## 数学代写|数学分析代写Mathematical Analysis代考|MATH2050

statistics-lab™ 为您的留学生涯保驾护航 在代写数学分析Mathematical 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学分析Mathematical Analysis代写方面经验极为丰富，各种代写数学分析Mathematical Analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|数学分析代写Mathematical Analysis代考|Power Series

Let $a_k, k=0,1,2, \ldots$, be a sequence of real numbers. The series of functions
$$a_0+a_1 x+a_2 x^2+\ldots+a_k x^k+\ldots$$
is called power series with coefficients $a_0, a_1, a_2, \ldots, a_k, \ldots$
A power series satisfies one of the following:
(i) the series converges only at $x=0$;
(ii) the series converges at any $x \in \mathbb{R}$;
(iii) there exists a real number $\varrho>0$ such that the series converges for $|x|<\varrho$ and does not converge for $|x|>\varrho$.

In particular, the convergence set of the power series (1.24), i.e. the set of points $x \in \mathbb{R}$ at which (1.24) converges, is an interval centred at the origin, namely: just ${0}$ in case (i), the whole $\mathbb{R}$ in case (ii), and an interval between $-\varrho, \varrho$ in case (iii). To prove these claims let us begin with the following result.

Theorem 1 If the power series (1.24) converges at some $\xi \neq 0$, it converges totally on any closed, bounded interval contained in $(-|\xi|,|\xi|)$.
Proof The convergence of the numerical series
$$a_0+a_1 \xi+a_2 \xi^2+\ldots+a_k \xi^k+\ldots$$
implies that the sequence $a_k \xi^k$ is infinitesimal as $k \rightarrow+\infty$, and hence bounded. Put equivalently, there exists $M>0$ such that
$$\left|a_k \xi^k\right| \leq M, \quad \forall k \in \mathbb{N} .$$

## 数学代写|数学分析代写Mathematical Analysis代考|Taylor Series

Let $f(x)$ be a real function defined on an interval $(a, b)$ in $\mathbb{R}$ and let $x_0 \in(a, b)$ be a point. We seek to establish whether there exists a power series centred at $x_0$ that converges on $(a, b)$ to $f$, which is usually phrased by saying that $f$ can be expanded in power series around $x_0$ on the interval $(a, b)$.
The first result in this direction goes as follows.
Theorem 1 If the power series
$$\sum_{k=0}^{\infty} a_k\left(x-x_0\right)^k$$
has convergence radius $\varrho>0$, its sum $f(x)$ is differentiable infinitely many times for $\left|x-x_0\right|<Q$. and for any $m \in \mathbb{N}$ the mth derivative equals
$$f^{(m)}(x)=\sum_{k=m}^{\infty} k(k-1) \cdots(k-m+1) a_k\left(x-x_0\right)^{k-m}$$
Furthermore, $f$ admits a series expansion of the form
$$f(x)=\sum_{k=0}^{\infty} \frac{f^{(k)}\left(x_0\right)}{k !}\left(x-x_0\right)^k$$
Proof Formula (1.36) arises by repeatedly applying Theorem 6 of the previous section (in particular, the theorem on differentiating power series). Putting $x=x_0$ in (1.36), all terms after the first vanish, and $f^{(m)}\left(x_0\right)=m ! a_m$ for every $m \in \mathbb{N}$. Substituting $a_k=f^{(k)}\left(x_0\right) / k$ ! in (1.35) gives (1.37).

By Theorem 1 we know that if $f$ can be expanded in power series around $x_0$ on $(a, b)$, then on some neighbourhood of $x_0$ inside $(a, b)$, of the form $\left|x-x_0\right|<\varrho$, we necessarily have that
(i) $f$ is differentiable infinitely many times when $\left|x-x_0\right|<\varrho$;

# 数学分析代考

## 数学代写|数学分析代写Mathematical Analysis代考|Power Series

$$a_0+a_1 x+a_2 x^2+\ldots+a_k x^k+\ldots$$

(i) 该级数仅收敛于 $x=0$;
(ii) 该系列收敛于任何 $x \in \mathbb{R}$;
(iii) 存在实数 $\varrho>0$ 使得该系列收敛于 $|x|<\varrho$ 并且不收敛于 $|x|>\varrho$.

$$a_0+a_1 \xi+a_2 \xi^2+\ldots+a_k \xi^k+\ldots$$

$$\left|a_k \xi^k\right| \leq M, \quad \forall k \in \mathbb{N}$$

## 数学代写|数学分析代写Mathematical Analysis代考|Taylor Series

$$\sum_{k=0}^{\infty} a_k\left(x-x_0\right)^k$$

$$f^{(m)}(x)=\sum_{k=m}^{\infty} k(k-1) \cdots(k-m+1) a_k\left(x-x_0\right)^{k-m}$$

$$f(x)=\sum_{k=0}^{\infty} \frac{f^{(k)}\left(x_0\right)}{k !}\left(x-x_0\right)^k$$

(i) $f$ 可微分无数次当 $\left|x-x_0\right|<\varrho$;

## 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|数学分析代写Mathematical Analysis代考|MATH2060

statistics-lab™ 为您的留学生涯保驾护航 在代写数学分析Mathematical 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学分析Mathematical Analysis代写方面经验极为丰富，各种代写数学分析Mathematical Analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|数学分析代写Mathematical Analysis代考|Uniform Convergence and Monotonicity

In this section we shall discuss two classical results regarding uniform convergence under a monotonicity hypothesis. The first theorem (by Dini) assumes monotonicity in the parameter $k$, the second one supposes monotonicity in the variable $x$.

Theorem 1 (Dini) Let $I=[a, b]$ be a closed and bounded interval and consider a sequence $f_k: I \rightarrow \mathbb{R}$ of continuous functions, monotone in $k$ (for instance, increasing: $f_k(x) \leq f_{k+1}(x)$ for any $\left.k \in \mathbb{N}, x \in I\right)$, and pointwise convergent on $[a, b]$ to some continuous function $f$. Then $f_k$ converges uniformly to $f$ on $[a, b]$.
Proof Consider, for example, an increasing sequence $f_k$, that is to say $f_k(x) \leq$ $f_{k+1}(x) \leq f(x)$ for any $k \in \mathbb{N}$ and any $x \in I=[a, b]$.

