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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

Let $A = (𝑎_{𝑖 𝑗})$ be the adjacency matrix of graph $𝐺$ . The corresponding Kirchhoff matrix $K = (𝑘_{𝑖 𝑗})$ is obtained from $A$ by replacing in $- A$ each diagonal entry by the degree of its corresponding vertex; i.e., the $𝑖$ th diagonal entry is identified with the degree of the $𝑖$ th vertex. It is well known that

\begin{matrix} (1) & \det K (𝑖 | 𝑖) = the number of spanning trees of𝐺, \end{matrix}

where $K (𝑖 | 𝑖)$ is the $𝑖$ th 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 $𝐶_{𝑖} = \underset{𝑗}{S} 𝐶_{𝑖 (𝑗)}$ . It is obvious that the collection of Hamiltonian cycles is a subset of $𝐶_{𝑖}$ . Note that the cardinality of $𝐶_{𝑖}$ is $𝑘_{𝑖 𝑖} \det K (𝑖 | 𝑖)$ . Let $\overset{b}{𝑋} = {{\hat{𝑥}}_{1}, . . ., {\hat{𝑥}}_{𝑛}}$ .

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

Define multiplication for the elements of $\overset{b}{𝑋}$ by

\begin{matrix} (2) & {\hat{𝑥}}_{𝑖} {\hat{𝑥}}_{𝑗} = {\hat{𝑥}}_{𝑗} {\hat{𝑥}}_{𝑖}, {\hat{𝑥}}_{𝑖}^{2} = 0, 𝑖, 𝑗 = 1, . . ., 𝑛 . \end{matrix}

Let ${\hat{𝑘}}_{𝑖 𝑗} = 𝑘_{𝑖 𝑗} {\hat{𝑥}}_{𝑗}$ and ${\hat{𝑘}}_{𝑖 𝑗} = - \underset{𝑗 ̸ = 𝑖}{P} {\hat{𝑘}}_{𝑖 𝑗}$ . Then the number of Hamiltonian cycles $𝐻_{𝑐}$ is given by the relation [8]

\begin{matrix} (3) & (Q_{}^{𝑛} {\hat{𝑥}}_{𝑗}) 𝐻_{𝑐} = \frac{1}{2} {\hat{𝑘}}_{𝑖 𝑗} \end{matrix}

The task here is to express (3) in a form free of any ${\hat{𝑥}}_{𝑖}$ , $i equals 1 comma period period period comma n$ . 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 $K sub n sub 1 n sub 2$ 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 $K sub n sub 1 n sub 2$ . 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 $K sub n sub 1 n sub 2$ 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 $K sub n sub 1 n sub 2$ . 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 $the set 0 comma 1$ . 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 $K sub n sub 1 period period period n sub p$ .

The conditions $𝑎_{𝑖 𝑗} = 𝑎_{𝑗 𝑖}$ , $i comma j equals 1 comma period period period comma n$ , 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 $𝑝, 𝑞 \in 𝑃$ and $𝑛 \in 𝜔$ we write $(𝑞, 𝑛) \leq (𝑝, 𝑛)$ if $𝑞 \leq 𝑝$ and $𝐴_{𝑞, 𝑛} = 𝐴_{𝑝, 𝑛}$ .

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

...

\end{notation}

Let $B = (𝑏_{𝑖 𝑗})$ be an $𝑛 \times 𝑛$ matrix. Let $n = {1, . . ., 𝑛}$ . Using the properties of (2), it is readily seen that

Lemma 3.1.

\begin{matrix} (4) & \underset{𝑖 \in n}{Q} (\underset{}{P} 𝑏_{𝑖 𝑗} {\hat{𝑥}}_{𝑖}) = (\underset{}{Q} {\hat{𝑥}}_{𝑖}) per \end{matrix}

where $per B$ is the permanent of $B$ .

Let $\overset{b}{𝑌} = {{\hat{𝑦}}_{1}, . . ., {\hat{𝑦}}_{𝑛}}$ . Define multiplication for the elements of $\overset{b}{𝑌}$ by

\begin{matrix} (5) & {\hat{𝑦}}_{𝑖} {\hat{𝑦}}_{𝑗} + {\hat{𝑦}}_{𝑗} {\hat{𝑦}}_{𝑖} = 0, 𝑖, 𝑗 = 1, . . ., 𝑛 . \end{matrix}

Then, it follows that

Lemma 3.2.

\begin{matrix} (6) & \underset{𝑖 \in n}{Q} (\underset{}{P} 𝑏_{𝑖 𝑗} {\hat{𝑦}}_{𝑗}) = (\underset{}{Q} {\hat{𝑦}}_{𝑖}) \det \end{matrix}

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

\begin{matrix} (7) & \underset{𝑗 \in n}{P} 𝑏_{𝑖 𝑗} {\hat{𝑦}}_{𝑗} = \underset{𝑗 \in n}{P} 𝑏_{𝑖 𝑗}^{(𝜆)} {\hat{𝑦}}_{𝑗} + (𝑏_{𝑖 𝑖} - 𝜆_{𝑖}) {\hat{𝑦}}_{𝑖} \hat{𝑦} \end{matrix}

where

\begin{matrix} (8) & 𝑏_{𝑖 𝑖}^{(𝜆)} = 𝜆_{𝑖}, 𝑏_{𝑖 𝑗}^{(𝜆)} = 𝑏_{𝑖 𝑗}, 𝑖 ̸ = 𝑗 . \end{matrix}

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

Theorem 3.3.

\begin{matrix} (9) & \det B = P_{𝑙 = 0}^{𝑛} \underset{𝐼_{𝑙} \subseteq 𝑛}{P} \underset{𝑖 \in 𝐼_{𝑙}}{Q} (𝑏_{𝑖 𝑖} - 𝜆_{𝑖}) \det B^{(𝜆)} (𝐼_{𝑙} | 𝐼_{𝑙}), \end{matrix}

where $𝐼_{𝑙} = {𝑖_{1}, . . ., 𝑖_{𝑙}}$ and $the lamda-th power of normal B of open paren I sub l divides I sub l close paren$ is the principal submatrix obtained from $B^{(𝜆)}$ by deleting its $i sub 1 comma period period period comma i sub l$ rows and columns.

