# All Questions

**2**

votes

**0**answers

53 views

### Normal-like coordinates for weakly differentiable metrics

Let $(M,g)$ be a Riemannian $W^{2,p}$ metric, with $p>n/2$. Thus $g$ is at least continuous. At any point $P\in M$, do there exist local coordinates $x^i$ such that $g$ can be decomposed as $g_{ij} ...

**4**

votes

**0**answers

107 views

### What is a stable $(n,1)$-category?

My question in its simplest form is whether we have any understanding of stable $(n,1)$-categories, by analogy with stable $(\infty,1)$-categories and abelian categories (the latter being like "stable ...

**1**

vote

**1**answer

67 views

### Entropy on a draw from a random distribution.

Suppose I am attempting to calculate the entropy of a continuous, normally distributed random variable $X$, from the distribution $\mathcal{N}(\mu, \sigma)$. This is easy to to do - I just calculate
...

**3**

votes

**1**answer

219 views

### Theorem 7b of Serre's “Propriétés galoisiennes des points d'ordre fini des courbes elliptiques”

Could someone please point me towards a proof of the statement in the second paragraph, in the proof of Theorem 7b of Serre's Propriétés galoisiennes...? The statement is as follows:
Let $F$ and $F'$ ...

**0**

votes

**0**answers

54 views

### A normal form theorem for presentations of finite $p$-groups of nilpotency class $2$?

When constructing examples of nonabelian finite $p$-groups with abelian automorphism group (and certain other desired properties), the authors of papers like http://arxiv.org/pdf/1304.1974v1.pdf leave ...

**0**

votes

**0**answers

41 views

### characterizing a sigma field generated by a field [on hold]

Given that $\mathcal{F}$ is a field (but not a $\sigma$-field), is there any characterization we can give of $\sigma(\mathcal{F})$, where $\sigma(\cdot)$ is the generating $\sigma$-field of a ...

**7**

votes

**3**answers

498 views

### Is it consistent with ZFC (or ZF) that every definable family of sets has at least one definable member?

I consider definability to mean one of either cases:
Definability without parameters (in the language of set theory), or
Definability from ordinals and a real (in the same language).
So my ...

**0**

votes

**1**answer

107 views

### sum over primes involving divisor function (variation of the Titchmarsh divisor problem)

This question was also asked on MSE.
Does there exist an asymptotic estimate for the following sum over primes
$$
\sum_{p\leq x} \frac{\tau(p-1)}{p}\;,
$$
where $\tau(n)=\sum_{d|n}1$ is the divisor ...

**2**

votes

**0**answers

214 views

### Periodicity with irrational numbers [migrated]

Recently, I invented the following theorem and found a proof,
it seems strange since it is very counter-intuitive to me.
The proof is long and non-conceptual.
Is there a place or a branch of math ...

**1**

vote

**0**answers

73 views

### Simply-connected 4-manifolds can be blown up and down to complex projective planes. How about non-simply-connected ones?

There is a nice theorem that for every simply connected, closed, smooth, oriented manifold $M$, we have for some $m \in \mathbb{N}$:
$$M \# \left(\mathop{\#}^m \left(\mathbb{C}\mathbb{P}^2 \# ...

**0**

votes

**0**answers

74 views

### About the Intersection of the nested sequence of Chebyshev centers of weakly compact convex sets

Let $K_0$ be a weakly compact convex subset of a Banach space $X$. For each $n\in\mathbb{N}$, let $K_n$ be the set of Chebyshev centers of the set $K_{n-1}$. Suppose $K_0$ has a normal structure. Is ...

**0**

votes

**0**answers

40 views

### Integration by parts for multidimensional Lebesgue-Stieltjes Integrals

I am concerned with the following problem:
I am wondering if there exists any sort of integration by parts formula for a multidimensional Lebesgue-Stieltjes integral. In my case the integral is given ...

**0**

votes

**0**answers

28 views

### Numerical analysis- Runge Kutta [on hold]

i have:
y'(x)= sin(y); y(0)=1
i need to calculate the function values with runge kutta.
my problem is that i need to choose the h (=dx) such that the error will be in order of 0.0001.
how i choose ...

**10**

votes

**0**answers

137 views

### A hypergeometric puzzle

$$
143\,\sqrt {3}\;{\mbox{$_2$F$_1$}\left(\frac{1}{2},\frac{1}{2};\,1;\,{\frac {3087}{8000}}\right)}=
40\,\sqrt {5}\;
{\mbox{$_2$F$_1$}\left(\frac{1}{3},\frac{2}{3};\,1;\,{\frac ...

**8**

votes

**2**answers

444 views

### Prove that the Dirichlet eta function is monotonic

Let us consider $\eta(p):= \sum\limits_{n=1}^\infty \frac{(-1)^{n+1}}{n^p}$ for $p>0$. Has anyone come along with an elementary proof that $\eta(x)$ is monotonically increasing on this set? By ...

**5**

votes

**1**answer

108 views

### Ham sandwich theorem for discrete measures - reference request

A discrete version of the ham sandwich theorem states as follows (see for instance "Common Hyperplane Medians for Random Vectors" - Hill):
For every $\mu_1,...,\mu_n$ discrete (i.e., purely atomic) ...

