数学代写|matlab代写|Difference Sets

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• (Generalized) Linear Models 广义线性模型
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|matlab代写|Difference Sets

In this section, we will show how to construct difference sets, and how difference sets can be used to construct block designs. As we will show, difference sets yield block designs with more of a variety of parameters than designs that result from Hadamard matrices. Specifically, block designs that result from difference sets do not need to be symmetric.

Suppose $G$ is an abelian group of order $v$ with identity 0 , and let $D$ be a subset of order $k$ in $G$. If every nonzero element in $G$ can be expressed as the difference of two elements in $D$ in exactly $\lambda$ ways with $\lambda<k$, then $D$ is called a difference set in $G$, described by the parameters $(v, k, \lambda)$.
Example 2.4 The set $D={0,1,2,4,5,8,10}$ is a $(15,7,3)$ difference set in $Z_{15}$.

Example 2.5 The set $D={1,2,4}$ is a $(7,3,1)$ difference set in $\mathbb{Z}{7}$. Also, note that if we take the elements in $\mathbb{Z}{7}$ one at a time, and add these elements to each of the elements in $D$ (i.e., if we form the sets $i+D$ for $i=0,1, \ldots, 6$ ), the seven resulting sets are the blocks in the block design in Example $2.1$ (with 0 represented by 7 in the blocks in Example 2.1). Thus, the $(7,3,1)$ difference set $D={1,2,4}$ in $\mathbb{Z}_{7}$ can be used to construct the $(7,7,3,3,1)$ block design in Example 2.1.

The fact that a block design results from adding each of the elements in $\mathbb{Z}_{7}$ to each of the elements in the difference set $D$ in Example $2.5$ is guaranteed by the following theorem.

Theorem $2.6$ Let $D=\left{d_{1}, d_{2}, \ldots, d_{k}\right}$ be $a(v, k, \lambda)$ difference set in $a$ group $G=\left{g_{1}, g_{2}, \ldots, g_{v}\right}$. Consider $g_{i}+D$ for $i=1,2, \ldots, v$ defined as follows.
$$g_{i}+D=\left{g_{i}+d_{1}, g_{i}+d_{2}, \ldots, g_{i}+d_{k}\right}$$
These sets are the blocks in $a(v, v, k, k, \lambda)$ block design.
Proof. Clearly there are $v$ objects in the design. Also, the $v$ blocks $g_{i}+D$ for $i=1,2, \ldots, v$ are distinct, for if $g_{i}+D=g_{j}+D$ for some $i \neq j$, then $\left(g_{i}-g_{j}\right)+D=D$. We can then find $k$ differences of elements in $D$ that are equal to $g_{i}-g_{j}$, contradicting the assumption that $\lambda<k$. Now, if we add an element in $D$ to each of the elements in $G$, the result will be $G$. Thus, each element in $G$ will appear exactly $k$ times among the elements $g_{i}+d_{j}$ for $i=1,2, \ldots, v$ and $j=1,2, \ldots, k$. Therefore, each element in $G$ will appear in exactly $k$ blocks. Also, by construction, each block will contain exactly $k$ objects. It remains to be shown only that each pair of elements in $G$ appears together in exactly $\lambda$ blocks. Let $x, y \in G$ be distinct. If $x$ and $y$ appear together in some block $g+D$, then $x=g+d_{i}$ and $y=g+d_{j}$ for some $i, j$. Then $x-y=d_{i}-d_{j}$, and so $x-y$ is the difference of two elements in $D$. Since $D$ is a $(v, k, \lambda)$ difference set in $G$, then $x-y$ can be written as the difference of two elements in $D$ in exactly $\lambda$ ways. Since $x=g+d_{i}=h+d_{i}$ implies $g=h$, the difference $d_{i}-d_{j}$ cannot come from more than one block. Thus, $x$ and $y$ cannot appear together in more than $\lambda$ blocks. On the other hand, suppose $x-y=d_{i}-d_{j}$ for some $i, j$. Then $x=g+d_{i}$ for some $g \in G$, and $y=x-\left(d_{i}-d_{j}\right)=\left(x-d_{i}\right)+d_{j}=g+d_{j}$. So $x$ and $y$ appear together in the block $g+D$. Therefore, $x$ and $y$ must appear together in at least $\lambda$ blocks. With our previous result, this implies that $x$ and $y$ must appear together in exactly $\lambda$ blocks.

