}}(t))-f(\tilde u(y+t\nu))}h=\frac{\wt{D}v_y}{\abs{
\wt{D}u_y}}(t)\qquad\abs{\wt{D}u_y}\text{-a.e. in }\mathbf{R}\]
for $\mathcal{H}_{n-1}$-almost every $y\in \pi_\nu$. We claim that
\begin{equation}
\frac{\langle \wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle
}}(y+t\nu)=\frac{\wt{D}u_y}
{\abs{\wt{D}u_y}}(t)\qquad\abs{\wt{D}u_y}\text{-a.e. in }\mathbf{R}
\end{equation}
for $\mathcal{H}_{n-1}$-almost every $y\in\pi_\nu$. In fact,
by~\eqref{sum-ali} and \eqref{delta-l} we get
\begin{multline*}
\int_{\pi_\nu}\frac{\wt{D}u_y}{\abs{\wt{D}u_y}}\cdot\abs{\wt{D}u_y
}\,d\mathcal{H}_{n-1}(y)=\int_{\pi_\nu}\wt{D}u_y\,d\mathcal{H}_{n-1}(y)\\
=\langle \wt{D}u,\nu\rangle =\frac
{\langle \wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle}}\cdot
\abs{\langle \wt{D}u,\nu\rangle }=\int_{\pi_\nu}\frac{
\langle \wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }}
(y+\cdot \nu)\cdot\abs{\wt{D}u_y}\,d\mathcal{H}_{n-1}(y)
\end{multline*}
and \eqref{far-d} follows from \eqref{sum-Di}. By the same argument it
is possible to prove that
\begin{equation}
\frac{\langle \wt{D}v,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle
}}(y+t\nu)=\frac{\wt{D}v_y}{\abs{\wt{D}u_y}}(t)\qquad\abs{
\wt{D}u_y}\text{-a.e. in }\mathbf{R}\end{equation}
for $\mathcal{H}_{n-1}$-almost every $y\in \pi_\nu$. By \eqref{far-d}
and \eqref{E_SXgYy} we get
\[
\lim_{h\to 0}\frac{f(\tilde u(y+t\nu)+h\dfrac{\langle \wt{D}
u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }}(y+t\nu))-f(\tilde
u(y+t\nu))}{h}
=\frac{\langle \wt{D}v,\nu\rangle }{\abs{\langle
\wt{D}u,\nu\rangle }}(y+t\nu)\]
for $\mathcal{H}_{n-1}$-almost every $y\in\pi_\nu$, and using again~\eqref{detK1}, \eqref{detK2} we get
\[
\lim_{h\to 0}\frac{f(\tilde u(x)+h\dfrac{\langle
\wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }}(x))-f(\tilde
u(x))}{h}=\frac{\langle \wt{D}v,\nu\rangle }{\abs{\langle \wt{D}u,\nu
\rangle }}(x)
\]
$\abs{\langle \wt{D}u,\nu\rangle}$-a.e. in $\mathbf{R}^n$.

Since the function $\abs{\langle \wt{D}u,\nu\rangle }/\abs{\wt{D}u}$
is strictly positive $\abs{\langle \wt{D}u,\nu\rangle }$-almost everywhere,
we obtain also
\begin{multline*}
\lim_{h\to 0}\frac{f(\tilde u(x)+h\dfrac{\abs{\langle
\wt{D}u,\nu\rangle }}{\abs{\wt{D}u}}(x)\dfrac{\langle \wt{D}
u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }}(x))-f(\tilde u(x))}{h}\\
=\frac{\abs{\langle \wt{D}u,\nu\rangle }}{\abs{\wt{D}u}}(x)\frac
{\langle \wt{D}v,\nu\rangle }{\abs{\langle
\wt{D}u,\nu\rangle }}(x)
\end{multline*}
$\abs{\langle \wt{D}u,\nu\rangle }$-almost everywhere in $\mathbf{R}^n$.

