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数学代写|数论代写Number Theory代考|Factoring Primes in a Monogenic Number Field

Let $K$ be an algebraic number field. Recall that $K$ is said to be monogenic (Definition 7.1.5) if there exists $\theta \in O_K$ such that
$$O_K=\mathbb{Z}+\mathbb{Z} \theta+\cdots+\mathbb{Z} \theta^{n-1},$$
where $[K: \mathbb{Q}]=n$. The next theorem shows how to factor $\langle p\rangle$ (with $p$ a rational prime) into prime ideals in a monogenic number field. It was originally proved by Dedekind [3] in 1878.

Theorem 10.3.1 Let $K=\mathbb{Q}(\theta)$ be an algebraic number field of degree $n$ such that
$$O_K=\mathbb{Z}+\mathbb{Z} \theta+\cdots+\mathbb{Z} \theta^{n-1} .$$
Let $p$ be a rational prime. Let
$$f(x)=\operatorname{irr}_{\mathbb{Q}} \theta \in \mathbb{Z}[x] .$$
Let ${ }^{-}$denote the natural map $: \mathbb{Z}[x] \rightarrow \mathbb{Z}_p[x]$, where $\mathbb{Z}_p=\mathbb{Z} / p \mathbb{Z}$. Let
$$\bar{f}(x)=g_1(x)^{e_1} \cdots g_r(x)^{e_r},$$
where $g_1(x), \ldots, g_r(x)$ are distinct monic irreducible polynomials in $\mathbb{Z}_p[x]$ and $e_1, \ldots, e_r$ are positive integers. For $i=1,2, \ldots, r$ let $f_i(x)$ be any monic polynomial of $\mathbb{Z}[x]$ such that $\bar{f}_i=g_i$. Set
$$P_i=\left\langle p, f_i(\theta)\right\rangle, i=1,2, \ldots, r .$$
Then $P_1, \ldots, P_r$ are distinct prime ideals of $O_K$ with
$$\langle p\rangle=P_1^{e_1} \cdots P_r^{e_r}$$
and
$$N\left(P_i\right)=p^{\operatorname{deg} f_i}, i=1,2, \ldots, r .$$

数学代写|数论代写Number Theory代考|Some Factorizations in Cubic Fields

Example 10.4.1 We factor $\langle 5\rangle$ as a product of prime ideals in $O_K$, where $K=$ $\mathbb{Q}(\sqrt[3]{2})$. Set $\theta=\sqrt[3]{2}$. We have seen in Example 7.1.6 that $\left{1, \theta, \theta^2\right}$ is an integral basis for $K=\mathbb{Q}(\theta)$ so that $K$ is monogenic. The minimal polynomial of $\theta$ over $\mathbb{Q}$ is $x^3-2$. We have
$$x^3-2=(x+2)\left(x^2+3 x+4\right)(\bmod 5),$$
where $x+2$ and $x^2+3 x+4$ are irreducible $(\bmod 5)$. Hence, by Theorem 10.3.1, we have
$$\langle 5\rangle=P Q,$$
where
$$P=\langle 5, \theta+2\rangle, Q=\left\langle 5, \theta^2+3 \theta+4\right\rangle$$
are distinct prime ideals with
$$N(P)=5, N(Q)=5^2=25 .$$
As a check on the calculation in Example 10.4.1 we compute $P Q$ directly.

We have
\begin{aligned} P Q & =\langle 5, \theta+2\rangle\left\langle 5, \theta^2+3 \theta+4\right\rangle \ & =\left\langle 25,5(\theta+2), 5\left(\theta^2+3 \theta+4\right), \theta^3+5 \theta^2+10 \theta+8\right\rangle \ & =\left\langle 25,5(\theta+2), 5\left(\theta^2+3 \theta+4\right), 5 \theta^2+10 \theta+10\right\rangle \ & =\langle 5\rangle\left\langle 5, \theta+2, \theta^2+3 \theta+4, \theta^2+2 \theta+2\right\rangle \ & =\langle 5\rangle \end{aligned}
as
$$1=1 \cdot 5+(2 \theta+2)(\theta+2)-2\left(\theta^2+3 \theta+4\right) .$$

数学代写|数论代写Number Theory代考|Factoring Primes in a Monogenic Number Field

$$O_K=\mathbb{Z}+\mathbb{Z} \theta+\cdots+\mathbb{Z} \theta^{n-1},$$

$$O_K=\mathbb{Z}+\mathbb{Z} \theta+\cdots+\mathbb{Z} \theta^{n-1} .$$

$$f(x)=\operatorname{irr}_{\mathbb{Q}} \theta \in \mathbb{Z}[x] .$$

$$\bar{f}(x)=g_1(x)^{e_1} \cdots g_r(x)^{e_r},$$

$$P_i=\left\langle p, f_i(\theta)\right\rangle, i=1,2, \ldots, r .$$

$$\langle p\rangle=P_1^{e_1} \cdots P_r^{e_r}$$

$$N\left(P_i\right)=p^{\operatorname{deg} f_i}, i=1,2, \ldots, r .$$

数学代写|数论代写Number Theory代考|Some Factorizations in Cubic Fields

\begin{aligned} \operatorname{ord}_P(\langle\gamma\rangle+A B) & =\min \left(\operatorname{ord}_P(\langle\gamma\rangle), \operatorname{ord}_P(A B)\right) \ & =\min \left(\operatorname{ord}_P(\gamma), 0\right) \ & =0 \ & =\operatorname{ord}_P(A) \end{aligned}

