All Questions
153,430
questions
4
votes
1
answer
185
views
Petersson-product of the cusp part of the theta series
Who can help me solving this problem: $Q:\mathbb{Z}^{2k}\to \mathbb{Z}$ is any positive definite integer -valued quadratic form in $2k$ variables, then it is well known, that the thetaseries $$\...
- 305
7
votes
1
answer
435
views
Is an infinite-dimensional "Lebesgue measure" uniquely determined by a set of positive finite measure?
Let $\mu$ be a probability measure on a subset $C \subset \mathbb{R}^\infty$ of the space of sequences, and assume, for simplicity, that $C$ is closed and convex.
We say that $\mu$ admits shifts if ...
- 3,970
1
vote
0
answers
70
views
Name for generalization of bivariate weighted-homogeneous polynomials
A polynomial $f = \sum_j c_j X^{\alpha_j}Y^{\beta_j}\in\mathbb K[X,Y]$ is said weighted-homogeneous if there exist $p$, $q$ and $d$ (where $p$ and $q$ are not both $0$) such that $p\alpha_j+q\beta_j=d$...
- 456
0
votes
1
answer
87
views
Intersection graphs
Does anybody know of a paper which proves that finding the maximum independent set in geometric intersection graphs is NP hard? Even general intersection graphs?
- 903
2
votes
0
answers
333
views
continuity with respect to weak-${\ast}$ topology
Let $V:=V([0,1],R)$ be the space of all cadlag functions defined on $[0,1]$ of bounded variation. Thus any element $v\in V$ determines a signed measure $\nu$ on $[0, 1]$ given by the formula $\nu([0, ...
- 1,837
20
votes
1
answer
1k
views
What makes the amenability of Thompsons group $F$ such a tricky problem?
An open problem that seems to get a lot of attention every once in a while is the amenability of Thompsons group $F$.
The problem seems to generate both proofs and disproofs at a fairly high rate, ...
- 2,488
3
votes
1
answer
932
views
A good reference for uniformization theorem for compact and non-compact Riemann surface
I am looking for a good reference for the uniformization theorem for Riemann surfaces, which states that each simply connected Riemannian surface is conformally equivalent to the complex plane $\...
- 1,861
8
votes
2
answers
2k
views
Applications of the Small and Great Theorems of Picard
I just presented the two famous theorems of Picard (Small and Great) in a graduate course, but I have not managed to discover a good number of interesting applications.
List of applications (rather ...
- 2,873
0
votes
1
answer
306
views
Is the kernel of a map between finite dimensional vector bundles still of finite type?
I'm not sure whether the level of this question is suitable for Mathoverflow.
Let $M$ be a smooth manifold, $E$ and $F$ are finite dimensional (smooth) vector bundles on $M$. Let $\phi: E\rightarrow ...
- 8,707
1
vote
0
answers
193
views
exponential and anisotropic torus
Let $F$ be a local p-adic field and $G$ a semisimple simply connected group over $F$, $\mathfrak{g}$ its Lie algebra. Let $T$ a maximal anisotropic torus of $G$, split over an etale extension of $F$ ...
- 3,432
2
votes
1
answer
85
views
Constructivity of zeros demanded by topological degree
Let $f : S^{n - 1} \to S^{n - 1}$ be a smooth map from the unit vectors of $\mathbb{R}^n$ to themselves. If $f$ has nonzero degree, then we know that any smooth map $g : D^n \to \mathbb{R}^n$ ...
- 5,652
3
votes
1
answer
148
views
Characterizing space that preserves positive-definiteness property
Given a symmetric positive-definite matrix $\Sigma$, consider the space $\mathcal{D}$ of diagonal matrices such that $\forall D\in\mathcal{D}$, the matrix $\Sigma-D\Sigma^{-1}D$ is positive definite. ...
- 133
28
votes
11
answers
2k
views
Combinatorial databases
At one point, I remember being excited by seeing the website Encyclopedia of Combinatorial Structures as an extension of Sloane's Online Integer Sequence Database site. Unfortunately, the site (ECS) ...
