1
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0answers
60 views

Expectation of couples surviving after some time.

There are $2m$ persons forming $m$ couples who live together at a given time. Suppose that at some later time, the probability of each person being alive is $p$, independently of other persons. At ...
1
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0answers
53 views

Endomorphism ring of semisimple modules

Can someone help me to prove the following statement, or introduce some reference about it? Statement: Suppose that $M$ is a semisimple $R$-module of finite type. There are division rings $D_1, ... ...
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0answers
47 views

Four dimensional integral calculation

I do some calculation in QED, but I can not calculate such integral $$ I(a_1,a_2,m_1,m_2)=\int\frac{ d\mathbf{x} d\mathbf{y}}{(1+\mathbf{x}^2)(1+\mathbf{y}^2)((\mathbf{x}+a_1 ...
1
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0answers
46 views

How to solve the 2-dim linear differential equation?

Since for any periodic equation $A(t)$, the differential equation $x'(t)=A(t)x(t)$ has the principal matrix solution. It's trivial for one-dim case, we have $x=\exp(\int_{t_{0}}^{t}a(s)ds)$ where ...
1
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0answers
131 views

Is my answer for the composite relation correct and not the textbook's?

Note: This example is from Discrete Mathematics and Its Applications [7th ed, example 5 pg 593] Here is how my textbook's way of representing a relation with a matrix And the definition of a ...
1
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0answers
42 views

Characterisation of subgroups of finite index of $SL_2(\mathbb Z)$

Is there a complete characterisation of subgroups of finite index of $SL_2(\mathbb Z)$?
1
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0answers
21 views

How is quasiconcavity and the sign of cross partial related?

Suppose $f(x,y)$ is differentiable. If $f(x,y)$ is quasiconcave, is the cross partial nonnegative? Or, if $f(x,y)$ is strictly quasiconcave, is the cross partial strictly positive? My initial ...
1
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0answers
29 views

why and when t0 use norm instead of abs and vice versa

What is the difference between the norm and abs of an expression.. as far i understand does ||a - z|| mean norm and |a-z| abs , but what is the difference?
1
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0answers
87 views

Identify subsets of $\mathbb{N}$ with their characteristic functions

If $G\subseteq2^{\mathbb{N}}$ is comeager then exist is a partition $\mathbb{N}=A_0\cup A_1$, $A_0\cap A_1=\emptyset$, and sets $B_i \subseteq A_i$ for $i \in \{0,1\}$, such that for $A \subseteq ...
1
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0answers
20 views

If $X \sim N(\mu,\sigma^2)$, then $\int^t_sxf(x)dx=\sigma [f(s)-f(t)]+ \mu [F(t)-F(s)] $?

Here is my work, kindly let me know if this is correct: \begin{align*}\int^t_sxf(x)dx=&\int^{\frac{t-\mu}{\sigma}}_{\frac{s-\mu}{\sigma}}(\sigma z+\mu)\frac{\phi(z)}{\sigma}\sigma dz \\=& ...
1
vote
0answers
41 views

Solution of a partial integro-differential equation

I want to find a solution for the following equation $$ \partial_y u(x,y) + \partial_x u(x,y) + \int_{0}^{x} r(x-x') u(x',y) dx' $$ $$ u(x,0)=0 \quad u(0,y)=\delta(y) $$ in $(x,y) \in ...
1
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0answers
134 views

Probability that the median of three numbers is in a certain range?

I'm given numbers $a_{1}$ to $a_{n}$ with the caveat that all of those numbers are distinct and $n$ is a multiple of 4. How can I prove that if I select three of those numbers at random, there is an ...
1
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0answers
33 views

Fibers of polynomial as submanifold

Let $f\in \mathbb{R}[X_1,...,X_n]$ be a homogeneous polynomial of degree $d$. Let $F_{a}=\{(x_1,...,x_n)\in\mathbb{R}^n : f(x_1,...,x_n)=a\}$. For which $a$ $F_a$ is a submanifold in $\mathbb{R}^n$ ? ...
1
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0answers
77 views

Are split exact sequences exact in the opposite direction?

In an abelian category, let $$0\longrightarrow A \overset{f}{\longrightarrow}B\overset{g}{\longrightarrow}C\longrightarrow0$$ be a split short exact sequence with $\ell f=1_A,gr=1_C$. Is the sequence ...
1
vote
0answers
29 views

Distribution of the increments of a Compound Poisson process

Let $X_t$ be a compound Poisson process defined as $X_t = \sum_{i=1}^{N_t} D_i$, where $D_i$ are i.i.d. and $D_i \sim Exp(\mu)$ and $N_t$ is a Poisson process with parameter $\lambda$. As usual the ...
1
vote
0answers
55 views

Show the series converges uniformly

Let $f(x) = \sum_{n=1}^\infty \frac{(-1)^n}{\sqrt n} \arctan\left(\frac{x}{\sqrt n}\right)$. Show that $f(x)$ converges uniformly. First, it is easy to see that the series converges for every $x$ by ...
1
vote
0answers
41 views

Matrix convexity!

