0
votes
3answers
83 views

Need pointers on how to do this trigonometric proof

$$ \cos x = \cos y + \cos^3 y$$ $$\sin x = \sin y - \sin^3 y$$ Prove that $\sin {(x - y)} = \pm \frac{1}{3}$. I need a little hint, not a complete answer.
1
vote
0answers
59 views

Coding Theory and Justesen Codes

I wonder if Justesen codes can have transmission rate $R=\frac{1}{1+\epsilon}$ for any constant $\epsilon>0$?
1
vote
3answers
144 views

Solve the equation : $2013x+\sqrt[4]{(1-x )^7}=\sqrt[4]{(1+x )^7}$.

Solve the equation : $2013x+\sqrt[4]{(1-x )^7}=\sqrt[4]{(1+x )^7}$. Show that it has percisely one root: $x=0$.
0
votes
2answers
82 views

In an inner product space over $\mathbb R$, prove $ (u,w)=0 \Leftrightarrow \left \| u+w \right \|=\left \| u-w \right \| $

Let $V$ be an inner product space over field $F$ and $u,w\in V$. Prove that if $F=\mathbb{R}$ then: $$ (u,w)=0 \Leftrightarrow \left \| u+w \right \|=\left \| u-w \right \| $$ Is it also true for ...
3
votes
3answers
92 views

Definition of the fundamental group

Why are the elements of the fundamental group of a space equivalence classes? Why isn't the group defined to be the set of all possible loops at a base point with the product operation of paths? What ...
4
votes
1answer
72 views

Explicitly writing out a differential 2-form

In Tu's An Introduction to Manifolds, one question asks: At each point $p\in \mathbb{R}^3$, define a bilinear function $\omega_p$ on $T_p(\mathbb{R}^3)$ by: ...
0
votes
1answer
116 views

Show mapping involving tensor product is well defined.

Let $R$ be a subring of $S$, let $N$ be a left $R$-module and let $\iota : N \to S \otimes_RN$ be the $R$-module homomorphism defined by $\iota(n) = 1 \otimes n$. Suppose that $L$ is any left ...
0
votes
1answer
51 views

Why do $UU^* = I$ and $U^*U = I$ hold on different spaces for the unitary matrix $U$ of a polar decomposition?

The following is from Lang $SL_2$. Consider the polar decomposition of a matrix A. We let $P_A = (A^*A)^{1/2}$ and set $U$ s.t. we have $$UP_Av = Av.$$ Then, it follows that $U\colon ...
3
votes
1answer
88 views

Bessel equation relation

Define Bessel function as: $$J_a(x)=\sum_{n=0}^{\infty}{{(-1)^n}\over{\Gamma(a+n+1)n!}}\left({{x}\over{2}}\right)^{a+2n}.$$ Where $a$ not an integer. Need to show ...
2
votes
0answers
47 views

Being a morphism of quasiprojective varieties is a local property

Let $X,Y$ be quasiprojective varieties and $\phi \colon X \to Y$ be a map. Suppose there exists a cover $\mathscr U = \{U_i\}_{i \in I}$ of open subsets $U_i \subset X$ such that for every $i \in ...
5
votes
2answers
126 views

why $\frac{d}{dy}$ can pass through integral w.r.t. $x$?

When I calculate integration of multivariables, many books use the following step without proofing. I want to know that why it is true: $$\frac{d}{dy}\left[\int^a_b f(x,y)dx\right]_{y=k}=\int^a_b ...
1
vote
1answer
267 views

Perfect matching in a graph

Assuming I have a bipartite graph with the following property: for each subgroup of nodes $s \subseteq {V} $ : $$ \sum_{v\epsilon N(S),z\epsilon N(N(S)) }{} {(v,z) \geq 2\left \| S \right \|} $$ ...
1
vote
1answer
75 views

An inequality-Is it possible?

I want to know that is it possible to show that $$ \int_{0}^{T}\Bigr(a(t )\Bigr)^{\frac{p+1}{2p}}dt\leq C\Bigr(\int_{0}^{T}a(t)dt\Bigr)^{\frac{p+1}{2p}} $$ for some $C>0$ where $a(t)>0$ and ...
1
vote
1answer
64 views

Approximating a function

I'm sorry if this question in not well formed... I would like to perform a computation of the following function: f() = -2*X1 -1*X2 +0*X3 + 1*X4 +2*X5 (The ...
3
votes
2answers
310 views

Prove the exsistence of 3 zero points of a function

Assuming $a<b$, function f(x) is continous at [a,b], and we have $\int_a^bf(x)dx=\int_a^bxf(x)dx=\int_a^bx^2f(x)dx=0$ Prove that $\exists \ x_1,x_2,x_3\text {(different from each other)} ...
1
vote
3answers
69 views

How to solve a system of equations

I have this system of equations: 3x²+7y²=55 and 2x²+7xy=60 Is there a method of solving [x,y] without using x²=t, y²=z?
0
votes
2answers
142 views

Intuition for orthogonal vectors in $\Bbb R^n$

Two vectors in $\Bbb R^n$ are orthogonal iff their dot product is $0$. I'm aware that the dot product can be defined in other spaces, but to keep things simple let's restrict ourselves to $\Bbb R^n$. ...
3
votes
1answer
427 views

Loop space suspension/adjunction

Let be $X,Y$ Hausdorff spaces. I denote with $\langle \cdot, \cdot \rangle$ the basepoint preserving homotopy classes of maps, with $\Sigma$ the reduced suspesion and with $\Omega$ the loop space. How ...
5
votes
1answer
208 views

What does $\mathrm d^2 x$ exactly mean?

