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Physics of Biological Oscillators: New Insights into Non-Equilibrium and Non-Autonomous Systems (Understanding Complex Systems)
by Aneta Stefanovska Peter V. E. McClintockThis book, based on a selection of invited presentations from a topical workshop, focusses on time-variable oscillations and their interactions. The problem is challenging, because the origin of the time variability is usually unknown. In mathematical terms, the oscillations are non-autonomous, reflecting the physics of open systems where the function of each oscillator is affected by its environment. Time-frequency analysis being essential, recent advances in this area, including wavelet phase coherence analysis and nonlinear mode decomposition, are discussed. Some applications to biology and physiology are described.Although the most important manifestation of time-variable oscillations is arguably in biology, they also crop up in, e.g. astrophysics, or for electrons on superfluid helium. The book brings together the research of the best international experts in seemingly very different disciplinary areas.
Physics of Complex Systems: Discovery in the Age of Gödel
by Dragutin T. Mihailović Darko Kapor Siniša Crvenković Anja MihailovićThis book analyses the physics of complex systems to elaborate the problems encountered in teaching and research. Inspired by the of Kurt Gödel (including his incompleteness theorems) it considers the concept of time, the idea of models and the concept of complexity before trying to assess the state of physics in general. Using both general and practical examples, the idea of information is discussed, emphasizing its physical interpretation, debates ideas in depth using examples and evidence to provide detailed considerations on the topics. Based on the authors’ own research on these topics, this book puts forward the idea that the application of information measures can provide new results in the study of complex systems. Helpful for those already familiar with the concepts who wish to deepen their critical understanding, Physics of Complex Systems will be extremely valuable both for people that are already involved in complex systems and also readers beginning their journey into the subject. This work will encourage readers to follow and continue these ideas, enabling them to investigate the various topics further.
The Physics of Computing
by Luca GammaitoniThis book presents a self-contained introduction to the physics of computing, by addressing the fundamental underlying principles that involve the act of computing, regardless of the actual machine that is used to compute. Questions like “what is the minimum energy required to perform a computation?”, “what is the ultimate computational speed that a computer can achieve?” or “how long can a memory last”, are addressed here, starting from basic physics principles. The book is intended for physicists, engineers, and computer scientists, and it is designed for self-study by researchers who want to enter the field or as the main text for a one semester course at advanced undergraduate or graduate level. The theoretical concepts presented in this book are systematically developed from the very beginning, which only requires basic knowledge in physics and mathematics.
The Physics of Laser Plasmas and Applications - Volume 1: Physics of Laser Matter Interaction (Springer Series in Plasma Science and Technology)
by Hideaki TakabeThe series of books discusses the physics of laser and matter interaction, fluid dynamics of high-temperature and high-density compressible plasma, and kinetic phenomena and particle dynamics in laser-produced plasma. The book (Vol.1) gives the physics of intense-laser absorption in matter and/or plasma in non-relativistic and relativistic laser-intensity regime. In many cases, it is explained with clear images of physics so that an intuitive understanding of individual physics is possible for non-specialists. For intense-laser of 1013-16 W/cm2, the laser energy is mainly absorbed via collisional process, where the oscillation energy is converted to thermal energy by non-adiabatic Coulomb collision with the ions. Collisionless interactions with the collective modes in plasma are also described. The main topics are the interaction of ultra-intense laser and plasma for the intensity near and over 1018W/cm2. In such regime, relativistic dynamics become essential. A new physics appears due to the relativistic effects, such as mass correction, relativistic nonlinear force, chaos physics of particle motions, and so on. The book provides clearly the theoretical base for challenging the laser-plasma interaction physics in the wide range of power lasers. It is suitable as a textbook for upper-undergraduate and graduate students as well as for readers who want to understand the whole physics structure about what happen when an intense-laser irradiates any materials including solids, gas etc. Explaining the physics intuitively without complicated mathematics, it is also a valuable resource for engineering students and researchers as well as for self-study.
Physics of Liquid Matter (Soft and Biological Matter)
by Paola Gallo Mauro RovereThis book offers a didactic and a self-contained treatment of the physics of liquid and flowing matter with a statistical mechanics approach. Experimental and theoretical methods that were developed to study fluids are now frequently applied to a number of more complex systems generically referred to as soft matter. As for simple liquids, also for complex fluids it is important to understand how their macroscopic behavior is determined by the interactions between the component units. Moreover, in recent years new and relevant insights have emerged from the study of anomalous phases and metastable states of matter. In addition to the traditional topics concerning fluids in normal conditions, the authors of this book discuss recent developments in the field of disordered systems in condensed and soft matter. In particular they emphasize computer simulation techniques that are used in the study of soft matter and the theories and study of slow glassy dynamics. For these reasons the book includes a specific chapter about metastability, supercooled liquids and glass transition.The book is written for graduate students and active researchers in the field.
