The examples tag has no wiki summary.

**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 ...

**22**

votes

**19**answers

3k views

**33**

votes

**19**answers

6k views

### Wonderful applications of the Vandermonde determinant

This semester I am assisting my mentor teaching a first-year undergraduate course on linear algebra in Peking University, China. And now we have come to the famous Vandermonde determinant, which has ...

**31**

votes

**7**answers

6k views

### Is there a measure zero set which isn't meagre?

A subset of ℝ is meagre if it is a countable union of nowhere dense subsets (a set is nowhere dense if every open interval contains an open subinterval that misses the set).
Any countable set ...

**4**

votes

**1**answer

344 views

### Categories with products that preserve quotients

It is well known that in the category of all topological spaces, quotient maps aren't preserved by products (this follows from the simpler fact that $X\times (-):Top\to Top$ doesn't preserve ...

**9**

votes

**3**answers

1k views

### Concavity of $\det^{1/n}$ over $HPD_n$.

One of my beloved theorems in matrix analysis is the fact that the map $H\mapsto (\det H)^{1/n}$, defined over the convex cone $HPD_n$ of Hermitian positive definite matrices, is concave. This is ...

**127**

votes

**67**answers

40k views

### Awfully sophisticated proof for simple facts [closed]

It is sometimes the case that one can produce proofs of simple facts that are of disproportionate sophistication which, however, do not involve any circularity. For example, (I think) I gave an ...

**4**

votes

**0**answers

295 views

### Example of a Grothendieck pretopology satisfying a weak saturation condition

Recall that a singleton Grothendieck pretopology (henceforth 'singleton pretopology') on a category $C$ is a collection of maps $J$ containing the isomorphisms, closed under composition and stable ...

**13**

votes

**3**answers

812 views

### Can a module be an extension in two really different ways?

(Edit: I've realized that there was an error in my reasoning when I was convincing myself that these two formulations are equivalent. Hailong has given a beautiful affirmative answer to my first ...

**5**

votes

**3**answers

439 views

### Nonmetrizable uniformities with metrizable topologies

I'm looking for such pathological examples of uniform spaces which are not metrizable, but whose underlying topology is metrizable. Willard in his General Topology text constructs such a uniformity ...

**1**

vote

**0**answers

603 views

### Example of smooth, proper but non-projective curve over an affine, connected base?

Would someone please give an example of a smooth, proper but non-projective curve $C/S$, where $S$ is affine and connected? I believe that whatever your example, $C/S$ must have genus $1$, admit no ...

**1**

vote

**2**answers

419 views

### Weakly initial sets - examples and nonexamples

A weakly initial set in a category C is a set of objects I of C such that every object a of C has at least one arrow from an object contained in I.
The question is then, does Fields have a weakly ...

**5**

votes

**2**answers

620 views

### Canonical geometric examples

The proofs without words post has some great entries. I'm interested in a similar concept: examples where a problem in math or physics is accompanied by a geometric figure that illuminates some key ...

**2**

votes

**2**answers

248 views

### Is a compactly generated Hausdorff space functionally Hausdorff?

Question is the title. I suspect the answer is no, without some further conditions (clearly, normal is sufficient). Pointers to counterexamples would be appreciated, but not necessary.

**0**

votes

**1**answer

168 views

### Geometric explanation of an orbit space: Integer action on the affine line

Let $k$ be a field of char $0$ and let $\mathbb{Z}$ act on $\mathbb{A}^1_k$ by the action induced by $G\to\mathrm{Aut}_k(k[X]), n\mapsto X+n$. It is rather easy to show that the orbit space ...

**96**

votes

**59**answers

16k views

### Jokes in the sense of Littlewood: examples? [closed]

First, let me make it clear that I do not mean jokes of the
"abelian grape" variety. I take my cue from the following
passage in A Mathematician's Miscellany by J.E. Littlewood
(Methuen 1953, p. 79):
...

