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## Dave Feldman

Dave Feldman

207-801-5709 | dfeldman@coa.edu | faculty website

David Feldman received a B.A. in Physics from Carleton College in 1991 and a Ph.D. in Physics from the University of California at Davis in 1998. From 1991-1993, he was a teacher of 9th and 10th grade physics and mathematics at The McCallie School in Chattanooga, TN. As a graduate student at UC Davis, Dave received several awards in recognition of both teaching and scholarship: The Dissertation Year Fellowship; The Chancellor's Teaching Fellowship; and he was nominated for the Outstanding Graduate Student Teaching Award.

Dave joined the faculty at College of the Atlantic in 1998. He served as Associate Dean for Academic Affairs from 2003-2007. At COA Dave has taught over twenty different courses in physics, mathematics, and computer science. Among these classes is Introduction to Chaos and Fractals, an introductory course for students with a preparation in algebra. Dave has authored at textbook based on this class, which is scheduled to be published by Oxford University Press in 2011.

Dave has recently become interested in teaching about renewable energy and energy conservation. With Anna E. Demeo, a lecturer at COA in engineering, he has developed and team-taught an introductory course on the physics and mathematics of sustainable energy. Anna and Dave received a $95,000 grant from the Maine Space Grant Consortium Research and Higher Education Program to support the development of the class. Anna and Dave have also received an $18,000 grant from the Environmental Education Program of the Environmental Protection Agency to develop and teach a workshop on sustainable energy for area elementary school teachers. This workshop will be held in the summer of 2011.

From 2004-2008, Dave gave a week-long series of lectures at the China Complex Systems Summer School (CSSS), co-sponsored by the Santa Fe Institute and the Institute of Theoretical Physics at the Chinese Academy of Sciences in Beijing. His lectures provided students with a broad introduction to complex systems, including dynamical systems, information theory, and computation theory. From 2006-2008 he was co-director of the China CSSS, collaborating with colleagues at the Chinese Academy to oversee all academic and logistical aspects of the program. He was PI on a $116,000 grant from the U.S. National Science Foundation that partially supported the CSSS.

Dave's research interests lie in the fields of statistical mechanics and nonlinear dynamics. In particular, his research has examined how one might measure "complexity" or pattern in a mathematical system, and how such complexity is related to disorder. This work belongs to the constellation of research topics often referred to as "chaos and complex systems." In his research, Dave uses both analytic and computational techniques. Dave has authored research papers in journals including Physical Review E, Chaos, and Advances in Complex Systems. He has recently begun a research project looking at trends in extreme precipitation events in Maine.

His other interests include ultimate frisbee, hockey, cooking, travel, and gardening. He is married to Doreen Stabinsky; they have three excellent cats.

B.A. Carleton College, 1991

Ph.D. Physics, University of California, Davis, 1998

### Courses Taught

ES1024Calculus I

The goal of this sequence of courses is to develop the essential ideas of single-variable calculus: the limit, the derivative, and the integral. Understanding concepts is emphasized over intricate mathematical maneuverings. The mathematics learned are applied to topics from the physical, natural, and social sciences. There is a weekly lab/discussion section. Evaluations are based on homework, participation in class and lab, and tests.Level: Introductory. Prerequisites: Precalculus or the equivalent or signature of the instructor. Class limit: 20. Lab fee: none. Meets the following degree requirements: QR

ES3012Calculus II

This course is the continuation of Calculus I. It begins by considering further applications of the integral. We then move to approximations and series; we conclude the course with a brief treatment of differential equations. The mathematics learned are applied to topics from the physical, natural, and social sciences. There is a weekly lab/discussion section. Evaluations are based on homework, participation in class and lab, and tests.Level: Intermediate. Prerequisites: Calculus I or the equivalent. Class limit: 20. Lab fee $10. Meets the following degree requirements: QR ES

ES3028Calculus III: Multivariable Calculus

The functions studied in Calculus I and II are one-dimensional. But the universe of everyday experience is, at minimum, three-dimensional. In this course we explore how Calculus can be extended so as to apply to functions of more than one variable, and thus apply to the three-dimensional world. We will begin by reviewing vectors and functions of several variables. We will then learn about partial derivatives and gradients and how apply these tools to multivariable optimization. Turning our attention to integral calculus, we will next cover double and triple integrals and their applications. We will conclude with a treatment of line integrals, flux integrals, the divergence and curl of a vector field, and Green's, and Stokes's theorems. Evaluation will be based on class participation and lengthy weekly problem sets.Level: Intermediate. Prerequisites: Calculus II or the equivalent or signature of instructor. Lab fee: none. Meets the following degree requirements: QR

