0
votes
0answers
2 views

Coordinate vector equation

I have the following bases which are bases of $\mathbb{R}^3$ $$B = ((1,1,1), (0,1,1), (0,0,1))$$ $$C = ((1,2,3), (-1,0,1), (1,0,1))$$ I need to find if this equation is correct $$[(1,2,3)]_B = ...
0
votes
0answers
3 views

At which sample size should you approx. a niid variable by z-statistic instead of t-statistic?

I have two datasets with both 50 samples ranging from 1965-2014. (The data contains the returns for W. Buffet over the 50 years and the returns from the S&P500, also over the 50 years.) I ...
0
votes
0answers
2 views

Closure under splitting?

Say there is a regular language $L\subseteq \lbrace 0,1\rbrace^\ast$ Would the language $L_1 =\lbrace w\in\lbrace 0,1\rbrace^\ast | w0\in L\rbrace $ also be regular? (i.e. $L = L_1\circ\lbrace ...
0
votes
0answers
3 views

Radius of convergence

Is this sequence of functions monotone? $$f_{n+1}(x)=\varphi(x)f_n(x)+\frac{1}{\varphi(x) f_n(x)}, \forall x \in[0,+\infty)$$ Where $\varphi:[0,+\infty)\to \mathbb{R}$, $1/2\leq \varphi(x) <1$ ...
2
votes
0answers
13 views

Differentiating both sides of an inequality with monotonic functions

If $f(x)\le g(x)$ for all real $x$ for monotonic functions $f$ and $g$ (say, both increasing), does it follow that $f'(x)\le g'(x)$? (Note: I've seen several questions asking the same thing without ...
0
votes
0answers
10 views

removable singularity iff null principal part of laurent series

A function $f:\Omega \to \mathbb C$ that is holomorphic on $\Omega \setminus {a}$is said to have a removable singularity at $a$ if there is an holomorphic function $g:\Omega \to \mathbb C$ such that ...
1
vote
0answers
12 views

Zariski closure of an infinite cyclic group of diagonal matrices

Suppose that $\Gamma=\{$exp $kX\mid k\in\mathbb{Z}\}$ where $X\in\mathfrak{gl}(n,\mathbb{R})$ is a diagonal matrix. How do we prove that the Zariski closure of $\Gamma$ must contain exp $tX$ for all ...
0
votes
0answers
6 views

Non-asymptotic error bound

I am looking for a sufficiently tight estimate of the following over the interval $[-T,T]$: $| \exp(t) - (1+t/n)^n|$. This is of course $o_n(1)$. What I am looking for is a non-asymptotic estimate ...
1
vote
0answers
13 views

Change of basis formula proof

So I know that this involves using the chain rule, but is the following attempt at a proof correct. Let $M$ be an $n$-dimensional manifold and let $(U,\phi)$ and $(V,\psi)$ be two overlapping ...
-2
votes
0answers
22 views

what is the legendre symbol of (18/43)

I know how to find legendre symbol and what a legendre symbol is but I am confused in using it for big numbers like 43 can any one help me and give a step by step answer.
0
votes
0answers
13 views

Finding normal curve given the minimum and maximum - is it possible?

I have a quick question regarding the normal distribution, or really any kind of distribution as it can also be skewed if need be. I was wondering if it were possible to let the curve have a minimum ...
1
vote
0answers
8 views

Specific requirements for Runge's theorem to hold

This question is exercise 8.2 in Conway's Functions of One Complex Variable I. It states: Let $\mathbb{D}\subset\mathbb{C}$ be the open unit disk, and let $K=\{z\in\mathbb{D}: \frac{1}{4}\leq ...
1
vote
0answers
11 views

Proof of a result on closed subgroups of Galois group

Let $M\supseteq K$ be an algebraic normal extension. Then its Galois group is profinite. I have been told that in this hypothesis, if $H\leq G$, then $H''=H\iff \bar H=H$, where ...
2
votes
0answers
11 views

question about direct sum of vector fields and preservation under quotient spaces

Hello all I was given this question in linear algebra it is two parts and asks to prove or give a counterexample. We are given a vector space V and a subspace of it W and the quotient map $ \pi : V ...
5
votes
1answer
37 views

Why $\lim$ of $\cos(f)$ equals to $\cos$ of $\lim(f)$?

Let $$\lim_{n\rightarrow \infty}\left(\cos\left(\frac{n\pi}{n+1}\right) \right) = \cos\left(\lim_{n\rightarrow \infty}\left(\frac{n\pi}{n+1} \right)\right)$$ Why the $\cos(x)$ function can be ...
1
vote
1answer
15 views

Conjugates of an $r$-cycle in $S_n$

How many conjugates does a cycle of length $r$ have in the permutation group $S_n$? I tried to find them but failed.
3
votes
0answers
17 views

Integral depending on a parameter

Task: find all values of the parameter, such that integral converges. $$\int_0^{+\infty} \frac{dx}{1+x^a \sin^2x}$$ I tried a lot and i used Cauchy and Weierstrass method but it was useless
0
votes
0answers
13 views

Growing slower than exponential?

