All Questions
152,881
questions
10
votes
4
answers
653
views
For which kinds of group $G$, can we identify a square element efficiently?
For a group $(G,\star)$, an element $x\in G$ is said to be square if there is $y\in G$ such that $x=y\star y$.
My question is: For which kinds of group $G$, can we decide whether $x\in G$ is a ...
1
vote
1
answer
941
views
Algorithm to find a $k$-partite graph
Is there any algorithm which finds any $k$-partite graph of a given graph which is known to be a $k$-partite graph?
For example, you are given a graph $G$ with vertices $V$ and edges $E$, and you ...
4
votes
3
answers
461
views
Deriving the functor $ \int_{\Gamma} F(-,-)$
Suppose that $C$, $D$, and $E$ are combinatorial model categories, so that for any category $\Gamma$, the functor categories $C^{\Gamma}$, $D^{\Gamma}$, and $E^{\Gamma}$ have both the projective and ...
1
vote
0
answers
126
views
Which object related to families of algebraic varieties over a scheme $ S $ corresponds to the tensor product of vector bundles?
I asked yesterday on math.stackexchange.com that if the fiber product of two vector bundles seen in general as the fiber product of two families of special algebraic varieties over a scheme $ S $ ...
5
votes
0
answers
99
views
Finitely generated submodules of projectives lie inside f. g. projectives?
Let $R$ be a (not necessarily commutative) ring.
If $M$ is a finitely generated submodule of a projective module $P$, is there a finitely generated projective submodule $P'$ such that
$M \subseteq P'...
8
votes
1
answer
398
views
Homeomorphism/ homotopy types of non-negatively curved manifolds
A (special case of a) theorem of Gromov says for any $n\in \mathbb{N}$ there exists a constant $C(n)$ such that for any smooth connected closed $n$-dimensional Riemannian manifold with non-negative ...
6
votes
1
answer
246
views
Triangulations of convex surfaces
Let $M$ be a smooth closed positively curved surface in Euclidean 3-space, $T$ be a geodesic triangulation of $M$, and $E$ be the edge graph of the convex hull of vertices of $T$.
It is easy to see ...
15
votes
2
answers
1k
views
Does foundation/regularity have any categorical/structural consequences, in ZF?
(Prompted by reflection on this old answer, and its suggestion of the “harmlessness” of the axiom of regularity.)
In ZFC, one may justify the axiom of foundation (AF, aka the axiom of regularity) as ...
2
votes
0
answers
54
views
Symmetrized tensors of Lie algebras
Is there a simple formula (or maybe a table) to decompose the vector space $V^{\otimes{n}}$ of a (semisimple) Lie algebra with respect to the irreps of the symmetric group $S^n$? Example (I use ...
0
votes
1
answer
3k
views
Summation of $\log n/ \log(\log n)$
Given $h>0$, I would like to estimate the following summation by some function $f(N)$:
$$
S_N=\sum_{n=2}^{N} \frac{\log n}{\log^h(\log n)}=O(f(N)).
$$
Obviously, we see that
$$
S_N>\sum_{n=2}^...
3
votes
1
answer
318
views
Klarner's theorem
Klarner's theorem (http://mathworld.wolfram.com/KlarnersTheorem.html) says in a special case that you cannot tile a $10 \times 10$-board with $1\times 4$-tiles (that can also be rotated and used as $4 ...
0
votes
1
answer
177
views
Vanishing bilinear forms
For a symmetric or antisymmetric bilinear form $\varphi$ on a vector space $V$, if $\varphi(x,y)=0$ then also $\varphi(y,x)=0$ ($x,y\in V$).
I was wondering if this is also a necessary condition for ...
3
votes
2
answers
435
views
Evaluating the integral $\int_1^x \log^k t \frac{\sqrt{t-1}}{t^2} dt$
I am trying to evaluate the integral
$$
I_k(x)=\int_1^x \log^k t \frac{\sqrt{t-1}}{t^2} dt
$$
with $x$ tending to infinity.
In fact, I wish to have an estimate
$$
\sum_{k=0}^\infty \frac{1}{\log^k x} ...
1
vote
1
answer
109
views
Indecomposable monoids
Let $M$ be a commutative reduced and cancellative monoid and $K(M)$ its group of quotients.
We say that $M$ is indecomposable if for every divisor-closed submonoids $M_1$ and $M_2$, $M=M_1\oplus M_2$...
2
votes
0
answers
217
views
Known Methods for "Mutexing" Antiparallel Arcs in Graphs
I recently faced the problem of calculating shortest paths in undirected graphs in the presence of negative edge weights; I could not find any applicable algorithms via online search.
Transforming the ...
10
votes
2
answers
7k
views
About the Fourier transform of the logarithm function
I want to calculate / simplify:
$$\mathcal{F} (\ln(|x|)\mathcal{F(f)}(x))=\mathcal{F} (\ln(|x|)) \star f$$
where $\mathcal{F}$ is the Fourier transform ($\mathcal[f](\xi)=\int_{\mathbb R}f(x)e^{ix\...