Remark 3.1. Let $M$ be an $𝑛 \times 𝑛$ matrix. The convention $normal M of open paren normal n divides normal n close paren equals 1$ 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 $\det (B - 𝑥 I) = P_{𝑙 = 0}^{𝑛} (- 1)^{𝑙} 𝑏_{𝑙} 𝑥^{𝑙}$ . Then

\begin{matrix} (10) & 𝑏_{𝑙} = \underset{𝐼_{𝑙} \subseteq n}{P} \det B (𝐼_{𝑙} | 𝐼_{𝑙}) . \end{matrix}

Let

\begin{matrix} (11) & K (𝑡, 𝑡_{1}, . . ., 𝑡_{𝑛}) = (\begin{matrix} 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 \end{matrix}), \end{matrix}

\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

\begin{matrix} (12) & 𝐷_{𝑖} = \underset{𝑗 \in n}{P} 𝑎_{𝑖 𝑗} 𝑡_{𝑗}, 𝑖 = 1, . . ., 𝑛 . \end{matrix}

Set

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

Then

\begin{matrix} (13) & 𝐷 (𝑡_{1}, . . ., 𝑡_{𝑛}) = \underset{𝑖 \in n}{P} 𝐷_{𝑖} \det K (𝑡 = 1, 𝑡_{1}, . . ., 𝑡_{𝑛}; 𝑖 | 𝑖), \end{matrix}

where $normal K of open paren t equals 1 comma t sub 1 comma period period period comma t sub n semicolon i divides i close paren$ is the $𝑖$ th principal submatrix of $normal K of open paren t equals 1 comma t sub 1 comma period period period comma t sub n close paren$ .

Theorem 3.3 leads to

\begin{matrix} (14) & \det K (𝑡_{1}, 𝑡_{1}, . . ., 𝑡_{𝑛}) = \underset{𝐼 \in n}{P} (- 1)^{| 𝐼 |} 𝑡^{𝑛 - | 𝐼 |} \underset{𝑖 \in 𝐼}{Q} 𝑡_{𝑖} \underset{𝑗 \in 𝐼}{Q} (𝐷_{𝑗} + 𝜆_{𝑗} 𝑡_{𝑗}) \det A^{(𝜆 𝑡)} (\overline{𝐼} | \overline{𝐼}) . \end{matrix}

Note that

\begin{matrix} (15) & \det K (𝑡 = 1, 𝑡_{1}, . . ., 𝑡_{𝑛}) = \underset{𝐼 \in n}{P} (- 1)^{| 𝐼 |} \underset{𝑖 \in 𝐼}{Q} 𝑡_{𝑖} \underset{𝑗 \in 𝐼}{Q} (𝐷_{𝑗} + 𝜆_{𝑗} 𝑡_{𝑗}) \det A^{(𝜆)} (\overline{𝐼} | \overline{𝐼}) = 0 . \end{matrix}

Let $𝑡_{𝑖} = {\hat{𝑥}}_{𝑖}, 𝑖 = 1, . . ., 𝑛$ . Lemma 3.1 yields

\begin{matrix} (\underset{}{P} 𝑎_{𝑙_{𝑖}} 𝑥_{𝑖}) \det \\ (16) & = (\underset{}{Q} {\hat{𝑥}}_{𝑖}) \underset{𝐼 \subseteq n - {𝑙}}{P} (- 1)^{| 𝐼 |} per \end{matrix}

\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.

\begin{matrix} (17) & 𝐻_{𝑐} = \frac{1}{2 𝑛} P_{𝑙 = 0}^{𝑛} (- 1)^{𝑙} 𝐷_{𝑙}, \end{matrix}

where

\begin{matrix} (18) & 𝐷_{𝑙} = \underset{𝐼_{𝑙} \subseteq n}{P} 𝐷 (𝑡_{1}, . . ., 𝑡_{𝑛}) 2 |_{𝑡_{𝑖} = {\begin{smallmatrix} 0, & if 𝑖 \in 𝐼_{𝑙} \\ 1, & otherwise \end{smallmatrix}, 𝑖 = 1, . . ., 𝑛} . \end{matrix}

4 Application

We consider here the applications of Theorems 5.1 and 5.2 to a complete multipartite graph $K sub n sub 1 period period period n sub p$ . It can be shown that the number of spanning trees of $K sub n sub 1 period period period n sub p$ may be written

\begin{matrix} (19) & 𝑇 = 𝑛^{𝑝 - 2} Q_{𝑖 = 1}^{𝑝} (𝑛 - 𝑛_{𝑖})^{𝑛_{𝑖} - 1} \end{matrix}

where

\begin{matrix} (20) & 𝑛 = 𝑛_{1} + \cdot \cdot \cdot + 𝑛_{𝑝} . \end{matrix}

It follows from Theorems 5.1 and 5.2 that

\begin{aligned} 𝐻_{𝑐} & = \frac{1}{2 𝑛} P_{𝑙 = 0}^{𝑛} (- 1)^{𝑙} (𝑛 - 𝑙)^{𝑝 - 2} \underset{𝑙_{1} + \cdot \cdot \cdot + 𝑙_{𝑝} = 𝑙}{P} Q_{𝑖 = 1}^{𝑝} ( \end{aligned}

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

and

\begin{aligned} 𝐻_{𝑐} & = \frac{1}{2} P_{𝑙 = 0}^{𝑛 - 1} (- 1)^{𝑙} (𝑛 - 𝑙)^{𝑝 - 2} \underset{𝑙_{1} + \cdot \cdot \cdot + 𝑙_{𝑝} = 𝑙}{P} Q_{𝑖 = 1}^{𝑝} ( \end{aligned}

The enumeration of $𝐻_{𝑐}$ in a $𝐾_{𝑛_{1} \cdot \cdot \cdot 𝑛_{𝑝}}$ 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 $K sub n sub 1 n sub 2 n sub 3$ graph may be written

\begin{aligned} 𝐻_{𝑐} = & \frac{𝑛_{1}! 𝑛_{2}! 𝑛_{3}!}{𝑛_{1} + 𝑛_{2} + 𝑛_{3}} \underset{𝑖}{P} [((( \\ + . \end{aligned}

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 $e is greater than 0$ ) returns on input $y is a member of open brace 0 comma 1 close brace to the k-th power$ 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 $𝐻 : ℜ^{𝑛} \to ℜ^{𝑛}$ 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 $𝐵 𝐻 (𝑧) : ℜ^{𝑛} \to ℜ^{𝑛}$ , called the B-derivative of $𝐻$ at $𝑧$ , such that

\lim_{𝑣 \to 0} \frac{𝐻 (𝑧 + 𝑣) - 𝐻 (𝑧) - 𝐵 𝐻 (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 $𝐵 𝐻 (𝑧)$ is said to be strong if

\lim_{(𝑣, 𝑣^{'}) \to (0, 0)} \frac{𝐻 (𝑧 + 𝑣) - 𝐻 (𝑧 + 𝑣^{'}) - 𝐵 𝐻 (𝑧) (𝑣 - 𝑣^{'})}{‖ 𝑣 - 𝑣^{'} ‖} = 0 .

