Showing papers in "Transactions of the American Mathematical Society in 1999"

Model theory of difference fields

Abstract: A difference field is a field with a distinguished automorphism σ. This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory on formulas, and in particular on difference equations. We show that an arbitrary formula may be reduced into one-dimensional ones, and analyze the possible internal structures on the one-dimensional formulas when the characteristic is 0.

Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum

Abstract: We discuss results where the discrete spectrum (or partial information on the discrete spectrum) and partial information on the potential q of a one-dimensional Schrodinger operator H = -(d^(2)/(dx^(2)) + q determine the potential completely. Included are theorems for finite intervals and for the whole line. In particular, we pose and solve a new type of inverse spectral problem involving fractions of the eigenvalues of H on a finite interval and knowledge of q over a corresponding fraction of the interval. The methods employed rest on Weyl m-function techniques and densities of zeros of a class of entire functions.

An approach to symmetrization via polarization

Abstract: We prove that the Steiner symmetrization of a function can be approximated in L p (R n ) by a sequence of very simple rearrangements which are called polarizations. This result is exploited to develop elementary proofs of many inequalities, including the isoperimetric inequality in Euclidean space. In this way we also obtain new symmetry results for solutions of some varia- tional problems. Furthermore we compare the solutions of two boundary value problems, one of them having a "polarized" geometry and we show some point- wise inequalities between the solutions. This leads to new proofs of well-known functional inequalities which compare the solutions of two elliptic or parabolic problems, one of them having a "Steiner-symmetrized" geometry. The method also allows us to investigate the case of equality in the inequalities. Roughly speaking we prove that the equality sign is valid only if the original problem has the symmetrized geometry.

Banach spaces with the daugavet property

Abstract: A Banach space X is said to have the Daugavet property if every operator T: X -* X of rank 1 satisfies 11 Id +Tl= 1 + flTIl. We show that then every weakly compact operator satisfies this equation as well and that X contains a copy of t1. However, X need not contain a copy of L1. We also study pairs of spaces X C Y and operators T: X -* Y satisfying I I J + T I I -_ 1-4- 1f T I I, where J: X -* Y is the natural embedding. This leads to the result that a Banach space with the Daugavet property does not embed into a space with an unconditional basis. In another direction, we investigate spaces where the set of operators with f Id +Tll 1 + Il T l I is as small as possible and give characterisations in terms of a smoothness condition.

Periodic traveling waves and locating oscillating patterns in multidimensional domains

Abstract: We establish the existence and robustness of layered, time-periodic solutions to a reaction-diffusion equation in a bounded domain in Rn, when the diffusion coefficient is sufficiently small and the reaction term is periodic in time and bistable in the state variable. Our results suggest that these patterned, oscillatory solutions are stable and locally unique. The location of the internal layers is characterized through a periodic traveling wave problem for a related one-dimensional reaction-diffusion equation. This one-dimensional problem is of independent interest and for this we establish the existence and uniqueness of a heteroclinic solution which, in constant-velocity moving coodinates, is periodic in time. Furthermore, we prove that the manifold of translates of this solution is globally exponentially asymptotically stable.

Conformal iterated function systems with applications to the geometry of continued fractions

Abstract: In this paper we obtain some results about general conformal iterated function systems. We obtain a simple characterization of the packing dimension of the limit set of such systems and introduce some special systems which exhibit some interesting behavior. We then apply these results to the set of values of real continued fractions with restricted entries. We pay special attention to the Hausdorff and packing measures of these sets. We also give direct interpretations of these measure theoretic results in terms of the arithmetic density properties of the set of allowed entries.

Sums of squares of regular functions on real algebraic varieties

Abstract: Let V be an affine algebraic variety over R (or any other real closed field R). We ask when it is true that every positive semidefinite (psd) polynomial function on V is a sum of squares (sos). We show that for dimV ≥ 3 the answer is always negative if V has a real point. Also, if V is a smooth non-rational curve all of whose points at infinity are real, the answer is again negative. The same holds if V is a smooth surface with only real divisors at infinity. The “compact” case is harder. We completely settle the case of smooth curves of genus ≤ 1: If such a curve has a complex point at infinity, then every psd function is sos, provided the field R is archimedean. If R is not archimedean, there are counter-examples of genus 1.

