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数学代写|组合数学代写Combinatorial Mathematics代考|Distance-regular graphs revisited

simple graph (Chapter 1, Section 1.1), where $X$ is the vertex set and $R$ is the edge set. For $x, y \in X$, let $\partial(x, y)$ be the length of a shortest path connecting $x$ and $y$. If there is no such path, then we set $\partial(x, y)=\infty$. We let $d=\operatorname{Max}{\partial(x, y) \mid x, y \in X}$ denote the maximum of the distances between vertices in $X$, and call $d$ the diameter of $\Gamma$. In what follows, we assume that the diameter satisfies $d<\infty$, that is, $\Gamma$ is a connected graph. Let $S^X$ be the symmetric group on $X$. An element $\sigma \in S^X$ is said to be an automorphism of $\Gamma$ if $\left(x^\sigma, y^\sigma\right) \in R$ for any edge $(x, y) \in R$. The set of all automorphisms of $\Gamma$ will be denoted by Aut( $\Gamma$ ). The set $\operatorname{Aut}(\Gamma)$ forms a subgroup of $S^X$, and is called the automorphism group of the graph $\Gamma$. It acts on $X \times X$ by $(x, y)^\sigma=\left(x^\sigma, y^\sigma\right)(\sigma \in \operatorname{Aut}(\Gamma))$. For $0 \leq i \leq d$, we set
$$R_i={(x, y) \in X \times X \mid \partial(x, y)=i}$$
Each $R_i$ is invariant under the action of $\operatorname{Aut}(\Gamma)$. Namely, Aut $(\Gamma)$ acts on each $R_i$. We say that $\Gamma$ is a distance-transitive graph if the action of Aut( $\Gamma$ ) on $R_i$ is transitive. That is, we call $\Gamma$ a distance-transitive graph if there exists $\sigma \in \operatorname{Aut}(\Gamma)$ such that $x^{\prime}=x^\sigma, y^{\prime}=y^\sigma$ for any $x, y, x^{\prime}, y^{\prime} \in X$ satisfying $\partial(x, y)=\partial\left(x^{\prime}, y^{\prime}\right)$.

We now forget about the group action, and return again to a finite connected simple graph $\Gamma=(X, R)$. Recall that $d$ denotes the diameter of $\Gamma$. For $x, y \in X$ and $i, j \in{0,1, \ldots, d}$, let
$$p_{i, j}(x, y)=\left|\left{z \in X \mid(x, z) \in R_i,(z, y) \in R_j\right}\right| .$$
If $\Gamma$ is a distance-transitive graph, then for any $i, j, k \in{0,1, \ldots, d}$ there exists a constant $p_{i, j}^k$ such that
$$p_{i, j}^k=p_{i, j}(x, y) \quad\left((x, y) \in R_k\right)$$

数学代写|组合数学代写Combinatorial Mathematics代考|Q-polynomial schemes revisited

As a preparation for Section $6.4$, where P- and Q-polynomial schemes will be discussed, we review the definition of Q-polynomial schemes briefly following the manner of the previous subsection, where P-polynomial schemes are discussed. Especially, the definition in terms of Terwilliger algebras will play an important role later. Let $\mathfrak{X}=\left(X,\left{R_i\right}_{0 \leq i \leq d}\right)$ be a symmetric association scheme and let $E_0, E_1, \ldots, E_d$ be the primitive idempotents of the Bose-Mesner algebra $\mathfrak{A}$ of $\mathfrak{X}$. By $\left(4^{\prime \prime}\right)$ in Chapter 2 , Section 2.3, we have
$$E_i \circ E_j=\frac{1}{|X|} \sum_{k=0}^d q_{i, j}^k E_k$$
with respect to the Hadamard product of $\mathfrak{A}$. Note that Krein numbers $q_{i, j}^k$ are nonnegative real (Theorem 2.26, Corollary 2.37). Then the following conditions (i) , (ii) $^{\prime}$, and (iii) ${ }^{\prime}$ on $q_{i, j}^k$ are equivalent, and a symmetric association scheme satisfying these conditions is called a Q-polynomial scheme. For the proof of the equivalence, the proof in the previous subsection is valid if we replace the ordinary matrix product by the Hadamard product and the intersection number $p_{i, j}^k$ by the Krein number $q_{i, j}^k$. We have: (i) ${ }^{\prime}$
$$q_{1, j}^k \begin{cases}=0, & \text { if } 1<|k-j|, \ \neq 0, & \text { if } 1=|k-j|\end{cases}$$ (ii) ${ }^{\prime} \quad E_i$ is a polynomial of degree $i$ in $E_1$ with respect to the Hadamard product ( $0 \leq$ $i \leq d)$ $\left(\right.$ (iii) $^{\prime}$ $$q_{i, j}^k \begin{cases}=0, & \text { if } i<|k-j| \text { or } i>k+j, \ \neq 0, & \text { if } i=|k-j| \text { or } i=k+j\end{cases}$$
If we set $b_i^=q_{1, i+1}^i, a_i^=q_{1, i}^i, c_i^=q_{1, i-1}^i$, then (i) ${ }^{\prime}$ is rewritten as (i) $$E_1 \circ E_j=\frac{1}{|X|}\left(b_{j-1}^ E_{j-1}+a_j^* E_j+c_{j+1}^* E_{j+1}\right) \quad(0 \leq j \leq d),$$
and $b_{i-1}^* c_i^* \neq 0(1 \leq i \leq d)$, where $b_{-1}^$ is indeterminate and $c_{d+1}^=1, E_{-1}=0, E_{d+1}=0$. Moreover, if we set $m=b_0^=q_{1,1}^0$, we have $$m=a_0^+b_0^=c_i^+a_i^+b_i^=c_d^+a_d^ \quad(1 \leq i \leq d-1)$$
(Proposition 2.24).

