# All Questions

**4**

votes

**1**answer

145 views

### Equitably distributed curve on a sphere

Let $\gamma=\gamma(L)$ be a
simple (non-self-intersecting) closed curve of length $L$
on the unit-radius sphere $S$.
So if $L=2\pi$, $\gamma$ could be a great circle.
I am seeking the most equitably ...

**-1**

votes

**0**answers

61 views

### Are there any special properties of graph eigenvalues of perfect matchings? [on hold]

Say if the graph is just a perfect matching of its vertices OR if it is an union of a few perfect matchings OR its an union of a few perfect matchings and a part of another?
Anything if one further ...

**4**

votes

**0**answers

87 views

### Is this $S$-birational map an open immersion on its domain of definition?

My question is about a claim on the bottom of p. 121 of the book "Neron models" by Bosch, Lutkebohmert, and Raynaud, so I will freely use the general terminology recalled in this book, but will ...

**1**

vote

**0**answers

48 views

### When the completed filtration of a process increases slowly

If $\mathcal{F}_t$ is the filtration of the evaluation process on $C_T$ (continuous function on $[0,T]$). Can we find some law of continuous process $\mathbb{P}$ so that for $t\leq T$
...

**-4**

votes

**0**answers

29 views

### Irreflexivity of relations on sets [on hold]

How can I know if the relations:
xy >= 1
and
x=y+1 or x=y-1
Are irreflexive on Z(all integers)?
Thank you!

**4**

votes

**2**answers

237 views

### Isomorphism problem for two radical extensions

Let $n\geq 2$ and let $a,b\in{\mathbb Q}$. Suppose that both the
polynomials $A=X^n-a$ and $B=X^n-b$ are irreducible. We want to know whether
( * ) there is a root $\alpha$ of $A$ and a root $\beta$ ...

**13**

votes

**0**answers

138 views

### Large almost equilateral sets in finite-dimensional Banach spaces

Question: Does there exist a function $C:~(0,1)\to
\mathbb{R}$ such that for each $\varepsilon\in(0,1)$ every Banach space
$X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point
set ...

**3**

votes

**1**answer

220 views

### Does this function have any exponential growth?

Has anyone seen any function of the following type?
$$
g(x):=\sum_{n=0}^\infty \frac{x^n}{n!}\exp\left(-\frac{a^n}{x}\right),\quad a>1,x\ge 0.
$$
The question is whether for some constant ...

**-2**

votes

**0**answers

77 views

### What is the characterization of a graph Laplacian? [on hold]

Given a matrix, what properties must it have so that its ensured that there exists a graph whose Laplacian it would be? (...may be you can consider weighted and unweighted cases separately...)
And ...

**0**

votes

**0**answers

16 views

### the ratio between product of two trace functions maximization

Consider the following Optimization
[\begin{array}{l}
\mathop {\max }\limits_{\bf{X}} \,\frac{{trace\left( {{\bf{XA}}} \right)trace\left( {{\bf{XB}}} \right)}}{{trace\left( {{\bf{XC}}} ...

**0**

votes

**0**answers

72 views

### Do they have the same limit?

Suppose $a(\cdot)\in L^p$ and is symmetric and $b(\cdot)\in L^q$, where $1/p+2/q=2$, $p,q\ge 1$. Consider the quantity $Q_T=$
$$
...

**0**

votes

**0**answers

34 views

### Random Matrix Theory: question on a quantity on page 87 in book of Bai and Silverstein 2010

Anyone else is reading carefully the book of Bai and Silverstein 2010, titled "Spectral Analysis of Large Dimensional Random Matrices"?
On page 87 of this book, when they state the final step in the ...

**4**

votes

**2**answers

437 views

### Mayer-Vietoris sequence for topological K-theory

I'm reading the paper Loop groups and twisted K-theory I by Freed, Hopkins, and Teleman. They give some examples of computing (twisted) K groups using the Mayer-Vietoris sequence.
I'm a bit ...

**0**

votes

**1**answer

62 views

### A decomposition of incompressible vector fields

In Andrew, Majda- Vorticity and incompressible flow page 93, there is a theorem which is not proved:
Take a smooth incompressible (free divergence) vector field $v$ in $\mathbb{R}^2$.
Call $w$ its ...

**2**

votes

**1**answer

131 views

### A question about generalized Dyck words

I am interested in counting the following. How many words using $n-1$ copies of $u$ and ${n \choose 2} - n+1$ copies of $d$ begin with $uu$ and, in general, the $k^{th}$ $u$ is among the first ${k ...

**2**

votes

**2**answers

143 views

### Integral domains with totally ordered spectra

In my research I ended up trying to prove some properties of integral domains such that their spectrum is a totally ordered poset. Are there some nice (ubiqitous/natural) examples of such domains, ...

**0**

votes

**0**answers

119 views

### Self homotopy equivalence

Given $X$, a simply connected CW-complex of finite type, ${\rm aut}(X)$ denotes the set of its self homotopy equivalences, that are maps $f: X\rightarrow X$ which admits a homotopy inverse (i.e., ...

