Amsart Mit

8/11/2019 Amsart Mit

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SAMPLE PAPER FOR THE AMSTEX OPTIONAND THE AMSART DOCUMENTSTYLE

FILE NAME: TESTART.TEX

DAVID BELAR, LENORE EIFRIG, AND SHAFI N AAT ANEN

Dedicated to the memory of Parfon Samuelson

Abstract. This paper is a sample illustrating the use of the amstex option

in LA

TEX, and the American Mathematical Society preprint documentstyle,mn amsart . The le used to prepare this sample is testart.tex .

Contents

1. Introduction 21.1. Acknowledgment 22. Top matter instructions 22.1. Title 32.2. Author 32.3. Addresses 32.4. First-page footnotes 33. Enumeration of Hamiltonian paths in a graph 44. Main Theorem 55. Application 76. Secret Key Exchanges 87. Review 98. One-Way Complexity 13Step 1 15Step 2 169. Various font features of the mn amstex option 189.1. Bold versions of special symbols 189.2. Poor mans bold 1810. Compound symbols and other features 1910.1. Multiple integral signs 19

10.2. Over and under arrows 1910.3. Dots 20

Date : October 23, 1991.1991 Mathematics Subject Classication. Primary 05C38, 15A15; Secondary 05A15, 15A18.Key words and phrases. K -graph, random pair, negligible set, nonatomic ccc order.Research of the rst author was supported in part by NSF grant CCR-87-10433 and DARPA

Contract N00019-89-J-1988.Research of the second author was supported in part by NSF grant CCR-86-75257 and DARPA

Contract N00019-89-J-1988.This paper is in nal form and no version of it will be submitted for publication elsewhere.

1


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2 DAVID BELAR, LENORE EIFRIG, AND SHAFI N AAT ANEN

10.4. Accents in math 2010.5. Superscripted accents 20

10.6. Dot accents 2010.7. Roots 2110.8. Boxed formulas 2110.9. Extensible arrows 2110.10. mn\overset , mn\underset , and mn \sideset 2110.11. The mn \text command 2110.12. Operator names 2210.13. mn\mod and its relatives 2210.14. Fractions and related constructions 2310.15. Continued fractions 2310.16. Smash 2410.17. The cases environment 2410.18. Matrix 2410.19. The mn Sb and mn Sp environments 2610.20. Commutative diagrams 2610.21. Big-g-g-g delimiters 26Appendix A. Examples of multiple-line equation structures 28A.1. Split 28A.2. Multline 33A.3. Gather 34A.4. Align 36A.5. Align and split within gather 37A.6. Alignat 39

1. Introduction

This paper illustrates the use of the amsart documentstyle, as well as the useof features from the amstex option. It consists of extracts from published papers,interspersed with sample T EX coding, instructions to authors, and comments aboutthe use of particular commands.

Sections 9 and 10 are devoted to examples of the commands described in the

AMS -LATEX users guide. Appendix A gives comprehensive examples of the displayenvironments mn align , mngather , mnsplit , mn multline , and mn alignat .1.1. Acknowledgment. It is a pleasure to thank the referee for his valuable sug-gestions which resulted in an improvement of the manuscript.

2. Top matter instructions

The term top matter will be used to mean the title, author, abstract and otherpreliminary information. All such information should be typed at the beginning of a document, between \begin{document} and \maketitle .1 See the examples atthe beginning of this paper. Some of the top matter information will print in a

1Actually, as stated in the L A TEX manual, the top matter information can also be typed before\begin{document} ; but the placement recommended here segregrates the information nicely in itsown area.


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AMSTEX/AMSART SAMPLE PAPER 3

footnote on the rst page, or at the end of the document, but such placement isdone automatically by L ATEX.

2.1. Title. The \title command is used as described in the L ATEX manual, butit has an additional option like sectioning commands, to specify the text of therunning head. Thus

\title[The Selberg Trace Formula]{Some Explicit Cases of the Selberg Trace\\Formula for Vector-Valued Functions}

would put the text The Selberg Trace Formula into the running head on right-hand pages, and the text in curly braces would print as the full title on the rstpage.

2.2. Author. Like the \title command, the \author command has a squarebracket option used to specify the left-hand running head. If there are two or more

authors, the American Mathematical Society practice is to use initials instead of full rst and middle names in the running head.

In addition, in the amsart documentstyle, a separate \author command shouldbe used for each author. This makes it easier to group commands like \addressand \thanks with the associated \author command. Cf. the example top mattersection in this le.

2.3. Addresses. Provide an address for each author by using the \address com-mand following the \author command. Note that no abbreviations are used inaddresses. Indicate the normal line breaks that would be made on a mailing labelusing \\ ; in the current version of the amsart documentstyle, these \\ s will bereplaced with commas, since author addresses are set in paragraph form, but otherdocumentstyles or a future version of the amsart documentstyle might treat theinformation differently.

The addresses should be entered in the same order as the author names on thetitle page. In the amsart documentstyle, address information prints at the end of the paper, following the references.

If the current address of an author is no longer the institution where the re-search being described took place, the current address can be provided throughthe \curraddr command. Electronic mail addresses should also be given, whenapplicable, using \email .

For multiple authors with assorted current addresses and e-mail addresses, square-bracket options can be used to indicate those individuals to which an address, cur-rent address, or e-mail address pertains. For example, for two authors at the sameinstitution, with different e-mail addresses, you would enter the information as

% Address of A. Author and B. Colleague:\address{North Carolina State University\\Raleigh,North Carolina 27695}\email[A. Author]{auth@@abel.ncsu.edu}\email[B. Colleague]{bcolleague@@fourier.ncsu.edu}

2.4. First-page footnotes. Use \thanks for acknowledgments of grant supportor other research support and similar information. This information will print as afootnote on the rst page. If it pertains to only one author out of two or more, thenrefer to rst author or second author, as shown in the examples in this paper.


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Papers published in proceedings of conferences are often abstracts or preliminaryversions, and review journals such as Mathematical Reviews will be more interested

in reviewing the nal version. In such a case, use \thanks to produce a rst-page footnote saying The nal [detailed] version of this paper will be [has been]submitted for publication elsewhere. Proceedings papers that are to be consid-ered for review by Mathematical Reviews should include the following statement:This paper is in nal form and no version of it will be submitted for publicationelsewhere.

Subject classication numbers ( \subjclass ) are part of the top matter andwill appear as footnotes at the bottom of the rst page. Subject classications(\subjclass ) are required for submissions to the American Mathematical Society.Use the 1980 Mathematics Subject Classication (1985 Revision) that appears inannual indexes of Mathematical Reviews beginning in 1984. (Note: Give a completeve-digit number; the two-digit prexes from the Contents are insufficient.)

An optional key words and phrases footnote may be added using the \keywordscommand.

An abstract should be typed immediately after the \maketitle command, usingthe standard abstract environment. The heading Abstract. will be addedautomatically.

3. Enumeration of Hamiltonian paths in a graph

Let A = ( a ij ) be the adjacency matrix of graph G. The corresponding Kirchhoff matrix K = ( k ij ) is obtained from A by replacing in A each diagonal entry bythe degree of its corresponding vertex; i.e., the ith diagonal entry is identied withthe degree of the ith vertex. It is well known that

det K (i|i) = the number of spanning trees of G, i = 1 , . . . , n (3.1)where K (i|i) is the ith principal submatrix of K .

