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

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• 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 $i$.

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。