Group actions on graphs related to Krishnan-Sunder subfactors
Bina
Bhattacharyya
433-463
Abstract: We describe the principal graphs of the subfactors studied by Krishnan and Sunder in terms of group actions on Cayley-type graphs. This leads to the construction of a tower of tree algebras, for every positive integer $k$, which are symmetries of the Krishnan-Sunder subfactors of index $k^2$. Using our theory, we prove that the principal graph of the irreducible infinite depth subfactor of index 9 constructed by Krishnan and Sunder is not a tree, contrary to their expectations. We also show that the principal graphs of the Krishnan-Sunder subfactors of index 4 are the affine A and D Coxeter graphs.
The Laplacian MASA in a free group factor
Allan
M.
Sinclair;
Roger
R.
Smith
465-475
Abstract: The Laplacian (or radial) masa in a free group factor is generated by the sum of the generators and their inverses. We show that such a masa $\mathcal{B}$is strongly singular and has Popa invariant $\delta(\mathcal{B}) = 1$. This is achieved by proving that the conditional expectation $\mathbb{E} _{\mathcal{B}}$ onto $\mathcal{B}$ is an asymptotic homomorphism. We also obtain similar results for the free product of discrete groups, each of which contains an element of infinite order.
The co-area formula for Sobolev mappings
Jan
Maly;
David
Swanson;
William
P.
Ziemer
477-492
Abstract: We extend Federer's co-area formula to mappings $f$ belonging to the Sobolev class $W^{1,p}(\mathbb{R}^n;\mathbb{R}^m)$, $1 \le m < n$, $p>m$, and more generally, to mappings with gradient in the Lorentz space $L^{m,1}(\mathbb{R}^n)$. This is accomplished by showing that the graph of $f$ in $\mathbb{R}^{n+m}$is a Hausdorff $n$-rectifiable set.
The radius of metric regularity
A.
L.
Dontchev;
A.
S.
Lewis;
R.
T.
Rockafellar
493-517
Abstract: Metric regularity is a central concept in variational analysis for the study of solution mappings associated with ``generalized equations'', including variational inequalities and parameterized constraint systems. Here it is employed to characterize the distance to irregularity or infeasibility with respect to perturbations of the system structure. Generalizations of the Eckart-Young theorem in numerical analysis are obtained in particular.
The Orevkov invariant of an affine plane curve
Walter
D.
Neumann;
Paul
Norbury
519-538
Abstract: We show that although the fundamental group of the complement of an algebraic affine plane curve is not easy to compute, it possesses a more accessible quotient, which we call the Orevkov invariant.
Linear systems of plane curves with a composite number of base points of equal multiplicity
Anita
Buckley;
Marina
Zompatori
539-549
Abstract: In this article we study linear systems of plane curves of degree $d$ passing through general base points with the same multiplicity at each of them. These systems are known as homogeneous linear systems. We especially investigate for which of these systems, the base points, with their multiplicities, impose independent conditions and which homogeneous systems are empty. Such systems are called non-special. We extend the range of homogeneous linear systems that are known to be non-special. A theorem of Evain states that the systems of curves of degree $d$ with $4^h$ base points with equal multiplicity are non-special. The analogous result for $9^h$ points was conjectured. Both of these will follow, as corollaries, from the main theorem proved in this paper. Also, the case of $4^{h}9^{k}$ points will follow from our result. The proof uses a degeneration technique developed by C. Ciliberto and R. Miranda.
Base loci of linear series are numerically determined
Michael
Nakamaye
551-566
Abstract: We introduce a numerical invariant, called a moving Seshadri constant, which measures the local positivity of a big line bundle at a point. We then show how moving Seshadri constants determine the stable base locus of a big line bundle.
Formulas for tamely ramified supercuspidal characters of $\operatorname{GL}_3$
Tetsuya
Takahashi
567-591
Abstract: Let $F$ denote a $p$-adic local field of residual characteristic $p\ne3$. This article gives formulas, valid on the regular elliptic set, for the irreducible supercuspidal characters of $\operatorname{GL}_3(F)$ which correspond to characters of a ramified Cartan subgroup. In the case in which $F$ does not contain cube roots of unity, i.e., the case in which ramified cubic extensions of degree $3$ over $F$ cannot be Galois, base change results concerning ``simple types" due to Bushnell and Henniart (1996) are used in the proofs.