\det\bold K(i|i)=\text{ the number of spanning trees of $G$},Let C i ( j ) be the set of graphs obtained from G by attaching edge ( vi vj ) to each

spanning tree of G. Denote by C i = 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 kii det K (i|i).Let X = {x1 , . . . , xn }.$\widehat X=\{\hat x_1,\dots,\hat x_n\}$Dene multiplication for the elements of X byx i x j = x j x i , x2i = 0, i, j = 1 , . . . , n . (3.2)Let kij = kij x j and kij = j = i kij . Then the number of Hamiltonian cycles H cis given by the relation [ ? ]

n

j =1

x j H c = 12

kij det K (i|i), i = 1 , . . . , n . (3.3)The task here is to express (3.3) in a form free of any x i , i = 1 , . . . , n . The resultalso 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 acomplete graph K n and in a complete bipartite graph K n 1 n 2 can only be foundfrom rst combinatorial principles [? ]. One wonders if there exists a formula which


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can be used very efficiently to produce K n and K n 1 n 2 . Recently, using Lagrangianmethods, Goulden and Jackson have shown that H c can be expressed in terms of the

determinant and permanent of the adjacency matrix [ ? ]. However, the formula of Goulden and Jackson determines neither K n nor K n 1 n 2 effectively. In this paper,using an algebraic method, we parametrize the adjacency matrix. The resultingformula also involves the determinant and permanent, but it can easily be appliedto K n and K n 1 n 2 . In addition, we eliminate the permanent from H c and showthat H c can be represented by a determinantal function of multivariables, eachvariable with domain {0, 1}. Furthermore, we show that H c can be written bynumber of spanning trees of subgraphs. Finally, we apply the formulas to a completemultigraph K n 1 ...n p .

The conditions aij = aji , i, j = 1 , . . . , n , are not required in this paper. Allformulas can be extended to a digraph simply by multiplying H c by 2.

4. Main Theorem

Notation . For p, q P and n we write (q, n) ( p, n) if q p and Aq,n = A p,n .\begin{notation} For $p,q\in P$ and $n\in\omega$...\end{notation}

We will make liberal use of Cichons Diagram [ ? ]:

cov(L) non(K) cf(K) cf(L)

add(

L)

add(

K)

cov(

K)

non(

L)

(4.1)

\begin{equation}\begin{CD}\cov(\cal L) @>>> \non(\cal K) @>>> \cf(\cal K) @>>> \cf(\cal L)\\@VVV @AAA @AAA @VVV\\\add(\cal L) @>>> \add(\cal K) @>>> \cov(\cal K) @>>> \non(\cal L)\end{CD}\end{equation}

Let B = ( b ij ) be an n n matrix. Let n = {1 , . . . , n }. Using the properties of (3.2), it is readily seen thatLemma 4.1.

in jnbij x i =

inx i per B (4.2)

where per B is the permanent of B .

Let Y = {y1 , . . . , yn }. Dene multiplication for the elements of Y byyi yj + yj yi = 0 , i, j = 1, . . . , n . (4.3)Then, it follows that


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Lemma 4.2.

in jn bij yj = in yi det B . (4.4)

Note that all basic properties of determinants are direct consequences of Lemma 4.2.Write

jnbij yj =

jnb( )ij yj + ( bii i )yi y (4.5)

where

b( )ii = i , b( )ij = bij , i = j. (4.6)

Let B ( ) = ( b ( )ij ). By (4.4) and (4.5), it is straightforward to show the followingresult:

Theorem 4.3.

det B =n

l= 0 I ln iI l

(b ii i )det B ( ) (I l |I l ), (4.7)

where I l = {i1 , . . . , i l} and B ( ) (I l |I l ) is the principal submatrix obtained from B ( )by deleting its i1 , . . . , i l rows and columns.Remark 4.1. Let M be an n n matrix. The convention M (n |n ) = 1 has beenused in (4.7) and hereafter.

Before proceeding with our discussion, we pause to note that Theorem 4.3 yieldsimmediately a fundamental formula which can be used to compute the coefficientsof a characteristic polynomial [ ? ]:

Corollary 4.4. Write det( B xI ) =nl= 0 (1 )l b l x l . Then

bl =I ln

det B (I l |I l ). (4.8)

Let

K (t , t 1 , . . . , t n ) =

D1 t a12 t2 . . . a1n tna21 t1 D2t . . . a2n tn. . . . . . . . . . . . . . . . . . . . . .an 1 t1 an 2 t2 . . . Dn t

, (4.9)

\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}

where

D i =jn

a ij t j , i = 1 , . . . , n . (4.10)

Set

D (t1 , . . . , t n ) = t

det K (t , t 1 , . . . , t n )|t = 1 .


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Then

D (t1 , . . . , t n ) =in

D i det K (t = 1 , t 1 , . . . , t n ; i

|i), (4.11)

where K (t = 1 , t 1 , . . . , t n ; i|i) is the ith principal submatrix of K (t = 1 , t 1 , . . . , t n ).Theorem 4.3 leads todet K (t 1 , t 1 , . . . , t n ) =

In

(1 ) | I | t n | I |iI

t i jI

(D j + j t j )det A ( t ) (I |I ).(4.12)Note that

det K (t = 1 , t 1 , . . . , t n ) =In

(1 ) | I |iI

t i jI

(D j + j t j )det A ( ) (I |I ) = 0 .(4.13)Let t i = x i , i = 1, . . . , n . Lemma 4.1 yields

ina l i x i det K (t = 1 , x 1 , . . . , x n ; l|l)

=in

x i I n { l }(1) | I | per A ( ) (I |I )det A ( ) (I{l}|I{l}). (4.14)\begin{multline}\biggl(\sum_{\,i\in\bold n}a_{l _i}x_i\biggr)\det\bold K(t=1,x_1,\dots,x_n;l |l )\\=\biggl(\prod_{\,i\in\bold n}\hat x_i\biggr)\sum_{I\subseteq\bold n-\{l \}}(-1)^{|I|}\per\bold A^{(\lambda)}(I|I)\det\bold A^{(\lambda)}(\overline I\cup\{l \}|\overline I\cup\{l \}).

\label{sum-ali}\end{multline}By (3.3), (4.4), and (4.5), we have

Proposition 4.5.

H c = 12n

n

l=0

(1)l D l , (4.15)where

D l =I ln

D (t1 , . . . , t n )t i = 0

, if iI l1, otherwise , i=1 ,...,n

. (4.16)

5. Application

We consider here the applications of Theorems 6.1 and 6.2 to a complete mul-tipartite graph K n 1 ...n p . It can be shown that the number of spanning trees of K n 1 ...n p may be written

T = n p 2 p

i=1(n n i )n i 1 (5.1)


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wheren = n1 +

+ n p . (5.2)

It follows from Theorems 6.1 and 6.2 that

H c = 12n

n

l=0

(1)l (n l) p 2l1 + + lp = l

p

i =1

n ili

[(n l) (n i li )]n i l i (n l)2 p

j =1(n i li )2 .

(5.3)

... \binom{n_i}{l _i}\\and

H c = 12

n 1

l=0

(1)l (n l) p 2l1 + + lp = l

p

i=1

n ili

[(n l) (n i li )]n i l i 1 l pn p

[(n l) (n p l p)].(5.4)

The enumeration of H c in a K n 1 n p graph can also be carried out by Theorem 8.2or 8.3 together with the algebraic method of (3.2). Some elegant representationsmay be obtained. For example, H c in a K n 1 n 2 n 3 graph may be written

H c = n1!n2!n3!n1 + n2 + n3 i

n1i

n2n3 n1 + i

n3n3 n2 + i

+n1 1

i n2 1

n3 n1 + i n3 1

n3 n2 + i.

(5.5)

6. Secret Key Exchanges

Modern cryptography is fundamentally concerned with the problem of secureprivate communication. A Secret Key Exchange is a protocol where Alice andBob, having no secret information in common to start, are able to agree on acommon secret key, conversing over a public channel. The notion of a Secret KeyExchange protocol was rst introduced in the seminal paper of Diffie and Hellman[? ]. [? ] presented a concrete implementation of a Secret Key Exchange protocol,dependent on a specic assumption (a variant on the discrete log), specially tailoredto yield Secret Key Exchange. Secret Key Exchange is of course trivial if trapdoorpermutations exist. However, there is no known implementation based on a weakergeneral assumption.