Physics of Oscillations and Waves: With Use Of Matlab And Python (Undergraduate Texts In Physics Ser.)
by Arnt Inge VistnesIn this textbook a combination of standard mathematics and modern numerical methods is used to describe a wide range of natural wave phenomena, such as sound, light and water waves, particularly in specific popular contexts, e.g. colors or the acoustics of musical instruments. It introduces the reader to the basic physical principles that allow the description of the oscillatory motion of matter and classical fields, as well as resulting concepts including interference, diffraction, and coherence. Numerical methods offer new scientific insights and make it possible to handle interesting cases that can’t readily be addressed using analytical mathematics; this holds true not only for problem solving but also for the description of phenomena. Essential physical parameters are brought more into focus, rather than concentrating on the details of which mathematical trick should be used to obtain a certain solution. Readers will learn how time-resolved frequency analysis offers a deeper understanding of the interplay between frequency and time, which is relevant to many phenomena involving oscillations and waves. Attention is also drawn to common misconceptions resulting from uncritical use of the Fourier transform. The book offers an ideal guide for upper-level undergraduate physics students and will also benefit physics instructors. Program codes in Matlab and Python, together with interesting files for use in the problems, are provided as free supplementary material.
Physics of Self-Organization and Evolution
by Werner Ebeling Rainer FeistelThis thoroughly updated version of the German authoritative work on self-organization has been completely rewritten by internationally renowned experts and experienced book authors to also include a review of more recent literature. It retains the original enthusiasm and fascination surrounding thermodynamic systems far from equilibrium, synergetics, and the origin of life, representing an easily readable book and tutorial on this exciting field. The book is unique in covering in detail the experimental and theoretical fundamentals of self-organizing systems as well as such selected features as random processes, structural networks and multistable systems, while focusing on the physical and theoretical modeling of natural selection and evolution processes. The authors take examples from physics, chemistry, biology and social systems, and include results hitherto unpublished in English. The result is a one-stop resource relevant for students and scientists in physics or related interdisciplinary fields, including mathematical physics, biophysics, information science and nanotechnology.
The Physics of the Dark Photon: A Primer (SpringerBriefs in Physics)
by Marco Fabbrichesi Emidio Gabrielli Gaia LanfranchiThis book is about the dark photon which is a new gauge boson whose existence has been conjectured. Due to its interaction with the ordinary, visible photon, such a particle can be experimentally detected via specific signatures. In this book, the authors review the physics of the dark photon from the theoretical and experimental point of view. They discuss the difference between the massive and the massless case, highlighting how the two phenomena arise from the same vector portal between the dark and the visible sector. A review of the cosmological and astrophysical observations is provided, together with the connection to dark matter physics. Then, a perspective on current and future experimental limits on the parameters of the massless and massive dark photon is given, as well as the related bounds on milli-charged fermions. The book is intended for graduate students and young researchers who are embarking on dark photon research, and offers them a clear and up-to-date introduction to the subject.
