1
vote
0answers
11 views

definable curves in definable sets

Suppose I have an UNBOUNDED subset X of $R^n$ ($R$=real numbers), definable in the o-minimal structure $R_{an, exp}$. It is possible to find an UNBOUNDED, ANALYTIC and DEFINABLE curve (i.e. ...
0
votes
0answers
10 views

Ergodic, non-atomic measure on the circle which are $\times 2$ and $\times \frac12$ invariant

There any many ergodic, $T$-invariant, non-atomic measures on the space $X = [0,1)$, where $Tx = 2x \pmod 1$ is the doubling map. My question is: are any such measures also $T^{-1}$-invariant? BYO ...
0
votes
0answers
12 views

$\int_{\mathbb{R}\setminus\left[-1,1\right]}\hat{f}^{2}\left(\xi\right)\left(\xi^{2}-1\right)d\xi $

Assume that $\hat{f}\in L^2(\mathbb{R})$, how can we compute this integral $$\int_{\mathbb{R}\setminus\left[-1,1\right]}\hat{f}^{2}\left(\xi\right)\left(\xi^{2}-1\right)d\xi $$ where $\hat{f}$ is ...
0
votes
0answers
23 views

Generating-bijective groups

We may say that two finitely generated groups $G$ and $H$ are generating-bijective when there exist homomorphisms $\phi:G\rightarrow H$ and $\psi:H\rightarrow G$ such that, for each ordered generating ...
0
votes
1answer
60 views

How to write $\mathbb{C}[G/U_-]=\oplus_{\lambda} V_{\lambda}$ explicitly?

Let $G=GL_n$ and $U_-$ the set of all lower unipotent triangular matrices. Then by Gauss Decomposition, we have $G = U_-B$, where $B$ is the set of all upper triangular matrices. The group $U_-$ acts ...
0
votes
0answers
31 views

Free cocompact action of discrete group gives a covering map [migrated]

I'd like to find a short proof of the following seemingly basic fact encountered on the second page of Atiyah's paper "Elliptic operators, discrete groups, and von Neumann algebras." ...
5
votes
0answers
51 views

Chern-Simons form and Rarita-Schwinger operator

The Rarita-Schwinger (RS) operator naturally generalizes the Dirac operator and in Physics it describes particles with spin-3/2. I was wondering if there exists any reference concerning the ...
0
votes
0answers
12 views

Example of a Schur-nontrivial group with no abelian subgroup of the form $H\times H$?

A group $G$ is Schur-nontrivial if the Schur multipler $H^2(G,U(1))$ is not the trivial group. I am trying to find an example of a Schur-nontrivial group which does not contain a subgroup of the form ...
-1
votes
0answers
10 views

Sequences of random variables converging in probability to the same limit a.s [migrated]

Let $(X_n)_{n \geq 1}$ and $(Y_n)_{n \geq 1}$ be two sequences of random variables s.t. $X_n$ converges to X and $Y_n$ to $Y$ both in probability. Furthemore, $X$ = $Y$ a.s. How can I prove that, for ...
0
votes
0answers
26 views

Differential categories vs McBride's notion of derivative

Has anyone done an analysis to see if Blute, Cockett, and Seely's differential categories suffice to provide a notion of 1-hole context in the symmetric monoidal setting?
0
votes
0answers
27 views

Expected value and variance of a stochastic process

I would like to ask if there is a way to find the expected value and the variance of the following process $$ dv_t=(a-be^{\alpha v_t})dt+\sigma dW_t, \quad v_t=v_0 $$ where $a\in (-\infty,+\infty), ...
0
votes
0answers
21 views

Morse function on slicing disk complement determines ribbon?

It is well-known that given a ribbon knot and the corresponding slicing disk in the 4-ball, the distance function (maybe squared) to the origin defines a Morse function in the complement of the ...
-4
votes
0answers
16 views

Defining a function on a subset but it's domain is the whole set [on hold]

Under what conditions would it be acceptable to define a function only on a subset of it's domain? Let $\{\mathbf{u}_1,\mathbf{u}_2\}$ be a basis for $\mathbb{R}^2=\mathbb{R}_1\times\mathbb{R}_2$ (in ...
-5
votes
0answers
44 views

I've read on the internet that 43 is congruent to 8 modulo 5+3w, can you explain? [on hold]

Initially, I wanted to determine whether 19 is a cube modulo 43. Reading online, I've come across this computation that includes a step that I don't understand. The step in question states that (43 ...
1
vote
0answers
115 views

What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$? [on hold]

Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$. If $L$ is a number field ...
0
votes
0answers
24 views

Eigenvalues of signed networks [on hold]

A signed graph is a graph where edges can be positive or negative. A cycle in a signed network is called balanced cycle when it has an even number of negative edges otherwise it is called unbalanced ...
-3
votes
0answers
26 views

Discrete circular distribution [on hold]

Having a distribution of a discrete number N of angular values [0:360], not necessarily all adjacent, ordered in time. How to determine the maximum and the minimum of this distribution? Please look ...
2
votes
0answers
37 views