Extending partial automorphisms and the profinite topology on free groups

Abstract: A class of structures C is said to have the extension property for partial automorphisms (EPPA) if, whenever C1 and C2 are structures in C, C1 finite, C1 ⊆ C2, and p1, p2, . . . , pn are partial automorphisms of C1 extending to automorphisms of C2, then there exist a finite structure C3 in C and automorphisms α1, α2, . . . , αn of C3 extending the pi. We will prove that some classes of structures have the EPPA and show the equivalence of these kinds of results with problems related with the profinite topology on free groups. In particular, we will give a generalisation of the theorem, due to Ribes and Zalesskĭı stating that a finite product of finitely generated subgroups is closed for this topology.

Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations

Abstract: We prove a local existence and uniqueness theorem for abstract parabolic problems of the type x = Ax-{-f(t, x) when the nonlinearity f satisfies certain critical conditions. We apply this abstract result to the Navier-Stokes and heat equations. ©1999 American Mathematical Society.

Inverse eigenvalue problems on directed graphs

Abstract: The differential operators iD and −D2 − p are constructed on certain finite directed weighted graphs. Two types of inverse spectral problems are considered. First, information about the graph weights and boundary conditions is extracted from the spectrum of −D2. Second, the compactness of isospectral sets for −D2 − p is established by computation of the residues of the zeta function.

Even valuations on convex bodies

Abstract: The notion of eveiL valuation is introduced as a natural generalization of volume on cornpact convex subsets of Euclidean space. A recent characterization theorem for volume leads in turn to a connection between even valuations on compact convex sets and continuous functions on Grassmannians. This connection can be described in part using generating distributions for symmetric compact convex sets. We also explore some consequences of these characterization results in convex and integral geometry. Recent interest in volume as a valuation on compact conlvex sets stems from Hilbert's Third Problem, which is actually an ancient problem recast in modern terms. Hilbert asked if two polytopes P and Q can be each cut into a finite number of pieces P.,... , Pn and Q ,... , Qm with each Pi congruent to Qi by a rigid Euclidean motion, provided that P and Q have the same volume [18]. This question was answered in the negative by Max Dehn [2, 4, 29], who found a functionlal on polytopes that is invariant under dissections over rigid motions, while varying in value among polytopes of equal volume. In other words, the Dehn invariant is a "simple rigid motion invariant valuation" on polytopes that is not equal to volume (under any normalization). Dehn's solution left open the question of exactly what conditions oil P and Q imply equidissectability over the group of rigid motions, although this problem was solved by Hadwiger in the case where only translations (and no rotations nor reflections) are permitted (see [2, 16, 17, 26, 29]). In the course of studying this and related problems, Hadwiger discovered a characterization of Euclidean volume as a continuous rigid motion invariant simple valuation on compact convex sets, that is, a continuous rigid motion invariant valuation that vanishes on convex sets of less than full dimension. This result led in turn to a complete characterization of all continuous rigid motion inivariant valuations on compact convex sets in 1R' as consisting of a real (n 4)-dimensional vector space spanned by the intrinsic volumes (or Quermassintegrals) [16] (also [20, 21, 31]). Since many standard functionals and integral operators can be interpreted as invariant valuations (such as intrinsic volumes, mean projections, Crofton and kinematic formulas), what came to be known as Hadwiger's chtracterization theorem proved to be a valuable tool for generating quick and effortless proofs of many formulas and equations in integral geometry. Unfortunately Hadwiger's original proof was long and difficult [161 While seeking a shorter proof of Hadwiger's volume characterization, the author discovered Received by the editors June 24, 1996 and, in revised form, September 29, 1997. 1991 Mathematics Subject Classification. Primary 52A22, 52A38, 52A39, 52B45. Research supported in part by NSF grants #DMS 9022140 to MSRI and #DMS 9626688 to the author. ? 1999 American Mathematical Society 71 This content downloaded from 207.46.13.52 on Mon, 24 Oct 2016 04:15:57 UTC All use subject to http://about.jstor.org/terms

