# All Questions

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 ...
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 ...
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### 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 ...
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### 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 ...
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 $n$ ...
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 ...
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### 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 ...
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### 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?
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 $d\geq 2$ ...
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### 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 ...
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### 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 ...
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, ...
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### 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 ...
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?
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 ...
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 ...
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 ...
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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 $\{ p(n) | n \in \mathbb{Z} ... 0answers 26 views ### Derived Deformations of associative algebras Let$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$as follows: ... 0answers 53 views ### Can someone explain some of the steps in this paper clearly? I'm having trouble understanding the steps this paper makes to come to the conclusion$p_{f}(d) \sim e^d\sqrt{d}$Marek Wolf, First occurrence of a given gap between consecutive primes, preprint, ... 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? ... 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$$$\frac an=\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_k} ... 3answers 165 views +50 ### Terminology for polygons As you may know term "polygon" might mean few different things and its meaning has to guessed from context. By some reason I have to use few of these meaning in one place. So I converge to the ... 2answers 32 views ### Connectivity of weighted graph and zero Laplacian eigenvalues Given an undirected graph G, and let V denote its set of vertices and E its set of edges. Suppose that there are no edges connecting the same vertex, and no more than one edge connecting any ... 1answer 261 views ### No limit shape for random Young diagrams under z-measure? In their paper Random partitions and the Gamma kernel (Advances in Mathematics 194 (2005) 141–202), Borodin and Olshanski state that: An important difference between the Plancherel measures and ... 0answers 20 views ### How possibly does a \alpha-stable process jump at this stopping time? Lemma 2.3.2 of [Applebum2009] states that, If X is a Levy process and let \Delta X(t) = X(t) - X(t-), then \Delta X(t) = 0 almost surely for a fixed t>0. There is also a warning that, ... 0answers 49 views ### Proof of a Fourier pair with Bessel functions? How can we prove that the Fourier transform of the function$$ 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} $$... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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' ? 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 ... 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 ... 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 ... 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$$ ... 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 ... 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 ... 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$D: \mathscr{A}^{op} \times \mathscr{B}\to Set$. ... 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$Mor_{C}(d,e)=\{0\}\ \forall d\in D$; ... 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 ... 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 ... 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, ... 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 ... 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 ... 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 ... 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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