Questions tagged [examples]

For questions requesting examples of a certain structure or phenomenon

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An example of a finite group with some specific permutable subgroups

The following question is about finite groups. Let $G$ be a finite group and let $H, K \leqslant G$. We say that $H$ permutes with $K$ if $HK = KH$ and in this case $HK \leqslant G$. The Symbol $\pi ...
user28083's user avatar
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11 votes
1 answer
624 views

Periodic function $f$ for which $f(x^2)$ is periodic too

There is the following question which was asked multiple times on Math.SE (e.g. here and here) without any final result: Question: Is there a periodic function $f:\Bbb R \to\Bbb R$ of smallest ...
M. Winter's user avatar
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0 votes
1 answer
79 views

Topology generated by complete and incomplete uniformities [closed]

Does there exist a topology which can be induced simultaneously by a complete and an incomplete uniformity?
Jave's user avatar
  • 195
16 votes
2 answers
999 views

Examples of triality in mathematics

There are tons of interesting examples of duality in mathematics (Poincaré duality, Verdier duality, Stone duality, s-duality, Tannaka duality, Koszul duality, Spanier-Whitehead duality ... ). What ...
2 votes
0 answers
183 views

Right adjoint completions

Forgive me if this question is not well thought out. I don't know how else to ask it. The nlab page on completion gives some examples of completions which are left adjoints. These completions are "...
24 votes
8 answers
4k views

When does a metric space have "infinite metric dimension"? (Definition of metric dimension)

Definition 1 A subset $B$ of a metric space $(M,d)$ is called a metric basis for $M$ if and only if $$[\forall b \in B,\,d(x,b)=d(y,b)] \implies x = y \,.$$ Definition 2 A metric space $(M,d)$ has &...
Chill2Macht's user avatar
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5 votes
0 answers
222 views

In search for examples concerning pushforward of nef divisors and lc-trivial fibrations

My question is motivated by ideas around the moduli b-divisor of an lc-trivial fibration (see for instance the following paper by Ambro https://arxiv.org/pdf/math/0308143.pdf). In such a setup, one ...
Stefano's user avatar
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1 vote
0 answers
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Is it possible that a convex cone and its closure both induce vector lattices?

Given a convex cone $P\subset X$ where $X$ is a $K$-vector space, $K=\mathbb{R}\text{ or }\mathbb{C}$ is a field. Suppose that $P$ satisfies positive element stipulations. (1) $X=P-P$. (2) $P\cap-P=...
Henry.L's user avatar
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10 votes
1 answer
432 views

Example of Banach spaces with non-unique uniform structures

While it is known that compact Hausdorff spaces admit unique uniform structures, it is further shown by Johson and Lindenstrauss's result that Banach spaces are characterized by their uniform ...
Henry.L's user avatar
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4 votes
1 answer
166 views

Semi-metrizable spaces with countable chain condition

Note that $X$ is semi-metrisable iff $X$ is first countable and semi-stratifiable. Definition A topological space $(X,\tau)$ is called semi-metric if there exists a function $g:\omega\times X\to\tau$...
Paul's user avatar
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3 answers
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For every monotonically-decreasing non-negative function $ f $, does there exist a function $ g $ so that $ f g $ is integrable? [closed]

Let $ f $ be a monotonically-decreasing non-negative function satisfying $ \displaystyle \lim_{x \to \infty} f(x) = 0 $. Is it true that the following claim holds? Claim: There exists a function $ ...
Spinorbundle's user avatar
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4 votes
2 answers
195 views

A result on spaces with countable pseudocharacter and countable tightness

There is a statement as follows: If a Hausdorff (regular, Tychonoff) space $X$ has countable pseudocharacter and countable tightness, then the closure of any set $Y\subset X$ of cardinality $\le \...
Paul's user avatar
  • 601
17 votes
2 answers
3k views

Consequences of the Birch and Swinnerton-Dyer Conjecture?

Before asking my short question I had made some research. Unfortunately I did not find a good reference with some examples. My question is the following What are the consequences of the Birch and ...
8 votes
2 answers
537 views

Non-trivial examples of Stably diffeomorphic 4-manifolds

I am looking for some non-trivial examples of (smooth) 4-mflds $M,N$ such that $M$ and $N$ are STABLY diffeomorphic. I.e. $$M\sharp_n (S^2\times S^2) \cong N \sharp_r (S^2\times S^2)$$ for $r,n$ not ...
Luigi M's user avatar
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40 votes
10 answers
4k views

Phenomena of gerbes

What is your favourite example of Gerbes? I would like to know Where do we find Gerbes in "nature"? The examples could vary from String theory to Galois theory. For example my favourite examples of ...
tttbase's user avatar
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7 votes
0 answers
435 views

Is there a list of examples of orthogonal spectra?

Schwede's symmetric spectra book project provides point-set models of many important spectra as symmetric spectra, including (in §I.1) the sphere spectrum, Eilenberg-Mac Lane spectra, several Thom ...
Arun Debray's user avatar
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2 votes
2 answers
185 views

Is there a known construction for heavy topologies of all sizes?

