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American Mathematical Society

January 1995

1 Introduction

This paper contains examples of various features from AMS-L^{A}TEX.

2 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 by the degree of its corresponding vertex; i.e., the ith diagonal entry is identified with the degree of the ith vertex. It is well known that

det K(i|i) = the number of spanning trees of G, i = 1, …, n  (1)

where K(i|i) is the ith principal submatrix of K.

\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} = ⋃_{j} C_{i(j)}. It is obvious that the collection of Hamiltonian cycles is a subset of C_{i}. Note that the cardinality of C_{i} is k_{ii} det K(i|i). Let X̂ = {x̂_{1}, …, x̂_{n}}.

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

Define multiplication for the elements of X̂ by

x̂_{i}x̂_{j} = x̂_{j}x̂_{i}, x̂_{i}^{2} = 0, i, j = 1, …, n.  (2)

Let k̂_{ij} = k_{ij}x̂_{j} and k̂_{ij} = − ∑_{j≠i} k̂_{ij}. Then the number of Hamiltonian cycles H_{c} is given by the relation [8]

(∏_{j=1}^{n} x̂_{j})H_{c} = \frac{1}{2}k̂_{ij} det K̂(i|i),  i = 1, …, n.  (3)

The task here is to express (3) in a form free of any x̂_{i}, 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 K_{n} and in a complete bipartite graph K_{n_{1}n_{2}} can only be found from first combinatorial principles [4]. One wonders if there exists a formula which can be used very efficiently to produce K_{n} and K_{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 [3]. 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 resulting formula also involves the determinant and permanent, but it can easily be applied to K_{n} and K_{n_{1}n_{2}}. In addition, we eliminate the permanent from H_{c} and show that H_{c} can be represented by a determinantal function of multivariables, each variable with domain {0, 1}. Furthermore, we show that H_{c} can be written by number of spanning trees of subgraphs. Finally, we apply the formulas to a complete multigraph K_{n_{1}...n_{p}}.

The conditions a_{ij} = a_{ji}, i, j = 1, …, n, are not required in this paper. All formulas can be extended to a digraph simply by multiplying H_{c} by 2.

3 Main Theorem

Notation. For p, q ∈ P and n ∈ ω we write (q, n) ≤ (p, n) if q ≤ p and A_{q,n} = A_{p,n}.

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

Let B = (b_{ij}) be an n × n matrix. Let n = {1, …, n}. Using the properties of (2), it is readily seen that

Lemma 3.1.

∏_{i∈n}(∑_{j∈n} b_{ij}x̂_{i}) = (∏_{i∈n} x̂_{i}) per B  (4)

where per B is the permanent of B.

Let Ŷ = {ŷ_{1}, …, ŷ_{n}}. Define multiplication for the elements of Ŷ by

ŷ_{i}ŷ_{j} + ŷ_{j}ŷ_{i} = 0, i, j = 1, …, n.  (5)

Then, it follows that

Lemma 3.2.

∏_{i∈n}(∑_{j∈n} b_{ij}ŷ_{j}) = (∏_{i∈n} ŷ_{i}) det B.  (6)

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

∑_{j∈n} b_{ij}ŷ_{j} = ∑_{j∈n} b_{ij}^{(λ)}ŷ_{j} + (b_{ii} − λ_{i})ŷ_{i}ŷ  (7)

where

b_{ii}^{(λ)} = λ_{i}, b_{ij}^{(λ)} = b_{ij}, i ≠ j.  (8)

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

Theorem 3.3.

det B = ∑_{l=0}^{n} ∑_{I_{l}⊆n} ∏_{i∈I_{l}}(b_{ii} − λ_{i}) det B^{(λ)}(I_{l}|I_{l}),  (9)

where I_{l} = {i_{1}, …, i_{l}} and B^{(λ)}(I_{l}|I_{l}) is the principal submatrix obtained from B^{(λ)} by deleting its i_{1}, …, i_{l} rows and columns.

Remark 3.1. Let M be an n × n matrix. The convention M(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 [9]:

Corollary 3.4. Write det(B − xI) = ∑_{l=0}^{n}(−1)^{l}b_{l}x^{l}. Then

b_{l} = ∑_{I_{l}⊆n} det B(I_{l}|I_{l}).  (10)

Let

K(t, t_{1}, …, t_{n}) = (\begin{matrix}D_{1}t & & −a_{12}t_{2} & & … & & & −a_{1n}t_{n} \\ −a_{21}t_{1} & & D_{2}t & & … & & & −a_{2n}t_{n} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ −a_{n1}t_{1} & & −a_{n2}t_{2} & & … & & & D_{n}t \end{matrix}) ,  (11)

\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} = ∑_{j∈n} a_{ij}t_{j}, i = 1, …, n.  (12)

Set

D(t_{1}, …, t_{n}) = \frac{δ}{δt} det K(t, t_{1}, …, t_{n})|_{t=1} .

Then

D(t_{1}, …, t_{n}) = ∑_{i∈n} D_{i} det K(t = 1, t_{1}, …, t_{n}; i|i),  (13)

where K(t = 1, t_{1}, …, t_{n}; i|i) is the ith principal submatrix of K(t = 1, t_{1}, …, t_{n}). Theorem 3.3 leads to

det K(t_{1}, t_{1}, …, t_{n}) = ∑_{I∈n}(−1)^{|I|}t^{n−|I|} ∏_{i∈I} t_{i} ∏_{j∈I}(D_{j} + λ_{j}t_{j}) det A^{(λt)}(\overline{I}|\overline{I}).

(14)

Note that

det K(t = 1, t_{1}, …, t_{n}) = ∑_{I∈n}(−1)^{|I|} ∏_{i∈I} t_{i} ∏_{j∈I}(D_{j} + λ_{j}t_{j}) det A^{(λ)}(\overline{I}|\overline{I}) = 0.

(15)

Let t_{i} = x̂_{i}, i = 1, …, n. Lemma 3.1 yields

(∑_{i∈n} a_{l_{i}}x_{i}) det K(t = 1, x_{1}, …, x_{n}; l|l)

= (∏_{i∈n} x̂_{i}) ∑_{I⊆n−\{l\}}(−1)^{|I|} per A^{(λ)}(I|I) det A^{(λ)}(\overline{I} ∪ \{l\}|\overline{I} ∪ \{l\}). (16)

\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.