2
votes
0
answers
169
views
Examples of symplectic manifolds which are twisted $T^n$ bundles over $T^n$
I'm looking for certain (higher-dimensional) analogues of the Kodaira-Thurston manifolds, i.e. I want to know whether in $\dim_\mathbb{R}X\geq6$ we have examples of symplectic manifolds satisfying the ...
- 3,157
7
votes
0
answers
170
views
Checking if a multiplication table represents a group [duplicate]
Given an $n \times n$ multiplication table,
can one check if it represents a group in $o(n^3)$ time?
All properties can be checked by mindless try-all possibilities loops:
Whether there is an ...
- 149k
4
votes
1
answer
215
views
Noncommutative version of Littlewood's First Principle
There are definitely noncommutative analogues for Lusin's theorem and Egoroff's theorem (found in Blackadar for example). I'm curious if there is a version of the first principle:
Every Lebesgue
...
- 165
11
votes
2
answers
567
views
Orthogonal polynomial under linear transformation
Let $M_n(x) = x^n$ be the standard monomials. The binomial formula allows one to expand $M_n(ax+b)$ as a linear combination of $M_k(x)$, for $k \leq n$, giving
$$
M_n(ax+b) = (ax+b)^n = \sum_{k=0}^n \...
- 141
1
vote
1
answer
57
views
$P_{x}(T_{A}<\infty)<P_{x}(T_{B}<\infty)$ imply $Cap_{N}(A)<Cap_{N}(B)$, where $Cap_{N}$ is Newtonian capacity
We start a Brownian motion at $x\in [B(0,r)]^{c}$, where $B(0,r)$ is a large enough ball that contains compact sets $A$ and $B$. In other words, the B.M. starts on the exterior of $A$ and $B$.
Then ...
- 651
4
votes
1
answer
423
views
At what level of the analytic hierarchy do Cohen reals lie?
In his doctoral thesis titled "Three models of ordinal computability", Benjamin Seyfferth proved the following theorems:
i) A set $\mathtt A$ of reals is Ordinal Turing Machine-enumerable if and only ...
- 6,024
2
votes
0
answers
156
views
Relationship between tangent spaces and tangent categories for smooth topoi
Let $\mathscr E$ be a smooth topos, where I am using the terminology of nLab. In particular that imples the existence of a line object $R$ and an "infinitesimally thickened point", which is an object ...
- 1,991
2
votes
1
answer
307
views
Bound for sums of bounded multiplicative functions that are zero at primes
Let $h:\mathbb{N}\rightarrow\mathbb{C}$ be a bounded multiplicative function with $h(p)=0$. The motivation for this question is just a general enquiry and, since I suppose it has already been ...
- 2,470
0
votes
2
answers
214
views
Newtonian capacity of sphere equals its hitting probability by Brownian motion?
Do we have $Cap_{N}(S(0,r))=P_{x}(T_{S(0,r)}<\infty)$ for $x\in [B(0,r)]^{c}$, where $B(0,r)$ is a ball centered at the origin ?
I know for $x=0$, they are both equal to 1. How can I go about ...
- 651
-4
votes
2
answers
283
views
Does the Laplacian commutes with the indicator function [closed]
We define the laplacian operator $\Delta$ with the Neumann boundary conditions on the space $H^2(\Omega)$, where $\Omega$ is an open set of $\mathbb{R}^n$ with a smooth boundary $\partial\Omega$, and ...
4
votes
1
answer
135
views
Counting a Modified Class of Standard Young Tableau
Let $\lambda=(\lambda_1,\cdots,\lambda_n)$ be a partition, with $|\lambda|:=N$. Attach an extra box to $\lambda$ to the right end of the $r$'th row. In coordinate form, the last box on row $r$ has ...
- 4,902
0
votes
1
answer
168
views
Integer-valuedness of a polynomial determined by output of first n integers? [closed]
An integer-valued polynomial is a polynomial $p(x)$ such that $\forall x \in \mathbb{Z}, p(x) \in \mathbb{Z}$.
Theorem: For any $n$-degree polynomial $p$, if $p(x) \in \mathbb{Z}$ for all $x \in \{0, ...
