## 数学代写|组合学代写Combinatorics代考|MATH482

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|Statement A—Object Existence and Detection

The first statement describes an extremely spartan setting in which the logical relationship between object existence and sensor detection is modeled but without specifying either the object state or measurement.

Statement $\mathbf{A}$. At most one object exists and, if it does exist, the sensor may or may not generate one measurement.

Let $N$ and $M$ denote the number of objects that exist and the number of measurements, respectively. Philosophers wrestle with the nature of existence, but here existence is strictly about counting: the object is said to exist if $N=1$ and not to exist if $N=0$. The number of objects is assumed to be a random ${0,1}$ integer, an assumption that justifies calling $\chi=\operatorname{Pr}{N=1}$ the object existence probability. The statement that at most one object exists is equivalent to $\operatorname{Pr}{N=0}=1-\chi$ and $\operatorname{Pr}{N \geq 2}=0$. By definition (see Appendix A), the GF for $N$ is
$$G_{N}(z)=\sum_{n=0}^{\infty} \operatorname{Pr}{N=n} z^{n}=1-\chi+\chi z,$$
where $z$ is the indeterminate variable of the GF for $N$. The linear equation (1.1) perfectly characterizes the statement that there is at most one object, but it says nothing about the number of measurements. It is the GF of the marginal distribution of $N$ for the joint random variable $(N, M)$.

## 数学代写|组合学代写Combinatorics代考|Statement B—Gridded Measurements

Statement A is “data starved,” meaning that most problems are formulated in a larger context in which objects have states and sensors produce measurements of the state. Statement B adds a point measurement to Statement A, which is otherwise unchanged. Statement $\mathrm{C}$ of the next subsection adds both a measurement and an object state.

The added measurement information is mapped into a counting problem. The mapping is done with a gridded measurement space. The same mapping works intuitively for continuous spaces, but technical details (see Appendix B) get in the way of the flow of ideass, so only griddéd spācês aree treaatēd héré. Laterr chapteers in the book do not use gridded spaces.

Statement B. At most one object exists and, if it does exist, the sensor may or may not generate a random measurement $Y$.

The measurement space $\mathcal{Y}$ is partitioned into a finite number of non-overlapping grid cells labeled $1, \ldots, R$. The measurement $Y$ is conceptualized as the nonnegative integer-valued random vector $Y_{1: R}=\left(Y_{1}, \ldots, Y_{R}\right)$, where $Y_{r}$ is the number of measurements in cell $r$. The random number of measurements $M$ is
$$M=\sum_{r=1}^{R} Y_{r}$$

## 数学代写|组合学代写Combinatorics代考|Statement A—Object Existence and Detection

$$G_{N}(z)=\sum_{n=0}^{\infty} \operatorname{Pr} N=n z^{n}=1-\chi+\chi z$$

## 数学代写|组合学代写Combinatorics代考|Statement B—Gridded Measurements

$$M=\sum_{r=1}^{R} Y_{r}$$

## 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|MATH393

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|Likelihood Functions and Assignments

The probability distributions of the measurement and object processes determine the joint object-measurement distribution. The distribution is complicated when measurements and objects can be combined in more than one way without violating constraints imposed by the application. The allowed combinations are called feasible combinations. The joint likelihood function is the sum of the probabilities of the events defined by the feasible combinations. Even when the feasible combinations are describable in simple down-to-earth language, the sums themselves can be large, elaborate expressions whose summands are cumbersome and unwieldy, even tedious. In short, any inherent simplicity of the underlying joint likelihood function is obscured in a veritable haze of formulae in the enumerated sums.

The feasible combinations for point objects are assumed to be those that satisfy the “at most one measurement per object per scan” rule. The rule is a model of the output of the sensor signal processor; it was first stated explicitly in [3]. If there is more than one sensor, the rule is applied to each one. Given the rule, it is meaningful to define a “label,” or indicator variable that specifies whether or not a given measurement corresponds to an object, and if it does, which one. In this language, the sensor measurement set is unlabeled. The rule systematizes the problem but does nothing to clear away the formulaic haze.

GFs clear away the haze by encoding joint likelihood functions into astonishingly concise, exactly equivalent algebraic expressions. They transform complicated sums in the likelihood function into products of algebraic factors. The resulting GFs are often so small that they fit on a single line of text, sometimes with room to spare. The expressive economy of GFs extends to Bayesian statistics. As will be seen.

