All Questions
153,397
questions
14
votes
1
answer
957
views
Area of the minimal surface of a non-planar quadrilateral in 3d
Consider a non-planar quadrilateral in three dimensions, i.e. four points $x_1,\dots,x_4$ in $\mathbb{R}^3$ that do not lie on a plane and connected by straight lines. Then, by general theory of ...
9
votes
1
answer
634
views
Embedding proper algebraic spaces
Does every proper algebraic space (over a field, say) admit a closed immersion into a smooth proper algebraic space?
Remark: Of course, if we say "projective" instead of "proper" then the answer is ...
22
votes
3
answers
2k
views
How mirror of quintic was originally found?
In the 90-91 pager
"A PAIR OF CALABI-YAU MANIFOLDS AS AN EXACTLY SOLUBLE SUPERCONFORMAL THEORY",
Candelas, de la Ossa, Green, and Parkes, brought up a family of Calabi-Yau 3-folds, canonically ...
5
votes
2
answers
320
views
Sheaf cohomology on non paracompact topological spaces
I have some confusion on the subject of sheaf cohomology on non-paracompact topological spaces, i hope you can help me.
My reference is Godement's book "Topologie algebrique et theorie dex faisceaux"....
4
votes
1
answer
289
views
Is $p$ is square modulo $F_p$ when $p=4k+1 > 5$?
$F_n$ are the Fibonacci numbers.
In On computing factors of cyclotomic polynomials p.1 for odd square-free $n>1$ the cyclotomic polynomial $\Phi_n(x)$
satisfies:
$$ 4 \Phi_n(x)=A_n(x)^2 - (-1)^{(n-...
3
votes
0
answers
324
views
Abel-Prym map for Prym-Tyurin varieties
Let $(J,\Theta)$ be the Jacobian of a smooth projective curve $C$, and let $i:P\hookrightarrow J$ be an abelian subvariety of $J$ such that $i^*\Theta\equiv e\Xi$ for some principal polarization $\Xi$ ...
4
votes
1
answer
165
views
Proper Model Category
Let R be a commuative ring. Consider the category of simplicial R-modules with the projective model stucture. Can someone give me a precise reference which proves that this model category is proper? ...
0
votes
1
answer
861
views
Sufficient and necessary condition for BIBO stability
I am looking for a reference for the proof of the next claim:
"BIBO—bounded input bounded output—stability.
We claim that a necessary and sufficient condition for a system described
by a linear, ...
2
votes
0
answers
193
views
computation with Hilbert scheme of $n$ points on $\mathbb C^2$ [closed]
How can we show that
$$\sum_{n = 0}^\infty q^n \operatorname{char}_T S^n(\mathbb C[x,y])=
\prod_{p_1,p_2\geq 0}\frac{1}{1-t_1^{p_1}t_2^{p_2}q}$$
where $\operatorname{char}_T V$ denotes the character ...
1
vote
1
answer
172
views
Are generically trivial finite unramified morphisms trivial
Let $S$ be a smooth affine variety over $\mathbb C$ and let $f:X\to S$ be a finite unramified morphism.
Suppose that $X(K(S))$ is non-empty. (This means that $X\to S$ has a section generically. It ...
0
votes
2
answers
450
views
Can simply or not simply connected maximally symmetric (Semi-)Riemannian manifold be completely classified?
A m-dimensional completed and connected (Semi-)Riemannian manifold which has $m(m+1)/2$ independent global Killing vector fields is called maximally symmetric space.
Then what are all possibilities ...
9
votes
1
answer
927
views
Space of Borel measurable maps
That's a question from MSE (here) that did not receive any answer for some days. I migrate it to MO.
Let $X$ and $Y$ be two standard Borel spaces and consider the set $M(X,Y)$ of measurable maps $f: ...
3
votes
0
answers
213
views
Unitary dual of $Sp_4(\mathbb{R})$
Do we know the unitary dual of $Sp_4(\mathbb{R})$? If so, can someone provide me any references? How about other rank 2 real groups? Thank you!
5
votes
1
answer
210
views
Interpretation for a condition in fluid dynamics
I have been working with some mathematical models in biology and fluid mechanics. My problem is about
the interpretation of a condition that I found for a vector
representing the velocity of a fluid. ...
17
votes
6
answers
3k
views
Does the linear automorphism group determine the vector space?
I was recently thinking about what it means to put structure on a set. It seems to me that, in my area (representation theory), the two main ways of imposing structure on a set $X$ are:
...
12
votes
0
answers
1k
views
How much algebraic geometry do I need to study complex geometry?
As one can deduce from the questions I have asked on MO, I'm interested in complex geometry. I am aware that there are many facets to the field, some of which I am more comfortable with than others. ...
3
votes
1
answer
2k
views
A.e. pointwise convergence of L2 functions - counterexample for generalization of Carleson's thm
Let $f_n \in L^2[0,1]$ be an orthonormal sequence and let $c_n \in \mathbb C$ be such that $\sum_{n = 1}^{\infty} |c_n|^2 < \infty$. Does this imply that the sequence $\sum_{n = 1}^{\infty}c_nf_n$ ...
