1
vote
1answer
45 views

a question on quasi-invariant measures (with respect to the irrational rotations) on the unit circle

Fix a $\sigma$-finite atom-less measure $\mu$ on the unit circle, which is quasi-invariant and ergodic under the rotation $T$ of the angle $2\pi\theta$, $\theta$ irrational. By a well-known result of ...
1
vote
0answers
10 views

Examples of combinatorial/probabilistic proofs of theorems in linear algebra

Are there any examples of combinatorial/probabilistic proofs of theorems in linear algebra? Motivation: I see here, the inverse is true.
0
votes
0answers
3 views

What is the orthogonal of an intersection?

Background I have been introduced to the notion of orthogonal complement of a subset of a (pre)hilbert space. Given $X$ a (pre)hilbert space and $A\subseteq X$, one defines $A^\perp:=\{x\in X:x\perp ...
0
votes
1answer
356 views

Problem with Newton's Method in solving a System of Equations

I'm trying to use Newton's method to solve the following system of equations, where f and g are functions of x and y. (h,a,f,c,d,b and k are just constants). $f(y,x)=\left[\begin{array}{c} y^{1}\\ ...
0
votes
0answers
4 views

Moebius band and Viviani Frill

I find a common rule that unites generation of Viviani Frill and the the Moebius Band. $$ \phi =\theta $$ where $ \phi,\theta $ are spherical coordinates. Please comment if this way looking at it ...
-1
votes
0answers
44 views

Why do we care about representaion of primes?

Im currently trying to figure out the genesis of quadratic reprocity by using Cox and Lemmermeyers books. I also got a copy of some works of Fermat but it is in German. It seems like there is some ...
0
votes
0answers
2 views

Regularity of Lebesgue outer measure

The terminology in this area is somewhat confusing, my question is how to prove: Given $E \subseteq \mathbb{R}$, there exists a Lebesgue measurable set $A$ such that $E \subseteq A$ and $\lambda^*(E) ...
0
votes
0answers
9 views

Equivalent condition for continuity of a function

Let $g: [0,+\infty) \rightarrow \mathbb{R}$ be a continuous function and let $f: [0,+\infty) \rightarrow \mathbb{R}$ be defined by \begin{equation} f(t) = \inf \{ s \geq 0 \,|\, g(s) > t\}. ...
1
vote
0answers
35 views

Topological space inducing a space such that it is a subspace of that space?

If $\;Y\subseteq X^2$ and $\tau$ is any topology on $Y$, is it possible to induce topologies in $X$ from $\tau$? Are there ways to define topologies on $X$ from $\tau$ so that $(Y,\tau)$ become a ...
2
votes
2answers
31 views

Integration without complex analysis on rational-improper integral

Evaluate: $$\int_{0}^{\infty} \frac{1}{x^6 + 1} \,\mathrm dx$$ Without the use of complex-analysis. With complex analysis it is a very simple problem, how can this be done WITHOUT complex analysis? ...
1
vote
0answers
11 views

A question about independence of sigma algebras (generated by random variables)

Let $X_1, X_2, \ldots$ i.i.d random variables. Is it possible that $$\{X_{n+1} \in B\} \in \sigma({X_1, \ldots, X_n})$$ for some $B$? I want to show that $\sigma(X_{n+1})$ and $\sigma(X_1, \ldots, ...
0
votes
1answer
14 views

Stuck on basics: How to prove that {subst($\alpha$,s)} is well defined?

So I feel like this is a really basic point that I'm missing and I can't really manage to prove that: So I have a substitution function $s: Var \rightarrow WFF$ and a subst function: $WFF \times ...
2
votes
2answers
89 views

Understanding infinity

I want to understand in a greater depth the concept of infinity. Can someone give me any reference/ text from where I can study and understand about the concept of infinity in mathematics? I would be ...
1
vote
0answers
8 views

Mobius transformations — $L^{-1} \times M\lbrace0,1,\infty\rbrace\times L$

$$L^{-1} \times M\lbrace0,1,\infty\rbrace\times L$$ I cannot seem to get the right answer when I multiple them out. $$(1-i)z \cdot (1-z) \cdot \frac{z}{1-i}$$ What do you get when you multiple ...
0
votes
0answers
8 views

Coming up with a condition for index selection from a set for a specific problem

In my research I have come across the following problem. I have two sets of real numbers, say $\{a_i\},\ \{b_i\},\ i=1,2,\cdots,\ n$. Let $S$ be a given set such that $S\subset \{1,2,\cdots,\ n\}$. ...
0
votes
0answers
15 views

Where is the maximum of the Gaussian?

The maximum of the envelope $e^{-(\frac{x^2}{2\sigma^2})}$ is supposed to be at $x=0$. Why is this the case?
4
votes
2answers
21 views

Sufficient conditions to have $f' = O(f(x)/x)$.