**1**

vote

**1**answer

67 views

### An identity for the exponential of a martingale

I am trying to understand a Lemma in Olav Kallenberg's book "Foundations of Modern Probability" (Lemma 26.19 in the second edition or 23.19 in the first edition).
The part of the lemma that I do not ...

**0**

votes

**1**answer

82 views

### Symmetric Zero-Diagonal Matrices

Consider matrices with entries in a field $F$ of characteristic $2$. Let $\Omega$ denote the $2n\times2n$ matrix $\left[\begin{array}{ll}0&1_n\\1_n&0\end{array}\right]$. Then $X^t\Omega X$ is ...

**4**

votes

**1**answer

227 views

### The space of varieties between two given varieties

Let $\mathbf{P} = \mathbf{P}^n(k)$ be the $n$-dimensional projective space over a field $k$, let $A, B$ be projective varieties in $\mathbf{P}$ such that $A \subset B$. Now define
$V(A,B)$ to be the ...

**2**

votes

**0**answers

103 views

### State-of-the-art for the descent principle in relation to surfaces over a number field

I'll start with some motivating remarks (edit: as pointed out in the comments, these motivational remarks do not hold for surfaces: there is an example of a conic bundle surface over a real quadratic ...

**6**

votes

**1**answer

306 views

### Is Gauss sum a p-adic measure?

Let $\Gamma$ be Galois group of cyclotomic $\mathbb{Z}_p$ extension over $\mathbb{Q}$. Consider the function $G$ which sends each finite order character $\chi$ of $\Gamma$ to the Gauss sum $G(\chi)$, ...

**3**

votes

**0**answers

179 views

### Reference request: Beilinson-Bernstein for finite-dimensional reps and category O

I think I’ve once been told that under the Beilinson-Bernstein correspondence, finite-dimensional representations of a semisimple Lie algebra $\mathfrak{g}$ correspond to (twisted) D-modules on $G/B$ ...

**3**

votes

**0**answers

119 views

### Subgroup cliques in the Paley graph

It is a famous open problem to estimate non-trivially, for a prime $p\equiv 1\pmod 4$, the largest size of a subset $A\subset{\mathbb F}_p$ such that the difference of any two elements of $A$ is a ...

**3**

votes

**0**answers

71 views

### Operator theory of initial-value ODE problems

The theory of elliptic boundary value problems is usually treated from the perspective of functional analysis, and the theory of operators between Hilbert spaces.
In contrast to that, the theory of ...

**-1**

votes

**0**answers

58 views

### Confusion about the projected component in an irreducible space in the tensor product decomposition using Littlewood-Richardson?

The regular representation of the symmetric group can be formulated in terms of an abstract tensor, where the action of the symmetric group elements is by means of permuting the indices. Given an ...

**1**

vote

**0**answers

18 views

### Differential operator with codimension 2 singularity in the domain

The soft version of the question is as follows: suppose I have a linear operator, and I know it is a 'nice' differential operator on its domain minus a singular set of codimension two. Does the ...

**-1**

votes

**0**answers

64 views

### Deciding whether a space finitely covered by a simple space is again simple [on hold]

Recall that a path-connected $X$ space is said to be simple (or, in Hatcher's terminology, abelian), if the action of $\pi_1$ on $\pi_n$ is trivial for all $n \geq 1$. I've been trying to come to ...

**0**

votes

**0**answers

52 views

### Speed up Linear programming

I have a linear programming problem like this:
minimize $c^t X$
under the constraint that $AX \ge b$.
I will need to solve this linear programming problem online many times. I need it to be as fast ...

**-2**

votes

**0**answers

113 views

### What is geometry? [closed]

My question is "How could we define the notion of Geometry ?", or maybe more precisely
"How could we define the notion of a "Geometric Theory ?"
Gérard Lang

**-2**

votes

**0**answers

67 views

### How can this statement of the link between Hamiltonian and symplectic matrices be made more rigorous? [closed]

I quote the a textbook, which says the following:
It is easily checked that the exponential of a Hamiltonian matrix
$$
g=exp(\phi\cdot\mathbf{T})
$$
is a symplectic matrix; Lie group ...

**3**

votes

**1**answer

375 views

### the existence of a real polynomial satisfying the following property

It is easy to verify that
$$ \frac{t}{2}\leq \frac{1}{4}+\frac{1}{4}t^2\leq \frac{t}{1-(1-t^2)^2}-\frac{t}{2}
\quad \quad 0<t\leq1$$
I want to ask if there exist a real polynomial $h(t)$ such ...

**1**

vote

**0**answers

46 views

### presence of turbulent phenomena in systems of linear pde?

Are there linear systems of PDE that are known to have solutions which exhibit turbulence, or can turbulence be firmly classified as a fundamentally non-linear phenomenon, similar to solitons or shock ...

