All Questions
15,616 questions
6
votes
3
answers
2k
views
A simple infinite dimensional optimization problem
I'd be grateful for a reference for the following result, which I believe to be true, and
should be well-known.
Let the continuous functions $f_0,f_1,\cdots,f_n: [0,1]\rightarrow [0,\infty)$ be ...
- 60.5k
3
votes
4
answers
3k
views
Reference request for category theory works which quickly prove the theorem which generalises the 1st isomorphism theorem for groups/rings/...
One aspect of category theory that caught my eye is that it can give simultaneously prove the 1st isomorphism theorem for groups/rings/fields/vector spaces/... Yet whenever I look up works on category ...
- 331
1
vote
0
answers
268
views
Learning statistical mechanics for non-particle phenomena
I'm interested in various areas of complex systems, and I often come across articles like these:
http://arxiv.org/PS_cache/cond-mat/pdf/0106/0106096v1.pdf
http://arxiv.org/abs/cond-mat/9804180
The ...
- 2,383
3
votes
0
answers
313
views
Proof of Saito criterion
Does it exist another proof of saito's criterion for free divisors, other than the one in "K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo ...
- 671
1
vote
1
answer
448
views
Characterisation of coherent sheaves on an algebraic variety
The Wikipedia article on coherent sheaves makes the following claim (without any reference), which I had trouble proving or finding a reference for: on an algebraic variety X (or I guess possibly even ...
- 481
6
votes
2
answers
1k
views
Introduction to the Podles Sphere
I am just looking for a basic introduction to the Podles sphere and its topology. All I know is that it's a q-deformation of $S^2$.
- 3,377
21
votes
2
answers
1k
views
Most squares in the first half-interval
It is well known that if $p$ is an odd prime, exactly one half of the numbers $1, \dots, p-1$ are squares in $\mathbb{F}_p$. What is less obvious is that among these $(p-1)/2$ squares, at least one ...
- 8,467
5
votes
6
answers
913
views
Topological results from geometry
Hi people,
I'm interested in results, such as the Gauß-Bonnet theorem, Fàry-Milnor theorem or classification theorems for manifolds, which give topological properties from geometric considerations. ...
- 14.4k
7
votes
1
answer
730
views
Example sought of an atomic domain R such that R[t] is not atomic
Recall that an integral domain $R$ is atomic if every nonzero nonunit admits at least one factorization into irreducible elements. (Indeed, hard-core factorization theorists have replaced the word "...
- 19.6k
25
votes
3
answers
5k
views
Conceptual understanding of the Gross-Zagier theorem.
The Gross-Zagier paper "Heegner points and derivatives of $L$-series", is really computational and hard to plow through. It seems it is futile to read it as such and one must look for a more ...
- 1,141
15
votes
3
answers
2k
views
Survey articles on homotopy groups of spheres
Are there general surveys or introductions to the homotopy groups of spheres? I'm interested especially in connections to low-dimensional geometry and topology.
- 8,467
1
vote
2
answers
3k
views
unit sphere is weak dense in the unit ball
As I remember the following is true:
Fact: for every infinite-dimensional normed space $X$
the unit sphere $S$ is weak-dense in the unit ball $B$.
Please help me find a reference.
Thanks in ...
- 29.7k
5
votes
2
answers
640
views
Trying to find a 1949 Russian Paper on Transportation Theory
I am a research student in transportation theory. I have difficulty in obtaining this paper:
L. V. KANTOROVICH and M. K. GAVURIN, "The application of mathematical methods
in problems of freight flow ...
CommunityBot
- 1
5
votes
0
answers
600
views
Barsotti--Weil formula over separably closed fields
Let $S$ be a noetherian scheme and let $A$ be an abelian scheme on $S$ with dual $A^\vee$. The generalised Barsotti Weil formula states that there is a canonical and functorial (in $S$ and $A$) ...
- 4,015
19
votes
1
answer
901
views
Locus of equal area hyperbolic triangles
Henry Segerman and I recently considered the following question:
Given a fixed area $A < \pi$ and two fixed points in the upper half-plane model for hyperbolic $2$-space, what is the locus of ...
- 28.1k
14
votes
2
answers
1k
views
Dyer-Lashof based spectral sequence for homotopy classes of maps between infinite loop spaces (spectra).
The homology of an infinite loop space, which represents a spectrum, is an algebra over the Dyer-Lashof algebra (see for example Cohen-Lada-May's Springer volume, or for part of the story the more ...
- 498
4
votes
1
answer
373
views
The geometry of closure of orbit of Borel subgroup in G/B × G/B.
Let $G$ be a reductive group, let $B$ be one of its Borel subgroups, and $T$ be a torus in $B$. $G/B$ is its flag variety. Let $y,w$ be two T-fixed points in $G/B$. Let $\mathcal{O}_{y,w}$ be the $B$-...
- 27.8k
5
votes
2
answers
332
views
Characterization of combinatorial manifolds in terms of links
I need to reference the following result. Do you know a good source?