Non-Haken hyperbolic 3-manifolds without nonorientable surfaces

It is well known that there exist infinitely many (non homeomorphic) non-Haken closed hyperbolic 3-manifolds. These can be obtained for example doing Dehn surgery on the figure eight knot complement. ...
-1
votes
0answers
15 views

Necessity of expansiveness for existence absolutely continuous invariant measures for piecewise smooth maps of an interval

A map $\tau:[0,1]\to[0,1]$ is piecewise smooth (or $C^r$) if there is a partition of $[0,1]$ into intervals, $[0,1]=\cup I_n$, (which can be either finite or countable) such that the restriction of ...
4
votes
0answers
59 views

Are the canonical maps from $\Omega^1_k(C^\infty(M))$ into $\Omega^1(M)$ and into $\Omega^1_k(C^\infty(M))^{**}$ compatible?

Let $M$ be a smooth manifold and let $A=C^\infty(M)$. In this question, it is observed that the map $\Omega^1_k(A)\to \Omega^1(M)$ from the Kähler differentials of $A$ to the 1-forms of $M$ is not an ...
0
votes
0answers
15 views

Inverse of a correlation matrix that has arcsin elements

Let $C$ be a correlation matrix whose off-diagonal elements are defined as follows: $C(i,j)=arcsin(\sqrt{k^{|i-j|}}$ where $k<1$. Can the inverse of the correlation matrix be derived analytically? ...
0
votes
1answer
61 views

Is there an entire solution for the Van der pol equation?

Is there a non constant entire function $\gamma(t)=(x(t),y(t)): \mathbb{C} \to \mathbb{C}^{2}$ which satisfy the following Vander pol dififferential equation? $$\begin{cases}\dot{x}=y-x^{3}\\\dot ...
21
votes
1answer
413 views

Complex manifold with subvarieties but no submanifolds

I previously asked this question on MSE and offered a bounty but received no responses. There are examples of compact complex manifolds with no positive-dimensional compact complex submanifolds. ...
2
votes
0answers
138 views

Is this Grothendieck trace map an isomorphism?

Let $A$ be a commutative ring and let $S := \operatorname{Spec}(A)$. Let $$ g : Y \to X $$ be a proper, birational morphism of separated schemes of finite type over $S$, where $X$ is affine and ...
0
votes
0answers
28 views

Fourier tranform of the Euclidean norm [migrated]

where can I find the Fourier transform of the power of the Euclidean norm?, that is: $$\mathcal{F}[\|x\|^{p}](\omega) = \int_{\mathbb{R}^{d}}\exp(-2\pi i \langle\omega, x\rangle) \|x\|^{p} dx$$ ...
3
votes
1answer
75 views

Minimal zero-dimensional Hausdorff spaces

A topological space $(X,\tau)$ is said to be zero-dimensional Hausdorff (zdH) if for $x\neq y\in X$ there is $C\subseteq X$ clopen (closed and open) such that $x\in C$, but $y\notin C$. We say a zdH ...
-3
votes
0answers
133 views

Can the work of Hardy & Ramanujan about partitions shed light on Hardy-Littlewood's k-tuple conjecture? [on hold]

If I'm not mistaken, Hardy and Ramanujan produced an asymptotic formula for the number of partitions of an integer that was later shown to be an exact formula by Selberg. But Hardy also formulated ...
6
votes
0answers
102 views

When does a CAT(0) group contain a rank one isometry

Let $G$ be a CAT(0) group which acts on the CAT(0) space $X$ properly and cocompactly via isometry. Let $g \in G$ be a hyperbolic isometry of $X$. Then $g$ is called $\textbf{rank one}$ if no axis of ...
5
votes
1answer
213 views

Rotation invariance of an integral

Consider the integral depending on 2 parameters $$f(\tau,x):=\int_{-\infty}^{+\infty}\frac{dp}{\sqrt{p^2+1}}e^{-\sqrt{p^2+1}\tau+ipx},$$ where $\tau >0,x\in \mathbb{R}$. This integral absolutely ...
1
vote
0answers
88 views

When is the sum of complemented subspaces complemented?

Let $X$ be a Banach space. Question. Suppose $X_1,...,X_n$ are complemented subspaces of $X$. When is the sum $X_1+...+X_n$ complemented? Further, suppose we know some projections $P_1,...,P_n$ onto ...
8
votes
0answers
127 views

Rationality of a certain real algebraic variety

Let $A_n$ denote the vector space of $n\times n$ antisymmetric matrices over ${\mathbb{Q}}$, where $n$ is even. Let $A_n^*\subset A_n$ denote the affine ${\mathbb{Q}}$-subvariety of invertible ...
0
votes
0answers
49 views

Milnor numbers and mixed multiplicities

section 6 of the link Teissier showed that Milnor numbers of a hypersurface $(X,0)$ with isolated singulraity at 0 is same as mixed multiplicities of the Hilbert polynomial of the filtration ...
1
vote
1answer
72 views

Does similarity imply symmetric similarity?

