All Questions
152,885
questions
1
vote
2
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287
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Asymptotics on prime divisors
Somewhat inspired from the Zsigmondy's theorem is my question. Suppose Let $a_{1}> a_{2}> \cdots > a_{k}$ be nonzero integers, with $k \geq 2$. Let $a(n) : = a_{1}^{n}+\cdots+a_{k}^{n}$ for $...
3
votes
1
answer
186
views
Rational homogenous spaces and symmetric spaces
What are the complex rational homogenous spaces $G/P$ ($G$ a semi-simple complex Lie group, $P$ a parabolic subgroup) such that the set of real points $(G/P)(\mathbb R)$ is a (compact) riemannian ...
3
votes
0
answers
67
views
Polynomials connected with Gale's condition and cyclic polytopes
I was reading about cyclic polytopes and (I think naturally) became interested in polynomials $f(X) \in \mathbb{R}[X]$ which have a factorization of the form $f(X)=g(X)g(X+1)$ for some $g(X) \in \...
6
votes
5
answers
5k
views
What is an extragradient method?
I've searched Google, but it seems that only research journal papers appear in search results, where some new, improved, or specialized extragradient method is discussed. I've also searched Wikipedia ...
4
votes
2
answers
308
views
reference help indecomposable representations of SL(2,R)
Let $\mathfrak{g}$ be the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, $K=SO(2)$ the maximal compact subgroup of $SL_2(\mathbb{R})$. Then the classification of irreducible admissible $(\mathfrak{g},K)$-...
1
vote
0
answers
138
views
symmetric theta structures and arithmetic subgroups
A symmetric theta structure is a theta structure that commutes with (a lift of) the natural involution $\imath: A \to A$ an an abelian variety. For simplicity I will assume that $A$ is a surface.
Now,...
1
vote
2
answers
216
views
A special class of random variables
I'm interested in classes C of $R^1$-valued random variables which possess the following properties:
1) the sum of two independent random variables from class C belongs to class C;
2) for any $\...
2
votes
1
answer
254
views
How can I prove that the negative biased triangular kernel is positive semidefinite
How can I prove that the following triangular kernel function defined in $[0, 1] \subset R^1$
$k(x, x') = (1 - 2|x-x'|)$
is a positive semidefinite function?
It turns out to be psd function when ...
0
votes
0
answers
201
views
A certain Acyclic Partition of a digraph
Has the following object been defined in the literature? What is it called? And what literature studies it? Are there other characterizations of this? What properties are known?
Let $G$ be a directed ...
3
votes
0
answers
98
views
Assumption of equal prior message probabilities in the standard proofs of the converse of Shannon's theorem
One of the first steps in the standard proofs for the (weak) converse of the Shannon's theorem (a.k.a. noisy-channel coding theorem) for the discrete memoryless sources is the assumption that messages ...
6
votes
1
answer
974
views
What does the defect of a block measure?
In the context of decomposition matrices for Hecke algebras of finite Coxeter groups at a root of unity (such as the tables at the end of the book "Hecke algebras at a root of unity" by Geck-Jacon or ...
5
votes
1
answer
319
views
Classes in $H^3(G; \mathbb{Z})$ that restrict to zero on abelian subgroups
Let $G$ be a finite $p$-group. Is it possible to have a nonzero class in $H^3(G; \mathbb{Z})$ that restricts to zero in $H^3(A; \mathbb{Z})$ for every abelian subgroup $A \subset G$? If so, what is a ...
4
votes
2
answers
240
views
Invariant planes of a nilpotent matrix with two Jordan blocks of size two
Describe all the invariant 2-dimensional subspaces of $\mathbb{C}^4$ (or $\mathbb{R}^4$) of the nilpotent map
$$
N = \begin{pmatrix}
0 & 1 & & \\
0 & 0 & & \\
& & 0 &...
7
votes
2
answers
608
views
Twist in identification with singular cohomology
Let $X$ be a smooth projective variety over $\mathbb{Q}$ and $$V = H^m(X(\mathbb{C}), \mathbb{Q} \cdot (2\pi i)^r)$$ Then I've seen people write the comparison with complex cohomology (an isomorphism ...
1
vote
1
answer
335
views
Algorithm for Polynomial Reduction in a Quotient Ring
Any reference or suggestion for the following problem would be greatly appreciated.
I am working on the quotient ring $Q=R[X_1,\dots,X_n]/<f_1,\dots,f_k>$. Given polynomials $p$ and $q$ I want ...
5
votes
1
answer
238
views
When are the congruence lattices nicer?
This is a purely idle question, but one I'm increasingly interested the more thought I put into it:
For $\mathcal{A}$ a universal algebra (that is, nonempty set together with some named functions), a ...
