1
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0answers
31 views

Multisection of a Power Series Proof

Suppose that $$H_{N,k}(x)=\frac{x^ke^{\frac{-x}{N}}}{N^{k-1}k!\sum_{n=0}^{N-1}{w_N^{-nk}e^{\frac{w_N^nx}{N}}}}=\sum_{n=0}^\infty{A_n\frac{x^n}{n!}}$$ where $k\lt N, w_N=e^{\frac{2i\pi}{N}}$, and ...
1
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0answers
26 views

Quotient is isomorphic exercise

Suppose $G$ is solvable, $N \vartriangleleft G$. Let $f \in Hom(G,H)$. We have a normal series $\{e\}=G_0 \vartriangleleft G_1 \vartriangleleft ... \vartriangleleft G_n = G$ with $G_{i+1}/G_i$ ...
1
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0answers
15 views

Finding probability of getting into a university based on 3 factors

How would I go about calculating probability of getting into a college (CU Boulder) using a data set containing GPA (0.00-4.00), ACT Composite score (0-36), Class rank percentile (0-100), and whether ...
1
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0answers
56 views

Solutions of an EDO in tempered distribution space being smooth out of the origin.

For $a\in\mathbb{C}$, let us consider the following differential equation in $\mathcal{S}'(\mathbb{R})$, the set of tempered distributions on $\mathbb{R}$: $$xT''+2T'+(a-x)T=0.$$ For $T\in\mathcal{S}'(...
1
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0answers
46 views

Are you allowed to substitute for imaginary numbers on a real integral?

So, lets assume I am give this integral: $$\int_{a}^b f(x)\text{d}x$$ where $a,b,c\in\Bbb{R}$ and $f:\Bbb{R}\to\Bbb{R}$ Are you allowed to make the following substitution? Substitute $x=ci$ ...
1
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0answers
28 views

A group with an element of finite order and another element of infinite order?

I have a (probably) stupid question. I'm in an intro to group theory class and I am trying to better grasp the concept of element order. My question is if I have a group and it has an element of order ...
1
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0answers
26 views

Show that there exists k such that $\mu(E_k) \geq m/n$.

Suppose that $\mu(X)=1$. Let $E_1, E_2, \dotsc, E_n$ be measurable subsets of $X$, and each point of $X$ belongs to at least $m$ of these sets. Show that there exists $k$ such that $\mu(E_k) \geq ...
1
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0answers
24 views

Order statistics and transformations

Assume random variables X$_1$, ... , X$_n$ and Y$_1$, ..., Y$_n$ are U(0,a)-distributed. Show that Z$_n$ = n*$log\frac{max(Y_{(n)},X_{(n)})}{min(Y_{(n)},X_{(n)})}$ has an Exp(1) Distribution. I've ...
1
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0answers
37 views

In how many different ways can you order in line the letters of the word AAAAABBB?

In how many different ways can you order in line the letters of the word AAAAABBB?I was thinking - I have $8!$ for the whole letters including repeats, and then because each word repeats $5!$ for the ...
1
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0answers
38 views

Proof for a periodic function

I have to solve the following exercise: The function $f : \mathbb{R} \rightarrow \mathbb{R}$ is a periodic function with $P = 2\pi$ so that $f(x) = f(x + 2\pi)$ is true for all $x \in \mathbb{R}$. ...
1
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0answers
42 views

Bias of two estimators

I hope someone can help me. I have some trouble calculating the bias of two estimators.Unluckily it is really urgent because I hold a presentation next week. The topic is nonparametric local ...
1
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0answers
56 views

integral vs. residue at infinity

I have an issue with residues at infinity. I am computing the integral $\displaystyle{\int_{C_3^+(0)} \dfrac{e^{3z}}{z^2(z^2+2z+2)} dz} $ Since all three poles ($0$ of order 2, $1\pm i$ of order 1) ...
1
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0answers
43 views