Let $F$ be an infinite field. If two $n×n$ matrices $A,B$ are similar in $M_n(F)$, then are they also similar via a symmetric matrix (that is, is there a symmetric matrix $Q∈GL_n(F)$ such that ...
7
votes
2answers
257 views

Link between the hairy ball theorem and the fundamental theorem of algebra

I read in the book "Concepts of modern mathematics" by Ian Stewart that it was possible to proof the fundamental theorem of algebra using the hairy ball theorem (complete reference to the page is in ...
0
votes
0answers
78 views

$z^n-t^m=x^3+y^3$ and Vojta's more general abc conjecture

In A more general abc conjecture, p. 7 Paul Vojta conjectures: If $x_0,\ldots x_{n-1}$ are nonzero coprime integers satisfing $x_0 + \cdots x_{n-1}=0$ $$ \max\{|x_0|,\ldots |x_{n-1}|\} \le C ...
6
votes
3answers
157 views

Logarithms of matrices in the disk-algebra

It is easy to see that within the disk algebra $A(D)$ $$\Delta(z):= \begin{pmatrix} 1&0\\z&1 \end{pmatrix}\; \begin{pmatrix} 1&1\\0&1 \end{pmatrix}= \begin{pmatrix} ...
2
votes
0answers
34 views

What is the densest bipartite graph with unique Hamiltonian cycle?

In a prior post regarding perfect matching, it was stated that the densest graph with a perfect matching cannot have more than $n^2$ edges, if graph has $2n$ vertices. Analogously, what is the ...
-3
votes
0answers
17 views

Integral of error function (erf) and gaussian [on hold]

answer given in mathoverflow for integral of erf and gaussian, what will be the right hand side limits if the lefthand side limits are not from -inf to +inf but from a to b.
0
votes
0answers
23 views

Quadratic characteristic and constancy

Consider a change of measure on $\mathcal{F}_{t}$ defined by the restriction of two probability measures of the form \begin{align} \frac{dQ_{t}(\theta)}{dP_{t}}=\exp^{ \theta A_{t}-\kappa(\theta) ...
-4
votes
0answers
36 views

GEOMETRY PROVE CHORDS CONGRUENT [on hold]

Attached is the picture of the proof. My teacher gave us some hints which were: draw radii, use triangles, and use addition property.
7
votes
0answers
135 views

Holomorphic contractibility of GL(H)?

Kuiper's theorem is well-known to give the triviality of the homotopy groups of ${\rm GL}(\mathcal{H})$ for $\mathcal{H}$ a (separable) infinite-dimensional complex Hilbert space. Work of Palais later ...
0
votes
0answers
41 views

Upper bound on the norm of the inverse of matrices with zero limit

Posted here too, with no answer yet: http://math.stackexchange.com/questions/1766281/upper-bound-on-the-norm-of-the-inverse-of-matrices-with-zero-limit Let $\{L(\sigma)\}_{\sigma}$ be a family of ...
3
votes
1answer
111 views

Transfer map in group cohomology

Let $H$ be a subgroup of a finite group $G$, and let $M$ be a $G$-module. Are there any simple conditions on $H,G$ and $M$ which would ensure that the transfer map $H^p(H,M)\to H^p(G,M)$ is the zero ...
4
votes
1answer
103 views

Upper Bound for the Difference of Even Probability and Odd Probability in Hypergeometric Distribution

Let $X$ be a random variable following the hypergeometric distribution with parameters $N,K,n$, where \begin{equation} Pr(X=k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}. \end{equation} To ...
3
votes
3answers
219 views

What is the group of automorphisms of $l^{\infty}$?

What is the group of automorphisms of $l^{\infty}$? I think it would be the permutations of the integers. Is this right?
0
votes
0answers
26 views

Lowest upper bound of resultant [on hold]

Given two polynomials $f(x)=X^N+1$ and $g(x)= \sum_{i=0}^{N-1} s_i x^i$, where $s_i$'s are $\eta$-bit integers, let $res=resultant(f(x),g(x))$. What is the upper bound of $\log_2 res$?. Below my ...
3
votes
0answers
81 views

Auslander-Reiten-Quivers of representation-finite algebras having different 3-dimensional forms

I am looking for references, where I can find (pictures of) connected Auslander-Reiten-Quivers of representation-finite $k$-algebras ($k$ is a (preferably, but not necessarily finite) field) with one ...
2
votes
2answers
138 views

Henkin semantics for Second-order Logic

I know that the natural numbers can be categorically characterized in second-order logic with the standard semantics. However, I could not find an example of a non-standard Henkin structure (one that ...
2
votes
0answers
48 views

Functions of form $f(z)/f(z^*)$

I am doing my research in mathematical physics, and in the process I am getting functions of complex variable $z$ of form $F(z) = \frac{f(z)}{f(z^*)}$ In my case $f$ doesn't have any interesting ...
0
votes
0answers
29 views

How to numerically evaluate a integral whose limits are functions of x (using Gauss quadrature rule)? [on hold]

I am trying to numerically evaluate an integral $\int_q^1 \ln (\sum_i \alpha_ix_i) dq$, in which $\ln (\sum_i \alpha_i x_i)$ is related to $q$ via the following: $z_i=(1-q)\frac{\alpha_ix_i}{\ln ...

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