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5
votes
1answer
163 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
vote
0answers
89 views

Examples of languages that are in P and are not in CFL [closed]

Any examples of languages that are in P(polynomial time to recognize it) and are not in CFL(context-free language)?The more the better.
12
votes
1answer
171 views

Nontrivial upper bounds on proof-theoretic ordinals of strong theories: do we have any?

Motivated by Consistency of Analysis (second order arithmetic) and Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?, I have the following question: Are there any examples of strong ...
3
votes
3answers
477 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 ...
5
votes
1answer
276 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
votes
1answer
113 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, ...
4
votes
1answer
374 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 ...
12
votes
9answers
1k views

What are your favorite concrete examples of limits or colimits that you would compute during lunch?

(The title was initially "What are your favorite concrete examples that you would compute on the table during lunch to convince a working mathematician that the notions of limits and colimits are not ...
9
votes
5answers
757 views

Examples of ubiquitous objects that are hard to find?

I've been wrestling with a certain research problem for a few years now, and I wonder if it's an instance of a more general problem with other important instances. I'll first describe a general ...
12
votes
7answers
645 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 ...
2
votes
1answer
181 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 ...
4
votes
3answers
345 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 ...
2
votes
1answer
277 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 ...
28
votes
5answers
728 views

are there natural examples of classical mechanics that happens on a symplectic manifold that isn't a cotangent bundle?

I'm curious about just how far the abstraction to a symplectic formalism can be justified by appeal to actual physical examples. There's good motivation, for example, for working over an arbitrary ...
7
votes
2answers
190 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 ...
14
votes
1answer
750 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 ...
1
vote
1answer
69 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 ...
5
votes
1answer
415 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 ...
12
votes
2answers
458 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 ...
14
votes
10answers
1k views

An example of a proof that is explanatory but not beautiful? (or vice versa)

This question has a philosophical bent, but hopefully it will evoke straightforward, mathematical answers that would be appropriate for this list (like my earlier question about beautiful proofs ...
6
votes
1answer
280 views

Generating functions with all non-zero coefficients equal to one

Inspired by this question, I have been wondering if there are any useful generating functions with all non-zero coefficients equal to one. Obviously, the trivial generating function $\frac{1}{1-x}$ ...
1
vote
0answers
274 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. ...
8
votes
4answers
407 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 ...
3
votes
1answer
143 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 ...
12
votes
1answer
422 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 ...
7
votes
7answers
915 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 ...
2
votes
1answer
189 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 ...
2
votes
2answers
221 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 ...
6
votes
2answers
885 views

Good examples of random variables whose image is not a measurable set?

Are their simple/natural examples of real-valued Borel-measurable random variables whose image is not a Borel set? Something that occurs "naturally"? I am teaching Doob's lemma (for two real-valued ...
1
vote
2answers
139 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 ...
4
votes
1answer
210 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 ...
6
votes
1answer
186 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 ...
8
votes
2answers
617 views

what's the idea behind Carleman estimate

A standard Carleman-type estimate is of the form $$ \sum_{|\alpha|<m}{\tau^{2(m-|\alpha|-1)}\int{|D^{\alpha}u|^{2}e^{2\tau\phi}}dx}\leq K\int{|Pu|^{2}e^{2\tau\phi}dx},\quad u\in C_{0}^{\infty} $$ ...
8
votes
3answers
575 views

What's the use of Malgrange preparation theorem?

The Malgrange preparation theorem,which is the $C^{\infty}$ version of the classical Weierstrass preparation theorem,says that if $f(t,x)$ is a $C^{\infty}$ function of $(t,x)\in\mathbb{R}^{n+1}$ near ...
2
votes
1answer
212 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 ...
2
votes
2answers
363 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 ...
7
votes
0answers
166 views

what's the role of $H^{p}(\mathbb{R}^{n})$ in modern (harmonic) analysis?

The classical theory of $H^p$,due to it's heavy dependence on the complex function theory(such as Blaschke products), seemed to have an insurmountable obstacle barrying its extension to several ...
4
votes
0answers
146 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 ...
9
votes
4answers
937 views

A map inducing isomorphisms on homology but not on homotopy

As a consequence of the Whitehead theorem, Spanier's Algebraic Topology book has on 7.6.25 the following theorem: A weak homotopy equivalence induces isomorphisms of the corresponding integral ...
1
vote
0answers
114 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 ...
3
votes
1answer
233 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 ...
2
votes
2answers
394 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 ...
20
votes
4answers
1k views

Elementary applications of Krein-Milman

This is a cross-post from MSE: http://math.stackexchange.com/questions/139754/elementary-applications-of-krein-milman. I'm starting to suspect that the question just doesn't really have a great ...
9
votes
2answers
400 views

Has there been any application of tensor species?

Joyal's combinatorial species, endofunctors in the category of finite sets with bijections $\mathbf B$ have found numerous applications. One generalisation is given by so-called "tensor species" ...
1
vote
0answers
113 views

uniqueness of a limit of a pseudo convergent set

Is there an example of valued field in which any pseudo convergent set has a limit and such that this limit is unique?
39
votes
35answers
6k views

Examples of theorems with proofs that have dramatically improved over time

I am looking for examples of theorems that may have originally had a clunky, or rather technical, or in some way non-illuminating proof, but that eventually came to have a proof that people consider ...
3
votes
2answers
670 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 ...
3
votes
1answer
192 views

When is the Freudenthal compactification an ANR?

Let $X$ be a locally compact metric ANR (or, if preferred, a locally compact simplicial complex). If needed, assume that $X$ has finitely many ends or is of finite dimension. My question is: What ...
13
votes
2answers
754 views

Does such an infinite index subgroup exist?

Notation: If $G$ is a countable group and $H$ is a subgroup, for $g\in G$, let $|\mathcal{O}_{gH}|$ be the size of the $H$-orbit of $gH$ in the $H$-set $G/H$. Does there exist a countable group $G$ ...
3
votes
1answer
290 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 ...