Core Course Requirements for the Ph.D. Degree

Each department and program may have identified math courses that will meet the math requirement:

Biomedical Engineering: ENAS 500 or ENAS 505
Chemical Engineering: ENAS 500 or ENAS 505
Electrical Engineering: No math requirement
Environmental Engineering: ENAS 500 or ENAS 505 or F&ES 758 or F&ES 781 or STAT 530 or STAT 660
Mechanical Engineering: ENAS 500 or PHYS 506 or ENAS 902

The core courses for each department and program are as follows:

Biomedical Engineering: Physiological Systems (ENAS 550), Physical and Chemical Basis of Biosensing (ENAS 510). One of these courses may be taken in the second year.

Chemical Engineering: Classical and Statistical Thermodynamics (ENAS 521), Energy, Mass, and Momentum Processes (ENAS 603), Chemical Reaction Engineering (ENAS 602).

Electrical Engineering (Microelectronics track): 2 of the following 4: Photonics and Optical Electronics (ENAS 511), Heterojunction Devices (ENAS 718), Solid State Physics I (ENAS 850), Semiconductor Silicon Devices and Technology (ENAS 986).

Electrical Engineering (System and Signals track): Linear Systems (ENAS 902), Stochastic Processes (ENAS 502).

Electrical Engineering (Computer Engineering track): Introduction to VLSI System Design (ENAS 875) and Advanced Topics in Computer Engineering (ENAS 921).

Environmental Engineering: Water Chemistry (ENAS 638), Biological Processes in Environmental Engineering (ENAS 641), Environmental Physicochemical Processes (ENAS 642).

Mechanical Engineering: Please refer to the Mechanical Engineering section in the Qualification Procedure on the forms and guides page.

Below are the frequently offered Engineering & Applied Science graduate courses. For the most up-to-date course offerings and lecturer information, visit Yale University’s Online Course Information site.

ENAS 500/APHY 500, Mathematical Methods I 
Vector analysis in three dimensions (2 weeks), linear algebra (4 weeks), functions of a complex variable (4 weeks), topics at the discretion of the instructor (3 weeks), e.g., (1) specific examples to reinforce the material already presented and (2) new topics (to choose among: Fourier series in one and more dimensions, Laplace transformations, Fourier integrals in one and more dimensions, optimization, elements of ODE).

ENAS 501, Mathematical Methods II 
Special functions, the Laplace transformations, Fourier series, Fourier integrals, and partial differential equations including separation of variables, methods of characteristics, variational techniques, and a brief discussion of numerical methods.

ENAS 502, Stochastic Processes
Elements of set and measure theory. Probability distributions, moments, characteristic functions. The central limit theorem. Basic properties of random processes. Stationarity and ergodicity. Correlation functions and power spectra. Linear and nonlinear operations on random processes.

ENAS 503, Probabilistic Networks, Algorithms, and Applications
This course examines probabilistic and computational methods for the statistical modeling of complex data. The emphasis is on the unifying framework provided by graphical models, a formalism that merges aspects of graph theory and probability theory. Graphical models: Markov random fields, Bayesian networks, and factor graphs. Algorithms: filtering, smoothing, belief-propagation, sum-product, and junction tree. Variational techniques: mean-field and convex relaxations. Markov processes on graphs: MCMC, factored HMMs, and Glauber dynamics. Some statistical physics techniques: cavity and replica methods. Applications to error-correcting codes, computer vision, bio-informatics, and combinatorial optimization.

ENAS 505, Advanced Engineering Mathematics
A beginning graduate-level introduction is given to ordinary and partial differential equations, vector and tensor analysis, and linear algebra. Laplace transform, series expansion, Fourier transform, and matrix methods are given particular attention. Applications to problems frequently encountered by chemical, biomedical, and environmental engineers are stressed throughout.

ENAS 506, Ethics and Professional Development for Biomedical Engineers and Scientists
A seminar class that explores ethical issues, frameworks for understanding issues, and boundaries of honorable execution of science and engineering through relevant reading of a broad variety of historical nonfiction, novels, case studies, newspaper and magazine articles, and other resource material. Lively but reasoned and respectful debate is encouraged and expected. Essentials of the practice of science are also addressed. Short writing exercises are used to foster good writing, thinking, and communication skills. Acquired skills are applied to ethical issues of science and engineering in the news.

ENAS 508, Responsible Conduct of Research 
Required for first-year students. Presentation and discussion of topics and best practices relevant to responsible conduct of research including academic fraud and misconduct, conflict of interest and conflict of commitment, data acquisition and human subjects, use and care of animals, publication practices and responsible authorship, mentor/trainee responsibilities and peer review, and collaborative science.

