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  Exercises in Applied Mathematics With a View toward Information Theory, Machine Learning, Wavelets, and Statistical Physics

  Daniel Alpay

  This text presents a collection of mathematical exercises with the aim of guiding readers to study topics in statistical physics, equilibrium thermodynamics, information theory, and their various connections. It explores essential tools from linear algebra, elementary functional analysis, and probability theory in detail and demonstrates their applications in topics such as entropy, machine learning, error-correcting codes, and quantum channels. The theory of communication and signal theory are also in the background, and many exercises have been chosen from the theory of wavelets and machine learning. Exercises are selected from a number of different domains, both theoretical and more applied. Notes and other remarks provide motivation for the exercises, and hints and full solutions are given for many. For senior undergraduate and beginning graduate students majoring in mathematics, physics, or engineering, this text will serve as a valuable guide as theymove on to more advanced work.

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  And I Saw Sequences of Petals and Leaves: My Life as the One They Call Fibonacci

  Daniele C. Struppa

  In this captivating historical novel, Daniele Struppa skillfully weaves a fictional autobiography, bringing Fibonacci to life with vivid details of his upbringing and adult years in Medieval Europe. As we explore the historical context of Fibonacci's time, we delve into the intriguing aspects of a bygone era, painting a compelling picture of a man whose contributions to mathematics continue to resonate today.

  From his groundbreaking work on congruent numbers to the famous numerical sequence that bears his name, the author invites readers to imagine the creative sparks that ignited Fibonacci's mathematical innovations. When historical evidence is elusive, accuracy and passion are seamlessly combined, offering plausible scenarios grounded in documented facts. A meticulously crafted apparatus of notes distinguishes fact from fiction, providing readers with a clear guide to navigate this enthralling reconstruction of Fibonacci's life.

  Step into the medieval world of Leonardo Fibonacci, one of the most celebrated mathematicians in history, and discover the man behind the mathematical genius. Mathematicians and curious readers alike will appreciate the allure of Fibonacci's mathematical brilliance.

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  Platonism, De Re, and (Philosophy of) Mathematical Practice

  Marco Panza

  The chapter advances a reformulation of the classical problem of the nature of mathematical objects (if any), here called “Plato’s problem,” in line with the program of a philosophy of mathematical practice. It then provides a sketch of a platonist solution, following the same perspective. This solution disregards as nonsensical the question of the existence of abstract, and specifically mathematical, objects, by rather focusing on the modalities of our access to them: objects (in general, both concrete and abstract) are regarded as individual contents that we have (or can have) a de re epistemic access to. The question of the existence of mathematical objects is then replaced by that of the modalities of our de re epistemic access to individual mathematical contents.

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  Extreme Development of Dragon Fruit Agriculture with Nighttime Lighting in Southern Vietnam

  Shenyue Jia, Son V. Nghiem, Seung-Hee Kim, Laura Krauser, Andrea E. Gaughan, Forest R. Stevens, Menas Kafatos, and Khanh D. Ngo

  Dragon fruit is widely grown in Southeast Asia and other tropical or subtropical regions. As a high-value cash crop ideal for exportation, dragon fruit cultivation has boomed during the past decade in southern Vietnam. Light supplementing during the winter months using artificial lighting sources is a widely adopted cultivation technique to boost productivity in the major dragon fruit planting regions of Vietnam. The application of electric lighting at night leads to a significant increase of nighttime light (NTL) observable by satellite sensors. The strong seasonality signal of NTL in dragon fruit cultivation enables identifying dragon fruit plantations using NTL images. We employed Visible Infrared Imaging Radiometer Suite (VIIRS) Day/Night Band (DNB) monthly nighttime imagery from 2012 to 2019 to extract the growing area of dragon fruit in Bình Thuận Province, the largest dragon fruit growing region of Vietnam. The Breakpoint for Additive Seasonal Trend (B-FAST) analysis was applied to calculate the seasonality of NTL inside the dragon fruit plantations and distinguish them from the background. The results indicated that the dragon fruit cultivation strongly increased after 2014 and reached a plateau after 2017. In recent years, dragon fruit cultivation has experienced a slight decrease due to market fluctuations. We applied a buffer analysis over the largest dragon fruit cultivation area in Bình Thuận to analyze the spatial trend of the expansion of dragon fruit planting. Our results suggest that the dragon fruit cultivation of Bình Thuận has expanded to cover most inter-hill plains, reaching a spatial extent capacity due to the topographical constraints, and thus has begun to encroach into the low-elevation foothill area. In the case of emergency lock-down orders in February 2020 during the COVID-19 pandemic, NTL used for dragon fruit cultivation changed heterogeneously in space and time, driven by market price and shipping limitations far away from the local restrictions. Under the dual rural-urban hot spot situation with strong and contemporary developments of both dragon fruit agriculture and the urban tourism industry, building structures were detected densely in the city and gradually dispersed well into the rural landscape in Bình Thuận. The outcomes of this study will be valuable for local policymakers to better understand of the available area for dragon fruit cultivation and achieve better-coordinated cultivation planning against future fluctuations of the global market while providing insights and new understanding into the dual hot-spot developments valuable for planning rural-urban change strategies.