1
vote
0
answers
71
views
Bogomol’nyi’s Formula for the Critical Action
I'm studying Aigner's paper 'Existence of the Ginzburg-Landau Vortex Number' (2001) and I have some difficulties to prove the equality (3.1) , which is
$(|F_A|^2+|d_A\phi|^2+|\frac12(1-|\phi|^2)|^2)...
1
vote
0
answers
192
views
A question on a Fourier transform
(This may sound too simple a question to be asked in MO. But it is a research problem but not an exercise, so I post it here. If it is no appropriate, I will move it to Stackexchange.)
Let $0 < a &...
0
votes
1
answer
278
views
Creating topological spaces with portals [closed]
I'm trying to rigorously describe an object that I'm calling a "portal". The situation is easiest to describe in two dimension.
I start with a line segment $pq$ in $\mathbb{R}^2$. I want to remove ...
15
votes
1
answer
795
views
Is the heat kernel more spread out with a smaller metric?
Suppose M is a smooth manifold, and we have two Riemannian metrics on M, say g and h, with g bigger than h (i.e. for every tangent vector at every point, the norm according to g is bigger than the ...
0
votes
0
answers
802
views
How to solve definite integral involving exponential function
I am trying to get a closed form for the following definite integral:
$$f(\theta)= \int_\frac{\pi}{2}^\pi \frac{1}{\sqrt{1-\alpha^2 \cos^2\theta}}\exp\left(C_2\cos\theta-C_1\sqrt{1-\alpha^2 \cos^2\...
5
votes
2
answers
815
views
A central limit theorem for a trigonometric series involving primes
In some recent work I found I needed to prove a central limit theorem for
the interesting series:
$\sum_{n=1}^\infty \cos (u \log p_n) $
where u is a random variable uniform on the interval $[0,2\...
4
votes
3
answers
674
views
Is $\lceil \frac{n}{\sqrt{3}} \rceil > \frac{n^2}{\sqrt{3n^2-5}}$ for all $n > 1$?
An equivalent inequality for integers follows:
$$(3n^2-5)\left\lceil n/\sqrt{3} \right\rceil^2 > n^4.$$
This has been checked for n = 2 to 60000. Perhaps there is some connection to the ...
6
votes
1
answer
390
views
Differential of homological atiyah-Hirzebruch Spectral sequence for K-homology
The first non vanishing differential $d_3$ of the cohomological Atiyah-Hirzebruch spectral sequence for computing (Complex) Topological $K$-theory out of ordinary cohomology has a ...
10
votes
2
answers
1k
views
Number of paths through infinite trees with given "growth rates"
(Preface: This may be a naive or easy question for experts....)
Consider an infinite tree, rooted on the left, where each node has two children; the number of nodes at each level (distance from the ...
1
vote
0
answers
81
views
Is there a unique tilted measure with specified marginals?
Suppose $\mathcal{A},\mathcal{B}$ are finite sets and $\mu_{A,B}(a,b)$ is a probability measure on the product set $\mathcal{A}\times \mathcal{B}$ so that $\mu_{A,B}(a,b)>0$ for each $a\in \mathcal{...
10
votes
2
answers
618
views
When does a cubic surface pass through five lines?
The set of 5-tuples of lines in $\mathbf{P}^3$ is parametrized by the 20-dimensional product of Grassmannians $G(2,4)^{\times 5}$. The set of cubic surfaces is parametrized by a 19-dimensional ...
9
votes
1
answer
1k
views
Obtaining non-normal varieties by pushout
In his answer to this MO question, Karl Schwede claimed that every non-normal variety can be obtained by an appropriate pushout diagram, as sketched in that answer. This would give substance to the ...
0
votes
1
answer
112
views
All $2$-designs arising from the action of the affine linear group on the field of prime order
Let $p$ be a prime and $\mathbb{Z}_p$ denote as usual the field of order $p$. Let $AL(p)$ be the affine linear group $\{x\mapsto ax+b \;|\; a\in \mathbb{Z}_p\setminus \{0\}, b\in\mathbb{Z}_p\}$. For a ...
1
vote
1
answer
242
views
Nonlinear system of equations whereas most of the equations are linear. How to minimise operation?
Let us say we have a n * n system of equations like KU=F where K is a n*n matrix and U and F are n*1 vectors. K and F are defined and the final goal is to find U values.
K is a sparse banded matrix ...
0
votes
0
answers
636
views
A question on tangent bundle (and second tangent bundle)
Let $M$ be a $n$ dimensional manifold and $p:TM\to M$ be the projection map. Then $\ker Dp$ is a $n$ dimensional vector bundle on $TM$, as a sub bundle of $TT(M)$.
For what type of manifolds, $...
0
votes
1
answer
101
views
Expectation of exp(-1/(ax^2)) when x is a standard normal variable and a>0 is a parameter [closed]
I would like to know if the mean value of $\exp(-1/(ax^2)) $ when $x \sim N(0,1)$ and $a>0$ is a parameter is known.