Lemma 6.1. There exists a smooth function $psi sub 0 of z$ defined for $| 𝑧 | > 1 - 2 𝑎$ satisfying the following properties:

(i) $psi sub 0 of z$ is bounded above and below by positive constants $c sub 1 is less than or equal to psi sub 0 of z is less than or equal to c sub 2$ .

(ii) If $the absolute value of z is greater than 1$ , then $psi sub 0 of z equals 1$ .

(iii) For all $𝑧$ in the domain of $psi sub 0$ , $hat hat 01 sub 0 l n psi sub 0 is greater than or equal to 0$ .

(iv) If $1 - 2 𝑎 < | 𝑧 | < 1 - 𝑎$ , then $hat hat 01 sub 0 l n psi sub 0 is greater than or equal to c sub 3 is greater than 0$ .

Proof. We choose $psi sub 0 of z$ to be a radial function depending only on $𝑟 = | 𝑧 |$ . Let $normal italic h of r is greater than or equal to 0$ be a suitable smooth function satisfying $normal italic h of r is greater than or equal to c sub 3$ for $1 - 2 𝑎 < | 𝑧 | < 1 - 𝑎$ , and $normal italic h of r equals 0$ for $| 𝑧 | > 1 - \frac{𝑎}{2}$ . The radial Laplacian

{^^01}_{0} \ln 𝜓_{0} (𝑟) = (\frac{𝑑^{2}}{𝑑 𝑟^{2}} + \frac{1}{𝑟} \frac{𝑑}{𝑑 𝑟}) \ln 𝜓_{0} (𝑟)

has smooth coefficients for $r is greater than 1 minus 2 a$ . Therefore, we may apply the existence and uniqueness theory for ordinary differential equations. Simply let $l n psi sub 0 of r$ be the solution of the differential equation

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

with initial conditions given by $l n psi sub 0 of 1 equals 0$ and $l n psi sub 0 prime of 1 equals 0$ .

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 $𝐷_{𝜈} (𝑎) = {𝑧 | | 𝑧 - 𝑧_{𝜈} | \leq (1 - 2 𝑎) 𝛿}$ . We define a smooth weight function $hat hat 08 sub 0 times z$ for $𝑧 \in 𝐶 - \underset{𝜈}{S} 𝐷_{𝜈} (𝑎)$ by setting ${^^08}_{0} (𝑧) = 1$ when $𝑧 \underset{𝜈}{S} 𝐷_{𝜈}$ and ${^^08}_{0} (𝑧) = 𝜓_{0} ((𝑧 - 𝑧_{𝜈}) / 𝛿)$ when $𝑧$ is an element of $𝐷_{𝜈}$ . It follows from Lemma 6.1 that $hat hat 08 sub 0$ satisfies the properties:

(i) $hat hat 08 sub 0 times z$ is bounded above and below by positive constants $c sub 1 is less than or equal to hat hat 08 sub 0 times z is less than or equal to c sub 2$ .

(ii) $hat hat 01 sub 0 l n hat hat 08 sub 0 is greater than or equal to 0$ for all $𝑧 \in 𝐶 - \underset{𝜈}{S} 𝐷_{𝜈} (𝑎)$ , the domain where the function $hat hat 08 sub 0$ is defined.

(iii) ${^^01}_{0} \ln {^^08}_{0} \geq 𝑐_{3} 𝛿^{- 2}$ when $(1 - 2 𝑎) 𝛿 < | 𝑧 - 𝑧_{𝜈} | < (1 - 𝑎) 𝛿$ .

Let $𝐴_{𝜈}$ denote the annulus $𝐴_{𝜈} = {(1 - 2 𝑎) 𝛿 < | 𝑧 - 𝑧_{𝜈} | < (1 - 𝑎) 𝛿}$ , and set $𝐴 = \underset{𝜈}{S} 𝐴_{𝜈}$ . The properties (2) and (3) of $hat hat 08 sub 0$ may be summarized as ${^^01}_{0} \ln {^^08}_{0} \geq 𝑐_{3} 𝛿^{- 2} 𝜒_{𝐴}$ , where $𝜒_{𝐴}$ is the characteristic function of $𝐴$ .

Suppose that $𝛼$ is a nonnegative real constant. We apply Proposition 3.5 with $^^08 (𝑧) = {^^08}_{0} (𝑧) 𝑒^{𝛼 {| 𝑧 |}^{2}}$ . If $𝑢 \in 𝐶_{0}^{\infty} (𝑅^{2} - \underset{𝜈}{S} 𝐷_{𝜈} (𝑎))$ , assume that $𝒟$ is a bounded domain containing the support of $𝑢$ and $𝐴 \subset 𝒟 \subset 𝑅^{2} - \underset{𝜈}{S} 𝐷_{𝜈} (𝑎)$ . A calculation gives

R_{𝒟} {| \overline{\partial} 𝑢 |}^{2} {^^08}_{0} (𝑧) 𝑒^{𝛼 {| 𝑧 |}^{2}} \geq 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 $hat hat 08 sub 0$ , then yields

R_{𝒟} {| \overline{\partial} 𝑢 |}^{2} 𝑒^{𝛼 {| 𝑧 |}^{2}} \geq 𝑐_{6} 𝛼 R_{𝒟} {| 𝑢 |}^{2} 𝑒^{𝛼 {| 𝑧 |}^{2}} + 𝑐_{7} 𝛿^{- 2} R_{𝐴} {| 𝑢 |}^{2} 𝑒^{𝛼 {| 𝑧 |}^{2}} .

Let $𝐵 (𝑋)$ be the set of blocks of $hat hat 03 sub X$ and let $𝑏 (𝑋) = | 𝐵 (𝑋) |$ . If $𝜙 \in 𝑄_{𝑋}$ then $𝜙$ is constant on the blocks of $hat hat 03 sub X$ .

\begin{matrix} (24) & 𝑃_{𝑋} = {𝜙 \in 𝑀 | {^^03}_{𝜙} = {^^03}_{𝑋}}, 𝑄_{𝑋} = {𝜙 \in 𝑀 | {^^03}_{𝜙} \geq {^^03}_{𝑋}} . \end{matrix}

If $hat hat 03 sub phi is greater than or equal to hat hat 03 sub X$ then $hat hat 03 sub phi equals hat hat 03 sub Y$ for some $𝑌 \geq 𝑋$ so that

𝑄_{𝑋} = \underset{𝑌 \geq 𝑋}{S} 𝑃_{𝑌} .