H_{c} = \frac{1}{2n} ∑_{l=0}^{n}(−1)^{l}D_{l},  (17)

where

D_{l} = ∑_{I_{l}⊆n} D(t_{1}, …, t_{n})2|_{t_{i}=\{ \begin{matrix}0, & if i∈I_{l} \\ 1, & otherwise \end{matrix} , i=1,...,n}.  (18)

4 Application

We consider here the applications of Theorems 5.1 and 5.2 to a complete multipartite 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} ∏_{i=1}^{p}(n − n_{i})^{n_{i}−1}  (19)

where

n = n_{1} + ⋯ + n_{p}.  (20)

It follows from Theorems 5.1 and 5.2 that

\begin{aligned}H_{c} & = \frac{1}{2n} ∑_{l=0}^{n}(−1)^{l}(n − l)^{p−2} ∑_{l_{1}+···+l_{p}=l} ∏_{i=1}^{p} \binom{n_{i}}{l_{i}} \\ & · [(n − l) − (n_{i} − l_{i})]^{n_{i}−l_{i}} · [(n − l)^{2} − ∑_{j=1}^{p}(n_{i} − l_{i})^{2}]. \end{aligned}  (21)

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

and

\begin{aligned}H_{c} & = \frac{1}{2} ∑_{l=0}^{n−1}(−1)^{l}(n − l)^{p−2} ∑_{l_{1}+···+l_{p}=l} ∏_{i=1}^{p} \binom{n_{i}}{l_{i}} \\ & · [(n − l) − (n_{i} − l_{i})]^{n_{i}−l_{i}} (1 − \frac{l_{p}}{n_{p}} ) [(n − l) − (n_{p} − l_{p})]. \end{aligned}  (22)

The enumeration of H_{c} in a K_{n_{1}···n_{p}} graph can also be carried out by Theorem 7.2 or 7.3 together with the algebraic method of (2). Some elegant representations may be obtained. For example, H_{c} in a K_{n_{1}n_{2}n_{3}} graph may be written

\begin{aligned}H_{c} = & \frac{n_{1}! n_{2}! n_{3}!}{n_{1} + n_{2} + n_{3}} ∑_{i} [\binom{n_{1}}{i}\binom{n_{2}}{n_{3} − n_{1} + i}\binom{n_{3}}{n_{3} − n_{2} + i} \\ & + \binom{n_{1} − 1}{i}\binom{n_{2} − 1}{n_{3} − n_{1} + i}\binom{n_{3} − 1}{n_{3} − n_{2} + i}] . \end{aligned}  (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 Secret Key Exchange protocol was first introduced in the seminal paper of Diffie and Hellman [1]. [1] presented a concrete implementation of a Secret Key Exchange protocol, dependent on a specific assumption (a variant on the discrete log), specially tailored to yield Secret Key Exchange. Secret Key Exchange is of course trivial if trapdoor permutations exist. However, there is no known implementation based on a weaker general assumption.

The concept of an informationally one-way function was introduced in [5]. We give only an informal definition here:

Definition 5.1. A polynomial time computable function f = {f_{k}} is informationally one-way if there is no probabilistic polynomial time algorithm which (with probability of the form 1−k^{−e} for some e > 0) returns on input y ∈ {0, 1}^{k} a random element of f^{−1}(y).

In the non-uniform setting [5] show that these are not weaker than one-way functions:

Theorem 5.1 ([5] (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 programming have been extended to the case of convex quadratic programs [11, 13], certain linear complementarity problems [7, 10], and the nonlinear complementarity problem [6]. The connection between these algorithms and the classical Newton method for nonlinear equations is well explained in [7].

6 Review

We begin our discussion with the following definition:

Definition 6.1. A function H: ℜ^{n} → ℜ^{n} is said to be B-differentiable at the point z if (i) H is Lipschitz continuous in a neighborhood of z, and (ii) there exists a positive homogeneous function BH(z): ℜ^{n} → ℜ^{n}, called the B-derivative of H at z, such that

lim_{v→0} \frac{H(z + v) − H(z) − BH(z)v}{∥v∥} = 0.

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

lim_{(v,v′)→(0,0)} \frac{H(z + v) − H(z + v′) − BH(z)(v − v′)}{∥v − v′∥} = 0.

Lemma 6.1. There exists a smooth function ψ_{0}(z) defined for |z| > 1 − 2a satisfying the following properties:

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

(ii) If |z| > 1, then ψ_{0}(z) = 1.

(iii) For all z in the domain of ψ_{0}, Δ_{0} ln ψ_{0} ≥ 0.

(iv) If 1 − 2a < |z| < 1 − a, then Δ_{0} ln ψ_{0} ≥ c_{3} > 0.

Proof. We choose ψ_{0}(z) to be a radial function depending only on r = |z|. Let h(r) ≥ 0 be a suitable smooth function satisfying h(r) ≥ c_{3} for 1 − 2a < |z| < 1 − a, and h(r) = 0 for |z| > 1 − \frac{a}{2}. The radial Laplacian

Δ_{0} ln ψ_{0}(r) = (\frac{d^{2}}{dr^{2}} + \frac{1}{r} \frac{d}{dr} ) ln ψ_{0}(r)

has smooth coefficients for r > 1 − 2a. Therefore, we may apply the existence and uniqueness theory for ordinary differential equations. Simply let ln ψ_{0}(r) be the solution of the differential equation

(\frac{d^{2}}{dr^{2}} + \frac{1}{r} \frac{d}{dr} ) ln ψ_{0}(r) = h(r)

with initial conditions given by ln ψ_{0}(1) = 0 and ln ψ′_{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 Φ_{0}(z) for z ∈ C − ⋃_{ν} D_{ν}(a) by setting Φ_{0}(z) = 1 when z ∉ ⋃_{ν} D_{ν} and Φ_{0}(z) = ψ_{0}((z − z_{ν})/δ) when z is an element of D_{ν}. It follows from Lemma 6.1 that Φ_{0} satisfies the properties:

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

(ii) Δ_{0} ln Φ_{0} ≥ 0 for all z ∈ C − ⋃_{ν} D_{ν}(a), the domain where the function Φ_{0} is defined.

(iii) Δ_{0} ln Φ_{0} ≥ c_{3}δ^{−2} when (1 − 2a)δ < |z − z_{ν}| < (1 − a)δ.

Let A_{ν} denote the annulus A_{ν} = {(1 − 2a)δ < |z − z_{ν}| < (1 − a)δ}, and set A = ⋃_{ν} A_{ν}. The properties (2) and (3) of Φ_{0} may be summarized as Δ_{0} ln Φ_{0} ≥ c_{3}δ^{−2}χ_{A}, where χ_{A} is the characteristic function of A.

Suppose that α is a nonnegative real constant. We apply Proposition 3.5 with Φ(z) = Φ_{0}(z)e^{α|z|^{2}}. If u ∈ C_{0}^{∞}(R^{2} − ⋃_{ν} D_{ν}(a)), assume that D is a bounded domain containing the support of u and A ⊂ D ⊂ R^{2} − ⋃_{ν} D_{ν}(a). A calculation gives

∫_{D} |\overline{∂}u|^{2} Φ_{0}(z)e^{α|z|^{2}} ≥ c_{4}α ∫_{D} |u|^{2} Φ_{0}e^{α|z|^{2}} + c_{5}δ^{−2} ∫_{A} |u|^{2} Φ_{0}e^{α|z|^{2}}.

The boundedness, property (1) of Φ_{0}, then yields

∫_{D} |\overline{∂}u|^{2} e^{α|z|^{2}} ≥ c_{6}α ∫_{D} |u|^{2} e^{α|z|^{2}} + c_{7}δ^{−2} ∫_{A} |u|^{2} e^{α|z|^{2}}.

Let B(X) be the set of blocks of Λ_{X} and let b(X) = |B(X)|. If ϕ ∈ Q_{X} then ϕ is constant on the blocks of Λ_{X}.