ENAS 509, Electronic Materials: Fundamentals and Applications
Survey and review of fundamental issues associated with modern microelectronic and optoelectronic materials. Topics include band theory, electronic transport, surface kinetics, diffusion, materials defects, elasticity in thin films, epitaxy, and Si integrated circuits.

ENAS 510, Physical and Chemical Basis of Bioimaging and Biosensing
Basic principles and technologies for imaging and sensing the chemical, electrical, and structural properties of living tissues and biological macromolecules. Topics include magnetic resonance spectroscopy, MRI, positron emission tomography, and molecular imaging with MRI and fluorescent probes.

ENAS 511, Physics and Devices of Optical Communication
A survey of the enabling components and devices that constitute modern optical communication systems. Focus on the physics and principles of each functional unit, its current technological status, design issues relevant to overall performance, and future directions. Permission of the instructor required.  

ENAS 513, Introduction to Analysis
Foundations of real analysis, including metric spaces and point set topology, infinite series, and function spaces.

ENAS 514, Real Analysis
The Lebesgue integral, Fourier series, applications to differential equations.

ENAS 517/MB&B 517/PHYS 517, Methods and Logic in Interdisciplinary
This half-term IGPPEB class is intended to introduce students to integrated approaches to research. Each session is led by faculty with complementary expertise and discusses papers that use different approaches to the same topic (for example, physical and biological or experiment and theory). Counts as 0.5 credit toward graduate course requirements. Required for students in IGPPEB.

ENAS 518/MB&B 635, Mathematical Methods in Biophysics
Applied mathematical methods relevant to analysis and interpretation of biophysical and biochemical data, including statistics and error analysis, differential equations, linear algebra, and Fourier transforms. The class covers both analytical and numerical implementations of these topics. Prerequisites: MATH 120a or b and MB&B 300a or equivalents, or permission of the instructors.

ENAS 521, Classical and Statistical Thermodynamics
A unified approach to bulk-phase equilibrium thermodynamics, bulk-phase irreversible thermodynamics, and interfacial thermodynamics in the framework of classical thermodynamics, and an introduction to statistical thermodynamics. Both the activity coffecient and the equations of state are used in the description of bulk phases. Emphasis on classical thermodynamics of multicomponents, including concepts of stability and criticality, curvature effect, and gravity effect. The choice of Gibbs free energy function covers applications to a broad range of problems in chemical, environmental, biomedical, and petroleum engineering. The introduction includes theory of Gibbs canonical ensembles and the partition functions, fluctuations, and Boltzmann’s statistics, Fermi-Dirac and Bose-Einstein statistics. Application to ideal monatomic and diatomic gases is covered.

ENAS 525, Optimization I
A problem-based introduction to linear programs and their generalizations. Includes theory, algorithms, uses and connections to economic reasoning. Optimality conditions for linear and nonlinear programs. Solution methods for linear, integer, and nonlinear programs. Solution concepts for games. Computation of Nash equilibria and Brouwer fixed points.

ENAS 530, Optimization Techniques
Fundamental theory and algorithms of optimization, emphasizing convex optimization. The geometry of convex sets, basic convex analysis, the principle of optimality, duality. Numerical algorithms: steepest descent, Newton’s method, interior point methods, dynamic programming, unimodal search.

ENAS 534, Biomaterials
Introduction to materials, classes of materials from atomic structure to physical properties. Major classes of materials: metals, ceramics and glasses, and polymers, addressing their specific characteristics, properties, and biological applications. Throughout the presentation of the synthesis, characterization, and properties of the classes of materials, a connection is made to the selection of materials for use in specific biological applications by matching the material’s properties to those necessary for success in the application. Case studies address the successes and failures of particular materials from each of the classes in biological applications.

ENAS 535, Tissue/Biomaterial Interactions
The course addresses the interactions between tissues and biomaterials, with an emphasis on the importance of molecular- and cellular-level events in dictating the performance and longevity of clinically relevant devices. In addition, specific areas such as biomaterials for tissue engineering and the importance of stem/progenitor cells, and biomaterial-mediated gene and drug delivery are addressed.

ENAS 541/MB&B 523/PHYS 523, Biological Physics
An introduction to the physics of several important biological phenomena, including molecular motors, protein folding, bacterial locomotion, and allostery. The material and approach are positioned at the interface of the physical and biological sciences. Required for students in IGPPEB.

