2
votes
0answers
11 views

A specific embedding of semisphere on $R^2$.

I was playing with piece of paper which has the form of semisphere, to be more precise we may assume that it satisfies $x^2+y^2+z^2=1$ for nonnegative $z$. I tried to make it flat without stretching ...
0
votes
0answers
6 views

process and renewal equation

The renewal equation is: $Z=z+F*Z$ and $Z(t)=z(t)+\int_0^t Z(t-u)F(du)$ Let $A(t)=\sum_0^{\infty} F^{*n}(t)$ the renewal function How to show $A(t)<\infty$? Thank you
0
votes
2answers
14 views

Show that f is measurable.

Let $a > 0, b \geq 0$ and the function $f: \mathbb{R} \to \mathbb{R}$ $$f(x) = \left\{\begin{matrix} 1, & |x| \leq a \\ b & |x| > a \end{matrix}\right.$$ show that it is measurable. ...
0
votes
0answers
2 views

Cuts in planar graphs

I am currently trying to prove the correspondence between cuts of a planar graph $G$ and the even sets of its dual, $G^*$. An even set $D\subseteq E$ is such that all vertices of $G^*$ are incident ...
0
votes
1answer
3 views

A sequence in a Hausdorff space and in a space that is not Hausdorff.

Let $X$ be a topological space and $\{x_n\}_{n=1}^{\infty}$ a sequence in $X$. Show that if $X$ is Hausdorff, $x_n \rightarrow x \:$, $x_n \rightarrow y \:$ implies $x=y$. Give an ...
0
votes
0answers
7 views

Ex ODE: $y'=4t \sqrt y- \lambda(y-(1+t^2)^2)$

How to solve the following equation? $y'=4t \sqrt y- \lambda(y-(1+t^2)^2)$ $y(0)=a$ Show those cases where a numerical method will solve this equation exactly. $(a,\lambda) \in {\Re}^2$
0
votes
0answers
12 views

Expected number of red balls in an urn | a specific ball being in it

This is a follow-up on this question. We toss balls into urns. Denote with $x$ the number of balls in an urn. And $x_r$ denotes the number of red balls. The share of red balls among the balls is ...
0
votes
0answers
3 views

How are the embeddings of a subfield of a Galois extension $K$ related to the embeddings of $K$?

Suppose we have a Galois extension $K/\mathbb{Q}$, then all embeddings of $K$ into $\mathbb{C}$ (or $\mathbb{R}$) are determined by the Galois group $G=\text{Gal}(K/\mathbb{Q})$. That is if we let ...
0
votes
1answer
24 views

$|\sin(\sin( \cdots \sin(x)\cdots))|$ ($N$ times) is always $\leq|\sin( \cdots \sin(1)\cdots)|$ ($N-1$ times)

Is this inequality always true? $$\left\lvert\underbrace{\sin(\sin( \cdots \sin}_{N\text{ times}}(x)\cdots))\right\rvert\le\left\lvert\underbrace{\sin(\sin( \cdots \sin}_{N-1\text{ ...
0
votes
0answers
11 views

Fourier transform on forced linear harmonic oscillator

I'm having trouble solving the following equation: $\ddot x + \omega^2 x = a \sin \Omega t$, where $t >0$, $\omega \neq \Omega$, $x(0+) = 0 = \dot x(0+)$. It is asked to be done by Fourier ...
1
vote
0answers
20 views

Realation between Matrices ?!

The answer to the following question could be trivial. Let $A_1, A_2$ be symmetric $n\times n$ matrices, $x=(x_1,\ldots,x_n)\in \mathbb{R}^n$. If the maximum is taken for over ($\|x\|=1,\, and ...
2
votes
3answers
30 views

If $n_{1}+n_{2}+n_{3}+n_{4}+n_{5} = 20.$ Then number of such distinct arrangements of $(n_{1},n_{2},n_{3},n_{4},n_{5})$

Let $n_{1}<n_{2}<n_{3}<n_{4}<n_{5}$ be the positive integers such that $n_{1}+n_{2}+n_{3}+n_{4}+n_{5} = 20$ Then number of such distinct arrangements of ...
0
votes
0answers
21 views

How to find out the following probability?

