## 数学代写|运筹学作业代写operational research代考|Merits and Demerits of Graphical Solutions

statistics-lab™ 为您的留学生涯保驾护航 在代写运筹学operational research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。我们的专家在代写运筹学operational research代写方面经验极为丰富，各种代写运筹学operational research相关的作业也就用不着说。

## 数学代写|运筹学作业代写operational research代考|Merits and Demerits of Graphical Solutions

Graphical solutions are easier to understand and reproduce. Also a pictorial view is always a better representation. Thus graphical solutions have gained prominence in Operations Research.
However, graphical solutions have certain limitations such as :

1. Limited to the problems of two decision variables only.
2. Accuracy can not be obtained.
3. Some times it is difficult to represent certain expressions, particularly in the case of non linear expressions.

Graphical Solution Procedure

1. Assuming $x_1$ and $x_2$ (the decision variables) to be represented on $X$ and $Y$ axes respectively check the conditions of variables and accordingly prepare the graph sheet.

If $x_1 \geq 0, x_2 \geq 0$, your graph will be in first quadrant, if $x_1$ is unrestricted and $x_2 \geq 0$, the solution will be I and II quadrants, if $x_1 \geq 0$ and $x_2$ is unrestricted the solution will be in I and IV quadrants. If both are unrestricted, the solution may be any where (in any quadrant).
[It is better to leave three units at the bottom and also at the left side so that your graphical solution will be clearly represented].

Divide the scale approximately on $X$ and $Y$ axes such that you can represent all its values. It is always better to have same scale on both axes.

Assume the constraints as equations and find any two points for each equation so that the equation can be represented as a straight line on graph.
[It is always easy to assume $x_2=0$ to find $x_1$ (say a) and $x_1=0$ to find $x_2$ (say b) and thus you draw line connecting $(a, 0)$ and $(0, b)]$

Similarly, draw all the constraint lines.

Shade the appropriate areas as given by the constraints. If the constraint is $\leq$ type shade the area towards origin. If the constraint is $\geq$ type shade the area away from origin. If the constraint is ‘ $=$ ‘ type, do not shade any area, and the line itself is the region.

## 数学代写|运筹学作业代写operational research代考|Redundant Constraint

As we have seen the structure of an LPP is composed of three main parts viz objective function, set of constraint and the conditions of variables. With reference to these parts, we report the following types of solutions.

1. Solutions.
2. Feasible solutions.
3. Basic feasible solutions.
4. Optimal solutions.
5. Solutions : All those values of variable which satisfy the conditions given are solutions: Thus if the conditions are $x_1 \geq 0, x_2>0$ (non-negative), then all the values in first quadrant (covered by positive $X$-axis and positive $Y$-axis) are the solutions. Similarly, $x_1 \geq 0\left(X\right.$-axis) and $x_2$ unrestricted yields the solutions in first and fourth quadrants of graph, and so on.
6. Feasible Solution : All the solutions which satiffy all the conditions of variables as well as all the constraints are feasible solutions.

We can notice here that if all the constraints are exact type, we may get ‘point solution’. One exact and another inequality constraint will yield line of solutions’. All the inequality constraints will generate an area of feasible solutions often referred to as ‘feasible region’.

Basic Feasible Solutions : The values of variables represented by the points along the border lines of feasible region are basic feasible solutions.

If $m$ non identical equation with ‘ $n$ ‘ variable $(m<n)$ exist in a problem, then keeping $(n-m)$ variables constant, usually at zero, the values of the variables yield a solution called ‘basic solution’. As it satisfies all the constraints it can be called a region, it can also be called a ‘basic feasible solution’. Therefore each selection of $(n-m)$ variables from $n$ variables gives raise to ‘ $\left(n_{C_{n-m}}\right)$ ‘ basic feasible solutions’ (BFS)

Of all these BFS, the one with which we start working out the problem is ‘initial basic feasible solution’. (IBFS).

Most commonly, the IBFS will be choosen to start at the worst case of the solution set so as not to miss to examine any solution. Thus a solution with zero profit or nil production or the values of decision variables as zeros (i.e., origin) in graphical solution will be IBFS.

Optimal Solution : The solutions which satisfy all the conditions of variables, all the constraints and the objective function are ‘optimal basic feasible solutions (OBFS)’ or simply ‘optimal solutions’.

# 运筹学代考

## 数学代写|运筹学作业代写operational research代考|Merits and Demerits of Graphical Solutions

[最好在底部和左侧留下三个单元，以便您的图形解决方案能够清楚地表示]。

[我们总是很容易假设$x_2=0$找到$x_1$(假设a)， $x_1=0$找到$x_2$(假设b)，因此你画一条线连接$(a, 0)$和 $(0, b)]$

## 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。

## 会计代写|财务管理代写Financial Management代考|Mutually Exclusive Alternatives and Capital Rationing

statistics-lab™ 为您的留学生涯保驾护航 在代写财务管理Financial Management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。我们的专家在代写财务管理Financial Management代写方面经验极为丰富，各种代写财务管理Financial Management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

## 会计代写|财务管理代写Financial Management代考|Mutually Exclusive Alternatives and Capital Rationing

We now consider briefly two common occurrences that often complicate investment selection. The first is known as mutually exclusive alternatives. Frequently, there is more than one way to accomplish an objective, and the investment problem is to select the best alternative. In this case, the investments are said to be mutually exclusive. Examples of mutually exclusive alternatives abound, including the choice of whether to build a concrete or a wooden structure, whether to drive to work or take the bus, and whether to build a 40-story or a 30 -story building. Even though each option gets the job done and may be attractive individually, it does not make economic sense to do more than one. If you decide to take the bus to work, driving to work as well could prove a difficult feat. When confronted with mutually exclusive alternatives, then, it is not enough to decide if each option is attractive individually; you must determine which is best. Mutually exclusive investments are in contrast to independent investments, where the capital budgeting problem is simply to accept or reject a single investment.

