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159,065 questions
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Can topologies induce a metric? (revised)
This is a revised version of a question I already posted, but which patently was ill posed. Please give me another try.
For comparison's sake, the axioms of a metric:
Axiom A1: $(\forall x)\ d(x,x) =...
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Number of independent distances between n points in d-dimensional Euclidean space?
There are $\binom{n}{2}$ distances between $n$ points in $\mathbb{R}^d$. Not all of them can be chosen freely if $n$ exceeds the number $n_d = d + 1$. If $n = n_d$ we obviously have $\binom{d+1}{2}$ ...
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easy(?) probability/diff eq. question
I've been wondering about this ever since I was a little kid and I used to ride in the back of the car and my mom would speed like hell towards a green light, only to slam on the brakes when she ...
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Is there any value in studying divisors with coefficients in a ring R?
As a rule, the various groups and quotients of the divisor group on a variety have coefficients in $\mathbb{Z}$. That is, you take $\mathbb{Z}$-linear combinations of Weil divisors or Cartier ...
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Can a singular Deligne-Mumford stack have a smooth coarse space?
Let XX be a Deligne-Mumford stack and let XX \to X be a coarse moduli space. Suppose that X is smooth. Is XX smooth? If not, what is an example? What if XX is of finite type over C (the complex ...
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Global fields: What exactly is the analogy between number fields and function fields?
Note: This comes up as a byproduct of Qiaochu's question "What are examples of good toy models in mathematics?"
There seems to be a general philosophy that problems over function fields are easier to ...
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An inequality relating the factorial to the primorial.
Let [a,b] = {k integer | a < k <= b}. Further let
Comp[a,b] = product_{c in [a,b]} c composite;
Fact[a,b] = product_{k in [a,b]} k integer;
Prim[a,b] = product_{p in [a,b]} p prime.
...
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Simplicial and cubical decompositions of low valence
Every surface can be triangulated in such a way that at most 7 trianlges meet at one vertex. Every surface can be decomposed in squares such that at every vertex at most 5 suqares meet. For surfaces ...
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distribution of non-solvable group orders
let $M$ be the set of natural numbers such that there is a group of this order, which is not solvable. what is the minimal distance $D$ of two numbers in $M$?
the examples $660$ and $672$ show $D \...
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Killing Chern classes
Let $G$ be a compact connected Lie group and let $E\to B$ be a principal $G$-bundle. Suppose $a$ is a rational cohomology class of $E$ such that its pullback $b$ under an orbit inclusion map $G\to E$ ...
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Logical problems in category theory [duplicate]
Possible Duplicate:
Set theory for category theory beginners
It is frustrating to hear people speak of Yoneda embedding, category of all categories/functors, n-categories, infinity categories and ...
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Examples of the varying strengths of topological invariants
In my first algebraic topology class, I remember being told that the simplest reason for homology was to distinguish spaces. For example, if is X=circle and a Y= wedge of a circle and a 2-sphere then ...
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Geometry Vs Arithmetic of schemes
Let's suppose we have a Scheme $X$ over the the field $k$, where such a field can be though to be either $\mathbb{C}$ or a finite field $\mathbb{F}_q$. Then having this in mind,
Where do we find some ...
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Newton Puiseux expansions for singular surfaces?
Is there a theory of Newton-Puiseux type expansions
which works to parameterize singular surface germs $F(x,y,z) =0$?
Ideally, each branch would be the image of map of the form the
$x = u^m, y = ...
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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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Explicit expression of an alternating polynomial in characteristic $2$?
Although the question is easy to pose, I think some background will help to motivate it, so I'll start with it.
Consider variables $X=(X_1, \ldots, X_n)$ over a field $K$ and the elementary symmetric ...
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Galois theory and rational points on elliptic curves
I am in search of a concrete example [a concrete elliptic curve in Weierstrass form] of how Galois theory helps to find rational points on an elliptic curve. Chapter VI of Silverman and Tate discusses ...
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How unhelpful is graph minors theorem?
