All Questions
153,397
questions
5
votes
2
answers
534
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^{...
- 507
3
votes
2
answers
332
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 $n$ ...
- 551
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 $N\setminus O$.
...
- 1,327
2
votes
0
answers
214
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 ?
...
- 3,696
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\...
- 1,025
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 \...
- 22.7k
3
votes
1
answer
378
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 ...
- 31
10
votes
1
answer
642
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 ...
- 535
10
votes
1
answer
890
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.
...
- 2,277
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 $\...
- 5,551
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 ...
- 9,312
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 ...
- 2,074
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 ...
- 28.8k
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 $SU(n)$. ...
- 2,069
2
votes
3
answers
505
views
What's the relation between fusion and coproduct?
For an irreducible finite depth finite index subfactor $(N \subset M)$, there is a structure of fusion category given by the even part of its principal graph. The simple objects $(X_i)_{i \in I}$ of ...
- 25.9k
5
votes
1
answer
1k
views
A example on Fourier tranform of a continous compactly supported function
I am trying to find a continuous compactly supported function $f$ such that the Fourier transform $f^{ft}$ and derivative $(f^{ft})'$ of the $f^{ft}$ decay faster than exponential rates, that is
$$|f^{...
- 71
1
vote
1
answer
180
views
The coproduct on the 2-boxes space of the group-subgroup subfactor planar algebras
Let $(H \subset G)$ be an inclusion of finite groups.
Let the subfactor $(\mathcal{R} \rtimes H \subset \mathcal{R} \rtimes G)$ with $\mathcal{R}$ the hyperfinite ${\rm II}_1$ factor, and its planar ...
- 25.9k
4
votes
1
answer
236
views
Are infinite groups "locally topologizable"?
Does every infinite group admit a Hausdorff topology such that the multiplication and inverse are continuous at $1$ but $1$ is not an isolated point?
The question is inspired by and related to this ...
- 3,912
0
votes
1
answer
144
views
$I$ is principal if and only if $ID_M$ is principal for all $M \in Max(D)$
In a semmi-quasi local domain $D$ ( i.e. $D$ has finitely many maximal ideals),
an ideal $I$ is principal if and only if $ID_M$ is principal for all $M \in Max(D)$.
[See Comm. Rings by Kaplansky, ...
- 41
-5
votes
1
answer
735
views
Why calculus textbooks do not include the natural integration constants in the tables of integrals? [closed]
The formulas for integrals in the textbooks usually define indefinite integral up to a constant term. Yet the natural integration constant for antiderivative can be fixed from the following formula ...
- 9,312
6
votes
1
answer
1k
views
Arthur's refinement of parameters for unitary automorphic representations
In his work on the classification of automorphic representations of a group $G$, Arthur has conjectured that the parameterization of such representations involves a homomorphism $\rho : SL_2 \times ...
- 1,063
10
votes
2
answers
484
views
Is there a hyperplane avoiding two independent sets?
Let $V$ be a vector space over a field with $5$ elements, $A,B \subseteq V$ independent subsets. Must there be a subspace of $V$ of codimension 1 disjoint from $A \cup B$?
- 11.2k
18
votes
1
answer
2k
views
Homology theory represented by Madsen-Tillmann spectra
The generalized homology theory of the Thom spectrum $MO=\varinjlim\Sigma^nMTO_n$ is bordism theory:\begin{equation*}\pi_k(MO\wedge X)=\Omega^O_k(X)\end{equation*}These groups form the ring of (...
- 1,189
8
votes
1
answer
472
views
Separation of lattice points on the Mordell elliptic curve
Consider the Mordell equation x^3 – y^2 = k, where x is a non-square positive integer and y^2 is the perfect square nearest to x^3. Noam Elkies (see http://www.math.harvard.edu/~elkies/hall.html) ...
- 81
4
votes
1
answer
892
views
Standard conjectures on positive characteristic
In this MO answer of M. Bondarko, he says:
"the Hodge conjecture implies all the Grothendieck's standard conjectures over base fields of characteristic 0..."
and in Remarks on Grothendieck's ...
