All Questions

0
votes
1answer
109 views

$f$ is differentiable twice, bounded and has a minimum on $x_0$, prove that there's a point $y\in\mathbb R$ such that $f''(y)=0$

Suppose $f:\mathbb R\to \mathbb R$ is differentiable and there's a constant $c>0$ such that $f'(x)>c$ for all $x\in(a,\infty)$. Prove that $\displaystyle\lim_{x\to\infty}f(x)=+\infty$ Suppose $...
3
votes
0answers
271 views

Interesting Metrics

To preface, this question might come off as a bit silly, but it would serve me well to understand the concepts, and to actually get a good answer for this. How can I design an ideal metric for ...
1
vote
1answer
353 views

Angle between the shortest and biggest diagonals of a Decagon.

I've been doing some geometry lately and approached this problem. I need to get an angle between the biggest and shortest diagonals of a Decagon (10 sided polygon). As the book says I will get only 1 ...
1
vote
2answers
120 views

Equation $1+x^8y^4+x^4y^8-x^2y^4-x^6y^6-x^4y^2=0$

How to prove that the following equation: $$1+x^8y^4+x^4y^8-x^2y^4-x^6y^6-x^4y^2=0$$ has for solution(in real numbers): $|x|=|y|=1~$ only. Any hint would be appreciated.
2
votes
1answer
77 views

Using calculus of residues, prove that $\int z^2log[(z+1)/(z-1)]dz$

Using calculus of residues, how can it be proven that $$ \int z^2\log\left[\frac{z+1}{z-1}\right]\;dz $$ taken round the circle $\left\vert z\right\vert=2$ has the value $\frac{4\pi i}{3}$?
0
votes
0answers
32 views

Finding the co-efficient

I am trying to find the co-efficient of $\frac{1}{z}$ in the expansion of $$\frac{(1+z^2)^{2n}}{z^{2n+1}}$$ I proceeded like this - $$\frac{1}{z^{2n+1}}[1+(2n)z^2+\frac{(2n)(2n-1)z^4}{2!}+\dots+\...
0
votes
1answer
131 views

Example of non-Abelianness of symmetric group for graphs

I know that for $n \ge 3$, $S_n$ is non-Abelian. I would like to work out an example in terms of graphs so to make it sure that I understand it right. A symmetric group of graphs of four vertices, $G$...
1
vote
1answer
52 views

Generalized Poincaré Inequality

In my numeric script it states for a bounded domain $\Omega \subset\mathbb{R^n}$ and any map $0 < a_0 \leq a(x) \leq a_1 < \infty $ for $x\in \Omega$ it exists a $\gamma>0$ such that the ...
-1
votes
1answer
145 views

How many triangles?

The problem is the following: Please include your steps. Thanks!
1
vote
1answer
144 views

Prove that $ \left\{\frac{\binom{n+k}{k} }{(n+k)^k}\right\}_{n=1}^{\infty} $ converges to $\frac{1}{k!}$.

Prove that $$ \left\{\frac{\binom{n+k}{k}}{(n+k)^k}\right\}_{n=1}^{\infty} $$ converges to $\frac{1}{k!}$, where $$\binom{n+k}{k} =\frac{(n+k)!}{n!k!}. $$ I'm not even sure how to approach this ...
1
vote
1answer
32 views

Differential problem in Gausses law

From the Gauss law we know, $\nabla \cdot \vec{E} = \rho / \epsilon_0 $. We have given that, $\vec{E}= kr^3 \hat {r}$ Now I have problem to get the identified part. Can you please elaborate that?
3
votes
1answer
483 views

Projection onto Birkhoff Polytope

Suppose we would like to compute the Euclidean projection of an arbitrary matrix $A$ onto the Birkhoff polytope, the set of doubly-stochastic matrices. Under some conditions on $A$, Sinkhorn's ...
3
votes
1answer
991 views

