# All Questions

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### Does existence of mean of arithmetic function imply it is bounded except on a set of density zero?

Let f(n) be an arithmetic function whose mean is finite.Is f bounded outside a set of density zero?
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### Let a, b, c>0, such that a+b+c=1, prove that:

Let a, b, c>0, such that a+b+c=1, prove that: $$\frac{a}{(b+c)^2}+\frac{b}{(a+c)^2}+\frac{c}{(a+b)^2}\ge\frac{9}{4}$$
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### Derivative problem(I think, that is Implicit function theorem)

I have a function: $$F(x,y) = 2x^4 + 3y^3 +5xy$$ And input $x$ and output $y$ we know that this relation $F(x,y) = 10$ confirms. We know, that this happens when x = 1 and y = 1. By small change of ...
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### System of Equation

Solve the system of equation: $$-x_1+x_2+x_3=a$$ $$x_1-x_2+x_3=b$$ $$x_1+x_2-x_3=c$$ I have tried by taking an augmented matrix of above system of equation and reduced into echelon form but doesn't ...
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### Simple groups and irreducible characters of degree 3

The only simple finite groups admitting an irreducible character of degree 3 are $\mathfrak{A}_5$ and $PSL(2,7)$. That seems to be a result coming from Blichfelt's work on $GL(3,\mathbb{C})$, which I ...
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### Condition in a theorem of Hall

There is a well-celebrated theorem of Hall, which characterizes solvable groups according to the existence of Hall-$\pi$ subgroups. In this theorem, I was wondering whether it can be stated in a ...
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### Bounding $L_p$ norms on a convergent $L_1$ sequence

I've encountered a prelim problem on $L_p$ spaces that I'm pretty stuck on. Suppose $1 < p < \infty$ and $f_n \in L_1([0,1]) \cap L_p([0,1])$, with $||f_n||_p$ bounded above by some constant $M$...
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### How to prove that a space is not a differential manifold?

Given a box （the surface of a cubic） in R^3 space, can I give a smooth structure on it to make it a differential manifold?
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### Lets a, b, c>0 such that a+b+c=6, prove that:

Let a, b, c>0 such that a+b+c=6, prove that: $$\sum_{cyc} \frac{a^7+b^7}{a^5+b^5}\ge12$$
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### Challenging linear algebra question - permutatoin matrices

If you take powers of a permutation, why is some $$P^k = I$$ Find a 5 by 5 permutation $$P$$ so that the smallest power to equal I is $$P^6 = I$$ (This is a challenge question, Combine a 2 ...
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### Reference of what metric can be placed on manifold?

I just read some conclusion that $T^2$ can't be placed metric with positive curvature at all points. I don't know why is so . And what book introduce about this ? I mean about what metric can be ...
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### What is the algebraic structure of $\Bbb Q_p/\Bbb Z_p$?

I am curious about the algebraic structure of $\Bbb Q_p/\Bbb Z_p$. Is there any result in this direction? Thanks!
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### A complex no. as the limit of integration

I have come across an integral equation, integral (f(x)dx) limits from 0 to i=sqrt(-1) and f(x) is a real function. Now, I know the value of integral(f(x)dx) limits from 0 to real number a. Also, f(x) ...
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### The $n$-th root of $(1+q^n)^2$

Let $0<q<1$ be rational. I am suspecting that $\sqrt[n]{(1+q^n)^2}$ is irrational. Can someone please help me to prove or to disprove this? $n=1$ and $n=2$ are simple cases. I am interested ...
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### Are there at least $3$ groups of order $16$ that has element of order $8$?

Are there at least $3$ groups of order $16$ that has element of order $8$? I know that probably the simplest way of doing this problem is looking at the element structure of the abelian groups of ...