## SAMPLE PAPER FOR THE IJMART CLASS

^{∗}

BY

American Mathematical Society Technical Support

Electronic Products and Services P. O. Box 6248

Providence, RI 02940 USA

e-mail: [email protected]^{∗∗}

AND

The Hebrew University Magnes Press^{†}

AND

Israel Journal of Mathematics Editorial Board^{††}

P.O. Box 39099 Jerusalem 91390

Israel

ABSTRACT

This is a test file forijmartclass based on thetestmath.texfile from the amsmathdistribution.

Contents

1. Introduction . . . 2

∗Version 2.0, 1999/11/15

∗∗Even e-mail addresses can have footnotes!

†This is the copyright owner of the style

††This entry is inserted just to show how to typeset several authors with the same address

Received on MONTH, YEAR

1

2. Enumeration of Hamiltonian paths in a graph . . . 2

3. Main Theorem . . . 3

4. Application . . . 6

5. Secret Key Exchanges . . . 7

6. Review . . . 8

7. One-Way Complexity . . . 14

8. Named Propositions . . . 21

9. Various font features of theamsmathpackage . . . 22

10. Compound symbols and other features . . . 23

Appendix A. Examples of multiple-line equation structures . 33 References . . . 49

1. Introduction

This paper demonstrates the use of ijmartclass. It is based ontestmath.tex
fromAMS-L^{A}TEX distribution. The text is (slightly) reformatted according to
the requirements of theijmartstyle.

2. Enumeration of Hamiltonian paths in a graph

LetA= (a_{ij}) be the adjacency matrix of graphG. The corresponding Kirchhoff
matrixK= (k_{ij}) is obtained fromAby replacing in−Aeach diagonal entry by
the degree of its corresponding vertex; i.e., theith diagonal entry is identified
with the degree of theith vertex. It is well known that

(1) detK(i|i) = the number of spanning trees ofG, i= 1, . . . , n whereK(i|i) is theith principal submatrix ofK.

\det\mathbf{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 (v_{i}v_{j}) to
each spanning tree of G. Denote by C_{i} = S

jC_{i(j)}. It is obvious that the
collection of Hamiltonian cycles is a subset of C_{i}. Note that the cardinality of
C_{i} isk_{ii}detK(i|i). LetXb ={xˆ_{1}, . . . ,xˆ_{n}}.

$\wh X=\{\hat x_1,\dots,\hat x_n\}$

Define multiplication for the elements ofXb by

(2) xˆixˆj = ˆxjxˆi, xˆ^{2}_{i} = 0, i, j= 1, . . . , n.

Let ˆkij =kijxˆj and ˆkij =−P

j6=ikˆij. Then the number of Hamiltonian cycles Hc is given by the relation [7]

(3)

^{n}
Y

j=1

ˆ xj

Hc= 1

2

ˆkijdetK(i|i),b i= 1, . . . , n.

The task here is to express (3) in a form free of any ˆxi,i= 1, . . . , 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 Kn and in a complete bipartite graph Kn1n2 can only be
found from first combinatorial principles [3]. One wonders if there exists a
formula which can be used very efficiently to produceK_{n} andK_{n}_{1}_{n}_{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
[2]. However, the formula of Goulden and Jackson determines neitherK_{n} nor
Kn_{1}n_{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 Kn and Kn_{1}n_{2}. In addition, we
eliminate the permanent from Hc and show that Hc can be represented by
a determinantal function of multivariables, each variable with domain {0,1}.

Furthermore, we show thatHc can be written by number of spanning trees of
subgraphs. Finally, we apply the formulas to a complete multigraphKn_{1}...n_{p}.

The conditions aij =aji, i, j = 1, . . . , n, are not required in this paper. All
formulas can be extended to a digraph simply by multiplyingH_{c} by 2.

3. Main Theorem

Notation: Forp, q∈P andn∈ω we write (q, n)≤(p, n) ifq≤pandAq,n= Ap,n.

\begin{notation} For $p,q\in P$ and $n\in\omega$

...

\end{notation}

LetB= (bij) be an n×nmatrix. Let n={1, . . . , n}. Using the properties of (2), it is readily seen that

Lemma 3.1:

(4) Y

i∈n

X

j∈n

b_{ij}xˆ_{i}

=

Y

i∈n

ˆ
x_{i}

perB where perB is the permanent ofB.

LetYb ={yˆ1, . . . ,yˆn}. Define multiplication for the elements ofYb by
(5) yˆ_{i}yˆ_{j}+ ˆy_{j}yˆ_{i}= 0, i, j= 1, . . . , n.

Then, it follows that Lemma 3.2:

(6) Y

i∈n

X

j∈n

b_{ij}yˆ_{j}

=

Y

i∈n

ˆ
y_{i}

detB.

Note that all basic properties of determinants are direct consequences of Lemma 3.2. Write

(7) X

j∈n

bijyˆj =X

j∈n

b^{(λ)}_{ij} yˆj+ (bii−λi)ˆyiyˆ
where

(8) b^{(λ)}_{ii} =λi, b^{(λ)}_{ij} =bij, i6=j.

Let B^{(λ)} = (b^{(λ)}_{ij} ). By (6) and (7), it is straightforward to show the following
result:

Theorem 3.3:

(9) detB=

n

X

l=0

X

I_{l}⊆n

Y

i∈Il

(bii−λi) detB^{(λ)}(Il|Il),

whereIl={i1, . . . , il}andB^{(λ)}(Il|Il)is the principal submatrix obtained from
B^{(λ)} by deleting itsi_{1}, . . . , i_{l} rows and columns.

Remark 3.1(convention): LetMbe ann×nmatrix. The conventionM(n|n) = 1 has been used in (9) and hereafter.

Before proceeding with our discussion, we pause to note that Theorem 3.3 yields immediately a fundamental formula which can be used to compute the coefficients of a characteristic polynomial [8]:

Corollary 3.4: Writedet(B−xI) =Pn

l=0(−1)^{l}blx^{l}. Then

(10) bl= X

I_{l}⊆n

detB(Il|Il).