I am learning radiometry and one of the equation is radiance which is given as the radiant flux per unit projected area per unit solid angle. In equation: $$L = {d^2\Phi \over {cos(\theta)dAd\omega}} ...
5
votes
2answers
47 views

$f(z)$ and $f(z+z^2)$ have the same singularities at 0

Let $f:\mathbb{C}\rightarrow\mathbb{C}$ be an analytic function in a punctured neighborhood of 0 then $f(z)$ and $h(z)=f(z+z^2)$ have the same singularity at $z_0=0$. I was able to show that every ...
2
votes
1answer
47 views

How many distinct copies of $P_m$ are in $K_n$?

Let $K_n$ be the complete graph of order $n$ and $P_m$ a path with $m$ distinct vertices, $1 \leq m \leq n$. Question: How many distinct copies of $P_m$ are contained in $K_n$? Given that a ...
4
votes
1answer
83 views

What is the probability that the roots of $P(x) = \frac{1}{4} x^2 +Ux+V^2$ are real?

Assume that $U$ and $V$ are independent normally distributed random variables, each with mean $0$ and variance $1$. Find the probability that the roots of the polynomial $P(x) = \frac{1}{4} ...
2
votes
3answers
85 views

Integral of $\frac{2}{x^3-x^2}$

How can I integrate $\dfrac2{x^3-x^2}$? Can someone please give me some hints? Thanks a lot!
3
votes
1answer
74 views

Can I write $(x^p)^{q} = x^{pq}$, where either of $p$ or $q$ are rational.

I am confused with a very basic algebra question about the following law of exponents. We know that $(x^n)^{m} = x^{nm}$, holds true for real $x$ and integer exponents $n, m$. I want to know whether ...
1
vote
1answer
183 views

Counterexample for finite dimensional weak convergence

Could you give an explicit construction of a sequence $\mathbb P_n$ of probability measures on $C[0,\infty]$ which converges in the sense of finite dimensional distributions BUT does not converge ...
2
votes
1answer
52 views

Kronecker-Weber does not apply

Let $K=\mathbb{Q}(\sqrt{D})\neq \mathbb{Q}$. Show that $K$ has an abelian extension that is not contained in $K(\theta)$ for any root of unity $\theta$. Hint: Find $u \in K $ such that $K(\sqrt{u})$ ...
4
votes
1answer
362 views

How to better understand where the circles and lines go under fractional linear transformations?

Today I encountered the transformation $f(z) = \frac{z}{z-1}$. It has the following property: As the point $z$ makes a counter-clockwise revolution around the unit circle beginning at $1$, the point ...
1
vote
1answer
178 views

Question regarding a Jacobian

Suppose I have these two pairs of variables: \begin{equation} u = g_1(x,y), \qquad v = g_2(x,y), \end{equation} \begin{equation} x = h_1(u,v), \qquad y = h_2(u,v). \end{equation} If my jacobian of $x$ ...
1
vote
1answer
54 views

Research and application of causal inference

I have been reading Pearl's book to understand how Bayesian networks and causal discovery might work. Other than Pearl, I haven't yet found a rigorous, systematic approach to causal inference from ...
0
votes
1answer
83 views

Why is this homotopy an isotopy?

I am trying to follow the proof of Willem's quantitative deformation lemma and I get everything except the justification for the (iv) property which states: $\eta_t(u)$ is a homeomorphism for ...
2
votes
2answers
70 views

a matrix with determinant $1$, what can be said about the column $(a \space c)^T$?

$\begin{pmatrix}a&b\\c&d\end{pmatrix}$ be a matrix with determinant $1$, what can be said about the column $(a \space c)^T$? from the condition we have $ad-bc=1$, so do I have to conclude ...
1
vote
3answers
85 views

Is $\alpha^{\beta}$ a cardinal?