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  L’applicabilité des Mathématiques

  Daniele Molinini and Marco Panza

  "Les mathématiques s'appliquent avec succès au monde qui nous entoure et nous aident à raisonner sur les phénomènes empiriques (tant naturels que sociaux), c'est-à-dire sur des faits qui font l'objet d'expéeriences par l'observation et l'expérimentation. Cela ne fait aucun doute."

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  Earthquake Precursors in the Atmosphere and Ionosphere

  Sergey Pulinets, Dimitar Ouzounov, Alexander Karelin, and Kyrill Boyarchuk

  This book discusses how the increased emanation of radon and other gases from the Earth’s crust in the vicinity of active tectonic faults triggers a chain of physical processes and chemical reactions in the atmospheric boundary layer and the Earth’s ionosphere over an earthquake area several days/hours before strong seismic shocks occur. It presents the two main concepts involved in this mechanism: atmosphere ionization and the global electric circuit. The Lithosphere-Atmosphere-Ionosphere Coupling (LAIC) concept is strongly supported by experimental data showing the atmospheric and ionospheric precursors for major recent earthquakes including 2004 Sumatra; 2008 Sichuan, China; 2011 Tohoku, Japan; and 2015 Nepal. The book not only addresses the theoretical considerations but also includes information on experimental techniques used for precursor observations based on the space-borne systems. Providing practical methods of precursor identification and interpretation, it is an excellent textbook for graduate courses in geophysics, earthquake science, atmospheric physics and remote sensing. Moreover, it offers a wealth of information for scientists and experts from governmental and international agencies working in the fields of natural-disaster mitigation, response and recovery.

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  On the Representation of Boolean Magmas and Boolean Semilattices

  Peter Jipsen, M. Eyad Kurd-Misto, and James Wimberley

  A magma is an algebra with a binary operation ·, and a Boolean magma is a Boolean algebra with an additional binary operation · that distributes over all finite Boolean joins. We prove that all square-increasing (x ≤ x²) Boolean magmas are embedded in complex algebras of idempotent (x = x²) magmas. This solves a problem in a recent paper [3] by C. Bergman. Similar results are shown to hold for commutative Boolean magmas with an identity element and a unary inverse operation, or with any combination of these properties.

  A Boolean semilattice is a Boolean magma where · is associative, commutative, and square-increasing. Let SL be the class of semilattices and let S(SL⁺) be all subalgebras of complex algebras of semilattices. All members of S(SL⁺) are Boolean semilattices and we investigate the question of which Boolean semilattices are representable, i.e., members of S(SL⁺). There are 79 eight-element integral Boolean semilattices that satisfy a list of currently known axioms of S(SL⁺). We show that 72 of them are indeed members of S(SL⁺), leaving the remaining 7 as open problems.

