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

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1
vote
1answer
119 views

Fourier approximation error in L^2 for piecewise continuous functions

Let $u:[0,2\pi)\to \mathbb{R}$ be the step function $$u(x) = \begin{cases} 1 & \text{if } x \in [0,\pi), \\ 0 & \text{if } x \in [\pi,2\pi) \end{cases}$$ By a direct computation, one ...
3
votes
1answer
257 views

Classifying space for fibrations with Eilenberg-MacLane space as fibers

The following result seems to be frequently quoted: Consider the fibration $K(\pi,n)=\Omega K(\pi,n+1)\to PK(\pi,n+1)\to K(\pi,n+1)$. Let $B$ be any topological space (which is not too pathologic). ...
8
votes
2answers
527 views

Are there nontrivial involutions of $S^7\times S^7$ with fixed point set homeo to $S^7$?

The group $\mathbb{Z}_2$ acts on $S^7\times S^7$ by switching the coordinates with fixed point set $\Delta(S^7\times S^7)\cong S^7$. I want to know whether there are some other $\mathbb{Z}_2$ actions ...
2
votes
1answer
63 views

Proof for power-law tail of Poisson-Dirichlet distribution (Pitman-Yor process & Zipf's law)

I'm trying to understand the motivation of using Pitman-Yor (PY) processes in language modeling, in particular Teh's hierarchical LM based on PY processes. A motivation frequently stated in research ...
10
votes
4answers
703 views

Soft question: beginners reference to moduli spaces

What is a geometrically intuitive yet reasonably general first introduction to the theory of Moduli spaces? (Possibly introducing stacks also)? I'm looking for something which really gets the pictures ...
3
votes
3answers
176 views

Reference request for a treatment of Schwinger–Dyson equations

Is there a treatment of Schwinger–Dyson equations with no mention of Green's functions? Is there perhaps a purely algebraic analog?
0
votes
0answers
77 views

Lyapunov stability, nonlinear system

Please, is there any reference for proposition below or does it perhaps follow from a standard fact? I've got it for some other problem but I actually do not know how to comment it in my article. ...
1
vote
0answers
110 views

Thickness of the category of perfect complexes with finite length homology

Let $R$ be a commutative Noetherian local ring and let $D(R)$ be the derived category of $R$-modules. Recall that a chain complex $C_\bullet$ of modules over $R$ is called perfect if it is isomorphic ...
4
votes
2answers
137 views

open question on intersecting rectangles - reference request

In the now-inactive (and very nice) "Tricki" website that describes various proof techniques, there is an article (written I think by Timothy Gowers) that describes the following Ramsey-type problem: ...
4
votes
2answers
138 views

Weight multiplicity formulae for $(\mathfrak g,B)$-irreps

Let $G$ be a complex reductive Lie group, $B$ a Borel subgroup, with which to define "dominant weight". Let $\lambda$ be an integral weight, not necessarily dominant, but nonetheless giving a ...
3
votes
0answers
95 views

Lyapunov stability of linear system

Consider a linear ODE system $$\dot x_k=\sum_{j=1}^ma_{kj}(t)x_j,\qquad k=1,\ldots, m,\quad a_{kj}(t)\in C[0,\infty).\quad (1)$$ Proposition. Suppose that $$\sup_{t\ge ...
2
votes
0answers
148 views

Dissection of a polygon into convex polygons

Problem: for a fixed integer $m\geqslant 3$ find all $n$ such that no $n$-gon can be dissected into convex $m$-gons. I would be very grateful for any information on this problem. Remark 1. There ...
0
votes
0answers
47 views

Models for events where position and time are correlated

Apologies in advance if this question is not sufficiently research-level: What are the standard models that are used to describe phenomena in which events that occur at the same time are likely to be ...
5
votes
0answers
93 views

Explicit generators for homotopy groups of Lie groups

I would like to know explicit formulas for generators of the infinite cyclic summands in the homotopy groups of Lie groups, in the form of continuous (or smooth if possible) maps $S^n\to G$. It is ...
1
vote
3answers
299 views

smooth connected affine scheme over Z has good reduction almost everywhere

Let $f(x_1,\ldots,x_n)\in\mathbf{Z}[x_1,\ldots,x_n]$ be a polynomial. Assume that the variety cut out by $f$ is smooth and connected (so irreducible) over $\overline{\mathbf{Q}}$. Where can I find a ...
0
votes
1answer
94 views

Models of BL$\forall$

What results are known about the construction of models for a theory $T$ of the logic BL$\forall$ for languages of higher cardinality? The construction for the countable case relies on 1) The fact ...
1
vote
0answers
131 views

Which “concrete” morphisms of varieties and motives induce bijections of their lower Chow groups?