Finally, since
\begin{align*}
&\frac{\abs{\langle \wt{D}u,\nu\rangle }}{\abs{\wt{D}u}}
\frac{\langle \wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle}}
=\frac{\langle \wt{D}u,\nu\rangle }{\abs{\wt{D}u}}
=\left\langle \frac{\wt{D}u}{\abs{\wt{D}u}},\nu\right\rangle
        \qquad\abs{\wt{D}u}\text{-a.e. in }\mathbf{R}^n\\
&\frac{\abs{\langle \wt{D}u,\nu\rangle }}{\abs{\wt{D}u}}
\frac{\langle \wt{D}v,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle}}
=\frac{\langle \wt{D}v,\nu\rangle }{\abs{\wt{D}u}}
=\left\langle \frac{\wt{D}v}{\abs{\wt{D}u}},\nu\right\rangle
        \qquad\abs{\wt{D}u}\text{-a.e. in }\mathbf{R}^n
\end{align*}
and since both sides of \eqref{alimo}
are zero $\abs{\wt{D}u}$-almost everywhere
on $\abs{\langle \wt{D}u,\nu\rangle }$-negligible sets, we conclude that
\[
\lim_{h\to 0}\frac{f\left(
\tilde u(x)+h\left\langle \dfrac{\wt{D}
u}{\abs{\wt{D}u}}(x),\nu\right\rangle \right)-f(\tilde u(x))}h
=\left\langle \frac{\wt{D}v}{\abs{\wt{D}u}}(x),\nu\right\rangle,
\]
$\abs{\wt{D}u}$-a.e. in $\mathbf{R}^n$.
Since $\nu$ is arbitrary, by Remarks \ref{r:dif} and~\ref{r:dif0}
the restriction of $f$ to
the affine space $T^u_x$ is differentiable at $\tilde u(x)$ for $\abs{\wt{D}
u}$-almost every $x\in \mathbf{R}^n$ and \eqref{quts} holds.\qed
\end{step}

It follows from \eqref{sum-Di}, \eqref{detK1}, and \eqref{detK2} that
\begin{equation}\label{Dt}
D(t_1,\dots,t_n)=\sum_{I\in\mathbf{n}}(-1)^{\abs{I}-1}\abs{I}
\prod_{i\in I}t_i\prod_{j\in I}(D_j+\lambda_jt_j)\det\mathbf{A}^{(\lambda)}
(\overline I|\overline I).
\end{equation}
Let $t_i=\hat x_i$, $i=1,\dots,n$. Lemma 1 leads to
\begin{equation}\label{Dx}
D(\hat x_1,\dots,\hat x_n)=\prod_{i\in\mathbf{n}}\hat x_i
\sum_{I\in\mathbf{n}}(-1)^{\abs{I}-1}\abs{I}\per \mathbf{A}
^{(\lambda)}(I|I)\det\mathbf{A}^{(\lambda)}(\overline I|\overline I).
\end{equation}
By \eqref{H-cycles}, \eqref{sum-Di}, and \eqref{Dx},
we have the following result:
\begin{thm}\label{thm-H-param}
\begin{equation}\label{H-param}
H_c=\frac{1}{2n}\sum^n_{l =1}l (-1)^{l -1}A_{l}
^{(\lambda)},
\end{equation}
where
\begin{equation}\label{A-l-lambda}
A^{(\lambda)}_l =\sum_{I_l \subseteq\mathbf{n}}\per \mathbf{A}
^{(\lambda)}(I_l |I_l )\det\mathbf{A}^{(\lambda)}
(\overline I_{l}|\overline I_l ),\abs{I_{l}}=l .
\end{equation}
\end{thm}

It is worth noting that $A_l ^{(\lambda)}$ of \eqref{A-l-lambda} is
similar to the coefficients $b_l $ of the characteristic polynomial
of~\eqref{bl-sum}. It is well known in graph theory that the
coefficients 
$b_l $ can be expressed as a sum over certain subgraphs. It is
interesting to see whether $A_l $, $\lambda=0$, structural properties
of a graph.

We may call \eqref{H-param} a parametric representation of $H_c$. In
computation, the parameter $\lambda_i$ plays very important roles. The
choice of the parameter usually depends on the properties of the given
graph. For a complete graph $K_n$, let $\lambda_i=1$, $i=1,\dots,n$.
It follows from \eqref{A-l-lambda} that
\begin{equation}\label{compl-gr}
A^{(1)}_l =\begin{cases} n!,&\text{if }l =1\\
0,&\text{otherwise}.\end{cases}
\end{equation}
By \eqref{H-param}
\begin{equation}
H_c=\frac 12(n-1)!.
\end{equation}
For a complete bipartite graph $K_{n_1n_2}$, let $\lambda_i=0$, $i=1,\dots,n$.
By \eqref{A-l-lambda},
\begin{equation}
A_l =
\begin{cases} -n_1!n_2!\delta_{n_1n_2},&\text{if }l =2\\
0,&\text{otherwise }.\end{cases}
\label{compl-bip-gr}
\end{equation}
Theorem~\ref{thm-H-param}
leads to
\begin{equation}
H_c=\frac1{n_1+n_2}n_1!n_2!\delta_{n_1n_2}.
\end{equation}

