# All Questions

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### Expressing surds in different forms

I know this might get taken down for being a dumb question but I'm not exactly a genius when it comes to maths. So the question is I need to express 6/√2 in the form of a√b and a and b both need to be ...
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### Markov Chain: reversible probability

Considering the price of a share $X_n$ that evolves every $n$ day that increases of one euro with probability $0<p<1$ or decreases of one euro of probability $q=1-p$. Assume that $X_0=10$ and ...
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### The diagonal of an infinite list.

My original question was closed and stack exchange suggested "Please improve the question by providing additional context, which ideally includes your thoughts on the problem ", so here is my attempt ...
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### Induced morphism on rings of regular functions injective iff original morphism has dense image.

Let $X \to Y$ be a morphism of affine varieties over a field $k$. How do i see that the induced morphism $k[Y] \to k[X]$ on rings of regular functions is injective if and only if the original morphism ...
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### A Problem of Tom Apostol about Prime Distribution

Let $s_n$ denote the sum of the first $n$ primes. Prove that for each $n$ there exists an integer whose square lies between $s_n$ and $s_{n + 1}$.
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### For any positive integer $n$, is there an irreducible representation of degree $n$?

I want to prove a problem: Prove that $Z(M_n(\Bbb{C}))=\{\lambda I\mid \lambda\in \Bbb{C}\}$. If $M\in Z(M_n(\Bbb{C}))$, let $\varphi:G\to GL_n(\Bbb{C})$ be an irreducible matrix representation of ...
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### Hints to compute if exists $\lim_{n\to\infty}\sum_{k=1}^n\sigma(k^2)/\sum_{k=1}^n\sigma(k)$, which $\sigma(n)=\sum_{d\mid n}d$, and other question

I would like receive hints at least for one of the following problems, these are going from experiments. Can you provide to me hints for at least one of the following problems? I will try put the ...
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### Calculate b such that $P(-b<X<b) =0.9$

Given the t-distribution X~t(14). How can I calculate b such that the following holds: $P(-b<X<b) =0.9$. I would start with $P(-b<X<b) =P(X<b)-P(X<-b)$ but then I dont know how ...
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### What exactly is the meaning of $P\left(\bigcap_{n=1}^{\infty}A_i\right)$ for sets $A_1, A_2, \ldots$?

If we have sets $A_1, A_2, \ldots$, then when we write $$P\left(\bigcap_{n=1}^{\infty}A_i\right)$$ what does this actually mean? Does it mean the probability of the intersection of infinite sets? ...
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### non-trivial solution for equation

Let $A_{mxn}$ and $B_{nxm}$. while $B\neq0$ and $A\neq0$. also, assume that $AB = 0$ Does the homogenic system $Bx = 0$, has a non-trivial solution? if yes, prove it. otherwise give a ...
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### Jacobian of the determinant?

I'm supposed to do this exercise: Consider the application $F:{M}_{2\times2}\left(\mathbb{R}\right)\longrightarrow\mathbb{R}$ F\left[\left(\begin{array}{cc} a & b\\ c ...
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### $\small{\left( {\begin{array}{*{20}{c}} I & {{B^{\frac{1}{2}}}} \\ {{B^{\frac{1}{2}}}} & A \\ \end{array}} \right) \ge 0 \Rightarrow A \ge B}$

Let $A, B \in M_n$ be positive definite, and $\left( {\begin{array}{*{20}{c}} I & {{B^{\frac{1}{2}}}} \\ {{B^{\frac{1}{2}}}} & A \\ \end{array}} \right) \ge 0$ Why does $A \ge B$ ?
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### Exists $f \in L^1(\mathbb{R})$ where $\lim_{r \to 0} {1\over{r}} \int_{x-r}^{x+r} f(y)\,dy = \infty$?

If $E \subset \mathbb{R}$ has measure $0$, does there exist $f \in L^1(\mathbb{R})$ such that, for every $x \in E$,$$\lim_{r \to 0} {1\over{r}} \int_{x-r}^{x+r} f(y)\,dy = \infty?$$What if $E$ has ...
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### Show that there exist $A>0$ (not depend on $m$) such that $|u_{m}|>A$

Let $(u_{m})_{m≥1}$ be a strictely positive and convergent sequence. Then show that there exist $A>0$ (not depend on $m$) such that $$|u_{m}|>A$$ for all $m≥1$. I think to exploite the fact ...
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### Exponential generating function and number of balls

Use exponential generating functions to determine the number $a_n$ of ordered choices of $n$ balls such that there are $2$ or $4$ red balls, an even number of green balls, and an arbitrary number ...
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### Can a mixing process be non-stationary?

I was always under the impression that a mixing process is ergodic and an ergodic process is necessarily stationary, so that a mixing process is stationary. I have come across a paper discussing ...
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### Limit of cos under radical using l'Hôpital

I'm trying to find $$\lim_{x\downarrow 0}\frac{\sqrt{1-\cos x}}{x}$$ using l'Hôpital's rule but I seem to be stuck in a loop. I have tried applying l'Hôpital several times but the derivatives always ...
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### Definitions of “linearity” in different branches of mathematics

Linearity is a ubiquitous concept in mathematics; however, each branch of mathematics appears to have its own definition of what a linear map (function, functional, functor, operator, or whatsoever) ...
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### IMO 1997/5 with solution

Find all pairs $(a, b)$ of integers $a, b \geqslant 1$ that satisfy the equation $$a ^ {b ^ 2} = b ^ a. \qquad \qquad (*)$$
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### If $A$ is positive definite then any principal submatrix of $A$ is positive definite

If $A$ is positive definite then any principal submatrix of $A$ is positive definite. Proof; In the proof i dont understand about $j_i$'s.. Can some one interpret the proof in a simpler ...
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### Real Analysis - Uniform Convergence of a Function

I am given that: For $n \in \mathbb{N}$, define $f_n: \mathbb{R} \to \mathbb{R}$ by $$f_n(x)=\frac{x^{4n}}{4+x^{4n}}.$$ I need to determine whether the sequence $(f_n)$ converges uniformly on ...
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### Study Group Graph Theory

I am confused about how to model this question in a graph theory perspective. What is the minimum number of sub-groups I would need to split $n$ people into subgroups such that every person in each ...
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### Basic estimates for geometric quantities in an asymptotically flat manifold.

We consider a manifold $M$ with a metric $g$. Let $N \subset M$. Let $h$ be the restriction of $g$ to the surface $N$ and let $\epsilon$ be the restriction of the flat metric $\delta$ to it. Let $\nu$ ...
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### Solve: $\int \dfrac{s^{2} + \sqrt{s}}{s^{2}}ds$, step by step please.

I need a step by step example to be able to understand how to solve this problem. I am just now learning the substitution method but I don't know how to apply it here, or if it is applicable at all. ...
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### Question involving multivariable calculus.

The temperature in a neighbourhood of the origin is given by a function $$T(x,y) = T_0+e^y \sin x$$ A heat fleeing particle is placed at the origin at time t = 0. find the differential equations on ...
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### Using 2 dice to generate number in range

I've been thinking about this but can't seem to figure it out. I need to pick a random integer between 1 to 50 (inclusive) in such a way that each of the integer in it would be equally likely. I will ...
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### aligning a matrix to reference matrix

Assuming X$_0$ as a matrix which represent some sort of transformation between TWO different coordinate system. Now, as a function of time the matrix which has three column vectors evolves in to X'. ...
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### Sketch of a possible equivalence with Riemann hypothesis

From Robin's equivalence (see [1]) and the following trigonometrics identitites, I ask to me if it is feasible write vagues equivalences using this strong result and if these equivalences will be ...