**4**

votes

**2**answers

65 views

### Measure on hyperspace of compact subsets

For a Polish space $X$, let $K(X)$ be the set of compact subsets of $X$. Given the topology with basis $\{K\in K(X):K\subset U_0, K\cap U_1\neq\emptyset,\ldots,K\cap U_n\neq\emptyset\}$ for open sets ...

**4**

votes

**1**answer

139 views

### Sites for seeking possible collaborations

As a material scientist, I have recently constructed algorithms for solving ground state of arbitrary cluster interactions models and prepared publications in the field of physics and material ...

**3**

votes

**0**answers

136 views

### Category of modules over commutative monoid in symmetric monoidal category

Let $\left(\cal{C},\otimes ,I\right)$ be a symmetric monoidal category (not necessarily closed) and $A$ a commutative monoid in $\cal{C}$. In his DAG III (page 95), Lurie writes:
In many cases, ...

**2**

votes

**2**answers

86 views

### Question on the number of equilibria

Let $f: C \to C$ be a smooth function and $C$ be a compact set, subset of $\mathbb{R}^n$.
We assume that all the fixed points are hyperbolic. Is it true that the number of fixed points is finite or ...

**4**

votes

**1**answer

140 views

### Totally disconnected subspaces

This question is motivated by this one, where no simple solution within ZFC seems to exist. Let me ask a weaker question then.
Suppose that $K$ is a compact, Hausdorff, non-metrizable space. Does it ...

**1**

vote

**0**answers

116 views

### which sections of elliptic curves are conjugate?

Suppose you have a relative elliptic curves $f : E\rightarrow S$ (say $S$ is connected). Then suppose you have two sections $g,g' : S\rightarrow E$, corresponding to two sections $g_*,g'_*$ to the map ...

**0**

votes

**0**answers

76 views

### Characterization convex function [closed]

Let be $f:[0,1]\rightarrow\mathbb{R}$ a continuous function such that far all $a,b\in [0,1]$ with $a<b$
$$f\left(\frac{a+b}{2}\right)\leq\frac{1}{b-a}\int_a^b f(x)\,dx.$$
How to prove that $f$ is ...

**13**

votes

**1**answer

203 views

### Flag complexes that are shellable but not vertex decomposable

As the title suggests, I was wondering if anyone can point me to any examples in the literature to flag complexes that are shellable but not vertex decomposable.
It is well-known that if a ...

**14**

votes

**1**answer

259 views

### Is every elementary absolute geometry Euclidean or hyperbolic?

Absolute geometry is any one that satisfies Hilbert's axioms of plane geometry without the axiom of parallels. It is well-known that it is either the Euclidean or a hyperbolic plane. For an elementary ...

**-3**

votes

**0**answers

81 views

### Tripet prime reciprocals series [closed]

Does any body know if the series of reciprocals of triplet primes of form
p,p+2,p+6 or p, p+4,p+6 converges or diverges. Could this be used as a proof of infinity of twin primes

**0**

votes

**1**answer

50 views

### Which Hyperspace Topologies Yield Topological Lattices?

At least on a continuum, the binary operations of intersection and union are Vietoris-continuous. But the Vietoris topology only applies the the collection of NONEMPTY closed subsets, and this means ...

**-1**

votes

**0**answers

28 views

### Mean exit time / first passage time for a general symmetric Markov chain [closed]

Suppose I have a Markov chain as depicted in the following figure:
where $N$ is even. State 0 and $N$ are the two sinks of the chain. The transition probabilities have the following property: ...

**9**

votes

**1**answer

331 views

### Can a division algebra have degree divisible by its characteristic?

I apologize in advance if this is easy, but I've tried Googling, and had no luck.
I'm currently working on a proof, and I realized in the course of writing that this proof will break if out there ...

**3**

votes

**2**answers

164 views

### Morse theory Vs degree theory

I asked this question on http://math.stackexchange.com but no unswers!
I have this paragraph from K.C. Chang Infinite dimensional Morse theory
In comparison with degree theory, which has proved ...

**5**

votes

**0**answers

150 views

### Distributivity of group topologies on $\Bbb Z$

Let $\mathcal L$ be the set of all group topologies on $\Bbb Z$.
It is known that $(\mathcal L,\subseteq)$ is a modular complete lattice.
Is $(\mathcal L,\subseteq)$ distributive?

**0**

votes

**1**answer

74 views

### Estimates on gamma- functions [closed]

I need a special inequality related to a fractional derivative problem.
Let k∈ℕ ,0<α<1 , 0<β<1.Consider :
A=[Γ(1-α)Γ(1+k-β)/Γ(2-β-α+k)].(1-α)
On what conditions (on k ,β and α) A is less ...

**0**

votes

**0**answers

27 views

### Beta distribution - changes in multiple time points

Let's say I have a set of daily data (assume iid) that I know is beta distributed (between 0 and 1). I can estimate the parameters of the distribution and calculate the tails etc. This would tell me ...

**-1**

votes

**0**answers

64 views

### I need help in understanding O(nlogn) question [closed]

I wish I could think of a better way to word my question. Maybe some one here could offer s suggestion for that, as well.
On to my question. Before I do, this is a class question that has been asked, ...