Thus by Möbius inversion

| 𝑃_{𝑌} | = \underset{𝑋 \geq 𝑌}{P} 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 $v to the k-th power$ defined by

𝑣_{𝑖}^{𝑘} = {\begin{matrix} 1 & if𝑖 \in {^^03}_{𝑘}, \\ 0 & otherwise. \end{matrix}

\[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) = \underset{ℬ \subseteq 𝒜}{P} (- 1)^{| ℬ |} 𝑡^{\dim 𝑇 (ℬ)} .

In order to compute $𝑅^{''}$ recall the definition of $𝑆 (𝑋, 𝑌)$ from Lemma 3.1. Since $𝐻 \in ℬ$ , $𝒜_{𝐻} \subseteq ℬ$ . Thus if $𝑇 (ℬ) = 𝑌$ then $ℬ \in 𝑆 (𝐻, 𝑌)$ . Let $𝐿^{''} = 𝐿 (𝒜^{''})$ . Then

\begin{aligned} 𝑅^{''} & = \underset{𝐻 \in ℬ \subseteq 𝒜}{P} (- 1)^{| ℬ |} 𝑡^{\dim 𝑇 (ℬ)} \\ = \underset{𝑌 \in 𝐿^{''}}{P} \underset{ℬ \in 𝑆 (𝐻, 𝑌)}{P} (- 1)^{| ℬ |} 𝑡^{\dim 𝑌} \\ = - \underset{𝑌 \in 𝐿^{''}}{P} \underset{ℬ \in 𝑆 (𝐻, 𝑌)}{P} (- 1)^{| ℬ - 𝒜_{𝐻} |} 𝑡^{\dim 𝑌} \\ = - \underset{𝑌 \in 𝐿^{''}}{P} 𝜇 (𝐻, 𝑌) 𝑡^{\dim 𝑌} \\ = - 𝜒 (𝒜^{''}, 𝑡) . \end{aligned}

Corollary 6.3. Let $(𝒜, 𝒜^{'}, 𝒜^{''})$ be a triple of arrangements. Then

𝜋 (𝒜, 𝑡) = 𝜋 (𝒜^{'}, 𝑡) + 𝑡 𝜋 (𝒜^{''}, 𝑡) .

Definition 6.2. Let $(𝒜, 𝒜^{'}, 𝒜^{''})$ be a triple with respect to the hyperplane $𝐻 \in 𝒜$ . Call $𝐻$ a separator if $𝑇 (𝒜) ̸ \in 𝐿 (𝒜^{'})$ .

Corollary 6.4. Let $(𝒜, 𝒜^{'}, 𝒜^{''})$ be a triple with respect to $𝐻 \in 𝒜$ .

(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 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) = \underset{𝐶 \in Cham (𝒜)}{S} 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 $pi of open paren hat hat 08 sub l comma t close paren equals 1$ , and (ii) is a consequence of Corollary 3.4.

Figure 1: $𝑄 (𝒜_{1}) = 𝑥 𝑦 𝑧 (𝑥 - 𝑧) (𝑥 + 𝑧) (𝑦 - 𝑧) (𝑦 + 𝑧)$

Figure 2: $𝑄 (𝒜_{2}) = 𝑥 𝑦 𝑧 (𝑥 + 𝑦 + 𝑧) (𝑥 + 𝑦 - 𝑧) (𝑥 - 𝑦 + 𝑧) (𝑥 - 𝑦 - 𝑧)$

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 $(𝑥, 𝑦) \in 𝑆_{𝑋, 𝑌}$ , then

⟨ 𝜎_{𝑗} (𝑥^{'}, 𝑦) ⟩_{𝑗 = 1}^{\infty} = ⟨ 𝜎_{𝑗} (𝑥, 𝑦) ⟩_{𝑗 = 1}^{\infty} = ⟨ 𝜎_{𝑗} (𝑥, 𝑦^{'}) ⟩_{𝑗 = 1}^{\infty} .

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 $i equals 0$ . Assume it holds for $i minus 1$ , in particular $[𝜎_{𝑗} (𝑥^{'}, 𝑦)]_{𝑗 = 1}^{𝑖 - 1} = [𝜎_{𝑗} (𝑥, 𝑦^{'})]_{𝑗 = 1}^{𝑖 - 1}$ . Then one of $[𝜎_{𝑗} (𝑥^{'}, 𝑦)]_{𝑗 = 𝑖}^{\infty}$ and $[𝜎_{𝑗} (𝑥, 𝑦^{'})]_{𝑗 = 𝑖}^{\infty}$ 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 $𝑖$ th 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 $𝑖$ th 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, $⟨ 𝜎_{𝑗} (𝑥, 𝑦) ⟩_{𝑗 = 1}^{\infty} ̸ = ⟨ 𝜎_{𝑗} (𝑥^{'}, 𝑦) ⟩_{𝑗 = 1}^{\infty}$ .

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 $𝑦 \in 𝑆_{𝑌}$ , the multiset1 of codewords ${𝜎_{𝜙} (𝑥, 𝑦) : 𝑥 \in 𝑆_{𝑋 | 𝑌} (𝑦)}$ is prefix free.

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

7 One-Way Complexity

$C hat sub 1 of open paren X divides Y close paren$ , 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

{\hat{𝐶}}_{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 $𝑦 \in 𝑆_{𝑌}$ .

We can now prove a continuity theorem.

Theorem 7.1. Let $subset of normal R to the n-th power$ be an open set, let $u is a member of B V of open paren semicolon normal R to the m-th power close paren$ , and let

\begin{matrix} (26) & 𝑇_{𝑥}^{𝑢} = {𝑦 \in R^{𝑚} : 𝑦 = \tilde{𝑢} (𝑥) + ⟨ \frac{𝐷 𝑢}{| 𝐷 𝑢 |} (𝑥), 𝑧 ⟩ for some 𝑧 \in R^{𝑛}} \end{matrix}

for every $𝑥 \in \ 𝑆_{𝑢}$ . Let $𝑓 : R^{𝑚} \to R^{𝑘}$ be a Lipschitz continuous function such that $f of 0 equals 0$ , and let $𝑣 = 𝑓 (𝑢) : \to R^{𝑘}$ . Then $v is a member of B V of open paren semicolon normal R to the k-th power close paren$ and

\begin{matrix} (27) & 𝐽 𝑣 = {(𝑓 (𝑢^{+}) - 𝑓 (𝑢^{-})) \otimes 𝜈_{𝑢} \cdot ℋ_{𝑛 - 1} |}_{𝑆_{𝑢}} . \end{matrix}