This question is a continuation of Varieties with Chow groups supported in positive codimension: examples and properties? What examples are known of morphisms of varieties and Chow motives (say, over ...
5
votes
0answers
76 views

Bound on the number of lattice points in d-dimensional ball

The following paper states that the number of lattice points in a $d$-dimensional ball of radius $R$ is $V_d R^d + O(R^\alpha)$ where $\alpha = d - 2$ and $V_d$ is the volume of the unit ...
2
votes
2answers
158 views

Standard homology result on double complexes

Suppose you have got a double complex in an abelian category with objects $(A_{rs},d_{rs})$ such that $A_{rs} = 0$ for $r < 0$ or $s < 0$. Suppose furthermore that the rows ...
1
vote
0answers
52 views

Reference needed for Hilbert-Schmidt result regarding basis of $V \subset H$

I am seeking a reference that says: If $V \subset H \subset V^*$ is a Gelfand triple with all spaces Hilbert spaces and if $V \subset H$ is a compact embedding, then there is a basis of $V$ which ...
0
votes
0answers
121 views

extension of function in an abstract metric space

my question is the following.(Maybe my title is not quite proper for this question): Let $(E,d)$ be a Polish space (or a separable metric space), let $\xi: E\to R_+$ be a Lipschitz function. Now set ...
-1
votes
1answer
61 views

Maximum size of set of points with distance bounded from below

I am interesting in finding a reference for a result of the following type: Suppose $D \subset \Bbb{R}^n$ is a bounded open set and $\delta>0$. Then the size $M$ of a family of points $F = ...
5
votes
3answers
274 views

A. Markov's papers?

A. Markov published several papers on his chains, starting in 1906, so it is written, in the journal: (1) Извѣстія Физико-математического общества при Казанском университете I am surprised by the ...
3
votes
2answers
297 views

Textbook for Partial Differential Equations with a viewpoint towards Geometry

I don't know whether I should ask this question here or not but I asked this question on MSE but didn't get any answer so I am posting it here. Though similar questions have been asked at ...
6
votes
1answer
113 views

How to “lift” a transitive group action on a manifold?

Let $M=G/H$ be a homogeneous manifold, with $G$ connected Lie group. Suppose that $\widetilde{M}$ is a covering of $M$. QUESTION: is there a general prescription to obtain a Lie group ...
6
votes
1answer
784 views

Why is the inverse of a bijective rational map rational?

Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map ...
2
votes
0answers
119 views

Varieties with Chow groups supported in positive codimension: examples and properties?

In their 1983 paper "Remarks on correspondences and algebraic cycles" Bloch and Srinivas proved several interesting properties of smooth proper varieties (over universal domains) whose Chow groups of ...
8
votes
2answers
938 views

The impact of large cardinals in mathematics [closed]

What are the main applications of large cardinals in ordinary mathematics, and what is the philosophy behind using them. In particular: Question 1. What is the philosophy behind accepting large ...
2
votes
2answers
179 views

A question about transitivity

Recently in something that I'm studying, I needed to know if the following map is transitive: $\sigma: M^{\mathbb{N}}\to M^{\mathbb{N}}$ the unilateral shift, where $M$ is a uncountable compact metric ...
1
vote
0answers
107 views

Why $D^b(S)\cong D^b_{\text{Car}}(\text{cosq}(X\rightarrow S))$?

Let $p: X\rightarrow S$ be a map between topological spaces and we can construct the simplicial space $\text{cosq}(X\rightarrow S)$ where $X_0=X$, $X_1=X\times_S X$ and $$ X_n=\underbrace{ X\times_S ...
0
votes
0answers
20 views

Wavelet transform stability to deformations

I've come across the following claim in a paper of Mallat: "High frequency instabilities [of a signal representation] to deformations can be avoided by grouping frequencies into dyadic packets in ...
1
vote
0answers
86 views

Morita Equivalence of Full Corners in $C^*$-algebras

Suppose $\mathcal{A}$ is a $C^*$-algebra with a unique normalized trace and $p \in \mathcal{A}$ is a projection so that $\mathcal{B} = p\mathcal{A}p$ is a full corner. Does $\mathcal{B}$ have a ...
12
votes
2answers
574 views

Special values of $\zeta$ outside the real line and the critical strip

The values of Riemann's function at the integers have been extensively studied. I was wondering, is there anything interesting known (or conjectured) to happen arithmetically outside the real line ...
1
vote
0answers
125 views

Strong Dependence

I don't know if this definition has been already given. Suppose $X$ and $Y$ are two random variables over finite alphabets $\mathcal{X}$ and $\mathcal{Y}$. We say $Y$ is strongly dependent on $X$ if ...
6
votes
1answer
176 views

Do the terms of an iid sequence whose law has infinite expected value necessarily exceed the partial sums of the sequence infinitely often?

