All Questions
152,891
questions
2
votes
0
answers
191
views
Generalized metric on spacetimes
I read many articles about space-times. Most authors consider these spaces as warped product manifolds $I\times M$ where $I$ is an open connected interval of the real line and $M$ is a Riemannian ...
2
votes
0
answers
168
views
(Reference) Asymptotics of hitting probability by Brownian motion
The problem is: Given compact set A with positive finite volume (eg. ball,cube), what happens to $P_{x}(T_{A}>t)$ as $t\to \infty$, where $T_{A}=inf_{t>0}(B_{t}\in A)$ and x is in the "exterior" ...
1
vote
1
answer
383
views
Is the Cassels-Tate pairing defined for elliptic curves over function fields?
The Cassels-Tate pairing is typically defined for elliptic curves (or abelian varieties) over number fields, but it seems like it should be defined for elliptic curves over function fields as well. ...
2
votes
1
answer
161
views
Is a rigid cycle a chordal graph?
There are two relevant questions:
(1) We know an edge set $C$ is a rigid cycle in $\mathcal{G}_2(n)$ if and only if $|E(C)|=2|V(C)|−2$ and $|F|≤2|V(F)|−3$ for every proper subset $F$ of $E(C)$. Thus, ...
4
votes
3
answers
819
views
Gauss Codes that produce classical knots as opposed to virtual knots
I have been doing some research in Gauss codes and have been reading Kauffman's paper Virtual Knot Theory. In section 3.3, Theorem 2, he states that
If $K$ is a virtual knot whose underlying Gauss ...
4
votes
1
answer
274
views
Extreme unit linear functional not norming a vector
If $E$ is a non-reflexive Banach space then there exists a linear functional $\lambda \in E^*$ of norm one such that $\lambda v < 1$ for all $v \in E$ of norm one. However, in the only non-...
23
votes
1
answer
1k
views
Geodesics in finite groups
It seems that I can generalize a result from compact, connected Lie groups to finite groups, but in order to do so, I need to have some kind of geodesics on finite groups.
Below is a proposition for ...
7
votes
1
answer
891
views
About the hypothesis of Zorn's lemma
The proofs I know of Zorn's lemma give the following refinement:
Let $(X,<)$ be a partially ordered set such that every well-ordered
subset of $X$ has an upperbound. Then $X$ has a maximal ...
7
votes
0
answers
777
views
What is the decomposition group at $p$ in the Galois group unramified outside $\ell?$
Let $p\ne\ell$ be two prime numbers, and let $K_{\ell}$ be the maximal extension of $\mathbb Q$ ramified only at $\ell$ and $\infty$ (i.e. it is the fixed subfield of $\overline{\mathbb Q}$ by all the ...
0
votes
0
answers
114
views
What is the proper Zariski-closed subset in these examples for Vojta's more general abc conjecture?
In A more general abc conjecture, p. 7 Paul Vojta conjectures:
If $x_0,\ldots x_{n-1}$ are nonzero coprime integers satisfing $x_0 + \cdots x_{n-1}=0$
$$ \max\{|x_0|,\ldots |x_{n-1}|\} \le C \prod_{...
1
vote
1
answer
418
views
Poisson approximation of random sub-graphs
I add the edges of $G(n)$ the complete graph on $n$ vertices one by one, at random and without replacement, and denote by $G(n,m)$ the resulting Erdos Renyi random graph process. At step $m$ in the ...
0
votes
2
answers
118
views
Inner Product of Given Sum Positive Sequence
Let $$A = \Big\{(a_1,a_2,\dots)\ \Big|\ a_i\ge 0, \sum_{i=1}^\infty a_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum_{i=1}^\infty a_ib_i\ \bigg|\ (a_i)_{i=1}^\infty,\, (b_i)_{i=1}^\infty \in A,\,\sup\...
9
votes
3
answers
1k
views
Poincaré duality for (co)homology of Lie algebras?
Let $R$ be a commutative ring and $\mathfrak{g}$ a Lie $R$-algebra that has an $R$-module basis with $n$ elements.
In Algebra, Geometry, and Software Systems by Joswig & Takayama on p.200, it ...
2
votes
0
answers
104
views
Regularity of the Minkowski functionnal of a convex
Let $K$ be a convex compact set in $\mathbb{R}^2$ with $0 \in \overset{\circ}{K}$. The Minkowski functional associated to K is:
\begin{align*}
\varphi_K(x):= \inf \left\{t>0 \; : \; tx \in K \...
2
votes
0
answers
392
views
relations between nori motives and pure motives
The category of Nori motives $NMM$ is defined by tools of graph category. I think one can similarly define a category $EM$ as the graph category of $P(k)$ (the graph of projective smooth k-varieties) ...
1
vote
0
answers
135
views
Can we define log-convex operators?
Let $I\subset\mathbb{R}$. A function $f:I\rightarrow\mathbb{R}$, is said to be log-convex if $\log f$ is convex or equivalently for all $x,y\in I$ and $\alpha\in [0,1]$
$$f(\alpha x+(1-\alpha)y)\leq [...