When investments are independent, all three figures of merit introduced earlier-the NPV, BCR, and IRR-will generate the same investment decision, but this is no longer true when the investments are mutually exclusive. In all of the preceding examples, we implicitly assumed independence.

A second complicating factor in many investment appraisals is known as capital rationing. So far, we have implicitly assumed that sufficient money is available to enable the company to undertake all attractive opportunities. In contrast, under capital rationing, the decision maker has a fixed investment budget that may not be exceeded. Such a limit on investment capital may be imposed externally by investors’ unwillingness to supply more money, or it may be imposed internally by senior management as a way to control the amount of investment dollars each operating unit spends. In either case, the investment decision under capital rationing requires the analyst to rank the opportunities according to their investment merit and accept only the best.
Both mutually exclusive alternatives and capital rationing require a ranking of investments, but here the similarity ends. With mutually exclusive investments, money is available, but for technological reasons only certain investments can be accepted; under capital rationing, a lack of money is the complicating factor. Moreover, even the criteria used to rank the investments differ in the two cases, so the best investment among mutually exclusive alternatives may not be the best under conditions of capital rationing. The Appendix to this chapter discusses these technicalities and indicates which figures of merit are appropriate under which conditions.

## 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。

## 会计代写|财务管理代写Financial Management代考|Uneven Cash Flows

statistics-lab™ 为您的留学生涯保驾护航 在代写财务管理Financial Management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。我们的专家在代写财务管理Financial Management代写方面经验极为丰富，各种代写财务管理Financial Management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

## 会计代写|财务管理代写Financial Management代考|Uneven Cash Flows

Perceptive readers may have noticed a problem with what we have covered to this point: All of the cash flows in all of the examples can be described using the four variables defined earlier. What happens when the cash flows are not so well behaved? What happens when they are more erratic? To illustrate the problem, let’s modify our container pier example a bit and assume that Pacific Container now estimates it will take time to ramp up to full capacity and that first-year cash flows will be only $\$ 3.5$million, not$\$7.5$ million as originally projected.

And we are now stuck because it is no longer possible to describe the investment’s cash flows purely in terms of the original four variables: nper, pv, pmt, and fv.

Fortunately, spreadsheets offer a simple, elegant solution to this problem involving two new functions known as =IRR and $=$ NPV. Table 7.3 shows an Excel spreadsheet illustrating their use. The numbers on the left are the revised container pier cash flows. The icons for the two new functions appear to the right. Looking first at the IRR icon, note that the prompt replaces the usual PV, PMT, and FV variables with a new variable called values. The values point the spreadsheet to a range of cells containing the investment’s cash flows. Here, the cash flows are in cells B3 through B13, and the values appear in the formula as B3:B13. All you need to do to calculate the IRR of an arbitrary list of numbers, then, is to enter the relevant range containing the numbers into the $=$ IRR function.

The $=$ NPV function is similar. It calls for an interest rate and a range of cash flows containing at least one nonzero value and returns the net present value of the cash flows. Here I have entered the cash flows in the range B4 through B13. “But wait,” you exclaim, “why did you omit the cash flow in B3 from this range?” The answer is that, by definition, the =NPV function calculates the net present value of the specified range as of one period before the first cash flow. Had I entered “=NPV $(\mathrm{C} 15, \mathrm{~B} 3: \mathrm{B} 13)$ )” the computer would have calculated the NPV of the investment as of time minus 1 . To avoid this error, I calculated the NPV of the cash flows from years 1 through 10 , which by definition the computer will calculate as of time 0 , and then I added the

## 会计代写|财务管理代写Financial Management代考|A Few Applications and Extensions

Think of it: Although the holder will receive a total of $\$ 100$, the present value is less than$\$9$. Why? Because if the investor put $\$ 8.33$in a bank account today yielding 12 percent a year, he could withdraw approximately$\$1$ in interest every year forever without touching the principal $(12 \% \times \$ 8.33=\$0.9996)$. Consequently, $\$ 8.33$today has approximately the same value as$\$1$ a year forever.
This suggests the following simple formula for the present value of a perpetuity. Letting $A$ equal the annual receipt, $r$ the discount rate, and $P$ the present value,
$$P=\frac{A}{r}$$
and
$$r=\frac{A}{P}$$
To illustrate, suppose a share of preferred stock sells for $\$ 980$and promises an annual dividend of$\$52$ forever. Then, its IRR is 5.3 percent (52/980). Because the equations are so simple, perpetuities are often used to value longlived assets and in many textbook examples.
Equivalent Annual Cost

## 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。

## FINN3120 Financial Management课程简介

Many of the demands placed on IT professionals are driven by an organization’s financial managers and their focus not only on the “bottom line,” but also on various “Return on Investment” measures. This course introduces students to some of the basic terminology, concepts and tools used by financial managers in making business decisions. Topics covered include: the relationship between an organization’s accounting function and the finance function; financial reporting principles and the financial statements; using financial statements in practice; determining relevant items in making short-term business decisions; the Time Value of Money principles; short- and long-term decision making; and the uses and misuses of accounting data in managing people and controlling processes.

## PREREQUISITES

Week 1: Course Overview – Financial Reporting, Managerial Accounting and Corporate Finance; The Three Major Financial Statements
Week 2: The Income Statement, Revenue Recognition, and Analysis of Monetary Assets
Week 3: Choices within GAAP: Inventory Costing and Depreciation;The Statement of Cash Flows
Week 4: Managerial Accounting – Decision-Making: Cost Behavior
Week 5: Managerial Accounting – Decision-Making Tools
Week 6: Managerial Accounting – Responsibility Accounting:Performance Evaluation and Control
Week 7: Managerial Accounting: Relevant Cost Information and Typical Decisions
Week 8: Corporate Finance and the Time Value of Money (TVM)
Week 9: Applications of TVM to Personal and Business Decisions
Week 10: Corporate Finance: Capital Budgeting
Teaching methods: The course will be primarily lecture format, combined with discussion of the assigned problems and cases. To obtain the maximum benefit from each class, the student should complete the relevant reading and homework assignments prior to class. Students should also attempt to solve the problems before the relevant session meets. Only those marked “Turn-In” (the cases) will form the basis of your course grade.