Let

(11) K(t, t_{1}, . . . , t_{n}) =

D_{1}t −a12t_{2} . . . −a1nt_{n}

−a_{21}t_{1} D_{2}t . . . −a_{2n}t_{n}
. . . .

−a_{n1}t_{1} −a_{n2}t_{2} . . . D_{n}t

,

\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

(12) D_{i}=X

j∈n

a_{ij}t_{j}, i= 1, . . . , n.

Set

D(t_{1}, . . . , t_{n}) = δ

δt detK(t, t_{1}, . . . , t_{n})|_{t=1}.
Then

(13) D(t1, . . . , tn) =X

i∈n

DidetK(t= 1, t1, . . . , tn;i|i),

whereK(t= 1, t_{1}, . . . , t_{n};i|i) is theith principal submatrix ofK(t= 1, t_{1}, . . . , t_{n}).

Theorem 3.3 leads to (14) detK(t1, t1, . . . , tn) =X

I∈n

(−1)^{|I|}t^{n−|I}^{|}Y

i∈I

ti

Y

j∈I

(Dj+λjtj) detA^{(λt)}(I|I).

Note that (15)

detK(t= 1, t1, . . . , tn) =X

I∈n

(−1)^{|I|}Y

i∈I

ti

Y

j∈I

(Dj+λjtj) detA^{(λ)}(I|I) = 0.

Letti= ˆxi, i= 1, . . . , n. Lemma 3.1 yields (16)

X

i∈n

a_{l}_{i}x_{i}

detK(t= 1, x_{1}, . . . , x_{n};l|l)

=

Y

i∈n

ˆ xi

X

I⊆n−{l}

(−1)^{|I|}perA^{(λ)}(I|I) detA^{(λ)}(I∪ {l}|I∪ {l}).

\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}

By (3), (6), and (7), we have Proposition 3.5:

(17) Hc = 1

2n

n

X

l=0

(−1)^{l}Dl,
where

(18) Dl= X

Il⊆n

D(t1, . . . , tn)2|

ti=n0,ifi∈I_{l}

1,otherwise , i=1,...,n.

4. Application

We consider here the applications of Theorems 5.1 and 5.2 to a complete mul-
tipartite graphKn1...np. It can be shown that the number of spanning trees of
K_{n}_{1}_{...n}_{p} may be written

(19) T =n^{p−2}

p

Y

i=1

(n−ni)^{n}^{i}^{−1}
where

(20) n=n1+· · ·+np.

It follows from Theorems 5.1 and 5.2 that Hc = 1

2n

n

X

l=0

(−1)^{l}(n−l)^{p−2} X

l_{1}+···+lp=l
p

Y

i=1

ni

l_{i}

·[(n−l)−(ni−li)]^{n}^{i}^{−l}^{i}·

(n−l)^{2}−

p

X

j=1

(ni−li)^{2}

. (21)

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

and

Hc= 1 2

n−1

X

l=0

(−1)^{l}(n−l)^{p−2} X

l_{1}+···+l_{p}=l
p

Y

i=1

ni

l_{i}

·[(n−l)−(ni−li)]^{n}^{i}^{−l}^{i}

1− lp

np

[(n−l)−(np−lp)].

(22)

The enumeration of Hc in a Kn1···np graph can also be carried out by The-
orem 7.2 or 7.3 together with the algebraic method of (2). Some elegant rep-
resentations may be obtained. For example, H_{c} in a K_{n}_{1}_{n}_{2}_{n}_{3} graph may be
written

Hc= n1!n2!n3!
n_{1}+n_{2}+n_{3}

X

i

n1

i

n2

n_{3}−n_{1}+i

n3

n_{3}−n_{2}+i

+

n_{1}−1
i

n_{2}−1
n3−n1+i

n_{3}−1
n3−n2+i

. (23)

5. Secret Key Exchanges

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 Se- cret Key Exchange protocol was first introduced in the seminal paper of Diffie and Hellman [1]. [1] presented a concrete implementation of a Secret Key Ex- change 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 [4].

We give only an informal definition here:

Definition 5.1(one way): A polynomial time computable functionf ={fk} 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∈ {0,1}^{k} a random element off^{−1}(y).

In the non-uniform setting [4] show that these are not weaker than one-way functions:

Theorem 5.1 ([4] (non-uniform)): The existence of informationally one-way functions implies the existence of one-way functions.

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 Theorem 5.1 that

Theorem 5.2 (non-uniform): Weak SKE implies the existence of a one-way function.

More recently, the polynomial-time, interior point algorithms for linear pro- gramming have been extended to the case of convex quadratic programs [10, 12], certain linear complementarity problems [6, 9], and the nonlinear complemen- tarity problem [5]. The connection between these algorithms and the classical Newton method for nonlinear equations is well explained in [6].

6. Review

We begin our discussion with the following definition:

Definition 6.1: A function H: <^{n} → <^{n} is said to be B-differentiable at the
pointzif (i)H is Lipschitz continuous in a neighborhood ofz, and (ii) there ex-
ists a positive homogeneous functionBH(z) :<^{n} → <^{n}, called theB-derivative
ofH atz, such that

v→0lim

H(z+v)−H(z)−BH(z)v

kvk = 0.

The functionH isB-differentiable in setSif it is B-differentiable at every point inS. The B-derivativeBH(z) is said to bestrong if

lim

(v,v^{0})→(0,0)

H(z+v)−H(z+v^{0})−BH(z)(v−v^{0})
kv−v^{0}k = 0.

Lemma 6.1: There exists a smooth function ψ0(z) defined for |z| > 1−2a satisfying the following properties:

(i) ψ0(z)is bounded above and below by positive constants c1≤ψ0(z)≤ c2.

(ii) If|z|>1, thenψ0(z) = 1.

(iii) For allz in the domain ofψ0, ∆0lnψ0≥0.

