This tag is used if a reference is needed in a paper or textbook on a specific result.

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8
votes
2answers
445 views

What is an ordered structure, in general?

This is basically a reference request, but the post is going to be relatively long (and a little bit verbose): I apologize in advance for that. Premise. There are several examples of "ordered ...
1
vote
1answer
89 views

Exponential sum estimates similar to the one for $\sum_p (\log p) e(p \alpha)$, but for different sequences

Obtaining a non-trivial estimate for $\sum_p (\log p) e(p \alpha)$ over the minor arcs is one of the estimates required for obtaining the ternary Goldbach for $n$ sufficiently large via the circle ...
1
vote
0answers
42 views

“Harmonic oscillator” with $p$-Laplacian

I wonder if there is any literature on the eigenvalue problem for the "$p$-harmonic oscillator" $$-(|u'|^{p-2}u')'(x)+(x^2-\lambda) |u(x)|^{p-2} u(x)=0$$ in $L^p(\mathbb R)$, $p\in(1,\infty)$. Are ...
2
votes
1answer
50 views

Complexity of recognizing equivalent translation surfaces

"A translation surface is a union of polygons with pairs of parallel edges identified by translation, up to cut and paste equivalence." I take that succinct (and not fully precise) definition ...
1
vote
1answer
84 views

Characterisation of the square root of the Laplacian as a Dirichlet to Neumann mapping

I am looking for a (classical and/or oldest) reference giving the characterisation of the operator $(-\Delta)^{\frac 12}$ as the Dirichlet to Neumann map $w_y$ where $w$ is the harmonic extension on ...
2
votes
0answers
203 views

Why is the normalization of a general fiber the general fiber of the normalization?

Suppose $X \rightarrow Y$ is a map of reduced connected projective schemes of finite type over an algebraically closed field of characteristic 0, where $Y$ is a smooth connected curve. Let $Z ...
3
votes
1answer
97 views

Homological characterisation of standardly stratified algebras using Ext

Let A be a finite dimensional algebra and $S_1,S_2,...,S_n$ the simple $A$-modules and $P_1,..,P_n$ the indecomposable projective $A$-modules. For $i=1,...,n$, define the standard module $\Delta_i$ as ...
2
votes
0answers
104 views

Integral cohomology of elementary abelian groups

Let $p$ be a prime. I am looking for a reference or a short proof for the fact that cohomology groups $H^i((\mathbb{Z}/p\mathbb{Z})^n,\, \mathbb{Z}),\, i>0,$ have exponent $p$ (i.e., that they are ...
4
votes
0answers
143 views

Does the cohomology comparison part of GAGA hold over the reals?

If $k=\mathbb{C}$, $X$ is a projective (or even proper) scheme over $k$ and $F$ a coherent sheaf on $X$, then there is an isomorphism $$ H^p(X,F)\rightarrow H^p(X^{an}, F^{an}) $$ (and there is also ...
2
votes
2answers
215 views

Maximal size of minimal generating set

Let $G$ be a finite group. Denote by $D(G)$ the maximal size of a minimal generating set, (minimal in the sense of inclusion). I vaguely remember seeing recently something on $D(G)$. Can anyone refer ...
5
votes
0answers
100 views

When is a Hermitian matrix of the form $g^*g$ for some matrix $g$

I tried asking this question on Math StackExchange and didn't get any replies. I was hoping that maybe someone here could help, sorry for duplicating. I'm trying to figure out some properties of ...
5
votes
1answer
262 views

Good references for K-theory of modular curves?

The title says it. I am looking for a good exposition on the K-theory of the curves $X_{i}(N)$, $Y_{i}(N)$, where $i\in\{0,1\}$. I have some background in $K$-theory and also some background in ...
4
votes
0answers
208 views

The original proof of Szemerédi's Theorem

Nowadays there are plenty of different proofs of the celebrated Szemerédi's Theorem but for historical reasons I would like to read and understand the original proof. The proof is very tricky and, for ...
2
votes
1answer
53 views

Bounding exceedance probabilities for correlated normal variables

Suppose $y\sim N(0,\Sigma)$ is an $n-$dimensional vector. I'm interested in an upper bound for $\Pr(\max_{1\leq i\leq n} y_i > k)$ for $k$ large. I know a little about $\Sigma$: ...
0
votes
1answer
90 views

Finding many disjoint sub-trees with many leaves

Let $T$ be a rooted binary tree with $L$ leaves, and let $\ell$ be a natural number smaller than $L$. The question is what is the maximal number of disjoint rooted sub-trees with at least $\ell$ ...
1
vote
0answers
139 views

Looking for Uehara, Massey article

Not sure if this is the right place to ask this kind of a question. But I cannot find the following article: Uehara, Hiroshi; Massey, W.S. The Jacobi identity for Whitehead products. Algebraic ...
0
votes
0answers
110 views

Path integral methods

Are there detailed expositions of the path integral methods in (mathematical) physics other than Feynman-Hibbs and Glimm-Jaffe?
27
votes
0answers
659 views

Grothendieck's “List of classes of structures”

In Lawvere's article Comments on the Development of Topos Theory, the author writes: Similarly, Grothendieck and others unerringly recognized which kinds of mathematical structures are 'preserved ...
0
votes
1answer
155 views

If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, is $n$ squarefree?