数学代写|数论代写Number Theory代考|Norm of a Fractional Ideal

$$A=\frac{1}{\alpha} I .$$

$$N(A)=\frac{N(I)}{N(\langle\alpha\rangle)},$$

$$A=\frac{1}{\alpha} I=\frac{1}{\beta} J$$

$$\beta I=\alpha J,$$

$$\langle\beta\rangle I=\langle\alpha\rangle J,$$

$$N(\langle\beta\rangle) N(I)=N(\langle\beta\rangle I)=N(\langle\alpha\rangle J)=N(\langle\alpha\rangle) N(J),$$

$$\frac{N(I)}{N(\langle\alpha\rangle)}=\frac{N(J)}{N(\langle\beta\rangle)} .$$

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。

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数学代写|随机过程代写Stochastic Porcess代考|Non-ageing property of geometric distribution

For a geometric r.v. $X$, we have
\begin{aligned} \operatorname{Pr}{X & =s+r \mid X \geq s}=\frac{q^{s+r} p}{q^s} \ & =q^r p=\operatorname{Pr}{X=r} . \end{aligned}
This property, called non-ageing (or memoryless) property, characterizes geometric distribution among all distributions of discrete non-negative integral r.v.’s.
Note: If $Y_1, Y_2, \ldots$, is a sequence of independent Bernoulli r.v.s, then
$$X_i=\min \left{i, Y_i=1\right}-1 \text { is a geometric r.v. }$$
Example 1(e). Logarithmic Series Distribution:
The r.v. $X$ has logarithmic series distribution if
\begin{aligned} & p_k=\operatorname{Pr}{X=k}=\frac{\alpha q^k}{k}, k=1,2,3, \ldots \ & \alpha=-1 /(\log p) \ & 0<q=1-p<1 . \end{aligned}
The p.g.f. of $X$ is
\begin{aligned} P(s) & =\sum_{k=1}^{\infty} p_k s^k=\sum_{k=1}^{\infty} \frac{\alpha q^k}{k} s^k \ & =-\alpha \log (1-s q)=\frac{\log (1-s q)}{\log (1-q)} . \end{aligned}

数学代写|随机过程代写Stochastic Porcess代考|Determination of $\left{p_k\right}$ from a given $P(s)$

From the above examples, we see how a single generating function $P(s)$ may be used to represent a whole set of probabilities
$$p_k=\operatorname{Pr}{X=k}, \quad k=0,1,2, \ldots$$
In these examples we were concerned with the problem of finding $P(s)$ for a given set of $p_k$ ‘s. In many cases the reverse problem arises: to determine $p_k$ from a given p.g.f. $P(s)$. Many situations arise, where it is easier to find the p.g.f. $P(s)$ of a variable rather than the probability distribution $\left{p_k\right}$ of the variable. One proceeds to find first the p.g.f. $P(s)$ and then to find the probability $p_k$ from the function $P(s)$. Even without finding the $p_k$ ‘s one can find the moments of the distribution from $P(s)$.
Again, $p_k$ can be (uniquely) determined from $P(s)$ as follows:
$p_k$ can be found from $P(s)$ by applying (1.2), i.e.
$$p_k=\frac{1}{k !}\left[\frac{d^k P(s)}{d s^k}\right]_{s=0} ;$$
$p_k$ is also given by the coefficient of $s^k$ in the expansion of $P(s)$ as a power series in $s$.
When $P(s)$ is of the form $P(s)=U(s) / V(s)$, it may be convenient to expand $P(s)$ in a power series in $s$ first by decomposing $P(s)$ into partial factions. Suppose that $s_1, \ldots, s_r$ are the distinct roots of $V(s)$, i.e. $V(s)=\left(s-s_1\right) \ldots\left(s-s_r\right)$ apart from a constant factor $c$, which, for simplicity, we take to be equal to $1)$, then $P(s)$ can be decomposed into partial fractions as
$$P(s)=\frac{a_1}{s_1-s}+\cdots+\frac{a_r}{s_r-s},$$
where $a_i$ ‘s can be determined. It may be verified that
$$a_i=-U\left(s_i\right) / V^{\prime}\left(s_i\right) .$$

