# All Questions

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### Is quotient module finitely generated?

Suppose $R$ be any ring containing left ideal $I$. Then $I$ is submodule of $R$, so $R/I$ is R-module. My question is, is $R/I$ always a finitely generated?
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### Optimization: Finding line connecting non-pareto-optimal allocation in Edgeworth Box to PO allocation

Two people, A and B, with respective utility functions of: $$U_a(X_a,Y_a) = X_a^2 Y_a\\ U_b(X_b,Y_b) = X_b Y_b^2$$ Total $X$ (that is, $X_a+X_b$) is fixed at $36$. Total $Y$ ($Y_a+Y_b$) is fixed ...
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### How to calculate perimeter of Polygon with missing the length of one side?

I have following sides(PQRST) of a Polygon where PQ=13, QR=22, RS=8, ST=?, PT= 10 ... i need to find out PT? i don't have any angle i just have the shape? And for calculating perimeter i need to find ...
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### Exact values cot sec cosec

If $\cot(x) = -12/5$ where $x$ is in $[\pi/2,\pi]$, find $\cos(x +\pi/3)$. What trig identity should you use? And how to bring it back to $\cos(x +\pi/3)$?
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### How can I show that $a^p \equiv a \pmod p$ for any $a$ and prime $p$ in $\mathbb Z$?

Assuming $p$ is prime, how do I show that $a^p \equiv a \pmod p$ for all $a$ (where $a$, $p$ $\in \mathbb Z$)? I think I can handle the case where $p$ doesn't divide $a$, because then I can use ...
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### Conditions on integral operator to be in $L^{2}$

Suppose we have an open set $\Omega$ in $\mathbb{R}^{n}$ and for every $x\in\Omega$ a function $T\left(x,\cdot\right)\in L^{2}\left(\Omega\right)$. If for $f\in L^{2}\left(\Omega \right)$ we consider ...
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### Representation theorem on normal spaces

I only know the representation theorem on locally compact Hausdorff spaces. I heard that there is a normal space version in the book of Dunford & Schwartz. However I cannot find where it is. Can ...
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### Why does $\sum_{n\neq0}\:\lvert\frac{a_n}{n}\rvert \leq \sqrt{\sum_{n\neq0}\frac{1}{n^2}}\sqrt{\sum_{n\neq0}|a_n|^2}$?

My question is: Why does $$\sum_{n\neq0}\:\lvert\frac{a_n}{n}\rvert \leq \sqrt{\sum_{n\neq0}\frac{1}{n^2}}\sqrt{\sum_{n\neq0}|a_n|^2},$$ where $a_n$ is some complex number, $n$ an integer going from ...
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### Maximal area of a monochromatic combinatorial rectangle

I'm stuck on this one and would appreciate any help: Let $M$ be a $2^n \times 2^n$random matrix of $0$'s and $1$'s (for each entry, there is probability of $\frac{1}{2}$ that it will be $0$, same ...
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### homotopy between two finite sets of distinct cardinality

Let us discuss this problem: Let $A$ and $B$ be two great circles in a sphere $S^2=\{x^2+y^2+z^2=1\}$ such that $A$ intersects $B$ in two and only two points (recall that any two great circles ...
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### Proving that $\dfrac{\sec\theta\cdot\sin\theta}{\tan\theta+\cot\theta}=\sin^2\theta$

The question is: Prove that: $$\dfrac{\sec\theta\cdot\sin\theta}{\tan\theta+\cot\theta}=\sin^2\theta$$ My proof is shown below. If anyone has an alternate proof please, please post it. Thanks!
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### How can I solve Dynamic Problem?

I tried to solve this Dynamic Problem , But I do not know how to begin solving it. A minivan starts from rest on the road whose constant radius of curvature is 40 m and whose bank angle is 10°. The ...
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### Deep Neural Network - Lower Dimension Problems

Deep Neural Networks have been used mostly for high dimensional problems such as image processing and speech recognition, as it has shown to be superior to neural networks trained with the ...
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### Need help finding closed form of finite product

Is there a closed form for this product? $$\prod\limits_{k=1}^n (n+k)$$ I checked it on wolfram alpha but it uses something called the pochhammer symbol. Does anyone else know of a nice explicit ...
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### If $S$ is an isometry, why is $\sqrt{S^{*}S}$ a positive and hence self adjoint operator?

I am trying to show that $S$ being an isometry leads to the fact that all singular values of $S$ equal 1. I know a key part of the proof is showing that $\sqrt{S^{*}S}$ is self adjoint so that I can ...
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### Integral $\int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx$

Hi I'm trying to solve this integral Fourier Transform $$\int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx=\sqrt{2\pi|k|}(1+i) (-1+\text{sgn}(k))$$ where sgn(k)$=1$ for k>1 and $-1$ for k<1. I am ...
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### R is countable using Zorn's Lemma?

I used Zorn's Lemma to cook up a proof that $\mathbb{R}$ is countable, and now I can't find a flaw in it. Can anyone help? Here it is... Denote by $\mathcal{A}$ the set of countable subsets of ...
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### If $R$ is a positive operator and $R^2 = T^{*}T$, why does this mean we can write $R$ as $\sqrt{T^{*}T}$?

If $R$ is a positive operator and $R^2 = T^{*}T$, why does this mean we can write $R$ as $\sqrt{T^{*}T}$? Should I be thinking of the square root of an operator as in the way I think about it when it ...
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### Square root of negative integer

Can I write: $-\sqrt(2)$ = $\sqrt(-2)$ and vice versa? Or, say, we have, $(-\sqrt{(x - 4)}$ Can this be changed into $(\sqrt{(4 - x)}$ by taking the minus sign inside the square root? How?
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### Show that $A, B \in L(V)$ Are Similar

The following question comes from Halmos' FDVS, page 95. I am okay with the $(\rightarrow)$ part, but I am not 100% sure with the converse. Specifically, can I "neatly" stack the vectors in the ...
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### Inverse Square root of a rectangular matrix

I am trying to compute the inverse square root ($X^{-1/2}$) of a $n \times p$ matrix with $n > p$. I was wondering if we can compute it via SVD just as we do it for square diagonalizable matrices ...
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### Projective and Injective Modules

Let $M$ be a free $\mathbb{Z}$-module, is $\text{Hom}_{\mathbb{Z}}(M,\mathbb{Q})$ injective or projective $\mathbb{Z}$-module? very thanks
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### How large do my $2$ primes need to be to “guarantee” a longevity of security for my RSA-encrypted plaintext?

I am currently attempting to learn RSA. Most of the literature I am using is at least a few years old, if not older. Given the advancements in computing and improvements in attacking RSA, I am wanting ...
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### rate of change/water poured into cone

Water is poured into an inverted cone at a rate of π-units per second. If the radius of the base of the cone is r and its height is 2r, what is the rate at which the depth of the water is changing ...
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### why is $\sqrt[3]{31}$ so close to $\pi$?

$\sqrt[3]{31}$ is about $3.14138$. Why is this so close to pi?
### show that $\bigtriangledown [f(r)]=f'(r)\frac {\mathbf{r}}{r}$
Let $\mathbf{r} = xi+yi+zk$, write $r= ||\mathbf{r}||$ and let $f:\mathbb{R}\to\mathbb{R}$ be a function of class $C^1$ So from what I know, we can derive the function at least once and we know ...