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

**8**

votes

**4**answers

197 views

### Advanced Differential Geometry Textbook

I tried this post on StackExchange with no luck. Hopefully the experts at MathOverflow can help.
In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual ...

**1**

vote

**0**answers

29 views

### Doubling theorem for Alexandrov spaces

Is there a user friendly exposition of the notion of boundary of an Alexandrov space with curvature bounded from below and of the Doubling theorem?
The only reference I am aware of is the original ...

**2**

votes

**0**answers

38 views

### Roller's problem on median groups

At the end of his dissertation Poc Sets, Median Algebras and Group Actions, Martin Roller asks
A group $G$ is called median if it acts freely and transitively on a median algebra. This is ...

**1**

vote

**2**answers

119 views

### Automorphism group of the affine groups AGL(n,q), ASL(n,q)

I have a question. The automorphism group of the linear groups $GL(n,q)$, the group of linear transformations of $V = \mathbb{F}_q^n$, and $SL(n,q)$, the subgroup of $GL(n,q)$ consisting of elements ...

**4**

votes

**1**answer

91 views

### symmetric measurable 2-cocycles on compact abelian groups vanish?

Is the following result true? If it is, could you plese give me a reference for it? Thanks in advance!
Let $(G, \mu)$ be any compact abelian group with Haar measure $\mu$ (The case I am interested ...

**5**

votes

**0**answers

176 views

### Why do Kashiwara and Schapira require a base ring of finite global dimension?

In the book "Sheaves on Manifolds" by Kashiwara and Schapira, they work always with sheaves of $R$-modules, where $R$ is a ring of finite global dimension.
Why do they do this, what care ...

**5**

votes

**0**answers

139 views

### Relationship between the syntomic cohomology of Kato and of Fontaine-Messing

Fix a prime $p$ and let $X$ be a $\mathbb{Z}_{p}$-scheme. Write $X_{n}:=X\otimes\mathbb{Z}/p^{n}$ and $\phi:X_{1}\rightarrow X_{1}$ for the absolute Frobenius. Let $X\hookrightarrow Z$ be a (suitable) ...

**1**

vote

**0**answers

98 views

### “GraphI Individualization” referece request [on hold]

Context: I am studying Weisfeiler Lehman method(WL method) and have clear idea about 1 and 2 dimensional WL method. I was wondering about the individualization process described below-
...

**6**

votes

**0**answers

74 views

### Recursions which define polynomials?

Let $k$ be a positive integer and let
$$h(n,k,q)=\frac{1-(1+q^{k})q^{2k(n-1)+1}+q^{2}}{1-q^{2n-1}}h(n-1,k,q)-\frac{(1-q^{k(2n-3)})(1-q^{2k(n-1)})q^2}{(1-q^{2n-1})(1-q^{2n-3})}h(n-2,k,q)$$
with ...

**10**

votes

**2**answers

382 views

### Continued Fractions from Digit Streams

let $x=\sum_{i=1}^{\infty}\delta_i2^{-i},\ \delta_i\in\{0,1\}$.
Is there an algorithm that converts the sequence $(\delta_0,\ \delta_1,\ ...)$ of the binary digits of $x$ to the sequence ...

**27**

votes

**3**answers

612 views

### The coupon collector's earworm

I thank Nicolas Dupont for the following question
(and for permission to disseminate it further):
I have a playlist with, say, $N$ pieces of music.
While using the shuffle option (each such ...

**4**

votes

**1**answer

170 views

### Derivatives of theta functions at zero

Let $L$ be a line bundle over complex elliptic curve, $\deg L = k>0$. Theta functions
$$
\theta_s(z;\tau)_k=\sum_{r\in \mathbb{Z}} e^{\pi i [(\frac{s}{k} + r)^2 k \tau + 2kz(\frac{s}{k}+r)]}, ...

**1**

vote

**1**answer

39 views

### Locality of homogeneous pseudo-differential operator

Let $P$ be a polynomial in several variables, and let $P(D)$ be the corresponding differential operator. Obviously, $P(D)$ is a local operator, in the sense that I need only to know the function $u$ ...

**11**

votes

**1**answer

339 views

### How big is the lattice of all functions?

Define the lattice $(\mathcal{L},\prec)$ as the set of all function $f:\mathbb{N}\rightarrow\mathbb{N}$ satisfying $f(n)\leq f(n+1)\leq f(n)+1$, where two functions are considered equal if they differ ...

**3**

votes

**0**answers

55 views

### Natural transformations of $A_\infty$-functors (between dg-categories) are “directed homotopies” (reference?)

Let $\mathbf A$ and $\mathbf B$ be dg-categories over a field, viewed as $A_\infty$-categories. The $A_\infty$-category (actually, dg-category) of strictly unital $A_\infty$-functors $\mathbf A \to ...

**4**

votes

**1**answer

155 views

+50

### Fair surfaces - general mathematical theory

Fairness measures for surfaces are, in general, functionals containing more complicated terms thatn the usual bending energy, and may depend not only on the mean curvature but also on principal ...