(iv) If1−2a <|z|<1−a, then∆0lnψ0≥c3>0.

Proof. We chooseψ_{0}(z) to be a radial function depending only onr=|z|. Let
h(r)≥0 be a suitable smooth function satisfyingh(r)≥c3 for 1−2a <|z|<

1−a, andh(r) = 0 for|z|>1−^{a}_{2}. The radial Laplacian

∆0lnψ0(r) =
d^{2}

dr^{2} +1
r

d dr

lnψ0(r)

has smooth coefficients forr >1−2a. Therefore, we may apply the existence and uniqueness theory for ordinary differential equations. Simply let lnψ0(r) be the solution of the differential equation

d^{2}
dr^{2} +1

r d dr

lnψ0(r) =h(r)
with initial conditions given by lnψ_{0}(1) = 0 and lnψ^{0}_{0}(1) = 0.

Next, let D_{ν} 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_{ν} = {z | |z−z_{ν}| < δ}. Suppose that D_{ν}(a) denotes the smaller concentric
disk D_{ν}(a) ={z| |z−z_{ν}| ≤ (1−2a)δ}. We define a smooth weight function
Φ0(z) forz∈C−S

νDν(a) by setting Φ0(z) = 1 whenz /∈S

νDν and Φ0(z) = ψ0((z−zν)/δ) whenzis an element ofDν. It follows from Lemma 6.1 that Φ0

satisfies the properties:

(i) Φ_{0}(z) is bounded above and below by positive constants c_{1} ≤Φ_{0}(z)≤
c_{2}.

(ii) ∆0ln Φ0≥0 for allz∈C−S

νDν(a), the domain where the function Φ0 is defined.

(iii) ∆0ln Φ0≥c3δ^{−2} when (1−2a)δ <|z−zν|<(1−a)δ.

Let Aν denote the annulus Aν = {(1−2a)δ < |z−zν| < (1−a)δ}, and set A = S

νAν. The properties (2) and (3) of Φ0 may be summarized as

∆0ln Φ0≥c3δ^{−2}χA, whereχAis the characteristic function of A.

Suppose thatαis a nonnegative real constant. We apply Proposition 3.5 with
Φ(z) = Φ0(z)e^{α|z|}^{2}. Ifu∈ C_{0}^{∞}(R^{2}−S

νDν(a)), assume that D is a bounded
domain containing the support ofuandA⊂ D ⊂R^{2}−S

νDν(a). A calculation gives

Z

D

∂u

2Φ0(z)e^{α|z|}^{2} ≥c4α
Z

D

|u|^{2}Φ0e^{α|z|}^{2}+c5δ^{−2}
Z

A

|u|^{2}Φ0e^{α|z|}^{2}.
The boundedness, property (1) of Φ0, then yields

Z

D

∂u

2e^{α|z|}^{2} ≥c6α
Z

D

|u|^{2}e^{α|z|}^{2}+c7δ^{−2}
Z

A

|u|^{2}e^{α|z|}^{2}.

LetB(X) be the set of blocks of ΛX and letb(X) =|B(X)|. Ifφ∈QX then φis constant on the blocks of ΛX.

(24) P_{X} ={φ∈M |Λ_{φ}= Λ_{X}}, Q_{X} ={φ∈M |Λ_{φ}≥Λ_{X}}.

If Λφ≥ΛX then Λφ= ΛY for some Y ≥X so that QX= [

Y≥X

PY.

Thus by M¨obius inversion

|P_{Y}|= X

X≥Y

µ(Y, X)|Q_{X}|.

Thus there is a bijection fromQX toW^{B(X)}. In particular |QX|=w^{b(X)}.
Next note thatb(X) = dimX. We see this by choosing a basis forX consist-
ing of vectorsv^{k} defined by

v^{k}_{i} =

1 ifi∈Λk, 0 otherwise.

\[v^{k}_{i}=

\begin{cases} 1 & \text{if $i \in \Lambda_{k}$},\\

0 &\text{otherwise.} \end{cases}

\]

Lemma 6.2: LetAbe an arrangement. Then χ(A, t) = X

B⊆A

(−1)^{|B|}t^{dim}^{T}^{(B)}.

In order to compute R^{00} recall the definition of S(X, Y) from Lemma 3.1.

SinceH∈ B,AH ⊆ B. Thus ifT(B) =Y thenB ∈S(H, Y). LetL^{00}=L(A^{00}).

Then

R^{00}= X

H∈B⊆A

(−1)^{|B|}t^{dim}^{T}^{(B)}

= X

Y∈L^{00}

X

B∈S(H,Y)

(−1)^{|B|}t^{dim}^{Y}

=− X

Y∈L^{00}

X

B∈S(H,Y)

(−1)^{|B−A}^{H}^{|}t^{dim}^{Y}

=− X

Y∈L^{00}

µ(H, Y)t^{dim}^{Y}

=−χ(A^{00}, t).

(25)

Corollary 6.3: Let(A,A^{0},A^{00})be a triple of arrangements. Then
π(A, t) =π(A^{0}, t) +tπ(A^{00}, t).

Definition 6.2: Let (A,A^{0},A^{00}) be a triple with respect to the hyperplaneH∈ A.

CallH aseparator ifT(A)6∈L(A^{0}).

Corollary 6.4: Let(A,A^{0},A^{00})be a triple with respect toH ∈ A.

(i) IfH is a separator then

µ(A) =−µ(A^{00})
and hence

|µ(A)|=|µ(A^{00})|.
(ii) IfH is not a separator then

µ(A) =µ(A^{0})−µ(A^{00})
and

|µ(A)|=|µ(A^{0})|+|µ(A^{00})|.

Proof. It follows from Theorem 5.1 thatπ(A, t) has leading term
(−1)^{r(A)}µ(A)t^{r(A)}.

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

The conclusion follows by comparing coefficients of the leading terms on both
sides of the equation in Corollary 6.3. IfH is a separator thenr(A^{0}) < r(A)
and there is no contribution fromπ(A^{0}, t).