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  Qué podría haber sido la universalidad para Euclides

  Marco Panza

  Las proposiciones geométricas de los Elementos de Euclides son universales. Pero, ¿en qué sentido lo son? ¿Tratan las proposiciones geométricas acerca de una cierta totalidad de ítems geométricos (posiblemente objetos)? Ello resultaría sugerido al leer tales proposiciones como sigue: 'Para todo segmento x, construir esto y esto'; 'Todos los triángulos son así y así'. ¿Tratan estas proposiciones de cualquier elemento de tal totalidad? Ello resultaría sugerido al leerlas del siguiente modo: 'Dado cualquier segmento, construir esto y esto'. 'Cualquier triángulo es así y así'. ¿Se refieren a esquemas (en el sentido lógico) de cualquier elemento de tal totalidad? Como parecería estar sugerido al leerlas de la siguiente manera: 'Dado el (un) segmento AB, construir esto y esto', 'El (un) triángulo ABC es así y así'. Considero que todas estas interpretaciones, estén ellas apoyadas o no en consideraciones filológicas, entran en conflicto con un hecho crucial: que no se proporciona ninguna condición global de identidad para los ítems relevantes, e incluso no hubiera podido proporcionarse en el entramado conceptual de la geometría de Euclides. Todo lo que se proporciona son condiciones locales de identidad, que dependen de representaciones diagramáticas de estos ítems. Luego, ningún sentido claro estaría disponible para las afirmaciones universales referidas, en un sentido u otro, a una totalidad fija de ítems geométricos. Posiblemente, el modo en que hoy concebimos a tal totalidad se encuentra condicionado, o al menos es diferente del modo en que lo griegos concebían una totalidad fija de ítems geométricos. Sin embargo, la pregunta que estoy planteando es una pregunta para nosotros, no para ellos. No estoy preguntando si Euclides concebía a una totalidad de ítems geométricos en algún sentido compatible con la ausencia de condiciones globales de identidad para sus elementos. Estoy preguntando si existe un modo para nosotros de comprender lo que él podría haber tomado como una afirmación universal, dado que no podemos adscribirle nuestra concepción de una totalidad fija de ítems geométricos. No quiero una respuesta desde la filología, simplemente porque esta es una pregunta que la filología no puede responder, puesto que no trata de lo que está escrito, sino de cómo nosotros entendemos lo que está escrito. Sin embargo,

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  Quaternionic de Branges Spaces and Characteristic Operator Function

  Daniel Alpay, Fabrizio Colombo, and Irene Sabadini

  This work contributes to the study of quaternionic linear operators. This study is a generalization of the complex case, but the noncommutative setting of quaternions shows several interesting new features, see e.g. the so-called S-spectrum and S-resolvent operators. In this work, we study de Branges spaces, namely the quaternionic counterparts of spaces of analytic functions (in a suitable sense) with some specific reproducing kernels, in the unit ball of quaternions or in the half space of quaternions with positive real parts. The spaces under consideration will be Hilbert or Pontryagin or Krein spaces. These spaces are closely related to operator models that are also discussed.

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  Commutative Doubly-Idempotent Semirings Determined by Chains and by Preorder Forests

  Natanael Alpay and Peter Jipsen

  A commutative doubly-idempotent semiring (cdi-semiring) (S,V,·,0,1) is a semilattice (S,V,0) with x V 0 = x and a semilattices (S,·,1) with identity 1 such that x0 = 0, and x(y V z) = xy V xz holds for all x, y, z ϵ S. Bounded distributive lattices are cdi-semirings that satisfy xy = x ^ y, and the variety of cdi-semirings covers the variety of bounded distributive lattices. Chajda and Länger showed in 2017 that the variety of all cdi-semirings is generated by a 3-element cdi-semiring. We show that there are seven cdi-semirings with a V-semilattice of height less than or equal to 2. We construct all cdi-semirings for which their multiplicative semilattice is a chain with n + 1 elements, and we show that up to isomorphism the number of such algebras is the nth Catalan number C_n = (1/(n+1)) (2n/n ) . We also show that cdi-semirings with a complete atomic Boolean V-semilattice on the set of atoms A are determined by singleton-rooted preorder forests on the set A. From these results we obtain efficient algorithms to construct all multiplicatively linear cdisemirings of size n and all Boolean cdi-semirings of size 2n.