Suppose, by contradiction, that $f_k$ does not converge uniformly to $f$ on $[a, b]$. This means there exists $\varepsilon_0>0$ such that for any $v \in \mathbb{N}$ we can find $k>v$ and $x \in[a, b]$ for which
$$\left|f_k(x)-f(x)\right|=f(x)-f_k(x) \geq \varepsilon_0 .$$
Hence for any $v=h \in \mathbb{N}$, there exist $k_h \rightarrow+\infty$ and $x_h \in[a, b]$ such that
$$f\left(x_h\right)-f_{k_h}\left(x_h\right) \geq \varepsilon_0 .$$
But the monotonicity of $f_k$ in $k$ forces $f_{k_h} \geq f_i$ when $k_h \geq i$. So we obtain
$$f\left(x_h\right)-f_i\left(x_h\right) \geq \varepsilon_0, \quad \forall h \in \mathbb{N}, \quad \forall i \leq k_h .$$
The sequence $x_h$, being bounded, admits a subsequence $x_{h_j}$ converging to a point $x_0$ of the interval $[a, b]$. Taking the limit as $j \rightarrow+\infty$ in
$$f\left(x_{h_j}\right)-f_i\left(x_{h_j}\right) \geq \varepsilon_0, \quad \forall h_j \in \mathbb{N}, \quad \forall i \leq k_{h_j},$$
due to the continuity of $f$ and $f_i$ we have
$$f\left(x_0\right)-f_i\left(x_0\right) \geq \varepsilon_0 \quad \forall i \in \mathbb{N} .$$
Taking the limit when $i \rightarrow+\infty$ we reach the contradiction $0 \geq \varepsilon_0$.

## 数学代写|数学分析代写Mathematical Analysis代考|Series of Functions

If $f_k$ is a sequence of real functions defined on the subset $I$ of $\mathbb{R}$, we indicate by $s_k$ the sequence of partial sums
\begin{aligned} & s_1=f_1 \ & s_2=f_1+f_2 \ & \ldots \ldots \ldots \ & s_k=f_1+f_2+\ldots+f_k \ & \ldots \ldots \ldots \ldots \end{aligned}
The sequence of functions $s_k$ is called series (of functions) with general term $f_k$, and we shall also use for it the expression
$$f_1+f_2+\ldots+f_k+\ldots$$
If, for any $x \in I$, the numerical series with general term $f_k(x)$
$$f_1(x)+f_2(x)+\ldots+f_k(x)+\ldots$$
is convergent, i.e. if the sequence $s_k(x)$ converges (it has finite limit) for every $x \in I$, one says that the series of functions (1.18) converges pointwise on $I$.

When the sequence of functions $s_k$ converges uniformly on $I$, we say the series of functions (1.18) converges uniformly on $I$. In either case, the limit of $s_k$ as $k \rightarrow+\infty$ is called sum of the series of general term $f_k$, and we denote it by
$$\sum_{k=1}^{\infty} f_k$$
At times, (1.19) also indicates the series of general term $f_k$, apart from its sum. Often, as in the case of numerical series, one uses distinct summation indices for a series’ general term and (for example) the sequence of partial sums of a convergent series:
\begin{aligned} s_k & =\sum_{i=1}^k f_i, \quad \forall k \in \mathbb{N} \ f & =\lim {k \rightarrow+\infty} s_k=\lim {k \rightarrow+\infty} \sum_{i=1}^k f_i=\sum_{i=1}^{\infty} f_i \end{aligned}

# 数学分析代考

## 数学代写|数学分析代写Mathematical Analysis代考|Uniform Convergence and Monotonicity

$$\left|f_k(x)-f(x)\right|=f(x)-f_k(x) \geq \varepsilon_0 .$$

$$f\left(x_h\right)-f_{k_h}\left(x_h\right) \geq \varepsilon_0 .$$

$$f\left(x_h\right)-f_i\left(x_h\right) \geq \varepsilon_0, \quad \forall h \in \mathbb{N}, \quad \forall i \leq k_h .$$

$$f\left(x_{h_j}\right)-f_i\left(x_{h_j}\right) \geq \varepsilon_0, \quad \forall h_j \in \mathbb{N}, \quad \forall i \leq k_{h_j},$$

$$f\left(x_0\right)-f_i\left(x_0\right) \geq \varepsilon_0 \quad \forall i \in \mathbb{N} .$$

## 数学代写|数学分析代写Mathematical Analysis代考|Series of Functions

$$s_1=f_1 \quad s_2=f_1+f_2 \ldots \ldots . \quad s_k=f_1+f_2+\ldots+f_k \ldots \ldots \ldots \ldots$$

$$f_1+f_2+\ldots+f_k+\ldots$$

$$f_1(x)+f_2(x)+\ldots+f_k(x)+\ldots$$

$$\sum_{k=1}^{\infty} f_k$$

$$s_k=\sum_{i=1}^k f_i, \quad \forall k \in \mathbb{N} f \quad=\lim k \rightarrow+\infty s_k=\lim k \rightarrow+\infty \sum_{i=1}^k f_i=\sum_{i=1}^{\infty} f_i$$

## 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|数学分析代写Mathematical Analysis代考|MATH307

statistics-lab™ 为您的留学生涯保驾护航 在代写数学分析Mathematical 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学分析Mathematical Analysis代写方面经验极为丰富，各种代写数学分析Mathematical Analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|数学分析代写Mathematical Analysis代考|Sequences of Functions: Pointwise and Uniform Convergence

Let $I$ be a set of real numbers and $f_k: I \rightarrow \mathbb{R}$ a sequence of real functions defined on $I$. One says that $f_k$ converges to the function $f: I \rightarrow \mathbb{R}$ pointwise on $I$ whenever
$$\lim {k \rightarrow+\infty} f_k(x)=f(x), \quad \forall x \in I .$$ In other words, if for any $\varepsilon>0$ and any $x \in I$ there exists $v{\varepsilon, x} \in \mathbb{N}$ such that
$$\left|f_k(x)-f(x)\right|<\varepsilon, \quad \forall k>v_{\varepsilon, x} .$$
In general, given $\varepsilon>0$, the number $v_{\varepsilon, x}$ depends on the point $x$; if, instead, this number is independent of $x$, one speaks of uniform convergence.

