1
vote
1answer
139 views

Maximal ideals of $L^1(R)$ and Gelfand transforms

I've been trying to understand something about the Gelfand transform. As I understand it, for a complex Banach algebra $A$, we consider the set $$\Delta = \{\phi \in A^* : \phi(x*y) = ...
4
votes
3answers
149 views

When would I NOT use a double implication when proving equalities between sets?

I am currently studying how to prove set equalities. For this, there is an example in my book which goes this way: Prove the following: $A-(B\cup C) = (A-B)\cap (A-C)$ We need to prove the ...
1
vote
1answer
160 views

Finitely generated free modules of infinite rank.

We know that for general modules over a commutative ring with $1$, you can't always extract a basis from a generating set. This makes me think that maybe there should be free modules of infinite ...
1
vote
1answer
204 views

What does the complement mean as it relates to Boolean Algebra?

For a lattice to be a Boolean Algebra it must be a distributive lattice and contain complements. What does the word complement mean?
2
votes
4answers
555 views

Space of bounded functions is reflexive if the domain is finite

Let $C_b(X)$ be a space of bounded continuous functions on a locally compact space $X$ equipped with the supremum norm. How to show that $C_b(X)$ is reflexive if and only if $X$ is finite?
1
vote
1answer
45 views

Prove that if $n \in \Bbb N$, $f: I_n \to B$, and $f$ is onto, then $B$ is finite and $|B|\leq n$.

Prove that if $n \in \Bbb N$, $f: I_n \to B$, and $f$ is onto, then $B$ is finite and $|B|\leq n$. Notes on notation: For each natural number $n$, $I_n = \{i \in \mathbb{Z} \mid i \leq n\}$. $A ...
5
votes
0answers
98 views

Representation theorem for continuous process of finite variation

There is a martingale representation theorem If $M$ is a continuous $L^2$-martingale, there is a Brownian motion $B$ and a cadlag adapted function $\sigma$ such that $$ M_t = M_0 + \int_0^t ...
1
vote
2answers
450 views

Closure of a metric space.

I have a problem about proving the following. Can you please help me? First of all, boundary of A is the set of points that for every r>0 we can find a ball B(x,r) such that B contains points from ...
3
votes
1answer
590 views

If a 3x3 matrix is diagonalizable and has eigenvalues 1,2 but has 2 eigenvectors with eigenvalue 2, would we…

If a $3 \times 3$ matrix is diagonalizable and has eigenvalues $1$ and $2$ but has two eigenvectors with eigenvalue $2$, would we have the eigenvalue matrix $$\begin{pmatrix} 1 & 0 & 0 \\ 0 ...
7
votes
0answers
120 views

Take an m x n grid, and in each box pick two opposite corners at random to connect. What can be said about the resulting pattern?

Inspired by the upcoming book 10 PRINT CHR$(205.5+RND(1)); : GOTO 10 by Nick Montfort et al., whose title derives from this particular example of emergent behavior. Here's an example: (Note that ...
2
votes
1answer
207 views

Equators and meridians on a discrete torus

Consider the 4 × 4 grid graph: Now torify it, i.e. connect its opposing vertices: How can one tell the difference between a “meridian” and an “equator”? The ...
0
votes
2answers
158 views

Showing that $(X^*)^{**}=(X^{**})^{*}$, where $X$ is a Banach space

Let $X$ be a Banach space. Let $X^*$ denote the dual space . Would you help me, How to show that $(X^*)^{**}=(X^{**})^*$?
2
votes
2answers
195 views

Finding angles of two vectors using simultaneous equations

In a physics problem, I am asked to find the resulting angle of two velocity vectors using each velocity vector's components. For the x-component, I have $m_av_{1ax} = m_av_{2ax} + m_bv_{2bx}$. ...
1
vote
1answer
83 views

What's so great about continuous embeddings?

I often see something like "$V \subset H$ is continuous embedding", meaning that the inclusion map $i:V \to H$ defined $i(v) = v \in H$ for $v \in V$ is continuous: so $$|i(v)|_H \leq C|v|_V$$ holds ...
15
votes
1answer
497 views

Given a group $ G $, how many topological/Lie group structures does $ G $ have?

Given any abstract group $ G $, how much is known about which types of topological/Lie group structures it might have? Any abstract group $ G $ will have the structure of a discrete topological group ...
4
votes
1answer
92 views

Proofs whose length depends on the input

This may be a question from proof theory, but I'm not sure, since I don't know any proof theory. What I will be asking about is what happens, if the length of a proof isn't fixed: I'm going to present ...
2
votes
2answers
380 views

Solving differential equation with power series

$$\begin{cases} w''=(z^2-1)w \\ w(0)=1 \\ w'(0)=0 \end{cases}$$ I tried the following: Let $$w(z)=\sum_{j=0}^{\infty}w_j z^j$$ $$\implies w''(z)=\sum_{j=0}^{\infty}j(j-1)w_j z^{j-2}$$ $$\implies ...
5
votes
3answers
124 views

Evaluation of a definite integral.

