数学代写|组合学代写Combinatorics代考|Finite State Machines

statistics-lab™ 为您的留学生涯保驾护航 在代写组合学Combinatorics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合学Combinatorics代写方面经验极为丰富，各种代写组合学Combinatorics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|组合学代写Combinatorics代考|Finite State Machines

A “finite state machine” is simply a device that can be in any one of a finite number of situations and is able to move from one situation to another. The classic example (and motivation for the subject) is the digital computer. If no peripherals are attached, then the state at any instant is what is stored in the machine. You may object that this fails to take into account what instruction the machine is executing. Not so; that information is stored temporarily in parts of the machine’s central processing unit. We can expand our view by allowing input and output to obtain a finite state machine with $\mathrm{I} / \mathrm{O}$.

By formalizing the concept of a finite state machine, computer scientists hope to capture the essential features of some aspects of computing. In this section we’ll study a very restricted formalization. These restricted devices are called “finite automata” or “finite state machines.” The input to such machines is fed in one symbol at a time and cannot be reread by the machine.
Turing Machines
A Turing machine, introduced by A.M. Turing in 1937, is a more flexible concept than a finite automaton. It is equipped with an arbitrarily long tape which it can reposition, read and write. To run the machine, we write the input on a blank tape, position the tape in the machine and turn the machine on. We can think of a Turing machine as computing a function: the input is an element of the function’s domain and the output is an element of the function’s range, namely the value of the function at that input. The input and/or the output could be nothing. In fact, the domain of the function is any finite string of symbols, where each symbol must be from some finite alphabet; eg. ${0,1}$. Of course, the input might be something the machine wasn’t designed to handle, but it will still do something.

How complicated a Turing machine might we need to build? Turing proved that there exists a “universal” Turing machine $\mathcal{U}$ by showing how to construct it. If $\mathcal{U}$ ‘s input tape contains

$\mathrm{D}(\mathcal{T})$, a description of any Turing machine $\mathcal{T}$ and

the input $I$ for the Turing machine $\mathcal{T}$,then $\mathcal{U}$ will produce the same output that would have been obtained by giving $\mathcal{T}$ the input $I$. This says that regardless of how complicated an algorithm we want to program, there is no need to build more than one Turing machine, namely the universal one $\mathcal{U}$. Of course, it might use a lot of time and a lot of tape to carry out the algorithm, so it might not be practical. Suprisingly, it can be shown that $\mathcal{U}$ will, in some sense, be almost as fast as the the Turing machine that it is mimicking. This makes it possible to introduce a machine independent measure of the complexity of a function.

数学代写|组合学代写Combinatorics代考|Finite State Machines and Digraphs

Consider a finite state machine that receives input one symbol at a time and enters a new state based on that symbol. We can represent the states of the machine by vertices in a digraph and the effect of the input $i$ in state $s$ by a directed edge that connects $s$ to the new state and contains $i$ and the associated output in its name. The following example should clarify this.

Example 6.17 Binary addition We would like to add together two nonnegative binary numbers and output the sum. The input is given as pairs of digits, one from each number, starting at the right ends (units digits) of the input. The pair 22 marks the end of the input. Thus to add 010 and 110 you would input the four pairs $00,11,01$ and 22 in that order. In other words,
\begin{aligned} & A_n A_{n-1} \cdots A_1 \ & \text { the sum problem } \frac{+B_n B_{n-1} \cdots B_1}{C_{n+1} C_n C_{n-1} \cdots C_1} \ & \text { becomes } A_1 B_1, \ldots, A_{n-1} B_{n-1}, A_n B_n, 22 \text {. } \ & \end{aligned}
The output is given as single digits with 2 marking the end of the output, so the output for our example would be 00012. (The sum is backwards, $C_1 \ldots C_{n-1}, C_n, C_{n+1}, 2$, because the first output is the units digit.) We have two internal states: carry (C) and no carry (N) You should verify that the adder can be described by the table in Figure 6.11. The entry $\left(o, s_2\right)$ in position $\left(s_1, i\right)$ says that if the machine is in state $s_1$ and receives input $i$, then it will produce output $o$ and move to state $s_2$. It is called the state transition table for the machine. Note that being in state $\mathrm{C}$ (carry) and receiving 22 as input causes two digits to be output, the carry digit and the termination digit 2 .
We can associate a digraph $(V, E, \varphi)$ with the tabular description, where $V={\mathrm{N}, \mathrm{C}}$, each edge is a 4-tuple $e=\left(s_1, i, o, s_2\right), \varphi(e)=\left(s_1, s_2\right)$ and $i$ and $o$ are the associated input and output, respectively. In drawing the picture, a shorthand is used: the label $00,22: 1$, 12 on the edge from $\mathrm{C}$ to $\mathrm{N}$ in Figure 6.11 stands for the two edges $(\mathrm{C}, 00,1, \mathrm{~N})$ and $(\mathrm{C}, 22,12, \mathrm{~N})$.