Precisely, we say that $f_k$ converges uniformly on $I$ to $f$ if, for any $\varepsilon>0$, there exists $v_{\varepsilon} \in \mathbb{N}$ such that
$$\left|f_k(x)-f(x)\right|<\varepsilon, \quad \forall k>v_{\varepsilon}, \quad \forall x \in I .$$
Equivalently, $f_k$ converges uniformly on $I$ to $f$ if, for any $\varepsilon>0$, there exists $v_{\varepsilon} \in \mathbb{N}$ such that
$$\sup \left{\left|f_k(x)-f(x)\right|: x \in I\right}<\varepsilon, \quad \forall k>v_{\varepsilon} .$$
Another way to express the same is saying that $f_k$ converges uniformly on $I$ to $f$ if the following condition on the limit of a numerical sequence holds
$$\lim _{k \rightarrow+\infty} \sup \left{\left|f_k(x)-f(x)\right|: x \in I\right}=0 .$$

## 数学代写|数学分析代写Mathematical Analysis代考|First Theorems on Uniform Convergence

Let us start by describing the continuity property of the uniform limit of continuous functions. Suppose $f_k: I \subseteq \mathbb{R} \rightarrow \mathbb{R}$ is a sequence of continuous functions on the subset $I$ of $\mathbb{R}$, and assume that $f_k$ converges uniformly on $I$ to the function $f: I \rightarrow \mathbb{R}$; we shall prove that $f$ is continuous on $I$. Observe that this result does not hold if we only assume that $f_k$ converges to $f$ pointwise. This is what happens, for instance, in Example 2 of the previous section, where the discontinuous function (1.2) is the pointwise limit of the sequence of continuous functions (1.1).
Theorem (Continuity of Limits) Let $f_k: I \subseteq \mathbb{R} \rightarrow \mathbb{R}$ be a sequence of continuous functions that converges uniformly on I to the function $f$. Then $f$ is continuous.

Proof Let us verify that $f$ is continuous at $x_0$, for any given $x_0 \in I$. By the uniform convergence hypothesis, given $\varepsilon>0$, there exists $v$ such that
$$\left|f_k(x)-f(x)\right|<\varepsilon, \quad \forall k>v, \quad \forall x \in I .$$
Let us choose $k_0>v$; then clearly for any $x \in I$ we have
\begin{aligned} \left|f(x)-f\left(x_0\right)\right| & \leq\left|f(x)-f_{k_0}(x)\right|+\left|f_{k_0}(x)-f_{k_0}\left(x_0\right)\right|+\left|f_{k_0}\left(x_0\right)-f\left(x_0\right)\right| \leq \ & <\varepsilon+\left|f_{k_0}(x)-f_{k_0}\left(x_0\right)\right|+\varepsilon . \end{aligned} Because of the continuity of $f_{k_0}$ it is possible to find $\delta>0$ such that
$$x \in I, \quad\left|x-x_0\right|<\delta \Rightarrow\left|f_{k_0}(x)-f_{k_0}\left(x_0\right)\right|<\varepsilon$$
and so for $x \in I,\left|x-x_0\right|<\delta$, we obtain
$$\left|f(x)-f\left(x_0\right)\right|<3 \varepsilon .$$
More generally, we have the following result.

## 数学代写|数学分析代写Mathematical Analysis代考|Sequences of Functions: Pointwise and Uniform Convergence

$$\lim k \rightarrow+\infty f_k(x)=f(x), \quad \forall x \in I .$$

$$\left|f_k(x)-f(x)\right|<\varepsilon, \quad \forall k>v_{\varepsilon, x} .$$

$$\left|f_k(x)-f(x)\right|<\varepsilon, \quad \forall k>v_{\varepsilon}, \quad \forall x \in I .$$

## 数学代写|数学分析代写Mathematical Analysis代考|First Theorems on Uniform Convergence

$$\left|f_k(x)-f(x)\right|<\varepsilon, \quad \forall k>v, \quad \forall x \in I .$$

$$\left|f(x)-f\left(x_0\right)\right| \leq\left|f(x)-f_{k_0}(x)\right|+\left|f_{k_0}(x)-f_{k_0}\left(x_0\right)\right|+\left|f_{k_0}\left(x_0\right)-f\left(x_0\right)\right| \leq \quad<\varepsilon+\mid f_{k_0}$$ 因为连续性 $f_{k_0}$ 有可能找到 $\delta>0$ 这样
$$x \in I, \quad\left|x-x_0\right|<\delta \Rightarrow\left|f_{k_0}(x)-f_{k_0}\left(x_0\right)\right|<\varepsilon$$

$$\left|f(x)-f\left(x_0\right)\right|<3 \varepsilon .$$

## 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|数学分析代写Mathematical Analysis代考|MATH2241

statistics-lab™ 为您的留学生涯保驾护航 在代写数学分析Mathematical 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学分析Mathematical Analysis代写方面经验极为丰富，各种代写数学分析Mathematical Analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|数学分析代写Mathematical Analysis代考|Solving the Constraint Equations

The constraint equations Eq. (14), may be concisely written as
$$\mathbf{U}{\mathbf{A}}=\mathbf{h} .$$ As a rule, the system is underdetermined due to a large number of elementary subnetworks and limited data on reaction rates known from experiments or literature that can be used in formulating constraint equations. Consequently, Eq. (16) is not expected to provide a unique solution and a suitable solution needs to be selected by utilizing an optimization procedure with an appropriate objective function. A clue to defining an objective function is readily provided by employing the stability analysis of stoichiometric networks as outlined in Sect. 2. More specifically, for chemical oscillators the emergence of oscillations via Hopf bifurcation is implied by dominance of the chosen (leading) unstable subnetwork. Therefore we can postulate that the contributions of the elementary subnetworks other than the leading unstable subnetwork should be as small as possible at the oscillatory instability. Thus the objective function to be minimized may be taken as the sum of the contributions of all subnetworks involved in the constraint equations other than the unstable dominant one, whose contribution is used as a free bifurcation parameter, which is varied until a Hopf bifurcation is found. Since the constraint equations are constructed to be linear, a linear programming solver [14] was used for solving the constrained system Eq. (16) by minimizing $$f(\mathbf{a})=\sum{k=1}^{p} \alpha_{k}^{u v} .$$
In general, the set of all admissible solutions of Eq. (16) with non-negative components of $\mathbf{a}$ is restricted to a set which may be a convex bounded polytope or a convex unbounded polytope, which arises by shifting the non-negative cone (if it exists) of the homogeneous subsystem of Eq. (16) in the space of a due to $\mathbf{b}$ and has a set of apexes in some directions but extends without bounds in other directions. The minimal solution sits in one of the apexes.

## 数学代写|数学分析代写Mathematical Analysis代考|Discussion and Conclusions

The approach outlined above has been applied to the glucose oxidase-catalase reaction [11] and the Belousov-Zhabotinsky reaction [15]. However, main applications are expected in identifying kinetic parameters in models of biological oscillating systems, such as circadian clocks [7]. Also, when temperature dependence of the rate coefficients is of interest, the input experimental information at two (or more) different temperatures needs to be provided and results subsequently fitted to Arrhenius law.