In my real analysis course I was given this exercise: Calculate $\displaystyle{\int_0^1e^{x^2}dx}$. What I did was to write $\displaystyle{e^{x^2}=\sum_{n=0}^\infty\dfrac{x^{2n}}{n!}}$ and conclude ...
2
votes
1answer
179 views

Definite integral involving hyperbolic cosine

I have had no experience so far with hyperbolic functions so any help will be appreciated. This is on the chapter of complex integration but I would especially appreciate it if you could turn this ...
0
votes
1answer
256 views

Omega limit set of a system

The system of ODE is $$\begin{cases} \dot{x}=-y(1-x^{2}) \\ \dot{y}=x+y(1-x^{2}) \end{cases} \tag{$\ast$}$$ Claim: $\forall p\in\left\{ (x,y)\in\mathbb{R}^{2} : |x|<1,\ x^{2}+y^{2}>0\right\} ...
4
votes
1answer
277 views

A regular graph of degree k is connected if and only if the eigenvalue k has multiplicity one

I saw this sentence in Wikipedia: A regular graph of degree k is connected if and only if the eigenvalue k has multiplicity one I couldn't find a proof to that statement - can someone address me to ...
1
vote
1answer
118 views

Is $ \int_0^b (a-x^m)^{1/n} dx $ solvable?

Is there a solution for the integral $$ \int_0^b (a-x^m)^{1/n} dx? $$ WolframAlpha doesn't help a lot: $$ \frac1a x(1-x^m)^{\frac1n+1}{}_2F_1\left(1,\frac1n+1+\frac1m; ...
1
vote
1answer
38 views

calculating correlation

If one fair six-sided die is rolled, suppose that $X$ is the total number of even numbers shown and $Y$ is the total number of fives shown. How can I go about calculating the correlation exactly in ...
3
votes
2answers
601 views

Is the conjugation map always an isomorphism?

Given a group $G$ with $a\in G$ fixed, define $\phi: G \to G$ by $\phi(x) = axa^{-1}, x \in G$, I am wondering when $\phi$ is an isomorphism. I think that it is always an isomorphism because it is ...
2
votes
1answer
183 views

Coin tossing questions

I had an exam today and I would like to know for sure that I got these answers correct. A fair coin will be tossed repeatedly. What is the probability that in 5 flips we will obtain exactly 4 ...
6
votes
3answers
735 views

Can distance between two closed sets be zero?

Is given metric space $(M, d)$. Let $A\cap B = \emptyset; \,\,\text{dist}(A,B):=\inf\{d(x,y):x\in A, y\in B\}$. $A, B$ are both closed sets. Is it possible that $\text{dist}(A,B)=0$? The first ...
1
vote
1answer
74 views

What is meant by “orbit” in this question?

I was reading "Prove that Anosov Automorphisms are chaotic," and the answer and a few of the comments talked about orbits. I'm curious what is meant by "orbits" in the given context. Is it analogous ...
3
votes
2answers
149 views

Computation of a certain integral

I would like to compute the following integral. This is for a complex analysis course but I managed to around some other integrals using real analysis methodologies. Hopefully one might be able to do ...
3
votes
2answers
294 views

Integers represented by the polynomial

Can every sufficiently large integer be written in the form $a^{100} + b^{101} + c^{102} + d^{103} + e^{104}$ for some non-negative integers $a$, $b$, $c$, $d$ and $e$? I'm only 15 so if u could ...
1
vote
2answers
128 views

$\mathbb{R}P^3$ is homeomorphic to the lens space $L(2,1)$

Show that the $3$-dimensional real projective space $\mathbb{R}P^3$ is homeomorphic to the lens space $L(2,1)$. (I am not sure but the problem is probably from the book Knots and Links which is ...
1
vote
1answer
485 views

Convergence in $\ell^p$ norm provided it weakly converges.