Let $\mu$ be a probability measure on $(0,\infty)$, and let $(\mathbf X_n)_1^\infty$ be a sequence of independent $\mu$-distributed random variables. Fix $\kappa > 0$, and consider A) $\int x \; ...
1
vote
1answer
156 views

pencil of quadrics consisting of singular quadrics

A pencil $l$ of quadrics in $\mathbb{P}^4_{<x_0,\cdots,x_4>}$ consists of singular quadrics only if: (1) quadrics in $l$ have a common singular point; or (2) quadrics in $l$ contain a common ...
0
votes
0answers
184 views

Holomorphic vector bundles over $\mathbb{CP}^1$ and elliptic curves

Can anyone suggest a good exposition of the classification of holomorphic vector bundles over $\mathbb{CP}^1$? Also does there exist any analytic or more elementary proof of Atiyah's classification of ...
10
votes
0answers
186 views

Derivation of Blattner's conjecture in the Beilinson-Bernstein picture

On the last page of Schmid's article "Discrete Series", he says "In the Beilinson-Bernstein picture, discrete series modules are attached to closed $K$-orbits in $X$... the $K_{\mathbb R}$-structure ...
6
votes
1answer
394 views

Is there a generalized Birkhoff ergodic theorem?

Is there a Birkhoff ergodic theorem for two measure preserving transformations $T$ and $S$ where $S\circ T= T \circ S$ so that $\frac{1}{n+1}\frac{1}{m+1}\sum_{i=0}^{n}\sum_{j=0}^{m}f \circ T^{i}\circ ...
10
votes
1answer
600 views

Is there an analog of the Birch/Swinnerton-Dyer conjecture for abelian varieties in higher dimensions?

I am wondering if there is a multi-dimensional analog of the Birch/Swinnerton-Dyer (BSD) conjecture. The recent famous result inching toward resolution of that conjecture is: Bhargava, Manjul, and ...
5
votes
0answers
94 views

Reference request: Optimal $L^p$-decay for nonhomogenous heat equation in $\mathbb R^d$

Let $u$ be a classical solution for the nonhomogeneous heat equation in $\mathbb R_+ \times\mathbb R^d$: $$ \begin{cases} \partial_tu(t,x)-\Delta u(t,x) = f(t,x), \\ u(0,x)=u_0(x). \end{cases} $$ ...
4
votes
2answers
182 views

The unit tangent bundle of 2- or 4-manifolds as a principal $S^{1}$- or $S^{3}$-bundle

What type of obstructions have been studied so that the unit tangent bundle of a Riemannian 2-(4-)manifold have a structure of a principal $S^{1}$-($S^{3}$-)bundle?
3
votes
0answers
117 views

Conics over number fields

I am looking for a reference for the following fact. Let $k$ be a number field and let $S$ be a finite set of places of $k$ of even cardinality. Then there exists a unique conic $C$ over $k$ such ...
2
votes
0answers
116 views

Questions about transformation or integral transformation

I have asked several mathematicians about the following questions,but all of them think they are good questions,but can not give a complete answer.Now I have to come here to ask mathematicians all ...
4
votes
1answer
170 views

$C^0$ estimate for solutions of elliptic PDE with Neumann BC

I am interested in a reference for (or counterexample to) a particular $C^0$ estimate for solutions of the Laplace equation with Neumann boundary conditions. More precisely, let $(M,g)$ be a ...
4
votes
5answers
185 views

Books on the analysis of hyperbolic partial differential equations

Most of the present books on pde analysis deal with the elliptic partial differential equations. Is there some book related to rigorous analysis with hyperbolic pdes, and especially hyperbolic systems ...
2
votes
0answers
53 views

Name of a difference of continuants

I am getting ready to publish the manuscript http://arxiv.org/pdf/1408.4631v2.pdf and I am trying to do due diligence on a quantity I study before it gets published. (This is cross-posted from ...
2
votes
2answers
88 views

Background for Kierstead terms

I was looking at some slides of John Longley's here, where he mentions "the Kierstead functional" $$\lambda f.f(\lambda x.f(\lambda y.x)) \ ,$$ (where $f$ should be of type $2$, and $x,y$ of ground ...
1
vote
0answers
72 views

Bounds re Asymptotic Formula for the Sum of Largest Prime Factors

I have a reference request related to the result : $\sum_{n=2}^{x} P(n)$ ~ $\frac{\pi^2}{12}\frac{x^{2}}{log(x)}$ as $x \rightarrow \infty$ where $P(n)$ is the largest prime factor of the positive ...
10
votes
2answers
371 views

on the center of a Lie group

I'm trying to set straight my various pieces of knowledge about the center of a compact Lie group, and I'm running in circles... First some definitions: • Let $G$ be compact, simple, and simply ...