10
votes
1
answer
1k
views
Higher vector spaces
As far as I know there are different ways to categorify the notion of vector space/module. These appear (for example) when trying to find extended TQFTs. There are at least two ways (presented at $2$-...
11
votes
2
answers
604
views
Nim and the Sierpinski Gasket
(I discovered this in high school, sent it off to a journal, never heard back, and forgot about it. I've never found anyone else who appeared to know about it; the combinatorial game theorists I've ...
2
votes
1
answer
213
views
Rational functions and polynomials evaluated on a set of points
Let $S$ be a collection of points on the real line.
Let $\{x_i\}_{i=1}^n$ take values in $S$.
Consider a polynomial $p(x_1,x_2,\dots,x_n)$ over $\mathbb R[x_1,x_2,\dots,x_n]$ of degree $d$ which ...
12
votes
3
answers
733
views
The continuum hypothesis for packing shapes without overlapping
Consider the finite cross $C$ (=union of line segments $\overline{(0, -1)(0, 1)}$ and $\overline{(-1, 0)(1, 0)}$) and the unit half-circle $H$. It is easy to see that we may pack continuum-many ...
4
votes
0
answers
228
views
Sobolev spaces of maps between manifolds and the Palais-Smale Condition
I'm currently reading some papers by Uhlenbeck on harmonic maps. She mentions the following facts:
Let $M^m$ and $N^n$ be compact Riemannian manifolds, $N$ embedded isometrically into Euclidean space....
-1
votes
1
answer
145
views
Analytic extension of the exterior Newtonian potential into the domain
I just want to know for a domain with smooth boundary whether exterior Newtonian potential has singular analytic extension into the domain or into a part of the domain.
Definition of Newtonian ...
19
votes
1
answer
807
views
Is the regularity of finitely generated rings decidable?
Q: Is there an algorithm to decide whether a given finitely generated (over $\mathbb{Z}$) commutative ring is regular?
I mean by regular that the localization at every prime ideal is a regular local ...
0
votes
2
answers
237
views
Form of the Shannon capacity for Heptagon?
Is the $0$-error capacity of $7$-cycle:
$(1)$ known to be of form $7^q$ for some $q\in \mathbb Q$?
3
votes
1
answer
401
views
Average entropy of quantum system in bipartite pure state for finite temperature
[I got halfway through writing this when I found the paper that answers the question in (essentially) the affirmative. I'll post it anyways in case anyone is interested.]
Background: If a random ...
2
votes
2
answers
468
views
A question from the proof of affine algebraic group is a linear
In (some version of) the proof of the fact that any affine algebraic group is a linear algebraic group, there is an important step as follows (for example in Borel's book "Linear Algebraic Groups", ...
10
votes
1
answer
2k
views
On unramified p-adic groups
Let G be a reductive group over a local field F. Let O be the ring of integers of F.
The following are equivalent (and groups satisfying these conditions are called unramified):
(a) G is quasisplit ...
15
votes
2
answers
6k
views
Linearly constrained eigenvalue problem
Suppose I'd like to:
\begin{align}
\mathop{\text{min}}_\mathbf{x} && \mathbf{x}^T\mathbf{A}\mathbf{x} \\
\text{subject to:} && \mathbf{x}^T \mathbf{M} \mathbf{x} = 1\\
&& \...
6
votes
1
answer
265
views
Isotropy of Apollonian disk-packing
Is there any sense in which the "epsilon-tail" of an Apollonian disk-packing (by which I mean the union of the disks of radius less than epsilon) exhibits more and more statistical isotropy as epsilon ...
7
votes
2
answers
927
views
Does the Gamma function preserve integers?
Does the Gamma function $\Gamma: \mathbb{C} \to \mathbb{C}$ preserve the Kummer ring $\mathbb{Z}[\exp(2\pi\imath/m)]$? And if not, then what about the Gaussian integers $\mathbb{Z}[\imath]$ or the ...
3
votes
1
answer
130
views
Conditions conformal mapping to be expansive
Let $\Omega' \subset \Omega$ be simply connected domains in the complex plane. The Riemann mapping theorem tells us that there a biholomorphism $f: \Omega' \to \Omega$. What conditions guarantee that ...
2
votes
1
answer
154
views
Coding for channels with concentrated error
Can we implement a reduced-error transmission over a channel with error frequency having $\liminf<\frac{1}{2}$ and $\limsup>\frac{1}{2}$?
We know that if a channel with error flips (in the ...
3
votes
1
answer
511
views
Points with pairwise integer distances in the plane
Consider $n>3$ points with pairwise integer distances in the plane! What is the relationship between these $n(n-1)/2$ integers? Do we have a theorem or result about these points? Does there exist a ...
12
votes
1
answer
812
views
Is there any relationship between the Euler class and the Vandermonde determinant?
Several Wikipedia articles claim that the relationship between the Euler class $e(V)$ and the top Pontryagin class $p_k(V)$ of an oriented $2k$-dimensional real vector bundle $V$ corresponds, via the ...
11
votes
3
answers
955
views
Why do $12$ and $120$ occur very often in the denominators of $\zeta(-n)$ for odd $n$?