The concept of an informationally one-way function was introduced in [ ? ]. Wegive only an informal denition here:

Denition 6.1. A polynomial time computable function f = {f k} is information-ally one-way if there is no probabilistic polynomial time algorithm which (withprobability of the form 1 k e for some e > 0) returns on input y {0, 1}k arandom element of f 1(y).

In the non-uniform setting [ ? ] show that these are not weaker than one-wayfunctions:

Theorem 6.1 ([? ] (non-uniform)) . The existence of informationally one-way func-tions implies the existence of one-way functions.


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We will stick to the convention introduced above of saying non-uniform beforethe 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 andthe non-uniform models.It now follows from Theorem 6.1 that

Theorem 6.2 (non-uniform) . Weak SKE implies the existence of a one-way func-tion.

More recently, the polynomial-time, interior point algorithms for linear program-ming have been extended to the case of convex quadratic programs [ ? , ?], certainlinear complementarity problems [ ? , ?], and the nonlinear complementarity prob-lem [? ]. The connection between these algorithms and the classical Newton methodfor nonlinear equations is well explained in [ ? ].

7. Review

We begin our discussion with the following denition:

Denition 7.1. A function H : n n is said to be B-differentiable at the pointz if (i) H is Lipschitz continuous in a neighborhood of z, and (ii) there exists apositive homogeneous function BH (z) : n n , called the B-derivative of H atz, such that

limv0

H (z + v) H (z) BH (z)vv

= 0.

The function H is B-differentiable in set S if it is B-differentiable at every point inS . The B-derivative BH (z) is said to be strong if

lim(v,v )(0 ,0)

H (z + v) H (z + v ) BH (z)(v v )v

v

= 0 .

Lemma 7.1. There exists a smooth function 0(z) dened for |z| > 12a satisfying the following properties :(i) 0(z) is bounded above and below by positive constants c1 0(z) c2 .(ii) If |z| > 1, then 0(z) = 1 .(iii) For all z in the domain of 0 , 0 ln 0 0.(iv) If 12a < |z| < 1 a , then 0 ln 0 c3 > 0.

Proof. We choose 0(z) to be a radial function depending only on r = |z|. Leth(r ) 0 be a suitable smooth function satisfying h(r ) c3 for 12a < |z| < 1a,and h(r ) = 0 for |z| > 1 a2 . The radial Laplacian 0 ln 0(r ) =

d2

dr 2 +

1r

ddr

ln 0(r )

has smooth coefficients for r > 1 2a. Therefore, we may apply the existence anduniqueness theory for ordinary differential equations. Simply let ln 0(r ) be thesolution of the differential equation

d2

dr 2 +

1r

ddr

ln 0(r ) = h(r )

with initial conditions given by ln 0(1) = 0 and ln 0(1) = 0.Next, let D be a nite collection of pairwise disjoint disks, all of which are

contained in the unit disk centered at the origin in C . We assume that D =


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{z| |z z | < }. Suppose that D (a) denotes the smaller concentric disk D (a) ={z| |z z | (1 2a) }. We dene a smooth weight function 0(z) for z C D (a) by setting 0(z) = 1 when z /

D and 0(z) = 0((z z )/ )when z is an element of D . It follows from Lemma 7.1 that 0 satises theproperties:

(i) 0(z) is bounded above and below by positive constants c1 0(z) c2 .(ii) 0 ln 0 0 for all z C D (a), the domain where the function 0is dened.(iii) 0 ln 0 c3 2 when (1 2a) < |z z | < (1 a) .

Let A denote the annulus A = {(1 2a) < |z z | < (1 a) }, and setA = A . The properties (2) and (3) of 0 may be summarized as 0 ln 0 c3 2 A , where A is the characteristic function of A. Suppose that is a nonnegative real constant. We apply Proposition 4.5 with

(z) = 0(z)e | z |2

. If u

C

0 (R2

D (a)), assume that

D is a bounded

domain containing the support of u and A D R D ( ). A calculationgives

D |u |20(z)e | z | 2 c4 D |u|20e | z | 2 + c5 2 A |u|20e | z | 2 .The boundedness, property (1) of 0 , then yields

D |u |2e | z | 2 c6 D |u|2e | z | 2 + c7 2 A |u|2e | z | 2 .Let B(X ) be the set of blocks of X and let b(X ) = |B (X )|. If QX then is constant on the blocks of X .

P X = {M | = X }, QX = {M | X }. (7.1)If X then = Y for some Y X so that

QX =Y X

P Y .

Thus by M obius inversion

|P Y | =X Y

(Y, X )|QX |.

Thus there is a bijection from QX to W B (X ) . In particular |QX | = wb(X ) .Next note that b(X ) = dim X . We see this by choosing a basis for X consistingof vectors vk dened by

vki =1 if ik ,

0 otherwise.\[v^{k}_{i}=\begin{cases} 1 & \text{if $i \in \Lambda_{k}$},\\0 &\text{otherwise.} \end{cases}\]

Lemma 7.2. Let A be an arrangement. Then (A, t ) =

BA

(1) |B| tdim T (B) .


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In order to compute R recall the denition of S (X, Y ) from Lemma 4.1. SinceH B , AH B . Thus if T (B ) = Y then B S (H, Y ). Let L = L(A). Then

R =H BA

(1) |B| tdim T (B)

=Y L BS (H,Y )

(1) |B| tdim Y

= Y L BS (H,Y )

(1) |BA H | tdim Y

= Y L

(H, Y )tdim Y

= (A, t ).

(7.2)

Corollary 7.3. Let (A, A, A) be a triple of arrangements. Then (A, t ) = (A, t ) + t (A, t ).

Denition 7.2. Let (A, A, A) be a triple with respect to the hyperplane H A.Call H a separator if T (A)L(A).Corollary 7.4. Let (A, A, A) be a triple with respect to H A.

(i) If H is a separator then

(A) = (A)and hence

|(A)| = |(A)|.(ii) If H is not a separator then

(

A) = (

A)

(

A)

and

|(A)| = |(A)|+ |(A)|.Proof. It follows from Theorem 6.1 that (A, t ) has leading term

(1)r (A ) (A)t r (A ) .The conclusion follows by comparing coefficients of the leading terms on both sidesof the equation in Corollary 7.3. If H is a separator then r(A) < r (A) and thereis no contribution from (A, t ).

The Poincare polynomial of an arrangement will appear repeatedly in thesenotes. It will be shown to equal the Poincare polynomial of the graded algebraswhich we are going to associate with A. It is also the Poincare polynomial of thecomplement M (A) for a complex arrangement. Here we prove that the Poincarepolynomial is the chamber counting function for a real arrangement. The comple-ment M (A) is a disjoint union of chambers

M (A) =C Cham( A )

C.

The number of chambers is determined by the Poincare polynomial as follows.

Theorem 7.5. Let AR be a real arrangement. Then |Cham( AR )| = (AR , 1).


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Figure 1. Q(A1) = xyz (x z)(x + z)(y z)(y + z)

Figure 2. Q(A2) = xyz(x + y + z)(x + y z)(x y + z)(x y z)

Proof. We check the properties required in Corollary 7.4: (i) follows from (l , t ) =1, and (ii) is a consequence of Corollary 4.4.

Theorem 7.6. Let be a protocol for a random pair (X, Y ). If one of (x , y)and (x, y ) is a prex of the other and (x, y )S X,Y , then

j (x , y) j =1 = j (x, y )j =1 = j (x, y )

j =1 .

