0
votes
0answers
12 views

Definition of an inverse-powerseries

Let $t(q)=\sum_{n=0}^{\infty}t_n q^n$ be a complex powerseries convergent for all $|q|<1$. Assume $t_0=0$ and $t_1\neq0$. Not it says Let $q(t)$ be the local inverse of $t(q)$ with $q(0)=0$. ...
1
vote
0answers
11 views

Homotopy type of mapping space

In Ralph Cohen's notes on the topology of fiber bundles (pp.63) he claims that the space of all $G$-equivariant maps from $P$ to $EG$ denoted by Map$^G(P,EG)$ is aspherical, where $EG$ is the total ...
1
vote
1answer
12 views

Proof of unicity of decomposition of a representation

I'm studying representation theory and in the book the author makes the following proposition with the following proof: Proposition: For any representation $V$ of a finite group $G$, there is a ...
-1
votes
2answers
36 views

Probability density function for product and minimum of i.i.d. $U(0,1)$ random variables

If $U$ and $Y$ and $Z$ are i.i.d. $U(0,1)$ random variables, find the pdf for $A= U \times Y$ and $B = \min \{ U,Y,Z\}$.
0
votes
0answers
44 views

Can anyone solve this geometric construction problem?

I remember when I was in high school, one of my all-time favorite books was College Geometry by Nathan Altshiller-Court. Some of its problems kept me wondering for days and even weeks. Now after about ...
3
votes
1answer
21 views

Can $\mathbb{Z}/6\mathbb{Z}$ act freely and properly discontinuously on $\Sigma_4$?

Let $\Sigma_m$ denote the closed connected orientable surface of genus $m$. Let $N_m$ denote the closed connected non-orientable surface of genus $m$. I was wondering which cyclic groups could act ...
0
votes
2answers
25 views

$f$ is factored into many same degree irreducible polynomials.

I met a problem when I study about Galois field and do this exercise. Hopefully, someone can help me. Suppose that $L/K$ is normal extension and $f$ is an irreducible polynomial in $K[X]$. Prove that ...
1
vote
2answers
97 views

Proof verification: Every Euclidean space is complete

To prove this, I would like to use induction. For $n=1$ it is easy to prove that $\mathbb{R}$ is complete. For $n=k$ we assume it is true. For $n=k+1$, we have to show that $\mathbb {R}^{k+1}$ is ...
1
vote
1answer
17 views

Tangent space to the intersection of two manifolds

Let $M, N \subset \mathbb{R}^n$ be two manifolds such that, for every $p \in M \cap N$, $(T_pM)^\bot \cap (T_pN)^\bot = \{0\}$. How do I determine the tangent space of $M \cap N$? I found some places ...
3
votes
1answer
27 views

How to reverse a certain formula

I've got this formula which gives me the experience required for a specific level. $25X^2-25$ Which gives me a nice experience table like this Experience from level X to X+1 From level 4 to 5, 600 ...
0
votes
1answer
27 views

$S_4/H \simeq S_3$ where $H$ is a normal subgroup

Prove that the group of permutations of four symbols $S_4$ contains a normal subgroup H such that the quotient group $S_4/H$ is isomorphic to the group of permutations of three symbols $S_3$. ...
0
votes
2answers
65 views

Preferential Attachment and salton similarity in directed networks

Preferential Attachment similarity between two nodes in an undirected graph is the degree of the first node multiplied by the degree of the second node. But what about directed graphs? Which degree ...
0
votes
1answer
17 views

On a linear non-homogeneous system of differential equations.

