### 数学代写|数论作业代写number theory代考|Estimates on the Growth of the Total Asymptotic Density

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

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数论作业代写number theory代考|Estimates on the Growth of the Total Asymptotic Density

The study of the limit law as $m \rightarrow \infty$ for the maximal leading parameter $\mathcal{K}{\max }^{[m]}$ is a more difficult problem. This parameter is connected with the total asymptotic density of resonances $\operatorname{Ad}\left(H{\Upsilon[m]}\right)$ and so, due to Theorem 3.1, with the size $V\left(\Upsilon^{[m]}\right)$ of the random set $\Upsilon_{m}^{[m]}$. A simple version of this connection is given by the inequality $\mathcal{K}_{\max }^{[m]} \geq \frac{m}{V\left(\Upsilon^{[m]}\right)}$ a.s. (see Corollary $5.1$ (iii)). A more precise dependence in the deterministic case can be seen from [7, formula (3.6)].

The following theorem describes the rate of growth as $m \rightarrow \infty$ of the total asymptotic densities $\operatorname{Ad}\left(H_{\Upsilon[m]}\right)$ and of the sizes $V\left(\Upsilon^{[m]}\right)$, which according to Theorem $3.1$ are connected by $\operatorname{Ad}\left(H_{\left.\Upsilon^{[m]}\right]}\right)=\frac{V\left(\Upsilon^{[m]}\right)}{\pi}$ a.s.
Theorem $5.3$ Let $r>0$ and $\Upsilon^{[m]} \in \Theta\left(m, \mathbb{B}{r}\right), m \in \mathbb{N}$. Then, for any $t \in \mathbb{R}$, $$\liminf {m \rightarrow \infty} \mathbb{P}\left{\frac{V\left(\Upsilon^{[m]}\right)}{r}>\frac{36}{35} m+\frac{2 \sqrt{87}}{35} t \sqrt{m}\right} \geq 1-\Phi(t)$$
where $\Phi(t)=(2 \pi)^{-1 / 2} \int_{-\infty}^{t} e^{-s^{2} / 2} \mathrm{~d}$ s (the standard normal distribution function). In particular, the following estimate is valid for the asymptotic density $\operatorname{Ad}\left(H_{\Upsilon[m]}\right)$ of resonances
$$\lim {m \rightarrow \infty} \mathbb{P}\left{\operatorname{Ad}\left(H{\Upsilon(m \mid)}\right)>\frac{m r}{\pi}\right} \rightarrow 1 \text { as } m \rightarrow \infty$$
Proof For convenience of the notation, we replace each process $\Upsilon^{[m]}$ by the process $\widetilde{\Upsilon}^{[m]}=\left{\xi_{j}\right}_{j=1}^{m}$ defined in (5.1). This does not influence the estimates below.
Let
$m_{}=2\lfloor m / 2\rfloor$, i.e., $m_{}=m$ if $m$ is even, and $m_{}=m-1$ if $m$ is odd. Then, from the definition of $V(\cdot)$, we have $$V\left(\widetilde{\Upsilon}^{[m]}\right) \geq 2 \mathcal{S}{m{} / 2}, \quad \text { where } \mathcal{S}{m}=\sum{j=1}^{m}\left|\xi_{2 j-1}-\xi_{2 j}\right|$$
The $\mathbb{R}{+}$-valned random variahles $\lambda{j}:=\frac{\left|\xi_{2 j-1}-\xi_{2 j}\right|}{2 r}$ are i.i.d. with the first two moments given by
$\mathbb{E}\left(\lambda_{1}\right)=18 / 35, \quad \mathbb{E}\left(\lambda_{1}^{2}\right)=3 / 10 \quad$ (see [31] for the general formula).
Hence, the variance of $\lambda_{j}$ is $\operatorname{Var} \lambda_{j}=\left(\frac{\sqrt{87}}{35 \sqrt{2}}\right)^{2}$. Applying the Central Limit Theorem, we get

