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

**2**

votes

**0**answers

106 views

### The Karoubi model structure on Cat

I am looking for any kind of informations about the Karoubi model structure on $\bf Cat$. I discovered the presence of this structure a few months ago on the Joyal Lab and now I would like to use it ...

**2**

votes

**1**answer

80 views

### Interpretation of riemannian geodesics in probability

Good morning everybody. My question is, as maybe already hinted in the title, rather philosopic.
We know that geometric properties of a riemannian manifold can be interpreted in terms of certain ...

**6**

votes

**1**answer

177 views

### Some polytopes in $\mathbb R^n$ whose vertices have coordinates 1, -1 or 0

Let $n$ and $k$ be positive integers with $k\leq n$.
Let $P(n,k)$ be the convex hull in $\mathbb R^n$ of the $2^k {n \choose k}$ vectors whose exactly $k$ coordinates belong to $\{\pm 1\}$ all the ...

**14**

votes

**0**answers

240 views

### Partitions of ${\rm Sym}(\mathbb{N})$ induced by convergent, but not absolutely convergent series

Let $(a_n) \subset \mathbb{R}$ be a sequence such that the series
$\sum_{n=1}^\infty a_n$ converges, but does not converge absolutely.
Then there is a partition of the symmetric group ${\rm ...

**1**

vote

**0**answers

35 views

### Non-Convex Polygons with “Antipodal Visibility”

by "antipodal visibility" of planar, simple polygons I mean the following property:
if two points $p$ and $q$ on the polygon's boundary divide the polygon's boundary into two polylines of equal ...

**3**

votes

**0**answers

74 views

### Counting Problems where Labeled is Known but Unlabeled is Not

Cayley's formula states that the number of labeled trees on $n$ vertices is $n^{n-2}$. There are many nice proofs of this compact formula.
To contrast, counting unlabeled trees is considerably ...

**2**

votes

**1**answer

90 views

### A quadratic algebra with four generators and four relations

Algebra that I'm going to describe pop-up in my research, it looks completely elementary, but I don't know any appropriate references.
Let $k$ be an algebraically closed field of characteristic ...

**4**

votes

**0**answers

94 views

### On the cohomology ring of the Hilbert scheme of points on k3 or abelian surfaces

There are many results on the cohomology of the Hilbert scheme of points of a surface.
Gottsche calcaluted the Betti numbers and Nakajima got the generators of the cohomology. Also
there are results ...

**1**

vote

**1**answer

111 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

**1**answer

240 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). ...

**6**

votes

**1**answer

372 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

**1**answer

59 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

**4**answers

622 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

**0**answers

43 views

### Good book on calculus of variation for a beginner [duplicate]

I am an undergraduate student and is about to start calculus of variation first time, could you please suggest me some books to start calculus of variation from basics ??

**3**

votes

**3**answers

156 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

**0**answers

73 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

**0**answers

104 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

**2**answers

135 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

**2**answers

132 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

**0**answers

92 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

**0**answers

138 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

**0**answers

43 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 ...

**1**

vote

**0**answers

111 views

### Runge-Kutta with all nodes at n+1 or zero weights otherwise [migrated]

So, lets say for the family of the explicit Runge-Kutta methods:
$$y_{n+1} = y_n + \sum_{i=1}^s b_i k_i$$
where,
$$k_1 = hf(t_n, y_n)$$
$$k_2 = hf(t_n+c_2h, y_n+a_{21}k_1)$$
$$\vdots$$
$$k_s = ...

**5**

votes

**0**answers

90 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

**3**answers

294 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

**1**answer

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

**0**answers

128 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

**0**answers

69 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

**2**answers

151 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

**0**answers

50 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

**0**answers

114 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

**1**answer

57 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

**3**answers

259 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

**2**answers

262 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

**1**answer

110 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 ...

**4**

votes

**1**answer

469 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

**0**answers

118 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

**2**answers

891 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

**2**answers

172 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

**0**answers

104 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

**0**answers

18 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

**0**answers

79 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 ...

**10**

votes

**2**answers

463 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

**0**answers

121 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

**1**answer

170 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

**1**answer

151 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

**0**answers

174 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

**0**answers

182 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

**1**answer

376 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

**1**answer

588 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 ...