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  Michele Sce's Works in Hypercomplex Analysis

  Fabrizio Colombo, Irene Sabadini, and Daniele C. Struppa

  "This book presents English translations of Michele Sce’s most important works, originally written in Italian during the period 1955-1973, on hypercomplex analysis and algebras of hypercomplex numbers. Despite their importance, these works are not very well known in the mathematics community because of the language they were published in. Possibly the most remarkable instance is the so-called Fueter-Sce mapping theorem, which is a cornerstone of modern hypercomplex analysis, and is not yet understood in its full generality."

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  Weakening Relation Algebras and FL²-algebras

  Nikolaos Galatos and Peter Jipsen

  FL²-algebras are lattice-ordered algebras with two sets of residuated operators. The classes RA of relation algebras and GBI of generalized bunched implication algebras are subvarieties of FL²-algebras. We prove that the congruences of FL²-algebras are determined by the congruence class of the respective identity elements, and we characterize the subsets that correspond to this congruence class. For involutive GBI-algebras the characterization simplifies to a form similar to relation algebras.

  For a positive idempotent element p in a relation algebra A, the double division conucleus image p/A/p is an (abstract) weakening relation algebra, and all representable weakening relation algebras (RWkRAs) are obtained in this way from representable relation algebras (RRAs). The class S(dRA) of subalgebras of {p/A/p∶ A ϵ RA; 1 ≤ p² = p ϵ A} is a discriminator variety of cyclic involutive GBI-algebras that includes RA. We investigate S(dRA) to find additional identities that are valid in all RWkRAs. A representable weakening relation algebra is determined by a chain if and only if it satisfies 0 ≤ 1, and we prove that the identity 1 ≤ 0 holds only in trivial members of S(dRA).

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  Shortcut to Superconductivity: Superconducting Electronics via COMSOL Modeling

  Armen Gulian

  "This accessible textbook offers a novel, concept-led approach to superconducting electronics, using the COMSOL Multiphysics software to help describe fundamental principles in an intuitive manner.

  Based on a course taught by the author and aimed primarily at engineering students, the book explains concepts effectively and efficiently, uncovering the “shortcut” to understanding each topic, enabling readers to quickly grasp the underlying essence. The book is divided into two main parts; the first part provides a general introduction to key topics encountered in superconductivity, illustrated using COMSOL simulations based on time-dependent Ginzburg-Landau equations and avoiding any deeply mathematical derivations. It includes numerous worked examples and problem sets with tips and solutions."

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  Measure and Continuity in Aristotle’s Physics V,3 (and Neighbourhoods)

  Marco Panza

  "As surprising as it might appear prima facie, it is not so much after a little reflexion. Whatever role measure could have played in Aristotle’s time in physics and mathematics, was depending on the content of the available mathematical theories, which is something Aristotle had less to say on, or even about which he never tried to say something. His attitude toward mathematics was always unequivocal: he certainly tried (without much success, by the way, as shown by the quite ambiguous claims of Metaphysics M 1-3) to understand its epistemic nature by so continuing, in a quite different direction, a reflexion already initiated by Plato, but he always took it as a datum, as an established portion of knowledge that is not to be contradicted. He never tried to improve or clarify it. What he says about measure seems to confirm this attitude. To clarify this point, a short survey of the mathematical conceptions of measure at his time, and of the way they differ from our present ones, is in order."

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  Note sull’evoluzione delle nozioni di continuità e di continuo in matematica

  Marco Panza

  Lo scopo di questo saggio è di fornire alcune linee direttive relative alla maniera in cui le nozioni matematiche di continuità e di continuo si sono gradualmente trasformate, a partire dalle concezioni di Aristotele, fino alle concezioni oggi in uso nelle presentazioni elementari dell'analisi reale, risalenti alla seconda metà dell'ottocento.

