-1
votes
1answer
38 views

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

Prove that $\forall m\in \mathbb Z$, $$\sum_{n=-\infty}^{\infty} \frac{1}{n+m+\frac{1}{2}}\equiv 0$$
3
votes
2answers
615 views

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 ...
1
vote
2answers
190 views

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 ...
2
votes
2answers
243 views

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 \begin{equation} I: \mathcal{A} \to \mathbb{R} \end{equation} the functional that needs ...
4
votes
2answers
383 views

“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 ...
2
votes
1answer
37 views

$f(x) = \inf_{y \in Y} c(x,y) - \inf_{\xi \in X} c(\xi,y) - f(\xi) \Rightarrow f$ 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 ...
1
vote
3answers
290 views

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 ...
2
votes
1answer
199 views

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 ...
1
vote
1answer
249 views

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 ...
1
vote
0answers
61 views

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 ...
8
votes
3answers
4k views

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 ...
4
votes
2answers
71 views

Calculate:$\int \frac{1}{(x+1)^\frac{3}{4}(x+2)^{\frac{5}{4}}}\ dx$

Calculate following integration $$\int \frac{1}{(x+1)^\frac{3}{4}(x+2)^{\frac{5}{4}}}\ dx$$
1
vote
2answers
65 views

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 $a|d$?
0
votes
1answer
59 views

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 ...
1
vote
1answer
92 views

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 ...
1
vote
1answer
212 views

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.
4
votes
2answers
104 views

Intersection of two localizations

Let $A$ be a commutative ring with unity. If $\mathfrak p,\mathfrak q\in \operatorname{Spec} (A)$ is it true the following equality $$A_\mathfrak p\cap A_\mathfrak q= A_{\mathfrak p\cup \mathfrak ...
2
votes
1answer
164 views

A problem with lambda calculus notation and semantics for function-valued functions

I would like to understand how to use the $\lambda$-notation to write usual (set-theoretic) functions, and if it is possible at all. Here are my naïve attempts. Assume that all variables are ...
0
votes
1answer
125 views

How to do complex integration. E.g. $ \int_\frac{\pi}{2}^{\frac{\pi}{2} + i} \cos(2z) \; \mathrm{d}z $

For my homework assignment I've been given a number of complex integrals to solve. I've already asked for help on a specific example here, but I was somewhat dissatisfied with the answers. The answers ...
4
votes
4answers
1k views

How to calculate the number of pieces in the border of a puzzle?

Is there any way to calculate how many border-pieces a puzzle has, without knowing it's width-height ratio? I guess it's not even possible, but I am trying to be sure about it. Thanks for your help! ...
0
votes
2answers
96 views

Checking the existence of a solution for a set of linear equality and ineaulity equations

I would like to know if there is a method to check the existence of the solution for a given set of linear equations composed with both equalities and inequalities? I'm not interested in the solution, ...
13
votes
2answers
529 views

Proving that $\pi=\sum\limits_{k=0}^{\infty}(-1)^{k}\left(\frac{2^{2k+1}+(-1)^{k}}{(4k+1)2^{4k}}+ \frac{2^{2k+2}+(-1)^{k+1}}{(4k+3)2^{4k+2}}\right)$

Long time ago I've been playng with formulas for $\pi$ and found that one above in the title which can also be expressed as \begin{align*} ...
4
votes
3answers
133 views

Prove $P(x)>0, \forall \space x>1$

$P(x)$ is a polynomial function such that $P(1)=0;$ $$P'(x)>P(x), \forall \space x>1 ;$$ Then prove that $$P(x)>0, \forall\space x>1;$$
2
votes
2answers
75 views

Prove that $f_{\beta'} = (f_{\beta})^{-1}$

I'm stuck on this problem for few days and can't find the solution.Hope some one here can help me solve this. I'm so grateful for any any help: Let $V$ be a finite-dimensional vector space ...
0
votes
1answer
61 views

Homogeneous Systems and Symmetric Matrices?

I’m having trouble showing that given a symmetric matrix $A$, the homogenous system $x’ = Ax$ will be stable (ie it has a fundamental matrix whose entries all remain bounded as $t \rightarrow ...
0
votes
1answer
354 views

Lipschitz constant for optimization of multivariate function

I intend to implement an optimization algorithm which requires the computation of the Lipschitz constant. My function is a multivariate function with more than 50 variables. I am wondering whether ...
4
votes
1answer
139 views

Solve in the positive real number the system of equation.

Solve in the positive real numbers, the system of equations : $$(2x)^{2013} + (2y)^{2013} + (2z)^{2013} =3 $$ and $$xy + yz + zx + 2xyz = 1.$$
2
votes
2answers
128 views

Why a “set” whose elements is contingent on the truth value of a proposition is a set?

Let $P$ be a proposition (such as Goldbach's conjecture) that we can state precisely, but that we do not necessarily know to be true or false. Now define $$ S = \begin{cases} ...
4
votes
2answers
579 views

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, ...
0
votes
1answer
90 views

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 ...
2
votes
2answers
76 views

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?
4
votes
2answers
1k views

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 ...
0
votes
5answers
153 views

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 ...
0
votes
2answers
400 views

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 $Z_{p^2} \rtimes_{\alpha} Z_p$, ...
1
vote
1answer
97 views

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 ...
2
votes
2answers
95 views

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 ...
0
votes
1answer
52 views

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 ...
1
vote
0answers
86 views

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 ...
1
vote
4answers
100 views

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, ...
1
vote
4answers
99 views

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 $(1,1,3)$
1
vote
0answers
36 views

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 ...
2
votes
2answers
87 views

$ \int_0^i\!e^z\,\mathrm{d}z $

I have a function that I need to integrate. I can't figure out how to do so though. $$ \int_0^i\!e^z\,\mathrm{d}z $$ My intuition is telling me that the answer is $ e^i - 1 = \cos(1) + i\sin(1) - 1 ...
-1
votes
1answer
759 views

Does uniform convergence of a sequence of functions imply uniform continuity of its limit?

Assume that $f_n$ converges to $f$ uniformly on $C$ and that each $f_n$ is uniformly continuous on $C$. Prove that $f$ is uniformly continuous on $C$.
4
votes
2answers
345 views

How fast can you determine if vectors are linearly independent?

Let us suppose you have $m$ real-valued vectors of length $n$ where $n \geq m$. How fast can you determine if they are linearly independent? In the case where $m = n$ one way to determine ...
1
vote
3answers
176 views

How do I find $\left|\langle a,b\mid a^2=b^3=e\rangle\right|$?

Suppose $G$ is a group satisfying $G=\langle a,b\mid a^2=b^3=e\rangle$. Find $|G|$.
3
votes
2answers
669 views

When finding root, does Newton's method fail if the function is non-differentiable?

According to wikipedia's description, the Newton's method finding a root presumes a differentiable function. Then, will it fail when encountering non-differentiable function? For example, can it find ...
1
vote
1answer
45 views

Using an ideal as an index set

In Algorithmic Algebra by Mishra, the sum and intersections of submodules of $M$ (an $S$-module) are defined using an ideal $I \subseteq S$ as an index set for a family of submodules, $\mathcal M = ...
0
votes
0answers
138 views

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 ...
0
votes
0answers
82 views

What is the MLE if the variance is heteroskedasticity

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 ...
1
vote
1answer
77 views

Why is this fact about the totient function true? [duplicate]

$\displaystyle \sum_{k<n}_{gcd(k,n)=1}k = \frac{1}{2} n \phi(n)$ 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 ...

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