Domain theory and differential calculus (functions of one variable)

Published versio

Geometrical Theory of Analytic Functions

The book contains papers published in the Mathematics Special Issue, entitled "Geometrical Theory of Analytic Functions". Fifteen papers devoted to the study concerning complex-valued functions of one variable present new outcomes related to special classes of univalent functions, differential equations in view of geometric function theory, quantum calculus and its applications in geometric function theory, operators and special functions associated with differential subordination and superordination theories and starlikeness, and convexity criteria

An Analysis of the Theory of Functions of One Real Variable

Few undergraduates are aware that the Riemann integral taught in introductory calculus courses has only limited application-essentially this integral can be used only to integrate continuous functions over intervals. The necessity to integrate a broader class of functions over a wider range of sets that arises in many applications motivates the theory of abstract integration and functional analysis. The founder of this theory was the French mathematician Henri Lebesgue, who in 1902 defined the Lebesgue measure of subsets of the real line. The purpose of this project is to elucidate the theory of abstract measure spaces and of important spaces of functions (a critical example of which are Banach spaces), and extend the application of this theory. Developing the tools for doing so has been the focus of my advisor Professor Dmitry Khavinson and me over the past three years. The primary goal of the thesis is to make this highly formal and abstract material accessible to an undergraduate having only a year of coursework in advanced calculus. These concepts are typically introduced at the graduate level, but the ideas require only a familiarity with the analytic style of proof learned as an undergraduate. It would be advantageous to expose advanced undergraduates to this material since these ideas form the foundation for how mathematical research is done at the professional level. The addition of interesting and practical examples (which are scarce in the standard graduate texts) will help to make the concepts more familiar and down-to-earth. The motivation for a new theory of integration came from the Riemann integral\u27s apparent inability to operate on functions that fail to be continuous. For example, the Riemann integral of the function that assigns the value 1 to rational numbers and 0 to irrational numbers can be evaluated over the interval [0, 1] with equally valid justification to be 0 or 1. This is because the definition of the Riemann integral depends on partitioning the domain of the function to be integrated, and finding the maximum and minimum values of the function over each partition. The Lebesgue integral, on the other hand, partitions the range of the function to be integrated and then considers the length of the Jason Reed and Dmitry Khavinson pre-image of each partition as well as the maximum and minimum values of the function of the partition. The utility of this change of perspective arises when we refine what is meant by length in the aforementioned pre-image. The Riemann integral requires that the domain consist of intervals of real numbers (where length makes sense), while the Lebesgue integral can be used with a much broader class of sets. Lebesgue modified the notion of length by defining the measure of a set E to be the smallest possible total length of all collections of intervals that cover E. Using this ingenious method, Lebesgue constructed a theory of integration which forms the most useful example of all general integration theories. The theory has important applications in many areas of science and engineering as well as probability and statistics. Our approach to the subject has emphasized theory developed in H.L. Royden\u27s classic text, Real Analysis. My project has included analysis of each concept in the text, and I have developed for each major subject a collection of problems solved and applications of major theorems that were explored. The result has been comprehension of many of the foundational ideas in the field. We have used a number of supplemental texts to gain depth of understanding where Royden\u27s text provides only a survey, such as the Riesz Representation theorem, and to extend important ideas, such as the consideration of complex-valued (in addition to real-valued) measures. The synthesis has been a comprehensive paper which describes the theoretical directions the research has taken, the major results and theorems with proof, and applications and examples which are worked out in detail. The final record of my research will be divided into the following six sections: Lebesgue measure, Lebesgue integral, relationship between differentiation and Lebesgue integration, Banach space theory, abstract measure theory, and general integration theory. The analysis encompasses discussion of the main ideas (what it means for a set function to be a measure, how an integral can be defined in a coherent way with respect to a measure, when the derivative of an integral of a function is the function itself, different ideas about what it means for a sequence of functions to converge to a function, what are the properties of Banach spaces and why they are useful, etc.), as well as important ideas and theorems that interrelate these concepts (i.e., when we can interchange the limit of a sequence of functions and the integral, how we can represent a bounded linear functional, the structure of certain spaces of integrable functions)

Looking backward: From Euler to Riemann

We survey the main ideas in the early history of the subjects on which Riemann worked and that led to some of his most important discoveries. The subjects discussed include the theory of functions of a complex variable, elliptic and Abelian integrals, the hypergeometric series, the zeta function, topology, differential geometry, integration, and the notion of space. We shall see that among Riemann's predecessors in all these fields, one name occupies a prominent place, this is Leonhard Euler. The final version of this paper will appear in the book \emph{From Riemann to differential geometry and relativity} (L. Ji, A. Papadopoulos and S. Yamada, ed.) Berlin: Springer, 2017

Entire slice regular functions

Entire functions in one complex variable are extremely relevant in several areas ranging from the study of convolution equations to special functions. An analog of entire functions in the quaternionic setting can be defined in the slice regular setting, a framework which includes polynomials and power series of the quaternionic variable. In the first chapters of this work we introduce and discuss the algebra and the analysis of slice regular functions. In addition to offering a self-contained introduction to the theory of slice-regular functions, these chapters also contain a few new results (for example we complete the discussion on lower bounds for slice regular functions initiated with the Ehrenpreis-Malgrange, by adding a brand new Cartan-type theorem). The core of the work is Chapter 5, where we study the growth of entire slice regular functions, and we show how such growth is related to the coefficients of the power series expansions that these functions have. It should be noted that the proofs we offer are not simple reconstructions of the holomorphic case. Indeed, the non-commutative setting creates a series of non-trivial problems. Also the counting of the zeros is not trivial because of the presence of spherical zeros which have infinite cardinality. We prove the analog of Jensen and Carath\'eodory theorems in this setting

Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics

We revisit the Mittag-Leffler functions of a real variable $t$, with one, two and three order-parameters $\{\alpha, \beta, \gamma\}$, as far as their Laplace transform pairs and complete monotonicty properties are concerned. These functions, subjected to the requirement to be completely monotone for $t>0$, are shown to be suitable models for non--Debye relaxation phenomena in dielectrics including as particular cases the classical models referred to as Cole-Cole, Davidson-Cole and Havriliak-Negami. We show 3D plots of the response functions and of the corresponding spectral distributions, keeping fixed one of the three order-parameters.Comment: 22 pages, 6 figures, Second Revised Versio

Differential Calculus: From Practice to Theory

Differential Calculus: From Practice to Theory covers all of the topics in a typical first course in differential calculus. Initially it focuses on using calculus as a problem solving tool (in conjunction with analytic geometry and trigonometry) by exploiting an informal understanding of differentials (infinitesimals). As much as possible large, interesting, and important historical problems (the motion of falling bodies and trajectories, the shape of hanging chains, the Witch of Agnesi) are used to develop key ideas. Only after skill with the computational tools of calculus has been developed is the question of rigor seriously broached. At that point, the foundational ideas (limits, continuity) are developed to replace infinitesimals, first intuitively then rigorously. This approach is more historically accurate than the usual development of calculus and, more importantly, it is pedagogically sound. The text also incorporates curated activities from the TRansforming Instruction in Undergraduate Mathematics Instruction via Primary Historical Sources (TRIUMPHS) project to provide students with ample opportunities to develop relevant competencies.https://knightscholar.geneseo.edu/oer-ost/1031/thumbnail.jp