ENAS 549, Biomedical Data Analysis
The course focuses on the analysis of biological and medical data associated with applications of biomedical engineering. It provides basics of probability and statistics, and analytical approaches for determination of quantitative biological parameters from noisy, experimental data. Programming in Matlab to achieve these goals is a major portion of the course. Applications include Michaelis-Menten enzyme kinetics, Hodgkin Huxley, neuroreceptor assays, receptor occupancy, MR spectroscopy, PET neuroimaging, brain image segmentation and reconstruction, and molecular diffusion.

ENAS 550/C&MP 550/MCDB 550, Physiological Systems
The course develops a foundation in human physiology by examining the homeostasis of vital parameters within the body, and the biophysical properties of cells, tissues, and organs. Basic concepts in cell and membrane physiology are synthesized through exploring the function of skeletal, smooth, and cardiac muscle. The physical basis of blood flow, mechanisms of vascular exchange, cardiac performance, and regulation of overall circulatory function are discussed. Respiratory physiology explores the mechanics of ventilation, gas diffusion, and acid-base balance. Renal physiology examines the formation and composition of urine and the regulation of electrolyte, fluid, and acid-base balance. Organs of the digestive system are discussed from the perspective of substrate metabolism and energy balance. Hormonal regulation is applied to metabolic control and to calcium, water, and electrolyte balance. The biology of nerve cells is addressed with emphasis on synaptic transmission and simple neuronal circuits within the central nervous system. The special senses are considered in the framework of sensory transduction. Weekly discussion sections provide a forum for in-depth exploration of topics. Graduate students evaluate research findings through literature review and weekly meetings with the instructor.

ENAS 551, Biomedical Engineering I: Quantitative Physiology
Demonstration of the use of engineering analysis and synthesis in problems in the life sciences and medicine; focus on modeling of molecular physiological processes and design of artificial organs. The lectures in the course are coordinated with the sequence of lectures in ENAS 550a to illustrate how engineering analysis can be used to understand physiological processes. In addition, the course presents elements of pharmacokinetics, heat and mass transfer in physiological systems, hemodialysis, drug delivery, and tissue engineering.

ENAS 553, Immuno-Engineering
An advanced class that introduces immunology principles and methods to engineering students. The course focuses on biophysical principles and biomaterial applications in understanding and engineering immunity. The course is divided into three parts. The first part introduces the immune system: organs, cells, and molecules. The second part introduces biophysical characterization and quantitative modeling in understanding immune system interactions. The third part focuses on intervention, modulation, and techniques for studying the immune system with emphasis on applications of biomaterials for intervention and diagnostics.

ENAS 554, Continuum Biomechanics
This course is designed to enable students to learn advanced and state of the art methods of continuum and computational biomechanics, especially related to the need to formulate new theories of soft tissue growth, remodeling, disease progression, healing, and aging. Emphasis will be placed on ensuring that the mechanics is driven by advances in the vascular mechanobiology.

ENAS 555, Vascular Mechanics
This course is designed to enable students to apply methods of continuum biomechanics to study diverse vascular conditions and treatments, including hypertension, atherosclerosis, aneurysms, vein grafts, and tissue engineered constructs from an engineering perspective. Emphasis will be placed on ensuring that the mechanics is driven by advances in the vascular mechanobiology.

ENAS 557, Biomechanics
An introduction to the application of mechanical engineering principles to biological materials and systems. Topics include ligaments, tendons, bones, muscles; joints, gait analysis; exercise physiology. The basic concepts are directed toward an understanding of the science of orthopaedic surgery and sports medicine.

ENAS 562, Digital Systems Testing and Design for Testability  
Introduction to the fundamental concepts, algorithms, and design techniques for testing digital systems. Covered topics include test issues and economics, fault modeling, logic and fault simulation, test generation algorithms for combinational and sequential circuits, testability analysis, design for testability, built-in self-test, delay fault test, functional test, case studies (memory test, FPGA test, system-on-chip test, etc.). Lab work consists of projects employing logic and fault simulation, automatic test pattern generation, and design for testability software tools.

ENAS 563, Fault Tolerant Computer Systems
This course provides an in-depth overview of the theory and practice of fault tolerant systems. Sources of defects as well as hardware and software fault tolerance techniques to mititgate their effects are reviewed. Case studies are used to demonstrate the practical applications of the theory presented in the lectures.

ENAS 564, Tissue Engineering
Introduction to the major aspects of tissue engineering, including materials selection, scaffold fabrication, cell sources, cell seeding, bioreactor design, drug delivery, and tissue characterization. Class sessions include lectures and hands-on laboratory work.

