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Sample Paper for the amsmath Package

File name: testmath.tex

American Mathematical Society

Version 2.0a, 2023/08/24

1 Introduction

This paper contains examples of various features from AMS-LA TEX.

2 Enumeration of Hamiltonian paths in a graph

Letbe the adjacency matrix of graph. The corresponding Kirchhoff matrixis obtained fromby replacing ineach diagonal entry by the degree of its corresponding vertex; i.e., theth diagonal entry is identified with the degree of theth vertex. It is well known that

(1) det K ( 𝑖 | 𝑖 ) = the number of spanning trees of

𝐺

, 𝑖 = 1 , . . . , 𝑛

whereis theth principal submatrix of.

\det\mathbf{K}(i|i)=\text{ the number of spanning trees of $G$},

Letbe the set of graphs obtained fromby attaching edgeto each spanning tree of. Denote by. It is obvious that the collection of Hamiltonian cycles is a subset of. Note that the cardinality ofis. Let.

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

Define multiplication for the elements ofby

(2) 𝑥 ^ 𝑖 𝑥 ^ 𝑗 = 𝑥 ^ 𝑗 𝑥 ^ 𝑖 , 𝑥 ^ 𝑖 2 = 0 , 𝑖 , 𝑗 = 1 , . . . , 𝑛 .

Letand. Then the number of Hamiltonian cyclesis given by the relation [8]

(3)

(

Q 𝑗 = 1 𝑛 𝑥 ^ 𝑗

)

𝐻 𝑐 = 1 2 𝑘 ^ 𝑖 𝑗 det K b ( 𝑖 | 𝑖 ) , 𝑖 = 1 , . . . , 𝑛 .

The task here is to express (3) in a form free of any,. 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 graphand in a complete bipartite graphcan only be found from first combinatorial principles [4]. One wonders if there exists a formula which can be used very efficiently to produceand. Recently, using Lagrangian methods, Goulden and Jackson have shown thatcan be expressed in terms of the determinant and permanent of the adjacency matrix [3]. However, the formula of Goulden and Jackson determines neithernoreffectively. 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 toand. In addition, we eliminate the permanent fromand show thatcan be represented by a determinantal function of multivariables, each variable with domain. Furthermore, we show thatcan be written by number of spanning trees of subgraphs. Finally, we apply the formulas to a complete multigraph.

The conditions,, are not required in this paper. All formulas can be extended to a digraph simply by multiplyingby 2.

3 Main Theorem

Notation. Forandwe writeifand.

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

...

\end{notation}

Letbe anmatrix. Let. Using the properties of (2), it is readily seen that

Lemma 3.1.

(4) Q 𝑖 ∈ n

(

P 𝑗 ∈ n 𝑏 𝑖 𝑗 𝑥 ^ 𝑖

)

(

Q 𝑖 ∈ n 𝑥 ^ 𝑖

)

per B

whereis the permanent of.

Let. Define multiplication for the elements ofby

(5) 𝑦 ^ 𝑖 𝑦 ^ 𝑗 + 𝑦 ^ 𝑗 𝑦 ^ 𝑖 = 0 , 𝑖 , 𝑗 = 1 , . . . , 𝑛 .

Then, it follows that

Lemma 3.2.

(6) Q 𝑖 ∈ n

(

P 𝑗 ∈ n 𝑏 𝑖 𝑗 𝑦 ^ 𝑗

)

(

Q 𝑖 ∈ n 𝑦 ^ 𝑖

)

det B .

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

(7) P 𝑗 ∈ n 𝑏 𝑖 𝑗 𝑦 ^ 𝑗 = P 𝑗 ∈ n 𝑏 𝑖 𝑗 ( 𝜆 ) 𝑦 ^ 𝑗 + ( 𝑏 𝑖 𝑖 − 𝜆 𝑖 ) 𝑦 ^ 𝑖 𝑦 ^

where

(8) 𝑏 𝑖 𝑖 ( 𝜆 ) = 𝜆 𝑖 , 𝑏 𝑖 𝑗 ( 𝜆 ) = 𝑏 𝑖 𝑗 , 𝑖 ̸ = 𝑗 .

Let. By (6) and (7), it is straightforward to show the following result:

Theorem 3.3.

(9) det B = P 𝑙 = 0 𝑛 P 𝐼 𝑙 ⊆ 𝑛 Q 𝑖 ∈ 𝐼 𝑙 ( 𝑏 𝑖 𝑖 − 𝜆 𝑖 ) det B ( 𝜆 ) ( 𝐼 𝑙 | 𝐼 𝑙 ) ,

whereandis the principal submatrix obtained fromby deleting itsrows and columns.

Remark 3.1. Letbe anmatrix. The conventionhas 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. Then

(10) 𝑏 𝑙 = P 𝐼 𝑙 ⊆ n det B ( 𝐼 𝑙 | 𝐼 𝑙 ) .

Let

(11) K ( 𝑡 , 𝑡 1 , . . . , 𝑡 𝑛 ) = ( 𝐷 1 𝑡 − 𝑎 12 𝑡 2 . . . − 𝑎 1 𝑛 𝑡 𝑛 − 𝑎 21 𝑡 1 𝐷 2 𝑡 . . . − 𝑎 2 𝑛 𝑡 𝑛 − 𝑎 𝑛 1 𝑡 1 − 𝑎 𝑛 2 𝑡 2 . . . 𝐷 𝑛 𝑡 ) ,

\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\

-a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\

\hdotsfor[2]{4}\\

-a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix}

where

(12) 𝐷 𝑖 = P 𝑗 ∈ n 𝑎 𝑖 𝑗 𝑡 𝑗 , 𝑖 = 1 , . . . , 𝑛 .

Set

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

Then

(13) 𝐷 ( 𝑡 1 , . . . , 𝑡 𝑛 ) = P 𝑖 ∈ n 𝐷 𝑖 det K ( 𝑡 = 1 , 𝑡 1 , . . . , 𝑡 𝑛 ; 𝑖 | 𝑖 ) ,

whereis theth principal submatrix of.

Theorem 3.3 leads to

(14) det K ( 𝑡 1 , 𝑡 1 , . . . , 𝑡 𝑛 ) = P 𝐼 ∈ n ( − 1 ) | 𝐼 | 𝑡 𝑛 − | 𝐼 | Q 𝑖 ∈ 𝐼 𝑡 𝑖 Q 𝑗 ∈ 𝐼 ( 𝐷 𝑗 + 𝜆 𝑗 𝑡 𝑗 ) det A ( 𝜆 𝑡 ) ( 𝐼 ‾ | 𝐼 ‾ ) .

Note that

(15) det K ( 𝑡 = 1 , 𝑡 1 , . . . , 𝑡 𝑛 ) = P 𝐼 ∈ n ( − 1 ) | 𝐼 | Q 𝑖 ∈ 𝐼 𝑡 𝑖 Q 𝑗 ∈ 𝐼 ( 𝐷 𝑗 + 𝜆 𝑗 𝑡 𝑗 ) det A ( 𝜆 ) ( 𝐼 ‾ | 𝐼 ‾ ) = 0 .