数学代写|组合数学代写Combinatorial Mathematics代考|Distance-regular graphs revisited

$$R_i=(x, y) \in X \times X \mid \partial(x, y)=i$$

\left 缺少或无法识别的分隔符

$$p_{i, j}^k=p_{i, j}(x, y) \quad\left((x, y) \in R_k\right)$$

数学代号|组合数学代与Combinatorial Mathematics代考|Q-polynomial schemes revisited

\left 缺少或无法识别的分隔符 是一个对称关联方宣，让 $E_0, E_1, \ldots, E_d$ 是 Bose-Mesner 代数的原始昌等元 A的 $\mathfrak{X}$. 经过 $\left(4^{\prime \prime}\right)$ 在第 2 章 $2.3$ 节中，我们有
$$E_i \circ E_j=\frac{1}{|X|} \sum_{k=0}^d q_{i, j}^k E_k$$

$$q_{1, j}^k{=0, \quad \text { if } 1<|k-j|, \neq 0, \quad \text { if } 1=|k-j|$$ (二) ${ }^{\prime} \quad E_i$ 是次数的多项式 $i$ 在 $E_1$ 关于 Hadamard 产品 $(0 \leq i \leq d)\left((\text { H })^{\prime}\right.$ $q_{i, j}^k{=0, \quad$ if $i<|k-j|$ or $i>k+j, \neq 0, \quad$ if $i=|k-j|$ or $i=k+j$

$c_{d+1} \overline{\bar{d}}i 1, E{-1}=0, E_{d+1}=0$. 此外，如果我们设置 $m=b_0^{=} q_{1,1}^0$ ， 我们有
$$m=a_0^{+} b_0^{=} c_i^{+} a_i^{+} b_i^{-} c_d^{+} a_d \quad(1 \leq i \leq d-1)$$
(提宲 2.24)。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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数学代写|组合数学代写Combinatorial Mathematics代考|Association schemes obtained from tight spherical t-designs

The contents here are due to Delsarte, Goethals, and Seidel [163]. We faithfully follow [163] in the discussion below (see also [32, Section 7.2]).

Let $A$ be a subset of $[-1,1)$ and let $X$ be an $A$-code on $S^{n-1}$. For $\alpha, \beta \in A^{\prime}(=A \cup{1})$ and for $x, y \in X$, we define
\begin{aligned} & v_\alpha(x)=|{z \in X \mid x \cdot z=\alpha}|, \ & p_{\alpha, \beta}(x, y)=|{z \in X \mid x \cdot z=\alpha, z \cdot y=\beta}| . \end{aligned}
By definition, we have $p_{\alpha, \alpha}(x, x)=v_\alpha(x)$ and $v_1(x)=1$. If for each $\alpha \in A^{\prime}, v_\alpha(x)$ is a constant independent of the choice of $x \in X$, then the $A$-code $X$ is said to be distanceinvariant. Moreover, for any $\alpha, \beta, \gamma \in A^{\prime}$, if $p_{\alpha, \beta}(x, y)$ is a constant independent of the choice of $x, y \in X$ such that $x \cdot y=y$, an association scheme is attached to the $A$-code $X$. To be more precise, let $A^{\prime}=\left{\alpha_0(=1), \alpha_1, \ldots, \alpha_s\right}$, and let $R_i=\left{(x, y) \in X \times X \mid x \cdot y=\alpha_i\right}$. Then
$$X \times X=R_0 \cup R_1 \cup \cdots \cup R_{\mathrm{S}}$$
holds, and $\left(X,\left{R_i\right}_{0 \leq i \leq s}\right)$ becomes an association scheme. The reference [163] gives a condition when an $A$-code $X$ on $S^{n-1}$ has this good property that an association scheme is attached to it (see also [32]). Now, let us define some more notation. Let
$$x^i=\sum_{\ell=0}^i f_{i, \ell} G_{\ell}^{(n)}(x)$$
be the Gegenbauer expansion of the monomial $x^i$. Also, let
$$F_{i, j}(x)=\sum_{\ell=0}^{\min {i, j}} f_{i, \ell} f_{j, \ell} G_{\ell}^{(n)}(x) .$$

数学代写|组合数学代写Combinatorial Mathematics代考|Connections of spherical designs with group theory, number theory, modular forms

(a) $t$-Designs obtained as orbits of finite groups
Let us consider what kinds of spherical $t$-designs there are. The most natural way of the construction is to consider orbits of a finite group $G$ in the orthogonal group $\mathrm{O}(n)$. Namely, for $x \in S^{n-1}$ we consider the orbit $x^G$ of $x$ by $G$ as follows:
$$X=x^G=\left{x^g \mid g \in G\right} \subset S^{n-1} .$$
There are many possibilities for $G$. We expect that if we take larger finite subgroups $G$ in $O(n)$, then we may get better designs. This topic was already treated in [32, Chapter 6] in a detailed way, so here we just mention the points that we think important, leaving the details to the book [32]. The most important finite groups are: real reflection groups (including Weyl groups and more generally Coxeter groups), the Conway group Co.0 in the 24-dimensional space and their various subgroups, and Clifford groups. The research in this direction was started by Sobolev in the 1960s (cf. [412]). See also Sidelnikov [428, 429]. There are some works on properties of finite groups such that orbits become $t$-designs (for details, see [32]). Here we just mention some important facts that we believe interesting.

For each $n \geq 3$, among spherical $t$-designs that are obtained as an orbit of a finite group $G$ of $O(n)$, those with large $t$ are not yet found. We remark the following.

