Questions tagged [examples]
For questions requesting examples of a certain structure or phenomenon
554 questions
11
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4
answers
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A map inducing isomorphisms on homology but not on homotopy
As a consequence of the Whitehead theorem, Spanier's Algebraic Topology book has on 7.6.25 the following theorem:
A weak homotopy equivalence induces isomorphisms of the corresponding integral ...
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11
votes
6
answers
5k
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Can we have A={A} ?
Does there exist a set $A$ such that $A=\{A\}$ ?
Edit(Peter LL): Such sets are called Quine atoms.
Naive set theory By Paul Richard Halmos On page three, the same question is asked.
Using the ...
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102
votes
17
answers
16k
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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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1
vote
0
answers
233
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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 ...
- 157
2
votes
2
answers
227
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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 ...
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votes
3
answers
1k
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Does the "continuous locus" of a function have any nice properties?
Suppose $f:\mathbf{R}\to\mathbf{R}$ is a function. Let $S=\{x\in \mathbf{R}|f\text{ is continuous at }x\}$. Does $S$ have any nice properties?
Here are some observations about what $S$ could be:
$S$ ...
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54
votes
30
answers
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What are examples of good toy models in mathematics?
This post is community wiki.
A comment on another question reminded me of this old post of Terence Tao's about toy models. I really like the idea of using toy models of a difficult object to ...
- 1,362
13
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1
answer
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Explicit examples of Azumaya algebras
I'm trying to understand the Brauer group of a scheme better. I know how to compute $\text{Br}(X)$ as an abstract group in some cases, but don't have a good idea of what the individual Azumaya ...
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1
vote
0
answers
90
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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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7
votes
7
answers
3k
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Gelfand representation and functional calculus applications beyond Functional Analysis
I think it is fair to say that the fields of Operator Algebras, Operator Theory, and Banach Algebras rely on Gelfand representation and functional calculus in a crucial way.
I am curious about ...
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8
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1
answer
284
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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 ...
- 22.3k
73
votes
30
answers
9k
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What are some examples of ingenious, unexpected constructions?
Many famous problems in mathematics can be phrased as the quest for a specific construction. Often such constructions were sought after for centuries or even millennia and later proved impossible by ...
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26
votes
5
answers
10k
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Locally compact Hausdorff space that is not normal
What is a good example of a locally compact Hausdorff space that is not normal? It seems to be well-known that not all locally compact Hausdorff spaces are normal (and only a weaker version of Urysohn'...
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135
votes
43
answers
38k
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What are the most attractive Turing undecidable problems in mathematics?
What are the most attractive Turing undecidable problems in mathematics?
There are thousands of examples, so please post here only the most attractive, best examples. Some examples already appear on ...
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3
votes
1
answer
242
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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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2
votes
0
answers
110
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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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7
votes
1
answer
501
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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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8
votes
1
answer
357
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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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9
votes
2
answers
738
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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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4
votes
1
answer
2k
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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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votes
10
answers
4k
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Hopf algebras examples
Following Richard Borcherds' questions 34110 and 61315, I'm looking for interesting examples of Hopf algebras for an introductory Hopf algebras graduate course.
Some of the examples I know are well-...
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4
votes
2
answers
235
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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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2
votes
1
answer
203
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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}...
- 91.8k
1
vote
0
answers
198
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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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votes
22
answers
15k
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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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9
votes
2
answers
700
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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 ...
- 356
72
votes
13
answers
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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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5
votes
2
answers
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Is beauty at the high school level even possible? [closed]
This question is a follow up to 74841, and follows from a suggestion by Gian-Carlo Rota that beauty as judged by the educated public differs from that experienced by mathematicians (he gives Euclidean ...
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50
votes
12
answers
7k
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Combinatorial results without known combinatorial proofs
Stanley likes to keep a list of combinatorial results for which there is no known combinatorial proof. For example, until recently I believe the explicit enumeration of the de Brujin sequences fell ...
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19
votes
6
answers
2k
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Nonfree projective module over a regular UFD?
What is the simplest example of a domain $R$ which is regular (in particular Noetherian) and factorial which admits a finitely generated projective module that is not free?
In fact I'll be at least ...
- 4,713
4
votes
3
answers
904
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Examples of (non-normal) unibranched rings?
For a local integral domain $R$ the following are equivalent:
a) The integral closure of $R$ in its fraction field (i.e., the normalization of $R$) is again local.
b) The henselization of $R$ is ...