数学代写|随机过程代写Stochastic Porcess代考|Non-ageing property of geometric distribution

\begin{aligned} \operatorname{Pr}{X & =s+r \mid X \geq s}=\frac{q^{s+r} p}{q^s} \ & =q^r p=\operatorname{Pr}{X=r} . \end{aligned}

$$X_i=\min \left{i, Y_i=1\right}-1 \text { is a geometric r.v. }$$

r.v. $X$具有对数级数分布
\begin{aligned} & p_k=\operatorname{Pr}{X=k}=\frac{\alpha q^k}{k}, k=1,2,3, \ldots \ & \alpha=-1 /(\log p) \ & 0<q=1-p<1 . \end{aligned}
$X$的p.g.f.是
\begin{aligned} P(s) & =\sum_{k=1}^{\infty} p_k s^k=\sum_{k=1}^{\infty} \frac{\alpha q^k}{k} s^k \ & =-\alpha \log (1-s q)=\frac{\log (1-s q)}{\log (1-q)} . \end{aligned}

数学代写|随机过程代写Stochastic Porcess代考|Determination of $\left{p_k\right}$ from a given $P(s)$

$$p_k=\operatorname{Pr}{X=k}, \quad k=0,1,2, \ldots$$

$$p_k=\frac{1}{k !}\left[\frac{d^k P(s)}{d s^k}\right]_{s=0} ;$$
$p_k$也由$P(s)$展开为$s$的幂级数时的$s^k$的系数给出。

$$P(s)=\frac{a_1}{s_1-s}+\cdots+\frac{a_r}{s_r-s},$$

$$a_i=-U\left(s_i\right) / V^{\prime}\left(s_i\right) .$$

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。

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数学代写|随机过程代写Stochastic Porcess代考|Processes with continuous time

Let $\xi(t), t \geqslant 0$, be a homogeneous Markov branching process with continuous time (parameter). Let $\mathscr{X}$ denote as above the phase space of the process $\xi(t)$ which is an $m$-dimensional lattice of vectors with non-negative integer-valued components. The transition probabilities of the process $\xi(t)$ will be denoted by $p_t(x, y)$ and, as before, in place of $p_t\left(e_i, y\right), p_t\left(x, e_j\right)$ and $p_t\left(e_i, e_j\right)$ we shall write $p_t(i, y), p_t(x, j), p_t(i, j)$ respectively. We shall assume that the transition probabilities satisfy the condition
$$\lim {t \downarrow 0} p_t(x, y)=\delta(x, y) .$$ First we shall discuss Kolmogorov’s differential equations for branching processes. In accordance with the general theory of homogeneous Markov processes the limits $$\lim {t \downarrow 0} \frac{p_t(x, y)-\delta(x, y)}{t}=q(x, y)$$
exist. We shall consider only regular branching processes, i. e. we shall assume that the conditions
$$-q(x, y)<\infty, \quad \sum_{y \in \mathcal{X}} q(x, y)=0$$
are satisfied.

数学代写|随机过程代写Stochastic Porcess代考|Moments (continuous time)

Assume that
$$\sum_{x \in \mathscr{X}} q(i, x) x^k=\alpha_i^k \neq \infty \quad(k, i=1, \ldots, m) .$$
Since the functions $Q(i, w)$ are analytic in the domain $|w|<1$, one can differentiate equations (32) in this domain. We then obtain:
$$\frac{d a_j^k(t, w)}{d t}=\sum_{r=1}^m Q_j^r\left(g_t(w)\right) a_r^k(t, w), \quad a_j^k(0, w)=\delta_j^k,$$
where
$$Q_j^k(t, w)=\frac{\partial Q(j, w)}{\partial w_k}=\sum_{x \in \mathscr{X}} q(j, x) x^k w^{x-e_k}$$
and
$$a_j^k(t, w)=\frac{\partial g_t(j, w)}{\partial w_k} .$$
Assume that the components of the vector $w$ are positive and $w_k \uparrow 1$. Then, by Lebesgue’s theorem,
$$\lim {w \uparrow 1} a_j^k(t, w)=\lim {w \uparrow 1} \mathrm{E}j \xi^k(t) w^{\xi(t)}=\mathrm{E}_j \xi^k(t)=a_j^k(t),$$ and, by Dini’s theorem, $g_t(w) \rightarrow 1$ uniformly in $t$. Approaching the limit as $w \uparrow 1$ in $$a_j^k(t, w)=\delta_j^k+\int_0^t \sum{r=1}^m Q_j^r\left(g_s(w)\right) a_r^k(s, w) d s,$$
we obtain
$$a_j^k(t)=\delta_j^k+\int_0^t \sum_{r=1}^m \alpha_j^r a_r^k(s) d s .$$

