All Questions
1,619,964
questions
0
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0
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8
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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 ...
0
votes
0
answers
3
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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{...
- 131
0
votes
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 +...
- 1,031
0
votes
0
answers
25
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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
$$\lim_{x\to\infty}\left(\frac{1+x^2}...
- 1,325
1
vote
0
answers
29
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Notation used within paper on commutative rings
I was reading through this paper and I came across some rather confusing notation in the proof of Theorem 4. It says
Proof. ...$Y$ is finite, and it is clear that the image of the restriction of any ...
- 43
1
vote
0
answers
27
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Exercise 2.5.8 Pedersen's Analysis Now
I'm trying to do the following problem, and I'm getting hung up on one part. Here is the problem:
Let $f: X \longrightarrow \mathfrak{X}$ be a continuous map from a compact Hausdorff space $X$ into a ...
- 1,059
-4
votes
0
answers
15
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Calculate G(s) = Y (s)/U(s) from the block diagram in Figure 4.
Calculate G(s) = Y (s)/U(s) from the block diagram in Figure 4.
2
votes
1
answer
12
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Depiction of convergence in categorical sense
It is well-known that for a product space $\Pi_{\alpha \in A} X_{\alpha}$, a sequence of points $\{\mathbf{x}_n\}$ converges if and only if for each $\alpha \in A$, $\{\pi_{\alpha}(\mathbf{x}_n)\}$ ...
- 112
0
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0
answers
24
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$[G:H]=4$ where $H\neq 1$ means that $G$ is not simple.
The question is as follows:
Suppose $G$ is a finite group with a nontrivial subgroup of index $4$. Prove that $G$ is not simple.
I am on the hunt for a non-trivial normal subgroup. This proof ...
-1
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0
answers
14
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Alternative form for $\sum^{\infty}_{n=1} \sin(ny)\sin(ny')$
I have this expression $$\sum^{\infty}_{n=1} \sin(ny)\sin(ny')=\frac{\pi}{2}\delta (y-y')$$
I want to know if it possible to express this in a different manner when I have $y'=0.$ I am asking this ...
- 1
-1
votes
0
answers
6
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Can LM algorithm be used to optimize complex number problems?
In the electromagnetic field, there are NxN points on the incident plane. By changing the phase of the electric field at these points, the desired electric field distribution is generated in the ...
- 1
2
votes
0
answers
32
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Prove that if $f: A \to B$ is an injective function and $h: A \to C$ is any function, then there always exists a function $g: B \to C$ with $h = gof$.
I'm not an incredibly well versed proof writer and I would like some help with this proof. I really want to know if I'm along the right track or if I need to be going in a different direction. The ...
- 21
2
votes
0
answers
13
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Given a representation of $C_0(Y)$ on $C_0(X)$, trying to find a continuous function that satisfies a certain condition.
Let $X$ and $Y$ be locally compact Hausdorff, consider an algebra homomorphism $\phi:C_0(Y) \rightarrow C_b(X)$. We have a representation, which is an algebra homomorphism, $\pi$ of $C_0(Y)$ on $C_0(X)...
- 405
-1
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0
answers
10
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Show that for associated Legendre function
Show that $P_n^m(-x)=(-1)^n(-1)^mP_n^m(x)$
I have tried to use the associated Legendre function. I also used Rodrigue's formula for $P_n^m$ and plug the value back in the associated Legendre function. ...
- 1
-1
votes
0
answers
29
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Why are the eigenvalues of this matrix incorrect?
I am trying to compute the eigenvalues and eigenvectors of $$
\left[\begin{array}{llll}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0
...
- 2,435
0
votes
0
answers
9
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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 "$\nu$&...
- 1,254
0
votes
0
answers
16
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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 ...
- 354
0
votes
0
answers
17
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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 ...
- 101
1
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0
answers
14
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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)=...
- 5,843
-1
votes
0
answers
15
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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?
$$\begin{align}
\mathsf P(R, H \mid S) & = \frac{\mathsf P(R,H,S)}{\mathsf P(...
- 99
-2
votes
0
answers
8
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image of proper morphism is proper
This is ex 4.4 chp II Algebraic Geometry by Robin Hartshorne. Even the given hint is of no help to me. Now the ultimate reference in algebraic geometry is EGA. It is very encyclopedic and has no ...
- 1
-1
votes
1
answer
17
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Short exact sequence of Lie Algebra representations
We have the following SES of a complex Lie Algebra representation,
$$0\to \mathbb{C} \to W \to \mathbb{C} \to 0$$ Now the easy claim would be: $(\rho,W)$ is a two dimension representation and $\rho(x)$...
