8
votes
3answers
213 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 ...
1
vote
0answers
15 views

Faithul map and (minimal) tensor product of $C^*$-algebras

Let $f$ be a faithful state on a $C^*$-algebra $A$, i.e. $f(a^*a)=0$ implies $a=0$. in general, call a mapping $T:A \to B$ between $C^*$-algebras faithful if $T(a^*a)=0$ implies $a=0$. How to prove ...
2
votes
1answer
65 views

map between $C^*$-algebras with special properties, why is $f(a^2)=f(a)^2 $ for all $a\in A$?

Let $A$ and $B$ unital $C^\ast$-algebras, $f:A\to B$ a linear, bounded map such that $f(a^*)=f(a)^*$ for all $a\in A$, $f(1_A)=1_B$ and $f(a)f(b)=0$ for all $a,b\in A_{sa}$ with $ab=0$. Follows ...
0
votes
0answers
17 views

reference for groupoid cohomology

In nLab (groupoid cohomology) says: "Under the homotopy hypothesis theorem, plain (non-internal) groupoid cohomology is the same as the cohomology of homotopy 1-types." Are there references for ...
0
votes
1answer
32 views

How to write a matrix with some constraints?

I want to write down an $m\times n$ $0/1$ matrix such that every row is distinct and every column is distinct. If I want to write with only rows or columns distinct, I could just pick $m$ or $n$ ...
0
votes
0answers
44 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 ...
5
votes
2answers
84 views

Random suborbits of a rotation

Let $u_n = x + n\alpha \pmod 1$ with $\alpha$ irrational. We know that $(u_n)_{n \geq 0}$ is dense in $\mathbb{R}/\mathbb{Z}$ (equivalently $(u_n)_{n \geq 0}$ visits every open interval infinitely ...
0
votes
0answers
17 views

How many techniques are there to test colliniarity of n points?

How many techniques are there to test coliniariry of n points? For example, suppose we have 4 points A, B , C, D. How many ways can it be tested that they are collinear?
6
votes
1answer
511 views

Green's function of the Ornstein-Uhlenbeck operator

The Ornstein-Uhlenbeck operator $L$ is given by $$ Lu = \Delta u- \frac{1}{2}x\cdot \nabla u. $$ Is there a known closed form expression of the Green's function of $L$ on $\mathbb R^d$ (for $d\geq 2$ ...
-1
votes
0answers
20 views

is graph coloring problem in general np-complete?(solvable)

graph coloring problem Hi, i tried to find an algorithm for this problem and i want to make sure. i found it with this knowledge. 1.is graph coloring problem in general to find the ...
0
votes
0answers
18 views

Box counting dimension of the graphs of functions on $\mathbb R \rightarrow \mathbb R$

Generally speaking box counting techniques are applied to fractals defined by some iterative process, but what about functions? Has the concept of box counting dimension been investigated on the graph ...
3
votes
4answers
3k views

Linear Regression Coefficients W/ X, Y swapped

Let's say I have a linear regression model of the form $ y = B_x x + I_x + \epsilon $, where $B_x$ is the beta coefficient of the $x$ term, $I_x$ is the intercept term and $\epsilon$ is additive, ...
-1
votes
0answers
45 views

Problem about the group theory in dummit [on hold]

I am struggling with Problem 43 of 3.1 of Dummit's algebra book. The problem is: Assume $P=\{A_i\}$ is any partition of $G$ with the property that a "quotient operation" is defined as follows: to ...
5
votes
2answers
476 views

flat metrics on the 2-sphere with conical singularities

Consider some flat metric on $S^2$ with a fixed finite number of conical singularities $p_1,\ldots,p_n$. What is the moduli space of such metrics up to isometry? In particular what is its dimension?
6
votes
2answers
375 views

On combinatorial and cellular model categories and infinity categories

I am looking for a counterexample. Let me first give the set-up. When you work with model categories, it is extremely common to assume they are cofibrantly generated. For me, this means the definition ...
0
votes
1answer
97 views

A particular interesting elliptic curve

Given the elliptic curve $E:y^2=x^3-4x+4$. (a) How to prove that the group of rational points $E(\mathbb{Q})$ is generated by $P=(2,2)$. (b) If we consider the piece of curve on the region ...
2
votes
2answers
216 views

Can the projective line be provided with a ring structure?