## 数学代写|组合学代写Combinatorics代考|A First Look at Generating Functions

Multiple object tracking problems are combinatorial problems in probabilistic clothing. To make the point and also show that GFs are valuable models in tracking, three hasic descriptive statements, or hypotheses, ahout ohjects and measurements are examined in this section. Elements of these statements appear in many tracking filters, frequently in complicated settings involving multiple objects and measurements, which is the focus of the book. The goal of this section is simple, yet remarkable-it is to show that these statements are perfectly encoded by a mathematical expression called a GF. After reading the examples, readers will understand the aptness of Wilf’s aphorism [1, p. 1], “A generating function is a clothesline on which we hang up a sequence of numbers for display.”

The first hypothesis is Statement A. It concerns the uncertainties of object existence and detection. It is interpreted as a counting problem and modeled as a GF. The second is Statement B and it extends Statement A by including the possibility of a measurement. Careful attention is paid to the critical step that maps the measurement into an equivalent counting problem. The third hypothesis is Statement $\mathrm{C}$, where Statement B is extended by including the possibility of object state and it too is mapped into a counting problem. Since Statement $\mathrm{C}$ is about two entities, object and measurement, it brings the discussion to a point where Bayesian inference is possible.

The following subsections discuss, in turn, the encoding of Statements A-C into joint GFs. Considerable insight into GF methods and the AC way of thinking is gained by using the GFs to explore the close relationships between the problems. For example, a simple algebraic procedure reduces the $\mathrm{GF}$ of Statement $\mathrm{C}$ to the $\mathrm{GF}$ of B, and the GF of B to that of A.

## 数学代写|组合学代写Combinatorics代考|Likelihood Functions and Assignments

GFs 通过将联合似然函数编码为令人惊讶的简洁、完全等价的代数表达式来消除雾霾。它们将似然函数中的复杂和转换为代数因子的乘积。生成的 GF 通常非常小，以至于它们可以放在一行文本中，有时还有空间。GF 的表达经济延伸到贝叶斯统计。正如将要看到的。

## 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|CS519

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 Benefits of Analytic Combinatorics to Tracking

The $\mathrm{AC}$ approach makes it easy to understand what distinguishes the different filters by comparing the individual algebraic factors of the GFs of the likelihood functions. Each factor has a specific combinatorial interpretation, and their product uniquely determines the fundamental structure of the filter. Using a chemistry analogy, factors are “elements” and their product is the “molecular formula” of the filter.
$\mathrm{AC}$ also gives a simple explanation of why it is that two seemingly similar filters may have very different computational complexities. As shown in Appendix A, the mathematical form of a filter is found by taking derivatives of the filter’s GF. The complexity of a filter is determined by the number of distinct terms in the derivative. As calculus students discover, the derivatives of similar looking functions can have very different numbers of terms. Simply by counting the number of terms in a derivative, practicing engineers can see in detail how a proposed change in the model alters the likelihood function and how that, in turn, affects the filter complexity.
AC enables approximations to be computed for very high computational complexity tracking filters using established classical applied mathematics. Further discussion is given in Sect. $6.3$ of Chap. 6, but it suffices here to say that the derivatives of the GF are written as Cauchy integrals (this is an exact equivalence), and then the saddle point method is applied to compute numerical approximations to the integrals. The saddle point method is a standard tool used in applied mathematics and physics for asymptotic analysis. The topic is the subject of ongoing work.

## 数学代写|组合学代写Combinatorics代考|Sensor and Object Models in Tracking

Single-object tracking filters estimate an object’s state using sensor data collected over a sequence of (non-overlapping) time intervals called scans. What constitutes the state of an object depends on the application, but it often comprises kinematic properties such as position and velocity. Object states are modeled as points, that is, objects appear as point sources in the sensor output. This modeling assumption has practical implications, e.g., in some surveillance applications, it requires that objects be neither too close nor too far from the sensor. ${ }^{2}$ Multiple object tracking filters estimate the multiobject state, which comprises the state of every object. The number of objects is stipulated in some filters and estimated in others, in which case the number of objects is part of the multiobject state.

Object motion is only partially predictable when some agent (e.g., a human pilot) is controlling them. This well-known and thorny modeling problem arises in other fields, too (e.g., control theory). It is treated here by assuming that object motion is governed by a random process whose probability distribution models the many sources of uncertainty in the object motion. Satisfying the assumption in practice is often a nontrivial exercise in model development – the model must incorporate not only the statistical nature of the inherent and unavoidable variability in object motion due to “system noise,” but it must also incorporate plausible models of the various unknown deterministic inputs from a controller (e.g., the pilot mentioned above).

