# Questions tagged [random-functions]

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80
questions

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### What is convergence in distribution of random variables taking values in a non-metrizable product space?

Let $F = (F(x) : x \in \mathbf{R}^n)$ be a family of $\mathbf{R}^k$-valued random variables indexed by $\mathbf{R}^n$ (to be clear there is a single probability space $(\Omega,\Sigma,\mathbf{P})$ such ...

1
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1
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49
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### Lower bounding the infimum of a random process

Let $X_{t}=\sum_{i=1}^n(1+s\cdot w_i)t_i\sin(t_i)$ where $t\in T=[-\pi/2,\pi/2]^n/\{\vec 0\}$, $w_i$ are iid standard gaussian variables, $s$ is a scalar denoting the strength of Gaussian noise.
How ...

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2
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88
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### the infimum of a random process

Let $X_{t}=\sum_{i=1}^n(1+s\cdot w)\sin(t_i)$ where $t\in T=[-\pi/2,\pi/2]^n/\{\vec 0\}$, $w\sim\mathbb{N}(0,1)$, $s$ is a scalar denoting the strength of Gaussian noise. How to find the condition on $...

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85
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### Lipschitz maximal inequality for random process

I am confused on the Lemma 5.7 (Lipschitz maximal inequality) here. Let me first restate the definition and the lemma:
Def
$\{X_t\}_{t\in T}$ is called Lipschitz for metric $d$ on $T$ if there exists ...

2
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73
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### Fourier expansion of random functions

Consider a random mapping $f:\{0,1\}^n \to \{0,1\}^n$, .i.e, a function such that for each $x \in \{0,1\}^n$, $f(x) \in \{0,1\}^n$ is chosen uniformly at random.
My question is what would the fourier ...

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1
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66
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### Difference between $P(f(x,w)>0)→1$ at any $x$ and $P(\inf(f(x,w))>0)\to1$ when dimension grows

Let $T:=[-1,1]^{n-1}\times (0,1]$. Let
$$f_n(x_1,\cdots,x_n,w_1,\cdots,w_n):=g(x_1,w_1)+\cdots+g(x_n,w_n)=\sum_{i=1}^ng(x_i,w_i),$$
where
(i) $w_1,\cdots,w_n$ are i.i.d. Gaussian random variables
(ii) ...

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1
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92
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### Positivity of linear combination of gaussian variables

Consider a collection of independent standard Gaussian variables $w_i$ for $i = 1, 2, \ldots, N$. Define its linear combination $f:=\sum_{i=1}^Na_iw_i+b_i$, where $a_i=pb_i$ ($p$ is a fixed parameter),...

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153
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### concentration of random field to its expectation function

Question
Given a random field $X(t)$ where the parameter space $T\subset\mathbb{R}_N$. Is there result regarding the concentration of the random field? For example
$\mathbb{P}\{\|X(t)-\mathbb{E}\{X(t)\...

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83
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### Expected Number of roots in $\mathbb D (0;r)$

In the literature about roots of random polynomials there are many results about the expected number of real roots of a complex polynomial building on Kac formula especially in the the asymptotic ...

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44
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### Diameter of the range of composition of random maps on the circle

My questions are related to the paper https://hal.science/hal-03933493v1 (accepted with corrections in Ergodic Theory and Dynamical Systems).
I fix an irrational number $\theta \in [0,1[$. I define ...

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1
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73
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### Does the following expectation-based inequality hold?

Let $\mathcal{F}$ be the space of all functions that uniformly and independently map the alphabet $\mathcal{X}$ to the set $\{1,2,\ldots,A\}$. Let $p(x|y)$ be an arbitrary conditional probability ...

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69
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### Distribution of zeros and angles of a function with additive coloured noise

Let us consider some real-variable function
$$
f(t) = f_0(t) + \xi(t),
$$
where $f_0$ - some "regular" (a continuously differentiable function without any noise [one can consider $f_0 = \...