数学代写|matlab代写|Difference Sets with Maple

In this section, we will show how Maple can be used to construct initial blocks in generalized difference sets and corresponding block designs. We will consider the design resulting from the initial blocks in Example 2.8.
We will begin by defining the primitive polynomial $f(x)=x^{2}+x+2$ in $\mathbb{Z}_{3}[x]$ used to construct the elements in the finite field $F$.
$>f:=x \rightarrow x^{n} 2+x+2:$
$>\operatorname{Primitive}(f(x)) \bmod 3$;
true
Next, recall that since $v=4 t+1=9$ implies $t=2$, there will be 2 initial blocks. We assign the value of this parameter next.
$$\mathrm{t}:=2 \text { : }$$
Because the field elements are the objects that will fill the blocks, we need to store these elements in a way so that they can be recalled. We will do this by storing these elements in a vector. We first create the following vector with the same number of positions as the number of field elements.
$>$ field : = Vector $[$ row $](4 * t+1)$;
field $:=\left[\begin{array}{lllllllll}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]$
We can then use the following commands to generate and store the field elements in the vector field. (Note the bracket [] syntax for accessing the positions in field.)
$>$ for i from 1 to $4 * t$ do
$>$ field $[i]:=\operatorname{Powmod}(x, i, f(x), x) \bmod 3:$
$>$ od:
$>$ for i from 1 to $4 * t$ do
$>$ field $[i]:=\operatorname{Powmod}(x, i, f(x), x) \bmod 3:$
$>$ od:
$>$ field $[4 * t+1]:=0:$
$>$ field $[4 * t+1]:=0:$
We can view the entries in the vector field by entering the following command.
\begin{aligned} &>\text { field; } \ &\qquad\left[\begin{array}{lllllllll} x & 2 x+1 & 2 x+2 & 2 & 2 x & x+2 & x+1 & 1 & 0 \end{array}\right] \end{aligned}

数学代写|matlab代写|Difference Sets with MATLAB

In this section, we will show how MATLAB can be used to construct initial blocks in generalized difference sets and corresponding block designs. We will consider the design resulting from the initial blocks in Example 2.8.

We will begin by declaring the variable $x$ as symbolic, and defining the primitive polynomial $f(x)=x^{2}+x+2 \in \mathbb{Z}{3}[x]$ used to construct the elements in the finite field $F$. $\gg$ syms $x$ $>>f=@(x) x^{\sim} 2+x+2$ $f=$ Q(x) $x^{\sim} 2+x+2$ Next, as in Section $1.6$, to verify that $f(x)$ is primitive in $\mathbb{Z}{3}[x]$, we use the user-written function Primitive, which we have written separately from this MATLAB session and saved as the M-file Primitive.m.
$\Rightarrow$ Primitive $(f(x), 3)$
ans $=$
TRUE
Recall that since $v=4 t+1=9$ implies $t=2$, there will be 2 initial blocks. We assign the value of this parameter next.
$$t=2$$
Because the field elements are the objects that will fill the blocks, we need to store these elements in a way so that they can be recalled. We will do this by storing these elements in a vector. In the following commands, we generate and store the field elements in the vector field. As in Section 1.6, to construct the field elements, we use the user-written function Powmod, which we have written separately from this MATLAB session and saved as the M-file Powmod.m.
$>$ for $i=1: 4 * t$
field $(i)=\operatorname{Powmod}(x, i, f(x), x, 3)$;
end$>$ field $(4 * t+1)=0 ;$

数学代写|matlab代写|Difference Sets

g_{i}+D=\left{g_{i}+d_{1}, g_{i}+d_{2}, \ldots, g_{i}+d_{k}\right}g_{i}+D=\left{g_{i}+d_{1}, g_{i}+d_{2}, \ldots, g_{i}+d_{k}\right}

>F:=X→Xn2+X+2:
>原始⁡(F(X))反对3;
true

>为了一世=1:4∗吨

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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。