# Tagged Questions

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

**1**

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**1**answer

73 views

### Convex subcomplexes of CAT(0) cubical complexes

Is the following statement true? If so, can anyone provide a reference?
Let $X$ be a CAT(0) cubical complex, and let $Y$ be a connected
subcomplex of $X$. Then the following are equivalent:
...

**7**

votes

**3**answers

334 views

### Ring of differential operators of a quotient ring

All rings are assumed to have unity.
Let $k$ be a field. Recall the definition of Grothendieck's ring of ($k$-linear) differential operators $D(R;k)$ of a commutative $k$-algebra $R$:
...

**2**

votes

**0**answers

83 views

### Under what conditions do time averages of ergodic transformations satisfy a central limit theorem?

Let $(X, \mu)$ be a probability space and $T:X\rightarrow X $ an ergodic transformation, i.e. $T$ is measure preserving and the only $T$ invariant subspaces have either measure $0$ or measure $1$ ...

**1**

vote

**3**answers

285 views

### What are Carnot groups?

I'm trying to learn the Pansu differentiability theorem and I need to know what Carnot groups are. Can someone please explain what Carnot groups are? An introductory reference would be greatly ...

**5**

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**0**answers

154 views

### Core model for supercompact cardinals and iteration trees

I have a few somehow related questions:
Question 1. What do we expect for an inner model $\mathcal{K}$ to be a core model for a supercompact cardinal? What properties should it have, and what ...

**5**

votes

**1**answer

346 views

### Why is this group called “The Holomorph of a group”

Many years ago I found in google the notation "Holomorph of group". It is the semi direct product of $G$ with $Aut(G)$. Why is the term "Holomorph" used here, while it is usually used for complex ...

**2**

votes

**1**answer

129 views

### Survey on functional equations and inequalities

Where can I find a comprehensive survey monograph on functional equations and inequalities from sketch to current research trends with some focus on applications (both inside and outside mathematics)?
...

**6**

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**0**answers

183 views

### Does anyone know this determinant?

The following determinant arises in a combinatorial enumeration problem. I wonder if anyone has seen it before in any context or knows how to evaluate it. I tried computing it for small $n$ but didn't ...

**4**

votes

**2**answers

273 views

### Has this construction, which builds a symmetric multicategory from a commutative monoid, been described or studied anywhere, and if so, where?

Whenever $R$ is a commutative ring, write $R[x^{(n)}]$ for the set of all $p \in R[x]$ such that $p$ is a monic polynomial of degree $n$. Then $R[x^{(n)}]$ is not closed under sums, nor does it ...

**5**

votes

**2**answers

217 views

### Embedding $G$ in a $Z(G)$ extension of $\operatorname{Aut}G$

This question follows up a question I asked on math.SE. This is a refinement and a reference request.
For what groups $G$ does there exist a $Z(G)$-extension of $\operatorname{Aut}G$ (call it ...

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**0**answers

129 views

### What is the state of the art of visualizing bifurcations for “difficult” dynamical systems?

This question is related to my other recent question on MO (although I am not confident that the dynamical system described in that other question is actually "difficult," in the sense that I will ...

**0**

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**0**answers

40 views

### representing quasicrystal as tilings and appearing frequencies of each tile

Quasicrystal can be fully represented either using projection method or tilings with constraints. For the latter, is there some sort of study on the "appearing frequency" of each tile or even ...

**5**

votes

**2**answers

303 views

### TM and abstract algebra

Usually, during lectures Turing Machines are firstly introduced from an informal point of view (for example, in this way: http://en.wikipedia.org/wiki/Turing_machine#Informal_description) and then ...

**5**

votes

**1**answer

196 views

### Is there a standard name for this poset

I've run into the following poset and I would expect it has a standard name. Let $n\geq k\geq 0$. Then $P_{n,k}$ consists of all $k$-element subsets of $\{1,\ldots,n\}$ ordered by $X\leq Y$ if ...

**2**

votes

**1**answer

106 views

### Ergodicity for the mean of a linear process without finite second moment

Suppose that $\{X_k:k\in\mathbb Z\}$ is a linear process, i.e. a sequence of random variables such that
$$
X_k=\sum_{j=0}^\infty\psi_j\varepsilon_{k-j}
$$
for each $k\in\mathbb Z$, where ...

**2**

votes

**2**answers

113 views

### Positive solutions of linear systems with a diagonally dominant matrix

Given a real linear system ($\mathbf{A}\mathbf{x} = \mathbf{b}$), is there any result regarding the positiveness of the solution $\mathbf{x}^*$ considering that $\mathbf{A}$ is diagonally dominant? ...

**5**

votes

**1**answer

143 views

### Reference for Mod 2 cohomology of $BZ_{2r}$ in terms of Stiefel-Whitney Classes

I was hoping for an explicit reference to the description of the mod 2 cohomology of a cyclic group $C_{2r}=\langle t \rangle$ of even order in terms of Stiefel-Whitney classes, i.e., that
...

**1**

vote

**1**answer

169 views

### A calculus question related to the nonnegative definite functions

I am looking for some sufficient conditions for an even, continuous, nonnegative, non increasing function $f(x)$ on $R$ such that
$$
\int_0^\infty \cos(xz) f(z) d z \ge 0 \qquad\text{for all $x\ge ...

**1**

vote

**1**answer

165 views

### Heat kernel upper bound on compact Riemannian manifold

Let $M$ be a compact Riemannian manifold without a boundary. Let $p_t(x,y)$ be the heat kernel. I am looking for a reference for the result: there exists a constant $C$ such that
$$|p_t(x,y)| \leq C$$
...

**0**

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**0**answers

25 views

### majorizing measures for small ball probabilities

This is a reference request. Can Majorizing measures be used to estimate small ball probabilities? Any help would be appreciated.
Thank you.

**0**

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**0**answers

30 views

### Covering number of the range of a function

I have come across the need to know a bound on a certain curious quantity: the covering number of the range of a continuous function $f: D \rightarrow \mathbb{R}^n$, where $D \subseteq \mathbb{R}^m$. ...

**2**

votes

**0**answers

69 views

### Solution of a linearly constrained quadratic programming problem [closed]

What is the solution of the following optimization problem:
\begin{align}
&\min{\mathbf{p}^\mathrm{T} \mathbf{B} \mathbf{p}}\\
&\text{subject to}: \mathbf{0}\leq{\mathbf{p}}\leq \mathbf{1}.
...

**2**

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**0**answers

77 views

### Buildings associated to generalized $BN$ pairs

I'll begin by asking a general question, and then specializing to the situation I really care about.
Let $G$ be a group and let $(B, N)$ be a $BN$-pair in $G$ (see, for instance, page 39 of Tits' ...

**2**

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**0**answers

98 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

**0**answers

129 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

**1**answer

107 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 ...

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**2**answers

374 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 ...

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vote

**2**answers

95 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

**0**answers

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

**0**answers

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 ...

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**0**answers

53 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

**3**answers

567 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**

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190 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

**2**answers

564 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

**0**answers

89 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

**4**answers

359 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

**1**answer

107 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

**3**answers

132 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

**1**answer

55 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. ...

**5**

votes

**0**answers

303 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

**1**answer

143 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

**1**answer

198 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

**3**answers

863 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

**1**answer

160 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

**1**answer

147 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

**1**answer

90 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

**1**answer

88 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 ...

**21**

votes

**2**answers

677 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

**2**answers

220 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

**1**answer

226 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 ...