There are certain caveats that must be taken care of to obtain the solution of Eq. (16). Some of the parameters $\mathbf{x}^{f v}$ and $\mathbf{k}^{f v}$ that are not available from measurements must be assigned fixed values chosen heuristically. It may happen that such a choice violates solvability of the system Eq. (16). In this case, an effective way of resolving the problem is to find incompatible constraint equations and remove them. Likewise, some constraint equations may be linear combinations of others, which causes the linear programming solver to fail. Both incompatible and linearly dependent equations can be removed by applying singular value decomposition [14]. Another limitation is the linearity of constraint equations. In future work a nonlinear constrained optimization [1] should be considered.

## 数学代写|数学分析代写Mathematical Analysis代考|Solving the Constraint Equations

$$\mathbf{U A}=\mathbf{h} .$$

$$f(\mathbf{a})=\sum k=1^{p} \alpha_{k}^{u v} .$$

## 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|数学分析代写Mathematical Analysis代考|MATH2050C

statistics-lab™ 为您的留学生涯保驾护航 在代写数学分析Mathematical 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学分析Mathematical Analysis代写方面经验极为丰富，各种代写数学分析Mathematical Analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|数学分析代写Mathematical Analysis代考|Determination of Parameters

Our major goal is to use the SNA approach to networks to estimate unknown parameters in a system, for which we have or assume a detailed mechanism implying power law kinetics, and are able to perform experiments leading the emergence of oscillations via Hopf bifurcation. Typically, some of the rate coefficients are known from previous research, thus our aim is to determine a subset of rate coefficients. Parameter determination can be based on choosing an appropriate subset of Eq. (6) with $\mathbf{v}{s}=\mathbf{v}\left(\mathbf{x}{s}\right)$ consistent with our experiments and already known parameters. These equations can be used for finding various unknown quantities including rate coefficients and steady state concentrations, given that other quantities are available, such as rate coefficients known from independent experiments or taken from literature, experimentally measured steady state concentrations, known inflow rate and inflow concentrations at a distinct dynamical instability. The major instabilities are: (i) emergence of oscillations at a Hopf hifurcation or (ii) switch to another steady state at a saddle-node bifurcation. To preserve linearity, we choose a subset of rate equations such that, upon substituting the known quantities, the rate expressions are either linear in a particular unknown or fully determined. We call such equations constraint equations.

In order to have a compact form of the constraint equations, let us arrange the order of elementary subnetworks in $\mathbf{E}$, the order of species in $\mathbf{x}{s}$ and the order of reactions in $\mathbf{v}{s}$ and $\mathbf{k}$ as follows (below we drop the subscript $s$ so that $\mathbf{x}$ and $\mathbf{v}$ now denote steady state quantities):
$$\boldsymbol{\alpha}=\left(\boldsymbol{\alpha}^{f v}, \boldsymbol{\alpha}^{u v}\right), \mathbf{x}=\left(\mathbf{x}^{f v}, \mathbf{x}^{u v}, \mathbf{x}^{i v}\right), \mathbf{k}=\left(\mathbf{k}^{f v}, \mathbf{k}^{u v}, \mathbf{k}^{i v}\right), \mathbf{v}=\left(\mathbf{v}^{f v}, \mathbf{v}^{u v}, \mathbf{v}^{i v}\right)$$

## 数学代写|数学分析代写Mathematical Analysis代考|Formulation of the Constraint Equations

After identifying fixed, unknown and implied quantities we can finally set up the constraint equations by selecting certain equations from Eq. (6) and rearranging them to obtain linear equations in a standard matrix form. First, we express $\mathbf{E}$ in terms of three blocks containing edge(s) involved in the unstable dominant subnetwork, nondominant subnetworks not to be used in the constraint equations and the remaining edges that will be part of the constraints, respectively:
$$\mathbf{E}=\left[\mathbf{E}^{u d s}, \mathbf{E}^{n d s}, \mathbf{E}^{u v}\right]$$
Equation (6) can be then rewritten as
$$\mathbf{E}^{u d s} \boldsymbol{\alpha}^{u d s}+\mathbf{E}^{n d s} \boldsymbol{\alpha}^{n d s}+\mathbf{E}^{u v} \boldsymbol{\alpha}^{u v}=\mathbf{v} .$$
Depending on the available input data, we choose constraint equations based on those rates $v_{j}$ which are either known entirely or expressed as a linear function of either $k_{j}^{u v}$ or $x_{i}^{u v}$. As a result, the remaining equations are all included in the term $\mathbf{E}^{n d s} \boldsymbol{\alpha}^{n d s}$ representing edges that have no contribution to the selected $v_{j}$ ‘s and are removed. The constraint equations then read
$$\hat{\mathbf{E}}^{u v} \boldsymbol{\alpha}^{u v}=\hat{\mathbf{v}}-\hat{\mathbf{E}}^{u d s} \boldsymbol{\alpha}^{u d s},$$
where $\hat{\mathbf{v}}=\left(\mathbf{v}^{f v}, \mathbf{v}^{u v k}, \mathbf{v}^{u v x}\right)$ is the set of rates used as constraints and $\hat{\mathbf{E}}^{u v}, \hat{\mathbf{E}}^{u d s}$ retain only rows corresponding to $\hat{\mathbf{v}}$. Since $\mathbf{v}^{u v k}$ depends linearly on $\mathbf{k}^{u v}$ and $\mathbf{v}^{u v x}$ on $\mathbf{x}^{u v}$, these terms can be moved to the 1.h.s. of Eq.