数学代写|组合数学代写Combinatorial Mathematics代考|Association schemes obtained from tight spherical t-designs

$$v_\alpha(x)=|z \in X| x \cdot z=\alpha\left|, \quad p_{\alpha, \beta}(x, y)=\right| z \in X|x \cdot z=\alpha, z \cdot y=\beta| .$$

\left 缺少或无法识别的分隔符 $\quad$.然后
$$X \times X=R_0 \cup R_1 \cup \cdots \cup R_{\mathrm{S}}$$

$$x^i=\sum_{\ell=0}^i f_{i, \ell} G_{\ell}^{(n)}(x)$$

$$F_{i, j}(x)=\sum_{\ell=0}^{\min i, j} f_{i, \ell} f_{j, \ell} G_{\ell}^{(n)}(x) .$$

数学代写|组合数学代写Combinatorial Mathematics代考|Connections of spherical designs with group theory, number theory, modular forms

(一种) $t$-作为有限群轨道获得的设计

\left 缺少或无法识别的分隔符

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Combinatorics, 数学代写, 组合学, 组合数学

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数学代写|组合数学代写Combinatorial Mathematics代考|The Assmus–Mattson theorem and its extensions

We introduce a famous theorem by Assmus and Mattson on a construction of designs from codes. Combinatorial 5-designs, which are not directly related to 5-transitive Mathieu groups, were first found by this method [6]. Let $N={1,2, \ldots, n}$ and $F_2={0,1}$. Each element of $F_2^n$ corresponds to a subset of $N$ as follows. For $\boldsymbol{u}=\left(u_1, u_2, \ldots, u_n\right) \in F_2^n$, the subset $\left{i \mid u_i=1,1 \leq i \leq n\right}$ of $N$ is called the support of $\boldsymbol{u}$ and is denoted by $\overline{\boldsymbol{u}}$. If $\boldsymbol{u}$ has weight $m$, then $\overline{\boldsymbol{u}}$ is an $m$-subset of $N$. By this correspondence, the set of codewords in $C$ of weight $m$ is identified with a subset of the set $N^{(m)}$ consisting of $m$-subsets of $N$. This means, in the language of association schemes, a subset in the Hamming scheme $H(n, 2)$ can be described in terms of the Johnson scheme $J(n, m)$.

Theorem 4.1 (Assmus and Mattson (1969) [6]). Let $\mathrm{C}$ be an $[n, k, \delta]$ code over $F_2$. Name$l y, C$ is a $k$-dimensional subspace of $F_2^n$ with minimum distance $\delta$. Let $t<\delta$. Moreover suppose the following condition holds.

There exist at most $\delta$ – $t$ positive integers in ${1,2, \ldots, n-t}$ which arise as the weights of codewords in $C^{\perp}$.
Then the following (1), (2) hold:
(1) $\left{\overline{\boldsymbol{u}} \in N^{(m)} \mid \boldsymbol{u} \in C^{\perp}, w(\boldsymbol{u})=m\right}$ forms a t-design in the Johnson scheme J(n,m);
(2) $\left{\overline{\boldsymbol{u}} \in N^{(m)} \mid \boldsymbol{u} \in C, w(\boldsymbol{u})=m\right}$ forms a t-design in the Johnson scheme J(n,m).
In (1), (2), we only consider the case that there exists a codeword such that $w(\boldsymbol{u})=m$.

数学代写|组合数学代写Combinatorial Mathematics代考|t-Designs in regular semilattices

First, we give definitions and basic facts on semilattices.
Definition $4.13$ (Poset). A partial order on a set $L$ is a binary relation $\leq$ on $L$ satisfying the following (1)-(3):
(1) reflexivity: $a \leq a$;
(2) transitivity: if $a \leq b$ and $b \leq c$, then $a \leq c$;
(3) antisymmetry: if $a \leq b$ and $b \leq a$, then $a=b$;
where $a, b, c \in L$. If a binary relation $\leq$ is a partial order on $L$, a pair $(L, \leq)$ is called a partially ordered set or a poset.

Definition 4.14 (Meet semilattice). Let $L$ be a poset. For a pair $a, b$ in $L$, an element of $L$, denoted by $a \wedge b$, is called the meet of $a$ and $b$ if it satisfies the following (1), (2):

(1) $a \wedge b \leq a, a \wedge b \leq b$;
(2) if $c \leq a$ and $c \leq b$ for $c \in L$, then $c \leq a \wedge b$.
Note that the meet $a \wedge b$ is uniquely determined if it exists. The poset $L$ is called a meet semilattice if the meet exists for every pair of elements of $L$.

Definition 4.15 (Join semilattice). Let $L$ be a poset. For a pair $a, b$ in $L$, an element of $L$, denoted by $a \vee b$, is called the join of $a$ and $b$ if it satisfies the following (1), (2):
(1) $a \vee b \geq a, a \vee b \geq b$
(2) if $c \geq a$ and $c \geq b$ for $c \in L$, then $c \geq a \vee b$.
Note that the join $a \vee b$ is uniquely determined if it exists. The poset $L$ is called a join semilattice if the join exists for every pair of elements of $L$. The poset $L$ is called a lattice if it is a meet lattice and a join lattice.

In what follows, a semilattice means a meet semilattice. Assume that for a semilattice $L$, there exists a unique element $u$ such that $u \leq x$ for all $x$. We denote this element by 0 .

数学代写|组合数学代写Combinatorial Mathematics代考|The Assmus-Mattson theorem and its extensions

$\boldsymbol{u}=\left(u_1, u_2, \ldots, u_n\right) \in F_2^n$, 子集 $\backslash$ left 缺少或无法识别的分隔符

$m$ ，然右 $\bar{u}$ 是一个 $m$-子集 $N$. 通过这种对应，代码字集在 $C$ 重量 $m$ 被识别为集合的一个子集 $N^{(m)}$ 包含由…组成 $m$-子集的 $N$. 这 意味着，在关联方案的语言中，汉明方客中的一个子集 $H(n, 2)$ 可以用约翰逊方客来苗述 $J(n, m)$.

(1) \left 缺少或无法识别的分隔符 在 Johnson 方案 $\mathrm{J}(\mathrm{n}, \mathrm{m})$ 中形成 $\mathrm{t}$ 设计；
(2) \left 缺少或无法识别的分隔符 在 Johnson 方案 $\mathrm{J}(\mathrm{n}, \mathrm{m})$ 中形成 $\mathrm{t}$ 设计。

数学代写|组合数学代写Combinatorial Mathematics代考|t-Designs in regular semilattices

1）自反性: $a \leq a$ ；
（2）传递性: if $a \leq b$ 和 $b \leq c$ ，然后 $a \leq c$;
（3）反对称性: 如果 $a \leq b$ 和 $b \leq a$ ，然后 $a=b$;

(2) :
(1) $a \wedge b \leq a, a \wedge b \leq b$;
（2）如果 $c \leq a$ 和 $c \leq b$ 为了 $c \in L$ ，然后 $c \leq a \wedge b$.