This example is slightly deficient. We tacitly assumed that everyone (and the machine!) somehow knew that the machine should start in state $\mathrm{N}$. We should really indicate this by labeling $\mathrm{N}$ as the starting state.

组合学代考

数学代写|组合学代写Combinatorics代考|Finite State Machines

“有限状态机”只是一种设备，它可以处于有限数量的情况中的任何一种，并且能够从一种情况移动到另一种情况。典型的例子(也是这门课的动机)是数字计算机。如果没有连接外设，则任何时刻的状态都是存储在机器中的状态。你可能会反对说这没有考虑到机器正在执行什么指令。不是这样;这些信息被临时存储在机器中央处理单元的某些部分中。我们可以通过允许输入和输出来扩展我们的视图，通过$\mathrm{I} / \mathrm{O}$来获得一个有限状态机。

$\ mathm {D}(\mathcal{T})$，任意图灵机$\mathcal{T}$的描述

数学代写|组合学代写Combinatorics代考|Finite State Machines and Digraphs

\begin{aligned} & A_n A_{n-1} \cdots A_1 \ & \text { the sum problem } \frac{+B_n B_{n-1} \cdots B_1}{C_{n+1} C_n C_{n-1} \cdots C_1} \ & \text { becomes } A_1 B_1, \ldots, A_{n-1} B_{n-1}, A_n B_n, 22 \text {. } \ & \end{aligned}

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

数学代写|组合学代写Combinatorics代考|Coloring Graphs

statistics-lab™ 为您的留学生涯保驾护航 在代写组合学Combinatorics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合学Combinatorics代写方面经验极为丰富，各种代写组合学Combinatorics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|组合学代写Combinatorics代考|Coloring Graphs

Example 6.5 Register allocation Optimizing compilers use a variety of techniques to produce faster code. One obvious way to produce faster code is to keep variables in registers so that memory references are eliminated. Unfortunately, there are often not enough registers available to do this, so choices must be made. For simplicity, assume that the registers and variables are all the same size. Suppose that, by some process, we have gotten a list of variables that we would like to keep in registers.

Can we keep them in registers? If the number of variables does not exceed the number of available registers, we can obviously do it. This sufficient condition is not necessary: We may have two variables that are only used in two separate parts of the program. They could share a register.
This suggests that we can define a binary relation among variables. We could say that two variables are “compatible” if they may share a register. Alternatively, we could say that two variables “conflict” if they cannot share a register. Two variables are either compatible or in conflict, but not both. Thus we can derive one relation from the other and it is rather arbitrary which we focus on. For our purposes, the conflict relation is better.

Construct a simple graph whose vertices are the variables. Two variables are joined by an edge if and only if they conflict. A register assignment can be found if and only if we can find a function $\lambda$ from the set of vertices to the set of registers such that whenever ${v, w}$ is an edge $\lambda(v) \neq \lambda(w)$. (This just says that if $v$ and $w$ conflict they must have different registers assigned to them.) This section studies such ““vertex labelings” $\lambda$.

数学代写|组合学代写Combinatorics代考|Planar Graphs

Recall that, drawing a graph in the plane without edges crossing is called embedding the graph in the plane. Any graph that can be embedded in the plane can be embedded in the sphere (i.e., the surface of a ball) and vice versa. The idea is simple: Cut a little hole out of the sphere in such a way that you don’t remove any of the graph, then, pretending the sphere is a rubber sheet, stretch it flat to form a disc. Conversely, any map on the plane is bounded, so we can cut a disc containing a map out of the plane and curve it around to fit on a sphere. Thus, studying maps on the plane is equivalent to studying maps on the sphere.

Sometimes fairly simple concepts in mathematics lead to a considerable body of research. The research related to planar graphs is among the most accessible such bodies for someone without extensive mathematical training. Here are some of the research highlights and what we’ll be doing about them.

1. The earliest is probably Euler’s relation, which we’ll discuss soon. If the sphere is cut along the edges of an embedded connected graph, we obtain pieces called faces. Euler discovered that the number of vertices and faces together differed from the number of edges by a constant. This has been extended to graphs embedded in other surfaces and to generalizations of graphs in higher dimensions. The result is an important number associated with a generalized surface called its Euler characteristic.
2. The four color problem has already been mentioned in the section on chromatic polynomials. As noted there, it has been generalized to other surfaces. We’ll use Euler’s relation to prove that five colors suffice on the plane.
3. A description of those graphs which can be drawn in the plane was obtained some time ago by Kuratowski: A graph is planar if and only if it does not “contain” either
• $K_5$, the five vertex complete graph, or
• $K_{3,3}$, the graph with $V=\left{a_1, a_2, a_3, b_1, b_2, b_3\right}$ and all nine edges of the form $\left{a_i, b_j\right}$.
We say that $G$ contains $H$ if, except for labels we can obtain $H$ from $G$ by repeated use of the three operations:
(a) delete an edge,
(b) delete a vertex that lies on no edges and
(c) if $v$ lies only on the edges $e_1=\left{v, a_1\right}$ and $e_2=\left{v, a_2\right}$, delete $v, e_1$ and $e_2$ and add the edge $\left{a_1, a_2\right}$.