$$\mathbb{P}\left{\frac{\mathcal{S}_{m}}{2 r}-\frac{18}{35} m \leq t \sqrt{m} \frac{\sqrt{87}}{35 \sqrt{2}}\right} \rightarrow \Phi(t)$$
as $m \rightarrow \infty$. This implies $(5.4)$ and, in turn, (5.5)
Acknowledgements IK is grateful to Richard Froese for inspiring and educative discussions about resonances and random spectral theory, to Jürgen Prestin for the hospitality of the University of Lübeck, to Baris Evren Ugurcan and the Hausdorff Research Institute for Mathematics (HIM) of the University of Bonn for their hospitality during the trimester program “Randomness, PDEs and Nonlinear Fluctuations” at HIM in 2020. IK was supported by the VolkswagenStiftung project “Modeling, Analysis, and Approximation Theory toward applications in tomography and inverse problems” and, during the workshop “Analytical Modeling and Approximation Methods” (Berlin, 04-08.03.2020), by the VolkswagenStiftung project “From Modeling and Analysis to Approximation”.

## 数学代写|数论作业代写number theory代考|Green’s Functions and Euler’s Formula

An astute student in a sophomore differential equations course can compute the set of eigenvalues of the Dirichlet boundary value problem
$$-\Delta_{D} f=-f^{\prime \prime}, \quad f(0)=f(1)=0$$
Indeed, the underlying linear operator $-\Delta_{D}$ is the positive, self-adjoint operator in the Hilbert space $L^{2}((0,1) ; d x)$,
$$\begin{gathered} \left(-\Delta_{D} f\right)(x)=-f^{\prime \prime}(x) \text { for a.e. } x \in(0,1), \ f \in \operatorname{dom}\left(-\Delta_{D}\right)=\left{g \in L^{2}((0,1) ; d x) \mid g, g^{\prime} \in A C([0,1]) ; g(0)=g(1)=0\right. \ \left.g^{\prime \prime} \in L^{2}((0,1) ; d x)\right} \end{gathered}$$
$(A C([0,1])$ denotes the set of absolutely continuous functions on $[0,1])$, with purely discrete spectrum,

$$\sigma\left(-\Delta_{D}\right)=\left{\lambda_{k}=(k \pi)^{2}\right}_{k \in \mathbb{N}^{*}}$$
The eigenspace of each $\lambda_{k}$ is one-dimensional and spanned by the normalized eigenfunctions
$$\begin{array}{ll} -\Delta_{D} u_{k}=(k \pi)^{2} u_{k}, & u_{k}(x)=2^{1 / 2} \sin (k \pi x), 0 \leq x \leq 1 \ \left|u_{k}\right|_{\left.L^{2}((0,1) ; d x)\right)}=1, & k \in \mathbb{N} \end{array}$$
In particular, the collection $\left{2^{1 / 2} \sin (k \pi x)\right}_{k \in \mathbb{N}}$ forms a complete orthonormal basis in $L^{2}((0,1) ; d x)$. Since $0 \notin \sigma\left(-\Delta_{D}\right),\left(-\Delta_{D}\right)^{-1}$ exists and is explicitly given by
$$\left(\left(-\Delta_{D}\right)^{-1} f\right)(x)=\int_{0}^{1} d y K_{1}(x, y) f(y), \quad f \in L^{2}((0,1) ; d x)$$
where $K_{1}(\cdot, \cdot)$ denotes the Green’s function for $-\Delta_{D}$ given by
$$K_{1}(x, y)= \begin{cases}x(1-y), & 0 \leq x \leq y \leq 1 \ y(1-x), & 0 \leq y<x \leq 1\end{cases}$$
This operator $\left(-\Delta_{D}\right)^{-1}$ is a bounded, self-adjoint, compact operator with eigenvalues $\left{\lambda_{k}^{-1}\right}_{k=1}^{\infty}$ and associated eigenfunctions $\left{u_{k}\right}_{k=1}^{\infty}$. Moreover, as discussed below, $\left(-\Delta_{D}\right)^{-1}$ is a trace class operator and this implies
$$\sum_{k \in \mathbb{N}} \lambda_{k}^{-1}=\int_{0}^{1} d x K_{1}(x, x)=\frac{1}{6}$$
from which the solution to the famous “Basel problem” quickly emerges:
$$\sum_{k \in \mathbb{N}} \frac{1}{k^{2}}=\frac{\pi^{2}}{6} .$$

