1
vote
0answers
6 views

Average minimum number of random k-sparse vectors in GF(2) to span the whole space?

What is the average minimum required number of independent $k$-sparse (having at most $k$ non-zero elements) random vectors belonging to $\mathbb{F}_2^n$ to span the whole space of $\mathbb{F}_2^n$? ...
0
votes
0answers
6 views

Decomposition of flat homogeneous Kahler manifolds

In a paper I am reading, it is claimed that a flat homogeneous Kahler manifold is a Kahler product $\mathbb C ^k \times T_1 \times \cdots \times T_s $ where $\mathbb C ^k $ is considered with its ...
2
votes
0answers
17 views

Does $\omega_C\simeq N_{C/S}$ always happen on Enriques surfaces?

Let $S$ be an Enriques surface and $C\subset S$ a smooth irreducible curve of genus $g$. Consider the condition $$\omega_C\simeq N_{C/S}$$ For example, when $g=1$ then $\omega_C=\mathcal{O}_C$ and ...
-5
votes
0answers
20 views

Translation services for math texts [on hold]

I'm looking for translation services that has expertise in translating math texts for popular world languages. The problem with common translation services is that such translations are not of very ...
1
vote
0answers
12 views

Why do we matter about simplicity of the spectrum in Oseledets' theorem?

Oseledets' theorem is a fundamental result in Ergodic theory (see for example here, or Chapter 4 of Lectures on Lyapunov Exponents by Marcelo Viana). The simplicity of the spectrum has been studied ...
3
votes
1answer
28 views

Style guide for referring to past work

Has anyone written or expressed a coherent position on how to refer to mathematical results (theorems, proofs) by past authors? Even if there are no hard and fast rules, I find it helpful to have a ...
0
votes
1answer
29 views

Finding functional equations that a given function satisfies

Suppose we're given a function, for example a function $f:\mathbb{C}\rightarrow \mathbb{C}$ such that $f(x)=ax+b $ with $a,b \in \mathbb{C} $. I would like to know which functional equations are ...
0
votes
0answers
7 views

How can I approximate this in terms of Gauss-Hermite abscissa and weights?

I am having the following expression. This is the PDF of Nakagami-Lognormal Distribution. I want to express in terms of Gauss-Hermite abscissas and weights. How can I do it? ...
0
votes
0answers
6 views

Is the equicontinuous weak-star topology locally convex on the dual of an LF-space?

The Banach-Dieudonné theorem states that if $X$ is a metrizable locally convex Hausdorff space then the equicontinuous weak-* topology on $X'$ coincides with the topology of precompact convergence and ...
3
votes
0answers
34 views

Smoothing a continuous section in 1-jet bundle

Here is a question I encountered when reading the book "Convex Integration Theory by D.Spring". My question lies in the second paragraph to the proof of theorem 4.2($C^{0}$-dense $h$-principle). I ...
-6
votes
0answers
39 views

What are singular value of $A$? [on hold]

Let $ A = \left( {\begin{array}{*{20}{c}} {x + (\frac{3}{4} + y)i}&1&1\\ 0&{(x - \frac{5}{4}) + iy}&1\\ 0&0&{(x + \frac{3}{4}) + iy} \end{array}} \right)$, and $x,y\in ...
3
votes
0answers
44 views

Historical perspectives on CAT(0) spaces

Does there exist a survey on the early developments of CAT(k) spaces, with the first motivations and the first problems considered? I looked at Bridson and Haefliger's book On metric spaces of ...
1
vote
0answers
55 views

Geometry of Rogers-Ramanujan continued fraction

I'd like to understand the underlying geometry of the Rogers-Ramanujan continued fraction from the point of view of integrable systems (ideally Toda type theories). The generating function $R(z) = ...
2
votes
1answer
179 views

Langlands program vs Shimura-Tanayama-Weil conjecture

Edward Frenkl said that "we can see Langlands program as a generalization of Shimura-Tanayama-Weil conjecture in the case of elliptic curves " I hope I'm not distorting his phrase, can someone ...
2
votes
1answer
25 views

On the eigenvalues' distribution of random unitary

Fix an integer $d$, let $\mathbb{U}_d$ be the $d\times d$ unitary group. For any $U\in \mathbb{U}_d$, define $\Omega(U)$ be the length of the smallest arc containing all the eigenvalues of $U$ on the ...
4
votes
0answers
38 views

Poincare-Hopf theorem for polytopes?

Is there an analogue of Poincare-Hopf theorem for polytopes? I want to apply it in the following situation. I have a polytope in $R^n$ and a smooth explicitly given vector field in $R^n$. I want to ...
1
vote
0answers
14 views

Maximum principle of the gradient of harmonic extension under weak regularity assumption

I have a question which is very likely to be trivial, but I'm stuck on it! Suppose $f \in W^{1, 2}(B_2(0))$ and $\|{\nabla f\|}_{L^{\infty}(B_2(0)\setminus B_1(0))} < \infty$. Consider then the ...
0
votes
0answers
18 views

Minimal permuted inner products

Fix $n\in\Bbb N$. Denote $P$ to be $2n+c$ smallest consecutive primes all bigger than $n^{\alpha\log^{\beta}(n)}$ for some constant $c>0$ and $\alpha,\beta\geq0$. Pick $2n$ random (might not be ...
1
vote
1answer
37 views

embeds in $ L(c_{0},\ell_{1}) $

Let $ c_{0}:=\lbrace x:\mathbb{N}\rightarrow \mathbb{R} :\lim_{j\rightarrow\infty} x_{j}=0 \rbrace$ denote the usual Banach sequence spaces. Given Banach spaces $X,Y$ let $L(X,Y)$ denote the Banach ...
2
votes
1answer
69 views

Can there be a numerical system in which logarithms can be expressed in terms of exponentials in closed form?

The invention of complex numbers allowed to express trigonometric functions through hyperbolic ones in closed form. Is there possible an extension of real/complex numbers in which logarithms and ...
0
votes
0answers
36 views

Banach-Mazur distance from finite-dimensional subspaces of $\ell_p$ to the Hilbert space

I am reading a paper http://www.math.tamu.edu/~johnson/TF3.4.pdf by Bill Johnson and Andrzej Szankowski and having trouble grasping why $d_n(Z_m) \leq d_n(\ell_{p_{m+1}} ) = n^{|p_{m+1}-2|}$ in the ...
2
votes
1answer
53 views

Definable curves in definable sets

Suppose that I have an unbounded subset $X \subset \mathbb{R}^n$, definable in the $o$-minimal structure $\mathbb{R}_{an, exp}$. Is it possible to find an unbounded, analytic and definable curve (i.e. ...
3
votes
1answer
40 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
43 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
2answers
110 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." ...
6
votes
0answers
75 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
1answer
28 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
35 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
37 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), ...
1
vote
1answer
44 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 ...
-6
votes
0answers
54 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
122 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
33 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
1answer
59 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. ...
0
votes
0answers
19 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
64 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
16 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? ...
1
vote
1answer
75 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 ...
23
votes
1answer
550 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
149 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
85 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
141 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
115 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 ...
6
votes
1answer
218 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
94 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
141 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 ...

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