0
votes
0answers
2 views

Infinite Set Proof (Countable and Uncountable )

I can't figure out this problem, I have to prove that Q × Q is denumerable, but I have no idea how to do it. Thanks
0
votes
0answers
2 views

Mathematical Induction.

We know that because of commutative law we have, $a+b=b+a$ and similarly we have associative property and likewise distributive. But isn't there a need to prove following laws by mathematical ...
0
votes
0answers
4 views

probability of the empty set for arbitrary probability measures

I have a probability space $(\Omega, \mathcal{P}(\Omega), P)$. I want to know the probability of the empty set $\{\}$. Intuitively, I would say this probability is zero. It certainly is for the ...
0
votes
0answers
16 views

Group actions: Why do we place the condition that $S$ be finite in the following theorem?

Theorem. Let $G$ be a group, $S$ be a $G$-set, and $S$ be finite, then $$|S|= \sum_{a \in A} [G : G_a],$$ where $A$ is a subset of $S$containing exactly one element from each orbit $[a]$. Here, $G_a$ ...
0
votes
0answers
5 views

Tychonoff spaces with small weight

Let $\kappa$ be an infinite cardinal. Is there a Tychonoff space $(X,\tau)$ such that $|X| = 2^\kappa$ and $(X,\tau)$ has weight $\kappa$ (= a basis consisting of $\kappa$ elements)?
0
votes
1answer
11 views

Is split-complex $j=i+2\epsilon$?

In matrix representation imaginary unit $$i=\begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix}$$ dual numbers unit $$\epsilon=\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}$$ ...
0
votes
2answers
12 views

Neat Diophantine Equation Question

After some fairly tedious work including studying multiple different cases separately, I have found all the solutions to $$a^n+1=b^2 $$ where $a$, $b$, $n$ can take on the value of any integer, be it ...
0
votes
1answer
8 views

Construct a procedure which determines the location of the shadow of a rectangluar box.

I drew a 3d rectangular box on a coordinate plan consisting of x, y, and z. A procedure is to be created that will determine the location of the shadow of the box on one of the coordinate planes. I ...
0
votes
0answers
7 views

Direct sum of two spaces

Let $\alpha_1=[1,1,0,1]$, $\alpha_2=[1,0,1,1], \alpha_3=[1,1,1,1],\alpha_4=[0,1,1,1]$ be a vectors from $\mathbb{R}^4$ let $U=span(\alpha_1, \alpha_2) \ and \ W=span(\alpha_3, \alpha_4)$ Check that ...
2
votes
0answers
6 views

Am I upper bounding or lower bounding my probability distribution function?

I came across a probability distribution function in my work, it is however difficult to find in closed form, therefore I am looking to either upper bound or lower bound it. Assuming $a,b,T$ are ...
0
votes
0answers
8 views

Existence & uniqueness of a second order ODE

Details on the model can be found here under III titled "Will The Valve Hold?". ...
2
votes
1answer
10 views

Does there exist a 4D torus with a spherical cross-section, analogous to a circle for the 3D case?

I don't mean to be a bother. It seems as though the answer may be obvious, but then, seemingly simple math questions can have surprising answers. I should also like any pointers re: the general ...
0
votes
1answer
25 views

Determinant: Continuity?

Building on the previous thread: Determinant: Definitions? Presumptions: Differential Geometry, Functional Analysis Given a vector space $V$. Consider an endomorphism $T:V\to V$. Define its ...
1
vote
0answers
11 views

A sum involving binomial coefficients and a simple fraction

Let $a_1$ and $a_2$ be real numbers. Let $n_1$ and $n_2$ be positive integers. Finally let $\theta$ be a real number which is different from a negative integer. By the generalizing the result from ...
1
vote
1answer
21 views

What can we say about the set X?

We have a certain set $X$ for which is valid: $\forall U\subset X:[ U\neq X ]\rightarrow U\nsim X$. What can we say about $X$? I think we've got to use the axiom of choice here. My first guess would ...
1
vote
0answers
9 views

Statistics question involving exponential distribution and (maybe) gamma function

This is from a past stat exam that I am studying for my final tomorrow (lol). I believe this might have to do with gamma function. Could someone guide me through step by step of how to do this? An ...
0
votes
1answer
5 views

how to find out intial direction and angle of collision

i have a problem in my game. I have a wall where a ball hit to a wall from anywhere. i need to give it to the direction according to collision law. Let suppose if a ball thrown from x == 0 and y == 0 ...
0
votes
0answers
12 views

C*-algebras: States?

I'd like to better understand states on C*-algebras. I suppose basic facts about functional analysis. (C*-algebras, spectral theory, functional calculus, Banach-Alaoglu, etc.) What properties ...
2
votes
0answers
14 views

My question pair about diagonalization

Let A = $\begin{bmatrix}1 & 1 & 4\\0 & 3 & -4\\0&0&-1\end{bmatrix}$. Is the matrix $A$ diagonalizable? If so find a matrix $P$ that diagonalizes $A$. Can you write $A$ as a ...
0
votes
1answer
30 views

What is my maths score as a percentage?