Let $\alpha , \beta$ cardinals. Is $\alpha ^{\beta}$, defined as the set of all functions $f:\beta\to \alpha$, a cardinal? I do this question because an autor of a text book says that the ...
5
votes
2answers
157 views

Problem from Hardy's _Pure Mathematics_

If $a$, $b$, $x$, $y$ are rational numbers such that $$(ay-bx)^2+4(a-x)(b-y) = 0 $$ then either (i) $x = a, y = b$ or (ii) $1-ab$ and $1-xy$ are squares of rational numbers. (Math. Trip. 1903) ...
2
votes
1answer
51 views

Restriction to $\mathbb{R}^{d-1}$ as an operator on $L^2(\mathbb{R}^d)$

Identify $\mathbb{R}^{d-1}$ with $\mathbb{R}^{d-1}\times \{0\}\subseteq \mathbb{R}^d$. Is there a bounded operator $T: L^2(\mathbb{R}^d)\rightarrow L^2(\mathbb{R}^{d-1})$ such that $T(\phi)=\phi ...
3
votes
1answer
186 views

Improper integral depending of one parameter

I would like to show that $$\int_0^{\infty}\frac{\sin x}{x}e^{-nx} \, dx=\arctan\frac{1}{n}$$ Any help is appreciate!
4
votes
2answers
142 views

If $a,b,c > 0$ satisfy $a^2+b^2+c^2=3$ then $\frac{a}{b+c+3}+\frac{b}{a+c+3}+\frac{c}{a+b+3} \geq \frac{3}{5}$

Given $a,b,c > 0$ satisfying the condition $$a^2+b^2+c^2=3,$$ prove that $$\frac{a}{b+c+3}+\frac{b}{a+c+3}+\frac{c}{a+b+3} \geq \frac{3}{5}.$$ Thank you all
3
votes
1answer
596 views

Matrices with trace zero.

I would like to show that every trace zero square matrix is similar to one with zero diagonal elements. This question has been asked before, and has had an answer by Don Antonio. And my problem is ...
2
votes
2answers
92 views

Determining Convergence of Power Series

Flip a fair coin until you get the first "head". Let X represent the number of flips before the first head appears. Calculate E[X]. So I solved this problem and you get a power series: $E[X] = ...
1
vote
1answer
65 views

Analytic extension of functions in Hardy spaces

This is a problem I came across in a direct scattering problem. I have a function $a(s)$ that is of the form$$ a(s)=\int_0^{\infty}e^{is\xi}A(\xi)d\xi $$ where $A(\xi)\in L^1\cap L^2$. Then is it ...
3
votes
1answer
600 views

Steady State Solution Non-Linear ODE

I'm working through some problems studying for a numerical methods course, but I'm stuck on how to answer the following question analytically. It says to find the steady state solution of the ...
1
vote
4answers
4k views

What is a fraction in which the greatest common factor of the numerator and the denominator is 1?

What is this fraction: A fraction in which the greatest common factor of the numerator and the denominator is 1?
3
votes
1answer
146 views

Find $A=\sum\limits_{n=1}^{\infty}\frac{\log 2^n}{3^n}$

I can't find $$A=\sum_{n=1}^{\infty}\frac{\log 2^n}{3^n}$$ Please help me?
3
votes
1answer
45 views

Showing a sequence of functions converges to a Gaussian

Suppose we have a thrice continuously differentiable function such that the following hold: $f(0)=1$, $f^\prime(0)=0$, and $f^{\prime\prime}(0)=-1$. There exists $\alpha>0$ such that $\vert ...
1
vote
1answer
81 views

product of positive semidefinite matrices

If $A$ and $B$ are both Hermitian positive semidefinite matrices, then whether $vec\{A\}vec^{T}\{B\}$ is Hermtian positive semidefinite? (vec{A} means column vectorization of the matrix $A$)
1
vote
2answers
67 views

If the points $x_1,x_2,\ldots,x_n$ are distinct,then…

I am stuck on the following problem that says: If the points $x_1,x_2,\ldots,x_n$ are distinct,then for arbitrary real values $y_1,y_2,\ldots,y_n$, prove that the degree of the unique ...
0
votes
2answers
123 views

Are “prime factorization” and “integer factorization” the same?

Are "prime factorization" and "integer factorization" the same? If not, what is the difference?
2
votes
3answers
318 views

Intuition of Empty Set in Ordered Pair

This question is inspired by Exercise 1.44 Mathematical Proofs, 2nd Ed, by Gary Chartrand et al. Given that $A = \{\emptyset, \color{green}{\{{\emptyset}\}}\}$, I understand that $A \times ...
6
votes
3answers
389 views

Show that $\int_0^\infty e^{-(x-u/x)^2}(1-\frac{u}{x^2})dx$ converges uniformly on $u\in [\delta,L]$

I would like to show that $$\int_0^\infty e^{-(x-u/x)^2}\left(1-\frac{u}{x^2}\right)dx$$ converges uniformly on, say, $u\in [\delta,L]$, for arbitrarily small $\delta>0$ and arbitrarily large ...
0
votes
1answer
62 views

Non-convergent series of convergent integral

I'm trying to find a series representation for a integral, but I think there's something I'm missing, as even though the algebraic manipulations I'm doing are valid (I think!), the series ...
2
votes
1answer
50 views

I need an equation for some data points.

My data points are (97.57,6.14), (90.54,7.03), (81.99,8.55), (71.47,10.52), (56.5,14.97) and (31.88,24.62). I'm trying to find the nonlinear equation that describes these points, but I'm having ...

15 30 50 per page