## 数学代写|数论作业代写number theory代考|Bernoulli Polynomials and Bernoulli Numbers

For properties of Bernoulli polynomials and Bernoulli numbers, we refer the reader to the standard sources [1] [Chap. 23] and [27] [Sects. 9.6 and 9.7] as well as the online Digital Library of Mathematical Functions https://dlmf.nist.gov/ and the accompanying book [46].

The Bernoulli polynomials $\left{B_{n}(x)\right}_{n \in \mathbb{N}{0}}$ can be defined through the generating function $$\frac{z e^{x z}}{e^{z}-1}=\sum{n \in \mathbb{N}{0}} B{n}(x) \frac{z^{n}}{n !}, \quad|z|<2 \pi, x \in \mathbb{R}$$
For the record, the first few Bernoulli polynomials are given by
\begin{aligned} &B_{0}(x)=1, \quad B_{1}(x)=x-\frac{1}{2}, \quad B_{2}(x)=x^{2}-x+\frac{1}{6}, \quad B_{3}(x)=x^{3}-\frac{3}{2} x^{2}+\frac{1}{2} x, \ &B_{4}(x)=x^{4}-2 x^{3}+x^{2}-\frac{1}{30}, \text { etc. } \end{aligned}
Among several properties that these polynomials satisfy, we will make use of the fact that
$$\int_{0}^{1} d x B_{2 n}(x)=0, \quad n \geq 1$$
which follows from the identities (see [1][23.1.8 and 23.1.11])

Green’s Functions and Euler’s Formula for $\zeta(2 n)$
33
$$B_{n}(1-x)=(-1)^{n} B_{n}(x), \quad n \geq 1,$$
and
$$\int_{0}^{x} d u B_{n}(u)=\frac{B_{n+1}(x)-B_{n+1}(0)}{n+1}, \quad n \geq 1 .$$
We also recall the Fourier cosine series expansion of $B_{2 n}(x)$ (see [1] [23.1.18] or [4] [Theorem 12.19]), which is given by
$$\frac{(-1)^{n-1}(2 n) !}{2^{2 n-1} \pi^{2 n}} \sum_{k \in \mathbb{N}} \frac{\cos (2 k \pi x)}{k^{2 n}}=B_{2 n}(x), \quad 0 \leq x \leq 1, n \in \mathbb{N}$$
The Bernoulli numbers $\left{B_{n}\right}_{n \in \mathbb{N}{0}}$ are defined as $B{n}=B_{n}(0)$. For example,
\begin{aligned} &B_{0}=1, B_{1}=-1 / 2, B_{2}=1 / 6, B_{4}=-1 / 30, B_{6}=1 / 42, B_{8}=-1 / 30 \ &B_{10}=5 / 66, \text { etc., } B_{2 n+1}=0 \text { for } n \geq 1 \end{aligned}
We will see in Sect. 4 , that the Green’s function $K_{n}(\cdot, \cdot)$, associated with the $n^{t h}$ power of the operator $-\Delta_{D}$ defined in (1.1), can be given explicitly in terms of Bernoulli polynomials.

