Questions tagged [examples]
For questions requesting examples of a certain structure or phenomenon
544
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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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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 \...
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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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Roller's problem on median groups
At the end of his dissertation Poc Sets, Median Algebras and Group Actions, Martin Roller asks
A group $G$ is called median if it acts freely and transitively on a median algebra. This is ...
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Example for an integral, rectifiable varifold with unbounded first variation
I'm just looking for an example of an integral, rectifiable varifold, which has no locally bounded first variation.
Recapitulation
for every $m$-rectifiable varifold $\mu$ exists a $m$-rectifiable ...
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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 ...
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Order-independent properties arising naturally in mathematics
The motivation for the following question comes from finite model theory,
but it is not a technical question about this field,
and it is particularly directed at people working in other fields.
It ...
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Examples of exotic modules for the additive group
Let $k$ be an algebraically closed field of positive characteristic $p > 0$, and let $X$ be an intedeterminate over $k$. I am interested in the additive group scheme $\mathbb{G}_a$, that is, the ...
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A "concrete" example of a one-sided Hopf algebra
I came to know from the paper Left Hopf Algebras by Green, Nichols and Taft that one may consider a Hopf algebra whose antipode satisfies only the left (resp. right) antipode condition.
To be more ...
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Explicit computations with crystalline cohomology
I am currently studying crystalline cohomology and while all the talk about crystalline topoi is nice, I would like to see some explicit computations. What are some references on this subject which ...
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What category without initial object do you care about?
Recently I have been listening to some constructions that have been designed to accommodate categories without an initial object. The speaker has given some idea of a category or two that he cares ...
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Non-real constants
Constants are usually real numbers e.g. e, pi, gamma etc. Can you give examples of special constants that are not real? e.g. complex or p-adic constants.
A real number in base10 can be viewed as the ...
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Examples of "Unusual" Classifications
When one says "classification" in math, usually one of a handful of examples springs to mind:
-Classification of Finite Simple Groups with 18 infinite families and 26 sporadic examples (assuming one ...
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Fibrations with isomorphic fibers, but not Zariski locally trivial
(I posted this same question on MSE. Sorry if it is too elementary.)
I am looking for examples of fibrations $f:X\to Y$ where the fibers are all isomorphic, but $f$ is not Zariski locally trivial. In ...
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Good example of a non-continuous function all of whose partial derivatives exist
What's a good example to illustrate the fact that a function all of whose partial derivatives exist may not be continuous?
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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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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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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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What's an illustrative example of a tame algebra?
A finite-dimensional associative $\mathbf{k}$-algebra $\mathbf{k}Q/I$ is of tame representation type if for each dimension vector $d\geq 0$, with the exception of maybe finitely many dimension vectors ...
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Example of non-closed convex hull in a CAT(0) space
this is related to this question but is simpler, and hopefully is well-known. There are a number of references that say that the convex hull of a collection of points in a CAT(0) space need not be ...
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Connection between bi-Hamiltonian systems and complete integrability
As I understand, the lack of indication on how to obtain first integrals in Arnol'd-Liouville theory is a reason why we are interested in bi-Hamiltonian systems.
Two Poisson brackets
$\{ \cdot,\cdot \}...
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Given a sequence defined on the positive integers, how should it be extended to be defined at zero?
This question is inspired by a lecture Bjorn Poonen gave at MIT last year. I have ideas of my own, but I'm interested in what other people have to say, so I'll make this community wiki and post my ...
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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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Examples of complicated parametric Jordan curves
For test purposes I need parametric Jordan curves that are complicated in the sense of having many inflection points and ideally no symmetries.
When doing online search I always land at complex ...
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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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The rank of a not necessarily finitely generated module.
This question is motivated by this one. The main point of the question (was) to try to weaken the notion of rank. After the answers and comments, it seems this is not a good way to do it, but perhaps ...
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Example of a smooth family of projective surfaces with non-vanishing integrals of Todd classes
Motivation:
Let $\pi\colon S \rightarrow B$ be smooth projective morphism of relative dimension 2 over a smooth projective scheme $B$. If the stucture sheaves of the fibres do not have higher ...