Let. Lemma 3.1 yields

(

P 𝑖 ∈ n 𝑎 𝑙 𝑖 𝑥 𝑖

)

det K ( 𝑡 = 1 , 𝑥 1 , . . . , 𝑥 𝑛 ; 𝑙 | 𝑙 ) (16) =

(

Q 𝑖 ∈ n 𝑥 ^ 𝑖

)

P 𝐼 ⊆ n − { 𝑙 } ( − 1 ) | 𝐼 | per A ( 𝜆 ) ( 𝐼 | 𝐼 ) det A ( 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) 𝐻 𝑐 = 1 2 𝑛 P 𝑙 = 0 𝑛 ( − 1 ) 𝑙 𝐷 𝑙 ,

where

(18) 𝐷 𝑙 = P 𝐼 𝑙 ⊆ n 𝐷 ( 𝑡 1 , . . . , 𝑡 𝑛 ) 2 | 𝑡 𝑖 = { 0 , if 𝑖 ∈ 𝐼 𝑙 1 , otherwise , 𝑖 = 1 , . . . , 𝑛 .

4 Application

We consider here the applications of Theorems 5.1 and 5.2 to a complete multipartite graph. It can be shown that the number of spanning trees ofmay be written

(19) 𝑇 = 𝑛 𝑝 − 2 Q 𝑖 = 1 𝑝 ( 𝑛 − 𝑛 𝑖 ) 𝑛 𝑖 − 1

where

(20) 𝑛 = 𝑛 1 + ⋅ ⋅ ⋅ + 𝑛 𝑝 .

It follows from Theorems 5.1 and 5.2 that

𝐻 𝑐 = 1 2 𝑛 P 𝑙 = 0 𝑛 ( − 1 ) 𝑙 ( 𝑛 − 𝑙 ) 𝑝 − 2 P 𝑙 1 + ⋅ ⋅ ⋅ + 𝑙 𝑝 = 𝑙 Q 𝑖 = 1 𝑝

(

𝑛 𝑖 𝑙 𝑖

)

⋅ [ ( 𝑛 − 𝑙 ) − ( 𝑛 𝑖 − 𝑙 𝑖 ) ] 𝑛 𝑖 − 𝑙 𝑖 ⋅

[

( 𝑛 − 𝑙 ) 2 − P 𝑗 = 1 𝑝 ( 𝑛 𝑖 − 𝑙 𝑖 ) 2

]

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

and

𝐻 𝑐 = 1 2 P 𝑙 = 0 𝑛 − 1 ( − 1 ) 𝑙 ( 𝑛 − 𝑙 ) 𝑝 − 2 P 𝑙 1 + ⋅ ⋅ ⋅ + 𝑙 𝑝 = 𝑙 Q 𝑖 = 1 𝑝

(

𝑛 𝑖 𝑙 𝑖

)

⋅ [ ( 𝑛 − 𝑙 ) − ( 𝑛 𝑖 − 𝑙 𝑖 ) ] 𝑛 𝑖 − 𝑙 𝑖 ( 1 − 𝑙 𝑝 𝑛 𝑝 ) [ ( 𝑛 − 𝑙 ) − ( 𝑛 𝑝 − 𝑙 𝑝 ) ] .

The enumeration ofin agraph 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,in agraph may be written

𝐻 𝑐 = 𝑛 1 ! 𝑛 2 ! 𝑛 3 ! 𝑛 1 + 𝑛 2 + 𝑛 3 P 𝑖 [

(

𝑛 1 𝑖

)

(

𝑛 2 𝑛 3 − 𝑛 1 + 𝑖

)

(

𝑛 3 𝑛 3 − 𝑛 2 + 𝑖

)

(

𝑛 1 − 1 𝑖

)

(

𝑛 2 − 1 𝑛 3 − 𝑛 1 + 𝑖

)

(

𝑛 3 − 1 𝑛 3 − 𝑛 2 + 𝑖

)

] .

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 functionis informationally one-way if there is no probabilistic polynomial time algorithm which (with probability of the formfor some) returns on inputa random element of.

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 functionis said to be B-differentiable at the pointif (i)is Lipschitz continuous in a neighborhood of, and (ii) there exists a positive homogeneous function, called the B-derivative ofat, such that

lim 𝑣 → 0 𝐻 ( 𝑧 + 𝑣 ) − 𝐻 ( 𝑧 ) − 𝐵 𝐻 ( 𝑧 ) 𝑣 ‖ 𝑣 ‖ = 0 .

The functionis B-differentiable in setif it is B-differentiable at every point in. The B-derivativeis said to be strong if

lim ( 𝑣 , 𝑣 ′ ) → ( 0 , 0 ) 𝐻 ( 𝑧 + 𝑣 ) − 𝐻 ( 𝑧 + 𝑣 ′ ) − 𝐵 𝐻 ( 𝑧 ) ( 𝑣 − 𝑣 ′ ) ‖ 𝑣 − 𝑣 ′ ‖ = 0 .

Lemma 6.1. There exists a smooth functiondefined forsatisfying the following properties:

(i)is bounded above and below by positive constants.

(ii) If, then.

(iii) For allin the domain of,.

(iv) If, then.

Proof. We chooseto be a radial function depending only on. Letbe a suitable smooth function satisfyingfor, andfor. The radial Laplacian

^^01 0 ln 𝜓 0 ( 𝑟 ) = ( 𝑑 2 𝑑 𝑟 2 + 1 𝑟 𝑑 𝑑 𝑟 ) ln 𝜓 0 ( 𝑟 )

has smooth coefficients for. Therefore, we may apply the existence and uniqueness theory for ordinary differential equations. Simply letbe the solution of the differential equation

( 𝑑 2 𝑑 𝑟 2 + 1 𝑟 𝑑 𝑑 𝑟 ) ln 𝜓 0 ( 𝑟 ) = ℎ ( 𝑟 )

with initial conditions given byand.

Next, letbe a finite collection of pairwise disjoint disks, all of which are contained in the unit disk centered at the origin in. We assume that. Suppose thatdenotes the smaller concentric disk. We define a smooth weight functionforby settingwhenandwhenis an element of. It follows from Lemma 6.1 thatsatisfies the properties:

(i)is bounded above and below by positive constants.

(ii)for all, the domain where the functionis defined.

(iii)when.

Letdenote the annulus, and set. The properties (2) and (3) ofmay be summarized as, whereis the characteristic function of.

Suppose thatis a nonnegative real constant. We apply Proposition 3.5 with. If, assume thatis a bounded domain containing the support ofand. A calculation gives

R 𝒟 | ∂ ‾ 𝑢 | 2 ^^08 0 ( 𝑧 ) 𝑒 𝛼 | 𝑧 | 2 ≥ 𝑐 4 𝛼 R 𝒟 | 𝑢 | 2 ^^08 0 𝑒 𝛼 | 𝑧 | 2 + 𝑐 5 𝛿 − 2 R 𝐴 | 𝑢 | 2 ^^08 0 𝑒 𝛼 | 𝑧 | 2 .

The boundedness, property (1) of, then yields

R 𝒟 | ∂ ‾ 𝑢 | 2 𝑒 𝛼 | 𝑧 | 2 ≥ 𝑐 6 𝛼 R 𝒟 | 𝑢 | 2 𝑒 𝛼 | 𝑧 | 2 + 𝑐 7 𝛿 − 2 R 𝐴 | 𝑢 | 2 𝑒 𝛼 | 𝑧 | 2 .

Letbe the set of blocks ofand let. Ifthenis constant on the blocks of.

(24) 𝑃 𝑋 = { 𝜙 ∈ 𝑀 | ^^03 𝜙 = ^^03 𝑋 } , 𝑄 𝑋 = { 𝜙 ∈ 𝑀 | ^^03 𝜙 ≥ ^^03 𝑋 } .

Ifthenfor someso that

𝑄 𝑋 = S 𝑌 ≥ 𝑋 𝑃 𝑌 .