Densities of idempotent measures and large deviations

Abstract: Considering measure theory in which the semifield of positive real numbers is replaced by an idempotent semiring leads to the notion of idempotent measure introduced by Maslov. Then, idempotent measures or integrals with density correspond to supremums of functions for the partial order relation induced by the idempotent structure. In this paper, we give conditions under which an idempotent measure has a density and show by many examples that they are often satisfied. These conditions depend on the lattice structure of the semiring and on the Boolean algebra in which the measure is defined. As an application, we obtain a necessary and sufficient condition for a family of probabilities to satisfy the large deviation principle as defined by Varadhan.

Nonlinear equations and weighted norm inequalities

Abstract: We study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities. In particular, we obtain explicit criteria for the solvability of the Dirichlet problem −∆u = v u + w, u ≥ 0 on Ω, u = 0 on ∂Ω, on a regular domain Ω in Rn in the “superlinear case” q > 1. The coefficients v, w are arbitrary positive measurable functions (or measures) on Ω. We also consider more general nonlinear differential and integral equations, and study the spaces of coefficients and solutions naturally associated with these problems, as well as the corresponding capacities. Our characterizations of the existence of positive solutions take into account the interplay between v, w, and the corresponding Green’s kernel. They are not only sufficient, but also necessary, and are established without any a priori regularity assumptions on v and w; we also obtain sharp two-sided estimates of solutions up to the boundary. Some of our results are new even if v ≡ 1 and Ω is a ball or half-space. The corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so-called 3G-inequality by an elementary “integration by parts” argument. This also gives a new unified proof for some classical inequalities including the Carleson measure theorem for Poisson integrals and trace inequalities for Riesz potentials and Green potentials.

A Littlewood-Richardson rule for factorial Schur functions

Abstract: We give a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial Schur functions. It is a special case of a more general rule which also gives the coefficients in the expansion of a skew factorial Schur function. Applications to Capelli operators and quantum immanants are also given.

A Combinatorial Proof of Bass’s Evaluations of the Ihara-Selberg Zeta Function for Graphs

Abstract: We derive combinatorial proofs of the main two evaluations of the Ihara-Selberg Zeta function associated with a graph. We give three proofs of the first evaluation all based on the algebra of Lyndon words. In the third proof it is shown that the first evaluation is an immediate consequence of Amitsur's identity on the characteristic polynomial of a sum of matrices. The second evaluation of the Ihara-Selberg Zeta function is first derived by means of a sign-changing involution technique. Our second approach makes use of a short matrix-algebra argument.

Resolutions of monomial ideals and cohomology over exterior algebras

Abstract: This paper studies the homology of finite modules over the exterior algebra E of a vector space V . To such a module M we associate an algebraic set VE(M) ⊆ V , consisting of those v ∈ V that have a non-minimal annihilator in M . A cohomological description of its defining ideal leads, among other things, to complementary expressions for its dimension, linked by a ‘depth formula’. Explicit results are obtained for M = E/J , when J is generated by products of elements of a basis e1, . . . , en of V . A (infinite) minimal free resolution of E/J is constructed from a (finite) minimal resolution of S/I, where I is the squarefree monomial ideal generated by ‘the same’ products of the variables in the polynomial ring S = K[x1, . . . , xn]. It is proved that VE(E/J) is the union of the coordinate subspaces of V , spanned by subsets of { e1, . . . , en } determined by the Betti numbers of S/I over S.

