# All Questions

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### Prove that $\forall m\in \mathbb Z$, $\sum_{n=-\infty}^{\infty} \frac{1}{n+m+\frac{1}{2}}\equiv 0$

Prove that $\forall m\in \mathbb Z$, $$\sum_{n=-\infty}^{\infty} \frac{1}{n+m+\frac{1}{2}}\equiv 0$$
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### Weierstrass factorization of sine, and related questions

So the idea is that you can represent a function as a product of its zeroes, and there are some fundamental factors that often crop up. I am interested in, give this is the WF of sine : Is it ...
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### Find the circle about the origin containing all roots of $X^7+X^2+1=0$

We need to find the circle about the origin containing all roots of $X^7+X^2+1=0$, I am not getting any hint where to start from. It is a problem of Munkres General topology section $56$, chapter ...
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### In calculus of variation: why are minimizing sequences bounded?

Assume the usual variational setting: Let $\mathcal{A} \subset W^{1,q}$ be the set of admissible functions and $$I: \mathcal{A} \to \mathbb{R}$$ the functional that needs ...
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### “Randomize” output of a Linear Feedback Shift Register for the same taps?

I'm using a (Galois) LFSR to sample a large array, ensuring that each entry is only visited once. I simply skip past the entries that exceed the array length. With the same taps then the array entry ...
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### $f(x) = \inf_{y \in Y} c(x,y) - \inf_{\xi \in X} c(\xi,y) - f(\xi) \Rightarrow f$ is upper semicontinuous

Let $X, Y$ be metric spaces. Given $c: X \times Y \mapsto \mathbb{R}$ continuous, define $$f(x) = \inf_{y \in Y} \left( c(x,y) - \inf_{\xi \in X} (c(\xi,y) - f(\xi)) \right).$$ Then is $f$ upper ...
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### Quadratic congruence equation with even modulus

I've been thinking for a while about how to go about solving the equation $x^2+3x+8 \equiv 0 \pmod{144}$ and similar ones. When the modulus is odd it's not too tricky, but when it is I can't see quite ...
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### Linear Fractional Transformation help

I am given this problem from a past test that I am trying to figure out, I have tried finding the conjugate and going about it. But i am not getting the right transformation. Please help out. Show ...
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### Radius of Convergence and Uniform Convergence

I am working on a question involving the function below and was hoping to have some help please. $$f(x)=\sum_{n=1}^{\infty} \frac{1}{n^3 9^n}(x-2)^{2n}$$ I am asked to: (a) Determine the ...
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### Question on Mc Carthy's nowhere differentiable function

Mc Carthy's function is a simple example of a nowhere differentiable but everwhere continuous function. Its construction here contains very little detail and I have some questions (highlighted in ...
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### Simplifying Catalan number recurrence relation

While solving a problem, I reduced it in the form of the following recurrence relation. $C_{0} = 1, C_{n} = \displaystyle\sum_{i=0}^{n - 1} C_{i}C_{n - i - 1}$ However ...
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### Calculate:$\int \frac{1}{(x+1)^\frac{3}{4}(x+2)^{\frac{5}{4}}}\ dx$

Calculate following integration $$\int \frac{1}{(x+1)^\frac{3}{4}(x+2)^{\frac{5}{4}}}\ dx$$
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### Number theory $a|d$

Suppose in $\mathbb{N}^** \mathbb{N}^*$ the equation $(E): x^2+y^2+xy-13x=0$. We set $x=ad$ and $y=bd$ and $d=GCD(x,y)$. How can I prove that $a|d$?
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### Upper Triangular Matrices of Monotone Vectors

I am looking for references to the following problem (I'm actually interested in general $n$, but will use $n=3$ as an example): consider a finite set, for example, N = {1,2,3}, and the associated ...
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### Conditions for loops to be homotopic

I am looking for a list of necessary and sufficient conditions for two loops on a (compact connected orientable) surface to be homotopic that could be made into a purely combinatorial definition of ...
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### Improper integral evaluates to $-\pi^2/12$

$$\int_0^1 \frac{\ln x}{1+x} \mathrm{d}x=-\int_0^1 \frac{\ln(1+x)}{x}\mathrm{d}x=-\frac{\pi^2}{12}$$ Please, help give me proper hints to solve. I was not even able to equate the first two.
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### Random walk on lollipop graph

