Spectral pairs in cartesian coordinates

Let ^d have finite positive Lebesgue measure, and let () be the corresponding Hilbert space of -functions on . We shall consider the exponential functionse on given bye (x)=e ^i2·x . If these functions form an orthogonal basis for (), when ranges over some subset in ^d , then we say that (, ) is a spectral pair, and that is a spectrum. We conjecture that (, ) is a spectral pair if and only if the translates of some set by the vectors of tile ^d. In the special case of =I^d, the d-dimensional unit cube, we prove this conjecture, with =I^d, for d3, describing all the tilings by I^d, and for all d when is a discrete periodic set. In an appendix we generalize the notion of spectral pair to measures on a locally compact abelian group and its dual.Acknowledgements and Notes, Work supported by the National Science Foundation.Dedicated to the memory of Irving E. Segal     

Subsets with Restricted Movement

Let G be a finite permutation group on a set with no fixed points in and let m and k be integers with 0 < m < k. For a finite subset of the movement of is defined as move() = max_gG| ^g \ |. Suppose further that G is not a 2-group and that p is the least odd prime dividing |G| and move() m for all k-element subsets of . Then either || k + m or k (7m – 5) / 2, || (9m – 3)/2. Moreover when || > k + m, then move() m for every subset of .     

The commutant of rationally cyclic subnormal operators and rational approximation

Let denote the set of analytic bounded point evaluations forR ^q (K, ). Assume that . In this paper, we first show that if is a finitely connected domain and if the evaluation map fromR ^q (K, )L () toH () is surjective, then | is absolutely continuous with respect to harmonic measure for . This generalizes Olin and Yang's corresponding result for polynomials and the proof we present here is simpler. We also provide an example that shows this absolute continuity property fails in general when is an infinitely connected domain. In the second part, we then offer a solution to a problem of Conway and Elias.     

Equivalence between the dilation and lifting properties of an ordered group through multiplicative families of isometries. A version of the commutant lifting theorem on some lexicographic groups

Let be a locally compact abelian ordered group. has the dilation property if a special extension of the Naimark dilation theorem holds for and it has the commutant lifting property if a natural extension of the Sz.-Nagy — Foias commutant lifting theorem holds for .We prove that these two conditions are equivalent and we give another necessary and sufficient condition in terms of unitary extensions of multiplicative families of partial isometries.A version of the commutant lifting theorem is given for the groups ⁿ and × ⁿ with the lexicographic order and the natural topologies.Both authors were partially supported by the CDCH of the Universidad Central de Venezuela, and by CONICIT grant G-97000668.     

Extension of Locally Defined Indefinite Functions on Ordered Groups

We give a definition of -indefinite function of archimedean type, on an interval of an ordered group with an archimedean point. We say that has the indefinite extension property if every continuous -indefinite function of archimedean type, on an interval of , can be extended to a continuous -indefinite function on the whole group .We show that if a group is semi-archimedean and it has the indefinite extension property, then with the lexicographic order and the product topology has the indefinite extension property. As a corollary it is obtained that the groups and with the lexicographic order and the usual topologies, have the indefinite extension property.To Professor José R. León, for his kind encouragement.Both authors were supported in part by the CDCH of the Univ. Central de Venezuela and by CONICIT grant G-97000668. Both authors were visitors at IVIC during the realization of this paper.Submitted: August 8, 2002 Revised: January 30, 2003     

The normal subgroup lattice of 2-transitive automorphism groups of linearly ordered sets

Using combinatorial and model-theoretic means, we examine the structure of normal subgroup lattices N(A()) of 2-transitive automorphism groups A() of infinite linearly ordered sets (, ). Certain natural sublattices of N(A()) are shown to be Stone algebras, and several first order properties of their dense and dually dense elements are characterized within the Dedekind-completion of (, ). As a consequence, A() has either precisely 5 or at least 2²¹ (even maximal) normal subgroups, and various other group- and lattice-theoretic results follow.     

New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.     

Semi-groupes sous-Markoviens engendrés par des opérateurs matriciels

We characterize generators of sub-Markovian semigroups onL ^p () by a version of Kato's inequality. This will be used to show (under precise assumptions) that the semigroup generated by a matrix operatorA=(A _ij )_1i,jn on (L ^p ()) ⁿ is sub-Markovian if and only if the semigroup generated by the sum of each rowA _i 1+...+A _in (1in), is sub-Markovian. The corresponding result on (C ₀(X)) ⁿ characterizes dissipative operator matrices.

