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-LATEX distribution. The text is (slightly) reformatted according to the requirements of theijmartstyle.
2. Enumeration of Hamiltonian paths in a graph
LetA= (aij) be the adjacency matrix of graphG. The corresponding Kirchhoff matrixK= (kij) 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 Ci(j) be the set of graphs obtained from G by attaching edge (vivj) to each spanning tree of G. Denote by Ci = S
jCi(j). It is obvious that the collection of Hamiltonian cycles is a subset of Ci. Note that the cardinality of Ci iskiidetK(i|i). LetXb ={xˆ1, . . . ,xˆn}.
$\wh X=\{\hat x_1,\dots,\hat x_n\}$
Define multiplication for the elements ofXb by
(2) xˆixˆj = ˆxjxˆi, xˆ2i = 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 produceKn andKn1n2. Recently, using Lagrangian methods, Goulden and Jackson have shown that Hc can be expressed in terms of the determinant and permanent of the adjacency matrix [2]. However, the formula of Goulden and Jackson determines neitherKn nor Kn1n2 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 Kn1n2. 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 multigraphKn1...np.
The conditions aij =aji, i, j = 1, . . . , n, are not required in this paper. All formulas can be extended to a digraph simply by multiplyingHc 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
bijxˆi
=
Y
i∈n
ˆ xi
perB where perB is the permanent ofB.
LetYb ={yˆ1, . . . ,yˆn}. Define multiplication for the elements ofYb by (5) yˆiyˆj+ ˆyjyˆi= 0, i, j= 1, . . . , n.
Then, it follows that Lemma 3.2:
(6) Y
i∈n
X
j∈n
bijyˆj
=
Y
i∈n
ˆ yi
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
Il⊆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 itsi1, . . . , il 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)lblxl. Then
(10) bl= X
Il⊆n
detB(Il|Il).
Let
(11) K(t, t1, . . . , tn) =
D1t −a12t2 . . . −a1ntn
−a21t1 D2t . . . −a2ntn . . . .
−an1t1 −an2t2 . . . Dnt
,
\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) Di=X
j∈n
aijtj, i= 1, . . . , n.
Set
D(t1, . . . , tn) = δ
δt detK(t, t1, . . . , tn)|t=1. Then
(13) D(t1, . . . , tn) =X
i∈n
DidetK(t= 1, t1, . . . , tn;i|i),
whereK(t= 1, t1, . . . , tn;i|i) is theith principal submatrix ofK(t= 1, t1, . . . , tn).
Theorem 3.3 leads to (14) detK(t1, t1, . . . , tn) =X
I∈n
(−1)|I|tn−|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
alixi
detK(t= 1, x1, . . . , xn;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)lDl, where
(18) Dl= X
Il⊆n
D(t1, . . . , tn)2|
ti=n0,ifi∈Il
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 Kn1...np may be written
(19) T =np−2
p
Y
i=1
(n−ni)ni−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
l1+···+lp=l p
Y
i=1
ni
li
·[(n−l)−(ni−li)]ni−li·
(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
l1+···+lp=l p
Y
i=1
ni
li
·[(n−l)−(ni−li)]ni−li
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, Hc in a Kn1n2n3 graph may be written
Hc= n1!n2!n3! n1+n2+n3
X
i
n1
i
n2
n3−n1+i
n3
n3−n2+i
+
n1−1 i
n2−1 n3−n1+i
n3−1 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,v0)→(0,0)
H(z+v)−H(z+v0)−BH(z)(v−v0) kv−v0k = 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 chooseψ0(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−a2. The radial Laplacian
∆0lnψ0(r) = d2
dr2 +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
d2 dr2 +1
r d dr
lnψ0(r) =h(r) with initial conditions given by lnψ0(1) = 0 and lnψ00(1) = 0.
Next, let Dν 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 Dν = {z | |z−zν| < δ}. Suppose that Dν(a) denotes the smaller concentric disk Dν(a) ={z| |z−zν| ≤ (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) Φ0(z) is bounded above and below by positive constants c1 ≤Φ0(z)≤ c2.
(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−2χA, whereχAis the characteristic function of A.
Suppose thatαis a nonnegative real constant. We apply Proposition 3.5 with Φ(z) = Φ0(z)eα|z|2. Ifu∈ C0∞(R2−S
νDν(a)), assume that D is a bounded domain containing the support ofuandA⊂ D ⊂R2−S
νDν(a). A calculation gives
Z
D
∂u
2Φ0(z)eα|z|2 ≥c4α Z
D
|u|2Φ0eα|z|2+c5δ−2 Z
A
|u|2Φ0eα|z|2. The boundedness, property (1) of Φ0, then yields
Z
D
∂u
2eα|z|2 ≥c6α Z
D
|u|2eα|z|2+c7δ−2 Z
A
|u|2eα|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) PX ={φ∈M |Λφ= ΛX}, QX ={φ∈M |Λφ≥ΛX}.