Resolutions of ideals of quasiuniform fat point subschemes of $\mathbf P^2$
Brian
Harbourne;
Sandeep
Holay;
Stephanie
Fitchett
593-608
Abstract: The notion of a quasiuniform fat point subscheme $Z\subset\mathbf P^2$is introduced and conjectures for the Hilbert function and minimal free resolution of the ideal $I$ defining $Z$ are put forward. In a large range of cases, it is shown that the Hilbert function conjecture implies the resolution conjecture. In addition, the main result gives the first determination of the resolution of the $m$th symbolic power $I(m;n)$ of an ideal defining $n$ general points of $\mathbf P^2$ when both $m$ and $n$ are large (in particular, for infinitely many $m$ for each of infinitely many $n$, and for infinitely many $n$ for every $m>2$). Resolutions in other cases, such as ``fat points with tails'', are also given. Except where an explicit exception is made, all results hold for an arbitrary algebraically closed field $k$. As an incidental result, a bound for the regularity of $I(m;n)$ is given which is often a significant improvement on previously known bounds.
Hyperplane arrangements and linear strands in resolutions
Irena
Peeva
609-618
Abstract: The cohomology ring of the complement of a central complex hyperplane arrangement is the well-studied Orlik-Solomon algebra. The homotopy group of the complement is interesting, complicated, and few results are known about it. We study the ranks for the lower central series of such a homotopy group via the linear strand of the minimal free resolution of the field $\mathbf{C}$ over the Orlik-Solomon algebra.
Test ideals and base change problems in tight closure theory
Ian
M.
Aberbach;
Florian
Enescu
619-636
Abstract: Test ideals are an important concept in tight closure theory and their behavior via flat base change can be very difficult to understand. Our paper presents results regarding this behavior under flat maps with reasonably nice (but far from smooth) fibers. This involves analyzing, in depth, a special type of ideal of test elements, called the CS test ideal. Besides providing new results, the paper also contains extensions of a theorem by G. Lyubeznik and K. E. Smith on the completely stable test ideal and of theorems by F. Enescu and, independently, M. Hashimoto on the behavior of $F$-rationality under flat base change.
On partitioning the orbitals of a transitive permutation group
Cai
Heng
Li;
Cheryl
E.
Praeger
637-653
Abstract: Let $G$ be a permutation group on a set $\Omega$ with a transitive normal subgroup $M$. Then $G$ acts on the set $\mathrm{Orbl}(M,\Omega)$ of nontrivial $M$-orbitals in the natural way, and here we are interested in the case where $\mathrm{Orbl}(M,\Omega)$ has a partition $\mathcal P$ such that $G$ acts transitively on $\mathcal P$. The problem of characterising such tuples $(M,G,\Omega,\mathcal P)$, called TODs, arises naturally in permutation group theory, and also occurs in number theory and combinatorics. The case where $\vert\mathcal P\vert$ is a prime-power is important in algebraic number theory in the study of arithmetically exceptional rational polynomials. The case where $\vert\mathcal P\vert=2$ exactly corresponds to self-complementary vertex-transitive graphs, while the general case corresponds to a type of isomorphic factorisation of complete graphs, called a homogeneous factorisation. Characterising homogeneous factorisations is an important problem in graph theory with applications to Ramsey theory. This paper develops a framework for the study of TODs, establishes some numerical relations between the parameters involved in TODs, gives some reduction results with respect to the $G$-actions on $\Omega$ and on $\mathcal P$, and gives some construction methods for TODs.
A generalized Minkowski problem with Dirichlet boundary condition
Oliver
C.
Schnurer
655-663
Abstract: We prove the existence of hypersurfaces with prescribed boundary whose Weingarten curvature equals a given function that depends on the normal of the hypersurface.