ENAS 570/C&MP 560/MCDB 560, Cellular and Molecular Physiology: Molecular Machines in Human Disease 
The course focuses on understanding the processes that transfer molecules across membranes at the cellular, molecular, biophysical, and physiological levels. Students learn about the different classes of molecular machines that mediate membrane transport, generate electrical currents, or perform mechanical displacement. Emphasis is placed on the relationship between the molecular structures of membrane proteins and their individual functions. The interactions among transport proteins in determining the physiological behaviors of cells and tissues are also stressed. Molecular motors are introduced and their mechanical relationship to cell function is explored. Students read papers from the scientific literature that establish the connections between mutations in genes encoding membrane proteins and a wide variety of human genetic diseases.

ENAS 575/CPSC 575, Computational Vision and Biological Perception
An overview of computational vision with a biological emphasis. Suitable as an introduction to biological perception for computer science and engineering students, as well as an introduction to computational vision for mathematics, psychology, and physiology students. Prerequisites: MATH 120a or b and CPSC 112a or b, or permission of the instructor.

ENAS 580,Seminars in Biomedical Engineering
The course is designed to provide graduate students in Biomedical Engineering with a broad perspective of research topics in their field, with a particular focus on topics directed toward clinically oriented research. Students attend a series of lectures by speakers from both inside and outside the Yale BME research community covering the areas of biomaterials/tissue engineering, drug delivery systems, biomechanics, and bioimaging. The week after each lecture, students gather to address questions posed by the lecturing faculty and the course organizers, with discussion led by the students themselves. In addition, each student picks a topic related to one of the lectures given during the term and submits an extended written analysis.

ENAS 585, Fundamentals of Neuroimaging
The neuroenergetic and neurochemical basis of several dominant neuroimaging methods, including fMRI. Topics range from technical aspects of different methods to interpretation of the neuroimaging results. Controversies and/or challenges for application of fMRI and related methods in medicine are identified.

ENAS 600, Computer-Aided Engineering
Aspects of computer-aided design and manufacture including reasons for increased use of CAD/CAM, the computer’s role in the mechanical engineering design and its manufacturing process, hardware and software elements of typical commercial systems, and computer graphics and drafting.

ENAS 601, Materials Chemistry
The approach is chemical and molecular and, of course, includes nanomaterials. We follow the Fahlman text outline on solid-state chemistry, metals, semiconducting materials, organic “soft” materials, and nanomaterials for two-thirds of the course. The last third of the course focuses on materials characterization by microscopy and spectroscopy and includes some surface characterization techniques. There are problem sets, at least one paper on a particular material or characterization technique, and both a midterm and final exam.

ENAS 602, Chemical Reaction Engineering
Applications of physical-chemical and chemical-engineering principles to the design of chemical process reactors. Ideal reactors treated in detail in the first half of the course, practical homogeneous and catalytic reactors in the second.

ENAS 603, Energy, Mass, and Momentum Processes
Application of continuum mechanics approach to the understanding and prediction of fluid flow systems that may be chemically reactive, turbulent, or multiphase.

ENAS 605, Colloidal Chemical Engineering
A graduate-level introduction to modern colloid science as practiced by engineers. Topics include self-assembly in solution and at surfaces, surface chemistry, the electric double layer, colloidal forces, and polymers. Applications to problems frequently encountered by chemical, biomedical, and environmental engineers are stressed throughout.

ENAS 606, Polymer Physics
A graduate-level introduction to the physics and physical chemistry of macromolecules. This course covers the static and dynamic properties of polymers in solution, melt and surface adsorbed states and their relevance in industrial polymer processing, nanotechnology, materials science, and biophysics. Starting from basic considerations of polymerization mechanisms, control of chain architecture, and a survey of polymer morphology, the course also extensively addresses experimental methods for the study of structure and dynamics via various scattering (light, x-ray, neutron) and spectroscopic methods (rheology, photon correlation spectroscopy) as integral components of polymer physics.

ENAS 608, Surface and Surface Processes
The chemistry and physics of solid surfaces. Emphasis on fundamental aspects of the following areas of surface science: surface crystallography and reconstruction; kinetics of gas-solid interactions; adsorption; heterogeneous catalysis by transition metal surfaces; oxidation and corrosion; and nucleation and growth of thin films by physical and chemical vapor deposition.

ENAS 610, Biomolecular Engineering
A survey of biomolecular engineering laboratory methods and strategies. An advanced workshop on a broad range of concepts at the interface of applied mathematics, biology, biophysical chemistry, and chemical engineering whose express purpose is developing novel molecular tools, materials, and approaches based on biological building blocks and machinery. Topics include understanding and modeling the physicochemical properties that confer function in biological systems, low- and high-resolution protein engineering, and the design of synthetic interactomes.