Hardy spaces, BMO, and boundary value problems for the Laplacian on a smooth domain in ^{}

Abstract: We study two different local Hp spaces, 0 < p ≤ 1, on a smooth domain in Rn, by means of maximal functions and atomic decomposition. We prove the regularity in these spaces, as well as in the corresponding dual BMO spaces, of the Dirichlet and Neumann problems for the Laplacian. 0. Introduction Let Ω be a bounded domain in R, with smooth boundary. The L regularity of elliptic boundary value problems on Ω, for 1 < p < ∞, is a classical result in the theory of partial differential equations (see e.g. [ADN]). In the situation of the whole space without boundary, i.e. where Ω is replaced by R, the results for L, 1 < p < ∞, extend to the Hardy spaces H when 0 < p ≤ 1 and to BMO. Thus it is a natural question to ask whether the L regularity of elliptic boundary value problems on a domain Ω has an H and BMO analogue, and what are the H and BMO spaces for which it holds. This question was previously studied in [CKS], where partial results were obtained and were framed in terms of a pair of spaces, hr(Ω) and h p z(Ω). These spaces, variants of those defined in [M] and [JSW], are, roughly speaking, the “largest” and “smallest” h spaces that can be associated to a domain Ω. Our purpose here is to substantially extend the previous results by determining those h spaces on Ω which are particularly applicable to boundary value problems. These spaces allow one to prove sharp results (preservation of the appropriate h spaces) for all values of p, 0 < p ≤ 1, as well as the preservation of corresponding spaces of BMO functions. 0.1. Motivation and statement of results. There are two approaches to defining the appropriate Hardy spaces on Ω. Recall that the spaces H(R), for p < 1, are spaces of distributions. Thus one approach is to look at the problem from the point of view of distributions on Ω. If we denote by D(Ω) the space of smooth functions with compact support in Ω, and by D′(Ω) its dual, we can consider the space of distributions in D′(Ω) which are the restriction to Ω of distributions in H(R) (or in h(R), the local Hardy spaces defined in [G].) These spaces were studied in [M] (for arbitrary open sets) and in [CKS] (for Lipschitz domains), where they were denoted hr(Ω) (the r stands for “restriction”.) While one is able to prove regularity results for the Dirichlet problem for these spaces when p is near 1 (see [CKS]), these spaces are no longer appropriate when p Received by the editors September 5, 1996 and, in revised form, March 20, 1997. 1991 Mathematics Subject Classification. Primary 35J25, 42B25; Secondary 46E15, 42B30. c ©1999 American Mathematical Society

Chern classes for singular hypersurfaces

Abstract: We prove a formula expressing the Chern-Schwartz-MacPherson class of a hypersurface in a nonsingular variety as a variation on another definition of the homology Chern class of singular varieties, introduced by W. Fulton; and we discuss the relation between these classes and others, such as Mather's Chern class and the ,l-class we introduced in previous work. 0. INTRODUCTION There are several candidates for a notion of homology 'Chern class' of a (possibly singular) algebraic variety X, agreeing with the class obtained as dual of the total Chern class of the tangent bundle of X if X is nonsingular. One such notion was introduced by R. MacPherson [MacPherson], agreeing (as it was later understood) with a class introduced earlier by M.-IJ. Schwartz, and enjoying good functorial properties. A different class was defined by W. Fulton ([Fulton], 4.2.6), in a different and somewhat more general setting. The main purpose of this paper is to prove a precise relation between the Schwartz-MacPherson and the Fulton class of a hypersurface X of a nonsingular variety. We denote these two classes by csM(X), CF (X) respectively. A summary of the context and essential definitions is given in ?1, where we present several different statements of the result. The version which expresses most directly the link between the two classes mentioned above goes as follows. Fulton's class for a subscheme X of a nonsingular variety M is defined by CF(X) :m= c(TMIx) n s(X, M), that is, by capping the total Chern class of the ambient space against the Segre class of the subscheme (it is proved in [Fulton], 4.2.6 that this definition does not depend on the choice of the ambient variety M). For a hypersurface X of a nonsingular variety M, let Y be the 'singular subscheme' of X (locally defined by the partial derivatives of an equation of X). For an integer k > 0 we can consider the k-th thickening X(k) of X along Y, and then consider its Fulton class CF(X(k)). It is easily seen that CF (X(k)) is a polynomial in k (with coefficients in the Chow group A,*(X)), so it can be formally evaluated at negative k's. We can then define a class (* C*(X) :_ CF(X( 1))The main result of this paper is Theorem. c* (X) is equal to the Chern-Schwartz-MacPherson class of X. Received by the editors June 3, 1997. 1991 Mathematics Subject Classification. Prinmary 14C17, 32S60. Supported in part by NSF grant DMS-9500843. @ 1999 American Mathbemiatical Society 3989 This content downloaded from 157.55.39.136 on Thu, 01 Dec 2016 05:45:16 UTC All use subject to http://about.jstor.org/terms

