Questions tagged [examples]
For questions requesting examples of a certain structure or phenomenon
555 questions
5
votes
1
answer
472
views
Measures which exhibit the "uncorrelated implies independent" property
Let $X$ be a topological linear space, and let $X^*$ be its dual space. Suppose that $X$ is complete and Hausdorff, and $X^*$ separates points. Let $Y$ be another such space, and let $f : X \to Y$ be ...
- 1,107
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 ...
- 8,998
3
votes
2
answers
800
views
Is there a non-trivial example for a 1-homogeneous function satisfying a specific inequality of second order?
Let $\mathbb{R}^n$ be the $n$-dimensional real vector space with Cartesian coordinates $x=(x^1,\ldots, x^n)\in \mathbb{R}^n$. I'm searching for a non-trivial example of a function $A:\mathbb{R}^n \...
CommunityBot
- 1
18
votes
12
answers
5k
views
What are some fundamental "sources" for the appearance of pi in mathematics?
I thought it might be fun to ask this question as a way of celebrating Pi Day. One way in which people popularize pi is that they say that even though it's defined in terms of properties of a circle, ...
- 456
8
votes
3
answers
2k
views
The harmonic (series) beetle: live illustrations of mathematical theorems
In my analysis class I use the following problem to illustrate the divergence
of the harmonic series (consider this as a hint for solving it).
Exercise.
A beetle creeps along a 1-meter infinitely ...
- 129
8
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 ...
CommunityBot
- 1
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:...
- 3,779
5
votes
1
answer
393
views
Not quite adjoint functors
What are standard and/or natural examples of pairs of functors $F:C\leftrightarrows D:G$ and unnatural bijections $\hom_D(Fx,y)\to\hom_C(x,Gy)$ for all $x$ and $y$? Can one do this so that the ...
- 2,754
16
votes
6
answers
6k
views
Fundamental group of the line with the double origin.
In the simplest cases, the fundamental group serves as a measure of the number of 2-dimensional "holes" in a space. It is interesting to know whether they capture the following type of "hole".
This ...
CommunityBot
- 1
2
votes
1
answer
730
views
Semitransitive relations
By a digraph, let us mean an ordered pair $(X,r)$ with $r : X \times X \rightarrow B,$ where $X$ is a set and $B = \{\mathrm{False}, \mathrm{True}\}.$
Then supposing $\mathbb{X} =(X,r)$ is a digraph, ...
- 236k
6
votes
1
answer
668
views
Examples of "nice" properties of algebraic extensions of $\mathbb{Q}$
I am writing a short survey of some "nice'' properties of algebraic extensions of $\mathbb{Q}$. Let's say a property (P) is nice if
every finite extension of $\mathbb{Q}$ satisfies (P), and
if $K \...
- 50.6k
17
votes
7
answers
1k
views
Examples of toposes for analysts
I've read that toposes are extremely important in modern mathematics, but I find the definitions and examples given on the nLab page a little too abstract to understand.
Can you provide some examples ...
CommunityBot
- 1
1
vote
1
answer
535
views
Examples of Quot schemes
I'm studying Quot schemes, that I denote with $Quot_{N,X,P}$, with $N \in \mathbb{Z}$, $X \subset \mathbb{P}^d$ and $P \in \mathbb{Q}[t]$. So, I'm looking for explicit examples of Quot schemes. Could ...
- 35.5k
33
votes
4
answers
11k
views
Counterexample for the Open Mapping Theorem
I would like to ask a counterexample for the classical theorem in functional analysis: the open mapping theorem in the case that $Y$ is Banach, but $X$ is not Banach to show that the completeness of X ...
CommunityBot
- 1
2
votes
1
answer
412
views
Vanishing Cech cohomology
Let $X$ be a manifold such that $dim(X)=n$. It is well-know that if $\mathcal{F}$ is a coherent sheaf $H^m(X,\mathcal{F})=0$ for all $m >n$ (where I denote with $H(-)$ Cech cohomology). But is ...
CommunityBot
- 1
3
votes
3
answers
925
views
A non-trivial probability measure on $2^{\mathbb R}$
Consider the measurable space $2^{\mathbb R}$, equipped with the tensor-product $\sigma$-algebra. Famously, this space has a measurable structure which is not generated by a topology (see this answer)....
CommunityBot
- 1
5
votes
2
answers
1k
views
Example for the Sobolev embedding theorem when p=n.
Let $\Omega$ be a bounded domain in $\mathbb R^2$. By the Sobolev embedding theorem, if $k>\frac np$ (in this case $k>\frac 2p$) then
$u\in W^{k,p}(U) \implies u\in C^{k-[\frac 2 p]-1,\gamma}(...