The $L^p$ Dirichlet problem and nondivergence harmonic measure
Cristian
Rios
665-687
Abstract: We consider the Dirichlet problem \begin{displaymath}\left\{ \begin{array}{rcl} \mathcal{L} u & = & 0\quad\text{ in }D, u & = & g\quad\text{ on }\partial D \end{array}\right.\end{displaymath} for two second-order elliptic operators $\mathcal{L}_k u=\sum_{i,j=1}^na_k^{i,j}(x)\,\partial_{ij} u(x)$, $k=0,1$, in a bounded Lipschitz domain $D\subset\mathbb{R} ^n$. The coefficients $a_k^{i,j}$ belong to the space of bounded mean oscillation ${{BMO}}$ with a suitable small ${{BMO}}$ modulus. We assume that ${\mathcal{L}}_0$ is regular in $L^p(\partial D, d\sigma)$ for some $p$, $1<p<\infty$, that is, $\Vert Nu\Vert _{L^p}\le C\,\Vert g\Vert _{L^p}$ for all continuous boundary data $g$. Here $\sigma$ is the surface measure on $\partial D$ and $Nu$ is the nontangential maximal operator. The aim of this paper is to establish sufficient conditions on the difference of the coefficients $\varepsilon^{i,j}(x)=a^{i,j}_1(x)-a^{i,j}_0(x)$ that will assure the perturbed operator $\mathcal{L}_1$ to be regular in $L^q(\partial D,d\sigma)$ for some $q$, $1<q<\infty$.
Estimations $L^p$ des solutions de l'équation des ondes sur certaines variétés coniques
Hong-Quan
Li;
Noël
Lohoue
689-711
Abstract: We prove R. Strichartz's $L^p$ estimates for solutions of the wave equation on some conical manifolds. RÉSUMÉ. On prouve des estimations $L^p$ pour les solutions de l'équation des ondes, analogues aux estimations de R. Strichartz, sur certaines variétés coniques.
From local to global behavior in competitive Lotka-Volterra systems
E.
C.
Zeeman;
M.
L.
Zeeman
713-734
Abstract: In this paper we exploit the linear, quadratic, monotone and geometric structures of competitive Lotka-Volterra systems of arbitrary dimension to give geometric, algebraic and computational hypotheses for ruling out non-trivial recurrence. We thus deduce the global dynamics of a system from its local dynamics. The geometric hypotheses rely on the introduction of a split Liapunov function. We show that if a system has a fixed point $p\in\operatorname{int}{{\mathbf R}^n_+}$ and the carrying simplex of the system lies to one side of its tangent hyperplane at $p$, then there is no nontrivial recurrence, and the global dynamics are known. We translate the geometric hypotheses into algebraic hypotheses in terms of the definiteness of a certain quadratic function on the tangent hyperplane. Finally, we derive a computational algorithm for checking the algebraic hypotheses, and we compare this algorithm with the classical Volterra-Liapunov stability theorem for Lotka-Volterra systems.
Axiom A flows with a transverse torus
C.
A.
Morales
735-745
Abstract: Let $X$ be an Axiom A flow with a transverse torus $T$ exhibiting a unique orbit $O$ that does not intersect $T$. Suppose that there is no null-homotopic closed curve in $T$ contained in either the stable or unstable set of $O$. Then we show that $X$ has either an attracting periodic orbit or a repelling periodic orbit or is transitive. In particular, an Anosov flow with a transverse torus is transitive if it has a unique periodic orbit that does not intersect the torus.
Exponential averaging for Hamiltonian evolution equations
Karsten
Matthies;
Arnd
Scheel
747-773
Abstract: We derive estimates on the magnitude of non-adiabatic interaction between a Hamiltonian partial differential equation and a high-frequency nonlinear oscillator. Assuming spatial analyticity of the initial conditions, we show that the dynamics can be transformed to the uncoupled dynamics of an infinite-dimensional Hamiltonian system and an anharmonic oscillator, up to coupling terms which are exponentially small in a certain power of the frequency of the oscillator. The result is derived from an abstract averaging theorem for infinite-dimensional analytic evolution equations in Gevrey spaces. Refining upon a similar result by Neishtadt for analytic ordinary differential equations, the temporal estimate crucially depends on the spatial regularity of the initial condition. The result shows to what extent the strong resonances between rapid forcing and highly oscillatory spatial modes can be suppressed by the choice of sufficiently smooth initial data. An application is provided by a system of nonlinear Schrödinger equations, coupled to a rapidly forcing single mode, representing small-scale oscillations. We provide an example showing that the estimates for partial differential equations we derive here are necessarily different from those in the context of ordinary differential equations.