Proof. We show by induction on i that

j (x , y) ij =1 = j (x, y )ij =1 = j (x, y )

ij =1 .

The induction hypothesis holds vacuously for i = 0. Assume it holds for i1, in par-ticular [ j (x , y)]i 1j =1 = [j (x, y )]i 1j =1 . Then one of [j (x , y)]j = i and [j (x, y )]j = i isa prex of the other which implies that one of i (x , y) and i (x, y ) is a prex of theother. If the ith message is transmitted by P X then, by the separate-transmissionsproperty and the induction hypothesis, i (x, y) = i (x, y ), hence one of i (x, y )and i (x , y) is a prex of the other. By the implicit-termination property, nei-ther i (x, y ) nor i (x , y) can be a proper prex of the other, hence they must bethe same and i (x , y) = i (x, y) = i (x, y ). If the ith message is transmittedby P Y then, symmetrically, i (x, y) = i (x , y) by the induction hypothesis and


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the separate-transmissions property, and, then, i (x, y) = i (x, y ) by the implicit-termination property, proving the induction step.

If is a protocol for (X, Y ), and ( x, y), (x , y) are distinct inputs in S X,Y , then,by the correct-decision property, j (x, y ) j =1 = j (x , y) j =1 .

Equation (7.2) dened P Y s ambiguity set S X | Y (y) to be the set of possible X values when Y = y. The last corollary implies that for all y S Y , the multiset 2 of codewords { (x, y) : xS X | Y (y)} is prex free.

8. One-Way Complexity

C 1(X |Y ), the one-way complexity of a random pair ( X, Y ), is the number of bits P X must transmit in the worst case when P Y is not permitted to transmitany feedback messages. Starting with S X,Y , the support set of ( X, Y ), we deneG(X |Y ), the characteristic hypergraph of (X, Y ), and show that

C 1(X |Y ) = log (G(X |Y )) .Let (X, Y ) be a random pair. For each y in S Y , the support set of Y , Equa-

tion (7.2) dened S X | Y (y) to be the set of possible x values when Y = y. Thecharacteristic hypergraph G(X |Y ) of (X, Y ) has S X as its vertex set and the hy-peredge S X | Y (y) for each y S Y .We can now prove a continuity theorem.Theorem 8.1. Let

R n be an open set, let uBV (;R m ), and let

T ux = y R m : y = u (x ) + Du

|Du |(x ), z for some z

R n (8.1)

for every x\S u . Let f : R m R k be a Lipschitz continuous function such that f (0) = 0 , and let v = f (u) : R k . Then v BV (; R k ) and Jv = ( f (u+ ) f (u )) u H\ |S . (8.2)

In addition, for |Du |-almost every x the restriction of the function f to T ux is differentiable at u(x) and Dv = (f |T ux )(u)

Du

|Du | |Du |. (8.3)

Before proving the theorem, we state without proof three elementary remarkswhich will be useful in the sequel.Remark 8.1. Let : ]0, + [ ]0, + [ be a continuous function such that (t) 0as t 0. Then lim

h 0+g((h)) = L limh 0+

g(h) = L

for any function g : ]0, + [ R .Remark 8.2. Let g : R n R be a Lipschitz continuous function and assume that

L(z) = limh 0+

g(hz ) g(0)h

exists for every z Q n and that L is a linear function of z. Then g is differentiableat 0.2A multiset allows multiplicity of elements. Hence, {0, 01, 01} is prex free as a set, but not

as a multiset.


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Remark 8.3. Let A: R n R m be a linear function, and let f : R m R be afunction. Then the restriction of f to the range of A is differentiable at 0 if andonly if f (A) : R

n

R is differentiable at 0 and(f |Im( A ) )(0) A = (f (A))(0) .

Proof. We begin by showing that v BV (; R k ) and|Dv |(B ) K |Du |(B ) B B (), (8.4)

where K > 0 is the Lipschitz constant of f . By (4.11) and by the approximationresult quoted in 4, it is possible to nd a sequence ( uh ) C 1(; R m ) convergingto u in L1(; R m ) and such that

limh + |uh |dx = |Du |().

The functions vh = f (uh ) are locally Lipschitz continuous in , and the denitionof differential implies that |vh | K |uh | almost everywhere in . The lowersemicontinuity of the total variation and (4.11) yield

|Dv |() liminf h + |Dv h |() = lim inf h + |vh |dxK liminf h + |uh |dx = K |Du |(). (8.5)

Since f (0) = 0, we have also

|v|dx K |u|dx;therefore u

BV (; R k ). Repeating the same argument for every open set A

,we get (8.4) for every B B (), because |Dv |, |Du | are Radon measures. To proveLemma 7.1, rst we observe that

S v S u , v(x) = f (u(x)) x\S u . (8.6)In fact, for every > 0 we have

{y B (x) : |v(y) f (u(x)) | > } {y B (x) : |u(y) u(x)| > /K },hence

lim0+

|{y B (x) : |v(y) f (u(x)) | > }|n

= 0

whenever x \S u . By a similar argument, if x S u is a point such that thereexists a triplet ( u+ , u , u ) satisfying (4.12), (4.13), then(v+ (x) v (x)) v = ( f (u+ (x)) f (u (x))) u if xS v

and f (u (x)) = f (u+ (x)) if xS u \S v . Hence, by (1.8) we getJv (B ) = B S v (v+ v ) v dH\ = B S v (f (u+ ) f (u )) u dH\

= B S u (f (u+ ) f (u )) u dH\and Lemma 7.1 is proved.


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To prove (8.6), it is not restrictive to assume that k = 1. Moreover, to simplifyour notation, from now on we shall assume that = R n . The proof of (8.6)

is divided into two steps. In the rst step we prove the statement in the one-dimensional case ( n = 1), using Theorem 6.2. In the second step we achieve thegeneral result using Theorem 8.1.

Step 1. Assume that n = 1. Since S u is at most countable, (4.5) yields that

|Dv |(S u \S v ) = 0, so that (5.1) and (5.3) imply that Dv = Dv + Jv is the Radon-Nikodym decomposition of Dv in absolutely continuous and singular part withrespect to |Du |. By Theorem 6.2, we haveDv

|Du |(t) = lim

s t +Dv ([t, s [)

|Du |([t, s [),

Du

|Du |(t) = lim

s t +Du ([t, s [)

|Du |([t, s [)

|Du |-almost everywhere in R . It is well known (see, for instance, [ ? , 2.5.16]) thatevery one-dimensional function of bounded variation w has a unique left contin-uous representative, i.e., a function w such that w = w almost everywhere andlims t w(s) = w(t) for every t

R . These conditions imply

u(t) = Du (], t [), v(t) = Dv (], t [) tR (8.7)and

v(t) = f (u(t)) tR . (8.8)Let t R be such that |Du |([t, s [) > 0 for every s > t and assume that the limitsin (5.4) exist. By (5.5) and (7.1) we get

v(s) v(t)|Du |([t, s [)

= f (u(s)) f (u(t))

|Du |([t, s [)

=f (u(s)) f (u(t) +

Du

|Du |(t)|Du |([t, s [))

|Du |([t, s [)

+f (u(t) +

Du

|Du |(t)|Du |([t, s [)) f (u(t))

|Du |([t, s [)for every s > t . Using the Lipschitz condition on f we nd

v(s) v(t)|Du |([t, s [)

f (u(t) +Du

|Du |(t)|Du |([t, s [)) f (u(t))

|Du |([t, s [)

K u(s) u(t)|Du |([t, s [)

Du

|Du |(t) .


