All Questions

33
votes
4answers
4k views

What does “linearly disjoint” mean for abstract field extensions?

All definitions I've seen for the statement "$E,F$ are linearly disjoint extensions of $k$" are only meaningful when $E,F$ are given as subfields of a larger field, say $K$. I am happy with the ...
2
votes
2answers
198 views

Decomposition result for multivariate polynomial

Let $k$ be a positive integer greater than $1$ and suppose that $F \in \mathbb{Z}[x_{1}, \ldots, x_{k}]$. Can we always find a natural number $n(k)$ and $f_{1}, \ldots f_{n(k)} \in \mathbb{Z}[x]$ ...
11
votes
4answers
2k views

intuitionistic interpretation of classical logic

Basically intuitionistic logic is classical logic minus the law of the excluded middle, i.e. $\neg A\vee A$ is not necessarily valid for all formulas. So I would take this to mean that classical logic ...
29
votes
22answers
8k views

Origins of Mathematical Symbols/Names

I'm not sure if this has been asked. I'll explain the question by an example. Fields are often denoted by the letter k, which comes from the German word Körper, meaning body (like corpse, corporeal). ...
22
votes
2answers
3k views

Why is the decomposition theorem awesome?

I saw the statement of the decomposition theorem for perverse sheaves sometime ago. I know that (modulo most of the details) it implies some big theorems in algebraic geometry and gives new proofs for ...
5
votes
4answers
7k views

Badiou and Mathematics [closed]

Does anyone have an opinion on Alain Badiou's use of set theory? Is there anything interesting mathematically there? Also could anyone shed any light on the comment in the Wikipedia article link text ...
7
votes
3answers
1k views

Holomorphic and antiholomorphic forms of projective space

For $\mathbb{CP}^1$ the bundles of holomorphic and antiholomorphic forms are equal to the $\mathcal{O}(-2)$ and $\mathcal{O}(2)$ respectively. Do the holomorphic and antiholomorphic bundles of $\...
10
votes
2answers
855 views

Adding a random real makes the set of ground model reals meager

This is a question about forcing. I have seen the following fact mentioned in multiple places, but have not been able to find a proof: if a random real is added to a transitive model of ZFC, then in ...
7
votes
5answers
2k views

Just starting with [combinatorial] game theory

I have recently become interested in game theory by way of John Conway's on Numbers and Games. Having virtually no prior knowledge of game theory, what is the best place to start?
13
votes
4answers
638 views

Is there a natural family of languages whose generating functions are holonomic (i.e. D-finite)?

Let $L$ be a language on a finite alphabet and let $L_n$ be the number of words of length $n$. Let $f_L(x) = \sum_{n \ge 0} L_n x^n$. The following are well-known: If $L$ is regular, then $f_L$ is ...
37
votes
5answers
6k views

Proof assistants for mathematics

This question is related to (maybe even the same in intent as) Question 1017, but none of the answers seem to address what I'm looking for. There are a lot of resources available for people who want ...
72
votes
15answers
11k views

What's a nice argument that shows the volume of the unit ball in $\mathbb R^n$ approaches 0?

Before you close for "homework problem", please note the tags. Last week, I gave my calculus 1 class the assignment to calculate the $n$-volume of the $n$-ball. They had finished up talking about ...
3
votes
3answers
1k views

Mathematical definition of running [closed]

This will be a tad hard to explain, so bear with me. Taking into account only the legs what would be an accurate definition of the position of the upper legs, lower legs and feet with respect to time? ...
3
votes
1answer
395 views

Convex n-polytope general position vectors to general position vectors of tetrahedron

I asked this question in a comment to this question, but got no response. I thought that perhaps it needed more exposure, so I made it a question in itself. Define a set of general position vectors $...
12
votes
0answers
530 views

References for a certain generalization of Hochschild cohomology?

Let $C$ be an algebra. Let $E = C^{\otimes 2n}$ be the tensor product (over the ground field) of $2n$ copies of $C$. [EDIT: Or better, $E = C\otimes C^{op}\otimes C\otimes C^{op}\cdots\otimes C \...
10
votes
2answers
3k views

Is there a meaningful difference between biased and unbiased composition?