In my maths module, there are 2 phases. In the 1st phase, there are 2 tests. In the 2nd phase there is 1 test. Phase 1's total weight is 60% and phase 2's total weight is 40%. This is what I ...
2
votes
1answer
9 views

example which doest not satify Lipchitz condition but has unique solution

$y'=1+\sqrt y , y(0)=0 $ Show that this IVP does not satify Lipchitz condition but has a unique solution. I have shown the first way, like this: Let $f(x,y)=1+\sqrt y $.Then $\frac ...
0
votes
0answers
7 views

Which are the best book for doing a project on graph theory, exspecially the topic spanning trees and its applications?

I want the best reference books in post graduate level or high. Can any one help me to select a book?
4
votes
2answers
28 views

Calculating Triple Integral

I have task : find volume of body limited by surface $(\frac{x}{a})^{2/3} + (\frac{y}{b})^{2/3} + (\frac{z}{c})^{2/3}$ = 1. I know that this task is about triple integral. But i have confused by such ...
2
votes
0answers
11 views

Proposed proof that $H^{1}(\mathbb{R})$ is closed under multiplication.

My idea is to approximate with functions in $C^{\infty}(\mathbb{R})$ with compact support. Let $u,v \in H^{1}(\mathbb{R})$. Since $C^{\infty}_0(\mathbb{R})$ is dense in $H^1(\mathbb{R})$ there ...
0
votes
0answers
6 views

Is the integral of $\det(I+f(x)t)$ on the unit ball a polynomial?

Let $B$ be the unit closed ball of $\mathbb{R}^n$, $f:\mathbb{R}^n\to \mathbb{R}^n$ be a linear application and $t\in \mathbb{R}$ a parameter. Is there an easy way to see that the (Lebesgue) integral ...
0
votes
0answers
10 views

$c_o$ is not isometric to $c_0 \oplus c_0$

$c_0$ is the Banach space of sequences converging to zero and $c_0 \oplus c_0$ is its algebraical direct sum with itself equipped with the norm $||(\xi,\eta)|| := ||\xi||+||\eta||$. How to prove that ...
0
votes
0answers
4 views

Contrability to zero state

We have LTI system $\dot{x}(t)=Ax(t)+Bu(t)$, where the matrix $A$ is Hurwitz and $(A,B)$ is controllable. I need to show that $\forall \beta >0$ and $\forall$ initial states $x(0)$, $\exists$ an ...
0
votes
0answers
15 views

Convergence of $\sum_{n=1}^{\infty} a^{-\frac{c}{c+1}}\prod_{k=1}^{n} \frac{a}{a+k^{c}}$ as $a\to\infty$

It looks to me, by doing numerical simulations, that $$ v_{c} = \lim_{a\rightarrow \infty}\sum_{n=1}^{\infty} a^{-\frac{c}{c+1}}\prod_{k=1}^{n} \frac{a}{a+k^{c}} $$ converges to some value bigger than ...
0
votes
0answers
14 views

Convert the following system to a first order system:

Really having a hard time with this.....Convert the following system to a first order system: $$\frac{d^2x}{dt^2} -3\frac{dy}{dt}+x=\sin(t)\\ \frac{d^2y}{dt^2} -t\frac{dx}{dt} - ye^t =t^2$$
0
votes
0answers
5 views

determine how much probability increase with an added condition

Suppose there are $N$ people and $N$ prizes, and only $M$ out of $N$ are valuable. Every time one person is picked randomly, then he pick one prize randomly as well (this prize/person is then removed ...
0
votes
0answers
17 views

Where can Gaussian Elimination be used?

I have searched for this and came to know about it that it is traditionally used to solve linear equations, finding determinant, rank of matrix, inverse of matrix. There was a problem on codechef: ...
1
vote
1answer
15 views

linear programming problem - how much additional resources should I buy?

I have the following linear optimization problem: Maximize $$\sum_{i=1}^{n}x_{i}B_{i}$$ subject to the constraints $$a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n \le l_1$$ $$...$$ ...
4
votes
2answers
23 views

Two-sheeted covering of the Klein bottle by the torus

Prove that there is a two-sheeted covering of the Klein bottle by the torus. OK, so we take the the polygonal representation of the torus and draw a line in the middle as follows: Then there are ...
0
votes
1answer
13 views

a question about general and particular solutions

We have a $3\times 6$ matrix $A$ with rank $3$ (this is all the information we have, no matrix given). Here comes the questions: What is the number of free variables in the solution to the system ...
1
vote
0answers
11 views

Need help with holomorphic functions on a domain interval removed.

I want to prove that for a region $\Omega$ with interval $I=[a,b]\subset\Omega$, if $f$ is continuous in $\Omega$ and $f\in H(\Omega-I)$, then actually $f\in H(\Omega)$. Is this problem related to ...
0
votes
0answers
15 views

Determine the number of even subsets

Let's say, that a set of edges $F \subseteq E$ is even, if all the vertices in the multigraph $H \subseteq G$ have even degrees. Prove, that $E$ has exactly $2^{n(E)}$ even subsets, where ...
1
vote
1answer
13 views

For $(l^2,\|\cdot\|_2)$ and $e_n=(0,0,.,1,0,.)$ and a bounded linear functional $\Phi$ find $p\geq 1$ where $\sum_{n=1}^\infty |b_n|^p$ converges?