Consider $2^{cn}+a(n)$ with $c$ being an exponential growth rate. Am I right that $$ 2^{cn}+a(n)\sim 2^{cn} $$ only, if $a(n)$ grows slower than exponential? And if yes.. when does $a(n)$ grow ...
1
vote
0answers
6 views

Subgraph isomorphism problem

Subgraph isomorphism problem is an NP-hard problem. However, if the subgraph size is constant (assume $k$), then it can be polynomial time solvable. The most easiest way is that: Randomly obtain $k$ ...
1
vote
1answer
30 views

How many person does the $n$th person know?

Arrange $N$ different people in a row. For each $k$ from $1$ to $N$, define $P(k)$ be a $k$th person in that row. Then $P(k)$ ($k$ is from $1$ to $N-1$) knows exactly $k$ people ...
1
vote
1answer
12 views

Joint probability distribution (over unit circle)

A couple of two continuous random variables $(X,Y)$ is distributed uniformly over the closed unity circle (so $-1\leq x \leq 1$ , $y$ analog). $U$ is defined as the distance from $O$ to the point ...
0
votes
0answers
22 views

Is a polyhedron an affine manifold?

I was reading the definition of an affine manifold (https://www.wikiwand.com/en/Affine_manifold) and was wondering if a polyhedron is an affine manifold. Could you also provide any hints to the proof ...
0
votes
0answers
6 views

Principal value methods for fourier laplace etc.

I recently saw this here: $$\int_{-\infty}^{\infty} \frac{1}{\omega^2} e^{j\omega t} d\omega$$ and I was unable to understand how such an integral could be computed. I want to learn about this ...
1
vote
0answers
9 views

Upper-hemicontinuity of product maps on compact metric spaces.

Let $X$ and $\{Y_i\}_{i\in I}$ be compact metric spaces (where $I$ an index set of possibly uncountable cardinality). Let $\Gamma_i$ be a compact valued, upper hemicontinuous (UHC) correspondence from ...
3
votes
1answer
96 views

What is the name of this paradox?

What is the name of the mathematical paradox which is arises from the following? If we imagine a point on a two-dimensional coordinate system (line graph), which moves from the positive part of the ...
1
vote
0answers
16 views

Subgroup of group is normal

This question came in the exam today, unfortunately I couldn't answer it. The question said: Proof whether or not this is a true statement, stating the reason. Subgroup of group is normal I ...
-1
votes
0answers
13 views

Simple interest ( Instalments)

A sum of Rs 10 is lent to be returned in 11 monthly instalments of Re 1 each, interest being simple. The rate of interest is? 100/11 % 10 % 11 % 240/11 % What I understand is this - Rs 1 for 11 ...
1
vote
1answer
9 views

Distributing infinite supply of $n$ distinct objects into $k$ identical urns

I have $n$ distinct objects, namely {$n_{1\le i \le n}$} with an infinite supply of each of them, and I have $k$ identical, indistinguishable urns to place the objects in. Each urn will contain ...
0
votes
1answer
12 views

Is this legitim to see polynomial growth?

Let $(a_n)$ be any monotoninally in creasing sequence where $a_n$ is a real number dependent on $n$. Of course it is $a_n\leq a_n^2$ for all $n\geq 1$. Does this means that any increasing sequence ...
0
votes
2answers
12 views

First order nonlinear ordinary differential equations

In my exercise I am stuck in a problem given below: $\ln\left(\frac{dy}{dx} \right) = x-y+1$ Although I could solve it if it was a linear equations. But ln() is a nightmare for me. Can anyone help me ...
0
votes
3answers
18 views

Proof of the construction of Dirac Delta

The Dirac Delta function pops up in a wide variety of applications, especially in applications that require Laplace and Fourier transforms. But my question is: what's the proof that the distribution ...
1
vote
0answers
9 views

Boxcar average algorithm of the specified width.

Ok, I need to write a java algorithm which simulates the SMOOTH function written in IDL where the SMOOTH function is given by $$ R_i = \begin{cases} \displaystyle \frac{1}{w} \sum_{j = ...
0
votes
0answers
13 views

Behavior of eigenvalues of certain matrices

I am trying to analyze the behavior of the 2 highest eigenvalues of matrices of this form : Symmetric $n*n$ matrices that contains only : $1/k$ (for fixed k), -1,1 and 0. My hope is to find some ...
3
votes
2answers
49 views

Solving differential equation $x''(t)=x^6$.