(I have asked a similar question in MSE around a week ago, but did not receive any responses. I have therefore cross-posted it to this site, hoping to get some answers.) An odd perfect number $N$ is ...
2
votes
1answer
92 views

Blaschke Condition for hyperbolic lattices

For $r$, $s$, small positive integers, do the complex numbers on the unit disc (without the hyperbolic metric) corresponding to the vertices of the hyperbolic tiling with Schläfli symbol $\{r,s\}$ ...
3
votes
1answer
96 views

Duistermaat-Heckman integral formula on compact manifold with boundary

Let a compact Lie group $G$ acts on a closed symplectic manifold $(M,\omega)$. If the action is Hamiltonian with $\mu$ the moment map, then the integral $$\int_M e^{i\mu (X)+\omega}$$ is equal to the ...
0
votes
0answers
33 views

Stochastic dominance for subsets

The subsets of a set $N=\{1,2,\ldots,n\}$ form a lattice, with larger sets being higher up, and a subset $B$ connected to another subset $A=B\cup\{x\}$ (for any $x\not\in B$) higher up by a "pipe". ...
0
votes
0answers
19 views

A Statement about a General Property of Negative Cycle Detection Algorithms

in this paper from 1999, the authors Boris Cherkassky and Andrew Goldberg state in the abstract that "The negative cycle problem is to find a negative length cycle in a network or to prove that ...
7
votes
2answers
205 views

The Picard number of the Kummer surface of an abelian surface

Let $A$ be an abelian surface and $\text{Km}(A)$ be the Kummer surface of $A$. If I remember correctly, the Picard number $\rho(\text{Km}(A))$ is equal to $16+\rho(A)$. Does anyone know any ...
0
votes
0answers
93 views

A priori estimate for diffraction problem for linear elliptic PDEs

I am looking for a reference to show how to obtain a priori estimate of the solution $u\in H^1$ and $u\in C^{2,\alpha}$ to the diffraction problem of linear elliptic equation. I looked at ...
2
votes
2answers
140 views

Commutator 2-forms on Lie groups

Let $G$ be a compact Lie group and $\mathfrak g$ its Lie algebra. For any $f$ in the dual space $\mathfrak g^*$, we can define a skew-symmetric bi-linear form on $\mathfrak g$ by $(A,B)\mapsto ...
4
votes
2answers
174 views

How many minimal surfaces do we have if the metric in the target space is not flat

It is trivial that there are a lot of minimal surfaces in the flat $R^3$: for example, for any point, any 2-plane containing this point, and any two othogonal vectors in this plane, and any ...
5
votes
0answers
93 views

Gleason’s Theorem for non-separable Hilbert spaces: On a theorem of Solovay

Gleason’s theorem plays an important role in the foundations of quantum mechanics. On the positive side it demonstrates how the probabilistic structure of quantum theory follows from its logical ...
5
votes
1answer
353 views

Anything about $\prod_{n \ge 1} (1 + n^{-n})$?

Sophomore's dream is especially the statement that the sum, let me call it $s$, of the (convergent) real series $\sum_{n \ge 1} n^{-n}$ is equal to the (improper) integral $\int_0^1 x^{-x} dx$. A few ...
4
votes
1answer
153 views

Reference for a Heat Process in a Wedge

I would like to ask about an explicit suggestion/reference for the following type of heat processes: Roughly, assume we have a "wedge" $W$ of the following form - a domain in $\mathbb{R}^n$ with a ...
3
votes
2answers
149 views

Bellman-Ford for Matching Problems?

I am looking for a simple way of calculating minimum-weight perfect matchings in complete graphs with an even number of vertices. I know that there are implementations that are based on Edmond's ...
6
votes
1answer
281 views

Embedding of real trees into $\ell_1(\Gamma)$

It seems plausible that any real tree or ${\mathbb{R}}$-tree in the sense of the definition in https://en.wikipedia.org/wiki/Real_tree admits an isometric embedding into the Banach space ...
3
votes
0answers
70 views

Classification of representation-finite algebras up to stable equivalence of Morita type

assume K is an algebraically closed field. I wanted to ask if there is a classification of the representation-finite K-algebras up to stable equivalence of Morita type at least for some small numbers ...
8
votes
0answers
162 views

A question about strongly compact cardinals

Is the following equiconsistent with the existence of a strongly compact cardinals: For every $\lambda > \kappa$ there exists a $\lambda$-strongly compact embedding $j: V \to M$ with the ...
2
votes
1answer
131 views

Additive functors to abelian groups: “additional structure” and functors induced by “additive pseudo-functors”; references?

