Questions tagged [examples]

For questions requesting examples of a certain structure or phenomenon

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An example of module which is square-free, CS, NOT C3, and NOT nonsingular

Let $M$ be a right $R$-module ($R$ has unity). Recall that $M$ is called square-free if $M$ does not contain two nonzero isomorphic submodules with zero intersection. $M$ is called CS if every ...
Hussein Eid's user avatar
12 votes
2 answers
667 views

Examples of non-polynomial comonads on Set?

Question: What are examples of comonads on $\mathbf{Set}$ that are not polynomial? Background: polynomial functors and comonads on Set A functor $F\colon\mathbf{Set}\to\mathbf{Set}$ is called ...
David Spivak's user avatar
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9 votes
1 answer
331 views

Natural set-theoretic principles implying the Ground Axiom

The Ground Axiom states that the set-theoretic universe is not a set-forcing extension of an inner model. By Reitz, it is first-order expressible and easy to force over any given ZFC model with class-...
Monroe Eskew's user avatar
  • 17.9k
0 votes
0 answers
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Diophantine-like approximation of dynamical subsystems

For $\alpha\in [0,1)$ irrational we know that there exists a sequences $\{ q_n \}_{n=1}^\infty\subseteq \mathbb{N}$ and $\{ p_n \}_{n=1}^\infty\subseteq \mathbb{Z}$ such that $$ \Big\vert \alpha-\frac{...
Keen-ameteur's user avatar
6 votes
0 answers
170 views

Is $L^2(I,\mathbb Z)$ homeomorphic to the Hilbert space?

I am somehow puzzled by the subset $G:=L^2(I,\mathbb Z)$ of $H:=L^2(I,\mathbb R)$ of all integer valued functions on $I=[0,1]$ (in fact I mentioned as an example in this old MO question). Some simple ...
Pietro Majer's user avatar
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0 votes
1 answer
274 views

Finding examples of functions which are infinite or undefined with current extensions of the expected value?

Preliminaries Consider the expectations desribed in this paper, which is an extension of the Lebesgue density theorem; this paper which is an extension of the Hausdorff measure, using Hyperbolic ...
Arbuja's user avatar
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1 vote
0 answers
203 views

Results that hold for the complex numbers but not for algebraically closed fields of characteristic zero

When a result is stated for the field of complex numbers it can usually be extended to a result for an algebraically closed field of characteristic zero. I would like to see a list of results that ...
Béla Fürdőház 's user avatar
2 votes
2 answers
151 views

Hardy space inclusion in the right-half plane

I'm looking for an example of a function $u \in H_2$ such that $u \notin H_\infty$, where $H_p$ is the Hardy space on the right-half plane. Since this notation is perhaps not standard, here is a ...
Laurent Lessard's user avatar
1 vote
0 answers
72 views

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 ...
Keen-ameteur's user avatar
3 votes
1 answer
221 views

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) $\{0\}$...
user_1789's user avatar
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2 votes
0 answers
94 views

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(...
Jianxing's user avatar
7 votes
1 answer
481 views

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&...
jkjfgk's user avatar
  • 73
8 votes
1 answer
327 views

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 ...
Noah Schweber's user avatar
9 votes
2 answers
630 views

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 ...
HASouza's user avatar
  • 283
3 votes
1 answer
1k views

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 ...
null's user avatar
  • 217
2 votes
1 answer
150 views

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}...
Pavel Gubkin's user avatar
4 votes
2 answers
193 views

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 ...
Pavel Gubkin's user avatar
0 votes
0 answers
167 views

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 ...
9 votes
2 answers
670 views

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 ...
Isaac's user avatar
  • 2,331
0 votes
1 answer
73 views

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)} \...
Motaka's user avatar
  • 291
9 votes
0 answers
193 views

Natural cotransformations and "dual" co/limits

$\DeclareMathOperator{\id}{\mathrm{id}}\DeclareMathOperator{\Hom}{\mathrm{Hom}}\DeclareMathOperator{\UnCoNat}{\mathrm{UnCoNat}}\DeclareMathOperator{\UnNat}{\mathrm{UnNat}}\DeclareMathOperator{\CoNat}{\...
Emily's user avatar
  • 9,727
2 votes
1 answer
166 views

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 $...
Cla's user avatar
  • 665
4 votes
1 answer
145 views

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 ...
Cla's user avatar
  • 665
4 votes
1 answer
148 views

$\ast$-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 ...
Max Demirdilek's user avatar
0 votes
1 answer
121 views

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 ...
XL _At_Here_There's user avatar
8 votes
1 answer
369 views

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 \...
Dat Minh Ha's user avatar
  • 1,432
2 votes
1 answer
324 views

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:...
Nik Pronko's user avatar
3 votes
1 answer
496 views

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 ...
8 votes
1 answer
243 views

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 ...
Max Demirdilek's user avatar
13 votes
0 answers
195 views

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 ...
varkor's user avatar
  • 7,889
16 votes
4 answers
6k views

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 ...
8 votes
2 answers
613 views

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 &...
varkor's user avatar
  • 7,889
11 votes
1 answer
297 views

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 ...
TopologicalDynamitard's user avatar
11 votes
1 answer
505 views

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 ...
Michael Albanese's user avatar
6 votes
2 answers
406 views

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/...
5 votes
1 answer
151 views

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 ...
rschwieb's user avatar
  • 1,593
6 votes
1 answer
151 views

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 ...
wadsc's user avatar
  • 63
103 votes
17 answers
15k views

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 ...
2 votes
1 answer
117 views

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 $...
Usa's user avatar
  • 119
80 votes
22 answers
14k views

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 ...
0 votes
0 answers
58 views

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?
Manfred Weis's user avatar
  • 12.5k
7 votes
2 answers
201 views

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 ...
varkor's user avatar
  • 7,889
3 votes
0 answers
146 views

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 ...
Mikhail Bondarko's user avatar
0 votes
0 answers
99 views

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 ...
Turbo's user avatar
  • 13.5k
1 vote
0 answers
57 views

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 ...
Nep Nep's user avatar
  • 111
0 votes
1 answer
112 views

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. ...
Mattheus Pereira's user avatar
49 votes
30 answers
7k views

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 ...
12 votes
1 answer
496 views

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(...
Evan O'Dorney's user avatar
72 votes
13 answers
10k views

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 ...
1 vote
1 answer
254 views

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,\...
Motaka's user avatar
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