Now, we consider an asymmetrical approach. Theorem \ref{thm-main} leads to
\begin{multline}
\det\mathbf{K}(t=1,t_1,\dots,t_n;l |l )\\
=\sum_{I\subseteq\mathbf{n}-\{l \}}
(-1)^{\abs{I}}\prod_{i\in I}t_i\prod_{j\in I}
(D_j+\lambda_jt_j)\det\mathbf{A}^{(\lambda)}
(\overline I\cup\{l \}|\overline I\cup\{l \}).
\end{multline}

By \eqref{H-cycles} and \eqref{sum-ali} we have the following asymmetrical
result:
\begin{thm}\label{thm-asym}
\begin{equation}
H_c=\frac12\sum_{I\subseteq\mathbf{n}-\{l \}}
(-1)^{\abs{I}}\per\mathbf{A}^{(\lambda)}(I|I)\det
\mathbf{A}^{(\lambda)}
(\overline I\cup\{l \}|\overline I\cup\{l \})
\end{equation}
which reduces to Goulden--Jackson's formula when $\lambda_i=0,i=1,\dots,n$
\cite{mami:matrixth}.
\end{thm}


\section{Various font features of the \pkg{amsmath} package}
\label{s:font}
\subsection{Bold versions of special symbols}

In the \pkg{amsmath} package \cn{boldsymbol} is used for getting
individual bold math symbols and bold Greek letters---everything in
math except for letters of the Latin alphabet,
where you'd use \cn{mathbf}.  For example,
\begin{verbatim}
A_\infty + \pi A_0 \sim
\mathbf{A}_{\boldsymbol{\infty}} \boldsymbol{+}
\boldsymbol{\pi} \mathbf{A}_{\boldsymbol{0}}
\end{verbatim}
looks like this:
\[A_\infty + \pi A_0 \sim \mathbf{A}_{\boldsymbol{\infty}}
\boldsymbol{+} \boldsymbol{\pi} \mathbf{A}_{\boldsymbol{0}}\]

\subsection{``Poor man's bold''}
If a bold version of a particular symbol doesn't exist in the
available fonts,
then \cn{boldsymbol} can't be used to make that symbol bold.
At the present time, this means that
\cn{boldsymbol} can't be used with symbols from
the \fn{msam} and \fn{msbm} fonts, among others.
In some cases, poor man's bold (\cn{pmb}) can be used instead
of \cn{boldsymbol}:
%  Can't show example from msam or msbm because this document is
%  supposed to be TeXable even if the user doesn't have
%  AMSFonts.  MJD 5-JUL-1990
\[\frac{\partial x}{\partial y}
\pmb{\bigg\vert}
\frac{\partial y}{\partial z}\]
\begin{verbatim}
\[\frac{\partial x}{\partial y}
\pmb{\bigg\vert}
\frac{\partial y}{\partial z}\]
\end{verbatim}
So-called ``large operator'' symbols such as $\sum$ and $\prod$
require an additional command, \cn{mathop},
to produce proper spacing and limits when \cn{pmb} is used.
For further details see \textit{The \TeX book}.
\[\sum_{\substack{i<B\\\text{$i$ odd}}}
\prod_\kappa \kappa F(r_i)\qquad
\mathop{\pmb{\sum}}_{\substack{i<B\\\text{$i$ odd}}}
\mathop{\pmb{\prod}}_\kappa \kappa(r_i)
\]
\begin{verbatim}
\[\sum_{\substack{i<B\\\text{$i$ odd}}}
\prod_\kappa \kappa F(r_i)\qquad
\mathop{\pmb{\sum}}_{\substack{i<B\\\text{$i$ odd}}}
\mathop{\pmb{\prod}}_\kappa \kappa(r_i)
\]
\end{verbatim}

