Inseparable extensions of algebras over the Steenrod algebra with applications to modular invariant theory of finite groups
Mara
D.
Neusel
4689-4720
Abstract: We consider purely inseparable extensions $\textrm{H}\hookrightarrow \sqrt[\mathscr{P}^*]{\textrm{H}}$ of unstable Noetherian integral domains over the Steenrod algebra. It turns out that there exists a finite group $G\le\textrm{GL}(V)$ and a vector space decomposition $V=W_0\oplus W_1\oplus\dotsb\oplus W_e$ such that $\overline{\textrm{H}}=(\mathbb{F}[W_0] \otimes\mathbb{F}[W_1]^p\otimes\dotsb\otimes\mathbb{F}[W_e]^{p^e})^G$ and $\overline{\sqrt[\mathscr{P}^*]{\textrm{H}}}=\mathbb{F}[V]^G$, where $\overline{(-)}$ denotes the integral closure. Moreover, $\textrm{H}$ is Cohen-Macaulay if and only if $\sqrt[\mathscr{P}^*]{\textrm{H}}$ is Cohen-Macaulay. Furthermore, $\overline{\textrm{H}}$ is polynomial if and only if $\sqrt[\mathscr{P}^*]{\textrm{H}}$ is polynomial, and $\sqrt[\mathscr{P}^*]{\textrm{H}}=\mathbb{F}[h_1,\dotsc,h_n]$ if and only if $\displaystyle \textrm{H}=\mathbb{F}[h_1,\dotsc,h_{n_0},h_{n_0+1}^p,\dotsc,h_{n_1}^p, h_{n_1+1}^{p^2},\dotsc,h_{n_e}^{p^e}],$ where $n_e=n$ and $n_i=\dim_{\mathbb{F}}(W_0\oplus\dotsb\oplus W_i)$.
Groupoid cohomology and extensions
Jean-Louis
Tu
4721-4747
Abstract: We show that Haefliger's cohomology for étale groupoids, Moore's cohomology for locally compact groups and the Brauer group of a locally compact groupoid are all particular cases of sheaf (or Cech) cohomology for topological simplicial spaces.
A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory
G.
C.
Bell;
A.
N.
Dranishnikov
4749-4764
Abstract: We prove an asymptotic analog of the classical Hurewicz theorem on mappings that lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite-dimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension. As applications we find upper bound estimates for the asymptotic dimension of nilpotent and polycyclic groups in terms of their Hirsch length. We are also able to improve the known upper bounds on the asymptotic dimension of fundamental groups of complexes of groups, amalgamated free products and the hyperbolization of metric spaces possessing the Higson property.
On neoclassical Schottky groups
Rubén
Hidalgo;
Bernard
Maskit
4765-4792
Abstract: The goal of this paper is to describe a theoretical construction of an infinite collection of non-classical Schottky groups. We first show that there are infinitely many non-classical noded Schottky groups on the boundary of Schottky space, and we show that infinitely many of these are ``sufficiently complicated''. We then show that every Schottky group in an appropriately defined relative conical neighborhood of any sufficiently complicated noded Schottky group is necessarily non-classical. Finally, we construct two examples; the first is a noded Riemann surface of genus $3$ that cannot be uniformized by any neoclassical Schottky group (i.e., classical noded Schottky group); the second is an explicit example of a sufficiently complicated noded Schottky group in genus $3$.
On the characterization of the kernel of the geodesic X-ray transform
Eduardo
Chappa
4793-4807
Abstract: Let $\overline{\Omega}$ be a compact manifold with boundary. We consider covariant symmetric tensor fields of order two that belong to a Sobolev space $H^{k}(\overline{\Omega}), k \geq 1$. We prove, under the assumption that the metric is simple, that solenoidal tensor fields that belong to the kernel of the geodesic X-ray transform are smooth up to the boundary. As a corollary we obtain that they form a finite-dimensional set in $H^{k}$.
On $C^\infty$ and Gevrey regularity of sublaplacians
A.
Alexandrou
Himonas;
Gerson
Petronilho
4809-4820
Abstract: In this paper we consider zero order perturbations of a class of sublaplacians on the two-dimensional torus and give sufficient conditions for global $C^\infty$ regularity to persist. In the case of analytic coefficients, we prove Gevrey regularity for a general class of sublaplacians when the finite type condition holds.