- 2,834
7
votes
2
answers
397
views
Natural $\Pi^1_2$ (or worse) classes of structures?
(To clarify, my interest is mainly lightface, that is, $\Pi^1_2$ instead of $\bf \Pi^1_2$, although it doesn't particularly matter.)
This is just an idle curiosity. In logic, I find myself frequently ...
- 7,145
17
votes
3
answers
3k
views
Nonseparable example in dimension theory?
Could you give me an example of a complete metric space with covering dimension $> n$ all of which closed separable subsets have covering dimension $\le n$?
The question closely related to this ...
CommunityBot
- 1
4
votes
1
answer
459
views
Uncountable Reduced ring $R$ with $R[x]$ has only a countable number of maximal left ideals
The question is following:
Is there an uncountable reduced ring (i.e., a ring with no non-zero nilpotent element) $R$ (with identity) such that
$R[x]$ has only a countable number of maximal left ...
- 1,371
13
votes
1
answer
2k
views
Learning a little Motivic Cohomology
Simply because I find it interesting, I have spent some time studying motivic cohomology from the lectures by Mazza, Voevodsky and Weibel. However, I'm finding it hard to tell if the theory is ...
- 3,555
6
votes
3
answers
435
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} \...
1
vote
1
answer
101
views
Existence of a moderate uniform structure on $\Bbb R$
A moderate uniform structure $\mathcal U$ on $\Bbb R$ is one for which
$\forall U\in \mathcal U, \exists n\in \Bbb N,\quad U^n=\Bbb R^2$
but
$ \not\exists n\in \Bbb N,\forall U\in \mathcal U,\quad U^...
- 39.1k
14
votes
2
answers
685
views
Occurrences of D. H. Lehmer's 10-th degree polynomial
Salem numbers and Lehmer's minimum height problem are venerated not only in number theory and diophantine analysis, where they are considered naturally interesting for their own sake, but also in ...
- 188k
2
votes
3
answers
1k
views
Algebraic structures of greater cardinality than the continuum?
Are there interesting algebraic structures whose cardinality is greater than the continuum? Obviously, you could just build a product group of $\beth_2$ many groups of whole numbers to get to such a ...
CommunityBot
- 1
29
votes
3
answers
7k
views
Non finitely-generated subalgebra of a finitely-generated algebra
Ok, I feel a little bit ashamed by my question.
This afternoon in the train, I looked for a counter-example:
— $k$ a field
— $A$ a finitely generated $k$-algebra
— $B$ a $k$-subalgebra of $A$ that ...
- 1,313
6
votes
4
answers
4k
views
Interesting examples of flasque sheaves?
Does anyone know any interesting examples of flasque sheaves? Ideally, I would like to see one that both arises naturally and is geometric in some sense. On the other hand, I know so few examples ...
- 47.5k
13
votes
2
answers
2k
views
Request: A Serre fibration that is not a Dold fibration
A Serre fibration has the homotopy lifting property with respect to the maps $[0,1]^n \times \{0\} \to [0,1]^{n+1}$. A Dold fibration $E \to B$ has the weak covering homotopy property: lifts with ...
- 27.5k
1
vote
0
answers
566
views
Homotopy theory of schemes
I have seen the notion of Homotopy come up in several contexts in schemes. For example, the book "Lectures on Motivic Cohomology" by Mazza, Weibel and Voevodsky uses this language to some extent. I.e. ...
- 33
14
votes
1
answer
1k
views
Examples of polynomial rings $A[x]$ with relatively large Krull dimension
If $A$ is a commutative ring we have the estimate
$$
\dim (A)+1 \le \dim (A[x])\le 2\dim (A)+1
$$
for the Krull dimension, with $\dim (A)+1 = \dim (A[x])$ for Noetherian rings.
I am looking for nice ...
CommunityBot
- 1
1
vote
1
answer
1k
views
Classification Problems [closed]
I was thinking about the famous question in philosophy of mathematics: "When are two proofs the same?" and I was wondering if we could somehow "classify" proofs by establishing some sort of functorial ...
CommunityBot
- 1
8
votes
4
answers
458
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 ...
CommunityBot
- 1
4
votes
1
answer
227
views
A group 3-cocycle, trivial on a pair of generating subgroups?
I'm looking for an example of the following situation:
A group $G$ generated by finite subgroups $H$ and $K$,
a non-trivial 3-cocycle $\omega \in H^3(G, \mathbb{k}^\times)$
such that
the ...
- 16.2k
4
votes
1
answer
1k
views
On using field extensions to prove the impossiblity of a straightedge and compass construction
Let $z \in \mathbb{C}$. Consider the following statements:
The point $z$ can be constructed with straightedge and compass starting from the points $\{ 0,1\}$.
