1
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0answers
137 views

Pear Orchard (word problem)

A farmer has $70$ acres on which to plant a pear orchard. three neighboring farms with similar soil conditions already established orchards. one of these orchards has $250$ tress planted per acre ...
1
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0answers
38 views

Formula of arc length

In some notes that I am reading there is the following: $$(\delta s)^2=(\delta x)^2+(\delta y)^2 \Rightarrow \left (\frac{\delta s}{\delta x}\right )^2=1+\left (\frac{\delta y}{\delta x}\right ...
1
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0answers
31 views

What kind of topological sorting exists in graph theory and what are their graphic plotting?

I know that there is at least two kinds of topological sorting: "by rank" and by"level" a level of a vertex is the maximal length of a path with x as an extremity. a rank of a vertex is the maximal ...
1
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0answers
46 views

How to calculate Laurent series coefficients?

I have the following Laurent development for $g(z) = \sin(1/z^3)$ : which is defined for all complex except 0. I have this formula to calculate the coefficient, but we have no example at all how ...
1
vote
0answers
88 views

Mean squared error consistency of estimator

Given is the following distribution: $f_\theta(x)=\frac{1}{\theta}$ if $0<x\leq\theta$, and $0$ otherwise; $\theta<0$. I need to show that the maximum likelihood estimator of $\theta$, ...
1
vote
0answers
13 views

Showing $\int_\Omega f(h(w))dP=\int_S f(y)d(P\circ h^{-1})$

Why does the following hold ? If $h:\Omega\to S$ then; $\displaystyle\int_\Omega f(h(w))dP=\int_S f(y)d(P\circ h^{-1})$ I know the change of variable formula: ...
1
vote
0answers
38 views

Proof of the Künneth Formula

I would like to proof the following version of the Künneth formula: $0\to \oplus_{p+q=n} H_p(C_*)\otimes H_q(C_*')\to H_n(C_*\otimes C_*')\to \oplus_{p+q=n-1}Tor(H_p(C_*),H_q(C_*'))\to 0$ is an short ...
1
vote
0answers
16 views

Is the map $\mathbb R^n\longrightarrow \mathscr{D}^\prime(\mathbb R^n)$, $x\longmapsto k(x, \cdot)$, continuous?

Recall, $a\in C^\infty(\mathbb R^n\times \mathbb R^n)$ is a symbol of order $m\in\mathbb R$ if for every $\alpha, \beta\in\mathbb N_0^n$ there is $C_{\alpha, \beta}>0$ such that ...
1
vote
0answers
33 views

Relation between orbitals and cosets of point stabilizer

Let $G$ be a group acting on a set $\Omega$. Then there exists a natural action of $G$ on $\Omega \times \Omega$ given by $(\alpha, \beta)^x = (\alpha^x, \beta^x)$. The orbits on $\Omega\times \Omega$ ...
1
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0answers
47 views

How to prove an exponential function preserve the positive semi-definite property?

If $f(x) \in \Re$ has the positive definite property, $\sum_i\sum_j a_i a_j f(x) \geq 0$ for $a_i,a_j \in \Re$, then $e^{f(x)}$ has the positive definite property. How can i prove it? And, i guess a ...
1
vote
0answers
30 views

Image of a ray under a complex map

Let $f(z) = \frac{1}{2}( z+ z^{-1})$ for $z \in \mathbb{C}\setminus \{0\}$. We want to see what the image of a ray in the complex plane is under this map. So let $z = re^{i\theta}$. Then: ...
1
vote
0answers
17 views

Discuss the recurrence of such a Markov chain

$\{X_n:n\ge0\}$ is a Markov chain in $\{0,1,2,\ldots\}$ whose transition probability satisfy $p_{01}=1, p_{i,i+1}+p_{i,i-1}=1$ and $$ p_{i,i+1}=(\frac{i+1}{i})^\alpha p_{i,i-1}, ...
1
vote
0answers
23 views

Difference of statistics of order of a exponential distribution

Let $X_1, ..., X_n$ be a random sample of a exponential distribution with mean $1$. Is there an easy way to show the following: $$(n - i + 1)(X_{(i)} - X_{(i-1)}) \stackrel{idd}{\sim} ...
1
vote
0answers
37 views

Calculating remaining balance [Monthly compounding question with rate= 0.01]

Suzanne opens a line of credit at a local bank, and immediately borrows 1870 dollars. 6 months later, she repays 1060 dollars. 5 months after the repayment, she borrows 570 more dollars. 6 months ...
1
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0answers
45 views

Is it possible for a Lebesgue integral of an unmeasurable function to be finite?

