All Questions
152,874
questions
10
votes
2
answers
707
views
The height of an orbit under rational self-maps
I have this basic question on which, strangely enough, the algebraic dynamics literature appears to be silent. But the question does not appear to be totally trivial or uninteresting to me - am I ...
3
votes
0
answers
313
views
Is there a way to metricize the notion of $C^\infty$ convergence of pointed Riemannian manifolds?
A sequence of pointed Riemannian manifolds $(M_n,p_n,g_n)$ is said to converge $C^\infty$ to pointed Riemannian manifold $(M,p,g)$ if for each positive radius $R$ there exists sequence of embeddings $...
4
votes
0
answers
198
views
Are these groups isomorphic (Cancellation in torsionfree, virtually Abelian groups)
I wondered whether it is possible to find two finitely generated, virtually Abelian, torsionfree groups $G,H$ that are not isomorphic but that become isomorphic after crossing with $\mathbb{Z}$. I ...
8
votes
1
answer
327
views
Smoothing of piecewise Euclidean Riemannian metrics
Let $M$ be a smooth closed manifold and $T$ be a triangulation of $M$. Endow each simplex of $T$ with the Euclidean metric making it a regular simplex; this gives a piecewise Euclidean metric $g_0$ on ...
0
votes
0
answers
91
views
Class of integrable 0/1-functions "with no null sets."
I am looking for (the name of) a class of functions from $\mathbf{R}^2$ $(\mathbf{R}^n)$ to {0, 1} that are integrable.
Let $f$ be in this class and $E$ be the set of all points where $f$ is equal to ...
2
votes
1
answer
272
views
Abelian subgroup of PSL(3,q)
Hello
I know that there are abelian subgroups of order (q^2-1)/d (cyclic group) and (q-1)^2/d and q^2(elementary abelian) and (q-1)q/d in the group PSL(3,q), where d=(3,q-1).
Also I saw in a text ...
2
votes
1
answer
579
views
Picture of a 3 dimensional amoeba.
On Wikipedia there some pictures of two dimensional amoebas (Thanks to Oleg Alexandrov for the pictures and the Matlab code he gives to build them). I was wondering if somewhere there are pictures ...
1
vote
0
answers
88
views
constant field of a divisor
Let $D$ be a divisor on a smooth, projective algebraic variety $X$ of dimension n over a field $k$ and let $D_j$ be an irreducible component of $D$.
I came across the expression the constant field ...
3
votes
2
answers
567
views
What logic is modelled by generalized boolean algebra?
While it is well known that classical propositional logic is modelled by booleaan algebras, I have never heard of the logic modelled by generalized boolean algebras (GBA) defined by M.Stone (author of ...
23
votes
6
answers
10k
views
Metrization of weak convergence of signed measures
Edit: Changed from "Hausdorff" to "metric" spaces.
Let $\mathcal{M}(\Omega)$ denote the space of signed regular Borel measures on a compact metric space $\Omega$. By Riesz-Markov, ...
3
votes
1
answer
120
views
Second difference
Is there an elementary example of a function f, such that $|f(x+t)+f(x-t)-2f(x)|/|t|^a\le C$, where $a>1$, such that $f$ is not $C^1$?
6
votes
3
answers
2k
views
What is the Weitzenböck formula for the $\bar\partial$-Laplacian
It is well-known that the Weitzenböck formula for the real Laplacian is
$$\frac12 Δ|∇f|2=|Hessf|2+⟨∇f,∇Δf⟩+Ricci(∇f,∇f)$$
where $Hess$ denotes the Hessian tensor of $f$. and $\nabla f$ denotes the ...
6
votes
4
answers
2k
views
Probability that randomly chosen integers from a restricted set of natural numbers are coprime
We know that the probability $P(k)$ of $k$ randomly chosen integers $(k \ge 2)$ from the set of natural number are coprime is
$$
P(k) = \frac{1}{\zeta(k)}.
$$
I am looking at a special case of ...
7
votes
2
answers
946
views
Another proof of the bidisc and the ball are biholomorphically inequivalent?
Does this outline of a proof work?
Consider the ball and the bidisc in $\mathbb{C}^2$. Give each space its Bergman metric. To show that the ball and the bidisc are not holomorphic, it is enough to ...
4
votes
0
answers
285
views
Translation of an article by Wolfgang Schmidt on normality for real numbers in different bases.
I would greatly appreciate a pointer to a translation from German into English of the article by Wolfgang Schmidt, Über die Normalität von Zahlen zu verschiedenen Basen, from Acta Arithmetica VII, ...
4
votes
1
answer
280
views
Is there a good way to estimate the Fourier transform of $\frac{1}{\lambda-iP(\xi)}$
Assume that P is a real valued strong elliptic polynomial, then what do we know about the following
$$
K(\lambda,x)=\int{\frac{e^{ix\xi}}{\lambda-iP(\xi)}}d\xi,\quad \lambda\in \mathbb{R}\0
$$
The ...
1
vote
0
answers
243
views
Factorization of permutations.
