Mathematics Questions, Problems & Solutions : TransWikia.com ~ Page 2

The topology generated by open intervals of rational numbers

Let$B = { mathbb{R} } cup { (a,b) capmathbb {Q} , alt b , a,b inmathbb{Q}}$ Thus, a set $V in B$...

Asked on 01/07/2022 by roslavets

1 answer

If the solutions of $X'=AX$ have a constant norm then $A$ is skew symmetric.

I have a differential equation $X'=AX$ where $Ainmathcal M_n(Bbb R)$. The question is to prove that if all the solutions have a constant norm then $A$ is skew-symmetric matrix. What...

Asked on 01/07/2022 by As soon as possible

3 answer

An interesting identity involving the abundancy index of divisors of odd perfect numbers

Let $sigma(x)$ denote the sum of divisors of the positive integer $x$. A number $y$ is said to be perfect if $sigma(y)=2y$. Denote the abundancy index...

Asked on 01/07/2022

1 answer

Prove the series converges almost everywhere

Question: Given Lebesgue integrable $f: mathbb{R}rightarrow [0,infty)$, prove the following series converges almost everywhere on $mathbb{R}$:$$varphi(x) = lim_{krightarrow infty} sum_{t=-k}^k f(t+x)$$ Attempt: Towards a contradiction suppose...

Asked on 01/07/2022 by Christopher Rose

1 answer

Prove that the functional in $C_c^0(Omega)$ is a Radon measure

Let $Omega subset mathbb{R}^n$ be an arbitrary open set and $(x_n)_{n inmathbb{N}} subset Omega$ a sequence. Let $(a_n)_{n inmathbb{N}} subset mathbb{C}$ be a sequence such that...

Asked on 01/07/2022

1 answer

How can I evaluate ${lim_{hto 0}frac{cos(pi + h) + 1}{h}}$?

I'm supposed to evaluate the following limit using the cosine of a sum and one of the "special limits" which are ${lim_{xto 0}frac{sin(x)}{x}=1}$ and ${lim_{xto 0}frac{1-cos(x)}{x}=0}$. The limit...

Asked on 01/07/2022 by DCdaKING

4 answer

Estimate $f(b)$ using Taylor Expansion for $f'(x) = cos(x^2)$

I am using Taylor Expansion for the following problem, but for some reason I am getting wrong solutions from a program I am running it on. Can someone please help...

Asked on 01/07/2022 by brucemcmc

1 answer

If $f ∈ C^∞(M)$ has vanishing first-order Taylor polynomial at $p$, is it a finite sum of $gh$ for $g, h ∈ C^∞(M)$ that vanish at $p$?

This is 11-4(a) in Lee's "Introduction to Smooth Manifolds": Let $M$ be a smooth manifold with or without boundary and $p$ be a point of $M$. Let...

Asked on 01/07/2022 by Fred Akalin

1 answer

$ sum_{n=1}^infty csc^2(omegapi n)= frac{A}{pi} +B $

$$ sum_{n=1}^infty csc^2(omegapi n)= frac{A}{pi} +B $$ if $omega =-frac{1}{2}+frac{sqrt{3}}{2}i$ find $frac{A^2}{B^2}$My Attempt$$ sum_{n=1}^infty csc^2(omegapi n)= sum_{n=1}^infty csch^2(iomegapi n)= 4sum_{n=1}^infty big(e^{pi n big( frac{i}{2} +...

Asked on 01/07/2022 by hwood87

2 answer

Show that $f$ is a strong contraction when $f$ is continuously differentiable.

Let $f: [a,b] to R$ be a differentiable function of one variable such that $|f'(x)| le 1$ for all $xin [a,b]$. Prove that $f$ is a...

Asked on 01/07/2022

2 answer

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