Grading criteria: There will be six graded case submissions. Performance on five of the six required cases will constitute $75 \%$ of the course grade ( $15 \%$ each). Students may choose not to submit one of the six cases, in which event the five cases submitted will determine the course grade. On the other hand, students may submit all six cases and only the five best will be used to determine the course grade. The remaining $25 \%$ of the course grade will be determined by class participation ( $5 \%$ ) and evaluations of each group member’s contribution to his/her group (20\%). An unexcused absence from more than one class will cause the final grade to be lowered one letter grade.

## FINN3120 Financial Management HELP（EXAM HELP， ONLINE TUTOR）

Answers to odd-numbered problems appear at the end of the book. Answers to even-numbered problems and additional exercises are available in the Instructor Resources within McGraw-Hill’s Connect (see the Preface for more information).

1. Looking at Table 6.4, why do public utilities have such a low times-interestearned ratio? Why is the ratio for information technology companies so high?

What is operating leverage? How, if at all, is it similar to financial leverage? If a firm has high operating leverage would you expect it to have high or low financial leverage? Explain your reasoning.

1. Explain why increasing financial leverage increases the risk borne by shareholders.
2. Explain how a company can incur costs of financial distress without ever going bankrupt. What is the nature of these costs?

1. One recommendation in the chapter is that companies with promising investment opportunities should strive to maintain a conservative capital structure. Yet many promising small businesses are heavily indebted.
a. Why should most companies with promising investment opportunities strive to maintain conservative capital structures?
b. Why do you suppose many promising small businesses fail to follow this recommendation?

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供charlotte.edu FINN3120 Financial Management财务管理课程的代写代考辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

statistics-lab™ 为您的留学生涯保驾护航 在代写傅里叶分析Fourier analysis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。我们的专家在代写傅里叶分析Fourier analysis代写方面经验极为丰富，各种代写傅里叶分析Fourier analysis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

To start with, we briefly discuss the relation between the space of continuous functions and its dual space.

Let $X$ be a locally compact Hausdorff topological space. We denote by $\mathfrak{C}{\infty}(X, \mathbb{C})$ (resp. $\left.\mathfrak{C}{\infty}(X, \mathbb{R})\right)$ the set of all the complex-valued (resp. real-valued) continuous functions vanishing at infinity. ${ }^1$ It is a Banach space with respect to the norm of uniform convergence: $$|f|_{\infty}=\sup {x \in X}|f(x)|, \quad f \in \mathbb{C}{\infty}$$
Let $\mu$ be a complex-valued regular measure on $(X, \mathcal{B}(X))$, the total variation $|\mu|$ of which is finite. $\mathcal{B}(X)$ is the Borel $\sigma$-field on $X$. The completion of such a measure with respect to $|\mu|$ is called a Radon measure. The set of all the Radon measures is denoted by $\mathfrak{M}(X) . \mathfrak{M}(X)$ is a Banach space, the norm of which is given by the total variation $|\mu|_{\mathscr{M}(X)}=|\mu|$. In particular, the set of all the positive (real-valued) Radon measures is denoted by $\mathfrak{M}{+}(X)$. For any $\mu \in \mathfrak{M}(X)$, we define a linear functional $\Lambda\mu$ on $\mathbb{C}{\infty}(X, \mathbb{C})$ by $$\Lambda\mu f=\int_X f(x) d \mu, \quad f \in \mathbb{C}{\infty}(X, \mathbb{C}) .$$ Then $\Lambda\mu$ is bounded; i.e. $\Lambda_\mu \in \mathbb{C}{\infty}(X, \mathbb{C})^{\prime}$. Conversely, for any $\Lambda \in \mathbb{C}{\infty}(X, \mathbb{C})^{\prime}$, there exists a measure $\mu_{\Lambda} \in \mathfrak{M}(X)$ which satisfies
$$\Lambda f=\int_X f(x) d \mu_{\Lambda}, \quad f \in \mathfrak{C}{\infty}(X, \mathbb{C})$$ and $|\Lambda|=|\mu|$. Such a measure $\mu$ is uniquely determined. Thus the two Banach spaces $\mathfrak{C}{\infty}(X, \mathbb{C})^{\prime}$ and $\mathfrak{M}(X)$ are isomorphic to each other. This result is called the Riesz-Markov-Kakutani theorem. ${ }^2$

## 数学代写|傅里叶分析代写Fourier analysis代考|Fourier Coefficients of Measures

The space $\mathbb{C}(\mathbb{T}, \mathbb{C})$ of complex-valued continuous function on $\mathbb{T}$ is a Banach space with the uniform convergence norm. By the Riesz-Markov-Kakutani theorem, the Banach space $\mathfrak{M}(\mathbb{T})$ of complex-valued Radon measures on $T$ (norm is given by the total variation) is isomorphic to the dual space of $\mathbb{C}(\mathbb{T}, \mathbb{C})$. From now on, the measurable space $(T, \mathcal{B}(\mathbb{T}))$ is identified with $([-\pi, \pi), \mathcal{B}([-\pi, \pi)))$. (cf. Appendix A.)

