# All Questions

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### Prove that $e^(2pi*i/5)$ is not in the 7-th cyclotomic field.

Let $\xi_n = e^\frac{2\pi i}{5}$. Prove that $\xi_5 \notin \Bbb{Q}(\xi_7)$ where $\Bbb{Q}(\xi_7)$ is the 7-th cyclotomic field. How would I approach this question? I'm having a difficult time coming ...
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### For which $s$ is defined $\frac{1}{\zeta(s)}\sum_{n=1}^\infty\frac{n}{ \left( \zeta(s) \right) ^n}$, where $\zeta(s)$ is the Riemann Zeta function?

If there are no mistakes in my words, if one of the factors $\sum_{n=1}^\infty a_n$ of a convolution product or Cauchy product converges, and the other $\sum_{n=1}^\infty b_n$ corverges absolutely ...
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### Proofing that the Lucas numbers come closer to the Phi rounded numbers then the Fibonacci numbers.

Morning everyone, Bit of background, I'm a mid level programmer with very limited mathematics skills. As part of an assessment for a new role I've been asked to complete a technical task which ...
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### Forcing Eigenvector elments to zero

I have a large sparse eigenvalue problem of the form: $A\mathbf{x}=\lambda\mathbf{x}$. The problem resembles an electromagnetic problem. Is there a general way of manipulating the system matrix to ...
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### How to determine if a triangle is inside another triangle without any intersecting sides

This question is for getting the right logic down for a programming task. I need to be able to determine if a triangle is located inside another without any sides intersecting each other. The two ...
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### What does the function domain with letter C stand for?

I am reading a mathematics textbook on the subject of numerical analysis. In one theory the author says let us assume $f$ to be a function in $C^{n+1}[a,b]$. I understand that $[a, b]$ is the ...
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### Integral of bounded function with limit zero at $\pm \infty$

Very simple question here, I almost feel bad for asking it.. Lets say we have a function bounded between $0$ and $1$. This function is high dimensional: $0<f(X) \le1, ~~~ X \in \mathbb{R}^D$ Now, ...
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### Sequence converging to a closed set

How can I prove that A sequence generated by a particular algorithm converges to a set? i.e irrespective of the starting point the sequence converges to any of the points in a set, or oscillates ...
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### Relation between $v$ and column space of matrix $A=(I+vv^T)$

Consider the identity matrix with a symmetric rank-one update, i.e., $A=I+vv^T$. Is there any relation between $v$ and the column space of $A$.
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### Find $\int_0^{\infty} \frac{dx}{1+e^x}$

$$\int_1^\infty\frac{dx}{1+e^x}$$ $$\lim_{M\to\infty}\int_1^M\frac{e^xdx}{e^x(1+e^x)} \\ u= 1 + e^x \\ du = e^x dx \\ \lim_{M\to\infty} \int_{1+e}^{1+e^M} \frac{du}{(u-1)u}$$ I then found the ...
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### $star(v)$ for a 0 simplex

Suppose you have a simplicial complex and a vertex $v$ which is not connected to any other vertex. Is $st(v)$ just the empty set? If you're looking at the inside of a simplex you don't look at ...
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### $x\cos(x)=-\frac{1}{2} \sin(x) + 2\sum_{n=2}^{\infty} \frac{(-1)^n n \sin(nx)}{n^2-1}$ for $x\in (-\pi,\pi)$

I am trying to establish the following $x\cos(x)=-\frac{1}{2} \sin(x) + 2\sum_{n=2}^{\infty} \frac{(-1)^n n \sin(nx)}{n^2-1}$ for $x\in (-\pi,\pi)$ The right sight looks the the Fourier expansion of ...
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### How to explain polynomial coefficients by minimezed Error function?

