2
votes
1answer
28 views

Inequality in matrix norm

Let $\|\cdot\|$ be matrix norm on $M_n$.Why does $\|A\|_2 \le \|A\|^{\frac{1}{2}} \|A^*\|^{\frac{1}{2}}$? ($\|A\|_2 = \displaystyle\max_{\|x\|_2 = 1} \|Ax\|_2$)
0
votes
1answer
15 views

Power Series - differentiation and absolute convergence

I am having problems with the following exercise: Ex. 1. Let $f(x) = \sum_{n=1}^{\infty} \frac{(x-1)^n}{n}$ (i) Find the convergence interval. Here I let $f(x) = \sum_{n=1}^{\infty} ...
2
votes
0answers
19 views

Limits of integral ratio: which theorems to use?

I "guessed" the limit $$ \lim_{n\rightarrow+\infty} \frac{\int_{a}^{b} f(x,y)\cdot g(x,y)^ndy}{\int_{a}^{b} h(x,y)\cdot g(x,y)^ndy}=\frac{f(x,a)}{h(x,a)}.$$ where $f,h>0$ and $f,g,h\in ...
-1
votes
0answers
28 views

John b.Conway chapter 2 section 2 exercise 4 [on hold]

Show that an idempotent is compact if and only if it has finite rank.
1
vote
0answers
26 views

Who first used the notation $\mathcal{O}_K$ for ring of integers?

I think this is a standard notation since almost every author uses it, but who came up with the notation? After all, what does $\mathcal{O}$ in $\mathcal{O}_K$ stand for? Thanks in advance.
0
votes
1answer
16 views

Closed subscheme of an affine scheme.

This is a portion of the exercise 2.3.11 (b) of Hartshorne. Suppose $Y$ is a closed subscheme of $X=\operatorname{Spec}A$. I would like to show that there are $f_{i}\in A$ so $D(f_{i})$ cover $X$ and ...
2
votes
3answers
25 views

For $n > 2, n \in \mathbb{Z}$, show the sum of integers coprime to $n$ in the range $[1,n-1]$ is equal to $\frac{1}{2}n \phi(n)$

For $n > 2, n \in \mathbb{Z}$, show the sum of integers coprime to $n$ in the range $[1,n-1]$ is equal to $\frac{1}{2}n \phi(n)$ Firstly $\phi(n)$ is Euler's totient function, the number of ...
3
votes
1answer
39 views

Subgroups of a finite $p$-group

Let $G$ be a group such that $|G|=p^n$ for some $p$ prime and $n\in\mathbb{N}$. I want to prove that if $k\le n$ then $G$ has a normal subgroup of order $p^k$. I want to use induction on $k$. If ...
0
votes
0answers
6 views

Maximal p-subgroup of inertia group.

We know from the theory that if $\mathbb{L}$ is a finite Galois extension of the local field $\mathbb{K}$ then the ramification group $G_1$ is a $p$-group where $p$ is the characteristic of the ...
6
votes
2answers
159 views

Infinite sum of reciprocals of pentagonal numbers

How do i find this sum: $$\sum_{n=1}^\infty \frac{1}{p(n)}$$ where $p(n)=\dfrac{n(3n-1)}{2}$ is the $n$th pentagonal number? Thank you for your help.
-2
votes
1answer
25 views

show that $f$ is concave iff $-f$ is convex.

let $f : S \subset \Bbb R^n \to \Bbb R$ be a function. then, I want to show that $f$ is concave iff $-f$ is convex. definition of convexity: $x,y\in S$ and $\alpha \in (0,1)$ $f(\alpha ...
6
votes
0answers
35 views

Does $M_n(R_1)\cong M_n(R_2)$ imply $R_1\cong R_2$?

Let $R_1,R_2$ be two rings with identity. If for some $n\in\mathbb N$, $M_n(R_1)$ and $M_n(R_2)$ are isomorphic as rings, can we deduce that $R_1\cong R_2$? I can prove it when both $R_1,R_2$ are ...
-6
votes
0answers
19 views

John B.conway chapter 2 sec 4 exe 3 [on hold]

If T∈B_00(H,K) show that T*∈B_00(K,H) and dim(ranT)=dim(ran T*)
4
votes
1answer
34 views

Is it possible to give the unit square a smooth structure?

At the beginning of Lee's "Introduction to Smooth Manifolds", Lee gives the example of a the square and the circle being homeomorphic as an intuitive motivation for smoothness not being invariant ...
0
votes
0answers
8 views

Calculus of variational in 2 dimension with constraints

Let $S$ be a 2D region and its boundary is $\partial S$. $u(x,y)$ is defined in S. The functional is of the following type: $J[u] = \int_S F(x,y,u,u_{x},u_{y}) \mathrm{d}s + \int_{\partial S} ...
-3
votes
2answers
33 views

Manufacturing problem, exponential distribution

A manufacturing process produces $92%$ good chips (G) and $8%$ bad chips (B). The lifetime, in seconds, of chips is exponentially distributed $E(\lambda)$.For good chips, ...
0
votes
0answers
11 views