- 111
0
votes
0
answers
171
views
The Diophantine equation $x^2 + bxy + cy^2 = p^z_1 \cdots p^{z_k}$
Let $b,c \in \mathbb{Z}$ and let $p_1,\ldots,p_k$ be given primes. Is there an effective algorithm to find all the solutions of the Diophantine equation $$x^2 + bxy + cy^2 = p_1^{z_1} \cdots p_k^{z_k}$...
- 11
1
vote
0
answers
83
views
Projection from a polytope to an affine space
Let $P\subseteq \mathbf{R}^n$ be some polytope defined by an intersection of half spaces with corresponding hyperplanes $H_k$, and let $A\subseteq \mathbf{R}^n$
be some affine space, with $A\cap P \...
- 445
6
votes
1
answer
439
views
Framed version of braided monoidal category
The operad $\mathcal{D}_2$ of little $2$-disks is an operad whose $n$-th space is a $K(PB_n,1)$ where $PB_n$ denotes the pure braid group on $n$-strands. Algebras over $\mathcal{D}_2$ have a ...
- 2,640
2
votes
1
answer
767
views
A question about Skorokhod metric
I have a question related to the Skorokhod distance.
Let $\Omega:=D([0,1],R)$ be the space of cadlag functions $x$ defined on $[0,1]$. Let $\Lambda$ be the collection of non-decreasing continuous ...
- 1,837
6
votes
1
answer
1k
views
Jackson's theorem for partial sum of Fourier series
There is a classical theorem of Jackson stating that the $N$-th partial sum $S_N f$ of the Fourier series of a Lipschitz continuous function $f$ (which is periodic with period 1) satisfies
$$
|f(x) - ...
- 2,199
3
votes
0
answers
334
views
What is the right categorical framework for diagonal approximations, cup and cap products and identities between them?
I wanted to know what is the right categorical analog for the diagonal approximation for the singular chain complex $\Delta: C\rightarrow C\otimes C$ as a morphism in the homotopy category of chain ...
- 621
22
votes
3
answers
1k
views
Two (other) rings...are they isomorphic?
Consider the local rings
$$R = \mathbb{C}[[x,y,z,w]]/\langle xyz+xyw+xzw+yzw\rangle$$
and
$$S = \mathbb{C}[[x,y,z,w]]/\langle xyz+xyw+xzw+yzw+xyzw\rangle.$$
Is $R$ isomorphic to $S$?
Some context:...
- 2,527
10
votes
1
answer
1k
views
Białynicki-Birula theory for non-complete varieties
I would like to know to which extent the theory developed for smooth projective varieties in the following articles
A. Białynicki-Birula, Some theorems on actions of algebraic groups.
Ann. of ...
- 22.7k
1
vote
0
answers
171
views
Does some square of the first Chern class preserved by conifold transition?
Let $X$ be a smooth projective 3-fold or a symplectic 6-manifold.
Suppose $Y$ is a conifold transition on a single nullhomologous
Lagrangian sphere $S^{3}$ in $X$. Then there is a exact sequence $0\to
...
- 153
2
votes
1
answer
267
views
Smooth curves in a Frechet space
Is the space $C^{\infty}([0,1];C^{\infty}(S^1))$ equal with the space $C^{\infty}([0,1]\times S^1)$ ? I am interested in characterizing the smooth curves in the space $C^{\infty}(S^1)$ where $S^1$ is ...
- 21
1
vote
0
answers
43
views
countably-infinite-index subgroup of a strongly complete profinite group
If $H$ is a strongly complete profinite group and $K$ is a dense countably-infinite-index subgroup, then I'm assuming a proper finite-index subgroup of $K$ could still be dense in $H$. Is there any ...
- 2,005
2
votes
1
answer
451
views
How does Azuma's Inequality result from Pinelis Inequality?
According to [1]
Let $(\mathcal{X},||\cdot||)$ be a separable Banach space and let
$S(\mathcal{X})$ denote the class of all sequences
$f=(f_j)=(f_0,f_1,...)$ of Bochner-integrable random ...
- 619
6
votes
1
answer
329
views
Realization of second Stiefel-Whitney class
I hope this is not too trivial.
