0
votes
0answers
89 views

Computing Hochschild Cohomology [on hold]

Let A be my noncommutative ring. I have computed a $A^e$ projective resolution and taken $Hom_{A^e}(.,A)$ so I am ready to compute kernels and images to find the hochschild cohomology groups. ...
12
votes
1answer
512 views

Why is it so hard to prove Toeplitz' conjecture?

I'm a layman in mathematics, so please excuse me in advance for anything in this question that may be inappropriate :D. Well: Four years ago, I was reading (and working to solve the puzzles on) ...
4
votes
0answers
88 views

Product of a Schubert polynomial and a double Schubert polynomial

Let $S_u(x)$ be a Schubert polynomial and let $S_v(x;y)$ be a double Schubert polynomial. Then their product can be expressed in terms of the double Schubert polynomials as ...
1
vote
0answers
27 views

Modularity in a graph - definition of the random component

This question concerns the definition of modularity in a graph. Consider a simple, undirected, unweighted graph $G$ with vertices in set $V$ and edges in set $E$ . Between any two vertices there is ...
3
votes
0answers
162 views

When does composing polynomials reduce the degree?

Let $\mathbb{F}$ be the field of size 2. Say I have functions $p_1,\ldots,p_m : \mathbb{F}^n \to \mathbb{F}$ each with high degree $\approx n$ as (multilinear) polynomials, and another function $s : ...
0
votes
1answer
98 views

Symmetries of non-Riemannian curvature tensor

The curvature tensor, $R_{ab}{}^c{}_d$, can be obtained from a connection which not necessarily is a metric connection. By construction it is antisymmetric in the first two indices, since roughly ...
1
vote
1answer
58 views

Sampling from random unimodular matrices of a particular type?

Is there a nice way to parametrize unimodular matrices of form $$\begin{bmatrix} a1& a2& 0& 0\\ b1& b2& a1& a2\\ c1& c2& b1& b2\\ 0& 0& c1& c2 ...
1
vote
0answers
89 views

Finding a lower bound in terms of given integers

Given four positive integers $n,$ $m,$ $l$ and $k \geq 2.$ I want to find a lower bound for this expression $$|\sqrt[k]{n}+\sqrt[k]{m}-\sqrt[k]{l}|$$ in terms of these integers. Many thanks
4
votes
2answers
83 views

Existence and characterization of transitive matrices?

We call a matrix $M \in \mathbb{R}^{d \times d}$ transitive if it satisfies the following: For any three vectors $u, v, w$ in $\mathbb{R}^d$. If $u^T M v > 0$ and $v^T M w > 0$ then $u^TMw ...
2
votes
1answer
64 views

Do the values of the differential of a function on a Lie group with a single maximum span the Lie algebra?

Consider: a vector field $X=\nabla \phi$ on a compact, semi-simple, connected matrix Lie group $G$ where $\phi$ as a smooth scalar field on $G$ possessing only a single maxima which topologically is a ...
0
votes
0answers
62 views

Proofs needed for observations regarding prime-partitionable numbers

Below is the definition of a prime-partitionable integer taken from W. Holsztynski, R. F. E. Strube, Paths and circuits in finite groups, Discr. Math. 22 (1987) 263-272 and is apparently the same in ...
1
vote
0answers
34 views

Optimization question: maximize quadratic objective with semidefinite constraints

I recently encountered the following optimization problem: $\max \|AX\|_F^2$ subject to: $X\succeq0$ and $Xb_i\leq c_i$ for a collection of $T$ conditions: $i=1,\ldots,T$. Matrices $A$ and $X$ are ...
15
votes
3answers
208 views

Lower bounding the probability that $\gcd(t,N)≤B$, for a random $t$ and fixed (large) $N$

$\newcommand{\Prb}[1]{\mathcal{P}_{#1}}$ I have the following number theory problem, related to Odlyzko's improvement on Shor’s factoring algorithm (see this cstheory.sx question for details). Let ...
6
votes
0answers
79 views

Why do the model structures on dg-algebras and on dg-categories are not compatible?

First we talk about dg-algebras. According to this n-lab page, we write $dgAlg$ for the category of cochain dg-algebras in non-negative degree over a field $k$ of characteristic $0$. Write ...
3
votes
1answer
119 views

Are automorphism groups of polarized varieties of finite type

It is "well-known" that the stack of polarized varieties is an algebraic stack with quasi-compact and separated diagonal. In particular, if $(X,L)$ and $(Y,M)$ are polarized schemes over a scheme ...
-4
votes
0answers
31 views

Calculating matix derivatives with MATLAB or MATHEMATICA? [on hold]

I'd like to calculate the following derivative, \begin{equation} \frac{d||(f(C)\cdot f(C)^{+}-I)\cdot u||^2)}{dC} \end{equation} Where $C$ is a matrix of dimension $n\times k$ (s.t $k < n$). And ...
1
vote
1answer
39 views

roots in a root system which have nonzero coefficients with respect to each simple root