组合学代考

数学代写|组合学代写Combinatorics代考|Planar Graphs

$K_5$，五顶点完全图，或

$K_{3,3}$，包含$V=\left{a_1, a_2, a_3, b_1, b_2, b_3\right}$和形式$\left{a_i, b_j\right}$的所有九条边的图形。

(a)删除一条边;
(b)删除不位于任何边上的顶点
(c)如果$v$仅位于$e_1=\left{v, a_1\right}$和$e_2=\left{v, a_2\right}$边，则删除$v, e_1$和$e_2$边，并添加$\left{a_1, a_2\right}$边。

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

数学代写|组合学代写Combinatorics代考|Paths and Subgraphs

statistics-lab™ 为您的留学生涯保驾护航 在代写组合学Combinatorics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合学Combinatorics代写方面经验极为丰富，各种代写组合学Combinatorics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|组合学代写Combinatorics代考|Paths and Subgraphs

An important concept for describing the structure of a graph is the concept of a path.
Definition 5.5 Path, trail, walk and vertex sequence Let $G=(V, E, \varphi)$ be a graph. Let $e_1, e_2, \ldots, e_{n-1}$ be a sequence of elements of $E$ (edges of $G$ ) for which there is a sequence $a_1, a_2, \ldots, a_n$ of distinct elements of $V$ (vertices of $G$ ) such that $\varphi\left(e_i\right)=\left{a_i, a_{i+1}\right}$ for $i=$ $1,2, \ldots, n-1$. The sequence of edges $e_1, e_2, \ldots, e_{n-1}$ is called a path in $G$. The sequence of vertices $a_1, a_2, \ldots, a_n$ is called the vertex sequence of the path. (Note that since the vertices are distinct, so are the edges.) If we require that $e_1, \ldots, e_{n-1}$ be distinct, but not that $a_1, \ldots, a_n$ be distinct, the sequence of edges is called a trail. If we do not even require that the edges be distinct, it is called a walk.

Note that the definition of a path requires that it not intersect itself (i.e., have repeated vertices), while a trail may intersect itself. Although a trail may intersect itself, it may not have repeated edges, but a walk may. If $P=\left(e_1, \ldots, e_{n-1}\right)$ is a path in $G=(V, E, \varphi)$ with vertex sequence $a_1, \ldots, a_n$ then we say that $P$ is a path from $a_1$ to $a_n$. Similarly for a trail or a walk.

In the graph of Figure 5.2 (p. 123), the sequence $c, d, g$ is a path with vertex sequence $A, C, B, D$. If the graph is of the form $G=(V, E)$ with $E \subseteq \mathcal{P}_2(V)$, then the vertex sequence alone specifies the sequence of edges and hence the path. Thus, in Figure 5.1 (p. 122), the vertex sequence MN, SM, SE, TM specifies the path ${\mathrm{MN}, \mathrm{SM}},{\mathrm{SM}, \mathrm{SE}},{\mathrm{SE}, \mathrm{TM}}$.

Note that every path is a trail and every trail is a walk, but not conversely. However, we can show that, if there is a walk between two vertices, then there is a path. This rather obvious result can be useful in proving theorems, so we state it as a theorem.
Theorem 5.2 Suppose $u \neq v$ are vertices in $G=(V, E, \varphi)$. The following are equivalent:
(a) There is a walk from $u$ to $v$.
(b) There is a trail from $u$ to $v$.
(c) There is a path from $u$ to $v$.
Furthermore, given a walk from $u$ to $v$, there is a path from $u$ to $v$ all of whose edges are in the walk.

数学代写|组合学代写Combinatorics代考|Trees

Trees play an important role in a variety of algorithms. We have already met decision trees in Chapter 3 . In this section, we define trees precisely and look at some of their properties. We study trees further in Section 6.1 and Chapter 9.

Definition 5.9 (Free) Tree If $G$ is a connected graph without any cycles then $G$ is called a tree. (If $|V|=1$, then $G$ is connected and hence is a tree.) A tree is also called a free tree.

The graph of Figure $5.2($ p. 123) is connected but is not a tree. The subgraph of this graph induced by the edges ${a, e, g}$ is a tree. If $G$ is a tree, then $\varphi$ is an injection since if $e_1 \neq e_2$ and $\varphi\left(e_1\right)=\varphi\left(e_2\right)$, then $\left{e_1, e_2\right}$ induces a cycle. Because of this, we can think of a tree as a simple graph when we are not interested in names of the edges.

It’s natural to ask how many trees can be formed using an $n$-set $V$ for the vertices. In Example 5.10 (p. 143), we’ll prove that the answer is $n^{n-2}$. Another proof is given in Exercise 5.4.12.
Since the notion of a tree is so important, it will be useful to have some equivalent definitions of a tree. We state them as a theorem

Theorem 5.4 Definitions of tree If $G$ is a connected graph, the following are equivalent.
(a) $G$ is a tree.
(b) $G$ has no cycles.
(c) For every pair of vertices $u \neq v$ in $G$, there is exactly one path from $u$ to $v$.
(d) Removing any edge from $G$ gives a graph which is not connected.
(e) The number of vertices of $G$ is one more than the number of edges of $G$.