I need some help with the following problem : $1<p < \infty$ , let $x_n$ be a sequence in $\ell^p$ and also $x\in \ell^p$ . I am interested in showing $$\lim_{n\to \infty} \|x_n-x\|_p\to0$$ ...
5
votes
1answer
125 views

Tarski's Undefinability Theorem Reference

There are many books articles that look to explain Godel's Incompleteness Theorems for laymen. Does anyone know of some good material (free online is most appreciated) that attempts to do the same ...
1
vote
2answers
121 views

The equation $\left(\frac{a+b}{2}\right)^{x+y}=a^xb^y$

Let $a, b \in (0, \infty)$, $a<b$. Prove that the equation $$\left(\frac{a+b}{2}\right)^{x+y}=a^xb^y$$ has at least one solution in $(a, b)$. Some suggestions? Thanks.
3
votes
1answer
76 views

Existence and uniqueness of solutions to difference equations

I know about existence and uniqueness of solutions to differential equations, but when it comes to difference equations, I am struggling to find a reference. I am looking for conditions under which, ...
0
votes
1answer
98 views

Question on Tensor Products

I am working on the following problem: Let $R$ be a PID and let $a,b \in R$ be such that $\gcd(a,b) = 1$. Prove that there are $s,t \in R$ such that $sa+tb = 1$, that the $R$-module $R/\langle a ...
1
vote
3answers
301 views

Probability of 2 sevens before 6 evens

In successive rolls of a pair of fair dice, what is the probability of getting 2 sevens before 6 even numbers? Assume that on the nth roll, the game ends. So this means we roll a seven on the nth ...
0
votes
1answer
100 views

Another version of PP

Prove the following version of the pigeonhole principle. Let $m$ and $n$ be positive integers. If $m$ objects are distributed in some way among $n$ containers, then at least one container must hold at ...
1
vote
1answer
73 views

Are these linear programming constraints correct?

The problem is: Beth works a maximum of $20$ hours/week programming computers and tutoring math. She receives $\$25$/hour for programming and $\$20$/hour for tutoring. She works between $3$ and $8$ ...
1
vote
1answer
61 views

Indecomposable L-module

I have the following exercice which I have be trying to solve: Let L be a Lie algebra and $r:L\rightarrow gl_3(F)$ a representation of L such that $im(r)=t_3(F)$ (the upper triangular matrices). Show ...
4
votes
0answers
64 views

quotient group scheme

assume I have a group $G$ over a field of char 0 and $H$ a closed subgroup. When is it true that the group $N(H)/H$ is representable? If $G$ has nice properties, like to be reductive or unipotent is ...
0
votes
1answer
262 views

Compact Set for Dummies

Can any one tell me in simple words what is a compact set? I read the definition of Compact set, but do not get it. BTW, I do not know topology. In particular, is the probability simplex, $W\ge0, ...
0
votes
2answers
161 views

Markov inequality proof step

Hej, I am studying a proof for the Markov inequality, and there is a certain step, which I don't understand: $\mathbb{E}(X \cdot \mathbb{I}_A) \ge \mathbb{E}(a \mu \mathbb{I_A})$ where ...
0
votes
2answers
72 views

How to generate a continuous random variable with the density $f(x) = 1.5x^2, -1 < x < 1$?

Let $U$ be a Standard Uniform random variable. Show all the steps required to generate a continuous random variable with the density $f(x) = 1.5x^2$, $-1 < x < 1$. I'm not looking for exact ...
0
votes
0answers
53 views

Linear Application that is open in a TVS

Let $T: E \to F$ be a linear map between topological vector spaces $E$, $F$. If for each nonempty open set $G$, the interior of $T(G)$ is non-empty, then, $T$ is open. Proof: $$\mathrm{Int}(T(G))= ...
1
vote
1answer
183 views

Sine series of $\pi/2$

I'm studying Fourier series and came across this peculiar problem. I just studied (along with proper reasoning) that if $f(x)$ is an even function, then the fourier series has only Cosine terms and if ...
2
votes
1answer
90 views

How would you define a geodesic in $\operatorname{SO}(2n)$ and $\operatorname{SO}(2n+1)$?

Consider me a beginner. I am trying to find the intrinsic diameter of $\operatorname{SO}(n)$ and $\operatorname{SO}(2n+1)$, but I am unsure how to define a metric between two different points in ...
3
votes
1answer
146 views

Knots in $S^1\times S^2$

Is there any special study of knots in this particular 3-manifold? A more targeted / simple question: What are some nontrivial examples of knots $S^1\subset S^1\times S^2$, and is there convenient ...
1
vote
1answer
132 views

Generating random commuting hermitian matrices

How can I generate random commuting hermitian matrices ? EDIT: Another question: given a certain hermitian matrix, how can I generate a random hermitian matrix which commutes with it?
1
vote
1answer
99 views

How to assure quality control to a 4% level of significance based on a 200 item sample with 24 defects?

For quality control purposes we have to accept or reject a large shipment of items. In order to do so we collect a sample of 200 items, 24 of which are defective. The manufacturer claims that at most ...
1
vote
1answer
80 views

Definition of exponential distribution and its relation to Poisson

I want to understand the Poisson and exponential distributions correctly. Would this be correct "If $X$ follows a Poisson distribution, then $T$ measures the probability that you have to wait $t_a$ ...

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