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

If Λ_{ϕ} ≥ Λ_{X} then Λ_{ϕ} = Λ_{Y} for some Y ≥ X so that

Q_{X} = ⋃_{Y ≥X} P_{Y} .

Thus by Möbius inversion

|P_{Y} | = ∑_{X≥Y} μ(Y, X) |Q_{X}| .

Thus there is a bijection from Q_{X} to W^{B(X)}. In particular |Q_{X}| = w^{b(X)}.

Next note that b(X) = dim X. We see this by choosing a basis for X consisting of vectors v^{k} defined by

v_{i}^{k} = \begin{cases}1 & if i ∈ Λ_{k}, \\ 0 & otherwise. \end{cases}

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

Lemma 6.2. Let A be an arrangement. Then

χ(A, t) = ∑_{B⊆A}(−1)^{|B|}t^{dim T (B)}.

In order to compute R′′ recall the definition of S(X, Y) from Lemma 3.1. Since H ∈ B, A_{H} ⊆ B. Thus if T (B) = Y then B ∈ S(H, Y). Let L′′ = L(A′′).

Then

R′′ = ∑_{H∈B⊆A}(−1)^{|B|}t^{dim T (B)}

= ∑_{Y ∈L′′} ∑_{B∈S(H,Y)}(−1)^{|B|}t^{dim Y}

= − ∑_{Y ∈L′′} ∑_{B∈S(H,Y)}(−1)^{|B−A_{H}|}t^{dim Y}

= − ∑_{Y ∈L′′} μ(H, Y)t^{dim Y}

= −χ(A′′, t).

(25)

Corollary 6.3. Let (A, A′, A′′) be a triple of arrangements. Then

π(A, t) = π(A′, t) + tπ(A′′, t).

Definition 6.2. Let (A, A′, A′′) be a triple with respect to the hyperplane H ∈ A. Call H a separator if T (A) ∉ L(A′).

Corollary 6.4. Let (A, A′, A′′) be a triple with respect to H ∈ A.

(i) If H is a separator then

μ(A) = −μ(A′′)

and hence

|μ(A)| = |μ(A′′)| .

(ii) If H is not a separator then

μ(A) = μ(A′) − μ(A′′)

and

|μ(A)| = |μ(A′)| + |μ(A′′)| .

Proof. It follows from Theorem 5.1 that π(A, t) has leading term

(−1)^{r(A)}μ(A)t^{r(A)}.

The conclusion follows by comparing coefficients of the leading terms on both sides of the equation in Corollary 6.3. If H is a separator then r(A′) < r(A) and there is no contribution from π(A′, t).

The Poincaré polynomial of an arrangement will appear repeatedly in these notes. It will be shown to equal the Poincaré polynomial of the graded algebras which we are going to associate with A. It is also the Poincaré polynomial

Figure 1: Q(A_{1}) = xyz(x − z)(x + z)(y − z)(y + z)

Figure 2: Q(A_{2}) = xyz(x + y + z)(x + y − z)(x − y + z)(x − y − z)

of the complement M(A) for a complex arrangement. Here we prove that the Poincaré polynomial is the chamber counting function for a real arrangement. The complement M(A) is a disjoint union of chambers

M(A) = ⋃_{C∈Cham(A)} C.

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

Theorem 6.5. Let A_{R} be a real arrangement. Then

|Cham(A_{R})| = π(A_{R}, 1).

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

Theorem 6.6. Let ϕ be a protocol for a random pair (X, Y). If one of σ_{ϕ}(x′, y) and σ_{ϕ}(x, y′) is a prefix of the other and (x, y) ∈ S_{X,Y}, then

⟨σ_{j}(x′, y)⟩_{j=1}^{∞} = ⟨σ_{j}(x, y)⟩_{j=1}^{∞} = ⟨σ_{j}(x, y′)⟩_{j=1}^{∞}.

Proof. We show by induction on i that

⟨σ_{j}(x′, y)⟩_{j=1}^{i} = ⟨σ_{j}(x, y)⟩_{j=1}^{i} = ⟨σ_{j}(x, y′)⟩_{j=1}^{i}.

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

If ϕ is a protocol for (X, Y), and (x, y), (x′, y) are distinct inputs in S_{X,Y}, then, by the correct-decision property, ⟨σ_{j}(x, y)⟩_{j=1}^{∞} ≠ ⟨σ_{j}(x′, y)⟩_{j=1}^{∞}.

Equation (25) defined 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^{1} of codewords {σ_{ϕ}(x, y) : x ∈ S_{X|Y} (y)} is prefix free.

7 One-Way Complexity

Ĉ_{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 transmit any feedback messages. Starting with S_{X,Y}, the support set of (X, Y), we define G(X|Y), the characteristic hypergraph of (X, Y), and show that

Ĉ_{1}(X|Y) = ⌈ log χ(G(X|Y))⌉ .

Let (X, Y) be a random pair. For each y in S_{Y}, the support set of Y, Equation (25) defined S_{X|Y} (y) to be the set of possible x values when Y = y. The characteristic hypergraph G(X|Y) of (X, Y) has S_{X} as its vertex set and the hyperedge S_{X|Y} (y) for each y ∈ S_{Y}.

We can now prove a continuity theorem.

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

Theorem 7.1. Let Ω ⊂ R^{n} be an open set, let u ∈ BV (Ω; R^{m}), and let

T_{x}^{u} = \{y ∈ R^{m} : y = ũ(x) + ⟨\frac{Du}{|Du|}(x), z⟩ for some z ∈ R^{n}\} (26)

for every x ∈ Ω\S_{u}. Let f: R^{m} → R^{k} be a Lipschitz continuous function such that f(0) = 0, and let v = f(u): Ω → R^{k}. Then v ∈ BV (Ω; R^{k}) and

Jv = (f(u^{+}) − f(u^{−})) ⊗ ν_{u} · H_{n−1}|_{S_{u}} .  (27)

In addition, for |D̃u|-almost every x ∈ Ω the restriction of the function f to T_{x}^{u} is differentiable at ũ(x) and

D̃v = ∇(f|_{T_{x}^{u}})(ũ)\frac{D̃u}{|D̃u|} · |D̃u| .  (28)

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 as t → 0. Then

lim_{h→0^{+}} g(ω(h)) = L ⇔ lim_{h→0^{+}} g(h) = L

for any function g: ]0, +∞[ → R. Remark 7.2. Let g: R^{n} → R be a Lipschitz continuous function and assume that

L(z) = lim_{h→0^{+}} \frac{g(hz) − g(0)}{h}

exists for every z ∈ Q^{n} and that L is a linear function of z. Then g is differentiable at 0. Remark 7.3. Let A: R^{n} → R^{m} be a linear function, and let f: R^{m} → R be a function. Then the restriction of f to the range of A is differentiable at 0 if and only 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 (Ω; R^{k}) and

|Dv| (B) ≤ K |Du| (B) ∀B ∈ B(Ω),  (29)

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

lim_{h→+∞} ∫_{Ω} |∇u_{h}| dx = |Du| (Ω).