Thus by Möbius inversion

| 𝑃 𝑌 | = P 𝑋 ≥ 𝑌 𝜇 ( 𝑌 , 𝑋 ) | 𝑄 𝑋 | .

Thus there is a bijection fromto. In particular.

Next note that. We see this by choosing a basis forconsisting of vectorsdefined by

𝑣 𝑖 𝑘 = { 1 if

𝑖 \in {^^03}_{𝑘}

, 0 otherwise.

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

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

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

Lemma 6.2. Letbe an arrangement. Then

𝜒 ( 𝒜 , 𝑡 ) = P ℬ ⊆ 𝒜 ( − 1 ) | ℬ | 𝑡 dim 𝑇 ( ℬ ) .

In order to computerecall the definition offrom Lemma 3.1. Since,. Thus ifthen. Let. Then

𝑅 ′ ′ = P 𝐻 ∈ ℬ ⊆ 𝒜 ( − 1 ) | ℬ | 𝑡 dim 𝑇 ( ℬ ) = P 𝑌 ∈ 𝐿 ′ ′ P ℬ ∈ 𝑆 ( 𝐻 , 𝑌 ) ( − 1 ) | ℬ | 𝑡 dim 𝑌 = − P 𝑌 ∈ 𝐿 ′ ′ P ℬ ∈ 𝑆 ( 𝐻 , 𝑌 ) ( − 1 ) | ℬ − 𝒜 𝐻 | 𝑡 dim 𝑌 = − P 𝑌 ∈ 𝐿 ′ ′ 𝜇 ( 𝐻 , 𝑌 ) 𝑡 dim 𝑌 = − 𝜒 ( 𝒜 ′ ′ , 𝑡 ) .

Corollary 6.3. Letbe a triple of arrangements. Then

𝜋 ( 𝒜 , 𝑡 ) = 𝜋 ( 𝒜 ′ , 𝑡 ) + 𝑡 𝜋 ( 𝒜 ′ ′ , 𝑡 ) .

Definition 6.2. Letbe a triple with respect to the hyperplane. Calla separator if.

Corollary 6.4. Letbe a triple with respect to.

(i) Ifis a separator then

𝜇 ( 𝒜 ) = − 𝜇 ( 𝒜 ′ ′ )

and hence

| 𝜇 ( 𝒜 ) | = | 𝜇 ( 𝒜 ′ ′ ) | .

(ii) Ifis not a separator then

𝜇 ( 𝒜 ) = 𝜇 ( 𝒜 ′ ) − 𝜇 ( 𝒜 ′ ′ )

and

| 𝜇 ( 𝒜 ) | = | 𝜇 ( 𝒜 ′ ) | + | 𝜇 ( 𝒜 ′ ′ ) | .

Proof. It follows from Theorem 5.1 thathas 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. Ifis a separator thenand there is no contribution from.

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. It is also the Poincaré polynomial of the complementfor a complex arrangement. Here we prove that the Poincaré polynomial is the chamber counting function for a real arrangement. The complementis a disjoint union of chambers

𝑀 ( 𝒜 ) = S 𝐶 ∈ Cham ( 𝒜 ) 𝐶 .

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

Theorem 6.5. Letbe a real arrangement. Then

| Cham ( 𝒜 R ) | = 𝜋 ( 𝒜 R , 1 ) .

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

Figure 1:

Figure 2:

Theorem 6.6. Letbe a protocol for a random pair. If one ofandis a prefix of the other and, then

⟨ 𝜎 𝑗 ( 𝑥 ′ , 𝑦 ) ⟩ 𝑗 = 1 ∞ = ⟨ 𝜎 𝑗 ( 𝑥 , 𝑦 ) ⟩ 𝑗 = 1 ∞ = ⟨ 𝜎 𝑗 ( 𝑥 , 𝑦 ′ ) ⟩ 𝑗 = 1 ∞ .

Proof. We show by induction onthat

⟨ 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. Assume it holds for, in particular. Then one ofandis a prefix of the other which implies that one ofandis a prefix of the other. If theth message is transmitted bythen, by the separate-transmissions property and the induction hypothesis,, hence one ofandis a prefix of the other. By the implicit-termination property, neithernorcan be a proper prefix of the other, hence they must be the same and. If theth message is transmitted bythen, symmetrically,by the induction hypothesis and the separate-transmissions property, and, then,by the implicit-termination property, proving the induction step.

Ifis a protocol for, and,are distinct inputs in, then, by the correct-decision property,.

Equation (25) defined’s ambiguity setto be the set of possiblevalues when. The last corollary implies that for all, the multiset1 of codewordsis prefix free.

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

7 One-Way Complexity

, the one-way complexity of a random pair, is the number of bitsmust transmit in the worst case whenis not permitted to transmit any feedback messages. Starting with, the support set of, we define, the characteristic hypergraph of, and show that

𝐶 ^ 1 ( 𝑋 | 𝑌 ) = ⌈ log 𝜒 ( 𝐺 ( 𝑋 | 𝑌 ) ) ⌉ .

Letbe a random pair. For eachin, the support set of, Equation (25) definedto be the set of possiblevalues when. The characteristic hypergraphofhasas its vertex set and the hyperedgefor each.

We can now prove a continuity theorem.

Theorem 7.1. Letbe an open set, let, and let

(26) 𝑇 𝑥 𝑢 = { 𝑦 ∈ R 𝑚 : 𝑦 = 𝑢 ~ ( 𝑥 ) + ⟨ 𝐷 𝑢 | 𝐷 𝑢 | ( 𝑥 ) , 𝑧 ⟩ for some 𝑧 ∈ R 𝑛 }

for every. Letbe a Lipschitz continuous function such that, and let. Thenand

(27) 𝐽 𝑣 = ( 𝑓 ( 𝑢 + ) − 𝑓 ( 𝑢 − ) ) ⊗ 𝜈 𝑢 ⋅ ℋ 𝑛 − 1 | 𝑆 𝑢 .

In addition, for-almost everythe restriction of the functiontois differentiable atand

(28) 𝐷 e 𝑣 = ∇ ( 𝑓 | 𝑇 𝑥 𝑢 ) ( 𝑢 ~ ) 𝐷 e 𝑢 | 𝐷 e 𝑢 | ⋅ | 𝐷 e 𝑢 | .

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

Remark 7.1. Letbe a continuous function such thatas. Then

lim ℎ → 0 + 𝑔 ( 𝜔 ( ℎ ) ) = 𝐿 ⇔ lim ℎ → 0 + 𝑔 ( ℎ ) = 𝐿

for any function.

Remark 7.2. Letbe a Lipschitz continuous function and assume that

𝐿 ( 𝑧 ) = lim ℎ → 0 + 𝑔 ( ℎ 𝑧 ) − 𝑔 ( 0 ) ℎ

exists for everyand thatis a linear function of. Thenis differentiable at 0.

Remark 7.3. Letbe a linear function, and letbe a function. Then the restriction ofto the range ofis differentiable at 0 if and only ifis differentiable at 0 and

∇ ( 𝑓 | Im ( 𝐴 ) ) ( 0 ) 𝐴 = ∇ ( 𝑓 ( 𝐴 ) ) ( 0 ) .

Proof. We begin by showing thatand

(29) | 𝐷 𝑣 | ( 𝐵 ) ≤ 𝐾 | 𝐷 𝑢 | ( 𝐵 ) ∀ 𝐵 ∈ B ( ) ,

whereis the Lipschitz constant of. By (13) and by the approximation result quoted in §3, it is possible to find a sequenceconverging toinand such that

lim ℎ → + ∞ R | ∇ 𝑢 ℎ | 𝑑 𝑥 = | 𝐷 𝑢 | ( ) .