组合学代考

数学代写|组合学代写Combinatorics代考|Paths and Subgraphs

(a)从$u$步行到$v$。
(b)从$u$到$v$有一条线索。
(c)有一条从$u$到$v$的路径。

数学代写|组合学代写Combinatorics代考|Trees

(a) $G$是一棵树。
(b) $G$没有周期。
(c)对于$G$中的每一对顶点$u \neq v$，从$u$到$v$都有一条路径。
(d)去掉$G$上的任何一条边，得到一个不连通的图。
(e) $G$的顶点数比$G$的边数多一个。

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

数学代写|组合学代写Combinatorics代考|Backtracking

statistics-lab™ 为您的留学生涯保驾护航 在代写组合学Combinatorics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合学Combinatorics代写方面经验极为丰富，各种代写组合学Combinatorics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|组合学代写Combinatorics代考|Backtracking

In many computer algorithms it is necessary to systematically inspect all the vertices of a decision tree. A procedure that systematically inspects all the vertices is called a traversal of the tree. How can we create such a procedure? One way to imagine doing this is to walk around the tree. An example is shown in Figure 9.2 (p. 249), where we study the subject in more depth. “Walking around the tree” is not a very good program description. We can describe our traversal more precisely by giving an algorithm. Here is one which traverses a tree whose leaves are associated with functions and lists the functions in the order of their rank.

Theorem 3.5 Systematic traversal algorithm The following procedure systematically visits the leaves in a tree from left to right by “walking” around the tree.

1. Start: Mark all edges as unused and position yourself at the root.
2. Leaf: If you are at a leaf, list the function.
3. Decide case: If there are no unused edges leading out from the vertex, go to Step 4; otherwise, go to Step 5.
4. Backtrack: If you are at the root, STOP; otherwise, return to the vertex just above this one and go to Step 3.
5. Decision: Select the leftmost unused edge out of this vertex, mark it used, follow it to a new vertex and go to Step 2 .

数学代写|组合学代写Combinatorics代考|Listing Gray coded subsets

Example 3.13 Listing Gray coded subsets In Example 3.12 we looked at a Gray code for listing all elements of an $n$ element set. Since there are only two decisions at each vertex, the entries in the decision sequence will be 0 or 1 (but they usually do not equal the entries in the Gray code). Here is the code with $s_i$ being the decision sequence and $g_i$ the Gray code.
Procedure GraySubsets $(n)$
\text { For } \begin{aligned} i & =1 \text { to } n: \quad / * \text { Set up first leaf } * / \ s_i & =0 \ g_i & =0 \end{aligned}
End for
End for
Goto ENDCASE

End if
empty = TRUE
Label ENDCASE
End for
End while
End
The statement $g_i=1-g_i$ changes 0 to 1 and vice versa. Since the Gray code changes only one entry, we are able to move down without changing any of the $g_j$ values.

组合学代考

学代写|组合学代写Combinatorics代考|Backtracking

Backtrack:如果你在根，停止;否则，返回到这个顶点的上方，然后转到步骤3。

数学代写|组合学代写Combinatorics代考|Listing Gray coded subsets

GraySubsets $(n)$
\text { For } \begin{aligned} i & =1 \text { to } n: \quad / * \text { Set up first leaf } * / \ s_i & =0 \ g_i & =0 \end{aligned}
End for
End for

empty = TRUE

End for
while结束

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

数学代写|组合学代写Combinatorics代考|Basic Concepts of Decision Trees

statistics-lab™ 为您的留学生涯保驾护航 在代写组合学Combinatorics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合学Combinatorics代写方面经验极为丰富，各种代写组合学Combinatorics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|组合学代写Combinatorics代考|Basic Concepts of Decision Trees

Decision trees provide a method for systematically listing a variety of functions. We’ll begin by looking at a couple of these. The simplest general class of functions to list is the entire set $\underline{k}$. We can create a typical element in the list by choosing an element of $n$ and writing it down, choosing another element (possibly the same as before) of $n$ and writing it down next, and so on until we have made $k$ decisions. This generates a function in one line form sequentially: First $f(1)$ is chosen, then $f(2)$ is chosen and so on. We can represent all possible decisions pictorially by writing down the decisions made so far and then some downward edges indicating the possible choices for the next decision.

The lefthand part of Figure 3.1 illustrates this way of generating a function in $2^3$ sequentially. It’s called the decision tree for generating the functions in $2^3$. Each line in the left hand figure is labeled with the choice of function value to which it corresponds. Note that the labeling does not completely describe the corresponding decision – we should have used something like “Choose 1 for the value of $f(2)$ ” instead of simply “1” on the line from 1 to 11 . At the end of each line is the function that has been built up so far. We’ve omitted commas, so 211 means $2,1,1$.