\section{Compound symbols and other features}
\label{s:comp}
\subsection{Multiple integral signs}
\cn{iint}, \cn{iiint}, and \cn{iiiint} give multiple integral signs
with the spacing between them nicely adjusted,  in both text and
display style.  \cn{idotsint} gives two integral signs with dots
between them.
\begin{gather}
\iint\limits_A f(x,y)\,dx\,dy\qquad\iiint\limits_A
f(x,y,z)\,dx\,dy\,dz\\
\iiiint\limits_A
f(w,x,y,z)\,dw\,dx\,dy\,dz\qquad\idotsint\limits_A f(x_1,\dots,x_k)
\end{gather}

\subsection{Over and under arrows}
Some extra over and under arrow operations are provided in
the \pkg{amsmath} package.  (Basic \LaTeX\ provides
\cn{overrightarrow} and \cn{overleftarrow}).
\begin{align*}
\overrightarrow{\psi_\delta(t) E_t h}&
=\underrightarrow{\psi_\delta(t) E_t h}\\
\overleftarrow{\psi_\delta(t) E_t h}&
=\underleftarrow{\psi_\delta(t) E_t h}\\
\overleftrightarrow{\psi_\delta(t) E_t h}&
=\underleftrightarrow{\psi_\delta(t) E_t h}
\end{align*}
\begin{verbatim}
\begin{align*}
\overrightarrow{\psi_\delta(t) E_t h}&
=\underrightarrow{\psi_\delta(t) E_t h}\\
\overleftarrow{\psi_\delta(t) E_t h}&
=\underleftarrow{\psi_\delta(t) E_t h}\\
\overleftrightarrow{\psi_\delta(t) E_t h}&
=\underleftrightarrow{\psi_\delta(t) E_t h}
\end{align*}
\end{verbatim}
These all scale properly in subscript sizes:
\[\int_{\overrightarrow{AB}} ax\,dx\]
\begin{verbatim}
\[\int_{\overrightarrow{AB}} ax\,dx\]
\end{verbatim}

\subsection{Dots}

Normally you need only type \cn{dots} for ellipsis dots in a
math formula.  The main exception is when the dots
fall at the end of the formula; then you need to
specify one of \cn{dotsc} (series dots, after a comma),
\cn{dotsb} (binary dots, for binary relations or operators),
\cn{dotsm} (multiplication dots), or \cn{dotsi} (dots after
an integral).  For example, the input
\begin{verbatim}
Then we have the series $A_1,A_2,\dotsc$,
the regional sum $A_1+A_2+\dotsb$,
the orthogonal product $A_1A_2\dotsm$,
and the infinite integral
\[\int_{A_1}\int_{A_2}\dotsi\].
\end{verbatim}
produces
\begin{quotation}
Then we have the series $A_1,A_2,\dotsc$,
the regional sum $A_1+A_2+\dotsb$,
the orthogonal product $A_1A_2\dotsm$,
and the infinite integral
\[\int_{A_1}\int_{A_2}\dotsi\]
\end{quotation}

\subsection{Accents in math}

Double accents:
\[\Hat{\Hat{H}}\quad\Check{\Check{C}}\quad
\Tilde{\Tilde{T}}\quad\Acute{\Acute{A}}\quad
\Grave{\Grave{G}}\quad\Dot{\Dot{D}}\quad
\Ddot{\Ddot{D}}\quad\Breve{\Breve{B}}\quad
\Bar{\Bar{B}}\quad\Vec{\Vec{V}}\]
\begin{verbatim}
\[\Hat{\Hat{H}}\quad\Check{\Check{C}}\quad
\Tilde{\Tilde{T}}\quad\Acute{\Acute{A}}\quad
\Grave{\Grave{G}}\quad\Dot{\Dot{D}}\quad
\Ddot{\Ddot{D}}\quad\Breve{\Breve{B}}\quad
\Bar{\Bar{B}}\quad\Vec{\Vec{V}}\]
\end{verbatim}
This double accent operation is complicated
and tends to slow down the processing of a \LaTeX\ file.