The functionsare locally Lipschitz continuous in, and the definition of differential implies thatalmost everywhere in. The lower semicontinuity of the total variation and (13) yield

| 𝐷 𝑣 | ( ) ≤ lim inf ℎ → + ∞ | 𝐷 𝑣 ℎ | ( ) = lim inf ℎ → + ∞ R | ∇ 𝑣 ℎ | 𝑑 𝑥 ≤ 𝐾 lim inf ℎ → + ∞ R | ∇ 𝑢 ℎ | 𝑑 𝑥 = 𝐾 | 𝐷 𝑢 | ( ) .

Since, we have also

R | 𝑣 | 𝑑 𝑥 ≤ 𝐾 R | 𝑢 | 𝑑 𝑥 ;

therefore. Repeating the same argument for every open set, we get (29) for every, because,are Radon measures. To prove Lemma 6.1, first we observe that

(31) 𝑆 𝑣 ⊂ 𝑆 𝑢 , 𝑣 ~ ( 𝑥 ) = 𝑓 ( 𝑢 ~ ( 𝑥 ) ) ∀ 𝑥 ∈ \ 𝑆 𝑢 .

In fact, for everywe have

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

hence

lim 𝜌 → 0 + | { 𝑦 ∈ 𝐵 𝜌 ( 𝑥 ) : | 𝑣 ( 𝑦 ) − 𝑓 ( 𝑢 ~ ( 𝑥 ) ) | > 𝜀 } | 𝜌 𝑛 = 0

whenever. By a similar argument, ifis a point such that there exists a tripletsatisfying (14), (15), then

( 𝑣 + ( 𝑥 ) − 𝑣 − ( 𝑥 ) ) ⊗ 𝜈 𝑣 = ( 𝑓 ( 𝑢 + ( 𝑥 ) ) − 𝑓 ( 𝑢 − ( 𝑥 ) ) ) ⊗ 𝜈 𝑢 if 𝑥 ∈ 𝑆 𝑣

andif. Hence, by (1.8) we get

𝐽 𝑣 ( 𝐵 ) = R 𝐵 ∩ 𝑆 𝑣 ( 𝑣 + − 𝑣 − ) ⊗ 𝜈 𝑣 𝑑 ℋ 𝑛 − 1 = R 𝐵 ∩ 𝑆 𝑣 ( 𝑓 ( 𝑢 + ) − 𝑓 ( 𝑢 − ) ) ⊗ 𝜈 𝑢 𝑑 ℋ 𝑛 − 1 = R 𝐵 ∩ 𝑆 𝑢 ( 𝑓 ( 𝑢 + ) − 𝑓 ( 𝑢 − ) ) ⊗ 𝜈 𝑢 𝑑 ℋ 𝑛 − 1

and Lemma 6.1 is proved.

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

Step 1

Assume that. Sinceis at most countable, (7) yields that, so that (19) and (21) imply thatis the Radon-Nikodým decomposition ofin absolutely continuous and singular part with respect to. By Theorem 5.2, we have

𝐷 e 𝑣 | 𝐷 e 𝑢 | ( 𝑡 ) = lim 𝑠 → 𝑡 + 𝐷 𝑣 ( [ 𝑡 , 𝑠 [ ) | 𝐷 e 𝑢 | ( [ 𝑡 , 𝑠 [ ) , 𝐷 e 𝑢 | 𝐷 e 𝑢 | ( 𝑡 ) = lim 𝑠 → 𝑡 + 𝐷 𝑢 ( [ 𝑡 , 𝑠 [ ) | 𝐷 e 𝑢 | ( [ 𝑡 , 𝑠 [ )

-almost everywhere in. It is well known (see, for instance, [12, 2.5.16]) that every one-dimensional function of bounded variationhas a unique left continuous representative, i.e., a functionsuch thatalmost everywhere andfor every. These conditions imply

(32) 𝑢 ^ ( 𝑡 ) = 𝐷 𝑢 ( ] − ∞ , 𝑡 [ ) , 𝑣 ^ ( 𝑡 ) = 𝐷 𝑣 ( ] − ∞ , 𝑡 [ ) ∀ 𝑡 ∈ R

and

(33) 𝑣 ^ ( 𝑡 ) = 𝑓 ( 𝑢 ^ ( 𝑡 ) ) ∀ 𝑡 ∈ R .

Letbe such thatfor everyand assume that the limits in (22) exist. By (23) and (24) we get

𝑣 ^ ( 𝑠 ) − 𝑣 ^ ( 𝑡 ) | 𝐷 e 𝑢 | ( [ 𝑡 , 𝑠 [ ) = 𝑓 ( 𝑢 ^ ( 𝑠 ) ) − 𝑓 ( 𝑢 ^ ( 𝑡 ) ) | 𝐷 e 𝑢 | ( [ 𝑡 , 𝑠 [ ) = 𝑓 ( 𝑢 ^ ( 𝑠 ) ) − 𝑓 ( 𝑢 ^ ( 𝑡 ) + 𝐷 e 𝑢 | 𝐷 e 𝑢 | ( 𝑡 ) | 𝐷 e 𝑢 | ( [ 𝑡 , 𝑠 [ ) ) | 𝐷 e 𝑢 | ( [ 𝑡 , 𝑠 [ ) + 𝑓 ( 𝑢 ^ ( 𝑡 ) + 𝐷 e 𝑢 | 𝐷 e 𝑢 | ( 𝑡 ) | 𝐷 e 𝑢 | ( [ 𝑡 , 𝑠 [ ) ) − 𝑓 ( 𝑢 ^ ( 𝑡 ) ) | 𝐷 e 𝑢 | ( [ 𝑡 , 𝑠 [ )

for every. Using the Lipschitz condition onwe find

| 𝑣 ^ ( 𝑠 ) − 𝑣 ^ ( 𝑡 ) | 𝐷 e 𝑢 | ( [ 𝑡 , 𝑠 [ ) − 𝑓 ( 𝑢 ^ ( 𝑡 ) + 𝐷 e 𝑢 | 𝐷 e 𝑢 | ( 𝑡 ) | 𝐷 e 𝑢 | ( [ 𝑡 , 𝑠 [ ) ) − 𝑓 ( 𝑢 ^ ( 𝑡 ) ) | 𝐷 e 𝑢 | ( [ 𝑡 , 𝑠 [ ) | ≤ 𝐾 | 𝑢 ^ ( 𝑠 ) − 𝑢 ^ ( 𝑡 ) | 𝐷 e 𝑢 | ( [ 𝑡 , 𝑠 [ ) − 𝐷 e 𝑢 | 𝐷 e 𝑢 | ( 𝑡 ) | .

By (29), the functionis continuous and converges to 0 as. Therefore Remark 7.1 and the previous inequality imply

𝐷 e 𝑣 | 𝐷 e 𝑢 | ( 𝑡 ) = lim ℎ → 0 + 𝑓 ( 𝑢 ^ ( 𝑡 ) + ℎ 𝐷 e 𝑢 | 𝐷 e 𝑢 | ( 𝑡 ) ) − 𝑓 ( 𝑢 ^ ( 𝑡 ) ) ℎ | 𝐷 e 𝑢 | -a.e. in R .