Let f be defined on $\mathbb{R}^d,$ suppose $\int_{\mathbb{R}^d} |f(x)|\,dx<\infty$. Is f then measurable? My question boils down to "Do I have to check that a given function is measurable, or can ...
1
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0answers
34 views

The $2$-norm of $A^*A$ is equal to the square of the norm of $A$

Show that $\|A^*A\|_2 = (\|A\|_2)^2$, where $A^*$ is the tranpose of $A$. My approach: $$\|A^*A\|_2 = \sqrt{\lambda_{\max} [(A^*A)^*(A^*A)]}= \lambda_{\max} (A^*A)=\|A\|_2^2$$ It that right?
1
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0answers
34 views

Dimensions of various lie groups by counting parameters of matrix representations

Show that the groups $\text{GL}(N,R), \text{GL}(N.C), \text{SL}(N,R), \text{SL}(N,C), O(N), SO(N), U(N),SU(N)$ have dimension $N^2, 2N^2, N^2-1, 2(N^2-1), \frac{1}{2}N(N-1), \frac{1}{2}N(N-1), N^2$ ...
1
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0answers
44 views

$G$ has exactly $n$ different subgroups of prime order $p$, then total number of elements of $p$ is $m(p-1)$

$A$ and $B$ are two distinct subgroups of group $G$ s.t. $o(A)=o(B)=p=prime$, show that $A\cap B=\{e\}$. Deduce that if $G$ has exactly $n$ different subgroups of prime order $p$, then total ...
1
vote
0answers
29 views

Creating a grid of coloured points

I wish to create a grid of $31\times31$ points with coordinates of the form $(\frac k{30};\frac n{30})$ within the unit square, and give each point a colour based on the 2-adic value of both its ...
1
vote
0answers
22 views

Binary operations and proofs

$\text{binary}(a)$ = binary representation of a base 10 number $a$ Are the following statements correct? If yes, where can I find the proofs? (1) $\text{binary}(a\times b)=\text{binary}(a)\times ...
1
vote
0answers
31 views

The expansion of the product of the sum of polynomials

Guys, I want to expand the product of the sum of polynomials into the form of $\sum\limits_{}^{} {a{x^b}} $, $\begin{array}{l} I = \left( {\frac{{\lambda _1^0}}{{0!}}{x^0} + \frac{{\lambda ...
1
vote
0answers
30 views

Can a non-homogeneous equation be solved by substitution normally done in homogeneous ones?

I have got a question in my material that's, $(y+3x^2)\frac {dx}{dy}=x$ For solving it I did following, $\frac {dy}{dx}= \frac {y}{x} + 3x$ and then substituted $y=vx$ i.e $\frac {dy}{dx} = v + ...
1
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0answers
26 views

Any way to simplify this integral?

$$\int_{-\infty}^{\infty}e^{-y\left(x\right)}\left(\log\int_{-\infty}^{\infty}e^{-y\left(z\right)}N\left(x-z,v\right)dz\right)dx $$ where $y(x)\equiv\sum_{i=1}^{n}c_{i}x^{i}$, $n$ is even and ...
1
vote
0answers
45 views

(Very) Twisted bundles

Suppose that $M$ is a smooth manifold. Over each coordinate chart $(U,\varphi)$ $M$ looks like euclidean space. In particular the tangent bundle $TM$ is trivial over $M$. So you the covering of $M$ by ...
1
vote
0answers
59 views

Proof that $\sum_{i=1}^\infty \mathbb{I}_{ \Gamma_i}$ is a Poisson Random Measure on $[0,\infty)$

For $X_i \sim \text{i.i.d}~ \text{expo(1)}$ defined on $E=[0,\infty)$, and $\Gamma_i = \sum_{i=1}^nX_i$. I want to show that $N=\sum_{i=1}^\infty \mathbb{I}_ {\Gamma_i} $ is a homogenous Poisson ...
1
vote
0answers
44 views

How to show the failure of the homotopy extension property for the pair $I=[0,1]$ and $A=\left\{ 0,1,\frac{1}{2},\frac{1}{3}…\right\}$?