For $(l^2,\|\cdot\|_2)$ and $e_n=(0,0,...,1,0,...)$ and a bounded linear functional $\Phi$ find a value of $p\geq 1$ where $\sum_{n=1}^\infty |b_n|^p$ converges for $b_n=\Phi(e_n)$? Ok so since ...
0
votes
0answers
5 views

Density of this quantity for a Geometric Brownian Motion?

If we define $X_T = X_t e^{(\mu-\frac{1}{2}\sigma^2 ) (T-t) + \sigma W_{T-t}}$ where $W_{T-t}$ is a classical weiner process. How would we go about deriving the density and expectation for $X_{max} - ...
1
vote
1answer
22 views

How to show that the following relation is not an equivalence relation?

We have the relation $\sim$ in $\mathbb{R}^n$: $x\sim y \leftrightarrow d(x,y)\in \mathbb{Q}$, where $d(x,y)=\sqrt{\sum^n_{i=1}(x_i-y_i)^2}$. How do you prove that this isn't an equivalence relation ...
0
votes
0answers
11 views

CDF of RVs taking infinite values

How can we define the CDF of a RV that takes positive infinite values with a tagged probability? Thanks in advance
2
votes
1answer
30 views

If $X$ is Hausdorff, then so is $E$

Let $q:E \to X$ be a covering map. If $X$ is Hausdorff, then so is $E$. OK, suppose $X$ is Hausdorff and let $x,y \in E$ with $x\neq y$. Let $V$ denote the evenly covered neighbourhood for $q(x)$, ...
0
votes
1answer
14 views

Inverse Laplace Transformation

I was solving a problem but I am stuck at it. Here is the question : $\frac{7s^2+9s+3}{(s^2-12s+40)(s^2+9)}$ Find inverse Laplace transform. I performed these operation : ...
4
votes
1answer
53 views

Show that $\{e^{in}: n\in\Bbb N\}$ is Dense in the Unit Circle

This problem gave me fits when I was in grad school. Looking back at it now, it still escapes me. The problem is from Conway's Functions of One Complex Variable. I'm looking for a proof from basic ...
1
vote
1answer
7 views

Odd coefficient in $M\in \mathcal{M}_n(\Bbb{Z})$ satisfies $n\le m\le n²-n+1$.

Let $M\in \mathcal{M}_n(\Bbb{Z})$ I would like to prove that all odd coefficient of $M$ satisfies $n\le m\le n²-n+1$. In fact I don't see why $m$ is necessary bigger than $n$. I can only prove ...
0
votes
1answer
13 views

Fourier transform, quadratic function

I'm trying to compute this convolution: $\frac{2 \alpha}{\alpha ^2 + 4 \pi ^2 x^2} * \frac{2 \beta}{\beta ^2 + 4 \pi ^2 x^2}$ I know that the Fourier transform of a convolution of two functions is ...
0
votes
0answers
9 views

Prove existence of (Nash) equilibrium

My question is about proving the existence of Nash equilibrium for a game involving two players. $x$ is player 1's strategy and $y$ is player 2's strategy; both strategies are continuous. For each ...
0
votes
0answers
5 views

Reference for work on abelian divisible groups $G$ such that for every $n \in \mathbb N , g \in G , \exists$ unique $x \in G$ such that $g=x^n$

Is there any work or reference in the literature about those abelian divisible groups $G$ such that for every $n \in \mathbb N , g \in G , \exists$ unique $x \in G$ such that $g=x^n$ ; I think then I ...
0
votes
1answer
23 views

Why $\left\|\sum_{i=1}^nx_i\right\|^2\leq n\sum_{i=1}^n\|x_i\|^2$

Why $$\left\|\sum_{i=1}^nx_i\right\|^2\leq n\sum_{i=1}^n\|x_i\|^2$$ for arbitrary norm on the inner product space over the real field? My attempt $$\left\|\sum_{i=1}^nx_i\right\|^2 = ...
1
vote
1answer
20 views

Mechanics question involving integration.

a particle P moves so that its position vector r satisfies the differential equation $$\frac{dr}{dt}= c \times r,$$ where c is a constant vector. Show that P moves with constant speed on a circular ...
1
vote
2answers
34 views

Prove $\lim_{x \to 0} \frac{e^{\sin(x)} - e^{\tan (x)}}{e^{\sin (2x)}-e^{\tan (2x)}} = \frac{1}{8}$

Here's a nice little problem. $$\lim_{x \to 0} \frac{e^{\sin(x)} - e^{\tan (x)}}{e^{\sin (2x)}-e^{\tan (2x)}}$$ What's the quickest way to do this? One line solutions will be applauded :D Cheers, my ...

15 30 50 per page