     

On monotone extensions of boundary data

Summary A functionf C (), is called monotone on if for anyx, y the relation x – y ₊ ^s impliesf(x)f(y). Given a domain with a continuous boundary and given any monotone functionf on we are concerned with the existence and regularity ofmonotone extensions i.e., of functionsF which are monotone on all of and agree withf on . In particular, we show that there is no linear mapping that is capable of producing a monotone extension to arbitrarily given monotone boundary data. Three nonlinear methods for constructing monotone extensions are then presented. Two of these constructions, however, have the common drawback that regardless of how smooth the boundary data may be, the resulting extensions will, in general, only be Lipschitz continuous. This leads us to consider a third and more involved monotonicity preserving extension scheme to prove that, when is the unit square [0, 1]² in ², strictly monotone analytic boundary data admit a monotone analytic extension.Research supported by NSF Grant 8922154Research supported by DARPA: AFOSR #90-0323     

Relaxation of a Quadratic Functional Defined by a Nonnegative Unbounded Matrix

We study the lower semicontinuous envelope in L^p(), F, of a functional F of the form F(u)=A uudx where A=A(x) is not strictly elliptic and not bounded. We prove that F; may also be written as F;(u)= Buudx with B=AP A for a matrix P which is the matrix of an orthogonal projection. In the one-dimensional case, we characterize the domain of F and we explicit the matrix P.     

A-stable spline-collocation methods of multivalue type

In this paper the general classV of spline-collocation methods presented by Mülthei is investigated. The methods ofV approximate solutions of first order initial value problems. ClassV contains as subclass the methods of so-called multivalue type, and in particular contains the generalized singly-implicit methods treated by Butcher.Any multivalue type representativeU V yields a matrix valued function corresponding toU, which characterizes the region of absolute stability ofU. If a sequence (U()) of multivalue type representatives ofV tending to some singlevalue type representative V is considered, it can easily be seen by the structure of , that the sequence of the greatest eigenvalues of the (.,) tends to the stability function corresponding to . This fact allows one to construct one-parameter families of A-stable methods of multivalue type.     

Nonlocal problems in the theory of the motion equations of Kelvin-Voight fluids

For the motion equations of Kelvin-Voight fluids one proves: 1) a global theorem for the existence and uniqueness of a solution (v;{u_e}) of the initial-boundary value problem on the semiaxis t R⁺ from the class W ¹ (R⁺); W ₂ ² () H()) with initial condition v_o(x) W ₂ ² () H() when the right-hand side f(x, t) L(R ⁺; L₂()); 2) a global theorem for the existence and uniqueness of a solution (v; {u_l}) on the entire axisR from the classW ¹ (R; W ₂ ² () H()) when the right-hand side f(x, t) L(R; L₂()); 3) a global theorem for the existence of at least one solution (v; {u_l}), periodic with respect to t with period , from the class W ¹ (R ⁺; W ₂ ² () H()) when the right-hand side f(x, t) L(R ⁺; L₂()) is periodic with respect to t with period , and a local uniqueness theorem for such a solution; 4) a theorem for the existence and uniqueness in the small of a solution (v; {u_l}), almost periodic with respect to t R, from V. V. Stepanov's class S ¹ (R; W ₂ ² ()H()) when the right-hand side f(x, t) S(R; L₂()) is almost periodic with respect to t; 5) the linearization principle (Lyapunov's first method) is justified in the theory of the exponential stability of the solutions of an initial-boundary value problem in the space H() and conditions are given for the exponential stability of a stationary and periodic solution, with respect to t R, of the system (1).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 181, pp. 146–185, 1990.     

On Compensated Compactness for Nonlinear Elliptic Problems in Perforated Domains

We consider a sequence of Dirichlet problems for a nonlinear divergent operator A: W _m ¹( _s ) [W _m ¹( _s )]^* in a sequence of perforated domains _s . Under a certain condition imposed on the local capacity of the set \ _s , we prove the following principle of compensated compactness: , where r _s(x) and z _s(x) are sequences weakly convergent in W _m ¹() and such that r _s(x) is an analog of a corrector for a homogenization problem and z _s(x) is an arbitrary sequence from whose weak limit is equal to zero.     

Homogenization of Some Elasticity Problems with Rapidly Oscillating Convex Constraints on Displacements

The problem of homogenization is considered for an elastic body occupying a perforated domain = obtained from a fixed domain and an -contraction of a 1-periodic domain .     