A very interesting Robertson-Seymour (graphs minors) theorem says:
Any infinite collection of graphs $C$ with the property that if $G\in C $ then its minors also are has the form $\{$graphs $G$ ...
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Are all coproducts of 1 in a topos distinct ?
Inspired by the two solutions to Harry's question
Can a topos ever be an abelian category?
I was wondering whether all coproducts of 1 in a topos are distinct up to isomorphism? That is $1 + 1 + \...
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Stable ∞-categories as spectral categories
Let C be a stable ∞-category in the sense of Lurie's DAG I. (In particular I do not assume that C has all colimits.) Then C does have all finite colimits, the suspension functor on C is an ...
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Yoneda embedding target
I'm learning about representable functors from Vistoli notes thanks to Charles Siegel's answer.
I see that any category $\mathcal C$ can be embedded into $\text{Hom}\\,(\mathcal C^{op}, \mathcal Set)$...
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General construction for internal hom in a presheaf category
I was reading about the internal hom functor for simplicial sets, and the construction is very "localized" (nothing to do with localization, just the english word). It seems like there should be a ...
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categorical homotopy colimits
let $hTop_*$ denote the homotopy category of pointed spaces. I believe that it has no pushouts, in general. the reason is that you can't expect the involved homotopies to be compatible. can anyone ...
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Smoothness of Symmetric Powers
Here's something that's been bothering me, and that's come up again for me recently while reading some stuff about Hilbert schemes of points (Nakajima's lectures, specifically):
Let $C$ be an ...
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A result on prime numbers [closed]
Hi everyone
This is my first post...
I do mathematics from home... ie., not attached with any institution...
I have deduced some results...
$\lim \inf_{n\to\infty} \frac{d_n}{\log p_n} = 0$
and, ...
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Why forgetful functors usually have LEFT adjoint?
for forgetful functors, we can usually find their left adjoint as some "free objects", e.g. the forgetful functor: AbGp -> Set, its left adjoint sends a set to the "free ab. gp gen. by it". This ...
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Any relationship of frobenius homomorphism and frobenius category?
I did not understand number theory or characteristic p-algebraic geometry at all. I just know a little about frobenius homomorphism between two schemes. On the other hand, when I learned something on ...
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number of irreducible representations over general fields
For a finite group, there are finitely many irreducible representations of complex numbers.
What if the field is changed to some other fields? Like real numbers, p-adic field, finite field?
In ...
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How can we use the bounded convergence theorem in this proof of the Riesz Representation Theorem?
I'm studying the proof of the Riesz Representation Theorem as it appears in Ch. 6 of Royden's Real Analysis. When I looked on the web I noted there are a few different theorems that go by the name "...
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Recursively dependent types?
Is there such a thing as "recursively dependent types"? Specifically, I would like a dependent type theory containing a type $A(x)$ which depends on a variable $x: A(z)$, where $z$ is a particular ...
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about Hilbert and Siegel modular varieties (forms)
It seems to me that Hilbert modular varieties (forms) are generalization from Q to totally real fields. While Siegel modular varieties (forms) are generalization from 1 dimensional to higher ...
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Any reason why K_23(Z) has order 65520?
I'm rereading my notes and they mention that $K_{23}(\mathbb Z) = \mathbb Z/(65520)$
This looks like a good point to stop and ask whether there is any explanation for this $K$-group of integers (23 ...
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Formulas for vector fields on Grassmannians?
The Wikipedia article on (real) Grassmannians gives a simple argument that the Euler characteristic satisfies a recurrence relation $$\chi G_{n,r} = \chi G_{n-1,r-1} + (-1)^r \chi G_{n-1,r}$$. This ...
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Does the product (by an object) in an abelian category ever have a right adjoint?
This is a follow-up to this question. Since an abelian category cannot be cartesian closed, clearly the hom functor is not right adjoint to the product (by an object). However, does the product (by ...
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If a quadratic form is positive definite on a convex set, is it convex on that set?
Consider a real symmetric matrix $A\in\mathbb{R}^{n \times n}$. The associated quadratic form $x^T A x$ is a convex function on all of $\mathbb{R}^n$ iff $A$ is positive semidefinite, i.e., if $x^T A ...