Definition 6.1 For $\mu \in \mathfrak{M}(\mathbb{T})$,
$$\hat{\mu}(n)=\frac{1}{\sqrt{2 \pi}} \int_{-\pi}^\pi e^{-i n x} d \mu(x),{ }^3 \quad n \in \mathbb{Z}$$
are called the Fourier coefficients of $\mu$.
Fourier coefficients of a measure are also called Fourier-Stieltjes coefficients in order to distinguish them from Fourier coefficients of a function. In particular, the Fourier coefficients of a measure $\mu_f=f d x$ defined by a function $f \in \mathfrak{L}^1(\mathbb{T}, \mathbb{C})$ are given by
$$\hat{\mu}f(n)=\frac{1}{\sqrt{2 \pi}} \int{-\pi}^\pi e^{-i n x} f(x) d x, \quad n \in \mathbb{Z} .$$
These are nothing other than usual Fourier coefficients of $f$.
Theorem 6.1 For any $f \in \mathbb{C}(\mathbb{T}, \mathbb{C})$ and $\mu \in \mathfrak{M}(\mathrm{T})$,
$$\int_{-\pi}^\pi f(x) d \mu=\lim {n \rightarrow \infty} \sum{j=-(n-1)}^{n-1}\left(1-\frac{|j|}{n}\right) \hat{f}(j) \hat{\mu}(-j) .$$
Proof Consider first a trigonometric polynomial
$$P(x)=\frac{1}{\sqrt{2 \pi}} \sum_{j=-(n-1)}^{n-1} a_j e^{i j x}$$
as a special case of $f$.

# 傅里叶分析代写

$$|f|{\infty}=\sup x \in X|f(x)|, \quad f \in \mathbb{C} \infty$$ 让 $\mu$ 是一个复值的常规措施 $(X, \mathcal{B}(X)$ ， 总变异 $|\mu|$ 其中是有限的。 $\mathcal{B}(X)$ 是宝来 $\sigma$-场上 $X$. 此类措施的完 成 $|\mu|$ 称为氡测量。所有 Radon 测量值的集合表示为 $\mathfrak{M}(X) . \mathfrak{M}(X)$ 是 Banach 空间，其范数由总变差 给出 $|\mu|{\mathscr{M}(X)}=|\mu|$. 特别是，所有正 (实值) Radon 测量值的集合表示为 $\mathfrak{M}+(X)$. 对于任何 $\mu \in \mathfrak{M}(X)$ ，我们定义一个线性泛函 $\Lambda \mu$ 在 $C \infty(X, \mathbb{C})$ 经过
$$\Lambda \mu f=\int_X f(x) d \mu, \quad f \in \mathbb{C} \infty(X, \mathbb{C}) .$$

$$\Lambda f=\int_X f(x) d \mu_{\Lambda}, \quad f \in \mathfrak{C} \infty(X, \mathbb{C})$$

## 数学代写|傅里叶分析代写Fourier analysis代考|Fourier Coefficients of Measures

$$\hat{\mu} f(n)=\frac{1}{\sqrt{2 \pi}} \int-\pi^\pi e^{-i n x} f(x) d x, \quad n \in \mathbb{Z}$$

$$\int_{-\pi}^\pi f(x) d \mu=\lim n \rightarrow \infty \sum j=-(n-1)^{n-1}\left(1-\frac{|j|}{n}\right) \hat{f}(j) \hat{\mu}(-j) .$$

$$P(x)=\frac{1}{\sqrt{2 \pi}} \sum_{j=-(n-1)}^{n-1} a_j e^{i j x}$$

## 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|傅里叶分析代写Fourier analysis代考|Summability Kernels on R

statistics-lab™ 为您的留学生涯保驾护航 在代写傅里叶分析Fourier analysis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。我们的专家在代写傅里叶分析Fourier analysis代写方面经验极为丰富，各种代写傅里叶分析Fourier analysis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|傅里叶分析代写Fourier analysis代考|Summability Kernels on R

Definition 5.2 A family of continuous functions $\left{k_\lambda: \mathbb{R} \rightarrow \mathbb{R}\right} \quad(\lambda \in(0, \infty)$, or $\lambda \in \mathbb{N}$ ) is called a summability kernel on $\mathbb{R}$ if it satisfies:
(i) $\int_{-\infty}^{\infty} k_\lambda(x) d x=1$ for all $\lambda$,
(ii) $\left|k_\lambda\right|_1=O(1)$ as $\lambda \rightarrow \infty$,
(iii) $\lim {\lambda \rightarrow \infty} \int{|x|>\delta}\left|k_\lambda(x)\right| d x=0$ for any $\delta>0$.
If a function $f \in \mathfrak{Q}^1(\mathbb{R}, \mathbb{R})$ satisfies
$$\int_{-\infty}^{\infty} f(x) d x=1$$
a summability kernel can be made based upon $f$. That is, if we define
$$k_\lambda(x)=\lambda f(\lambda x)$$
then $\left{k_\lambda\right}$ is a summability kernel. In fact, the condition (i) is verified by changing the variables: $y=\lambda x$. (ii) is satisfied, since
$$\left|k_\lambda\right|_1=\int_{-\infty}^{\infty}\left|k_\lambda(x)\right| d x=\int_{-\infty}^{\infty}|f(y)| d y=|f|_1$$
for every $\lambda>0$. It is also easy to check (iii), since
$$\int_{|x|>\delta}\left|k_\lambda(x)\right| d x=\int_{|y|>\lambda \delta}|f(y)| d y \rightarrow 0 \quad \text { as } \quad \lambda \rightarrow \infty .$$
For instance, if we define functions $A(x)$ and $G(x)$ by
$$A(x)=\frac{1}{2} e^{-|x|}, \quad G(x)=\frac{1}{\sqrt{\pi}} e^{-x^2},$$
the integrals of them over $\mathbb{R}$ are equal to 1 . So it is possible to make summation kernels based upon them. The kernels based upon $A$ and $G$ are called the Abel summability kernel and the Gauss summability kernel, respectively. ${ }^8$

## 数学代写|傅里叶分析代写Fourier analysis代考|Inverse Fourier Transforms

We now try to look for a method of inverse Fourier transforms. Given any integrable function, is it possible to find some function, the Fourier transform of which is exactly equal to it? We already know the positive answer to this question in the frameworks of $\mathfrak{Q}^2$ (Plancherel’s Theorem 4.3, p. 72), $\Xi$ (Theorem 4.2, p. 68) and $\Xi^{\prime}$ (Theorem 4.5 , p. 84). But how about in the case of $\mathfrak{}^1$ ?

