# All Questions

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### new profit / old profit ratio of a merchant

A merchant was selling his goods at 20% profit. When he allowed a discount of 5p per rupee on sale, his sale improved in the ratio 8:5. What is the new profit/old profit ratio? 1 Rupee= 100p
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### Show that the sum of (outdeg(v)-indeg(v))=0

Let $G = (V,E,\Phi)$ a directed graph. Let $outdeg(v)=\#\{e \in E| source(e) = v\}$ and $indeg(v)=\#\{e \in E| sink(e) = v\}$. Show that $$\sum \limits_{v \in V}(outdeg(v)-indeg(v)) = 0$$ Can you ...
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### How can I prove $\pi=e^{3/2}\prod_{n=2}^{\infty}e\left(1-\dfrac{1}{n^2}\right)^{n^2}$?

I am interested about some infinite product representations of $\pi$ and $e$ like this. Last week I found this formula on internet ...
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### optimal control, semismooth newton, bounded norm

I'm solving an optimal control problem (Poisson's equation with dirichlet BVP) $F(y,u) :=\frac{1}{2}\int_{\Omega} (y-y_d)^2 dx + \frac{\lambda}{2} \int_{\Omega} u^2 dx$ with finite element method. ...
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### Do we really need the constraint qualification?

I can't keep my fingers off Nocedal/Wright's Numerical Optimization (1999,1E) and I apologize. But maybe YOU can shed light on the question: Why does a point $x \in \mathbb{R}^n$ need to satisfy the ...
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### moments of scaled gamma distributed random var

I have a question about the distribution of a scaled $\Gamma$-random variable. I read the following. Suppose $x$ is $\Gamma(\alpha, \beta)$ distributed $y = a + b * x$ Now we can derive the ...
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### Second Order Non linear Differential Equation

I have arrived at a differential equation and I need to solve for x. $d^2x \over dE^2$+$Hx$ =$a$($1$+$J\over x^4$ -$1 \over {2x^2}$) Thank you
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### Algorithm for identifying Markov chain communicating classes

Let $P$ be a transition matrix of a time-homogeneous Markov chain with at least one closed communication class. Is there an algorithm / optimization problem that outputs the identification of the ...
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### derivative of a recursive vector-valued function

I have a recursive vector-valued function $$\mathbf{y}(t)=\mathbf{W}\mathbf{y}(t-1).$$ To compute the derivative of $\mathbf{y}(t)$ with respect to $\mathbf{W}$, do I need to use the product rule? ...
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### Construction of a matrix over $\{-1,0,1\}$

Let $Z=(z_{ij})$ be a $(n,n)$-matrix, for which: $z_{ij} \in \mathbb{R};$ $z_{ij}= -z_{ji}$ for $i,j=1, \dots , n;$ $\sum_{j=1}^n z_{ij} = 0$ for $i=1, \dots , n.$ Please help me ...
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### Limit of given expression

Let $\sum a_k=s$. I want to show that $$\lim\limits_{x\to 1^-}(1-x)\sum\limits_{k=1}^{\infty}\frac{ka_kx^k}{1-x^k}=s$$ where $x\in(0,1)$. Thanks for your helps.
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### Identification of $H$ with $H^{*}$ relativ the Killing-form

Let $H$ be a maximal toral subalgebra of a semisimple Lie Algebra $L$. The identification of $H^{*}$ and H relativ the Killing-form says, that to $\phi\in H^{*}$ corresponds the unique element ...
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### What exactly is Hensel doing for us in this result?

I'm reading a paper where the author appeals to Hensel's lemma, but it is not clear to me quite how it is meant to be applied (or, for that matter, which version!). My commutative algebra background ...
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### Can any one help me solve this integral ???

![i cannot able to solve this integral ,can any one able to solve this integral and i used integral technique but i cannot able to solve this equation the integral is with respect to x ...
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### Wavelet transform and taking out of frequencies

We use a scaled wavelet and move it across the signal taking out frequencies so that they need not to be processed with a differently scaled wavelet. How does this show up in the math behind wavelet ...
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### Composition of Relations solving p 0 σ and σ 0 p [on hold]

Explain me difference between p 0 σ and σ 0 p and how to get the answer.
$F(x)=x^2-5x+3$. Solve $f'(x)=-1$. $F (x)=1/x^2$. Calculate $f'(2)$. I have tried $2x-5=2(-1)-5=-3$ for the first question. The second question I'm not sure of. Explain how these are done not just ...
Does there exist an infinite closed shift-invariant $X \subset \{0,1\}^\Bbb Z$ with zero topological entropy? How to think of an example? Will periodic points of shift$|_X$ have zero entropy?