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1
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2answers
393 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 ...
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2answers
596 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
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2answers
238 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.
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1answer
167 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 ...
92
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59answers
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
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1answer
759 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 ...
38
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7answers
4k 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 ...
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17answers
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
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2answers
396 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 ...
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9answers
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 ...
26
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14answers
4k 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
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3answers
485 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
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3answers
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
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12answers
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
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5answers
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
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3answers
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
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1answer
770 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 ...
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7answers
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 ...
13
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3answers
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 $$ ...
5
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16answers
2k 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
3answers
983 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
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1answer
286 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
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13answers
1k 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 ...
11
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4answers
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
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13answers
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
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1answer
403 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 ...
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3answers
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 ...
2
votes
1answer
362 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 ...
10
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6answers
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
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10answers
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
2answers
502 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
9answers
675 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
1answer
1k 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
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2answers
468 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
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1answer
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
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3answers
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 ...
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4answers
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
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6answers
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
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3answers
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 ...
8
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0answers
224 views

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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2answers
953 views

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 ...
5
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4answers
4k views

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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12answers
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, ...
1
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0answers
986 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
1answer
586 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 ...
1
vote
1answer
432 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! ...
2
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1answer
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?
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4answers
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. ...
11
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1answer
452 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 ...
10
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2answers
2k views

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 ...