By (22),for every; moreover, applying the same argument to the functions,, we get

𝐷 e 𝑣 | 𝐷 e 𝑢 | ( 𝑡 ) = lim ℎ → 0 𝑓 ( 𝑢 ~ ( 𝑡 ) + ℎ 𝐷 e 𝑢 | 𝐷 e 𝑢 | ( 𝑡 ) ) − 𝑓 ( 𝑢 ~ ( 𝑡 ) ) ℎ | 𝐷 e 𝑢 | -a.e. in R

and our statement is proved.

Step 2

Let us consider now the general case. Letbe such that, and let. In the following, we shall identifywith, and we shall denote bythe variable ranging inand bythe variable ranging in. By the just proven one-dimensional result, and by Theorem 3.3, we get

for-almost every. We claim that

(34) ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | ( 𝑦 + 𝑡 𝜈 ) = 𝐷 e 𝑢 𝑦 | 𝐷 e 𝑢 𝑦 | ( 𝑡 ) | 𝐷 e 𝑢 𝑦 | -a.e. in R

for-almost every. In fact, by (16) and (18) we get

R 𝜋 𝜈 𝐷 e 𝑢 𝑦 | 𝐷 e 𝑢 𝑦 | ⋅ | 𝐷 e 𝑢 𝑦 | 𝑑 ℋ 𝑛 − 1 ( 𝑦 ) = R 𝜋 𝜈 𝐷 e 𝑢 𝑦 𝑑 ℋ 𝑛 − 1 ( 𝑦 ) = ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ = ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | ⋅ | ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | = R 𝜋 𝜈 ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | ( 𝑦 + ⋅ 𝜈 ) ⋅ | 𝐷 e 𝑢 𝑦 | 𝑑 ℋ 𝑛 − 1 ( 𝑦 )

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

(35) ⟨ 𝐷 e 𝑣 , 𝜈 ⟩ | ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | ( 𝑦 + 𝑡 𝜈 ) = 𝐷 e 𝑣 𝑦 | 𝐷 e 𝑢 𝑦 | ( 𝑡 ) | 𝐷 e 𝑢 𝑦 | -a.e. in R

for-almost every. By (24) and (25) we get

lim ℎ → 0 𝑓 ( 𝑢 ~ ( 𝑦 + 𝑡 𝜈 ) + ℎ ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | ( 𝑦 + 𝑡 𝜈 ) ) − 𝑓 ( 𝑢 ~ ( 𝑦 + 𝑡 𝜈 ) ) ℎ = ⟨ 𝐷 e 𝑣 , 𝜈 ⟩ | ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | ( 𝑦 + 𝑡 𝜈 )

for-almost every, and using again (14), (15) we get

lim ℎ → 0 𝑓 ( 𝑢 ~ ( 𝑥 ) + ℎ ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | ( 𝑥 ) ) − 𝑓 ( 𝑢 ~ ( 𝑥 ) ) ℎ = ⟨ 𝐷 e 𝑣 , 𝜈 ⟩ | ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | ( 𝑥 )

-a.e. in.

Since the functionis strictly positive-almost everywhere, we obtain also

lim ℎ → 0 𝑓 ( 𝑢 ~ ( 𝑥 ) + ℎ | ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | | 𝐷 e 𝑢 | ( 𝑥 ) ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | ( 𝑥 ) ) − 𝑓 ( 𝑢 ~ ( 𝑥 ) ) ℎ = | ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | | 𝐷 e 𝑢 | ( 𝑥 ) ⟨ 𝐷 e 𝑣 , 𝜈 ⟩ | ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | ( 𝑥 )

-almost everywhere in.

Finally, since

| ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | | 𝐷 e 𝑢 | ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | = ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | 𝐷 e 𝑢 | = ⟨ 𝐷 e 𝑢 | 𝐷 e 𝑢 | , 𝜈 ⟩ | 𝐷 e 𝑢 | -a.e. in R 𝑛 | ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | | 𝐷 e 𝑢 | ⟨ 𝐷 e 𝑣 , 𝜈 ⟩ | ⟨ 𝐷 e 𝑢 , 𝜈 ⟩ | = ⟨ 𝐷 e 𝑣 , 𝜈 ⟩ | 𝐷 e 𝑢 | = ⟨ 𝐷 e 𝑣 | 𝐷 e 𝑢 | , 𝜈 ⟩ | 𝐷 e 𝑢 | -a.e. in R 𝑛

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

lim ℎ → 0 𝑓 ( 𝑢 ~ ( 𝑥 ) + ℎ ⟨ 𝐷 e 𝑢 | 𝐷 e 𝑢 | ( 𝑥 ) , 𝜈 ⟩ ) − 𝑓 ( 𝑢 ~ ( 𝑥 ) ) ℎ = ⟨ 𝐷 e 𝑣 | 𝐷 e 𝑢 | ( 𝑥 ) , 𝜈 ⟩ ,

-a.e. in. Sinceis arbitrary, by Remarks 7.2 and 7.3 the restriction ofto the affine spaceis differentiable atfor-almost everyand (26) holds.

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

(36) 𝐷 ( 𝑡 1 , . . . , 𝑡 𝑛 ) = P 𝐼 ∈ n ( − 1 ) | 𝐼 | − 1 | 𝐼 | Q 𝑖 ∈ 𝐼 𝑡 𝑖 Q 𝑗 ∈ 𝐼 ( 𝐷 𝑗 + 𝜆 𝑗 𝑡 𝑗 ) det A ( 𝜆 ) ( 𝐼 ‾ | 𝐼 ‾ ) .

Let,. Lemma 1 leads to

(37) 𝐷 ( 𝑥 ^ 1 , . . . , 𝑥 ^ 𝑛 ) = Q 𝑖 ∈ n 𝑥 ^ 𝑖 P 𝐼 ∈ n ( − 1 ) | 𝐼 | − 1 | 𝐼 | per A ( 𝜆 ) ( 𝐼 | 𝐼 ) det A ( 𝜆 ) ( 𝐼 ‾ | 𝐼 ‾ ) .

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

Theorem 7.2.

(38) 𝐻 𝑐 = 1 2 𝑛 P 𝑙 = 1 𝑛 𝑙 ( − 1 ) 𝑙 − 1 𝐴 𝑙 ( 𝜆 ) ,

where

(39) 𝐴 𝑙 ( 𝜆 ) = P 𝐼 𝑙 ⊆ n per A ( 𝜆 ) ( 𝐼 𝑙 | 𝐼 𝑙 ) det A ( 𝜆 ) ( 𝐼 ‾ | 𝐼 ‾ ) , | 𝐼 𝑙 | = 𝑙 .

It is worth noting thatof (39) is similar to the coefficientsof the characteristic polynomial of (10). It is well known in graph theory that the coefficientscan be expressed as a sum over certain subgraphs. It is interesting to see whether,, structural properties of a graph.

We may call (38) a parametric representation of. In computation, the parameterplays very important roles. The choice of the parameter usually depends on the properties of the given graph. For a complete graph, let,. It follows from (39) that

(40) 𝐴 𝑙 ( 1 ) = { 𝑛 ! , if 𝑙 = 1 0 , otherwise .

By (38)

(41) 𝐻 𝑐 = 1 2 ( 𝑛 − 1 ) ! .

For a complete bipartite graph, let,. By (39),

(42) 𝐴 𝑙 = { − 𝑛 1 ! 𝑛 2 ! 𝛿 𝑛 1 𝑛 2 , if 𝑙 = 2 0 , otherwise .

Theorem 7.2 leads to

(43) 𝐻 𝑐 = 1 𝑛 1 + 𝑛 2 𝑛 1 ! 𝑛 2 ! 𝛿 𝑛 1 𝑛 2 .