Total flux estimates for a finite element approximation of the Dirichlet problem using the boundary penalty method

Summary This paper considers a fully practical piecewise linear finite element approximation of the Dirichlet problem for a second order self-adjoint elliptic equation,Au=f, in a smooth region< ⁿ (n=2 or 3) by the boundary penalty method. Using an unfitted mesh; that is ^h , an approximation of with dist (, ^h )Ch ² is not in general a union of elements; and assuminguH ⁴ () we show that one can recover the total flux across a segment of the boundary of with an error ofO(h ²⁾. We use these results to study a fully practical piecewise linear finite element approximation of an elliptic equation by the boundary penalty method when the prescribed data on part of the boundary is the total flux.Supported by a SERC research studentship     

Nonbranching weak and starshaped strong solutions for Hele-Shaw dynamics

We consider multidimensional weak and strong Hele-Shaw dynamics (t) of an advancing/receding viscous fluid injected/removed through a single finite point into/from a bounded domain (0). A class of weak solutions is shown to preserve local uniqueness in both directions. Then we also consider strong solutions (t), and show that if (0) is starshaped with respect to a small ball centered on the point of injection, then the evolution (t) exists for all time.Received: 5 March 2004     

Convergence of the highest derivatives in projection methods

Assume that for the approximate solution of an elliptic differential equation in a bounded domain , under a natural boundary condition, one applies the Galerkin method with polynomial coordinate functions. One gives sufficient conditions, imposed on the exact solutionu ^*, which ensure the convergence of the derivatives of order k of the approximate solutions, uniformly or in the mean in or in any interior subdomain. For example, ifu ^*W^k ₂, then the derivatives of order k converge in L₂(), where is an interior subdomain of . Somewhat weaker statements are obtained in the case of the Dirchlet problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 70, pp. 11–18, 1977.The author expresses his gratitude to Yu. K. Dem'yanovich for drawing his attention to [10].     

Some properties of the riesz charge associated with a δ-subharmonic function

The first property is a refinement of earlier results of Ch. de la Vallée Poussin, M. Brelot, and A. F. Grishin. Let w=u–v with u, v superharmonic on a suitable harmonic space (for example an open subset of R ⁿ ), and let [w]=[u]–[v] denote the associated Riesz charge. If w0, and if E denotes the set of those points of at which the lim inf of w in thefine topology is 0, then the restriction of [w] to E is 0. Another property states that, if e denotes a polar subset of such that the fine lim inf of |w| at each point of e is finite, then the restriction of [w] to e is 0.     

Extremal feedback perturbations for families of closed operators

Let T- S, be a family of not necessarily bounded semi-Fredholm operators, where T and S are operators acting between Banach spaces X and Y, and where S is bounded with D(S) D(T). For compact sets , as well as for certain open sets , we investigate existence and minimal rank of bounded feedback perturbations of the form F=BE such that min.ind (T-S+F)=0 for all . Here B is a given operator from a linear space Z to Y and E is some operator from X to Z.We give a simple characterization of that situation, when such regularizing feedback perturbations exist and show that for compact sets the minimal rank never exceeds max { min.ind (T-S) }+1. Moreover, an example shows that the minimal rank, in fact, may increase from max {...} to max {...}+1, if the given B enforces a certain structure of the feedbachk perturbation F.However, the minimal rank is equal to max { min.ind (T-S) }, if is an open set such that min.ind (T-S) already vanishes for all but finitely many points . We illustrate this result by applying it to the stabilization of certain infinite-dimensional dynamical systems in Hilbert space.     

Boundary-value problems for two-dimensional canonical systems

The two-dimensional canonical systemJy=–Hy where the nonnegative Hamiltonian matrix functionH(x) is trace-normed on (0, ) has been studied in a function-theoretic way by L. de Branges in [5]–[8]. We show that the Hamiltonian system induces a closed symmetric relation which can be reduced to a, not necessarily densely defined, symmetric operator by means of Kac' indivisible intervals; of. [33], [34]. The formal defect numbers related to the system are the defect numbers of this reduced minimal symmetric operator. By using de Branges' one-to-one correspondence between the class of Nevanlinna functions and such canonical systems we extend our canonical system from (0, ) to a trace-normed system on which is in the limit-point case at ±. This allows us to study all possible selfadjoint realizations of the original system by means of a boundaryvalue problem for the extended canonical system involving an interface condition at 0.