11
votes
0
answers
1k
views
Total spaces of tangent/cotangent bundles in a course where all varieties are quasi-projective
$\def\PP{\mathbb{P}}$In a course where all varieties are quasi-projective (as in Shafarevich Volume I), I am trying to figure out whether I can justify talking about the total spaces of the tangent ...
6
votes
1
answer
384
views
Reference request: Wasserstein metric spaces for non linear weights/mobility?
There is a very nice theory of gradient flows in metric spaces by Ambrosio, Gigli and Savaré. One particularly important application is the quadratic Wasserstein setting, where the metric space in ...
2
votes
1
answer
213
views
Tower-of-squares sequence divides linear recurrent A001921 sequence?
Let $(a_n)$ be the A001921 sequence
$$
a_0 = 0,\ a_1 = 7, \quad a_{n+2} = 14a_{n+1} - a_n + 6.
$$
Let $(b_k)$ be the (almost)"tower-of-squares" sequence defined by
$$
b_0=2, \quad b_{k+1}=2b_k^...
7
votes
1
answer
423
views
On a claim of Deligne about representations of Weil-Deligne groups
In Deligne's article "Les constantes des equations fonctionelles des fonctions L", we find the following claim:
Proposition 8.9 (ibid.): Suppose $(V, \rho, N )$ and $(V', \rho', N')$ ...
19
votes
0
answers
603
views
Coarse moduli spaces of stacks for which every atlas is a scheme
Let $X = [P/G]$ be a smooth finite type separated DM-stack over $\mathbb C$ given as the quotient of a smooth quasi-projective scheme $P$ by the action of a smooth (finite type separated) reductive ...
24
votes
1
answer
3k
views
When did "Betti cohomology" come to be used the way it is today? (and how is it used)
This is sort of a mixture of a math and history question.
First the math part: thinking about it, I do not actually know how to properly use the term "Betti cohomology". I know I should, but I don't....
1
vote
1
answer
335
views
Center of a compact group
Suppose $G$ is a compact Lie group, we know the center of $G$ is a compact Abelian subgroup, so it must be isomorphic to a direct product of finite abelian subgroup and a torus.
Now suppose the ...
13
votes
1
answer
375
views
Homotopy groups of $MO(2)$
Have there been any computations of the higher homotopy groups of $MO(2)$, the Thom space of the universal $O(2)$-bundle? Thom himself noted in his landmark 1954 paper that
$$
\pi_1(MO(2))=0,\quad \...
4
votes
1
answer
612
views
Are there sets which are computable in one model, but uncomputable in another?
Suppose we have two models of set theory, $U$ and $V$ which have the same $\Bbb N$. Is it possible that there is a set $A\subseteq\Bbb N$ such that, in $U$, this set is computable, i.e. there is a ...
6
votes
2
answers
282
views
Computing certain integrals over high-dimensional polyhedra
Let $\delta>0$ be a small real number and consider the $k$-dimensional region consisting of points for which
$$\delta\leq x_1\leq x_2\leq\ldots \leq x_k$$
and
$$x_1+\ldots+x_k\leq 1.$$
I am ...
3
votes
1
answer
101
views
Length preserving rewriting system with NP-complete $u\to v$ problem
My question is related to Computational complexity of the word problem for semi-Thue systems with certain restrictions.
Is there a finite length-preserving string rewriting system $R$ (over say $\{0,...
9
votes
2
answers
2k
views
Resolving ADE singularities by blowing up
Let's say we have a finite subgroup $\Gamma \subseteq SL(2,\mathbb{C})$ and consider the quotient variety $\mathbb{C}^2/\Gamma$, which will have one of the well-known ADE or du Val surface ...
6
votes
1
answer
623
views
Why does a flat degeneration induce equality in the K-Theory?
Let $Z_1$ and $Z_2$ be two closed subschemes of a smooth, complex, algebraic variety $X$. We will be interested in the Grothendieck ring of coherent sheaves of $X$, i.e. isomorphism classes of such ...
5
votes
1
answer
321
views
Largest area of a compactly supported positive definite function
Consider a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$, supported on $[-1,1]$, of positive type. Assume $f(0) = 1$; what is the "largest area" $\int f\,dx$ that can be achieved?
To be ...
5
votes
0
answers
188
views
Asymptotics of the quantum exponential
Let $\epsilon$ be an $N$th root of unity, and $q=\epsilon e^h$ where $h<0$.
I am trying to give a derivation of the lead term of
$$(z;q)_{\infty}=\prod_{n=1}^{\infty}(1-zq^n),$$ as $h\rightarrow 0^...
8
votes
2
answers
3k
views
The relation between group cohomology and the cohomology of the classifying space
We know that the Borel group cohomology (group cohomology of measurable functions) of a group $G$, ${\cal H}_B^d(G,Z)$, is given by the cohomology of the classifying space: ${\cal H}_B^d(G,Z)=H^d(BG,Z)...
2
votes
0
answers
375
views
Are there good properties of the divided power completion map?
Let $Y \to X$ be a closed immersion of smooth schemes over, say, the ${\rm Spec}(\mathbb{Z}_p)$. The completion map $$X_{/Y}\to X$$ is an ind-closed immersion (sometimes called pseudo-closed immersion)...