Let $\mathcal{A}$ be a small additive category. Consider the category $PreSh(\mathcal{A})$ of all additive functors from $\mathcal{A}^{op}$ into abelian groups; note that this category is abelian and ...
9
votes
0answers
168 views

Algebraic K-theory of a ring.

I started to learn some algebraic $K$-theory and its relation to geometric topology problems. My question is : What is the list of rings such that all their algebraic $K$-theory groups are known ? I ...
3
votes
1answer
127 views

Is it possible for a random nowhere dense closed set to have a positive probability of hitting any given point?

Given a compact metrisable topological space $X$, we write $\mathcal{N}(X)$ for the set of non-empty closed nowhere dense subsets of $X$, which is a Polish space under the topology induced by the ...
4
votes
1answer
188 views

Mazur's Galois Deformations paper for non-residually irreducible case

In Barry Mazur's paper introducing Galois deformations, he hints at having a general theory for representations which are not residually Schur, but with more complicated statements. Does anyone know ...
1
vote
0answers
43 views

References for a minor variant of the Rayleigh quotient

I believe this variant of the Rayleigh quotient inequality must be well known but I could not find references for it online. It's proof is straightforward. Let $\mathbf{A}\in\mathbb{R}^{n\times n}$ ...
0
votes
0answers
65 views

Grobner basis for a general algebra

Let $R$ be a quotient of the polynomial ring $\mathbb{C}[x_1,\dots , x_n]$. We fix a $\mathbb{C}^*$ action on $R$ which preserve homogenous components and the multiplication. (The geometric analogue ...
4
votes
0answers
76 views

Kruskal-Katona for homocyclic groups?

I need a version of the Kruskal-Katona theorem (better still, of the Lovasz "approximate" version thereof) for the elementary abelian / homocyclic groups, in the following spirit: What is the ...
4
votes
0answers
163 views

Lie group structure over diffeomorphisms group

Let $M$ be a smooth manifold. Is it true that for every subgroup $G$ of $diff(M)$ there is at most one Lie group structure on $G$ such that the natural left $G$-action on $M$ is smooth? Edit: by "Lie ...
6
votes
1answer
269 views

Encyclopedia of properties of nonnegative matrices

I'd like to buy a book that contains more or less all known properties of elementwise nonnegative nonnegative matrices, i.e. matrices $A$ such that $A_{i,j}\geq 0$ for all $i,j$. Chapter 8 in ...
8
votes
1answer
146 views

Is the modularisation of a unitary fusion category always unitary?

Suppose $\mathcal{C}$ is a unitary ribbon fusion category. Also assume that its symmetric centre has trivial twist and trivial pivotal structure, i.e. is tannakian. Thus, the Müger/Bruguières ...
2
votes
0answers
134 views

On the use of the term “field of sets” in Maharam's papers

I am reading some papers by D. Maharam, and feel a little bit confused about her use of the term "field of sets". Nowadays, I think the term is standardly used to mean a pair $(X, \mathscr{F})$ for ...
13
votes
1answer
207 views

Reference request: Morita bicategory

I have two closely related questions: Who first recognized that Morita equivalence was equivalence in the bicategory of Rings, Bimodules, and Intertwiners? I've heard this bicategory called the ...
6
votes
1answer
171 views

Darboux property of non-atomic sigma-additive nonnegative measures equivalent to the AC?

A result commonly, and probably erroneously, attributed to W. Sierpiński is that every non-atomic, countably additive, nonnegative measure $\mu: \Sigma \to \bf R$, where $\Sigma$ is a sigma-algebra on ...
3
votes
0answers
113 views

Hamiltonian on the torus

In discrete models like Ising we have Hamiltonians of the form $$H(\sigma)=\frac{1}{N}\sum_{i=1}^{N}J_{ij}\sigma_{i}\sigma_{j},$$ where $\sigma_{i}=\pm 1$ , $J_{ij}$ interaction coefficients and N ...
0
votes
1answer
73 views

Boundary behaviour of a second order pde with characteristics

Good morning everybody. My question is inspired from the following fact: Consider $\mathbb R^3$ endowed with coordinates $(x,y,z)$. Of course if we were to solve the second order pde $\partial_x^2 ...
5
votes
1answer
270 views

reference request for mod p and p-adic K-theory

Is there a good reference that explains mod p K-theory and p-adic or p-complete K- theory? All I know about K-theory is the topological K-theory of "vector bundles and k-theory" in Switzer's book ...