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

det K ( 𝑡 = 1 , 𝑡 1 , . . . , 𝑡 𝑛 ; 𝑙 | 𝑙 ) (44) = P 𝐼 ⊆ n − { 𝑙 } ( − 1 ) | 𝐼 | Q 𝑖 ∈ 𝐼 𝑡 𝑖 Q 𝑗 ∈ 𝐼 ( 𝐷 𝑗 + 𝜆 𝑗 𝑡 𝑗 ) det A ( 𝜆 ) ( 𝐼 ‾ ∪ { 𝑙 } | 𝐼 ‾ ∪ { 𝑙 } ) .

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

Theorem 7.3.

(45) 𝐻 𝑐 = 1 2 P 𝐼 ⊆ n − { 𝑙 } ( − 1 ) | 𝐼 | per A ( 𝜆 ) ( 𝐼 | 𝐼 ) det A ( 𝜆 ) ( 𝐼 ‾ ∪ { 𝑙 } | 𝐼 ‾ ∪ { 𝑙 } )

which reduces to Goulden–Jackson’s formula when[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 ∞ + 𝜋 𝐴 0 ∼ A

(

+

𝜋

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:

∂ 𝑥 ∂ 𝑦

|

|

|

∂ 𝑦 ∂ 𝑧

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

\pmb{\bigg\vert}

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

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

P Q 𝜅 𝜅 𝐹 ( 𝑟 𝑖 )

P

P

P

Q

Q

Q

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.

(46) R R 𝐴 𝑓 ( 𝑥 , 𝑦 ) 𝑑 𝑥 𝑑 𝑦 R R R 𝐴 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) 𝑑 𝑥 𝑑 𝑦 𝑑 𝑧 (47) R R R R 𝐴 𝑓 ( 𝑤 , 𝑥 , 𝑦 , 𝑧 ) 𝑑 𝑤 𝑑 𝑥 𝑑 𝑦 𝑑 𝑧 R ⋅ ⋅ ⋅ R 𝐴 𝑓 ( 𝑥 1 , . . . , 𝑥 𝑘 )

9.2 Over and under arrows

Some extra over and under arrow operations are provided in the amsmath package. (Basic LA TEX provides \overrightarrow and \overleftarrow).

= = =

\begin{align*}

\overrightarrow{\psi_\delta(t) E_t h}&

=\underrightarrow{\psi_\delta(t) E_t h}\\

\overleftarrow{\psi_\delta(t) E_t h}&

=\underleftarrow{\psi_\delta(t) E_t h}\\

\overleftrightarrow{\psi_\delta(t) E_t h}&

=\underleftrightarrow{\psi_\delta(t) E_t h}

\end{align*}

These all scale properly in subscript sizes:

R 𝑎 𝑥 𝑑 𝑥

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

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, the regional sum, the orthogonal product, and the infinite integral

R 𝐴 1 R 𝐴 2 ⋅ ⋅ ⋅

9.4 Accents in math

Double accents:

^ ^ 𝐻 ^^14 ^^14 𝐶 ~ ~ 𝑇 ^^13 ^^13 𝐴 ^^12 ^^12 𝐺 _ _ 𝐷 𝐷 ^^15 ^^15 𝐵 ^^16 ^^16 𝐵 → → 𝑉

\[\Hat{\Hat{H}}\quad\Check{\Check{C}}\quad

\Tilde{\Tilde{T}}\quad\Acute{\Acute{A}}\quad

\Grave{\Grave{G}}\quad\Dot{\Dot{D}}\quad

\Ddot{\Ddot{D}}\quad\Breve{\Breve{B}}\quad

\Bar{\Bar{B}}\quad\Vec{\Vec{V}}\]

This double accent operation is complicated and tends to slow down the processing of a LA 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 LA TEX:

𝑄 𝑅

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

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{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}

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:

\leftarrow -

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 asor, without affecting the placement of limits. Examples:

_{*}^{*}

Q_{*}^{*}

𝑘

)

P^{'}

0 ≤ 𝑖 ≤ 𝑚 𝐸 𝑖 𝛽 𝑥

\[\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,, andhave predefined control 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:

(48)

\lim

‾ 𝑛 → ∞ 𝒬 ( 𝑢 𝑛 , 𝑢 𝑛 − 𝑢 # ) ≤ 0 (49)

\lim

_ 𝑛 → ∞ | 𝑎 𝑛 + 1 | / | 𝑎 𝑛 | = 0 (50) ( 𝑚 𝑖 𝜆 ⋅ ) * ≤ 0 (51) 𝑝 ∈ 𝑆 ( 𝐴 ) 𝐴 𝑝 ≤ 0

\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:

(52) 𝑥 ≡ 𝑦 + 1 ( mod 𝑚 2 ) (53) 𝑥 ≡ 𝑦 + 1 mod 𝑚 2 (54) 𝑥 ≡ 𝑦 + 1 ( 𝑚 2 )

\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:

P 𝛾 ∈ ^^00 𝐶 𝐼 𝛾 = 2 𝑘 −

P

𝑘 1

)

2 𝑘 − 1 +

(

𝑘 2

)

2 𝑘 − 2 + ⋅ ⋅ ⋅ + ( − 1 ) 𝑙

(

𝑘 𝑙

)

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

\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:

(56) \over: 𝑛 + 1 2 \overwithdelims:

⟨

𝑛 + 1 2

⟩

(57) \atop: 𝑛 + 1 2 \atopwithdelims:

(

𝑛 + 1 2

)

(58) \above: 𝑛 + 1 2 \abovewithdelims:

[

𝑛 + 1 2

)

\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

(59) 1 2 + 1 2 + 1 2 + 1 2 + 1 2 + ⋅ ⋅ ⋅

can be obtained by typing

\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\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:, with the radical extending to encompass the depth of the subscript.

9.16 The ‘cases’ environment

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

(60) 𝑃 𝑟 − 𝑗 = { 0 if

P

is odd , 𝑟 ! ( − 1 ) ( 𝑟 − 𝑗 ) / 2 if

𝑟 - 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 \text and the embedded math.

9.17 Matrix

Here are samples of the matrix environments, \matrix, \pmatrix, \bmatrix, \Bmatrix, \vmatrix and \Vmatrix:

(61) 𝜗 𝜚 𝜑 𝜛 ( 𝜗 𝜚 𝜑 𝜛 ) [ 𝜗 𝜚 𝜑 𝜛 ] { 𝜗 𝜚 𝜑 𝜛 } | 𝜗 𝜚 𝜑 𝜛 | ‖ 𝜗 𝜚 𝜑 𝜛 ‖

\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: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 ( ^^08 ) = ‖ 𝜑 ( 𝜑 1 , 𝜀 1 ) 0 . . . 0 𝜑 𝑘 𝑛 2 ( 𝜑 2 , 𝜀 1 ) 𝜑 ( 𝜑 2 , 𝜀 2 ) . . . 0 𝜑 𝑘 𝑛 1 ( 𝜑 𝑛 , 𝜀 1 ) 𝜑 𝑘 𝑛 2 ( 𝜑 𝑛 , 𝜀 2 ) . . . 𝜑 𝑘 𝑛 𝑛 − 1 ( 𝜑 𝑛 , 𝜀 𝑛 − 1 ) 𝜑 ( 𝜑 𝑛 , 𝜀 𝑛 ) ‖

\[W(\Phi)= \begin{Vmatrix}

\dfrac\varphi{(\varphi_1,\varepsilon_1)}&0&\dots&0\\

\dfrac{\varphi k_{n2}}{(\varphi_2,\varepsilon_1)}&

\dfrac\varphi{(\varphi_2,\varepsilon_2)}&\dots&0\\

\hdotsfor{5}\\

\dfrac{\varphi k_{n1}}{(\varphi_n,\varepsilon_1)}&

\dfrac{\varphi k_{n2}}{(\varphi_n,\varepsilon_2)}&\dots&

\dfrac{\varphi k_{n\,n-1}}{(\varphi_n,\varepsilon_{n-1})}&

\dfrac{\varphi}{(\varphi_n,\varepsilon_n)}

\end{Vmatrix}\]

