Expansiveness of algebraic actions on connected groups
Siddhartha
Bhattacharya
4687-4700
Abstract: We study endomorphism actions of a discrete semigroup $\Gamma$ on a connected group $G$. We give a necessary and sufficient condition for expansiveness of such actions provided $G$ is either a Lie group or a solenoid.
Quaternionic algebraic cycles and reality
Pedro
F.
dos Santos;
Paulo
Lima-Filho
4701-4736
Abstract: In this paper we compute the equivariant homotopy type of spaces of algebraic cycles on real Brauer-Severi varieties, under the action of the Galois group $Gal({\mathbb C} / {\mathbb R})$. Appropriate stabilizations of these spaces yield two equivariant spectra. The first one classifies Dupont/Seymour's quaternionic $K$-theory, and the other one classifies an equivariant cohomology theory ${\mathfrak Z}^*(-)$ which is a natural recipient of characteristic classes $KH^*(X) \to {\mathfrak Z}^*(X)$ for quaternionic bundles over Real spaces $X$.
Complete second order linear differential operator equations in Hilbert space and applications in hydrodynamics
N.
D.
Kopachevsky;
R.
Mennicken;
Ju.
S.
Pashkova;
C.
Tretter
4737-4766
Abstract: We study the Cauchy problem for a complete second order linear differential operator equation in a Hilbert space ${\mathcal H}$ of the form \begin{displaymath}{\mathcal D}(F)\subset{\mathcal D}(B),\quad {\mathcal D}(F)\subset {\mathcal D}(K). \end{displaymath} We also suppose that $F$ and $B$ are bounded from below, but the operator coefficients are not assumed to commute. The main results concern the existence of strong solutions to the stated Cauchy problem and applications of these results to the Cauchy problem associated with small motions of some hydrodynamical systems.
Geometric aspects of frame representations of abelian groups
Akram
Aldroubi;
David
Larson;
Wai-Shing
Tang;
Eric
Weber
4767-4786
Abstract: We consider frames arising from the action of a unitary representation of a discrete countable abelian group. We show that the range of the analysis operator can be determined by computing which characters appear in the representation. This allows one to compare the ranges of two such frames, which is useful for determining similarity and also for multiplexing schemes. Our results then partially extend to Bessel sequences arising from the action of the group. We apply the results to sampling on bandlimited functions and to wavelet and Weyl-Heisenberg frames. This yields a sufficient condition for two sampling transforms to have orthogonal ranges, and two analysis operators for wavelet and Weyl-Heisenberg frames to have orthogonal ranges. The sufficient condition is easy to compute in terms of the periodization of the Fourier transform of the frame generators.
Second order parabolic equations in Banach spaces with dynamic boundary conditions
Ti-Jun
Xiao;
Jin
Liang
4787-4809
Abstract: In this paper, we exhibit a unified treatment of the mixed initial boundary value problem for second order (in time) parabolic linear differential equations in Banach spaces, whose boundary conditions are of a dynamical nature. Results regarding existence, uniqueness, continuous dependence (on initial data) and regularity of classical and strict solutions are established. Moreover, several examples are given as samples for possible applications.
On peak-interpolation manifolds for $\boldsymbol{A}\boldsymbol{(}\boldsymbol{\Omega}\boldsymbol{)}$ for convex domains in $\boldsymbol{\mathbb{C}}^{\boldsymbol{n}}$
Gautam
Bharali
4811-4827
Abstract: Let $\Omega$ be a bounded, weakly convex domain in ${\mathbb{C} }^n$, $n\geq 2$, having real-analytic boundary. $A(\Omega)$ is the algebra of all functions holomorphic in $\Omega$ and continuous up to the boundary. A submanifold $\boldsymbol{M}\subset \partial \Omega$ is said to be complex-tangential if $T_p(\boldsymbol{M})$ lies in the maximal complex subspace of $T_p(\partial \Omega)$ for each $p\in\boldsymbol{M}$. We show that for real-analytic submanifolds $\boldsymbol{M}\subset\partial \Omega$, if $\boldsymbol{M}$ is complex-tangential, then every compact subset of $\boldsymbol{M}$ is a peak-interpolation set for $A(\Omega)$.
An analogue of continued fractions in number theory for Nevanlinna theory
Zhuan
Ye
4829-4838
Abstract: We show an analogue of continued fractions in approximation to irrational numbers by rationals for Nevanlinna theory. The analogue is a sequence of points in the complex plane which approaches a given finite set of points and at a given rate in the sense of Nevanlinna theory.
Global Strichartz estimates for solutions to the wave equation exterior to a convex obstacle
Jason
L.
Metcalfe
4839-4855
Abstract: In this paper, we show that certain local Strichartz estimates for solutions of the wave equation exterior to a convex obstacle can be extended to estimates that are global in both space and time. This extends the work that was done previously by H. Smith and C. Sogge in odd spatial dimensions. In order to prove the global estimates, we explore weighted Strichartz estimates for solutions of the wave equation when the Cauchy data and forcing term are compactly supported.