Let $n,k$ be positive integers such that $3n=2k$ and $N = \lfloor \alpha n\rfloor$ for some constant $0<\alpha<1$. Let $S_{3n}$ denote the permutation group of order $3n$. Consider the following ...
4
votes
1
answer
991
views
Formal criterion of flatness
Let $k$ be a field, $S$ and $R$ be local $k$-algebras with residue field $k$ and $\phi:S\to R$ be a local homomorphism. Then $\phi$ induces (obviously) a natural transformation of "functors of points" ...
2
votes
2
answers
519
views
When does a G-invariant one to one map between two closed algebraic G-set descend to a one to one map on the G.I.T quotient ?
I do not know much about Geometric Invariant Theory. My question is the following:
Let $X$ and $Y$ be two complex affine or projective varieties. Let $G$ be a reductive group which acts on both $X$ ...
0
votes
1
answer
195
views
lattice of subalgebras of a finite commutative algebra
(I) Suppose A is a finite commutative local algebra. Must every lattice of local subalgebras of A be a distributive lattice ?
By a subalgebra of A we mean an algebra contained in A that shares the ...
1
vote
0
answers
206
views
convergence of sets and limit of an integral
Let $X\subset\mathbb{R}$ and $Y\subset\mathbb{R}$ be compact sets.
Let $f:X\times Y\rightarrow\mathbb{R}$ be a $C^{1}$ function.
Let $s:Y\rightarrow X$ be a function (not necessarily continuous).
...
5
votes
0
answers
280
views
characteristic polynomial of an endomorphism of on $\mathcal{O}_X$-module
I've seen several times people looking at the characteristic polynomial of some
$f \in \mathrm{End}_{\mathcal{O}_X}(\mathcal{E})$
where $\mathcal{E}$ is an $\mathcal{O}_X$-module on let's say some ...
5
votes
2
answers
1k
views
A question on the Picard group
Let $X$ be a simply connected smooth projective variety, whose Picard group is generated by the classes of the irreducible codimension 1 loci $D_1, \ldots, D_k$. Let $E_1, \ldots, E_r$ be other ...
12
votes
1
answer
1k
views
Is an ultrafinitist Hilbert's program doomed?
Hilbert's program is popularly understood as an attempt to justify infinitary mathematics with a finitary consistency proof. Godel's Second Theorem is usually considered as showing this is not ...
2
votes
1
answer
373
views
Zero and Negative Gromov-Witten invariants in genus 0
I'm working on a project and I've used the Picard-Fuchs equation at a maximally unipotent monodromy point for a certain 1-dimensional family of Calabi-Yau 3-folds to calculate the A-model Yukawa ...
1
vote
1
answer
204
views
Growth constant limit for sum of products of two binomial coefficients
For the integer sequence {1,13,314,9368,312411,11163022, ...}, each term is given by the function
$f(n)=\sum_{k=0}^n{1\over2n+3k-1}{2n+3k-1\choose k}{6n-6k-3\choose2n-2k-2}$. Is there a method to ...
25
votes
3
answers
3k
views
Research trends in geometry of numbers?
Geometry of numbers was initiated by Hermann Minkowski roughly a hundred years ago. At its heart is the relation between lattices (the group, not the poset) and convex bodies. One of its fundamental ...
4
votes
0
answers
345
views
cech cohomology in topos
Hi,
The following result seems to be well known, but I can't come up with a proof.
Suppose that $C$ is a topos and that $F\to G$ is an effective epimorphism in $C$. If $P$ is
any abelian sheaf on $C$...
1
vote
0
answers
121
views
Is it possible to use BCH formula on Riemannian exp, or just for a 'good' approximation?
Given 3 points A, B, C on a Riemannian manifold, if we already have the geodesics between A->B and A->C (fully determined by their correspondent initial directions v0(A->B) and v0(A->C)), is it ...
1
vote
0
answers
335
views
Elliptic curves with almost prime conductor
Let $E/\mathbb{Q}$ be an elliptic curve having a rational 3-torsion point. Then $E$ can be given an affine equation of the type $$y^{2} = x^{3} + (ax + b)^{2}$$ for $a, b, D \in \mathbb{Q}$. Has there ...
1
vote
2
answers
468
views
When is PSU(2,q^2) = PSL(2,q) ?
The context for this question is from page 284 - 287 of Berger's paper: http://pdn.sciencedirect.com/science?_ob=MiamiImageURL&_cid=272332&_user=209810&_pii=S0021869398976785&_check=y&...
11
votes
2
answers
3k
views
Good examples of random variables whose image is not a measurable set?
Are their simple/natural examples of real-valued Borel-measurable random variables whose image is not a Borel set? Something that occurs "naturally"?
I am teaching Doob's lemma (for two real-valued ...
1
vote
0
answers
391
views
Interesting examples of minimal action on torus
Edit 1:This is a cross post on MSE. See math.stackexchange.com/q/289595/12952
Edit 2:I originally asked for finite group actions as I thought that will be easier. But as pointed out by Victor minimal ...
13
votes
2
answers
1k
views
Is the derived category of abelian groups a subcategory of the stable homotopy category?
