# All Questions

**10**

votes

**0**answers

153 views

### How long can a cycle of antichains in a finite partial order be?

Suppose that $X$ is a finite partially ordered set. Then a subset $A\subseteq X$ is said to be an antichain if there do not exist elements $a,b\in A$ with $a<b$. Let $\mathcal{A}_{X}$ be the set of ...

**3**

votes

**2**answers

108 views

### About supersolvable Lie algebras

A colleague of mine asked me the question below, and since I could not answer it, I thought I might have more luck on MO.
In Encyclopedia of Mathematics, a finite dimensional Lie algebra $L$ over a ...

**2**

votes

**1**answer

72 views

### Real slices of Minkowski space, using a complex quadratic form

Ordinary Minkowski space is $\mathbb{R}^{3,1}:=(\mathbb{R}^4,\phi)$
where $\phi:\mathbb{R}^4\rightarrow\mathbb{R}$
is a quadratic form of signature $(3,1)$.
Lying within this is a hyperboloid model ...

**2**

votes

**0**answers

159 views

### References for 'Theory of $p$-adic Galois Representations by Fontaine & Ouyang'

Presently I am reading the 'Theory of $p$-adic Galois Representations by Fontaine & Ouyang'. I am finding it difficult for eg. the initial sections on $l$-adic geometric representation of finite ...

**0**

votes

**1**answer

68 views

### Derivatives of radial functions can be bounded by derivatives in terms of radial distance?

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$,
and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$.
Prove or disprove the following.
Given any ...

**0**

votes

**1**answer

331 views

### Will this be a case of self plagiarism or will it annoy the referee? [on hold]

About 2 months ago, I uploaded a fairly long paper (P1) to arXiv and it is currently under review.
Now, I am writing a second paper (P2) on a somewhat different topic. But quite unexpectedly, it ...

**3**

votes

**1**answer

167 views

### Model structure on non-negative differential graded algebras with homological grading

I was wondering if there exists a model structure on the category of non-negative differential graded algebras with homological grading. To be more precise: Let $Ch_{k}$ the model category of ...

**0**

votes

**1**answer

137 views

### Consequences of Serre's property FA

Proposition 21 of Serre's Trees:
Let G be a group with property FA. If G is contained in an amalgam then G is contained in a conjugate of one of the amalgam's factors.
Can anybody help with this ...

**0**

votes

**1**answer

103 views

### Schatten $p$-classes for small $p$

Suppose $\mathcal H$ is a separable Hilbert space and $T$ is a compact self-adjoint operator on $\mathcal H$. Let $\{e_n\}$ be an orthonormal basis for $\mathcal H$.
Fix $1<p<2$.
Does ...

**3**

votes

**0**answers

72 views

### Localized at $p$ integral representations of finite elementary $p$-groups

Let $C_p$ be a cyclic group of prime order $p$.
Let $F=C_p^n=C_p\times\dots\times C_p$ ($n$ times).
I would like to to classify finite dimensional representations of $F$ over ${\mathbb{Z}}$.
However, ...

**2**

votes

**0**answers

103 views

### Find the intersection between two convex hulls, in this specific case

We work over $\mathbb{R}^K$. Let $V$ be the set of vectors whose coordinates take values $0$ or $1$, or equivalently the corners of the unit cube $[0,1]^K$.
Let $d:\{0, \ldots, K\} \to \mathbb{R}_+$ ...

**1**

vote

**1**answer

83 views

### Is the endpoint map smooth

Given $a,b \in \mathfrak{su}(n)$ and (with $U_0 = I$ taken) the following ODE:
$\frac{d U_t}{dt} = (a + w(t)b)U_t$
consider the "fixed time" endpoint map $V_T: L^2([0,T]) \rightarrow SU(n)$ for ...

**0**

votes

**0**answers

43 views

### Alternating projections and trace preserving maps

Let $\{\Pi_i\}_{i=1}^N$, $\Pi_i\in \mathbb{C}^{n\times n}$ be a set of orthogonal projections. By the Von Neumann alternating projection theorem, it holds that
$$
...

**5**

votes

**2**answers

165 views

### Continuous map from connected subset of R^n to one of the real zero of an odd degree polynomial whose coefficients are polynoms of the variables

Let take a real multivariate polynomial $P(x_1, \ldots, x_n, y)$ such as the degree of P relatively to the variable $y$ is odd. Thus, for each $X = x_1,\ldots,x_n \in\mathbb{R}^n$, the univariate ...

**0**

votes

**1**answer

63 views

### Reference for measures of commutativity needed

I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if A and B are two n×n Hermitian matrices, and [A,B]=C.
I'd like a function μ:Cn×n→[0,∞) ...

**3**

votes

**3**answers

215 views

### Difference between Gieseker semistable and slope semistable

Let $X$ be a projective reduced (not necessarily irreducible) curve over an algebraically closed field and $\mathcal{F}$ be a pure coherent sheaf on $X$. Is it true that $\mathcal{F}$ is Gieseker ...