There is a field extension $K / \mathbb{...
- 1,785
31
votes
3
answers
4k
views
Algebras over the little disks operad
Hello,
The so-called "recognition principle" of Boardman-Vogt and May leaves me unsatisfied.
My problem is the following:
The "recognition principle" says that every "group-like" algebra over the ...
CommunityBot
- 1
2
votes
1
answer
296
views
Methods to tell if a magma has idempotents
(Disclaimer: below, when I say "compact" I mean "compact Hausdorff.")
I asked a version of this question on math stackexchange (https://math.stackexchange.com/questions/305186/left-continuous-magmas-...
CommunityBot
- 1
3
votes
2
answers
438
views
What is a good example of a hyperspace where the base space is non-Hausdorff?
Let $X$ be a topological space, and let $\operatorname{CL}(X)$ be its hyperspace. That is, $\operatorname{CL}(X)$ is the set of closed subsets of $X$, equipped with the minimal topology so that the ...
- 2,550
1
vote
2
answers
406
views
Understanding the left-separated spaces
A space $X$ is called left-separated if it can be well-ordered in such a way that every initial segment is closed in $X$.
Could someone post some left-separated space to help me understand such ...
- 73.2k
5
votes
1
answer
459
views
Example of a non-closed cocomplete symmetric monoidal category
Background
By a cocomplete symmetric monoidal category $C$ I mean a symmetric monoidal category whose underlying category is cocomplete and such that $- \otimes X : C \to C$ is cocontinuous for all $X ...
CommunityBot
- 1
6
votes
1
answer
318
views
Hamiltonian polar action with Lagrangian section
I am looking for examples of Hamiltonian polar isometric actions of a compact Lie group on a Kahler-Einstein (or perhaps just Kahler) manifold, that admits a Lagrangian section.
Recall that an ...
- 108k
62
votes
6
answers
6k
views
Simplest examples of nonisomorphic complex algebraic varieties with isomorphic analytifications
If they are not proper, two complex algebraic varieties can be nonisomorphic yet have isomorphic analytifications. I've heard informal examples (often involving moduli spaces), but am not sure of the ...
CommunityBot
- 1
3
votes
2
answers
1k
views
Function with all but mixed second partial derivatives twice differentiable?
Let $f(x,y)$ be a a real valued function on an open subset of $\mathbf{R}^2$ with continuous partial derivatives $\frac{\partial^2 f}{\partial x^2}$ and $\frac{\partial^2}{\partial y^2}$. Is $f$ twice ...
- 555
5
votes
1
answer
477
views
Clarification and intuition request for rationally equivalent algebraic cycles
I am having some difficulty lining up the definition and my intuition for rational equivalence of cycles. My intuition is based off of the idea that two cycles being rationally equivalent is analogous ...
- 23k
4
votes
0
answers
276
views
Interesting commensurated subgroups of countable groups
Let $G$ be a group and let $K$ be a subgroup. Say $K$ is commensurated in $G$ if $gKg^{-1} \cap K$ has finite index in $K$ for all $g \in G$. Commensurated subgroups are an inherent feature of ...
- 4,728
1
vote
0
answers
169
views
Algebraic properties of the semiring of open subsets.
Does anyone know of a useful general topological application of the algebraic properties of the semiring of open subsets of some topological space? Or examples of any such nontrivial properties at all?...
- 3,513
26
votes
1
answer
2k
views
Important open questions in the field of Tropical geometry
What are some of the important unanswered questions in the field of tropical geometry?
CommunityBot
- 1
13
votes
2
answers
1k
views
Families of curves for which the Belyi degree can be easily bounded
I know (edit: three) families of smooth projective connected curves over $\bar{\mathbf{Q}}$ for which the Belyi degree is not hard to bound from above.
The modular curves $X(n)$. They are ...
- 9,497
14
votes
9
answers
2k
views
Examples of noncommutative analogs outside operator algebras?
Theo's question made me wonder if there are other "noncommutative analogs" outside of operator algebras. Some noncommutative analogs from operator algebras include:
A $C^\ast$-algebra is a ...
CommunityBot
- 1
4
votes
1
answer
502
views
examples of Chow rings of surfaces
Can somone provide me (articles/literature) with examples of Chow rings of surfaces?
(e.g. here: http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf Chapter 9)
What I want is a list of (smooth ...
CommunityBot
- 1
2
votes
2
answers
1k
views
Infinite domain with finite number of prime ideals(elements)
While trying to prove one property of commutative rings with units I can't prove one fact without assuming existence of infinitely many different prime ideals or elements. I tried to test if it was ...
- 306