Questions tagged [examples]

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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 ...
Noah Schweber's user avatar
8 votes
1 answer
394 views

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 \...
Dat Minh Ha's user avatar
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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
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8 votes
2 answers
263 views

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 ...
Seirios's user avatar
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8 votes
2 answers
1k views

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 ...
Elgrimm's user avatar
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8 votes
1 answer
257 views

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 ...
Max Demirdilek's user avatar
8 votes
4 answers
452 views

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 ...
user34458's user avatar
8 votes
1 answer
418 views

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 ...
Christopher Drupieski's user avatar
8 votes
1 answer
744 views

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 ...
Ender Wiggins's user avatar
8 votes
0 answers
286 views

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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7 votes
9 answers
3k views

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 ...
7 votes
11 answers
1k views

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 ...
7 votes
10 answers
1k views

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 ...
7 votes
3 answers
2k views

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 ...
Brenin's user avatar
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7 votes
4 answers
8k views

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?
Dyke Acland's user avatar
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7 votes
7 answers
3k views

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 ...
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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7 votes
2 answers
628 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 &...
varkor's user avatar
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7 votes
3 answers
822 views

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 ...
Mike Pierce's user avatar
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7 votes
2 answers
1k views

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 ...
Suresh Venkat's user avatar
7 votes
1 answer
3k views

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 \}...
Gjergji Zaimi's user avatar
7 votes
7 answers
632 views

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 ...
7 votes
1 answer
491 views

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&...
jkjfgk's user avatar
  • 73
7 votes
3 answers
649 views

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 ...
Manfred Weis's user avatar
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7 votes
1 answer
2k views

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 $...
Andres Abella's user avatar
7 votes
2 answers
1k views

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 ...
Sándor Kovács's user avatar
7 votes
1 answer
460 views

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 ...
user'''''''s user avatar
7 votes
1 answer
603 views

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, ...
Zach H's user avatar
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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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7 votes
3 answers
2k views

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 ...
7 votes
2 answers
212 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 ...
varkor's user avatar
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7 votes
1 answer
275 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 ...
Maxime Ramzi's user avatar
  • 13.3k
7 votes
1 answer
273 views

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 $r$ ...
Joseph Van Name's user avatar
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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7 votes
0 answers
383 views

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 ...
Ignasi Mundet i Riera's user avatar
7 votes
0 answers
512 views

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 ...
domenico fiorenza's user avatar
6 votes
14 answers
5k views

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 ...
6 votes
4 answers
1k views

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.
Anweshi's user avatar
  • 7,272
6 votes
5 answers
2k views

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. ...
Henry Wegener's user avatar
6 votes
5 answers
775 views

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 ...
Boyu Zhang's user avatar
6 votes
5 answers
3k views

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:...
Andrew Critch's user avatar
6 votes
3 answers
563 views

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 ...
solbap's user avatar
  • 3,938
6 votes
1 answer
455 views

Satellite knot example

Can someone provide me with an example of a satellite knot with symmetry group which is neither cyclic nor dihedral?
Jason's user avatar
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6 votes
1 answer
446 views

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 ...
KotelKanim's user avatar
  • 2,007
6 votes
1 answer
2k views

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 ...
Andrew Parker's user avatar
6 votes
3 answers
2k views

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 ...
Chris Aholt's user avatar
6 votes
1 answer
1k views

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 ...
Dario's user avatar
  • 643
6 votes
3 answers
431 views

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} \...
Najdorf's user avatar
  • 731
6 votes
4 answers
2k views

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 -...
miforbes's user avatar
  • 1,088
6 votes
1 answer
172 views

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 ...
Max Demirdilek's user avatar

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