# All Questions

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### Proof of an inequality with entropy and mutual information.

Entropy of a random variable (a) is (h) : $H(a) = h$. Mutual information of (a) and (b) is (3h/4) : $I(a;b) = 3h/4$. Mutual information of (a) and (c) is (3h/4) : $I(a;c) = 3h/4$. It is needed to ...
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### $b^{\frac{m}{n}}=(b^{\frac{1}{n}})^m=(b^m)^{\frac{1}{n}}$ when $b \in \mathbb W$ and $n$ is Even.

The following property, known as Rational number property, is taken from the book (I am following now a days) College Algebra by Raymond A Barnett and Micheal R Ziegler I restate, ...
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### Use determinants to calculate the area bounded by 3 vectors

I have seen the proof of why the area of the parallelogram created by 2 vectors $u = \left(\begin{matrix} u_1\\ u_2 \end{matrix}\right)$ and $v = \left(\begin{matrix}v_1 \\ v_2 \end{matrix}\right)$ ...
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### linear transform of functions

I'm trying to understand what a vector of functions is, from trying to understand how to solve linear homogeneous differential equations. It seems that functions can be manipulated as vectors as ...
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### Different Inverse Approach

As it is known, we use inverse (Gauss Elm, Jordan...) or pseudo-inverse methods (LU, SVD, Chol, QR...) to solve linear equation namely $A*x=b$ when $A$ is $[n,n]$ and $b$ is $[1,n]$ matrix. These all ...
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### A variation of Lévy's characterization of Brownian motion

It is shown here, without using stochastic calculus, that if $W_t$ is a standard Brownian motion, then $$f(W_t)-\frac{1}{2}\int_0^t f''(W_s)ds$$ is a martingale, where $f\in C^2$ and compactly ...
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### How find function $f: \Bbb Q_{+} \to \Bbb Z$ such that $f(\frac{1}{x}) = f(x)$ and $(x+1)f(x-1)= xf(x)$

How to find all function $f: \Bbb Q_{+} \to \Bbb Z$ such that $f(\frac{1}{x}) = f(x)$ and $(x+1)f(x-1)= xf(x)$ for $x> 1$?
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### Steiner Triple System

A Steiner Triple System, denoted by $STS(v),$ is a pair $(S,T)$ consisting of a set $S$ with $v$ elements, and a set $T$ consisting of triples of $S$ such that every pair of elements of $S$ appear ...
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### How find the length of the third of the chord for of a circle of radius r?

Three chord of a circle of radius $r$ are the sides of the triangle inscribed in the circle. Two of these chords have a length $\frac{r}{2}$ and $r\sqrt{3}$. How find the length of the third of the ...
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### Is it true that $\sup_{y'\in Y'}h_d(T,U^{-1}(y'))=0$, i.e. $h(T')=h(T)$? (Bowen, Topological entropy)

First I have to give the background to my question: Let $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and on it the map $T\colon X\to X$ which describes the following dynamics: For $x\in X$, which is a ...
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### Probability - very difficult combinatorial question - don't have the theoretical background

Combinatorics - a (very) old question (which I hope I have remembered correctly) from a Cambridge Math Tripos. "A student sits 6 examination papers, each worth 100 marks. In how many possible ways can ...
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### How to show that $p(t|x,\mathbf x,\mathbf t)= \int p(t|x,\mathbf w)p(\mathbf w|\mathbf x, \mathbf t)d\mathbf w$

The following paragraph is approximately cited from Bishop's book, Pattern Recognition and Machine Learning. In carve fitting problem, we have training data $\mathbf x$ and $\mathbf t$, along ...
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### Determinitic finite automaton (DFA) that accepts natural numbers divisible by 6

I'm new to Formal Systems and Automata and I'm working on some exercises to get familiar with the concepts. I want to create a DFA that accepts natural numbers divisible by 6. I know that a number ...
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### Compatibility of direct product and quotient in group theory

This question came to me when I tried comparing direct product and quotients of groups with products and quotients of natural numbers. When we divide a number by another and multiply the result with ...
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### Is a limit point compact subspace of a Hausdorff space necessarily closed?

I think the answer should be "no", but I can't give a counter-example.
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### How to finish proof of ${1 \over 2}+ \sum_{k=1}^n \cos(k\varphi ) = {\sin({n+1 \over 2}\varphi)\over 2 \sin {\varphi \over 2}}$

I'm trying to prove the identity $${1 \over 2}+ \sum_{k=1}^n \cos(k\varphi ) = {\sin({n+1 \over 2}\varphi)\over 2 \sin {\varphi \over 2}}$$ What I've done so far: From geometric series ...
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### complex numbers find greatest value of z

I've to sketch the complex number $z$ such that it satisfy both the inequality $|(z-2i)|\le2$ and $0\le \arg(z+2)\le 45\deg$ I was able to sketch and shade the region that satisfies both ...
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### Is $T_n(R) \cong T_n(R)^{op}$?

I am working on the following problem: Let $R$ be a commutative ring, and $T_n(R)$ be the ring of $n \times n$ upper triangular matrices. Is $T_n(R) \cong T_n(R)^{op}$? I have already shown ...
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### Trigonometric limits as x approaches infinity. Need help.

I know how to solve limits with polynomials but i'm not sure how to tackle these limit problems involving trig and the constant, e. I can't seem to find any similar examples either. Any help? Thanks. ...
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### Optimization of the surface area of a open rectangular box to find the cost of materials

A rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of the box is twice its width. Material for the base costs ten dollars per square meter and for the ...
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### How to prove a cube minus a cube is never a cube (in whole numbers) [duplicate]

How to prove $x^3-y^3\neq z^3$ where $x$, $y$, and $z$ are whole numbers (integers greater than zero)?
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### Proving strong stability of semigroup

$X$ is the Hilbert space $L^{2}(0,\infty)$ and let $T(t):X\to X$ with $t\ge 0$ be defined by $(T(t)f)(\zeta):=f(t+\zeta)$. I want to prove that the $C_{0}$-semigroup $(T(t))_{t\ge 0}$ is strongly ...