Questions about soundness and completeness

I am doing a module about introduction to symbolic logic and I seem to understand most of it, apart from problems of the following kind involving that involve soundness and completeness: The ...
3
votes
2answers
67 views

Residues and poles proof

Let the degree of the polynomials $P(z)=a_0+a_1z+a_2z^2+\cdots+a_nz^n$ $a_n\neq0$ and $Q(z)=b_0+b_1z+b_2z^2+\cdots+b_mz^m$ $b_m\neq 0$ be such that $m\geq n+2$. Show that if all the zeros of ...
1
vote
1answer
30 views

If $f:\mathbb{R}\to\mathbb{R}^2$ is of class $C^1$, show that $f$ does not carry $\mathbb{R}$ onto $\mathbb{R}^2$

If $f:\mathbb{R}\to\mathbb{R}^2$ is of class $C^1$, show that $f$ does not carry $\mathbb{R}$ onto $\mathbb{R}^2$. In few words I have to show that $f(\mathbb{R})$ contains no open set of ...
0
votes
1answer
24 views

Continuity in closed sets

Please help me, I have being trying this for days now. Let $f:F \to \mathbb{R}$ be a function on a closed set $F$. Show that $f$ is continuous if and only if $A=\{x \in F; f (x) \leq c\}$ and $B=\{x ...
2
votes
2answers
75 views

Innocent-looking Diophantine equation with smallest solution of the order $10^{50}$?

Recently someone mentioned to me that there is a diophantine equation that looks very simple and innocent, but the smallest solution involves numbers of the order $10^{50}$ or something like this. The ...
0
votes
1answer
25 views

Geometric proof that (symmetry w/r to $x$ and $y$ axes) $\implies$ (symmetry w/r to origin)

I'm trying to prove that reflecting a point about the x and y axes is equivalent to reflecting it about the origin. Is my proof valid? How could I improve it? Proof: Take a point $a$ in the first ...
-3
votes
0answers
14 views

Simplify Boolean Expression xyz' + x'yz' + y'z' [on hold]

without using k-map, Simplify Boolean Expression xyz' + x'yz' + y'z' I have tried this xy(z+z') + x'yz' + y'z' xy(1) + x'yz' + y'z' xy + z'(x'y+y') I am pretty much stuck
0
votes
0answers
10 views

can we extend Whitney’s theorem to multi graphs?

Here is an introduction to the Whitney’s theorem: https://www.math.hmc.edu/~kindred/cuc-only/math104/lectures/lect09-slides-handout.pdf But it doesn't say whether we can use this theorem for multi ...
2
votes
1answer
40 views

Function defined by integral

this question is driving me nuts, I can't think about an easy solution. Let $F(x)=\int_{0}^{x} \sqrt{1+t^3}\,dt. $ Evaluate $\int_{0}^{2} x\,F(x)\,dx$ in terms of $F(2)$. I know that the derivative ...
7
votes
4answers
171 views

Solve trigonometric integral

Please help me to solve the following integral: $$\int_{-\pi/2}^{\pi/2} \frac{\sin^{2014}x}{\sin^{2014}x+\cos^{2014}x} dx.$$ I have tried a lot, but no results. I only transformed this integral to the ...
0
votes
0answers
13 views

Closed subsets of empty interior are of 1st category

In a metric space is it true that closed sets with empty interior are of 1st category? I.e., that it can be represented as a at most countable union of meager sets? Thanks
3
votes
3answers
70 views

How to show algebraically that $x^3 +3x +1$ is injective?

How to show algebraically that $$x^3 +3x +1$$ is injective? Working with the usual method of assuming that $f(c)=f(d)$ and then seeing if $c=d$. I've tried several approaches, including factoring ...
1
vote
1answer
16 views

Limit of a Function with Parameter

Given that $a\ne-1, \lim\limits_{x \to 0} f(x) = L$, prove by limit definition ($\epsilon, \delta)$ that $\lim\limits_{x \to \infty} f(\frac{a+1}{2x}) = L$. I would greatly appreciate any thoughts ...
0
votes
1answer
7 views

Energy integral is convex for non-uniform diffusion equation in $\Omega\subset\Bbb R^n$

I'm having trouble proving that a certain integral that is a function of time, is a convex function. Let $\Omega\subset \Bbb R^n$ be a bounded Lipschitz domain, and let $u: ...
0
votes
0answers
19 views

Calculate the expectation of psi(x) with a gamma density x∼Gamma(α,β )

Suppose I have a Gamma distributed random Variable x∼Gamma(α,β). Now I want to calculate the following expectation values (integrals): E[psi(x)] with psi(x) being the digamma function Many thanks ...
3
votes
0answers
30 views

Showing that a recursively defined sequence is decreasing.