数学代写|随机过程代写Stochastic Porcess代考|Processes with continuous time

$$\lim {t \downarrow 0} p_t(x, y)=\delta(x, y) .$$首先，我们将讨论柯尔莫哥洛夫分支过程的微分方程。根据齐次马尔可夫过程的一般理论，极限$$\lim {t \downarrow 0} \frac{p_t(x, y)-\delta(x, y)}{t}=q(x, y)$$

$$-q(x, y)<\infty, \quad \sum_{y \in \mathcal{X}} q(x, y)=0$$

数学代写|随机过程代写Stochastic Porcess代考|Moments (continuous time)

$$\sum_{x \in \mathscr{X}} q(i, x) x^k=\alpha_i^k \neq \infty \quad(k, i=1, \ldots, m) .$$

$$\frac{d a_j^k(t, w)}{d t}=\sum_{r=1}^m Q_j^r\left(g_t(w)\right) a_r^k(t, w), \quad a_j^k(0, w)=\delta_j^k,$$

$$Q_j^k(t, w)=\frac{\partial Q(j, w)}{\partial w_k}=\sum_{x \in \mathscr{X}} q(j, x) x^k w^{x-e_k}$$

$$a_j^k(t, w)=\frac{\partial g_t(j, w)}{\partial w_k} .$$

$$\lim {w \uparrow 1} a_j^k(t, w)=\lim {w \uparrow 1} \mathrm{E}j \xi^k(t) w^{\xi(t)}=\mathrm{E}j \xi^k(t)=a_j^k(t),$$，根据迪尼定理，$g_t(w) \rightarrow 1$均匀分布于$t$。接近$$a_j^k(t, w)=\delta_j^k+\int_0^t \sum{r=1}^m Q_j^r\left(g_s(w)\right) a_r^k(s, w) d s,$$中的$w \uparrow 1$的极限 我们得到 $$a_j^k(t)=\delta_j^k+\int_0^t \sum{r=1}^m \alpha_j^r a_r^k(s) d s .$$

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。

Posted on Categories:Stochastic Porcesses, 数学代写, 随机过程

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数学代写|随机过程代写Stochastic Porcess代考|Finite-Dimensional Homogeneous Processes with Independent Increments

In this section we shall discuss homogeneous processes with independent increments with values in $\mathscr{R}^m$. The characteristic function of such a process is of the form
\begin{aligned} \mathrm{E} e^{i(z, \xi(t))} & =\exp {t K(z)} \ & =\exp \left{t\left[i(a, z)-\frac{1}{2}(B z, z)+\int\left(e^{i(z, x)}-1-\frac{i(z, x)}{1+|x|^2}\right) \Pi(d x)\right]\right}, \end{aligned}
where $z \in \mathscr{R}^m$ and $(z, y)$ is the scalar product in $\mathscr{R}^m$. In formula (1) $a \in \mathscr{R}^m, B$ is a non-negative symmetric linear operator in $\mathscr{R}^m$, and the measure $\Pi$ is defined on Borel sets and is such that
$$\int \frac{|x|^2}{1+|x|^2} \Pi(d x)<\infty .$$
Analogously to the one-dimensional case the function $K(z)$ appearing in (1) is called the cumulant of the process; it completely determines the marginal distributions of the processes. The processes under consideration are assumed to be separable and thus have no discontinuities of the second kind. Sample functions of the processes are assumed to be continuous from the right.

One can associate uniquely a homogeneous Markov process $\left{\mathscr{F}, \mathscr{N}, \mathrm{P}_x\right}$ with a homogeneous process with independent increments $\xi(t)$. This Markov process is of the form: the set of functions of the type $x_t=\xi(s+t)-\xi(s)+x$, $s \geqslant 0, x \in \mathscr{R}^m$, where $\xi(\cdot)$ are various sample functions of the process $\xi(t)$, is chosen as the set $\mathscr{F}$; the set $\mathscr{N}$ is defined in the usual manner as the minimal $\sigma$-algebra containing all the cylinders in $\mathscr{F}$. For each cylinder $A$ we have
$$\mathrm{P}x{A}=\mathrm{P}{x+\xi(\cdot) \in A}$$ (the probability on the r.h.s. is defined on the same probability space on which the process $\xi(t)$ is defined). The process is homogeneous Markov in view of the relation \begin{aligned} \mathrm{P}{x+\xi(t+s) \in A \mid \xi(u), u \leqslant s} & =\mathrm{P}{x+\xi(s)+\xi(t+s)-\xi(s) \in A \mid \xi(u), u \leqslant s} \ & =\mathrm{P}{y+\xi(t+s)-\xi(s) \in A}{y=x+\xi(s)} \ & =\mathrm{P}{y+\xi(t) \in A}_{y=x+\xi(s)}=\mathrm{P}_{x(s)}{x(t) \in A} . \end{aligned}

数学代写|随机过程代写Stochastic Porcess代考|Resolvent, characteristic and generating operators