Hi i am trying to prove expected Hitting time on the Lollipop graph. It is a graph on $n$ vertices with clique on $n/2$ vertices and path joined to this. Let vertex $i$ be a vertex on the clique, ...
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### Bounding the smallest eigenvalue of an ergodic Markov Chain

I am trying to prove that the smallest eigenvalue of an ergodic Markov chain is greater than -1. Can we do that using proof by contradiction, i.e. assuming the smallest eigenvalue being -1, etc.? The ...
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### An ideal, $I$, is maximal iff $R/I$ is a field

The first line of the proof given in my book says that the ideals of $R/I$ are in bijective correspondence with the ideals of $R$ lying between $I$ and $R$. What is the bijection?
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### Pseudo Random Number Generation on the GPU

Idea For a Path Tracing application I need to generate good quality pseudo random numbers in the closed range [0~1]. Because I'm doing this on the GPU (HLSL Shader Model 5) I can only use 32bit ...
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### Two easy questions for arithmetic progression

I was not able to solve either of these, I kept hitting many mistakes and it would be much appreciated if the solution to these two could be provided, thanks a lot in advance. If the sum of all the ...
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### Nonabelian group of order $p^3$ and semidirect products

Let G be a nonabelian group of order $p^3$ where p is an odd prime. Suppose that G contains an element of order $p^2$. Then G is isomorphic to the semidirect product $Z_{p^2} \rtimes_{\alpha} Z_p$, ...
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### Evaluating integral

Math question please take a look at the picture, Evalulate the $$\int_{\Gamma} z^2\:dz$$ where $\Gamma$ is the parabola arc $$y=x^2$$ running from $(0,0)$ to $(1,1)$. when I look at the solution, I ...
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### Find the line passing thought the point $p=(1,2,0)$, paralel to the plane…

Find the line passing thought the point $p=(1,2,0)$, paralel to the plane $P=\{x,y,z \mid x+2y-z=-4\}$ and crossing the line $L=\{(x,y,z):x+2y=2, y+z=4\}$ So I've tried to put the equation of plane ...
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### Representing Solutions to Homogeneous System Proof Help

I’m attempting to prove the following theorem If $x_1, \space \ldots \space, x_n$ is a linearly independent set of solutions to the $n$ x $n$ system $x’ = \space A(t)x$, then the general ...
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### A question about a projection of a variety

Let $\mathbb K$ be an algebraically closed field (of characteristic zero) and $H$ an irreducible variety in $\mathbb K ^n$. Let $t \in \mathbb K [x_1,\ldots,x_n]$ and let $T:= \mathsf V ( t )$ be the ...
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### Software for solving geometry problems symbolically

I've got Maple and it's excellent when it comes to solving math problems algebraically, but is there a counterpart for geometry problems? Such software would allow me to compose drawings in 2D, ...
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### Find the equation of plane containing line described by

Please help me in this really easy task Find the equation of plane containing line described by $x+3y-2z=1$, $2x-y+2z=3$, containing point $(1,1,3)$
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### Three body problem with point interactions

I've studied the HVZ theorem for the three body problem interacting with regular potentials. I'd like to extend this result to the three body problem with point interactions (delta potentials). Is ...
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### Bott periodicity

I studied (using Morse theory) Bott periodicity theorem for the unitary group $U(n)$: $\pi_{k}(U)=\pi_{k+2}(U)$. Do you know some interesting application of this result? Can this theorem help you to ...
The normal distribution of MLE is start $({x_1\ldots x_n})$ random sample from $N(u,\sigma^2)$ This family of distributions has two parameters: $\theta = (\mu, \sigma^2)$, so we maximize the ...
$\displaystyle \sum_{k<n}_{gcd(k,n)=1}k = \frac{1}{2} n \phi(n)$ This is a homework problem. I would ideally like to get to the final proof on my own. But at the moment I can't even decide how to ...