ENAS 611, Separation Processes
Theory and design of separation processes for multicomputer and/or multiphase mixtures via equilibrium and rate phenomena. Included are single-stage and cascaded absorption, adsorption, extraction, distillation, filtration, and crystallization processes.

ENAS 615, Synthesis of Nanomaterials
This course focuses on the synthesis and engineering of nanomaterials, a primary frontier for the development of new and improved materials with new properties. We also introduce different types of nanomaterials, unique properties at the nanoscale, measurement and important applications of nanomaterials (including biomedical, electronic, and energy applications). Synthesis methods covered include gas phase and high vacuum techniques (CVD, MOCVD) as well as wet chemistry techniques such as reduction of metal salts, sonochemistry, and sol gel methods. Taking sample applications, we discuss the properties necessary for each, and how to control these properties through synthesis control, such as by using templating methods. This course is directed to chemistry, biology, and engineering students.

ENAS 626, Chemical Engineering Process Control
Transient regime modeling and simulations of chemical processes. Conventional and state-space methods of analysis and control design. Applications of modern control methods in chemical engineering.

ENAS 628, Sensors and Biosensors
The course provides students with the knowledge of basic integrated analog blocks and how to combine these circuits into sensory systems for biomedical applications. Target areas are in physiology, brain-machine interfaces, neural recording and stimulation, imaging and bioimaging. Lectures include details on operational amplifiers, voltage amplifiers, current mode circuits, analog-to-digital converters, photo-transduction circuits, layout, simulation, and design of VLSI circuits and systems.

ENAS 639, Management of Water Resources and Environmental Systems    
Management tools to analyze problems related to water resources and environmental systems. A focus on characterizing, defining, and solving natural and water resources (quality, location, treatment) and environmental problems (soil, water, air pollution, risks) implementing Operation Research (OR) methods. Topics include introduction to OR methods and its role in natural resources and water resources,  environmental systems, economic criteria and optimization criteria. Management modeling refers to application of linear programming (e.g. river contamination), integer programming and fixed charge problems (e.g. solid waste disposal and renovation), non-linear programming (e.g. optimal water blending), goal programming, and Analytic Hierarchy Processes (AHP) (selection of preferable membrane treatment systems; selection of preferable wastes treatment method). Main principles of multi-objective optimization are presented.

ENAS 640/F&ES 60109, Aquatic Chemistry
A detailed examination of the principles governing chemical reactions in water. Emphasis is on developing the ability to predict the aqueous chemistry of natural and perturbed systems based on a knowledge of their biogeochemical setting. Focus is on inorganic chemistry, and topics include elementary thermodynamics, acid-base equilibria, alkalinity, speciation, solubility, mineral stability, redox chemistry, and surface complexation reactions. Illustrative examples are taken from the aquatic chemistry of estuaries, lakes, rivers, wetlands, soils, aquifers, and the atmosphere. A standard software package used to predict chemical equilibria may also be presented.

ENAS 641, Biological Processes in Environmental Engineering
Fundamental aspects of microbiology and biochemistry, including stoichiometry, kinetics, and energetics of biochemical reactions, microbial growth, and microbial ecology, as they pertain to biological processes for the transformation of environmental contaminants; principles for analysis and design of aerobic and anaerobic processes including suspended- and attached-growth systems, for treatment of conventional and hazardous pollutants in municipal and industrial wastewaters and in groundwater.

ENAS 642, Environmental Physicochemical Processes
Fundamental and applied concepts of physical and chemical (“physicochemical”) processes relevant to water quality control. Topics include chemical reaction engineering, overview of water and wastewater treatment plants, colloid chemistry for solid-liquid separation processes, physical and chemical aspects of coagulation, coagulation in natural waters, filtration in engineered and natural systems, adsorption, membrane processes, disinfection and oxidation, disinfection by-products.

ENAS 645/F&ES 96007, Industrial Ecology
Industrial ecology is an organizing concept that is increasingly applied to define various interactions of today’s technological society with both natural and altered environments. Technology and its potential for modification and change are central to this topic, as are implications for government policy and corporate response. The course discusses how industrial ecology is being applied in corporations to minimize the environmental impacts of products, processes, and services, and shows how industrial ecology serves as a technological framework for science, policy, and management in government and society.