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 complementM(A) for a complex arrangement. Here we prove that the Poincar´e polynomial is the chamber counting function for a real arrangement.

The complementM(A) is a disjoint union of chambers

M(A) = [

C∈Cham(A)

C.

The number of chambers is determined by the Poincar´e polynomial as follows.

Theorem 6.5: LetARbe a real arrangement. Then

|Cham(AR)|=π(AR,1).

Proof. We check the properties required in Corollary 6.4: (i) follows from π(Φl, t) = 1, and (ii) is a consequence of Corollary 3.4.

Theorem 6.6: Letφbe a protocol for a random pair(X, Y). If one ofσφ(x^{0}, y)
andσφ(x, y^{0})is a prefix of the other and(x, y)∈SX,Y, then

hσ_{j}(x^{0}, y)i^{∞}_{j=1}=hσ_{j}(x, y)i^{∞}_{j=1}=hσ_{j}(x, y^{0})i^{∞}_{j=1}.

Figure 2. Q(A2) =xyz(x+y+z)(x+y−z)(x−y+z)(x−y−z) Proof. We show by induction onithat

hσ_{j}(x^{0}, y)i^{i}_{j=1}=hσ_{j}(x, y)i^{i}_{j=1}=hσ_{j}(x, y^{0})i^{i}_{j=1}.

The induction hypothesis holds vacuously for i = 0. Assume it holds for i−
1, in particular [σj(x^{0}, y)]^{i−1}_{j=1} = [σj(x, y^{0})]^{i−1}_{j=1}. Then one of [σj(x^{0}, y)]^{∞}_{j=i} and
[σj(x, y^{0})]^{∞}_{j=i} is a prefix of the other which implies that one of σi(x^{0}, y) and
σi(x, y^{0}) is a prefix of the other. If theith message is transmitted byP_{X} then,
by the separate-transmissions property and the induction hypothesis,σi(x, y) =
σi(x, y^{0}), hence one of σi(x, y) and σi(x^{0}, y) is a prefix of the other. By the
implicit-termination property, neither σi(x, y) nor σi(x^{0}, y) can be a proper
prefix of the other, hence they must be the same and σi(x^{0}, y) = σi(x, y) =
σi(x, y^{0}). If theith message is transmitted byPYthen, symmetrically,σi(x, y) =
σ_{i}(x^{0}, y) by the induction hypothesis and the separate-transmissions property,
and, then,σ_{i}(x, y) =σ_{i}(x, y^{0}) by the implicit-termination property, proving the
induction step.

If φ is a protocol for (X, Y), and (x, y), (x^{0}, y) are distinct inputs inSX,Y,
then, by the correct-decision property, hσj(x, y)i^{∞}_{j=1}6=hσj(x^{0}, y)i^{∞}_{j=1}.

Equation (25) definedP_{Y}’s ambiguity setS_{X|Y}(y) to be the set of possibleX
values whenY =y. The last corollary implies that for ally∈S_{Y}, the multiset^{1}
of codewords{σ_{φ}(x, y) :x∈S_{X|Y}(y)}is prefix free.

1 A multiset allows multiplicity of elements. Hence,{0,01,01}is prefix free as a set, but not as a multiset.

7. One-Way Complexity

Cˆ1(X|Y), the one-way complexity of a random pair (X, Y), is the number of
bitsP_{X} must transmit in the worst case whenP_{Y} is not permitted to transmit
any feedback messages. Starting withSX,Y, the support set of (X, Y), we define
G(X|Y), thecharacteristic hypergraph of (X, Y), and show that

Cˆ1(X|Y) =dlogχ(G(X|Y))e.

Let (X, Y) be a random pair. For each yin SY, the support set of Y, Equa-
tion (25) definedS_{X|Y}(y) to be the set of possiblexvalues when Y =y. The
characteristic hypergraph G(X|Y) of (X, Y) has SX as its vertex set and the
hyperedgeS_{X|Y}(y) for eachy∈SY.

We can now prove a continuity theorem.

Theorem 7.1: LetΩ⊂R^{n} be an open set, let u∈BV(Ω;R^{m}), and let
(26) T_{x}^{u}=

y∈R^{m}:y= ˜u(x) +
Du

|Du|(x), z

for somez∈R^{n}

for everyx∈Ω\S_{u}. Letf:R^{m}→R^{k} be a Lipschitz continuous function such
thatf(0) = 0, and letv=f(u) : Ω→R^{k}. Thenv∈BV(Ω;R^{k})and

(27) J v= (f(u^{+})−f(u^{−}))⊗ν_{u}· H_{n−1}
_{S}

u. In addition, for

Due

-almost everyx∈Ωthe restriction of the functionf toT_{x}^{u}
is differentiable atu(x)˜ and

(28) Dve =∇(f|_{T}u

x)(˜u) Due Due

· Due

.

Before proving the theorem, we state without proof three elementary remarks which will be useful in the sequel.

Remark 7.1: Let ω: ]0,+∞[ → ]0,+∞[ be a continuous function such that ω(t)→0 ast→0. Then

lim

h→0^{+}g(ω(h)) =L⇔ lim

h→0^{+}g(h) =L
for any functiong: ]0,+∞[→R.

Remark 7.2: Letg:R^{n} → R be a Lipschitz continuous function and assume
that

L(z) = lim

h→0^{+}

g(hz)−g(0) h

exists for everyz∈Q^{n} and thatLis a linear function ofz. Theng is differen-
tiable at 0.

Remark 7.3: LetA:R^{n}→R^{m}be a linear function, and letf:R^{m}→R be a
function. Then the restriction off to the range ofAis differentiable at 0 if and
only iff(A) :R^{n} →Ris differentiable at 0 and

∇(f|_{Im(A)})(0)A=∇(f(A))(0).