Resolvent, characteristic and generating operators. Consider a resolvent of a Markov process associated with a process with independent increments (hereafter we shall refer to it as the resolvent of the process $\xi(t))$. Formula (3) implies
$$\mathbf{R}\lambda f(x)=\int_0^{\infty} e^{-\lambda t} \mathrm{E} f(x+\xi(t)) d t .$$ Let $F(t, A)=\mathrm{P}{\xi(t) \in A}$. where $$\mathbf{R}\lambda f(x)=\frac{1}{\lambda} \int f(x+y) F_\lambda(d y),$$
$$F_\lambda(A)=\lambda \int_0^{\infty} e^{-\lambda t} F(t, A) d t .$$
The function $F_\lambda(A)$ can be conveniently defined by means of the Fourier transform
$$\Phi_\lambda(z)=\int e^{i(z, y)} F_\lambda(d y) .$$
Utilizing (5) we obtain
$$\Phi_\lambda(z)=\lambda \int_0^{\infty} e^{-\lambda t} e^{t K(z)} d t=\frac{\lambda}{\lambda-K(z)} .$$

Analogously to the one-dimensional case, it follows that $\Phi_\lambda(z)$, for $\lambda>0$, is the characteristic function of an infinitely divisible distribution since
$$\Phi_\lambda(z)=\exp \left{\int_0^{\infty} e^{-\lambda t} \frac{e^{t \mathbf{K}(z)}-1}{t} d t\right},$$
so that
$$\Phi_\lambda(z)=\lim {\varepsilon \rightarrow 0} \exp \left{\int{\varepsilon}^{\infty} e^{-\lambda t} \frac{e^{t K(z)}-1}{t} d t\right}=\lim {\varepsilon \rightarrow 0} \exp \left{\int{\varepsilon}^{\infty} \frac{e^{-\lambda t} e^{t K(z)}}{t} d t-\int_{\varepsilon}^{\infty} \frac{e^{-\lambda t}}{t} d t\right} ;$$
the function $\int_{\varepsilon}^{\infty}\left(e^{-\lambda t} e^{t K(z)} / t\right) d t$ is positive definite since $e^{t K(z)}$ is such a function. The compound function $\exp {\Phi(z)-\Phi(a)}$, where $\Phi(z)$ is positive definite, is infinitely divisible and finally the limit of infinitely divisible functions is also infinitely divisible. The infinite divisability of $\Phi_\lambda(z)$ implies the existence of $a_\lambda$, $B_\lambda$ and $\Pi_\lambda$ such that
$$\Phi_\lambda(z)=\exp \left{K_\lambda(z)\right},$$
where
$$K_\lambda(z)=i\left(a_\lambda z\right)-\frac{1}{2}\left(B_\lambda z, z\right)+\int\left(e^{i(z, x)}-1-\frac{i(z, x)}{1+|x|^2}\right) \Pi_\lambda(d x) .$$

数学代写|随机过程代写Stochastic Porcess代考|Finite-Dimensional Homogeneous Processes with Independent Increments

\begin{aligned} \mathrm{E} e^{i(z, \xi(t))} & =\exp {t K(z)} \ & =\exp \left{t\left[i(a, z)-\frac{1}{2}(B z, z)+\int\left(e^{i(z, x)}-1-\frac{i(z, x)}{1+|x|^2}\right) \Pi(d x)\right]\right}, \end{aligned}

$$\int \frac{|x|^2}{1+|x|^2} \Pi(d x)<\infty .$$

数学代写|随机过程代写Stochastic Porcess代考|Resolvent, characteristic and generating operators

$$\mathbf{R}\lambda f(x)=\int_0^{\infty} e^{-\lambda t} \mathrm{E} f(x+\xi(t)) d t .$$让$F(t, A)=\mathrm{P}{\xi(t) \in A}$。在哪里$$\mathbf{R}\lambda f(x)=\frac{1}{\lambda} \int f(x+y) F_\lambda(d y),$$
$$F_\lambda(A)=\lambda \int_0^{\infty} e^{-\lambda t} F(t, A) d t .$$

$$\Phi_\lambda(z)=\int e^{i(z, y)} F_\lambda(d y) .$$

$$\Phi_\lambda(z)=\lambda \int_0^{\infty} e^{-\lambda t} e^{t K(z)} d t=\frac{\lambda}{\lambda-K(z)} .$$

$$\Phi_\lambda(z)=\exp \left{\int_0^{\infty} e^{-\lambda t} \frac{e^{t \mathbf{K}(z)}-1}{t} d t\right},$$

$$\Phi_\lambda(z)=\lim {\varepsilon \rightarrow 0} \exp \left{\int{\varepsilon}^{\infty} e^{-\lambda t} \frac{e^{t K(z)}-1}{t} d t\right}=\lim {\varepsilon \rightarrow 0} \exp \left{\int{\varepsilon}^{\infty} \frac{e^{-\lambda t} e^{t K(z)}}{t} d t-\int_{\varepsilon}^{\infty} \frac{e^{-\lambda t}}{t} d t\right} ;$$

$$\Phi_\lambda(z)=\exp \left{K_\lambda(z)\right},$$

$$K_\lambda(z)=i\left(a_\lambda z\right)-\frac{1}{2}\left(B_\lambda z, z\right)+\int\left(e^{i(z, x)}-1-\frac{i(z, x)}{1+|x|^2}\right) \Pi_\lambda(d x) .$$