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By (8.4), the function s |Du |([t, s [) is continuous and converges to 0 as s t.Therefore Remark 8.1 and the previous inequality imply

Dv

|Du |(t) = lim

h 0+

f (u(t) + h Du

|Du |(t)) f (u(t))

h |Du |-a.e. in R .By (5.4), u(x) = u(x) for every x

R \S u ; moreover, applying the same argumentto the functions u (t) = u(t), v (t) = f (u (t)) = v(t), we get

Dv

|Du |(t) = lim

h 0

f (u(t) + hDu

|Du |(t)) f (u(t))

h |Du |-a.e. in Rand our statement is proved.

Step 2. Let us consider now the general case n > 1. Let

R n be such that

| | = 1, and let = {y R n : y , = 0}. In the following, we shall identifyR n with R , and we shall denote by y the variable ranging in and byt the variable ranging in R . By the just proven one-dimensional result, and byTheorem 4.3, we get

limh 0

f (u(y + t ) + hDu y

|Du y |(t)) f (u(y + t ))

h =

Dv y

|Du y |(t) |Du y |-a.e. in R

for H\ -almost every y . We claim thatDu,

|Du,

|

(y + t ) =Du y

|Du y

|

(t) |Du y |-a.e. in R (8.9)for H\ -almost every y . In fact, by (4.14) and (4.16) we get

Du y|Du y | |Du y |dH\ () = D H\ ()= Du, = Du,

| Du, | |Du, |= Du, | Du, |(y + ) |Du y |dH\ ()

and (7.1) follows from (4.11). By the same argument it is possible to prove that

Dv,

| Du, |(y + t ) =

Dv y

|Du y |(t) |Du y |-a.e. in R (8.10)

for

H\ -almost every y

. By (7.1) and (7.2) we get

limh 0

f (u(y + t ) + h Du, | Du, |

(y + t )) f (u(y + t ))h

= Dv, | Du, |

(y + t )

for H\ -almost every y , and using again (4.12), (4.13) we get

limh 0

f (u(x) + h Du, | Du, |

(x)) f (u(x))h

= Dv, | Du, |

(x)


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| Du, |-a.e. in R n .Since the function

|Du,

|/

|Du

|is strictly positive

|Du,

|-almost everywhere,

we obtain also

limh 0

f (u(x) + h | Du, ||Du |

(x) Du, | Du, |

(x)) f (u(x))h

= | Du, ||Du |

(x) Dv, | Du, |

(x)

| Du, |-almost everywhere in R n .Finally, since| Du, |

|Du |Du,

| Du, |= Du,

|Du |= Du|Du |, |Du |-a.e. in R n

| Du, ||Du

|

Dv,

|Du,

|= Dv,

|Du

|=

Dv

|Du

|, |Du |-a.e. in R n

and since both sides of (8.8) are zero |Du |-almost everywhere on | Du, |-negligiblesets, we conclude that

limh 0

f u(x) + h Du|Du |(x), f (u(x))h

= Dv|Du |(x), ,|Du |-a.e. in R n . Since is arbitrary, by Remarks 8.2 and 8.3 the restriction of f to the affine space T ux is differentiable at u(x) for |Du |-almost every x R n and(8.1) holds.

It follows from (4.11), (4.12), and (4.13) that

D (t1 , . . . , t n ) =I n

(1) | I | 1|I | iI t i jI (D j + j t j )det A ( ) (I |I ). (8.11)Let t i = x i , i = 1, . . . , n . Lemma 1 leads to

D (x1 , . . . , xn ) =in

x iI n

(1) | I | 1|I |per A ( ) (I |I )det A ( ) (I |I ). (8.12)By (3.3), (4.11), and (8.12), we have the following result:

Theorem 8.2.

H c = 12n

n

l=1

l(1) l 1A( )l , (8.13)

where

A( )l =I ln

per A ( ) (I l |I l )det A (( ) (I l |I l ), |I l | = l. (8.14)

It is worth noting that A ( )l of (8.14) is similar to the coefficients bl of the char-acteristic polynomial of (4.8). It is well known in graph theory that the coefficientsbl can be expressed as a sum over certain subgraphs. It is interesting to see whetherAl , = 0, structural properties of a graph.

We may call (8.13) a parametric representation of H c . In computation, theparameter i plays very important roles. The choice of the parameter usually


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depends on the properties of the given graph. For a complete graph K n , let i = 1,i = 1 , . . . , n . It follows from (8.14) that

A(1)l =n!, if l = 10, otherwise .

(8.15)

By (8.13)

H c = 12

(n 1)!. (8.16)For a complete bipartite graph K n 1 n 2 , let i = 0, i = 1, . . . , n . By (8.14),

Al = n1!n2! n 1 n 2 , if l = 20, otherwise . (8.17)Theorem 8.2 leads to

H c = 1

n1 + n2 n1!n2! n 1 n 2 . (8.18)

Now, we consider an asymmetrical approach. Theorem 4.3 leads to

det K (t = 1 , t 1 , . . . , t n ; l|l)=

I n { l }

(1) | I |iI

t ijI

(D j + j t j )det A ( ) (I{l}|I{l}). (8.19)

By (3.3) and (4.14) we have the following asymmetrical result:

Theorem 8.3.

H c = 12

I n { l}

(1) | I | per A ( ) (I |I )det A ( ) (I{l}|I{l}) (8.20)which reduces to GouldenJacksons formula when i = 0, i = 1, . . . , n [? ].

9. Various font features of the mn amstex option

9.1. Bold versions of special symbols. In the mn amstex option mn \boldsymbolis used for getting individual bold math symbols and bold Greek letterseverythingin math except for letters of the Latin alphabet, where youd use mn \bold . Forexample,

A_\infty + \pi A_0 \sim\bold{A}_{\boldsymbol{\infty}} \boldsymbol{+}\boldsymbol{\pi} \bold{A}_{\boldsymbol{0}}

looks like this:

A + A 0 A + A

0

9.2. Poor mans bold. If a bold version of a particular symbol doesnt exist inthe available fonts, then mn \boldsymbol cant be used to make that symbol bold.At the present time, this means that mn \boldsymbol cant be used with symbolsfrom the msam and msbm fonts, among others. In some cases, poor mans bold(mn \pmb ) can be used instead of mn \boldsymbol :

xy

yz


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\[\frac{\partial x}{\partial y}\pmb{\Bigg\vert}

\frac{\partial y}{\partial z}\]So-called large operator symbols such as and require an additional com-mand, mn \mathop , to produce proper spacing and limits when mn \pmb is used.For further details see The T E Xbook .

i


20/41


10.3. Dots. Normally you need only type mn \dots for ellipsis dots in a mathformula. The main exception is when the dots fall at the end of the formula;

then you need to specify one of mn \dotsc (series dots, after a comma), mn \dotsb(binary dots, for binary relations or operators), mn \dotsm (multiplication dots), ormn \dotsi (dots after an integral). For example, the input

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\[\int_{A_1}\int_{A_2}\dotsi\].

producesThen we have the series A1 , A2 , . . . , the regional sum A1 + A2 + ,the orthogonal product A1A2 , and the innite integral

A 1 A 2 10.4. Accents in math. Double accents:H C T A `G D D B B V

\[\Hat{\Hat{H}}\quad\Check{\Check{C}}\quad\Tilde{\Tilde{T}}\quad\Acute{\Acute{A}}\quad\Grave{\Grave{G}}\quad\Dot{\Dot{D}}\quad\Ddot{\Ddot{D}}\quad\Breve{\Breve{B}}\quad\Bar{\Bar{B}}\quad\Vec{\Vec{V}}\]

This double accent operation is complicated and tends to slow down the processing

of a LATEX le. If your document contains many double accents, you can usemn \accentedsymbol in the preamble of your document to help speed things up.

mn \accentedsymbol is used like mn\newcommand :\accentedsymbol{\Ahathat}{\Hat{\Hat A}}

10.5. Superscripted accents. For superscripting accent characters, use mn \sphat ,mn \spcheck , mn\sptilde , mn\spdot , mn\spddot , mn\spdddot , mn\spbreve .