ENAS 648, Environmental Transport Processes
Analysis of transport phenomena governing the fate of chemical and biological contaminants in environmental systems. Emphasis on quantifying contaminant transport rates and distributions in natural and engineered environments. Topics include distribution of chemicals between phases; diffusive and convective transport; interfacial mass transfer; contaminant transport in groundwater, lakes, and rivers; analysis of transport phenomena involving particulate and microbial contaminants.

ENAS 649/MGT 611, Policy Modeling
Building on earlier course work in quantitative analysis and statistics, Policy Modeling provides an operational framework for exploring the costs and benefits of public policy decisions. The techniques employed include “back of the envelope” probabilistic models, Markov processes, queuing theory, and linear/integer programming. With an eye toward making better decisions, these techniques are applied to a number of important policy problems. In addition to lectures, assigned articles and text readings, and short problem sets, students are responsible for completing a take-home midterm exam and a number of cases. In some instances, it is possible to take a real problem from formulation to solution, and compare the student’s own analysis to what actually happened. Prerequisites: Decision Analysis and Game Theory, Data Analysis and Statistics, or a demonstrated proficiency in quantitative methods.

ENAS 658, MEMS Design
An introduction to the broad field of microelectromechanical systems (MEMS), using examples and design projects drawn from real-world MEMS applications. Topics include material properties, microfabrication technologies, structural behavior, sensing techniques, actuation schemes, fluid behavior, simple electronic circuits, and feedback systems. Student teams design complete microsystems to meet a set of specifications based on realistic microfabrication processes. Emphasis on modeling and simulation in the design process.

ENAS 660, Green Engineering and Sustainability
The course focuses on a green engineering design framework, the Twelve Principles of Green Engineering, highlighting the key approaches to advancing sustainability through engineering design. The class begins with discussions on sustainability, metrics, general design processes, and challenges to sustainability. The current approach to design, manufacturing, and disposal is discussed in the context of examples and case studies from various sectors. This provides a basis for what and how to consider when designing products, processes, and systems to contribute to furthering sustainability. The fundamental engineering design topics to be addressed include toxicity and benign alternatives, pollution prevention and source reduction, separations and disassembly, material and energy efficiencies and flows, systems analysis, biomimicry, and life cycle design, management, and analysis.

ENAS 704, Theoretical Fluid Dynamics
Derivation of the equations of fluid motion from basic principles. Potential theory, viscous flow, flow with vorticity. Topics in hydrodynamics, gas dynamics, stability, and turbulence.

ENAS 708, Fundamentals of Combustion
Review of relevant aspects of chemical thermodynamics and chemical kinetics. Explosion and oxidation of fuels. Laminar premixed fuels. Detonations. Diffusion flame and droplet burning.

ENAS 711, Biomedical Microtechnology and Nanotechnology
Principles and applications of micro- and nanotechnologies for biomedicine. Approaches to fabricating micro- and nanostructures. Fluid mechanics, electrokinetics, and molecular transport in microfluidic systems. Integrated biosensors and microTAS for laboratory medicine and point-of-care uses. High-content technologies including DNA, protein microarrays, and cell-based assays for differential diagnosis and disease stratification. Emerging nanobiotechnology for systems medicine. Prerequisites: CHEM 112a, 114a, or 118a, and ENAS 194a or b.

ENAS 747, Applied Numerical Methods I
The derivation, analysis, and implementation of various numerical methods. Topics include root-finding methods, numerical solution of systems of linear and nonlinear equations, eigenvalue/eigenvector approximation, polynomial-based interpolation, and numerical integration. Additional topics such as computational cost, error analysis, and convergence are addressed in a variety of contexts.

ENAS 748, Applied Numerical Methods II
The derivation, analysis, and implementation of numerical methods for the solution of ordinary and partial differential equations, both linear and nonlinear. Additional topics such as computational cost, error estimation, and stability analysis are studied in several contexts throughout the course. ENAS 747a is not a prerequisite.

ENAS 752, Solidification and Phase Transformations   
This graduate level course covers solidification phenomena with a focus on metallic systems. We will be covering thermodynamics including, thermodynamic functions, Gibbs free energy, solution models (regular, ideal), chemical potential, equilibrium in heterogeneous systems, which will allow us to create and understand phase diagrams. Kinetic aspects like diffusion, viscosity, and its connection through Stokes-Einstein will be covered. Nucleation theory, homogeneous vs. heterogeneous, steady state, transient, activation energies, multicomponent systems, topological and chemical fluctuations will be discussed in detail. Growth processes will be considered such as diffusion limited, interfacial limited and phase transitions solid=>solid or liquid => liquid such as spinodal composition. Vitrification processes will be discussed with focus on time temperature transformation diagrams, glass transition, and structural relaxation.