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Equations for Integrable Systems
So, let's say we have a symplectic variety over $\mathbb{C}$, $M$, of dimension $2n$, and $f_1,\ldots,f_n$ Poisson commuting functions with $df_1\wedge\ldots\wedge df_n$ generically nonzero. Further ...
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Categorifications of the Fibonacci Fusion Ring arising from Conformal Field Theory
I was reading about realizations of the "Fibonacci" fusion ring $X \otimes X = X \oplus 1$ in Fusion Categories of Rank 2 by Victor Ostrik. Apparently, there are two of them and they arise in various ...
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When does local invertibility imply invertibility?
Generally, local invertibility does not imply invertibility. However, for differentiable functions from $\mathbb{R}$ to $\mathbb{R}$ then surjectivity and local invertibility do imply invertibility.
...
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Choosing a fast computer algebra system that works in characteristic p?
Hi all,
I want to compute in $\mathbb{F}_q (x)((y))$ i.e. a Laurent series ring over the rational functions over $\mathbb{F}_q$. The computations are fairly basic, but they involve raising to the qth ...
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on chern classes and Riemann Roch theorem for torsion-free sheaves on singular (possibly multiple) curve
I'm looking for a definition of Chern class (at least the first one) for a torsion-free sheaf $F$ (not necessarily locally free) on a singular curve (for simplicity can assume all the singularities ...
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Explicit computation of induced modules of semidirect products with the symmetric group
I've gotten stuck in a project I have been working on, essentially on the following combinatorial question about the symmetric group.
One can obtain a 1-dimensional representation $M^n_c$ of the ...
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Moduli spaces of coherent sheaves on K3s
Reading 2007 paper A tour of theta dualities on moduli spaces of sheaves by Alina Marian and Dragos Oprea.
Why is any moduli space of coherent sheaves on a K3 surface deformation equivalent to a ...
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Schemes of Representations of Groups
Let $G$ be a group, say finitely presented as $\langle x_1,\ldots,x_k|r_1,\ldots,r_\ell\rangle$. Fix $n\geq 1$ a natural number. Then there exists a scheme $V_G(n)$ contained in $GL(n)^k$ given by ...
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Iwasawa and Cartan Decompositions.
Consider the tome of Bruhat and Tits: Groupes réductifs sur un corps local : I. Données radicielles valuées. Publications Mathématiques de l'IHÉS, 41 (1972), p. 5-251. (available on NUMDAM). I am ...
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Does a regular neighborhood always exist for a properly embedded surface in a 3-manifold?
Can someone please clarify if there always exist regular neighborhoods for a properly embedded surface in a 3-manifold? More precisely, if $F$ is a properly embedded surface in a 3-manifold $M$ and I ...
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subspace topology for functors
let $X : Ring \to Set$ be a functor (a Z-functor in the language of demazure, gabriel) and $V \subseteq X$ a locally closed subfunctor. assume that $U \subseteq V$ is an open subfunctor. does then ...
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How many L-values determine a modular form?
Suppose $f$ and $g$ are two newforms of certain levels, weights etc. If we know that
L(f,n)=L(g,n) for all sufficiently large $n$, can we conclude that $f=g$?
Same question when the forms have the ...
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Cohomology rings and 2D TQFTs
There is a "folk theorem" (alternatively, a fun and easy exercise) which asserts that a 2D TQFT is the same as a commutative Frobenius algebra. Now, to every compact oriented manifold $X$ we can ...
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A GAGA question
A GAGA question.
Say I have a ``quasi-projective'' (*) subvariety X over the complex
numbers within a smooth complex algebraic variety Z.
True or False: The analytic and algebraic closure ...
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Pimsner-Popa Bases
Let $N\subset M$ be a finite index $II_1$-subfactor. Let $B=\{b_i\}$ be a finite orthonormal (Pimsner-Popa) basis for $M$ over $N$. Let $d=[M\colon N]^{1/2}$. It is well known that $B_1=\{d b_{i_1} ...
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