Given $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$, if $\mathsf{rank}(M-Q_i)=\mathsf{rank}(Q_i)$ where $i\in\{1,2\}$ with $Q_i\in\Bbb R_{\geq0}^{n\times n}$, then if $\forall ...
1
vote
0answers
64 views

Borel sigma algebra, generated by borel functions, representation of a point

Please let me ask a question about $\sigma$-algebra. Let $E$ be a Hausdorff space and $\mathcal{B}(E)$ denotes its Borel $\sigma$-algebra. Let $(u_{n})_{n \in \mathbb{N}}$ be a family of ...
1
vote
0answers
36 views

To show the function holomorphic in $\mathbb{C}$

Let $f \in H(\mathbb{C} \setminus \mathbb{R})$ and $f$ continuous in $\mathbb{C}$. Then to show that $f ∈ H(\mathbb{C})$. I thought I should apply Morera's theorem. SO I need to show that for all ...
1
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0answers
23 views

Alternative proof of the differential equation

Prove that: If $y=(x^2-1)^n$, then $y^{n+2}(1-x^2)-2xy^{n+1}+n(n+1)y^n=0$, where $y^n$ is nth derivative of $y$ My solution: Taking log on both sides; $\log y=n \log (x^2-1)$ $\implies ...
1
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0answers
53 views

Generalized Riesz theorem of operator value function

I am reading a book Gustafson-Rao's Numerical Range and come across a problem that I really don't understand. In theorem 2.1-2 of the book, it asserts that for an operator valued function ...
1
vote
0answers
39 views

Limit points of a subset of a topological space.

Determine the set of limit points of $A=\{1/n+1/m : n,m \in \mathbb Z^+\}$ in the standard topology on $\mathbb R$. I think that the limit points of $A$ is $A'=\{0\}$. Am I correct? How would I prove ...
1
vote
0answers
27 views

Prove $m_*( \bigcup_{i=1}^\infty I_i ) = \sum_{i=1}^\infty \ell ( I_i )$ if $I_i \cap I_j = \emptyset \forall i,j$.

Define $m_*(E) = \inf_{E \subset \bigcup_{i=1}^\infty J_i} \sum_{i=1}^\infty \ell(J_i)$ where $\bigcup_{i=1}^\infty J_i$ is an open cover of $E$ (i.e. an outer measure in $\mathbb{R}$). Let $\ell$ ...
1
vote
0answers
73 views

According to Liouville's theorem, why is the measure on an energy-surface different from the measure on the phase space in general

I recently read Khinchin's derivation of Liouville's theorem. I was able to follow the math for the most part, however I was hoping for an intuitive understanding about why the form of the measure on ...
1
vote
0answers
37 views

Significance of orthonormal basis in wavelet analysis

I've recently been looking into wavelet analysis and I have the question: What is the importance of wavelets having an orthogonal basis, say as opposed to a bi-orthogonal basis or otherwise? I'm ...
1
vote
0answers
16 views

Calculating gain ratio from a dB value

In a practice problem I have: power gain = $\log_{10}(\frac{db}{20})$ The final answer for the ratio is 1. The dB value is $-3$. When I do $\log_{10}(\frac{3}{20})$ I get $-0.823$. Just wondering ...
1
vote
0answers
32 views

Continuity at a point II

For a function $f$ to be continuous at a point $x_0$, is it necessary that there exists a ball centered at $x_0$ such that $f$ is continuous at all points within this ball? Put another way, does $f$ ...
1
vote
0answers
65 views

Basis pursuit denoising

The Lagrangian form of basis pursuit denoising $\min_{w} ||w||_{\ell_{1}} + \lambda ||Aw-x||_{\ell_{2}}^{2}$ can be solved using proximal gradient descent. Proximal methods also can be used to ...
1
vote
0answers
35 views

orthogonal polynomials: explicit representation

Consider a sequence of orthogonal polynomials $P_0(x) = 1$, $P_1(x) = x$, and recursively $P_{n}(x) = (a_n x + b_n) P_{n-1}(x) + c_n P_{n-2}(x)$ for some sequences of real constants $a_n$, $b_n$, ...
1
vote
0answers
47 views

$SO(3)$ has a subgroup $U(1) \times U(1)$?

I am wondering - and asking you - whether there is a subgroup $U(1) \times U(1)$ of the Lie group $SO(3)$. Equivalently, I can reformulate it from a geometrical point of view: does there exist a torus ...
1
vote
0answers
33 views

Convolution of which distribution will give a uniform distribution?