ENAS 777, Introduction to Robot Analysis
Intended for graduate students in robotics. Fundamental topics in robot kinematics and dynamics. Topics include coordinate frames and transformations, forward and inverse kinematic solutions to open and closed chain manipulators, kinematic structure and solutions, statics and dynamics of serial and parallel chain manipulators. Special topics introduced according to the research interests of enrolled students. Course includes a significant final project.

ENAS 787, Intermolecular and Surface Forces
Modern materials science often exploits the fact that atoms located at surfaces or in thin layers behave differently from bulk atoms to achieve new or greatly altered material properties. The course provides an in-depth discussion of intermolecular and surface forces, which determine the mechanical and chemical properties of surfaces. In a first part, we discuss the fundamental principles and concepts of forces between atoms and molecules. Part two generalizes these concepts to surface forces. Part three then gives a variety of examples. The course is of interest to students studying thin-film growth, surface coatings, mechanical and chemical properties of surfaces, soft matter including biomembranes, and colloidal suspensions.

ENAS 802, Nano and Microsystem Technology
Cross-disciplinary laboratory experiments covering microfabrication, silicon micromachining, MEMS device fabrication and characterization, scanned probe microscopy, electron microscopy, microfluidics, lab-on-a-chip system. Students fabricate MEMS, BioMEMS, and microfluidic devices in a cleanroom environment.

ENAS 806, Photovoltaic Energy  
Survey of photovoltaic energy devices, systems, and applications, including review of optical and electrical properties of semiconductors. Topics include solar radiation, solar cell design, performance analysis, solar cell materials, device processing, photovoltaic systems, and economic analysis.

ENAS 812/NSCI 612, Molecular Transport and Intervention in the Brain
A graduate-level seminar on mechanisms and rates of movement of molecules in the brain and the design of novel drug delivery systems. Topics include mathematical methods for modeling diffusion and flow processes, diffusion in the brain interstitium, fluid flows in the brain and spinal cord, the blood-brain barrier, microdialysis measurements, controlled release systems, microfluidic approaches for drug delivery. Weekly readings are assigned from neuroscience and engineering texts; current papers from the literature are used to guide discussion each week.

ENAS 821, Physics of Medical Imaging
The physics of image formation with special emphasis on techniques with medical applications. Concepts that are common to different types of imaging are emphasized, along with an understanding of how information is limited by the basic physical phenomena involved. Mathematical concepts of image analysis, the formation of images by ionizing radiation, ultrasound, NMR, and other energy forms, and methods of evaluating image quality.

ENAS 825, Physics of Magnetic Resonance Spectroscopy in Vivo
The physics of chemical measurements performed with nuclear magnetic resonance spectroscopy, with special emphasis on applications to measurement studies in living tissue. Concepts that are common to magnetic resonance imaging are introduced. Topics include safety, equipment design, techniques of spectroscopic data analysis, and metabolic modeling of dynamic spectroscopic measurements.

ENAS 836, Biophotonics and Optical Microscopy
A review of linear and nonlinear optical microscopies and other biophotonics applications. Topics include wide-field techniques, linear and nonlinear laser scanning microscopy, fundamentals of geometrical and physical optics, optical image formation, laser physics, single molecule techniques, fluorescence correlation spectroscopy, and light scattering. Discussion of fluorescence and the underlying physics of light-matter interactions that provide biologically relevant signals.

ENAS 850 and 851/PHYS 548 and 549/APHY 548 and 549, Solid State Physics I and II
A two-term sequence covering the principles underlying the electrical, thermal, magnetic, and optical properties of solids, including crystal structures, phonon, energy bands, semiconductors, Fermi surfaces, magnetic resonance, phase transitions, and superconductivity.

ENAS 866, MOS Device Physics and Technology
Topics include basic MOS device physics, science and technology of thermal SiO2, interface properties of MOS structures, experimental techniques to probe MOS parameters, hot-carrier effects, radiation effects, channel mobility and carrier transport in MOS inversion layers, scaling of MOS devices, low-temperature properties of MOS devices, SOI device physics and technology, advanced gate dielectrics, MOS devices with wide-bandgap semiconductors, nonvolatile memory devices, ferroelectric memory devices, single-electron MOS transistors, and other MOS topics of current interest.