The following conditions on an $n$-dimensional simplicial complex $S$ are equivalent:
a) $S$ is an $n$ manifold;
b) The link of ...
- 5,009
11
votes
1
answer
654
views
Nonseparable Hilbert spaces as quotients of spaces of bounded functions
Is the following result true: the Hilbert space $\ell^{2}\left(2^{\Gamma}\right)$ is a quotient of $\ell^{\infty}\left(\Gamma\right)$ for any
uncountable $\Gamma$ ? [I think it is, but cannot remember ...
- 31.5k
10
votes
3
answers
2k
views
A Reference for Schubert's Theorem
Schubert's Theorem in Knot Theory says that any knot can be uniquely decomposed as the connected sum of prime knots.
Unfortunately the original paper is in German.
Does anyone know a good english ...
- 22.1k
2
votes
2
answers
353
views
Reference request: given a divisor d of N, how quickly can I obtain the largest factor of N coprime to d?
This is quite likely to be a solved problem, perhaps even a standard exercise. However, being a non-[number theorist], I don't know where to look. A quick perusal of the basic starting references ...
- 453
4
votes
1
answer
1k
views
Lifting Etale Morphisms
I am trying to find a reference for the following theorem:
Let $R$ be a complete DVR, and let $Y$ be a scheme projective and flat over $R$. Suppose that $X_0 \longrightarrow Y_0$ is a finite etale ...
- 8,296
7
votes
2
answers
1k
views
Reference for the existence of a Shapovalov-type form on the tensor product of integrable modules
Shapovalov and Jantzen showed us how to construct a nice inner product on finite dimensional representations of a semi-simple Lie algebra, by simply giving the highest weight vector inner product 1 ...
- 44.7k
4
votes
0
answers
390
views
Is there a reference that treats principal homogeneous spaces for (say) group varieties using schemes?
I was wondering if anyone could recommend a reference that discusses principal homogeneous spaces for general finite type group schemes over a field $k$ entirely in the language of schemes (or even ...
- 3,380
2
votes
1
answer
647
views
Reference: Countable Models of (Non-)Euclidean Geometry
Has there been a survey written on the model theory of first-order (non-)Euclidean geometry in the spirit of Hilbert and Tarski? I'm especially interested in two aspects of the model theory:
...
- 25.7k
15
votes
1
answer
1k
views
If the tensor product of two representations are crystalline, are the original representations crystalline?
Let $K$ be a finite extension of the $p$-adic numbers. Suppose that $V$ and $W$ are two (finite dimensional, $p$-adic) continuous representations of $G_K$. Suppose that $V \otimes W$ is crystalline. ...
CommunityBot
- 1
6
votes
1
answer
967
views
Chow Ring of Moduli Space of Abelian Varieties
Is there a good reference for the structure of the Chow ring of $\mathcal{A}_g$, the moduli space of complex principally polarized abelian varieties? More generally, references for the intersection ...
- 6,229
8
votes
0
answers
1k
views
Is there a deep relationship between models and étale cohomology ? If so, why, and is it made precise somewhere ?
Let me recall two theorems :
Let $K$ be a field, $\overline{K}$ be
a separable closure of $K$ with
absolute Galois group
$G_K:=Gal(\overline{K}/K)$, and let
$\ell$ be a prime that is ...
- 373
12
votes
1
answer
3k
views
Perverse Sheaves - MacPherson Lecture Notes
I keep running across papers that refer to a set of lecture notes by Robert MacPherson at MIT during the fall of 1993 on Perverse Sheaves. There might also be a set of notes from lectures in Utrecht ...
- 186
4
votes
1
answer
246
views
Name for an inequality of isoperimetric type
I want to know if the following fact has a standard name and/or reference
Let $X$ be a subset of $\mathbb R^2$ and $B$ be a disc of the same area as $X$.
Set $X_\epsilon$ to be the $\epsilon$-...
- 85.6k
4
votes
1
answer
262
views
Criterion for being a simple vector
1) I was wondering if there exists a criterion (of (a) combinatorial or (b) geometric nature) for a sum of simple vectors $V\in\wedge^k(\mathbb R^n)$,
$V=e_{a_{11}}\wedge\cdots\wedge e_{a_{1k}} + \...
- 156k
3
votes
4
answers
2k
views
Equivalent definitions of Gaussian curvature
I'm trying to find out more about geometry of surfaces and, in particular, Gaussian curvature. I understand that it can be defined in terms of the principal curvatures (extrinsically) and also ...
- 27.5k
2
votes
1
answer
1k
views
Several question on Affine Lie algebra
These questions might be elementary for I just started to learn affine Kac-Moody algebra.
It is well known that if we consider finite dimensional Lie algebra, we have the folloing projection:
$R(\...
CommunityBot
- 1
1
vote
0
answers
1k
views
Covariance matrix formula interpretation - what am I missing?
