# All Questions

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### A second question about a proof of Banach-Alaoglu

I have another question about the proof of Banach-Alaoglu using nets. The proof proceeds by considering a universal net into the closed unit ball of $X^\ast$, let's call the ball $S$ and the net ...
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### Complex function to plane with non-negative real axis removed must be constant

I'm trying to prove that if I have a holomorphic function $f: \mathbb{C} \to \mathbb{C}\backslash A$ where A is the non-negative real axis then $f$ must be constant. My thoughts so far are to ...
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### Determining the number of solutions of the equation $y^2 = x^3 + x^2 + 5$ in $\mathbb{Z}/p^k \mathbb{Z}$ in function of $k$.

I found a question that asked me to discuss the number of solutions of the equation $y^2 = x^3 + x^2 + 5$ in $\mathbb{Z}/p^k \mathbb{Z}$ in function of $k$. I would like to use the multivariate ...
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### Implementing Equation on current data

I am trying to implement Personality, Gender, and Age in the Language of Social Media equation. I have 5 patterns and one list of 100 text = 900 words. The result of find a Match in the 900 to the ...
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### Derived series of a Lie algebra

I've been studying semisimple Lie algebras and solvability and was wondering if someone could explain to me the meaning of the derived series of a Lie algebra L and this part: $$L^{(1)}=[LL]$$ I don't ...
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### Finding coefficients of the min polynomial of an $n\times n$

Given an $n\times n$ matrix, for ease assume this matrix is over the $F_m$. What we know about min poly is the the non-zero components of the min polynomial for this case, ie if there is $x^2$, or ...
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### Calculating this integral: $\int_{\partial_{10}(1-2i)}\frac{39!\require{cancel}\cancel{z^z}z^z}{(z+4)^{42}}dz$

Please take a look at $$\int_{\partial B_{10}(1-2i)}\frac{39!\require{cancel}\cancel{z^z}ze^z}{(z+4)^{42}}dz$$ At a first glance, this looks like a case for Cauchy's differentiation formula, which ...
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### Prove that every symmetric matrix can be diagonalized using similarity transformation even if it has repeated eigenvalue

Prove that every symmetric matrix can be diagonalized using similarity transformation even if it has repeated eigenvalue by showing that the Jordan form of a symmetric matrix has no Jordan block of ...
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### Normal Closure, Normal Interior (Core) and Lattice Theory in Groups

In D. J. S. Robinson's, A Course in the Theory of Groups it is written (on page 16): "If $X$ is a nonempty subset of a group $G$, the normal closure of $X$ in $G$ is the intersection of all the ...
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### Hinge point in quadratic program (bilateral constraint)

My question itself is possibly quite simple and I guess that if someone can answer me they probably does not need a wall of text that is my background to the problem, but I figured I should provide as ...
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### Weighted sum of $\cos(nx)$ series

This is a follow up question to Prove $\frac{1}{2} + \cos(x) + \cos(2x) + \dots+ \cos(nx) = \frac{\sin(n+\frac{1}{2})x}{2\sin(\frac{1}{2}x)}$ for $x \neq 0, \pm 2\pi, \pm 4\pi,\dots$ I am looking ...
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### moduli of lattices

Consider the set $M$ of all (rank $g$) lattices in $g$-dimensional complex affine space $C^g$. Does M identify in some way with Siegel upper half space $H_g$? Let's say a lattice has CM if it has ...
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### What is real periodic function

I would like to know what is real periodic function. I understand what is periodic function, but I do not understand what is "real" periodic function.
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### Constructing a map sending a generator of $H_n(M)$ to given generators of $H_n(M, M-U_i)$.

