# All Questions

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### find a point in 3D space

Suppose we have $3$ fixed points $P_1, P_2, P_3$ in $3$-D space, their coordinates are $(x_i, y_i, z_i)$ for $i=1,2,3$. The problem is to find a point $P$ so that the distances from $P$ to ...
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### How to prove the inequality: $\frac{(1+x)^2}{2x^2+(1-x)^2}+\frac{(1+y)^2}{2y^2+(1-y)^2}+\frac{(1+z)^2}{2z^2+(1-z)^2}\leq 8$

Prove that: $$\frac{(1+x)^2}{2x^2+(1-x)^2}+\frac{(1+y)^2}{2y^2+(1-y)^2}+\frac{(1+z)^2}{2z^2+(1-z)^2}\leq 8$$ subject to the constraints: $$x,y,z >0$$ and $$x+y+z=1.$$
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### Possible to use exact differential to prove Euler's test?

I know that when we want to prove an equation is exact differential, we use $$\frac{d^2Q}{dxdy}=\frac{d^2Q}{dydx}$$ But I wonder why? $$dQ=\frac{dQ}{dx}dx+\frac{dQ}{dy}dy$$ Then differential again ...
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### Locus in the complex plane given an equation

I have the question Let $a$ and $e$ be two positive real numbers, with $0 < e < 1$. Describe the locus of the points $z$ in the complex plane which satisfy $|z - ae| + |z + ae| = 2a$. I ...
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### convergence ratio of the serie $e^{xn}$

How can I determine the values of $x$ such that the series converge: $$\sum_{n=0}^\infty e^{xn}$$ I'm really lost in this problem, please help.
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### Cannot follow proof that between two reals there is a rational.

I am following a textbook and as is good practice, I am only skipping things I can do. this is self-learning I always struggle with chapter 1, the one that builds the "axioms", hate it. Anyway, I am ...
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### Math Shock in graduate program

People call it Culture shock but I call it Math Shock... let me explain my Problem... First I am graduate student in a good university in USA ( I get scholarship from my country). Before I lived in ...
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### Factoring $s^2+4s+13$

I was looking at an example, and it was factored as follow: $$s^{2}+4s+13 = (s+2)^{2}+9$$ How can we do that?
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### Is Fourier transform of a $L^{1}$ integrable function is $L^{1}$ integrable?

Let $f:\mathbb R \to \mathbb R$ such that $$f(x)= \frac{\sin \pi x}{x (x^{2}-1)}$$ for $x\in \mathbb R - \{ 0, -1, 1 \}$ and $f(x):= \pi$ for $x=0$ and $f(x)=-\frac{\pi}{2}$ for $x= -1, 1$. Let ...
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### harder expected value probability question

I have a question on expected value. I have the solutions for it but they havent explained exactly what they have done, and i am a bit confused whilst revising for an exam in a few days. Here is the ...
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### Let $G$ be a finite group and let $H$ be a solvable subgroup of $G$.

Let $G$ be a finite group and let $H$ be a solvable subgroup of $G$. Let $N=N_G(H)$ and assume that $N/H$ is a nonabelian simple group. Prove that $N=N_G(N)$. $N=N_G(N)$. That means ...
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### Show that $\max_{x\in[a,b]}|f'(x)|\geq\frac{4}{(b-a)^2}\int_a^b|f(x)|dx$ [duplicate]

Let $f:[a,b]\to\mathbb{R}$ be differentiable and $f'$ is continuous. Suppose $f(a)=f(b)=0$, show that $$\max_{x\in[a,b]}|f'(x)|\geq\frac{4}{(b-a)^2}\int_a^b|f(x)|dx$$ My approach. For any $x$, by ...
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### Algebra 2-Factoring sum of cubes by grouping

Factor the sum of cubes: $81x^3+192$ After finding the prime factorization of both numbers I found that $81$ is $3^4$ and $192$ is $2^6 \cdot 3$. The problem is I tried grouping and found $3$ is ...
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### How prove this $4A^5+2A^3+A=7I$ [duplicate]

Let $A_{n\times n}$ is Hermitian matrix,and $A_{n\times n}\neq I_{n\times n}$,where $I_{n\times n}$ is Identity matrix, prove or disprove $$4A^5+2A^3+A=7I$$ my try: if such this condition: then ...
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### Anyone has a good recommendation of a free pdf book on group theory?

Anyone has a good recommendation of a free pdf book on group theory? I am specially interested in its application for computer science, however, I do not want it to be less mathematically rigorous ...
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### Eigenvalues of $\frac{d^2}{dx^2}$ in $C^2(\mathbb{R})$

Consider the eigenvalue problem \left\{ \begin{array}{l} \Phi \in C^{2}(\mathbb{R}) \ \text{and bounded }\\ -\Phi^{''}(x)=\lambda\Phi(x), \ x\in \mathbb{R}. \end{array} \right. ...
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### Prove a metric space is compact, if every infinite subset in it has a limit point.

This is an exercise in W. Rudin's book. Actually my question is, what is an open cover of a metric space? Since an open set is embedded in a certain metric space, how can it cover those points which ...
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### Convolution of distribution and Poisson kernel

I know that for a general tempered distribution (see here) $f$ the convolution $f\star P_t$ is not meaningful. Where $P_t$ is the Poisson kernel (see here) which is given by ...
118 views

### Trigonometry or inequality problem

Today, I saw this question: If $x,y,z \in [0,\frac\pi 2]$, $x+y+z=\frac{3\pi}{4}$ and $\sec^2(x)\sec^2(y)\sec^2(z)=8$, calculate $E=\tan x\tan y+\tan y\tan z+\tan z\tan x$ My first thought was ...
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### Application of convergence of Fibonacci series

'There are infinite prime numbers' is a fact that can be deduced by 'reciprocal of primes diverges' statement, so from this can we deduce the fact that --> 'there are finite Fibonacci numbers in ...
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### Show that if $a_n$ is descending and $∑a_n$ is convergent then $n∗a_n→0$

Show that if $a_n$is descending and $\sum a_n$ is convergent then $n*a_n \to 0$. Does $\sum a_n$ has to be convergent? I see that, but I can't prove it. I think that it need not to be convergent, ...