**4**

votes

**1**answer

118 views

### Hyperfunctions supported at a point

Is it true that the space of hyperfunctions on $\mathbb{R}^n$ supported at 0 coincides with the space of Schwartz distributions supported at 0?
More explicitly, is it true that any hyperfunction ...

**-4**

votes

**0**answers

165 views

### Has Frucht's theorem been successfully used in inverse Galois theory? [closed]

Logically, one can associate to any finite extension $K$ of $\mathbb{Q}$ a directed graph describing it. Can such a graph be used together with Frucht's theorem asserting that every finite group is ...

**3**

votes

**0**answers

81 views

### Does this notion of “$\mathcal{F}$-digraph” appear in the literature?

By a digraph, I mean a quiver with no multiple edges. So in particular:
Loops are okay.
An infinite set of vertexes is okay.
Furthermore, I will tend to identify each digraph with its underlying ...

**9**

votes

**2**answers

623 views

### How should a mathematician approach the physics literature concerning percolation?

I would like to read some of the physics literature on two-dimensional percolation, however in attempting this I have run into two problems. (1) Physics papers on percolation are (relatively) hard ...

**4**

votes

**2**answers

247 views

### Algebras for probability monad

What is the Eilenberg-Moore category for the non-finitary probability distribution monad is, that is, the monad $D \colon \mathbf{Set} \to \mathbf{Set}$ defined by
$$
DX = \left\{ p \in [0,1]^X \ ...

**3**

votes

**1**answer

96 views

### Scaling properties of the Hölder estimate for heat equation

Lately, I have been interested in scaling properties of parabolic equations, and this question is related to an earlier one I asked about Harnack constants.
Let $Q(R) := Q(R^2,R) = B(0, R) \times ...

**1**

vote

**0**answers

80 views

### Gluing two diffeomorphisms and then smoothing

This question did not get an adequate answer on math.stackexchange.
Let $M_1,M_2$ be two $n$-dimensional closed manifolds and suppose that $M_i=\bar{U}_i^+\cup \bar{U}_i^-$ where $\bar{U}_i^\pm$ are ...

**1**

vote

**1**answer

62 views

### Maximum size of a union of incomparable chains

The following question was asked on math.stackexchange, where it received no answers.
http://math.stackexchange.com/questions/1392669/maximum-size-of-a-union-of-incomparable-chains
Let ...

**1**

vote

**0**answers

58 views

### On (universal) additive functors making a given complex contractible: examples?

Let $M=(M^i)$ be a (cohomological) complex of objects of some additive category $A$ (I am mostly interested in "short" complexes; yet one may also consider an unbounded $M$). I am interested in those ...

**-1**

votes

**0**answers

47 views

### Orbit closures of symmetric bilinear form [migrated]

Let $A$ and $B$ be two real symmetric matrices in $M_n(\mathbb{R})$. I would like to learn about necessary and sufficient conditions for knowing when $B \in \overline{GL_n(\mathbb{R})\cdot A}$; where:
...

**16**

votes

**1**answer

311 views

### Is there a reference for “computing $\pi$” using external rays of the Mandelbrot set?

I was recently reminded of the following cute fact which I will state as a proposition to fix notation:
Proposition
Given $\epsilon > 0$, let $c = -3/4 + \epsilon i \in \mathbb{C}$ and $q_c(z) = ...

**1**

vote

**0**answers

43 views

### Monotone version of one-dimensional Whitney extension theorem

Is there a version of the Whitney extension theorem that would extend a monotone $C^\infty$ function on a compact subset of $\mathbb R$ (satisfying the usual Whitney's compatibility conditions) to a ...

**1**

vote

**0**answers

116 views

### Was this particular case of the tube formula known before Weyl and Hotelling?

The tube formula is a really nice result in differential geometry which relates the volume of the tubular neighborhood of a submanifold to its intrinsic geometry. It has been proved by Weyl in 1939 ...

**4**

votes

**1**answer

102 views

### Approximation of topological dynamical systems?

I'm trying to find references to approximations of topological dynamical systems in the following sense:
A topological dynamical system $(X, f)$ consists of a topological space (typically compact ...

**2**

votes

**1**answer

87 views

### Heat equation: impact of the diffusion coefficient on the Harnack constant

Consider the heat equation
$$
u_t - div[a(x,t) \nabla u] =0,\quad (x,t) \in B(r) \times [-r^2, 0] \subset \mathbb R^{d+1}
$$
for a Hölder continuous coefficient $a(x,t)$ satisfying
$$
0<C_o \le ...

**2**

votes

**1**answer

67 views

### Higher dimensional analogue of Ahlfors covering surface theory

It is well known that Ahlfors covering surface theory in one dimensional is very powerful in dealing with many problems. I wonder whether there exists some generalization of this theory into higher ...

**1**

vote

**0**answers

153 views

### Stone-Cech compactification of $\mathbb{R}^n$ and smooth functions

I am currently attending a course where we are now covering the Stone-Cech compactification. Today we proved in some detail that extensions of bounded smooth functions on $\mathbb{R}^n$ to ...