The Physics of Wall Street: A Brief History of Predicting the Unpredictable
by James Owen WeatherallA look inside the world of &“quants&” and how science can (and can&’t) predict financial markets: &“Entertaining and enlightening&” (The New York Times). After the economic meltdown of 2008, Warren Buffett famously warned, &“beware of geeks bearing formulas.&” But while many of the mathematicians and software engineers on Wall Street failed when their abstractions turned ugly in practice, a special breed of physicists has a much deeper history of revolutionizing finance. Taking us from fin-de-siècle Paris to Rat Pack–era Las Vegas, from wartime government labs to Yippie communes on the Pacific coast, James Owen Weatherall shows how physicists successfully brought their science to bear on some of the thorniest problems in economics, from options pricing to bubbles. The crisis was partly a failure of mathematical modeling. But even more, it was a failure of some very sophisticated financial institutions to think like physicists. Models—whether in science or finance—have limitations; they break down under certain conditions. And in 2008, sophisticated models fell into the hands of people who didn&’t understand their purpose, and didn&’t care. It was a catastrophic misuse of science. The solution, however, is not to give up on models; it&’s to make them better. This book reveals the people and ideas on the cusp of a new era in finance, from a geophysicist using a model designed for earthquakes to predict a massive stock market crash to a physicist-run hedge fund earning 2,478.6% over the course of the 1990s. Weatherall shows how an obscure idea from quantum theory might soon be used to create a far more accurate Consumer Price Index. The Physics of Wall Street will change how we think about our economic future. &“Fascinating history . . . Happily, the author has a gift for making complex concepts clear to lay readers.&” —Booklist
Physics with Excel and Python: Using the Same Data Structure Volume I: Basics, Exercises and Tasks
by Dieter MergelThis book is intended to serve as a basic introduction to scientific computing by treating problems from various areas of physics - mechanics, optics, acoustics, and statistical reasoning in the context of the evaluation of measurements. After working through these examples, students are able to independently work on physical problems that they encounter during their studies. For every exercise, the author introduces the physical problem together with a data structure that serves as an interface to programming in Excel and Python. When a solution is achieved in one application, it can easily be translated into the other one and presumably any other platform for scientific computing. This is possible because the basic techniques of vector and matrix calculation and array broadcasting are also achieved with spreadsheet techniques, and logical queries and for-loops operate on spreadsheets from simple Visual Basic macros. So, starting to learn scientific calculation with Excel, e.g., at High School, is a targeted road to scientific computing. The primary target groups of this book are students with a major or minor subject in physics, who have interest in computational techniques and at the same time want to deepen their knowledge of physics. Math, physics and computer science teachers and Teacher Education students will also find a companion in this book to help them integrate computer techniques into their lessons. Even professional physicists who want to venture into Scientific Computing may appreciate this book.
Physics with Photons Using the ATLAS Run 2 Data: Calibration and Identification, Measurement of the Higgs Boson Mass and Search for Supersymmetry in Di-Photon Final State (Springer Theses)
by Stefano ManzoniThe work presented in this book is based on the proton-proton collision data from the Large Hadron Collider at a centre-of-mass energy of 13 TeV recorded by the ATLAS detector in 2015 and 2016. The research program of the ATLAS experiment includes the precise measurement of the parameters of the Standard Model, and the search for signals of physics beyond the SM. Both these approaches are pursued in this thesis, which presents two different analyses: the measurement of the Higgs boson mass in the di-photon decay channel, and the search for production of supersymmetric particles (gluinos, squarks or winos) in a final state containing two photons and missing transverse momentum. Finally, ATLAS detector performance studies, which are key ingredients for the two analyses outlined before, are also carried out and described.
Physics—Problems, Solutions, and Computer Calculations: Vol. 1 Mechanics, Properties of Matter, and Heat
by Wan Muhamad Wan HassanKnowledge of and skill in physics are essential foundations for studies in science and engineering. This book offers students an introduction to the basic concepts and principles of physics. It covers various topics specifically related to physical mechanics, the properties of matter, and heat. Each chapter begins with a summary of concepts, principles, definitions, and formulae to be discussed, as well as ending with problems and solutions that illustrate the specific topic. Steps are detailed to help build reasoning and understanding. There are 300 worked problems and 100 exercises in the book, as well as 306 figures to help the reader visualize the processes being addressed. Computer calculations and solutions are carried out using wxMaxima to give insight and help build computational skills. The book is aimed at first-year undergraduate students studying introductory physics, and would also be useful for physics teachers in their instruction, particularly the exercises at the end of each chapter.
Physik für Studierende der Biowissenschaften, Chemie und Medizin
by Gerhard RufaDieses Lehrbuch bietet eine klar strukturierte Einführung in die Physik, wobei die Grundlagen, wie sie für Studierende mit Physik im Nebenfach benötigt werden, in einer leicht verständlichen Weise dargestellt werden. Die physikalischen Begriffe und Gesetzmäßigkeiten werden anhand vieler Abbildungen sowie durch Beispiele aus dem täglichen Leben, den Biowissenschaften, der Chemie und der Medizin erläutert. Vorkenntnisse aus der Physik werden nicht vorausgesetzt, sodass Studierende sich selbstständig das physikalische Grundlagenwissen erarbeiten können und ein Verständnis der abstrakten physikalischen Begriffe und Gesetzmäßigkeiten erhalten.Ergänzt wird dieses Lehrbuch von dem separaten Übungsbuch Physik für Studierende der Biowissenschaften, Chemie und Medizin, das eine Vielzahl von Übungsserien und Testserien mit ausführlichen Lösungen und Erklärungen enthält. Es hilft Studierenden, sich anhand ausgewählter physikalischer Probleme Schritt für Schritt die Physik und auch die erforderliche Mathematik zu erarbeiten. Beide Bücher eignen sich somit hervorragend als Ergänzung zur Vorlesung sowie für die Klausurvorbereitung.