An extension of the Dold-Kan equivalence gives an adjunction between the stable homotopy category and the (unbounded) derived category of abelian groups $SH \rightleftarrows D(Ab)$.
Question 1: Is ...
8
votes
2
answers
2k
views
Pair correlation for the Riemann zeros and $(\zeta^\prime(s)/\zeta(s))^\prime$
Added Background: The pair correlation of the zeros of the Riemann zeta function is influenced by the the derivative of the logarithmic derivative $(\zeta^\prime(s)/\zeta(s))^\prime$; see for example ...
1
vote
0
answers
117
views
Is there a standard name for functions of the form $x^\alpha p(x)$, where $p(x)$ is a polynomial?
Is there any existing standard terminology for functions of the form $x^\alpha p(x)$, where $p(x)$ is a polynomial and $\alpha$ is e.g. a complex number? I haven't been able to come up with any good ...
1
vote
1
answer
237
views
Analytic curve on Riemann surface
Suppose there is a closed analytic curve $C$ on a Riemann surface $S$, that is the image of a map $\gamma$ from the equator $E$ of the Riemann sphere to the surface $S$ which is a restriction of a ...
1
vote
0
answers
645
views
Prime numbers characterization
When, in one endeavour, I investigated prime numbers, I came up with a formula that characterizes primes, and the job was done in essentially this way:
First i defined a function $sr$ (sum of ...
0
votes
1
answer
661
views
asymptotic or approximate formula for a combination expression
Let 0<=p<=1, I want the value of q1 and q2 where
$q1=\sum_{k=0}^n [C(n,k)p^k(1-p)^{n-k}*\sum_{i=0}^{k-1} C(m,i)p^i(1-p)^{m-i}]$
$q2=\sum_{k=0}^n [C(n,k)p^k(1-p)^{n-k}*\sum_{i=k}^{m} C(m,i)p^...
7
votes
1
answer
422
views
Constructing expanders in Z/pZ
Fix a positive integer $k>0$. For $p>k$ a prime, let $A_p$ be a subset of the finite field $\mathbb{Z}/p\mathbb{Z}$ of size $k$ which contains a primitive element.
Define $G_p$ to be the (di)...
4
votes
0
answers
188
views
Is there a flow criterion for a graph being planar?
Given the set od propagators (say momenta flows) on a Feynman diagram (flow network), I would like to decide whether this diagram is planar or not.
I know that non-planar diagrams manifest different "...
2
votes
2
answers
624
views
Proof of Arnold Conjecture for monotone symplectic manifolds
I have a question about the proof of the Arnold conjecture for monotone symplectic manifolds as it is explained in http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf: Namely the author on page 32 ...
2
votes
2
answers
392
views
Mersenne primes problem
Well, I need someone here with programming skills (because I have none of it) to check if this problem that I am proposing is at least true for the known Mersenne primes, and here is the list of the ...
8
votes
1
answer
760
views
Question about local description of the branch locus
Let $\pi:Y\to X$ be a dominant, finite morphism of nonsingular varieties over an algebraically closed field $\Bbbk$. Assume furthermore that for all $Q\in Y$, with $P=\pi(Q)$, we have
$$\mathcal O_{Y,...
3
votes
1
answer
439
views
euler characteristic + normal crossings divisor
Let $D$ be a normal crossing divisors on a projective smooth algebraic variety $X$ over a field $k$ of characteristic zero. Put $n=\dim X$ and denote by $D_i$ the irreducible components of $D$.
I'm ...
3
votes
1
answer
285
views
Generator of a generated $C_0$ semigroup
Consider a $C_0$-semigroup $S_t:\mathscr{B(H)} \to \mathscr{B(H)}$ with generator $U$. Now define $P_t:\mathscr{B_1(H)} \to \mathscr{B_1(H)}$ where $P_t(\rho)=S_t\rho S_t^*$. How can I prove $P_t$ ...
7
votes
1
answer
364
views
Adams-Novikov spectral sequence at p = 2
Does anyone know of any computer calculations of the E2-term of the Adams-Novikov spectral sequence at p=2?
I'd love to get my hands on this data.
10
votes
1
answer
928
views
Factoriality vs $\mathbf{Q}$-factoriality for threefolds hypersurfaces with isolated singularities
Let $X \subset \mathbf{P}^4$ be a complex threefold hypersurface with isolated singularities.
We denote as usual by $\textrm{Cl}(X)$ the group of Weil divisors modulo linear equivalence and by $\...
5
votes
3
answers
978
views
Selecting $k$ integers from an interval $[0, N]$ to maximize the minimum difference between pairwise sums
I have an optimization problem where I need to select $k$ integers over the interval $[0, N]$ s.t. I maximize the minimum difference between any pairwise sum of the $k$ integers (where we also include ...
1
vote
2
answers
1k
views
What are the symmetric and anti-symmetric representations of $6\times6$ of $SU(6)$ in $SU(3)\times SU(2)$?
A 6-dimensional ( fundamental) representation of $SU(6)$ becomes (3,2) representation in $SU(3)\times SU(2)$. We can decompose $6\times 6$ of $SU(6)$ into 21-dimensional symmetric and 15-dimensional ...