The functions v_{h} = f(u_{h}) are locally Lipschitz continuous in Ω, and the definition of differential implies that |∇v_{h}| ≤ K |∇u_{h}| 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→+∞} ∫_{Ω} |∇v_{h}| dx

≤ K lim inf_{h→+∞} ∫_{Ω} |∇u_{h}| dx = K |Du| (Ω).

(30)

Since f(0) = 0, we have also

∫_{Ω} |v| dx ≤ K ∫_{Ω} |u| dx;

therefore u ∈ BV (Ω; R^{k}). Repeating the same argument for every open set A ⊂ Ω, we get (29) for every B ∈ B(Ω), because |Dv|, |Du| are Radon measures. To prove Lemma 6.1, first we observe that

S_{v} ⊂ S_{u},  ṽ(x) = f(ũ(x)) ∀x ∈ Ω\S_{u}.  (31)

In fact, for every ε > 0 we have

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

hence

lim_{ρ→0^{+}} \frac{|\{y ∈ B_{ρ}(x) : |v(y) − f(ũ(x))| > ε\}|}{ρ^{n}} = 0

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

(v^{+}(x) − v^{−}(x)) ⊗ ν_{v} = (f(u^{+}(x)) − f(u^{−}(x))) ⊗ ν_{u} if x ∈ S_{v}

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

\begin{aligned}Jv(B) = ∫_{B∩S_{v}}(v^{+} − v^{−}) ⊗ ν_{v} dH_{n−1} & = ∫_{B∩S_{v}}(f(u^{+}) − f(u^{−})) ⊗ ν_{u} dH_{n−1} \\ & = ∫_{B∩S_{u}}(f(u^{+}) − f(u^{−})) ⊗ ν_{u} dH_{n−1} \end{aligned}

and Lemma 6.1 is proved.

To prove (31), it is not restrictive to assume that k = 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 S_{u} is at most countable, (7) yields that |D̃v| (S_{u}\S_{v}) =

0, so that (19) and (21) imply that Dv = D̃v + Jv is the Radon-Nikodým decomposition of Dv in absolutely continuous and singular part with respect to |D̃u|. By Theorem 5.2, we have

\frac{D̃v}{|D̃u|}(t) = lim_{s→t^{+}} \frac{Dv([t, s[)}{|D̃u| ([t, s[)}, \frac{D̃u}{|D̃u|}(t) = lim_{s→t^{+}} \frac{Du([t, s[)}{|D̃u| ([t, s[)}

|D̃u|-almost everywhere in R. It is well known (see, for instance, [12, 2.5.16]) that every one-dimensional function of bounded variation w has a unique left continuous representative, i.e., a function ŵ such that ŵ = w almost everywhere and lim_{s→t^{−}} ŵ(s) = ŵ(t) for every t ∈ R. These conditions imply

û(t) = Du(]−∞, t[),  v̂(t) = Dv(]−∞, t[) ∀t ∈ R  (32)

and

v̂(t) = f(û(t)) ∀t ∈ R.  (33)

Let t ∈ R be such that |D̃u| ([t, s[) > 0 for every s > t and assume that the limits in (22) exist. By (23) and (24) we get

\frac{v̂(s) − v̂(t)}{|D̃u| ([t, s[)} = \frac{f(û(s)) − f(û(t))}{|D̃u| ([t, s[)}

= \frac{f(û(s)) − f(û(t) + \frac{D̃u}{|D̃u|}(t) |D̃u| ([t, s[))}{|D̃u| ([t, s[)}

+ \frac{f(û(t) + \frac{D̃u}{|D̃u|}(t) |D̃u| ([t, s[)) − f(û(t))}{|D̃u| ([t, s[)}

for every s > t. Using the Lipschitz condition on f we find

|\frac{v̂(s) − v̂(t)}{|D̃u| ([t, s[)} − \frac{f(û(t) + \frac{D̃u}{|D̃u|}(t) |D̃u| ([t, s[)) − f(û(t))}{|D̃u| ([t, s[)}|

≤ K |\frac{û(s) − û(t)}{|D̃u| ([t, s[)} − \frac{D̃u}{|D̃u|}(t)| .

By (29), the function s → |D̃u| ([t, s[) is continuous and converges to 0 as s ↓ t. Therefore Remark 7.1 and the previous inequality imply

\frac{D̃v}{|D̃u|}(t) = lim_{h→0^{+}} \frac{f(û(t) + h\frac{D̃u}{|D̃u|}(t)) − f(û(t))}{h} |D̃u| -a.e. in R.

By (22), û(x) = ũ(x) for every x ∈ R\S_{u}; moreover, applying the same argument to the functions u′(t) = u(−t), v′(t) = f(u′(t)) = v(−t), we get

\frac{D̃v}{|D̃u|}(t) = lim_{h→0} \frac{f(ũ(t) + h\frac{D̃u}{|D̃u|}(t)) − f(ũ(t))}{h}  |D̃u| -a.e. in R

and our statement is proved.

Step 2

Let us consider now the general case n > 1. Let ν ∈ R^{n} be such that |ν| = 1, and let π_{ν} = {y ∈ R^{n} : ⟨y, ν⟩ = 0}. In the following, we shall identify R^{n} with π_{ν} ×R, and we shall denote by y the variable ranging in π_{ν} and by t the variable ranging in R. By the just proven one-dimensional result, and by Theorem 3.3, we get

lim_{h→0} \frac{f(ũ(y + tν) + h\frac{D̃u_{y}}{|D̃u_{y}|}(t)) − f(ũ(y + tν))}{h} = \frac{D̃v_{y}}{|D̃u_{y}|}(t)  |D̃u_{y}| -a.e. in R

for H_{n−1}-almost every y ∈ π_{ν}. We claim that

\frac{⟨D̃u, ν⟩}{|⟨D̃u, ν⟩|}(y + tν) = \frac{D̃u_{y}}{|D̃u_{y}|}(t)  |D̃u_{y}| -a.e. in R  (34)

Sample paper for the amsmath package 16 for H_{n−1}-almost every y ∈ π_{ν}. In fact, by (16) and (18) we get

∫_{π_{ν}} \frac{D̃u_{y}}{|D̃u_{y}|} · |D̃u_{y}| dH_{n−1}(y) = ∫_{π_{ν}} D̃u_{y} dH_{n−1}(y)

= ⟨D̃u, ν⟩ = \frac{⟨D̃u, ν⟩}{|⟨D̃u, ν⟩|} · |⟨D̃u, ν⟩| = ∫_{π_{ν}} \frac{⟨D̃u, ν⟩}{|⟨D̃u, ν⟩|}(y + ·ν) · |D̃u_{y}| dH_{n−1}(y)

and (24) follows from (13). By the same argument it is possible to prove that

\frac{⟨D̃v, ν⟩}{|⟨D̃u, ν⟩|}(y + tν) = \frac{D̃v_{y}}{|D̃u_{y}|}(t) |D̃u_{y}| -a.e. in R (35)

for H_{n−1}-almost every y ∈ π_{ν}. By (24) and (25) we get

lim_{h→0} \frac{f(ũ(y + tν) + h\frac{⟨D̃u, ν⟩}{|⟨D̃u, ν⟩|}(y + tν)) − f(ũ(y + tν))}{h} = \frac{⟨D̃v, ν⟩}{|⟨D̃u, ν⟩|}(y + tν)

for H_{n−1}-almost every y ∈ π_{ν}, and using again (14), (15) we get

lim_{h→0} \frac{f(ũ(x) + h\frac{⟨D̃u, ν⟩}{|⟨D̃u, ν⟩|}(x)) − f(ũ(x))}{h} = \frac{⟨D̃v, ν⟩}{|⟨D̃u, ν⟩|}(x)

|⟨D̃u, ν⟩|-a.e. in R^{n}.

