0
votes
0answers
2 views

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?
0
votes
0answers
11 views

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: $$\frac{a}{(b+c)^2}+\frac{b}{(a+c)^2}+\frac{c}{(a+b)^2}\ge\frac{9}{4}$$
0
votes
0answers
7 views

Limit of the given expression.

For $x>0$,$$lim{x\to 0}((\sin(x))^{\frac{1}{x}}+(\frac{1}{x})^{\sin(x)})$$ is?. So now I calculated limits individually. Let $lim{x\to 0} ((\sin(x))^{1/x})=y$ thus I took log to get $\frac{1}{x}(1-(...
0
votes
0answers
4 views

maximize a sum of unit fractions (without containing a subset of sum 1)

Let $ u \ge 2 $ be fixed. Then consider: $ S(u)=\max\left\lbrace \sum_{i=1}^{u+1} \frac{c_i}{t_i} \, \middle| \, 2 \le t_1 \le t_2-1 \le \ldots \le t_{u+1}-1, \, t_i \in \mathbb{N}, \, c_i \in \...
0
votes
0answers
7 views

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 ...
1
vote
4answers
29 views

How to solve $x<\frac{1}{x+2}$

Need some help with: $$x<\frac{1}{x+2}$$ This is what I have done: $$Domain: x\neq-2$$ $$x(x+2)<1$$ $$x^2+2x-1<0$$ $$x_{1,2} = \frac{-2\pm\sqrt{4+4}}{2} = \frac{2 \pm \sqrt{8}}{2} = \frac{...
-1
votes
0answers
8 views

Convex subset and linear equalities

Let S denote the set of $(a,b,c)$ $\in$ ${\mathbb{R^3}}$ which satisfies the following equalities: $-2a+b+c \leq 4 $ $a-2b+c \leq 1 $ $2a+2b-c \leq 5 $ $ a \geq 1 $ $ b \geq 2 $ $ c \geq 3 $ ...
0
votes
0answers
8 views

Converse of Lagrange Theorem for p-groups.

I need clarification on my understanding of Sylow Theorems. Can I say that a finite p-group will have a subgroup for each prime power? If the above is valid, can I say then that p-groups satisfy the ...
0
votes
0answers
18 views

Is finding the extreme points of a differentiable function by first derivative always correct?

I came across this question where i was asked to find the local minimum and local maximum of the function $$y=\sec x + 2\ln(|\cos x|),$$ domain of $x$ being $$(0,2\pi)\setminus \left\{\frac\pi2 , ...
0
votes
1answer
9 views

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 ...
1
vote
0answers
7 views

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 ...
1
vote
1answer
11 views

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 ...
0
votes
0answers
11 views

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 $M$...
0
votes
1answer
12 views

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?
0
votes
0answers
19 views

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: $$\sum_{cyc} \frac{a^7+b^7}{a^5+b^5}\ge12$$
0
votes
0answers
10 views

Solving vector equation 2

Using vector method, show that the vector equation $$\bar{x}\times \bar{a}+(\bar{x}.\bar{b})\bar{c}=\bar{d}$$ is satisfied if $$\bar{x}=\lambda \bar{a}+\bar{a}\times \frac{\bar{a}\times (\bar{d}\...
1
vote
3answers
20 views

If $G$ isn't abelian and $|G|=p^{3}$ then $Z(G)=G'$

Let $G$ a finite group and $G' \subseteq G$ the smallest normal subgroup of $G$ such that $G/G'$ is abelian. Prove that if $G$ isn't abelian and $|G|=p^{3}$ then $Z(G)=G'$ My attempt: If $G$ is not ...
1
vote
0answers
9 views

topology of compact convergence, closed sets

Let $H(\mathbb{D})$ be the vector space of all analytic functions on the unit disk. Then the topology induced by uniform convergence on compact subsets is metrizable. Thus the following topology ...
1
vote
1answer
24 views

How are sets “detached” from their structure?

This question is best asked with an example. Consider the real numbers. However we construct the real numbers, the "final product" so to speak, is not just a set, but it is a complete ordered field. ...
0
votes
0answers
6 views

An algorithm to find the general classical solution to a linear gradient system in partial derivatives

I'm looking for a book where the algorithm to construct the general solution for system $$\nabla u(x,y) = \vec a(x,y)\cdot \nabla v(x,y)$$ is given. Could ypu please advice me some source?
1
vote
0answers
16 views

If $H$ and $G/H$ are p-groups then $G$ is a p-group.

Please verify: If $H$ is a p-group, then $|H| = p^r$, for some integer $r$. If $G/H$ is a p-group, then $|G/H| = p^s$, for some integer $s$. But the cardinality of a quotient set is the index, then $[...
0
votes
0answers
10 views

About some good references for self study

I'm willing to start a self study of Hardy spaces, Bergman spaces and Bloch spaces. I would like to know good books on the subject. Since I'm going to study on my own, would be great to find one that ...
1
vote
2answers
13 views

Show that the circle drawn on a focal chord of a parabola $y^2=4ax$, as a diameter touches the directrix

Question: Show that the circle drawn on a focal chord of a parabola $y^2=4ax$, as a diameter touches the directrix. Let the parabola be $y^2=4ax$ Let the focal chord be $y = m(x-a) $ Subbing ...
2
votes
0answers
12 views

Why is this Isometry an rotation?

i need a little help. Did someone have an idea how to prove this? Thanks in advance. Be $\Phi$ an direct isometry of the euclidean Space $\mathbb{R}^3$ with $\Phi (\left(\begin{eqnarray} 2\\0 \\1 \...
0
votes
0answers
7 views

How many eigenvectors can a real, symmetric, sparse matrix have?