-1
votes
1
answer
378
views
Solution to simple first-order partial differential equations [closed]
Is there a general solution for first-order partial differential equations of the form
$$m(x) \partial_x f(x,y) = n(y) \partial_y f(x,y)$$
for given $m(x),n(y)$ and reasonable boundary conditions (...
0
votes
0
answers
265
views
PBW proof proposal
One version of the PBW theorem states:
$\omega $:$\mathfrak {S} \mapsto \mathfrak {E} $ is an isomorphism of algebras.
I am curious if this is a possible proof for the PBW theorem, part is taken ...
3
votes
1
answer
476
views
Existence of finite nonabelian groups satisfying certain identities
Is there a finite nonabelian group satisfying all of the following identities?
$$
(x^py^p)^2 = (y^px^p)^2, \quad p = 2,3,5,7,11,\ldots (\text{primes})
$$
I thank you all in advance.
3
votes
1
answer
743
views
Lp estimate for resolvent of Laplace operator
Consider for $1<p<\infty$ operator $A_p:L_p(0,1)\to L_p(0,1), \ D(A_p)=\{u\in W^2_p(0,1): u'(0)=u'(1)=0\}, \ A_pu=u''$, i.e. $L_p$-realisation of the Laplace operator with Neumann bcd on the ...
4
votes
1
answer
667
views
$\mathcal{H}$-polyhedron under a linear map
Let $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$ be a (bounded) polyhedron for $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$, $n,m > 0$.
Moreover, let $M \colon \mathbb{R}^n \to \...
12
votes
1
answer
715
views
If the generating function summation and zeta regularized sum of a divergent series exist, do they always coincide?
One could assign a value to divergent series by means of several summation methods. One summation method we could consider is the generating function method. Let's sum, for example, the fibonacci ...
0
votes
0
answers
128
views
uniform continuity of a function in ultrametric spaces
Consider $[0,1]$ with the metric $d_1(x,y)=\left\{\begin{array}{cc}
0&x=y,\\
\max\{x,y\}&x\ne y.
\end{array}\right.$. Moreover let $(M,d_2)$ be an ultrametric space. Let
$f:(M,d_2)\to([0,1],...
12
votes
1
answer
503
views
Is a generic closed orientable hyperbolic 3-manifold Haken?
My question is as follows:
"Is a generic closed orientable hyperbolic 3-manifold Haken?"
Of course the word 'generic' can be interpreted in many ways, and the answer might depend on the way how one ...
1
vote
0
answers
178
views
Laplacian mapping on various function spaces
I have a question related to a certain elliptic operator on $R^N$ but I think i can clarify my confusion if I just consider the Laplacian $\Delta$ on the unit ball in $R^N$.
If $ 1 <p< \infty$...
4
votes
0
answers
172
views
Ultracoproducts of C(X)-algebras
Let $X$ be a metrizable compact topological space, let $\mathcal U$ be an ultrafilter, and denote by $X^{\mathcal U}$ the ultracopower of $X$ with respect to $\mathcal U$.
As a C$^*$-algebraist, I ...
4
votes
1
answer
374
views
A family Mersenne composite numbers?
I believe that the number
$$2^{2^{2t+1}+2t-1}-1$$
is composite for all positive integer $t$. I tested this for many $t$'s, but so far I didn't get a proof. Any idea?
4
votes
3
answers
750
views
Basic Questions about Radon Transforms
I am currently working on a problem that may be interpreted as recovering an unknown function from its Radon transform.
Unfortunately I don't have any background in Radon transform, but need to ...
5
votes
1
answer
346
views
what is the stabilization of pointed sets?
Given a(n $\infty$-)category, there is a process called "stabilitazion" which spits out a stable $\infty$-category (as one can read about in either Higher Algebra or the nlab).
The famous example is ...
8
votes
1
answer
374
views
Second homology of mapping class group of genus 3
In a survey paper of Korkmaz it is stated that $H_2(\mathrm{Mod}_3)$ is either $\Bbb Z$ or $\Bbb Z \oplus \Bbb Z_2$, but I was not able to find out a precise computation of this group (resolving the ...
2
votes
1
answer
141
views
Characteristic polynomials of reductive subgroup over C
Can any one provide a hint to prove the following statement? :
Let $H$ be a complex reductive subgroup (not necessarily connected) contained in $SO(n,\mathbb{C)})$. Consider the map $H \rightarrow ...
3
votes
0
answers
367
views
Metric on the set of subsets of the rational primes
Note: this is a revision of an earlier post. It was kindly pointed out that my initial proposed metric was in fact not a metric, so this is a revised version.
I was thinking how to say that two sets ...
10
votes
1
answer
385
views
Repetend digit graphs for $1/n$ in base $b$
Here is a decimal expansion of $\frac{1}{34}$:
$$(1/34)_{10}=0.02941176470588235\overline{2941176470588235}\ldots$$
And here is a graphical representation of the 16-digit
"repetend," as a directed ...