Stationary isothermic surfaces and uniformly dense domains
R.
Magnanini;
J.
Prajapat;
S.
Sakaguchi
4821-4841
Abstract: We establish a relationship between stationary isothermic surfaces and uniformly dense domains. A stationary isothermic surface is a level surface of temperature which does not evolve with time. A domain $\Omega$ in the $N$-dimensional Euclidean space $\mathbb{R}^N$ is said to be uniformly dense in a surface $\Gamma\subset\mathbb{R}^N$ of codimension $1$ if, for every small $r>0,$ the volume of the intersection of $\Omega$ with a ball of radius $r$ and center $x$ does not depend on $x$ for $x\in\Gamma.$ We prove that the boundary of every uniformly dense domain which is bounded (or whose complement is bounded) must be a sphere. We then examine a uniformly dense domain with unbounded boundary $\partial\Omega$, and we show that the principal curvatures of $\partial\Omega$ satisfy certain identities. The case in which the surface $\Gamma$ coincides with $\partial\Omega$ is particularly interesting. In fact, we show that, if the boundary of a uniformly dense domain is connected, then (i) if $N=2$, it must be either a circle or a straight line and (ii) if $N=3,$ it must be either a sphere, a spherical cylinder or a minimal surface. We conclude with a discussion on uniformly dense domains whose boundary is a minimal surface.
$W^{2,p}$--estimates for the linearized Monge--Ampère equation
Cristian
E.
Gutiérrez;
Federico
Tournier
4843-4872
Abstract: Let $\Omega \subseteq \mathbb{R}^n$ be a strictly convex domain and let $\phi \in C^2(\Omega)$ be a convex function such that $\lambda \leq$ det$D^2\phi \leq\Lambda$ in $\Omega$. The linearized Monge-Ampère equation is $\displaystyle L_{\Phi}u=\textrm{trace}(\Phi D^2u)=f,$ where $\Phi = ($det$D^2\phi)(D^2\phi)^{-1}$ is the matrix of cofactors of $D^2\phi$. We prove that there exist $p>0$ and $C>0$ depending only on $n,\lambda,\Lambda$, and $\textrm{dist}(\Omega^\prime,\Omega)$ such that $\displaystyle \Vert D^2u\Vert _{L^p(\Omega^\prime)}\leq C(\Vert u\Vert _{L^\infty(\Omega)}+\Vert f\Vert _{L^n(\Omega)})$ for all solutions $u\in C^2(\Omega)$ to the equation $L_{\Phi}u=f$.
Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators
Andrea
Pascucci;
Sergio
Polidoro
4873-4893
Abstract: We prove a global Harnack inequality for a class of degenerate evolution operators by repeatedly using an invariant local Harnack inequality. As a consequence we obtain an accurate Gaussian lower bound for the fundamental solution for some meaningful families of degenerate operators.
The intersection of a matroid and a simplicial complex
Ron
Aharoni;
Eli
Berger
4895-4917
Abstract: A classical theorem of Edmonds provides a min-max formula relating the maximal size of a set in the intersection of two matroids to a ``covering" parameter. We generalize this theorem, replacing one of the matroids by a general simplicial complex. One application is a solution of the case $r=3$ of a matroidal version of Ryser's conjecture. Another is an upper bound on the minimal number of sets belonging to the intersection of two matroids, needed to cover their common ground set. This, in turn, is used to derive a weakened version of a conjecture of Rota. Bounds are also found on the dual parameter--the maximal number of disjoint sets, all spanning in each of two given matroids. We study in detail the case in which the complex is the complex of independent sets of a graph, and prove generalizations of known results on ``independent systems of representatives" (which are the special case in which the matroid is a partition matroid). In particular, we define a notion of $k$-matroidal colorability of a graph, and prove a fractional version of a conjecture, that every graph $G$ is $2\Delta(G)$-matroidally colorable. The methods used are mostly topological.
Semifree symplectic circle actions on $4$-orbifolds
L.
