0
votes
0answers
3 views

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
0
votes
1answer
7 views

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 ...
1
vote
0answers
27 views

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 ...
0
votes
0answers
4 views

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. ...
0
votes
0answers
5 views

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 ...
0
votes
0answers
4 views

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 ...
0
votes
1answer
15 views

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
0
votes
0answers
6 views

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 ...
1
vote
0answers
11 views

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? ...
0
votes
0answers
18 views

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 ...
0
votes
0answers
15 views

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.
1
vote
0answers
5 views

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 ...
3
votes
0answers
25 views

Is $\pi(n)$ a Rational Function?

Are there some two-variable polynomials $P(n,\log n)$ and $Q(n,\log n)$ which we have the bellow equation for prime counting function $\pi(n)$ for $n \in \mathbb{n}$? $$\pi(n) = \Bigl{\lfloor} ...
1
vote
0answers
10 views

construction of a path of quadratic variation

Consider a path $x: [0,1] \to \mathbb R$. it's $p$-variation on an interval is defined by $$V_{p}(x, [a, b]) = \lim_{|\Pi| \to 0} \sum_{i=1}^{n}|x(t_{i}) - x(t_{i-1})|^{p}$$ where $\Pi = \{a= ...
0
votes
0answers
16 views

Factorizations of Finite Abelian Groups

Every finite abelian group $G$ can be uniquely written as $$\mathbb{Z}/{d_1\mathbb{Z}} \times \mathbb{Z}/{d_2\mathbb{Z}} \times \cdots \times \mathbb{Z}/{d_r\mathbb{Z}},$$ where $d_i$ divides ...
0
votes
0answers
6 views

Determine Similarity Algorithm Used

For a known set the comparison values, and result scores, is it possible to determine the similarity comparison algorithm that is being used? When using a Similarity function for SQL Servers ...
0
votes
0answers
11 views

Show that $W$ is a Gaussian process

I have the following problem: I want to prove that the vector $(W(1_{[t_0,t_1]}),...,W(1_{[t_{n-1},t_n]}))$ is normally distributed with mean $0$ and covariance matrix ...
0
votes
1answer
18 views

If $a|(p+1)$ for all but finitely many $p=3 (\text{ mod } 4)$ then $a$ divides $4$

I have the following question: Let $a$ be an integer such that $a$ divides $p+1$ for all but finitely many primes $p=3 \text{ mod } 4$ Can we conclude that $a$ must divide $4$? How we can prove ...
0
votes
0answers
11 views

Logical consequence and resolution rule

Which of this are false? a) If some formula H results from premises D, then H could be derived from D with using (reapetedly) resolution rule. b) If some formula H results from premises D, then we ...
0
votes
0answers
20 views

Is $\nabla \chi^2 \cdot \nabla^2 (\nabla \chi^2) = 0$ if $\nabla^2 \chi = 0$? [on hold]

The title says it. Is $\nabla \chi^2 \cdot \nabla^2 (\nabla \chi^2) = 0$ if $\nabla^2 \chi = 0$?
1
vote
1answer
9 views

Find the integral closure of an integral domain in its field of fractions

Let $k$ be a field and let $R = k[x,y]/(x^2-y^2+y^3)$. Note that $R$ is an integral domain. Let $F$ be the field of fractions of $R$. How to determine the integral closure of $R$ in $F$? I have ...
-2
votes
0answers
18 views

Solutions of the following differential equation

$$\frac{-2q}{k}+z^2+2zp-2zN+(p-N)^2=0$$ What is the solution of this differential equation? Where $N$ is a constant and $p$ and $q$ are the usual notations.
-5
votes
0answers
22 views

How to find shaded area of square? [on hold]

How to find shaded area of square if it has four circles in it each having 2 meter square area?
0
votes
0answers
22 views

Help with Fourier transform of product

I was reading this article in wikipedia, and I supposed $f,g \in L^1(\mathbb{R^n})$ such that their product $f \cdot g$ are in $L^1(\mathbb{R^n})$ too. So let $h=f \cdot g$, and ...
-1
votes
0answers
9 views

Find how many elements

Find how many elements in a group of order 30 has the order 5,and explain the reasons. Cant do it. Any ideas?
2
votes
0answers
18 views

Unique ways to distribute k1, k2, .. colored balls into n boxes uniquely

Example: Uniquely distribute 2 Red Balls and 4 Blue Balls into 3 boxes: [B][BB][RRB] [B][BBB][RR] [B][R][RBBB] [B][RB][RBB] [BB][R][RBB] [BBB][R][RB] Answer: ...
0
votes
0answers
19 views

Distribution of the minimum of two exponential random variables

$X$ and $Y$ are two exponential random variables with rate 1 and 2. lets define random variable $Z$ such that: $z_i = min(x_i,y_i)$, where $i =1,2,3,...N$. Let $V$ be another random variable and ...
0
votes
0answers
4 views

Ideals of the operator algebra

Let A be a Banach algebra. Is there any relation ship between two-sided closed ideals of A and two-sided closed ideals of the operator algebra B(A)? Is there any characterization for ideals of B(A)?
4
votes
0answers
14 views

$\sum_{k=1}^n \binom{n}{a_1,a_2, \cdots , a_k} \binom mk \binom{k}{b_1,b_2, \cdots , b_l}= m^n,$

