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Let $X=(X_1,\dots,X_n)$ be a random variable on $(R^{n},B^{n},P)$, calculate the conditional disribution of X given $\sigma(f), f(x)=x^TD^{-1}x$

Let $X=(X_1,\dots,X_n)$ be a random variable on $(R^{n},B^{n},P)$, and follow the n-dimensional multivariate norm distribution $N(0,D)$. Let$$f(x)=x^TD^{-1}x,x\in R^{n}.$$ Calculate the conditional ...
James Lee's user avatar
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3 views

Covariance of the product of two random variables with another random variable

Let X, Y and V be three binomial random variables, that are NOT independent from each other. My objective is to find an expression for the covariance \begin{align}\operatorname{Cov}(XY,V)&\end{...
CafféSospeso's user avatar
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0 answers
10 views

Number of natural numbers with an odd sum of sum of divisors till that number

Find the number of natural numbers $n<2023$ such that $\sum_{i=1}^n \sigma_0(i)$ is odd, where $\sigma_0(x)$ denotes the number of divisors of $x$. First, I know that $\sigma_0(x) = (n_1 +1)(n_2 +...
Sahaj Satish Sharma's user avatar
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25 views

When is the power rule for limits acceptable to use?

Suppose you wish to calculate the following limit: $$\lim_{x\to\infty}\frac{(1+x^2)^{-a}}{e^{-bx}}$$ for real numbers $a,b>0$. My instinct here is to rewrite as $$\lim_{x\to\infty}\left(\frac{1+x^2}...
wrb98's user avatar
  • 1,325
1 vote
0 answers
29 views

Notation used within paper on commutative rings

I was reading through this paper and I came across some rather confusing notation in the proof of Theorem 4. It says Proof. ...$Y$ is finite, and it is clear that the image of the restriction of any ...
AVP Neelam's user avatar
1 vote
0 answers
27 views

Exercise 2.5.8 Pedersen's Analysis Now

I'm trying to do the following problem, and I'm getting hung up on one part. Here is the problem: Let $f: X \longrightarrow \mathfrak{X}$ be a continuous map from a compact Hausdorff space $X$ into a ...
Isochron 's user avatar
  • 1,059
-4 votes
0 answers
15 views

Calculate G(s) = Y (s)/U(s) from the block diagram in Figure 4.

Calculate G(s) = Y (s)/U(s) from the block diagram in Figure 4.
student 1's user avatar
2 votes
1 answer
12 views

Depiction of convergence in categorical sense

It is well-known that for a product space $\Pi_{\alpha \in A} X_{\alpha}$, a sequence of points $\{\mathbf{x}_n\}$ converges if and only if for each $\alpha \in A$, $\{\pi_{\alpha}(\mathbf{x}_n)\}$ ...
SalutaFungo's user avatar
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0 answers
24 views

$[G:H]=4$ where $H\neq 1$ means that $G$ is not simple.

The question is as follows: Suppose $G$ is a finite group with a nontrivial subgroup of index $4$. Prove that $G$ is not simple. I am on the hunt for a non-trivial normal subgroup. This proof ...
Chris Christopherson's user avatar
-1 votes
0 answers
14 views

Alternative form for $\sum^{\infty}_{n=1} \sin(ny)\sin(ny')$

I have this expression $$\sum^{\infty}_{n=1} \sin(ny)\sin(ny')=\frac{\pi}{2}\delta (y-y')$$ I want to know if it possible to express this in a different manner when I have $y'=0.$ I am asking this ...
Gusklin's user avatar
-1 votes
0 answers
6 views

Can LM algorithm be used to optimize complex number problems?

In the electromagnetic field, there are NxN points on the incident plane. By changing the phase of the electric field at these points, the desired electric field distribution is generated in the ...
chong er's user avatar
2 votes
0 answers
32 views

Prove that if $f: A \to B$ is an injective function and $h: A \to C$ is any function, then there always exists a function $g: B \to C$ with $h = gof$.

I'm not an incredibly well versed proof writer and I would like some help with this proof. I really want to know if I'm along the right track or if I need to be going in a different direction. The ...
Verv14's user avatar
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2 votes
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13 views

Given a representation of $C_0(Y)$ on $C_0(X)$, trying to find a continuous function that satisfies a certain condition.