Physik und Finanzen
by Volker ZiemannDieses Buch führt Physikstudenten in die Konzepte und Methoden der Finanzwissenschaft ein. Obwohl die Finanzwissenschaft als recht weit von der Physik entfernt wahrgenommen wird, teilt sie eine Reihe gemeinsamer Methoden und Ideen, die in der Regel mit Rauschen und Unsicherheiten zu tun haben. Durch die Gegenüberstellung der wichtigsten Methoden mit Anwendungen in der Physik und im Finanzwesen werden sowohl die Unterschiede als auch die Gemeinsamkeiten deutlich, was den Studierenden ein tieferes Verständnis der zugrunde liegenden Ideen vermittelt. Darüber hinaus lernen sie eine Reihe nützlicher mathematischer und rechnerischer Werkzeuge kennen, wie stochastische Differentialgleichungen, Pfadintegrale, Monte-Carlo-Methoden und grundlegende Kryptologie. Jedes Kapitel endet mit einer Reihe sorgfältig ausgearbeiteter Übungen, mit denen die Leser ihr Verständnis überprüfen können.
Physiologically Based Pharmacokinetic (PBPK) Modeling and Simulations: Principles, Methods, and Applications in the Pharmaceutical Industry
by Sheila Annie PetersPhysiologically Based Pharmacokinetic (PBPK) Modeling and Simulations The first book dedicated to the emerging field of physiologically based pharmacokinetic modeling (PBPK) Now in its second edition, Physiologically Based Pharmacokinetic (PBPK) Modelling and Simulations: Principles, Methods, and Applications in the Pharma Industry remains the premier reference book throughout the rapidly growing PBPK user community. Using clear and concise language, author Sheila Annie Peters connects theory with practice as she explores the vast potential of PBPK modeling for improving drug discovery and development. This fully updated new edition covers key developments in the field of PBPK modelling and simulations that have emerged in recent years. A brand-new section provides case studies in different application areas of PBPK modelling, including drug-drug interaction, genetic polymorphism, renal impairment, and pediatric extrapolation. Additional chapters address topics such as model-informed drug development (MIDD) and expose readers to a wide range of current applications in the field. Throughout the book, substantially revised chapters simplify complex topics and offer a balanced view of both the opportunities and challenges of PBPK modelling. Providing timely and comprehensive coverage of one of the most exciting new areas of pharmaceutical science, this book: Describes the principles behind physiological modeling of pharmacokinetic processes, inter-individual variability, and drug interactions for small molecule drugs and biologics Features a wealth of new figures and case studies of the applications of PBPK modelling along the value chain in drug discovery and development Reflects the latest regulatory guidelines on the reporting of PBPK modelling analysis Includes access to a new companion website containing code, datasets, explanations of case examples in the text, and discussion of key developments in the field Contains a brief overview of the field, end-of-chapter keywords for easy reference, and an extensive bibliography Physiologically Based Pharmacokinetic (PBPK) Modeling and Simulations: Principles, Methods, and Applications in the Pharmaceutical Industry, Second Edition is an indispensable single-volume resource for beginning and intermediate practitioners across the pharmaceutical sciences in both industry and academia.
The Pi-Theorem
by L. P. YarinThis volume presents applications of the Pi-Theorem to fluid mechanics and heat and mass transfer. The Pi-theorem yields a physical motivation behind many flow processes and therefore it constitutes a valuable tool for the intelligent planning of experiments in fluids. After a short introduction to the underlying differential equations and their treatments, the author presents many novel approaches how to use the Pi-theorem to understand fluid mechanical issues. The book is a great value to the fluid mechanics community, as it cuts across many subdisciplines of experimental fluid mechanics.