In addition, for $| \overset{e}{𝐷} 𝑢 |$ -almost every $𝑥 \in$ the restriction of the function $𝑓$ to $T sub x to the u-th power$ is differentiable at $\tilde{𝑢} (𝑥)$ and

\begin{matrix} (28) & \overset{e}{𝐷} 𝑣 = \nabla ({𝑓 |}_{𝑇_{𝑥}^{𝑢}}) (\tilde{𝑢}) \frac{\overset{e}{𝐷} 𝑢}{| \overset{e}{𝐷} 𝑢 |} \cdot | \overset{e}{𝐷} 𝑢 | . \end{matrix}

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

Remark 7.1. Let $omega colon close bracket 0 comma positive normal infinity open bracket right arrow close bracket 0 comma positive normal infinity open bracket$ be a continuous function such that $omega of t right arrow 0$ as $t right arrow 0$ . Then

\lim_{ℎ \to 0^{+}} 𝑔 (𝜔 (ℎ)) = 𝐿 \Leftrightarrow \lim_{ℎ \to 0^{+}} 𝑔 (ℎ) = 𝐿

for any function $g colon close bracket 0 comma positive normal infinity open bracket right arrow normal R$ .

Remark 7.2. Let $g colon normal R to the n-th power right arrow normal R$ be a Lipschitz continuous function and assume that

𝐿 (𝑧) = \lim_{ℎ \to 0^{+}} \frac{𝑔 (ℎ 𝑧) - 𝑔 (0)}{ℎ}

exists for every $z is a member of normal Q to the n-th power$ and that $𝐿$ is a linear function of $𝑧$ . Then $𝑔$ is differentiable at 0.

Remark 7.3. Let $𝐴 : R^{𝑛} \to R^{𝑚}$ be a linear function, and let $f colon normal R to the m-th power right arrow normal R$ be a function. Then the restriction of $𝑓$ to the range of $𝐴$ is differentiable at 0 if and only if $f of A colon normal R to the n-th power right arrow normal R$ is differentiable at 0 and

\nabla ({𝑓 |}_{Im (𝐴)}) (0) 𝐴 = \nabla (𝑓 (𝐴)) (0) .

Proof. We begin by showing that $v is a member of B V of open paren semicolon normal R to the k-th power close paren$ and

\begin{matrix} (29) & | 𝐷 𝑣 | (𝐵) \leq 𝐾 | 𝐷 𝑢 | (𝐵) \forall 𝐵 \in B (), \end{matrix}

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

\lim_{ℎ \to + \infty} R_{} | \nabla 𝑢_{ℎ} | 𝑑 𝑥 = | 𝐷 𝑢 | () .

The functions $𝑣_{ℎ} = 𝑓 (𝑢_{ℎ})$ are locally Lipschitz continuous in , and the definition of differential implies that $| \nabla 𝑣_{ℎ} | \leq 𝐾 | \nabla 𝑢_{ℎ} |$ almost everywhere in . The lower semicontinuity of the total variation and (13) yield

\begin{aligned} | 𝐷 𝑣 | () \leq \underset{ℎ \to + \infty}{\lim \inf} | 𝐷 𝑣_{ℎ} | () & = \underset{ℎ \to + \infty}{\lim \inf} R_{} | \nabla 𝑣_{ℎ} | 𝑑 𝑥 \\ \leq 𝐾 \underset{ℎ \to + \infty}{\lim \inf} R_{} | \nabla 𝑢_{ℎ} | 𝑑 𝑥 = 𝐾 | 𝐷 𝑢 | () . \end{aligned}

Since $f of 0 equals 0$ , we have also

R_{} | 𝑣 | 𝑑 𝑥 \leq 𝐾 R_{} | 𝑢 | 𝑑 𝑥;

therefore $u is a member of B V of open paren semicolon normal R to the k-th power close paren$ . Repeating the same argument for every open set $𝐴 \subset$ , we get (29) for every $𝐵 \in B ()$ , because $| 𝐷 𝑣 |$ , $| 𝐷 𝑢 |$ are Radon measures. To prove Lemma 6.1, first we observe that

1 lines Line 1: open paren 31 close paren S sub v subset of S sub u comma v tilde of x equals f of u tilde of x for all x is a member of backslash S sub u period

In fact, for every $epsilon is greater than 0$ we have

{𝑦 \in 𝐵_{𝜌} (𝑥) : | 𝑣 (𝑦) - 𝑓 (\tilde{𝑢} (𝑥)) | > 𝜀} \subset {𝑦 \in 𝐵_{𝜌} (𝑥) : | 𝑢 (𝑦) - \tilde{𝑢} (𝑥) | > 𝜀 / 𝐾},

hence

\lim_{𝜌 \to 0^{+}} \frac{| {𝑦 \in 𝐵_{𝜌} (𝑥) : | 𝑣 (𝑦) - 𝑓 (\tilde{𝑢} (𝑥)) | > 𝜀} |}{𝜌^{𝑛}} = 0

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

(𝑣^{+} (𝑥) - 𝑣^{-} (𝑥)) \otimes 𝜈_{𝑣} = (𝑓 (𝑢^{+} (𝑥)) - 𝑓 (𝑢^{-} (𝑥))) \otimes 𝜈_{𝑢} if 𝑥 \in 𝑆_{𝑣}

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

\begin{aligned} 𝐽 𝑣 (𝐵) = R_{𝐵 \cap 𝑆_{𝑣}} (𝑣^{+} - 𝑣^{-}) \otimes 𝜈_{𝑣} 𝑑 ℋ_{𝑛 - 1} & = R_{𝐵 \cap 𝑆_{𝑣}} (𝑓 (𝑢^{+}) - 𝑓 (𝑢^{-})) \otimes 𝜈_{𝑢} 𝑑 ℋ_{𝑛 - 1} \\ = R_{𝐵 \cap 𝑆_{𝑢}} (𝑓 (𝑢^{+}) - 𝑓 (𝑢^{-})) \otimes 𝜈_{𝑢} 𝑑 ℋ_{𝑛 - 1} \end{aligned}

and Lemma 6.1 is proved.