Let $X$ and $Y$ be locally compact Hausdorff, consider an algebra homomorphism $\phi:C_0(Y) \rightarrow C_b(X)$. We have a representation, which is an algebra homomorphism, $\pi$ of $C_0(Y)$ on $C_0(X)...
3j iwiojr3's user avatar
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0 answers
10 views

Show that for associated Legendre function

Show that $P_n^m(-x)=(-1)^n(-1)^mP_n^m(x)$ I have tried to use the associated Legendre function. I also used Rodrigue's formula for $P_n^m$ and plug the value back in the associated Legendre function. ...
Kylie's user avatar
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0 answers
29 views

Why are the eigenvalues of this matrix incorrect?

I am trying to compute the eigenvalues and eigenvectors of $$ \left[\begin{array}{llll} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 ...
random_0620's user avatar
  • 2,435
0 votes
0 answers
9 views

Deviation with respect of the Mode instead from the Mean Value

Deviation with respect of the Mode instead from the Mean Value I am trying to figure out if the following calculations make sense. If I try to make a deviation measure from the Mode value "$\nu$&...
Joako's user avatar
  • 1,254
0 votes
0 answers
16 views

Best approximation of a line by a rational line

Given an arbitrary line L in the plane, how can one find a line M with integer coefficients that best approximates L. I'm open to reasonable interpretations of ``best approximates". As a second ...
sitiposit's user avatar
  • 354
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0 answers
17 views

Looking for the name and reference to a table which represents all positive integer numbers

What is the name of the table is made up on the bottom row of all ODD numbers out to infinity. The row above that is the 2 to the power of the index of that row times the ODD number at the bottom of ...
mazecreator's user avatar
1 vote
0 answers
14 views

Let $f(x) = \tanh(x) - \frac{1}{x}$ find $\lim_{t \to 0} t f^{-1}(t) $

Consider the following function \begin{align} f(x) = \tanh(x) - \frac{1}{x} \end{align} We are interested in finding the following limit \begin{align} \lim_{t \to 0} t f^{-1} \left( \sqrt{1-t} \right)=...
Boby's user avatar
  • 5,843
-1 votes
0 answers
15 views

Bayes Rule with 3 random variables, conditioned on 1

This is a variation of the question found here. Is this a legitimate way to work with the conditioning on one variable? $$\begin{align} \mathsf P(R, H \mid S) & = \frac{\mathsf P(R,H,S)}{\mathsf P(...
Omortis's user avatar
  • 99
-2 votes
0 answers
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image of proper morphism is proper

This is ex 4.4 chp II Algebraic Geometry by Robin Hartshorne. Even the given hint is of no help to me. Now the ultimate reference in algebraic geometry is EGA. It is very encyclopedic and has no ...
bnb's user avatar
  • 1
-1 votes
1 answer
17 views

Short exact sequence of Lie Algebra representations

We have the following SES of a complex Lie Algebra representation, $$0\to \mathbb{C} \to W \to \mathbb{C} \to 0$$ Now the easy claim would be: $(\rho,W)$ is a two dimension representation and $\rho(x)$...
Donky Dang's user avatar
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0 answers
24 views

Describe the equivalence classes

If $\sim$ on $\mathbb{R}^3$ is defined by: $(x,y,z) \sim (x',y',z') \Leftrightarrow 3x - y +2z = 3x' -y' +2z'$ How are the equivalence classes described? According to the definition: $[(x,y,z)] = \{ (...
LianLi's user avatar
  • 21
-5 votes
0 answers
16 views

Laplace problem [closed]

Solve $tY'' +2Y' +tY = 0$, $Y(0^+) = 1$, $Y(7r) = 0$.
Obaydul Islam's user avatar
-1 votes
1 answer
30 views

Diagonalization problem, linear algebra

Probe that if $A$ is an invertible matrix, then $\frac{\chi_A(\lambda)}{\det(A)} = (-\lambda)^n \chi_{A^{-1}} (\lambda^{-1})$, where $\chi_A(\lambda)$ is the characteristic polynomial of the matrix $A$...
Estiven's user avatar
-1 votes
0 answers
9 views

Gaussian supremum bound

Let $x_1,...,x_n$ be normal Gaussian variables and $y_1,...,y_n$ be i.i.d. random unit vectors. Prove that: \begin{align*} \mathbb{E}[\sup_{||u|| \leq 1} \sum_{i=1}^n x_i |\langle y_i, u \rangle|] \...
PepeHands's user avatar
2 votes
0 answers
19 views

Can the existence of a transitive closure for every set be proven in ZFC - Replacement?