Proof. We begin by showing thatv∈BV(Ω;R^{k}) and
(29) |Dv|(B)≤K|Du|(B) ∀B∈B(Ω),

where K > 0 is the Lipschitz constant of f. By (13) and by the approxima-
tion result quoted in §3, it is possible to find a sequence (uh) ⊂ C^{1}(Ω;R^{m})
converging touinL^{1}(Ω;R^{m}) and such that

h→+∞lim Z

Ω

|∇uh|dx=|Du|(Ω).

The functionsvh=f(uh) are locally Lipschitz continuous in Ω, and the defini- tion of differential implies that|∇vh| ≤K|∇uh|almost everywhere in Ω. The lower semicontinuity of the total variation and (13) yield

|Dv|(Ω)≤lim inf

h→+∞|Dv_{h}|(Ω) = lim inf

h→+∞

Z

Ω

|∇v_{h}|dx

≤Klim inf

h→+∞

Z

Ω

|∇uh| dx=K|Du|(Ω).

(30)

Sincef(0) = 0, we have also Z

Ω

|v|dx≤K Z

Ω

|u|dx;

therefore u ∈ BV(Ω;R^{k}). Repeating the same argument for every open set
A⊂Ω, we get (29) for everyB ∈B(Ω), because|Dv|,|Du|are Radon measures.

To prove Lemma 6.1, first we observe that

(31) Sv⊂Su, ˜v(x) =f(˜u(x)) ∀x∈Ω\Su.

In fact, for everyε >0 we have

{y∈B_{ρ}(x) :|v(y)−f(˜u(x))|> ε} ⊂ {y∈B_{ρ}(x) :|u(y)−u(x)|˜ > ε/K},
hence

lim

ρ→0^{+}

|{y∈Bρ(x) :|v(y)−f(˜u(x))|> ε}|

ρ^{n} = 0

wheneverx∈Ω\S_{u}. By a similar argument, ifx∈S_{u}is a point such that there
exists a triplet (u^{+}, u^{−}, νu) satisfying (14), (15), then

(v^{+}(x)−v^{−}(x))⊗νv= (f(u^{+}(x))−f(u^{−}(x)))⊗νu ifx∈Sv

and f(u^{−}(x)) =f(u^{+}(x)) if x∈Su\Sv. Hence, by (1.8) we get
J v(B) =

Z

B∩Sv

(v^{+}−v^{−})⊗νvdH_{n−1}=
Z

B∩Sv

(f(u^{+})−f(u^{−}))⊗νudH_{n−1}

= Z

B∩S_{u}

(f(u^{+})−f(u^{−}))⊗ν_{u}dH_{n−1}
and Lemma 6.1 is proved.

To prove (31), it is not restrictive to assume thatk= 1. Moreover, to simplify
our notation, from now on we shall assume that Ω =R^{n}. The proof of (31)
is divided into two steps. In the first step we prove the statement in the one-
dimensional case (n= 1), using Theorem 5.2. In the second step we achieve the
general result using Theorem 7.1.

Step 1: Assume that n = 1. Since Su is at most countable, (7) yields that

Dve

(Su\Sv) = 0, so that (19) and (21) imply that Dv = Dve +J v is the Radon-Nikod´ym decomposition of Dv in absolutely continuous and singular part with respect to

Due

. By Theorem 5.2, we have Dve

Due

(t) = lim

s→t^{+}

Dv([t, s[)

Due

([t, s[)

, Due Due

(t) = lim

s→t^{+}

Du([t, s[)

Due

([t, s[)

Due

-almost everywhere inR. It is well known (see, for instance, [11, 2.5.16])
that every one-dimensional function of bounded variation whas a unique left
continuous representative, i.e., a function ˆwsuch that ˆw=walmost everywhere
and lim_{s→t}^{−}w(s) = ˆˆ w(t) for every t∈R. These conditions imply

(32) u(t) =ˆ Du(]−∞, t[), v(t) =ˆ Dv(]−∞, t[) ∀t∈R

and

(33) ˆv(t) =f(ˆu(t)) ∀t∈R.

Lett ∈ R be such that Due

([t, s[) >0 for every s > t and assume that the limits in (22) exist. By (23) and (24) we get

ˆ

v(s)−v(t)ˆ

Due

([t, s[)

= f(ˆu(s))−f(ˆu(t))

Due

([t, s[)

=

f(ˆu(s))−f(ˆu(t) + Due Due

(t) Due

([t, s[))

Due

([t, s[)

+

f(ˆu(t) + Due Due

(t) Due

([t, s[))−f(ˆu(t))

Due

([t, s[) for everys > t. Using the Lipschitz condition onf we find

ˆ

v(s)−v(t)ˆ

Due

([t, s[)

−

f(ˆu(t) + Due Due

(t) Due

([t, s[))−f(ˆu(t))

Due

([t, s[)

≤K ˆ

u(s)−ˆu(t)

Due

([t, s[)

− Due Due

(t) .

By (29), the functions→ Due

([t, s[) is continuous and converges to 0 ass↓t.

Therefore Remark 7.1 and the previous inequality imply

Dve Due

(t) = lim

h→0^{+}

f(ˆu(t) +h Due Due

(t))−f(ˆu(t)) h

Due

-a.e. in R.

By (22), ˆu(x) = ˜u(x) for everyx∈R\Su; moreover, applying the same argu-
ment to the functionsu^{0}(t) =u(−t),v^{0}(t) =f(u^{0}(t)) =v(−t), we get

Dve Due

(t) = lim

h→0

f(˜u(t) +h Due Due

(t))−f(˜u(t)) h

Due

-a.e. in R and our statement is proved.