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。

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数学代写|交换代数代写Commutative Algebra代考|Basic definitions

Suppose $(M,+)$ is an abelian group. For any $m \in M$ and any integer $n$, one can make sense of $n \bullet m$. If $n$ is a positive integer, this means $m+\cdots+m$ ( $n$ times); if $n=0$ it means 0 , and if $n$ is negative, then $n \bullet m=-(-n) \bullet m$. Thus we have defined a function $\bullet: \mathbb{Z} \times M \rightarrow M$ which enjoys the following properties: for all $n, n_1, n_2 \in \mathbb{Z}, m, m_1, m_2 \in M$, we have
(ZMOD1) $1 \bullet m=m$.
(ZMOD2) $n \bullet\left(m_1+m_2\right)=n \bullet m_1+n \bullet m_2$.
(ZMOD3) $\left(n_1+n_2\right) \bullet m=n_1 \bullet m+n_2 \bullet m$.
(ZMOD4) $\left(n_1 n_2\right) \bullet m=n_1 \bullet\left(n_2 \bullet m\right)$
It should be clear that this is some kind of ring-theoretic analogue of a group action on a set. In fact, consider the slightly more general construction of a monoid $(M, \cdot)$ acting on a set $S$ : that is, for all $n_1, n_2 \in M$ and $s \in S$, we require $1 \bullet s=s$ and $\left(n_1 n_2\right) \bullet s=n_1 \bullet\left(n_2 \bullet s\right)$.

For a group action $G$ on $S$, each function $g \bullet: S \rightarrow S$ is a bijection. For monoidal actions, this need not hold for all elements: e.g. taking the natural multiplication action of $M=(\mathbb{Z}, \cdot)$ on $S=\mathbb{Z}$, we find that $0 \bullet: \mathbb{Z} \rightarrow{0}$ is neither injective nor surjective, $\pm 1 \bullet: \mathbb{Z} \rightarrow \mathbb{Z}$ is bijective, and for $|n|>1, n \bullet: \mathbb{Z} \rightarrow \mathbb{Z}$ is injective but not surjective.

数学代写|交换代数代写Commutative Algebra代考|Finitely presented modules

One of the major differences between abelian groups and nonabelian groups is that a subgroup $N$ of a finitely generated abelian group $M$ remains finitely generated, and indeed, the minimal number of generators of the subgroup $N$ cannot exceed the minimal number of generators of $M$, whereas this is not true for nonabelian groups: e.g. the free group of rank 2 has as subgroups free groups of every rank $0 \leq r \leq \aleph_0$. (For instance, the commutator subgroup is not finitely generated.)
Since an abelian group is a $\mathbb{Z}$-module and every $R$-module has an underlying abelian group structure, one might well expect the situation for $R$-modules to be similar to that of abelian groups. We will see later that this is true in many but not all cases: an $R$-module is called Noetherian if all of its submodules are finitely generated. Certainly a Noetherian module is itself finitely generated. The basic fact here which we will prove in $\S 8.7$ – is a partial converse: if the ring $R$ is Noetherian, any finitely generated $R$-module is Noetherian. Note that we can already see that the Noetherianity of $R$ is necessary: if $R$ is not Noetherian, then by definition there exists an ideal $I$ of $R$ which is not finitely generated, and this is nothing else than a non-finitely generated $R$-submodule of $R$ (which is itself generated by the single element 1.) Thus the aforementioned fact about subgroups of finitely generated abelian groups being finitely generated holds because $\mathbb{Z}$ is a Noetherian ring.
When $R$ is not Noetherian, it becomes necessary to impose stronger conditions than finite generation on modules. One such condition indeed comes from group theory: recall that a group $G$ is finitely presented if it is isomorphic to the quotient of a finitely generated free group $F$ by the least normal subgroup $N$ generated by a finite subset $x_1, \ldots, x_m$ of $F$.

数学代写|交换代数代写Commutative Algebra代考|Basic definitions

(zmod1) $1 \bullet m=m$．
(zmod2) $n \bullet\left(m_1+m_2\right)=n \bullet m_1+n \bullet m_2$．
(zmod3) $\left(n_1+n_2\right) \bullet m=n_1 \bullet m+n_2 \bullet m$．
(zmod4) $\left(n_1 n_2\right) \bullet m=n_1 \bullet\left(n_2 \bullet m\right)$

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。

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数学代写|交换代数代写Commutative Algebra代考|The basic formalism

