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

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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

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