A definition of multiplication on the projective $1$-points $(a:b)$ of $P_K^1$ with $a$ and $b$ elements of a field $K$ ( e.g. the real or rational numbers ) can be given by mimicking the ...
44
votes
3answers
6k views

Is a “non-analytic” proof of Dirichlet's theorem on primes known or possible?

It is well-known that one can prove certain special cases of Dirichlet's theorem by exhibiting an integer polynomial $p(x)$ with the properties that the prime divisors of $\{ p(n) | n \in \mathbb{Z} ...
1
vote
0answers
26 views

Derived Deformations of associative algebras

Let $k$ be a field (if necessary for my question, we can assume its characteristic to be zero). In the non derived context, we can then define deformations of an associative algebra $S$ as follows: ...
1
vote
0answers
53 views

Can someone explain some of the steps in this paper clearly?

I'm having trouble understanding the steps this paper makes to come to the conclusion $p_{f}(d) \sim e^d\sqrt{d}$ Marek Wolf, First occurrence of a given gap between consecutive primes, preprint, ...
5
votes
1answer
259 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
0answers
23 views

Egyptian fractions similar to Erdos-Straus conjecture

It is known that the Erdos-Straus conjecture is about writing $4/n$ as three unit fractions. My question is that if it is known that if $a>4$ $$ \frac an=\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_k} ...
3
votes
3answers
165 views
+50

Terminology for polygons

As you may know term "polygon" might mean few different things and its meaning has to guessed from context. By some reason I have to use few of these meaning in one place. So I converge to the ...
0
votes
2answers
32 views

Connectivity of weighted graph and zero Laplacian eigenvalues

Given an undirected graph $G$, and let $V$ denote its set of vertices and $E$ its set of edges. Suppose that there are no edges connecting the same vertex, and no more than one edge connecting any ...
11
votes
1answer
261 views

No limit shape for random Young diagrams under z-measure?

In their paper Random partitions and the Gamma kernel (Advances in Mathematics 194 (2005) 141–202), Borodin and Olshanski state that: An important difference between the Plancherel measures and ...
0
votes
0answers
20 views

How possibly does a $\alpha$-stable process jump at this stopping time?

Lemma 2.3.2 of [Applebum2009] states that, If $X$ is a Levy process and let $\Delta X(t) = X(t) - X(t-)$, then $\Delta X(t) = 0$ almost surely for a fixed $t>0$. There is also a warning that, ...
3
votes
0answers
49 views

Proof of a Fourier pair with Bessel functions?

How can we prove that the Fourier transform of the function $$ f(x) = \begin{cases} (a^2-x^2)^{c/2} BesselJ[c,b\sqrt{a^2-x^2}] & \text{for }x^2 < a^2\\ 0 & \text{otherwise} \end{cases} $$ ...
8
votes
1answer
270 views

Does “cardinal arithmetic is well-defined” imply axiom of choice?

Let me quickly explain what I mean with my question. Let $(\kappa_i)_{i\in I}$ be a collection of cardinal numbers, indexed by elements of some set $I$. We can try to define $\sum\limits_{i\in ...
4
votes
4answers
216 views

Deceptive linear algebra problem

Does anyone recognize this problem? There are $2p$ equations and $2p$ unknowns, and it feels like a classic, but I've never encountered it before: Given $d_1, d_2, \ldots d_p$, find $a_1, a_2, \ldots ...
5
votes
0answers
135 views

How algebraic is the holonomy map?

Let $G$ be either a compact, simple, simply-connected Lie group, or a simply-connected complex reductive group (so either $SU(n)$ or $SL(n,\mathbb{C})$, for instance), or even complex affine. I don't ...
-1
votes
0answers
79 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 ...
5
votes
2answers
155 views

Lindel's theorem for semisimple simply connected G

Let $k$ be a field. $G/k$ be a simply connected semisimple algebraic group. Let $X/k$ be a smooth affine $k$-scheme. Question: Is every principal $G$ bundle on $X\times {\mathbb A}^1$ a pull back ...
0
votes
1answer
88 views

Is it possible to represent non-linear ranking type constraints as equivalent linear constraints?