ENAS 875, Introduction to VLSI System Design
Chip design. Provides background in integrated devices, circuits, and digital subsystems needed for design and implementation of silicon logic chips. Historical context, scaling, technology projections, physical limits. CMOS fabrication overview, complementary logical circuits, design methodology, computer-aided design techniques, timing, and area estimation. Case studies of recent research and commercial chips. Objectives of the course are (1) to give students the ability to complete the course project (design of a digital CMOS subsystem chip through layout), and (2) to understand the directions that future chip technologies may take. Selected projects are fabricated and packaged for testing by students. Prerequisite: circuits at the level of introductory physics and computer programming.

ENAS 880/NSCI 523, Imaging Drugs in the Brain
Seminar course to explore the uses of PET, SPECT, and fMRI to study the mechanisms of action, and long term effects of drugs (legal and illegal) on brain function. Basic research will be the main focus, augmented by two class periods allotted to uses of imaging in drug development by Pharma. Syllabus will be comprised of review articles, book chapters and journal articles. Some class periods will begin with short lecture to cover methodological concepts followed by discussion of reading material. Topics include basic understanding of imaging technology (physics, biochemistry and mathematics) as it relates to imaging of drugs, receptors, neurotransmitters; understanding the primary outcomes of imaging experiments; imaging experiment design; recent findings related to drug abuse; common neurophysiological pathways of addictive drugs – how to image reward; uses of imaging in drug development – what do drug companies want to measure.

ENAS 902, Linear Systems
Background linear algebra; finite-dimensional, linear-continuous, and discrete dynamical systems; state equations, pulse and impulse response matrices, weighting patterns, transfer matrices. Stability, Lyapunov’s equation, controllability, observability, system reduction, minimal realizations, equivalent systems, McMillan degree, Markov matrices. Recommended for all students interested in robotics, systems, and information sciences.

ENAS 912, Biomedical Image Processing and Analysis.  
A study of the basic computational principles related to processing an analysis of biomedical images (e.g., magnetic resonance, computed X-ray tomography, fluorescence microscopy). Basic concepts and techniques related to discrete image representation, multidimensional frequency transforms, image enhancement/restoration, image segmentation, and image registration.

ENAS 920, Programming for Image Analysis
Topics include using scripting languages for visualization, introduction to scripting languages, in particular Tcl, introduction to the Visualization Toolkit (Tcl) and local extensions, designing graphical user interfaces using Tk, introduction to Object Oriented programming (using [Incr Tcl]), using compiled languages to implement additional algorithms, intoduction to C++ programming, extending VTK by implementing additional image processing algorithms, an overview of the Insight Toolkit (ITK), and advanced software engineering techniques. Prerequisite: ENAS 912a, or permission of the instructor.

ENAS 921, Advanced Topics in Computer Engineering
Review of current topics and principles of modern computing systems, including concepts from computer architecture, computer-aided design, reconfigurable computing, VLSI design and testing, as well as hardware security. Reading material is based on recent research papers and other similar sources. Laboratory work consists of the completion of a project using computer-aided design and test tools as well as reconfigurable or custom hardware design platforms. Prerequisite: permission of the instructor.

ENAS 936, Systems and Control
Design of feedback control systems with applications to engineering, biological, and economic systems. Topics include stat-space representation, stability, controllability, and observability of discrete-time systems; system identification; optimal control of systems with multiple outputs.

ENAS 944, Digital Communications Systems
An introduction to the rapidly expanding field of mobile and fixed, voice and data communications systems. A review of analog and digital signals and their time and frequency domain representations. Topics include modulation methods, including amplitude; frequency and time division multiplexing for continuous and discrete/digital signals; an overview of modern voice and data communications networks; and an overview of information theory, including entropy, the quantification of information, data rates, coding, and compression. Examples and demonstrations are drawn from radio, telephone, television, computer, cellular, and satellite communications networks.

ENAS 986, Semiconductor Silicon Devices and Technology
Introduction to integrated circuit technology, theory of solid state devices, and principles of device design and fabrication. Laboratory involves the fabrication and analysis of semiconductor devices, including Ohmic contacts, Schottky diodes, p-n junctions, MOS capacitors, MOSFETS, and integrated circuits.

ENAS 990, Special Investigations
Faculty-supervised individual projects with emphasis on research, laboratory, or theory. Students must define the scope of the proposed project with the faculty member who has agreed to act as supervisor, and submit a brief abstract to the director of graduate studies for approval.

ENAS 991/MB&B 591/PHYS 991, Integrated Workshop
This required course for students in IGPPEB involves hands-on laboratory modules with students working in pairs. A biology student is paired with a physics or engineering student; a computation/theory student is paired with an experimental student. The modules are devised so that a range of skills are acquired, and students learn from each other.