- 253
0
votes
1
answer
85
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)} \...
- 12.7k
10
votes
0
answers
228
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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}{\...
- 11.8k
2
votes
1
answer
198
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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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votes
1
answer
605
views
Nontrivial upper bounds on proof-theoretic ordinals of strong theories: do we have any?
Motivated by Consistency of Analysis (second order arithmetic) and Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?, I have the following question:
Are there any examples of strong ...
- 2,031
20
votes
3
answers
2k
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Can a module be an extension in two really different ways?
(Edit: I've realized that there was an error in my reasoning when I was convincing myself that these two formulations are equivalent. Hailong has given a beautiful affirmative answer to my first ...
- 2,821
73
votes
10
answers
11k
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Riemannian surfaces with an explicit distance function?
I'm looking for explicit examples of Riemannian surfaces (two-dimensional Riemannian manifolds $(M,g)$) for which the distance function d(x,y) can be given explicitly in terms of local coordinates of ...
- 2,821
4
votes
1
answer
148
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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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votes
3
answers
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Can integration spoil real-analyticity?
Is there an example of a function $f:(a,b)\times(c,d)\to\mathbb{R}$, which is real analytic in its domain, integrable in the second variable, and such that the function
$$ g:(a,b)\to\mathbb{R},\qquad ...
- 31.5k
15
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7
answers
2k
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Examples of rational families of abelian varieties.
I'd like to know examples of non-trivial families of abelian varieties over rational bases (e.g. open subschemes of the projective line P^1).
One can generate many examples as Jacobians of rational ...
- 42.9k
3
votes
0
answers
271
views
Seek "typical examples" for the structure of spaces with two-sided Ricci bounds
By a 1990 paper of Michael Anderson, the following is true:
Theorem. Let the metric space $(X,d,p)$ be a pointed Gromov-Hausdorff limit of a sequence of complete pointed Riemannian manifolds $(M_i,...
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6
votes
2
answers
447
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Common/well-known results with natural and/or useful reformulations
$\DeclareMathOperator{\pp}{\mathbb{P}}$My aim here is to have a collection of "natural" not-so-common reformulations/extensions of common/well-known results such that
the reformulation/...
- 1,701
156
votes
52
answers
24k
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Experimental mathematics leading to major advances
I would like to ask about examples where experimentation by computers has led to major mathematical advances.
A new look
Now as the question is five years old and there are certainly more examples of ...
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0
votes
1
answer
125
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Examples of real-time transcendental number and superlinear-time trancsendental number
Computation model is defined as Hartmanis and Stearns 4, it is well known that Liouvilles constant
$$C_L=\sum_{i=1}^{\infty} 10^{-i!}$$ is computable in real time or linear time 1, 5 especially ...
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76
votes
34
answers
7k
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Dimension leaps
Many mathematical areas have a notion of "dimension", either rigorously or naively, and different dimensions can exhibit wildly different behaviour. Often, the behaviour is similar for "nearby" ...
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8
votes
1
answer
437
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Noetherian but not strongly Noetherian
What are some examples of Tate rings $R$ (i.e. Huber rings with with topologically nilpotent units) which are Noetherian but not strongly Noetherian ($R$ is strongly Noetherian iff for all $n \in \...
- 466
2
votes
1
answer
332
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Are there "pathological convex sets" over ultravalued fields of char 2?
In their book Topological Vector Spaces (2nd ed.) Lawrence Narici and Edward Beckenstein generalise convex sets for TVS over ultravalued field $K$ as $K$-convex sets. The definition goes as following:...
- 519
13
votes
8
answers
3k
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Applications of the notion of of Gromov-Hausdorff distance
I am looking for applications of the notion of Gromov-Hausdorff convergence to prove theorems that a priori have nothing to do with it. Examples that I am aware of (thanks to wikipedia and google):
...
- 2,600
18
votes
3
answers
2k
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Eta-products and modular elliptic curves
Recently the elliptic curve $E:y^2+y=x^3-x^2$ of conductor $11$ (which appears in my answer) became my favourite elliptic over $\bf Q$ because the associated modular form
$$
F=q\prod_{n>0}(1-q^n)^2(...
- 2,821
8
votes
3
answers
482
views
Problems and algorithms requiring non-bipartite matching
While the importance of the non-bipartite matching problem itself from an algorithmic and complexity point of view is well known, applications of non-bipartite matching are hard to find.
I did an ...
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