I'm reading a paper that outlines the calculation of a covariance matrix like the following:
$C=\displaystyle\sum^{N_b}_{i=1}\vec{x}_i\vec{x}_i^T$
What is the order of this matrix? My interpretation ...
- 111
4
votes
3
answers
470
views
Is there a nice way to characterise the derived equivalence induced by a flop?
Hello, I'm interested in the case when all varieties are projective threefolds over the complex numbers.
Start with a flopping contraction $f:Y \rightarrow X$, with corresponding flop $f^+: Y^+ \...
- 301
9
votes
2
answers
7k
views
Constant curvature manifolds
In two different books I found these two related statements.
The book by Jost defines a ``locally symmetric space" as one for which the curvature tensor is constant and which is geodesically complete....
CommunityBot
- 1
4
votes
0
answers
352
views
"Cholesky decomposition" X=YY* for p-adic matrices?
Let $E/F$ be a quadratic extension of $p$-adic fields. Consider $M_n(E)$ with the unitary (aka 2nd kind) involution $X \mapsto \sigma(X)^{tr}$, where $\sigma(X)$ denotes the entry-wise application of $...
- 932
6
votes
2
answers
976
views
References for modular polynomials
I am teaching a graduate "classical" course on modular forms. I try to achieve the most elementary level for presenting modular polynomials. Serge Lang's "Elliptic functions" cover the topic quite ...
- 1,694
14
votes
1
answer
503
views
Is a polynomial group law on $\mathbb{R}^n$ automatically nilpotent?
I was told that a polynomial group law on (all of) $\mathbb{R}^n$ gives automatically a nilpotent (Lie, of course) group.
Is it true? Where can I find a proof?
A counterexample for open subsets of $...
- 22.6k
2
votes
1
answer
2k
views
What is Extreme/Extremal vector according to some weights
I know this might be a very elementary question. But I could not find the original definition of Extreme(or Extremal)vectors of some weights $\lambda$(fixed the $w\in W$,where $W$ is Weyl group)
I am ...
- 44.7k
0
votes
1
answer
1k
views
What is Taft algebra?
What is a Taft algebra? Is there any references about the original conception?
CommunityBot
- 1
14
votes
2
answers
3k
views
How many ways are there to prove flag variety is a projective variety?
I am looking for references talking about different ways to prove flag variety $G/B$ is projective variety. Now I have some in mind:
There is a proof in Humphreys Linear algebraic groups, he first ...
- 5,515
8
votes
0
answers
521
views
Skew polynomial algebra
When I was a very little hare, a big grey wolf told me about the following skew polynomial algebra, which I never understood. My question is whether the following construction is a part of some bigger ...
- 12.3k
7
votes
1
answer
826
views
Weight filtration for smooth analytic manifolds
In his ICM 2002 talk (Topology of singular algebraic varieties, available also on arXiv) B. Totaro says on p. 3 (of the arXiv version): "Using the method of Guillen and Navarro Aznar I was able to ...
- 35.2k
8
votes
2
answers
429
views
Is there a source for a diagrammatic description of the induction functor C->Z(C)?
Suppose that C is a fusion category (over the complex numbers) and that Z(C) is its Drinfel'd center. By definition an object in Z(C) consists of an object V in C together with a collection of half-...
- 246
5
votes
2
answers
346
views
Reference request: A theorem by S. Garrison
A theorem by S. Garrison states that if $G$ is a finite solvable group and $|cd(G)| = 4$ then $dl(G)\leq |cd(G)|$ (the Taketa inequality, which is conjectured to hold for all finite solvable groups). ...
- 10.7k
9
votes
3
answers
2k
views
Is there a "primitive-recursively enumerable" set whose complement is not such?
Call a subset of $\mathbb{N}$ primitive-recursively enumerable (p-r.e.) if it is empty or an image of a primitive recursive function. I feel like a lot must be known about the poset of such sets ...
- 1,185
7
votes
1
answer
799
views
Liftability of Enriques Surfaces (from char. p to zero)
Let $k$ be an algebraically closed field of characteristic $p > 0$, $X$ a variety over $k$.
We say $X$ lifts to characteristic zero, if there exists a local ring $R$ containing $\mathbb Z$ with ...
- 22.6k
4
votes
1
answer
358
views
Prime-ness checking for polynomial ideals over ACFs( algebraically closed fields).
Let $f_1,\ldots f_m \in k[X]$ have degrees bounded by $l$. and $I(\bar{f})$ be the ideal generated by $\bar{f}$.
If $I(\bar{f})$ is not a prime ideal then its non-primeness is witnessed by polynomials ...
- 44.4k
10
votes
1
answer
2k
views
Sum of difference moduli vs. sum of modulus differences
This is a failed attempt of mine at creating a contest problem; the failure is in the fact that I wasn't able to solve it myself.
Let $x_1$, $x_2$, ..., $x_n$ be $n$ reals. For any integer $k$, ...
- 61.9k