# All Questions

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### How to approximate $w(x)$? [Konno-Yamazaki (1991)]

I am going through Konno and Yamazaki's paper: Mean-Absolute Deviation Portfolio Optimization and Its Applications to Tokyo Stock Market. On page 524, it states that ...
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### Homeomorphism from $U\subset \mathbb{R}^2 \rightarrow U$ which permutes points.

I'm trying to solve the following problem: Let U be a connected open set, $U \subset \mathbb{R^2}$, and consider $p_1,p_2,...,p_n \in U$ and $q_1,q_2,...,q_n \in U$. Show that exists an ...
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### Integral representation of the modified Bessel function involving $\sinh(t) \sinh(\alpha t)$

I've come across this peculiar integral representation for $K_\alpha(x)$: $\frac{\alpha}{x}K_\alpha(x) = \int_0^\infty dt \sinh(t) \sinh(\alpha t) e^{-x \cosh(t)}$ How does it come about? Are there ...
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### Prove that exists $\delta > 0$ such that, if $(x,y) \in S$ complies with $||(x,y)|| < \delta$, so $f(x,y)\le f(0,0)$

Let $f,g: \mathbb{R}^2 \rightarrow \mathbb{R}$ two functions, $f,g \in C^2$ in all plane. Let $S=${$(x,y) \in \mathbb{R}^2: g(x,y)=0$}. We assume that $g(0,0)= \frac{\partial g}{\partial x}(0,0)=0$ ...
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### Algorithm: making querries on trees

You are given a tree with $N$ $(1 \le N \le 10^5)$ vertices and $N - 1$ edges. Weight of every edge won't exceed 200. Design an algorithm to do $Q$ $(1 \le Q \le 10^5)$ operations of two types as fast ...
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### congruence with exponential, $8^{2012} + 2012^8$

I have an assignment where the following statement is: Determine the lowest positive integer that is congruent with the statement mod 12: $$8^{2012} + 2012^8$$ How can I solve this? I have totally ...
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### $X$,$Y$,$Z$ mutually independent implies $X+Y$ independent of $Z$

Supposing $X$, $Y$ and $Z$ and mutually independent real random variables, how can we prove that $X+Y$ and $Z$ are independent from the definition? If not from the definition, using $\sigma$-algebras? ...
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### Using similar triangles, find l?

I am attempting to render some 3D shapes and am having trouble with some of the math. I'm hoping someone can help me out here. Given two arbitrary points in space, P and C, I need to find l. I have L ...
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### total differential of product of scalar & vector functions

I've probably made mathematical mincemeat out of this but, suppose I have a product of scalar and vector functions, such as the momentum $\mathbf{p} = m \mathbf{v}$. To keep it reasonably simple but ...
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### Galois group of the extension

Let $K$ be the splitting field over $\mathbb{Q}$ w.r.t. the polynomial $x^7 - 10x ^5+15x+5$ I think its Galois group is the symmetric group $S_7$. I tried to prove it using a theorem which says: "If ...
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### Solve $\lim\limits _{n\to \infty }\lim\limits_{x\to \:0}\left(1+\sin^2x+\sin^22x+\ldots+\sin^2nx\right)^{1/(n^3x^2)}$

I've no clue how to solve this limit. I tried to solve the inner limit first, but the fact that I have another variable there(n) really makes things difficult(never had to solve such a limit before). ...
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### Solving Gaussian probability equation

I have two Gaussian random variables $X_1,X_2$ with distributions $(1,\sigma_1^2), (1,\sigma_2^2)$ Respectively. and with symmetric covariance $cov(X_1,X_2) = n\sigma_1^2$ noting that $n\gt 0$. And I ...
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### Find $a$, $b$ such that the ellipse $(x/a)^2 + (y/b)^2 = 1$ passes through $(\sqrt 2, 2)$ and has minimum area

I am working on a problem in which, for $a$, $b \gt 0$, we let $(x/a)^2 + (y/b)^2 = 1$ describe an ellipse. I am required to use the method of Lagrange multipliers and the corresponding second ...
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### Image of generators of subalgebra of coordinate ring

Let $f_1, ..., f_n$ be the generators of a subalgebra of the coordinate ring $k[X]$ of the algebraic variety $X$. Is $p := f_1 \times...\times f_n : X \rightarrow k^n$ a surjection onto the variety ...
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### Number of 5-digit numbers such that the sum of their digits is even

I found the same question on this site and many others...where everywhere the answer $45000$ is written ...while I have no problem with this but I am having problems with the logical answer...as ...
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### Efficiently find $x(k)$ where $x$ is given by $Ax=b$ and $A$ is tridiagonal

Say $A$ is a $n\times n$ ($n$ odd) real matrix that is tridiagonal (but need not be symmetric). What is the most efficient way to compute the value of $x(\frac{n+1}{2})$ (informally, the 'middle ...
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### Predicate Logic Proof

$:\forall x[((Px \rightarrow Qx)\rightarrow Px)\rightarrow Px]$ Stuck with this question, any help would be greatly appreciated.
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### finding an entire function with certain property

I am trying to find all entire function f(z)= u+iv satisfying : $|f(z)|^2$= $g(x)^2$ + $h(y)^2$ , where g and h are differentiable functions of one variable. I tried to differentiate both sides, and ...
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### Can I think of convergence as an independent mathematical object?

Recently I studied convergence of sequence of real numbers in an introductory analysis course. Can I think of convergence as an independent mathematical object just as we think of numbers and ...
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### Efficient volume partition for a set of particles

I am dealing with a set of $N$ dimensionless (point) particles in a box. The box has a certain volume $V$. I need to assign a volume to each particle, whose position within the box changes over time, ...
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### Probability Situation

A well known medical research organization conducted a study on a certain brand of test strips to see if a person has a certain disease. They found that $32\%$ of the people using the test strips ...
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### Expected values with power functions

Let $X_1,\dots,X_T$ be a sequence of i.i.d. random variables. I want to calculate the expected value of the following function: $$\left (\sum_{t=1}^T X_t^a\right)^{(1/a)}$$ as a function of $a$ ...
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### all possible, unique mixtur combination with constraint

I have a mathematical problem and don't know how to solve it, can you recommend a solution/approach or Tags to look further into Problem: I have N variables eg. 1-100 and would like to determine ...
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### $L^1$ convergence of a martingale, conditional expectation

I am trying to prove the following: Let our space be $(\Omega, \mathcal{F}, P, \{ \mathcal{F}_n \}_{n \in \mathbb{N}})$. Let $\{X_n \}_{n \in \mathbb{N}}$ be a martingale (adapted to the filtration ...
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### Center of normalizer of cyclic group

Given a Group of order $pm$ with $p$ the least prime dividing the order of $G$ and p does not divide m. Is it true that a $p$-sylow subgroup is contained in the center of its normalizer. I think yes, ...
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### Analyticical solution of least square problem

could anyone explain: a) $||{Ax-b}||^2$ (there is also a lowered 2): what does this two 2's mean? b) why is the solution: $x =(A^TA)^{-1} A^Tb$ is? Thank you very much:)
I'm having some trouble with the following problem: Let $N(f;y) = |f^{-1}(y)|$ be number of $x \in [a,b]$ such that $f(x) = y$. Show if $f$ is of bounded variation, then \int_\mathbb{R} N(f;y) ...