**-4**

votes

**0**answers

36 views

### Packing a box into a bag [on hold]

If we have a bag dim 60x82, what would bee the max size box to put into this bag, without tearing the bag, of course?
So the max height of the box would be... how to calc this?
Thanks,
S.

**-1**

votes

**0**answers

63 views

### If $N = q^k n^2$ is an odd perfect number with Euler prime $q$, can $\sigma(n^2)$ be divisible by $(q+1)/2$?

If $N = q^k n^2$ is an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$), then can $\sigma(n^2)$ be divisible by $(q+1)/2$?
I think ...

**-2**

votes

**0**answers

56 views

### Formulating conditional constraints in optimization [on hold]

In an optimization problem, is it possible to formulate the following conditional constraint:
If $y > x_1$ then $x_2 \ge y - x_1$, else $x_2=0$.
Here are $x_1$ and $x_2$ are to be determined by ...

**2**

votes

**0**answers

52 views

### Second eigenvalue of biased reflected random walk

Let $Z_n$ be a reflected random walk on the non-negative integers with negative drift. That is,
$Z_n$ is non-negative and moves one to the right w.p. $p<1/2$. It moves one to the left w.p. $1-p$, ...

**2**

votes

**1**answer

95 views

### Positive rational numbers as sum of unit fractions [duplicate]

Let $U = \{\frac{1}{n}: n\in\mathbb{N}, n>0\}$ be the set of unit fractions. For integers $m,n>0$ there is always a finite subset $S\subseteq U$ such that $\frac{m}{n} = \sum_{u\in S} u$, see ...

**4**

votes

**2**answers

158 views

### Find the expansion of the exact solution (beyond Taylor)

In a paper by Kitagawa & Ueda Squeezed spin states they give an argument that the minimum variance in one-axis twisting Hamiltonian scales like $V_{min} \propto S^{-2/3}$. I will shortly describe ...

**7**

votes

**0**answers

114 views

### Complete list of exceptions and efficient algorithm for Waring's problem

2 weeks ago, Samir Siksek http://arxiv.org/abs/1505.00647 proved more than 150-years-old conjecture that every positive integer other than 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, ...

**2**

votes

**0**answers

45 views

### Rank of the augmented jacobian matrix

I'm struggling to understand the proof of the following theorem found in 'Solving the Likelihood Equations'.
Suppose that $V$ is a complete intersection, i.e. its defining ideal $P$ can be generated ...

**-1**

votes

**0**answers

35 views

### Which spaces admit bump functions? [migrated]

Let me first fix terminology. Let $X$ be a topological space and $A\subseteq B\subseteq X$ be subsets.
Let's say X admits a bump function relative to $A\subseteq B$ if there's a continuous function ...

**0**

votes

**0**answers

127 views

### Are del-Pezzo surfaces complete intersections?

Let $X_k$ be $\mathbb{CP}^2$ blown up at $k$-points (where $k$ is from $0$
to $8$). I think it is known that $X_k$ can be embedded in $\mathbb{CP}^n$
for some $n$.
$\textbf{Question:}$ Can $X_k$ ...

**0**

votes

**1**answer

80 views

### topology of setwise convergence of measures

It is well known that if $X$ is, say, compact and metric, then the set of probability measures on the Borel subsets of $X$ endowed with the usual topology of weak convergence of measures has as a ...

**10**

votes

**1**answer

240 views

### Graph $G$ with $\omega(G) = 2$ but $\chi(G) \geq \aleph_0$

Given an infinite cardinal $\kappa$, is there a graph $G$ that has no clique consisting of more than 2 points, but $\chi(G) = \kappa$?

**5**

votes

**1**answer

119 views

### What's the relation between spin model for subfactors theory and physics?

In the sense of subfactor theory, a spin model is a commuting square of the form
$$\begin{matrix}
\Delta &\subset & M_n(\mathbb{C})\cr
\cup &\ &\cup\cr
\mathbb{C} &\subset ...

**0**

votes

**0**answers

11 views

### Copula theory on discrete random variables [closed]

How can I find the joint pmf on two discrete random variables using the copula theory

**3**

votes

**1**answer

85 views

### Measurability and continuity for general topological spaces

Let $(X,\tau)$ be a topological space. We call $S\subseteq X$ saturated if $S=\bigcap\{U\in\tau: U\supseteq S\}$. Let $\sigma(X,\tau)$ be the $\sigma$-algebra generated by $\tau\cup\{K\subseteq X: K ...

**0**

votes

**0**answers

14 views

### Approximation of a generlized hypergeometric function for large parameters

I found the following approximation in the literature:
If z is fixed and $|ph(1−z)|<π$. Assuming that none of a1,a2,…,ap is a nonpositive integer, $p<q$, $r\rightarrow+\infty$. For each ...

**2**

votes

**0**answers

56 views

### Newton polyhedron and product of ideals

Let $I$ be an ideal generated by monomials $\underline{x}^{\underline{a}_1},\ldots ,\underline{x}^{\underline{a}_s}$ and
$J$ be the ideal generated by $\underline{x}^{\underline{b}_1},\ldots ...

**3**

votes

**1**answer

68 views

### Hyperelliptic generalization of Euler's formula

Are there any hyperelliptic generalizations of the following formula, first proved by Euler in 1782,
...