The -Lagrangian of a convex function

Abstract: At a given point p, a convex function f is differentiable in a certain subspace U (the subspace along which ∂f(p) has 0-breadth). This property opens the way to defining a suitably restricted second derivative of f at p. We do this via an intermediate function, convex on U . We call this function the U-Lagrangian; it coincides with the ordinary Lagrangian in composite cases: exact penalty, semidefinite programming. Also, we use this new theory to design a conceptual pattern for superlinearly convergent minimization algorithms. Finally, we establish a connection with the Moreau-Yosida regularization.

The Krein-Milman theorem in operator convexity

Abstract: We generalize the Krein-Milman theorem to the setting of matrix convex sets of Effros-Winkler, extending the work of Farenick-Morenz on compact C∗-convex sets of complex matrices and the matrix state spaces of C∗-algebras. An essential ingredient is to prove the non-commutative analogue of the fact that a compact convex set K may be thought of as the state space of the space of continuous affine functions on K. The Krein-Milman theorem is without doubt one of the cornerstones of functional analysis. With the rise of non-commutative functional analysis and related notions of convexity ([15], [10], [11]), the question naturally arises how to formulate a notion of extreme points for which the theorem remains true. Such a notion exists in the case of C∗-convexity, which has been studied by LoeblPaulsen ([15]), Hopenwasser-Moore-Paulsen ([12]), and, more recently, FarenickMorenz ([7], [8], [9], [17]). C∗-convexity is the natural extension of the classical scalar-valued convex combination to include C∗-algebra-valued coefficients. It therefore makes sense in a C∗-algebra and, more generally, for bimodules over C∗algebras. In particular, there is a rich class of such C∗-convex sets in the n × n complex matrices,Mn. The matrix state spaces of a C∗-algebra are another class of examples. In both these cases the C∗-convexity version of the Krein-Milman has been proven to hold by Farenick and Morenz ([9], [17]). The above two examples both fit in the framework of another non-commutative convexity theory, the theory of matrix convex sets, developed by Effros and the second author ([11], [21]). In this paper we develop a notion of extreme points in this context, and we prove a corresponding Krein-Milman result, including a minimality condition which shows that the result is indeed optimal. Even though the difference between extremality in C∗-convexity and matrix convexity might seem minor at first, we hope to convince the reader that our approach is the natural one. Moreover, our methods are seemingly new and different, the central idea being to prove the analogue of the fact that a compact convex set K can be represented as the state space of the space of continuous affine functions on K. We begin with a review of matrix convexity, followed by a treatment of extreme points in this context. We then prove our representation result, from which we proceed to prove the Krein-Milman theorem. We wish to thank Douglas Farenick and Phillip Morenz for making a copy of [9] available to us. Received by the editors January 22, 1997. 1991 Mathematics Subject Classification. Primary 47D20; Secondary 46A55, 46L89.

The diagonal subring and the Cohen-Macaulay property of a multigraded ring

Abstract: Let T be a multigraded ring defined over a local ring (A, m). This paper deals with the question how the Cohen-Macaulay property of T is related to that of its diagonal subring T∆. In the bigraded case we are able to give necessary and sufficient conditions for the Cohen-Macaulayness of T . If I1, . . . , Ir ⊂ A are ideals of positive height, we can then compare the CohenMacaulay property of the multi-Rees algebra RA(I1, . . . , Ir) with the CohenMacaulay property of the usual Rees algebra RA(I1 · · · Ir). We also obtain a bound for the joint reduction numbers of two m-primary ideals in the case the corresponding multi-Rees algebra is Cohen-Macaulay.

