The examples tag has no wiki summary.

**94**

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### Best online mathematics videos?

I know of two good mathematics videos available online, namely:
Sphere inside out (part I and part II)
Moebius transformation revealed
Do you know of any other good math videos? Share.

**153**

votes

**62**answers

79k views

### Proofs without words

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results?
(One could ask if this is of interest to mathematicians, and I ...

**32**

votes

**22**answers

6k views

### 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 ...

**5**

votes

**1**answer

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 ...

**6**

votes

**3**answers

650 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 ...

**1**

vote

**0**answers

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.

**86**

votes

**53**answers

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### What are your favorite instructional counterexamples?

Related: question #879, Most interesting mathematics mistake. But the intent of this question is more pedagogical.
In many branches of mathematics, it seems to me that a good counterexample can be ...

**13**

votes

**7**answers

2k views

### Examples of rational families of abelian varieties.

I'd like to know examples of non-trivial families of abelian varieties over rational bases (e.g. open subschemes of the projective line P^1).
One can generate many examples as Jacobians of rational ...

**111**

votes

**26**answers

12k views

### What are some reasonable-sounding statements that are independent of ZFC?

Every now and then, somebody will tell me about a question. When I start thinking about it, they say, "actually, it's undecidable in ZFC."
For example, suppose A is an abelian group such that every ...

**12**

votes

**1**answer

174 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 ...

**2**

votes

**2**answers

391 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 ...

**15**

votes

**12**answers

2k 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, ...

**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 ...

**3**

votes

**3**answers

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 ...

**12**

votes

**4**answers

2k views

### Locally compact Hausdorff space that is not normal

What is a good example of a locally compact Hausdorff space that is not normal? It seems to be well-known that not all locally compact Hausdorff spaces are normal (and only a weaker version of ...

**2**

votes

**5**answers

738 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 ...

**5**

votes

**1**answer

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 ...

**10**

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 ...

**2**

votes

**1**answer

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

**4**answers

609 views

### Examples of $G_\delta$ sets

Recall that a subset A of a metric space X is a $G_\delta$ subset if it can be written as a countable intersection of open sets. This notion is related to the Baire category theorem. Here are three ...

**4**

votes

**1**answer

376 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

**9**answers

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 ...

**111**

votes

**67**answers

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### 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 ...

**9**

votes

**5**answers

760 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 ...

**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
...

**9**

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 ...

**7**

votes

**7**answers

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 ...

**39**

votes

**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 ...

**5**

votes

**2**answers

5k views

### Simple bijection between reals and sets of natural numbers

Using the Cantor–Bernstein–Schröder theorem, it is easy to prove that there exists a bijection between the set of reals and the power set of the natural numbers. However, it turns out to be difficult ...

**12**

votes

**7**answers

646 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

**1**answer

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

**5**answers

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 ...

**6**

votes

**2**answers

832 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 ...

**37**

votes

**28**answers

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### Applications of the Chinese remainder theorem

As the title suggests I am interested in CRT applications. Wikipedia article on CRT lists some of the well known applications (e.g. used in the RSA algorithm, used to construct an elegant Gödel ...

**16**

votes

**5**answers

4k 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 ...

**2**

votes

**1**answer

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 ...

**24**

votes

**12**answers

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### 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

346 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 ...

**16**

votes

**11**answers

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### Examples of using physical intuition to solve math problems

For the purposes of this question let a "physical intuition" be an intuition
that is derived from your everyday experience of physical reality. Your
intuitions about how the spin of a ball affects ...

**28**

votes

**2**answers

2k views

### Are there pairs of highly connected finite CW-complexes with the same homotopy groups?

Fix an integer n. Can you find two finite CW-complexes X and Y which
* are both n connected,
* are not homotopy equivalent, yet
* $\pi_q X \approx \pi_q Y$ for all $q$.
In Are there two ...

**66**

votes

**41**answers

15k views

### What are the most attractive Turing undecidable problems in mathematics?

What are the most attractive Turing undecidable problems in mathematics?
There are thousands of examples, so please post here only the most attractive, best examples. Some examples already appear on ...

**12**

votes

**2**answers

725 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 ...

**3**

votes

**2**answers

673 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 ...

**7**

votes

**2**answers

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 ...

**68**

votes

**6**answers

4k views

### Counterexamples in algebraic topology?

In this thread
Books you would like to read (if somebody would just write them...),
I expressed my desire for a book with the title "(Counter)examples in Algebraic Topology".
My reason for doing so ...

**16**

votes

**3**answers

2k 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 ...

**36**

votes

**22**answers

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### What's a groupoid? What's a good example of a groupoid?

Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?

**5**

votes

**1**answer

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 ...

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votes

**5**answers

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### `Naturally occuring' $K(\pi, n)$ spaces, for $n \geq 2$.

[edited!] Given a group $\pi$ and an integer $n>1$, what are examples of Eilenberg-Maclane spaces $K(\pi, n)$ that can be constructed as "known" manifolds? (or if not a manifold, say some space ...

**71**

votes

**82**answers

57k views

### Do good math jokes exist? [closed]

Have a good joke? Share.
I know this is subjective, but the principle "should be of interest to mathematicians" trumps. (I hope.)