Rating system for 2 vs 2, 2 vs 1 and 1 vs 1 game

We play football table in such configurations: 2 vs 2 players 2 vs 1 1 vs 1 Team (consisting of 1 or 2 players) wins the single game when they score 10th goal. I would like to introduce a rating ...
3
votes
1answer
264 views

Properties of absolutely continuous functions

Let $u:[0,1]\to\mathbb{R}$ be a absolutely continuous function. It is know that $u'(x)$ exist almost everywhere and $u'\in L^1(0,1)$. Let $A=\{s\in [0,1]:\ u'(s)\ \mbox{exist and}\ u'(s)\neq 0 \}$ ...
0
votes
1answer
23 views

not complete solution in computation of singular cohomology

I know how to compute singular cohomology of single point. Then I want to compute cohomology of $\mathbb{Z}$. Consider $C_{k}(\mathbb{Z})=\oplus_{i\in Z} C_{k}(point)$. $$\text{Hom}(A\oplus B, G)= \...
8
votes
1answer
189 views

How many different values can that sum take?

Let $x_1,x_2,\dots,x_{100} $ be a permutation of $1,2,\dots,100.$ How many different values does the sum $ x_1+2x_2+\cdots+100x_{100}$ take?
3
votes
2answers
117 views

very strange phenomenon $f(x,y)=x^4-6x^2y^2+y^4$ integral goes wild

I am going over my lecture's notes in preparation for exam and I saw something a bit strange I would like someone to explain how it is possible. Look at the function $f(x,y) = x^4-6x^2y^2+y^4$ if we ...
0
votes
1answer
54 views

Find polynomials $u(x), v(x)$ satisfying $X^5 + X^2 = (X^3 + X + 1)u(x) + v(x)$

I'm not absolutely sure if I'm answering this correctly but here is the question. Let $F=GF(2)$. find polynomials $u(x), v(x) \in F[X]$ satisfying $X^5 + X^2 = (X^3 + X + 1)u(x) + v(x)$ What I think ...
2
votes
2answers
969 views

Quotient spaces and quotient groups: equivalence classes and cosets

(Throughout this post, I am talking about vector spaces.) I had the pleasure of doing Abstract Algebra two semesters early, however, I feel like some general context was lost in the process. While I ...
1
vote
0answers
100 views

about the Perron method for the Dirichlet problem

Consider $\Omega$ an open, convex and bounded set of $R^n$. Let $g: \partial \Omega \rightarrow R$ a function. Supose that $g$ is continuous except in one point. By the convexity of $\Omega $ we can ...
3
votes
1answer
2k views

Counting triplets with red edges in each pair

Given a tree having N vertices and N-1 edges where each edges is having one of either red(r) or black(b) color. I need to find how many triplets(a,b,c) of vertices are there, such that on the path ...
1
vote
1answer
42 views

What truly are length, area and volume? And considerations about divergence in normed spaces

All the "(?)" are parts when i'm not sure at all if what i'm saying is right or not, it's just my intuition. Part 1 In $\mathbb{R}$, we can define the length of a segment. In $\mathbb{R}^2$, the '...
0
votes
1answer
53 views

Random variables and integrals

Could someone please explain how this holds: $\displaystyle \int_{\mathbb{R^n}} f d\mu = \int_{\Omega}f(Y_n)d\mathbb{P}$ Does it use the following proposition? Furthermore how does $\mathbb{E}(f(...
0
votes
1answer
157 views

Convergent's numerators of the continued fraction for $\pi$

Call $C_{\pi} = \{ 1,3,22,333,355,… \}$, it´s the sequence of the numerators of convergents of the continued fraction for $\pi$, its OEIS' A002485, http://oeis.org/A002485. Let $n \in C_{\pi}$, such ...
0
votes
1answer
97 views

