Riesz Potentials and Liouville Operators on Fractals

An analogue to the theory of Riesz potentials and Liouville operators in R ⁿ for arbitrary fractal d-sets is developed. Corresponding function spaces agree with traces of Euclidean Besov spaces on fractals. By means of associated quadratic forms we construct strongly continuous semigroups with Liouville operators as infinitesimal generator. The case of Dirichlet forms is discussed separately. As an example of related pseudodifferential equations the fractional heat-type equation is solved.     

A random fractal in the ring of <Emphasis Type="Italic">p</Emphasis>-adic integers

In this article, we will give a construction of a random fractal in the ring of p-adic integers and examine an extent of the random fractals. Paying attention to an importance in statistical self similarity, we will perform measurement for the extent in terms of the Hausdorff dimension similarly to the typical fractal analysis in the Euclidean space. In our study, we will take a measure theoretic approach combined with the martingale theory based on Falconer’s method.     

A new approach to superfractals

V-variable fractals and superfractals have recently been introduced by Barnsley [Barnsley Michael, Hutchinson John, Stenflo Örjan. A fractal valued random iteration algorithm and fractal hierarchy. Fractals 2005;13(2):111–46 [MR2151094 (2006b:28014)]] to the world of mathematics and computer graphics. In this paper, we introduce superior iterates to study the role of contractive and non-contractive operators in relation to superfractals. A modified algorithm along with details of computer implementation is also provided to compute V-variable fractals. A brief discussion about the various aspects of the computed figures indicates usefulness of the study.     

Existence and data dependence of fixed points for multivalued operators on gauge spaces

The purpose of this note is to present some fixed point and data dependence theorems in complete gauge spaces and in hyperconvex metric spaces for the so-called Meir-Keeler multivalued operators and admissible multivalued aα-contractions. Our results extend and generalize several theorems of Espínola and Kirk [R. Espínola, W.A. Kirk, Set-valued contractions and fixed points, Nonlinear Anal. 54 (2003) 485-494] and Rus, Petru?el, and Sînt?m?rian [I.A. Rus, A. Petru?el, A. Sînt?m?rian, Data dependence of the fixed point set of some multivalued weakly Picard operators, Nonlinear Anal. 52 (2003) 1947-1959].     

Fundamental Sets of Fractal Functions

Fractal interpolants constructed through iterated function systems prove more general than classical interpolants. In this paper, we assign a family of fractal functions to several classes of real mappings like, for instance, maps defined on sets that are not intervals, maps integrable but not continuous and may be defined on unbounded domains. In particular, based on fractal interpolation functions, we construct fractal Müntz polynomials that successfully generalize classical Müntz polynomials. The parameters of the fractal Müntz system enable the control and modification of the properties of original functions. Furthermore, we deduce fractal versions of classical Müntz theorems. In this way, the fractal methodology generalizes the fundamental sets of the classical approximation theory and we construct complete systems of fractal functions in spaces of continuous and p-integrable mappings on bounded domains. This work is supported by the project No: SB 2005-0199, Spain.     

Special bases for solutions of a generalized Maxwell system in 3‐dimensional space

The aim of this paper is to study complete polynomial systems in the kernel space of conformally invariant differential operators in higher spin theory. We investigate the kernel space of a generalized Maxwell operator in 3‐dimensional space. With the already known decomposition of its homogeneous kernel space into 2 subspaces, we investigate first projections from the homogeneous kernel space to each subspace. Then, we provide complete polynomial systems depending on the given inner product for each subspace in the decomposition. More specifically, the complete polynomial system for the homogenous kernel space is an orthogonal system wrt a given Fischer inner product. In the case of the standard inner product in L² on the unit ball, the provided complete polynomial system for the homogeneous kernel space is a partially orthogonal system. Further, if the degree of homogeneity for the respective subspaces in the decomposed kernel spaces approaches infinity, then the angle between the 2 subspaces approaches π/2.     