Let $(X, \leq)$ be a partially ordered set. We denote by $X^{\vee}$ the order dual of $X$ : it has the same underlying set as $X$ but the inverse order relation: $x \preceq y \Longleftrightarrow y \leq x$.
Let $(X, \leq)$ and $(Y, \leq)$ be partially ordered sets. A map $f: X \rightarrow Y$ is isotone (or order-preserving) if for all $x_1, x_2 \in X, x_1 \leq x_2 \Longrightarrow f\left(x_1\right) \leq f\left(x_2\right) ; f$ is antitone (or order-reversing) if for all $x_1, x_2 \in X, x_1 \leq x_2 \Longrightarrow f\left(x_1\right) \geq f\left(x_2\right)$.
Exercise 2.1. Let $X, Y, Z$ be partially ordered sets, and let $f: X \rightarrow Y, g$ : $Y \rightarrow Z$ be functions. Show:
a) If $f$ and $g$ are isotone, then $g \circ f$ is isotone.
b) If $f$ and $g$ are antitone, then $g \circ f$ is isotone.
c) If one of $f$ and $g$ is isotone and the other is antitone, then $g \circ f$ is antitone.
Let $(X, \leq)$ and $(Y, \leq)$ be partially ordered sets. An antitone Galois connection between $\mathbf{X}$ and $\mathbf{Y}$ is a pair of maps $\Phi: X \rightarrow Y$ and $\Psi: Y \rightarrow X$ such that:
(GC1) $\Phi$ and $\Psi$ are both antitone maps, and
(GC2) For all $x \in X$ and all $y \in Y, x \leq \Psi(y) \Longleftrightarrow y \leq \Phi(x)$.
There is a pleasant symmetry in the definition: if $(\Phi, \Psi)$ is a Galois connection between $X$ and $Y$, then $(\Psi, \Phi)$ is a Galois connection between $Y$ and $X$.

If $(X, \leq)$ is a partially ordered set, then a mapping $f: X \rightarrow X$ is called a closure operator if it satisfies all of the following properties:
(C1) For all $x \in X, x \leq f(x)$.
(C2) For all $x_1, x_2 \in X, x_1 \leq x_2 \Longrightarrow f\left(x_1\right) \leq f\left(x_2\right)$.
(C3) For all $x \in X, f(f(x))=f(x)$.

数学代写|交换代数代写Commutative Algebra代考|Lattice properties

Recall that a partially ordered set $X$ is a lattice if for all $x_1, x_2 \in X$, there is a greatest lower bound $x_1 \wedge x_2$ and a least upper bound $x_1 \vee x_2$. A partially ordered set is a complete lattice if for every subset $A$ of $X$, the greatest lower bound $\wedge A$ and the least upper bound $\bigvee A$ both exist.
Lemma 2.4. Let $(X, Y, \Phi, \Psi)$ be a Galois connection.
a) If $X$ and $Y$ are both lattices, then for all $x_1, x_2 \in X$,
\begin{aligned} & \Phi\left(x_1 \wedge x_2\right)=\Phi\left(x_1\right) \vee \Phi\left(x_2\right), \ & \Phi\left(x_2 \vee x_2\right)=\Phi\left(x_1\right) \wedge \Phi\left(x_2\right) . \end{aligned}
b) If $X$ and $Y$ are both complete lattices, then for all subsets $A \subset X$,
\begin{aligned} & \Phi(\bigwedge A)=\bigvee \Phi(A), \ & \Phi(\bigvee A)=\bigwedge \Phi(A) . \end{aligned}
Exercise 2.2. Prove Lemma 2.4.
Complete lattices also intervene in this subject in the following way.
Proposition 2.5. Let $A$ be a set and let $X=\left(2^A, \subset\right)$ be the power set of $A$, partially ordered by inclusion. Let $c: X \rightarrow X$ be a closure operator. Then the collection $c(X)$ of closed subsets of $A$ forms a complete lattice, with $\wedge S=\bigcap_{B \in S} B$ and $\bigvee S=c\left(\bigcup_{B \in S} B\right)$

数学代写|交换代数代写Commutative Algebra代考|The basic formalism

a)如果 $f$ 和 $g$ 是等音的吗 $g \circ f$ 是等音的。
b)如果 $f$ 和 $g$ 是反调吗 $g \circ f$ 是等音的。
c)如果其中之一 $f$ 和 $g$ 一个是等音，另一个是反音 $g \circ f$ 是反调。

(gc1) $\Phi$ 和 $\Psi$ 两者都是反调地图吗
(GC2)对所有人 $x \in X$ 等等 $y \in Y, x \leq \Psi(y) \Longleftrightarrow y \leq \Phi(x)$．

(C1)所有人$x \in X, x \leq f(x)$。
(C2)所有人$x_1, x_2 \in X, x_1 \leq x_2 \Longrightarrow f\left(x_1\right) \leq f\left(x_2\right)$。
(C3)对所有人$x \in X, f(f(x))=f(x)$。

数学代写|交换代数代写Commutative Algebra代考|Lattice properties

a)如果$X$和$Y$都是格，则对于所有$x_1, x_2 \in X$，
\begin{aligned} & \Phi\left(x_1 \wedge x_2\right)=\Phi\left(x_1\right) \vee \Phi\left(x_2\right), \ & \Phi\left(x_2 \vee x_2\right)=\Phi\left(x_1\right) \wedge \Phi\left(x_2\right) . \end{aligned}
b)如果$X$和$Y$都是完全格，则对于所有子集$A \subset X$，
\begin{aligned} & \Phi(\bigwedge A)=\bigvee \Phi(A), \ & \Phi(\bigvee A)=\bigwedge \Phi(A) . \end{aligned}

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。

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数学代写|交换代数代写Commutative Algebra代考|The Category of Finitely Presented Modules

The category of finitely presented modules over A can be constructed from the category of free modules of finite rank over $\mathbf{A}$ by a purely categorical procedure.