In higher category theory, there are notions of biased and unbiased definitions of composition of $n$-morphisms (or, as a special case, tensor products of objects). In the biased framework, we define ...
76
votes
20answers
7k views

One-step problems in geometry

I'm collecting advanced exercises in geometry. Ideally, each exercise should be solved by one trick and this trick should be useful elsewhere (say it gives an essential idea in some theory). If you ...
6
votes
4answers
997 views

What is the name for the following categorical property?

Is there a name for those categories where objects posses a given structure and every bijective morphism determines an isomorphism between the corresponding objects? Examples of categories of that ...
3
votes
3answers
408 views

Nature of Invertible Sheaves in which there are no global sections.

EDIT: Let me try to make the question clearer. Consider the invertible sheaves $\mathcal{O}(d)$ over the projective space $\mathbb{P}^n$ where $d\in \mathbb{Z}$. Now, if $d>0$, among many ...
19
votes
2answers
1k views

Four Dimensional Origami Axioms

What are the axioms of four dimensional Origami. If standard Origami is considered three dimensional, it has points, lines, surfaces and folds to create a three dimensional form from the folded ...
15
votes
11answers
2k views

Chromatic number of graphs of tangent closed balls

The Koebe–Andreev–Thurston theorem gives a characterization of planar graphs in terms of disjoint circles being tangent. For every planar graph $G$ there is a disk packing whose graph is $G$. What ...
3
votes
1answer
389 views

How do Dehn functions of special linear and mapping class groups behave?

Hi, I apologize for the basic questions. I am looking for good references on the following problems: 1) What is known about the Dehn function of $SL_n(\mathbb{Z})$? 2) What is known about the Dehn ...
6
votes
1answer
717 views

Encoding fuzzy logic with the topos of set-valued sheaves

One of the canonical examples used by Barr & Wells in order to motivate the use of topoi is that we can construct a theory for fuzzy logic and fuzzy set theory as set-valued sheaves on a poset (...
3
votes
3answers
642 views

Reducible 3d torus bundles

Here reducible means that the mapping class for the fiber is a reducible auto-homeomorph in the sense of Nielsen-Thruston. So, could anyone give me a hint to classify them? In contrast, do you agree ...
5
votes
2answers
911 views

Elementary theory of finite fields

I read on Ax's article that the elementary theory of finite fields is decidable if one assumes the continuum hypothesis to be true. What about if one assumes the hypothesis to be false?
11
votes
8answers
2k views

less elementary group theory

Most of the group theory that is taught in introductory graduate classes is of the form $$(\mbox{number theory} + \mbox{ group actions} + \mbox{ orbit-stabilizer thm}) + \mbox{group axioms} \...
2
votes
3answers
881 views

Axiom systems and Information Theory

Is there a concept of "information" with respect to the axioms of a mathematical system? Suppose we have a universe U of theorems. Suppose an axiom system A=(a1,a2,...) has the universe U as the ...
24
votes
10answers
3k views

How can I really motivate the Zariski topology on a scheme?

First of all, I am aware of the questions about the Zariski topology asked here and I am also aware of the discussion at the Secret Blogging Seminar. But I could not find an answer to a question that ...
3
votes
2answers
390 views

Semilattices in atomless boolean algebras

Let S be a bounded semilattice without maximal elements. Can we always construct an atomless boolean algebra B, containing S as a subsemilattice, such that S is cofinal in B-{1}? That is, for every x&...
1
vote
1answer
299 views

Systems of conics

It seems well-known that the system of conics given by $\frac{x^2}{a^2}+\frac{y^2}{a^2-c^2}=1$ for $c>0$ fixed and $a \in (0,c)\cup(c,\infty)$ varying is orthogonal: whenever two of these curves ...
16
votes
5answers
2k views

Elliptic Curves over F_1?

Is there an notion of elliptic curve over the field with one element? As I learned from a previous question, there are several different versions of what the field with one element and what schemes ...
3
votes
5answers
1k views

Cardinality of Equivalence Classes of Cauchy Sequences

What's the cardinality of a single equivalence class of Cauchy sequences in ℚ? To clarify, I'm not asking for the cardinality of the real numbers, but for the cardinality of the set of Cauchy ...
10
votes
2answers
370 views

Subfields joining an algebraic element to another

Let $\alpha$ and $\beta$ be two algebraic numbers over $\mathbb Q$. Say that a subfield $\mathbb K$ of $\mathbb C$ joins $\alpha$ to $\beta$ iff $\beta \in {\mathbb K}[\alpha]$ but $\beta \not\in {\...
12
votes
4answers
2k views

Is a polynomial with 1 very large coefficient irreducible?