The spacing of the dots can be varied through use of a square-bracket option, 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:

(62) P 𝑃 ( 𝑖 , 𝑗 )

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:

(63) P 𝑃 ( 𝑖 , 𝑗 )

\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 R 0 𝑡 𝜀 𝐿 𝑥 , 𝑦 𝑥 ( 𝑠 ) 𝜑 ( 𝑥 ) 𝑑 𝑠

)

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

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

\biggr)

and now in \Large size:

(

E 𝑦 R 0 𝑡 𝜀 𝐿 𝑥 , 𝑦 𝑥 ( 𝑠 ) 𝜑 ( 𝑥 ) 𝑑 𝑠

)

{\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.

𝑓 ℎ , 𝜀 ( 𝑥 , 𝑦 ) = 𝜀 E 𝑥 , 𝑦 R 0 𝑡 𝜀 𝐿 𝑥 , 𝑦 𝜀 ( 𝜀 𝑢 ) 𝜑 ( 𝑥 ) 𝑑 𝑢 = ℎ R 𝐿 𝑥 , 𝑧 𝜑 ( 𝑥 ) 𝜌 𝑥 ( 𝑑 𝑧 ) + ℎ

(

1 𝑡 𝜀

(

E 𝑦 R 0 𝑡 𝜀 𝐿 𝑥 , 𝑦 𝑥 ( 𝑠 ) 𝜑 ( 𝑥 ) 𝑑 𝑠 − 𝑡 𝜀 R 𝐿 𝑥 , 𝑧 𝜑 ( 𝑥 ) 𝜌 𝑥 ( 𝑑 𝑧 )

)

= + ℎ

[

+ 1 𝑡 𝜀

(

E 𝑦 R 0 𝑡 𝜀 𝐿 𝑥 , 𝑦 𝑥 ( 𝑠 ) 𝜑 ( 𝑥 ) 𝑑 𝑠 − E 𝑥 , 𝑦 R 0 𝑡 𝜀 𝐿 𝑥 , 𝑦 𝜀 ( 𝜀 𝑠 ) 𝜑 ( 𝑥 ) 𝑑 𝑠

)

]

= ℎ 𝐿 b 𝑥 𝜑 ( 𝑥 ) + ℎ 𝜃 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}

Unnumbered version:

𝑓 ℎ , 𝜀 ( 𝑥 , 𝑦 ) = 𝜀 E 𝑥 , 𝑦 R 0 𝑡 𝜀 𝐿 𝑥 , 𝑦 𝜀 ( 𝜀 𝑢 ) 𝜑 ( 𝑥 ) 𝑑 𝑢 = ℎ R 𝐿 𝑥 , 𝑧 𝜑 ( 𝑥 ) 𝜌 𝑥 ( 𝑑 𝑧 ) + ℎ

[

1 𝑡 𝜀

(

E 𝑦 R 0 𝑡 𝜀 𝐿 𝑥 , 𝑦 𝑥 ( 𝑠 ) 𝜑 ( 𝑥 ) 𝑑 𝑠 − 𝑡 𝜀 R 𝐿 𝑥 , 𝑧 𝜑 ( 𝑥 ) 𝜌 𝑥 ( 𝑑 𝑧 )

)

= + ℎ

[

+ 1 𝑡 𝜀

(

E 𝑦 R 0 𝑡 𝜀 𝐿 𝑥 , 𝑦 𝑥 ( 𝑠 ) 𝜑 ( 𝑥 ) 𝑑 𝑠 − E 𝑥 , 𝑦 R 0 𝑡 𝜀 𝐿 𝑥 , 𝑦 𝜀 ( 𝜀 𝑠 ) 𝜑 ( 𝑥 ) 𝑑 𝑠

)

]

= ℎ 𝐿 b 𝑥 𝜑 ( 𝑥 ) + ℎ 𝜃 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:

| 𝐼 2 | = | R 0 𝑇 𝜓 ( 𝑡 ) { 𝑢 ( 𝑎 , 𝑡 ) − R 𝛾 ( 𝑡 ) 𝑎 𝑑 𝜃 𝑘 ( 𝜃 , 𝑡 ) R 𝑎 𝜃 𝑐 ( 𝜉 ) 𝑢 𝑡 ( 𝜉 , 𝑡 ) 𝑑 𝜉 } 𝑑 𝑡 | ≤ 𝐶 6 | | 𝑓 R | 𝑆 e 𝑎 , − − 1 , 0 𝑊 2 ( , ^^00 𝑙 ) | | | | 𝑢 | → ∘ 𝑊 2 𝐴 e ( ; ^^00 𝑟 , 𝑇 ) | | .

Some text after to test the below-display spacing.

Use of split within align:

(66) | 𝐼 1 | = | R 𝑔 𝑅 𝑢 𝑑 | ≤ 𝐶 3 [ R ( R 𝑎 𝑥 𝑔 ( 𝜉 , 𝑡 ) 𝑑 𝜉 ) 2 𝑑 ] 1 / 2 × [ R { 𝑢 𝑥 2 + 1 𝑘 ( R 𝑎 𝑥 𝑐 𝑢 𝑡 𝑑 𝜉 ) 2 } 𝑐 ] 1 / 2 ≤ 𝐶 4 | | 𝑓 | 𝑆 e 𝑎 , − − 1 , 0 𝑊 2 ( , ^^00 𝑙 ) | | | | 𝑢 | → ∘ 𝑊 2 𝐴 e ( ; ^^00 𝑟 , 𝑇 ) | | . (67) | 𝐼 2 | = | R 0 𝑇 𝜓 ( 𝑡 ) { 𝑢 ( 𝑎 , 𝑡 ) − R 𝛾 ( 𝑡 ) 𝑎 𝑑 𝜃 𝑘 ( 𝜃 , 𝑡 ) R 𝑎 𝜃 𝑐 ( 𝜉 ) 𝑢 𝑡 ( 𝜉 , 𝑡 ) 𝑑 𝜉 } 𝑑 𝑡 | ≤ 𝐶 6 | | 𝑓 R | 𝑆 e 𝑎 , − − 1 , 0 𝑊 2 ( , ^^00 𝑙 ) | | | | 𝑢 | → ∘ 𝑊 2 𝐴 e ( ; ^^00 𝑟 , 𝑇 ) | | .