(2) : (
1) $a \vee b \geq a, a \vee b \geq b$
(2) 如果 $c \geq a$ 和 $c \geq b$ 为了c $c \in L$ ，然后 $c \geq a \vee b$.

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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数学代写|组合数学代写Combinatorial Mathematics代考|Finite subsets on projective spaces (compact symmetric spaces of rank 1)

The study of finite subsets on the sphere is generalized to the study of finite subsets on the real projective space. The real projective space is obtained by identifying two antipodal points of the sphere. Therefore, we can regard the study of finite subsets on the real projective space as the study of antipodal subsets on the sphere. In this sense, some of the results mentioned in the previous sections can be regarded as the results on the real projective space. The projective spaces over the real and complex spaces are identified as the space of 1-dimensional subspaces of the vector space over each of these fields. In this sense, the study of equiangular lines has been made. See Delsarte, Goethals, and Seidel (1975) [162] and Koornwinder (1976) [286]. Besides real and complex fields, there are projective spaces over the quaternion (skew) field and the Cayley octonion. (Only the projective plane exists over the Cayley octonion.) These spaces are, together with the sphere, called compact symmetric spaces of rank 1 . It is shown by Cartan [124] (work in the 1920s) that these are the only connected compact symmetric spaces of rank 1. Also, H.-C. Wang (1952) [503] showed that the compact symmetric spaces of rank 1 are characterized as compact 2-point homogeneous symmetric spaces. Note that this last property essentially corresponds to the distance-transitive property, and therefore in the case of graphs, to the property of distance-regular graphs or Ppolynomial association schemes.

数学代写|组合数学代写Combinatorial Mathematics代考|Finite subsets on compact symmetric spaces of general ranks

The compact symmetric spaces of a general rank were classified by E. Cartin around 1920 Helgason (1962) [213], Wolf [519], mathematical dictionaries, etc.

Now, we will see that finite subsets (codes and designs) on these spaces are very much studied. Typical compact symmetric spaces of rank greater than 1 are real Grassmann spaces. Finite subsets there have been studied extensively by Shor, Sloane, Calderbank, Hardin, Rains, and others. This work came from the study of error correcting codes in quantum computers, indeed a very interesting topic. The reader is referred to $[427,118]$. For a simple explanation in Japanese, please see [32, Chapter 15, Section 15.6].

The study of finite subsets of real Grassmann spaces from the viewpoint of algebraic combinatorics was started by Bachoc, Coulangeon, and Nebe (2002) [10], and then further developed in Bachoc, Bannai, and Coulangeon (2004) [9]. There, the concepts of $t$-designs and tight $t$-designs were defined, and a similar theory to the case of those on the spheres has been developed, although the classification of tight $t$-designs is not yet within reach. In the case of compact rank 1 symmetric spaces, 1-variable Jacobi polynomials appear as spherical functions. In the case of real Gassmann space, multivariable orthogonal polynomials appear. In fact, in the general compact symmetric rank $r$, certain generalized Jacobi polynomials of $r$ variables appear (the reader is referred to Vretare (1984) [498] or further more general work by Koornwinder (cf. [285])).

Roughly speaking, there are two kinds of compact symmetric spaces. One class consists of the homogeneous spaces $G / H$ of simple Lie groups $G$ by subgroups $H$, and another class consists of the compact simple Lie groups $G$ themselves. (The second class is also interpreted as $(G \times G) / G$ by the diagonal subgroup isomorphic to $G$.) The real Grassmann space is one in the first class, and the work on the real Grassmann case is generalized for other symmetric spaces in the first class. (For the complex Grassmann case, see Y. Miura [350] [the Master’s thesis of Kyushu University (2004)] and Roy (2010) [406]. There was an unpublished generalization by Takanori Yasuda on this topic.)

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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数学代写|组合数学代写Combinatorial Mathematics代考|Finite subsets on spheres

In this section, we first explain that similar methods to the study of finite subsets of association schemes are used for the study of finite subsets of spheres. The origin of this study can be found in Delsarte, Goethals, and Seidel [163]. This will be described as Delsarte’s theory on spheres. See also the survey article [38].

The main purpose of the study that we call algebraic combinatorics on a sphere is to study “good” finite subsets of the sphere. What “good” means can be regarded as a part of the problem. Actually, what “good” is not unique and there are various viewpoints. Among them, roughly speaking, we will first treat the following two viewpoints: the viewpoint from coding theory and the viewpoint from design theory.

数学代写|组合数学代写Combinatorial Mathematics代考|Study of ffnite sets on the sphere from the viewpoint of coding theory

We divide this subsection into parts (a) to (e) as follows.
(a) Let $N$ be a natural number. Among all the $N$-element subsets of the sphere, find subsets with the property that the minimum value of the non-zero distances (i. e., the minimum distance) is the largest. Then classify such sets. (Such sets are called “optimal” codes.)

This problem is also called the “Tammes problem” [447]. This problem, in botany, originated from the problem to study the locations of the pollen grain on a pistil of flower. The classification of optimal codes for the 2-dimensional sphere $S^2$ (in the 3-dimensional Euclidean space) had been known for $N \leq 12$ and $N=24$ until relatively recently (for the details, see Ericson and Zinoviev [180]). This problem for $N=13$ was solved by Musin and Tarasov (2012) [363] and for $N=14$ also by Musin and Tarasov (2015) [364].
The following problem (b) is in a similar direction to problem (a).
(b) Suppose that a positive real number is given. Among all the subsets of the sphere, what is the largest size of them having the property that the minimum distance is greater than or equal to that given positive real number? Then also classify those subsets with this largest size.

In the unit sphere $S^{n-1}$ in the real Euclidean space $\mathbb{R}^n$, the problem of finding a subset whose Euclidean distance between the distinct points in it are at least 1 (or equivalently the geodesic distance on the sphere is at least $\pi / 3$, or the central angle is at least 60 degrees, or the minimum inner product is at most $1 / 2$ ) is well known as the problem of finding the kissing number $k(n)$.