$\zeta(-n) = - \dfrac{B_{n+1}}{n+1}$
$\zeta(-2n) = 0$
$\zeta(-1) = - \dfrac{1}{12}$
$\zeta(-3) = \dfrac{1}{120}$
$\zeta(-5) = - \dfrac{1}{252}$
$\zeta(-7) = \dfrac{1}{240}$
$\zeta(-9) = - \dfrac{...
7
votes
1
answer
427
views
Extending compact operators
Let $X$ be a separable, infinite-dimensional complex Banach space and $Y\subseteq X$ an infinite-dimensional closed subspace. Suppose $K:Y\to X$ is an arbitrary compact operator. I would like to ...
-1
votes
1
answer
135
views
terminology: "complex" and "sequence" in homological algebra
It appears that the terms "complex" and "sequence" are used synonymously in homological algebra.
But there seem to be collocations (in the linguistic sense) that prefer one of those words. For ...
2
votes
1
answer
179
views
Triangulations of translation surfaces whose edges are shorter than the diameter
Let $S$ be a compact translation surface (i.e. a surface endowed with a singular flat metric such that singular points are locally isometric to a cone of angle an integer multiple of $2\pi$, and that ...
5
votes
2
answers
1k
views
Isotypic components of the action of the symmetric group on polynomials
The polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ decomposes as a direct sum of isotypic components for the action of the symmetric group $S_n$. The isotypic component of the trivial representation is ...
2
votes
0
answers
127
views
Slice a compact C1 surface in R3 by a moving transverse plane. Does the length of the slice depend C1 on the plane?
To be more precise I am interested in questions similar to the one below
(I asked the question below on math.stackexchange last week but got not answer.)
I have a $C^1$ function $f:[0,1]^2 \to \...
2
votes
1
answer
166
views
$(LLP(Epi), Epi)$ is a WFS on any variety of algebras
This entry in the Joyal catlab claims without proof that in a category $\bf V$ which is a "variety of algebras" the two classes $(LLP(Epi), Epi)$ form a weak factorization system. I interpret this ...
2
votes
1
answer
899
views
Definition of primitive divisor of a Lucas sequence
If $\{a_n\}_{n=1}^\infty$ is a sequence of integers, then the definition of primitive divisor of one of its terms is quite natural:
Def. 1. A prime number $p$ is a primitive divisor of $a_n$ if $p \...
4
votes
1
answer
420
views
Applying Lemma 2.12 of Deligne's/ Milne's "Tannakian Categories" on an irreducible representation
since this is my first question here, I'm not very certain whether I state it properly. I'm thankful for any helpful remarks.
Currently I'm trying to understand the above-mentioned article which can ...
7
votes
1
answer
1k
views
Can the Cohen forcing collapse cardinals?
Let $\kappa$ be a regular cardinal, and let $\mathbb{P} = Add(\kappa,1)$ be the standard forcing notion for adding a new subset of $\kappa$ using partial function from $\kappa$ to $2$ with domain of ...
4
votes
0
answers
461
views
Towards an enhanced version of homological mirror symmetry for affine varieties
Let $X$ and $X^\vee$ be a mirror pair, homological mirror symmetry relates the symplectic geometry of $X$ to the complex geometry of $X^\vee$ via the equivalence of triangulated categories
$D^\pi\...
11
votes
2
answers
760
views
Is every order type of a PA model the \omega of some ZFC model?
Let $N$ be a model of first-order Peano arithmetic, and let $\sigma$ be its order-type. Does it follow that there is a (non-transitive, expect when $M$ is the standard model) $ZFC$-model $M$ such that ...
3
votes
1
answer
721
views
what would be the consequences on the distribution of primes of $\Lambda=\infty$?
It is widely believed that the quantity $\Lambda:=\lim\sup\dfrac{t_{n+1}-t_{n}}{2\pi/\log t_{n}}$, where $t_{n}$ is the imaginary part of the $n$-th non-trivial zero on the critical line of the ...
7
votes
0
answers
416
views
Moduli interpretation of Eisenstein series
Let $N \geq 11$ be an integer and consider the basis of Eisenstein series for $M_2(\Gamma_0(N))$ described in Theorem $4.6.2$ of Diamond--Shurman's book. Pick and Eisenstein series $F$ in this basis. ...
3
votes
1
answer
362
views
Is $L(\ell_2,\ell_2)$ dense in $L(\ell_2,c_0)$?
Let $\ell_2:=\{ x:\mathbb{N} \to \mathbb{R}: \sum_{j=1}^\infty x_j^2<\infty\}$ and
$c_0:=\{ x:\mathbb{N} \to \mathbb{R}: \lim_{j\to\infty}x_j=0,\, \sup_{j\in\mathbb{N}}|x_j|<\infty\}$ denote the ...
0
votes
1
answer
248
views
A hyperbolic partial differential equation (wave-like) with variable-dependent coefficient and possibly singular in one variable
First, I beg your pardon since the title of the question is a bit confusing I guess. I'm working on a physical equation of the wave-like form. Explicitly, it reads
$$\left[\left(\cos\phi\partial_{z}+\...