## 数学代写|数学分析代写Mathematical Analysis代考|Determination of Parameters

$$\boldsymbol{\alpha}=\left(\boldsymbol{\alpha}^{f v}, \boldsymbol{\alpha}^{u v}\right), \mathbf{x}=\left(\mathbf{x}^{f v}, \mathbf{x}^{u v}, \mathbf{x}^{i v}\right), \mathbf{k}=\left(\mathbf{k}^{f v}, \mathbf{k}^{u v}, \mathbf{k}^{i v}\right), \mathbf{v}=\left(\mathbf{v}^{f v}, \mathbf{v}^{u v}, \mathbf{v}^{i v}\right)$$

## 数学代写|数学分析代写Mathematical Analysis代考|Formulation of the Constraint Equations

$$\mathbf{E}=\left[\mathbf{E}^{u d s}, \mathbf{E}^{n d s}, \mathbf{E}^{u v}\right]$$

$$\mathbf{E}^{u d s} \boldsymbol{\alpha}^{u d s}+\mathbf{E}^{n d s} \boldsymbol{\alpha}^{n d s}+\mathbf{E}^{u v} \boldsymbol{\alpha}^{u v}=\mathbf{v}$$

$$\hat{\mathbf{E}}^{u v} \boldsymbol{\alpha}^{u v}=\hat{\mathbf{v}}-\hat{\mathbf{E}}^{u d s} \boldsymbol{\alpha}^{u d s},$$ 于 $\mathbf{k}^{u v}$ 和 $\mathbf{v}^{u v x}$ 上 $\mathbf{x}^{u v}$ ，这些项可以移至方程式的 1.hs。

## 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|数学分析代写Mathematical Analysis代考|MATH2060B

statistics-lab™ 为您的留学生涯保驾护航 在代写数学分析Mathematical 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学分析Mathematical Analysis代写方面经验极为丰富，各种代写数学分析Mathematical Analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|数学分析代写Mathematical Analysis代考|Theoretical Part

All spatially homogeneous isothermal chemical oscillators are based on stoichiometry and kinetics and fall within the formal mathematical description given below.

Let us assume a system involving $m$ reactions and a total number of species $n^{\text {tot }}$,
$$v_{1 j}^{L} A_{1}+\cdots+v_{n^{t a t} j}^{L} A_{n^{t o \omega}} \rightarrow v_{1 j}^{R} A_{1}+\cdots+v_{n^{t o j} j}^{R} A_{n^{t a t}}, j=1, \ldots, m,$$
where $\mathrm{A}{i}$ are the reacting species and $v{i j}^{L}, v_{i j}^{R}$ are left and right stoichiometric coefficients. Any reversible reaction is treated as a pair of forward and backward steps. In a spatially homogeneous system, such as a flow-through reactor, dynamics of $n \leq n^{t o t}$ species that are not inert products or in a pool condition are governed by a set of coupled mass balance equations which have the following pseudolinear form:
$$\frac{\mathrm{d} \mathbf{x}}{\mathrm{d} t}=\mathbf{N} \mathbf{v}(\mathbf{x}),$$
where $\mathbf{x}=\left(x_{1}, \ldots, x_{n}\right)$ is the vector of concentrations of the interacting dynamical species, $\mathbf{N}=\left{\Delta v_{i j}\right}=\left{v_{i j}^{R}-v_{i j}^{L}\right}$ is the $(n \times m)$ stoichiometric matrix and $\mathbf{v}=\left(v_{1}, \ldots, v_{m}\right)$ is the non-negative vector of reaction rates (fluxes) (All vectors are assumed being column vectors). The reaction rates are assumed to follow mass action kinetics,
$$v_{j}=k_{j} \prod_{i=1}^{n} x_{i}^{k_{i j}}=k_{j} \bar{v}{j},$$ where $\kappa{i j}=\partial \ln v_{j} / \partial \ln x_{i} \geq 0$ is the reaction order of species $i$ in reaction $j$ and $k_{j}$ is the corresponding rate coefficient, which may include fixed concentration(s) of pooled species and $\bar{v}_{j}$ is the reduced reaction rate.

## 数学代写|数学分析代写Mathematical Analysis代考|Identification of Dominant Subnetworks

As mentioned above, the instability induced by a negative principal minor reflects the susceptibility of the subnetwork to possessing an unstable steady state provided that the corresponding steady state concentrations are sufficiently small. Although there are special cases when more subtle criteria have to be applied to indicate oscillatory instability, $[4,6]$, generally the outlined features provide excellent guidelines in evaluating the potential of a reaction network to undergo a dynamical instability. A Hopf bifurcation represents the emergence of oscillations [10], which is of primary importance in this work.

When applying the SNA to oscillatory mechanisms of inorganic reactions that were discovered since the pioneering work of Belousov and Zhabotinsky [19], is has been found [5] that dominant subnetworks forming the core oscillator have only a few topological arrangements of their networks, which are called prototypes or motifs. They all possess an autocatalytic cycle, i.e. a cycle connecting species (denoted as type $\mathrm{X}$ ) of which at least one has a stoichiometric overproduction. In addition, there is a negative feedback loop involving a noncyclic species (denoted as type $\mathrm{Z}$ ) and a removal of a type $X$ species either by decomposition or via reaction with an inhibitory species (denoted as type Y).

However, many biochemical oscillators do not possess an autocatalytic cycle. Instead, their core oscillator possesses two type X-like species competing for a type Y-like species. In addition, there is a negative feedback loop involving type $\mathrm{Z}$ species, but all cycles present in the network are “ordinary” or nonautocatalytic cycles that do not provide for stoichiometric overproduction. Yet the network admits an instability leading to oscillations. Such a feature is called competitive autocatalysis. As with the cyclic autocatalysis, only a few basic motifs are expected to constitute dominant subnetworks of many biochemical networks.

## 数学代写|数学分析代写Mathematical Analysis代考|Theoretical Part

$$v_{1 j}^{L} A_{1}+\cdots+v_{n^{t a t} j}^{L} A_{n^{t \sigma \omega}} \rightarrow v_{1 j}^{R} A_{1}+\cdots+v_{n^{t o j} j}^{R} A_{n^{\text {tat }}}, j=1, \ldots, m,$$

$$\frac{\mathrm{d} \mathbf{x}}{\mathrm{d} t}=\mathbf{N v}(\mathbf{x})$$

$\mathbf{v}=\left(v_{1}, \ldots, v_{m}\right)$ 是反应速率 (通量) 的非负向量 (假设所有向量都是列向量)。假设反应速率遵循质量作用 动力学，
$$v_{j}=k_{j} \prod_{i=1}^{n} x_{i}^{k_{i j}}=k_{j} \bar{v} j,$$

## 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|数学分析代写Mathematical Analysis代考|MATH 212

statistics-lab™ 为您的留学生涯保驾护航 在代写数学分析Mathematical 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学分析Mathematical Analysis代写方面经验极为丰富，各种代写数学分析Mathematical Analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|数学分析代写Mathematical Analysis代考|Theoretical Part

All spatially homogeneous isothermal chemical oscillators are based on stoichiometry and kinetics and fall within the formal mathematical description given below.