If we consider crystallographic root systems, then for each $k$ such that $n \leq k \leq d-1$ where $d$ is the Coxeter number, it seems to be the case that there is exactly one root of height $k$ with ...
5
votes
0answers
129 views

Manifold approximations to $BO(3)$

We know that $BO(1) =\mathbb{R}P^\infty$ has closed, finite-dimensional manifold approximations $\mathbb{R}P^1\subset \mathbb{R}P^2\subset\cdots.$ Similarly $BO(2)$ can be approximated by closed, ...
0
votes
1answer
266 views

Particular case of Beal's Conjecture

Is it known that there exist no coprime positive integers $A$, $B$ and $C$ such that $A^3+B^4=C^3$? This is a particular case of Beal's Conjecture.
2
votes
1answer
89 views

Do the full subcategories have a simple structure in higher category theory?

Let $C \in Cat$ be an $(\infty,1)$-category. Let $P$ be the partially ordered subset of full subcategories of $C$. Is there a (canonical?) functor from the nerve of $P$ to $Cat$? I think the answer ...
0
votes
0answers
71 views

power expansion and matrix inverse

Consider the vector-valued function ($s$ complex): $$ f(s) = (I-A/s)^{-1} v. $$ Here $A$ is a real square matrix, $v$ a non null column vector. It is known that $A$ has one simple $0$ eigenvalue, and ...
3
votes
0answers
38 views

$n!$ computation in $\mathsf{BSS}$ model

It is well known that if $n!$ cannot be computed in $\mathsf{polylog}(n)$ ring ($\Bbb Z$) operations, then $\mathsf P\neq\mathsf{NP}$ in $\mathsf{BSS}$ model. Suppose if we assume $\mathsf ...
2
votes
2answers
172 views

Hausdorff Dimensions of Limit set of subgroups of SL(2,Z)

In a recent paper by Bourgain, Sarnak, Gamburd [1] talks about subgroups of $SL(2,\mathbb{Z})$. Let $\Lambda$ be a finitely generated non-elementary subgroup of $SL(2,\mathbb{Z})$ with Hausdorff ...
-1
votes
0answers
18 views

probability for having the exact same result in a given number of test with a given number of possible combinations [on hold]

The questions was: What is the probability for having exact the same page at least twice when printing my whole SSD (as bytes) on paper. We have 1117bytes per page leading to ~6,3e2834 possible ...
4
votes
3answers
337 views

Do cotangent bundles have “bounded geometry”?

I have often heard the phrase "a manifold $M$ has bounded geometry" thrown around without ever seeing a precise definition of what this means. Apparent examples are compact manifolds and ...
-1
votes
1answer
62 views

A question about Lebesgue density [on hold]

Is there a set $ A \subseteq \mathbf{R} $ such that the upper lebesgue density of $ A $ and the upper lebesgue density of $ \mathbf{R} \setminus A $ are equal to $ 1 $ at a fixed point? I would say ...
3
votes
1answer
80 views

almost diagonal Positive semidefinite Matrix

Consider the set $\mathcal{D}_n$ of $n$-dimensional positive semidefinite matrices. A matrix $M\in \mathcal{D}_n$ is called $\epsilon$ diagonal in trace distance if there is a diagonal matrix $D\in ...
1
vote
0answers
56 views

Examples for differential operators first order

Currently I am dealing with the problems which involve the general first order differential operator, i.e., for some open domain $\Omega\subset\mathbb{R}^n$ with certain regular bondary and a function ...
0
votes
0answers
47 views

Limit inferior of Borel functions [on hold]

Suppose $X$ is separabile metric and $F \colon X \times \mathbb{R}_+ \to [ 0 , 1 ]$ is Borel. Let $ f ( x ) = \liminf_{\varepsilon \to 0} F ( x , \varepsilon )$. Is $f$ Borel?
8
votes
2answers
187 views

Asymptotics of a recurrence relation

The sequence $(a_n)_{n \ge 0}$ satisfies, $a_0 = a_1 = 1$ and the recursion relation: $$a_n = \sum\limits_{k=0}^{[n/2]} \frac{a_k}{(n-2k)!}$$ where, $[x]$ is the nearest integer to $x$ not exceeding ...
1
vote
1answer
74 views

extension of holomorphic mappings

In 1971 Phillip.A.Griffith proved that Let $B_n^\ast=\{z\in\mathbb{C}^n:0<\|z\|\le 1\}$ be the punctured ball in $\mathbb{C}^n$,and $f:B_n^\ast\rightarrow M$ a holomorphic mapping into a compact ...
-1
votes
0answers
89 views

Virasoro-like algebras over the quaternions

The Virasoro algebra is a central extension of the Lie algebra of vector fields on $\mathbb{S}^1$. This central extension exists and is unique because $H^2(Vect (\mathbb{S}^1))$ is one-dimensional ...
2
votes
1answer
40 views

Is there a curve on a surface where an integrable function is pointwise bounded?