**-1**

votes

**0**answers

89 views

### Questions about the Moduli Space of Vector Bundles of rank 2 with trivial determinant [migrated]

Let $M_0$ be the moduli space of rank 2 semi-stable vector bundles over X with trivial determinant which is a singular projective variety of dimension $3g-3$. $M_0$ is constructed as the $GIT$ ...

**0**

votes

**0**answers

127 views

### Homotopical nilpotency [on hold]

For a connected grouplike space $G$, the homotopical nilpotency of $G$ is the invariant defined by Berstein and Ganea as follows: ${\rm Hnil} (G)$ is then the least integer $n$ such that the ...

**7**

votes

**1**answer

194 views

### Busy beaver function vs low Turing degrees

Let $BB(n)$ denote busy beaver function. It's well known that $BB(n)$ dominates all computable functions (I'm quite certain it includes partial computable functions too). However, I was wondering if ...

**3**

votes

**1**answer

100 views

### A (“Rice-like”) conjecture about the decidability of primitive recursive (PR) problems

Question: is the conjecture below true?
Consider decision problems in which the instance is (the PR index, definition,
or LOOP program of) a primitive recursive function.
Denote the PR function (with ...

**0**

votes

**0**answers

25 views

### proving that a complicated function is concave or strongly uni-modal

I am trying to prove the concave property for a complicated function during my research project (imperfect maintenance modelling for starter) which has the following form:
$\eta(t)= \alpha \beta ...

**2**

votes

**0**answers

25 views

### Example of joint cyclic and separating vector

Let $\mathcal{H}$ be a separate Hilbert space and $\mathcal{B}(\mathcal{H}) \subset
\mathcal{B}(\mathcal{H}) \otimes M_2(C)$ be a W$^*$-inclusion pairs. It is known that
this pair share a joint cyclic ...

**-1**

votes

**0**answers

42 views

### What is the difference between disturbance and noise for dynamic systems? [on hold]

In most references from dynamic system theory, the following linear continuous dynamic system is considered.
$$\frac{\text{d}x(t)}{\text{d}t}=Ax(t)+Bu(t)+Dd_{1}(t)\quad (1)$$
$$y(t)=Cx(t)+Ed_{2}(t) ...

**2**

votes

**1**answer

125 views

### Axiomatization of Degree Theory

I am reading Ruiz's Mapping Degree Theory, and find an axiomatization of degree theory of $\mathbb R^n$ in P38. It says that there exists a unique map $d(f,D,y)\in\mathbb Z$ satisfies
Normality
...

**-1**

votes

**0**answers

71 views

### Estimating the moments of a random variable

Suppose i wanted to estimate the expectation and variance of a random variable $X$. More over suppose i could write a variable $X$ as a sum of indicator random variables $X=\sum_{i=1}^{k} X_{i}$. Are ...

**4**

votes

**1**answer

100 views

### Averaging maps of Riemannian manifolds

Let $M$ be a compact Riemannian manifold. We know how to average functions $f\colon M\to {\mathbb R}$; the integral $\frac{\int_M f}{\int_M 1}$ returns a value in ${\mathbb R}$. If intead $f\colon ...

**3**

votes

**1**answer

98 views

### Resolvent of a triangular matrix

Suppose $A$ is a triangular matrix. What is the most efficient known algorithm to compute the polynomial (in $x$) matrix $(xI-A)^{-1}$?
Of course, $(xI-A)^{-1}= N(x)/p_A(x)$, where $p_A$ is the ...

**3**

votes

**0**answers

40 views

### Behaviour of Markov type under uniform homeomorphism of spheres

A metric space $(X,d_X)$ has Markov type $p$ (with $p \in [1,2]$), if, for every stationary Markov chain $\{Z_n\}_{n=0}^\infty$ on $Y$ (a finite space) and every mapping $f:Y \to X$, one has
$$
...

**2**

votes

**1**answer

51 views

### Estimating mean and variance of a distribution based on error-prone estimates of its cdf

Suppose I have some random variable $X$ taking values in $[a, b]$ with unknown distribution (I am happy to assume the distribution is smooth, though it would be nice to not have to).
I have a ...

**8**

votes

**1**answer

271 views

### Could there be an exact formula for the Ramsey numbers?

Let $R(k)$ denote the diagonal Ramsey number, i.e. the minimal $n$ such that every red-blue colouring of the edges of $K_n$ produces at least one monochromatic $K_k$.
Is it possible that there ...

**0**

votes

**0**answers

18 views

### Question on Stationary & Cointegration Test (Augmented Dickey Fuller & Engle Granger test) [on hold]

I'm performing the stationary and cointegration test on stock prices.
What I'm confused is
1) the difference between ADF stationary test and ADF cointegration test.
2) Also, in ADF stationary test, ...

**-2**

votes

**0**answers

27 views

### Mean time for the renewal process [on hold]

The system is as below.
Energy keeps coming at a node with a constant rate $\rho$.
Node has files of size exponential($\lambda$) to be transmitted. At time zero, say the energy at the node be zero. ...