ES381Chaos and Complex Systems

This course is a survey of a variety of modern topics in nonlinear dynamics: differential equations, finite difference equations, chaos, fractals, multifractals, boolean networks, and cellular automata. The survey will be conducted at a fairly advanced mathematical level, but the material will be covered with an applied emphasis. Numerical results and applications will be stressed rather than proofs. Evaluation will be based on class participation, weekly problem sets and a final project. Some computer work will be required, but no computer experience is necessary. The final project will provide students an opportunity to examine a particular topic or area of application in considerable depth.

Level: Advanced. Prerequisite: Calculus II or the equivalent. Lab fee $10. *ES* *QR*

ES3022Differential Equations

Differential equations are an application of calculus used to model a wide variety of physical and natural phenomena. The rate at which a cup of coffee cools, populations of predators and prey in ecosystems, the spread of disease, and the behavior of electric circuits, are all examples of systems that have been described with differential equations. This course is an introduction to ordinary differential equations, intended for students who have completed a single-variable calculus course. The course covers a variety of techniques for solving and understanding differential equations, including numerical and qualitative solution methods. Students will learn to solve and analyze differential equations using the python programming language. Students will also gain experience formulating mathematical models using differential equations. To do so, we will discuss general modeling principles and also consider several case studies. In addition to learning the mathematics of differential equations, a central goal of this course is to gain skills necessary for research in the mathematical, natural, and social sciences. This includes conceptualizing and framing a research question, conducing a literature review, giving a research presentation, and writing up results in a style appropriate for publication.Evaluation will be based on class participation, bi-weekly problem sets, and a term-long project culminating in a presentation and short research paper. Some computer work will be required, but no computer experience is necessary.

LEVEL: Intermediate. PREREQUISITES: Calculus II or the equivalent or permission of instructor. LAB FEE: none. MEETS THE FOLLOWING DEGREE REQUIREMENTS: ES, QR

ES1026Introduction to Chaos and Fractals

This course presents an elementary introduction to chaos and fractals. The main focus will be on using discrete dynamical systems to illustrate many of the key phenomena of chaotic dynamics: stable and unstable fixed and periodic points, deterministic chaos, bifurcations, and universality. A central result of this study will be the realization that very simple non-linear equations can exhibit extremely complex behavior. In particular, a simple deterministic system (i.e., physical system governed by simple, exact mathematical rules) can behave in a way that is unpredictable and random, (i.e., chaotic). This result suggests that there are potentially far-reaching limits on the ability of science to predict certain phenomena. Students in this class will also learn about fractals---self-similar geometric objects---including the Mandelbrot set and Julia sets. We will also read about and discuss the development of the field of chaos. In so doing, we will examine the nature of scientific communities, with a particular eye toward how changes in scientific outlooks occur. Throughout the course, students will be encouraged to explore the relations between chaos, fractals, and other areas of study such as literature, art, and cultural studies. Students who successfully complete this class should gain a quantitative and qualitative understanding of the basic ideas of chaos and fractals, a greater understanding of the cultural practice of science, and improved mathematical skills. Evaluation will be based on class and lab participation, weekly problem sets several short writing assignments and a finalLevel: Introductory. Prerequisite: A high school algebra course or signature of instructor. Lab fee: $20. Class limit: 15. Meets the following degree requirements: ES QR

ES539Introduction to Computer Science

This course is an intensive introduction to computer science for students with little or no programming experience. The primary goal for this course is to provide students with a solid foundation in Python, a modern, high-level, object-oriented programming language. A secondary goal is for students to gain an initial introduction to algorithmic approaches to interdisciplinary problem-solving. Constructing effective software involves considerable creativity and judgment, and there are general theoretical principles and practical considerations that inform and guide this construction. Students will gain an introduction to these general principles and will also gain experience applying these principles to practical problems. Students who successfully complete this class will: gain a solid, practical understanding of the core python language, including control statements, functions, simple data structures, and input/output; learn how to extend their knowledge of python or other languages; develop good programming techniques; and be able apply algorithmic thinking and programming skills to areas of their interest. This course is designed for students interested in using programming in a wide range of areas, including as a tool for research in biology, economics, statistics, and other mathematical sciences. Additionally, this class will help prepare students to write web applications or applications for mobile devices. This course is also well suited for students who do not have a particular area of programming application in mind, but who simply wish to experience the challenge and excitement of designing and implementing algorithms. Evaluation will be based on weekly programming exercises and a final programming project. Level: Introductory. Prerequisites: none. Lab Fee: none. Class size: 12. *QR* *ES*.