- 535
2
votes
0
answers
143
views
target category of extended field theory
An A-S TFT is a functor from $\text{Bord}_{<n−1,n>}(\mathcal{F})$ to $\text{Vect}$ where $\mathcal{F}$ denotes a set of background fields, eg a spin structure. An extended theory is a functor ...
- 1,189
10
votes
1
answer
392
views
Game on the tree [closed]
There's a problem from programming competition which already finished:
http://codeforces.com/contest/458/problem/F
Two weeks already passed but still nobody solved it yet - in fact you can see here ...
- 109
2
votes
1
answer
127
views
When does a hypergraph represent maximal independent sets?
Let $G = (V,E)$ be a simple graph. Then, we can view the set of maximal independent sets (or the set of maximal cliques) as a hypergraph $H = (V, E')$. This is quite a useful device when connecting ...
6
votes
1
answer
642
views
Who defined and who coined "module"?
The title of my Q. says it all:
QUESTION: Who defined and who coined: module?
Would it be Emmy Noether?
EDIT In view of @anon's and KConrad's answers, and as it could have been ...
- 7,472
8
votes
2
answers
540
views
How do I find coefficients of a product expansion
Any power series $f(t) = 1 + t \mathbb{Z}[[t]]$ can be uniquely expanded in the following two ways:
$$1 + \sum_{i=1}^\infty f_i t^i =
\prod_{i=1}^\infty (1-t^i)^{-n_i}$$
Here, the $f_i$ and $n_i$ ...
- 8,643
19
votes
2
answers
976
views
Who originated the standard symbols for Lie groups GL, SL, SU, etc.?
Who was first to use symbols GL, SL, O, SO, U, SU, Sp and their projective versions, and how did this notation become standard?
The notation appears in fairly modern form in Weyl's "The Classical ...
- 3,741
1
vote
1
answer
238
views
A possible minimal aperiodic set of corner Wang Tile
From one of my previous question Aperiodic set of corner Wang Tile (although it is put on hold), I realize there is a systematic way to construct aperiodic corner type of Wang tile from edge type ...
- 867
4
votes
1
answer
274
views
Higher Degree Data in a Cosimplicial Quasicategory and Delooping
If there is a short answer to this question and someone can write it here that'd be wonderful, but if it's longer, I'm also perfectly happy with a reference.
My question is regarding accessing data ...
- 10.1k
4
votes
0
answers
367
views
Extension of the product formula for valuations to a simultaneous completion
It is well known that $\mathbb{C}$ and $\mathbb{C}_p$ are "algebraically" isomorphic (that is, ignoring the topology), but an isomorphism depends on the axiom of choice and there is no canonical way ...
- 1,269
7
votes
0
answers
265
views
Toda Flow Embeddings
What are strategies for generating the following types of pictures:
Here's what's going on here. Take a toda flow in 3 variables. The equations of motion are:
$$\frac{d}{dt}a_k=2(b_k^2-b_{k-1}^2),$$
...
- 4,902
1
vote
1
answer
158
views
For a defined set $M$ (see problem) do there exist $a,b$ natural numbers so that $a,ab+1 \in M$
Let $\rho \in \mathbb{R}\setminus \mathbb{Q}$ be a irrational nuber, and
let $\varepsilon>0$ be arbitrarily small. Define $M=\{m \in \mathbb{N}: \exists k \in \mathbb{N}\hbox{ so that} |\rho m -k |&...
- 67
37
votes
13
answers
4k
views
Continuous relations?
What might it mean for a relation $R\subset X\times Y$ to be continuous, where $X$ and $Y$ are topological spaces? In topology, category theory or in analysis? Is it possible, canonical, useful?
I ...
- 852
9
votes
2
answers
626
views
Behavior of the spectrum of the Laplacian under pointed smooth convergence
The Laplacian on a compact Riemannian manifold has a discrete spectrum. For example on a circle of perimeter $L$ the $n$-th eigenvalue starting at $0$ is $-\lambda_n = -(2\pi/L)^2 n^2$.