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  Interrogating Regulatory Mechanisms in Signaling Proteins by Allosteric Inhibitors and Activators: A Dynamic View Through the Lens of Residue Interaction Networks

  Lindy Astl, Amanda Tse, and Gennady M. Verkhivker

  Computational studies of allosteric interactions have witnessed a recent renaissance fueled by the growing interest in modeling of the complex molecular assemblies and biological networks. Allosteric interactions in protein structures allow for molecular communication in signal transduction networks. In this chapter, we discuss recent developments in understanding of allosteric mechanisms and interactions of protein systems, particularly in the context of structural, functional, and computational studies of allosteric inhibitors and activators. Computational and experimental approaches and advances in understanding allosteric regulatory mechanisms are reviewed to provide a systematic and critical view of the current progress in the development of allosteric modulators and highlight most challenging questions in the field. The abundance and diversity of genetic, structural, and biochemical data underlies the complexity of mechanisms by which targeted and personalized drugs can combat mutational profiles in protein kinases. Structural and computational studies of protein kinases have generated in recent decade significant insights that allowed leveraging knowledge about conformational diversity and allosteric regulation of protein kinases in the design and discovery of novel kinase drugs. We discuss recent developments in understanding multilayered allosteric regulatory machinery of protein kinases and provide a systematic view of the current state in understanding molecular basis of allostery mediated by kinase inhibitors and activators. In conclusion, we highlight the current status and future prospects of computational biology approaches in bridging the basic science of protein kinases with the discovery of anticancer therapies.

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  Logics for Rough Concept Analysis

  Giuseppe Greco, Peter Jipsen, Krishna Manoorkar, Alessandra Palmigiano, and Apostolos Tzimoulis

  Taking an algebraic perspective on the basic structures of Rough Concept Analysis as the starting point, in this paper we introduce some varieties of lattices expanded with normal modal operators which can be regarded as the natural rough algebra counterparts of certain subclasses of rough formal contexts, and introduce proper display calculi for the logics associated with these varieties which are sound, complete, conservative and with uniform cut elimination and subformula property. These calculi modularly extend the multi-type calculi for rough algebras to a ‘nondistributive’ (i.e. general lattice-based) setting.

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  On the Structure of Generalized Effect Algebras and Separation Algebras

  Sarah Alexander, Peter Jipsen, and Nadiya Upegui

  Separation algebras are models of separation logic and effect algebras are models of unsharp quantum logics. We investigate these closely related classes of partial algebras as well as their noncommutative versions and the subclasses of (generalized) (pseudo-)orthoalgebras. We present an orderly algorithm for constructing all nonisomorphic generalized pseudoeffect algebras with n elements and use it to compute these algebras with up to 10 elements.

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  Introduction to Indefinite Inner Product Spaces, Schur Analysis, and Differential Equations: A Volume Dedicated to Heinz Langer

  Daniel Alpay and Bernd Kirstein

  "This volume is a tribute to Heinz Langer on the occasion of his eightieth birthday. Two earlier OT volumes (namely, volume 106 and volume 163; see [19] and [35] respectively) were dedicated to Heinz, the first one on the occasion of his sixtieth birthday, and the second one on the occasion of his retirement...The first named editor (DA) worked very hard on Heinz’ papers (and in particular on the above-mentioned series of papers with M.G. Krein) during his doctoral studies [1] at the Weizmann Institute in Israel."

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  Asymptotic Quasi-completeness and ZFC

  Mirna Džamonja and Marco Panza

  The axioms ZFC of first order set theory are one of the best and most widely accepted, if not perfect, foundations used in mathematics. Just as the axioms of first order Peano Arithmetic, ZFC axioms form a recursively enumerable list of axioms, and are, then, subject to Gödel’s Incompleteness Theorems. Hence, if they are assumed to be consistent, they are necessarily incomplete. This can be witnessed by various concrete statements, including the celebrated Continuum Hypothesis CH. The independence results about the infinite cardinals are so abundant that it often appears that ZFC can basically prove very little about such cardinals. However, we put forward a thesis that ZFC is actually very powerful at some infinite cardinals, but not at all of them. We have to move away from the first few and to look at limits of uncountable cardinals, such as N_w. Specifically, we work with singular cardinals (which are necessarily limits) and we illustrate that at such cardinals there is a very serious limit to independence and that many statements which are known to be independent on regular cardinals become provable or refutable by ZFC at singulars. In a certain sense, which we explain, the behavior of the set-theoretic universe is asymptotically determined at singular cardinals by the behavior that the universe assumes at the smaller regular cardinals. Foundationally, ZFC provides an asymptotically univocal image of the universe of sets around the singular cardinals. We also give a philosophical view accounting for the relevance of these claims in a platonistic perspective which is different from traditional mathematical platonism.