\subsection{Dot accents}
\cn{dddot} and \cn{ddddot} are available to
produce triple and quadruple dot accents
in addition to the \cn{dot} and \cn{ddot} accents already available
in \LaTeX:
\[\dddot{Q}\qquad\ddddot{R}\]
\begin{verbatim}
\[\dddot{Q}\qquad\ddddot{R}\]
\end{verbatim}

\subsection{Roots}

In the \pkg{amsmath} package \cn{leftroot} and \cn{uproot} allow you to adjust
the position of the root index of a radical:
\begin{verbatim}
\sqrt[\leftroot{-2}\uproot{2}\beta]{k}
\end{verbatim}
gives good positioning of the $\beta$:
\[\sqrt[\leftroot{-2}\uproot{2}\beta]{k}\]

\subsection{Boxed formulas} The command \cn{boxed} puts a box around its
argument, like \cn{fbox} except that the contents are in math mode:
\begin{verbatim}
\boxed{W_t-F\subseteq V(P_i)\subseteq W_t}
\end{verbatim}
\[\boxed{W_t-F\subseteq V(P_i)\subseteq W_t}.\]

\subsection{Extensible arrows}
\cn{xleftarrow} and \cn{xrightarrow} produce
arrows that extend automatically to accommodate unusually wide
subscripts or superscripts.  The text of the subscript or superscript
are given as an optional resp.\@ mandatory argument:
Example:
\[0 \xleftarrow[\zeta]{\alpha} F\times\triangle[n-1]
  \xrightarrow{\partial_0\alpha(b)} E^{\partial_0b}\]
\begin{verbatim}
\[0 \xleftarrow[\zeta]{\alpha} F\times\triangle[n-1]
  \xrightarrow{\partial_0\alpha(b)} E^{\partial_0b}\]
\end{verbatim}

\subsection{\cn{overset}, \cn{underset}, and \cn{sideset}}
Examples:
\[\overset{*}{X}\qquad\underset{*}{X}\qquad
\overset{a}{\underset{b}{X}}\]
\begin{verbatim}
\[\overset{*}{X}\qquad\underset{*}{X}\qquad
\overset{a}{\underset{b}{X}}\]
\end{verbatim}

The command \cn{sideset} is for a rather special
purpose: putting symbols at the subscript and superscript
corners of a large operator symbol such as $\sum$ or $\prod$,
without affecting the placement of limits.
Examples:
\[\sideset{_*^*}{_*^*}\prod_k\qquad
\sideset{}{'}\sum_{0\le i\le m} E_i\beta x
\]
\begin{verbatim}
\[\sideset{_*^*}{_*^*}\prod_k\qquad
\sideset{}{'}\sum_{0\le i\le m} E_i\beta x
\]
\end{verbatim}

\subsection{The \cn{text} command}
The main use of the command \cn{text} is for words or phrases in a
display:
\[\mathbf{y}=\mathbf{y}'\quad\text{if and only if}\quad
y'_k=\delta_k y_{\tau(k)}\]
\begin{verbatim}
\[\mathbf{y}=\mathbf{y}'\quad\text{if and only if}\quad
y'_k=\delta_k y_{\tau(k)}\]
\end{verbatim}

\subsection{Operator names}
The more common math functions such as $\log$, $\sin$, and $\lim$
have predefined control sequences: \verb=\log=, \verb=\sin=,
\verb=\lim=.
The \pkg{amsmath} package provides \cn{DeclareMathOperator} and
\cn{DeclareMathOperator*}
for producing new function names that will have the
same typographical treatment.
Examples:
\[\norm{f}_\infty=
\esssup_{x\in R^n}\abs{f(x)}\]
\begin{verbatim}
\[\norm{f}_\infty=
\esssup_{x\in R^n}\abs{f(x)}\]
\end{verbatim}
\[\meas_1\{u\in R_+^1\colon f^*(u)>\alpha\}
=\meas_n\{x\in R^n\colon \abs{f(x)}\geq\alpha\}
\quad \forall\alpha>0.\]
\begin{verbatim}
\[\meas_1\{u\in R_+^1\colon f^*(u)>\alpha\}
=\meas_n\{x\in R^n\colon \abs{f(x)}\geq\alpha\}
\quad \forall\alpha>0.\]
\end{verbatim}
\cn{esssup} and \cn{meas} would be defined in the document preamble as
\begin{verbatim}
\DeclareMathOperator*{\esssup}{ess\,sup}
\DeclareMathOperator{\meas}{meas}
\end{verbatim}