Step 2: Let us consider now the general casen >1. Letν ∈R^{n} be such that

|ν|= 1, and letπ_{ν} ={y∈R^{n} :hy, νi= 0}. In the following, we shall identify
R^{n} withπ_{ν}×R, and we shall denote byy the variable ranging inπ_{ν} and byt
the variable ranging inR. By the just proven one-dimensional result, and by
Theorem 3.3, we get

lim

h→0

f(˜u(y+tν) +h Due _{y}
Due y

(t))−f(˜u(y+tν))

h = Dve _{y}

Due y

(t)

Due _{y}

-a.e. inR forHn−1-almost everyy∈πν. We claim that

(34) hDu, νe i

hDu, νe i

(y+tν) = Due _{y}
Due y

(t) Due y

-a.e. in R

forHn−1-almost everyy∈πν. In fact, by (16) and (18) we get Z

πν

Due y

Due _{y}

· Due y

dH_{n−1}(y) =
Z

πν

Due ydH_{n−1}(y)

=hDu, νie = hDu, νie

hDu, νie

·

hDu, νie =

Z

π_{ν}

hDu, νie

hDu, νie

(y+·ν)· Due y

dHn−1(y) and (24) follows from (13). By the same argument it is possible to prove that (35) hDv, νie

hDu, νe i

(y+tν) = Dve y

Due _{y}

(t) Due y

-a.e. in R

forHn−1-almost everyy∈πν. By (24) and (25) we get

h→0lim

f(˜u(y+tν) +h hDu, νe i

hDu, νe i

(y+tν))−f(˜u(y+tν))

h = hDv, νie

hDu, νie

(y+tν)

forH_{n−1}-almost everyy∈πν, and using again (14), (15) we get

h→0lim

f(˜u(x) +h hDu, νie

hDu, νie

(x))−f(˜u(x))

h = hDv, νie

hDu, νe i

(x)

hDu, νe i

-a.e. in R^{n}.
Since the function

hDu, νe i /

Due

is strictly positive

hDu, νie

-almost every- where, we obtain also

lim

h→0

f(˜u(x) +h

hDu, νie Due

(x) hDu, νie

hDu, νie

(x))−f(˜u(x)) h

=

hDu, νie Due

(x) hDv, νie

hDu, νie

(x)

hDu, νe i

-almost everywhere in R^{n}.
Finally, since

hDu, νie Due

hDu, νie

hDu, νie

= hDu, νie Due

=

* Due Due

, ν +

Due

-a.e. inR^{n}

hDu, νie Due

hDv, νie

hDu, νie

= hDv, νie Due

=

* Dve Due

, ν +

Due

-a.e. inR^{n}

and since both sides of (33) are zero Due

-almost everywhere on

hDu, νe i - negligible sets, we conclude that

lim

h→0

f

˜u(x) +h

* Due Due

(x), ν +

−f(˜u(x))

h =

* Dve Due

(x), ν +

,

Due

-a.e. inR^{n}. Sinceν is arbitrary, by Remarks 7.2 and 7.3 the restriction of
f to the affine space T_{x}^{u} is differentiable at ˜u(x) for

Due

-almost everyx∈R^{n}
and (26) holds.

It follows from (13), (14), and (15) that (36) D(t1, . . . , tn) =X

I∈n

(−1)^{|I|−1}|I|Y

i∈I

ti

Y

j∈I

(Dj+λjtj) detA^{(λ)}(I|I).

Letti= ˆxi,i= 1, . . . , n. Lemma 1 leads to (37) D(ˆx1, . . . ,xˆn) =Y

i∈n

ˆ xi

X

I∈n

(−1)^{|I|−1}|I|perA^{(λ)}(I|I) detA^{(λ)}(I|I).

By (3), (13), and (37), we have the following result:

Theorem 7.2:

(38) Hc = 1

2n

n

X

l=1

l(−1)^{l−1}A^{(λ)}_{l} ,

where

(39) A^{(λ)}_{l} = X

Il⊆n

perA^{(λ)}(Il|Il) detA^{(λ)}(Il|Il),|Il|=l.

It is worth noting that A^{(λ)}_{l} of (39) is similar to the coefficients b_{l} of the
characteristic polynomial of (10). It is well known in graph theory that the
coefficientsb_{l}can be expressed as a sum over certain subgraphs. It is interesting
to see whetherA_{l},λ= 0, structural properties of a graph.

We may call (38) a parametric representation of Hc. In computation, the parameterλi plays very important roles. The choice of the parameter usually depends on the properties of the given graph. For a complete graph Kn, let

λi= 1, i= 1, . . . , n. It follows from (39) that

(40) A^{(1)}_{l} =

n!, ifl= 1 0, otherwise.

By (38)

(41) H_{c}= 1

2(n−1)!.

For a complete bipartite graphKn_{1}n_{2}, letλi= 0, i= 1, . . . , n. By (39),

(42) A_{l}=

−n1!n2!δn_{1}n_{2}, ifl= 2

0, otherwise.

Theorem 7.2 leads to

(43) Hc = 1

n1+n2

n1!n2!δn_{1}n_{2}.

Now, we consider an asymmetrical approach. Theorem 3.3 leads to
(44) detK(t= 1, t_{1}, . . . , t_{n};l|l)

= X

I⊆n−{l}

(−1)^{|I}^{|}Y

i∈I

ti

Y

j∈I

(Dj+λjtj) detA^{(λ)}(I∪ {l}|I∪ {l}).

By (3) and (16) we have the following asymmetrical result:

Theorem 7.3:

(45) Hc= 1 2

X

I⊆n−{l}

(−1)^{|I|}perA^{(λ)}(I|I) detA^{(λ)}(I∪ {l}|I∪ {l})
which reduces to Goulden–Jackson’s formula when λi= 0, i= 1, . . . , n[8].

8. Named Propositions

Here we discuss several propositions:

G¨odel Theorem 8.1 (First incompleteness theorem): For any consistent for- mal, computably enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true, but not provable in the theory, can be constructed. That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

G¨odel Theorem 8.2: For any formal recursively enumerable (i.e. effectively generated) theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.

Abel’s Lemma(Summation by parts): For any sequencesfk andgk n

X

k=m

fk(gk+1−gk) =fn+1gn+1−

n

X

k=m

gk+1(fk+1−fk)

Fermat’s last theorem: For anyn >2 the equation x^{n}+y^{n} =z^{n} has no
non-zero integer solutions.