**1**

vote

**1**answer

846 views

### Ringed and locally ringed spaces

A pair $(X,O_X)$ is a ringed space if $X$ is a topological space and $O_X$ is a sheaf of rings. If every stalk $O_{X,x}$ is a local ring, then we say that $(X,O_X)$ is a locally ringed space.
In the ...

**39**

votes

**7**answers

5k views

### Riemannian surfaces with an explicit distance function?

I'm looking for explicit examples of Riemannian surfaces (two-dimensional Riemannian manifolds $(M,g)$) for which the distance function d(x,y) can be given explicitly in terms of local coordinates of ...

**13**

votes

**17**answers

3k views

### What is your favorite isomorphism? [closed]

The other day I was trying to figure out how to explain why isomorphisms are important. I pulled Boyer's A History of Mathematics off the bookshelf and was surprised to find that isomorphism isn't ...

**2**

votes

**2**answers

435 views

### description of functions of conditionally negative type on a group

Recall that a kernel conditionaly of negative type on a set $X$ is a map $\psi:X\times X\rightarrow\mathbb{R}$ with the following properties:
1) $\psi(x,x)=0$
2) $\psi(y,x)=\psi(x,y)$
3) for any ...

**5**

votes

**9**answers

1k 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 ...

**30**

votes

**14**answers

5k views

### Explicit computations using the Haar measure

This question is somewhat related to my previous one on Grassmanians. The few times I've encountered the Haar measure in the course of my mathematical education, it's always been used in a very ...

**4**

votes

**3**answers

501 views

### Natural examples of finite dimensional spaces with interesting 2-type

Riemann surfaces provide interesting examples of 1-types - interesting as they have roles in diverse areas. However, apart from 2-dimensional lens spaces, I can't readily bring to mind natural ...

**6**

votes

**3**answers

644 views

### Is $\mathbb{A}²$ the universal smooth scheme which is a finite cover of $\mathbb{A}²/μ₂$?

One very handy (counter)example I often think about is the scheme $Spec(k[a,b,c]/(ab-c^2))$ (where $k$ is a field), which you may also know as $Spec(k[x^2,xy,y^2])$, as $\mathbb A^2/\mu_2$, or as the ...

**11**

votes

**12**answers

2k views

### On proving that a certain set is not empty by proving that it is actually large

It happens occasionally that one can prove that a given set is not empty by proving that it is actually large. The word "large" here may refer to different properties.
For example, one can prove that ...

**22**

votes

**5**answers

1k views

### Explicit elements of $K((x))((y)) \setminus K((x,y))$

In an answer to the popular question on common false beliefs in mathematics
Examples of common false beliefs in mathematics.
I mentioned that many people conflate the two different kinds of formal ...

**3**

votes

**3**answers

1k 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
...

**14**

votes

**1**answer

805 views

### Comodule exercises desired

This Question is inspired by a Quote of Moore's
"There are two ‘evil’ influences at work here:
1. we are toilet trained with algebras not coalgebras
2. some of us are addicted to manifolds and so ...

**10**

votes

**7**answers

3k views

### Can we have A={A} ?

Does there exist a set $A$ such that $A=\{A\}$ ?
Edit(Peter LL): Such sets are called Quine atoms.
Naive set theory By Paul Richard Halmos On page three, the same question is asked.
Using the ...

**14**

votes

**3**answers

1k views

### Eta-products and modular elliptic curves

Recently the elliptic curve $E:y^2+y=x^3-x^2$ of conductor $11$ (which appears in my answer) became my favourite elliptic over $\bf Q$ because the associated modular form
$$
...

**6**

votes

**16**answers

3k views

### Vector spaces without natural bases

Does anyone know any nice examples of vector spaces without a basis that is in some sense "natural".
To clarify what I mean, suppose we look at $\mathbb{R}^2$. We define $\mathbb{R}^2$ as pairs of ...

**0**

votes

**3**answers

1k views

### Intuitions/connections/examples for “eigen-*”

There are many concepts in mathematics that begin with the German word "eigen": eigenvector, eigenvalue, eigenspace, eigenstate, eigenfunction, eigensystem etc. (to name just the most important (?) ...