If Λφ≥ΛX then Λφ= ΛY for some Y ≥X so that QX= [
Y≥X
PY.
Thus by M¨obius inversion
|PY|= X
X≥Y
µ(Y, X)|QX|.
Thus there is a bijection fromQX toWB(X). In particular |QX|=wb(X). Next note thatb(X) = dimX. We see this by choosing a basis forX consist- ing of vectorsvk defined by
vki =
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|tdimT(B).
In order to compute R00 recall the definition of S(X, Y) from Lemma 3.1.
SinceH∈ B,AH ⊆ B. Thus ifT(B) =Y thenB ∈S(H, Y). LetL00=L(A00).
Then
R00= X
H∈B⊆A
(−1)|B|tdimT(B)
= X
Y∈L00
X
B∈S(H,Y)
(−1)|B|tdimY
=− X
Y∈L00
X
B∈S(H,Y)
(−1)|B−AH|tdimY
=− X
Y∈L00
µ(H, Y)tdimY
=−χ(A00, t).
(25)
Corollary 6.3: Let(A,A0,A00)be a triple of arrangements. Then π(A, t) =π(A0, t) +tπ(A00, t).
Definition 6.2: Let (A,A0,A00) be a triple with respect to the hyperplaneH∈ A.
CallH aseparator ifT(A)6∈L(A0).
Corollary 6.4: Let(A,A0,A00)be a triple with respect toH ∈ A.
(i) IfH is a separator then
µ(A) =−µ(A00) and hence
|µ(A)|=|µ(A00)|. (ii) IfH is not a separator then
µ(A) =µ(A0)−µ(A00) and
|µ(A)|=|µ(A0)|+|µ(A00)|.
Proof. It follows from Theorem 5.1 thatπ(A, t) has leading term (−1)r(A)µ(A)tr(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(A0) < r(A) and there is no contribution fromπ(A0, 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(x0, y) andσφ(x, y0)is a prefix of the other and(x, y)∈SX,Y, then
hσj(x0, y)i∞j=1=hσj(x, y)i∞j=1=hσj(x, y0)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
hσj(x0, y)iij=1=hσj(x, y)iij=1=hσj(x, y0)iij=1.
The induction hypothesis holds vacuously for i = 0. Assume it holds for i− 1, in particular [σj(x0, y)]i−1j=1 = [σj(x, y0)]i−1j=1. Then one of [σj(x0, y)]∞j=i and [σj(x, y0)]∞j=i is a prefix of the other which implies that one of σi(x0, y) and σi(x, y0) is a prefix of the other. If theith message is transmitted byPX then, by the separate-transmissions property and the induction hypothesis,σi(x, y) = σi(x, y0), hence one of σi(x, y) and σi(x0, y) is a prefix of the other. By the implicit-termination property, neither σi(x, y) nor σi(x0, y) can be a proper prefix of the other, hence they must be the same and σi(x0, y) = σi(x, y) = σi(x, y0). If theith message is transmitted byPYthen, symmetrically,σi(x, y) = σi(x0, y) by the induction hypothesis and the separate-transmissions property, and, then,σi(x, y) =σi(x, y0) by the implicit-termination property, proving the induction step.
If φ is a protocol for (X, Y), and (x, y), (x0, y) are distinct inputs inSX,Y, then, by the correct-decision property, hσj(x, y)i∞j=16=hσj(x0, y)i∞j=1.
Equation (25) definedPY’s ambiguity setSX|Y(y) to be the set of possibleX values whenY =y. The last corollary implies that for ally∈SY, the multiset1 of codewords{σφ(x, y) :x∈SX|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 bitsPX must transmit in the worst case whenPY 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) definedSX|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 hyperedgeSX|Y(y) for eachy∈SY.
We can now prove a continuity theorem.
Theorem 7.1: LetΩ⊂Rn be an open set, let u∈BV(Ω;Rm), and let (26) Txu=
y∈Rm:y= ˜u(x) + Du
|Du|(x), z
for somez∈Rn
for everyx∈Ω\Su. Letf:Rm→Rk be a Lipschitz continuous function such thatf(0) = 0, and letv=f(u) : Ω→Rk. Thenv∈BV(Ω;Rk)and
(27) J v= (f(u+)−f(u−))⊗νu· Hn−1 S
u. In addition, for
Due
-almost everyx∈Ωthe restriction of the functionf toTxu is differentiable atu(x)˜ and
(28) Dve =∇(f|Tu
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:Rn → R be a Lipschitz continuous function and assume that
L(z) = lim
h→0+
g(hz)−g(0) h
exists for everyz∈Qn and thatLis a linear function ofz. Theng is differen- tiable at 0.
Remark 7.3: LetA:Rn→Rmbe a linear function, and letf:Rm→R be a function. Then the restriction off to the range ofAis differentiable at 0 if and only iff(A) :Rn →Ris differentiable at 0 and
∇(f|Im(A))(0)A=∇(f(A))(0).
