The examples tag has no wiki summary.

**-2**

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**0**answers

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### Summation with 2 functions [on hold]

Solve for $L$ if you can please. I want to know how to solve $G(x_i,y_j)$ with double integration, where $G=(x-y)$. The equation is below as follows:
$L=\sum\limits_{i=1}^2 $ $\sum\limits_{j=1}^2 ...

**2**

votes

**1**answer

147 views

### Divergence invariant lifting of a vector field via a submersion

What is an example of a smooth submersion $P:S^{3}\to S^{2}$ for which the following statment is Not true:
For every vector field $X$ on $S^{2}$ there is a non vanishing vector field ...

**16**

votes

**4**answers

4k views

### Finite extension of fields with no primitive element

What is an example of a finite field extension which is not generated by a single element?
Background: A finite field extension E of F is generated by a primitive element if and only if there are a ...

**7**

votes

**0**answers

167 views

### Is Hironaka's example the only known deformation of Kähler manifolds with non-Kähler central fibre?

A well-known example in the deformation theory of compact complex manifolds is the one given by Hironaka in his 1962 paper An Example of a Non-Kählerian Complex-Analytic Deformation of Kählerian ...

**12**

votes

**3**answers

279 views

### Examples of compact complex non-Kähler manifolds which satisfy $h^{p,q} = h^{q,p}$

The existence of a Kähler metric on a compact complex manifold $X$ imposes restrictions on it's Dolbeault cohomology; namely, $h^{p,q}(X) = h^{q,p}(X)$ for every $p$ and $q$. I am looking for some ...

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**0**answers

42 views

### Example of A semigroup with a unique left invariant measure

I'm looking for an example of a locally compact semigroup that has a unique left invariant measure.
I know that every locally compact group will admit a unique left invariant measere namely the Haar ...

**4**

votes

**3**answers

780 views

### An example where GCD depends on the domain

First some notation. Given a domain $R$ and $x,a,b \in R$, I write $x=gcd(a,b)_R$ to mean that $x$ is one gcd of $a$ and $b$ in $R$.
I want to find an example of an GCD-domain $R$, a subdomain $S ...

**108**

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**75**answers

34k views

### 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.

**24**

votes

**4**answers

2k views

### Recent, elementary results in algebraic geometry

Next semester I will be teaching an introductory algebraic geometry class for a smallish group of undergrads. In the last couple weeks, I hope that each student will give a one-hour presentation. ...

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votes

**10**answers

853 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 ...

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**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 ...

**34**

votes

**24**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 ...

**177**

votes

**63**answers

89k 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 ...

**14**

votes

**2**answers

880 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 ...

**74**

votes

**6**answers

7k views

### Mistakes in mathematics, false illusions about conjectures

Since long time ago I have been thinking in two problems that I have not been able to solve. It seems that one of them was recently solved. I have been thinking a lot about the motivation and its ...

**27**

votes

**13**answers

5k views

### 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 ...

**3**

votes

**5**answers

262 views

### Existence of orientation preserving, finite order self homeomorphism on a genus 2 surface without fixed point

Let $M$ be a compact 2-manifold of genus 2. Does there exist an orientation preserving homeomorphism $f:M\to M$, so that $f^n=id$ for some integer $n$, and $f$ doesn't have fixed points?
Using ...

**80**

votes

**42**answers

16k 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 ...

**27**

votes

**14**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 ...

**11**

votes

**2**answers

387 views

### A sequence of subsets of an infinite group

Is there an infinite group $G$ such that there is not any sequence $(A_n)$ of its subsets such that always
$$A_n=A_n^{-1}, \quad A_{n+1}A_{n+1}\subsetneqq A_n$$
?
link

**4**

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**0**answers

169 views

### Examples of unproven but likely true existential sentence (in the sense of incompleteness)

Some examples of universal statements that are unproven but likely true include the Riemann hypothesis (all non-trivial zeros of the zeta function have real part 1/2) and the Goldbach conjecture (all ...

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**0**answers

108 views

### Is there a smooth rationally connected, proper variety $M$ over $\mathbb{C}$ such that c_1(L)([A]) = 1 which is not projective space?

Is there a smooth, rationally connected, projective variety $M$, which is not a projective space and is equipped with an ample line bundle $L$, such that the homology class of the curves $[A]$ which ...

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**33**answers

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### Dimension Leaps

Many mathematical areas have a notion of "dimension", either rigorously or naively, and different dimensions can exhibit wildly different behaviour. Often, the behaviour is similar for "nearby" ...

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**0**answers

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### Examples of functions with natural boundary that do not satisfy Fabry or Hadamard gap theorem condition

there are examples of lacunary functions with natural boundary that do not satisfy Fabry or Hadamard gap theorem condition.I want to know more examples of those functions,the more the ...

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**5**answers

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### 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 ...

**3**

votes

**1**answer

131 views

### Mumford-Ramanujam examples in characteristic p [and in Arakelov geometry]

For a compact Riemann surface $B$ of genus $\geq 2$, it is a consequence of the Narasimhan-Seshadri theorem that there exist rank-$2$ vector bundles $E \to B$ of degree zero, all of whose symmetric ...