Let us assume a system involving $m$ reactions and a total number of species $n^{\text {tot }}$,
$$v_{1 j}^{L} A_{1}+\cdots+v_{n^{t o t} j}^{L} A_{n^{t o x}} \rightarrow v_{1 j}^{R} A_{1}+\cdots+v_{n^{t o t} j}^{R} A_{n^{\text {tot }}}, j=1, \ldots, m,$$
where $\mathrm{A}{i}$ are the reacting species and $v{i j}^{L}, v_{i j}^{R}$ are left and right stoichiometric coefficients. Any reversible reaction is treated as a pair of forward and backward steps. In a spatially homogeneous system, such as a flow-through reactor, dynamics of $n \leq n^{\text {tot }}$ species that are not inert products or in a pool condition are governed by a set of coupled mass balance equations which have the following pseudolinear form:
$$\frac{\mathrm{d} \mathbf{x}}{\mathrm{d} t}=\mathbf{N} \mathbf{v}(\mathbf{x})$$
where $\mathbf{x}=\left(x_{1}, \ldots, x_{n}\right)$ is the vector of concentrations of the interacting dynamical species, $\mathbf{N}=\left{\Delta v_{i j}\right}=\left{v_{i j}^{R}-v_{i j}^{L}\right}$ is the $(n \times m)$ stoichiometric matrix and $\mathbf{v}=\left(v_{1}, \ldots, v_{m}\right)$ is the non-negative vector of reaction rates (fluxes) (All vectors are assumed being column vectors). The reaction rates are assumed to follow mass action kinetics,
$$v_{j}=k_{j} \prod_{i=1}^{n} x_{i}^{\kappa i j}=k_{j} \bar{v}{j}$$ where $\kappa{i j}=\partial \ln v_{j} / \partial \ln x_{i} \geq 0$ is the reaction order of species $i$ in reaction $j$ and $k_{j}$ is the corresponding rate coefficient, which may include fixed concentration(s) of pooled species and $\bar{v}{j}$ is the reduced reaction rate. In vector notation we have $\mathbf{k}=$ $\left(k{1}, \ldots, k_{m}\right)$ and $\overline{\mathbf{v}}(x)=\left(\bar{v}{1}, \ldots, \bar{v}{m}\right)$. For elementary reactions, $\kappa_{i j}=v_{i j}^{L}$. However; in general case power law terms may also be used for quasi-elementary steps with $\kappa_{i j} \neq v_{i j}^{L}$. The kinetic matrix $\left{\kappa_{i j}\right}$ is denoted as $\mathbf{K}$. In flow systems, the inflows and outflows are included as pseudoreactions of zeroth and first order, respectively; the rate coefficient corresponding to an inflow term ia $k_{j} \quad k_{0} x_{i}{ }^{i}$ and that for añ outfow is $k_{j}=k_{0}$, where $k_{0}$ is the flow rate and $x_{i 0}$ is the feed concentration of any inflowing species $i$.

## 数学代写|数学分析代写Mathematical Analysis代考|Identification of Dominant Subnetworks

As mentioned above, the instability induced by a negative principal minor reflects the susceptibility of the subnetwork to possessing an unstable steady state provided that the corresponding steady state concentrations are sufficiently small. Although there are special cases when more subtle criteria have to be applied to indicate oscillatory instability, $[4,6]$, generally the outlined features provide excellent guidelines in evaluating the potential of a reaction network to undergo a dynamical instability. A Hopf bifurcation represents the emergence of oscillations $\lfloor 10\rfloor$, which is of primary importance in this work.

When applying the SNA to oscillatory mechanisms of inorganic reactions that were discovered since the pioneering work of Belousov and Zhabotinsky [19], is has been found [5] that dominant subnetworks forming the core oscillator have only a few topological arrangements of their networks, which are called prototypes or motifs. They all possess an autocatalytic cycle, i.e. a cycle connecting species (denoted as type $\mathrm{X}$ ) of which at least one has a stoichiometric overproduction. In addition, there is a negative feedback loop involving a noncyclic species (denoted as type Z) and a removal of a type $\mathrm{X}$ species either by decomposition or via reaction with an inhibitory species (denoted as type Y).

However, many biochemical oscillators do not possess an autocatalytic cycle. Instead, their core oscillator possesses two type X-like species competing for a type Y-like species. In addition, there is a negative feedback loop involving type $\mathrm{Z}$ species, but all cycles present in the network are “ordinary” or nonautocatalytic cycles that do nnt provide for ntniehinmetris nverproduetion. Yot the network ndmitn n inntnhility leading to oscillations. Such a feature is called competitive autocatalysis. As with the cyclic autocatalysis, only a few basic motifs are expected to constitute dominant subnetworks of many biochemical networks.

## 数学代写|数学分析代写Mathematical Analysis代考|Theoretical Part

$$\frac{\mathrm{d} \mathbf{x}}{\mathrm{d} t}=\mathbf{N} \mathbf{v}(\mathbf{x})$$

$\mathbf{v}=\left(v_{1}, \ldots, v_{m}\right)$ 是反应速率 (通量) 的非负向量 (假设所有向量都是列向量)。假设反应速率遵循质量作用 动力学，
$$v_{j}=k_{j} \prod_{i=1}^{n} x_{i}^{\kappa i j}=k_{j} \bar{v} j$$

Veft {রkappa_{i j}\right} 表示为 $\mathbf{K}$. 在流动系统中，流入和流出分别作为零级和一级伪反应包括在内；流入项 ia 对 应的比率系数 $k_{j} \quad k_{0} x_{i}{ }^{i}$ 而对于 añ outfow 是 $k_{j}=k_{0}$ ，在哪里 $k_{0}$ 是流速和 $x_{i 0}$ 是任何流入物质的进料浓度 $i$.

## 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|数学分析代写Mathematical Analysis代考|MATH2241

statistics-lab™ 为您的留学生涯保驾护航 在代写数学分析Mathematical 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学分析Mathematical Analysis代写方面经验极为丰富，各种代写数学分析Mathematical Analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|数学分析代写Mathematical Analysis代考|Properties of the Switching Function

An analysis of the function $L(t)$ leads to the validity of the following lemma.
Lemma 2 There is such a value $t_{0} \in[0, T)$ that on the interval $\left(t_{0}, T\right]$ the switching function $L(t)$ is negative.