ES532Introduction to Linear Algebra

Through the study of linear algebra in this course, students will acquire powerful analytic techniques that are essential tools in almost any field of applied mathematics, including: physics, engineering, computer science, economics. Linear algebra is also commonly used in chemistry and mathematical biology. Our study of linear algebra will begin by abstracting and formalizing the idea behind solving familiar systems of linear equations. This will lead us to the study of matrices and determinants. We will study these mathematical objects both algebraically and geometrically, leading up to a general treatment of linear vector spaces. Additional topics covered will include: linear transformations; inner products and orthogonality; eigenvectors, eigenvalues, and their application. Where possible, applications to students' fields of interest will be emphasized. Students will leave this course with a solid foundation in the key ideas and techniques of linear algebra. Evaluation will be based on class participation and weekly problem sets. Level: Intermediate. Prerequisites: Signature of Instructor. *QR*

ES2026Introduction to Quantum Mechanics

What is quantum mechanics and why was it viewed as revolutionary when it was formulated in the early 1900s? How and why does quantum mechanics compel us to discard commonsense ideas about the nature of the physical world that are part of classical mechanics? Why is there not agreement on how to interpret quantum mechanics, and does quantum mechanics even need interpretation? This course is designed to introduce students to some of the central ideas of quantum mechanics so they can formulate scientifically grounded answers to the above questions. Our entry point will be to take a close look at a prototypical quantum system, spin-1/2 particles, and we will use this system to examine the experimental evidence for quantum behavior. We will then turn our attention to some of the well-known conundrums of quantum mechanics, such as wave/particle duality (via the two-slit experiment) and the Einstein-Podolsky-Rosen paradox. We will also look at the early development of quantum theory covering phenomena such as blackbody radiation, the photoelectric effect, and Bohr's model of the hydrogen atom. As part of this we will examine some of the early debates about quantum mechanics and learn about the scientists who played a key role in these debates. As time permits, we will look at recent developments in quantum mechanics, including Bell inequalities, delayed-choice experiments, and quantum cryptography. This course is designed to appeal to a wide range of students---both those whose interests lie outside of science as well as those who are concentrating in the sciences or mathematics. Students who successfully complete this course will have gained a solid understanding of the central ideas of quantum mechanics. This understanding should be mathematical and quantitative as well as conceptual. Students will also gain some experience with scientific reasoning and quantitative problem solving. Evaluation will be based on weekly problem sets, participation in a weekly discussion section, and a final presentation or paper.Level: Introductory/Intermediate. Prerequisites: Comfort working with algebra and familiarity with high school chemistry or physics. Class limit: 26. Lab fee: $10. Meets the following degree requirements: QR ES

ES3054Linear Algebra

Linear algebra is a foundational area of applied mathematics, finding widespread application in physics, engineering, economics, and across the sciences. The starting point for this course is to consider basic properties of matrices and techniques for solving systems of linear equations. Abstracting and formalizing the process of solving linear equations leads us to the notion of a vector space and related ideas, such as linear independence, dimension, and basis. The course then turns to further properties of matrices and vector spaces, including determinants, eigenvalues and eigenvectors, and linear transformations. As time permits, we will study various applications of linear algebra, such as Markov models, Fourier series, principal component analysis, and Leontief input-output models. Students who successfully complete this course will gain a solid introduction to the calculational techniques and key constructions and ideas of linear algebra. Additionally, this course serves as an introduction to formal mathematics. Students will learn to work at a level of generality and abstraction above that encountered in a typical introductory calculus sequence. Students will also be introduced to methods of mathematical proof and argument.This course will be taught in a "flipped" format. Students will be required to watch video lectures outside of class. Class sessions will be devoted exclusively to discussion and problem solving. Evaluation will be based on class participation and weekly problem sets.

