Questions tagged [examples]

For questions requesting examples of a certain structure or phenomenon

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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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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
582 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 &...
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10 votes
1 answer
192 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 ...
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11 votes
1 answer
340 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 ...
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5 votes
0 answers
132 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/...
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5 votes
1 answer
121 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 ...
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6 votes
1 answer
118 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 ...
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99 votes
16 answers
14k 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
98 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 $...
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77 votes
21 answers
13k 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
37 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?
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7 votes
2 answers
135 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 ...
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3 votes
0 answers
125 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 ...
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0 votes
0 answers
94 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 ...
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1 vote
0 answers
43 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 ...
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0 votes
1 answer
79 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. ...
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46 votes
30 answers
6k 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
419 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(...
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0 votes
0 answers
32 views

Length of shorter directed cycle relative to average with oppositely oriented counter-part

let $G(V,E)$ be a symmetric cyclic graph with positively weighted edges; derive now from $G$ a network $N(V,A)$ by replacing each undirected edge $e_{ij}$ of $G$ by a pair of antiparallel arcs $\...
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67 votes
11 answers
8k 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
220 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,\...
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8 votes
1 answer
340 views

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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2 votes
1 answer
132 views

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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2 votes
1 answer
126 views

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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1 vote
1 answer
144 views

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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4 votes
1 answer
220 views

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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8 votes
3 answers
287 views

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, ...
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0 votes
1 answer
130 views

What are examples of mathematical objects that are 'constructed out of' a range of other objects but fall out of them? [closed]

What are examples of mathematical objects that are somehow 'constructed out of' a whole range of other objects but fall out of them? One example that comes to my mind is that of ordinal numbers: $\...
2 votes
0 answers
66 views

Examples of multiplier Hopf algebras

A multiplier Hopf-algebra (introduced by Van Daele) is a pair $(A, \Delta)$ where $A$ is a non-degenerate algebra $A$ together with a non-degenerate algebra morphism $\Delta: A \to M(A \otimes A)$ ...
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2 votes
1 answer
183 views

Making coherent sheaves with nonvanishing higher Chern classes

Let $\mathcal{F}$ be a coherent sheaf on a variety $X$, and assume $\mathcal{F}$ has generic rank $n$. I expect (see e.g. here) that this actually puts no conditions on its Chern classes $c_1(\mathcal{...
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2 votes
0 answers
213 views

description of very ample bundle of Hirzebruch surface

I learned some basic properties of Hirzebruch surface mainly from Vakil's notes "the rising sea", section 20.2.9. the Hirzebruch surface is defined as $\mathbb{F}_n:=\operatorname{Proj} (\...
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3 votes
1 answer
79 views

On cospectral graphs

Is there examples of non-isomorphic cospectral graphs having Non-isomorphic automorphism groups? Isomorphic automorphism groups?
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3 votes
1 answer
256 views

Find strictly subharmonic function that vanishes at infinity

I am not sure about the term "strictly" subharmonic. What I want is a function $\psi\in C^{\infty}(\mathbb{R}^3)$ with $\Delta\psi>0$ and $\lim\limits_{|x|\rightarrow\infty}\psi(x)=0$. I ...
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  • 433
0 votes
1 answer
104 views

Uniform approximation of indicator function of a point

Fix $x \in \mathbb{R}$ and let $I_{[x]}$ be its indicator function. Does anyone know of a sequence of (obviously) discontinuous approximations $g_n$ to $I_{[x]}$ such that $g_n$ converge uniformly ...
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4 votes
1 answer
285 views

Push-out in the category of coherent sheaves over the complex projective plane

I'm trying to deal with an example of a rank two vector bundle over the complex projective plane which is non slope-stable (because the associated sheaf of sections has a coherent subsheaf of equal ...
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25 votes
1 answer
607 views

"Non-categorical" examples of $(\infty, \infty)$-categories

This title probably seems strange, so let me explain. Out of the several different ways of modeling $(\infty, n)$-categories, complicial sets and comical sets allow $n = \infty$, providing ...
1 vote
0 answers
103 views

Examples for certain class of projective varieties

I am looking for specific varieties that satisfy certain property. I call them symmetric varieties, I want to know what varieties are in it. It contains the projective spaces $\mathbb{P}^n$ for all $n$...
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11 votes
1 answer
645 views

Example of a PID with a residue field of finite characteristic and a residue field of characteristic 0?