Proof The functions $\psi_{1}^{}(t), \psi_{2}^{}(t), \psi_{3}^{}(t)$, as the components of the absolutely continuous solution $\psi_{}(t)$ to system (5), are absolutely continuous as well. Hence, by formula (7), the switching function $L(t)$ is also absolutely continuous, and therefore a continuous function. Due to formula (7) and the initial conditions of system (5), it takes the negative value at $t=T$ :
$$L(T)=-(\beta+\delta)<0 .$$

Then, the stability of the sign of the continuous function $L(t)$ yields the required fact. This completes the proof.
Corollary 1 From Lemma 2 and formula (6), it follows the relationship:
$$v_{}(t)=v_{\min }, \quad t \in\left(t_{0}, T\right] .$$ Now, we introduce positive constants: $$\alpha=\gamma_{2}^{-1}\left((\beta+\delta) \gamma_{1}-\delta \gamma_{2}\right), \quad \varepsilon=\alpha(\lambda-v)+\delta(\lambda-\mu)$$ and then also the following functions: \begin{aligned} g_{11}(t)=& v_{}(t)\left(\delta m_{}(t)+\beta l_{}(t)\right)+v \ g_{21}(t)=& v_{}(t) m_{}(t)\left(\gamma_{1}\left(\delta k_{}(t)-(\beta+\delta) l_{}(t)\right)+\delta(\mu-v)\right) \ g_{22}(t)=&(\lambda-\mu) \varepsilon^{-1} \gamma_{1}\left(\delta k_{}(t)-(\beta+\delta) l_{}(t)\right) \ &+\varepsilon^{-1}(\alpha(\lambda-v) \lambda+\delta(\lambda-\mu)(\lambda-v+\mu)) \ g_{31}(t)=& \gamma_{1} v_{}(t) m_{}(t), \quad g_{32}(t)=(\lambda-\mu) \gamma_{1} \varepsilon^{-1} \ g_{33}(t)=&\left(\gamma_{1} k_{}(t)-\gamma_{2} l_{}(t)\right)-(\lambda-\mu) \varepsilon^{-1} \gamma_{1}\left(\delta k_{}(t)-(\beta+\delta) l_{}(t)\right) \ &+\varepsilon^{-1}(\alpha(\lambda-v) \mu+\delta(\lambda-\mu) v) \end{aligned}

## 数学代写|数学分析代写Mathematical Analysis代考|Numerical Results

Further, only numerical investigation of optimal control $v_{*}(t)$ is possible. For the corresponding numerical calculations, the following values of the parameters and initial conditions of system (1) were used [3,11], as well as the control constraints (2):
$$\begin{array}{llll} \sigma=15.0 & \rho=3.6 & \beta=0.4 & \delta=0.005 \ \mu=0.01 & v=0.02 & \gamma_{1}=0.8 & \gamma_{2}=0.05 \ l_{0}=100.0 & k_{0}=40.0 & m_{0}=50.0 & \ v_{\min }=0.3 & T=100.0 & & \end{array}$$
The numerical calculations were carried out using the software “BOCOP 2.0.5” (see [1]), and are shown in Figs. 1 and $2 .$

In Fig. 3 the surface $\Phi(l, k)$ is presented. It can be seen that positive values of the function $\Phi(l, k)$ in formula (10) in the region of variation of the variables $l$ and $k$ provide the admissibility of singular control $v_{\text {sing }}^{*}(t)$.

Physical optimal control $\tilde{v}{}(t)$ according to Figs. 1 and 2 describes the situation when, first there is the period of the psoriasis treatment with greatest intensity. Next, it is followed by the period of the treatment with a smooth decrease in the dose of the used medication from the greatest intensity to the lower intensity. Then, there is a period of the psoriasis treatment with lower intensity, and finally the switching occurs to the period of the treatment with greatest intensity. Also, we emphasize that in all performed numerical calculations, the optimal concentration of keratinocytes $k{}(t)$ decreases to the end $T$ to the level that is the minimal for the entire period $[0, T]$ of the psoriasis treatment (see Figs. 1 and 2 ).

## 数学代写|数学分析代写Mathematical Analysis代考|Properties of the Switching Function

(7) ，切换函数 $L(t)$ 也是绝对连续的，因此是连续函数。由于公式 (7) 和系统 (5) 的初始条件，它在 $t=T$
$$L(T)=-(\beta+\delta)<0 .$$

$$v(t)=v_{\min }, \quad t \in\left(t_{0}, T\right] .$$

$$\alpha=\gamma_{2}^{-1}\left((\beta+\delta) \gamma_{1}-\delta \gamma_{2}\right), \quad \varepsilon=\alpha(\lambda-v)+\delta(\lambda-\mu)$$

$$g_{11}(t)=v(t)(\delta m(t)+\beta l(t))+v g_{21}(t)=\quad v(t) m(t)\left(\gamma_{1}(\delta k(t)-(\beta+\delta) l(t))+\delta(\mu-v)\right) g_{22}(t)$$

## 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|数学分析代写Mathematical Analysis代考|MATH2050C

statistics-lab™ 为您的留学生涯保驾护航 在代写数学分析Mathematical 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学分析Mathematical Analysis代写方面经验极为丰富，各种代写数学分析Mathematical Analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|数学分析代写Mathematical Analysis代考|Optimal Control Problem

On a given time interval $[0, T]$ we consider the nonlinear control system of differential equations:
$$\left{\begin{array}{l} l^{\prime}(t)=\sigma-\delta v(t) l(t) m(t)-\gamma_{1} l(t) k(t)-\mu l(t) \ k^{\prime}(t)=(\beta+\delta) v(t) l(t) m(t)+\gamma_{2} l(t) k(t)-\lambda k(t) \ m^{\prime}(t)=\rho-\beta v(t) l(t) m(t)-v m(t) \ l(0)=l_{0}, k(0)=k_{0}, m(0)=m_{0} ; l_{0}, k_{0}, m_{0}>0 \end{array}\right.$$