On the other ...
- 4,194
6
votes
5
answers
855
views
Approaching convex and discrete geometry from other disciplines
I would like to learn some convex and discrete geometry (number 52 in MSC2010). I thought that it would be interesting to approach it from some other parts of mathematics - either by learning ...
- 2,074
4
votes
1
answer
489
views
Nearby cycles and specialisation - properties
I am looking for reference for properties of nearby cycles - specifically, commutation with non-characteristic pull-back (good enough - commutation with pull-back to closed subvariety which is ...
- 5,492
5
votes
2
answers
2k
views
Plucker embedding and tautological/universal quotient bundle
Let $G$ be a Grassmannian and $Q$ the tautological/universal quotient bundle of $G$. As far as I understand, the associated tautological quotient line bundle for the Plucker embedding of the ...
- 503
2
votes
0
answers
281
views
A lifting problem
Let $E\overset{\pi'}{\longrightarrow} B'$ and $E\overset{\pi}{\longrightarrow} B$ be vector bundles.
For $i=0,1$, let $f_i$ be a fiber-preserving open embeddings of $\pi'$ into $\pi$, with $g_i$ the ...
- 21
15
votes
1
answer
615
views
Torsion-free group that is not of type F but is virtually of type F
Recall that a group $G$ is of type F if there exists a compact $K(G,1)$.
There are many examples of groups which are not of type F but which are virtually of type F, that is, they have finite-index ...
- 153
4
votes
1
answer
231
views
Reference for Elliptic PDE on $\mathbb{R}^d$
Could anyone suggest a textbook, article, or lecture notes that covers elliptic PDE theory (existence, uniqueness, regularity) on all of $\mathbb{R}^d$, as opposed to the Dirichlet or Neumann problem ...
- 379
4
votes
1
answer
453
views
Eigendecomposition of analytic Hermitian matrix-valued functions of several variables
If $A(t)$ is an analytic, Hermitian matrix-valued function of a real variable $t$, then it is known that there are analytic functions $\lambda_i(t)$ and $x_i(t)$ corresponding to the eigenvalues and ...
- 585
15
votes
4
answers
1k
views
Is the sequence of Apéry numbers a Stieltjes moment sequence?
Consider the sequence of Apéry numbers
$$
A_n = \sum_{k=0}^n \binom{n}{k}\binom{n+k}{k}\sum_{j=0}^k \binom{k}{j}^3
= \sum_{k=0}^n \binom{n}{k}^2\binom{n+k}{k}^2 .
$$
In an email, physicist Alan Sokal ...
- 40.2k
5
votes
1
answer
3k
views
Operator norm vs spectral radius for positive matrices
I believe the following statement should be true but somehow I don't see an argument:
For every integer $d>1$ there exists a constant $C=C(d)>1$ such that whenever $A$ is a $d \times d$ matrix ...
- 411
1
vote
1
answer
206
views
Semidirect products with braid groups and type $F_\infty$
Let $F$ be a group which is strongly type $F_\infty$ in the sense that every subgroup is of type $F_\infty$. Here, type $F_\infty$ means that the group admits a classifying space with compact skeleta.
...
- 1,540
4
votes
0
answers
304
views
Without Skolem–Mahler–Lech Theorem? [closed]
Using Skolem–Mahler–Lech theorem one can easily prove the $\displaystyle \lim_{n\to +\infty}\left|\Re\left(\frac{1+i\sqrt{7}}{2} \right)^n\right| =+\infty$.
Is there a "simple way" to prove this ...
- 41
1
vote
1
answer
71
views
Converting Dirichlet boundary conditions for E-L equations on a Lie group into an equivilent condition for EP equaiton
I need to find a specific geodesic of a right invariant Finsler geodesic on a Lie group ($SU(n)$) that connects $I$ to some desired $O$. These are Dirichlet boundary conditions for the E-L equations (...
- 2,069