Let $M$ be a compact oriented manifold and $x\in H^2(M, \mathbb{Z}/2\mathbb{Z})$. My question is that if there exists a real oriented vector bundle $V$ over $M$ such ...
- 61
2
votes
0
answers
116
views
request for any expository works in pointwise convergence of double Fourier series and especially a paper by Hardy
Quart. J. Math. Volume 37, Issue 1, Pages 53-79
On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters.
Hardy, G.H.
I am not ...
- 714
2
votes
1
answer
134
views
Skorokhod distance between $\omega, \omega\circ f_{\varepsilon}$ and $\omega, \omega\circ b_{\varepsilon}$
Let $\Omega:=D([0,1],R)$ be the space of cadlag functions $x$ defined on $[0,1]$. Let $\rho$ be the Skorokhod metric on $\Omega$, see e.g.
http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g
Now define ...
- 1,837
13
votes
1
answer
748
views
Maximum number of vectors in a hypercube satisfying given conditions
$\mathcal{C}$ is a collection of binary vectors of length $n$, i.e. $\mathcal{C}\subseteq\{0,1\}^n$. For arbitrary $x,y,z\in\mathcal{C}$ and $x\neq z$, $y\neq z$, there always holds that the Euclidean ...
- 147
5
votes
1
answer
424
views
Book about the history of mathematics for weather prediction
Can someone recommend a book about the history of mathematics being used for weather prediction, preferable one which covers recent developments?
- 1,161
2
votes
0
answers
98
views
Finding a general form of the density function when we have a four dimensional random variable
Consider a subject having time of the specific event $T_i$, which is a single sample from a
distribution $F_i$ with density $f_i$ and support
$[t_{\min},t_{\max}]$, for $i= 1,\ldots,n$. Let these ...
- 35
2
votes
2
answers
365
views
Speed and absence of non-constant bounded harmonic functions
For a (symmetric) random walks on countable groups generated by $\mu$, there is a "brute-force computation" argument of Avez (1974) that shows that if the entropy $h_\mu$ is trivial then there are no ...
- 4,342
1
vote
0
answers
331
views
A hard combinatorial identity
I asked this questions on the https://math.stackexchange.com/questions/1016713/ But I don't get answer.
I try to prove the following hypothesis
$$\sum_{i=0}^{min\{k, n-1\}}(-1)^i { n+i-1 \choose i}{{n+...
- 111
9
votes
3
answers
2k
views
Gerbes and Stacks
The definition of a gerbe on a smooth manifold that I know is that - after fixing an open cover $U_i$, a gerbe consists of the data of line bundles $L_{ij}$ on two-fold-intersections $U_{ij}$, ...
- 9,591
3
votes
1
answer
163
views
Strong and weak equivalence of C$^∗$-extensions by compacts
Let $A$ be a $C^*$-algebra. An extension of $A$ by the compact operators $K$ is an embedding $\epsilon$ of $A$ into the Calkin algebra $B(H)/K$.
Two embeddings $\epsilon_1$ and $\epsilon_2$ are ...
- 263
0
votes
0
answers
134
views
Degree of Map between Pseudomanifold
There are two different ways to define a degree of map.
Let $M$, $N$ be smooth, and $M$ is compact, $N$ is connected. If $f\in C(M,N)$, we define $\deg f$ by smooth approximation. [Milnor/Topology ...
- 424
5
votes
1
answer
554
views
Does the weak Hadwiger conjecture imply the Hadwiger conjecture?
For any cardinal $\kappa$, let $K_\kappa$ denote the complete graph on $\kappa$. We consider the following statements:
(H) If $G$ is a graph and $\chi(G) = \kappa$ then $K_\kappa$ is a minor of $G$.
...
- 45.5k
7
votes
2
answers
738
views
Are all unstable homotopy groups of $U(n)$ torsion?
The first few unstable homotopy groups of the unitary groups $U(n)$ were calculated by Borel-Hirzebruch, Toda, and Kervaire, and they are all torsion. There is a paper by Matsunaga (details below) in ...
- 8,508