Since the function |⟨D̃u, ν⟩| / |D̃u| is strictly positive |⟨D̃u, ν⟩|-almost everywhere, we obtain also

lim_{h→0} \frac{f(ũ(x) + h\frac{|⟨D̃u, ν⟩|}{|D̃u|}(x)\frac{⟨D̃u, ν⟩}{|⟨D̃u, ν⟩|}(x)) − f(ũ(x))}{h}

= \frac{|⟨D̃u, ν⟩|}{|D̃u|}(x)\frac{⟨D̃v, ν⟩}{|⟨D̃u, ν⟩|}(x)

|⟨D̃u, ν⟩|-almost everywhere in R^{n}.

Finally, since

\frac{|⟨D̃u, ν⟩|}{|D̃u|} \frac{⟨D̃u, ν⟩}{|⟨D̃u, ν⟩|} = \frac{⟨D̃u, ν⟩}{|D̃u|} = ⟨\frac{D̃u}{|D̃u|}, ν⟩  |D̃u| -a.e. in R^{n}

\frac{|⟨D̃u, ν⟩|}{|D̃u|} \frac{⟨D̃v, ν⟩}{|⟨D̃u, ν⟩|} = \frac{⟨D̃v, ν⟩}{|D̃u|} = ⟨\frac{D̃v}{|D̃u|}, ν⟩  |D̃u| -a.e. in R^{n}

and since both sides of (33) are zero |D̃u|-almost everywhere on |⟨D̃u, ν⟩|- negligible sets, we conclude that

lim_{h→0} \frac{f (ũ(x) + h ⟨\frac{D̃u}{|D̃u|}(x), ν⟩) − f(ũ(x))}{h} = ⟨\frac{D̃v}{|D̃u|}(x), ν⟩ ,

|D̃u|-a.e. in R^{n}. Since ν is arbitrary, by Remarks 7.2 and 7.3 the restriction of

f to the affine space T_{x}^{u} is differentiable at ũ(x) for |D̃u|-almost every x ∈ R^{n} and (26) holds.

It follows from (13), (14), and (15) that

D(t_{1}, …, t_{n}) = ∑_{I∈n}(−1)^{|I|−1} |I| ∏_{i∈I} t_{i} ∏_{j∈I}(D_{j} + λ_{j}t_{j}) det A^{(λ)}(\overline{I}|\overline{I}).  (36)

Let t_{i} = x̂_{i}, i = 1, …, n. Lemma 1 leads to

D(x̂_{1}, …, x̂_{n}) = ∏_{i∈n} x̂_{i} ∑_{I∈n}(−1)^{|I|−1} |I| per A^{(λ)}(I|I) det A^{(λ)}(\overline{I}|\overline{I}).  (37)

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

Theorem 7.2.

H_{c} = \frac{1}{2n} ∑_{l=1}^{n} l(−1)^{l−1}A_{l}^{(λ)},  (38)

where

A_{l}^{(λ)} = ∑_{I_{l}⊆n} per A^{(λ)}(I_{l}|I_{l}) det A^{((λ)}(\overline{I}_{l}|\overline{I}_{l}), |I_{l}| = l.  (39)

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

We may call (38) a parametric representation of H_{c}. In computation, the parameter λ_{i} plays very important roles. The choice of the parameter usually depends on the properties of the given graph. For a complete graph K_{n}, let λ_{i} = 1, i = 1, …, n. It follows from (39) that

A_{l}^{(1)} = \begin{cases}n!, & if l = 1 \\ 0, & otherwise. \end{cases}  (40)

By (38)

H_{c} = \frac{1}{2}(n − 1)!.  (41)

For a complete bipartite graph K_{n_{1}n_{2}}, let λ_{i} = 0, i = 1, …, n. By (39),

A_{l} = \begin{cases}−n_{1}!n_{2}!δ_{n_{1}n_{2}}, & if l = 2 \\ 0, & otherwise . \end{cases}  (42)

Theorem 7.2 leads to

H_{c} = \frac{1}{n_{1} + n_{2}} n_{1}!n_{2}!δ_{n_{1}n_{2}}.  (43)

Now, we consider an asymmetrical approach. Theorem 3.3 leads to

det K(t = 1, t_{1}, …, t_{n}; l|l)

= ∑_{I⊆n−\{l\}}(−1)^{|I|} ∏_{i∈I} t_{i} ∏_{j∈I}(D_{j} + λ_{j}t_{j}) det A^{(λ)}(\overline{I} ∪ \{l\}|\overline{I} ∪ \{l\}). (44)

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

Theorem 7.3.

H_{c} = \frac{1}{2} ∑_{I⊆n−\{l\}}(−1)^{|I|} per A^{(λ)}(I|I) det A^{(λ)}(\overline{I} ∪ \{l\}|\overline{I} ∪ \{l\})  (45)

which reduces to Goulden–Jackson’s formula when λ_{i} = 0, i = 1, …, n [9].

8 Various font features of the amsmath package

8.1 Bold versions of special symbols

In the amsmath package \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}

8.2 “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:

\frac{∂x}{∂y}|||\frac{∂y}{∂z}

\[\frac{\partial x}{\partial y}
\pmb{\bigg\vert}
\frac{\partial y}{\partial z}\]

So-called “large operator” symbols such as ∑ and ∏ require an additional command, \mathop, to produce proper spacing and limits when \pmb is used. For further details see The TEXbook.

∑_{\begin{matrix}i<B \\ i odd \end{matrix}} ∏_{κ} κF (r_{i})  ∑ ∑ ∑_{\begin{matrix}i<B \\ i odd \end{matrix}} ∏ ∏ ∏_{κ} κ(r_{i})

\[\sum_{\substack{i<B\\\text{$i$ odd}}}
\prod_\kappa \kappa F(r_i)\qquad
\mathop{\pmb{\sum}}_{\substack{i<B\\\text{$i$ odd}}}
\mathop{\pmb{\prod}}_\kappa \kappa(r_i)
\]

9 Compound symbols and other features

9.1 Multiple integral signs

\iint, \iiint, and \iiiint give multiple integral signs with the spacing between them nicely adjusted, in both text and display style. \idotsint gives two integral signs with dots between them.

∫∫_{A} f(x, y) dx dy  ∫∫∫_{A} f(x, y, z) dx dy dz  (46)

∫∫∫∫_{A} f(w, x, y, z) dw dx dy dz  ∫ ⋯ ∫_{A} f(x_{1}, …, x_{k})  (47)

9.2 Over and under arrows

Some extra over and under arrow operations are provided in the amsmath pack-

age. (Basic L^{A}TEX provides \overrightarrow and \overleftarrow).