To the right of the figure for generating the functions is the same structure without any labels. The dots $(\bullet)$ are called nodes or vertices and the lines connecting them are called edges. Sometimes it is more convenient to specify an edge by giving its two end points; for example, the edge from 1 to 12 in the figure can be described uniquely as $(1,12)$. The nodes with no edges leading down are called the leaves. The entire branching structure is called a tree or, more properly, an ordered rooted tree. The topmost node is called the root.

数学代写|组合学代写Combinatorics代考|Two hands of cards

Two hands of cards In Chapter 1 we studied various problems involving a hand of cards. Now we complicate matters by forming more than one hand on the same deal from the deck. How many ways can two 5 card hands be formed so that each hand contains 2 pairs (and a fifth card that has a different value)?

The problem can be solved by forming the first hand and then forming the second, since the number of choices for the second hand does not depend on what the first hand is as long as we know it is a hand with 2 pairs. We solved the two pair problem in Example 1.16 (p. 22), so we know that the first hand can be formed in 123,552 ways.

Forming the second hand is complicated by the fact that the first hand has used up some of the cards in the deck. As a result, we must consider different cases according to the amount of overlap between the first and second hands. We’ll organize the possibilities by using a decision tree. Let $P_i$ be the set of values of the pairs in the $i$ thhand; e.g., we might have $P_1={3, \mathrm{Q}}$ and $P_2={2,3}$, in which case the hands have one pair value in common. Our first decision will be the value of $\left|P_1 \cap P_2\right|$, which must be 0,1 or 2 . Our next decision will be based on whether or not the value of the unpaired card in the first hand is in $P_2$; i.e., whether or not a pair in the second hand has the same value as the nonpair card in the first hand. We’ll label the edges $\mathrm{Y}$ and $\mathrm{N}$ according as this is true or not. The decision tree is shown in Figure 3.4, where we’ve labeled the leaves $A-E$ for convenience.

We’ll prove that the number of hands that correspond to each of the leaves is
\begin{aligned} A & :\left(\begin{array}{l} 10 \ 2 \end{array}\right)\left(\begin{array}{l} 4 \ 2 \end{array}\right)^2(52-8-5)=63,180, \ B: & \left(\begin{array}{l} 3 \ 2 \end{array}\right) \times\left(\begin{array}{c} 10 \ 1 \end{array}\right)\left(\begin{array}{l} 4 \ 2 \end{array}\right)(52-8-4)=7,200, \ C: & 2 \times\left(\begin{array}{c} 10 \ 1 \end{array}\right)\left(\begin{array}{l} 4 \ 2 \end{array}\right)(52-8-3)=4,920, \ D: & 2\left(\begin{array}{l} 3 \ 2 \end{array}\right)(52-8-2)=252, \ E: & 52-8-1=43, \end{aligned}
giving a total of 75,595 choices for the second hand. Multiplying this by 123,552 (the number of ways of forming the first hand) we find that there are somewhat more than $9 \times 10^9$ possibilities for two hands. Of course, if the order of the hands is irrelevant, this must be divided by 2 .

组合学代考

数学代写|组合学代写Combinatorics代考|Two hands of cards

\begin{aligned} A & :\left(\begin{array}{l} 10 \ 2 \end{array}\right)\left(\begin{array}{l} 4 \ 2 \end{array}\right)^2(52-8-5)=63,180, \ B: & \left(\begin{array}{l} 3 \ 2 \end{array}\right) \times\left(\begin{array}{c} 10 \ 1 \end{array}\right)\left(\begin{array}{l} 4 \ 2 \end{array}\right)(52-8-4)=7,200, \ C: & 2 \times\left(\begin{array}{c} 10 \ 1 \end{array}\right)\left(\begin{array}{l} 4 \ 2 \end{array}\right)(52-8-3)=4,920, \ D: & 2\left(\begin{array}{l} 3 \ 2 \end{array}\right)(52-8-2)=252, \ E: & 52-8-1=43, \end{aligned}

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

数学代写|组合学代写Combinatorics代考|Some Basic Terminology

statistics-lab™ 为您的留学生涯保驾护航 在代写组合学Combinatorics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合学Combinatorics代写方面经验极为丰富，各种代写组合学Combinatorics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|组合学代写Combinatorics代考|Terminology for Sets

Except for the real numbers $(\mathbb{R})$, rational numbers $(\mathbb{Q})$ and integers $(\mathbb{Z})$, our sets are normally finite. The set of the first $n$ positive integers, ${1,2, \ldots, n}$ will be denoted by $n$.