Magnitude and Angle of Discrete Fourier Transform

I can't figure out how to get the magnitudes for periodic discrete Fourier transforms. For example if $x[n] = cos(\frac{\pi}{4}n + \frac{\pi}{2})$, I need to find and plot the magnitude $|X(e^{jw})|$...
1
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0answers
28 views

Find the bounded orbits of the system $x'=y$, $y'=x+x^2-y$

Consider $$ \frac{dx}{dt}=y,\quad \frac{dy}{dt}=x+x^2-y. $$ Describe the set of all bounded orbits. Hint, use $H(x,y)=\frac{y^2}{2}-\frac{x^2}{2}-\frac{x^3}{3}$. The equilibria are $$ O_1=(0,0),...
1
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0answers
18 views

Maurer-Cartan form left invariance

Let $G$ be a lie group and $\mathfrak{g}$ its lie algebra, that is, the $\mathbb R$-space of left invariant vector fields on $G$. Recall the isomorphism $\mathfrak{g}\simeq T_eG$. The Maurer-cartan ...
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0answers
26 views

Probability distribution of $g^h f f^h g$

We define an $k \times k$ complex matrix $M=[V \, \mathbf{0}]$, where matrix $V$ is $k \times (k-l)$ dimensional and is unitary, and $\mathbf{0}$ is the $k \times l$ zero matrix. Let vector $f$ be a $...
1
vote
0answers
21 views

Motivation for dual bases

I am encountering dual bases for the first time in the context of algebraic number theory, mainly in proofs regarding the existence of an integral basis for $\mathcal{O}_K$ and its ideals. I am ...
1
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0answers
51 views

Prey predator excercice

Consider the 2-dimensional system of non-linear ODEs, semplified instance of a predator-prey population model $\dot x=\alpha x (1-x)-xy$ $\dot y = y(x-y)-\beta y$ with $\alpha = 1 ,\beta = 1/2$ ...
1
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0answers
37 views

Big O Notation Clarification

Working through a textbook on algorithms (CLRS intro to algorithms) and just wanted to see if someone could help me understand one of the exercises at the end of a chapter. Problem: Is $n^{2 + 1} = ...
1
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0answers
31 views

Impossible form of a triangular number

Show that there are no positive integers $t,i,j$ with $j>i$ such that: $\displaystyle \frac{t(t+1)}2=\frac{2i(j-i)j(j+i)}3$ If possible provide an elementary proof. I believe the statement is ...
1
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0answers
33 views

Approximation of integrable function by polynomials

Assume $f\in \mathscr{R}(\alpha)$ on $[a,b]$, and prove that there are polynomials $P_n$ such that $$\lim \limits_{n\to \infty}\int_{a}^{b}|f-P_n|^2d\alpha=0.$$ Proof: Let $\varepsilon>0$ be given ...
1
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0answers
52 views

Something fishy in the Zeta function

Recently I came across the Riemmann representation of the Zeta function as follows: $$\zeta (s) = (2^s)(\pi^s-1) \sin(\frac {\pi s} 2) \Gamma(1-s) \zeta(1-s) .$$ Now, I went ahead to calculate the ...
1
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0answers
27 views

Determining an unknown parameter in an equation with modular arithmetics

Consider a machine which takes $x$ and $y$ as input, and gives $z$ as an output using the following equation: $z =\left[(x + k) \mod l\right] \oplus \left[(y + k) \mod l\right].$ $x, y, z, k,$ and $...
1
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0answers
44 views

Finding solution to matrix equation over GF(2) with minimal true variables

I am looking for a general way to find a solution to a system of equations in GF(2) such that the solution has the least amount of true variables. After Gaussian elimination I get a matrix like such: ...
1
vote
0answers
58 views

Is it possible to compute $\Gamma(\rho_k)$ or $\zeta(3+2\omega_k)$, when $\rho_k=\frac{1}{2}+i\omega_k$ is a nontrivial zero of Riemann zeta function?