I'm attempting to diagonalise a sparse, real and symmetric matrix H in MATLAB, using [F, E] = eigs(H, size(H,1)). This however ...
3
votes
1answer
21 views

Groups of order $25$

Please verify my solution that there are only two groups of order $25$ up to isomorphism. As $|G|$ is a prime squared, then $G$ is abelian. Since the Theorem of Finite Abelian Groups, $G$ is a direct ...
3
votes
2answers
46 views

If $x,y,z>0\;,$ and $xyz=1$ Then minimum value of $\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}$

If $x,y,z>0\;,$ and $xyz=1$ Then find the minimum value of $\displaystyle \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}$ $\bf{My\; Try::}$Using Titu's Lemma $$\frac{x^2}{y+z}+\frac{y^2}{z+x}+\...
2
votes
2answers
26 views

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 ...
0
votes
2answers
11 views

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 ...
3
votes
0answers
17 views

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!
3
votes
0answers
27 views

Only five solvable quintic equations of the form $x^5+ax^2+b=0$? What are their solutions?

According to Wikipedia there is only five solvable quintic equations of the form $x^5+ax^2+b=0$ (up to a scaling constant $s$). $$x^5-2s^3x^2-\frac{s^5}{5}=0 $$ $$ x^5-100s^3x^2-1000s^5=0 $$ $$x^5-...
0
votes
0answers
10 views

The discrete Laplacian

I am working on the $d$-dimensional integer lattice. Let $S$ be a random walk with increment distribution $p$. Given the distribution $p$ we can define the discrete Laplacian just as in Wikipedia is ...
0
votes
0answers
8 views

How to prove that probability for different initial conditions to yield similar trajectory is very small?

For $\epsilon > 0$, suppose $f$ is a chaotic function. Then, for any trajectory $\mathbf{x,y}$ how can I prove that we have $Probability\{f(\mathbf{x}) = f(\mathbf{y}) \}$ $\le \epsilon$ where $\...
0
votes
0answers
10 views

Find a point along the line created by two points based on distance

Let's say I have two people on two points on a 2 dimensional plane. Person B is 400 units away from Person A. But Person A wants to be always 950 units away from Person B, preferably moving back in a ...
1
vote
1answer
13 views

Improper rotation matrx in $2D$

The following is the related problem: Improper Rotations in Even Dimensions I want the simpler explanation. An improper rotation is rotation, followed by reflection in the plane perpendicular ...
2
votes
0answers
25 views

Where does the premise of this idea come from?

Let $x$ , $y$ be positive real numbers. Prove the inequality $$x^ y + y^x \ge 1$$ This is the solution provided by my textbook: Where does this first idea (proving that $a^b \ge \frac{a}{a+ b - ...
0
votes
1answer
11 views

Elimination and exchanging rows

Solve by elimination, exchanging rows when necessary $$ v + w = 0\\ u + v = 0\\ u + v + w = 1\\ $$ Which permutation matrix is required? answer is $$ P= \begin{bmatrix} 0 & 1 & 0 \\ 1 ...
-1
votes
0answers
15 views

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) ...
0
votes
0answers
34 views

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 ...
1
vote
1answer
20 views

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 ...
-1
votes
1answer
24 views

Simmetric group exercise of an exam

In $S_5$ let be $\sigma=(12435)$ and $\tau=(25)(34)$, and $H= <\sigma, \tau>$. Show that $N(<\sigma>) = H$ N.B. $N(<\sigma>)$ is normalizer in $H$, not in S5. Deduce that $H=&...
1
vote
1answer
13 views

Proving the congruence $((p-1)\ /\ 2)!^2 \equiv (p-1)!\ (\textrm{mod}\ p)$

We are given that $p$ is a prime congruent to $1$ modulo $4$. The proof for the congruence $$\left(\frac{p-1}{2}\right)!^2 \equiv (p-1)!\ (\textrm{mod}\ p)$$ is argued as follows: Proof. Since we ...
-3
votes
0answers
12 views

Producing factors L and U

Apply elimination to produce the factors L and U for the matrices below. $$ A= \begin{bmatrix} 3 & 1 & 1 \\ 1 & 3 & 1 \\ 1 & 1 & 3 \\ \end{bmatrix} $$ and $$ A= \begin{...
1
vote
1answer
16 views

Converting Fractional Coordinates to Cartesian

I'm confused about what I am reading online - different sites tell me different answers. Lets say I have a point pair in fractional coordinates, [xf,yf,zf]. I know that to convert them to their ...
-2
votes
0answers
18 views

Existence of (non) complete metric on an interval [on hold]

I am stuck with this problem. Can anyone help me out? Thank you in advance. Which one of these are correct. a) (0,1) with the usual topology admits a metric which is complete. b) (0,1) with the ...
0
votes
0answers
9 views

Clarification on a concept involving Hall subgroups

Suppose that $G$ is a finite group and $H\leq G$. If $Q \in$ Hall$_{\pi}(G)$ such that $Q \cap H \in$ Hall$_{\pi}(H)$. Is true that $Q \leq H$ when Hall$_\pi(H)$ $\subseteq$ Hall$_\pi(G)$
1
vote
1answer
17 views

What is the maximum length continuity of function $f$?

Let $f(x) = \int_{.25}^x {(\frac{t}{{{t^2} - 1}})} (\cos \frac{1}{{\sqrt t }})dt$. What is the maximum length continuity of function $f$?
-1
votes
0answers
12 views

Can i say that ( U(Zm) , * ) is isomorphic to ( Zk , +)

Can i say that ( U(Zm) , * ) is isomorphic to ( Zk , +) where k=phi(m) or to Za x Zb x Zc x....where abc..=k with the right combination of a,b,c... ?

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