The transitive closure of $A$, $TC(A)$, is a transitive set containing $A$ with the property that for every set $K$, if $K$ is transitive and $A \subseteq K$, then $TC(A) \subseteq K$. $HC_{\beth_\...
Hussein Aiman's user avatar
1 vote
1 answer
24 views

Relationship between $A$ and $B$ where $A D A^T = B D B^T$.

Let $C$ be a positive semidefinite matrix with eigendecomposition, $$C = A D A^T,$$ so that $A$ is orthonormal and $D$ is diagonal. If it can be shown that $$C = B D B^T,$$ what can be said about the ...
knrumsey's user avatar
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-1 votes
0 answers
16 views

how to find transfer function of y/u when u is not an input?

The block diagram of the problem. help find Y(s)/U(s)
student 1's user avatar
1 vote
1 answer
32 views

How to apply queuing theory to find the long run proportion of customers who leave the system?

I am trying to apply queuing theory / birth and death process to the following. Suppose customers arrive in a restaurant according to a Poisson process with rate $\lambda = 1$. Suppose there are $2$ ...
MilesToGo's user avatar
1 vote
0 answers
24 views

Relationship between compactness of a set and compactness of a linear operator

I am recently learning functional analysis and topology, and I am wondering whether there is some relationship between compactness of a set (or a topological space) and compactness of a linear ...
ZYX's user avatar
  • 100
2 votes
1 answer
62 views

Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$. If $\lim_{x\to{a}} f^{'}(x)$ exists, show that $f$ is differentiable at $a$

I was having trouble trying to show this proof and needed some guidance: Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$. If $\lim_{x\to{a}} f^{'}(x)$ exists, show that $f$ is ...
Muhammad Abdurrahman Ullah's user avatar
0 votes
0 answers
21 views

Isomorphism between Cartesian and Polar coordinates

I have a homework problem that is explicitly stated as follows: The set $C$ of the complex numbers can be defined in two different ways: a. As a set of all ordered pairs $(x,y)$ where $x,y∈R$ b. As a ...
requiemman's user avatar
0 votes
0 answers
4 views

Modelling a state space with a CTMC

I study a two-echelon inventory system that takes one-for-one inventory policies into account and is continuously reviewed, including one external supplier, one central warehouse, and several local ...
ningning's user avatar
-3 votes
0 answers
55 views

Who are some of the mathematicians working on Hodge theory in the U.S. right now? [closed]

I would like to know some mathematicians in the U.S. who are working on Hodge theory right now.
Jingzhang Liu's user avatar
-2 votes
0 answers
18 views

Integration with Gamma function equals to the limitation at infinity?

Suppose f(x) is continuous on [0,+∞) and $$ \lim_{x \to \infty} f(x)=1 $$ Prove the following equation $$ \displaystyle \lim_{n \to \infty}\frac{1}{n!}\int_{0}^{\infty}f(x)e^{-x}x^{n}dx=1 $$ How to ...
JR zhang's user avatar
0 votes
1 answer
54 views

Is it possible to "sort" a continuous function?

I was motivated for this question while seeking for a new sorting algorithm. Suppose a continuous function $f : [a, b] \to \mathbb{R}$ is given. I wanted to define the sorted version $g$ of $f$, which ...
Dannyu NDos's user avatar
  • 1,701
1 vote
1 answer
17 views

For $F:R^{n} \to R^{m}, m>n$, with local inverse $G$ at $x$ such that $G(F(x)) = x$, is $DG$ a left inverse of $DF$? Why/Why not?