Level: Intermediate. Prerequisites: Calculus II or the equivalent, and permission of instructor. Class Limit: 15. Lab Fee: none. Meets the following degree requirements: QR

ES1034Physics and Mathematics of Sustainable Energy

The aim of this course is to help students learn some basic physics and quantitative and analytical skills so that they can participate intelligently and responsibly in policy discussions, personal and community decisions, and ventures in the area of sustainable energy. We will begin with some basic physics, including: the definition of energy, the difference between energy and power, different forms of energy, and the first and second laws of thermodynamics. We will also provide students with a basic scientific and economic introduction tovarious alternative energy technologies. Along the way, students will gain mathematical skills in estimation and dimensional analysis, and will learn to use spreadsheets to assist in physical and financial calculations. There will also be a weekly lab to help students understand the physical principles behind different energy technologies and gain experience gathering and analyzing data.Students who successfully complete this course will be able to apply what they have learned to basic issues in sustainable energy. For example, they will be able to evaluate and analyze a proposed technology improvement by considering its dollar cost, carbon reduction, return to investment, payback time, and how all this might depend on, say, interest rates or the cost of electricity or gasoline. Students will also be able to analyze the potential of a technology or energy source to scale up. E.g., they will be able to consider not only the benefits to a homeowner of a solar installation, but to also analyze the degree to which solar power may contribute to Maine's energy needs.

This will be a demanding, introductory, class. Evaluation will be based on weekly problem sets, participation in class and lab, and a final project. At least one college-level class in mathematics or physical science is strongly recommended.

Level: Introductory. Prerequisites: None. Class limit: 15. Lab fee $50.00 Meets the following degree requirements: QR ES

ES1018Physics I: Mechanics and Energy

ES395Physics III: Introduction to Quantum Mechanics

This course is designed to introduce students to the two central ideas of quantum mechanics. First, the outcomes of experiments cannot be predicted exactly; one can only predict the probability of various outcomes. And second, these probabilities do not behave like normal probabilities; the probabilities interfere with each other in a manner that has no counterpart in our everyday experience with probabilities. We will develop these ideas by taking a close look at a prototypical quantum system: "spin-1/2" particles. We will carefully discuss the experimental evidence for quantum mechanics, and we will also look at some of the well-known conundrums of quantum mechanics, such as the two-slit experiment and the Einstein-Podolsky-Rosen paradox. Along the way, students will also be introduced to basic probability theory. We will conclude by looking at some of the applications and implications of quantum mechanics, such as: the Bohr atom, quantum computation, quantum cryptography, and the photoelectric effect. Quantum mechanics is an exciting, challenging topic which has made an impact in many different fields. As such, this course is designed to appeal to a wide range of students --- both those whose interests lie outside of science as well as those who are concentrating in the sciences or mathematics. Students who successfully complete this course will have gained a solid understanding of the central ideas of quantum mechanics. This understanding should be mathematical and quantitative as well as conceptual. Students will also gain some experience with scientific reasoning and quantitative problem solving. Evaluation will be based on class participation, weekly problem sets, and a final presentation or paper. Some computer work may be required, but no computer experience is necessary. Level: Introductory/Intermediate. Prerequisites: Familiarity with algebra and trigonometry and high school chemistry or physics. Physics I and II are not prer

ES543The Nature and Language of Mathematics

The Nature and Language of Mathematics is an introductory course designed to help students discover the connections between mathematics and other areas of human understanding. It is intended primarily for students with limited prior math experience. By exploring diverse mathematics topics, students will see the varied roles that mathematics play in our world. Topics covered will depend on student interest, and may include the following: graph theory, probability, estimation, logic, and linear equations. The majority of in-class work will take place in small groups, allowing students to be active, engaged learners. In addition, students will read several articles, and possibly a popular book or historical or sociological treatment of mathematics or mathematicians. Through this course, the student will be encouraged to understand the patterns, language, and logic that underlies what we call mathematics. Evaluation will be based on class participation and group work, weekly projects and assignments, and a final paper or project. Students may also be asked to present their research topic orally. Level: Introductory. Prerequisites: Permission of Instructor. *QR*

ES496Theory and Applications of Complex Networks

Network structures are ubiquitous in the world around us: communication networks, transportation networks, networks of friends and acquaintances, and biological networks, to name just a few. In this class, students will learn about the mathematical similarities and abstractions that under-lie these examples. Additional examples will be drawn from molecular biology (gene regulation and protein interaction networks), economics (trading networks, relations among firms, and strategic interactions on networks), computer science (computer networks and the world wide web), and ecology (food webs). The last decade has seen an explosion of work in the theory and applications of networks to an enormously wide range of problems.

Students who successfully complete this course will: gain a broad introduction to recent work in this field; understand the strengths and weaknesses of network modeling; and be able to apply networks and network analysis in a variety of settings. Evaluation will be based on several problem sets, three short literature reviews to be posted on the course blog, and a final project on a topic of the student's choosing.

