arXiv subjects

John A. Rock

Publications and source records attributed to John A. Rock.

9 recordsLinked to original sources

arXiv2022

Arbitrarily Close for Summer 2022 Analysis

The kernel of analysis, to me anyway, is the following idea: A point is arbitrarily close to a set if every neighborhood of the point intersects the set. Defining ``arbitrarily close'' in this way provides a foundation for classical results in calculus and real analysis dealing with convergence, limits, connectedness, limits, continuity, differentiation, integration, series, and more. This book contains: a thorough introduction to arbitrarily close; an approach to limits and convergence of sequences using arbitrarily close as a key first step; the topology of Euclidean spaces stemming from closed sets; an exploration of the properties of functions like domains, ranges, and images of sets and sequences; and an introduction to basic aspects of continuous functions between Euclidean spaces. Even so, ``arbitrarily close'' reaches deeper than discussed here. The idea is topological in nature and there is much more to explore.

math.HO↗

arXiv2022

Arbitrarily Close

Mathematicians tend to use the phrase "arbitrarily close" to mean something along the lines of "every neighborhood of a point intersects a set". Taking the latter statement as a technical definition for arbitrarily close leads to an alternative development of classic concepts in real analysis such as supremum, closure, convergence and limits of sequences, closure, connectedness, compactness, and continuity. The goal of this text is to provide readers with an introduction to real analysis by taking deliberate steps to parse these difficult concepts using arbitrarily close as the kernel.

math.HO↗

arXiv2016

A Lecture on Integration by Parts

Integration by parts (IBP) has acquired a bad reputation. While it allows us to compute a wide variety of integrals when other methods fall short, its implementation is often seen as plodding and confusing. Readers familiar with tabular IBP understand that, in particular cases, it has the capacity to significantly streamline and shorten computation. In this paper, we establish a tabular approach to IBP that is completely general and, more importantly, a powerful tool that promotes exploration and creativity.

math.HO↗

arXiv2015

A survey of complex dimensions, measurability, and the lattice/nonlattice dichotomy

The theory of complex dimensions of fractal strings developed by Lapidus and van Frankenhuijsen has proven to be a powerful tool for the study of Minkowski measurability of fractal subsets of the real line. In a very general setting, the Minkowski measurability of such sets is characterized by the structure of corresponding complex dimensions. Also, this tool is particularly effective in the setting of self-similar fractal subsets of $\mathbb{R}$ which have been shown to be Minkowski measurable if and only if they are nonlattice. This paper features a survey on the pertinent results of Lapidus and van Frankenhuijsen and a preliminary extension of the theory of complex dimensions to subsets of Euclidean space, with an emphasis on self-similar sets that satisfy various separation conditions. This extension is developed in the context of box-counting measurability, an analog of Minkowski measurability, which is shown to be characterized by complex dimensions under certain mild conditions.

math-ph↗

arXiv2013

Box-counting fractal strings, zeta functions, and equivalent forms of Minkowski dimension

We discuss a number of techniques for determining the Minkowski dimension of bounded subsets of some Euclidean space of any dimension, including: the box-counting dimension and equivalent definitions based on various box-counting functions; the similarity dimension via the Moran equation (at least in the case of self-similar sets); the order of the (box-)counting function; the classic result on compact subsets of the real line due to Besicovitch and Taylor, as adapted to the theory of fractal strings; and the abscissae of convergence of new classes of zeta functions. Specifically, we define box-counting zeta functions of infinite bounded subsets of Euclidean space and discuss results pertaining to distance and tube zeta functions. Appealing to an analysis of these zeta functions allows for the development of theories of complex dimensions for bounded sets in Euclidean space, extending techniques and results regarding (ordinary) fractal strings obtained by the first author and van Frankenhuijsen.

math-ph↗

arXiv2013

Multifractal analysis via scaling zeta functions and recursive structure of lattice strings

The multifractal structure underlying a self-similar measure stems directly from the weighted self-similar system (or weighted iterated function system) which is used to construct the measure. This follows much in the way that the dimension of a self-similar set, be it the Hausdorff, Minkowski, or similarity dimension, is determined by the scaling ratios of the corresponding self-similar system via Moran's theorem. The multifractal structure allows for our definition of scaling regularity and scaling zeta functions motivated by geometric zeta functions and, in particular, partition zeta functions. Some of the results of this paper consolidate and partially extend the results regarding a multifractal analysis for certain self-similar measures supported on compact subsets of a Euclidean space based on partition zeta functions. Specifically, scaling zeta functions generalize partition zeta functions when the choice of the family of partitions is given by the natural family of partitions determined by the self-similar system in question. Moreover, in certain cases, self-similar measures can be shown to exhibit lattice or nonlattice structure with respect to specified scaling regularity values. Additionally, in the context provided by generalized fractal strings viewed as measures, we define generalized self-similar strings, allowing for the examination of many of the results presented here in a specific overarching context and for a connection to the results regarding the corresponding complex dimensions as roots of Dirichlet polynomials. Furthermore, generalized lattice strings and recursive strings are defined and shown to be very closely related.

math-ph↗

arXiv2011

Partition zeta functions, multifractal spectra, and tapestries of complex dimensions

For a Borel measure and a sequence of partitions on the unit interval, we define a multifractal spectrum based on coarse Holder regularity. Specifically, the coarse Holder regularity values attained by a given measure and with respect to a sequence of partitions generate a sequence of lengths (or rather, scales) which in turn define certain Dirichlet series, called the partition zeta functions. The abscissae of convergence of these functions define a multifractal spectrum whose concave envelope is the (geometric) Hausdorff multifractal spectrum which follows from a certain type of Moran construction. We discuss at some length the important special case of self-similar measures associated with weighted iterated function systems and, in particular, certain multinomial measures. Moreover, our multifractal spectrum is shown to extend to a tapestry of complex dimensions for two specific atomic measures.

math-ph↗

arXiv2009

Fractal Strings and Multifractal Zeta Functions

For a Borel measure on the unit interval and a sequence of scales that tend to zero, we define a one-parameter family of zeta functions called multifractal zeta functions. These functions are a first attempt to associate a zeta function to certain multifractal measures. However, we primarily show that they associate a new zeta function, the topological zeta function, to a fractal string in order to take into account the topology of its fractal boundary. This expands upon the geometric information garnered by the traditional geometric zeta function of a fractal string in the theory of complex dimensions. In particular, one can distinguish between a fractal string whose boundary is the classical Cantor set, and one whose boundary has a single limit point but has the same sequence of lengths as the complement of the Cantor set. Later work will address related, but somewhat different, approaches to multifractals themselves, via zeta functions, partly motivated by the present paper.

math-ph↗

arXiv2008

Toward zeta functions and complex dimensions of multifractals

Multifractals are inhomogeneous measures (or functions) which are typically described by a full spectrum of real dimensions, as opposed to a single real dimension. Results from the study of fractal strings in the analysis of their geometry, spectra and dynamics via certain zeta functions and their poles (the complex dimensions) are used in this text as a springboard to define similar tools fit for the study of multifractals such as the binomial measure. The goal of this work is to shine light on new ideas and perspectives rather than to summarize a coherent theory. Progress has been made which connects these new perspectives to and expands upon classical results, leading to a healthy variety of natural and interesting questions for further investigation and elaboration.

math-ph↗