数学代写|组合数学代写Combinatorial Mathematics代考|Study of ffnite sets on the sphere from the viewpoint of coding theory

(a) 让 $N$ 是 个自然数。在所有的 $N$ – 球体的元拜子集，找到具有非雬距离 (即最小距离) 的最小值最大的属性的子集。然后对这 些集合进行分宩。（这样的集合称为“最优”代码。）

(b) 假设给定一个正实数。在球体的所有子集中，具有最小距离大于或等于给定正实数的性质的最大尺寸是多少? 然后也对具有最 大尺寸的那些子焦进行分类。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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数学代写|组合数学代写Combinatorial Mathematics代考|Solutions

14.1 Solutions for Chapter 1
1.1. $6^5 . \quad \mathbf{1 . 2} .4500 . \quad \mathbf{1 . 3 .} 3^4$. 1.4. $36 . \quad$ 1.5. $2^n . \quad$ 1.6. 72000 .
$14.2$ Solutions for Chapter 2
2.1. Digit $c_1$ of the three-digit number $c_1 c_2 c_3$ can be chosen arbitrarily from the set ${1,2, \ldots, 9}$. There are 9 possible choices of digit $c_2$ such that $c_2 \neq c_1$, and there are 8 possible choices of digit $c_3$ such that $c_3 \neq c_1$ and $c_3 \neq c_2$. By the product rule it follows that the number of positive integers with the given properties is equal to $9 \cdot 9 \cdot 8=648$.
2.2. 512. 2.3. $1320 . \quad 2.4 .2^n$. 2.5. (a) 20 , (b) $\left(k_1+1\right)\left(k_2+1\right) \ldots\left(k_m+1\right)$.
2.6. A positive integer $n$ has an odd number of divisors if and only if $n$ is a perfect square.
2.7. Every arrangement of teeth uniquely determines the 32-variation of elements 0 and 1 . Hence, the maximal possible number of citizens is $2^{32}$.
2.8. The number of permutations of the set ${1,2, \ldots, n}$ in which elements 1 and 2 are adjacent, and 1 is placed before 2 , is equal to the number of permutations of the set ${b, 3, \ldots, n}$, where $b$ is notation for 12 , i.e., $(n-1) !$. The number of permutations of the set ${1,2, \ldots, n}$, in which elements 1 and 2 are adjacent, and 1 comes after 2 is the same. Therefore, the number of permutations of the set ${1,2, \ldots, n}$ with adjacent elements 1 and 2 is $2(n-1) !$.
2.9. The number of permutations of the set ${1,2, \ldots n}$ in which element 2 is placed after element 1 (not necessarily in the adjacent position) is equal to the number of permutations in which 2 is placed before 1 . Since the total number of permutations of the $n$-set is equal to $n$ !, it follows that the number we are asking for is equal to $\frac{1}{2} n !$.
2.10. The number of permutations of the set ${1,2, \ldots, n}$ such that element 1 is placed at position $i$, and element 2 is placed at position $j$, where $i \neq j$, and $i, j \in{1,2, \ldots, n}$, is equal to the number of permutations of a set consisting of $n-2$ elements, i.e., $(n-2)$ !. The list of pairs of positions that can be occupied by 1 and 2 is the following: $(1, k+2),(2, k+3), \ldots,(n-k-1, n)$. Hence, there are $n-k-1$ such pairs. Elements 1 and 2 can occupy any of these pairs in 2 ways. Hence, the number of permutations that satisfy the given conditions is $2(n-k-1)(n-2) !$.

数学代写|组合数学代写Combinatorial Mathematics代考|Solutions for Chapter

3.1. $\left(\begin{array}{c}20 \ 14\end{array}\right) 3^{14} 2^{-6}$
3.2. $\left(\begin{array}{l}15 \ 12\end{array}\right)(\sqrt{2})^3(\sqrt[3]{3})^{12}=73710 \sqrt{2} . \quad$ 3.3. 17
3.4. The sum of all coefficients is the value of the polynomial for $x=1$ and is equal to 1 .
3.5. All terms that contain $\sqrt{2}$ vanish.
3.6. $n=7, k=2 . \quad$ 3.7. $n=14, k=6$.
3.8. By the binomial theorem it follows that
$$(1+x)^n=1+n x+\sum_{k=2}^n\left(\begin{array}{l} n \ k \end{array}\right) x^k \geqslant 1+n x, \quad \text { if } x \geqslant 0$$

3.9. Hint. Use the Bernoulli inequality, the binomial theorem and the fact that for $k \in{2,3, \ldots, n}$ the inequalities $\left(\begin{array}{l}n \ k\end{array}\right) \frac{1}{n^k}<\frac{1}{k !}<\frac{1}{2^{k-1}}$ hold.
3.10. From the binomial theorem it follows that
$$(2+\sqrt{3})^n+(2-\sqrt{3})^n=2\left[2^n+\left(\begin{array}{l} n \ 2 \end{array}\right) 2^{n-2} 3+\left(\begin{array}{c} n \ 4 \end{array}\right) 2^{n-4} 3^2+\cdots\right] .$$
Therefore, $(2+\sqrt{3})^n+(2-\sqrt{3})^n$ is an even positive integer. Since $0<$ $(2-\sqrt{3})^n<1$, it follows that $\left[(2+\sqrt{3})^n\right]=(2+\sqrt{3})^n+(2-\sqrt{3})^n-1$, i.e., $\left[(2+\sqrt{3})^n\right]$ is an odd positive integer.
3.11. For every positive integer $n$ it follows by the binomial theorem that
$$\left(n+\sqrt{n^2+1}\right)^n-\left(\sqrt{n^2+1}-n\right)^n=x_n \in \mathbb{N} .$$
For $n \geqslant 5$ we obtain
\begin{aligned} y_n &=\left(\sqrt{n^2+1}-n\right)^n=\frac{1}{\left(\sqrt{n^2+1}+n\right)^n} \ & \leqslant \frac{1}{(\sqrt{26}+5)^n}<10^{-n}=0 \cdot \underbrace{00 \ldots 0}_{n-1} 1 . \end{aligned}
Since $\left(n+\sqrt{n^2+1}\right)^n=x_n+y_n$, the statement follows.

