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.     

Products of random matrices and derivatives on p.c.f. fractals

We define and study intrinsic first order derivatives on post critically finite fractals and prove differentiability almost everywhere with respect to self-similar measures for certain classes of fractals and functions. We apply our results to extend the geography is destiny principle to these cases, and also obtain results on the pointwise behavior of local eccentricities on the Sierpiński gasket, previously studied by Öberg, Strichartz and Yingst, and the authors. We also establish the relation of the derivatives to the tangents and gradients previously studied by Strichartz and the authors. Our main tool is the Furstenberg-Kesten theory of products of random matrices.     

APPROXIMATION OPERATORS AND SELFSIMILARITY OVER P-ADIC FIELD

Fourier analysis on local fields has been developed since M. H. Taiblesoa, Some of its theory and technique are much similar to the classical ones while some are not and even have not appropriately mathematical toolS to deal with. Recently we find there are a few but interesting applications to fractals ,especial to self-similar functions of the p-adic analysis and such a setting seems to be natural. This note also includes a concept of a derivative and approximation operators.     

Electromagnetism on anisotropic fractal media

Basic equations of electromagnetic fields in anisotropic fractal media are obtained using a dimensional regularization approach. First, a formulation based on product measures is shown to satisfy the four basic identities of the vector calculus. This allows a generalization of the Green–Gauss and Stokes theorems as well as the charge conservation equation on anisotropic fractals. Then, pursuing the conceptual approach, we derive the Faraday and Ampère laws for such fractal media, which, along with two auxiliary null-divergence conditions, effectively give the modified Maxwell equations. Proceeding on a separate track, we employ a variational principle for electromagnetic fields, appropriately adapted to fractal media, so as to independently derive the same forms of these two laws. It is next found that the parabolic (for a conducting medium) and the hyperbolic (for a dielectric medium) equations involve modified gradient operators, while the Poynting vector has the same form as in the non-fractal case. Finally, Maxwell’s electromagnetic stress tensor is reformulated for fractal systems. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions in three different directions and reduce to conventional forms for continuous media with Euclidean geometries upon setting these each of dimensions equal to unity.     

Random point fields with Markovian refinements and the geometry of fractally disordered media

We give a general construction of the probability measure for describing stochastic fractals that model fractally disordered media. For these stochastic fractals, we introduce the notion of a metrically homogeneous fractal Hansdorff-Karathéodory measure of a nonrandom type. We select a classF[q] of random point fields with Markovian refinements for which we explicitly construct the probability distribution. We prove that under rather weak conditions, the fractal dimension D for random fields of this class is a self-averaging quantity and a fractal measure of a nonrandom type (the Hausdorff D-measure) can be defined on these fractals with probability 1. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 490–505, September, 2000.     

Kusuoka–Stroock Formula on Configuration Space and Regularities of Local Times with Jumps

In this paper, we first extend the classical Itô stochastic integral to the case of measurable fields of Hilbert spaces. Then, a Kusuoka–Stroock formula on configuration space is proved. Using this formula, we study the fractional regularities of local times with jumps in the sense of the Malliavin calculus.     

Crystallization of space: Space-time fractals from fractal arithmetic

Fractals such as the Cantor set can be equipped with intrinsic arithmetic operations (addition, subtraction, multiplication, division) that map the fractal into itself. The arithmetic allows one to define calculus and algebra intrinsic to the fractal in question, and one can formulate classical and quantum physics within the fractal set. In particular, fractals in space-time can be generated by means of homogeneous spaces associated with appropriate Lie groups. The construction is illustrated by explicit examples.     

On modeling with multiplicative differential equations

This work is aimed to show that various problems from different fields can be modeled more efficiently using multiplicative calculus, in place of Newtonian calculus. Since multiplicative calculus is still in its infancy, some effort is put to explain its basic principles such as exponential arithmetic, multiplicative calculus, and multiplicative differential equations. Examples from finance, actuarial science, economics, and social sciences are presented with solutions using multiplicative calculus concepts. Based on the encouraging results obtained it is recommended that further research into this field be vested to exploit the applicability of multiplicative calculus in different fields as well as the development of multiplicative calculus concepts.     

RESEARCH ANNOUNCEMENTS：On the Fractional Calculus of a Type of Weierstrass Functions

I Introduction In recent years, fractals have shown important applications in many fields. [1, 2] and [3] havedone some excellent initial and conclusion work on fractal and it's mathematical foundations.However, a fractal function: a type of Weierstrass functions defined bybecause of it's special fractal properties, [1,2, 4, 5] have given some detailed discussion about it'sgraph, fractal dimension, etc.     

Optimal control of a Vlasov–Poisson plasma by an external magnetic field

The aim of various technical applications (for example fusion research) is to control a plasma by magnetic fields in a desired fashion. In our model the plasma is described by the Vlasov–Poisson system that is equipped with an external magnetic field. We will prove that this model satisfies some basic properties that are necessary for calculus of variations. After that, we will analyze an optimal control problem with a tracking type cost functional with respect to the following topics: necessary conditions of first order for local optimality, derivation of an optimality system, sufficient conditions of second order for local optimality, uniqueness of the optimal control under certain conditions.     

