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

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2
votes
0answers
105 views

Terminology question for maps between posets

Let $P$ and $Q$ be two poset (partially ordered sets) and $\phi : P \to Q$ an order-preserving function. I would like to know whether there is a name and perhaps a different characterizations of such ...
0
votes
0answers
133 views

Does the proportion of non trivial zeroes of given real part increase on [0,1/2] for all L-functions?

Here what I call an L-function is either an element of the Selberg class or an automorphic L-function. For such a function $F$, $x\in [0,1/2]$ and $T\gt 0$, let's define $\delta_{F,T}(x)$ such that ...
2
votes
1answer
119 views

Local Biot-Savart law in $B(x_o,r) \subset \mathbb R^2$

Let $u: \mathbb R^2 \to \mathbb R^2$ and let $\omega = \text{curl } u$ be the 2D vorticity of $u$, where $u, \omega \in L^2(\mathbb R^2)$ and $\nabla \cdot u = 0$. The classical Biot-Savart law states ...
10
votes
2answers
390 views

What is the smallest 4-chromatic graph of girth 5?

It is known that the smallest 4-chromatic graph of girth 4 is the Grötzsch graph (11 vertices). What happens for girth 5? The Brinkmann graph (21 vertices) has chromatic number 4, girth 5 and is ...
1
vote
2answers
99 views

Reference for the proof of a neighbourhood characterisation of cofibrations

I am interested in a reference for the proof of the following theorem for $A,X$ being CGWH topological spaces. Let $A\subset X$ be a closed subspace, such that there exists a continuous $\phi : ...
4
votes
0answers
57 views

A “gradient” weak Harnack inequality for quasilinear elliptic equations

Suppose we are in the following loosely described setting: we have a non-negative supersolution $h$ of the following elliptic equation: \begin{equation} \Delta h + \|\nabla h\|^2 + f(x) \geq 0 ...
4
votes
0answers
85 views

Continuity of the curve-shortening flow with respect to the curve

The curve-shortening flow is an evolution equation for a smooth closed curve $\alpha$ inmersed in a Riemannian surface $M$. The version where $M$ is the Euclidean plane is illustrated for example in ...
1
vote
0answers
56 views

Stochastic Resonance in Infinite Dimensions

I'll ask this from the point of view of physics more than of theoretical mathematics. I'm searching for a mathematical discussion of stochastic resonance interpreted in a PDE sense. This is a good ...
1
vote
3answers
577 views

Brouwer vs. Cantor

Brouwer criticises Cantor e.g. in Intuitionistiche Mengenlehre. Is there a link or reference to some streamlined modern account of Brouwer's ideas?
2
votes
0answers
198 views

The kernel of $C^{*}(G)\to C_{r}^{*}(G)$

Let $G$ be a locally compact group. Put $I(G)=\ker: C^{*}(G) \to C_{r}^{*} (G)$, the kernel of the canonical morphism. What type of $C^{*}$ algebras can not be isomorphic to $I(G)$, for some ...
11
votes
2answers
589 views

What is the longest recorded gap between “proof” of a “theorem” and discovery that the result is false [duplicate]

I hope this question is not a duplicate. I am motivated by wondering when widely accepted results may be considered have a secure place in the mathematical literature. The question is intended to ...
4
votes
0answers
97 views

Unit eigenvalue of the linearized Poincare return map

Consider a surface $S$ and a vector field on the surface which has a closed orbit. The vector field on both sides of the closed orbit spirals towards it, which gives us that the linearized Poincare ...
7
votes
4answers
376 views

quasicrystal and penrose tiling, mathematical introduction

Starting to research on quasicrystal from material science, I want to know more about how to understand quasicrystal from a purely mathematical (especially tiling) perspective (probably start from ...
2
votes
1answer
109 views

Totally aperiodic sequence

Let $A$ be a finite set. Let $A^k$ be the set of words in the alphabet $A$ of length $k$ and $A^*$ be the set of infinite words. I was looking for an element $a = \lbrace a_n \rbrace_{n \in ...
1
vote
2answers
139 views

spaces of probability measures on a Polish space and the convergence

I want to read the topic "spaces of probability measures on a Polish space and the convergence". What is the best reference for that ?
2
votes
1answer
56 views

What is Known about Preprocessing for Stabbing Queries?