I rewrite my attempt at solving this system \begin{cases} x'(t) = 3x(t) + y(t) + e^{2t} \\y'(t) = - x(t) + y(t) + e^t\\ x(0) = 1 \\ y(0) = 0 \end{cases} I notice that the eigenvalue of the matrix ...
1
vote
0answers
13 views

Generalized function in term of the Dirac-$\delta$-function

Personal question : Could it possible for the distribution distribution $<tr(e^{it \sqrt{\Delta}},\varphi)>=<-i \ln(1-e^{it}), \varphi'>$ to be expressed in term of the ...
1
vote
0answers
4 views

Fourier series of piecewise-defined function and convergence

I'm learning about Fourier series and need help with the following problem: Consider the function $$g(x) = \begin{cases} x^{\frac{1}{3}}, & x \in [0, \frac{\pi}{2}] \\ ...
2
votes
1answer
63 views

Number of solutions to $x+y+z = n$

If $\beta(n)$ is the number of triples $(x, y, z)$ such that $x + y + z = n$ and $0 \le z \le y \le x$, find $\beta(n)$. Attempt: I think there are many cases to look at to find $\beta(n)$. We ...
4
votes
2answers
57 views

Solving $y^y = x$ for large $x$?

I was playing around with recurrence relations and notice that $\sqrt x$ has the fun property that it (and its negation) are the only functions $f(x)$ where $$\frac{x}{f(x)} = f(x).$$ That got me ...
0
votes
0answers
18 views

Limit of Jacobi elliptic functions

I would like to compute the limit when $k\rightarrow 0$ of $$\frac{1}{k}\,\left(\,Z(\frac{u}{k},k) - dn(\frac{u}{k},k)\,\right)$$ where $k$ is the modulus of the Jacobi elliptic function $dn$ ...
0
votes
1answer
30 views

Suppose $0<a<b$. Prove for all $n\geq 2$, $0< \sqrt[n]a< \sqrt[n]b$.

Let a and b be real numbers, and suppose $0<a<b$. Prove for all $n\geq 2$, $0< \sqrt[n]a< \sqrt[n]b$. Proof: Suppose there exists an $n\geq 2$ such that $0 \geq \sqrt[n]a ...
1
vote
1answer
10 views

Statement verification - Stable and unstable manifold theorem

Let $\dot{X} = f(X)$ have hyperbolic fixed point $\overline{X}$ and linearisation $\dot{X} = Df(\overline{X})X$. Then there exists a stable manifold $W^s_{\overline{X}}$ of dimension $d_s$ and an ...
0
votes
0answers
8 views

Appropriate Generalization of Statement about Pure Subgroups to Pure Submodules

I have been working in a book on Homology by Hilton & Stammbach, wherein they introduce the idea of a "pure sequence of Abelian groups", which is a short exact sequence of Abelian groups ...
1
vote
2answers
29 views

Minimizing an integral — Hilbert space

Find the real values of $a, b$ which minimize $$\int_1^{\infty} \left| \frac{1}{x^2} - a \frac{1}{x^3} - b\frac{1}{x^4}\right|^2 \; dx.$$ Hint : Work in an appropriate Hilbert space. Here is ...
0
votes
1answer
20 views

Find widest subset on which Fourier series can be integrated and derived term by term

As part of one problem I need to find the widest subset of $\mathbb{R}$ on which the obtained Fourier series can be integrated and derived term by term. I found that it has something to do with ...
1
vote
2answers
486 views

Cardinality of a set of matrices

Consider the set $S$ of $3\times3$ matrices with binary coefficients, that is, the coefficients are integers modulo $2$. Compute $|S|$. I am not sure what is this question trying to ask. Am I ...
-3
votes
0answers
51 views

Prove $\int_{-1}^{1} (1-x^2)dx = \int_{0}^{1}(\sqrt{x}-(-\sqrt{x}))dx$

I can calculate both integrals: $\int_{-1}^{1} (1-x^2)dx = \int_{0}^{1}(\sqrt{x}-(-\sqrt{x}))dx$ and verify they yield the same value $\frac{4}{3}$, but is there any way to prove they are the same ...
1
vote
1answer
87 views

Prove an inequality using complex analysis

If $f:\mathbb{D}\rightarrow\mathbb{D}$ is holomorphic then prove that $$\frac{|f(0)| - |z|}{1 + |f(0)||z|} \leq|f(z)| \leq\frac{|f(0)| + |z|}{1 - |f(0)||z|} $$ I have been wracking my brain for ...
3
votes
2answers
41 views

Why do we have to add a term for every exponent when integrating by partial fractons?