It describes the interactions of various types of cells in a human body with drug therapy of psoriasis $[3,10,11]$. In system $(1), l(t), k(t)$, and $m(t)$ are the concentrations of T-lymphocytes, keratinocytes and dendritic cells; $l_{0}, k_{0}, m_{0}$ are their initial conditions, respectively. The values $\sigma, \rho, \mu, \lambda, v, \gamma_{1}, \gamma_{2}, \delta, \beta$ are the given positive parameters of this system, which have the following meaning. The values $\sigma$ and $\rho$ are the appropriate inflow rates of T-lymphocytes and dendritic cells, $\mu$ and $v$ are the removal rates of these cells, respectively; $\lambda$ is the decay rate of keratinocytes. In addition, the rate of activation of keratinocytes due to T-lymphocytes is indicated by $\gamma_{1}$ and the rate of keratinocytes growth is denoted by $\gamma_{2}$. The value $\delta$ is the activation rate of T-lymphocytes by dendritic cells, $\beta$ is conversely the activation rate of dendritic cells due to T-lymphocytes. The interactions between T-lymphocytes and dendritic cells help to form keratinocytes through some cell biological procedures and thus the concentrations of both T-lymphocytes and dendritic cells are reduced by the terms $\delta v l m$ and $\beta v l m$, respectively. On the other hand, under mixing homogeneity, the combined interaction of T-lymphocytes and dendritic cells contributes to the growth of concentration of epidermal keratinocytes by the term $(\beta+\delta) \mathrm{vlm}$. Model (1) was kindly provided for analysis by Professor P. K. Roy (Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata, India).
In system $(1), v(t)$ is a control function that satisfies the constraints:
$$0<v_{\min } \leq v(t) \leq 1 .$$

## 数学代写|数学分析代写Mathematical Analysis代考|Pontryagin Maximum Principle

In order to analyze the optimal control $v_{}(t)$ and the corresponding optimal solution $\left(l_{}(t), k_{}(t), m_{}(t)\right)$, we apply the Pontryagin maximum principle [9]. Firstly, we write down the Hamiltonian
\begin{aligned} H\left(l, k, m, v, \psi_{1}, \psi_{2}, \psi_{3}\right)=&\left(\sigma-\delta v l m-\gamma_{1} l k-\mu l\right) \psi_{1} \ &+\left((\beta+\delta) v l m+\gamma_{2} l k-\lambda k\right) \psi_{2}+(\rho-\beta v l m-v m) \psi_{3} \end{aligned}
where $\psi_{1}, \psi_{2}, \psi_{3}$ are adjoint variables.
Secondly, we calculate the required partial derivatives:
\begin{aligned} &H_{l}^{\prime}\left(l, k, m, v, \psi_{1}, \psi_{2}, \psi_{3}\right)=v m\left(-\delta \psi_{1}+(\beta+\delta) \psi_{2}-\beta \psi_{3}\right) \ &\quad+k\left(\gamma_{2} \psi_{2}-\gamma_{1} \psi_{1}\right)-\mu \psi_{1} \ &H_{k}^{\prime}\left(l, k, m, v, \psi_{1}, \psi_{2}, \psi_{3}\right)=l\left(\gamma_{2} \psi_{2}-\gamma_{1} \psi_{1}\right)-\lambda \psi_{2} \ &H_{m}^{\prime}\left(l, k, m, v, \psi_{1}, \psi_{2}, \psi_{3}\right)=v l\left(-\delta \psi_{1}+(\beta+\delta) \psi_{2}-\beta \psi_{3}\right)-v \psi_{3}, \ &H_{v}^{\prime}\left(l, k, m, v, \psi_{1}, \psi_{2}, \psi_{3}\right)=\operatorname{lm}\left(-\delta \psi_{1}+(\beta+\delta) \psi_{2}-\beta \psi_{3}\right) \end{aligned}

Then, in accordance with the Pontryagin maximum principle, for the optimal control $v_{}(t)$ and the optimal solution $\left(l_{}(t), k_{}(t), m_{}(t)\right)$ there exists a vector-function $\psi_{}(t)=\left(\psi_{1}^{}(t), \psi_{2}^{}(t), \psi_{3}^{}(t)\right)$ such that:

• $\psi_{}(t)$ is a nontrivial solution of the adjoint system: \left{\begin{aligned} \psi_{1}^{ \prime}(t)=&-v_{}(t) m_{}(t)\left(-\delta \psi_{1}^{}(t)+(\beta+\delta) \psi_{2}^{}(t)-\beta \psi_{3}^{}(t)\right) \ &-k_{}(t)\left(\gamma_{2} \psi_{2}^{}(t)-\gamma_{1} \psi_{1}^{}(t)\right)+\mu \psi_{1}^{}(t), \ \psi_{2}^{ \prime}(t)=&-l_{}(t)\left(\gamma_{2} \psi_{2}^{}(t)-\gamma_{1} \psi_{1}^{}(t)\right)+\lambda \psi_{2}^{}(t), \ \psi_{3}^{* \prime}(t)=&-v_{}(t) l_{}(t)\left(-\delta \psi_{1}^{}(t)+(\beta+\delta) \psi_{2}^{}(t)-\beta \psi_{3}^{}(t)\right)+v \psi_{3}^{}(t), \ \psi_{1}^{}(T)=0, \psi_{2}^{}(T)=-1, \psi_{3}^{*}(T)=0 \end{aligned}\right.

## 数学代写|数学分析代写Mathematical Analysis代考|Optimal Control Problem

$\$ \$$Veft$$
l^{\prime}(t)=\sigma-\delta v(t) l(t) m(t)-\gamma_{1} l(t) k(t)-\mu l(t) k^{\prime}(t)=(\beta+\delta) v(t) l(t) m(t)+\gamma_{2} l(t) k(t)-\lambda k(t) m^{\prime}(t)
$$【正确的。 \ \$$

$$0<v_{\min } \leq v(t) \leq 1$$

## 数学代写|数学分析代写Mathematical Analysis代考|Pontryagin Maximum Principle

$$H\left(l, k, m, v, \psi_{1}, \psi_{2}, \psi_{3}\right)=\left(\sigma-\delta v l m-\gamma_{1} l k-\mu l\right) \psi_{1} \quad+\left((\beta+\delta) v l m+\gamma_{2} l k-\lambda k\right) \psi_{2}+(\rho-\beta v$$

$$H_{l}^{\prime}\left(l, k, m, v, \psi_{1}, \psi_{2}, \psi_{3}\right)=v m\left(-\delta \psi_{1}+(\beta+\delta) \psi_{2}-\beta \psi_{3}\right) \quad+k\left(\gamma_{2} \psi_{2}-\gamma_{1} \psi_{1}\right)-\mu \psi_{1} H_{k}^{\prime}(l,$$

• $\psi(t)$ 是伴随系统的非平凡解: $\$ \$V$ left {
$$\psi_{1}^{\prime}(t)=-v(t) m(t)\left(-\delta \psi_{1}(t)+(\beta+\delta) \psi_{2}(t)-\beta \psi_{3}(t)\right) \quad-k(t)\left(\gamma_{2} \psi_{2}(t)-\gamma_{1} \psi_{1}(t)\right)+\mu \psi_{1}$$
【正确的。
$\$ \

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