Given a set $A$ is there a known way to find a topological space $X$ such that $|A|=|X|<w(X)$? Here $w(X)$ is the weight of the topological space. This is clearly impossible for finite sets $A$. ...
Gorka's user avatar
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24 votes
8 answers
3k views

Applications of logic to group theory?

There seems to be an ever-growing literature on the first-order theory of groups. While I find this interaction between group theory and logic quite appealing, I was wondering the following: Are ...
Ganon's user avatar
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2 votes
1 answer
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An example of Guillemin Sternberg Conjecture

Guillemin Sternberg Conjecture(proved) says that for symplectic manifold $(M,\omega)$ with $[Q,R]=0$ condiction, with compact group action $G$, such that $\mu:M\to \mathfrak g^*$ is regular at $0$, ...
DLIN's user avatar
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15 votes
1 answer
534 views

Torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z^2$

Is there a torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z \oplus \Bbb Z$ (as groups)? Notice that $\Bbb Z$ is not cancellable, so $A \oplus \Bbb Z \...
Watson's user avatar
  • 1,702
14 votes
4 answers
1k views

Is the "Moebius Stairway" Graph Already Known?

It is a wellknown fact, that Moebius Ladder Graphs have $2n$ vertices, but nowhere could I find any hint of how to generalize them to Graphs with $2n+1$ vertices. Last week I had the idea of giving up ...
Manfred Weis's user avatar
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5 votes
2 answers
207 views

Confusion in some notations in Lie sub-algebras of exceptional Lie algebra

I was following Humphrey's Lie algebra for study, and came to study of Weyl groups of root systems. The book has stated orders of Weyl groups of exceptional Lie algebras, and there were no comments or ...
p Groups's user avatar
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4 votes
2 answers
287 views

more examples of non-weakly Lindelöf spaces

A space $X$ is called weakly-Lindelöf if every open cover $\mathcal{U}$ has a countable subcover $\mathcal{U'} \subseteq \mathcal{U}$ such that $\cup \mathcal{U}'$ is dense in $X$. This class seems ...
Henno Brandsma's user avatar
0 votes
3 answers
189 views

Connected $T_2$-spaces with only constant maps between them

If $f:\mathbb{R}\to\mathbb{Q}$ is continuous, then it is constant. Are there infinite connected $T_2$-spaces $X,Y$ such that the only continuous maps $f:X\to Y$ are the constant maps?
Dominic van der Zypen's user avatar
2 votes
1 answer
258 views

Worst Case Region for a Convex Hull Heuristic

I am currently implementing a heuristic algorithm for planar convex hulls hand would like to know, for which kind of strictly convex region it exhibits worst performance. I know that there are many ...
Manfred Weis's user avatar
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29 votes
0 answers
1k views

Is there a field $F$ which is isomorphic to $F(X,Y)$ but not to $F(X)$?

Is there a field $F$ such that $F \cong F(X,Y)$ as fields, but $F \not \cong F(X)$ as fields? I know only an example of a field $F$ such that $F$ isomorphic to $F(x,y)$ : this is something like $F=k(...
Watson's user avatar
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63 votes
7 answers
8k views

Theorems demoted back to conjectures

Many mathematicians know the Four Color Theorem and its history: there were two alleged proofs in 1879 and 1880 both of which stood unchallenged for 11 years before flaws were discovered. I am ...
6 votes
2 answers
551 views

Applications of isotropic quadratic forms

I will soon be teaching an introductory course on bilinear algebra and quadratic forms. I will likely spend most of the time and effort on positive definite quadratic forms and euclidean spaces. These ...
7 votes
2 answers
936 views

Second Stiefel-Whitney class is a square

I'm interested in examples of manifolds which are orientable and such that the second Stiefel-Whitney class is a square. (Of course the second Stiefel-Whitney class should be non-zero.) An easy ...
AlexE's user avatar
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13 votes
1 answer
1k views

A topology on $\Bbb R$ where the compact sets are precisely the countable sets

QUESTION. In there a topology on $\Bbb R$ where the compact subsets are precisely the countable subsets? I am trying to create a counterexample to a certain claim, and I found that what I need is a ...
Cauchy's user avatar
  • 233
3 votes
1 answer
169 views

p-Group satisfying the minimal condition on abelian subgroups

Are there examples of $p$-groups satisfying the minimal condition on abelian subgroups but do not satisfying the minimal condition on subgroups? Obviously such a group cannot be locally finite. I've ...
W4cc0's user avatar
  • 137
7 votes
3 answers
487 views

Where can I find explicit descriptions of principal $SL(2,\mathbb{C})$s?

I am interested in an explicit description of the principal homomorphism from $SL(2,\mathbb{C})$ to $G$, for each complex semisimple Lie group $G$. Does any one have specific references please? ...
Malkoun's user avatar
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10 votes
1 answer
472 views

Properties of the petit Zariski topos

What are some (intrinsically formulated) properties of the locally ringed topos $(\mathbf{Sh}(X),\mathcal{O}_X)$ for some scheme $X$, which do not hold for arbitrary locally ringed toposes? Is there, ...
HeinrichD's user avatar
  • 5,402
3 votes
1 answer
306 views

Simply connected 4-manifolds with boundary

I think I've encountered a question about 4-manifolds which maybe easy but I'm not familiar with. Can anyone give me an example of a simply connected 4-manifold $M$ (with boundary, of course) with $...
Ivy's user avatar
  • 123
33 votes
2 answers
1k views

can another topology be given to $\mathbb R$ so it has the same continuous maps $\mathbb R\rightarrow \mathbb R$?