(Own) Let $n,m$ be positive integers such that $m>n$. Prove that $$\sum_{k=1}^n \binom{n}{a_1,a_2, \cdots , a_k} \binom mk \binom{k}{b_1,b_2, \cdots , b_l}= m^n,$$ where $1 \le a_i \; (1 \le i \le ...
1
vote
1answer
32 views

Finding inverse

What would be the binary operator of an algebra $\langle \{1, \dots n\}, ? \rangle$ so that every element $k \in \{1 \dots n\}$ would have $k-1$ left-inverse elements? I have been trying various ...
-2
votes
2answers
50 views

Mathematical induction problem. Let $S_{n}=\left (3+\sqrt{5}\right)^{n}+\left(3-\sqrt{5}\right)^{n}$

Let $S_{n}=\left (3+\sqrt{5}\right)^{n}+\left(3-\sqrt{5}\right)^{n}$then, by mathematical induction, show that $S_{n}$ is an integer. Also, prove that the next integer greater than ...
0
votes
1answer
22 views

Verify $\frac {\partial B} {\partial T} =$ $\frac{c}{(e^\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}e^\frac{hf}{kT}$

Find an expression for $\frac {\partial B} {\partial T}$ applied to the Black-Body radiation law by Planck: $$B(f,T)=\frac{2hf^3}{c^2\left(e^\frac{hf}{kT}-1\right)}$$ The correct answer (I believe) ...
0
votes
1answer
16 views

Connecting a mathematical solution to a differential equation with it's physical solution

I have seen this question in a neuroscience course: It is given after the lecture with these and these slides. I have no background in physics. However, I do know how to solve a differential ...
0
votes
0answers
7 views

Convolution of measures on a measurable group is associative

I've come across a statement in Kallenberg's Foundations of Modern Probability which claims this and only tells me to use Fubini's theorem. I am not very familiar with this topic and the text doesn't ...
0
votes
1answer
11 views

Is my example of non equivalent maps correct?

We define two smooth maps $f: (\mathbb R, 0) \to (\mathbb R^2, 0)$ and $g: (\mathbb R, 0) \to (\mathbb R^2, 0)$ to be equivalent if there exist diffeomorphisms $\tau : \mathbb R \to \mathbb R$ and ...
1
vote
1answer
11 views

Question in proof from James Milne's Algebraic Number Theory

I'm having difficulty understanding a step in a proof from J.S. Milne's Algebraic Number Theory (link). Here $\zeta$ is a $p$th root of unity and $\mathfrak p = (1-\zeta^i)$ for any $1\leq i\leq p-1$ ...
1
vote
0answers
7 views

Derivative of sum of two functional derivatives with different ranges

I have a functional of the the following form, $(o<a<1)$ : $F(g(x)) = \int_0^a \! g(x) x \, \mathrm{d}x. + \int_a^1 \! (g(x)-k)x^2 \, \mathrm{d}x. $ I want to find $ \frac{\partial ...
1
vote
1answer
43 views

Sum of super exponentiation

$f(x,n)=x^{2^{1}}+x^{2^{2}}+x^{2^{3}}+...+x^{2^{n}}$ Example: $f(2,10)$ mod $1000000007$ = $180974681$ Calculate $\sum_{x=2}^{10^{7}} f(x,10^{18})$ mod $1000000007$. We know that $a^{b^{c}}$ mod ...
1
vote
0answers
13 views

Matrix of $f^p:\Lambda^p(E)\rightarrow \Lambda^p(E)$

Let $\Lambda^p (E)$ be the set of $p$-covariant exterior(alternative) tensors on linear space $E$ over field $K$ (dim$E=n$ and $ 0\leq p\leq n $ , $\Lambda^0E:=K$). We define linear map ...
0
votes
1answer
30 views

Is seven-dimensional cross product rotationally invariant?

For three-dimensional cross product, the following property holds true: \begin{equation} (R\mathbf x) \times (R \mathbf y)=R(\mathbf x \times \mathbf y) \end{equation} where $R\in SO(3)$. Is the ...
-2
votes
1answer
14 views

Statistics probability

A coffee shop sells 4 sizes with 4 different varieties. Customers can choose to add one or more syrups that come in 4 flavors. How many different coffees drinks can be made?
1
vote
1answer
26 views

Fermat's Theorem involving GP

Question:It is known by Fermat's Theorem that $n^p-n= M(p)$= a multiple of $p$,if $p$ is a prime number and $n$ is a prime to p.If $n$ is a prime number which divides neither $a,b$ nor $a+b$ ...
0
votes
0answers
19 views

Connectedness of circle without center line across it

Using a definition I saw in an old Russian book, a set in $\mathcal R^{n}$ is said to be connected if it cannot be represented as a disjoint union of two nonempty, separated sets. Separated, meaning ...
1
vote
3answers
29 views

Finding X from Exponential Equations

$$2^x \cdot 4^{1-x}= 8^{-x}$$ I wrote all the base numbers as a power of 2 but I'm not sure what to do after.
3
votes
0answers
38 views

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 ...
-6
votes
0answers
39 views

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 ...
0
votes
0answers
10 views

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 ...
-5
votes
0answers
25 views

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.
-2
votes
1answer
27 views

Calculus deriving functions for a given value

$F(x)=x^2-5x+3$. 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 ...
0
votes
1answer
8 views

Infinite closed shift-invariant set of binary sequences with zero entropy?

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?

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