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0
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72
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### Constructing k-wise independent variables over a general set

We have seen in class a polynomials based construction that builds in $O(n^k)$ time, $n$ random variables, $k$-wise independent, over a field with $n$ elements. More specifically, you generate all the ...

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76
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### Does $E(XU)\neq 0$ imply $E(f(X) U)\neq 0$ "almost always"?

Consider two non-orthogonal random variables
$$
(1) \quad E(XU)\neq 0,
$$
where $X$ can be a vector.
Can we claim that (1) implies that $U$ will be "generically" non-orthogonal to any ...

2
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133
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### Expected value of number of collisions for a matrix valued random function

Consider the function
$$f(x_1, x_2, \ldots, x_k) = S_1^{x_1} S_2^{x_2} \cdots S_k^{x_k}.$$
Each $x_1, x_2, \ldots, x_k \in \{0, 1\}$ and each $S_i \in \mathbb{F}_q^{n \times n}$ is a randomly chosen $...

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1
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84
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### On the growth of sample paths of Gaussian random fields

Consider a centered Gaussian random field on $\mathbb{R}^n$ with continuous covariance and a.s. continuous sample paths. What is known about the growth of the sample paths at infinity of such a random ...

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71
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### Distribution of multivariate polynomial evaluation

Let $R:\mathbb{F}^n \rightarrow \mathbb{F}^m$ be a multivariate quadratic map. Here $\mathbb{F}$ denotes the finite field of order $q$.
I am curious to know whether the distribution of $R(x)$ for ...

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277
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### Fastest sine of a large power of 2

What is the fastest known way to calculate $\sin(2^{n})$ for large integer $n$?
I only need the highest few bits to be correct. I suspect that the compute time required
scales with $n$ (and actually ...

1
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1
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199
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### Partial derivative of expectation and Stein's lemma

Currently, I am reading a paper about the Gaussian Process in Neural Network [1]. In the solution of the main result in this paper, the author applied Stein's lemma and claimed an equation about the ...

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1
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228
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### Construct a random vector as a function of another random vector

ASSUMPTION 1: there exists a continuous random vector $(X,Y,Z)$ such that
$$
\begin{cases}
p_1=\Pr(X\geq 0, Z\geq 0)\\
p_2=\Pr(Y\geq 0, Z< 0)\\
p_3=\Pr(X< 0, Y<0)\\
\end{cases}
$$
where $(p_1,...

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0
answers

68
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### Measurability of $\mathbb{R}^n$-Random Field

Let $(X_x)_{x\in [0,1]^d}$ be a collection of integrable random variable defined on a (common) probability space $(\Omega,\mathcal{F},\mathbb{P})$. Under what condition is the map:
$$
[0,1]^d\ni x \...

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3
answers

179
views

### Practical pseudorandom generators

It is known that existence of pseudorandom generators (PRGs) is equivalent to the existence of one-way functions. In turn, the latter is an open problem.
I am curious if someone developed kind of &...

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0
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36
views

### Get covariance from log-density function

Problem
Given a following log-density function
$$ \ln p(y| a, b) = a \cdot g(y) + b \cdot h(y) + k(a,b)$$
where $g(y), h(y), k(a,b)$ are difined function and $a,b$ are parameters.
Find $\Bbb Cov( g(Y),...

1
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1
answer

412
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### Approximate expectation of a random variable that is the logarithm of a function of a binomial

I want the expectation of the following random variable: $\log\left(\frac{X}{k-X}+\alpha \right)$ with $X \sim Bin_{(k-1),p}$ and $\alpha > 0$, Therefore I derived the Taylor Series:
\begin{...

3
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1
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195
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### Kac-Rice formula and Borell-TIS inequalities for gradient-flow of centered gaussian random field

Let $x\mapsto g(x)$ be a centered gaussian random field on $\mathbb R^m$. Let $x_0 \in \mathbb R^n$, and (assuming regularity conditions) consider the gradient-flow
$$
\dot{x}(t) = -\nabla g(x(t)), \;...