I need to find $\mathbb{E}_d[\mathbb{P}\left\{X\le\mu\right\}|\hspace{1mm}d]$ with \begin{equation} ...
1
vote
0answers
19 views

Does the product of two numbers with a primitive representation have a primitive representation?

I know the theorem that $n = x^2 + y^2, \, \textrm{gcd}(x, y) = 1 \iff p | n \implies p \equiv 1 \bmod 4$. We call an expression of $n$ in this form primitive. I'm trying to prove the statement. I've ...
0
votes
1answer
17 views

Affine varieties and their ideals (part2)

On wikipedia, they talk about varieties $V,W$ and the $I(V)$ and $I(W)$ as well as the quotient ideal, $$I(V):I(W) = I(V - W)$$ Can someone show me a quick proof of the identity?
1
vote
1answer
27 views

proving the volume of sphere by using tiny volumes

how can I prove the volume of sphere,by using many cones starting at the center of the sphere?it deosent HAVE to be cones, pyramids also work.
0
votes
1answer
24 views

Probability of infected but does not show symptoms of disease?

A person moving through a tuberculosis prone zone has a $50\%$ probability of becoming infected. However, only $30\%$ of infected people develop the disease. What percentage of people moving through a ...
1
vote
0answers
11 views

Linear Algebraic Groups with Same Lie Algebra (Soft Question)

Let $G$ and $H$ be two linear algebraic groups over an algebraically closed field $F$ (char 0 ) such that their lie algebras are isomorphic. Now what can we say about the relation between these two ...
2
votes
4answers
38 views

How many $3$-tuples $(a, b, c) \in M^3$ are there such $a+b+c$ is even?

The task is the following: $M= \left \{ 1,2, ... 99,100 \right \}$ How many $3$-tuples $(a, b, c) \in M^3$ are there such $a+b+c$ is even? I tried to solve it this way: There are only two ...
-2
votes
0answers
7 views

Lebesgue Spaces on the Boundary: Outward Normal Vector to ∂Ω

So basically, there is an outward normal vector from the boundary of an open set $\Omega$. What i need to do is to prove that this is actually a unit vector. Can anyone please help? Other sources like ...
0
votes
0answers
19 views

Solving modular equation conatining exponentiations

I encountered the following equation. Is there a general solution method for it? $$ b^c = x_1^{x_2} * d^{x_1}\mod a $$ Exponentiations and multiplications are done modulo $a. a, b, c, d$ are known ...
0
votes
1answer
9 views

Converse of realisation lemma for bisimplicial sets

Given two bisimplicial sets $X_{\bullet,\bullet}$ and $Y_{\bullet,\bullet}$, we have the result that if given a map $f:X_{\bullet,\bullet}\to Y_{\bullet,\bullet}$ such that the restriction ...
1
vote
0answers
15 views

moment of inertia and parallel axis theorem

A lamina with density $\delta \left ( x,y \right ) = x^{2}$ has the shape of the disk $\left \{ \left ( x,y \right )|x^{2}+y^{2}\leq 4 \right \}$. Find the moment of inertia of the lamina about the ...
0
votes
2answers
35 views

Integer value of the given radical

What is the value of $$\sqrt{2+\sqrt{5}-\sqrt{6-3\sqrt{5}+\sqrt{14-6\sqrt{5}}}}$$ I dont know how to simplify it?
0
votes
0answers
14 views

Simply connected linear algebraic group

Following is what I understand regarding the simply connected linear algebraic groups afer reading some definition in Hochschild's 'Basic Theory of Algebraic Groups' : (I don't know about fundamental ...
0
votes
1answer
20 views

Solution Set of System of ODE.