数学代写组合数学代写Combinatorial Mathematics代考|Solutions

$14.1$ 第 1 章的解决方案
1.1。 $6^5$. 1.2.4500. 1.3.34. 1.4.36. 1.5. $2^n . \quad 1.6 .72000$.
$14.2$ 第 2 章的解决方案
2.1。数字 $c_1$ 三位数的 $c_1 c_2 c_3$ 可以从堆合中任意选择 $1,2, \ldots, 9$. 有 9 种可能的数字选择 $c_2$ 这样 $c_2 \neq c_1$ ，并且有 8 种可能的数字选

2.2. 512. 2.3. $1320 . \quad 2.4 .2^n \cdot 2.5$. (a) 20 （b $\left(k_1+1\right)\left(k_2+1\right) \ldots\left(k_m+1\right)$.
2.6. 一个正整数 $n$ 有奇数个除数当且仅当 $n$ 是一个完美的正方形。
2.7. 牙齿的每一种排列都唯一地决定了元轸 0 和 1 的 32 种变化。因此，最大可能的公民人数是 $2^{32}$.
2.8. 集合的排列数 $1,2, \ldots, n$ 其中元㗑 1 和 2 相邻，并且 1 放在 2 之前，等于集合的排列数 $b, 3, \ldots, n$ ，在哪里 $b$ 是 12 的符号，

$(1, k+2),(2, k+3), \ldots,(n-k-1, n)$. 因此，有 $n-k-1$ 这样的对。元塐 1 和 2 可以以 2 种方式占据这些对中的任何 一个。因此，满足给定条件的排列数是 $2(n-k-1)(n-2) !$.

数学代写|组合数学代写Combinatorial Mathematics代考|Solutions for Chapter

3.1. $(2014) 3^{14} 2^{-6}$
3.2. $(1512)(\sqrt{2})^3(\sqrt[3]{3})^{12}=73710 \sqrt{2} . \quad$ 3.3. 17
3.4。所有系数的总和是多项式的值 $x=1$ 并且等于 1 。
3.5. 包含的所有条款 $\sqrt{2}$ 消失。
$$\text { 3.6. } n=7, k=2 \text {. 3.7. } n=14, k=6 \text {. }$$
3.8. 由二项式定理可知
$$(1+x)^n=1+n x+\sum_{k=2}^n(n k) x^k \geqslant 1+n x, \quad \text { if } x \geqslant 0$$
3.9. 暗二示。使用伯努利不等式、二项式定理和事实 $k \in 2,3, \ldots, n$ 不平等 $(n k) \frac{1}{n^k}<\frac{1}{k !}<\frac{1}{2^{k-1}}$ 抓住。
3.10。从二项式定理可以得出
$$(2+\sqrt{3})^n+(2-\sqrt{3})^n=2\left[2^n+(n 2) 2^{n-2} 3+(n 4) 2^{n-4} 3^2+\cdots\right] .$$

3.11。对于每个正整数 $n$ 它遭循二项式定理
$$\left(n+\sqrt{n^2+1}\right)^n-\left(\sqrt{n^2+1}-n\right)^n=x_n \in \mathbb{N}$$

$$y_n=\left(\sqrt{n^2+1}-n\right)^n=\frac{1}{\left(\sqrt{n^2+1}+n\right)^n} \quad \leqslant \frac{1}{(\sqrt{26}+5)^n}<10^{-n}=0 \cdot \underbrace{00 \ldots 0}_{n-1} 1$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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数学代写|组合数学代写Combinatorial Mathematics代考|Existence and non-existence of tight designs

Let $V={1,2, \ldots, v}$, and let $X=\left(\begin{array}{l}V \ d\end{array}\right)$ be the set of $d$-element subsets of $V$. We assume $1 \leq t \leq d \leq \frac{v}{2}$. Let $\left(X,\left{R_i\right}_{0 \leq i \leq d}\right)$ be the Johnson scheme $J(v, d)$. Recall that a 2 -design $Y \subset X$ is tight if $\left.|Y|=m_0+m_1+\cdots+m_e=\sum_{i=0}^e\left(\begin{array}{l}v \ i\end{array}\right)-\left(\begin{array}{c}v \ i-1\end{array}\right)\right)=\left(\begin{array}{c}v \ e\end{array}\right)$ holds. In general, a 2e-design $Y$ satisfies the Fisher type inequality $|Y| \geq\left(\begin{array}{l}v \ v\end{array}\right)$. This inequality was first obtained by Petrenjuk [395] for the case $e=2$. Afterwards, Ray-Chaudhuri and Wilson announced that it holds for any $e$ [514]. For the proof, see Ray-Chaudhuri and Wilson [400]. Delsarte (1974) also gave a proof, and as was written in the book of Hiroshi Nagao [366], Noda and Bannai obtained the same result independently in 1972 . If there exists a tight $2 e$-design, by using Theorem 3.16, we can show the following theorem.
Theorem 3.27. If there exists a tight $2 e$-design in the Johnson scheme $J(v, d)$, then all $e$ zeros of the polynomial
$$\Psi_e(x)=\sum_{i=0}^e(-1)^{e-i} \frac{\left(\begin{array}{c} v-e \ i \end{array}\right)\left(\begin{array}{c} k-i \ e-i \end{array}\right)\left(\begin{array}{c} k-1-i \ e-i \end{array}\right)}{\left(\begin{array}{c} e \ i \end{array}\right)}\left(\begin{array}{c} x \ i \end{array}\right)$$
are positive integers. This polynomial $\Psi_e$ is called the Wilson polynomial or the RayChaudhuri-Wilson polynomial.

When $e=1, Y$ is a tight 2-design if and only if $b=v(b=|Y|)$ if and only if $Y$ is a symmetric 2-design (Chapter 1, Section 1.3, Definition 1.37). There exist quite a few symmetric 2-designs and their classification seems to be almost impossible.

