Mathematics Questions, Problems & Solutions : TransWikia.com ~ Page 10

Establishing divisibility by a prime

Let $a,b,c$ belong to ${0,1,....p-1}$, where $p$ is an odd prime. We need to find the number of triplets $(a,b,c)$ such that $a^2-bc$ is divisible...

Asked on 01/03/2022

1 answer

Flabby representable sheaves

Let $S$ be a scheme. Consider some representable moduli functor $mathcal{M}:(Sch/S)^{op}rightarrow Set$ represented by some scheme $M$. Then for each $Vin (Sch/S)^{op}$, let define$$mathcal{M}^{glob}(V):=text{im}left(mathcal{M}(S)rightarrow...

Asked on 01/03/2022

1 answer

How fast does $lim_{ t to 0} E left[ |Z|^2 1_{B}(X,X+sqrt{t} Z) right]= E left[ |Z|^2 right] E[1_B(X)]$

Let $X in mathbb{R}^n $ and $Z in mathbb{R}^n $ be two independent standard normal random vectors. We are interested in the following quantity:begin{align}E...

Asked on 01/03/2022

1 answer

If $int_0^{10} g(x),dx = 8$, what is $int_0^{10} [g(x)+2],dx$?

Consider the function $g$ that is continuous on the interval $[−10, 10]$ and that $int_0^{10}g(x)dx=8$. What is $int_0^{10}[g(x)+2]dx$ equal to? So I tried just substituting the...

Asked on 01/03/2022 by Ayanoria

0 answer

Outermost Minimal Surface in a Manifold

An outermost minimal surface is a minimal surface which does not contain another minimal surface within it. Interestingly, outer minimal surfaces are always spheres, because the only other possibility...

Asked on 01/03/2022

0 answer

Show $mathbb{P}[X-m>alpha]leq frac{sigma^2}{sigma^2+alpha^2}$

I found this problem in an old statistics book: Suppose $X$ is a square integrable random variable with mean $m$ and variance $sigma^2$. For any ...

Asked on 01/01/2022

1 answer

If $f(AB)=f(A)f(B)$ then $f(A)=g(det(A))$

Let $mathbb{K}$ a field, $f: M_n(mathbb{K}) to mathbb{K}$ non constant sucht that: $f(AB)=f(A)f(B)$. Prove that there exist an endomorphism $g$ on the monoid $(mathbb{K}, cdot)$...

Asked on 01/01/2022

3 answer

Extend a measure to a Radon measure

A Radon measure $mu:mathcal{P}(mathbb{R}^n)to [0,infty]$ is an outer measure such thatEach Borel set $Einmathcal{B}(mathbb{R^n})$ is $mu$-measurable;For every $Fsubset mathbb{R}^n$, there exists $Ein mathcal{B}(mathbb{R^n})$ such that...

Asked on 01/01/2022 by inbrevi

0 answer

Are Normal variables constructed by CDF inverse of uniform variables indepdent?

Let $Phi$ be the CDF of the normal distribution, and let $u,v,ssimmathrm{Unif}[0,1]$ be iid uniform variables, then $X_1:= Phi^{-1}(u),Y_1:= Phi^{-1}(v)$ will be independent normal variables, therefore ...

Asked on 01/01/2022

1 answer

Removing superfluous functions from limits

Suppose you have a quotient of form, $$ frac{ f(x) + g(x) + h(x) + cdots}{ q(x) + r(x) + p(x)+cdots}$$ Consider an expression of sort, $$ lim_{x...

Asked on 01/01/2022

3 answer

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