\overrightarrow{ψ_{δ}(t)E_{t}h} = \underrightarrow{ψ_{δ}(t)E_{t}h}
\overleftarrow{ψ_{δ}(t)E_{t}h} = \underleftarrow{ψ_{δ}(t)E_{t}h}
\overleftrightarrow{ψ_{δ}(t)E_{t}h} = \underleftrightarrow{ψ_{δ}(t)E_{t}h}

\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:

∫\overrightarrow{AB} ax dx

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

9.3 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 series A_{1}, A_{2}, …, the regional sum A_{1} + A_{2} + ⋯, the orthogonal product A_{1}A_{2} ⋯, and the infinite integral

∫_{A_{1}} ∫_{A_{2}}⋯

9.4 Accents in math

Double accents:

ˆĤ ˇ Č ˜ T̃ ´ Á ` G̀ ˙ Ḋ ¨ D̈ ˘ B̆ ¯ B̄ V⃗

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

This double accent operation is complicated and tends to slow down the processing of a L^{A}TEX file.

9.5 Dot accents

\dddot and \ddddot are available to produce triple and quadruple dot accents in addition to the \dot and \ddot accents already available in L^{A}TEX:

\begin{matrix}... \\ Q \end{matrix}  \begin{matrix}.... \\ R \end{matrix}

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

9.6 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 the β:

\sqrt[β]{k}

9.7 Boxed formulas

The command \boxed puts a box around its argument, like \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} .

9.8 Extensible arrows

\xleftarrow and \xrightarrow produce arrows that extend automatically to accommodate unusually wide subscripts or superscripts. The text of the subscript or superscript are given as an optional resp. mandatory argument: Example:

0 ⟵_{ζ}^{α} F × △[n − 1] ⟶^{∂_{0}α(b)} E^{∂_{0}b}

\[0 \xleftarrow[\zeta]{\alpha} F\times\triangle[n-1]
  \xrightarrow{\partial_0\alpha(b)} E^{\partial_0b}\]
9.9 \overset, \underset, and \sideset

Examples:

X^{∗}  X_{∗}  X_{b}^{a}

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

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

\begin{aligned}{∗ \atop ∗} & ∏{∗ \atop ∗} \\ & k \end{aligned} \begin{matrix}∑ & ′ \\ 0≤i≤m \end{matrix} E_{i}βx

\[\sideset{_*^*}{_*^*}\prod_k\qquad
\sideset{}{'}\sum_{0\le i\le m} E_i\beta x
\]

9.10 The \text command

The main use of the command \text is for words or phrases in a display:

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

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

9.11 Operator names

The more common math functions such as log, sin, and lim have predefined con-

trol sequences: \log, \sin, \lim. The amsmath package provides \DeclareMathOperator

and \DeclareMathOperator* for producing new function names that will have the same typographical treatment. Examples:

∥f∥_{∞} = ess sup_{x∈R^{n}} |f(x)|

\[\norm{f}_\infty=
\esssup_{x\in R^n}\abs{f(x)}\]

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

\[\meas_1\{u\in R_+^1\colon f^*(u)>\alpha\}
=\meas_n\{x\in R^n\colon \abs{f(x)}\geq\alpha\}
\quad \forall\alpha>0.\]

\esssup and \meas would be defined in the document preamble as

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

The following special operator names are predefined in the amsmath package:

\varlimsup, \varliminf, \varinjlim, and \varprojlim. Here’s what they

look like in use:

\overline{lim}_{n→∞}Q(u_{n}, u_{n} − u^{#}) ≤ 0  (48)
\underline{lim}_{n→∞} |a_{n+1}| / |a_{n}| = 0  (49)
\underrightarrow{lim}(m_{i}^{λ}·)^{∗} ≤ 0  (50)
\underleftarrow{lim}_{p∈S(A)}A_{p} ≤ 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}

9.12 \mod and its relatives

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

x ≡ y + 1 (mod m^{2})  (52)
x ≡ y + 1 mod m^{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}

9.13 Fractions and related constructions

The usual notation for binomials is similar to the fraction concept, so it has a similar command \binom with two arguments. Example:

\begin{aligned}∑_{γ∈Γ_{C}} I_{γ} & = 2^{k} − \binom{k}{1}2^{k−1} + \binom{k}{2}2^{k−2} \\ & + ⋯ + (−1)^{l}\binom{k}{l}2^{k−l} + ⋯ + (−1)^{k} \\ & = (2 − 1)^{k} = 1 \end{aligned}  (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: \frac{n + 1}{2}  \overwithdelims: ⟨\frac{n + 1}{2}⟩  (56)
           \atop: {n + 1 \atop 2}  \atopwithdelims: \binom{n + 1}{2}  (57)
          \above: \frac{n + 1}{2}  \abovewithdelims: [\frac{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}

9.14 Continued fractions

The continued fraction

\frac{1}{\sqrt{2} + \frac{1}{\sqrt{2} + \frac{1}{\sqrt{2} + \frac{1}{\sqrt{2} + \frac{1}{\sqrt{2} + ⋯}}}}}  (59)

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.

9.15 Smash

In amsmath there are optional arguments t and b for 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 formula X_{j} = (1/\sqrt{λ_{j}})X′_{j} \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/\sqrt{λ_{j}})X′_{j}, with the radical extending to encompass the depth of the subscript j.

9.16 The ‘cases’ environment

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

P_{r−j} = \begin{cases}0 & if r − j is odd, \\ r! (−1)^{(r−j)/2} & if r − j is even. \end{cases}  (60)

\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 \text and the embedded math.

9.17 Matrix

Here are samples of the matrix environments, \matrix, \pmatrix, \bmatrix,

\vmatrix and \Vmatrix:

\begin{matrix}ϑ & ϱ \\ φ & ϖ \end{matrix} (\begin{matrix}ϑ & ϱ \\ φ & ϖ \end{matrix}) [\begin{matrix}ϑ & ϱ \\ φ & ϖ \end{matrix}] |\begin{matrix}ϑ & ϱ \\ φ & ϖ \end{matrix}| ∥\begin{matrix}ϑ & ϱ \\ φ & ϖ \end{matrix}∥  (61)

\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 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: ( \begin{matrix}a & b \\ c & d \end{matrix} ) 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) = ∥\begin{matrix}\frac{φ}{(φ_{1}, ε_{1})} & & & 0 & & & … & & & 0 \\ \frac{φk_{n2}}{(φ_{2}, ε_{1})} & & & \frac{φ}{(φ_{2}, ε_{2})} & & & … & & & 0 \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ \frac{φk_{n1}}{(φ_{n}, ε_{1})} & & & \frac{φk_{n2}}{(φ_{n}, ε_{2})} & & & … & & & \frac{φk_{n n−1}}{(φ_{n}, ε_{n−1})} & & & & \frac{φ}{(φ_{n}, ε_{n})} \end{matrix}∥

\[W(\Phi)= \begin{Vmatrix}
\dfrac\varphi{(\varphi_1,\varepsilon_1)}&0&\dots&0\\
\dfrac{\varphi k_{n2}}{(\varphi_2,\varepsilon_1)}&
\dfrac\varphi{(\varphi_2,\varepsilon_2)}&\dots&0\\
\hdotsfor{5}\\
\dfrac{\varphi k_{n1}}{(\varphi_n,\varepsilon_1)}&
\dfrac{\varphi k_{n2}}{(\varphi_n,\varepsilon_2)}&\dots&
\dfrac{\varphi k_{n\,n-1}}{(\varphi_n,\varepsilon_{n-1})}&
\dfrac{\varphi}{(\varphi_n,\varepsilon_n)}
\end{Vmatrix}\]

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.