I believe that I can to use the equation (1) (from reference [1]), a well known equation involving the Gamma function, Riemann zeta function and the theta function $\psi(x)=\sum_{n=1}^\infty e^{-n^2\...
1
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0answers
49 views

textbook on calculus of variation which focuses on the following topics

I need a textbook (or set of online lecture notes) on calculus of variation which focuses on the following topics "Variation of a functional, Euler-Lagrange equation, Necessary and sufficient ...
1
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0answers
41 views

Is exponential of GUE random matrix Haar random?

Consider the matrix exponential map $H \mapsto e^{i H t}$ acting on the Gaussian unitary ensemble (GUE) of Hermitian matrices. I would expect that for large $t$, the resulting measure on the unitary ...
1
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0answers
39 views

Inversion across a general ellipse

This paper is very useful in how it explains the mapping of any coordinates $(x,y)$ across an ellipse with the function $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ to $$\left(\frac{a^2b^2x}{a^2y^2+b^2x^2},\...
1
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0answers
20 views

Convergence result of a UI martingale

This Lemma is from Ethier and Kurtz's Markov Processes on page 205. My question is about the equation (5.47). I can prove it is a uniformly integrable martingale (bounded by 1), so converges to a ...
1
vote
0answers
52 views

Can every monad give rise to a monad transformer?

Can every monad give rise to a monad transformer? In the paper Calculating monad transformers with category theory by Oleksandr Manzyuk, one finds a construction of monad transformers as ...
1
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0answers
17 views

Constant Gauss curvature from bipolar projections.

Please help finding z-coordinate for constant positive and negative Gauss curvatures in Mongé form : $$ x= \sqrt{R^2 + T^2} + R \cos u ,\, y= R \sin u ,\, z= f(x,y). $$ $ R,T $ are constants. (...
1
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0answers
571 views

How to solve this integral to find the exact length of an equation in the polar plane?

I hope it is only because it's late and I've been studying for a calculus exam for several hours, but I cannot see how to solve this integral. The problem states: Find the exact length of the ...
1
vote
0answers
25 views

What determines the number of families to $1-4x-4(1-x^2)z = w^2$?

This is related to this post. First, we have, Theorem: "If $w_0, z_0$ is a solution to, $$1-4x-4(1-x^2)z = w^2\tag1$$ then, $$w = w_0+2(x^2-1)n$$ $$z = z_0+w_0\,n+(x^2-1)n^2$$ is also a ...
1
vote
0answers
60 views

Collatz Conjecture: Literature on Convergence

Does anyone know of a paper showing that if all n converge, they must converge to unity for n>0. Else, any literature related to convergence properties would be appreciated. Thanks, Jordan
1
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0answers
20 views

What process does this SDE weakly converge to?

So my question is motivated by the following: Note that the ODE $$ dy_t = 2sgn(y_t)\sqrt{|y_t|}$$ $$y_0 = 0$$$$ has no unique solution. However, consider the SDE as follows: $$ dy_t = 2sgn(y_t)\sqrt{...
1
vote
0answers
225 views

A planar graph $G$ is the dual of its dual if and only if $G$ is connected.

That is, prove that a planar graph is the dual of it's dual iff it is connected. I know that in order for this to be true, G must be isomorphic to it's dual (G'), but I'm not sure how connectedness ...
1
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0answers
69 views

Young diagram for $S_5$

I am trying to draw the Young diagram for $S_5$. I know the following pieces of information about $S_5$. The order of the group is $120$. The number of conjugacy classes and so partitions is $7$. ...
1
vote
0answers
54 views

If $G$ acts such that $|\mbox{fix}(g)|\le 2$ for $g \ne 1$ and $O_p(G) \ne 1$ and $|G_{\alpha}|$ odd. Assertions about Frobenius groups

Let $G$ be a finite group acting nonregularly and transitive on $\Omega$ such that each nontrivial element has at most two fixed points and $|\Omega| \ge 4$. I know three facts: i) If $1 \ne X \le G_{...
1
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0answers
41 views

Why is the cross-correlation an integral?