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  Enthymemathical Proofs and Canonical Proofs in Euclid’s Plane Geometry

  Abel Lassalle and Marco Panza

  Since the application of Postulate I.2 in Euclid’s Elements is not uniform, one could wonder in what way should it be applied in Euclid’s plane geometry. Besides legitimizing questions like this from the perspective of a philosophy of mathematical practice, we sketch a general perspective of conceptual analysis of mathematical texts, which involves an extended notion of mathematical theory as system of authorizations, and an audience-dependent notion of proof.

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  Multiparameter Assessment of Pre‐Earthquake Atmospheric Signals

  Dimitar Ouzounov, Sergey Pulinets, Jann-Yenq Liu, Katsumi Hattori, and Peng Han

  We apply interdisciplinary observation to study earthquake processes, their physics, and the phenomena that precede their energy release. Our approach is based on multisensor observations of short-term pre-earthquake phenomena preceding large earthquakes (M>6). The integrated satellite and terrestrial framework is our method for validation and is based on a sensor web of several physical and environmental parameters (satellite thermal infrared radiation (STIR), electron concentration in the ionosphere, air temperature, and relative humidity measurements) that were associated with earthquakes. The scientific rationale for multidisciplinary analysis is founded on the concept lithosphere-atmosphere-ionosphere coupling. To check the predictive potential of pre-earthquake signals we validate in retrospective and prospective modes. Our validation processes consist of two steps: (a) a retrospective analysis preformed over three different regions with high seismic activity (M 6.0 Napa of 2014, M 6.0 Taiwan of 2016, and M 7.9 Kumamoto, Japan of 2016); (b) testing of Molchan's error diagram (MED) for STIR and differential total electron content anomalous events over Japan and Taiwan. Our findings suggest that: (a) pre-earthquake signals (with 1-30 days time lag) follow a general temporal-spatial evolution pattern; (b) pre-earthquake atmospheric anomalies can provide short-term predictive information for the occurrence of major earthquakes in the tested regions.

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  Was Frege a Logicist for Arithmetic?

  Marco Panza

  The paper argues that Frege’s primary foundational purpose concerning arithmetic was neither that of making natural numbers logical objects, nor that of making arithmetic a part of logic, but rather that of assigning to it an appropriate place in the architectonics of mathematics and knowledge, by immersing it in a theory of numbers of concepts and making truths about natural numbers, and/or knowledge of them transparent to reason without the medium of senses and intuition.

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  Lithosphere–Atmosphere–Ionosphere–Magnetosphere Coupling—A Concept for Pre‐Earthquake Signals Generation

  Sergey Pulinets, Dimitar Ouzounov, Alexander Karelin, and Dmitry Davidenko

  "The physics of earthquake genesis at the latest stage of a seismic cycle has always been the subject of scientific interest and discussion. In such discussion and in model development, however, only solid Earth processes have been contemplated. Development of satellite technologies brought new insight to the problem: new types of anomalies in the atmosphere and ionosphere were discovered that appear a few days/weeks before the main shock within the zone of earthquake preparation. These anomalies were thoroughly investigated, and results of the research have demonstrated a high statistical confidence warrants their identification as short-term earthquake precursors. In this chapter we seek to clarify how information and energy are transported from underground to the upper and lower layers of the atmosphere, including near-Earth space. Owing to the interdisciplinary character of the model, it is very difficult to create a common code for all its physical and spatial domains. That is why the material presented here should be considered conceptual rather than modeling output. As an example, if the boundary layer plasmachemistry instrumentation is applied, for ionospheric anomalies the electromagnetism approach is adopted. This conceptual approach, however, proved to be valid not only for earthquakes but also for other natural and technological disasters where air ionization occurs."

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  Preface to Natural Hazards: Earthquakes, Volcanoes, and Landslides

  Ramesh Singh and Darius Bartlett

  A preface to Natural Hazards: Earthquakes, Volcanoes, and Landslides, edited by Ramesh Singh and Darius Bartlett, that gives an overview of the book and the contributions of each chapter.