1
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1
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128
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### Central limit theorem for chi-squared random field on $\mathbb R^p$

Let $X:x \mapsto X(x)$ be a centered stationary Gaussian process on the $\Omega:=\mathbb R^p$, such that $X(x) \overset{d}{=}X(x')$ for all $x,x' \in \Omega$. Set $\sigma^2 := \mbox{Var}(X(0)) = \...

1
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1
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233
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### Concentration inequality for the supremum of $L_2$ norm of a vector-valued Gaussian process with iid components

Let $\Omega$ be a compact subset of $\mathbb R^p$ and let $f_1,\ldots,f_k$ be zero mean identically distrubuted Gaussian processes on $\Omega$ such that $f_1(x),\ldots,f_k(x)$ are independent $x \in \...

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78
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### Concentration inequalities for gradient flows induced by random fields

Let $G=(G(x))_{x \in \mathbb R^m}$ be a conservative random field with values in $\mathbb R^m$, for large positive integer $m$. That is, there exists a scalar random field $g=(g(x))_{x \in \mathbb R^m}...

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1
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48
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### Emergence of non-power-law behaviour under infinite summing

Suppose $X_1,X_2,...$ is a sequence of random vectors in $\mathbb{R}^n$ s.t for all $k \in \mathbb{Z}^+$ and $u \in \mathbb{R}^n$ we have that $E [ \langle u, X_i \rangle ^k]$ is finite. (The $X_i$s ...

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418
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### Forcing, constructibility, and random functions

This question is in some ways an offshoot of my recent question about trying to explain forcing to someone (such as Scott Aaronson, whose questions have prompted my questions) encountering it for the ...

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0
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164
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### Two kinds of generating functions

Sorry for a possibly off-the-topic question, but I am afraid to gain the necessary overview to give an answer (supposed the question is not ill-posed) is beyond my capabilities.
In the course of ...

3
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1
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223
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### Reference for Function-Valued Random Variables?

Question: Are there any good references for facts about function-valued random variables? In particular for facts like the following:
Let $X$ be a topological space, $Y$ be a random variable with ...

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0
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74
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### Negative moments of Steinhaus random variables

Let $f_i, i=1, \ldots, n$ be independent Steinhaus random variables, i.e. variables which are uniformly distributed on the complex unit circle. Let $a \in R^n$.
1) Find $E\left(\sum_{i=1}^nf_i a_i\...

3
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1
answer

142
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### Proving anti-concentration for the operator norm of a random matrix

If $X$ is a random matrix then I would like to find $\theta >0$ and $\delta \in (0,1)$ s.t I can say,
$$\mathbb{P} \Bigg [ \Big \vert \Vert X \Vert - \mathbb{E} [ \Vert X \Vert ] \Big \vert > \...

3
votes

1
answer

109
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### About concentration of eigenvalues values of a random symmetric matrix in a specific interval

Given a random symmetric matrix $M$ and two numbers $\lambda_\min$ and $\lambda_\max$ how do we calculate the expected or high probability value of the fraction of its eigenvalues which lie in the ...

2
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0
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### Less regular version of the Gaussian free field

One can define (continuous) Gaussian free field as follows: one can consider some orthonormal basis $(\psi_k)_{k=1}^{\infty}$ in the Sobolev space $H^1(\Omega)$ (here $\Omega \subset \mathbb{R}^d$) ...

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166
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### Canonical embedding of Hilbert space in random $L^2$

This question is a followup of Canonical embedding of Hilbert space in $L^2$ space, where it was essentially shown that there is no canonical way to construct, from an abstract Hilbert space $H$, a ...

0
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1
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185
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### The distribution of the power of the sum of inner products of two independent complex normal vectors

If I have $\mathbf x_n=[x_0, x_1,... ,x_K]^T$ and $\mathbf y_n=[y_0, y_2, ..., y_K]^T$, where $x,y\sim\mathcal C\mathcal N(\mathbf 0,\sigma^2\mathbf I)$.
What is the distribution of the following ...