Analog of eigenvalue bound for a general bounded operator

It's known that for a matrix, $\max|λ|≤\sqrt{tr(A^*A)}=\sqrt{∑_{i,j=1}^n|A_{i,j}|^2}$ where $\lambda$ denotes its eigenvalue. I'm wondering whether there's an analog of this inequality for a general ...
3
votes
2answers
111 views

Confused about the use of variables w/ logical quantifiers

Sorry if this is a really dumb question, but... After reading How to Prove it, I've become a little confused. On page 70, an example stating something similar to this is provided: $[\exists x P(x) \...
3
votes
1answer
156 views

Double integral containing $e^{(b+ic)/z^2}$

I want to solve the two integrals \begin{aligned} I_3\,& = \int_{0}^{\infty} ze^{a/z^2 - z^2} dz\\ I_4\,& = \int_{0}^{\infty} \frac{1}{z}e^{a/z^2 - z^2} dz. \end{aligned} where \begin{aligned}...
1
vote
3answers
110 views

Initial Value Problem: $\frac {dy}{dx}=\frac {xy\sin x}{y+1}, y(0)=1 $

Initial Value Problem: $$\frac {dy}{dx}=\frac {xy\sin x}{y+1}, y(0)=1 $$ I know I'm supposed to separate the values and integrate. this is where I get stuck: $$y+\ln y = -x\cos x+\sin x+c$$ This ...
0
votes
1answer
24 views

Defining $\sin$ using inverse function as teh first step

We compute the length of the piece of the circle between $0$ and $\theta$ for $\theta < \frac{\pi}{2}$ by considering it as the graph of the function $g(y)=\sqrt{1 - y^2}$ as $y$ varies between $0$ ...
1
vote
1answer
102 views

Prove there's $x_0$ such that $f'(x_0)=0$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ differentiable at $\mathbb{R}$ and: $$\lim_{x\rightarrow \infty}\left( f(x)-f(-x) \right) = 0$$ Show there's $x_0$ such that $f'(x_0) = 0$. I tried to use ...
3
votes
2answers
117 views

What is the domain of $x^x$

I'm trying to figure out the domain of the function $y=x^x$. When I graph it, it appears to be defined on $[0, \infty)$, but then when I plug in individual negative numbers, for some of them I get ...
0
votes
1answer
482 views

Can the set L, of all even polynomials be a subspace of F[X]?

So I have the question Let F be a field and let L be the set of all polynomials f(x) element of F[X] satisfying the condition that deg(f) is even. Is L a subspace of F[X]? I would say that L is not a ...
1
vote
0answers
43 views

Influence of preconditioning on degenerate eigenvectors

I'm using a hierarchical decomposition of a sparse matrix $A$ as suggested here. I find that the method essentially finds eigenvectors using the QR algorithm. $A$ has some eigenvalues with ...
0
votes
2answers
134 views

$\log$ approximation for $\pi(x)$

It seems that a reasonable $\log$ approximation for $\pi(x)$ can be given, where $f(y, x) := \log\left(\dfrac{\log(x)}{\log\left(e(y - \lfloor y\rfloor) + x^{1/x}(1 - y + \lfloor y\rfloor)\right)}\...
0
votes
1answer
136 views

Locally convex space - closed sets

Assume we have a locally convex topology $\tau$ induced by semi-norms $\mathcal P$, on some real vectorspace $E$. Let $\sigma$ be the locally convex topology induced by the semi-norms $$ \mathcal Q :=...
0
votes
1answer
69 views

What ratio of students in a 20-student class with total average of 17.06 got at least 17?

The maximum score is at most 20. The class has 20 students. For math exam the average of the whole class is 17.06. What percent of the class got at least 17 out of 20 in the exam?
1
vote
1answer
139 views

Unable to solve system of equations in Lagrange multiplier problem.