When $e=2$, the non-trivial tight 4-designs are the Witt design 4- $(23,7,1)$ and its complementary design 4- $(23,16,52)$ only. (In the latter case, $d \leq \frac{v}{2}$ does not hold.) The classification was started by Noboru Ito $[245,246]$ and was almost completed by Enomoto, Ito, and Noda [179]. To be precise, the classification was completely solved by Bremner [111] and Stroeker [440] by determining a rational integral solution of the Diophantine equation $3 x^4-4 y^4-2 x^2+12 y^2-9=0$, which is related to an elliptic function. In the next subsection, we present the detailed proof by Noda.

数学代写|组合数学代写Combinatorial Mathematics代考|Classiffcation of tight 4-designs in Johnson schemes

Let $(V, \mathcal{B})$ be a tight 4-design in the Johnson scheme $J(v, k)$. Suppose $(V, \mathcal{B})$ is nontrivial. Assume $k \leq \frac{v}{2}$. Note that if $k>\frac{v}{2}$, the complementary design is also a tight 4-design. In this section, we prove the following theorem.

Theorem $3.32$ (Enomoto-Ito-Noda [179]). A non-trivial tight 4-design in the Johnson scheme is the 4- $(23,7,1)$ design or its complementary design 4-(23, 16, 52) only.

Remark 3.33. Noboru Ito $[245,246]$ started the proof of this theorem, and Enomoto, Ito, and Noda (1979) [179] almost completed the proof by correcting errors. In a part of the proof, a number theoretic result on the solution of a Diophantine equation was used (Bremner [111], Stroeker [440]). The proof given here is based on an unpublished note by Ryuzaburo Noda, which was written soon after [179]. We are grateful to Professor Noda, who permits us to use the contents of his note. It is similar to [179] that the problem is transformed into the Diophantine equation. However, compared to the proof combining three papers $[245,246,179]$, this proof is clearer and easier to read.
The proof of Theorem $3.32$ consists of steps $(\mathrm{A})-(\mathrm{K})$.
(A) Let $i, j$ be the cardinalities of intersections of two distinct blocks. Let $i<j$. Then $i, j$ are the roots of the following quadratic equation:
$$X^2-\left(\frac{2(k-1)(k-2)}{v-3}+1\right) X+\frac{k(k-1)^2(k-2)}{(v-2)(v-3)}=0 .$$
(B) We have
$$(v-2)(v-3) \mid 2 k(k-1)(k-2)$$
Proof. Since $b=\lambda_0=\left(\begin{array}{c}v \ 2\end{array}\right), \lambda_4=\frac{k(k-1)(k-2)(k-3)}{2(v-2)(v-3)}$ is an integer. Therefore, $2 \lambda_4=$ $\frac{k(k-1)(k-2)(k-3)}{(v-2)(v-3)}$ is an integer. On the other hand, by (A), $\frac{k(k-1)^2(k-2)}{(v-2)(v-3)}$ is an integer, and hence $\frac{k(k-1)^2(k-2)}{(v-2)(v-3)}-\frac{k(k-1)(k-2)(k-3)}{(v-2)(v-3)}=\frac{2 k(k-1)(k-2)}{(v-2)(v-3)}$ is also an integer.

数学代写|组合数学代写组合数学代考|紧密设计的存在与不存在

$$\Psi_e(x)=\sum_{i=0}^e(-1)^{e-i} \frac{\left(\begin{array}{c} v-e \ i \end{array}\right)\left(\begin{array}{c} k-i \ e-i \end{array}\right)\left(\begin{array}{c} k-1-i \ e-i \end{array}\right)}{\left(\begin{array}{c} e \ i \end{array}\right)}\left(\begin{array}{c} x \ i \end{array}\right)$$

数学代写|组合数学代写组合数学代考| Johnson方案中紧密的4-设计的分类

(A)设$i, j$为两个不同块的交集的基数。让$i<j$。那么$i, j$是下面的二次方程的根:
$$X^2-\left(\frac{2(k-1)(k-2)}{v-3}+1\right) X+\frac{k(k-1)^2(k-2)}{(v-2)(v-3)}=0 .$$
(B)我们有
$$(v-2)(v-3) \mid 2 k(k-1)(k-2)$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Combinatorics, 数学代写, 组合学, 组合数学

avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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数学代写|组合数学代写Combinatorial Mathematics代考|The deffnition of association schemes

Definition 2.1 (Association schemes). If a pair $\mathfrak{X}=\left(X,\left{R_i\right}_{0 \leq i \leq d}\right)$ of a finite set $X$ and a set $\left{R_0, R_1, \ldots, R_d\right}$ of subsets of the direct product $X \times X$ satisfies the following conditions (1), (2), (3), and (4), then $\mathfrak{X}=\left(X,\left{R_i\right}_{0 \leq i \leq d}\right)$ is called an association scheme of class $d$. In what follows, $R_i$ is called the (i-th) relation.
(1) $R_0={(x, x) \mid x \in X}$.
(2) $X \times X=R_0 \cup R_1 \cup \cdots \cup R_d$, and $R_i \cap R_j=\emptyset$ if $i \neq j$. In other words, $\left{R_0, R_1, \ldots, R_d\right}$ gives a partition of $X \times X$.
(3) Define ${ }^t R_i=\left{(x, y) \mid(y, x) \in R_i\right}$ for $R_i(0 \leq i \leq d)$. Then there exists $i^{\prime} \in{0,1, \ldots, d}$ such that ${ }^t R_i=R_{i^{\prime}}$.
(4) Fix $i, j, k \in{0,1, \ldots, d}$. Then $p_{i, j}(x, y)=\left|\left{z \in X \mid(x, z) \in R_i,(z, y) \in R_j\right}\right|$ is constant for any $(x, y) \in R_k$. In other words, the number is independent of the choice of $(x, y)$ in $R_k$, and depends only on $i, j, k$. The number is denoted by $p_{i, j}^k$ and called the intersection number.