We say two topologies $\tau$ and $\rho$ on $X$ are similar if the set of continuous functions $f:(X,\tau) \rightarrow (X,\tau)$ is the same as the set of continuous functions $f:(X,\rho)\rightarrow (X,...
Gorka's user avatar
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3 votes
0 answers
138 views

Are there any interesting examples of geometric triangulated categories with the Jordan-Holder property?

In this paper, Kuznetsov mentions that the following triangulated categories have the Jordan-Holder property $\mathbf{D}(\Bbb P^1)$ and $\mathbf{D}(\Bbb P^1/\Gamma)$ connected Calabi-Yau categories ...
Brian Fitzpatrick's user avatar
82 votes
17 answers
11k views

Examples of algorithms requiring deep mathematics to prove correctness

I am looking for examples of algorithms for which the proof of correctness requires deep mathematics ( far beyond what is covered in a normal computer science course). I hope this is not too broad.
12 votes
1 answer
867 views

What are types of coalgebras that are more naturally described by cooperads?

Some background. Let $\mathsf{C}$ be a symmetric monoidal category. An object $X \in \mathsf{C}$ has two operads "naturally" (the two constructions aren't functorial) associated to it: the operad of ...
Najib Idrissi's user avatar
2 votes
0 answers
658 views

Applications of the Weak and Weak$^*$ topologies to PDEs?

Chapter $3$ of Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis constructs and explains the Weak and Weak$^*$ topologies over a Banach Space $E$. The most ...
D1X's user avatar
  • 121
8 votes
3 answers
473 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 ...
Manfred Weis's user avatar
  • 12.6k
4 votes
0 answers
55 views

Looking for a Collection of Examples and Counter Examples for Assumptions about the Properties of Planar Euclidean TSP Instances?

Where can I find example and counter examples to seemingly plausible assumption about the properties of optimal solutions of planar euclidean TSP instances? The reason for asking is that the ...
Manfred Weis's user avatar
  • 12.6k
14 votes
1 answer
837 views

Examples of étale covers of arithmetic surfaces

Define an arithmetic scheme $X$ to be a separated, integral scheme, flat and finite type over $\mathbb{Z}$. I am interested in obtaining examples of finite étale covers of arithmetic schemes. I am ...
PrimeRibeyeDeal's user avatar
5 votes
1 answer
825 views

Uniquely ergodic and strongly mixing transformation

Is there an example of a non-trivial measure preserving transformation that is uniquely ergodic and strongly mixing (in the measure theoretic sense)? This was asked here, but with no answer.
George Shakan's user avatar
5 votes
1 answer
142 views

Example request: seriously deficient homogeneous spaces

In a previous post, I cite a dimension condition commonly satisfied by homogeneous spaces and claim that a counterexample must have deficiency at least $3$. For convenience, I reproduce the definition ...
jdc's user avatar
  • 2,984
1 vote
0 answers
61 views

Applications of systems with multiple time

A dynamical system with multiple time is an action of a group $\mathbb{Z}^d$ or $\mathbb{R}^d$ on a metric space. I am interested in informative examples and applications of such systems. I know ...
demolishka's user avatar
10 votes
1 answer
612 views

Problems which use S₄ → S₃

I need examples of problems which use, directly or indirectly, the homomorphism $S_4\to S_3$ in the solution (its kernel is $\mathbb{Z}_2\oplus\mathbb{Z}_2$). Obvious candidates: Lagrange resolvent (...
Anton Petrunin's user avatar
3 votes
0 answers
186 views

Groups with probability measures

Are there algebraic structures that integrate groups with probability measures? For instance, can the closure operation on a group be assigned a probability that says "how much" a member belongs to ...
Kasthuri's user avatar
  • 141
11 votes
2 answers
995 views

Densest Graphs with Unique Perfect Matching

Given a graph $G$ with $n$ vertices, that has a perfect matching $M$, what is the maximal number of edges that $G$ can have without contradicting the uniqueness of $M$? Are examples of such extremal ...
Manfred Weis's user avatar
  • 12.6k
1 vote
1 answer
62 views

Test Instances for Perfect Matchings in Graphs

Are there any graphs with a known set of perfect matchings and other predefined properties, such as vertex connectivity, which can be used for testing the implementation of matching algorithms? ...
Manfred Weis's user avatar
  • 12.6k
74 votes
1 answer
5k views

$R$ is isomorphic to $R[X,Y]$, but not to $R[X]$

Is there a commutative ring $R$ with $R \cong R[X,Y]$ and $R \not\cong R[X]$? This is a ring-theoretic analog of my previous question about abelian groups: In fact, in any algebraic category we may ...
Martin Brandenburg's user avatar

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