The problem: Find the right triangular prism of given volume and least area if the base is required to be a right triangle. As for parameters of the right triangular prism, $V$ is volume, $A$ is ...
5
votes
1answer
103 views

Name for a property in a brutally elementary presentation of a monad

For evil reasons of my own, I'm trying to give a presentation of a monad in primitive terms, assuming only the notion of a category. More honestly, I looked at this post and got intrigued by the ...
0
votes
1answer
588 views

Double sigma summation is in complexity calculation

Basically i was reading skiena and doing exercise of 2nd chapter.The result of my calculation got differed from the actual solution given on Solution site and there is one thing i don't understand how ...
1
vote
1answer
245 views

Morphism of rings and localization

Let $ \varphi : A \to B $ be a morphism of rings. Why are the two following assertions equivalent: $ 1) $ There exists a multiplicative subset $ S $ of the ring $ A $, and an ideal $ I $ of $ A $, ...
0
votes
1answer
50 views

Analogous for Weierstrass' theorem for functional series.

The Weierstrass' theorem for functional series say the following: Suppose $\{f_n\}$ is a sequence of functions defined on $E$, and suppose $$|f_n(x)|\leq M_n$$ for every $x\in E$ and $n=1,2,3,.....
0
votes
1answer
86 views

How to find the roots of $x^6 + x^5 +x^4 + x^3 +x^2 + x = n$ using trigonometric methods

Can all the roots (real or complex) of $x^6 + x^5 +x^4 + x^3 +x^2 + x = n$ be found using trigonometric methods? Many thanks to all of answers.
8
votes
1answer
758 views

Degree of ample bundle over projective curve is positive

(From Vakil's notes, Exercise 18.4.K) If $C$ is an integral projective curve over a field $k$, and $\mathscr{L}$ is an ample line bundle on $C$, why is the degree of $\mathscr{L}>0$? If $C$ is ...
2
votes
1answer
85 views

Integration of exponential functions: $\int_0^\infty e^{-({x^2}/{y^2})-y^2}\; dx$

How I am to solve this integral? I am not able to use any of the methods. $$\int_0^\infty e^{-({x^2}/{y^2})-y^2}\; dx$$
0
votes
1answer
21 views

Confusion related to proximal mapping [duplicate]

I was reading this paper http://machinelearning.wustl.edu/mlpapers/paper_files/NIPS2012_0388.pdf and I came across this part I didn't get how the third line came from the second line. Any ...
1
vote
1answer
107 views

Differentiability in a neighborhood of a strictly convex continuous function

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a strictly convex continuous function, and let $f$ be differentiable at the point $x_0\in \mathbb{R}$. Can we say that $f$ is differentiable on some ...
1
vote
1answer
317 views

Strong solution of stochastic differential equation

Consider the stochastic differenctial equation: $dX_t=\frac34 X_t^2 dt-X_t^{3/2}dW_t$. How to find a strong solution?
-3
votes
2answers
272 views

How to solve the SDE $dX_t = \frac{b-X_t}{T-t} \,dt + dW_t$?

SDE: $$dX_t=\frac{b-X_t}{T-t}dt+dW_t,t<T, \qquad X_0 = a$$ Answer: Let $b(t)=\frac{-1}{T-t},c(t)=\frac{b}{T-t},\sigma(t)=1$, then $$\begin{align*} X_t&=X_0\exp(\int_{0}^{t}b(s)ds)+\int_{0}^{...
1
vote
2answers
90 views

Travelling round a circle

A and B start running from the same point to run in opposite directions round a circular race course 4324 meters in circumference, A not starting till B has run 716 meters. They pass each other when A ...
1
vote
1answer
67 views

On the confinement to $[0;1]$ of the solution of $dX_{t}=(1-X_{t})X_{t}dB_{t}$

One considers the stochastic differential equation $$dX_{t}=(1-X_{t})X_{t}dB_{t},$$ with $B$ Brownian motion, and one assumes that $0\leq X_{0}\leq1 $. One wants to show that $\textbf{P}(X_{t}\in[0;1]...

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