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(\xi,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 | = | R 𝑔 𝑅 𝑢 𝑑 | ≤ 𝐶 3 [ R ( R 𝑎 𝑥 𝑔 ( 𝜉 , 𝑡 ) 𝑑 𝜉 ) 2 𝑑 ] 1 / 2 = × [ R { 𝑢 𝑥 2 + 1 𝑘 ( R 𝑎 𝑥 𝑐 𝑢 𝑡 𝑑 𝜉 ) 2 } 𝑐 ] 1 / 2 ≤ 𝐶 4 | | 𝑓 | 𝑆 e 𝑎 , − − 1 , 0 𝑊 2 ( , ^^00 𝑙 ) | | | | 𝑢 | → ∘ 𝑊 2 𝐴 e ( ; ^^00 𝑟 , 𝑇 ) | | . (67

^{'}

) | 𝐼 2 | = | R 0 𝑇 𝜓 ( 𝑡 ) { 𝑢 ( 𝑎 , 𝑡 ) − R 𝛾 ( 𝑡 ) 𝑎 𝑑 𝜃 𝑘 ( 𝜃 , 𝑡 ) R 𝑎 𝜃 𝑐 ( 𝜉 ) 𝑢 𝑡 ( 𝜉 , 𝑡 ) 𝑑 𝜉 } 𝑑 𝑡 | ≤ 𝐶 6 | | 𝑓 R | 𝑆 e 𝑎 , − − 1 , 0 𝑊 2 ( , ^^00 𝑙 ) | | | | 𝑢 | → ∘ 𝑊 2 𝐴 e ( ; ^^00 𝑟 , 𝑇 ) | | .

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(\xi,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:

R 𝑎 𝑏

{

R 𝑎 𝑏 [ 𝑓 ( 𝑥 ) 2 𝑔 ( 𝑦 ) 2 + 𝑓 ( 𝑦 ) 2 𝑔 ( 𝑥 ) 2 ] − 2 𝑓 ( 𝑥 ) 𝑔 ( 𝑥 ) 𝑓 ( 𝑦 ) 𝑔 ( 𝑦 ) 𝑑 𝑥

}

𝑑 𝑦 (68) = R 𝑎 𝑏

{

𝑔 ( 𝑦 ) 2 R 𝑎 𝑏 𝑓 2 + 𝑓 ( 𝑦 ) 2 R 𝑎 𝑏 𝑔 2 − 2 𝑓 ( 𝑦 ) 𝑔 ( 𝑦 ) R 𝑎 𝑏 𝑓 𝑔

}

𝑑 𝑦

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:

R 𝑎 𝑏

{

R 𝑎 𝑏 [ 𝑓 ( 𝑥 ) 2 𝑔 ( 𝑦 ) 2 + 𝑓 ( 𝑦 ) 2 𝑔 ( 𝑥 ) 2 ] − 2 𝑓 ( 𝑥 ) 𝑔 ( 𝑥 ) 𝑓 ( 𝑦 ) 𝑔 ( 𝑦 ) 𝑑 𝑥

}

𝑑 𝑦 = R 𝑎 𝑏

{

𝑔 ( 𝑦 ) 2 R 𝑎 𝑏 𝑓 2 + 𝑓 ( 𝑦 ) 2 R 𝑎 𝑏 𝑔 2 − 2 𝑓 ( 𝑦 ) 𝑔 ( 𝑦 ) R 𝑎 𝑏 𝑓 𝑔

}

𝑑 𝑦

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:

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

\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 : ℑ 𝑧 = ℑ 𝑎 , | 𝑧 − 𝑎 | < 𝑟 } , 𝑐 ( 𝑒 , 𝜃 , 𝑟 ) ≡ { ( 𝑥 , 𝑦 ) ∈ C : | 𝑥 − 𝑒 | < 𝑦 tan 𝜃 , 0 < 𝑦 < 𝑟 } , 𝐶 ( 𝐸 , 𝜃 , 𝑟 ) ≡ S 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:

(72) 𝛾 𝑥 ( 𝑡 ) = ( cos 𝑡 𝑢 + sin 𝑡 𝑥 , 𝑣 ) , (73) 𝛾 𝑦 ( 𝑡 ) = ( 𝑢 , cos 𝑡 𝑣 + sin 𝑡 𝑦 ) , (74) 𝛾 𝑧 ( 𝑡 ) = ( cos 𝑡 𝑢 + 𝛼 𝛽 sin 𝑡 𝑣 , − 𝛽 𝛼 sin 𝑡 𝑢 + cos 𝑡 𝑣 ) .

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 𝑡 𝑢 + sin 𝑡 𝑥 , 𝑣 ) , 𝛾 𝑦 ( 𝑡 ) = ( 𝑢 , cos 𝑡 𝑣 + sin 𝑡 𝑦 ) , 𝛾 𝑧 ( 𝑡 ) = ( cos 𝑡 𝑢 + 𝛼 𝛽 sin 𝑡 𝑣 , − 𝛽 𝛼 sin 𝑡 𝑢 + cos 𝑡 𝑣 ) .

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:

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

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.

Automatically numbered gather with split and align*:

(78) 𝜑 ( 𝑥 , 𝑧 ) = 𝑧 − 𝛾 10 𝑥 − 𝛾 𝑚 𝑛 𝑥 𝑚 𝑧 𝑛 = 𝑧 − 𝑀 𝑟 − 1 𝑥 − 𝑀 𝑟 − ( 𝑚 + 𝑛 ) 𝑥 𝑚 𝑧 𝑛

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.

𝜑 ( 𝑥 , 𝑧 ) = 𝑧 − 𝛾 10 𝑥 − 𝛾 𝑚 𝑛 𝑥 𝑚 𝑧 𝑛 = 𝑧 − 𝑀 𝑟 − 1 𝑥 − 𝑀 𝑟 − ( 𝑚 + 𝑛 ) 𝑥 𝑚 𝑧 𝑛

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:

(82) 𝑉 𝑖 = 𝑣 𝑖 − 𝑞 𝑖 𝑣 𝑗 , 𝑋 𝑖 = 𝑥 𝑖 − 𝑞 𝑖 𝑥 𝑗 , 𝑈 𝑖 = 𝑢 𝑖 , for

𝑖 ̸ = 𝑗

; (83) 𝑉 𝑗 = 𝑣 𝑗 , 𝑋 𝑗 = 𝑥 𝑗 , 𝑈 𝑗 𝑢 𝑗 + P 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 $i\ne j$;}\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

(

; 𝑉 𝑗 = 𝑣 𝑗 , 𝑋 𝑗 = 𝑥 𝑗 , 𝑈 𝑗 𝑢 𝑗 + P 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

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

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

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[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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[5] R. Impagliazzo, L. Levin, and M. Luby, Pseudo-random generation from one-way functions, Proc. 21st STOC (1989), ACM, New York, pp. 12–24.

[6] M. Kojima, S. Mizuno, and A. Yoshise, A new continuation method for complementarity problems with uniform p-functions, Tech. Report B-194, Tokyo Inst. of Technology, Tokyo, 1987, Dept. of Information Sciences.

[7] , A polynomial-time algorithm for a class of linear complementarity problems, Tech. Report B-193, Tokyo Inst. of Technology, Tokyo, 1987, Dept. of Information Sciences.

[8] C. J. Liu and Yutze Chow, On operator and formal sum methods for graph enumeration problems, SIAM J. Algorithms Discrete Methods 5 (1984), 384–438.

[9] M. Marcus and H. Minc, A survey of matrix theory and matrix inequalities, Complementary Series in Math. 14 (1964), 21–48.

[10] S. Mizuno, A. Yoshise, and T. Kikuchi, Practical polynomial time algorithms for linear complementarity problems, Tech. Report 13, Tokyo Inst. of Technology, Tokyo, April 1988, Dept. of Industrial Engineering and Management.

[11] R. D. Monteiro and I. Adler, Interior path following primal-dual algorithms, part II: Quadratic programming, August 1987, Working paper, Dept. of Industrial Engineering and Operations Research.

[12] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N.J., 1970.

[13] Y. Ye, Interior algorithms for linear, quadratic and linearly constrained convex programming, Ph.D. thesis, Stanford Univ., Palo Alto, Calif., July 1987, Dept. of Engineering–Economic Systems, unpublished.