Godinho
4919-4933
Abstract: A theorem of Tolman and Weitsman states that all symplectic semifree circle actions with isolated fixed points on compact symplectic manifolds must be Hamiltonian and have the same equivariant cohomology and Chern classes of $(\mathbb{C}P^1)^n$ equipped with the standard diagonal circle action. In this paper, we show that the situation is much different when we consider compact symplectic orbifolds. Focusing on $4$-orbifolds with isolated cone singularities, we show that such actions, besides being Hamiltonian, can now be obtained from either $S^2\times S^2$ or a weighted projective space, or a quotient of one of these spaces by a finite cyclic group, by a sequence of special weighted blow-ups at fixed points. In particular, they can have any number of fixed points.
The invariant factors of the incidence matrices of points and subspaces in $\operatorname{PG}(n,q)$ and $\operatorname{AG}(n,q)$
David
B.
Chandler;
Peter
Sin;
Qing
Xiang
4935-4957
Abstract: We determine the Smith normal forms of the incidence matrices of points and projective $(r-1)$-dimensional subspaces of $\operatorname{PG}(n,q)$ and of the incidence matrices of points and $r$-dimensional affine subspaces of $\operatorname{AG}(n,q)$ for all $n$, $r$, and arbitrary prime power $q$.
Open loci of graded modules
Christel
Rotthaus;
Liana
M.
Sega
4959-4980
Abstract: Let $A=\bigoplus_{i\in \mathbb{N}}A_i$ be an excellent homogeneous Noetherian graded ring and let $M=\bigoplus_{n\in \mathbb{Z}}M_n$ be a finitely generated graded $A$-module. We consider $M$ as a module over $A_0$ and show that the $(S_k)$-loci of $M$ are open in $\operatorname{Spec}(A_0)$. In particular, the Cohen-Macaulay locus $U^0_{CM}=\{{\mathfrak{p}}\in \operatorname{Spec}(A_0) \mid M_\mathfrak{p} \;$ is Cohen-Macaulay$\}$ is an open subset of $\operatorname{Spec}(A_0)$. We also show that the $(S_k)$-loci on the homogeneous parts $M_n$ of $M$ are eventually stable. As an application we obtain that for a finitely generated Cohen-Macaulay module $M$ over an excellent ring $A$ and for an ideal $I\subseteq A$ which is not contained in any minimal prime of $M$, the $(S_k)$-loci for the modules $M/I^nM$ are eventually stable.
Partitioning $\alpha$--large sets: Some lower bounds
Teresa
Bigorajska;
Henryk
Kotlarski
4981-5001
Abstract: Let $\alpha\rightarrow(\beta)_m^r$ denote the property: if $A$ is an $\alpha$-large set of natural numbers and $[A]^r$ is partitioned into $m$ parts, then there exists a $\beta$-large subset of $A$ which is homogeneous for this partition. Here the notion of largeness is in the sense of the so-called Hardy hierarchy. We give a lower bound for $\alpha$ in terms of $\beta,m,r$ for some specific $\beta$.
On almost one-to-one maps
Alexander
Blokh;
Lex
Oversteegen;
E.
D.
Tymchatyn
5003-5014
Abstract: A continuous map $f:X\to Y$ of topological spaces $X, Y$ is said to be almost $1$-to-$1$ if the set of the points $x\in X$ such that $f^{-1}(f(x))=\{x\}$ is dense in $X$; it is said to be light if pointwise preimages are zero dimensional. We study almost 1-to-1 light maps of some compact and $\sigma$-compact spaces (e.g., $n$-manifolds or dendrites) and prove that in some important cases they must be homeomorphisms or embeddings. In a forthcoming paper we use these results and show that if $f$ is a minimal self-mapping of a 2-manifold $M$, then point preimages under $f$ are tree-like continua and either $M$ is a union of 2-tori, or $M$ is a union of Klein bottles permuted by $f$.
Multiplier ideals of hyperplane arrangements
Mircea
Mustata
5015-5023
Abstract: In this note we compute multiplier ideals of hyperplane arrangements. This is done using the interpretation of multiplier ideals in terms of spaces of arcs by Ein, Lazarsfeld, and Mustata (2004).