11
votes
2
answers
849
views
Existence of subset of reals such that any real number is unique sum of exactly two elements of the subset
It is easy to see (using AC, of course) that there exist two sets $U\subset\mathbb{R}$ and $V\subset\mathbb{R}$ such that any real number $x$ can be represented as unique sum $x=u+v$, where $u\in U$ ...
5
votes
1
answer
124
views
Quasi-symmetric generalizations of classical symmetric functions
I am looking for quasi-symmetric versions of the classical $e_\lambda$ and $h_\lambda$ (the elementary and complete homogeneous symmetric functions). Is there some reference for this?
I am aware of ...
7
votes
0
answers
216
views
A funny kind of Ramsey number
A shorter version of this question was posted on Math Stack Exchange.
Let $V$ be a nonempty set. $(V,S)$ is a graph if $S\subseteq\binom V2,$ a triple system if $S\subseteq\binom V3,$
a quadruple ...
42
votes
3
answers
4k
views
The Origin(s) of Modular and Moduli
In mathematics and in physics, people use the terms "modular..." and "moduli space" very often. I was puzzled by the etymology, the origins and the similarity/equivalence/differences for these usages/...
0
votes
0
answers
180
views
How are incompleteness and independence proofs related?
(1) Some typical inclompleteness proofs use a kind of fixed point argument - for certain $\Phi$ you find a $\varphi$ with $\Phi(\mathrm{code}(\varphi))\leftrightarrow\varphi$.
(2) Some independence ...
6
votes
1
answer
235
views
Is any nonarchimedean field containing all roots of unity perfectoid?
Say $K$ is a complete nonarchimedean extension of $\mathbf{Q}_p$, i.e., it is the fraction field of a $p$-adically complete and $p$-torsionfree rank $1$ valuation ring. Assume that the residue field ...
0
votes
0
answers
164
views
Prime gap heuristics (follows up my question "Moments of merit")
I previously asked generally what people knew or conjectured concerning the moments of the probability distribution governing $M_n:= g_n/\ln(p_n)$, the normalized $n$th prime gap (or ``merit''). Greg ...
5
votes
1
answer
158
views
$S^{2}$-bundles over complex projective varieties
Is there an example of a smooth complex projective variety and an $S^{2}$-bundle over it which is not diffeomorphic to a complex projective variety?
5
votes
1
answer
435
views
Is it true that every vector bundle over a non compact smooth manifold is trivial at infinity?
Let $M$ be a non compact smooth manifold and suppose that $\pi:E\rightarrow M$ is a vector bundle over it. Is there a compact subset $K\subset M$ such that the restricted bundle $\pi|_U:E|_U\...
1
vote
0
answers
47
views
elliptic pde with supercritical advection term
Let $B$ denote the unit ball in $ R^N$ centered at the origin and consider
$$ -\Delta u(x) + \frac{ x \cdot \nabla u(x)}{|x|^\alpha} = f(x) \quad \mbox{ in } B$$ with $u=0$ on $ \partial B$. (or ...
5
votes
1
answer
533
views
Spectrum of the product of operators
Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space $F$.
Let $A,B\in \mathcal{B}(F)^+:=\left\{T\in \mathcal{B}(F);\,\langle Tx, x\...
7
votes
1
answer
380
views
Combinatorial/probabilistic statements having $F_{\text{un}}$/$F_q$ geometric interpetation
$\newcommand{\Fun}{F_\text{un}}$There was lots of "Fun with $\Fun$" (field with one element) in recent years.
One of the points is that it provides bridge between geometrical and ...
1
vote
0
answers
150
views
Factorially closed, finitely generated $k$-sub-algebra of $k[X_1,X_2,X_3]$ , where $k$ is algebraically closed field of positive characteristic
Let $S$ be a sub-ring of a commutative ring with unity $R$. Then $S$ is called factorially closed in $R$ if $a,b \in R$ and $ab \in S\setminus \{0\} \implies a,b \in S$.
Let $k$ be an algebraically ...
3
votes
1
answer
314
views
Is this the correct closed form for a series similar to $\zeta(2)$?
I hope this question is well received. I don't have a computer that can calculate very many terms for the infinite series: $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{(2n-1)^{2}},$$
but is it going to equal ...
1
vote
1
answer
155
views
Errors in Waksman's Solution to Cellular Automaton Firing Squad Problem?
Recently, a student and I have been working through Waksman's paper ``An Optimum Solution to the Firing Squad Synchronization Problem.'' The paper claims that for any value of $n$, the proposed ...
3
votes
0
answers
124
views
multiplicity of components in resolution of surface singularity
Let $X$ be a smooth compact algebraic surface, and $C=\bigcup E_i$ be a connected divisor where the $E_i$ are integral and the matrix $(E_i\cdot E_j)$ is negative definite. In the analytic category, ...
13
votes
3
answers
3k
views
Differentiability of Eigenvalues - Perturbation Theory
first, I have a general question. In perturbation theory, I saw perturbations in eigenvalues and eigenvectors of square, non-symmetric matrices and the calculations were all right but no one ever ...