The procedure to find some function, the Fourier transform of which is given is called spectral synthesis.

A vector-valued integration appearing in the next lemma (which corresponds to Lemma 5.1) is the one in the sense of Cauchy-Bochner. ${ }^{11}$

Lemma 5.3 Let $\mathfrak{X}$ be a Banach space, $\varphi: \mathbb{R} \rightarrow \mathfrak{X}$ a bounded continuous function and $\left{k_\lambda\right}$ a summability kernel. Then
$$\lim {\lambda \rightarrow \infty} \int{-\infty}^{\infty} k_\lambda(x) \varphi(x) d x=\varphi(0)$$
Proof Taking account of the condition (i) of summability kernels, we have
$$\int_{-\infty}^{\infty} k_\lambda(x) \varphi(x) d x-\varphi(0)=\int_{-\infty}^{\infty} k_\lambda(x)(\varphi(x)-\varphi(0)) d x=\int_{-\delta}^\delta+\int_{|x|>\delta}=I_1+I_2$$
for any $\delta>0$
$I_1$ can be evaluated as
$$\left|I_1\right| \leqq \operatorname{Max}{|x| \leqq \delta}|\varphi(x)-\varphi(0)| \cdot\left|k\tau\right|_1$$
Let $\varepsilon>0$ be any positive number. If we choose $\delta>0$ sufficiently small, the righthand side of (5.33) is less than $\varepsilon$. As for $I_2$, we obtain$\left|I_2\right| \leqq \sup {|x|>\delta}|\varphi(x)-\varphi(0)| \int{|x|>\delta}\left|k_\lambda(x)\right| d x$.

# 傅里叶分析代写

## 数学代写|傅里叶分析代写Fourier analysis代考|Summability Kernels on R

(i) $\int_{-\infty}^{\infty} k_\lambda(x) d x=1$ 对全部 $\lambda$,
(二) $\left|k_\lambda\right|1=O(1)$ 作为 $\lambda \rightarrow \infty$ ， (iii) $\lim \lambda \rightarrow \infty \int|x|>\delta\left|k\lambda(x)\right| d x=0$ 对于任何 $\delta>0$.

$$\int_{-\infty}^{\infty} f(x) d x=1$$

$$k_\lambda(x)=\lambda f(\lambda x)$$

$$\left|k_\lambda\right|1=\int{-\infty}^{\infty}\left|k_\lambda(x)\right| d x=\int_{-\infty}^{\infty}|f(y)| d y=|f|1$$ 每一个 $\lambda>0$. (iii) 也很容易检验，因为 $$\int{|x|>\delta}\left|k_\lambda(x)\right| d x=\int_{|y|>\lambda \delta}|f(y)| d y \rightarrow 0 \quad \text { as } \quad \lambda \rightarrow \infty .$$

$$A(x)=\frac{1}{2} e^{-|x|}, \quad G(x)=\frac{1}{\sqrt{\pi}} e^{-x^2}$$

## 数学代写|傅里叶分析代写Fourier analysis代考|Inverse Fourier Transforms

$$\lim \lambda \rightarrow \infty \int-\infty^{\infty} k_\lambda(x) \varphi(x) d x=\varphi(0)$$

$$\int_{-\infty}^{\infty} k_\lambda(x) \varphi(x) d x-\varphi(0)=\int_{-\infty}^{\infty} k_\lambda(x)(\varphi(x)-\varphi(0)) d x=\int_{-\delta}^\delta+\int_{|x|>\delta}=I_1+I_2$$

$I_1$ 可以评价为
$$\left|I_1\right| \leqq \operatorname{Max}|x| \leqq \delta|\varphi(x)-\varphi(0)| \cdot|k \tau|1$$ 让 $\varepsilon>0$ 是任何正数。如果我们选择 $\delta>0$ 足够小，(5.33) 的右边小于 $\varepsilon$. 至于 $I_2$ ，我们获得 $\left|I_2\right| \leqq \sup |x|>\delta|\varphi(x)-\varphi(0)| \int|x|>\delta\left|k\lambda(x)\right| d x$.

## 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。

## MATH4100 Fourier analysis课程简介

The goals for the course are to gain a facility with using the Fourier transform, both specific techniques and general principles, and learning to recognize when, why, and how it is used. Together with a great variety, the subject also has a great coherence, and the hope is students come to appreciate both.

## PREREQUISITES

Topics include: The Fourier transform as a tool for solving physical problems. Fourier series, the Fourier transform of continuous and discrete signals and its properties. The Dirac delta, distributions, and generalized transforms. Convolutions and correlations and applications; probability distributions, sampling theory, filters, and analysis of linear systems. The discrete Fourier transform and the FFT algorithm. Multidimensional Fourier transform and use in imaging. Further applications to optics, crystallography. Emphasis is on relating the theoretical principles to solving practical engineering and science problems.

## MATH4100 Fourier analysis HELP（EXAM HELP， ONLINE TUTOR）

Theorem $4.5 \mathfrak{F}$ is an automorphism of $\leftrightarrows(\mathbb{R})^{\prime}$. The inverse Fourier transform $\mathfrak{F}^{-1}$ is its inverse.