Intrinsic ultracontractivity of the Feynman-Kac semigroup for relativistic stable processes
Tadeusz
Kulczycki;
Bartlomiej
Siudeja
5025-5057
Abstract: Let $X_t$ be the relativistic $\alpha$-stable process in $\mathbf{R}^d$, $\alpha \in (0,2)$, $d > \alpha$, with infinitesimal generator $H_0^{(\alpha)}= - ((-\Delta +m^{2/\alpha})^{\alpha/2}-m)$. We study intrinsic ultracontractivity (IU) for the Feynman-Kac semigroup $T_t$ for this process with generator $H_0^{(\alpha)} - V$, $V \ge 0$, $V$ locally bounded. We prove that if $\lim_{\vert x\vert \to \infty} V(x) = \infty$, then for every $t >0$ the operator $T_t$ is compact. We consider the class $\mathcal{V}$ of potentials $V$ such that $V \ge 0$, $\lim_{\vert x\vert \to \infty} V(x) = \infty$ and $V$ is comparable to the function which is radial, radially nondecreasing and comparable on unit balls. For $V$ in the class $\mathcal{V}$ we show that the semigroup $T_t$ is IU if and only if $\lim_{\vert x\vert \to \infty} V(x)/\vert x\vert = \infty$. If this condition is satisfied we also obtain sharp estimates of the first eigenfunction $\phi_1$ for $T_t$. In particular, when $V(x) = \vert x\vert^{\beta}$, $\beta > 0$, then the semigroup $T_t$ is IU if and only if $\beta >1$. For $\beta >1$ the first eigenfunction $\phi_1(x)$ is comparable to $\displaystyle \exp(-m^{1/{\alpha}}\vert x\vert) \, (\vert x\vert + 1)^{(-d - \alpha - 2 \beta -1 )/2}.$
Ratio limit theorem for parabolic horn-shaped domains
Pierre
Collet;
Servet
Martinez;
Jaime
San Martin
5059-5082
Abstract: We prove that for horn-shaped domains of parabolic type, the ratio of the heat kernel at different fixed points has a limit when the time tends to infinity. We also give an explicit formula for the limit in terms of the harmonic functions.
Frequently hypercyclic operators
Frédéric
Bayart;
Sophie
Grivaux
5083-5117
Abstract: We investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators $T$ on separable complex $\mathcal{F}$-spaces: $T$ is frequently hypercyclic if there exists a vector $x$ such that for every nonempty open subset $U$ of $X$, the set of integers $n$ such that $T^{n}x$ belongs to $U$ has positive lower density. We give several criteria for frequent hypercyclicity, and this leads us in particular to study linear transformations from the point of view of ergodic theory. Several other topics which are classical in hypercyclicity theory are also investigated in the frequent hypercyclicity setting.
Partial hyperbolicity or dense elliptic periodic points for $C^1$-generic symplectic diffeomorphisms
Radu
Saghin;
Zhihong
Xia
5119-5138
Abstract: We prove that if a symplectic diffeomorphism is not partially hyperbolic, then with an arbitrarily small $C^1$ perturbation we can create a totally elliptic periodic point inside any given open set. As a consequence, a $C^1$-generic symplectic diffeomorphism is either partially hyperbolic or it has dense elliptic periodic points. This extends the similar results of S. Newhouse in dimension 2 and M.-C. Arnaud in dimension 4. Another interesting consequence is that stably ergodic symplectic diffeomorphisms must be partially hyperbolic, a converse to Shub-Pugh's stable ergodicity conjecture for the symplectic case.
Jointly hyponormal pairs of commuting subnormal operators need not be jointly subnormal
Raúl
E.
Curto;
Jasang
Yoon
5139-5159
Abstract: We construct three different families of commuting pairs of subnormal operators, jointly hyponormal but not admitting commuting normal extensions. Each such family can be used to answer in the negative a 1988 conjecture of R. Curto, P. Muhly and J. Xia. We also obtain a sufficient condition under which joint hyponormality does imply joint subnormality.
Inverse scattering with fixed energy and an inverse eigenvalue problem on the half-line
Miklós
Horváth
5161-5177
Abstract: Recently A. G. Ramm (1999) has shown that a subset of phase shifts $\delta_l$, $l=0,1,\ldots$, determines the potential if the indices of the known shifts satisfy the Müntz condition $\sum_{l\neq0,l\in L}\frac{1}{l}=\infty$. We prove the necessity of this condition in some classes of potentials. The problem is reduced to an inverse eigenvalue problem for the half-line Schrödinger operators.