I have formulated a linear program with binary indicator variables $z_i(a)$ which is equal to $1$ if the $i^{th}$ document is of rank $a$ and $0$ otherwise. The other variables in the linear ...
3
votes
1answer
172 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
1answer
4k views

Product of Positive Matrices

Is the product of non-negative definite matrices also non-negative definite? If not, let A and B be non-negative definite matrices, is '$\operatorname{tr}(A^T B) \ge0$' ?
71
votes
8answers
8k views

What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?

I know the following facts. (Don't assume I know much more than the following facts.) The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem. The ...
4
votes
0answers
113 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 ...
2
votes
1answer
205 views

Rank changes with matrix edits

Assume we have rank $r$ real matrix $M\in\{0,1\}^{n\times n}$ (not constrained to symmetric). Assume $W\in\{0,1\}^{n\times n}$ is rank $1$ real matrix. Case $1$: $M+W\in\{0,1\}^{n\times n}$. Could ...
2
votes
4answers
378 views

Integral points on a particular family of curves

This is a follow-up to this question (and comments thereon). Namely, it follows from Felipe Voloch's comment that for any $n>2$ there is a finite set of integral $(x, y),$ such that $$ ...
3
votes
1answer
655 views

Special automorphisms of extraspecial groups

Let $G$ be an extraspecial group of order $p^{2r+1}$ (where $p$ is an odd prime), and let $V$ be a faithful representation of $G$. Consider the normal subgroup $H$ of $Aut(G)$ consisting of all ...
0
votes
0answers
41 views

Decomposition of polynomial ring as $S_n$-module [migrated]

I want to whether there is a containment relation between the $S_n$-modules $\mathbb{C}S_n$ and $\mathbb{C}[x_1,\ldots ,x_n]$. Is it true that $\mathbb{C}[x_1,\ldots ,x_n]$ contains an isomorphic copy ...
3
votes
1answer
59 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 $D: \mathscr{A}^{op} \times \mathscr{B}\to Set$. ...
4
votes
1answer
254 views

On various “extension closures” and “orthogonals” in triangulated categories

A vague form of my question is the following one: for a class of objects $D$ of a triangulated category $C$ we consider the class $E$ of objects that satisfy $Mor_{C}(d,e)=\{0\}\ \forall d\in D$; ...
3
votes
2answers
178 views

Triangles in rigid Riemann surfaces

Edit: We thank Vladimir Matveev for his comment on this post which leeds us to revise the question as follows: Assume that $M_{g}$ is a compact Riemann surface with constant negative cuvature (That ...
0
votes
1answer
302 views

Number of Dyck paths with k returns and b peaks

The number of Dyck paths from the origin to $(2n,0)$ which touch the $x$-axis $k+2$ times ($k$ internal touches) is given by $$\frac{k}{2n-k}{2n-k \choose n}.$$ The number of Dyck paths from the ...
2
votes
1answer
186 views

Two questions about Whittaker functions

I am watching the video: Modeling p-adic Whittaker functions, Part I. I have two questions about Whittaker functions in the video. From 33:00 to 37:00, it is said that after changing of variables, ...
1
vote
0answers
33 views

asymptotic behavior of the solution of an ordinary differential equation

I am a civil engineer with basic mathematics skills and need help for the following - perhaps simple - problem. Consider the following autonomous system of two non-linear ordinary differential ...
0
votes
0answers
28 views

One parameter family of elliptic equations

Consider the following 2nd order nonlinear elliptic equation on $\mathbb{R}^n$: $$-\Delta \varphi + \sum_i a_i(x, \varepsilon)\partial_i \varphi + \varphi = N(\varphi),$$ where $N$ is a smooth ...
6
votes
1answer
288 views

Time averages and differentiability

Let $\varphi_t : M \rightarrow M$ be a smooth flow on a smooth manifold $M$. We may assume (although I'm not sure if this is important) that the flow preserves a smooth volume form on $M$. Given a ...
4
votes
1answer
169 views

Rellich's theorem from compact resolvent

On a compact Riemannian manifold, we know that the Laplacian $\Delta$ has compact resolvent. In proving this, one typical way is to use Rellich's theorem about the compact embedding of $H^1(M)$ into ...

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