In a concrete setting, I have the following problem: given a fixed set of simple objects (e.g. disks or, convex polygons with few vertices), I need to quickly report the objects that are hit (i.e. ...
6
votes
0answers
315 views

Large sets not containing arithmetic progressions of length 3 in intervals

Given a large enough natural number $N$, let $\Delta_N=\{A \subseteq [N, 2N]: A$ contains no arithmetic progressions of length $3 \},$ where for natural numbers $N<M$ we have $[N, M]=\{N, N+1, ..., ...
2
votes
1answer
150 views

Reference for “multi-monoidal categories”

I have attempted to find a definition of a monoidal category which incorporates $n$-fold tensor products instead of just binary tensor products. Definition. A "multi-monoidal category" consists of ...
4
votes
1answer
230 views

Schrodinger equation with magnetic vector potential

In many papers dealing with the Schrodinger equation with magnetic potential $$u_t=i(\nabla+iA(t,x))^2u$$ the authors say that this equation can be studied with Kato's methods for abstract evolution ...
11
votes
3answers
888 views

Non-abelian Grothendieck group

By general nonsense the forgetful functor from groups to monoids has a left adjoint. It maps a monoid $(X,\cdot,1)$ to the free group on $\{\underline{x} : x \in X\}$ modulo the relations ...
3
votes
1answer
165 views

Reference for puzzle on dividing piles and scoring products

There is a pile of $n$ items. Every time you divide a pile into two piles, you get a score being the product of the number of items in the two piles. Show that the sum of your scores at the end is ...
0
votes
1answer
159 views

Cauchy problem for an overdetermined system of PDE

This question is strictly related to this one. Let us consider the differential system with constant coefficients $$\left(\begin{array}{ccc} B_{11} & B_{12} & 0\\ ...
1
vote
1answer
92 views

Heat Kernel estimate at the level of the form

Let $(M,g)$ be a compact Riemannian manifold. It is known there exist Gaussian estimates of the heat kernel and its derivatives acting on functions on $M$. The kind of estimate I'm looking for could ...
5
votes
1answer
92 views

Number of regions of a hyperplane arrangement avoiding a generic hyperplane

Let $\mathcal{A}$ be an essential arrangement of hyperplanes in $\mathbb{R}^n$. Zaslavsky's theorem says that the number of regions of $\mathcal{A}$ is given by ...
23
votes
2answers
698 views

Forcing as a replacement of induction and diagonal arguments

Let me give some examples motivating the question. The use of forcing instead of induction: For this consider Cantor's theorem: Theorem 1. Any two countable dense linear orders $I, J$ without end ...
2
votes
2answers
242 views

Can homotopy pullbacks of spaces be checked on fibers?

As should be clear, I would like to know if it is true that a given commmutative square of spaces (i.e. simplicial sets) is a homotopy pullback iff the induced map on each homotopy fiber is a weak ...
2
votes
1answer
228 views

Is there a name for this cardinal?

Let $X$ be a set and $\omega$ be a family of its subsets. Consider the family $\mathcal{F}$ of subsets of $X$, such that any $A\in\mathcal{F}$ has a non-empty intersection with each element of ...
10
votes
1answer
433 views

Metric $d(A,B) = \mathbb P(\overline A\cup\overline B\mid A\cup B)$

I'm wondering where the relative probabilistic distance was first studied: $$d(A,B) =\mathbb P(\overline A\cup\overline B\mid A\cup B)$$ where $\overline A$ is the complement of $A$. A web search ...
1
vote
1answer
203 views

Error term for prime harmonic

What is known about the asymptotic behavior of $$ f(x)=\sum_{p\le x}\frac1p-\log\log x-B_1? $$ Of course by Mertens we know that $f(x)=o(1),$ but has more been proved (in terms of $O$ or ...
1
vote
2answers
197 views

Polar Coordinate Systems on Manifolds [closed]

Is there agreement on how to interpret $r$ and $\varphi$ on a manifold if a reference point and a reference direction are given, or, put differently, does the definition of a reference point and, of a ...
1
vote
2answers
71 views

Reference request: minimal (maximal) Lorentzian surfaces in $\mathbb{R}^{1,2}$

Let $R^{1,2}$ be the Minkowski 3-space, I would like to know any references about minimal (maximal) orientable Lorentzian surfaces in $\mathbb{R}^{1,2}$, including examples and maybe general theories, ...
0
votes
1answer
108 views

Heat Equation on LCA Group

I am reading The Laplacian on A Riemannian Manifold. In the book, author defines the heat equation on manifold. In the situation $\mathbb R$ and $\mathbb S$, people often use Fourier transformation ...
0
votes
1answer
78 views

Practical Algorithm for Comparing the Discrepancy of Point Sets (on Unit Hyper Spheres)