Recall that $|A|$ is the number of elements in the set $A$. When it is convenient to do so, we’ll assume that the elements of a set $A$ have been linearly ordered and denote the ordering by $a_1, a_2, \ldots, a_{|A|}$. Unless clearly stated otherwise, the ordering on a set of numbers is the numerical ordering. For example, the ordering on $n$ is $1,2,3, \ldots, n$.
If $A$ and $B$ are sets, we write $A-B$ for the set of elements in $A$ that are not in $B$ :
$$A-B={x \mid x \in A \text { and } x \notin B} .$$
(This is also written $A \backslash B$.)
If $A$ and $B$ are sets, recall from the previous chapter that the Cartesian product $A \times B$ is the set of all ordered pairs built from $A$ and $B$ :
$$A \times B={(a, b) \mid a \in A \text { and } b \in B}$$
We also call $A \times B$ the direct product of $A$ and $B$.
If $A=B=\mathbb{R}$, the real numbers, then $\mathbb{R} \times \mathbb{R}$, written $\mathbb{R}^2$ is frequently interpreted as coordinates of points in the plane. Two points are the same if and only if they have the same coordinates, which says the same thing as our definition of $(a, b)=\left(a^{\prime}, b^{\prime}\right)$. Recall that the direct product can be extended to any number of sets. How can $\mathbb{R} \times \mathbb{R} \times \mathbb{R}=\mathbb{R}^3$ be interpreted?

数学代写|组合学代写Combinatorics代考|What are Functions?

Definition 2.1 Function If $A$ and $B$ are sets, a function from $A$ to $B$ is a rule that tells us how to find a unique $b \in B$ for each $a \in A$. We write $f: A \rightarrow B$ to indicate that $f$ is a function from $A$ to $B$. We call the set $A$ the domain of $f$ and the set $B$ the codomain of $f$. To specify a function completely you must give its domain, codomain and rule. The set of all functions from $A$ to $B$ is written $B^A$, for a reason we will soon explain. Thus $f: A \rightarrow B$ and $f \in B^A$ say the same thing.

In calculus you dealt with functions whose codomains were $\mathbb{R}$ and whose domains were contained in $\mathbb{R}$; for example, $f(x)=1 /\left(x^2-1\right)$ is a function from $\mathbb{R}-{-1,1}$ to $\mathbb{R}$. You also studied functions of functions! The derivative is a function whose domain is all differentiable functions and whose codomain is all functions. If we wanted to use functional notation we could write $D(f)$ to indicate the function that the derivative associates with $f$. Can you see how to think of the integral as a function? This is a bit tricky because of the constant of integration. We won’t pursue it.

Definition 2.2 One-line notation When $A$ is ordered, a function can be written in oneline notation as $\left(f\left(a_1\right), f\left(a_2\right), \ldots, f\left(a_{|A|}\right)\right)$. Thus we can think of function as an element of $B \times B \times \ldots \times B$, where there are $|A|$ copies of $B$. Instead of writing $B^{|A|}$ to indicate the set of all functions, we write $B^A$. Writing $B^{|A|}$ is incomplete because the domain is not specified. Instead, only its size is given.

Example 2.1 Using the notation To get a feeling for the notation used to specify a function, it may be helpful to imagine that you have an envelope or box that contains a function. In other words, this envelope contains all the information needed to completely describe the function. Think about what you’re going to see when you open the envelope.
You might see
$$P={a, b, c}, \quad g: P \rightarrow 4, \quad g(a)=3, \quad g(b)=1 \quad \text { and } \quad g(c)=4 .$$
This tells you that the name of the function is $g$, the domain of $g$ is $P$, which is ${a, b, c}$, and the codomain of $g$ is $4={1,2,3,4}$. It also tells you the values in $4$ that $g$ assigns to each of the values in its domain. Someone else may have put
$$g:{a, b, c} \rightarrow 4, \quad \text { ordering: } a, b, c, \quad g=(3,1,4) .$$

组合学代考

数学代写|组合学代写Combinatorics代考|Terminology for Sets

$$A-B={x \mid x \in A \text { and } x \notin B} .$$
(这也写在$A \backslash B$。)

$$A \times B={(a, b) \mid a \in A \text { and } b \in B}$$

数学代写|组合学代写Combinatorics代考|What are Functions?

$$P={a, b, c}, \quad g: P \rightarrow 4, \quad g(a)=3, \quad g(b)=1 \quad \text { and } \quad g(c)=4 .$$

$$g:{a, b, c} \rightarrow 4, \quad \text { ordering: } a, b, c, \quad g=(3,1,4) .$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

数学代写|组合学代写Combinatorics代考|Permutations, ascents, and the Eulerian numbers

statistics-lab™ 为您的留学生涯保驾护航 在代写组合学Combinatorics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合学Combinatorics代写方面经验极为丰富，各种代写组合学Combinatorics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|组合学代写Combinatorics代考|Ascents, descents, and runs