Moreover, if the following condition holds, $\mathfrak{X}=\left(X,\left{R_i\right}_{0 \leq i \leq d}\right)$ is called a commutative association scheme.
(5) (Commutativity) For any $i, j, k \in{0,1, \ldots, d}, p_{i, j}^k=p_{j, i}^k$.
Also, if the following condition holds, $\mathfrak{X}=\left(X,\left{R_i\right}_{0 \leq i \leq d}\right)$ is called a symmetric association scheme.
(6) (Symmetry) For all $i \in{0,1, \ldots, d},{ }^t R_i=R_i$, (i. e., $i^{\prime}=i$ ).

数学代写|组合数学代写Combinatorial Mathematics代考|Bose–Mesner algebras

We start with the notation. For a finite set $X$, we consider $|X| \times|X|$-matrices whose rows and columns are indexed by the elements of $X$. Let $M_X(\mathbb{C})$ be the full matrix algebra of such matrices over the complex field. For $x, y \in X, M(x, y)$ denotes the $(x, y)$-entry of a matrix $M \in M_X(\mathbb{C})$. Let $I$ be the identity matrix of $M_X(\mathbb{C})$, and let $J$ be the matrix in $M_X(\mathbb{C})$ whose entries are all 1 . For a matrix $M \in M_X(\mathbb{C}),{ }^t M$ denotes the transpose of $M$. For any matrices $M_1, M_2$ in $M_X(\mathbb{C})$, we define the Hadamard product $M_1 \circ M_2$ by
$$\left(M_1 \circ M_2\right)(x, y)=M_1(x, y) M_2(x, y), \quad(x, y) \in X \times X .$$
Namely, the Hadamard product is the entry-wise product of matrices. (In elementary linear algebra, this product is forbidden.)

Let $\mathfrak{X}=\left(X,\left{R_i\right}_{0 \leq i \leq d}\right)$ be an association scheme. For each relation $R_i(0 \leq i \leq d)$, we define the matrix $A_i \in M_X(\mathbb{C})$ as follows:
$$A_i(x, y)= \begin{cases}1, & \text { if }(x, y) \in R_i, \ 0, & \text { if }(x, y) \notin R_i ;\end{cases}$$
$A_i$ is called the adjacency matrix of the relation $R_i$. Then by conditions (1), (2), (3), and (4) in Definition 2.1, we obtain the following conditions $\left(1^{\prime}\right),\left(2^{\prime}\right),\left(3^{\prime}\right)$, and $\left(4^{\prime}\right)$ :
(1 $\left.{ }^{\prime}\right) A_0=I$;
(2 $\left.2^{\prime}\right) A_0+A_1+\cdots+A_d=J$
$\left(3^{\prime}\right)$ for each $i(0 \leq i \leq d)$, there exists $i^{\prime} \in{0,1, \ldots, d}$ such that ${ }^t A_i=A_{i^{\prime}}$;
( $\left.4^{\prime}\right)$ for each $i, j(0 \leq i, j \leq d)$, there exist non-negative integers $p_{i, j}^k(0 \leq k \leq d)$ such that
$$A_i A_j=\sum_{k=0}^d p_{i, j}^k A_k .$$

数学代写|组合数学代写组合数学代考|关联方案的定义

(1) $R_0={(x, x) \mid x \in X}$.
(2) $X \times X=R_0 \cup R_1 \cup \cdots \cup R_d$，以及 $R_i \cap R_j=\emptyset$ 如果 $i \neq j$。换句话说， $\left{R_0, R_1, \ldots, R_d\right}$ 给出了 $X \times X$.
(3)定义 ${ }^t R_i=\left{(x, y) \mid(y, x) \in R_i\right}$ 为 $R_i(0 \leq i \leq d)$。那么就存在 $i^{\prime} \in{0,1, \ldots, d}$ 如此这般 ${ }^t R_i=R_{i^{\prime}}$
(4)修复 $i, j, k \in{0,1, \ldots, d}$。然后 $p_{i, j}(x, y)=\left|\left{z \in X \mid(x, z) \in R_i,(z, y) \in R_j\right}\right|$ 对于任何变量都是常数 $(x, y) \in R_k$。换句话说，数量与选择无关 $(x, y)$ 在 $R_k$，而只依赖于 $i, j, k$。这个数字用 $p_{i, j}^k$ 并被称为路口号。

(5)(交换性)对于任何$i, j, k \in{0,1, \ldots, d}, p_{i, j}^k=p_{j, i}^k$ .

(6)(对称性)对于所有$i \in{0,1, \ldots, d},{ }^t R_i=R_i$，(即，$i^{\prime}=i$).

.
(5)(交换性)对于所有 .
(6)(对称性)对于所有，(即，). .
(5)(交换性)对于任何 .
(6)(对称性)对于所有，(即，).

数学代写|组合数学代写combinatormathematics代考| Bose-Mesner algebras

. >

$$\left(M_1 \circ M_2\right)(x, y)=M_1(x, y) M_2(x, y), \quad(x, y) \in X \times X .$$

$$A_i(x, y)= \begin{cases}1, & \text { if }(x, y) \in R_i, \ 0, & \text { if }(x, y) \notin R_i ;\end{cases}$$
$A_i$称为关系$R_i$的邻接矩阵。然后通过定义2.1中的条件(1)(2)(3)(4)，我们得到了以下条件$\left(1^{\prime}\right),\left(2^{\prime}\right),\left(3^{\prime}\right)$和$\left(4^{\prime}\right)$:
(1 $\left.{ }^{\prime}\right) A_0=I$;
(2 $\left.2^{\prime}\right) A_0+A_1+\cdots+A_d=J$
$\left(3^{\prime}\right)$对于每个$i(0 \leq i \leq d)$，存在$i^{\prime} \in{0,1, \ldots, d}$使得${ }^t A_i=A_{i^{\prime}}$;
($\left.4^{\prime}\right)$对于每个$i, j(0 \leq i, j \leq d)$，存在非负整数$p_{i, j}^k(0 \leq k \leq d)$使得
$$A_i A_j=\sum_{k=0}^d p_{i, j}^k A_k .$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。