The following special operator names are predefined in the \pkg{amsmath}
package: \cn{varlimsup}, \cn{varliminf}, \cn{varinjlim}, and
\cn{varprojlim}. Here's what they look like in use:
\begin{align}
&\varlimsup_{n\rightarrow\infty}
       \mathcal{Q}(u_n,u_n-u^{\#})\le0\\
&\varliminf_{n\rightarrow\infty}
  \left\lvert a_{n+1}\right\rvert/\left\lvert a_n\right\rvert=0\\
&\varinjlim (m_i^\lambda\cdot)^*\le0\\
&\varprojlim_{p\in S(A)}A_p\le0
\end{align}
\begin{verbatim}
\begin{align}
&\varlimsup_{n\rightarrow\infty}
       \mathcal{Q}(u_n,u_n-u^{\#})\le0\\
&\varliminf_{n\rightarrow\infty}
  \left\lvert a_{n+1}\right\rvert/\left\lvert a_n\right\rvert=0\\
&\varinjlim (m_i^\lambda\cdot)^*\le0\\
&\varprojlim_{p\in S(A)}A_p\le0
\end{align}
\end{verbatim}

\subsection{\cn{mod} and its relatives}
The commands \cn{mod} and \cn{pod} are variants of
\cn{pmod} preferred by some authors; \cn{mod} omits the parentheses,
whereas \cn{pod} omits the `mod' and retains the parentheses.
Examples:
\begin{align}
x&\equiv y+1\pmod{m^2}\\
x&\equiv y+1\mod{m^2}\\
x&\equiv y+1\pod{m^2}
\end{align}
\begin{verbatim}
\begin{align}
x&\equiv y+1\pmod{m^2}\\
x&\equiv y+1\mod{m^2}\\
x&\equiv y+1\pod{m^2}
\end{align}
\end{verbatim}

\subsection{Fractions and related constructions}
\label{fracs}

The usual notation for binomials is similar to the fraction concept,
so it has a similar command \cn{binom} with two arguments. Example:
\begin{equation}
\begin{split}
\sum_{\gamma\in\Gamma_C} I_\gamma&
=2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}\\
&\quad+\dots+(-1)^l\binom{k}{l}2^{k-l}
+\dots+(-1)^k\\
&=(2-1)^k=1
\end{split}
\end{equation}
\begin{verbatim}
\begin{equation}
\begin{split}
[\sum_{\gamma\in\Gamma_C} I_\gamma&
=2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}\\
&\quad+\dots+(-1)^l\binom{k}{l}2^{k-l}
+\dots+(-1)^k\\
&=(2-1)^k=1
\end{split}
\end{equation}
\end{verbatim}
There are also abbreviations
\begin{verbatim}
\dfrac        \dbinom
\tfrac        \tbinom
\end{verbatim}
for the commonly needed constructions
\begin{verbatim}
{\displaystyle\frac ... }   {\displaystyle\binom ... }
{\textstyle\frac ... }      {\textstyle\binom ... }
\end{verbatim}

The generalized fraction command \cn{genfrac} provides full access to
the six \TeX{} fraction primitives:
\begin{align}
\text{\cn{over}: }&\genfrac{}{}{}{}{n+1}{2}&
\text{\cn{overwithdelims}: }&
  \genfrac{\langle}{\rangle}{}{}{n+1}{2}\\
\text{\cn{atop}: }&\genfrac{}{}{0pt}{}{n+1}{2}&
\text{\cn{atopwithdelims}: }&
  \genfrac{(}{)}{0pt}{}{n+1}{2}\\
\text{\cn{above}: }&\genfrac{}{}{1pt}{}{n+1}{2}&
\text{\cn{abovewithdelims}: }&
  \genfrac{[}{]}{1pt}{}{n+1}{2}
\end{align}
\begin{verbatim}
\text{\cn{over}: }&\genfrac{}{}{}{}{n+1}{2}&
\text{\cn{overwithdelims}: }&
  \genfrac{\langle}{\rangle}{}{}{n+1}{2}\\
\text{\cn{atop}: }&\genfrac{}{}{0pt}{}{n+1}{2}&
\text{\cn{atopwithdelims}: }&
  \genfrac{(}{)}{0pt}{}{n+1}{2}\\
\text{\cn{above}: }&\genfrac{}{}{1pt}{}{n+1}{2}&
\text{\cn{abovewithdelims}: }&
  \genfrac{[}{]}{1pt}{}{n+1}{2}
\end{verbatim}

