# All Questions

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### Every natural number greater than 1 is divisible by a prime number

Sorry if this question has been asked, but a couldn't find one using the method I need. I want to prove that every natural number greater than 1 is divisible by some prime number using the WOP. I ...
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### Is there an Artinian ring with exactly two prime ideals which their product is non-zero?

Is there an Artinian ring with exactly two prime ideals which their product is non-zero? Clearly these prime ideals could not be zero on the other hand the summation of them is equal to R.
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### Minimal immersion

Let $N$ be an $n$-dimensional manifold immersed in an $l$-dimensional manifold $L$ by immersion $\iota: N\to L$. And let $\iota$ be minimal immersion. On the other hand, let $\phi: M\to N$ be totally ...
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### group of elements whose orders are a power of some integer

I am stuck with the following problem, from C. Pinter's "A Book of Abstract Algebra", p. 153 ex. C 6: Let $G$ be an abelian group, and $H_p$ the subset of $G$ such that the order of every $x \in H_p$ ...
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### Colimits and epimorphisms.

I am working on a project, and I need to know the proof of this: Any functor which preserves all colimits preserves epimorphisms. So could you please tell me how or where I can find the proof ...
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### Nondegenerate bilinear map

Let $V$ be the vector space consists of all $n\times n$ real matrices, and $f$ is a nonvanishing linear function on $V$ such that $$f(AB)=f(BA),\ \forall\ A,B\in V.$$ Show that $g(A,B)=f(AB)$ is a ...
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### Find the volume of the region with triple integrals.

The volume of the region bounded by $y^2+z^2=1\ \text{and} \ z^2+x^2=1$ should be found. Cylindrical conditions are perhaps the most appropriate in this case but the limits of the integral are what ...
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### Rotation/Inner Product between eigenspaces of Hermitian matrices

Given two Hermitian, positive definite matrices $\mathbf{T_1}$ and $\mathbf{T_2}$, is it generally true that the matrix $\mathbf{G}_{ij} = \lvert\langle \mathbf{u}_i \mathbf{v}_j \rangle\rvert^2$ of ...
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### When $x$ goes to $0$ , what happens to $\sin\left(\frac{1}{x}\right)$ and $\cos\left(\frac{1}{x}\right)$?

When $x$ approaches $0$, do $\sin\frac{1}{x}$ and $\cos\frac{1}{x}$ converge or diverge? How do you show this?
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### question about cauchy sequences [closed]

For which values of $\lambda$ is the sequence $f_n : [0,1] \to \mathbb{R}$ with $f_n(x) = 2^{- \lambda n x }$ when $x \in [0,2^{-n}]$ and $0$ otherwise, Cauchy sequence in $L^7([0,1])$??
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### If $g$ is invariant under an ergodic map then it's almost everywhere constant

Let $(X,B(X),\mu)$ be a probability space, where $X$ is compact metrizable, and $B(X)$ are the Borel sets. Let $f:X\to X$ be a measurable function such that: i) $\forall A\in B(X)$ ...
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### A visual proof of - Curved surface area of a hemisphere = 2(Area of circle)

Suppose we have a circle with radius $r$ . So its area is $\pi r^2$. Now suppose we have a hemisphere of the same radius ie. $r$. Then its curved surface area is $2 \pi r^2$. Which means it is equal ...
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### Conditional expectation probability question

In my problem, $Y$~$Bin(4,p)$ and the conditional distribution of $X$ given that $Y=y$ Is $Poi(y)$. I have to work out $E(X|Y=y)$ and then use the formula $E(X)=E(E(X|Y))$ to find $E(X)$. For ...
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### Limit of series $4\left( \frac {1}{8}+\frac {1}{12}\right) +6\left( \frac {1}{24}+\frac {1}{36}\right) +\ldots$

How to find this serie $4\left( \dfrac {1}{8}+\dfrac {1}{12}\right) +6\left( \dfrac {1}{24}+\dfrac {1}{36}\right) +8\left( \dfrac {1}{64}+\dfrac {1}{96}\right) +\ldots$ I think it's telescopic, ...
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### What is the proof that $\sum \limits_{v \in V} deg(v) = 2|E|$?

My textbook gives $\sum \limits_{v \in V} deg(v) = 2|E|$ and has the proof If an edge is not a loop it gets counted twice b/c it's incident with 2 different vertices. If an edge is a loop, by ...
Let $M$ be an oriented, compact, connected $n$-dimensional smooth manifold with boundary. Then, is it true that $n$-th singular homology of M, that is $H_n(M)$, is vanish? I can't make counterexamples ...