Suppose there are two IID random variables x1 and x2. What should be the distribution of these random variables so that the distribution of x1-x2 is a uniform distribution?
1
vote
0answers
40 views

Problems in understanding the notion of “active” diffeomorphisms on a manifold

I have been studying differential geometry recently in the hope of gaining a deeper understanding of General Relativity (GR). I have run into an issue when trying to understand the notion of what is ...
1
vote
0answers
172 views

How to evaluate solid angle subtended by an ellipse at any arbitrary point on the vertical axis passing through the center

`Question: How to evaluate the exact value of solid angle subtended by an ellipse (or elliptical plane) at any arbitrary point lying on the vertical axis passing through the center. Standard equation ...
1
vote
0answers
72 views

Proof of Frobenous Coin Problem lower bounds (Chicken McNugget theorem)

For the Frobenius Coin Problem, where $n = 3$ http://en.wikipedia.org/wiki/Coin_problem Does anyone know a proof for Davidson's formula? The one that states that the lower bound for the Frobenius ...
1
vote
0answers
31 views

Supports of elements in $\ell_\infty$

It is idle curiosity that makes me ask so apologies for no motivation. Let $X$ be an infinite-dimensional, closed subspace of $\ell_\infty$. Can we find a non-zero element $x\in X$ such that ...
1
vote
0answers
66 views

Imaginary part of An Squre Root Integration

I am looking for a particular form of an integral which some simplified version of it has the following form $$ \Im\int_{0}^{\infty} \frac{\sqrt{1+u^4-u^6}}{u^5}du. $$ Could someone gives some idea ...
1
vote
0answers
54 views

Structure of the semidirect product decomposition

I'm looking at a complicated group that involves many semidirect products, and I realized that I have a fundamental confusion about how to use the structure of a semidirect product decomposition of a ...
1
vote
0answers
113 views

Definition of $\Omega$-algebra

I'm studying universal algebra. I have this definiton: given a signature $\Omega$, an $\Omega$-algebra is comprised of a "carrier" for the algebra and an "interpretation" for every operation symbol. ...
1
vote
0answers
33 views

Oscillation of a disc about a rod perpendicular to the disc but not through the centre.

A thin uniform circular disc of radius a and centre A, with density p, has a circular hole cut in it of radius b and centre B, where $AB = c < a−b$. The disc is free to oscillate in a vertical ...
1
vote
0answers
63 views

Why is it that the angle around any arbitrary closed curve is 2 pi?

I'm trying to motivate the idea of a solid angle and so I decided to relate it to the regular angle. And I realized I'm a little hazy on that as well. Why is it that the angle around any arbitrary ...
1
vote
0answers
139 views

What does it mean by density argument?

When I read some books, it states "a density argument", what does it mean? When applying to symmetric function, what does it mean?
1
vote
0answers
38 views

Coset state for non-abelian hidden subgroup problem

On page 14 of his classic paper, Quantum factoring, discrete logarithms and the hidden subgroup problem, Jozsa introduced a function $f$ for the non-abelian hidden subgroup representation of the graph ...
1
vote
0answers
66 views

Show that $1-(-1)^n +1/n$ is divergent.

My process was to take each of the two sequences and say that we know $(-1)^n$ is divergent because its two sub-sequences have different limits, and $\frac{1}{n}$ is convergent because it converges ...
1
vote
0answers
192 views

Probability of rolling n dice to match another set of dice, d, given r rolls (like yahtzee)

(Note: I will eventually code this, but i'm primarily interested in the math behind it) I'm trying to create a function in Java to calculate the probability of getting a desired outcome from n rolled ...
1
vote
0answers
45 views

solutions to $X^T X +BX= C$

I am trying to get a handle on solving the following matrix equation (for $X\in \mathbb{R}^{n\times n}$): $$X^T X + BX = C,$$ where $B, C\in \mathbb{R}^{n\times n}$. This form crops up in an ...
1
vote
0answers
33 views

Evaluating this limit - help

Let $g_n(x) = $ \begin{cases} \hfill x^n \sin(\frac{1}{x}) \hfill & \text{ if} \space x \neq 0 \\ \hfill 0 \hfill & \text{if} x= 0 \\ \end{cases} Then why is $g_2'(0) = ...
1
vote
0answers
73 views

Surfaces, vector fields, and the Lie bracket.

$\textbf{Theorem:}$ Suppose that $X_1, \dots, X_k$ are vector field on a manifold $M$ and at a point $p \in M$, we have that $X_1(p), \dots, X_k(p) \in T_p(M)$ are linearly independent, then the Lie ...
1
vote
0answers
64 views

Trace in finite Fields

(1) In $\mathbb{F}_{2^n}$ with odd n it should be shown that half of the Trace values are 0 and the other part is 1 with help of the additivity of the Trace. (2) Now n shall be even. Now the ...
1
vote
0answers
50 views

Have there been any attempts to unify statistics and decision theory into a single framework that refrains from estimating probabilities?

If I understand correctly: statistics, narrowly construed, is all about using data to estimate probabilities. decision theory can then be applied to those probabilities in order to predict which ...
1
vote
0answers
24 views

Finding Lifetime Incidence Based on Annual Incidence

This question should be simple, but I can't wrap my head around it. The thing I can't wrap my head around most of all is how to calculate lifetime incidence among a whole population based on a certain ...

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