## 数学代写|组合学代写Combinatorics代考|The Benefits of Analytic Combinatorics to Tracking

AC 能够使用已建立的经典应用数学计算非常高计算复杂度的跟踪滤波器的近似值。进一步的讨论在第 3 节中给出。6.3章。6，但这里只要说 GF 的导数写成柯西积分（这是精确等价的），然后应用鞍点法计算积分的数值近似就足够了。鞍点法是应用数学和物理学中用于渐近分析的标准工具。该主题是正在进行的工作的主题。

## 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|CS-E4555

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|Lexicographic Order

If $x$ and $y$ are $\mathcal{A}$-letters, then we shall write
$$x<\mathscr{A} y$$
to indicate that when the letters of $\mathcal{A}$ are listed in the given order, $x$ comes before $y$. Of course, when there is no danger of confusion, we shall drop the subscript $\mathcal{A}$ and write simply $x<y$. Thus for the alphabet $\mathcal{A}$ given in (1.1), the statements in (1.2) may simply be written as
\begin{aligned} &a<5 \ &5<b \ &b<x \ &x<c \end{aligned}
This given, if $w_{1}$ and $w_{2}$ are $\mathcal{A}$-words, we shall say that $w_{1}$ lexicographically precedes $w_{2}$ and write
$$w_{1}<\mathcal{A} w_{2}$$

if either

1. $w_{1}$ is an initial segment of $w_{2}$; or we have
2. $w_{1}=x_{1} x_{2} \cdots x_{h}, w_{2}=y_{1} y_{2} \cdots y_{k}$ with
$$\begin{gathered} x_{1}=y_{1}, \ x_{2}=y_{2}, \ \vdots \ x_{i-1}=y_{i-1} \ \text { and } \ x_{i}<\mathcal{A} y_{i} . \end{gathered}$$
Crudely speaking, $w_{1}<\mathcal{A} \quad w_{2}$ means that in the first position where $w_{1}$ and $w_{2}$ disagree, $w_{1}$ either has no letter at all or has a smaller letter than $w_{2}$.

## 数学代写|组合学代写Combinatorics代考|Listing Series and Algebraic Operations

One of the fundamental facts that shall guide us throughout this book is that certain basic constructions involving combinatorial objects may be translated into algebraic operations.

This is very fortunate since algebraic manipulations are very precise and almost mechanical, while object manipulations can be rather clumsy, difficult to describe, and ambiguous.

Languages are most suited for the algebraic treatment. To help translating operations on languages into algebraic operations, it is convenient to represent languages as sum of words. To this end, if $\mathcal{L}$ is a language, we shall set
$$s \mathcal{L}=\sum_{w \in \mathcal{L}} w$$
and refer to it as the listing series for $\mathcal{L}$.
We should emphasize that the expression on the right hand side of (1.7) is to be interpreted only as a formal sum, which is merely a convenient way to deal with all the words of $\mathcal{L}$ at the same time. For instance, if $\mathcal{L}$ consists of all the 2-letter words in the alphabet
$$\mathcal{A}={a, b, c}$$
then in the lexicographic order, we have
$$s \mathcal{L}=a a+a b+a c+b a+b b+b c+c a+c b+c c$$

## 数学代写|组合学代写Combinatorics代考|Lexicographic Order

$$x<\mathscr{A} y$$

$$a<5 \quad 5<b b<x \quad x<c$$

$$w_{1}<\mathcal{A} w_{2}$$

1. $w_{1}$ 是的初始段 $w_{2}$; 或者我们有
2. $w_{1}=x_{1} x_{2} \cdots x_{h}, w_{2}=y_{1} y_{2} \cdots y_{k}$ 和
$$x_{1}=y_{1}, x_{2}=y_{2}, \vdots x_{i-1}=y_{i-1} \text { and } x_{i}<\mathcal{A} y_{i} .$$
粗略地说， $w_{1}<\mathcal{A} \quad w_{2}$ 意味着在第一个位置 $w_{1}$ 和 $w_{2}$ 不同意， $w_{1}$ 要么根本没有字母，要么字母小于 $w_{2}$

## 数学代写|组合学代写Combinatorics代考|Listing Series and Algebraic Operations

$$s \mathcal{L}=\sum_{w \in \mathcal{L}} w$$

$$\mathcal{A}=a, b, c$$

$$s \mathcal{L}=a a+a b+a c+b a+b b+b c+c a+c b+c c$$

## 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。

## 英国补考|组合学代写Combinatorics代考|MATH 482

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

## 英国补考|组合学代写Combinatorics代考|The Empty Word

Frequently, for technical reasons, we have to make use of the special word consisting of no letters at all. This is usually referred to as the empty word or the null word and it is assigned zero length.

This word is often denoted by the symbol $\epsilon$. Sometimes we shall do so here as well. However we can easily see that this choice may lead to confusion if $\epsilon$ happens to be also one of the letters of the alphabet with which we work. To avoid this problem, we shall have to keep at hand an alternate symbol.

