# All Questions

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### 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 ...
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### 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
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### 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. ...
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### 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 ...
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### 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 ...
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### 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. $(a,\lambda) \in {\Re}^2$
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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: $$\lim_{x\to -3}\frac{x^2+9}{\sqrt{x^2+16}-5}$$ $$=\lim_{x\to -3}\frac{x^2+9}{\sqrt{x^2+16}-5}\times\frac{\sqrt{x^2+16}+5}{\sqrt{x^2+16}+5}$$ ...
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### A relation between a group and its subgroups

Let be $H$ a proper subgroup of finite group $G$. Who can we show that $G\not=\cup_{a \in G}aHa^{-1}$?
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### 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 ...
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### 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 ...
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### 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 ...
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 $$ord \langle \alpha, \beta \rangle ... 0answers 15 views ### consistency strength I am just beginning to read about consistency strength, and wondered if someone could clarify the relation between a two kinds of claims that I'm encountering. (1) A theory, T, proves the consistency ... 0answers 21 views ### How to calculate inverse laplace of e^{a\sqrt s}? I was using Laplace to find solutions for$$\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$$I used ... 1answer 14 views ### How polynomials are represented in matrix form for Univariate Polynomial. Represent this polynomial equation in matrix form$$P(x)=a_2 x^{2} +a_1x^{1} +a_0$$? 0answers 23 views ### Describe curves by words I am trying to describe the general aspect of the following curves My ideas: The curves are continuous, and defined for positive values of x, and giving negative values. Have logarithmic growth ... 1answer 13 views ### Find the equation of locus of a point which is at a distance 5 from A (4,-3) Find the equation of locus of a point which is at a distance 5 from A (4,-3). Could some explain how to solve this in detailed. 1answer 21 views ### Prove  \frac{4cos^{2}(2x)-4cos^2(x)+3sin^2(x)}{4cos^2(\frac{5\pi}{2} - x) -sin^22(x-\pi)} = \frac{8cos(2x)+1}{2(cos(2x)-1)} Question: Prove$$ \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)}$$My attempt starting with the bottom ... 1answer 35 views ### 4\times ABCDE = EDCBA: Four times a five digit integer is that integer backwards. A student gave me this puzzle the other day. Where A,B,C,D,E are distinct digits, and where A,E\ne0, what 5 digit integer satisfies the condition below?$$4\times ABCDE=EDCBA$$What I'm ... 0answers 17 views ### Can you estimate the difference of primes between numerator and denominator? Let p_n the nth twin prime, it is p_n is a prime number and 2+p_n is also a prime. It is well know that Brun's theorem states (unconditionally) that$$\mathcal{B}=\sum_{n\geq ...
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 ...