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Generating functions with all non-zero coefficients equal to one
I have been wondering if there are any useful generating functions with all non-zero coefficients equal to one. Obviously, the trivial generating function $\frac{1}{1-x}$ has significant applications, ...
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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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What are some interesting sequences of functions for thinking about types of convergence?
I'm thinking about the basic types of convergence for sequences of functions: convergence in measure, almost uniform convergence, convergence in Lp and point wise almost everywhere convergence. I'm ...
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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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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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Factoring $\frac{1}{1-rx}$ into an infinite products of polynomials
I am looking for examples of sequences of polynomials $(p_{k}(x))_{k=1}^{\infty}$ with positive integer coefficients where $p_{k}(0)=1$ for all $k\geq 1$ and where there is a positive integer $r$ ...
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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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Linear vs smooth actions of finite groups on spheres, euclidean spaces and closed disks
I would like to know examples (with references, if possible) of the following:
(1) a finite group $G$ acting effectively and smoothly on a sphere $S^n$ (any $n$) but admitting no effective linear ...
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Principal $G$-bundles as fully extended TQFTs, and $n$-representations
This is a follow up to this MO question: Fully dualizable objects in classical field theories
Assuming the notation there (which in turn come from Topological Quantum Field Theories from Compact Lie ...
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Applications of compactness [closed]
Similar to this question: Applications of connectedness I want to collect applications of compactness.
E.g.: compact + discrete => finite, which can be used to prove the finiteness of the ...
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Kähler manifold which is not algebraic
Can someone provide examples of Kähler manifolds which are not algebraic?
This question came to my mind seeing the post of Andrea Ferretti.
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Examples of Riemannian Submersions
Is there any example of a Riemannian submersion, which is no fibration?
As far as I know, a (any) submersion is locally, but not globally, given by a fibration. The converse holds globally. ...
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Existence of orientation preserving, finite order self homeomorphism on a genus 2 surface without fixed point
Let $M$ be a compact 2-manifold of genus 2. Does there exist an orientation preserving homeomorphism $f:M\to M$, so that $f^n=id$ for some integer $n$, and $f$ doesn't have fixed points?
Using ...
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Is very ampleness of a divisor on a curve determined entirely by degree and genus?
Edit: Apparently the answer is "no", so what is an example of two curves of genus g, and a divisor of degree d on each, such that one is very ample and the other is not?
Question as originally stated:...
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Weil divisors on non Noetherian schemes
Let X be an integral scheme that is separated (say over an affine scheme). Define a Weil divisor as a finite integral combination of height 1 points of X, where the height of a point of X is the ...
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Satellite knot example
Can someone provide me with an example of a satellite knot with symmetry group which is neither cyclic nor dihedral?
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A counter example in obstruction theory
Let $K$ denote a simplicial complex and $Y$ a path-connected topological space. Let us also denote by $K^n$ the $n$-skeleton of $K$. I would like to have an example for the following situation or a ...
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An example of a rank one projective R-Module that is not invertible
Let $R$ be a commutative noetherian ring. I know that an $R$-module is invertible iff it is finitely generated and locally free of rank one. I presume then that there are examples of non-finitely ...
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Examples of birational equivalence of a variety and a hypersurface
There's an algebraic geometry theorem (I.4.9 in Hartshorne) that says: any variety of dimension r (over an algebraically closed field) is birationally equivalent to a hypersurface in projective space ...
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Example of a triangulable topological manifold which does not admit a PL structure
I know there are some examples of manifolds which don't admit a PL structure (combinatorial triangulation), and that it has been recently proven that in dimension $n\geq5$ there are manifold which are ...
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Non-trivial integral forms of algebras
Suppose $\mathcal{A}$ is a $\mathbf{C}$-algebra then an integral form would be a subring $\mathcal{B} \subset \mathcal{A}$ such that the canonical map $\mathcal{B} \otimes_{\mathbf{Z}} \mathbf{C} \...
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Examples of Super-polynomial time algorithmic/induction proofs?
In combinatorics, one can sometimes get an algorithmic proof, which loosely has the following form:
-The proof moves through stages
-An invariant is shown to hold by induction from previous stages
-...
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Hopf monads in categorical probability theory
1. Context. According to [1], probability monads are arguably the most important concept in categorical probability theory. In [2] Fritz and Perrone argue that "in order for a monad to really ...