I'm facing difficulties regarding a statement which Hatcher makes in page 14 of his 'Algebraic Topology' book-that there's no continuous retraction from $I\times I\rightarrow ...
1
vote
0answers
27 views

Diagonalizing a matrix perturbatively

Say we have two matrices $H$ and $J$, both of which are diagonalizable. Let $\lambda$ be some small parameter. What's the best method for diagonalizing the sum $H + \lambda J$, working perturbatively ...
1
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0answers
65 views

System of differential equations - disagreement with paper

I have the following system of differential equations: $$A'-\frac{m}{r}A=(\epsilon-1)B \tag{1}$$ and $$B'+\frac{m+1}{r}B=-(\epsilon+1)A \tag{2}$$ where $A$ and $B$ are functions of $r$ and $A'$ and ...
1
vote
0answers
103 views

Bounding reachability of damped harmonic oscillator using barrier certificates

I'm trying to prove that, under certain conditions, a damped harmonic oscillator that starts on one side of the equilibrium remains on that side of the equilibrium. More precisely, consider the ...
1
vote
0answers
36 views

Convergence in $L_p$ does not imply “a.s.” convergence

I'm having a hard time understanding some examples about this that I found on stack exchange. So I'd like to ask for the simpler possible example where a sequence of random variables $\{T_n\}$ ...
1
vote
0answers
101 views

Show that the Cauchy problem $x''=x|x|,x(0)=a,x^{\prime}(0)=b$ ,has a unique solution for all $a,b\in \Bbb R$.

Show that the Cauchy problem $x''=x|x|,x(0)=a,x^{\prime}(0)=b$,has a unique solution for all $a,b\in \Bbb R$. (The Problem is from A text book on ordinary differential equation by Shair Ahmad and ...
1
vote
0answers
21 views

Name of operation: changing the root of a rooted tree

What is (if any) the name of the operation of changing the root of a rooted tree? Picking a vertex which is not the root, then reorienting the edges in such a way that the vertex becomes the root?
1
vote
0answers
53 views

A question about the proof of a theorem in representation theory

My question is about some parts of the proof of Theorem 8.1.10 from the book "A Course in the Theory of Groups" by Derek J.S. Robinson. To prove Theorem 8.1.10, we want to prove that there is a ...
1
vote
0answers
48 views

Proving that proper maps are compact

I have just proved the result that if, $f:X \to Y$ is compact then for any $B\subset Y$ the map $g: f^{-1}(B) \to B$ is also proper. So in particular we can say that the constant map,$f^{-1}(y) \to ...
1
vote
0answers
38 views

Doubt while proving that an irreducible matrix lie algebra representation implies an irreducible matrix lie group representation

Let $G$ be a connected matrix Lie group with Lie algebra $g$. Let $\Pi$ be a representation of $G$ acting on $V$ a finite dimensional vector space, and $\pi$ the associated representation of $g$. ...
1
vote
0answers
35 views

$\lim_{\alpha\rightarrow 0}\left(\alpha^{(m+1)/m-2/n}\int_0^{\tan^{-1}(\alpha^{-1})}\sin^{1/m}(\theta)\cos^{2(1-n)n-1/m}(\theta)\:d\theta\right)=$?