9
votes
2
answers
764
views
Rational points techniques on curves not using their Jacobian
Let $C/K$ be a curve of genus > 2 over a number field $K$ and suppose there exists a $p \in C(K)$. Then a recurring theme in studying $C(K)$ is using the map $C \to J(C)$ normalized by sending $p$ to ...
2
votes
0
answers
162
views
Irreducibility of $x^m-g(y)$
Let $g(y)\in \mathbb{C}[y]$, $ m\in \mathbb{Z}_{\ge 2}$. Are there some results on the irreducibility of $x^m-g(y)$ in $\mathbb{C}[x,y]$?
16
votes
1
answer
2k
views
Time in Girard's Geometry of Interaction
Jean-Yves Girard writes at the end of his paper
"Towards a Geometry of Interaction", page 105, that we have three intuitions about the nature of time:
time is logic modulo the order of rules,
time ...
5
votes
2
answers
532
views
Estimates on derivatives of Bessel function
In Duke, Friedlander and Iwaniec's Erratum on "Bounds for automorphic L–functions. II"
They have the following estimates for derivatives of Bessel functions: For $k \geq 2$
\begin{align}
& z^{l}J^{...
3
votes
2
answers
330
views
Real solutions for systems of monomial equations
I have a $K$ equations of the form $x_1^{a_{i1}} \cdots x_n^{a_{in}}=c_i$ where $a_{ij}$ are non-negative integer constants and $c_i$ are real constants -- i.e. each equation is a monomial in $n$ ...
2
votes
1
answer
93
views
Separating Differences of Open Sets
Has anyone ever considered something like the following separation axiom?
$(*)$ For any pair of open sets $O$ and $N$, there exist disjoint open sets containing $O\setminus N$ and $N\setminus O$.
...
2
votes
0
answers
213
views
Separating duality for TVS?
What is the modern concept (term) for "separating duality" (dualité séparante in french) in the sense of Bourbaki (TVS Ch II § 6) as explained in the following ?
...
17
votes
3
answers
2k
views
Finding integer points on elliptic curves via divisibility conditions like $(a+b)^2 \mid (2b^3+6ab^2-1)$
Is the following conjecture correct?
Conjecture. The divisibility condition $(\alpha+\beta)^2 \mid (2\beta^3+6\alpha\beta^2-1)$ has no solutions in positive integers $1 \le \beta < \alpha < 2\...
3
votes
0
answers
125
views
Non-linearly isomorphic non-equivalent $G-$representations?
Let $G$ be an algebraic group (or a group scheme) over a field $\Bbbk$, and let $V$ be an algebraic $G-$representation (I mean, corresponding to a homomorphism of $\Bbbk-$group schemes $G\rightarrow \...
3
votes
1
answer
376
views
reduction to np hard ordering problem
I am trying to show a reduction from a problem of ordering problem to an np-hard problem that has approximation poly-time algorithm.
My problem is: I have M auctions and in each auction I have N ...
10
votes
1
answer
638
views
What is the current state of the crystalline analogue of the Weil conjectures?
In "F-isocrystals on open varieties results and conjectures" Faltings says:
"Finally, we extend the theory of weights and show as much as possible of the crystalline analogue of the Weil ...
10
votes
1
answer
885
views
Morava $K(n)$'s are not $E_{\infty}$
I am looking for a reference/proof that shows that the Morava $K$-theory spectra, $K(n)$ are not $E_{\infty}$ ring spectra. I suspect that this should be a calculation but I can't quite get it right.
...
10
votes
2
answers
417
views
Slim Kurepa tree at a singular strong limit cardinal of uncountable cofinality
For a strong limit cardinal $\kappa$ the notion of $\kappa$-Kurepa tree is trivial: the full binary tree is a $\kappa$-Kurepa tree. Accordingly, we consider the following strengthening:
A slim $\...
3
votes
1
answer
489
views
Prove that these two definitions of "natural" integration constant coincide when both converge
These are two possible definitions of antiderivative (integral) incorporating a supposedly natural choice of an integration constant (see this question for further details).
The first one is based on ...
2
votes
0
answers
68
views
Any results on rayless simplicial complexes?
We define a closed ray in a topological space $X$ to be its closed subset homeomorphic to the real half-line $[0,\infty)\subseteq \mathbb{R}$. Call a topological space $X$ rayless if it does not ...
31
votes
2
answers
5k
views
A question about "Zariski dense" arguments
This question is a little basic, but I think it is consistent with the goals of MO.
My question is about a certain type of argument in algebraic geometry which exploits the abundance of dense sets ...
0
votes
1
answer
126
views
equation for geodesics of a right invairant Finsler metric on $SU(n)$ which are parallel to a linear affine distribution
I am looking for an equation analogous to the Euler-Poincare equations for a right invariant Finlser metric except I want the geodesics which are parallel to a linear affine distribution on $SU(n)$. ...