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

Step 1

Assume that $n equals 1$ . Since $𝑆_{𝑢}$ is at most countable, (7) yields that $| \overset{e}{𝐷} 𝑣 | (𝑆_{𝑢} \ 𝑆_{𝑣}) = 0$ , so that (19) and (21) imply that $𝐷 𝑣 = \overset{e}{𝐷} 𝑣 + 𝐽 𝑣$ is the Radon-Nikodým decomposition of $𝐷 𝑣$ in absolutely continuous and singular part with respect to $| \overset{e}{𝐷} 𝑢 |$ . By Theorem 5.2, we have

\frac{\overset{e}{𝐷} 𝑣}{| \overset{e}{𝐷} 𝑢 |} (𝑡) = \lim_{𝑠 \to 𝑡^{+}} \frac{𝐷 𝑣 ([𝑡, 𝑠 [)}{| \overset{e}{𝐷} 𝑢 | ([𝑡, 𝑠 [)}, \frac{\overset{e}{𝐷} 𝑢}{| \overset{e}{𝐷} 𝑢 |} (𝑡) = \lim_{𝑠 \to 𝑡^{+}} \frac{𝐷 𝑢 ([𝑡, 𝑠 [)}{| \overset{e}{𝐷} 𝑢 | ([𝑡, 𝑠 [)}

$| \overset{e}{𝐷} 𝑢 |$ -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 $\hat{𝑤}$ such that $\hat{𝑤} = 𝑤$ almost everywhere and $\lim_{𝑠 \to 𝑡^{-}} \hat{𝑤} (𝑠) = \hat{𝑤} (𝑡)$ for every $𝑡 \in R$ . These conditions imply

\begin{matrix} (32) & \hat{𝑢} (𝑡) = 𝐷 𝑢 (] - \infty, 𝑡 [), \hat{𝑣} (𝑡) = 𝐷 𝑣 (] - \infty, 𝑡 [) \forall 𝑡 \in R \end{matrix}

and

1 lines Line 1: open paren 33 close paren v hat of t equals f of u hat of t for all t is a member of normal R period

Let $𝑡 \in R$ be such that $| \overset{e}{𝐷} 𝑢 | ([𝑡, 𝑠 [) > 0$ for every $𝑠 > 𝑡$ and assume that the limits in (22) exist. By (23) and (24) we get

\begin{aligned} \frac{\hat{𝑣} (𝑠) - \hat{𝑣} (𝑡)}{| \overset{e}{𝐷} 𝑢 | ([𝑡, 𝑠 [)} & = \frac{𝑓 (\hat{𝑢} (𝑠)) - 𝑓 (\hat{𝑢} (𝑡))}{| \overset{e}{𝐷} 𝑢 | ([𝑡, 𝑠 [)} \\ = \frac{𝑓 (\hat{𝑢} (𝑠)) - 𝑓 (\hat{𝑢} (𝑡) + \frac{\overset{e}{𝐷} 𝑢}{| \overset{e}{𝐷} 𝑢 |} (𝑡) | \overset{e}{𝐷} 𝑢 | ([𝑡, 𝑠 [))}{| \overset{e}{𝐷} 𝑢 | ([𝑡, 𝑠 [)} \\ + \frac{𝑓 (\hat{𝑢} (𝑡) + \frac{\overset{e}{𝐷} 𝑢}{| \overset{e}{𝐷} 𝑢 |} (𝑡) | \overset{e}{𝐷} 𝑢 | ([𝑡, 𝑠 [)) - 𝑓 (\hat{𝑢} (𝑡))}{| \overset{e}{𝐷} 𝑢 | ([𝑡, 𝑠 [)} \end{aligned}

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

\begin{matrix} | \frac{\hat{𝑣} (𝑠) - \hat{𝑣} (𝑡)}{| \overset{e}{𝐷} 𝑢 | ([𝑡, 𝑠 [)} - \frac{𝑓 (\hat{𝑢} (𝑡) + \frac{\overset{e}{𝐷} 𝑢}{| \overset{e}{𝐷} 𝑢 |} (𝑡) | \overset{e}{𝐷} 𝑢 | ([𝑡, 𝑠 [)) - 𝑓 (\hat{𝑢} (𝑡))}{| \overset{e}{𝐷} 𝑢 | ([𝑡, 𝑠 [)} | \\ \leq 𝐾 | \frac{\hat{𝑢} (𝑠) - \hat{𝑢} (𝑡)}{| \overset{e}{𝐷} 𝑢 | ([𝑡, 𝑠 [)} - \frac{\overset{e}{𝐷} 𝑢}{| \overset{e}{𝐷} 𝑢 |} (𝑡) | . \end{matrix}

By (29), the function $𝑠 \to | \overset{e}{𝐷} 𝑢 | ([𝑡, 𝑠 [)$ is continuous and converges to 0 as $𝑠 ↓ 𝑡$ . Therefore Remark 7.1 and the previous inequality imply

\frac{\overset{e}{𝐷} 𝑣}{| \overset{e}{𝐷} 𝑢 |} (𝑡) = \lim_{ℎ \to 0^{+}} \frac{𝑓 (\hat{𝑢} (𝑡) + ℎ \frac{\overset{e}{𝐷} 𝑢}{| \overset{e}{𝐷} 𝑢 |} (𝑡)) - 𝑓 (\hat{𝑢} (𝑡))}{ℎ} | \overset{e}{𝐷} 𝑢 | -a.e. in R .

By (22), $\hat{𝑢} (𝑥) = \tilde{𝑢} (𝑥)$ for every $𝑥 \in R \ 𝑆_{𝑢}$ ; moreover, applying the same argument to the functions $𝑢^{'} (𝑡) = 𝑢 (- 𝑡)$ , $𝑣^{'} (𝑡) = 𝑓 (𝑢^{'} (𝑡)) = 𝑣 (- 𝑡)$ , we get

\frac{\overset{e}{𝐷} 𝑣}{| \overset{e}{𝐷} 𝑢 |} (𝑡) = \lim_{ℎ \to 0} \frac{𝑓 (\tilde{𝑢} (𝑡) + ℎ \frac{\overset{e}{𝐷} 𝑢}{| \overset{e}{𝐷} 𝑢 |} (𝑡)) - 𝑓 (\tilde{𝑢} (𝑡))}{ℎ} | \overset{e}{𝐷} 𝑢 | -a.e. in R

and our statement is proved.

