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9 votes
1 answer
514 views

Well known matrix inequality?

I suspect that the following matrix inequality is well known, but I can't find a reference or proof: Given $n \times n$ symmetric matrices $A,B$ such that $I_n \leq A,B$, is the following true? $${...
Hammerhead's user avatar
  • 1,171
2 votes
1 answer
174 views

$[J_F : P_FN_{E/F}J_E] = 2$ for quadratic extensions of global fields of characteristic 2?

Let $E/F$ be a quadratic extension of global fields. Denote by $J_F$ the idèle group of $F$, by $P_F$ the subgroup of principal idèles and by $N_{E/F} : J_E \to J_F$ the norm map between idèles. In ...
Bib-lost's user avatar
  • 267
7 votes
1 answer
498 views

Linear permutations commuting with $x\rightarrow x^{-1}$

Let $F = \operatorname{GF}(2^n)$ be a finite field. Define a permutation $\phi:F \rightarrow F$ by the formula $$ \phi(x) = x^{-1}, \ x\neq 0; \ \phi(0) =0. $$ We say that a permutations $\psi$ of $F$ ...
Mikhail Goltvanitsa's user avatar
9 votes
1 answer
171 views

Variant of mutual information

Given a discrete random variable $(X,Y)$, one can consider the smallest entropy of a random variable $Z$ such that $X$ and $Y$ are independent conditioned to $Z$. This quantity is akin to the mutual ...
alesia's user avatar
  • 2,582
2 votes
1 answer
75 views

CAS implementing free algebras with involution

Is there any software that easily allows to make symbolic computations with involutions and homomorphisms? I need to define a product in an associative algebra with an (abstract) involution and ...
Jose Brox's user avatar
  • 2,962
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 ...
Licheng Wang's user avatar
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 ...
MaJoR21's user avatar
  • 111
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 ...
Gaussler's user avatar
  • 295
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 $ ...
YoYo's user avatar
  • 325
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'...
user124388's user avatar
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 ...
asv's user avatar
  • 21.1k
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 ...
Mohammad Ghomi's user avatar
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 ...
Peter LeFanu Lumsdaine's user avatar
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 ...
Hauke Reddmann's user avatar
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}^...
user119197's user avatar
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 ...
Andreas Thom's user avatar
  • 25.3k
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 ...
user124321's user avatar
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} ...
toshi's user avatar
  • 130
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 $M_2$, $M=M_1\oplus M_2$...
Rajkarov's user avatar
  • 933
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 ...
Manfred Weis's user avatar
  • 12.6k
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\...
Bertrand's user avatar
  • 1,121
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 $u\in U$ ...
ar.grig's user avatar
  • 1,133
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 ...
Per Alexandersson's user avatar
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 ...
bof's user avatar
  • 11.5k
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/...
wonderich's user avatar
  • 10.3k
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 ...
მამუკა ჯიბლაძე's user avatar
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 ...
clueless's user avatar
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 ...
David Feldman's user avatar
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?
Nick L's user avatar
  • 6,923
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\...
user avatar
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 ...
Math604's user avatar
  • 1,363
5 votes
1 answer
534 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\...
Schüler's user avatar
  • 724
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 ...
Alexander Chervov's user avatar
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 ...
user avatar
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 ...
user124808's user avatar
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 ...
Andrew Penland's user avatar
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, ...
Hans Sachs's user avatar
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 ...
xmonetx's user avatar
  • 138
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 \...
T. Amdeberhan's user avatar
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 ...
burtonpeterj's user avatar
  • 1,689
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 ...
user avatar
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 ...
user avatar
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 ...
user43198's user avatar
  • 1,949
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 ...
Mircea's user avatar
  • 2,031
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{...
Julienne Franz's user avatar
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 $0\neq x_F\in X$ ...
ABB's user avatar
  • 3,898
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 ...
Steve Huntsman's user avatar
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 $...
Philippe Gaucher's user avatar
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 $\...
user43198's user avatar
  • 1,949
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 ...
Babs's user avatar
  • 73

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