Is there any way to evaluate the limit \begin{align}\lim_{\alpha\rightarrow ...
1
vote
0answers
54 views

Integrating $\int \frac{e^{\cos x} \cdot \sin x}{2 + \cos x}$

In an exam this morning, I was given a definite integral that can supposedly be solved using substitution: $$\int_a^b \frac{e^{\cos x} \cdot \sin x}{2 + \cos x} dx$$ (I forget what the limits were, ...
1
vote
0answers
57 views

If $0< 2\varepsilon < \sigma^2 < 1$ then $\prod\limits_{i = 1}^n (1 + \varepsilon + \sigma \xi _i )$ converges almost surely to $0$

I posted this question a few days ago and there were some errors in my post. I have fixed them and it should be all right now. Hope someone can help with my confusion. Let $(X_n)_{n\ge0}$ denote an ...
1
vote
0answers
29 views

Nilpotent differential operators of degree $2$?

I've reflecting about global solvability of a class of (pseudo-)differential operators. In the middle of my researchs a condition like $P^2=0$ appeared where $P$ is a differential operator. However, ...
1
vote
0answers
40 views

Size of group of roots of unity

Let $p$ be an odd prime and $G=(\mathbb{Z}/p^N\mathbb{Z})^{\times}$. Let $a=p^{N-1}$, $b=p-1$, $A=\{g\in G: g^{a}=1\}$, $B=\{g\in G: g^{b}=1\}$. Prove that $A,B$ are subgroups of size $a,b$ ...
1
vote
0answers
56 views

Showing Markov kernel properties

For $a\in\mathbb R$ and $\sigma^2>0$ let $\phi_{a,\sigma^2}$ be the Gauss distribution on $(\mathbb R, \mathcal B)$ with the expectation $a$ and variance $\sigma^2$. Show that ...
1
vote
0answers
68 views

Minimize the number of non-zero elements of a matrix

I have a matrix A, which is huge, and all its elements are non-zero. I want to perform the operation: $ U_1\otimes U_2 \otimes U_3...\cdot A \cdot U_1^{\dagger}\otimes U_2^{\dagger}\otimes ...
1
vote
0answers
34 views

Variance and conditional independence

Random variables $X_1,X_2,...$ are conditionaly independent, given a random variable $\Theta$ and their conditional distibutions satisfy $E(X_i|\Theta)=10\Theta$ and $Var(X_i|\Theta)=100\Theta^2$. $N$ ...
1
vote
0answers
31 views

Build polynomial orthogonal to set of other pre-defined polynomials

Basically. the question is simple. Is there any algorithm so I can build a polynomial, orthogonal for the set of pre-defined polynomials? I need the algorithm(like Gram-Schmidt) that would be ...
1
vote
0answers
37 views

Graph having bounded degree

A graph is said to have bounded degree if there exists $N \in \mathbb{N}$ such that, for every $x \in V$, one has $\sum\limits_{y \in V} A_{x,y} \le N$. Show that, in this case, for any $f \in ...
1
vote
0answers
24 views

Change of the sequence of differentation in physics?

Assume to have a quantity A which is calculated from the formula $A=\frac{dB}{dC}$. dC can be written as dC=dEdF so $A=\frac{dB}{dEdF}$. I assume that the differential of A is also ...
1
vote
0answers
61 views

Ordinal Numbers within Simple Type Theory

A little preamble: I recently came across the system of higher-order logic known as simple type theory, and I was immediately intrigued by it, as it seemed to be exactly what I was looking for as an ...
1
vote
0answers
18 views

Verbiage in book, …uniformly in $\theta$ belonging to compact subsets…

I am reading Statistical Estimation and Asymptotic Theory by Ibragimov and Has'minskii and was confused by the following passage: ... and that we have uniformly in $\theta$ belonging to the compact ...
1
vote
0answers
60 views

Prove that $\mathbb{Z}$-module is flat.

Let $p$ be prime and let $\mathbb{Z}_{(p)} \subset \mathbb{Q}$ denote the set of all fractions $n/q$ for which $p \nmid q$. Is it true that $\mathbb{Z}_{(p)}$ is flat as a $\mathbb{Z}$-module? ...
1
vote
0answers
35 views

How to sample multivariate random normals?

Suppose $\Omega=(0,1)$, $\mathcal{B}$ is the Borel $\sigma$-algebra on $(0,1)$ and $P$ the Lebesgue measure. Let $\Phi:\mathbb{R}\to(0,1)$ denote the standard normal CDF. Then, we can generate $X\sim ...

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