Mathematics Questions, Problems & Solutions : TransWikia.com ~ Page 31

Time continuity of the function in L1 norm i.e. $C([0,T];L^1) $

Let $fin L^1(mathbb{R}),$then its translation defined by $f(t,x):=f(x+t)$ belongs to $C([0,T];L^1(mathbb{R})).$ In addition if $fin BV(mathbb{R}),$ thenbegin{eqnarray}intlimits_{mathbb{R}} |f(x+t_1)-f(x+t_2)|dx &=& sumlimits_{imathbb{Z}}intlimits_{j=|t_1-t_2|i}^{|t_1-t_2|(i+1)}|f(x+t_1)-f(x+t_2)|dx\&=&sumlimits_{iin...

Asked on 12/20/2021

1 answer

stationary independent increments: f(t+s) = f(t) + f(s), therefore f(t)=ct

I don't get it... Why is the only solution to: $$f(t+s) = f(t) + f(s)$$ equal to: $$f(t) =...

Asked on 12/20/2021

2 answer

Let $Phi$ be standard Gaussian CDF and $u > 0$. What is good u-bound for $int_0^1Phi(u/r - ur)dr$ as a function of $u$?

Let the function $Phi(x) := (1/sqrt{2pi})int_{-infty}^x e^{-t^2/2}dt$ be the standard Gaussian CDF. For $u > 0$, define $I(u) := int_0^1 Phi(u/r-ur)dr$.Question. What are good upper-bounds for ...

Asked on 12/20/2021

1 answer

Diagonal of (self) product of doubly stochastic transition matrix

By doubly stochastic and transition, I mean each row sum and column sum of a matrix is 1 and each element of the matrix is in [0, 1]. Here, I...

Asked on 12/20/2021 by lovemath

1 answer

$mathbb R$ as continuum many of pairwise disjoint of Bernstein sets

The set $Bsubsetmathbb R$ is called Bernstein set if neither $B$ nor $mathbb Rsetminus B$ contains any perfect sets. Theorem: $mathbb R$ can be written as...

Asked on 12/20/2021 by 00GB

1 answer

How to evaluate the following limit: $lim_{xto 0}frac{12^x-4^x}{9^x-3^x}$?

How can I compute this limit$$lim_{xto 0}dfrac{12^x-4^x}{9^x-3^x}text{?}$$ My solution is here: $$lim_{xto 0}dfrac{12^x-4^x}{9^x-3^x}=dfrac{1-1}{1-1} = dfrac{0}{0}$$ I used L'H$hat{mathrm{o}}$pital's rule: begin{align*}lim_{xto 0}dfrac{12^xln12-4^xln4}{9^xln9-3^xln3}&=dfrac{ln12-ln4}{ln9-ln3}\ &=dfrac{ln(12/4)}{ln(9/3)}\...

Asked on 12/20/2021 by user811107

6 answer

Is my proof for $f$ is convex iff $f'$ is monotonically increasing correct?

I am following up on my previous question. My previous attempt for the proof was wildly incorrect (my question was how that proof was exactly my old proof was...

Asked on 12/20/2021

2 answer

Is difference of two projection matrices positive semi-definite or negative definite or indefinite?

Suppose $S$ is an $m times n$ matrix of full column rank and $W$ is an $m times m$ positive definite matrix. Let $R =...

Asked on 12/20/2021

2 answer

Directly Calculating Birthday Paradox Probabilites

I am trying to calculate the probability of at least 2 people sharing a birthday in a group of 4 people. I understand that calculating it as 1-P(no shared birthdays)...

Asked on 12/20/2021 by user553664

2 answer

Show that $mathfrak{m}_p$ is an ideal in $mathcal{O}_V.$

I'm working through Algebraic Geometry: A Problem Solving Approach by Garrity et al, and I have found myself stuck on Exercise 4.13.1, which is the section Points and Local Rings.Let...

Asked on 12/20/2021 by user525033

1 answer

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