Related papers: Calculus, fractals, and analysis on metric spaces

This work contains a set of lectures on defect structures, mainly in models described by scalar fields in diverse dimensions.

The purpose of this paper is to study the fractal phenomena in large data sets and the associated questions of dimension reduction. We examine situations where the classical Principal Component Analysis is not effective in identifying the…

In this paper, the fractal calculus of fractal sets and fractal curves are compared. The analogues of the Riemann-Liouville and the Caputo integrals and derivatives are defined for the fractal curves which are non-local derivatives. The…

This paper presents a review of the fractal approach for describing the large scale distribution of galaxies. We start by presenting a brief, but general, introduction to fractals, which emphasizes their empirical side and applications…

Our main goal in this long survey article is to provide an overview of the theory of complex fractal dimensions and of the associated geometric or fractal zeta functions, first in the case of fractal strings (one-dimensional drums with…

These are notes from a lecture course on symmetric spaces by the second author given at the University of Pittsburgh in the fall of 2010.

We define a new structure on a space endowed with convexities, and call it a fractoconvex structure (or, a space with fractoconvexity). We introduce two operations on a set of fractoconvexities and in a special case we show that they…

A short introduction to the mathematical methods and technics of differential algebras and modules adapted to the problems of mathematical and theoretical physics is presented.

We consider a special type of self-similar sets, called fractal squares, and give a brief review on recent results and unsolved issues with an emphasis on their topological properties.

Shape is one of the most important visual attributes to characterize objects, playing a important role in pattern recognition. There are various approaches to extract relevant information of a shape. An approach widely used in shape…

We introduce fractional flat space, described by a continuous geometry with constant non-integer Hausdorff and spectral dimensions. This is the analogue of Euclidean space, but with anomalous scaling and diffusion properties. The basic tool…

We survey old and new conjectures and results on various types of spherical maximal functions, emphasizing problems with a fractal dilation set.

This is a brief introduction to fractals, multifractals and wavelets in an accessible way, in order that the founding ideas of those strange and intriguing newcomers to science as fractals may be communicated to a wider public. Fractals are…

Here I share a few notes I used in various course lectures, talks, etc. Some may be just calculations that in the textbooks are more complicated, scattered, or less specific; others may be simple observations I found useful or curious.

The science of fractography revolves around the correlation between topographic characteristics of the fracture surface and the mechanisms and external conditions leading to their creation. While being a topic of investigation for…

We provide a rigorous study on dimensions of fractal interpolation function defined on a closed and bounded interval of $\mathbb{R}$ which is associated to a continuous function with respect to a base function, scaling functions and a…

In this article we extend on work which establishes an analology between one-way quantum computation and thermodynamics to see how the former can be performed on fractal lattices. We find fractals lattices of arbitrary dimension greater…

This report on the topics in the title was written for a lecture series at the Southwestern Center for Arithmetic Algebraic Geometry at the University of Arizona.It may serve as an introduction to certain conjectural relations between…

These are lecture notes that are based on the lectures from a class I taught on the topic of Randomized Linear Algebra (RLA) at UC Berkeley during the Fall 2013 semester.

Fractional dynamics is a field of study in physics and mechanics investigating the behavior of objects and systems that are characterized by power-law non-locality, power-law long-term memory or fractal properties by using integrations and…