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Let $p$ be odd prime number. Show that$$F(x)=sum_{k=1}^{p-1}left(frac{k}{p}right)x^k$$has at least $dfrac{p-1}{2}$ different complex roots $z$ with $|z|=1$,where $left(dfrac{k}{p}right)$ is Legendre's symbol.I...
Asked on 11/02/2021 by geromty
0 answerConsider a measure space $(X, mathcal{F}, mu)$ and let $f in L^1(X, mathcal{F}, mu)$ with $f(x) >0$ for all $x in X$ be a probability...
Asked on 11/02/2021 by Luiz Max Carvalho
0 answerIn the paper "The Complexity of Parallel Search" (Karp, Upfal and Wigderson; Journal of Computer and System Sciences 36, 225 - 253, 1988) in the appendix, the authors present the...
Asked on 11/02/2021
0 answerHi it's a little refinement to play with a hard inequality of Vasile Cirtoaje :Let $ageq b>0$ such that $a+b=1$ then we have :$$a^{2b}+b^{2a}leq a^{Big(frac{a(1-a)(frac{1}{2}-a)}{4}Big)^2}=f(a)$$It...
Asked on 11/02/2021 by Erik Satie
1 answerI know that the space $mathsf{C}^infty(M;N)$ of smooth maps from a closed (smooth) manifold $M$ to a (smooth) manifold $N$ is a Fréchet manifold. I have been...
Asked on 11/02/2021
0 answerWhile studying Determinants from text book Hoffman and Kunze, I have a in an argument in a theorem whose reasoning is not provided . Questions:1st question is in underlined...
Asked on 11/02/2021
2 answerGiven the roles generating functions and coefficient extraction play in solving recurrence relations, they are clearly analogous to the Laplace Transform and Inverse Laplace Transform. A hypothesis would then...
Asked on 11/02/2021 by user10478
1 answerI am not familiar with random matrices but I need to confirm the correctness of the inequality below.Let $xi_iin{pm 1}$ be independent random signs, and let$A_1,ldots,...
Asked on 11/02/2021 by Rockafellar
2 answer(Crossposted to Math Overflow) Suppose we have an Euler product over the primes $$F(s) = prod_{p} left( 1 - frac{a_p}{p^s} right)^{-1},$$ where each $a_p in mathbb{C}$. The...
Asked on 11/02/2021
0 answerLet $Omega subset mathbb{R}^n$ be a smooth domain, $u,v in H^1_0(Omega)$ with the usual inner product and let $M=M(x)$ be a $ntimes n$ matrix with...
Asked on 11/02/2021 by StopUsingFacebook
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