0
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1
answer

601
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### The distribution of the sum of inner products of two independent complex normal vectors

If I have $\mathbf x_n=[x_0, x_1,... ,x_K]^T$ and $\mathbf y_n=[y_0, y_2, ..., y_K]^T$, where $x,y\sim\mathcal C\mathcal N(\mathbf 0,\sigma^2\mathbf I)$.
What is the distribution of the following ...

2
votes

1
answer

247
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### Smooth transformation of a curve with fixed ends and length [duplicate]

I am simulating polymers of fixed length and fixed ends. I would like to search the phase space of all possible conformations quickly. Is there anyway I can generate efficiently a lot of (rather) ...

1
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1
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100
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### Expected roots of polynomials with randomness in coefficients

I need some results on expected roots of polynomial with random coefficients. I searched for $\textit{roots of polynomial with random coefficients}$ online and found some papers, but they are dealing ...

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88
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### Largest component and number of components of random mappings with bounded in-degree

Let $S$ be a finite set of size $n$ and consider all functions $f$ from $S$ to itself, such that the preimage $f^{-1} (k)$ obeys $0 \leq |f^{-1} (k)| \leq m$.
Let $F$ be chosen uniformly at random ...

3
votes

1
answer

343
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### Attractors in random dynamics

Let $\Delta$ be the interval $[-1,1]$, then we can consider the probability space $(\Delta , \mathcal{B}(\Delta),\nu)$, where $\mathcal{B}(\Delta)$ is the Borel $\sigma$-algebra and $\nu$ is equal ...

2
votes

0
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105
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### Average number of pieces of a random piecewise-linear function

Let $I$ be a (nonempty) compact interval in $\mathbb R$ and $a_1,b_1,\ldots,a_L,b_L \in \mathbb R$. Let $\varphi$ be a piecewise function with $T \ge 2$ pieces(for example $T=2$ for the choice $\...

1
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1
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120
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### Proof of variance bounds for transformed random variables

Given I have a random variable $a$ that can be realised in the domain $D$ and has a finite variance $\sigma^2$. Furthermore I have a function $f$ which is differentiable(hence continuous) with an ...

2
votes

2
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223
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### Substitute Concrete Value in Conditional Expectation

Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space.
Let $$ X, Y : \Omega \rightarrow \mathbb{R} $$ be random variables.
Furthermore, let
$$ f: \mathbb{R}^2 \rightarrow \mathbb{R} $$
be a $...

1
vote

0
answers

43
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### Random solute transport equation

After reading the article Probability density functions for solute transport in random fields, by M. Shvidler, K. Karasaki, see https://pubarchive.lbl.gov/islandora/object/ir%3A116072/datastream/PDF/...

-2
votes

1
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573
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### Can this criterion to indicate the randomness some numbers? [closed]

John Derbyshire in his book PRIME OBSESSION says on page 366:
CHAPTER 3
10.
"Here is an example of e turning up unexpectedly. Select a random number
between 0 and 1. Now select another and add ...

5
votes

1
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392
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### A question about the paper "The Condition Number of a Randomly Perturbed Matrix"

My question pertains to this paper by Terence Tao and Van Vu, https://arxiv.org/abs/math/0703307
Both my questions pertain to the argument presented in this paper in its section 6 (page 5). We are ...

0
votes

1
answer

225
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### 2D wave equation with gaussian boundary condition

Given the 2D wave equation in polar coordinates:
$$u_{\rho\rho}+\dfrac{1}{\rho}u_{\rho}+\dfrac{1}{\rho^2}u_{\theta\theta}=\dfrac{1}{a^2}u_{tt}$$
with $u=u(\rho,\theta,t),(\rho,\theta,t)\in [0,c]\times(...