Distribution theory on P.C.F. fractals

We construct a theory of distributions in the setting of analysis on post-critically finite self-similar fractals, and on fractafolds and products based on such fractals. The results include basic properties of test functions and distributions, a structure theorem showing that distributions are locally-finite sums of powers of the Laplacian applied to continuous functions, and an analysis of the distributions with point support. Possible future applications to the study of hypoelliptic partial differential operators are suggested.     

Derivations and Dirichlet forms on fractals

We study derivations and Fredholm modules on metric spaces with a local regular conservative Dirichlet form. In particular, on finitely ramified fractals, we show that there is a non-trivial Fredholm module if and only if the fractal is not a tree (i.e. not simply connected). This result relates Fredholm modules and topology, refines and improves known results on p.c.f. fractals. We also discuss weakly summable Fredholm modules and the Dixmier trace in the cases of some finitely and infinitely ramified fractals (including non-self-similar fractals) if the so-called spectral dimension is less than 2. In the finitely ramified self-similar case we relate the p-summability question with estimates of the Lyapunov exponents for harmonic functions and the behavior of the pressure function.     

Space-filling curves and geodesic laminations. II: Symmetries

In 2008, the author introduced a class of space-filling curves associated to fractals that satisfy the a special property. These structures admit geodesic laminations on the disc, which help to understand the geometrical and the dynamical properties of the space-filling curves. In the present article we study the relation between the symmetries of the laminations and the fractals. In particular we prove that the group of symmetries of the lamination is isomorphic to a subgroup of the full group of symmetries of the fractal. We extend the results to a larger class of fractals using the concept of sub-IFS.     

Second order Poincaré inequalities and CLTs on Wiener space

We prove infinite-dimensional second order Poincaré inequalities on Wiener space, thus closing a circle of ideas linking limit theorems for functionals of Gaussian fields, Stein's method and Malliavin calculus. We provide two applications: (i) to a new “second order” characterization of CLTs on a fixed Wiener chaos, and (ii) to linear functionals of Gaussian-subordinated fields.     

Space-filling curves and geodesic laminations. II: Symmetries

In 2008, the author introduced a class of space-filling curves associated to fractals that satisfy the a special property. These structures admit geodesic laminations on the disc, which help to understand the geometrical and the dynamical properties of the space-filling curves. In the present article we study the relation between the symmetries of the laminations and the fractals. In particular we prove that the group of symmetries of the lamination is isomorphic to a subgroup of the full group of symmetries of the fractal. We extend the results to a larger class of fractals using the concept of sub-IFS.     

A new sufficiency theorem for strong minima in the calculus of variations

Sufficiency for strong local optimality in the calculus of variations involves, in the classical theory, the strengthened condition of Weierstrass. A proof of sufficiency for strong minima, modifying this condition under certain uniform continuity assumptions on the functions delimiting the problem, is presented. The proof is direct in nature as it makes no use of fields, Hamilton–Jacobi theory, Riccati equations or conjugate points. Some examples illustrate clear advantages of the new sufficient condition over the classical one.     

On physical interpretations of fractional integration and differentiation

Is there a relation between fractional calculus and fractal geometry? Can a fractional order system be represented by a causal dynamical model? These are the questions recently debated in the scientific community. The author intends to answer these questions. In the first part of the paper, some recently suggested models are reviewed and no convincing evidence is found for any dynamic model of a fractional order system having been built with the help of fractals. Linear filters with lumped constant parameters have a very limited use as approximations of fractional order systems. The model suggested in the paper is a state-space representation with parameters as functions of the independent variable. Regularization of fractional differentiation is considered and asymptotic error estimates, as well as simulation results, are presented.     

Exploring the applications of fractional calculus: Hierarchically built semiflexible polymers

In this article we study, through extensions of the generalized Gaussian scheme, the dynamics of semiflexible treelike polymers under the influence of external forces acting on particular (say, charged) monomers. Semiflexibility is introduced following our previous work (Dolgushev and Blumen, 2009 [15]), a procedure which allows one to study treelike structures with arbitrary stiffness and branching. Exemplarily, we illustrate the procedure using linear chains and hyperbranched polymers modeled through Vicsek fractals, and obtain in every case the monomer displacement averaged over the structure. Anomalous behavior manifests itself in the intermediate time region, where the different fractal architectures show distinct scaling behaviors. These behaviors are due to the power law behavior of the spectral density and lead, for arbitrary pulling forces, based on causality and the linear superposition principle, to fractional calculus expressions, in accordance to former phenomenological fractional laws in polymer physics.     

Local Limit Theorems for Sequences of Simple Random Walks on Graphs

In this article, local limit theorems for sequences of simple random walks on graphs are established. The results formulated are motivated by a variety of random graph models, and explanations are provided as to how they apply to supercritical percolation clusters, graph trees converging to the continuum random tree and the homogenisation problem for nested fractals. A subsequential local limit theorem for the simple random walks on generalised Sierpinski carpet graphs is also presented.