Multivariate matrix refinable functions with arbitrary matrix dilation

Abstract: Characterizations of the stability and orthonormality of a multivariate matrix refinable function D with arbitrary matrix dilation M are provided in terms of the eigenvalue and 1-eigenvector properties of the restricted transition operator. Under mild conditions, it is shown that the approximation order of D is equivalent to the order of the vanishing moment conditions of the matrix refinement mask {Pa}. The restricted transition operator associated with the matrix refinement mask {Pa } is represented by a finite matrix (Ami-j)i,j, with A -Idet(M)I-1 E, P-j Q3 P,E and P,-j Q3 P, being the Kronecker product of matrices P,-j and P,. The spectral properties of the transition operator are studied. The Sobolev regularity estimate of a matrix refinable function 4 is given in terms of the spectral radius of the restricted transition operator to an invariant subspace. This estimate is analyzed in an example.

Summability of Fourier orthogonal series for Jacobi weight on a ball in ℝ

Abstract: Fourier orthogonal series with respect to the weight function (1 − |x|2)μ−1/2 on the unit ball in Rd are studied. Compact formulae for the sum of the product of orthonormal polynomials in several variables and for the reproducing kernel are derived and used to study the summability of the Fourier orthogonal series. The main result states that the expansion of a continuous function in the Fourier orthogonal series with respect to (1−|x|2)μ−1/2 is uniformly (C, δ) summable on the ball if and only if δ > μ + (d − 1)/2.

Connectedness properties of limit sets

Abstract: We study convergence group actions on continua, and give a criterion which ensures that every global cut point is a parabolic fixed point. We apply this result to the case of boundaries of relatively hyperbolic groups, and consider implications for connectedness properties of such spaces.