We wish to predict ${\bf{t}}$ from an observed $\bf{x}$.We shall fit the data using a polynominal function of the form$$y({\bf{x}},{\bf{w}})=w_0+w_1x+w_2x^2+...+w_Mx^M=\sum_{j=0}^{M}w_jx^j$$ where $M$ ...
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### Accurate summation of mixed-sign floating-point values

Due to the finite representation, floating-point addition loses significant bits. This is particularly noticeable when there is catastrophic cancellation, such that all the significant bits can ...
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### Sequence of integrable functions [on hold]

let $f_n\left(x\right)$ be a sequence of integrable functions defined on a closed interval. If $f_n\left(x\right)$ approaches to zero, and the derivatives of $f_n\left(x\right)$ are uniformly bounded ...
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### bounded but not convergent sequences

I am not sure that if this question has a positive answer...I am looking for a sequence of real numbers $(p_{n})_{n\geq 1}$ such that $-1<\lim _{n}\inf p_{n}\leq \lim_{n}\sup p_{n} <1$ (as ...
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### Number of ways of visiting N places

A tourist wants to visit $N$ cities, each numbered from $1$ to $N$, but he wants to visit them in a weird order. A weird order is such in which no city numbered $i$ is the $i$-th to visit in his ...
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### Series proof needed

I have following equations but I do not know the proof. Kindly provide the proof or give me some reference to look into. Here are the equations. 1- ...
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### How to use monotone convergence theorem to show that $\int \sum |f_n| = \sum \int |f_n|$

In https://math.la.asu.edu/~quigg/teach/courses/473/2009/lectures/11integral.pdf, pg 4 The monotone convergence theorem was used to state $$\int \sum |f_n| = \sum \int |f_n|$$ without proof. Can ...
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### Why is The Following equality true? (limit of a sum and integrals)

I saw the following equality: $$\lim_{n\to\infty} \sum_{k=1}^{n}\left[ \frac{1}{(1+\frac{k}{n})^3}\right]\dfrac{1}{n} = \int\limits_{0}^{1} \dfrac{1}{(1+x)^3}dx$$ Why don't we divide the integral by ...
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### Find the minimum $k$

Find the minimum $k$, which $\exists a,b,c>0$, satisfies $$\frac{kabc}{a+b+c}\geq (a+b)^2+(a+b+4c)^2$$ My Progress With the help of Mathematica, I found that when $k=100$, we can take ...
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### Let $M=A^{T}A$ be a positive semidefinite matrix. Is it true that for $i \neq j$, $|m_{ij}| \leq \frac{m_{ii}+m_{jj}}{2}$ [duplicate]

Let $M=A^{T}A$ be a symmetric positive semidefinite matrix. Is it true that for $i \neq j$, $|m_{ij}| \leq \frac{m_{ii}+m_{jj}}{2}$? Where $m_{ij}$ is an element of matrix $M$, and $i$ represents the ...
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### An optimisation problem

I have an optimisation probem given below $$argmax_{x_i \ \ \forall x_i=1,2...n} \sum_{i} S_ie^{-\alpha x_i}$$ subject to $$\sum x_i = 1$$ $$\sum C_i x_i \leq B$$ $$\sum x_i \geq 0$$ where ...
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### Mean value of periodic function

$f(t) = \begin{cases} A \sin\Omega t, & {-{T\over 2}\le t \le 0} \\ 0, & {0 \lt t \lt {T\over 2}} \end{cases}$ where $A, \Omega, T$ are constants If I want to calculate the mean value of ...
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### What's the integral of $\csc^2(\frac{1}{x})$?

It's known that $\int\csc^2(x)dx = -\cot(x) + C$, but I don't know how to integrate $\int\csc^2(\frac{1}{x})dx$ and it looks much harder. Can you help? Answer of this question may give some hints to ...
I have to show that in D-dimensional Euclidean space: \begin{align} \int d^D q=\frac{2\pi^{D/2}}{\Gamma(\frac{D}{2})}\int d q^{D-1} \end{align} By the way shouldn't it be $\int dq q^{D-1}$ if it ...
### What is the value of $k^2$
For all $f(x)$ and $g(x)$ functions that are differentiable in $\mathbb{R}$, and satisfy the following conditions: Condition A: $$f(1)=1,~f(3)=3.$$ Condition B: ...