I am trying to find the solution of the system $$\begin{bmatrix}x_1\\x_2\end{bmatrix}'= \begin{bmatrix}1&3\\3&1\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}$$. I am given that ...
2
votes
0answers
7 views

What is the rate of convergence in the uniform local limit theorem?

Let $(X_i)_i$ be a sequence of iid random variables, s.t. for some sequences $a_n, b_n$ the normalized sum $$Z_n=\frac{X_1+\dots+X_n}{b_n}-a_n$$ converges weakly to an $\alpha$-stable distributed ...
3
votes
3answers
38 views

Inverse of $f: \mathbb{Z}_{21} \times \mathbb{Z}_{10} \to \mathbb{Z}_{210}$ such that $f(a,b)= 10a +21b$

Let $f: \mathbb{Z}_{21} \times \mathbb{Z}_{10} \to \mathbb{Z}_{210}$ such that $$f(a,b)= 10a +21b.$$ We have that $f$ is an isomorphism, but how does one go about finding explicitly the inverse ...
2
votes
2answers
49 views

Taylor expanding $\frac{e^x}{x}$?

How can you taylor expand $$\frac{e^x}{x}$$ Can it be expanded at $x = 0$? Can it be expanded as $x \to 0$?
0
votes
0answers
11 views

Passage in the proof that $\overline{\mathbb{Q}_p}^{\text{alg}}$ is not (metric) complete

I need some hints on a passage in the proof given in Gouvêa p-Adic numbers of the well-known fact that $\overline{\mathbb{Q}_p}^{\text{alg}}$ is not (metric) complete. The passage regards what I ...
3
votes
1answer
14 views

Intuitive understanding into the mean curvature flow

I am trying to develop some intuition into the properties of mean curvature flow of a surface in $\mathbb{R}^3$. As an example, I am trying to understand what happens to a surface of revolution $$S = ...
1
vote
0answers
18 views

Every Hilbert space is isometrically isomorphic with $\ell^2$

Let $H$ be a hilbert space and let $\{u_\alpha\}_{\alpha \in A}$ be a orthornormal basis ($A$ is not supposed to be countable a priori). Then there is an isometric isomorphism between $H$ and ...
1
vote
1answer
24 views

what is the max possible combinations of 1 2 3 4 5 6 without repeating

Each number has to be used and only once in each set. I don't know how to put it but it can't cycle . here is my example 123456 Is the same as 234561 Same as 345612 This isn't for any homework or ...
0
votes
1answer
22 views

Isomorphism matrix problem

So the question asks: Recall that $U^{2\times 2}$ is the vector space of 2X2 upper triangular matrices. Which of the following functions are isomorphisms? A. The function T: $U^{2\times 2}$ to ...
0
votes
3answers
16 views

Proove that Unions and intersections of recursively enumerable sets are also recursively enumerable

How do I prove that Unions and intersections of recursively enumerable sets are also recursively enumerable?
1
vote
3answers
49 views

Is $\lim_{x\to -3}\frac{x^2+9}{\sqrt{x^2+16}-5} = \infty$?

It was asked in our test, and below is what I did: $$\lim_{x\to -3}\frac{x^2+9}{\sqrt{x^2+16}-5} $$ $$=\lim_{x\to -3}\frac{x^2+9}{\sqrt{x^2+16}-5}\times\frac{\sqrt{x^2+16}+5}{\sqrt{x^2+16}+5} $$ ...
0
votes
0answers
27 views

A relation between a group and its subgroups

Let be $H$ a proper subgroup of finite group $G$. Who can we show that $G\not=\cup_{a \in G}aHa^{-1}$?
0
votes
1answer
7 views

Prove that a small shift in the diagonal term leads to smaller spectral radius (for Perron-Frobenius theorem)

On Wikipedia, the proof for Perron Frobenius theorem in the strictly positive case has a confusing step: Suppose $T=A^m-\epsilon I$, where $\epsilon$ is smaller than the smallest diagonal term of ...
1
vote
1answer
21 views

Each alphabet of KANGAROO is replaced with number by 2 people,which alphabet is replaced with the same number?

