Bounded function with finitely many discontinuities is integrable $\overset{?}{\Rightarrow}$ density of continuous distribution function is not unique

The density function of the distribution function of a continuous random variable is not uniquely defined.
A new density function can be obtained by changing the value of the function at finite number of points to some non-negative value, without changing the integral of the function. We then get a new density function for the same continuous distribution.

Does this follow from the theorem-

A bounded function with finite number of discontinuities over an interval is Riemann integrable.

or is there a different theorem supporting the above claim? Is the theorem a sufficient justification?

Density of numbers with multiple factors near square root

Fix constants $ 1\leq \alpha<\beta$ . What is the density of the set of positive integers $ n$ with at least two factors between $ \alpha\sqrt{n}$ and $ \beta\sqrt{n}$ ?

(I am specifically interested when $ \alpha=\sqrt{2}$ and $ \beta=\sqrt{3}$ , and I am hoping the density is zero. I am not an expert in this field, so apologies in advance.)

Spectral density of $D + XX^T$

Let $ D$ be a fixed diagonal matrix with real entries, and $ X$ a random $ m\times n$ matrix. More precisely, the entries $ X_{ij}$ are real independent and identically distributed. It can be shown that the eigenvalues of $ XX^T$ have a density following the Marcenko-Pastur law with probability 1, in the limit that $ n,m\rightarrow\infty$ with a fixed ratio $ m/n$ , and this result depends only on the variance of the entries $ X_{ij}$ .

Can we say something about the spectral density of $ D + XX^T$ , also in the limit of large $ m,n$ ? Is an analytical expression (as in the Marcenko-Pastur law) known in this case? At least something numerically tractable?

Solving density matrix equations

I am trying to evaluate Numerical Integration of a density matrix element by solving coupled density matrix equations for different values of certain parameter L ( find the code in the link). My code is working fine for L=0, but not L =1,2,…

Please help me if there is any other way to do this numerical integration.

Definition of model functions and their density in $C^0(X^\text{an})$

I am (still) working through the paper Singular semipositive metrics in non-Archimedean geometry by Sebastien Boucksom, Charles Favre and Mattias Jonsson (J. Algebraic Geom. 25 (2016), 77-139, doi:10.1090/jag/656, arXiv:1201.0187).

Here is another question i have: In subsection 2.5 they define a model function as a continous $ \varphi$ on $ X^{\text{an}}$ s.t. there is a vertical Cartier Divisor $ D \in \text{Div}_{\mathbb{Q}}(\mathcal{X})$ , where $ \mathcal{X}$ is a model of $ X^{\text{an}}$ , with $ \varphi_D = \log \vert \mathcal{O}_{\mathcal{X}}(D) \vert = \varphi$ . The set of these functions is denoted by $ \mathcal{D}(X) = \mathcal{D}(X) _{\mathbb{Q}}$ . In Prop. 2.2 they then show, among other things, that $ \mathcal{D}(X) _{\mathbb{Z}}$ is stable under max. and seperates points. (I am assuming in the definition of $ \mathcal{D}(X) _{\mathbb{Z}}$ one just considers $ D \in \text{Div}_{\mathbb{Z}}(\mathcal{X})$ ?)

Here are my Questions: Is it clear, that if $ \mathcal{D}(X) _{\mathbb{Z}}$ seperates Points and is stable under max. (see Prop.2.2), that this is also true for $ \mathcal{D}(X) _{\mathbb{Q}}$ ?

Is there an isomorphism from $ \mathcal{D}(X) _{\mathbb{Q}}$ to $ \mathcal{D}(X) _{\mathbb{Z}}\underset{\mathbb{Z}}{\otimes}\mathbb{Q}$ ?

Is it immediate that the $ \mathbb{Q}$ -VS $ \mathcal{D}(X) _{\mathbb{Q}}$ fulfills the conditions of the Stone-WeierstraƟ Theorem(see Cor.2.3) and is thus dense in $ C^0(X^{\text{an}})$ ? From my understanding one would require that $ \mathbb{Q}$ -VS $ \mathcal{D}(X) _{\mathbb{Q}}$ is closed uner multiplication with elements from $ \mathbb{R}$ ?