The cross-correlation of continuous $f,g$ is: $$(f \star g)(\tau)=\int_{-\infty}^{\infty}f^*(t)g(t+\tau)dt$$ Why is it an integral? Why doesn't The cross-correlation of continuous $f,g$ is: $$(f \...
1
vote
0answers
33 views

Can someone give examples of where deviations from pure randomness can be handled by number theory?

In his excellent book, How Not to Be Wrong: The Power of Mathematical Thinking, Jordan Ellenberg makes the following statement on page 142: There are some deviations from pure randomness whose ...
1
vote
0answers
43 views

Inverted Sigmoid Curve

I want to find a function $f$ that has an inverted sigmoid-like curve and one other constraint: $f(0) = 1$ I've inverted the sigmoid like this: $1- \dfrac{1}{(1 + \exp(-x))}$ but don't know how to do ...
1
vote
0answers
63 views

Derivative, Calculus

Under what conditions we can write: $ \frac{\partial f}{\partial x} = g(f)$ where $f$ is assumed to be function of $x$ and $t$, for example.
1
vote
0answers
18 views

Finding the Joint Density from the Conditional Hazard Functions

Let $X_{1}<X_{2}<\dots<X_{n}$ be continuous nonnegative random variables where the hazard functions are $$\lambda_1(t)=\lambda*e^{\beta*g(t)}$$ $$\lambda_i(t|x_{i-1})=\lambda*e^{\beta*g(t)}*...
1
vote
0answers
27 views

how to prove this type of problems?

Let $p(x)=x^2+ax+b$ be a quadratic polynomial where $a\in\mathbb{R}$ and $b≠2$ has rational roots.suppose $p(0)^2\times p(1)^2\times p(2)^2$ are integer then prove that $a$ and $b$ are integers.
1
vote
0answers
22 views

Check uniform equicontinuity of a function family

I am struggling to prove or disprove that the following function family is uniformly equicontinuous. $$F = \{f \in C^1[0,1]: \forall x \text{ } |f(x)| + \sqrt x |f'(x)| \leq 1 \}$$ First I tried to ...
1
vote
0answers
38 views

Verify proof that $f:X\to Y$ a homeomorphism implies $f_*:H_p(X)\to H_p(Y)$ is an isomorphism

I have to prove that $f:X\to Y$ a homeomorphism implies $f_*:H_p(X)\to H_p(Y)$ is an isomorphism for all $p$ Where $H_p(X)$ is the $p$th homology group of $X$. To me this seems to come down to ...
1
vote
0answers
28 views

tight bound for a finite sum involving harmonic series

I want to a know tight bound of this quantity when $n$ is even $$\sum_{k=1}^{n/2}\sum_{m=n-k}^{n}\frac{1}{k(k+1)m}$$. I simplified the expression and it comes like $$H_n[1-\frac{1}{n/2+1}]-\sum_{k=1}...
1
vote
0answers
18 views

asymptotic normality of m-estimator

I' struggling with an argument in van der Vaart proof of theorem 5.21 at p.52: https://books.google.co.uk/books?id=UEuQEM5RjWgC&pg=PA36&lpg=PA36&dq=lemma+4.2+van+der+vaart+one-to-one+...
1
vote
0answers
28 views

Closed form for $\int_0^{\pi/2} \log\bigl( g(x) \bigr) f(n x) \,\mathrm{d}x$

Let $g_+(x) = \sin x$ and $g_-(x) = \cos x$ and similarly let $f_+(x)=\sin x$ be $f_-(x)=\cos x$. I was trying to study the integrals $$ I_{(\pm,\pm)}(a,n) = \int_0^{\pi/2} \log\bigl( g_{\pm}(ax) \...
1
vote
0answers
31 views

Why is this prime a bad choice for the ElGamal cryptosystem?

Using the ElGamal cryptosystem in $\mathbb{Z}_{p}^{\times}$, the proposed prime is $p = 2^{1947}\cdot 5 + 1$. The exercise asks me to show why this is a poor choice, and I can't quite do it. In my ...

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