I am asking for some sort of generalization to Perron's criterion which is not dependent on the index of the "large" coefficient. (the criterion says that for a polynomial $x^n+\sum_{k=0}^{n-1} a_kx^k\...
3
votes
4answers
960 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 ...
4
votes
2answers
745 views

Notation/name for “Artin-Schreier roots”?

If x is an element of a field K and n is a positive integer, we have both a symbol and a name for a root of the polynomial t^n - x = 0: we denote it by x^{1/n} and call it an nth root of x. Of course ...
11
votes
1answer
1k views

measurable sets not depending on even coordinates

Let $A\subset\{0,1\}^\omega$ be a measurable set (w.r.t. the usual borel sigma algebra) which does not depend on any even coordinate (that is, if $x\in A$ and $x$ and $y$ agree except on a finite ...
2
votes
3answers
278 views

Expressing field inclusions by polynomial equalities on coefficients

Let $A$ be the set of all quadruples $(a_0,a_1,a_2,a_3) \in {\mathbb Q}^4$ such that the polynomial $P=X^4+a_3X^3+a_2X^2+a_1X+a_0$ is irreducible and if $z$ is any root of $P$, then ${\mathbb Q}(z)$ ...
4
votes
2answers
713 views

a question about Gromov-Witten invariant

Do the Gromov-Witten invariants count the morphisms from a curve to a variety over $\mathbb{C}$?
1
vote
2answers
483 views

Homomorphism between exterior powers of a free module of finite rank

I´m looking for homomorphisms between exterior powers of a free module M of rank m ΛmR M → Λm-1R M Exactly, I´m looking for an explicit isomorphism M → Hom R (ΛmR M , Λm-1R M) I compare the ranks ...
16
votes
2answers
449 views

Are there piecewise-linear unknots that are not metrically unknottable?

A stick knot is a just a piecewise linear knot. We could define "stick isotopy" as isotopy that preserves the length of each linear piece. Are there stick knots which are topologically trival, but ...
4
votes
1answer
575 views

Coordinates on Teichmuller space

We know that every surface of genus ($g\geq 2$) admits a pair of pants decomposition. And there is the Fenchel Nielsen Coordinates on the Teichmuller space associated to such a decomposition where we ...
5
votes
3answers
2k views

Existence of projective resolutions in abelian categories

It is a standard result of elementary homological algebra that to every R-module $A$ there exists a projective resolution. It is often said that the category of R-modules has "enough projectives." ...
11
votes
2answers
782 views

Does sheafification preserve sheaves for a different topology?

Let $T_1$ and $T_2$ be two Grothendieck topologies on the same small category $C$, and let $T_3 = T_1 \cup T_2$ (by which I mean the smallest Grothendieck topology on $C$ containing $T_1$ and $T_2$). ...
5
votes
4answers
601 views

Higher-rank Borel sets

What are interesting, illustrative examples of Borel sets, situated in Borel hierarchy higher than $\Sigma^{0}_{2}$ /$\Pi^{0}_{2}$?
4
votes
1answer
572 views

Canonical basis for the extended quantum enveloping algebras

I am trying to understand some construction done by Lusztig in his book on quantum groups. Given some Cartan datum, let $U=U_q(\mathfrak{g})$ the standard quantized enveloping algebra of the Kac-Moody ...
12
votes
4answers
5k views

Number theory textbook with an algebraic perspective

Most of the number theory textbooks I've dealt with take a very classical approach to the subject. I'm looking for a textbook that's something like a first course in number theory for people who have ...
2
votes
1answer
119 views

Some equivalent statements about primitive algebras

I was reading a paper, and it said that the following were equivalent using the Axiom of Choice, but I tried working it out, and I wasn't sure how: an algebra $A$ is primitive; $A$ has a proper left ...
83
votes
11answers
11k views

Is it possible to capture a sphere in a knot?

You and I decide to play a game: To start off with, I provide you with a frictionless, perfectly spherical sphere, along with a frictionless, unstretchable, infinitely thin magical rope. This rope ...
0
votes
2answers
330 views

name for “solid” subset of a partially ordered set?

For P a partially ordered set, let S be a subset of P such that if: a,c\in S and b\in P and a<=b<=c then b\in S Is there a name for a subset with this property? The term "dense" subset is ...

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