A colleague of mine is interested in finding out how to show the following: Prove that the sequence $(a_n)$ defined by ...
0
votes
0answers
11 views

Decomposition of axial vector and vector representions of C$_{4v}$ group

Let $R$ be the orthogonal matrix corresponding to an operation in $O(3)$. If R is a proper rotation, then both vectors $\vec{V}$ and axial vectors $\vec{A}$ are transformed in the same way $$ \vec{V} ...
0
votes
0answers
24 views

Taking Analysis I, Abstract Algebra I, and Theoretical Linear Algebra [on hold]

S.E advisers, I am a college sophomore in US with a major in mathematics, and an aspiring algebraic number theorist and cryptographer. I wrote this email to seek your advice about taking the ...
1
vote
1answer
10 views

Finding explicit discrete valuation of ring of germs of analytic functions on $\mathbb{C}$

I found interesting problem set http://www.math.lsa.umich.edu/~kesmith/593hmwk2-2014.pdf and I noted Problem 3-3. And I found another version: Let $\mathcal{U}$ be the subset of all open sets of ...
3
votes
1answer
26 views

Invertible matrix over a ring and its eigenvalues

Eigenvalues and invertible matrices for fields and vector spaces: Let $K$ be a field (so $K^n$ is a $K$-vector space) and let $A \in K^{n\times n}$ be an $n\times n$-matrix. Then we have the following ...
0
votes
1answer
25 views

Finding area of rectangle [on hold]

Given that $ABCD$ is a rectangle. $AB=30\space m$, $CE=20\space m$ and $E$ is a point between CB, so we can get $CD=30$ & $DF=20$. $F$ is a point between AD How to find BE? Any clue to find that ...
2
votes
1answer
23 views

A combinatorial question about outer automorphisms of $S_6$

Quite possibly I'll solve this and post my answer below, but maybe others will post better answers before I get to that.$^\dagger$ The group of permuations of $\{a,b,c,d,e,f\}$ is generated by $15$ ...
0
votes
0answers
6 views

Applying the cycle algorithm for Hamiltonian graphs

I have this graph and the idea is to produce a planar graph. I first choose a Hamiltonian cycle c $$1,2, 3, 4, 5, 6$$ The edges not in the cycle c are $$(13),(14),(15),(24),(26),(46)$$ We will ...
1
vote
1answer
36 views

Im having trouble figuring this integral out can someone help? Not allowed to use polar coordinates.

$$c\int_{-1}^{1} \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\sqrt{1-x^2-y^2}dy \:dx=1 .$$ Find $c.$ I went with the substitution say $b=1-x^2$ in the first integral. Then I went with : $\cos t= {y \over ...
0
votes
1answer
15 views

Book for Module Theory

I want a book to cover the following topics in Module Theory: Modules ,Submodules,Quotient modules,Morphisms Exact sequences ,three lemma,four lemma,five lemma,Product and Co products,Free modules, ...
1
vote
0answers
13 views

Network theory and football?

I was reading the latest post on Azimuth, Network Theory in Turin, and I watched many of the lectures Baez posted on his site here. This might be a crazy question to ask considering it's not ...
1
vote
2answers
18 views

Complex analysis, residues and integrate

Let $C$ denote the circle $|z|=1$ oriented counterclockwise. Show that i)$\int_Cz^ne^{\frac{1}{z}}dz=\frac{2\pi i}{(n+1)!}$ for $n=0,1,2$ ii)$\int_C e^{z+\frac{1}{z}}dz=2\pi ...
1
vote
1answer
18 views

What are the summands in $\mathbb{Z}^n$?

I am interested in knowing what all the rank $k$ submodules of the $\mathbb{Z}$ module $\mathbb{Z}^n$ are that are also summands. I know that $\mathbb{Z}^k$ sitting in $\mathbb{Z}^n$ in the standard ...
3
votes
3answers
23 views

How to prove that this matrix is positive definite?

Let $\mathbf{A}=\begin{pmatrix}a^2+b^2 & b^2 & b^2 & ... & b^2 \\ b^2 & a^2+b^2 & b^2 & ... & b^2\\ \vdots & b^2 & \ddots & & b^2 \\ b^2 & \dots ...
1
vote
0answers
28 views

Show that this polynomial is irreducible

I want to show that this polynomial is irreducible over $Q$ . $f(x)= x^4 -2x^2 +8 x+1 $ I prove it by long method but i need easy and short method . My proof: I reduce it $mod 3 .$ I proved that ...
2
votes
0answers
19 views

simplification of a linear algebra equation

I have been trying to simplify this equation, but with no success at all: So far what I have done is the following: but I get stuck in the last part, what am I missing?
0
votes
0answers
15 views

Proving that $\sum_{i\geq0}f_i(n,m)x^i=\sum_{k=0}^n(-1)^k{n \choose k}\big((1+x)^{n-k}-1\big)^m$

Let $f_i(n,m)$ ($n,m\geq1i\geq0$) be the number of $m\times n$ matrices with entry of 0 and 1, so that there is in each row and column at least one 1 and total exactly i ones. I have to show that: ...
-1
votes
0answers
22 views

Quick MATLAB Question [on hold]

I have a script I have to execute with MATLAB. This script takes a bunch of numbers and produces a single output from those numbers and prints them to a new file. However, I am running into a problem ...
-3
votes
1answer
28 views

Variance of a special random walk [on hold]

I am trying to find the variance of the following special random walk: Suppose that $U=(U_1,U_2,...)$ is a sequence of independent random variables, each taking values $u$ (for up) and $d$ (for down) ...

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