Let $M$ be an orientable closed manifold of dimension $n$ covered by coordinate discs $\{ U_i : 1 \le i \le k\}$ such that for each $i$, $\bar{U_i}-U_i$ is homeomorphic to $S^{n-1}$, and suppose ...
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### Sum of normally distributed random variables

Let's say X~N(0,4) and Y~N(1,1). According to what law is this random variable distributed: Z=X+2Y. What confuses me is that I get different answers depending on the way I solve this (using ...
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### Orbit , trajectory, dynamical system

The orbit of $φ$ through $x_0$ is the set $O(x_0) \equiv \{φ_t(x_0) : −∞ < t < ∞\}$. This is also called the trajectory through $x_0$. Then, what is the difference between an orbit and a ...
I have to calculate: $$\iint _{R} \frac{x}{\sqrt{x^{2}+y^{2}}}dA$$ and R is this region: $$x^{2}+y^{2}=16; x^{2}+y^{2}=4; y = \sqrt{3}x; y=\frac{x}{\sqrt{3}}$$ so I used substitution: $$x = r\cos ... 0answers 35 views ### How find this t_{n} if such \sum_{k=1}^{n}\binom{2n+2}{2k+1}t_{k}=n(n+1) If the sequence \{t_{n}\} is such that$$\sum_{k=1}^{n}\binom{2n+2}{2k+1}t_{k}=n(n+1)$$and t_{1}=\dfrac{1}{2}, then prove or disprove$$t_{n}=(2n+1)\cdot (2^{2n-2})\cdot B_{2n}$$... 0answers 500 views ### Conditional binomial distribution Imagine I have two people, L and J shooting to a target. Both shoot 5 times, where the probability for L to hit the target is \frac 1 2 and the probability for J is \frac 1 4. Both independent. ... 0answers 45 views ### Expected number of neighbor nodes of a subset of nodes of a randomly constructed bipartite graph Suppose a right-regular bipartite graph with m left nodes and n right nodes (m>n and B=m/n is an integer) is constructed as follows: First, each successive B=m/n left nodes are connected ... 0answers 22 views ### Population confidence interval from sample SD and sample mean? A sample has:$$\text{a sample size } n = 70,\text{sample standard deviation } s = 184.43,\text{and a sample mean } \bar{x} = 564.15.$$Compute a 95% confidence interval for the population ... 0answers 33 views ### Kahler condition related to Ricci curvature formula of a Hermitian, holomorphic vector bundle over a complex manifold I read a local formula like this: Under some sort of Kahler condition,$$Ric(h)=-i\partial\bar\partial \log \det(h_{\alpha\bar\beta})$$where h_{\alpha\bar\beta} is the matrix of the Hermitian ... 0answers 99 views ### How to generilize the the following summation. While searching for a summation formula I come accross the following equation on wikipedia Equation$$\sum\limits_{k=1}^{n}{k^m z^k}=\left(z\frac{d}{dz}\right)^m\frac{z-z^{n+1}}{1-z}$$So I tried to ... 0answers 37 views ### What is the PDF of a function with 2 variables? The problem I'm working on involves finding the PDF of a function. In my case, my function is$$ f(r,\phi) = \frac{\cos{(2\phi)}}{r^2} \text{for } b > r > a$$The general recipe, as I've ... 0answers 47 views ### f'(x) to f(x) is it possible without knowing the value of f'(x) or f(x) I don't know much math and i got stuck at a problem: I'm not sure if it possible how to do it. I must use hermite interpolation for the following: 'Find the polynomial interpolating the function f ... 0answers 26 views ### Bayesian inference for dependent data Is it possible to use bayesian inference technique for data which does not follow the memoryless property? What is the likelihood function and prior in this case? 0answers 72 views ### Is this a geometric series? A geometric series is, in general, defined by:$$ \sum_{k=0}^{n-1}a\cdot r^{k}=a\cdot\dfrac{1-r^n}{1-r},\quad\quad \quad\quad \quad\quad r\neq1 $$If I have instead the following:$$ ...
I am reading Silverman's "The Arithmetic of Elliptic Curves". On page 10 he defines the function field of a projective variety $V$ over a field $K$ to be the function field of $V\cap\mathbb{A}^n$, ...