I understand that for any nonempty set $S$ of characteristics, there exists a PID $R$ such that the set of characteristics of residue fields of $R$ (i.e. quotients by of $R$ by maximal ideals -- I'm ...
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1 vote
0 answers
57 views

Multivarate "RKHS" Examples

I've been reading about RKHSs and Hilbert spaces of functions these days a bit these days and I haven't yet come across an example of a hilbert space $H$ whose elements are all functions $f:\mathbb{R}^...
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7 votes
1 answer
229 views

Interesting "epimorphisms" of $E_\infty$-ring spectra

$\newcommand{\Mod}{\mathbf{Mod}} \newcommand{\map}{\mathrm{map}_{E_\infty-A}}$ Suppose $i:A\to B$ is a map of $E_\infty$-ring spectra. It induces a functor of $\infty$-categories $\Mod_B\to\Mod_A$ by ...
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1 vote
1 answer
176 views

Examples of conjectures whose direct falsity implies different consequences than indirect falsity

Mathematics several times has statements of form $$\mathsf{Statement A}\implies\mathsf{Statement B}$$ where $\mathsf{Statement A}$ and $\mathsf{Statement B}$ are conjectures while the implication is ...
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3 votes
1 answer
239 views

Lagrange’s interpolation formula: Theoreme and Example [closed]

I would like to know where they come up with the formula of Lagrange interpolation (Lagrange’s interpolation formula),Lagrange_polynomial because I did some research, but I find a different definition ...
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  • 131
1 vote
1 answer
368 views

Known dense subset of Schwartz-like space and $C_c^{\infty}$?

After reading this question, which asked for some examples of commonly used (proper) dense subsets of $C_0^{\infty}(\mathbb{R}^n)$ with the $L^p$-norm I wonder. What are some "well-known" ...
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1 vote
1 answer
75 views

Folding a non-rectangular shape into a rectangle of uniform thickness

I think the following might be an interesting subproblem of this question: Question: For an odd number $n\ge 3$, is there a non-rectangular but still convex shape of area $A=1$, that can be folded (...
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42 votes
9 answers
5k views

What are some examples of proving that a thing exists by proving that the set of such things has positive measure?

Suppose we want to prove that among some collection of things, at least one of them has some desirable property. Sometimes the easiest strategy is to equip the collection of all things with a measure, ...
13 votes
1 answer
516 views

Is there an orientable prime manifold covered by a non-prime manifold?

A manifold is called prime if whenever it is homeomorphic to a connected sum, one of the two factors is homeomorphic to a sphere. Is there an example of a finite covering $\pi : N \to M$ of closed ...
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0 votes
1 answer
230 views

Examples of additive categories [closed]

I already this question here but I didn't get any satisfactory answer, so I will try in MO now. There are a lot of interesting and creative examples of categories, such as for example, the category ...
2 votes
1 answer
91 views

A diameter 2 arc-transitive graph whose complement is not arc-transitive?

A graph $G=(V,E)$ is arc-transitive if its symmetry group acts transitively on ordered pairs of adjacent vertices. In general, the complement of an arc-transitive graph is not arc-transitive. But I ...
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3 votes
1 answer
205 views

A connected topological space whose points cannot be connected by irreducible components

Does there exist a topological space $X$ with the following properties? $X$ is connected. The set of irreducible components of $X$ is locally finite. Not every pair of points in $X$ can be "connected ...
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