0
votes
1answer
130 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
1answer
72 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
0answers
56 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
0answers
90 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
1answer
78 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
0answers
37 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
2answers
162 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 ...
-11
votes
0answers
64 views

In Bedrock the citizens can rent cars from Wreck Rental for $40 down and $0.04 per mile. [on hold]

But they can also rent from Cutie Pie Rental for $0.20 per mile. After how many miles does Wreck Rental become a better deal?
-9
votes
0answers
47 views

The ordered pair (3,100) indicated that it costs $100 for a job that is 3 hours work. He charges $190 for 13 hours of work [on hold]

Find the slope between these two points. Slope = _____________ Find the y-intercept __________________ Write a linear equation to represent this scenario. Where x is the number of hours and y is the ...
0
votes
1answer
58 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,∞) ...
1
vote
2answers
171 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
0answers
35 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
0answers
60 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
0answers
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
0answers
44 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
1answer
93 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
2answers
154 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
0answers
109 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, ...
-8
votes
0answers
45 views

Store 5 numbers in 3 [on hold]

I am trying to store 3 variables in 5, it sounds stupid. But i need to know if it is possible. I have tried using methods like; a-b = c, and being able to work from that. But i cant figure it out. ...
2
votes
0answers
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
0answers
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
0answers
123 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
1answer
76 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
1answer
220 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 $\chi(G) = \kappa$?
4
votes
1answer
111 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
0answers
11 views

Copula theory on discrete random variables [on hold]

How can I find the joint pmf on two discrete random variables using the copula theory
3
votes
1answer
82 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
0answers
13 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
0answers
53 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
1answer
64 views

Hyperelliptic generalization of Euler's formula

Are there any hyperelliptic generalizations of the following formula, first proved by Euler in 1782, ...
0
votes
1answer
62 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
1answer
193 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
0answers
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
0answers
26 views

Probability of co-occurence [on hold]

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
0answers
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
0answers
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
0answers
29 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
0answers
43 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
0answers
54 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
0answers
54 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 ...
0
votes
0answers
58 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
0answers
48 views

inequality involving determinants and quadratic forms [on hold]

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: ...
7
votes
1answer
117 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
2answers
109 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
1answer
234 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
2answers
171 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 ...
7
votes
0answers
66 views

Kinematics of rolling knots

It is well known that there are trefoil knots without tritangent planes, and with 3d printers one can print these beautiful objects and make them roll on planes. (An ...
0
votes
0answers
24 views

Bounds on the moments of truncated sub-gaussian random variables

If $X$ is a centered sub-gaussian random variable, then there exists a constant $c$ such that $$ \mathbb{P}[|X|>t] \leq \exp(1-ct^2) $$ for all $t\geq 0$. Moreover, we know that the normalized ...
7
votes
0answers
210 views

Why does $Mf$ always support an $Mf$-orientation?

Let $f:X\to BGL_1(\mathbb{S})$ be a morphism of $E_n$-spaces and determine a principle $GL_1(\mathbb{S})$-bundle over $X$. Then it can be shown in the classical case that there is always a Thom ...
3
votes
1answer
141 views

Numbers represented by inhomogeneous forms

I have a family of Diophantine equations that I am trying to solve, and I am trying to figure out what methods could be used to prove existence of solutions. Unfortunately, the equations are ...

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