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Expressing one gradient in terms of the other

Suppose I have the following relation:$mathbf{v}=t|mathbf{k}|^{t-2}mathbf{k}$ where $mathbf{v}$ and $mathbf{k}$ are $d-$dimensional vectors and $tinmathbb{R}$. I want to express $nabla_{mathbf{v}}$ in terms of...

Mathematics Asked on 2 years ago

Show that $m({xin[0,1]:text{$x$ lies in infinitely many $E_j$}})geqfrac{1}{2}$ when $m(E_j)geqfrac{1}{2}$

Question: Suppose $E_1, E_2,ldots$ is a sequence of measurable subsets of $[0,1]$ with $m(E_j)geqdfrac{1}{2}$. Show that $m({xin[0,1]:text{$x$ lies in infinitely many $E_j$}})geqdfrac{1}{2}$, where $m$...

Mathematics Asked on 2 years ago

Stars and bars but with distinct objects

The total number of ways which 5 balls of different colors can be distributed among 3 persons so that each person gets at least one ball is..?My attempt: So, first...

Mathematics Asked on 2 years ago

Laurent Series and residue of $frac{z}{(z-1)(z-3)}$ around z = 3

As mentionned in the title, I'd like to get the function's Laurent series and after its residue, I have tried to separate the two denominators to get a partial fraction...

Mathematics Asked by Dimitri on 2 years ago

Frobenius Norm and Relation to Eigenvalues

I've been working on this problem, and I think that I almost have the solution, but I'm not quite there.Suppose that $A in M_n(mathbb C)$ has $n$ distinct...

Mathematics Asked by 100001 on 2 years ago

Club Election Voting Permutation Question

In a club election the number of contestants is one more than the number of maximum candidates for which a voter can vote. If the total number of ways is...

Mathematics Asked on 2 years ago

Finding a closed form to a minimum of a function

It's a try to find a closed form to the minimum of the function :Let $0<x<1$ then define :$$g(x)=x^{2(1-x)}+(1-x)^{2x}$$Denotes $x_0$ the abscissa of the...

Mathematics Asked on 2 years ago

Solve $x^2+3y = u^2$ and $y^2+3x=v^2$ in positive integers.

The question is from the pg - 59 from ' An Introduction to Diophantine Equations ' by Titu Andreescu , Dorin Andrica , Ion Cucurezeanu.Example 1 : Solve in positive...

Mathematics Asked by The Demonix _ Hermit on 2 years ago

Decomposition of $s+sqrt{-1}$ in $mathbb{Z}[sqrt{-1}]$

I am trying to prove that in $mathbb{Z}[sqrt{-1}]$, every non-unit divisor of $s+sqrt{-1}$ is not associate with an element of $mathbb{Z}$ (which means, it is not a...

Mathematics Asked on 2 years ago

Suppose $z$ and $omega$ are two complex numbers such that $|z|≤1$ and $|omega|≤1$ and $|z+iomega|=|z-iomega|=2$. Find $|z|$ and $|omega|$.

Suppose $z$ and $omega$ are two complex numbers such that $|z|≤1$ and $|omega|≤1$ and $|z+iomega|=|z-iomega|=2$. Find $|z|$ and $|omega|$. My attempt: I squared...

Mathematics Asked on 2 years ago

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