Thanks a lot guys.

Eigenvalue density of $S^TS + ee^T$

Let $ S$ be a random $ M\times N$ matrix with independently identically distributed entries. The Pastur-Marcenko law gives the spectral density of $ S^T S$ as $ N\rightarrow\infty$ with a fixed ratio $ \alpha=M/N$ :

$ $ \rho_{\textrm{MP}} (\lambda) = \left\{\begin{array}{ll} \frac{\alpha}{2 \pi \lambda \sigma^2} \sqrt{(b – \lambda) (\lambda – a)} + (1 – \alpha) \delta (\lambda) & a < \lambda < b\ 0 & \textrm{otherwise} \end{array}\right.$ $

where $ a,b=\left( 1 \pm \sqrt{\alpha} \right)^2 \sigma^2 / \alpha$ .

Let $ e_1$ be the Cartesian unit vector in a fixed direction and consider the rank-one updated matrix $ A = S^T S + e_1e_1^T$ . The spectral density of $ A$ has an analytical formula in the large $ N$ limit?

Probability density from standard domain (Typical Box principle and Chinese Remainder Theorem) – III

Pick a random pair $ (a,b)\in\mathbb Z_n^2\backslash\{0,0\}$ . Denote $ N_2(a,b,n)$ to be minimum $ \ell_r$ norm of vector $ (x,y)$ as $ (x,y)$ ranges over all non-zero integral solutions to $ (x,y)\equiv t(a,b)\bmod n$ where $ t\in\mathbb Z$ with $ 0<t<n$ .

Now let integers $ a,b$ be of size in $ (\sqrt m,m-\sqrt m)$ .

Let $ p_1,p_2$ be primes of similar size some integer $ m$ and let $ t_1$ attain $ N_2(a,b,p_1)$ and $ t_2$ attain $ N_2(a,b,p_2)$ . From Dirichlet pigeonhole we know $ N_2(a,b,p_1)$ and $ N_2(a,b,p_2)$ are less than $ \sqrt{2m}$ .

Let $ t$ attain $ N_2(a,b,p_1p_2)$ (note $ (t,p_1p_2)=1$ need not hold while $ (t_1,p_1)=(t_2,p_2)=1$ holds).

Then $ N_2(a,b,p_1p_2)$ seems to be typically $ m\sqrt 2$ .

  1. Is it possible for $ N_2(a,b,p_1p_2)$ to be smaller than $ \sqrt{2m}-\mu$ if $ N_2(a,b,p_1)$ and $ N_2(a,b,p_2)$ are greater than $ \sqrt{2m}-\mu $ at a $ \mu\in(0,\sqrt{2m})$ ?

  2. If so what is the probability?

This is the result from Akshay Venkatesh when $ n$ is prime. Then it is true as $ n\rightarrow\infty$ the distribution of $ N_2(a,b,n)/\sqrt{n}$ coincides with distribution of $ 1/\sqrt y$ where $ x+iy$ is picked at random with respect to hyperbolic measure from $ \{z:|z|\geq1,|\mathcal R(z)|\leq\frac12\}$ .

Given two probability density functions find a number that satisfies a given equation

I have a problem for which I either need a proof or a counterexample.

We are given two discrete random variables $ x_1$ and $ x_2$ in $ [0, n]$ where $ F_1(x)$ is the probability of $ x_1\leq x$ , and similarly $ F_2(x)$ is the probability of $ x_2 \leq x$ . My goal is to show that there always exists a number $ c > 0$ that satisfies $ $ \int_{0}^{\infty}x{f_2(x)}(1-F_1(x))\leq c\int_{c}^{\infty}f_1(x)F_2(x),$ $ where $ f_1(x)= \Pr[x_1=x]$ and $ f_2(x)=\Pr[x_2=x]$ .

Thanks for any help.