Numeration systems and Markov partitions from self similar tilings

Abstract: Using self similar tilings we represent the elements of RI as digit expansions with digits in RI being operated on by powers of an expansive linear map. We construct Markov partitions for hyperbolic toral automorphisms by considering a special class of self similar tilings modulo the integer lattice. We use the digit expansions inherited from these tilings to give a symbolic representation for the toral automorphisms. FRactals and fractal tilings have captured the imaginations of a wide spectrum of disciplines. Computer generated images of fractal sets are displayed in public science centers, museums, and on the covers of scientific journals. Fractal tilings which have interesting properties are finding applications in many areas of mathematics. For example, number theorists have linked fractal tilings of JR2 with numeration systems for JR2 in complex bases [16], [8]. We will see that fractal self similar tilings of R' provide natural building blocks for numeration systems of IR'. These numeration systems generalize the 1-dimensional cases in [14],[10],[11] as well as the 2-dimensional cases mentioned above. Our motivation for studying fractal tilings comes from ergodic theory. In [2] Adler and Weiss show that topological entropy is a complete invariant for metric equivalence of continuous ergodic automorphisms of the two-dimensional torus. Their method of proof is to construct a partition of the 2-torus which satisfies certain properties. The partition is called a Markov partition. By assigning each element of the partition a symbol, it is possible to assign each point in the 2-torus a bi-infinite sequence of symbols which corresponds to the orbit of the point. The objective is to represent the continuous dynamical system as a symbolic one in such a way that periodicity and transitivity is preserved in the representation. In [4] Bowen shows that every Anosov diffeomorphism has a Markov partition. In particular every hyperbolic toral automorphism has a Markov partition. His construction uses a recursive definition which deforms rectangles in the stable and unstable directions. While existence is shown, the proof does not indicate an efficient way to actually construct the partitions. In [5] Bowen shows that in the case of the 3-torus the boundary sets of the Markov partition have fractional Hausdorff dimension. In [3] Bedford constructs examples of Markov partitions for the 3-torus and describes the sets as crinkly tin cans. In [6] Cawley generalizes Bowen's results to higher dimensional tori. Since Markov partitions have properties which resemble those of self similar tilings and the boundary sets of these partitions have fractional dimension, it is Received by the editors October 2, 1996. 1991 Mathematics Subject Classification. Primary 58F03, 34C35, 54H20. ( 1999 Arnerican Mathematical Society 3315 This content downloaded from 157.55.39.170 on Tue, 20 Sep 2016 05:07:32 UTC All use subject to http://about.jstor.org/terms 3316 BRENDA PRAGGAS'TIS natural to suggest that an explicit finite construction of Markov partitions for hyperbolic toral automorphisms may be given using fractal self similar tilings. Moreover, there is a natural way to represent points in a tiled space using a symbolic system. The representations correspond to digit expansions. The set of sequences of digits used in these expansions will in turn generate a symbolic dynamical system which is metrically similar to the continuous system defined by the corresponding toral automorphism. 1. SUBDIVIDING AND SELF SIMILAR TILINGS In this first section we review the definition and properties of self similar tilings. Much of this information may be found in [17] and [10]. We then indicate how a self similar tiling of Rn provides a numeration system for Rt. Let X be a subset of RT which is the closure of its interior. Definition. A collection T of compact subsets of X is a tiling if it satisfies the following properties. (1) The union of the sets in T is equal to X. (2) Each set in T is the closure of its interior. (3) Each compact set in X ilntersects a finite number of sets in T. (In this case we say that T is locally finite.) (4) The interiors of the sets in T are mutually disjoint. The sets in T are called tiles. Suppose G is a subgroup of RW. A tiling T of X is G -finite if there exists a finite partition {Ij}GJ of T such that if T E Tj, then TS C {T+g: 9 EG}. The collection {T}jlj is a tile type partition for T. Suppose T is a G-finite tiling of X with a tile type partition {T}jgJ. If T', T2 E Tj for some j E J, then we will say that T1 and T2 are of the same type. This implies that there exists g E G such that T' + g = T2. We say that T2 is a G-translate of T'. This does not mean that if T', T2 E T and g E G such that T1 + g = T2, then T1 and T2 are of the same type. We will see that there will often exist tile type partitions which contain elements Tij and T1j2 such that every tile in Tj, is a G-translate of every tile in 2h. It follows that there are arbitrarily many different ways to define a tile type partition for a G-finite tiling. For now, let {Tj }jGJ define a tile type partition for T. Let 93 be the set of all finite unions of tiles in T. Let 93 be the set of all bounded subsets of X. Note that q3 C 93. Definition. Let Pl; p2 E q3, then there is a finite collection of tiles {Tk}kCK such that Pl = UkeK Tk. The sets P' and p2 have the same pattern if there exists g E G such that Pl + g = UkEK Tk + g = p2 and for each k E K, Tk + g is a tile in T of the same type as Tk. In particular tiles of the same type have the same pattern. If U E X, then there is a finite union of tiles P E q3 such that U C P. If g E G, then U and U + g have the same pattern if for some P E q3 such that U C P we have that P + g E v and P + g and P have the same pattern. If V is a subset of X and U E 93, then V contains the pattern of U if for some g E G, U + g C V and U + g has the same pattern as U. Let 0 be an expansive linear map on RI such that OX = X. To be expansive means that the eigenvalues for q all have modulus greater than 1. We adapt a This content downloaded from 157.55.39.170 on Tue, 20 Sep 2016 05:07:32 UTC All use subject to http://about.jstor.org/terms NUMERATION SYSTEMS AND MARKOVKOV PARTITIONS 3317 norm for R' which reflects the expansiveness of q as in [12]. More specifically, let {A j}j=j be the eigenvalues of q ordered so that 1 0 we have that the set {P E q4: diameter(P) 0 there exists R = R(r) > 0 such that every open ball of radius R contained in X contains the pattern of Br(x) n x for all x e X. Let G be a subgroup of R' such that bG = G. Definition. A G-finite tiling T of X with tile type partition {fT}.jEJ is subdividing with expansion map 0, if for each T E i, XT is a finite union of tiles and if T and T' are tiles of the same type, then XT and XT' have the same pattern. For each T E Ti we may think of the pattern of XT as defining a subdivision rule for Tj. A quasi-periodic subdividing tiling of T of X with expansion map 0 is called a self similar tiling. Example 1. Let 0 be the expansive map on R given by multiplication by 2 Note that A is a zero of x2 x 1. Let X+ = [0,oo). We construct a Z[A finite subdividing tiling T of X+ with expansion map q. Let TA = [0, 1] and TB = [0, A 1]We will partition T into two sets TA and TB. The tiles in TA will be Z[A]-translates of TA and the tiles in TB will be Z[A]-translates of TB. We define T by inductively defining TA and TB. Let TA E TA. If x E Z[A] and TA + X E TA, then o(TA+x) = [O,A]+Ax = ([O,1]+?x)U([1,A]+Ax) = (TA+Ax)U(TB+1+Ax). Let TA + AX E TA and let TB + 1 + Ax E TB. If y E Z[A] and TB + Y E TB, then q(TB+y) = [0,A2-A]+ Ay = [0,1]+Ay = TA+AY. This content downloaded from 157.55.39.170 on Tue, 20 Sep 2016 05:07:32 UTC All use subject to http://about.jstor.org/terms 3318 BRENDA PRAGGASTIS Let TA + Ay E TA. The tiling T is a Z[A ]-finite subdividing tiling of X+ with expansion map 0 and tile type partition {TA, JB}. We can represent the subdivision rules for T in terms of a substitution map. Let A represent a tile in WA and B represent a tile in TB. Define a substitution map 0 on the words in {A, B} by letting 0(A) = AB and letting 0(B) = A. We define 0 on each finite or infinite string of symbols in {A, B} by applying it to each symbol in the string. We see that T is represented by the fixed word w = ABAABABAABAAB....