For example, to integrate: $1/(1+x)^3$ you can write it as a partial fraction as $A/(1+x) + B/(1+x)^2 + C/(1+x)^3$ but i don't really understand why. I found that it is analogous to representing for ...
0
votes
1answer
25 views

Lebesgue measure as $\sup$ of measures of contained compact sets

I know, from Kolmogorov-Fomin's Элементы теории функций и функционального анализа, the definition of external measure of a bounded set $A\subset \mathbb{R}^n$ as $$\mu^{\ast}(A):=\inf_{A\subset ...
1
vote
1answer
22 views

Volume of intersection of the $n$-ball with a hyperplane

Let $\mathcal{B}_n$ be the $n$-ball of radius $r>0$ and centre $\mathbf{x}_0$, i.e., $\mathcal{B}_n=\{\mathbf{x}\in\mathbb{R}^n\colon \|\mathbf{x}-\mathbf{x}_0\| \leq r\}$. The volume of ...
-1
votes
0answers
9 views

Elliptic Curve Problem (finding a factor of n by using elliptic curve)

Let n be a a composite integer such that q is a factor of it. Consider the elliptic curve E defined by $y^2=x^3+3x+36$. The point $P=(0,6)$ is on the curve. Suppose it is given that the order of P mod ...
1
vote
3answers
37 views

Are all solutions to the ODE $ay''(t) + by'(t) + cy(t) = 0$ of the form $y(t)= \alpha e^{(\beta + i\gamma)t}$?

Let $a$ $b$ and $c$ be complex numbers. Consider the complex solution of the ODE $$ay''(t) + by'(t) + cy(t) = 0.$$ If there exist solutions to this, are they necessarily of the form $$y(t)= \alpha ...
3
votes
3answers
87 views

numbers from $1$ to $2046$

We have randomly taken $21$ integers from $1$ to $2046$. Show that we can take $a$, $b$ and $c$ from the previous $21$ integers in a way such that the following inequality holds \begin{equation} ...
1
vote
0answers
26 views

Probability generating functions of coin tosses

I have just came across a weird definition for the probability generating function of a random variable $N$ that denotes the integer value for the $n^{\mathrm{th}}$ toss on which the coin turned out ...
4
votes
1answer
51 views

number of ways to partition an integer.

A partition of a positive integer n is a way of writingn as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. For example, 4 ...
0
votes
3answers
30 views

How is the entropy of the normal distribution derived?

Wikipedia says the entropy of the normal distribution is $\frac{1}2 \ln(2\pi e\sigma^2)$ I could not find any proof for that, though. I found some proofs that show that the maximum entropy resembles ...
1
vote
2answers
23 views

If $A$ is a matrix with negative eigenvalues, then $\exists M$ : $A = -MM^T$

Let $A$ be a symmetric matrix with all its eigenvalues negative. Prove that there exists a matrix $M$ such that : $A = -MM^T$. Now, regarding my question, I have found another older question, that ...
2
votes
0answers
8 views

Distribution $<tr(e^{it \sqrt{\Delta}},\varphi)>=<-i \ln(1-e^{it}), \varphi '>$ in term of Dirac-$\delta$-function

Personal question : Could it possible for the distribution distribution $<tr(e^{it \sqrt{\Delta}},\varphi)>=<-i \ln(1-e^{it}), \varphi'>$ to be expressed in term of the ...
5
votes
0answers
30 views

The negation of the continuum hypothesis as an axiom [duplicate]

Suppose that one considers the formal theory "ZFC+~CH", or the set of all axioms of ZFC plus the negation of the continuum hypothesis. Then we have that there exists at least one set S whose ...
3
votes
0answers
23 views

Prove $(x+y)(y^2+z^2)(z^3+x^3) < \frac92$ for $x+y+z=2$

$x,y,z \geqslant 0$ and $x+y+z=2$, Prove $$(x+y)(y^2+z^2)(z^3+x^3) < \frac92$$ While numerical method can solve this problem, I am more interested in classical solutions. I tried this problem for ...
0
votes
0answers
25 views

Rational matrix existence problem.