Level: Intermediate/Advanced. Pre-requisites: One college-level mathematics course, Signature of instructor. Lab fee: $10. *ES**QR*

ES3052Thermodynamics

Thermodynamics is the area of physics concerned with the behavior of very large collections of particles. Examples include the water molecules in glass of water, the electrons in a wire, or the photons given off by a light bulb. Thermodynamics studies properties of collections of particles that are largely independent of the particles' detail, for example, the tendency for heat to flow from a hot object to a cold one.This course will begin with a treatment of the first law of thermodynamics and basic thermal physics. Topics to be covered include the conservation of energy, heat and work, the ideal gas, the equipartition of energy, heat capacities, and latent heat. We will then move to the second law of thermodynamics, beginning with a statistical definition of entropy. This will require learning some combinatorics (a mathematical technique for counting) and approximation methods for working with very large numbers. This statistical approach will enable us to understand the origin of the second law of thermodynamics, and will lead naturally to statistical definitions of temperature, pressure, and chemical potential. We will then turn our attention to two broad areas of application. The first of these is heat engines and refrigerators, including heat pumps. The second set of applications involve free energy and chemical equilibrium. Depending on student interest, we will cover batteries and fuel cells, phase transitions, adiabatic lapse rates in meteorology, and nitrogen fixation. Thermodynamics is a broadly applicable field of physics, and so this course should be of relevance to students whose interests are in almost any area of science or engineering, as well as those who wish to gain a general introduction to a field that is one of the pillars of modern physical science. Evaluation will be based on weekly problem sets and a final research paper, presentation, or lab project.

Level: Intermediate. Prerequisites: Calculus II and either a college-level physics or chemistry class. Course Limit: 20. Lab Fee: None. Meets the following degree requirements: QR ES

ES576Tutorial: Dynamical Systems

This course is a survey of dynamical systems, the field of applied mathematics that studies systems that change over time. The modern study of dynamical systems includes examining particular systems or areas of application, as well as looking at systems more broadly and abstractly to develop generally applicable tools for studying dynamical systems or to classify different sorts of behavior.

This course is intended for motivated students with strong math backgrounds who wish to gain an overview of dynamical systems and to discuss and debate the insights the study of dynamical systems holds for the physical, natural, and social sciences. Using both differential equations and difference equations as our main items of study, we will cover standard topics in dynamical systems, including phase space, bifurcation diagrams, chaotic behavior, sensitive dependence on initial conditions, strange attractors, embedding and attractor reconstruction, and Lyapunov exponents. A central theme that emerges from the study of dynamical systems is that there is a subtle relationship between order and disorder. Unpredictable behavior can arise from deterministic dynamical systems, and complex behavior can have simple origins. We shall see that predictability and unpredictability, simplicity and complexity, and order and disorder are not opposites, but often exist simultaneously in the same dynamical system. Evaluation will be based on participation in seminar-style class sessions, problems sets, and a final project and presentation.

Level: Intermediate. Prerequisites: Calculus II and permission of instructor. Experience writing simple computer programs (in any language) will be helpful, but not required. Class size: 5. Lab fee: none

ES559Tutorial: Theory and Applications of Complex Networks

Network structures are ubiquitous in the world around us: communication networks, transportation networks, networks of friends and acquaintances, and biological networks, to name just a few. In this tutorial students will learn about the mathematical similarities and abstractions that under-lie these examples. Additional examples may be drawn from molecular biology (gene regulation and protein interaction networks), economics (trading networks, relations among firms, and strategic interactions on networks), computer science (computer networks and the world wide web), and ecology (food webs), depending on students' interests. The last decade has seen an explosion of work in the theory and applications of networks to an enormously wide range of problems. Students who successfully complete this tutorial will: gain a broad introduction to recent work in this field; understand the strengths and weaknesses of network approaches; and be able to apply networks and network analysis in a variety of settings. In addition to learning about networks, a central goal of this tutorial is for students to gain skills necessary for research in the mathematical, natural, and social sciences. This includes conceptualizing and framing a research question, conducing a literature review, presenting results in a professional-style research talk, and writing up results in a style appropriate for publication. In the first part of the course we will focus on empirical descriptions of network structure, including algorithms for discovering communities or clusters. We will then turn our attention to dynamics of networks: how do networks form and grow, and how are these growth rules related to global structure? Finally, as time permits we will consider dynamics of processes that occur on networks. Evaluation will be based on participation in seminar-style class meetings, several short problem sets, and a project on a topic of the student's choosing. Level: Advanced. Pre-requisite