Characterization of Smoothness of Multivariate Refinable Functions in Sobolev Spaces

Abstract: Wavelets are generated from refinable functions by using multiresolution anlalysis. In this paper we investigate the srloothness properties of multivariate refinable functionis in Sobolev spaces. We characterize the optinlal smoothness of a multivariate refinable function in terms of the spectral radius of the corresponding transition operator restricted to a suitable finite dimensional invariant subspace. Several examples are provided to illustrate the general theory.

Lacunarity of self-similar and stochastically self-similar sets

Abstract: Let K be a self-similar set in Rdd, of Hausdorff dimension D, and denote by IK(c)l the d-dimensional Lebesgue measure of its 6-neighborhood. We study the limiting behavior of the quantity e-(d-D) IK(e)I as 6 -e 0. It turns out that this quantity does not have a limit in many interesting cases, including the usual ternary Cantor set and the Sierpinski carpet. We also study the above asymptotics for stochastically self-similar sets. The latter results then apply to zero-sets of stable bridges, which are stochastically self-similar (in the sense of the present paper), and then, more generally, to level-sets of stable processes. Specifically, it follows that, if Kt is the zero-set of a realvalued stable process of index a C (1,2], run up to time t, then e -/ SIIt(e)I converges to a constant multiple of the local time at 0, simultaneously for all t > 0, on a set of probability one. The asymptotics for deterministic sets are obtained via the renewal theorem. The renewal theorem also yields asymptotics for the mean 1E[IK(c) ] in the random case, while the almost sure asymptotics in this case are obtained via an analogue of the renewal theorem for branching random walks.

Existence and uniqueness for a degenerate parabolic equation with L1-data

Abstract: In this paper we study existence and uniqueness of solutions for the boundary-value problem, with initial datum in L1(Ω), ut = div a(x, Du) in (0,∞)×Ω, − ∂u ∂ηa ∈ β(u) on (0,∞)× ∂Ω, u(x, 0) = u0(x) in Ω, where a is a Caratheodory function satisfying the classical Leray-Lions hypothesis, ∂/∂ηa is the Neumann boundary operator associated to a, Du the gradient of u and β is a maximal monotone graph in R× R with 0 ∈ β(0).