\subsection{Continued fractions}
The continued fraction
\begin{equation}
\cfrac{1}{\sqrt{2}+
 \cfrac{1}{\sqrt{2}+
  \cfrac{1}{\sqrt{2}+
   \cfrac{1}{\sqrt{2}+
    \cfrac{1}{\sqrt{2}+\dotsb
}}}}}
\end{equation}
can be obtained by typing
\begin{verbatim}
\cfrac{1}{\sqrt{2}+
 \cfrac{1}{\sqrt{2}+
  \cfrac{1}{\sqrt{2}+
   \cfrac{1}{\sqrt{2}+
    \cfrac{1}{\sqrt{2}+\dotsb
}}}}}
\end{verbatim}
Left or right placement of any of the numerators is accomplished by using
\cn{cfrac[l]} or \cn{cfrac[r]} instead of \cn{cfrac}.

\subsection{Smash}

In \pkg{amsmath} there are optional arguments \verb"t" and \verb"b" for
the plain \TeX\ command \cn{smash}, because sometimes it is advantageous
to be able to `smash' only the top or only the bottom of something while
retaining the natural depth or height. In the formula
$X_j=(1/\sqrt{\smash[b]{\lambda_j}})X_j'$ \cn{smash}\verb=[b]= has been
used to limit the size of the radical symbol.
\begin{verbatim}
$X_j=(1/\sqrt{\smash[b]{\lambda_j}})X_j'$
\end{verbatim}
Without the use of \cn{smash}\verb=[b]= the formula would have appeared
thus: $X_j=(1/\sqrt{\lambda_j})X_j'$, with the radical extending to
encompass the depth of the subscript $j$.

\subsection{The `cases' environment}
`Cases' constructions like the following can be produced using
the \env{cases} environment.
\begin{equation}
P_{r-j}=
  \begin{cases}
    0&  \text{if $r-j$ is odd},\\
    r!\,(-1)^{(r-j)/2}&  \text{if $r-j$ is even}.
  \end{cases}
\end{equation}
\begin{verbatim}
\begin{equation} P_{r-j}=
  \begin{cases}
    0&  \text{if $r-j$ is odd},\\
    r!\,(-1)^{(r-j)/2}&  \text{if $r-j$ is even}.
  \end{cases}
\end{equation}
\end{verbatim}
Notice the use of \cn{text} and the embedded math.

\subsection{Matrix}

Here are samples of the matrix environments,
\cn{matrix}, \cn{pmatrix}, \cn{bmatrix}, \cn{Bmatrix}, \cn{vmatrix}
and \cn{Vmatrix}:
\begin{equation}
\begin{matrix}
\vartheta& \varrho\\\varphi& \varpi
\end{matrix}\quad
\begin{pmatrix}
\vartheta& \varrho\\\varphi& \varpi
\end{pmatrix}\quad
\begin{bmatrix}
\vartheta& \varrho\\\varphi& \varpi
\end{bmatrix}\quad
\begin{Bmatrix}
\vartheta& \varrho\\\varphi& \varpi
\end{Bmatrix}\quad
\begin{vmatrix}
\vartheta& \varrho\\\varphi& \varpi
\end{vmatrix}\quad
\begin{Vmatrix}
\vartheta& \varrho\\\varphi& \varpi
\end{Vmatrix}
\end{equation}
%
\begin{verbatim}
\begin{matrix}
\vartheta& \varrho\\\varphi& \varpi
\end{matrix}\quad
\begin{pmatrix}
\vartheta& \varrho\\\varphi& \varpi
\end{pmatrix}\quad
\begin{bmatrix}
\vartheta& \varrho\\\varphi& \varpi
\end{bmatrix}\quad
\begin{Bmatrix}
\vartheta& \varrho\\\varphi& \varpi
\end{Bmatrix}\quad
\begin{vmatrix}
\vartheta& \varrho\\\varphi& \varpi
\end{vmatrix}\quad
\begin{Vmatrix}
\vartheta& \varrho\\\varphi& \varpi
\end{Vmatrix}
\end{verbatim}

To produce a small matrix suitable for use in text, use the
\env{smallmatrix} environment.
\begin{verbatim}
\begin{math}
  \bigl( \begin{smallmatrix}
      a&b\\ c&d
    \end{smallmatrix} \bigr)
\end{math}
\end{verbatim}
To show
the effect of the matrix on the surrounding lines of
a paragraph, we put it here: \begin{math}
  \bigl( \begin{smallmatrix}
      a&b\\ c&d
    \end{smallmatrix} \bigr)
\end{math}
and follow it with enough text to ensure that there will
be at least one full line below the matrix.

