# Questions tagged [examples]

For questions requesting examples of a certain structure or phenomenon

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### Periodic tilings in finite type tiling spaces and substitution tiling spaces

I was reviewing the following statement from a survey by E. Arthur Robinson about tilings in $\mathbb{R}^d$ to better understand geometric tiling rather than tilings over symbols. I consider the ...

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### Functions that have their second derivatives as a lipschitz function [migrated]

What would be an example of a function that is not $C^3$ on $[0,1]$, but has its second derivative as a Lipschitz function?

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### Closed subset of unit ball with peculiar connected components

Let $n\geq 2$ and denote by $B\subset \mathbb{R}^n$ the closed unit ball.
Does there exist a closed subset $A\subset B$ containing $0\in \mathbb{R}^n$ with the following properties i,ii,iii?
i) $\{0\}$...

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### Real analytic periodic function whose critical points are fully denegerated

I have asked this question on MathStackExchange. My question: is there any non-constant real analytic function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ such that, $$\nabla f(x_0)=0 \Rightarrow \nabla^2 f(...

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### Non-homeomorphic connected one-dimensional Hausdorff spaces that have continuous bijections between them in both sides

I need to construct an example of two non-homeomorphic connected one-dimensional Hausdorff spaces that have continuous bijections between them in both sides. Spaces should have induced ("good&...

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### Example of trickiness of finite lattice representation problem?

I'm trying to come up with a good explanation for my students of why the finite lattice representation problem is difficult. I've already shown that the "greedy approach" to representing the ...

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### Torsion-free virtually free-by-cyclic groups

Is it known if there are any examples of a finitely generated group $G$ such that:
$G$ has a finite index subgroup $H$ which is free-by-cyclic
$G$ itself is not free-by-cyclic
$G$ is torsion-free
...

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### Examples of convergence in distribution not implying convergence in moments

It is well know that the convergence in distributions does not necessarily imply convergence in expectation, but implies convergence in expectation of bounded continuous functions.
Let $\{X_n\}$ be a ...

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### Existence of the special entire Hardy space function with infinitely many zeros in the strip

Question. Does there exist an entire function $h$ satisfying three following assertions:
$h$ belongs to the $H^2$ Hardy space in every horizontal upper half-plane;
$zh - 1$ belongs to $H^2(\mathbb{C}...

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### Existence of nonzero entire function with restrictions of growth

Question. Is there an entire function $F$ satisfying first two or all three of the following assertions:
$F(z)\neq 0$ for all $z\in \mathbb{C}$;
$1/F - 1\in H^2(\mathbb{C}_+)$ -- the classical Hardy ...

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### A zoo of derivations

Recall that given a $k$-algebra $A$, a derivation on $A$ is a $k$-linear morphism $d:A\to A$ such that $$d(ab)=d(a)b+ad(b).$$
The use of derivations is of paramount importance in mathematics. I think ...

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### Is there a nonpolynomial $C^\infty$ function $f$ such that $\sup_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every $q >1$?

The question is as in the title:
Is there a nonpolynomial $C^\infty$ function $f$ on $\mathbb{R}$ such that $\sup_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every natural ...

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### Example of two equivariant structures on the same coherent sheaf which do not differ by a grading shift

Suppose we have a variety $X$ with a $\mathbb{C}^*$-action. If $\mathcal F$ is a $\mathbb C^*$-equivariant coherent sheaf on $X$ and $m \in \mathbb Z$, we define the grading shift ${\mathcal F}\{m\}$ ...

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### On the measure of nonconvexity (MNC)

I'm actually working on the measure of nonconvexity and its application. Especially, the Eisenfeld–Lakshmikantham MNC defined - in a Banach space - by:
$$\alpha(A)=\sup_{b\in\operatorname{conv}(A)} \...

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### Natural cotransformations and "dual" co/limits

$\DeclareMathOperator{\id}{\mathrm{id}}\DeclareMathOperator{\Hom}{\mathrm{Hom}}\DeclareMathOperator{\UnCoNat}{\mathrm{UnCoNat}}\DeclareMathOperator{\UnNat}{\mathrm{UnNat}}\DeclareMathOperator{\CoNat}{\...

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### A stronger version of paracompactness

Given a topological space $(X,\tau)$, recall that a cover $\mathcal{U}$ of $X$ is locally finite if for every point $x\in \mathcal{U}$ has a neighborhood $U$ that intersects finitely many elements of $...

