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

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• 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，
$$v_{j}=k_{j} \prod_{i=1}^{n} x_{i}^{k_{i j}}=k_{j} \bar{v} j,$$

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