Suppose $f$ a nonnegative real-valued function, non-decreasing, $O(x^m)$ for some $m \in \mathbb{Z}_{\geqslant 0}$ and $C^1$, with $f'$ being monotonic and nonnegative. Are this sufficient conditions ...
0
votes
0answers
14 views

LU factorization of a modify matrix

Suppose you know $L$, $U$, decomposition LU of a matrix $M+I$ ($M+I=LU$). Lets $J$ a diagonal matrix whose elements are $0$ or $1$. Is there any relation between the factorization LU of $M+I$, and ...
2
votes
1answer
31 views

For elements of the intersection of C*-algebras with the same unit, can the spectra be distinct depending on the algebra?

Problem Given unital C*-algebras $1\in\mathcal{A}$ and $1'\in\mathcal{A}'$. Regard an element $A\in\mathcal{A}\cap\mathcal{A}'$. Can it happen that: ...
0
votes
0answers
11 views

Rings with right identity and left inverse.

Let R be a ring with a right identity e and for each non zero element a in R, there exist a left inverse......implies R is a division ring. Please help.
0
votes
3answers
404 views

Algebra clock problem

An absent-minded watch repairman connected the hour hand to the minute hand pinion and the minute hand to the hour hand pinion and set the clock at 6AM which was the correct time then. How soon after ...
0
votes
0answers
11 views

Switch integral and sum for Bessel function.

I haven't real knowledge in Bessel's function and I'd like to know how to switch integral and sum in these two equations. I've already tried a lot of ideas but nothing really works. The first one is : ...
5
votes
5answers
202 views

For every positive integer $n, n^2 + 4n + 3$ is not a prime

Prove: For every positive integer $n, n^2 + 4n + 3$ is not a prime. I tried to disprove the statement, which I could not using several number examples with constructive proof. However I am not sure ...
0
votes
1answer
18 views

Question regarding solution to the problem of computing the expected number of coin flips until getting 5 in a row.

I have a question for further explanation regarding the solution to the problem here Expected Number of Coin Tosses to Get Five Consecutive Heads. The most favorite solution by André Nicolas is: ...
1
vote
0answers
6 views

Etale morphism and reduced schemes

Let $f:X \to Y$ be an etale morphism of Noetherian schemes. Is it true that the induced morphism on the reduced schemes, i.e., $f_{\mathrm{red}}:X_{\mathrm{red}} \to Y_{\mathrm{red}}$ is etale as ...
0
votes
3answers
126 views

Function with zero description

Is there a nice expression (possibly differentiable outside $0$) for a function $f(x)$ that satisfies the following property other than the delta? $$f(x)=1\iff x=0$$ $$f(x)=0\iff x\neq0$$ Is it ...
1
vote
1answer
12 views

Find homotopy fibers of $\mathbb{R}P^1 \hookrightarrow \mathbb{R}P^\infty$ and $\mathbb{C}P^1 \hookrightarrow \mathbb{C}P^\infty$

I found a following tasks in my algebraic topology notes: Find homotopy fibers of $\mathbb{R}P^1 \hookrightarrow \mathbb{R}P^\infty$ and $\mathbb{C}P^1 \hookrightarrow \mathbb{C}P^\infty$. For a ...
2
votes
1answer
20 views

Multi Variable Partial Fraction Problem

The question is to develop $$(x-\alpha)(x-\beta)(x-\gamma)\over(x-a)(x-b)$$ into partial fractions. Someone challenged me to solve this question and said the answer is ...
0
votes
1answer
44 views

Convergence and limit of Muller sequence

The Muller sequence is given by the recursive definition: $U(n+1)=111-\frac{1130}{U(n)}+\frac{3000}{U(n)U(n-1)}$ with $U(0)=5.5$ and $U(1)=\frac{61}{11}$. This sequence is interesting in ...
1
vote
1answer
49 views

About fractional iterations and improper integrals

Let $g(x,0) = x$ and $g(x,t+1) = g(x,t) - \dfrac{1}{g(x,t)}$ for every real $t$. From the fact \begin{align} ...
1
vote
2answers
90 views

This set of matrices whose eigenvalues have non-zero real part is dense

I'm trying to prove the set of the matrices whose eigenvalues have non-zero real part is an dense subset of $M^n$, the set of square matrices with order $n$ which is identify with $\mathbb R^{n^2}$. ...
4
votes
1answer
168 views

Calculus and Real Analysis: open source lecture notes ready to be edited

I would like to collect a big list of good open source lecture notes for a course in calculus; real analysis. Such notes should be in .tex format, that is, ...
2
votes
1answer
56 views

Boundary of Boundary of a set?