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### When does the refinement of a paracompact topology remain paracompact?

Let $(X,\tau)$ be a Hausdorff paracompact space. Let $\tau'$ be the smallest $P$-topology refining $(X,\tau)$, i.e. the topology which has for base the $G_\delta$-subsets of $(X,\tau)$.
Is it true ...

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### $\ast$-autonomous categories with non-invertible dualizing object?

1. Definition
Firstly, recall the following nLab-definition of a $\ast$-autonomous category:
A $\ast$-autonomous category is a symmetric closed monoidal category $(C,\otimes,I,\multimap)$ with a ...

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### Examples of real-time transcendental number and superlinear-time trancsendental number

Computation model is defined as Hartmanis and Stearns 4, it is well known that Liouvilles constant
$$C_L=\sum_{i=1}^{\infty} 10^{-i!}$$ is computable in real time or linear time 1, 5 especially ...

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### Noetherian but not strongly Noetherian

What are some examples of Tate rings $R$ (i.e. Huber rings with with topologically nilpotent units) which are Noetherian but not strongly Noetherian ($R$ is strongly Noetherian iff for all $n \in \...

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### Are there "pathological convex sets" over ultravalued fields of char 2?

In their book Topological Vector Spaces (2nd ed.) Lawrence Narici and Edward Beckenstein generalise convex sets for TVS over ultravalued field $K$ as $K$-convex sets. The definition goes as following:...

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### Novel examples, proofs or results in mathematics from arithmetic billiards

The goal of the post is get a repository of mathematical results, proofs or examples by users of the site, arising from arithmetic billiards in number theory, analysis, geometry,….
Wikipedia has an ...

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### Cartesian monoidal star-autonomous categories

Disclaimer: This is a crosspost (see MathStackexchange). Apologies if cross-posting is frowned upon. However, it seems that on Stackexchange there are not many people familiar with star-autonomous ...

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### Examples and counterexamples to Lack's coherence observation

In Lack's A 2-categories companion, he states
There are general results asserting that any bicategory is biequivalent to
a 2-category, but in fact naturally occurring bicategories tend to be ...

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### Mathematicians learning from applications to other fields

Once upon a time a speaker at the weekly Applied Mathematics Colloquium at MIT (one of two weekly colloquia in the math department (but the other one is not called "pure")) said researchers ...

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### Existence of nontrivial categories in which every object is atomic

An object $X$ of a cartesian closed category $\mathbf C$ is atomic if $({-})^X \colon \mathbf C \to \mathbf C$ has a right adjoint (hence is also internally tiny). Intuitively, atomic objects are &...

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### Examples of continua that are contractible but are not locally connected at any point

A continuum is a compact, connected, metrizable space.
What are examples of continua that are contractible but nowhere locally connected, meaning that no point has a neighbourhood basis consisting of ...

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### Is there a closed manifold whose universal cover is $\mathbb{R}^n\setminus\{x_1, \dots, x_k\}$ for some $k > 1$?

There are many closed manifolds with universal cover homotopy equivalent to $\mathbb{R}^n$, they are precisely the closed aspherical manifolds. There are also many closed smooth manifolds with ...

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### Common/well-known results with natural and/or useful reformulations

$\DeclareMathOperator{\pp}{\mathbb{P}}$My aim here is to have a collection of "natural" not-so-common reformulations/extensions of common/well-known results such that
the reformulation/...

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### Finding non-inner derivations of simple $\mathbb Q$-algebras

What's a good example of a simple algebra over a field of characteristic $0$ which has a non-inner derivation but also has the invariant basis number property (IBN)?
I'm under the impression that when ...

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### Mañé's example of an attractor with no natural measure

I'm reading Milnor's notes on dynamical systems and in Lecture 3 he gives an example of an attractor with no natural measure, which he attributes to Mañé. I can find no other reference in which this ...

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### Theorems that are essentially impossible to guess by empirical observation

There are many mathematical statements that, despite being supported by a massive amount of data, are currently unproven. A well-known example is the Goldbach conjecture, which has been shown to hold ...

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### Example of maximal multicurve complex

in this paper we have :
" On the Teichmüller tower of mapping class groups By Allen Hatcher at Ithaca, Pierre Lochak at Paris and Leila Schneps."
Definition. The maximal multicurve complex $...

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### How would you have answered Richard Feynman's challenge?