9. Various font features of theamsmathpackage

9.1. Bold versions of special symbols. In theamsmathpackage\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

\mathbf. For example, A_\infty + \pi A_0 \sim

\mathbf{A}_{\boldsymbol{\infty}} \boldsymbol{+}

\boldsymbol{\pi} \mathbf{A}_{\boldsymbol{0}}

looks like this:

A_{∞}+πA_{0}∼A_{∞}+πA_{0}

9.2. “Poor man’s bold”. If a bold version of a particular symbol doesn’t exist in the available fonts, then \boldsymbol can’t be used to make that symbol bold. At the present time, this means that\boldsymbol can’t be used with symbols from the msam and msbm fonts, among others. In some cases, poor man’s bold (\pmb) can be used instead of\boldsymbol:

∂x

∂y

∂y

∂z

\[\frac{\partial x}{\partial y}

\pmb{\bigg\vert}

\frac{\partial y}{\partial z}\]

So-called “large operator” symbols such as P and Q require an additional command, \mathop, to produce proper spacing and limits when \pmb is used.

For further details seeThe TEXbook. X

i<B iodd

Y

κ

κF(r_{i}) XXX

i<B iodd

YYY

κ

κ(r_{i})

\[\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)

\]

10. Compound symbols and other features

10.1. Multiple integral signs. \iint, \iiint, and\iiiintgive multiple integral signs with the spacing between them nicely adjusted, in both text and display style. \idotsintgives two integral signs with dots between them.

Z Z

A

f(x, y)dx dy

Z Z Z

A

f(x, y, z)dx dy dz (46)

Z Z Z Z

A

f(w, x, y, z)dw dx dy dz Z

· · · Z

A

f(x1, . . . , xk) (47)

10.2. Over and under arrows. Some extra over and under arrow operations
are provided in theamsmathpackage. (Basic L^{A}TEX provides\overrightarrow
and\overleftarrow).

−−−−−−→

ψδ(t)Eth=ψδ(t)Eth

−−−−−−→

←−−−−−−

ψ_{δ}(t)E_{t}h=ψ_{δ}(t)E_{t}h

←−−−−−−

←−−−−→

ψδ(t)Eth=ψδ(t)Eth

←−−−−→

\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*}

These all scale properly in subscript sizes:

Z

−−→ AB

ax dx

\[\int_{\overrightarrow{AB}} ax\,dx\]

10.3. Dots. Normally you need only type \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\dotsc (series dots, after a comma), \dotsb (binary dots, for binary relations or operators), \dotsm (multiplication dots), or\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

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

produces

Then we have the seriesA_{1}, A_{2}, . . ., the regional sumA_{1}+A_{2}+

· · ·, the orthogonal productA_{1}A_{2}· · ·, and the infinite integral
Z

A_{1}

Z

A_{2}

· · · 10.4. Accents in math. Double accents:

ˆˆ

H Cˇˇ 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 pro-
cessing of a L^{A}TEX file.

10.5. Dot accents. \dddot and\ddddotare available to produce triple and quadruple dot accents in addition to the\dotand\ddotaccents already avail-

able in L^{A}TEX: ...

Q ....

R

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

10.6. Roots. In the amsmath package \leftroot and \uproot allow you to adjust the position of the root index of a radical:

\sqrt[\leftroot{-2}\uproot{2}\beta]{k}

gives good positioning of theβ:

√β

k

10.7. Boxed formulas. The command \boxed puts a box around its argu- ment, like \fboxexcept that the contents are in math mode:

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

Wt−F ⊆V(Pi)⊆Wt.

10.8. Extensible arrows. \xleftarrowand\xrightarrow produce arrows that extend automatically to accommodate unusually wide subscripts or super- scripts. The text of the subscript or superscript are given as an optional resp.

mandatory argument: Example:

0←^{α}−

ζ F× 4[n−1]−−−−→^{∂}^{0}^{α(b)} E^{∂}^{0}^{b}

\[0 \xleftarrow[\zeta]{\alpha} F\times\triangle[n-1]

\xrightarrow{\partial_0\alpha(b)} E^{\partial_0b}\]

10.9. \overset, \underset, and \sideset. Examples:

∗

X X

∗ a

X

b

\[\overset{*}{X}\qquad\underset{*}{X}\qquad

\overset{a}{\underset{b}{X}}\]

The command \sidesetis for a rather special purpose: putting symbols at the subscript and superscript corners of a large operator symbol such asPor Q, without affecting the placement of limits. Examples:

∗

∗

Y^{∗}

∗ k

X^{0}

0≤i≤m

Eiβx

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

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

\]

10.10. The\textcommand. The main use of the command\textis for words or phrases in a display:

y=y^{0} if and only if y^{0}_{k}=δ_{k}y_{τ(k)}

\[\mathbf{y}=\mathbf{y}’\quad\text{if and only if}\quad y’_k=\delta_k y_{\tau(k)}\]

10.11. Operator names. The more common math functions such as log, sin, and lim have predefined control sequences: \log, \sin, \lim. The amsmath package provides\DeclareMathOperatorand\DeclareMathOperator*for pro- ducing new function names that will have the same typographical treatment.

