Questions tagged [examples]
For questions requesting examples of a certain structure or phenomenon
538
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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=...
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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 ...
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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 $g:\omega\times X\to\tau$...
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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 $ ...
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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 \...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $A$. ...
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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 ...
- 359
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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 $0$, ...
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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 \...
- 1,682
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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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(...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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? ...
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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, ...
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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 $...
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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,...
- 1,825
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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
...
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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.
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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 ...
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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 ...
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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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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 ...
- 12.5k
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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 ...
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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.
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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 ...
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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 ...
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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 (...
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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 ...
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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 ...
- 12.5k
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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?
...
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$R$ 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 ...
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Rank versus free-rank of a module
Suppose $M$ is a finitely generated left module over a ring $R.$
We define the rank of $M$ as the minimal number of generators of $M.$
If in addition $M$ is free, then we define the free-rank of $...
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simple and non nuclear $C^*$-algebra
Is there an example of simple and non-nuclear(non-amenable) $C^*$-algebra?
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Vector field on a K3 surface with 24 zeroes
In https://mathoverflow.net/a/44885/4177, Tilman points out that one can use a $K3$ surface minus the zeroes of a generic vector field to build a nullcobordism for $24[SU(2)]$. Given that a) this is a ...
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Examples when one can use the the symmetric power $L$-functions to study topics related to the number theory
"The symmetric power $L$-functions are a powerful tool for studying algebraic or geometric objects through analytic methods." I read this sentence in the introduction of a Master thesis. I want to ...
- 1,193
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What are examples when the equality of some invariants is good enough in algebraic topology?
As far as my understanding goes, most of the tools of algebraic topology (homotopy groups, homology groups, cup product, cohomology operations, Hopf invariant, signature, characteristic classes, knot ...
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Actions on spaces with measured walls
In geometric group theory, the question of whether or not a group acts nicely on a CAT(0) cube complex, or equivalently on a median graph, is of interest. The same question for actions on spaces with ...
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$A$ is isomorphic to $A \oplus \mathbb{Z}^2$, but not to $A \oplus \mathbb{Z}$
Are there abelian groups $A$ with $A \cong A \oplus \mathbb{Z}^2$ and $A \not\cong A \oplus \mathbb{Z}$?