**4**

votes

**1**answer

289 views

### Example of a quasitopological group with discontinuous power map

A quasitopological group is a group $G$ with topology such that multiplication $G\times G\rightarrow G$ is continuous in each variable (i.e. all translations are continuous) and inversion ...

**5**

votes

**13**answers

2k views

### Applications of compactness

Similar to this question: Applications of connectedness I want to collect applications of compactedness.
E.g.: compact + discrete => finite, which can be used to prove the finiteness of the ...

**12**

votes

**4**answers

2k views

### What are your favorite finite non-commutative rings?

When you are checking a conjecture or working through a proof, it is nice to have a collection of examples on hand.
There are many convenient examples of commutative rings, both finite and infinite, ...

**12**

votes

**13**answers

2k views

### Applications of connectedness

In an «advanced calculus» course, I am talking tomorrow about connectedness (in the context of metric spaces, including notably the real line).
What are nice examples of applications of the idea ...

**0**

votes

**1**answer

413 views

### Must finite groups with isomorphic commutators and quotients be isomorphic?

Let G and H be finite groups. Let G' = [G,G] and H' = [H,H] be the corresponding derived groups (commutator subgroups) of G and H. I am looking for an example where G' is isomorphic to H' and G/G' is ...

**8**

votes

**3**answers

1k 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 ...

**1**

vote

**1**answer

372 views

### An example of a space which is locally relatively contractible but not contractible?

A space $X$ is called locally contractible it it has a basis of neighbourhoods which are themselves contractible spaces. CW complexes and manifolds are locally contractible. On the other hand, the ...

**12**

votes

**6**answers

2k 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 ...

**21**

votes

**10**answers

3k views

### Examples of non-abelian groups arising in nature without any natural action

It's said that most groups arise through their actions. For instance, Galois groups arise in Galois theory as automorphisms of field extensions. Linear groups arise as automorphisms of vector spaces, ...

**5**

votes

**2**answers

526 views

### Gaining intuition for how submodules behave

I'm studying elementary commutative algebra this semester, largely following Atiyah-MacDonald. I often find myself in a situation where I'm interested in whether some property of an R-module M is ...

**3**

votes

**9**answers

681 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 ...

**9**

votes

**1**answer

2k views

### Sequence that converge if they have an accumulation point

I am looking for classes of sequence, that converge iff they contain a converging sub-sequence.
The basic example of such sequences are monotone sequences of real numbers.
A more interesting ...

**5**

votes

**2**answers

477 views

### What is an example of a non-regular, totally path-disconnected Hausdorff space?

I need this for a counterexample: the multiplication in the fundamental group $\pi_1(\Sigma X_+)$, when it is equipped with the topology inherited from $\Omega \Sigma X_+$, fails to be continuous for ...

**1**

vote

**1**answer

1k views

### Example for deterministic function with unbounded total variation and bounded quadratic variation

It is well known that e.g. $sin(1/x)$ is of unbounded total variation (in the interval [0,1] assuming $f(0)=0$). (Preliminary numerical tests suggest that) it is also of unbounded quadratic variation. ...

**29**

votes

**3**answers

2k 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 ...

**16**

votes

**4**answers

1k views

### Example of a smooth morphism where you can't lift a map from a nilpotent thickening?

Definition. A locally finitely presented morphism of schemes $f\colon X\to Y$ is smooth (resp. unramified, resp. étale) if for any affine scheme $T$, any closed subscheme $T_0$ defined by a square ...

**17**

votes

**6**answers

1k views

### What can you do with a compact moduli space?

So sometime ago in my math education I discovered that many mathematicians were interested in moduli problems. Not long after I got the sense that when mathematicians ran across a non compact moduli ...

**3**

votes

**3**answers

1k views

### Examples of Banach spaces and their duals

There are many representation theorems which state that the dual space of a Banach space $X$ has a particularly concrete form. For example, if $X = C([0,1],\mathbb R)$ is the space of real-valued ...