Piaget’s Genetic Epistemology for Mathematics Education Research (Research in Mathematics Education)
by Paul Christian Dawkins Amy J. Hackenberg Anderson NortonThe book provides an entry point for graduate students and other scholars interested in using the constructs of Piaget’s genetic epistemology in mathematics education research. Constructs comprising genetic epistemology form the basis for some of the most well-developed theoretical frameworks available for characterizing learning, particularly in mathematics. The depth and complexity of Piaget’s work can make it challenging to find adequate entry points for learners, not least because it requires a reorientation regarding the nature of mathematical knowledge itself. This volume gathers leading scholars to help address that challenge. The main section of the book presents key Piagetian constructs for mathematics education research such as schemes and operations, figurative and operative thought, images and meanings, and decentering. The chapters that discuss these constructs include examples from research and address how these constructs can be used in research. There are two chapters on various types of reflective abstraction, because this construct is Piaget’s primary tool for characterizing the advancement of knowledge. The later sections of the book contain commentaries reflecting on the contributions of the body of theory developed in the first section. They connect genetic epistemology to current research domains such as equity and the latest in educational psychology. Finally, the book closes with short chapters portraying how scholars are using these tools in specific arenas of mathematics education research, including in special education, early childhood education, and statistics education.
Piano-Hinged Dissections: Time to Fold!
by Greg N. FredericksonA dissection involves cutting a polygon into pieces in such a way that those pieces form another polygon; for a hinged dissection, the pieces must be attached by hinges. A piano hinge is "a long narrow hinge with a pin running the entire length of its joint." So, unlike regular hinged dissections, which swing or twist (around single point of hinge), piano-hinged dissections fold along an edge. This book discusses the history, methods, and variations of these dissections and is rich with illustrations that clearly depict the cuts of the dissections and three-dimensional simulations of the dissections in the process of being folded. The downloadable resources include video recordings of select dissections being transformed accompanies the book.
Piazza: Catcher, Slugger, Icon, Star
by Greg W. PrinceA franchise and fan base in perpetual search of validation finally had its ticket punched as 2016 dawned. Mike Piazza, who held records in one hand and a city’s rapt attention in the other, gained election to the Hall of Fame. Within weeks of this long-awaited announcement, the ballclub with whom he chose to cast his eternal lot, the New York Mets, made a date to retire his number.In Piazza: Catcher, Slugger, Icon, Star, Greg W. Prince-cocreator of Faith and Fear in Flushing, "the blog for Mets fans who like to read,” and author of Amazin’ Again, the story of the 2015 National League champions-explores the parallel paths Piazza and the Mets set out on in the early 1990s and how their individual journeys merged into a mutual quest for transcendence. From marriage of convenience to lifetime bond to a state of baseball grace reached only once before in team history, Piazza examines how the stranger from Los Angeles became New York’s favorite son and why Mets fans continued to rally to Piazza’s cause years after he took his final swing for them.
The Picture Book of Quantum Mechanics
by Hans Dieter Dahmen Siegmund BrandtThe aim of this book is to explain the basic concepts and phenomena of quantum mechanics by means of visualization. Computer-generated illustrations in color are used extensively throughout the text, helping to establish the relation between quantum mechanics--wave functions, interference, atomic structure, and so forth--and classical physics--point mechanics, statistical mechanics, and wave optics. Even more important, by studying the pictures in parallel with the text, readers develop an intuition for such notoriously abstract phenomena as the tunnel effect excitation and decay of metastable states wave-packet motion within a well systems of distinguishable and indistinguishable particles free wave packets and scattering in 3 dimensions angular-momentum decomposition stationary bound states in various 3-dimensional potentials hybrid states Kepler motion of wave packets in the Coulomb field spin and magnetic resonance Illustrations from experiments in a variety of fields, including chemistry, and molecular, atomic, nuclear, and particle physics, underline the basic as well as the practical importance of quantum mechanics. In the present, fourth edition all computer graphics are presented in full color. It also contains additional physics topics such as hybridization. Praise for Previous Editions "The book is highly recommended as a complement to any standard textbook in quantum mechanics, but it will also be valuable to all of us who studied quantum mechanics without the pictures." -- International Journal of Quantum Chemistry "This book would be an excellent basis for the study of special topics in a quantum physics course. Most serious students of physics and all of their teachers will want to consider having this orderly and graphic outline of introductory quantum theory at their fingertips." -- American Journal of Physics "Their aim is the presentation of the 'principal ideas of wave mechanics' in such a way that students can build a quantum intuition out of their graphics." -- Scientific American "This is a unique book. It does not provide a complete course in quantum theory, but as a companion work of reference it should be quite useful to students in providing insights into the dynamical structure of the theory." -- Nature
PICTURING QUANTUM PROCESSES: A First Course in Quantum Theory and Diagrammatic Reasoning
by Bob Coecke Aleks KissingerThe unique features of the quantum world are explained in this book through the language of diagrams, setting out an innovative visual method for presenting complex theories. Requiring only basic mathematical literacy, this book employs a unique formalism that builds an intuitive understanding of quantum features while eliminating the need for complex calculations. This entirely diagrammatic presentation of quantum theory represents the culmination of ten years of research, uniting classical techniques in linear algebra and Hilbert spaces with cutting-edge developments in quantum computation and foundations. Written in an entertaining and user-friendly style and including more than one hundred exercises, this book is an ideal first course in quantum theory, foundations, and computation for students from undergraduate to PhD level, as well as an opportunity for researchers from a broad range of fields, from physics to biology, linguistics, and cognitive science, to discover a new set of tools for studying processes and interaction.