I have reading about the closure interior and boundary operators in a topological space. I have been thinking about the following: If $ b $ denotes boundary operator and $c$ , $i$ and $k$ denote ...
0
votes
2answers
30 views

Solutions of $z^6 + 1 = 0$

Solve: $$z^6 + 1 = 0$$ That lie in the top region of the plane. We know that: $$(z^2 + 1)(z^4 - z^2 + 1) = 0$$ $$z = -i, i$$ We need to solve: $$((z^2)^2 - (z)^2 + 1) = 0$$ $$z = \frac{1 \pm ...
0
votes
2answers
29 views

Boundary of two sets

If $b,c,k,i$ denotes the boundary,closure,complement and interior operator respectively and $A,B $ $\subseteq X$ where $X$ is a topological space.Suppose $bA=bB$ holds it does not imply $A=B $ ...
2
votes
2answers
31 views

If $AB=0$ prove that $rank(A)+rank(B)\leq n$

Let $A,B\in M_n(\mathbb{R})$ such that $AB=0$. Prove that $$rank (A)+rank(B)\leq n$$ From the given information, I only know that $rank(AB)=0$.
2
votes
3answers
38 views

Linear Algebra - elimination and linear systems

By given this matrix: \begin{pmatrix}1&1&1&0\\2&3&k&1\\3&k&5&1\end{pmatrix} I need to find, what are the values of k the system has infinity/single/no solution. So ...
6
votes
1answer
312 views

How much is cohomotopy dual to homotopy?

To what degree can we dualize theorems regarding homotopy into theorems about cohomotopy (or is there a good source that tries to do this)? For instance, is there some kind of Hurewicz theorem ...
3
votes
1answer
20 views

Prove dimension of sum of two subspaces

Let $U$ and $W$ be subspaces of $\mathbb{R^n}$ where $\dim(U)=n-1$, $\dim(W)=n-3$ and $n\geq 3$ Prove that $\dim(U\cap W)\geq n-3$ I used the property that both $U$ and $W$ are subspaces of ...
0
votes
2answers
38 views

Induction proof for natural numbers

I am unable to proceed with the below claim. $$2^{m} \times 2^{n} = 2^{m+n}$$ Could anyone let me know how to prove the above claim using induction proof? I was able to derive proof for odd natural ...
0
votes
1answer
14 views

Listing the elements of $U(\mathbb{Z}_{54})$.

The set of all integers modulo $q$ is denoted by $\mathbb{Z}_q$. When equipped with multiplication modulo $q$, has the structure of a commutative monoid, the identity element being equivalence class ...
0
votes
0answers
11 views

Proving integration techniques with intuition.

I've recently competed my A levels and now that I'm in the university I finally found the time to understand calculus on a intuitive level. So I've been reading up on books such as "Calculus with ...
-1
votes
0answers
15 views

homomorphism/ isomorphism

Let $f : G \to H$ be a homomorphism of groups. Let $K$ be a subgroup of $H$, and $A$ a subgroup of $G$. Show that (1) $f^{-1}(K)$ is a subgroup of $G$, (2) $f(A)$ is a subgroup of $H$, (3) if $G$ is ...
0
votes
0answers
8 views

Find the flux through a closed volume with the divergence theorem and using the definition

Given the vector field F(x,y,z)=(xy,xy,z) and $D= \{(x,y,z) \in R^3 : x^2 + y^2 + z^2 \le 4, x^2 + y^2 \le 1, z\ge 0 \}$ Find the flux through ...
2
votes
1answer
24 views

Integral domain without unity has prime characteristic?

By an integral domain, we mean here, a ring (not necessarily with unity) in which $ab=0$ implies $a=0$ or $b=0$. Question: If an integral domain without unity has positive characteristic, is it ...
-2
votes
1answer
8 views

Distribution of a Gaussian Random variable vector [on hold]

I read a slide on the internet which show that: If the random vector $w\sim N(0,I )$ then how can I prove: $x= A^{1/2}w+\bar{x}$ has the distribution $N(\bar{x},A)$ Here A is the covariance matrix ...
0
votes
1answer
26 views

Determination of area between two ellipses. [on hold]

Determination of area between two ellipses $x^2+2y^2=a^2$ and $2x^2+y^2=a^2$.
0
votes
1answer
20 views

Showing that the Brownian Bridge is Gaussian

Take $X_t = (1-t)B_{t/(1-t)}$ for $t\in[0, 1)$ where $B_t$ is a $1$-dimensional Brownian motion. I want to show that $X_t$ is Gaussian. I have actually never been able to find a precise definition ...
0
votes
0answers
52 views

Definite integral $\int_0^{2\pi}\frac{1}{\cos^2(x)}dx$

I encountered this very simple problem recently, but I got stuck on it because I think I am missing something. It is easy to see that indefinite integral $\int\frac{1}{\cos^2(x)}dx$ is $\tan(x)+C$. ...
1
vote
0answers
17 views

How to prove there exists a positive integer $k,d$ such $\sum_{i=1}^{2k}a_{id}=k-2014$

Let $\{a_{n}\}_{n=1}^{\infty}$ be a non-negative integer such that: for any postive integer $m,n$, we have: $$\sum_{i=1}^{2m}a_{in}\le m.$$ Show that there exist positive integers $k,d$ such that: ...

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