Fixed points for mixed monotone multivalued operators in Banach spaces with applications

In this paper, the existence and iterative approximation of fixed points for a class of systems of mixed monotone multivalued operator are discussed. We present some new fixed point theorems of mixed monotone operators and increasing operators which need not be continuous or satisfy a compactness condition. We also give some applications to differential inclusions with discontinuous right hand side in Banach spaces and to Hammerstein integral inclusions on RN.     

A Random Multivalued Uniform Boundedness Principle

We present a generalization of the Uniform Boundedness Principle valid for random multivalued linear operators, i.e., multivalued linear operators taking values in the space L₀(,Y) of random variables defined on a probability space with values in the Banach space Y. Namely, for a family of such operators that are continuous with positive probability, if the family is pointwise bounded with probability at least >0, then the operators are uniformly bounded with a probability that in each case can be estimated in terms of and the index of continuity of the operator. To achieve this result, we develop the fundamental theory of multivalued linear operators on general topological vector spaces. In particular, we exhibit versions of the Closed Graph Theorem, the Open Mapping Theorem, and the Uniform Boundedness Principle for multivalued operators between F-spaces.Mathematics Subject Classifications (2000) 46A16, 46A30, 47A06, 47B80, 60H25.     

Existence results for hemivariational inequalities with measures

We prove existence results for multivalued quasilinear elliptic problems of hemivariational inequality type with measure data right-hand sides. In case of L ¹-data, we study existence and enclosure behaviors of solutions by an appropriate sub-supersolution approach. The proofs of our results are based on general existence theory for multivalued pseudomonotone operators, and approximation-, truncation-, and special test function techniques.     

On solutions of differential inclusions in homogeneous spaces of functions

In this paper we study the solvability of some classes of differential inclusions with multivalued linear operators in homogeneous spaces of functions. These spaces include a large number of functional spaces like periodic functions and Bohr and Stepanov almost periodic functions. As an application, we consider some existence results for feedback control systems governed by degenerate differential equations of Sobolev type in a Banach space.     

Integrals,conditional expectations,and martingales of multivalued functions

Let (Ω, $\text{A}$, μ) be a finite measure space and $\text{X}$ a real separable Banach space. Measurability and integrability are defined for multivalued functions on Ω with values in the family of nonempty closed subsets of $\text{X}$. To present a theory of integrals, conditional expectations, and martingales of multivalued functions, several types of spaces of integrably bounded multivalued functions are formulated as complete metric spaces including the space L¹(Ω; $\text{X}$) isometrically. For multivalued functions in these spaces, multivalued conditional expectations are introduced, and the properties possessed by the usual conditional expectation are obtained for the multivalued conditional expectation with some modifications. Multivalued martingales are also defined, and their convergence theorems are established in several ways.     

Multivariational Inequalities and Operator Inclusions in Banach Spaces with Mappings of the Class (S)+

We prove theorems on the existence of solutions of variational inequalities and operator inclusions in Banach spaces with multivalued mappings of the class (S)₊. We justify the method of penalty operators for variational inequalities.     

Multivalued differential equations in Banach space. An application to control theory

This paper is concerned with the study of existence theorems for multivalued differential systems in infinite-dimensional Banach space: the method used is based on techniques (extension theorem for linear operator, compactly convergent sequences) developed earlier by the authors for multivalued differential systems defined inn-dimensional vector spaces. As an application, the authors consider a distributed-parameter control problem arising in mathematical physics, more specifically, in the study of heat transfer in solids.This work was performed under the auspices of the National Research Council of Italy (CNR).     

Calculus on fractals based upon local fields In memory of Founding Editor Professor M. T. Cheng with great respect and deep sorrow

The main contents in this note are: 1. introduction: 2. locally compact groups and local fields: 3. calculus on fractals based upon local fields: 4. fractional calculus and fractals: 5. fractal function spaces and PDE on fractals.     

Calculus on fractals based upon local fields

The main contents in this note are: 1. introduction: 2. locally compact groups and local fields: 3. calculus on fractals based upon local fields: 4. fractional calculus and fractals: 5. fractal function spaces and PDE on fractals.     