2
votes
2
answers
448
views
These polynomials are always either even or odd [duplicate]
The falling factorials are $(x)_n=x(x-1)\cdots(x-n+1)$ with $(x)_0:=1$. Define the forward shift $E$ and the discrete derivative $\delta=E-1$, respectively, by
$$Ef(x)=f(x+1) \qquad \text{and} \qquad \...
5
votes
2
answers
480
views
Gaussians at lattice points
Let $\epsilon > 0$. I would like to know if there exists $c < \infty$ such that for all $d \in \mathbb{N}$ the following holds. If $x \in \mathbb{R}^d$ let $N_x$ be the standard Gaussian ...
2
votes
1
answer
203
views
Flatness of Fano Contractions
In his 1984 paper Cone of Curves, Kawamata asks on p 629 if a Fano Contraction is flat. ( an extremal ray contraction is called a Fano Contraction if the dimension of the target is less than the ...
1
vote
0
answers
274
views
If $R$ is UFD , then does $R \cong R[X,Y]$ imply $R \cong R[X]$?
$R$ is isomorphic to $R[X,Y]$, but not to $R[X]$ shows that it is possible to have commutative ring $R$ with unity such that $R \cong R[X,Y]$ but $R \ncong R[X]$.
My questions are: Is it possible ...
3
votes
0
answers
195
views
Representability of Flattening stratification functor
Let $f:X \to Y$ be a projective morphism between noetherian scheme. Is there any know condition under which there exists a functorial stratification of $Y$ i.e., there exists a filtration by locally ...
8
votes
1
answer
230
views
Is there a non-atomic finite positive measure in the plane, of which uncountably many projections have atoms?
I would like to know whether or not there exists a finite probability measure $\mu$ on $\mathbb R^2$ which has no atoms, but such that there exists an uncountable set $A\subset \mathbb S^1$, such that ...
1
vote
0
answers
134
views
On the solution of Laplace equation with mixed boundary condition
Let $\Omega \subset \mathbb{R}^2$ be an annular (bounded and connected) domain with inner and outer boundary $\Gamma_1$ and $\Gamma_2$, respectively. It is known that the PDE system
$$
\begin{...
2
votes
1
answer
151
views
A particular separation example
Q1. Does there exist a separable Banach space $X$ satisfying in the following property?
1- $X^*$ is non separable.
2- For every countable subset $F\subset X^*$ there exists $0\neq x_F\in X$ ...
6
votes
0
answers
239
views
Dowker and neighborhood complexes: reference wanted
Let $R$ be a 0-1 matrix whose rows or columns are maximal.
Q1. Is there a name for such a matrix (or, e.g., a corresponding relation)?
From 0-1 matrix corresponding to an abstract simplicial ...
2
votes
2
answers
413
views
Euclidean model structure on multipointed $d$-spaces
I use the notation of this question. A non-decreasing continuous bijection from $[0,a]$ to $[0,b]$ where $a,b\geq 0$ are two real numbers is denoted by $[0,a] \cong^+ [0,b]$. If $\phi:[0,a]\to U$ and $...
1
vote
1
answer
267
views
Variation of global sections of line bundles
The underlying field is $\mathbb{C}$.
Let $\pi:\mathcal{C} \to \mathbb{A}^n$ be a flat family of projective curves (not necessarily smooth) of genus $g \ge 2$. Assume $\mathcal{C}$ is regular. Let $\...
6
votes
3
answers
1k
views
Where to find the results of Onishchik?
I would like to have a good reference where the results in
"Inclusion relations between transitive compact transformation groups" https://mathscinet.ams.org/mathscinet-getitem?mr=27:3740
can be ...
3
votes
0
answers
604
views
Diagonal elements of Hermitian matrices with given eigenvalues
Given real vectors $d = (d_1, \ldots, d_n)$ and $\lambda = (\lambda_1, \ldots, \lambda_n)$, where I will assume that their coefficients are arranged in non-increasing order, the Schur-Horn theorem ...
1
vote
0
answers
161
views
Integral model of etale covering
Let $K$ be a $p$-adic field whose residue field $k$ is algebraically closed. Let $X$ be a hyperbolic curve over $K$, what I mean by a curve is a smooth geometrically connected scheme of dimension one ...
3
votes
1
answer
498
views
Is the linear span of irrep matrices a complete matrix basis?
Let $G$ be a finite or compact group and $\rho: G \to \mathrm{U}(d)$ a $d$-dimensional unitary representation of $G$. If $\rho$ is irreducible then the following seems to be true:
$$
\mathrm{span}_\...
9
votes
0
answers
181
views
$p$-groups and the arithmetic of $p$
I cannot think of an example of a property of a $p$-group or a pro-$p$ group that depends on the arithmetic of the prime number $p$. To clarify l do not mean the size of $p$. Clearly lots of stuff ...
3
votes
1
answer
214
views
Real automorphisms of the "quaternionic" real group ${\rm SO}^*(4m)$
Let $m\ge 2$, and let $G={\rm SO}^*(4m)$ denote the "quaternionic" real form of the special orthogonal group ${\rm SO}(4m,\mathbb C)$ of type ${\sf D}_{2m}$.
Let $\tau\in{\rm Aut}_{\Bbb R}(G)$ be a ...