- 21
0
votes
0
answers
24
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Describe the equivalence classes
If $\sim$ on $\mathbb{R}^3$ is defined by:
$(x,y,z) \sim (x',y',z') \Leftrightarrow 3x - y +2z = 3x' -y' +2z'$
How are the equivalence classes described?
According to the definition:
$[(x,y,z)] = \{ (...
- 21
-5
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0
answers
16
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-1
votes
1
answer
30
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Diagonalization problem, linear algebra
Probe that if $A$ is an invertible matrix, then $\frac{\chi_A(\lambda)}{\det(A)} = (-\lambda)^n \chi_{A^{-1}} (\lambda^{-1})$, where $\chi_A(\lambda)$ is the characteristic polynomial of the matrix $A$...
- 9
-1
votes
0
answers
9
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Gaussian supremum bound
Let $x_1,...,x_n$ be normal Gaussian variables and $y_1,...,y_n$ be i.i.d. random unit vectors. Prove that:
\begin{align*}
\mathbb{E}[\sup_{||u|| \leq 1} \sum_{i=1}^n x_i |\langle y_i, u \rangle|] \...
- 61
2
votes
0
answers
19
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Can the existence of a transitive closure for every set be proven in ZFC - Replacement?
The transitive closure of $A$, $TC(A)$, is a transitive set containing $A$ with the property that for every set $K$, if $K$ is transitive and $A \subseteq K$, then $TC(A) \subseteq K$.
$HC_{\beth_\...
1
vote
1
answer
24
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Relationship between $A$ and $B$ where $A D A^T = B D B^T$.
Let $C$ be a positive semidefinite matrix with eigendecomposition,
$$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 ...
- 834
-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)
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$ ...
- 97
1
vote
0
answers
24
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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 ...
- 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 ...
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 ...
- 73
0
votes
0
answers
4
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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 ...
- 1
-3
votes
0
answers
55
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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.
-2
votes
0
answers
18
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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 $$
How to ...
- 1
0
votes
1
answer
54
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Is it possible to "sort" a continuous function?
I was motivated for this question while seeking for a new sorting algorithm.
Suppose a continuous function $f : [a, b] \to \mathbb{R}$ is given. I wanted to define the sorted version $g$ of $f$, which ...
- 1,701
1
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1
answer
17
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For $F:R^{n} \to R^{m}, m>n$, with local inverse $G$ at $x$ such that $G(F(x)) = x$, is $DG$ a left inverse of $DF$? Why/Why not?
Assuming both functions are differentiable in some neighboorhood, under which conditions, if any, is the Jacobian of $G$ the left inverse of the Jacobian of $F$, ie $(JG)(JF) = Id^{nxn}$?
Since we're ...
- 11
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0
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25
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Need help finding why there no equilibrium solutions to this function
Why does the function $\Bbb dy/\Bbb dx=\sin(x)\cos(y)+\cos(x)$ have no equilibrium solutions? I have already thought of the fact that for $\sin(x)$ to equal 0, $\cos(x)$ would equal 1 and that rules ...
0
votes
0
answers
19
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Eigenvalues of the logarithm
For $M \in B(I,1) \subset \operatorname{GL}_n(\mathbf{R})$, define
$$ \operatorname{log}(M) = \sum_{k=1}^\infty (-1)^{k+1} \frac{(M-I)^k}{k}. $$
Let $\Gamma$ denote the subset of $\operatorname{M}_n(\...
- 36
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votes
1
answer
55
views
$\inf\{f (x)+g(x)\} \leq \inf\{f (x)\}+\sup\{g(x)\} \leq \sup\{f (x)+g(x)\} $
I know that there are lots of similar questions on the website, but my question here is more specific, and I think my argument has some flaws.
If $f, g$ are both defined on the set $E$, then \begin{...
- 1,215
0
votes
1
answer
39
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How do I prove that this extension is integral?
Let $A$ be an integral domain, let $K$ be its field of fractions, and let $L$ be a finite extension of $K$. For a $\alpha \in L$, let $B=A[\alpha]$. Prove that exists a non-zero $a\in A$ such that the ...
0
votes
0
answers
24
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Why can't I define the tangent bundle this way?
What specifically breaks? Why do I need all the additional bits attached? What is gained by insisting on the distinction between tangent spaces at different points and double-tagging them in the ...
- 4,828
1
vote
0
answers
40
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Determine the sign of Gaussian curvature
I calculated the Gaussian curvature of a metric and obtained the following expression
$$
\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)...
- 31
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}\...
- 1,298
3
votes
0
answers
53
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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_{\...
- 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 ...
- 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.
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}+\...
- 854
0
votes
0
answers
12
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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^*\...
- 2,072