(AmBD ) (AmBD ) (AmBD ) (10.3)

(AmBD ). (AmBD ).. (AmBD )... (AmBD ) (10.4)

\begin{gather}(AmBD)\sphat\quad(AmBD)\spcheck\quad(AmBD)\sptilde\\

(AmBD)\spdot\quad(AmBD)\spddot\quad(AmBD)\spdddot\quad(AmBD)\spbreve\end{gather}

10.6. Dot accents. mn\dddot and mn \ddddot are available to produce triple andquadruple dot accents in addition to the mn \dot and mn \ddot accents alreadyavailable in LATEX: ...

Q ....

R\[\dddot{Q}\qquad\ddddot{R}\]


21/41


10.7. Roots. In the mn amstex option mn \leftroot and mn \uproot allow you toadjust the position of the root index of a radical:

\sqrt[\leftroot{-2}\uproot{2}\beta]{k}gives good positioning of the :

k10.8. Boxed formulas. The command mn \boxed puts a box around its argument,like mn\fbox except that the contents are in math mode:

\boxed{W_t-F\subseteq V(P_i)\subseteq W_t}

W t F V (P i )W t .10.9. Extensible arrows. @>>> and @>{\partial_0\alpha(b)}>E^{\partial_0b}\]

For users whose keyboards dont have < and > keys, @))) and @((( are available assynonyms.

10.10. mn \overset , mn \underset , and mn \sideset . Examples:

X X

aX b

\[\overset{*}{X}\qquad\underset{*}{X}\qquad\overset{a}{\underset{b}{X}}\]

The command mn \sideset is for a rather special purpose: putting symbols atthe subscript and superscript corners of a large operator symbol such as or ,without affecting the placement of limits. Examples:

k 0 i m

E i x

\[\sideset{_*^*}{_*^*}\prod_k\qquad

\sideset{}{}\sum_{0\le i\le m} E_i\beta x\]

10.11. The mn \text command. The main use of the command mn \text is forwords or phrases in a display:

y = y if and only if y k = k y (k )

\[\bold{y}=\bold{y}\quad\text{if and only if}\quady_k=\delta_k y_{\tau(k)}\]


22/41


10.12. Operator names. The more common math functions such as log, sin, andlim have predened control sequences: \log , \sin , \lim . The mn amstex option

provides mn \operatorname and mn \operatornamewithlimits for producing newfunction names that will have the same typographical treatment. Examples:

f = ess supxR n |f (x)|

\[\|f\|_\infty=\operatornamewithlimits{ess\,sup}_{x\in R^n}|f(x)|\]

meas1{uR1+ f (u) > }= meas n {xRn |f (x)| } > 0.\[\operatorname{meas}_1\{u\in R_+^1\:f^*(u)>\alpha\}=\operatorname{meas}_n\{x\in R^n\:|f(x)|\geq\alpha\}\quad \forall\alpha>0.\]

If you use a particular operator name often, you can save yourself some typing and

make your LATEX le more readable by dening an abbreviation in the preamblearea of your document, using mn \newcommand :\newcommand{\esssup}{\operatornamewithlimits{ess\,sup}}\newcommand{\meas}{\operatorname{meas}}

Some special operatornames are predened in the mn amstex option: mn \varlimsup ,mn \varliminf , mn\varinjlim , and mn \varprojlim . Heres what they look likein use:

lim n Q( \ , \ # ) (10.5)lim n |an +1 | / |an | = 0 (10.6)lim(m

i )0 (10.7)

lim

pS (A ) A p

0 (10.8)

\begin{align}&\varlimsup_{n\rightarrow\infty}

\cal{Q}(u_n,u_n-u^{\#})\le0\\&\varliminf_{n\rightarrow\infty}

\left|a_{n+1}\right|/\left|a_n\right|=0\\&\varinjlim (m_i^\lambda\cdot)^*\le0\\&\varprojlim_{p\in S(A)}A_p\le0\end{align}

10.13. mn \mod and its relatives. The commands mn \mod and mn \pod are vari-ants of mn \pmod preferred by some authors; mn \mod omits the parentheses, whereasmn \pod omits the mod and retains the parentheses. Examples:

x y + 1 (mod m2) (10.9)x y + 1 mod m2 (10.10)x y + 1 (m2) (10.11)

\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}


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10.14. Fractions and related constructions. In the mn amstex option, mn \frachas a square bracket option that can be used to specify the thickness of the fraction

line. For example:H (z + v) H (z) BH (z)v

v\[\frac[1.5pt]{H(z+v)-H(z)-BH(z)v}{\|v\|}\]

There is also mn \fracwithdelims , for stacked constructions with delimiters oneither side.

H (z + v) H (z) BH (z)vv

\[\fracwithdelims[]{H(z+v)-H(z)-BH(z)v}{\|v\|}\]

Because it is used fairly often, the construction \fracwithdelims()[0pt] has a

short form, mn \binom . Example:

C

I = 2 k k1

2k 1 +k2

2k 2

+ + ( 1)lkl

2k l + + ( 1)k= (2 1)k = 1

(10.12)

\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}

There are also abbreviations\dfrac \dbinom\tfrac \tbinom

for the commonly needed constructions{\displaystyle\frac ... } {\displaystyle\binom ... }{\textstyle\frac ... } {\textstyle\binom ... }

10.15. Continued fractions. The continued fraction

1

2 + 1 2 + 1

2 + 1 2 + 1 2 +

(10.13)


24/41


can be obtained by typing

\cfrac{1}{\sqrt{2}+\cfrac{1}{\sqrt{2}+\cfrac{1}{\sqrt{2}+

\cfrac{1}{\sqrt{2}+\cfrac{1}{\sqrt{2}+\dotsb

}}}}}

Left or right placement of any of the numerators is accomplished by usingmn \lcfrac or mn\rcfrac instead of mn \cfrac .

10.16. Smash. In mnamstex there are optional arguments t and b for the plain

TEX command mn \smash , because sometimes it is advantageous to be able tosmash only the top or only the bottom of something while retaining the naturaldepth or height. In the formula X j = (1 / j )X j mn\smash[b] has been used tolimit the size of the radical symbol.

$X_j=(1/\sqrt{\smash[b]{\lambda_j}})X_j$

Without the use of mn \smash[b] the formula would have appeared thus: X j =(1/ j )X j , with the radical extending to encompass the depth of the subscript j .10.17. The cases environment. Cases constructions like the following can beproduced using the mn cases environment.

P r j =0 if r j is odd,r ! (1)(r j ) / 2 if r j is even.

(10.14)

\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}

Notice the use of mn \text and the embedded math.

10.18. Matrix. Here are samples of the matrix environments, mn \matrix , mn \pmatrix ,mn \bmatrix , mn\vmatrix and mn \Vmatrix :

(10.15)


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\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{vmatrix}\vartheta& \varrho\\\varphi& \varpi\end{vmatrix}\quad\begin{Vmatrix}\vartheta& \varrho\\\varphi& \varpi\end{Vmatrix}

To produce a small matrix suitable for use in text, use the mn smallmatrixenvironment.

\begin{math}\bigl( \begin{smallmatrix}

a&b\\ c&d\end{smallmatrix} \bigr)

\end{math}

To show the effect of the matrix on the surrounding lines of a paragraph, we put ithere: a bc d and follow it with enough text to ensure that there will be at least onefull line below the matrix.

mn \hdotsfor{ number } produces a row of dots in a matrix spanning the given

number of columns:

W () =

(1 , 1) 0 . . . 0

kn 2(2 , 1)

(2 , 2) . . . 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .kn 1

(n , 1)kn 2

(n , 2) . . .

kn n 1(n , n 1)

(n , n )

\[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}\]

The spacing of the dots can be varied through use of a square-bracket option, forexample, \hdotsfor[1.5]{3} . The number in square brackets will be used as amultiplier; the normal value is 1.