Reading the autobiography of Richard Feynman, I struck upon the following paragraphs, in which Feynman recall when, as a student of the Princeton physics department, he used to challenge the students ...

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### Regular graphs without non-trivial $f$-factor

Question:
are there any known examples of $k$-regular graphs that have no regular $f$-faktor, $1\le f\lt k;\ k\ge 3$, resp., can their existence or nonexistence be proved?

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### Examples of 2-categories with multiple interesting proarrow equipment structures

Proarrow equipments (also known as framed bicategories) are identity-on-objects locally fully faithful pseudofunctors $({-})_* \colon \mathcal K \to \mathcal M$ for which every 1-cell $f_*$ in the ...

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### Bounded derived categories of which smooth projectives possess bounded t-structures whose hearts are categories of modules?

I am interested in $P$ that is smooth and proper over a field and such that the derived category of coherent sheaves $D^b(P)$ possesses a $t$-structure whose heart is the category of finitely ...

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### On simple examples of unimodularity

$w=z=x+ 1 =y−1$ provides $wz−xy=w^2−(w−1)(w+ 1) = 1$. Hence if $x,y$ are odd then $w,z$ are even and all four integers are close.
Is there elementary example where only $w$ is even and all four ...

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### Real (non-complex) Du Val singularities for quartics of high global Milnor number

I posted this in MathStackexchange and was advised to come here. As I am only looking for examples, I didn't feel MathOverflow was necessary.
I am looking for examples of specific quartic projective ...

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### Examples of non-proper profinite HNN extensions

We define a profinite HNN extension as the profinite completion of the abstract HNN extension. In the abstract case, the homomorphim of the base group to the HNN extension is always a monomorphism. ...

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### Taking a theorem as a definition and proving the original definition as a theorem

Gian-Carlo Rota's famous 1991 essay, "The pernicious influence of mathematics upon philosophy" contains the following passage:
Perform the following thought experiment. Suppose that you are ...

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### Finite Galois module whose Ш¹ is nonzero?

In algebraic number theory, we constantly make use of the nine-term Poitou-Tate sequence: Let $K$ be a number field and $M$ a finite $K$-Galois module. Then we have the nine-term exact sequence
$$
H^0(...

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### The use of computers leading to major mathematical advances II

I would like to ask about recent examples, mainly after 2015, where experimentation by computers or other use of computers has led to major mathematical advances.
This is a continuation of a question ...

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### Simple example of Hammerstein integral equation

I'm currently reading this paper (and working on a similar one). The main goal is to study the Hammerstein integral equation (in $\mathcal{C}(I,E))$:
$$x(t) = \int_{0}^{t} K(t,s)f\big(s,x(s)\big)ds,\...

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### Simple example of nontrivial simplicial localization

Does anyone has a simple example of a 1-category $\mathcal{C}$ and a collection of morphisms W such that the infinity-categorical / simplicial localization $\mathcal{C}\left[W^{-1}\right]$ is not a 1-...

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### Does a quaternion algebra exist over a number field that is split over some infinite real places, but not others?

Let $F$ be a totally real number field having at least two different real embeddings $\sigma_1 : F \hookrightarrow \mathbb{R}$ and $\sigma_2 : F \hookrightarrow \mathbb{R}$.
Does a quaternion algebra $...

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### An example of a $T_1$ space where all closed $G_\delta$ sets are zero-sets, but it isn't normal

In Engelking's General topology, in the exercises section, there is Ju. M. Smirnov's characterization of normal spaces:
A $T_1$ space is normal iff the following properties hold (both):
Every closed $...

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### Examples of $C^{k,1}$ functions which are not $C^{k+1}$?

I'm currently reading this paper and the authors define the set $C^{k,1}(\mathbb{R}^n)$ as consisting of all functions $f:\mathbb{R}^n\rightarrow \mathbb{R}$ having $k$ derivatives and for which:
$$
\|...

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### Example of a non-cocomplete model category of a realized limit sketch

Let $(\mathcal{E},\mathcal{S})$ be a realized limit sketch, i.e. a locally small category $\mathcal{E}$ with a class $\mathcal{S}$ of limit cones in it. It is not assumed that $\mathcal{E}$ is small, ...

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### Mechanical systems with their configuration space being a Lie group

Cross-posted from Physics.SE
In Marsden, Ratiu - Introduction To Mechanics And Symmetry there is a certain focus on reducing cotangent bundles of Lie groups. More precisely, if $G$ is a Lie group, ...