# All Questions

100,923 questions
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### Are injective Omega-spectra the S-local objects of symmetric spectra for some class S?

I am trying to read the Hovey-Shipley-Smith article as defining the stable model structure on symmetric spectra as a left Bousfield localization (as explained on nLab) of the projective level model ...
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### Tensored Over Abelian Groups?

Suppose I have an category additive category C (i.e. the hom sets are enriched in abelian groups and there are finite direct sums). Suppose further that C has cokernels. Then I can make C tensored ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 (...
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### 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 ...
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### 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?
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### 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} \...
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### 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 ...
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### 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 ...
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### 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&...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ${\mathbb Q}(z)$ ...
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### a question about Gromov-Witten invariant

Do the Gromov-Witten invariants count the morphisms from a curve to a variety over $\mathbb{C}$?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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." ...
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 $T_2$). ...
What are interesting, illustrative examples of Borel sets, situated in Borel hierarchy higher than $\Sigma^{0}_{2}$ /$\Pi^{0}_{2}$?