I have devised a simple geometric algorithm for generating a sequence of points on unit hyper spheres; that algorithm depends on a single real parameter, which I would like to optimize in order to get ...
3
votes
1answer
134 views

distribution discretization

Let $\mu$ be a distribution on $\mathbb{R}^n$. We partition $\mathbb{R}^n$ into small cubes congruent with $[0,\delta)^n$, parallel to the axes. In each cube, pick a point $x$ (for instance, the ...
5
votes
1answer
269 views

Itô's article “A measure-theoretic approach to Malliavin calculus”

Apart from citations all over the internet, the following paper appears to be off-the-grid. K. Itô, A measure-theoretic approach to Malliavin calculus, in 'New Trends in Stochastic Analysis', Proc. ...
5
votes
1answer
203 views

A hyperbolic group with a small profinite completion

Is there a finitely generated non-elementary word hyperbolic group the profinite completion of which is known (or conjectured) to be rather restricted, that is: abelian, pro-$p$, virtually ...
1
vote
1answer
111 views

basis of the lattice generated by the integer points inside a subspace of R^L

Consider $K$ linearly independent vectors $\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{a}_K \in \mathbb{Z}^L$, where $1 \leq K<L $. Hence, the span of $\lbrace\mathbf{a}_1, \mathbf{a}_2, ..., ...
4
votes
0answers
200 views

Reference Request for TQFTs

I had originally asked this question on Math StackExchange but have not obtained any answers, so I decided to post this here. (I have flagged the MSE post to be moved to MathOverflow, for the ...
9
votes
0answers
185 views

Is the quotient map of the action of homeomorphisms on embeddings well-behaved?

It is well known that if $M$ and $N$ are smooth manifolds, the diffeomorphisms $Diff(M)$ act continuously on the smooth embeddings $Emb^{C^\infty}(M,N)$ by precomposition, if both are given the ...
1
vote
2answers
148 views

Local structure of the quotient of a Lie group action

Suppose $M$ is a smooth manifold and a compact Lie group $G$ acts freely on $M$, then it is well known that $M/G$ has a manifold structure. Are there any results for the general case? (a) If the ...
1
vote
1answer
97 views

asymptotic for restricted partitions

Let $m$ and $n$ be two positive integers and denote by $P(n,m)$ the number of partitions of $n$ into $m$ non-negative integers. Is there an asymptotic formula for $P(n,m)$ ?? Any reference is ...
5
votes
0answers
261 views

Transforming a binary matrix into triangular form using permutation matrices

I am interested in the complexity of the following problem: Given an $m\times n$ binary matrix $M$, can we permute its rows/columns to obtain a triangular matrix? I am also interested in ...
0
votes
0answers
37 views

References about reality of minimal affinizations of quantum affine algebras

Let $U_q(\widehat{\mathfrak{g}})$ be the quantum affine algebra associated to a complex simple Lie algebra $\mathfrak{g}$. A simple module $M$ of $U_q(\widehat{\mathfrak{g}})$ is called real if $M ...
49
votes
4answers
4k views

Are there any serious investigations of whether “mathematicians do their best work when they're young”?

There is no shortage of anecdotes and conjectures on both sides of this widespread belief, but good supporting data either way is harder to find. Can anyone provide any references for serious ...
7
votes
1answer
345 views

Discovery and Study of Conic Sections in Ancient Greece

Is there anything known about what drew the attention of ancient greek mathematicians to conic sections and, what were the models they used to study conic sections? What I would like to know, is ...
1
vote
1answer
178 views

automorphism group of a given period

Maxim Kontsevich and Don Zagier defined the algebra of periods and conjectured that one can pass from a representation of a given period to another one using only three rules. Assuming this ...
1
vote
0answers
223 views

On the remainder term in Taylor's formula [closed]

(1) What are the main differences, in terms of "usefulness" while solving problems (even at research level), among Cauchy, Lagrange, and Schlömilch's forms of the remainder in Taylor's formula? Could ...
4
votes
0answers
163 views

What will be the consequences if second Hardy-Littlewood conjecture turns out to be true?

It is generally believed that the Second Hardy-Littlewood Conjecture is false. But it has not been proved (or disproved) yet. My question is, What would be the consequences if Second ...
12
votes
2answers
1k views

New research and re-discovering classic results in “basic” real analysis

Sometimes, it happens that researchers publish a new proof of an old well-known result in "basic real analysis" (I'm referring to what some American people may call "honors calculus"). For instance, ...
8
votes
1answer
215 views

Characterization of Frobenius complements

I have learned that Frobenius complements are characterized (among finite groups) by having a fixed point free complex representation. That is, a finite group $G$ is a Frobenius complement if and only ...