Step 2

Let us consider now the general case $n is greater than 1$ . Let $nu is a member of normal R to the n-th power$ be such that $| 𝜈 | = 1$ , and let $𝜋_{𝜈} = {𝑦 \in R^{𝑛} : ⟨ 𝑦, 𝜈 ⟩ = 0}$ . In the following, we shall identify $normal R to the n-th power$ with $𝜋_{𝜈} \times R$ , 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 \to 0} \frac{𝑓 (\tilde{𝑢} (𝑦 + 𝑡 𝜈) + ℎ \frac{\overset{e}{𝐷} 𝑢_{𝑦}}{| \overset{e}{𝐷} 𝑢_{𝑦} |} (𝑡)) - 𝑓 (\tilde{𝑢} (𝑦 + 𝑡 𝜈))}{ℎ} = \frac{\overset{e}{𝐷} 𝑣_{𝑦}}{| \overset{e}{𝐷} 𝑢_{𝑦} |} (𝑡) | \overset{e}{𝐷} 𝑢_{𝑦} | -a.e. in R

for $normal script cap H sub n minus 1$ -almost every $𝑦 \in 𝜋_{𝜈}$ . We claim that

\begin{matrix} (34) & \frac{⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩}{| ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ |} (𝑦 + 𝑡 𝜈) = \frac{\overset{e}{𝐷} 𝑢_{𝑦}}{| \overset{e}{𝐷} 𝑢_{𝑦} |} (𝑡) | \overset{e}{𝐷} 𝑢_{𝑦} | -a.e. in R \end{matrix}

for $normal script cap H sub n minus 1$ -almost every $𝑦 \in 𝜋_{𝜈}$ . In fact, by (16) and (18) we get

\begin{matrix} R_{𝜋_{𝜈}} \frac{\overset{e}{𝐷} 𝑢_{𝑦}}{| \overset{e}{𝐷} 𝑢_{𝑦} |} \cdot | \overset{e}{𝐷} 𝑢_{𝑦} | 𝑑 ℋ_{𝑛 - 1} (𝑦) = R_{𝜋_{𝜈}} \overset{e}{𝐷} 𝑢_{𝑦} 𝑑 ℋ_{𝑛 - 1} (𝑦) \\ = ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ = \frac{⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩}{| ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ |} \cdot | ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ | = R_{𝜋_{𝜈}} \frac{⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩}{| ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ |} (𝑦 + \cdot 𝜈) \cdot | \overset{e}{𝐷} 𝑢_{𝑦} | 𝑑 ℋ_{𝑛 - 1} (𝑦) \end{matrix}

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

\begin{matrix} (35) & \frac{⟨ \overset{e}{𝐷} 𝑣, 𝜈 ⟩}{| ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ |} (𝑦 + 𝑡 𝜈) = \frac{\overset{e}{𝐷} 𝑣_{𝑦}}{| \overset{e}{𝐷} 𝑢_{𝑦} |} (𝑡) | \overset{e}{𝐷} 𝑢_{𝑦} | -a.e. in R \end{matrix}

for $normal script cap H sub n minus 1$ -almost every $𝑦 \in 𝜋_{𝜈}$ . By (24) and (25) we get

\lim_{ℎ \to 0} \frac{𝑓 (\tilde{𝑢} (𝑦 + 𝑡 𝜈) + ℎ \frac{⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩}{| ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ |} (𝑦 + 𝑡 𝜈)) - 𝑓 (\tilde{𝑢} (𝑦 + 𝑡 𝜈))}{ℎ} = \frac{⟨ \overset{e}{𝐷} 𝑣, 𝜈 ⟩}{| ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ |} (𝑦 + 𝑡 𝜈)

for $normal script cap H sub n minus 1$ -almost every $𝑦 \in 𝜋_{𝜈}$ , and using again (14), (15) we get

\lim_{ℎ \to 0} \frac{𝑓 (\tilde{𝑢} (𝑥) + ℎ \frac{⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩}{| ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ |} (𝑥)) - 𝑓 (\tilde{𝑢} (𝑥))}{ℎ} = \frac{⟨ \overset{e}{𝐷} 𝑣, 𝜈 ⟩}{| ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ |} (𝑥)

$| ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ |$ -a.e. in $normal R to the n-th power$ .

Since the function $| ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ | / | \overset{e}{𝐷} 𝑢 |$ is strictly positive $| ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ |$ -almost everywhere, we obtain also

\begin{matrix} \lim_{ℎ \to 0} \frac{𝑓 (\tilde{𝑢} (𝑥) + ℎ \frac{| ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ |}{| \overset{e}{𝐷} 𝑢 |} (𝑥) \frac{⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩}{| ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ |} (𝑥)) - 𝑓 (\tilde{𝑢} (𝑥))}{ℎ} \\ = \frac{| ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ |}{| \overset{e}{𝐷} 𝑢 |} (𝑥) \frac{⟨ \overset{e}{𝐷} 𝑣, 𝜈 ⟩}{| ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ |} (𝑥) \end{matrix}

$| ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ |$ -almost everywhere in $normal R to the n-th power$ .

Finally, since

\begin{align} \frac{| ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ |}{| \overset{e}{𝐷} 𝑢 |} \frac{⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩}{| ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ |} = \frac{⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩}{| \overset{e}{𝐷} 𝑢 |} = ⟨ \frac{\overset{e}{𝐷} 𝑢}{| \overset{e}{𝐷} 𝑢 |}, 𝜈 ⟩ | \overset{e}{𝐷} 𝑢 | -a.e. in R^{𝑛} \\ \frac{| ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ |}{| \overset{e}{𝐷} 𝑢 |} \frac{⟨ \overset{e}{𝐷} 𝑣, 𝜈 ⟩}{| ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ |} = \frac{⟨ \overset{e}{𝐷} 𝑣, 𝜈 ⟩}{| \overset{e}{𝐷} 𝑢 |} = ⟨ \frac{\overset{e}{𝐷} 𝑣}{| \overset{e}{𝐷} 𝑢 |}, 𝜈 ⟩ | \overset{e}{𝐷} 𝑢 | -a.e. in R^{𝑛} \end{align}

and since both sides of (33) are zero $| \overset{e}{𝐷} 𝑢 |$ -almost everywhere on $| ⟨ \overset{e}{𝐷} 𝑢, 𝜈 ⟩ |$ -negligible sets, we conclude that

\lim_{ℎ \to 0} \frac{𝑓 (\tilde{𝑢} (𝑥) + ℎ ⟨ \frac{\overset{e}{𝐷} 𝑢}{| \overset{e}{𝐷} 𝑢 |} (𝑥), 𝜈 ⟩) - 𝑓 (\tilde{𝑢} (𝑥))}{ℎ} = ⟨ \frac{\overset{e}{𝐷} 𝑣}{| \overset{e}{𝐷} 𝑢 |} (𝑥), 𝜈 ⟩,

$| \overset{e}{𝐷} 𝑢 |$ -a.e. in $normal R to the n-th power$ . Since $𝜈$ is arbitrary, by Remarks 7.2 and 7.3 the restriction of $𝑓$ to the affine space $T sub x to the u-th power$ is differentiable at $\tilde{𝑢} (𝑥)$ for $| \overset{e}{𝐷} 𝑢 |$ -almost every $x is a member of normal R to the n-th power$ and (26) holds.

