0
votes
0answers
5 views

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 ...
0
votes
0answers
2 views

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 ...
0
votes
0answers
8 views

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 ...
0
votes
0answers
6 views

Covering number of the set of $n_1\times n_2$ matrices of rank at most $r$

What is the covering number of the set of $n_1\times n_2$ matrices of rank at most $r$? We know that the dimension of the set is $r(n_1+n_2-r)$. Thus, the covering number $N(\rho)\le C ...
0
votes
0answers
6 views

Proving a basis and dimension

Hi I'm currently looking at proofs in linear algebra and came across this one and I'm compketely baffled Suppose $ C_{ij} $ is the $2\times3$ matrix with $1$ in the $ i,j^{th} $ entry and zero ...
1
vote
1answer
14 views

Inner Product of center Z(G) of a Group

Let $G$ be a group and $Z(G)$ be its center. For $n\in \mathbb{N}$, define $$J_n=\{(g_1,g_2,...,g_n)\in Z(G)\times Z(G)\times\cdots\times Z(G): g_1g_2\cdots g_n=e\}.$$ Then $J_n$ is (1) not ...
1
vote
1answer
10 views

Proving a basis in linear algebra

So at the moment I'm trying to go through proofs and I came across this one: Suppose $ P_n $ is the vector space of all polynomials with degree less than or equal to n. Prove that $ \{1, x − 1, ...
0
votes
0answers
5 views

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 ...
0
votes
1answer
15 views

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 ...
1
vote
0answers
10 views

Show that for propositional logic $\vdash_i \neg \varphi \Leftrightarrow \vdash_c \neg \varphi$.

As the title says, where $\vdash_i$ is derivations in Intuitionistic logic and $\vdash_c$ is derivations in Classical logic. I am allowed to use a corollary that states that $\vdash_i \varphi ...
0
votes
0answers
7 views

Is the Shapely value of this voting game in the core?

Given a voting game where $v(1),v(2),v(3) = 0, v(1,2)= \frac{1} {3}, v(2,3) = \frac{5} {6}$, $v(1,3)= \frac{1} {6}$ and $v(1,2,3) = 1$ I know the Shapely coefficients for a 3 player game, for ...
0
votes
0answers
15 views

Integrability of $sgn(\sin(\frac{\pi}{x}))$ on $[0,1]$

Is $sgn(\sin(\frac{\pi}{x}))$ fit the basic setting of Riemann Integrability on $[0,1]$(bounded function, closed and bounded interval)? Since $sgn(x)$ is bounded for all $x$, so ...
0
votes
0answers
10 views

Metric compatibility of induced connection on submanifold of $\mathbb{R}^{n+1}$

Let $M\subset \mathbb{R}^{n+1}$ be a smooth submanifold with $\dim M =n $. Let $g$ be the induced metric on $M$ from the Standard metric on $\mathbb{R}^{n+1}$. Now, we define a connection on $TM$ by ...
1
vote
1answer
25 views

Is it possible to construct Hausdorff compact topology on every set?

I'd like to know if it's possible to construct Hausdorff compact topology on every set. Assume the axiom of choice if needed. Thanks for ideas.
0
votes
2answers
37 views

Find all positive inegers solution for $x^2-xy-y^2=1$

Find all positive inegers solution for the following diophantine equation $$x^2-xy-y^2=1$$ My work so far 1)$$x^2-xy-y^2-1=0$$ $$D=y^2+4(y^2+1)=5y^2+4=k^2, k \in \mathbb Z$$ 2)$$ ...
-2
votes
0answers
14 views

Finding the image of a vertical line under the function $f(z) = sin(z)$

My trouble is expressing an arbitrary vertical line as a complex function. I remember in class there was an example that included an angle. I think once I have this I should be able to do the rest. ...
-1
votes
2answers
34 views

Quick Question - Complex Roots of Polynomials?

I'm asked to solve for Z where $$\frac{z+i}{2z-i} = \frac{-1}{2} + i\frac{\sqrt 3}{2}$$ As a result i got $$2z = \sqrt{3}zi + \frac{i}{2} - i^2\frac{\sqrt 3}{2} - i$$ The answer is supposed to be ...
0
votes
0answers
8 views

How to lift an action along a covering

Took from Lawson-Michelsohn "Spin Geometry", pages 80-81. Let $E\to X$ be an oriented $n$-dimensional vector bundle over a manifold $X$, and let $P_{SO}E$ be the associated $SO(n)$-bundle. Assume ...
0
votes
3answers
33 views

How can I solve for the eigenvalues for this 3x3 matrix?