Definition 6.3.1. Let a permutation
$$\pi=\left(\begin{array}{ccccc} 1 & 2 & 3 & \cdots & n \ i_1 & i_2 & i_3 & \cdots & i_n \end{array}\right)$$
be given. The position $j$ in (the bottom line of) $\pi$ is called ascent if $i_ji_{j+1}$ we say that the position $j$ is a descent.
For example, in
$$\left(\begin{array}{llllll} 1 & 2 & 3 & 4 & 5 & 6 \ 3 & 1 & 4 & 6 & 2 & 5 \end{array}\right)$$
the positions $2,3,5$ are ascents, while 1,4 are descents.
The following definition is strongly related.
Definition 6.3.2. The consecutive maximal increasing subsequences of permutations are called runs.
In the above example, there are three runs: $3 ; 1,4,6 ;$ and 2,5 .
Next we study how the ascents, descents and runs are related. If we want to count the ascents and descents in a permutation, we go from left to the right and compare two consecutive elements: the first with the second, then the second with the third, and so on, and finally we compare the penultimate with the last element. Altogether, we have $n-1$ comparisons and in each position we have two alternatives: we can have an ascent or a descent. Hence,
$$\text { ascents }+ \text { descents }=n-1$$
On the other hand, a permutation always begins with a run and this run ends at the first descent. Then a new run follows until the next descent, and so on. This means that two neighboring runs are separated by a descent, so
$$\text { runs }=\text { descents }+1 .$$

数学代写|组合学代写Combinatorics代考|The definition of the Eulerian numbers

Definition 6.3.3. The number of $n$-permutations which contain exactly $k$ ascents is given by the Eulerian number with parameters $n$ and $k$, and it is denoted by $\left\langle\begin{array}{l}n \ k\end{array}\right\rangle^5$.

There is a table for the first Eulerian numbers at the end of the book. For now, let us see a simple example. All the permutations on the 3-set ${1,2,3}$ are listed here:
\begin{aligned} & \left(\begin{array}{lll} 1 & 2 & 3 \ 1 & 2 & 3 \end{array}\right),\left(\begin{array}{lll} 1 & 2 & 3 \ 2 & 1 & 3 \end{array}\right),\left(\begin{array}{lll} 1 & 2 & 3 \ 3 & 1 & 2 \end{array}\right), \ & \left(\begin{array}{lll} 1 & 2 & 3 \ 1 & 3 & 2 \end{array}\right),\left(\begin{array}{lll} 1 & 2 & 3 \ 2 & 3 & 1 \end{array}\right),\left(\begin{array}{lll} 1 & 2 & 3 \ 3 & 2 & 1 \end{array}\right) \text {. } \ & \end{aligned}
As it is easy to see, the numbers of ascents in these permutations are $2,1,1,1,1,0$, respectively. Therefore
$$\left\langle\begin{array}{l} 3 \ 0 \end{array}\right\rangle=1, \quad\left\langle\begin{array}{l} 3 \ 1 \end{array}\right\rangle=4, \quad\left\langle\begin{array}{l} 3 \ 2 \end{array}\right\rangle=1 .$$
(The numbers of runs are $1,2,2,2,2,3$, respectively, while the numbers of descents are $0,1,1,1,1,2$.)
If a permutation
$$\left(\begin{array}{ccccc} 1 & 2 & 3 & \cdots & n \ i_1 & i_2 & i_3 & \cdots & i_n \end{array}\right)$$
contains $k$ ascents, then its reverse
$$\left(\begin{array}{ccccc} 1 & 2 & 3 & \cdots & n \ i_n & i_{n-1} & i_{n-2} & \cdots & i_1 \end{array}\right)$$
contains $n-k-1$ ascents, so
$$\left\langle\begin{array}{l} n \ k \end{array}\right\rangle=\left\langle\begin{array}{c} n \ n-k-1 \end{array}\right\rangle$$
holds, which is similar to the symmetry of the binomial coefficients.

组合学代考

数学代写|组合学代写Combinatorics代考|Ascents, descents, and runs

$$\pi=\left(\begin{array}{lllllllll} 1 & 2 & 3 & \cdots & n i_1 & i_2 & i_3 & \cdots & i_n \end{array}\right)$$

$$\left(\begin{array}{lllllllllll} 1 & 2 & 3 & 4 & 5 & 63 & 1 & 4 & 6 & 2 & 5 \end{array}\right)$$

$$\text { ascents }+ \text { descents }=n-1$$

$$\text { runs }=\text { descents }+1 .$$

数学代写|组合学代写Combinatorics代考|The definition of the Eulerian numbers

$$\langle 30\rangle=1, \quad\langle 31\rangle=4, \quad\langle 32\rangle=1$$
(运行次数为 $1,2,2,2,2,3$ ，分别是，而下降的数量是 $0,1,1,1,1,2$.) 如果一个排列
$$\left(\begin{array}{lllllllll} 1 & 2 & 3 & \cdots & n i_1 & i_2 & i_3 & \cdots & i_n \end{array}\right)$$

$$\left(\begin{array}{lllllllll} 1 & 2 & 3 & \cdots & n i_n & i_{n-1} & i_{n-2} & \cdots & i_1 \end{array}\right)$$

$$\langle n k\rangle=\langle n n-k-1\rangle$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

数学代写|组合学代写Combinatorics代考|The Hankel determinants of the Fubini numbers

statistics-lab™ 为您的留学生涯保驾护航 在代写组合学Combinatorics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合学Combinatorics代写方面经验极为丰富，各种代写组合学Combinatorics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|组合学代写Combinatorics代考|The Hankel determinants of the Fubini numbers

To determine the Hankel determinants of the Fubini numbers, we use the just proven ordered Dobiński formula and the orthogonal polynomial theory tool we got to know in Subsection 2.10.3.