Examples:

kfk_{∞}= ess sup_{x∈R}n|f(x)|

\[\norm{f}_\infty=

\esssup_{x\in R^n}\abs{f(x)}\]

meas1{u∈R^{1}_{+}: f^{∗}(u)> α}= measn{x∈R^{n}: |f(x)| ≥α} ∀α >0.

\[\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.\]

\esssupand\meas would be defined in the document preamble as

\DeclareMathOperator*{\esssup}{ess\,sup}

\DeclareMathOperator{\meas}{meas}

The following special operator names are predefined in theamsmathpackage:

\varlimsup, \varliminf, \varinjlim, and \varprojlim. Here’s what they look like in use:

n→∞lim Q(un, un−u^{#})≤0
(48)

lim

n→∞

|an+1|/|an|= 0 (49)

lim−→(m^{λ}_{i}·)^{∗}≤0
(50)

lim←−

p∈S(A)

Ap≤0 (51)

\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}

10.12. \mod and its relatives. The commands \mod and\pod are variants of\pmodpreferred by some authors;\mod omits the parentheses, whereas\pod omits the ‘mod’ and retains the parentheses. Examples:

x≡y+ 1 (modm^{2})
(52)

x≡y+ 1 modm^{2}
(53)

x≡y+ 1 (m^{2})
(54)

\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}

10.13. Fractions and related constructions. The usual notation for bi- nomials is similar to the fraction concept, so it has a similar command\binom with two arguments. Example:

X

γ∈ΓC

Iγ= 2^{k}−
k

1

2^{k−1}+
k

2

2^{k−2}

+· · ·+ (−1)^{l}
k

l

2^{k−l}+· · ·+ (−1)^{k}

= (2−1)^{k}= 1
(55)

\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 ... }

The generalized fraction command \genfrac provides full access to the six TEX fraction primitives:

\over: n+ 1

2 \overwithdelims:

n+ 1 2

(56)

\atop: n+ 1

2 \atopwithdelims:

n+ 1 2

(57)

\above: n+ 1

2 \abovewithdelims:

n+ 1 2

(58)

\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}

10.14. Continued fractions. The continued fraction

(59) 1

√2 + 1

√2 + 1

√2 + 1

√2 + 1

√2 +· · · 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 using

\cfrac[l]or\cfrac[r]instead of\cfrac.

10.15. Smash. Inamsmaththere are optional argumentstandbfor the plain TEX command \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 formulaXj= (1/√

λj)X_{j}^{0} \smash[b]has been used to
limit the size of the radical symbol.

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

Without the use of\smash[b] the formula would have appeared thus: X_{j} =
(1/p

λ_{j})X_{j}^{0}, with the radical extending to encompass the depth of the subscript
j.

10.16. The ‘cases’ environment. ‘Cases’ constructions like the following can be produced using the casesenvironment.

(60) Pr−j =

0 ifr−j is odd,

r! (−1)^{(r−j)/2} ifr−j is even.

\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\textand the embedded math.

10.17. Matrix. Here are samples of the matrix environments,\matrix,\pmatrix,

\bmatrix,\Bmatrix,\vmatrixand\Vmatrix:

(61) ϑ %

ϕ $

ϑ %

ϕ $

! "

ϑ %

ϕ $

# (

ϑ %

ϕ $

)

ϑ %

ϕ $

ϑ %

ϕ $

\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}

To produce a small matrix suitable for use in text, use the smallmatrix environment.

\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 it here: ^{a b}_{c d}

and follow it with enough text to ensure that there will be at least one full line below the matrix.

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

W(Φ) =

ϕ

(ϕ1, ε1) 0 . . . 0
ϕk_{n2}

(ϕ2, ε1) ϕ

(ϕ2, ε2) . . . 0

. . . . ϕkn1

(ϕn, ε1)

ϕkn2

(ϕn, ε2) . . . ϕkn n−1

(ϕn, ε_{n−1})
ϕ
(ϕn, εn)

\[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, for example,\hdotsfor[1.5]{3}. The number in square brackets will be used as a multiplier; the normal value is 1.

10.18. The \substack command. The\substack command can be used to produce a multiline subscript or superscript: for example

\sum_{\substack{0\le i\le m\\ 0<j<n}} P(i,j) produces a two-line subscript underneath the sum:

(62) X

0≤i≤m 0<j<n

P(i, j)

A slightly more generalized form is thesubarrayenvironment which allows you to specify that each line should be left-aligned instead of centered, as here:

(63) X

0≤i≤m 0<j<n

P(i, j)

\sum_{\begin{subarray}{l}

0\le i\le m\\ 0<j<n

\end{subarray}}

P(i,j)

10.19. Big-g-g delimiters. Here are some big delimiters, first in\normalsize:

Ey

Z t_{ε}
0

Lx,y^{x}(s)ϕ(x)ds

\[\biggl(\mathbf{E}_{y}

\int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds

\biggr)

\]

and now in\Largesize:

## E

y## Z

tε0

## L

_{x,y}

^{x}

_{(s)}

## ϕ(x) ds

{\Large

\[\biggl(\mathbf{E}_{y}

\int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds

\biggr)

\]}

Appendix A. Examples of multiple-line equation structures

## Note: Starting on this page, vertical rules are added at the margins so that the positioning of various display elements with respect to the margins can be seen more clearly.

A.1. Split. The split environment is not an independent environment but should be used inside something else such asequationoralign.

If there is not enough room for it, the equation number for asplit will be shifted to the previous line, when equation numbers are on the left; the number shifts down to the next line when numbers are on the right.

fh,ε(x, y) =εEx,y

Z t_{ε}
0

L_{x,y}_{ε}_{(εu)}ϕ(x)du

=h Z

Lx,zϕ(x)ρx(dz) +h

1
t_{ε}

E_{y}

Z tε

0

L_{x,y}x(s)ϕ(x)ds−t_{ε}
Z

L_{x,z}ϕ(x)ρ_{x}(dz)

+ 1
t_{ε}

E_{y}

Z tε

0

L_{x,y}x(s)ϕ(x)ds−E_{x,y}
Z tε

0

L_{x,y}_{ε}_{(εs)}ϕ(x)ds

=hbLxϕ(x) +hθε(x, y), (64)

Some text after to test the below-display spacing.

\begin{equation}

\begin{split}

f_{h,\varepsilon}(x,y)

&=\varepsilon\mathbf{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(\mathbf{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(\mathbf{E}_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)}

\varphi(x)\,ds -\mathbf{E}_{x,y}\int_0^{t_\varepsilon}

L_{x,y_\varepsilon(\varepsilon s)}

\varphi(x)\,ds\biggr)\biggr]\\

&=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y),

\end{split}

\end{equation}