Proof The linearity and the injectivity are clear. The surjectivity can be shown as follows. If $T$ is an element of $\leftrightarrows(\mathbb{R})^{\prime}$, then
$$\hat{\tilde{T}}(\varphi)=\tilde{T}(\hat{\varphi})=T(\tilde{\hat{\varphi}})=T(\varphi) \text { for all } \varphi \in \Xi(\mathbb{R}) .$$
This shows that $T$ is the Fourier transform of $\tilde{T} \in \Xi(\mathbb{R})^{\prime}$. Similarly, the inverse of $\mathfrak{F}$ is given by the inverse Fourier transform, since
$$\left(\mathfrak{F}^{-1} \circ \mathscr{F}\right)(T)(\varphi)=T\left(\mathcal{F} \circ \mathcal{F}^{-1}\right)(\varphi)=T(\varphi) \text { for all } \quad T \in \widetilde{G}(\mathbb{R})^{\prime}, \varphi \in \mathbb{G}(\mathbb{R}) .$$
Hence $\mathfrak{\&}^{-1} \circ \mathfrak{F}=I$ (identity).
$\tilde{F}$ and $\tilde{F}^{-1}$ are continuous (in the strong topology) in view of Theorem 4.4.
Remark 4.3 We use the notation $\check{\varphi}$ (or $\check{T}$ ) which means
$$\check{\varphi}(x)=\varphi(-x), \quad \check{T}(\varphi)=T(\check{\varphi}) .$$

Theorem 5.2 (shift operator and convolution) If $f \in \mathfrak{Q}^1(\mathbb{R}, \mathbb{C})$ and $k: \mathbb{R} \rightarrow \mathbb{C}$ is continuous and integrable, then ${ }^3$
$$\int_{-\infty}^{\infty} k(x) \tau_x f d x=k * f .$$

Proof Assume first that $f$ is continuous and $\operatorname{supp} f$ is compact. In this case, the integration on the left-hand side is actually evaluated on some finite interval. Hence
$$\int_{-\infty}^{\infty} k(x) \tau_x f d x=\lim \sum_j\left(x_{j+1}-x_j\right) k\left(x_j\right) \tau_{x_j} f,$$
where the limit is taken with respect to $q^1$-norm as the decomposition of the interval of integration becomes finer and finer. On the other hand, we have
$$(k * f)(x)=\lim \sum_j\left(x_{j+1}-x_j\right) k\left(x_j\right) f\left(x-x_j\right) \quad \text { (uniform convergence). }$$
Comparing (5.7) and (5.8), the proof is finished in this special case.
We shall now turn to the general case: $f \in \mathfrak{q}^1$. There exists, for any $\varepsilon>0$, some continuous function $g$ with compact support which satisfies $|f-g|_1<\varepsilon$. Since
$$\int_{-\infty}^{\infty} k(x) \tau_x g d x=k * g$$ as observed above, it follows that
$$\int_{-\infty}^{\infty} k(x) \tau_x f d x-k * f=\int_{-\infty}^{\infty} k(x) \tau_x(f-g) d x+k *(g-f) .$$

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供unt.edu MATH4100 Fourier analysis傅里叶分析课程的代写代考辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

## 数学代写|数值分析代写numerical analysis代考|Inequality Constrained Optimization

statistics-lab™ 为您的留学生涯保驾护航 在代写数值分析numerical analysis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。我们的专家在代写数值分析numerical analysis代写方面经验极为丰富，各种代写数值分析numerical analysis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|数值分析代写numerical analysis代考|Inequality Constrained Optimization

Inequality constrained optimization is more complex, both in theory and practice. The theorem giving necessary conditions for inequality constrained optimization was only discovered in the middle of the twentieth century, while Lagrange used Lagrange multipliers in his Mécanique Analytique 151. The necessary conditions for inequality constrained optimization are called Kuhn-Tucker or Karush-Kuhn-Tucker conditions. The first journal publication with these conditions was a paper by Kuhn and Tucker in 149, although the essence of these conditions was contained in an unpublished Master’s thesis of Karush 139.

The work of Kuhn and Tucker was intended to build on the work of G. Dantzig and others [69] on linear programming:
$\min x c^T x \quad$ (8.6.7) subject to $$A \boldsymbol{x} \geq \boldsymbol{b}$$ where ” $\boldsymbol{a} \geq \boldsymbol{b}$ ” is understood to mean ” $a_i \geq b_i$ for all $i$ “. It was Dantzig who created the simplex algorithm in 1946 [69], being the first general-purpose and efficient algorithm for solving linear programs (8.6.7, 8.6.8). The simplex method can be considered an example of an active set method as it tracks which of the inequalities $(A x)_i \geq b_i$ is actually an equality as it updates the candidate optimizer $\boldsymbol{x}$. Since then there has been a great deal of work on alternative methods, most notably interior point methods that typically minimize a sequence of penalized problems such as $$c^T \boldsymbol{x}-\alpha \sum{i=1}^m \ln \left((A x)_i-b_i\right)$$
where $\alpha>0$ is a parameter that is reduced to zero in the limit. The first published interior point method was due to Karmarkar 138. Another approach is the ellipsoidal method of Khachiyan 120, which at each step $k$ minimizes $\boldsymbol{c}^T \boldsymbol{x}$ over $\boldsymbol{x}$ lying inside an ellipsoid centered at $\boldsymbol{x}_k$ that is guaranteed to be inside the feasible set ${\boldsymbol{x} \mid A x \geq \boldsymbol{b}}$. Khachiyan’s ellipsoidal method built on previous ideas of N.Z. Shor but was the first guaranteed polynomial time algorithm for linear programming. Karmarkar’s algorithm also guaranteed polynomial time, but was much faster in practice than Khachiyan’s method and the first algorithm to have a better time than the simplex method on average.