**-1**

votes

**0**answers

79 views

### Degrees of multilinear polynomials satisfying some constraints [on hold]

Let $t<\sqrt{n}$.
$\Bbb Z^t[x_1,\dots,x_n]=\{f\in\Bbb Z[x_1,\dots,x_n]: deg(f)\leq t$ and $f$ is multilinear$\}$.
Fix an ordering of $S=\{0,1\}^n.$
If $f\in\Bbb Z[x_1,\dots,x_n]$, let $f(S)$ be ...

**0**

votes

**0**answers

64 views

### Derivability of a function defined on the tangent bundle. Foundations of Finsler metrics

My question is linked to the foundations of Finsler metrics (with weak derivability assumptions).
Let $M$ be a manifold of dimension $n$, and $F$ is a function from the tangent bundle $TM$ to ...

**1**

vote

**1**answer

94 views

### ideal of maximal minors is cohen-macaulay?

Let $k$ be an algebraically closed field.
Let $A$ be an $m \times n$ matrix with linear forms $a_{ij} \in k[x_1, \ldots, x_p]_1$ as entries. Let $I$ be the ideal generated by the maximal minors of ...

**3**

votes

**1**answer

121 views

### On the size of centralizers in a non-abelian finite simple group

It is known that for a finite non-abelian simple group $G$ we have $|G|<|C_G(x)|^3$ for some involution $x$. Is there a better bound for the order of centralizer of a nontrivial element of $G$ (not ...

**2**

votes

**0**answers

101 views

### Example of symplectic 4-manifolds with no Lefschetz fibration structure?

I just read about Donaldson's result on existence of Lefschetz pencil structure on symplectic manifolds (Donaldson 1999). However, one has to blow up the base locus to get a Lefschetz fibration ...

**0**

votes

**0**answers

60 views

### Asymptotic analysis of a sum of complex summands using integral

I'm trying to find the exact asymptotics of a sum:
$$A = \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^{i} y^{2n-i} $$
as $n\rightarrow\infty$. Here $x,y$ are complex numbers, $|x|\leq1, ...

**2**

votes

**0**answers

85 views

### “simulteneous eigenvectors” under the full set of weighted Laplacians on a $g$-fold product of the Poincare half plane

This question is closely related to the following MO question Characterizing the real analytic Eisenstein series
Let $\mathfrak{h}=\{z=x+iy\in\mathbf{C}\}$ be the Poincare upper half plane endowed ...

**1**

vote

**1**answer

103 views

### Absolutely continuous functions

it is well known that if a function $f:[0,T]\to\mathbb{R}$ satisfies the inequality
$$\vert f(t)-f(s)\vert\leq \int_s^t{m(r) dr},$$
for $s<t$ and some $m\in L^1([0,T])$ then $f$ is absolutely ...

**1**

vote

**1**answer

121 views

### abelian p- subgroups of E_6(q)

Is there any result about maximal abelian p-subgroups of the exceptional group E_6(q), where q=p^a is prime power?

**8**

votes

**1**answer

172 views

### Non-abelian freeness of SU_2

The distribution of the trace of a random element of $SU_2$ is the Sato-Tate distribution.
The analogue of the Gaussian distribution in free probability theory is the Wigner semicircle distribution.
...

**3**

votes

**1**answer

85 views

### Dehn twist about an arbitrary curve

I need to know if there is an algortihm to write down a Dehn twist about an arbitrary curve on an orientable surface $S$ , as product of a set of generators of $MCG(S)$.
Since we have the conjugacy ...

**-4**

votes

**0**answers

42 views

### finding points on elliptic curve over finite field [on hold]

Find the points on the elliptic curve y^2 = x^3 + 2x + 2 in F17 (field of prime 17).
Do I have to guess a first point and then use an algorithm to spit out all other points?

**29**

votes

**3**answers

2k views

### A game of stones

How I arrived at this question is a rather long story having to do with the honors calculus class I am teaching. At this point it's sheer curiosity on my part. Here is the game.
...

**1**

vote

**0**answers

100 views

### Proper monomorphisms in complex analytic spaces

In the algebraic geometry of schemes, we know that monomorphisms which are universally closed (= every pullback is a closed map) and of finite type are closed immersions. See Gortz, Wedhorn "Algebraic ...

**3**

votes

**1**answer

175 views

### Example of proof using the generic matrix

There is a really nice proof of the Cayley-Hamilton Theorem using the generic matrix. I expose it briefly.
One defines the generic matrix $G:=(X_{ij})_{ij} ...

**0**

votes

**0**answers

41 views

### Bounding a ratio by its complement [on hold]

Given some integers $n > \beta + \alpha, \beta > \alpha$ and $\alpha > 0$, is there a real number $\delta$ for which $\frac{n-\alpha}{n-\beta} \geq (\frac{\beta}{\alpha})^{\delta}$, where ...

**1**

vote

**0**answers

91 views

### How to see that this pairing of line bundles is multiplicative?

Given a projective flat morphism $p: X \rightarrow Y$ of integral noetherian schemes of relative dimension one.
For a coherent sheaf $F$ on $Y$ we can define a line bundle $det(F)$ on $Y$ and for a ...