The examples tag has no wiki summary.

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### Krull rings and determinantal invariants

During another attempt to come to grips with Hillman's excellent book Algebraic Invariants of Links, I am having difficulty figuring out why Krull rings are the setting for Chapter 3- the natural ...

**11**

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**2**answers

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### Request: A Serre fibration that is not a Dold fibration

A Serre fibration has the homotopy lifting property with respect to the maps $[0,1]^n \times \{0\} \to [0,1]^{n+1}$. A Dold fibration $E \to B$ has the weak covering homotopy property: lifts with ...

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**4**answers

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### Good example of a non-continuous function all of whose partial derivatives exist

What's a good example to illustrate the fact that a function all of whose partial derivatives exist may not be continuous?

**15**

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**12**answers

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### 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, ...

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**0**answers

1k views

### Again about Bing's house with two rooms [duplicate]

Possible Duplicate:
How to show that the “bing’s house with two rooms” is contractible?
I don't know why my question is closed? here, I make my question clearly, when ...

**2**

votes

**1**answer

611 views

### Example of restriction of a finite morphism which is not finite

Every closed immersion is a finite morphism. Therefore, restriction of a finite morphism to a closed subset is always a finite morphism itself. Can you give an example of quasi-projective varieties ...

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**1**answer

455 views

### Example of inclusion which is not a finite morphism [closed]

Every closed immersion is a finite morphism. Can you give an example of quasi-projective varieties $X\subset Y$ such that inclusion $X\hookrightarrow Y$ is not finite? Same with Y projective?
Thanks!
...

**3**

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**1**answer

2k views

### How to show that the “bing's house with two rooms” is contractible? [closed]

I can't image this, Someone can give a clear illustration?

**10**

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**4**answers

1k views

### Regular spaces that are not completely regular

In the undergraduate toplogy course we were given examples of spaces that are $T_i$ but not $T_{i+1}$ for $i=0,\ldots,4$. However, no example of a space which is $T_3$ but not $T_{3.5}$ was given. ...

**12**

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**1**answer

455 views

### Tensor products and two-sided faithful flatness

Let $f: R \to S$ be a morphism of Noetherian rings (or more generally $S$ can just be an $R-R$ bimodule with a bimodule morphism $R \to S$). Suppose $f$ is faithfully flat on both sides, so $M \to M ...

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**2**answers

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### non principally polarized complex abelian varieties

I've read in (abstracts of) papers that there are abelian varieties over fields of positive characteristic that admit no prinicipal polarization. Apparently its not the easiest thing to find an ...

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**3**answers

806 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 ...

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**55**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 ...

**6**

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**2**answers

1k views

### Example of connected-etale sequence for group schemes over a Henselian field?

Can someone give a really concrete example of such a sequence? I am looking at several notes related with such things, but haven't seen any well-calculated example. And I'm really confused at this ...

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**8**answers

1k views

### Cryptomorphisms

I am curious to collect examples of equivalent axiomatizations of mathematical structures. The two examples that I have in mind are
Topological Spaces. These can be defined in terms of open sets, ...

**2**

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**0**answers

243 views

### Forgetting extra structure inducing Symmetries

This is a major edit of the original post after receiving helpful comments.
It is often the case when one adds additional structure to make a problem more tractable. When one attempts to forget this ...

**4**

votes

**1**answer

814 views

### On using field extensions to prove the impossiblity of a straightedge and compass construction

Let $z \in \mathbb{C}$. Consider the following statements:
The point $z$ can be constructed with straightedge and compass starting from the points $\{ 0,1\}$.
There is a field extension $K / ...

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**5**answers

683 views

### Is there a finitely complete category with terminal object but NO subobject classifier?

This came up today while thinking about topoi in seminar, as the title suggests my question is;
Is there a finitely complete category with terminal object but NO subobject classifier?
Hopefully ...

**6**

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**1**answer

1k views

### Connection between bi-Hamiltonian systems and complete integrability

As I understand, the lack of indication on how to obtain first integrals in Arnol'd-Liouville theory is a reason why we are interested in bi-Hamiltonian systems.
Two Poisson brackets
$\{ \cdot,\cdot ...

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**2**answers

3k views

### A finitely generated, locally free module over a domain which is not projective?

This is a followup to a previous question
What is the right definition of the Picard group of a commutative ring?
where I was worried about the distinction between invertible modules and rank one ...

**2**

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**4**answers

507 views

### Ternary relations that are not binary functions

By far the most prominent elementary relations that are not functions are binary and the most prominent elementary ternary relations are in fact binary functions.
"Elementary" shall mean "part of the ...

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**4**answers

565 views

### Kähler manifold which is not algebraic

Can someone provide examples of Kähler manifolds which are not algebraic?
This question came to my mind seeing the post of Andrea Ferretti.

**0**

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**1**answer

803 views

### Maximal subgroups of abelian groups and Q-algebras

Let $G$ be an abelian group which does not have a maximal subgroup. Does it follow that $G$ is a $\mathbb{Q}$-algebra?
It is easy to see that $\mathbb{Q}$-algebras do not admit any maximal subgroups. ...

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votes

**1**answer

697 views

### Example for an integral, rectifiable varifold with unbounded first variation

I'm just looking for an example of an integral, rectifiable varifold, which has no locally bounded first variation.
Recapitulation
for every $m$-rectifiable varifold $\mu$ exists a $m$-rectifiable ...

