Related papers: Some remarks about Cauchy integrals and fractal se…
In the paper, the authors analytically generalize the Catalan numbers in combinatorial number theory, establish an integral representation of the analytic generalization of the Catalan numbers by virtue of Cauchy's integral formula in the…
In this paper, we discussed the non-local derivative on the fractal Cantor set. The scaling properties are given for both local and non-local fractal derivatives. The local and non-local fractal differential equations are solved and…
Parabolic integro-differential non degenerate Cauchy problem is considered in the scale of H\"older spaces of functions whose regularity is defined by a radially O-regularly varying L\'evy measure. Existence and uniqueness and the estimates…
Recent numerical results on the fractal structure of two-dimensional quantum gravity coupled to $c=-2$ matter are reviewed. Analytic derivation of the fractal dimensions based on the Liouville theory and diffusion equation is also…
Some aspects of the connection between differential geometry and multidimensional soliton equations are discussed.
This article addresses the question of involutiveness and discusses the initial value problem for a class of overdetermined systems of partial differential equations which arise in the theory of integrable systems and are defined by…
Recently, Komatsu introduced the concept of poly-Cauchy numbers and polynomials which generalize Cauchy numbers and polynomials. In this paper, we consider the new concept of higher-order Cauchy numbers and polynomials which generalize…
We study dimensional properties of visible parts of fractal percolation in the plane. Provided that the dimension of the fractal percolation is at least 1, we show that, conditioned on non-extinction, almost surely all visible parts from…
This talk reviews some mathematical and physical ideas related to the notion of dimension. After a brief historical introduction, various modern constructions from fractal geometry, noncommutative geometry, and theoretical physics are…
An atomic random complex measure defined on the unit disk with Normally distributed moments is considered. An approximation to the distribution of the zeros of its Cauchy transform is computed. Implications of this result for solving…
We provide a general framework for fractional Hardy inequalities. Our framework covers, for instance, fractional inequalities related to the Dirichlet forms of some L\'evy processes, and weighted fractional inequalities on irregular open…
The concept of fractal index is introduced in connection with the idea of universal class $h$ of particles or quasiparticles, termed fractons, which obey fractal statistics. We show the relation between fractons and conformal field…
We study the convergence of certain subseries of the harmonic series corresponding to increasing sequences of integers whose digits in a certain base are not uniformly distributed. We also discuss the case of irregular sequences, where the…
We describe the fractal solid by a special continuous medium model. We propose to describe the fractal solid by a fractional continuous model, where all characteristics and fields are defined everywhere in the volume but they follow some…
We investigate the Cauchy problem for a 2x2-system of weakly coupled semi-linear fractional wave equations with polynomial nonlinearities posed in R+ x RN. Under appropriate conditions on the exponents and the fractional orders of the time…
An overview of some of the recent developments in the theory of valuations on convex sets and its generalizations to manifolds is given. The exposition is focused towards applications to integral geometry; several of such applications are…
We study the kinetics of random sequential adsorption of a mixture of particles with continuous distribution of sizes for different deposition rules. It appears in the long time limit the resulting system can be described using the fractal…
We present an analogue of the differential calculus in which the role of polynomials is played by certain ordered sets and trees. Our combinatorial calculus has all nice features of the usual calculus and has an advantage that the elements…
Various topics concerning the entanglement of composite quantum systems are considered with particular emphasis concerning the strict relations of such a problem with the one of attributing objective properties to the constituents. Most of…
We introduce a hierarchy of integrable PDEs in 2+1 dimensions arising from the commutation of 2-dimensional vector fields. We also solve the associated Cauchy problems, using the recently developed Inverse Scattering Transform for…