The examples tag has no wiki summary.

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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 ...

**2**

votes

**1**answer

204 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**

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**2**answers

248 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 ...

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951 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

**2**answers

153 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

**1**answer

227 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 ...

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votes

**1**answer

191 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**

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**2**answers

983 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**

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**3**answers

654 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

**1**answer

223 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**

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**2**answers

414 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 ...

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**0**answers

178 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 ...

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**0**answers

174 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**

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**4**answers

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### 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 ...

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**0**answers

123 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

**1**answer

251 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 ...

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**2**answers

434 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 ...

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**4**answers

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**

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**2**answers

423 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" ...

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**0**answers

125 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?

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**35**answers

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### 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 ...

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**2**answers

744 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**

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**1**answer

203 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 ...

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**2**answers

785 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 $G$ ...

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**1**answer

296 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 ...

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**12**answers

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### Excellent uses of induction and recursion

Can you make an example of a great proof by induction or construction by recursion?
Given that you already have your own idea of what "great" means, here it can also be taken to mean that the chosen ...

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**11**answers

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### Interesting conjectures “discovered” by computers and proved by humans?

There are notable examples of computers "proving" results discovered by mathematicians, what about the opposite:
Are there interesting conjectures "discovered" by computers and proved by humans?
...

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**3**answers

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### Arithmetic geometry examples

(This is inspired by Algebraic geometry examples.)
I want to collect here (counter)examples in arithmetic geometry.
Curves violating the Hasse principle: The Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$. ...

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**28**answers

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### What are some examples of ingenious, unexpected constructions?

Many famous problems in mathematics can be phrased as the quest for a specific construction. Often such constructions were sought after for centuries or even millennia and later proved impossible by ...

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**1**answer

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### Mean value property with fixed radius

Let $f$ be a continuous function defined on $\mathbb{R^n}$. It is well known that both the spherical mean value property (MVP) of $f$, i.e.
$$f(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}f,\ ...

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**5**answers

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### Easy and Hard problems in Mathematics [closed]

Modified question:
I would like to know some examples of problems in Mathematics, for pedagogical purposes, which do not involve difficult techiques to solve the problem but with a change of context ...

**3**

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**0**answers

193 views

### Seek “typical examples” for the structure of spaces with two-sided Ricci bounds

By a 1990 paper of Michael Anderson, the following is true:
Theorem. Let the metric space $(X,d,p)$ be a pointed Gromov-Hausdorff limit of a sequence of complete pointed Riemannian manifolds ...

**14**

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**2**answers

884 views

### A simplicial complex which is not collapsible, but whose barycentric subdivision is

Does anyone know of a simplicial complex which is not collapsible but whose barycentric subdivision is?
Every collapsible complex is necessarily contractible, and subdivision preserves the ...

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vote

**1**answer

214 views

### “Highly balanced” periodic functions

The function $f(x) = e^{2\pi ix}$ on the domain $\mathbb{R}/\mathbb{Z}$ has the property that, for every $n > 1$ and every $x$, $\displaystyle \sum_{i = 0}^{n-1} f(x + \frac{i}{n}) = 0$.
Other ...

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**1**answer

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### When are technical assumptions critical? [closed]

Apart from their technical statement and proof, a usual presentation of theorems is by leading up to them with a definite motivation or intuition, for example putting the results in the wider context ...

**8**

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**1**answer

327 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 ...

**14**

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**1**answer

633 views

### What are “good” examples of string manifolds?

Based on this mathoverlow question, I would like to have a similar list for the case of string manifolds. An $n$-dim. Riemannian manifold $M$ is said to be string, if the classifying map of its bundle ...

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**4**answers

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### What are “good” examples of spin manifolds?

I'm trying to get a grasp on what it means for a manifold to be spin. My question is, roughly:
What are some "good" (in the sense of illustrating the concept) examples of manifolds which are spin ...

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**3**answers

204 views

### two essentially different concretizaions

It is sometimes emphasized that a "concrete category" is not a property of a category $C$, but rather a structure, i.e. a faithful functor from $C$ to $Set$. Thus, When people talk about a concrete ...

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**1**answer

173 views

### Non-uniruled variety with level one Hodge structure.

I wonder if there exists one example of non-uniruled algebraic variety with level one Hodge structure.

**2**

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**2**answers

616 views

### Non-split groups

I am looking for a reference with definitions on what it means for an algebraic group to be split, quasi-split, and non-split. I would like to see some examples of the different "types".
Thanks,
Tom

**4**

votes

**2**answers

958 views

### Is beauty at the high school level even possible? [closed]

This question is a follow up to 74841, and follows from a suggestion by Gian-Carlo Rota that beauty as judged by the educated public differs from that experienced by mathematicians (he gives Euclidean ...

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**24**answers

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### Examples of seemingly elementary problems that are hard to solve?

I'm looking for a list of problems such that
a) any undergraduate student who took multivariable calculus and linear algebra can understand the statements, (Edit: the definition of understanding here ...

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**44**answers

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### An example of a beautiful proof that would be accessible at the high school level?

The background of my question comes from an observation that what we teach in schools does not always reflect what we practice. Beauty is part of what drives mathematicians, but we rarely talk about ...

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**28**answers

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### Examples where it's useful to know that a mathematical object belongs to some family of objects

For an expository piece I'm writing, it would be useful to have good examples of the following phenomenon:
(1) ${\cal X}$ is a parameterized family of somethings. (Varieties, schemes, manifolds, ...

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**3**answers

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### Wanted: example of a non-algebraic singularity

Given a finitely generated $\def\CC{\mathbb C}\CC$-algebra $R$ and a $\CC$-point (maximal ideal) $p\in Spec(R)$, I define the singularity type of $p\in Spec(R)$ to be the isomorphism class of the ...

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**0**answers

211 views

### Example of function with a certain behavior.

Let $f: R \rightarrow R$. Consider the following properties:
$(1)$ - There are positive constants $a$ and $r$ such that $\forall x, y$
$$|f(x)-f(y)|\leq a(1 + |x|^r+|y|^r)|x-y|.$$
$(2)$ - There is a ...

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votes

**1**answer

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### Examples of “inner products” of parallel morphisms in a dagger category

There is a very interesting abstract notion of the trace of an endomorphism $f : c \to c$ of an object $c$ in a braided monoidal category (although the symmetric case is easier): see, for example, ...

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424 views

### higher order structure by higher order derivatives

Anyone recall a structure determined by a 3rd order partial derivative?
not the general nth order of recent Baranovsky

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**2**answers

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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 ...