Solve the following differential equation $$x''(t)=x^6(t)$$ If I had $x'(t)$ instead of $x''(t)$ the exercise would have been easier for me. I would appreciate some help with this problem. Thank ...
1
vote
0answers
32 views

Question about your function,

I'm Xavier Vigan, a physical oceanographer. I've been using your $f(x)=\dfrac 12 \times \left(X+C-\sqrt{S+(X-C)^2}\right)$ function to calibrate quantile vs quantile plots. Because of the shape of ...
0
votes
1answer
16 views

Instalments ( simple interest)

The price of a T.V set worth Rs. 20,000 is to be paid in 20 instalments of Rs. 1000 each. If the rate of interest be 6% per annum, and the first instalments be paid at the time of purchase, then the ...
0
votes
2answers
26 views

Maximum of given expression?

Suppose $a,b,c>0$ and further that $a^{2} + b^{2} + c^{2}=2abc + 1 $. The problem is to find $\max \big(a-2bc\big) \big(b-2ca\big) \big(c-2ab\big) $. Give me some help. I've tried $X=a-2bc$, ...
0
votes
1answer
19 views

using matrix with cos/sin etc.

I need to check if the equation is linear independent so: $$ \alpha x^2 \cos x + \beta x + \gamma \sin x = 0 $$ I got 3 equations of it: $$\beta \pi/2 + \gamma = 0$$ $$\alpha \pi^2(-1) + \beta \pi = ...
0
votes
0answers
10 views

Translation:Bayes Classificator -> precise math?

I want to understand the most simple form of the Bayes classificator (see here) but I want to understand it in a really precise, clean, mathematical way. Math description of the setting: Let us ...
1
vote
1answer
13 views

Why boundary of a locally closed set is nowhere dense?

Let $X$ is locally closed , i.e. exist open $U$ S.t. $X=\overline{X} \cap U $ , and $bd (X) = \overline{X} \setminus \mathring{X} $. How can I show that $ bd(X) $ is nowhere dense? I read topics ...
1
vote
1answer
31 views

In $S_{3}$ what is the group generated by $(123)$?

In $S_{3}$ what is the group generated by $(123)$? Is there a way to find the elements of the group generated by $(123)$?
1
vote
2answers
31 views

definite integral of a complex function

I wonder if there is a way to evaluate this definite integral... $$\frac{2}{\pi}(\ln (2) + \int_{0}^{\infty}({\sqrt{\frac{1}{t^{4}} - \frac{4e^{-4t}}{(1 - e^{-4t})^{2}}}} - \frac{1}{t^2 + 1})dt)$$ ...
1
vote
0answers
11 views

Dimension of the restricted representation

My definition of restriction is: Let $H < G$, $\rho: G \rightarrow GL(V)$. The restriction of $\rho$ to $H$, $\rho: H \rightarrow GL(V)$. Its character is $$Res_H\chi(h)=\chi(h) \ \ \forall h \in ...
0
votes
2answers
19 views

If I have a matrix M=[A,B;0,C], how do I prove that rank(A)+rank(C)<=rank(M)?

. . . . . . . A . . B . . . . . . . 0 0 0 . . . 0 . 0 . C . 0 0 0 . . . If I have a matrix $M$ as displayed in the text above ($A$ ...
0
votes
2answers
19 views

$\Bbb{Z}_{2}(\alpha)$ as splitting field

i have problems with an exercise: let $\alpha$ be a root of the polynomial $X^{3}+X^{2}+1$ in $\Bbb{Z}_{2}$. Prove that $\Bbb{Z}_{2}(\alpha)$ is the splitting field of this polynomial over ...
0
votes
0answers
19 views

How to prove a point in a set is an extreme point of the set ?

Def: an extreme point of a set $K$ is the point that cannot be expresssed as a convex combination of other points in $K$. Apart from the definition, what else arguments can we use to prove that a ...
1
vote
0answers
14 views

Gradient w.r.t. boundary conditions in PDE

I am trying to solve the following problem. Suppose I have a field $\Phi(r)$, which is the solution to a partial differential equation: $\mathcal{L}\Phi(r) = s(r)$, as long as $r \neq r_0$ Here ...
0
votes
1answer
25 views

if the integrals of two function are equal then the functions are equal almost everywhere. true or false?

Actually I know that if the integral of a non negative function is equal to zero then that function is equal to zero almost everywhere. Can I use that to prove or is there a counter example for my ...
0
votes
1answer
20 views

Take the outcome of a draw in ELO formula

Is there any way to get the probability of a draw outcome using ELO formula as it only gives the Win probability ELO formula is given by $E = \frac{1}{1+10^\frac{d}{a}}$ where d is the difference in ...
2
votes
4answers
46 views

What is the idea behind a projection operator?

I know what a projection operator is, but I am unable to explain it in words without using mathematical symbols. Can any one help me?

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