26/41


10.19. The mn Sb and mn Sp environments. The mn Sb and mn Sp environmentscan be used to typeset several lines as a subscript or superscript: for example

\begin{equation}\sum\begin{Sb}

0\le i\le m\\ 0


27/41


Note: Starting on the following page, vertical rules are added atthe margins so that the positioning of various display elements

with respect to the margins can be seen more clearly.


28/41


Appendix A. Examples of multiple-line equation structures

The lines indicating the margins are to make the marginal spacing stand outmore clearly in some of the display examples in this appendix.

A.1. Split. The split environment is not an independent environment but shouldbe used inside something else such as equation or align .

If there is not enough room for it, the equation number for a split will beshifted to the previous line, when equation numbers are on the left; the numbershifts down to the next line when numbers are on the right.

f h, (x, y ) = E x ,y t 0 L x ,y ( u )(x ) du= h Lx,z (x)x (dz)

+ h1t E y

t

0 Lx ,y x (s )(x ) ds t L

x ,z(x )x (dz )

+ 1t

E y t 0 L x ,y x ( s )(x ) ds E x ,y t 0 L x ,y ( s )(x ) ds= h Lx(x) + h (x, y ),

(A.1)

Some text after to test the below-display spacing.\begin{equation}\begin{split}f_{h,\varepsilon}(x,y)&=\varepsilon\bold E_{x,y}\int_0^{t_\varepsilon}L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\&= h\int L_{x,z}\varphi(x)\rho_x(dz)\\&\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\bold E_{y}

\int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds-t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\

&\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon}\biggl(\bold E_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)}

\varphi(x)\,ds -\bold E_{x,y}\int_0^{t_\varepsilon}L_{x,y_\varepsilon(\varepsilon s)}\varphi(x)\,ds\biggr)\biggr]\\

&=h\widehat{L}_x\varphi(x)+h\theta_\varepsilon(x,y),\end{split}\end{equation}


29/41


Unnumbered version:

f h, (x, y ) = E

x ,y

t

0L

x ,y ( u )(x

)du

= h Lx,z (x)x (dz)+ h

1t

E y t 0 L x ,y x (s )(x ) ds t L x ,z(x )x (dz )+

1t

E y t 0 L x ,y x ( s )(x ) ds E x ,y t 0 L x ,y ( s )(x ) ds= h Lx(x) + h (x, y ),Some text after to test the below-display spacing.

\begin{equation*}

\begin{split}f_{h,\varepsilon}(x,y)&=\varepsilon\bold E_{x,y}\int_0^{t_\varepsilon}L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\&= h\int L_{x,z}\varphi(x)\rho_x(dz)\\&\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\bold E_{y}

\int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds-t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\

&\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon}\biggl(\bold E_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)}

\varphi(x)\,ds -\bold E_{x,y}\int_0^{t_\varepsilon}L_{x,y_\varepsilon(\varepsilon s)}\varphi(x)\,ds\biggr)\biggr]\\

&=h\widehat{L}_x\varphi(x)+h\theta_\varepsilon(x,y),\end{split}\end{equation*}


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If the option ctagsplt is included in the documentstyle options list, the equationnumbers for split environments will be centered vertically on the height of the

split :

|I 2| = T 0 (t) u(a, t ) a ( t ) dk(, t ) a c( )u t (, t ) d dtC 6 f S 1 ,0a, W 2(, l ) |u| W A2 (;r , T ) .

(A.2)

Some text after to test the below-display spacing.


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Use of split within align :

|I 1| = gRud C 3 xa g(x, t ) d 2 d 1/ 2 u2x + 1k xa cut d 2 c 1/ 2

C 4 f S 1,0

a, W 2(, l ) |u|

W A2 (;r , T ) .

(A.3)

|I 2| = T 0 (t) u(a, t ) a ( t ) dk(, t ) a c( )u t (, t ) d dt

C 6

f

S 1,0a,

W 2(,

l)

|u

|

W A

2 (;

r, T ) .

(A.4)

Some text after to test the below-display spacing.\begin{align}\begin{split}|I_1|&=\left|\int_\Omega gRu\,d\Omega\right|\\

&\le C_3\left[\int_\Omega\left(\int_{a}^xg(x\i,t)\,d\xi\right)^2d\Omega\right]^{1/2}\\

&\quad\times \left[\int_\Omega\left\{u^2_x+\frac{1}{k}\left(\int_{a}^x cu_t\,d\xi\right)^2\right\}c\Omega\right]^{1/2}\\

&\le C_4\left|\left|f\left|\widetilde{S}^{-1,0}_{a,-}W_2(\Omega,\Gamma_l)\right|\right|\left||u|\overset{\circ}\to W_2^{\widetilde{A}}

(\Omega;\Gamma_r,T)\right|\right|.\end{split}\label{eq:A}\\\begin{split}|I_2|&=\left|\int_{0}^T \psi(t)\left\{u(a,t)

-\int_{\gamma(t)}^a\frac{d\theta}{k(\theta,t)}\int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right|\\

&\le C_6\left|\left|f\int_\Omega\left|\widetilde{S}^{-1,0}_{a,-}W_2(\Omega,\Gamma_l)\right|\right|\left||u|\overset{\circ}\to W_2^{\widetilde{A}}(\Omega;\Gamma_r,T)\right|\right|.

\end{split}\end{align}


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Unnumbered align , with a number on the second split :

|I 1| = gRud C 3 xa g(x, t ) d 2 d1/ 2

u2x + 1k xa cut d 2 c1/ 2

C 4 f S 1,0a, W 2(, l ) |u|

W A2 (;r , T ) .

|I 2| = T 0 (t) u(a, t ) a ( t ) dk(, t ) a c( )u t (, t ) d dtC 6 f S

1 ,0a, W 2(, l ) |u| W A2 (;r , T ) . (A.4 )

Some text after to test the below-display spacing.\begin{align*}\begin{split}|I_1|&=\left|\int_\Omega gRu\,d\Omega\right|\\

&\le C_3\left[\int_\Omega\left(\int_{a}^xg(x\i,t)\,d\xi\right)^2d\Omega\right]^{1/2}\\

&\phantom{=}\times \left[\int_\Omega\left\{u^2_x+\frac{1}{k}\left(\int_{a}^x cu_t\,d\xi\right)^2\right\}c\Omega\right]^{1/2}\\

&\le C_4\left|\left|f\left|\widetilde{S}^{-1,0}_{a,-}W_2(\Omega,\Gamma_l)\right|\right|

\left||u|\overset{\circ}\to W_2^{\widetilde{A}}(\Omega;\Gamma_r,T)\right|\right|.

\end{split}\\\begin{split}|I_2|&=\left|\int_{0}^T \psi(t)\left\{u(a,t)

-\int_{\gamma(t)}^a\frac{d\theta}{k(\theta,t)}\int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right|\\

&\le C_6\left|\left|f\int_\Omega\left|\widetilde{S}^{-1,0}_{a,-}W_2(\Omega,\Gamma_l)\right|\right|\left||u|\overset{\circ}\to W_2^{\widetilde{A}}(\Omega;\Gamma_r,T)\right|\right|.