Assuming both functions are differentiable in some neighboorhood, under which conditions, if any, is the Jacobian of $G$ the left inverse of the Jacobian of $F$, ie $(JG)(JF) = Id^{nxn}$? Since we're ...
gaaaaaaaaaah's user avatar
0 votes
0 answers
25 views

Need help finding why there no equilibrium solutions to this function

Why does the function $\Bbb dy/\Bbb dx=\sin(x)\cos(y)+\cos(x)$ have no equilibrium solutions? I have already thought of the fact that for $\sin(x)$ to equal 0, $\cos(x)$ would equal 1 and that rules ...
Jake Degnan's user avatar
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0 answers
19 views

Eigenvalues of the logarithm

For $M \in B(I,1) \subset \operatorname{GL}_n(\mathbf{R})$, define $$ \operatorname{log}(M) = \sum_{k=1}^\infty (-1)^{k+1} \frac{(M-I)^k}{k}. $$ Let $\Gamma$ denote the subset of $\operatorname{M}_n(\...
Nilav's user avatar
  • 36
0 votes
1 answer
55 views

$\inf\{f (x)+g(x)\} \leq \inf\{f (x)\}+\sup\{g(x)\} \leq \sup\{f (x)+g(x)\} $

I know that there are lots of similar questions on the website, but my question here is more specific, and I think my argument has some flaws. If $f, g$ are both defined on the set $E$, then \begin{...
LJNG's user avatar
  • 1,215
0 votes
1 answer
39 views

How do I prove that this extension is integral?

Let $A$ be an integral domain, let $K$ be its field of fractions, and let $L$ be a finite extension of $K$. For a $\alpha \in L$, let $B=A[\alpha]$. Prove that exists a non-zero $a\in A$ such that the ...
Juan José Campos's user avatar
0 votes
0 answers
24 views

Why can't I define the tangent bundle this way?

What specifically breaks? Why do I need all the additional bits attached? What is gained by insisting on the distinction between tangent spaces at different points and double-tagging them in the ...
R. Burton's user avatar
  • 4,828
1 vote
0 answers
40 views

Determine the sign of Gaussian curvature

I calculated the Gaussian curvature of a metric and obtained the following expression $$ \kappa(r, Q) = \frac{2 \chi E^2 r^2 Q^2 \bigl[ r^2 \xi \!+\! m^2 Q^2 (r^2 \!+\! 2Q^2) \!+\! r^4(E^2\!-\!m^2)...
Soliton-104's user avatar
0 votes
2 answers
44 views

How to prove $\sum^{m}_{k=0}(-2)^k\binom{2n-k}{n}\binom{m}{k}=0$ for $m=1,3,5,\dots,2n-1$?

I would like to prove that $$ \sum^{m}_{k=0}(-2)^k\binom{2n-k}{n}\binom{m}{k}=0,\quad m=1,3,5,\cdots,2n-1,n\in\mathbb{N}^*. $$ Currently I have no idea of this: expanding $\displaystyle\binom{2n-k}{n}\...
Jianing Song's user avatar
  • 1,298
3 votes
0 answers
53 views

Natural way of inducing sequence of subgroups of $S_n$

Consider the trace map $tr: M_{n \times n} ( \mathbb{F}) \to \mathbb{F}$. It is well-known that this satisfies the property: $$tr(AB) = tr(BA)$$ This is equivalent to $tr(A_1 \cdots A_n) = tr(A_{\...
legionwhale's user avatar
  • 2,241
0 votes
0 answers
15 views

Decomposition of identity operator

Let $B$ be a Banach space and $I$ be an identity operator. Let $T$ be a bounded operator and its operator norm bounded above by $1$. Let $F$ be a compact operator on $B$. I try to prove that the ...
Hermi's user avatar
  • 1,388
-4 votes
0 answers
47 views

What is the indefinite integral of $1/(1-\ln x)$? [closed]

$$\int \frac{1}{1-\ln{x}} \,\Bbb dx$$ After using the reduction method to obtain the equation containing $e$, I expanded it through McGlaurin expansion.
Fanshuai Wang's user avatar
0 votes
1 answer
31 views

Inequality $\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\ge \frac{5}{6}\cdot\sqrt{a^2+b^2+c^2+7}.$

Let $a,b,c\ge 0: ab+bc+ca=1.$ Prove that $$\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\ge \frac{5}{6}\cdot\sqrt{a^2+b^2+c^2+7}.$$ Here is what I've done so far. By C-S $$\frac{1}{a+b}+\frac{1}{b+c}+\...
Anonymous's user avatar
  • 854
0 votes
0 answers
12 views

Finding a unitary matrix such that conjugation by that unitary matrix make a set of complex matrices real

Consider the set $\mathcal{X}_N$ to be the set of $N \times N$ Hermitian matrices whose off-diagonal elements are purely imaginary. Does there exist a unitary $U$ such that matrices of the set $U^*\...
khashayar's user avatar
  • 2,072

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