**0**

votes

**1**answer

64 views

### What is the maximal number of solutions of $\sum_{i = 1}^n 1/a_i^x - \sum_{i = 1}^m 1/b_i^x = 0$?

What is the maximal number of solutions of the following equation?
$\sum_{i = 1}^n 1/a_i^x - \sum_{i = 1}^m 1/b_i^x = 0$
where $x$ is the unknown and $n, m$, $a_i$'s, $b_i$'s are constant.
It ...

**7**

votes

**1**answer

199 views

### Verifying that $\epsilon^!$ is indeed the right adjoint of $\epsilon_*$ in the context of algebraic stacks

The question is about the last paragraph of Remark 12.4.3 in the book on algebraic stacks by Laumon and Moret-Bailly.
Let $S$ be a (quasi-separated) scheme and let $\mathscr{X}$ be an algebraic stack ...

**1**

vote

**0**answers

21 views

### Is there an equivalent form for Wishart to a power times a normal?

Lin described two equivalent characterizations of the multivariate t-distribution, viz.
As a normal vector divided by an independent chi. That is, $t = Z / \sqrt{\chi^2/v}$, where $Z$ is ...

**-1**

votes

**0**answers

32 views

### Probability of co-occurence [closed]

Of total N people, m people are good at mathematics and c people are good at computer science. What is the expected number of people good at both mathematics and computer science? Or what is the ...

**0**

votes

**0**answers

28 views

### Functional approximation that vanishes at infinity

I have a function $f(x)$ that I wish to approximate using a linear combination of basis functions
$$
\hat{f}(x) = \sum_{i=1}^k c_i \varphi_i(x).
$$
The approximation is done via an orthogonal ...

**1**

vote

**0**answers

37 views

### Projecting a convex partition onto a convex set

Say that $X$ and $Y$ are two convex regions in the plane, and suppose that $X \subset Y$. Further suppose that $Y$ is partitioned into disjoint convex subsets $Y_1 ,\dots, Y_n$. Is there a way of ...

**1**

vote

**0**answers

37 views

### Scaling of First-passage times for Random Walk on integer lattices

Consider simple symmetric random walk $S_{n} = (S_{n}^{(1)},\dots,
S_{n}^{(d)})$ on the d-dimensional integer lattice with starting point the origin.
Let $\tau_{N}$ be the first time $S_{n}$ exits ...

**0**

votes

**0**answers

48 views

### Global existence solutions NLS

Let's consider the following NLS in $\mathbb{R}^3$
$$i\partial_t\psi=-\Delta\psi+\vert\psi\vert^2\psi$$
How to prove that $H^2$-solutions are globally in time? Can someone suggest references to me?

**1**

vote

**0**answers

57 views

### Does the monodromy of such VHS have to be trivial

Consider a variation of polarized Hodge structure on a punctured disk. Suppose that connection preserves Hodge filtration (which is much stronger, than Griffiths transversality). Moreover assume that ...

**3**

votes

**0**answers

59 views

### Integrability of Continuous Tangent Subbundles

Are there any field of mathematics, except dynamical systems, where one needs to integrate continuous sub-bundles of the tangent space?
More specifically given a smooth manifold of $M$ and a ...

**1**

vote

**0**answers

65 views

### partial pullback-completion of a category

Let $\mathcal{C}$ be a (possibly enriched) category with all finite product, and $\mathbf{M}$ a class of morphisms.
Can one construct completion of $\mathcal{C}$ w.r.t. all pullbacks along morphisms ...

**0**

votes

**0**answers

48 views

### inequality involving determinants and quadratic forms [closed]

I'm interested in comparing $\det(x'\boldsymbol{A}x)$ and $\det(\boldsymbol{A})x'x$ where $\boldsymbol{A}$ is symmetric positive semidefinite, and $x$ is a free vector of constant. My argument is: ...

**8**

votes

**1**answer

121 views

### Specifying $L^p$ norms of derivatives

Given a sequence of positive numbers $\{a_n\}$ and $1 < p < \infty$, $p\neq 2$, is it possible to build a function $f\in C^\infty(\mathbb R)$ so that $\|f^{(n)}\|_{L^p(\mathbb R)} = a_n$?
For ...

**2**

votes

**2**answers

113 views

### Are there examples of functions in $L_1$ and $L_\infty$ whose Fourier series divergent (“weakly”)?

It is wellknown that there is a convergence in norm for Fourier series in $L_p$, if $1<p<\infty$, but are there some examples for pointwise divergence if $p=1,\infty$ in books, or somewhere? I ...

**4**

votes

**1**answer

247 views

### Flat Riemannian manifold

Is it true that a Riemannian manifold is flat, if and only if a coordinate transformation $f$ exists, such that the geodesics after transformation is in linear form ...

**4**

votes

**2**answers

173 views

### hyperbolic structure on Figure–8 knot complement

I was trying to understand the proof of the fact that there is a hyperbolic structure on Figure–8 knot complement initially from Thurston's notes and then from some online notes; but unfortunately I ...