\end{split}\tag{\theequation$$}\end{align*}


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A.2. Multline. Numbered version:

b

a b

a [f (x)2

g(y)2

+ f (y)2

g(x)2

]2f (x)g(x)f (y)g(y) dx dy= ba g(y)2 ba f 2 + f (y)2 ba g2 2f (y)g(y) ba fg dy (A.5)

To test the use of \label and \ref , we refer to the number of this equation here:(A.5).

\begin{multline}\label{eq:E}\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x) 2]

-2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\=\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2

\int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy\end{multline}

Unnumbered version: ba ba [f (x)2g(y)2 + f (y)2g(x)2]2f (x)g(x)f (y)g(y) dx dy= ba g(y)2 ba f 2 + f (y)2 ba g2 2f (y)g(y) ba fg dy

Some text after to test the below-display spacing.\begin{multline*}\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x) 2]


\int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy

\end{multline*}


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And now an unnumbered version numbered with a literal tag:

b

a b

a[f (x)2g(y)2 + f (y)2g(x)2]2f (x)g(x)f (y)g(y) dx dy

= ba g(y)2 ba f 2 + f (y)2 ba g2 2f (y)g(y) ba fg dy [a]Some text after to test the below-display spacing.

\begin{multline*}\tag*{[a]}\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x) 2]


\int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy\end{multline*}

The same display with \multlinegap set to zero. Notice that the space on theleft in the rst line does not change, because of the equation number, while thesecond line is pushed over to the right margin.

ba ba [f (x)2g(y)2 + f (y)2g(x)2]2f (x)g(x)f (y)g(y) dx dy= ba g(y)2 ba f 2 + f (y)2 ba g2 2f (y)g(y) ba fg dy [a]

Some text after to test the below-display spacing.{\setlength{\multlinegap}{0pt}\begin{multline*}\tag*{[a]}\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x) 2]


\int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy\end{multline*}}

A.3. Gather. Numbered version with \notag on the second line:

D(a, r ) {z C : |z a | < r }, (A.6)seg(a, r ) {z C : z = a , |z a | < r },

c(e,,r ) {(x, y )C : |x e | < y tan , 0 < y < r }, (A.7)C (E, , r )

eE c(e,,r ). (A.8)

\begin{gather}D(a,r)\equiv\{z\in\bold C\colon |z-a|


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Unnumbered version.D(a, r )

{z

C :

|z

a

|< r

},

seg(a, r ) {z C : z = a , |z a | < r },c(e,,r ) {(x, y )C : |x e | < y tan , 0 < y < r },

C (E, , r ) eE

c(e,,r ).

Some text after to test the below-display spacing.\begin{gather*}D(a,r)\equiv\{z\in\bold C\colon |z-a|


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A.4. Align. Numbered version: x (t) = (cos tu + sin tx,v ), (A.9) y (t) = ( u, cos tv + sin ty), (A.10)

z (t) = cos tu +

sin tv,

sin tu + cos tv . (A.11)

Some text after to test the below-display spacing.\begin{align}\gamma_x(t)&=(\cos tu+\sin tx,v),\\\gamma_y(t)&=(u,\cos tv+\sin ty),\\\gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv,

-\frac\beta\alpha\sin tu+\cos tv\right).\end{align}

Unnumbered version:

x (t) = (cos tu + sin tx,v ), y (t) = ( u, cos tv + sin ty),

z (t) = cos tu +

sin tv,

sin tu + cos tv .

Some text after to test the below-display spacing.\begin{align*}\gamma_x(t)&=(\cos tu+\sin tx,v),\\\gamma_y(t)&=(u,\cos tv+\sin ty),\\\gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv,

-\frac\beta\alpha\sin tu+\cos tv\right).\end{align*}


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A.5. Align and split within gather. When using the align environment withinthe gather environment, one or the other, or both, should be unnumbered (using

the * form), since having numbering for both the outer and inner environmentwould cause a conict.Automically numbered gather with split and align* :

(x, z ) = z 10 x m + n 2

mn xm zn

= z Mr 1x m + n 2

Mr (m + n ) xm zn(A.12)

0 = ( 0)2 ,

1 = 0 1 ,

2 = ( 1)2 ,

Here the split environment gets a number from the outer gather environment;numbers for individual lines of the align* are suppressed because of the star.

\begin{gather}\begin{split} \varphi(x,z)&=z-\gamma_{10}x-\sum_{m+n\ge2}\gamma_{mn}x^mz^n\\&=z-Mr^{-1}x-\sum_{m+n\ge2}Mr^{-(m+n)}x^mz^n\end{split}\\[6pt]\begin{align*}\zeta^0 &=(\xi^0)^2,\\\zeta^1 &=\xi^0\xi^1,\\\zeta^2 &=(\xi^1)^2,\end{align*}\end{gather}

The *-ed form of gather with the non- *-ed form of align .

(x, z ) = z 10 x m + n 2

mn xm zn

= z Mr 1

x m + n 2 Mr (m + n )

xm

zn

0 = ( 0)2 , (A.13)

1 = 0 1 , (A.14)

2 = ( 1)2 , (A.15)

Some text after to test the below-display spacing.


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\begin{gather*}\begin{split} \varphi(x,z)

&=z-\gamma_{10}x-\sum_{m+n\ge 2}\gamma_{mn}x^mz^n\\&=z-Mr^{-1}x-\sum_{m+n\ge 2}Mr^{-(m+n)}x^mz^n\end{split}\\[6pt]\begin{align} \zeta^0&=(\xi^0)^2,\\\zeta^1 &=\xi^0\xi^1,\\\zeta^2 &=(\xi^1)^2,\end{align}\end{gather*}


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A.6. Alignat. Numbered version:

V i = vi

q i vj , X i = xi

q i x j , U i = u i , for i = j ; (A.16)

V j = vj , X j = xj , U j u j +i= j

q i u i . (A.17)

Some text after to test the below-display spacing.\begin{alignat}{3}V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j,

& \qquad U_i & = u_i,\qquad \text{for $i\ne j$;}\label{eq:B}\\

V_j & = v_j, & \qquad X_j & = x_j,& \qquad U_j & u_j + \sum_{i\ne j} q_i u_i.

\end{alignat}Unnumbered version:

V i = vi q i vj , X i = xi q i x j , U i = u i , for i = j ;V j = vj , X j = xj , U j u j +

i= j

q i u i .

Some text after to test the below-display spacing.\begin{alignat*}3V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j,

& \qquad U_i & = u_i,\qquad \text{for $i\ne j$;} \\

V_j & = v_j, & \qquad X_j & = x_j,& \qquad U_j & u_j + \sum_{i\ne j} q_i u_i.

\end{alignat*}


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The most common use for mn alignat is for things like

x = y by (A.3) (A.18)x = y by (A.16) (A.19)

x + x = y + y by Axiom 1. (A.20)Some text after to test the below-display spacing.

\begin{alignat}{2}x& =y && \qquad \text {by (\ref{eq:A})}\label{eq:C}\\x& = y && \qquad \text {by (\ref{eq:B})}\label{eq:D}\\x+x & = y+y && \qquad \text {by Axiom 1.}\end{alignat}

The expanded version, mn xalignat :

x = y by (A.18) (A.21)

x = y by (A.19) (A.22)x + x = y + y by Axiom 1. (A.23)

Some text after to test the below-display spacing.\begin{xalignat}{2}x& =y && \text {by (\ref{eq:C})}\\x& = y && \text {by (\ref{eq:D})}\\x+x & = y+y && \text {by Axiom 1.}\end{xalignat}


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(D. Belar and L. Eifrig) Department of Electrical Engineering, University of Min-nesota, Minneapolis, Minnesota 55455

E-mail address , D. Belar: [email protected]

Current address , L. Eifrig: San Diego Supercomputer Center, P.O. Box 85608, San Diego,California 92138

E-mail address , L. Eifrig: [email protected]

(S. N aat anen) MIT Laboratory for Computer Science, 545 Technology Square, Cam-bridge, MA 02139

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