**1**

vote

**1**answer

151 views

### Examples of nontrivial local systems in Decomposition Theorem

There is a proper map $f: X \rightarrow Y$ of projective varieties. The Decomposition Theorem of Beilinson–Bernstein–Deligne-Gabber states that
$$Rf∗IC_X \cong \oplus_a ...

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votes

**2**answers

190 views

### How weird can Modular Tensor Categories be over non-algebraically closed fields?

I am trying to understand better the behaviour and character of modular tensor categories over non-algebraically closed fields. How weird can they be?
The reason I am interested in this is that my ...

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votes

**59**answers

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### 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):
...

**10**

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**3**answers

578 views

### Are all vector-space valued functors on sets free?

Let $\mathbf{Set}$ be the category of finite sets and functions between them, and let $\mathbf{Vect}$ be the category of finite-dimensional complex vector spaces and linear transformations between ...

**6**

votes

**16**answers

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### 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 ...

**60**

votes

**23**answers

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### What are the most misleading alternate definitions in taught mathematics?

I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in ...

**10**

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**1**answer

578 views

### Nondifferentiability set of an arbitrary real function

A theorem by Zygmunt Zahorski states that a necessary and sufficient condition for a subset of $\mathbb{R}$ to be the nondifferentiability set of a continuous real function is that it is the union of ...

**5**

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**2**answers

183 views

### smooth affine surfaces over algebraically closed fields with trivial l-torsion of the Brauer group

I am looking for examples of smooth affine surfaces over algebraically closed fields with trivial $\ell$-torsion of the Brauer group.
Related questions: Schemes with trivial brauer group and Brauer ...

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votes

**2**answers

215 views

### Example(s) of monoidal symmetric closed category with NNO without infinite coproducts?

The question is in the title, here is my motivation:
$\require{AMScd}$Let $(\mathcal C,\otimes,I)$ be a monoidal symmetric closed category. Then, the tensor product commutes with colimits, and if ...

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**27**answers

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### books well-motivated with explicit examples

It is ultimately a matter of personal taste, but I prefer to see a long explicit example, before jumping into the usual definition-theorem path (hopefully I am not the only one here). My problem is ...

**2**

votes

**0**answers

235 views

### Closed 4-manifolds with uncountably many differentiable structures

I know that $\mathbb{R}^4$ admits uncountably many differentiable structures and I was wandering what happen if we consider closed 4-manifolds. Are there any closed 4-manifolds with uncountably many ...

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votes

**10**answers

841 views

### Examples of “Unusual” Classifications

When one says "classification" in math, usually one of a handful of examples springs to mind:
-Classification of Finite Simple Groups with 18 infinite families and 26 sporadic examples (assuming one ...

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**0**answers

209 views

### Homeomorphisms of product spaces: an example

In the first of these lectures (http://www.mpim-bonn.mpg.de/node/4436) given by M. Freedman he says that there exists (compact metric) spaces $X$ and $Y$ such that $X\times S^{1}$ is homeomorphic to ...

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**6**answers

403 views

### Do you have examples of such “transitive” elements?

(I've asked the same question at the MSE, so far with no answers, so I thought I'd try it here as well. If there's some clash with any site rules, please let me know and I'll abide.)
Let $A$ be a set ...

**45**

votes

**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, ...

**4**

votes

**1**answer

191 views

### Unravelling some hypotheses on a variety

In Le group de Brauer II, Grothendieck states
Proposition 1.4.- Soit $X$ a préschéma noetherien. Supposon que les anneaux hensélisés stricts des anneaux locaux de $X$ soient factoriels, [...] ...

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**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 ...

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**3**answers

143 views

### A model with $\kappa$ many automorphism and a rigid element.

The following should be known, but I could not find an example.
Let $\kappa$ be an uncountable cardinal. Find a model $M$ of size $\kappa$ which has $\ge\kappa$ many automorphisms, but for some $m\in ...

**0**

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**1**answer

117 views

### Simple but topologizable [closed]

Do you have an example of an infinite simple group with at least 3 distinct group topologies on it?

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**7**answers

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### Simple show cases for the Yoneda lemma

I've been given a very simple motivating and instructive show case for the Yoneda lemma:
Given the category of graphs and a graph object $G$, seen as a quadruple $(V_G,\ E_G,\ S_G:E\rightarrow V,\ ...

**0**

votes

**1**answer

197 views

### Examples of groups such that order isomorphism of the subgroups of $G\times G$ and $H\times H$ does not imply isomorphism of $G$ and $H$

Let $G$ and $H$ be groups, $\operatorname{Sub}(G\times G)$ be the set of all subgroups of $G\times G$ and $\operatorname{Sub}(H\times H)$ be the set of all subgroups of $H\times H$. Assume there ...

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**33**answers

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### Experimental Mathematics

I would like to ask about examples where experimentation by computers have led to major mathematical advances.
Motivation
I am aware about a few such cases and I think it will be useful to gather ...

**5**

votes

**1**answer

184 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 ...

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**3**answers

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### 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 ...