Proof. We begin by showing thatv∈BV(Ω;Rk) 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) ⊂ C1(Ω;Rm) converging touinL1(Ω;Rm) 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→+∞|Dvh|(Ω) = lim inf
h→+∞
Z
Ω
|∇vh|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;Rk). 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∈Bρ(x) :|v(y)−f(˜u(x))|> ε} ⊂ {y∈Bρ(x) :|u(y)−u(x)|˜ > ε/K}, hence
lim
ρ→0+
|{y∈Bρ(x) :|v(y)−f(˜u(x))|> ε}|
ρn = 0
wheneverx∈Ω\Su. By a similar argument, ifx∈Suis 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−)⊗νvdHn−1= Z
B∩Sv
(f(u+)−f(u−))⊗νudHn−1
= Z
B∩Su
(f(u+)−f(u−))⊗νudHn−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 =Rn. 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 lims→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 functionsu0(t) =u(−t),v0(t) =f(u0(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 ∈Rn be such that
|ν|= 1, and letπν ={y∈Rn :hy, νi= 0}. In the following, we shall identify Rn withπν×R, and we shall denote byy the variable ranging inπν 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
dHn−1(y) = Z
πν
Due ydHn−1(y)
=hDu, νie = hDu, νie
hDu, νie
·
hDu, νie =
Z
πν
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)
forHn−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 Rn. 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 Rn. Finally, since
hDu, νie Due
hDu, νie
hDu, νie
= hDu, νie Due
=
* Due Due
, ν +
Due
-a.e. inRn
hDu, νie Due
hDv, νie
hDu, νie
= hDv, νie Due
=
* Dve Due
, ν +
Due
-a.e. inRn
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. inRn. Sinceν is arbitrary, by Remarks 7.2 and 7.3 the restriction of f to the affine space Txu is differentiable at ˜u(x) for
Due
-almost everyx∈Rn 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−1A(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 bl of the characteristic polynomial of (10). It is well known in graph theory that the coefficientsblcan be expressed as a sum over certain subgraphs. It is interesting to see whetherAl,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) Hc= 1
2(n−1)!.
For a complete bipartite graphKn1n2, letλi= 0, i= 1, . . . , n. By (39),
(42) Al=
−n1!n2!δn1n2, ifl= 2
0, otherwise.
Theorem 7.2 leads to
(43) Hc = 1
n1+n2
n1!n2!δn1n2.
Now, we consider an asymmetrical approach. Theorem 3.3 leads to (44) detK(t= 1, t1, . . . , tn;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 xn+yn =zn 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∞+πA0∼A∞+πA0
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(ri) XXX
i<B iodd
YYY
κ
κ(ri)
\[\sum_{\substack{i<B\\\text{$i$ odd}}}
\prod_\kappa \kappa F(r_i)\qquad
\mathop{\pmb{\sum}}_{\substack{i<B\\\text{$i$ odd}}}
\mathop{\pmb{\prod}}_\kappa \kappa(r_i)
\]
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 LATEX provides\overrightarrow and\overleftarrow).
−−−−−−→
ψδ(t)Eth=ψδ(t)Eth
−−−−−−→
←−−−−−−
ψδ(t)Eth=ψδ(t)Eth
←−−−−−−
←−−−−→
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 seriesA1, A2, . . ., the regional sumA1+A2+
· · ·, the orthogonal productA1A2· · ·, and the infinite integral Z
A1
Z
A2
· · · 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 LATEX 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 LATEX: ...
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]−−−−→∂0α(b) E∂0b
\[0 \xleftarrow[\zeta]{\alpha} F\times\triangle[n-1]
\xrightarrow{\partial_0\alpha(b)} E^{\partial_0b}\]
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
X0
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=y0 if and only if y0k=δkyτ(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 supx∈Rn|f(x)|
\[\norm{f}_\infty=
\esssup_{x\in R^n}\abs{f(x)}\]
meas1{u∈R1+: f∗(u)> α}= measn{x∈Rn: |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−→(mλi·)∗≤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 (modm2) (52)
x≡y+ 1 modm2 (53)
x≡y+ 1 (m2) (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= 2k− k
1
2k−1+ k
2
2k−2
+· · ·+ (−1)l k
l
2k−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)Xj0 \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: Xj = (1/p
λj)Xj0, 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 bc 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 ϕkn2
(ϕ2, ε1) ϕ
(ϕ2, ε2) . . . 0
. . . . ϕkn1
(ϕn, ε1)
ϕkn2
(ϕn, ε2) . . . ϕkn n−1
(ϕn, εn−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 tε 0
Lx,yx(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
yZ
tε0
L
x,yx(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 tε 0
Lx,yε(εu)ϕ(x)du
=h Z
Lx,zϕ(x)ρx(dz) +h
1 tε
Ey
Z tε
0
Lx,yx(s)ϕ(x)ds−tε Z
Lx,zϕ(x)ρx(dz)
+ 1 tε
Ey
Z tε
0
Lx,yx(s)ϕ(x)ds−Ex,y Z tε
0
Lx,yε(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}