**6**

votes

**0**answers

146 views

### rigidity of $\mathcal P(\omega_1) / NS$ under MA

In Woodin's book, Lemma 5.100 asserts that if $MA_{\omega_1}$ holds and there is an $\omega_2$-saturated ideal $I$ on $\omega_1$, then $\mathcal P(\omega_1)/I$ is a rigid boolean algebra, meaning it ...

**2**

votes

**0**answers

235 views

### A variant frobenius problem

From Sylvester's theorem we know that using only coins of sizes $a,b$, we can change exactly $\frac{(a-1)(b-1)}2$ different big coins up to $(a-1)(b-1)$.
Denote sets ...

**4**

votes

**2**answers

109 views

### Expected Cardinality of the First n Coefficients of a Continued Fraction

Is there anything known about the asymptotic expected growth of $card\{a_0,a_1,...,a_n\}$, where the $a_i$ are the first $n$ coefficient of a continued fraction $[a_0; a_1, ... ]$?

**2**

votes

**1**answer

212 views

### Training towards research on k3 surfaces

I am a graduate student learning basic algebraic geometry (from Hartshorne, Shafarevich). I'm planning to work in k3 surfaces (arithmetic and geometric properties, in my guide's words). I came to know ...

**2**

votes

**0**answers

87 views

### About a (new?) definition of transformation (anti.transformation) as a link between natural and dinatural transformations

This is not a hard topic, but I post here as "reference request" or because elementary aspects (but not previously vocalized) can be interesting too for researchers.
Given $F: \mathscr{A}\to ...

**3**

votes

**1**answer

127 views

### Paper of Boutot-Carayol in `Courbes modulaires et courbes de Shimura'

I am trying to obtain a copy of the following
J.-F. Boutot and H. Carayol, Uniformisation p-adique des courbes de Shimura: les
théorèmes de Čerednik et de Drinfel'd , Astérisque No. 196-197 ...

**1**

vote

**0**answers

46 views

### Reference/proof for parabolic Holder spaces property

Suppose that a function $u:[0,1]\times[0,T]\to\mathbb{R}$ belongs to the parabolic Holder space $C^{2+\alpha,1+\alpha/2}$, for $\alpha\in(0,1)$.
What can be said about $u_x=\partial_x u$?
I am not ...

**6**

votes

**0**answers

161 views

### Criteria for irreducibility using the location of complex roots

I would like to see criteria for the irreducibility of a polynomial over $\mathbb{Z}$ based (mainly) on the location of the roots of the polynomial in the complex plane. An example of such a criterion ...

**10**

votes

**2**answers

505 views

### What's a good introduction to category theory for someone doing analysis?

I do functional analysis, and diagrams are popping all over the place. It is about time I learned me some category theory.
Any recommendations?

**4**

votes

**0**answers

154 views

### Open questions in “Spin geometry”

This is a very naive question. I have the impression that the area of "Spin geometry" is not an active research field. Sure Spin geometry is used in many different branches of mathematics and physics ...

**2**

votes

**0**answers

93 views

### Generalizing disjointness

The following definition generalizes set-theoretic disjointess:
Definition 0. (Autonomy). Given a Lawvere theory $\mathsf{T}$, a $\mathsf{T}$-algebra $X$, and an indexed family $S$ of subalgebras ...

**5**

votes

**2**answers

111 views

### Reference to iterated logarithm law and Smirnov law of empirical CDF

I am reading V. Vapnik's "Statistical Learning Theory". The author layouts following two statistical laws related to empirical CDF. I am looking for reference about proofs on these two laws.
Let ...

**6**

votes

**1**answer

109 views

### Radon transform between affine grassmannians

Let $\overline{GR}(n,k)$ be the manifold of all affine k-dimensional subspaces in $R^n$, and let
$R:C^{\infty}_c(\overline{GR}(n,k))\to C^{\infty}_c(\overline{GR}(n,l))$, $0\le k<l\le n-1$, be the ...

**6**

votes

**0**answers

169 views

### The p-adic valuation of a linear recurrence

Let $(u_n)_{n \geq 0}$ be an integer-valued linear recurrence of order $k \geq 1$. Precisely,
$$u_n = a_1 u_{n-1} + \cdots + a_k u_{n - k} \quad \forall n \geq k ,$$
for some $a_1, \ldots, a_k \in ...

**1**

vote

**1**answer

64 views

### One question about the tensor product of $L^1(G)$ and a Banach space $A$

We know that the tensor product of $L^1(G)$ and a Banach space $A$ is isometric to $L^1(G, A)$, the space of all Bochner-integrable $A$-valued functions on a locally compact group $G$. I am looking ...

**0**

votes

**0**answers

46 views

### What results exist for functions with regionally fluctuant fractal dimension?

I'm interested in functions that have a varying fractal dimension at different scales and/or regions. Has this been investigated in detail? I'd be interested in results and references in this area of ...

**0**

votes

**1**answer

88 views

### Reference request: Strong Connectivity and the Incidence Matrix

Question: What would be a good reference for characterizations of strong connectivity of a digraph in terms of its incidence matrix?
Details: Consider a digraph $(V, E)$ with vertex set
$$V = ...