All Questions

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Mollifiers and smooth path connetction

Generaly it is about smooth path connection. I have a two smooth paths f,g and a points x,y,z of open subset of $R^n$ such that f is smooth path from a to y and g is smooth path from y to z. I define ...
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Volume integral $\int_V y \frac{dy}{dt} + z\frac{dz}{dt} dV$

How do I solve $$\int_V y \frac{dy}{dt} + z\frac{dz}{dt} dV$$ Where $V$ is the volume of a hollow sphere. Usually I would use spherical coordinates, but I don't know how to express $\frac{dy}{dt}$ ...
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prove that matrix is diagonal by matrix rank and eigenvalue rank

$A$ is matrix $9\times9$ with rank of $5$, there is rank$(A-3I)=5$, the matrix has another eigenvalue of 5. I need to prove that $A$ is diagonal and find the similar diagonal matrix of $A$. I'm stuck, ...
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Can a point $z$ which belongs to a closed set be a limit point of an open set which is disjoint from the closed set in topological space $X$?

Say $X$ be a topological space, and $U$ and $V$ are open and closed sets respectively. Furthermore, $U$ and $V$ are disjoint. Now there is a point $z \in V$. Is it possible for the point $z$ to be a ...
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How much faster will a task be in percentage terms?

This isn't necessarily a difficult question, but it's something I'm trying to set straight. The task I have in mind is mowing my lawn. Let's say I have a 16" lawnmower. I decide I'd rather have a ...
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Information Theory - Uniquely Decipherable code problem

Hi I'm having trouble with parts bii) and c) of the following problem. For bii) I feel I might need to apply Markov's inequality but I'm really not sure. Thanks!
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Sum of reciprocal of $p \log p$ where $p$ is a prime [duplicate]

Does $$\sum_{p\in Primes} \frac{1}{p \log p}$$ converge? Notice $$\sum_{n\in \mathbb{N}} \frac{1}{n \log^{1+\epsilon} n}$$ converges for all $\epsilon>0$, so one can wonder is the primes sparse ...
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Let $\mathbb{K}$ be a field of characteristic $p>0$ and $\mathbb{F} | \mathbb{K}$ a finite and separable extension.

Let $\mathbb{K}$ be a field of characteristic $p>0$ and $\mathbb{F}/ \mathbb{K}$ a finite and separable extension. Show that if $B=\{\alpha_1,\dots,\alpha_n\}$ is a basis, then ...
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Generalizing the Remainder Factor Theorem

Today, I spent most of my time developing a systematic procedure for finding remainder polynomial when higher degree polynomials are divided by some polynomial of degree $\leq$ the degree of the ...
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How to evaluate $\lim\limits_{x\to1}\frac{x^{m+1}-x^{n+1}+x^n-mx+m-1}{(x-1)^2}$?

I used substitution $t=x-1 \Rightarrow t\rightarrow 0$ After this, the limit is: $$\lim\limits_{t\to0}\frac{(t+1)^{m+1}-(t+1)^{n+1}+(t+1)^n-m(t+1)+m-1}{t^2},m,n \in \mathbb{N}$$ Then I used ...
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Injective linear endomorphism of hilbert space is bijective?

Is it true that an injective continuous endomorphism of a hilbert space is bijective? If not, are there conditions that imply this? I know this would follow from the rank nullity theorem in finite ...
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Is there an alternative to Taylor expansion of functions with more control over the error distribution?

As a physics student I see the Taylor series being (ab)used very often, mostly for the expansion of $\exp(x)$. The usual line of though is that the error (remainder) term of a high order is likely ...
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Existence of analytic function on a unit disc $\triangle$

Let $\triangle$ be the open unit disc. Then can there be analytic functions with the property (1) $f(\frac{3}{4})=\frac{3}{4}$ and $f'(\frac{2}{3})=3/4$ 2) $f(\frac{3}{4})= -\frac{3}{4}$ and ...
10 views