2
votes
0answers
17 views

Hochschild Cohomology of the Quantum Torus

I would like some advice on how to compute directly, or by a higher powered method the Hochschild Cohomology groups of the quantum torus using the stated complex I have found. I think there are ...
0
votes
0answers
9 views

Create solid torus using geometric algebra-or otherwise

I have created an algorithm for tracking a vessel's centerline in 3 dimensions, and traveling through that centerline. My supervisor asked me whether I could add a small theoretical section on ...
0
votes
0answers
12 views

examples of completely positive order zero maps to demonstrate a theorem

I'm interested explicit examples which can be used to demonstate the theorem: Theorem: Let $A$ and $B$ $C^*-$algebras and $\phi:A\to B$ be a completely postive map of order zero. Set ...
1
vote
0answers
11 views

Radicals of co-quasitriangular map

Let $B=\mathcal{O}_R\left(GL(n)\right)$ be a localization of the algebra $A(R)$ of functions on the quantum formal group corresponding to the matrix $R$ ["Quantization of Lie groups and Lie algebras", ...
1
vote
1answer
53 views

Is this kind of scheme integral?

Let $X\rightarrow Spec(R)$ an irreducible projective scheme over a dvr. Suppose the generic fiber $X_{\eta}$ is smooth (over the field $Frac(R)$) and irreducible. Is it true that $X$ is integral (i.e. ...
4
votes
1answer
63 views

Does the ring generated by the odd power sum symmetric functions have a name?

Let $\Lambda$ be the ring of symmetric functions and recall the power sum symmetric function $p_i = \sum x_1^i + x_2^i + \dots$ generate this ring. Let $\tilde\Lambda$ be the ring generated by the odd ...
3
votes
0answers
37 views

soft: Reference/ Suggested Read: Homological Algebraic techniques in PDEs

I was reading this article on wikipiedia and was interested by the apparent link between Homological Algebra and PDEs. What is an accessible reference which showcases the link between these topics? ...
1
vote
1answer
113 views

reference on Dirichlet theorem on primes in arithmetic progression

I appreciate if you could help me to find a reference (and a proof). Combining Dirichlet theorem on primes in arithmetic progression with Chebotarev densitiy theorem, we know that given two positive ...
-2
votes
0answers
33 views

Indecomposable commutative rings

Let $R$ be an indecomposable commutative ring. Can we say that $R=\bigoplus_{i\in I}R_i$ or $R=\prod_{i\in I}R_i$ where $R_i$ are commutative ring and $I$ is an infinite set?
4
votes
1answer
84 views

Integral formula for $\int_{0}^{\infty}e^{-3\pi x^{2}}((\sinh \pi x)/(\sinh 3\pi x))\,dx$ by Ramanujan

The following is a re-post from MSE because I did not get any answer even after offering a bounty. Towards the end of G. N. Watson's (one of the joint authors of famous book "A Course of Modern ...
3
votes
2answers
72 views

Reference for (co)limit-preserving functor $X\mapsto R^X$

Fix a commutative ring $R$. There's a contravariant functor from finite sets to finite $R$-algebras sending $X$ to $R^X$. Viewed as a covariant functor $\text{set}^{op}\to R\text{-alg}$, this functor ...
1
vote
0answers
26 views

Understanding the definition of an F-connected simplicial complex

I'm reading the classic paper "Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one" by Gromov-Schoen. In Section 6, they define the notion of F-connectedness ...
1
vote
0answers
25 views

About a close strucure on profunctors

Let $Prof$ the bicategory with profunctors (on small categories), arrows are like $D: \mathscr{A} \dashrightarrow \mathscr{B}$ and this mean that $D: \mathscr{A}^{op} \times \mathscr{B}\to Set$. ...
0
votes
0answers
69 views

Computing Hochschild Cohomology

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
417 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) ...
2
votes
0answers
30 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
19 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
86 views

When does composing polynomials reduce the degree?

Let $\mathbb{F}$ be the field of size 2. Say I have polynomials $p_1,\ldots,p_m : \mathbb{F}^n \to \mathbb{F}$ with high degree $\approx n$, and another polynomial $s : \mathbb{F}^m \to \mathbb{F}$ ...
0
votes
1answer
75 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
52 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
75 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
3
votes
2answers
62 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
59 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
29 views

Proofs needed for observations regarding prime-partitionable numbers

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, doi and apparently the same as in W. T. ...
1
vote
0answers
29 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 ...
14
votes
3answers
173 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 ...
5
votes
0answers
61 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
103 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
28 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 ...
0
votes
0answers
23 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 with nonzero ...
2
votes
0answers
90 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
246 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
84 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
67 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
37 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 ...
1
vote
2answers
147 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
312 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
71 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
50 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
182 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
65 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
57 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
37 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
246 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
21 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
51 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
38 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), ...

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