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**5**answers

3k views

### What is Yoneda's Lemma a generalization of?

What is Yoneda's Lemma a generalization of?
I am looking for examples that were known before category theory entered the stage resp. can be known by students before they start with category theory.
...

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**2**answers

506 views

### Examples of Completions and Algebraic Closures

It is widely known that the algebaric closure of the $p$-adic completion $\mathbb{Q}_p$ of $\mathbb{Q}$ isn't complete anymore. It's completion is complete and known as $\mathbb{C}_p$.
I have read ...

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**1**answer

196 views

### Motivation for Cosuspended Category Axioms

Today I was wondering about the axioms given by Bernhard Keller for Cosuspended Categories.
The axioms of a triangle feel very much like exactness, but not quite. The last axiom about the large ...

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**8**answers

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### What are examples illustrating the usefulness of Krull (i.e., rank > 1) valuations?

In modern valuation theory, one studies not just absolute values on a field, but also Krull valuations. The motivation is easy enough:
If $k$ is a field, a valuation ring of $k$ is a subring $R$ ...

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**34**answers

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### Experimental Mathematics

I would like to ask about examples where experimentation by computers have led to major mathematical advances.
A new look
Now as the question is five years old and there are certainly more examples ...

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**15**answers

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### Strong induction without a base case

Strong induction proves a sequence of statements $P(0)$, $P(1)$, $\ldots$ by proving the implication
"If $P(m)$ is true for all nonnegative integers $m$ less than $n$, then $P(n)$ is true."
for ...

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**20**answers

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### Nontrivial question about fibonacci numbers?

I'm looking for a nontrivial, but not super difficult question concerning Fibonacci numbers. It should be at a level suitable for an undergraduate course.
Here is a (not so good) example of the sort ...

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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,\ ...

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**2**answers

4k views

### Galoisian sets of prime numbers

The question is about characterising the sets $S(K)$ of primes which split completely in a given galoisian extension $K|\mathbb{Q}$. Do recent results such as Serre's modularity conjecture (as proved ...

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**4**answers

829 views

### Examples of Poisson Schemes

A Poisson Manifold is a real manifold $M$ along with a Lie bracket $[\cdot,\cdot]$ on $C^\infty(M)$ which is a derivation in each variable. Poisson manifolds are interesting for a few reasons, among ...

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**42**answers

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### 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 ...

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**5**answers

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### An example of two elements without a greatest common divisor

Is there an easy example of an integral domain and two elements on it which do not have a greatest common divisor? It will have to be a non-UFD, obviously.
"Easy" means that I can explain it to my ...

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**5**answers

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### non-abelian groups of prescribed order

Is there a construction that will give a non-abelian group of order $p^mr$ where $p$ is a prime, $r$ and $p$ are relatively prime and $m$ is an arbitrary non-negative integer? I suspect in this ...

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**1**answer

471 views

### obstruction to smooth lifting of smooth schemes

According to general theory, for a square zero thickening defined by an ideal I: SpecA -> SpecA', there is an obstruction of lifting a smooth scheme X over A to a smooth scheme over A' living in ...

**1**

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**1**answer

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### What are examples of theorems get extensions based on simple observation?

Here are some examples illustrate what I meant:
Bonnet-Myers:Bonnet in 1855 proved n=2 case, Myers in 1941 extended to any dimension using the same idea.
Bishop-Gromov Volume comparison: Bishop knew ...

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**3**answers

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### What are some interesting sequences of functions for thinking about types of convergence?

I'm thinking about the basic types of convergence for sequences of functions: convergence in measure, almost uniform convergence, convergence in Lp and point wise almost everywhere convergence. I'm ...

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**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 ...

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**11**answers

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### When does 'positive' imply 'sum of squares'?

Does anyone have examples of when an object is positive, then it has (or does not have) a square root? Or more generally, can be written as a sum of squares?
Example. A positive integer does not ...

**186**

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**65**answers

93k 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 ...

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votes

**3**answers

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### What manifolds are bounded by RP^odd?

Real projective spaces ℝPn have ℤ/2 cohomology rings ℤ/2[x]/(xn+1) and total Stiefel-Whitney class (1+x)n+1 which is 1 when n is odd, so it follows that odd dimensional ones are boundaries of compact ...

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**27**answers

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### Justifying a theory by a seemingly unrelated example

Here is a topic in the vein of Describe a topic in one sentence and Fundamental examples : imagine that you are trying to explain and justify a mathematical theory T to a skeptical mathematician ...

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**2**answers

875 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 ...

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**2**answers

432 views

### Can we distinguish the algebraic and continuous duals of a Banach space without choice (or HBT)?

The algebraic dual of a normed vector space is the space of all linear functionals to the ground field (either $\mathbb{R}$ or $\mathbb{C}$ for this question). The continuous dual is the subspace of ...

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**4**answers

739 views

### Examples of divisors on an analytical manifold

I am trying to understand divisors reading through Griffith and Harris but it is difficult to come up with any particular interesting example. I have browsed through Hartshone's book but everything is ...

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**29**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 ...

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**2**answers

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### Is there a poset with 0 with countable automorphism group?

Is there a poset P with a unique least element, such that every element is covered by finitely many other elements of P (and P is locally finite -- actually, per David Speyer's example, let's say that ...