\cn{hdotsfor}\verb"{"\textit{number}\verb"}" produces a row of dots in a matrix
spanning the given number of columns:
\[W(\Phi)= \begin{Vmatrix}
\dfrac\varphi{(\varphi_1,\varepsilon_1)}&0&\dots&0\\
\dfrac{\varphi k_{n2}}{(\varphi_2,\varepsilon_1)}&
\dfrac\varphi{(\varphi_2,\varepsilon_2)}&\dots&0\\
\hdotsfor{5}\\
\dfrac{\varphi k_{n1}}{(\varphi_n,\varepsilon_1)}&
\dfrac{\varphi k_{n2}}{(\varphi_n,\varepsilon_2)}&\dots&
\dfrac{\varphi k_{n\,n-1}}{(\varphi_n,\varepsilon_{n-1})}&
\dfrac{\varphi}{(\varphi_n,\varepsilon_n)}
\end{Vmatrix}\]
\begin{verbatim}
\[W(\Phi)= \begin{Vmatrix}
\dfrac\varphi{(\varphi_1,\varepsilon_1)}&0&\dots&0\\
\dfrac{\varphi k_{n2}}{(\varphi_2,\varepsilon_1)}&
\dfrac\varphi{(\varphi_2,\varepsilon_2)}&\dots&0\\
\hdotsfor{5}\\
\dfrac{\varphi k_{n1}}{(\varphi_n,\varepsilon_1)}&
\dfrac{\varphi k_{n2}}{(\varphi_n,\varepsilon_2)}&\dots&
\dfrac{\varphi k_{n\,n-1}}{(\varphi_n,\varepsilon_{n-1})}&
\dfrac{\varphi}{(\varphi_n,\varepsilon_n)}
\end{Vmatrix}\]
\end{verbatim}
The spacing of the dots can be varied through use of a square-bracket
option, for example, \verb"\hdotsfor[1.5]{3}".  The number in square brackets
will be used as a multiplier; the normal value is 1.

\subsection{The \cn{substack} command}

The \cn{substack} command can be used to produce a multiline
subscript or superscript:
for example
\begin{verbatim}
\sum_{\substack{0\le i\le m\\ 0<j<n}} P(i,j)
\end{verbatim}
produces a two-line subscript underneath the sum:
\begin{equation}
\sum_{\substack{0\le i\le m\\ 0<j<n}} P(i,j)
\end{equation}
A slightly more generalized form is the \env{subarray} environment which
allows you to specify that each line should be left-aligned instead of
centered, as here:\EditorialComment{Maybe ``\dots{} as below''?}
\begin{equation}
\sum_{\begin{subarray}{l}
        0\le i\le m\\ 0<j<n
      \end{subarray}}
 P(i,j)
\end{equation}
\begin{verbatim}
\sum_{\begin{subarray}{l}
        0\le i\le m\\ 0<j<n
      \end{subarray}}
 P(i,j)
\end{verbatim}


\subsection{Big-g-g delimiters}

Here are some big delimiters, first in \cn{normalsize}:
\[\biggl(\mathbf{E}_{y}
  \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
  \biggr)
\]
\begin{verbatim}
\[\biggl(\mathbf{E}_{y}
  \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
  \biggr)
\]
\end{verbatim}
and now in \cn{Large} size:
{\Large
\[\biggl(\mathbf{E}_{y}
  \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
  \biggr)
\]}
\begin{verbatim}
{\Large
\[\biggl(\mathbf{E}_{y}
  \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
  \biggr)
\]}
\end{verbatim}

\subsection{Acknowledgements}
\label{sec:acks}

The authors are grateful to \funding{NASA}{123456} and
\funding{NIH}{}.  They acknowledge \funding[the generous help of other
agencies]{Department of Defense}{Special project Secret Defense}.


\bibliography{aomsample}
\bibliographystyle{aomplain}

\end{document}
\endinput