I consider $\Sigma:=S^2$, as a unit radius sphere in $\mathbb R^3$ so that it has an induced metric, measure etc on it. Is the following statement true? For a large constant $K$, there exist ...
4
votes
2answers
266 views

What does it mean that the Hessian is proportional to the metric?

Let $(M,g)$ be a smooth manifold equipped with a metric tensor $g$, and $f\in C^\infty(M)$ a regular function (i.e., with nowhere vanishing differential). Denote by $\mathrm{Hess}_g(f):=\nabla df$ ...
0
votes
0answers
22 views

Can a relationship be constructed between the Coherence space and Phase space semantics of linear logic?

I'm not very familiar with linear logic, so please bear with me, i.e., please "read between the lines" to my underlying question if I don't formulate it rigorously correctly. To help model some of my ...
4
votes
0answers
57 views

Fock Space Proof of $(g(x)\phi^4)_2$ Mass Gap?

Is there a proof that does not depend on Euclidean methods? Is this a proof? : $V(g)$ can be written as $P+R$ where $P$ is non-negative and $R$ is $N$-bounded (and hence $(H_0+\lambda P)$-bounded). ...
0
votes
0answers
39 views

Calculation of fixed point set

Let $p,q$ be two distinct primes and let $G_p$ and $G_q$ be nontrivial $p$-group and $q$-group respectively. Let $G = G_p \rtimes G_q$. Embed $G_q \subset U(n_1), G \subset U(n) $ where $U(n_1), ...
5
votes
0answers
113 views

Pattern Avoidance in Poset Permutations

I am not sure if it is appropriate to use MathOverflow to publicize a conjecture, but I think this is an interesting question and I have no real ideas of how to solve it. A permutation on a ...
8
votes
3answers
249 views

“Most Similar Vector Problem” on an Integer Lattice?

I am currently working on problem that I think could be expressed as an integer lattice problem. Given $u \in \mathbb{R}^n$ and a bounded integer lattice $L = \mathbb{Z}^n \cap [-M,M]^n$ I would like ...
6
votes
4answers
377 views

SO$(4)$ (& SO$(n)$) characterization?

I believe it is the case that any finite subgroup of SO$(3)$ (the $3 \times 3$ orthogonal matrices of determinant $1$) is either a cyclic group $C_n$, or a dihedral group $D_n$, or one of the groups ...
7
votes
0answers
123 views

Modular factorization of Dedekind zeta functions

It is well known that for abelian number fields, the factorization of its Dedekind zeta function goes like this: $$\zeta_K=\zeta\prod_\chi L(s,\chi)$$ with the Dirichlet characters distinct and ...
3
votes
1answer
46 views

Number of samples needed as input to Bernoulli factory

Let $\{X_i\}$ denote an i.i.d. sequence of Bernoulli variables with parameter $p$. A Bernoulli factory is a procedure that generates events with probability $f(p)$ using the observations $\{X_i\}$, ...
3
votes
1answer
173 views

History of unstable formulas [on hold]

There are many equivalent definitions for stability, one of them being that being unstable is equivalent to the existence of a formula having the order property. While intuitively it makes sense that ...
3
votes
0answers
44 views
+100

Bounded input Bounded output stability for heat equation

This is a cross-post from Computational Science. I am interested in proving or obtaining a counterexample to the following conjecture. Let $\Omega\subset\mathbb{R}^d$ be a bounded open domain. Let ...
2
votes
0answers
33 views

Co-quasitriangular Hopf algebra - notation

In one article I found the following statement : If $A$ is a Hopf algebra over $k$ with co-quasitriangular structure $r$, then one can define map $r_{21}*r:A\otimes A\rightarrow k \ \ $ (...). ...
7
votes
1answer
138 views

Consistency strength of being strong cardinal and indestructible under collapses

What is the consistency strength of the following statement: $\kappa$ is a strong cardinals and it is indestructible under $Col(\kappa, <\theta),$ where $\theta> \kappa$ is some fixed ...
5
votes
1answer
142 views

Connes Embedding Conjecture and Fusion Categories

I was recently introduced to Connes' Embedding Conjecture (CEC) which states: Every separable type $II_{1}$ factor is embeddable into $R^{\omega}$. Where $\omega$ is a generic free ultrafilter on ...
-3
votes
0answers
51 views

Can somone explain a global cascade condition in plain english? [on hold]

Details here: https://en.wikipedia.org/wiki/Global_cascades_model#Global_cascades_condition If I have a hierarchical structure, what is the required number of nodes / clusters of nodes required to ...
6
votes
1answer
55 views

Analysis of the Laplacian of a random bipartite graph

My analysis of an engineering problem reduced to analysis of the Laplacian of a (random) bipartite graph. There are a few particular questions I am interested in, but not sure which direction to take ...
2
votes
0answers
25 views

Variance of a functional of transition probabilities using spectral gap of Markov chain

Consider an irreducible, aperiodic, time-reversible, discrete-time Markov chain on a finite state space $S$ whose Markov kernel is $K$ and unique stationary distribution is $\pi.$ Then, reversibility ...

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