Clearly, concatenating the empty word with any other word $w$ should give $w$ back. More precisely, we want
$$w \cdot \epsilon=\epsilon \cdot w=w$$
Thus $\epsilon$ behaves with respect to concatenation very much in the same manner the familiar number 1 behaves with respect to multiplication. Thus an alternate and very appropriate notation for the empty word could be the symbol “1” itself, assuming of course that it is not one of the letters of the alphabet we are working with. Using “1” for the null word, (1.4) could be written as
$$w \cdot 1=1 \cdot w=w,$$
and this is consistent with common use of 1 . We shall later see that this notation has further advantages.

## 英国补考|组合学代写Combinatorics代考|Initial Segments

If $w_{1}$ and $w$ are words, we shall say that $w_{1}$ is an initial segment of $w$ or a prefix of $w$ if for some word $w_{2}$, we have
$$w=w_{1} \cdot w_{2} .$$
Crudely speaking, $w_{1}$ is an initial segment of $w$ if $w_{1}$ is at the beginning of $w$. For instance, $a x$ is an initial segment of
axcde and also of $a x b b f$.
Note that the null word $\epsilon$ and $w$ itself are always prefixes of $w$.

## 英国补考|组合学代写Combinatorics代考|The Empty Word

$$w \cdot \epsilon=\epsilon \cdot w=w$$

$$w \cdot 1=1 \cdot w=w$$

## 英国补考|组合学代写Combinatorics代考|Initial Segments

$$w=w_{1} \cdot w_{2}$$

## 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|MATH 393

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 Combinatorial Structures

In our presentation, we shall use terms such as words, languages, etc., which have already a meaning in ordinary conversation. However, the meaning assigned to them here will be slightly different. Thus it is good to make this distinction precise before proceeding any further.

The basic terms are alphabet, letter, word, string, and language. They are defined as follows. First of all,
alphabet $=$ any finite collection of symbols.
Even though reordering the elements of a set does not change it, i.e.,
$${a, 5, b, x, c}={b, a, x, 5, c},$$
for instance, when we view these sets as alphabets, we assume that they are ordered from left to right in the order we put them down. Thus when we write the sentence
$$\text { Let } \mathcal{A}={a, 5, b, x, c} \text { be an alphabet, }$$
we shall mean not only that $\mathcal{A}$ consists of $a, 5, b, x$, and $c$ but also that as long as we work with $\mathcal{A}$,
\begin{aligned} &\text { a comes before } 5 \ &5 \text { comes before } b \ &b \text { comes before } x \ &x \text { comes before } c . \end{aligned}

## 数学代写|组合学代写Combinatorics代考|Concatenation

If $w_{1}$ and $w_{2}$ are $\mathcal{A}$-words by $w_{1} \cdot w_{2}$ or by simply $w_{1} w_{2}$, we mean the word of length $l\left(w_{1}\right)+l\left(w_{2}\right)$ obtained by placing $w_{1}$ and $w_{2}$ one after the other. Thus if $w_{1}=a c c d$ and $w_{2}=x x y$, then $w_{1} w_{2}$ is the 7-letter word
$$w_{1} w_{2}=\operatorname{accd} x x y .$$
It is customary to refer to this operation as concatenation. In fact, the same term is used when we juxtapose any number of words.

We should also mention that our dot notation $w_{1} \cdot w_{2}$ refers to the fact that concatenation has the mathematical character of multiplication.
Indeed, we can trivially see that for any triplet of words $w_{1}, w_{2}, w_{3}$, we have
$$\left(w_{1} \cdot w_{2}\right) \cdot w_{3}=w_{1} \cdot\left(w_{2} \cdot w_{3}\right) .$$
Clearly, it is immaterial whether we concatenate $w_{1}$ and $w_{2}$ and then concatenate the resulting word with $w_{3}$ or concatenate $w_{1}$ with the result of concatenating $w_{2}$ and $w_{3}$.
We might refer to (1.3) as associativity of word concatenation.

## 数学代写|组合学代写Combinatorics代考|Basic Combinatorial Structures

$$a, 5, b, x, c=b, a, x, 5, c$$

Let $\mathcal{A}=a, 5, b, x, c$ be an alphabet,

a comes before $5 \quad 5$ comes before $b b$ comes before $x \quad x$ comes before $c$.

## 数学代写|组合学代写Combinatorics代考|Concatenation

$$w_{1} w_{2}=\operatorname{accd} x x y .$$

$$\left(w_{1} \cdot w_{2}\right) \cdot w_{3}=w_{1} \cdot\left(w_{2} \cdot w_{3}\right) \text {. }$$

## 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。