# All Questions

**1**

vote

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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 ...