On the nonexistence of closed timelike geodesics in flat Lorentz 2-step nilmanifolds
Mohammed
Guediri
775-786
Abstract: The main purpose of this paper is to prove that there are no closed timelike geodesics in a (compact or noncompact) flat Lorentz 2-step nilmanifold $N/\Gamma ,$ where $N$ is a simply connected 2-step nilpotent Lie group with a flat left-invariant Lorentz metric, and $\Gamma$ a discrete subgroup of $N$ acting on $N$ by left translations. For this purpose, we shall first show that if $N$ is a 2-step nilpotent Lie group endowed with a flat left-invariant Lorentz metric $g,$ then the restriction of $g$ to the center $Z$of $N$ is degenerate. We shall then determine all 2-step nilpotent Lie groups that can admit a flat left-invariant Lorentz metric. We show that they are trivial central extensions of the three-dimensional Heisenberg Lie group $H_{3}$. If $\left( N,g\right)$ is one such group, we prove that no timelike geodesic in $\left( N,g\right)$ can be translated by an element of $N.$ By the way, we rediscover that the Heisenberg Lie group $H_{2k+1}$admits a flat left-invariant Lorentz metric if and only if $k=1.$
Quasiconformal groups, Patterson-Sullivan theory, and local analysis of limit sets
Petra
Bonfert-Taylor;
Edward
C.
Taylor
787-811
Abstract: We extend the part of Patterson-Sullivan theory to discrete quasiconformal groups that relates the exponent of convergence of the Poincaré series to the Hausdorff dimension of the limit set. In doing so we define new bi-Lipschitz invariants that localize both the exponent of convergence and the Hausdorff dimension. We find these invariants help to expose and explain the discrepancy between the conformal and quasiconformal setting of Patterson-Sullivan theory.
Notes on interpolation in the generalized Schur class. II. Nudel$'$man's problem
D.
Alpay;
T.
Constantinescu;
A.
Dijksma;
J.
Rovnyak
813-836
Abstract: An indefinite generalization of Nudel$'$man's problem is used in a systematic approach to interpolation theorems for generalized Schur and Nevanlinna functions with interior and boundary data. Besides known results on existence criteria for Pick-Nevanlinna and Carathéodory-Fejér interpolation, the method yields new results on generalized interpolation in the sense of Sarason and boundary interpolation, including properties of the finite Hilbert transform relative to weights. The main theorem appeals to the Ball and Helton almost-commutant lifting theorem to provide criteria for the existence of a solution to Nudel$'$man's problem.
Analytic models for commuting operator tuples on bounded symmetric domains
Jonathan
Arazy;
Miroslav
Englis
837-864
Abstract: For a domain $\Omega$ in $\mathbb{C} ^{d}$ and a Hilbert space $\mathcal{H}$of analytic functions on $\Omega$ which satisfies certain conditions, we characterize the commuting $d$-tuples $T=(T_{1},\dots ,T_{d})$ of operators on a separable Hilbert space $H$ such that $T^{*}$ is unitarily equivalent to the restriction of $M^{*}$ to an invariant subspace, where $M$ is the operator $d$-tuple $Z\otimes I$ on the Hilbert space tensor product $\mathcal{H} \otimes H$. For $\Omega$ the unit disc and $\mathcal{H}$ the Hardy space $H^{2}$, this reduces to a well-known theorem of Sz.-Nagy and Foias; for $\mathcal{H}$a reproducing kernel Hilbert space on $\Omega \subset \mathbb{C} ^{d}$ such that the reciprocal $1/K(x,\overline{y})$ of its reproducing kernel is a polynomial in $x$ and $\overline y$, this is a recent result of Ambrozie, Müller and the second author. In this paper, we extend the latter result by treating spaces $\mathcal{H}$ for which $1/K$ ceases to be a polynomial, or even has a pole: namely, the standard weighted Bergman spaces (or, rather, their analytic continuation) $\mathcal{H} =\mathcal{H} _{\nu }$ on a Cartan domain corresponding to the parameter $\nu$ in the continuous Wallach set, and reproducing kernel Hilbert spaces $\mathcal{H}$ for which $1/K$ is a rational function. Further, we treat also the more general problem when the operator $M$is replaced by $M\oplus W$, $W$ being a certain generalization of a unitary operator tuple. For the case of the spaces $\mathcal{H} _{\nu }$ on Cartan domains, our results are based on an analysis of the homogeneous multiplication operators on $\Omega$, which seems to be of an independent interest.