I know the eigenvalues are $0, 0, 6$ but I just don't understand how it was solved. To be more precise, please explain the last few steps. The matrix is $$\begin{bmatrix} 1 & 2 & 3 \\ 1 & ...
3
votes
1answer
42 views

If $BA = I$, prove that $AB = I$ (using determinants)

I've seen this problem around here, but I wanted to check if this particular solution is right. So, if $BA = I$, then $det(B)det(A) = 1$, meaning neither $det(B)$ or $det(A)$ are equal to $0$. ...
1
vote
0answers
16 views

Is there a Mobius (infinite) cylinder?

In order to understand the question of the title I need to understand another thing first. If we consider the Mobius band, locally, for a $U_i \subset S^1$, where $S^1$ is the base space, the bundle ...
2
votes
0answers
11 views

Abstraction and Genaralization

This is a (bit funny) ultra-soft question regarded to a type of thinking that is puzzling me. Suppose $a(p_{1}, p_{2}, p_{3}, p_{4}, ... , p_{n}, Q)$ denote: from the items $p_{i}$, find a common ...
1
vote
2answers
15 views

Partly solving an underdetermined system of equations

Assume that $A=\{a_{ij}\}$ is an $M\times N$ matrix where $M<N$ and $Ax=b$ where $x$ is the vector of unknown variables and $b$ is a known binary vector. Assume $a_{ij}$ values are also binary. ...
1
vote
4answers
28 views

If $a,A,b,B,c,C$ are non negative reals such that $a+A=b+B=c+C=k$ Prove that $aB+bC+cA \le k^2$

If $a,A,b,B,c,C$ are non negative reals such that $a+A=b+B=c+C=k$ Prove that $aB+bC+cA \le k^2$ I substituted $B=k-b,C=k-c,A=k-a$ and plugged them to get a quadratic of $k$ which I had to show ...
0
votes
0answers
14 views

Riemann Curvature tensor for surfaces

Let $M$ be a regular surface on $\mathbb{R}^3$. I am trying to express the Riemann's curvature tensor: $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ respect $R(\vec x_i,\vec x_j)\vec ...
0
votes
3answers
25 views

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 ...
2
votes
2answers
21 views

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, ...
-1
votes
1answer
16 views

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 ...
1
vote
1answer
14 views

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 $A$.
0
votes
2answers
40 views

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 ...
1
vote
1answer
14 views

$star(v)$ 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 ...
0
votes
0answers
7 views

$x\cos(x)=-\frac{1}{2} \sin(x) + 2\sum_{n=2}^{\infty} \frac{(-1)^n n \sin(nx)}{n^2-1}$ 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 ...
-1
votes
0answers
8 views

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 $M$ ...
1
vote
1answer
12 views

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 ...
-1
votes
0answers
20 views

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 ...
0
votes
0answers
9 views

covering number of cartesian product of manifolds

Let there be two manifolds $P$ and $Q$, with their covering numbers $N_P(\rho)$ and $N_Q(\rho$), respectively. Is it true that the covering number for the Cartesian Product $P\times Q$, $N_{P\times ...
1
vote
0answers
25 views

Functional equation in natural numbers $x+y|f(x)+f(y)$ [on hold]

Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that $x+y|f(x)+f(y)$ and for $x\ge1395$ we have $2f(x)\le x^3$. What I've tried so far: For $x=y$ we get $2x|2f(x)$ and $x|f(x)$. It's ...
0
votes
1answer
19 views

Express $e^{cos(i)}$ in the form $x+iy$

Does this seem good so far? Since cosine can be expressed as $\frac{e^{iz}+e^{-iz}}{2i}$, $cos(i) = \frac{e^{i^2}+e^{-{i^2}}}{2i} = \frac{e^{-1}+e}{2i}$ Then $e^{cos(i)} = e^{\frac{e^{-1}+e}{2i}} ...
0
votes
2answers
16 views

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 ...
2
votes
2answers
67 views

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 ...
0
votes
1answer
26 views

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- ...
0
votes
1answer
18 views

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 ...
1
vote
1answer
41 views

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 ...
2
votes
1answer
34 views

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 ...
-1
votes
0answers
10 views

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 ...
1
vote
0answers
15 views

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 ...
-1
votes
0answers
12 views

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 ...
1
vote
0answers
38 views

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 ...
0
votes
0answers
6 views

Integration over D-dimensional euclidean space

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 ...
0
votes
0answers
53 views

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: ...

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