9.18 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:

∑_{\begin{matrix}0≤i≤m \\ 0<j<n \end{matrix}} P (i, j)  (62)

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

∑_{\begin{matrix}0≤i≤m \\ 0<j<n \end{matrix}} P (i, j)  (63)

\sum_{\begin{subarray}{l}
        0\le i\le m\\ 0<j<n
      \end{subarray}}
 P(i,j)

9.19 Big-g-g delimiters

Here are some big delimiters, first in \normalsize:

(E_{y} ∫_{0}^{t_{ε}} L_{x,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 \Large size:

(E_{y} ∫_{0}^{t_{ε}} L_{x,y^{x}(s)}φ(x) ds)

{\Large
\[\biggl(\mathbf{E}_{y}
  \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
  \biggr)
\]}

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 as equation or align.

If there is not enough room for it, the equation number for a split 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.

\begin{aligned}f_{h,ε}(x, y) & = εE_{x,y} ∫_{0}^{t_{ε}} L_{x,y_{ε}(εu)}φ(x) du \\ & = h ∫ L_{x,z}φ(x)ρ_{x}(dz) \\ & + h[\frac{1}{t_{ε}} (E_{y} ∫_{0}^{t_{ε}} L_{x,y^{x}(s)}φ(x) ds − t_{ε} ∫ L_{x,z}φ(x)ρ_{x}(dz)) \\ & + \frac{1}{t_{ε}} (E_{y} ∫_{0}^{t_{ε}} L_{x,y^{x}(s)}φ(x) ds − E_{x,y} ∫_{0}^{t_{ε}} L_{x,y_{ε}(εs)}φ(x) ds)] \\ & = hL̂_{x}φ(x) + hθ_{ε}(x, y), \end{aligned} (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}

Unnumbered version:

f_{h,ε}(x, y) = εE_{x,y} ∫_{0}^{t_{ε}} L_{x,y_{ε}(εu)}φ(x) du

= h ∫ L_{x,z}φ(x)ρ_{x}(dz)

+ h[\frac{1}{t_{ε}} (E_{y} ∫_{0}^{t_{ε}} L_{x,y^{x}(s)}φ(x) ds − t_{ε} ∫ L_{x,z}φ(x)ρ_{x}(dz))

+ \frac{1}{t_{ε}} (E_{y} ∫_{0}^{t_{ε}} L_{x,y^{x}(s)}φ(x) ds − E_{x,y} ∫_{0}^{t_{ε}} L_{x,y_{ε}(εs)}φ(x) ds)]

= hL̂_{x}φ(x) + hθ_{ε}(x, y),

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*}

If the option centertags is included in the options list of the amsmath package, the equation numbers for split environments will be centered vertically on the height of the split:

\begin{aligned}|I_{2}| & = |∫_{0}^{T} ψ(t) \{u(a, t) − ∫_{γ(t)}^{a} \frac{dθ}{k(θ, t)} ∫_{a}^{θ} c(ξ)u_{t}(ξ, t) dξ\} dt| \\ & ≤ C_{6} ||f ∫_{Ω} |S̃_{a,−}^{−1,0}W_{2}(Ω, Γ_{l})|| ||u| →^{∘} W Ã ; 2(Ω; Γ_{r}, T)|| . \end{aligned}  (65)

Some text after to test the below-display spacing.

Use of split within align:

|I_{1}| = |∫_{Ω} gRu dΩ|

≤ C_{3} [∫_{Ω} (∫_{a}^{x} g(xß, t) dξ)^{2} dΩ]^{1/2}

× [∫_{Ω} \{u_{x}^{2} + \frac{1}{k} (∫_{a}^{x} cu_{t} dξ)^{2}\} cΩ]^{1/2}

≤ C_{4} ||f |S̃_{a,−}^{−1,0}W_{2}(Ω, Γ_{l})|| ||u| →^{∘} W Ã ; 2(Ω; Γ_{r}, T)|| .

(66)

\begin{aligned}|I_{2}| & = |∫_{0}^{T} ψ(t) \{u(a, t) − ∫_{γ(t)}^{a} \frac{dθ}{k(θ, t)} ∫_{a}^{θ} c(ξ)u_{t}(ξ, t) dξ\} dt| \\ & ≤ C_{6} ||f ∫_{Ω} |S̃_{a,−}^{−1,0}W_{2}(Ω, Γ_{l})|| ||u| →^{∘} W Ã ; 2(Ω; Γ_{r}, T)|| . \end{aligned}  (67)

Some text after to test the below-display spacing.

\begin{align}
\begin{split}\abs{I_1}&=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\
&\le C_3\left[\int_\Omega\left(\int_{a}^x
g(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\lvert \left\lvert f\left\lvert \wt{S}^{-1,0}_{a,-}
W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
\left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
(\Omega;\Gamma_r,T)\right\rvert\right\rvert.
\end{split}\label{eq:A}\\
\begin{split}\abs{I_2}&=\left\lvert \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\rvert\\
&\le C_6\left\lvert \left\lvert f\int_\Omega
\left\lvert \wt{S}^{-1,0}_{a,-}
W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
\left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
(\Omega;\Gamma_r,T)\right\rvert\right\rvert.
\end{split}
\end{align}

Unnumbered align, with a number on the second split:

|I_{1}| = |∫_{Ω} gRu dΩ|

≤ C_{3} [∫_{Ω} (∫_{a}^{x} g(xß, t) dξ)^{2} dΩ]^{1/2}

× [∫_{Ω} \{u_{x}^{2} + \frac{1}{k} (∫_{a}^{x} cu_{t} dξ)^{2}\} cΩ]^{1/2}

≤ C_{4} ||f |S̃_{a,−}^{−1,0}W_{2}(Ω, Γ_{l})|| ||u| →^{∘} W Ã ; 2(Ω; Γ_{r}, T)|| .

\begin{aligned}|I_{2}| & = |∫_{0}^{T} ψ(t) \{u(a, t) − ∫_{γ(t)}^{a} \frac{dθ}{k(θ, t)} ∫_{a}^{θ} c(ξ)u_{t}(ξ, t) dξ\} dt| \\ & ≤ C_{6} ||f ∫_{Ω} |S̃_{a,−}^{−1,0}W_{2}(Ω, Γ_{l})|| ||u| →^{∘} W Ã ; 2(Ω; Γ_{r}, T)|| . \end{aligned}  (67′)

Some text after to test the below-display spacing.