## 数学代写|数值分析代写numerical analysis代考|Proving the Karush–Kuhn–Tucker Conditions

To prove the Karush-Kuhn-Tucker conditions we need a constraint qualification to ensure that
$$T_{\Omega}(\boldsymbol{x})=\left{\boldsymbol{d} \mid \nabla g_i(\boldsymbol{x})^T \boldsymbol{d}=0 \text { for all } i \in \mathcal{E},\right.$$
(8.6.9) $\nabla g_i(\boldsymbol{x})^T \boldsymbol{d} \geq 0$ for all $i \in \mathcal{I}$ where $\left.g_i(\boldsymbol{x})=0\right}=C_{\Omega}(\boldsymbol{x})$.
A constraint $g_i(\boldsymbol{x}) \geq 0$ is called active at $\boldsymbol{x}$ if $g_i(\boldsymbol{x})=0$ and inactive at $\boldsymbol{x}$ if $g_i(\boldsymbol{x})>0$. Inactive inequality constraints at $x$ do not affect the shape of the feasible set $\Omega$ near to $\boldsymbol{x}$. We designate the set of active constraints by
$$\mathcal{A}(\boldsymbol{x})=\left{i \mid i \in \mathcal{E} \cup \mathcal{I} \text { and } g_i(\boldsymbol{x})=0\right} .$$
The equivalence (8.6.9) holds under a number of constraint qualifications, the most used of which is the Linear Independence Constraint Qualification (LICQ) for inequality constrained optimization:
(8.6.11) $\quad\left{\nabla g_i(\boldsymbol{x}) \mid i \in \mathcal{A}(\boldsymbol{x})\right}$ is a linearly independent set.
Weaker constraint qualifications that guarantee (8.6.9) include the MangasarianFromowitz constraint qualification (MFCQ):
$\left{\nabla g_i(\boldsymbol{x}) \mid i \in \mathcal{E}\right}$ is a linearly independent set, and there is $\boldsymbol{d}$ where $\nabla g_i(\boldsymbol{x})^T \boldsymbol{d}=0$ for all $i \in \mathcal{E}$, and
$$\nabla g_i(\boldsymbol{x})^T \boldsymbol{d}>0 \text { for all } i \in \mathcal{I} \cap \mathcal{A}(\boldsymbol{x}) .$$
With a suitable constraint qualification, we can prove the existence of Lagrange multipliers satisfying the Karush-Kuhn-Tucker conditions.

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Inequality Constrained Optimization

Kuhn 和 Tucker 的工作旨在建立在 G. Dantzig 和其他人 [69] 关于线性规划的工作之上: $\min x c^T x \quad(8.6 .7)$ 受制于
$$A \boldsymbol{x} \geq \boldsymbol{b}$$

$$c^T \boldsymbol{x}-\alpha \sum i=1^m \ln \left((A x)_i-b_i\right)$$

## 数学代写|数值分析代写numerical analysis代考|Proving the Karush–Kuhn–Tucker Conditions

为了证明 Karush-Kuhn-Tucker 条件，我们需要一个约束条件来确保
$$T_{\Omega}(\boldsymbol{x})=\left{\boldsymbol{d} \mid \nabla g_i(\boldsymbol{x})^T \boldsymbol{d}=0 \text { for all } i \在 \mathcal{E} 中，\对。$$
(8.6.9) $\nabla g_i(\boldsymbol{x})^T \boldsymbol{d} \geq 0$ 对于所有 $i \in \mathcal{I}$ 其中 $\left.g_i(\boldsymbol{x })=0\right}=C_{\Omega}(\boldsymbol{x})$。

$$\mathcal{A}(\boldsymbol{x})=\left{i \mid i \in \mathcal{E} \cup \mathcal{I} \text { and } g_i(\boldsymbol{x})=0\正确的} 。$$

(8.6.11) $\quad\left{\nabla g_i(\boldsymbol{x}) \mid i \in \mathcal{A}(\boldsymbol{x})\right}$ 是线性独立集。

$\left{\nabla g_i(\boldsymbol{x}) \mid i \in \mathcal{E}\right}$ 是线性独立集，有 $\boldsymbol{d}$ 其中 $\nabla g_i( \boldsymbol{x})^T \boldsymbol{d}=0$ 对于所有 $i \in \mathcal{E}$，并且
$$\nabla g_i(\boldsymbol{x})^T \boldsymbol{d}>0 \text { for all } i \in \mathcal{I} \cap \mathcal{A}(\boldsymbol{x}) 。$$

## 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|数值分析代写numerical analysis代考|Constrained Optimization

statistics-lab™ 为您的留学生涯保驾护航 在代写数值分析numerical analysis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。我们的专家在代写数值分析numerical analysis代写方面经验极为丰富，各种代写数值分析numerical analysis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|数值分析代写numerical analysis代考|Constrained Optimization

Constrained optimization can be represented most abstractly in terms of a feasible set, often denoted $\Omega \subseteq \mathbb{R}^n$ :
(8.6.1) $\min _x f(x) \quad$ subject to $x \in \Omega$
Solutions exist if $f$ is continuous and either $\Omega$ is a compact (closed and bounded) subset of $\mathbb{R}^n$, or if $\Omega$ is closed and $f$ is coercive. Usually $\Omega$ is represented by equations and inequalities:
(8.6.2) $\Omega=\left{\boldsymbol{x} \in \mathbb{R}^n \mid g_i(\boldsymbol{x})=0\right.$ for $i \in \mathcal{E}$, and $g_i(\boldsymbol{x}) \geq 0$ for $\left.i \in \mathcal{I}\right}$.
If $\mathcal{I}$ is empty but $\mathcal{E}$ is not empty, then we say (8.6.1) is an equality constrained optimization problem. If $\mathcal{I}$ is non-empty, we say (8.6.1) is an inequality constrained optimization problem.