On orbital partitions and exceptionality of primitive permutation groups
R.
M.
Guralnick;
Cai
Heng
Li;
Cheryl
E.
Praeger;
J.
Saxl
4857-4872
Abstract: Let $G$ and $X$ be transitive permutation groups on a set $\Omega$ such that $G$ is a normal subgroup of $X$. The overgroup $X$ induces a natural action on the set $\operatorname{Orbl}(G,\Omega)$ of non-trivial orbitals of $G$ on $\Omega$. In the study of Galois groups of exceptional covers of curves, one is led to characterizing the triples $(G,X,\Omega)$ where $X$fixes no elements of $\operatorname{Orbl}(G,\Omega)$; such triples are called exceptional. In the study of homogeneous factorizations of complete graphs, one is led to characterizing quadruples $(G,X,\Omega,\mathcal{P})$ where $\mathcal{P}$ is a partition of $\operatorname{Orbl}(G,\Omega)$ such that $X$ is transitive on $\mathcal{P}$; such a quadruple is called a TOD (transitive orbital decomposition). It follows easily that the triple $(G,X,\Omega)$ in a TOD $(G,X,\Omega,\mathcal{P})$is exceptional; conversely if an exceptional triple $(G,X,\Omega)$ is such that $X/G$ is cyclic of prime-power order, then there exists a partition $\mathcal{P}$ of $\operatorname{Orbl}(G,\Omega)$ such that $(G,X,\Omega,\mathcal{P})$ is a TOD. This paper characterizes TODs $(G,X,\Omega,\mathcal{P})$ such that $X^\Omega$ is primitive and $X/G$ is cyclic of prime-power order. An application is given to the classification of self-complementary vertex-transitive graphs.
Change of rings in deformation theory of modules
Runar
Ile
4873-4896
Abstract: Given a $B$-module $M$ and any presentation $B=A/J$, the obstruction theory of $M$ as a $B$-module is determined by the usual obstruction class $\mathrm{o}_{ \scriptscriptstyle{A}}^{\scriptscriptstyle{}}$ for deforming $M$ as an $A$-module and a new obstruction class $\mathrm{o}_{ \scriptscriptstyle{J}}^{\scriptscriptstyle{}}$. These two classes give the tool for constructing two obstruction maps which depend on each other and which characterise the hull of the deformation functor. We obtain relations between the obstruction classes by studying a change of rings spectral sequence and by representing certain classes as elements in the Yoneda complex. Calculation of the deformation functor of $M$ as a $B$-module, including the (generalised) Massey products, is thus possible within any $A$-free $2$-presentation of $M$.
A local limit theorem for closed geodesics and homology
Richard
Sharp
4897-4908
Abstract: In this paper, we study the distribution of closed geodesics on a compact negatively curved manifold. We concentrate on geodesics lying in a prescribed homology class and, under certain conditions, obtain a local limit theorem to describe the asymptotic behaviour of the associated counting function as the homology class varies.
Characterizations of regular almost periodicity in compact minimal abelian flows
Alica
Miller;
Joseph
Rosenblatt
4909-4929
Abstract: Regular almost periodicity in compact minimal abelian flows was characterized for the case of discrete acting group by W. Gottschalk and G. Hedlund and for the case of $0$-dimensional phase space by W. Gottschalk a few decades ago. In 1995 J. Egawa gave characterizations for the case when the acting group is $\mathbb{R}$. We extend Egawa's results to the case of an arbitrary abelian acting group and a not necessarily metrizable phase space. We then show how our statements imply previously known characterizations in each of the three special cases and give various other applications (characterization of regularly almost periodic functions on arbitrary abelian topological groups, classification of uniformly regularly almost periodic compact minimal $\mathbb{Z}$- and $\mathbb{R}$-flows, conditions equivalent with uniform regular almost periodicity, etc.).
The Perron-Frobenius theorem for homogeneous, monotone functions
Stéphane
Gaubert;
Jeremy
Gunawardena
4931-4950
Abstract: If $A$ is a nonnegative matrix whose associated directed graph is strongly connected, the Perron-Frobenius theorem asserts that $A$ has an eigenvector in the positive cone, $(\mathbb R^{+})^n$. We associate a directed graph to any homogeneous, monotone function, $f: (\mathbb R^{+})^n \rightarrow (\mathbb R^{+})^n$, and show that if the graph is strongly connected, then $f$ has a (nonlinear) eigenvector in $(\mathbb R^{+})^n$. Several results in the literature emerge as corollaries. Our methods show that the Perron-Frobenius theorem is ``really'' about the boundedness of invariant subsets in the Hilbert projective metric. They lead to further existence results and open problems.