\begin{align*}
\begin{split}\abs{I_1}&=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\
&\le C_3\left[\int_\Omega\left(\int_{a}^x
g(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\lvert \left\lvert f\left\lvert \wt{S}^{-1,0}_{a,-}
W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
\left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
(\Omega;\Gamma_r,T)\right\rvert\right\rvert.
\end{split}\\
\begin{split}\abs{I_2}&=\left\lvert \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\rvert\\
&\le C_6\left\lvert \left\lvert f\int_\Omega
\left\lvert \wt{S}^{-1,0}_{a,-}
W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
\left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
(\Omega;\Gamma_r,T)\right\rvert\right\rvert.
\end{split}\tag{\theequation$'$}
\end{align*}

A.2 Multline

Numbered version:

∫_{a}^{b}{∫_{a}^{b}[f(x)^{2}g(y)^{2} + f(y)^{2}g(x)^{2}] − 2f(x)g(x)f(y)g(y) dx} dy

= ∫_{a}^{b}{g(y)^{2} ∫_{a}^{b} f^{2} + f(y)^{2} ∫_{a}^{b} g^{2} − 2f(y)g(y) ∫_{a}^{b} fg} dy (68)

To test the use of \label and \ref, we refer to the number of this equation here: (68).

\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:

∫_{a}^{b}{∫_{a}^{b}[f(x)^{2}g(y)^{2} + f(y)^{2}g(x)^{2}] − 2f(x)g(x)f(y)g(y) dx} dy

= ∫_{a}^{b}{g(y)^{2} ∫_{a}^{b} f^{2} + f(y)^{2} ∫_{a}^{b} g^{2} − 2f(y)g(y) ∫_{a}^{b} 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]
 -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*}

A.3 Gather

Numbered version with \notag on the second line:

D(a, r) ≡ {z ∈ C: |z − a| < r},  (69)
seg(a, r) ≡ {z ∈ C: ℑz = ℑa, |z − a| < r},
c(e, θ, r) ≡ {(x, y) ∈ C: |x − e| < y tan θ, 0 < y < r},  (70)
C(E, θ, r) ≡ ⋃_{e∈E} c(e, θ, r).  (71)

\begin{gather}
D(a,r)\equiv\{z\in\mathbf{C}\colon \abs{z-a}<r\},\\
\seg(a,r)\equiv\{z\in\mathbf{C}\colon
\Im z= \Im a,\ \abs{z-a}<r\},\notag\\
c(e,\theta,r)\equiv\{(x,y)\in\mathbf{C}
\colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\
C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r).
\end{gather}

Unnumbered version.

D(a, r) ≡ {z ∈ C: |z − a| < r}, seg(a, r) ≡ {z ∈ C: ℑz = ℑa, |z − a| < r}, c(e, θ, r) ≡ {(x, y) ∈ C: |x − e| < y tan θ, 0 < y < r},

C(E, θ, r) ≡ ⋃_{e∈E} c(e, θ, r).

Some text after to test the below-display spacing.

\begin{gather*}
D(a,r)\equiv\{z\in\mathbf{C}\colon \abs{z-a}<r\},\\
\seg (a,r)\equiv\{z\in\mathbf{C}\colon
\Im z= \Im a,\ \abs{z-a}<r\},\\
c(e,\theta,r)\equiv\{(x,y)\in\mathbf{C}
 \colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\
C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r).
\end{gather*}

A.4 Align

Numbered version:

γ_{x}(t) = (cos tu + sin tx, v),  (72)
γ_{y}(t) = (u, cos tv + sin ty),  (73)
γ_{z}(t) = (cos tu + \frac{α}{β} sin tv, −\frac{β}{α} sin tu + cos tv) .  (74)

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:

γ_{x}(t) = (cos tu + sin tx, v),
γ_{y}(t) = (u, cos tv + sin ty),
γ_{z}(t) = (cos tu + \frac{α}{β} sin tv, −\frac{β}{α} 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*}

A variation:

x = y  by (84)  (75)
x′ = y′  by (85)  (76)
x + x′ = y + y′  by Axiom 1.  (77)

Some text after to test the below-display spacing.

\begin{align}
x& =y && \text {by (\ref{eq:C})}\\
x'& = y' && \text {by (\ref{eq:D})}\\
x+x' & = y+y' && \text {by Axiom 1.}
\end{align}

A.5 Align and split within gather

When using the align environment within the gather environment, one or the other, or both, should be unnumbered (using the * form); numbering both the outer and inner environment would cause a conflict.

Automically numbered gather with split and align*:

\begin{aligned}φ(x, z) & = z − γ_{10}x − γ_{mn}x^{m}z^{n} \\ & = z − Mr^{−1}x − Mr^{−(m+n)}x^{m}z^{n} \end{aligned}  (78)

\begin{aligned}ζ^{0} & = (ξ^{0})^{2}, \\ ζ^{1} & = ξ^{0}ξ^{1}, \\ ζ^{2} & = (ξ^{1})^{2}, \end{aligned}

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-\gamma_{mn}x^mz^n\\
&=z-Mr^{-1}x-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.

\begin{aligned}φ(x, z) & = z − γ_{10}x − γ_{mn}x^{m}z^{n} \\ & = z − Mr^{−1}x − Mr^{−(m+n)}x^{m}z^{n} \end{aligned}

\begin{aligned}ζ^{0} & = (ξ^{0})^{2}, & (79) \\ ζ^{1} & = ξ^{0}ξ^{1}, & (80) \\ ζ^{2} & = (ξ^{1})^{2}, & (81) \end{aligned}

Some text after to test the below-display spacing.

\begin{gather*}
\begin{split} \varphi(x,z)
&=z-\gamma_{10}x-\gamma_{mn}x^mz^n\\
&=z-Mr^{-1}x-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*}

A.6 Alignat

Numbered version:

V_{i} = v_{i} − q_{i}v_{j},  X_{i} = x_{i} − q_{i}x_{j},  U_{i} = u_{i},  for i ≠ j;  (82)
V_{j} = v_{j},  X_{j} = x_{j},  U_{j}u_{j} + ∑_{i≠j} q_{i}u_{i}.  (83)

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} = v_{i} − q_{i}v_{j},  X_{i} = x_{i} − q_{i}x_{j},  U_{i} = u_{i},  for i ≠ j;
V_{j} = v_{j},  X_{j} = x_{j},  U_{j}u_{j} + ∑_{i≠j} q_{i}u_{i}.

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

The most common use for alignat is for things like

x = y  by (66)  (84)
x′ = y′  by (82)  (85)
x + x′ = y + y′  by Axiom 1.  (86)

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}

References

[1] W. Diffie and E. Hellman, New directions in cryptography, IEEE Transactions on Information Theory 22 (1976), no. 5, 644–654.

[2] D. H. Fremlin, Cichon’s diagram, 1983/1984, presented at the Séminaire Initiation à l’Analyse, G. Choquet, M. Rogalski, J. Saint Raymond, at the Université Pierre et Marie Curie, Paris, 23e année.

[3] I. P. Goulden and D. M. Jackson, The enumeration of directed closed Euler trails and directed Hamiltonian circuits by Langrangian methods, European J. Combin. 2 (1981), 131–212.

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