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

\begin{matrix} (36) & 𝐷 (𝑡_{1}, . . ., 𝑡_{𝑛}) = \underset{𝐼 \in n}{P} (- 1)^{| 𝐼 | - 1} | 𝐼 | \underset{𝑖 \in 𝐼}{Q} 𝑡_{𝑖} \underset{𝑗 \in 𝐼}{Q} (𝐷_{𝑗} + 𝜆_{𝑗} 𝑡_{𝑗}) \det A^{(𝜆)} (\overline{𝐼} | \overline{𝐼}) . \end{matrix}

Let $𝑡_{𝑖} = {\hat{𝑥}}_{𝑖}$ , $i equals 1 comma period period period comma n$ . Lemma 1 leads to

\begin{matrix} (37) & 𝐷 ({\hat{𝑥}}_{1}, . . ., {\hat{𝑥}}_{𝑛}) = \underset{𝑖 \in n}{Q} {\hat{𝑥}}_{𝑖} \underset{𝐼 \in n}{P} (- 1)^{| 𝐼 | - 1} | 𝐼 | per A^{(𝜆)} (𝐼 | 𝐼) \det A^{(𝜆)} (\overline{𝐼} | \overline{𝐼}) . \end{matrix}

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

Theorem 7.2.

\begin{matrix} (38) & 𝐻_{𝑐} = \frac{1}{2 𝑛} P_{𝑙 = 1}^{𝑛} 𝑙 (- 1)^{𝑙 - 1} 𝐴_{𝑙}^{(𝜆)}, \end{matrix}

where

\begin{matrix} (39) & 𝐴_{𝑙}^{(𝜆)} = \underset{𝐼_{𝑙} \subseteq n}{P} per A^{(𝜆)} (𝐼_{𝑙} | 𝐼_{𝑙}) \det A^{(𝜆)} (\overline{𝐼} | \overline{𝐼}), | 𝐼_{𝑙} | = 𝑙 . \end{matrix}

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 $𝐴_{𝑙}$ , $lamda equals 0$ , 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 $lamda sub i equals 1$ , $i equals 1 comma period period period comma n$ . It follows from (39) that

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

By (38)

\begin{matrix} (41) & 𝐻_{𝑐} = \frac{1}{2} (𝑛 - 1)! . \end{matrix}

For a complete bipartite graph $K sub n sub 1 n sub 2$ , let $lamda sub i equals 0$ , $i equals 1 comma period period period comma n$ . By (39),

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

Theorem 7.2 leads to

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

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

\begin{matrix} \det K (𝑡 = 1, 𝑡_{1}, . . ., 𝑡_{𝑛}; 𝑙 | 𝑙) \\ (44) & = \underset{𝐼 \subseteq n - {𝑙}}{P} (- 1)^{| 𝐼 |} \underset{𝑖 \in 𝐼}{Q} 𝑡_{𝑖} \underset{𝑗 \in 𝐼}{Q} (𝐷_{𝑗} + 𝜆_{𝑗} 𝑡_{𝑗}) \det A^{(𝜆)} (\overline{𝐼} \cup {𝑙} | \overline{𝐼} \cup {𝑙}) . \end{matrix}

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

Theorem 7.3.

\begin{matrix} (45) & 𝐻_{𝑐} = \frac{1}{2} \underset{𝐼 \subseteq n - {𝑙}}{P} (- 1)^{| 𝐼 |} per A^{(𝜆)} (𝐼 | 𝐼) \det A^{(𝜆)} (\overline{𝐼} \cup {𝑙} | \overline{𝐼} \cup {𝑙}) \end{matrix}

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_{\infty} + 𝜋 𝐴_{0} \sim A_{\infty}

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{\partial 𝑥}{\partial 𝑦} | \frac{\partial 𝑦}{\partial 𝑧}

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

\pmb{\bigg\vert}

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

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

\underset{}{P} \underset{𝜅}{Q} 𝜅 𝐹 (𝑟_{𝑖}) \underset{}{P}

\[\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.

\begin{matrix} (46) & \underset{𝐴}{R R} 𝑓 (𝑥, 𝑦) 𝑑 𝑥 𝑑 𝑦 \underset{𝐴}{R R R} 𝑓 (𝑥, 𝑦, 𝑧) 𝑑 𝑥 𝑑 𝑦 𝑑 𝑧 \\ (47) & \underset{𝐴}{R R R R} 𝑓 (𝑤, 𝑥, 𝑦, 𝑧) 𝑑 𝑤 𝑑 𝑥 𝑑 𝑦 𝑑 𝑧 \underset{𝐴}{R \cdot \cdot \cdot R} 𝑓 (𝑥_{1}, . . ., 𝑥_{𝑘}) \end{matrix}

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} = \\ = \\ = \end{align}

\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 $A sub 1 comma A sub 2 comma period period period$ , the regional sum $𝐴_{1} + 𝐴_{2} + \cdot \cdot \cdot$ , the orthogonal product $𝐴_{1} 𝐴_{2} \cdot \cdot \cdot$ , and the infinite integral

R_{𝐴_{1}} R_{𝐴_{2}} \cdot \cdot \cdot

9.4 Accents in math

Double accents:

\hat{\hat{}} 𝐻 \overset{^^14}{\overset{^^14}{}} 𝐶 \tilde{\tilde{}} 𝑇 \overset{^^13}{\overset{^^13}{}} 𝐴 \overset{^^12}{\overset{^^12}{}} 𝐺 \overline{\overline{}} 𝐷 \overset{}{\overset{}{}} 𝐷 \overset{^^15}{\overset{^^15}{}} 𝐵 \overset{^^16}{\overset{^^16}{}} 𝐵 \vec{\vec{}} 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:

\overset{}{𝑄} \overset{}{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 $𝛽$ :

𝛽

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:

0 {\leftarrow -}_{}^{} 𝐹 \times △ [𝑛 - 1] \overset{\partial_{0} 𝛼 (𝑏)}{-} 𝐸^{\partial_{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:

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

\[\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 $P$ or $Q$ , without affecting the placement of limits. Examples:

\underset{𝑘}{_{*}^{*}Q_{*}^{*}}