For a general constrained optimization problem, first-order conditions can be given in terms of the tangent cone
$(8.6 .3)$
$$T_{\Omega}(\boldsymbol{x})=\left{\lim _{k \rightarrow \infty} \frac{\boldsymbol{x}_k-\boldsymbol{x}}{t_k} \mid \boldsymbol{x}_k \in \Omega, \boldsymbol{x}_k \rightarrow \boldsymbol{x} \text { as } k \rightarrow \infty \text {, and } t_k \downarrow 0 \text { as } k \rightarrow \infty\right}$$

Lemma 8.19 If $\boldsymbol{x}=x^$ minimizes $f(x)$ over $\boldsymbol{x} \in \Omega$ and $f$ is differentiable at $\boldsymbol{x}^$, then
(8.6.4) $\nabla f\left(x^\right)^T d \geq 0 \quad$ for all $d \in T_{\Omega}\left(x^\right)$.
Proof Suppose $x=x^* \in \Omega$ minimizes $f(x)$ over $x \in \Omega$ and $f$ is differentiable. Then for any $\boldsymbol{d} \in T_{\Omega}\left(\boldsymbol{x}^\right)$, there is a sequence $\boldsymbol{x}k \rightarrow \boldsymbol{x}^$ as $k \rightarrow \infty$ with $\boldsymbol{x}_k \in \Omega$ where $\boldsymbol{d}_k:=\left(\boldsymbol{x}_k-\boldsymbol{x}^\right) / t_k \rightarrow \boldsymbol{d}$ as $k \rightarrow \infty$. Since $f\left(\boldsymbol{x}^\right) \leq f\left(\boldsymbol{x}_k\right)=f\left(\boldsymbol{x}^+t_k \boldsymbol{d}_k\right)$, $$0 \leq \lim {k \rightarrow \infty} \frac{f\left(x^+t_k \boldsymbol{d}k\right)-f\left(x^\right)}{t_k}=\nabla f\left(x^\right)^T \lim {k \rightarrow \infty} \boldsymbol{d}k=\nabla f\left(x^\right)^T \boldsymbol{d} .$$ This holds for any $d \in T{\Omega}\left(x^\right)$ showing (8.6.4), as we wanted.
Constraint qualifications relate the tangent cone $T_{\Omega}(\boldsymbol{x})$ to the linearizations of the constraint functions:
\begin{aligned} & C_{\Omega}(\boldsymbol{x})=\left{\boldsymbol{d} \in \mathbb{R}^n \mid \nabla g_i(\boldsymbol{x})^T \boldsymbol{d}=0 \text { for all } i \in \mathcal{E},\right. \ & \left.\quad \nabla g_i(\boldsymbol{x})^T \boldsymbol{d} \geq 0 \text { for all } i \in \mathcal{I} \text { where } g_i(\boldsymbol{x})=0\right} . \end{aligned}
For equality constrained optimization $(\mathcal{I}=\emptyset)$, the LICQ (8.1.2) implies that $T_{\Omega}(\boldsymbol{x})=C_{\Omega}(\boldsymbol{x})$ as noted in Section 8.1.3.

## 数学代写|数值分析代写numerical analysis代考|Equality Constrained Optimization

The theory of Section 8.1.3 for Lagrange multipliers and equality constrained optimization (8.1.5) can be immediately turned into a numerical method. To solve
\begin{aligned} & \mathbf{0}=\nabla f(\boldsymbol{x})-\sum_{i \in \mathcal{E}} \lambda_i \nabla g(\boldsymbol{x}) \ & 0=g_i(\boldsymbol{x}), \quad i \in \mathcal{E} \end{aligned}
for $(\boldsymbol{x}, \boldsymbol{\lambda})$ with $\boldsymbol{\lambda}=\left[\lambda_i \mid i \in \mathcal{E}\right]$ we can apply, for example, Newton’s method. For unconstrained optimization, we can then perform a line search to ensure that the step improves the solution estimate. The issue in constrained optimization is that $f(\boldsymbol{x})$ alone is no longer suitable for measuring improvements. Constrained optimization problems have two objectives: staying on the feasible set, and minimizing $f(\boldsymbol{x})$. It may be necessary to increase $f(\boldsymbol{x})$ in order to return to the feasible set. Solving the Newton equations for $(8.6 .5,8.6 .6)$ gives a direction $\boldsymbol{d}$. Because of the curvature of the feasible set $\Omega$ for general functions $g_i$, moving in the direction $\boldsymbol{d}$ even if $\boldsymbol{x}$ is feasible may take the point $\boldsymbol{x}+s \boldsymbol{d}$ off the feasible set. This can be offset by having a second order correction step to move back toward the feasible set. This second order correction uses a least squares version of Newton’s method to solve $g(\boldsymbol{x})=\mathbf{0}$.

Since this is an under-determined system for $|\mathcal{E}|<n$, we find the solution $\delta \boldsymbol{x}$ for $\nabla g_i(\boldsymbol{x})^T \delta \boldsymbol{x}=-g_i(\boldsymbol{x}), i \in \mathcal{E}$, that minimizes $|\delta \boldsymbol{x}|_2$, which can be done using the QR factorization of $\left[\nabla g_i(\boldsymbol{x}) \mid i \in \mathcal{E}\right]$.

For line search algorithms, we can use a merit function to determine the quality of the result of the step. Often, merit functions of the form $\boldsymbol{x} \mapsto f(\boldsymbol{x})+\alpha \sum_{i \in \mathcal{E}}\left|g_i(\boldsymbol{x})\right|$ are used where $\alpha>\max _{i \in \mathcal{E}}\left|\lambda_i\right|$. A basic method for solving equality constrained optimization problems is shown in Algorithm 82.

If the second-order correction is skipped, then the Newton method may fail to give rapid convergence, as was noted by N. Maratos in his PhD thesis [170].

# 数值分析代考

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供uconn.edu MATH3510 numerical analysis数值分析的代写代考和辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。