In the word KANGAROO Bill and Bob replace the letters by digits, so that the resulting numbers are multiples of 11. They each replace different letters by different digits and the same letters by the ...
0
votes
0answers
5 views

Finding Kac-Moody algebras with given root strings

For two hypothetical real roots $\alpha$,$\beta$ and their hypothetical root strings $S(\alpha, \beta)$ and $S(\beta,\alpha)$ through each other, is there a general procedure (or method of "informed ...
0
votes
0answers
13 views

Subgroup of order $9$ and $4$ in $\langle \alpha, \beta \rangle$

Let $$\alpha = (1,2,3,4,5)(6,7,8)(9,10,11)$$ $$\beta = (1,2,3)(4,5)(6,9,7,10,8,11)$$ We have that $\langle \alpha \rangle \cap \langle \beta \rangle = \{id\}$. So $$ord \langle \alpha, \beta \rangle ...
0
votes
0answers
15 views

consistency strength

I am just beginning to read about consistency strength, and wondered if someone could clarify the relation between a two kinds of claims that I'm encountering. (1) A theory, T, proves the consistency ...
1
vote
0answers
21 views

How to calculate inverse laplace of $e^{a\sqrt s}$?

I was using Laplace to find solutions for $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}$$ with boundary conditions $$u(0,t)=1 \\ u(1,t)=1 \\ u(x,0)=1+ \sin \pi x$$ I used ...
0
votes
1answer
14 views

How polynomials are represented in matrix form for Univariate Polynomial.

Represent this polynomial equation in matrix form $$P(x)=a_2 x^{2} +a_1x^{1} +a_0$$ ?
0
votes
0answers
23 views

Describe curves by words

I am trying to describe the general aspect of the following curves My ideas: The curves are continuous, and defined for positive values of x, and giving negative values. Have logarithmic growth ...
0
votes
1answer
13 views

Find the equation of locus of a point which is at a distance $5$ from $A$ $(4,-3)$

Find the equation of locus of a point which is at a distance $5$ from $A$ $(4,-3)$. Could some explain how to solve this in detailed.
1
vote
1answer
21 views

Prove $ \frac{4cos^{2}(2x)-4cos^2(x)+3sin^2(x)}{4cos^2(\frac{5\pi}{2} - x) -sin^22(x-\pi)} = \frac{8cos(2x)+1}{2(cos(2x)-1)}$

Question: Prove $$ \frac{4cos^{2}(2x)-4cos^2(x)+3sin^2(x)}{4cos^2(\frac{5\pi}{2} - x) -sin^22(x-\pi)} = \frac{8cos(2x)+1}{2(cos(2x)-1)}$$ My attempt starting with the bottom ...
1
vote
1answer
35 views

$4\times ABCDE = EDCBA$: Four times a five digit integer is that integer backwards.

A student gave me this puzzle the other day. Where $A,B,C,D,E$ are distinct digits, and where $A,E\ne0$, what 5 digit integer satisfies the condition below? $$4\times ABCDE=EDCBA$$ What I'm ...
0
votes
0answers
17 views

Can you estimate the difference of primes between numerator and denominator?

Let $p_n$ the nth twin prime, it is $p_n$ is a prime number and $2+p_n$ is also a prime. It is well know that Brun's theorem states (unconditionally) that $$\mathcal{B}=\sum_{n\geq ...
2
votes
1answer
10 views

Example of inverse semigroup with at least two idempotent elements

We say that the semigroup $S$ is inverse semigroup if for any $s\in S$ there is a unique $t\in S$ such that $sts=s,\ tst=t$. Suppose that $E(S)=\{e:\ e\in S,\ e^2=e\}$ and define $$s\sim ...

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