We are looking for an orthogonal polynomial sequence such that the attached functional $L$ is such that
$$L\left(x^n\right)=F_n$$
The Meixner polynomials are the corresponding polynomials we are looking for. In Section 1.9 of [338], we see that for the $M_n(x ; \beta, c)$ polynomials
$$\sum_{x=0}^{\infty} \frac{(\beta)x}{x !} c^x M_n(x ; \beta, c) M_m(x ; \beta, c)=\frac{c^{-n} n !}{(\beta)_n(1-c)^\beta} \delta{n m}$$ for $\beta>0$ and $0<c<1$. Here
$$\delta_{n m}= \begin{cases}1, & \text { if } n=m \ 0, & \text { otherwise }\end{cases}$$
is the Kronecker delta symbol.
We read out the functional $L$ which is ${ }^3$
$$L(f(x))=\sum_{t=0}^{\infty} \frac{(\beta)t}{t !} c^t f(t) .$$ Setting $\beta=1$ and $c=\frac{1}{2}$ we have that $$L\left(x^n\right)=\sum{t=0}^{\infty} \frac{1}{2^t} t^n=2 F_n,$$
by the ordered Dobiński formula. The factor two is irrelevant here because it can be incorporated into $L$ and the orthogonality relation. The (1.9.4) recurrence in [338] for the normalized Meixner polynomials
$$p_n(x)=\left(\frac{c}{c-1}\right)^n(\beta)n M_n(x ; \beta, c)$$ is, in our particular choice of parameters, $$p{n+1}(x)=(x-(3 n+1)) p_n(x)-2 n^2 p_{n-1}(x),$$
meaning that
$$a_n=-(3 n+1), \quad \text { and } \quad b_n=2 n^2 \text {. }$$

数学代写|组合学代写Combinatorics代考|Fubini polynomials

It is worth it to introduce the Fubini polynomials because they will have interesting connections to permutations. The Fubini polynomials are defined as
$$F_n(x)=\sum_{k=0}^n k !\left{\begin{array}{l} n \ k \end{array}\right} x^k$$
for all $n \geq 0$.
The fourth Fubini polynomial, for instance, is
$$F_4(x)=24 x^4+36 x^3+14 x^2+x$$
In this section we study the main properties of these polynomials. First, we note that the exponential generating function of $F_n(x)$ can be deduced similarly as we did for $(6.5)$ :
$$\sum_{n=0}^{\infty} F_n(y) \frac{x^n}{n !}=\frac{1}{1-y\left(e^x-1\right)}$$

The $F_n(x)$ polynomials satisfy a recurrence that can be proven by the recursion of the Stirling numbers (similarly as we did with the Bell polynomials). Since
$$\left{\begin{array}{l} n \ k \end{array}\right}=k\left{\begin{array}{c} n-1 \ k \end{array}\right}+\left{\begin{array}{l} n-1 \ k-1 \end{array}\right}$$
we have that
$$F_n(x)=\sum_{k=1}^n k ! k\left{\begin{array}{c} n-1 \ k \end{array}\right} x^k+\sum_{k=1}^n k !\left{\begin{array}{c} n-1 \ k-1 \end{array}\right} x^k$$
For the first sum
$$\sum_{k=1}^n k ! k\left{\begin{array}{c} n-1 \ k \end{array}\right} x^k=x\left(\sum_{k=1}^n k !\left{\begin{array}{c} n-1 \ k \end{array}\right} x^k\right)^{\prime}=x F_{n-1}^{\prime}(x),$$
while for the second
$$\begin{gathered} \sum_{k=1}^n k !\left{\begin{array}{l} n-1 \ k-1 \end{array}\right} x^k=x\left(\sum_{k=1}^n(k-1) !\left{\begin{array}{l} n-1 \ k-1 \end{array}\right} x^k\right)^{\prime}=x\left(x F_{n-1}(x)\right)^{\prime}= \ x F_{n-1}(x)+x^2 F_{n-1}^{\prime}(x) . \end{gathered}$$
Finally,
$$F_n(x)=x\left[F_{n-1}(x)+(1+x) F_{n-1}^{\prime}(x)\right] .$$
Note that this yields the relation
$$F_n(x)=x\left((1+x) F_{n-1}(x)\right)^{\prime} .$$

组合学代考

数学代写|组合学代写Combinatorics代考|The Hankel determinants of the Fubini numbers

$$L\left(x^n\right)=F_n$$
Meixner 多项式就是我们正在寻找的相应多项式。在 [338] 的第 1.9 节中，我们看到对于 $M_n(x ; \beta, c)$ 多 项式

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。