Spectral properties and dynamics of quantized Henon maps
Brendan
Weickert
4951-4968
Abstract: We study a generalization of the Airy function, and use its properties to investigate the dynamics and spectral properties of the unitary operators on $L^2(\mathbf{R})$ of the form $U_c:=Fe^{i(q(x)+cx)}$, where $q$ is a real polynomial of odd degree, $c$ is a real number, and $F$ is the Fourier transform. We show that $U_c$ is a quantization of the classical Henon map \begin{align*}f_\lambda:\mathbf{R}^2 &\to \mathbf{R}^2 , (x,y) &\mapsto (y+q'(x)+c,-x), \end{align*} and show that for $c>0$ sufficiently large, $U_c$ has purely continuous spectrum. This fact has implications for the dynamics of $U_c$, which are shown to correspond when the condition is satisfied to the dynamics of its classical counterpart on $\mathbf{R}^2$.
Cross characteristic representations of even characteristic symplectic groups
Robert
M.
Guralnick;
Pham
Huu
Tiep
4969-5023
Abstract: We classify the small irreducible representations of $Sp_{2n}(q)$ with $q$ even in odd characteristic. This improves even the known results for complex representations. The smallest representation for this group is much larger than in the case when $q$ is odd. This makes the problem much more difficult.
Blakers-Massey elements and exotic diffeomorphisms of $S^6$ and $S^{14}$ via geodesics
C.
E.
Durán;
A.
Mendoza;
A.
Rigas
5025-5043
Abstract: We use the geometry of the geodesics of a certain left-invariant metric on the Lie group $Sp(2)$ to find explicit related formulas for two topological objects: the Blakers-Massey element (a generator of $\pi_6(S^3)$) and an exotic (i.e. not isotopic to the identity) diffeomorphism of $S^6$ (C. E. Durán, 2001). These formulas depend on two quaternions and their conjugates and we produce their extensions to the octonions through formulas for a generator of $\pi _{14}(S^{7})$ and exotic diffeomorphisms of $S^{14}$, thus giving explicit gluing maps for half of the 15-dimensional exotic spheres expressed as the union of two 15-disks.
Radon transforms on affine Grassmannians
Boris
Rubin
5045-5070
Abstract: We develop an analytic approach to the Radon transform $\hat f (\zeta)=\int_{\tau\subset \zeta} f (\tau)$, where $f(\tau)$ is a function on the affine Grassmann manifold $G(n,k)$ of $k$-dimensional planes in $\mathbb{R}^n$, and $\zeta$ is a $k'$-dimensional plane in the similar manifold $G(n,k'), \; k'>k$. For $f \in L^p (G(n,k))$, we prove that this transform is finite almost everywhere on $G(n,k')$ if and only if $\mathbb{R}^{n+1}$. It is proved that the dual Radon transform can be explicitly inverted for
On the representation of integers as linear combinations of consecutive values of a polynomial
Jacques
Boulanger;
Jean-Luc
Chabert
5071-5088
Abstract: Let $K$ be a cyclotomic field with ring of integers $\mathcal{O}_{K}$ and let $f$ be a polynomial whose values on $\mathbb{Z}$ belong to $\mathcal{O}_{K}$. If the ideal of $\mathcal{O}_{K}$ generated by the values of $f$ on $\mathbb{Z}$ is $\mathcal{O}_{K}$ itself, then every algebraic integer $N$ of $K$ may be written in the following form: \begin{displaymath}N=\sum_{k=1}^l\;\varepsilon_{k}f(k)\end{displaymath} for some integer $l$, where the $\varepsilon_{k}$'s are roots of unity of $K$. Moreover, there are two effective constants $A$ and $B$ such that the least integer $l$ (for a fixed $N$) is less than $A\,\widetilde{N}+B$, where \begin{displaymath}\widetilde{N}= \max_{\sigma\in Gal(K/\mathbb{Q} )} \; \vert \sigma (N) \vert.\end{displaymath}
Character sums and congruences with $n!$
Moubariz
Z.
Garaev;
Florian
Luca;
Igor
E.
Shparlinski
5089-5102
Abstract: We estimate character sums with $n!$, on average, and individually. These bounds are used to derive new results about various congruences modulo a prime $p$ and obtain new information about the spacings between quadratic nonresidues modulo $p$. In particular, we show that there exists a positive integer $n\ll p^{1/2+\varepsilon}$ such that $n!$ is a primitive root modulo $p$. We also show that every nonzero congruence class $a \not \equiv 0 \pmod p$can be represented as a product of 7 factorials, $a \equiv n_1! \ldots n_7! \pmod p$, where $\max \{n_i \vert i=1,\ldots, 7\}=O(p^{11/12+\varepsilon})$, and we find the asymptotic formula for the number of such representations. Finally, we show that products of 4 factorials $n_1!n_2!n_3!n_4!,$ with $\max\{n_1, n_2, n_3, n_4\}=O(p^{6/7+\varepsilon})$ represent ``almost all'' residue classes modulo p, and that products of 3 factorials $n_1!n_2!n_3!$ with $\max\{n_1, n_2, n_3\}=O(p^{5/6+\varepsilon})$ are uniformly distributed modulo $p$.