Fractal systems of central places based on intermittency of space-filling

The central place models are fundamentally important in theoretical geography and city planning theory. The texture and structure of central place networks have been demonstrated to be self-similar in both theoretical and empirical studies. However, the underlying rationale of central place fractals in the real world has not yet been revealed so far. This paper is devoted to illustrating the mechanisms by which the fractal patterns can be generated from central place systems. The structural dimension of the traditional central place models is d = 2 indicating no intermittency in the spatial distribution of human settlements. This dimension value is inconsistent with empirical observations. Substituting the complete space filling with the incomplete space filling, we can obtain central place models with fractional dimension D < d = 2 indicative of spatial intermittency. Thus the conventional central place models are converted into fractal central place models. If we further integrate the chance factors into the improved central place fractals, the theory will be able to explain the real patterns of urban places very well. As empirical analyses, the US cities and towns are employed to verify the fractal-based models of central places.     

Linear homeomorphisms of non-classical Hilbert spaces

Let K be a complete infinite rank valued field. In [4] we studied Norm Hilbert Spaces (NHS) over K i.e. K-Banach spaces for which closed subspaces admit projections of norm ≤ 1. In this paper we prove the following striking properties of continuous linear operators on NHS. Surjective endomorphisms are bijective, no NHS is linearly homeomorphic to a proper subspace (Theorem 3.7), each operator can be approximated, uniformly on bounded sets, by finite rank operators (Theorem 3.8). These properties together — in real or complex theory shared only by finite-dimensional spaces — show that NHS are more ‘rigid’ than classical Hilbert spaces.     

Anamorphoses and Flat Morphological Operators on Power Lattices

Flat morphological operators are operators on grey-level images derived from increasing set operators by a combination of thresholding and stacking. For analog grey-levels, they commute with anamorphoses or contrast mappings, that is, continuous increasing grey-level transformations; when the underlying set operator is upper semi-continuous, they also commute with thresholding. For bounded discrete grey-levels, commutation with increasing grey-level transformations and with thresholding is guaranteed, without any continuity conditions. In this paper we consider flat operators for images defined on an arbitrary space of points and taking their values in an arbitrary complete lattice. We study their commutation with increasing transformations of values. This requires some continuity requirements on the transformations of values or on the underlying set operator, which are expressed in terms of the lattice of values. We obtain as particular cases the known conditions for analog and discrete grey-levels, and also new conditions for other examples of values: multivalued vectors or any finite set of values.     

Analysis of orthogonality and of orbits in affine iterated function systems

We introduce a duality for affine iterated function systems (AIFS) which is naturally motivated by group duality in the context of traditional harmonic analysis. Our affine systems yield fractals defined by iteration of contractive affine mappings. We build a duality for such systems by scaling in two directions: fractals in the small by contractive iterations, and fractals in the large by recursion involving iteration of an expansive matrix. By a fractal in the small we mean a compact attractor X supporting Hutchinson’s canonical measure μ, and we ask when μ is a spectral measure, i.e., when the Hilbert space has an orthonormal basis (ONB) of exponentials . We further introduce a Fourier duality using a matched pair of such affine systems. Using next certain extreme cycles, and positive powers of the expansive matrix we build fractals in the large which are modeled on lacunary Fourier series and which serve as spectra for X. Our two main results offer simple geometric conditions allowing us to decide when the fractal in the large is a spectrum for X. Our results in turn are illustrated with concrete Sierpinski like fractals in dimensions 2 and 3. Research supported in part by the National Science Foundation DMS 0457491.     

Vector-valued Modulation Spaces and Localization Operators with Operator-valued Symbols

We study the short-time Fourier transformation, modulation spaces, Gabor representations and time-frequency localization operators, for functions and tempered distributions that have as range space a Banach or a Hilbert space. In the Banach space case the theory of modulation spaces contains some modifications of the scalar-valued theory, depending on the Banach space. In the Hilbert space case the modulation spaces have properties similar to the scalar-valued case and the Gabor frame theory essentially works. For localization operators in this context symbols are operator-valued. We generalize two results from the scalar-valued theory on continuity on certain modulation spaces when the symbol belongs to an L^p,q space and M^∞, respectively. The first result is true for any Banach space as range space, and the second result is true for any Hilbert space as range space.