## 数学代写|数论作业代写number theory代考|Estimates on the Growth of the Total Asymptotic Density

\liminf {m \rightarrow \infty} \mathbb{P}\left{\frac{V\left(\Upsilon^{[m]}\right)}{r}>\frac{36}{35} m+\ frac{2 \sqrt{87}}{35} t \sqrt{m}\right} \geq 1-\Phi(t)\liminf {m \rightarrow \infty} \mathbb{P}\left{\frac{V\left(\Upsilon^{[m]}\right)}{r}>\frac{36}{35} m+\ frac{2 \sqrt{87}}{35} t \sqrt{m}\right} \geq 1-\Phi(t)

\lim {m \rightarrow \infty} \mathbb{P}\left{\operatorname{Ad}\left(H{\Upsilon(m \mid)}\right)>\frac{m r}{\pi}\right } \rightarrow 1 \text { as } m \rightarrow \infty\lim {m \rightarrow \infty} \mathbb{P}\left{\operatorname{Ad}\left(H{\Upsilon(m \mid)}\right)>\frac{m r}{\pi}\right } \rightarrow 1 \text { as } m \rightarrow \infty

\mathbb{P}\left{\frac{\mathcal{S}_{m}}{2 r}-\frac{18}{35} m \leq t \sqrt{m} \frac{\sqrt{87 }}{35 \sqrt{2}}\right} \rightarrow \Phi(t)\mathbb{P}\left{\frac{\mathcal{S}_{m}}{2 r}-\frac{18}{35} m \leq t \sqrt{m} \frac{\sqrt{87 }}{35 \sqrt{2}}\right} \rightarrow \Phi(t)

## 数学代写|数论作业代写number theory代考|Green’s Functions and Euler’s Formula

−ΔDF=−F′′,F(0)=F(1)=0

\begin{聚集} \left(-\Delta_{D} f\right)(x)=-f^{\prime \prime}(x) \text { for ae } x \in(0,1), \ f \in \operatorname{dom}\left(-\Delta_{D}\right)=\left{g \in L^{2}((0,1) ; d x) \mid g, g^{\prime } \in A C([0,1]) ; g(0)=g(1)=0\对。\ \left.g^{\prime \prime} \in L^{2}((0,1) ; d x)\right} \end{聚集}\begin{聚集} \left(-\Delta_{D} f\right)(x)=-f^{\prime \prime}(x) \text { for ae } x \in(0,1), \ f \in \operatorname{dom}\left(-\Delta_{D}\right)=\left{g \in L^{2}((0,1) ; d x) \mid g, g^{\prime } \in A C([0,1]) ; g(0)=g(1)=0\对。\ \left.g^{\prime \prime} \in L^{2}((0,1) ; d x)\right} \end{聚集}
(一种C([0,1])表示绝对连续函数的集合[0,1])，具有纯离散谱，

\sigma\left(-\Delta_{D}\right)=\left{\lambda_{k}=(k \pi)^{2}\right}_{k \in \mathbb{N}^{*} }\sigma\left(-\Delta_{D}\right)=\left{\lambda_{k}=(k \pi)^{2}\right}_{k \in \mathbb{N}^{*} }

−ΔD在ķ=(ķ圆周率)2在ķ,在ķ(X)=21/2罪⁡(ķ圆周率X),0≤X≤1 |在ķ|大号2((0,1);dX))=1,ķ∈ñ

((−ΔD)−1F)(X)=∫01d是ķ1(X,是)F(是),F∈大号2((0,1);dX)

ķ1(X,是)={X(1−是),0≤X≤是≤1 是(1−X),0≤是<X≤1

∑ķ∈ñλķ−1=∫01dXķ1(X,X)=16

∑ķ∈ñ1ķ2=圆周率26.

## 数学代写|数论作业代写number theory代考|Bernoulli Polynomials and Bernoulli Numbers

∫01dX乙2n(X)=0,n≥1

33

∫0Xd在乙n(在)=乙n+1(X)−乙n+1(0)n+1,n≥1.

(−1)n−1(2n)!22n−1圆周率2n∑ķ∈ñ因⁡(2ķ圆周率X)ķ2n=乙2n(X),0≤X≤1,n∈ñ

## 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。