1. A finitely presented module $M$ is described by a triplet
$$\left(\mathrm{K}_M, \mathrm{G}_M, \mathrm{~A}_M\right),$$
where $\mathrm{A}_M$ is a linear map between the free modules of finite ranks $\mathrm{K}_M$ and $\mathrm{G}_M$. We have $M \simeq$ Coker $\mathrm{A}_M$ and $\pi_M: \mathrm{G}_M \rightarrow M$ is the surjective linear map with kernel $\operatorname{Im} \mathrm{A}_M$. The matrix of the linear map $\mathrm{A}_M$ is a presentation matrix of $M$.
2. A linear map $\varphi$ of the module $M$ (described by $\left(\mathrm{K}M, \mathrm{G}_M, \mathrm{~A}_M\right)$ ) to the module $N$ (described by $\left(\mathrm{K}_N, \mathrm{G}_N, \mathrm{~A}_N\right)$ ) is described by two linear maps $\mathrm{K}{\varphi}: \mathrm{K}M \rightarrow \mathrm{K}_N$ and $\mathrm{G}{\varphi}: \mathrm{G}M \rightarrow \mathrm{G}_N$ subject to the commutation relation $\mathrm{G}{\varphi} \circ \mathrm{A}M=\mathrm{A}_N \circ \mathrm{K}{\varphi}$.
1. The sum of two linear maps $\varphi$ and $\psi$ of $M$ to $N$ represented by $\left(\mathrm{K}{\varphi}, \mathrm{G}{\varphi}\right)$ and $\left(\mathrm{K}\psi, \mathrm{G}\psi\right)$ is represented by $\left(\mathrm{K}{\varphi}+\mathrm{K}\psi, \mathrm{G}{\varphi}+\mathrm{G}\psi\right)$.
The linear map $a \varphi$ is represented by $\left(a \mathrm{~K}{\varphi}, a \mathrm{G}{\varphi}\right)$.
2. To represent the composite of two linear maps, we compose their representations.
3. Finally, the linear map $\varphi$ of $M$ to $N$ represented by $\left(\mathrm{K}{\varphi}, \mathrm{G}{\varphi}\right)$ is null if and only if there exists a $Z_{\varphi}: \mathrm{G}M \rightarrow \mathrm{K}_N$ satisfying $\mathrm{A}_N \circ Z{\varphi}=\mathrm{G}_{\varphi}$.

数学代写|交换代数代写Commutative Algebra代考|Stability Properties

4.1 Proposition Let $N_1$ and $N_2$ be two finitely generated $\mathbf{A}$-submodules of an $\mathbf{A}$ module $M$. If $N_1+N_2$ is finitely presented, then $N_1 \cap N_2$ is finitely generated.
D We can follow almost word for word the proof of item $I$ of Theorem II-3.4 (necessary condition).
4.2 Proposition Let $N$ be an A-submodule of $M$ and $P=M / N$.

1. If $M$ is finitely presented and $N$ finitely generated, then $P$ is finitely presented.
2. If $M$ is finitely generated and $P$ finitely presented, then $N$ is finitely generated.
3. If $P$ and $N$ are finitely presented, then $M$ is finitely presented. More precisely, if $A$ and $B$ are presentation matrices for $N$ and $P$, we have a presentation matrix $D=$\begin{tabular}{|l|l|} \hline$A$ & $C$ \ \hline 0 & $B$ \ \hline \end{tabular}

D 1. We can suppose that $M=\mathbf{A}^p / F$ with $F$ finitely generated. If $N$ is finitely generated, it is of the form $N=\left(F^{\prime}+F\right) / F$ where $F^{\prime}$ is finitely generated, so $P \simeq \mathbf{A}^p /\left(F+F^{\prime}\right)$.

We write $M=\mathbf{A}^p / F$ and $N=\left(F^{\prime}+F\right) / F$. We have $P \simeq \mathbf{A}^p /\left(F^{\prime}+F\right)$, so $F^{\prime}+F($ and also $N)$ is finitely generated (Sect. 1).

数学代写|交换代数代写Commutative Algebra代考|The Category of Finitely Presented Modules

A上有限呈现模的范畴可以由$\mathbf{A}$上有限秩的自由模的范畴用纯范畴的方法构造。

$$\left(\mathrm{K}_M, \mathrm{G}_M, \mathrm{~A}_M\right),$$

MATLAB代写

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