Given large coprime $A,B$ with $\frac12<A/B<B/A<2$ can we find coprime $C,D\approx A^2$ with $\frac12<C/D<D/C<2$ such that for some $\frac a{a'},\frac b{b'},\frac c{c'},\frac ...
1
vote
1answer
27 views

Extending function to holomorphic

Good day, I am having some problems with this question. Our supervisor's notes gave an answer, but she doesn't know how to get to it (it was set by an old lecturer). Given a continuous function ...
0
votes
1answer
15 views

How do calculate not emty set levels of function $f$?

Let $f$ be defined as follows: $$f:\mathbb{R}^{2}\to\mathbb{R}:(x,y)\mapsto\begin{cases}\frac{xy^{2}}{x^{2}+y^{4}}&\text{if } (x,y)\neq (0,0)\\ 0&\text{if } (x,y)=(0,0)\end{cases}$$ I put ...
1
vote
1answer
36 views

Generalized eigenvector space with $\lambda_1=\lambda_2=\lambda_3=2$

Let be $$M:=\begin{pmatrix} 1 & 1 & 1 \\ -1 & 3 & 1 \\ 0 & 0 & 2 \end{pmatrix}$$ The characteristic polynomial would then be ...
0
votes
0answers
6 views

Bounding Problem /Conditions for the Lebsgue Integral of a Function depending on two parameters to be continuous

let $F(t)=\int_{E} f_t(x)$ for $t\in J \subseteq R$. Then some theorem says that $F$ is continuous if $1)$ $\forall t_0$ $f_t(x) \rightarrow f_{t_0}(x)$ as $t \rightarrow t_0$ almost everywhere on ...
4
votes
1answer
33 views

Density of a set of numbers.

Firstly, I introduce a notation. $\Bbb{N}$ denotes the set of natural numbers, $0$ included. For $E \subseteq \Bbb{N}$ and $n \in \Bbb{N}$, I denote by $$\pi_E(n) = |E \cap \{ 1, \dots , n\}|$$ and ...
0
votes
0answers
29 views

Characterization of prime homogeneous ideals

Let $R$ be a graded ring and $I$ ideal in $R$ and homogeneous. $I$ is prime if and only if for all $a, b\in R$ homogeneous such that $ab\in I$ then $a\in I$ or $b\in I$. Let $ab\in I$ and $a ...
1
vote
0answers
30 views

Perturbation of the principal eigenvector of a PSD matrix

I have a $n \times n$ PSD matrix $A$ and $\tilde{A}=A+E$ be its symmetric perturbation. Let $\|E\|_2=\epsilon.$ Let $(\lambda,u)$ be the principal eigenvalue, eigenvector pair of $A$ and ...
0
votes
0answers
25 views

If $F$ is closed and $g:\Bbb R\to\Bbb R$ show that $g(x)=\inf\{|x-a|:a\in F\}$ is continuous

First attempt: I want to show that because $g(x)$ is the composition of continuous functions then is continuous but I dont know how to show that if $$\min\{d_1,d_2\}=\frac12(d_1+d_2-|d_1-d_2|),\text{ ...
3
votes
0answers
18 views

Random Binary matrix

This is a question from Strang's "Linear Algebra and its Applications", right in the first chapter (I'm studying it by myself). I couldn't solve it, it isn't in the Solutions Manual, and my research ...
6
votes
1answer
48 views

Integral inequality :$\int_0^1(f'(x))^2dx\geq 32\int_0^1(f(x))^2dx + 16\left(\int_0^{\frac{1}{2}}f(x)dx-\int_{\frac{1}{2}}^1f(x)dx\right)^2$

Assume $f:[0,1]\to \mathbb{R}$ is differentiable and $f'$ is integrable. Given $f\left(\frac{1}{4}\right)=f\left(\frac{3}{4}\right)=f(1)-f(0)=0$, then prove that $$\int_0^1(f'(x))^2dx\geq ...

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