Picturing the Uncertain World: How to Understand, Communicate, and Control Uncertainty through Graphical Display
by Howard WainerIn his entertaining and informative book Graphic Discovery, Howard Wainer unlocked the power of graphical display to make complex problems clear. Now he's back with Picturing the Uncertain World, a book that explores how graphs can serve as maps to guide us when the information we have is ambiguous or incomplete. Using a visually diverse sampling of graphical display, from heartrending autobiographical displays of genocide in the Kovno ghetto to the "Pie Chart of Mystery" in a New Yorker cartoon, Wainer illustrates the many ways graphs can be used--and misused--as we try to make sense of an uncertain world.Picturing the Uncertain World takes readers on an extraordinary graphical adventure, revealing how the visual communication of data offers answers to vexing questions yet also highlights the measure of uncertainty in almost everything we do. Are cancer rates higher or lower in rural communities? How can you know how much money to sock away for retirement when you don't know when you'll die? And where exactly did nineteenth-century novelists get their ideas? These are some of the fascinating questions Wainer invites readers to consider. Along the way he traces the origins and development of graphical display, from William Playfair, who pioneered the use of graphs in the eighteenth century, to instances today where the public has been misled through poorly designed graphs. We live in a world full of uncertainty, yet it is within our grasp to take its measure. Read Picturing the Uncertain World and learn how.
Piece of Pi: Wit-Sharpening, Brain-Bruising, Number-Crunching Activities With Pi (Grades 6-8)
by Naila BokariThere are some topics or problems that have captured the interest of mathematicians for ages. Calculating pi is one of them. While students often encounter pi in the mathematics classroom when applying various formulas, rarely do they use or explore pi in other contexts. This marvelous infinite number we know as pi shows up in many fascinating and mysterious ways. It can be found everywhere, from astronomy and probability, to the physics of sound and light. It is one of the most important numbers that exists.Help your students discover the number that has intrigued mathematicians for centuries. Learn different ways pi has been calculated through the ages, use pi to figure out your hat size, perform a variety of experiments to estimate the value of pi, or relate pi to the alphabet. These interesting and exciting activities encourage higher order thinking and offer a complete overview of this important number while giving students practice in important math skills.This guide includes detailed lesson plans aligned to NCTM standards and reproducible student worksheets. Use them for Pi Day (March 14), as an enrichment or extension to your existing curriculum, or to challenge your ablest math students.Grades 6-8
Piece = Part = Portion: Fractions=decimals=percents/fracciones=decimales=porcentajes
by Scott GiffordNIMAC-sourced textbook
Piece-wise and Max-Type Difference Equations: Periodic and Eventually Periodic Solutions
by Michael A. RadinPiece-wise and Max-Type Difference Equations: Periodic and Eventually Periodic Solutions is intended for lower-level undergraduate students studying discrete mathematics. The book focuses on sequences as recursive relations and then transitions to periodic recursive patterns and eventually periodic recursive patterns. In addition to this, it will also focus on determining the patterns of periodic and eventually periodic solutions inductively. The aim of the author, throughout this book, is to get students to understand the significance of pattern recognition as a mathematical tool. Key Features Can provide possible topics for undergraduate research and for bachelor’s thesis Provides supplementary practice problems and some open-ended research problems at the end of each chapter Focusses on determining the patterns of periodic and eventually periodic solutions inductively Enhances students’ algebra skills before moving forward to upper level courses Familiarize students with the topics before they start undergraduate research by providing applications.