Overview of multicore, cluster, and supercomputer architectures; procedure and object oriented languages; parallel computing paradigms and languages; graphics and visualization of large data sets; validation and verification; and scientific software development. May not be offered every quarter; content may vary from one offering to another. Prerequisite: permission of instructor. AMATH Vector Calculus and Complex Variables 5 Emphasizes acquisition of solution techniques; illustrates ideas with specific example problems arising in science and engineering. Prerequisite: either a course in vector calculus or permission of instructor.

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Prerequisite: either a course in differential equations or permission of instructor. AMATH Introduction to Fluid Dynamics 4 Eulerian equations for mass-motion; Navier-Stokes equation for viscous fluids, Cartesion tensors, stress-strain relations; Kelvin's theorem, vortex dynamics; potential flows, flows with high-low Reynolds numbers; boundary layers, introduction to singular perturbation techniques; water waves; linear instability theory.

Prerequisite: either a course in partial differential equations or permission of instructor. Legendre transformation, Hamiltonian systems. Constraints and Lagrange multipliers. Space-time problems with examples from elasticity, electromagnetics, and fluid mechanics. Sturm-Liouville problems.

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Approximate methods. Offered: W, odd years. AMATH Networks and Combinatorial Optimization 3 Mathematical foundations of combinatorial and network optimization with an emphasis on structure and algorithms with proofs. Topics include combinatorial and geometric methods for optimization of network flows, matching, traveling salesmen problem, cuts, and stable sets on graphs.

Special emphasis on connections to linear and integer programming, duality theory, total unimodularity, and matroids. Offered: jointly with MATH Basic problem types and examples of applications; linear, convex, smooth, and nonsmooth programming. Optimality conditions. Saddlepoints and dual problems. Penalties, decomposition.

Overview of computational approaches. Desirable: optimization, e. Math , and scientific programming experience in Matlab, Julia or Python. Steepest descent, quasi-Newton methods. Quadratic programming and complementarity. Exact penalty methods, multiplier methods. Sequential quadratic programming. Cutting planes and nonsmooth optimization. Controllability, optimality, maximum principle.

Relaxation and existence of solutions. Techniques of nonsmooth analysis. AMATH Mathematical Analysis in Biology and Medicine 5 Focuses on developing and analyzing mechanistic, dynamic models of biological systems and processes, to better understand their behavior and function. Prerequisite: either courses in differential equations and statistics and probability, or permission of instructor.

Draws examples from molecular and cell biology, ecology, epidemiology, and neurobiology. Topics include reaction-diffusion equations for biochemical reactions, calcium wave propagation in excitable medium, and models for invading biological populations. Focuses on analyzing dynamics leading to functions of cellular components gene regulation, signaling biochemistry, metabolic networks, cytoskeletal biomechanics, and epigenetic inheritance using deterministic and stochastic models.

Prerequisite: either courses in dynamical systems, partial differential equations, and probability, or permission of instructor. AMATH Neural Control of Movement: A Computational Perspective 3 Systematic overview of sensorimotor function on multiple levels of analysis, with emphasis on the phenomenology amenable to computational modeling. Topics include musculoskeletal mechanics, neural networks, optimal control and Bayesian inference, learning and adaptation, internal models, and neural coding and decoding.

Offered: jointly with CSE AMATH Dynamics of Neurons and Networks 5 Covers mathematical analysis and simulation of neural systems - singles cells, networks, and populations - via tolls of dynamical systems, stochastic processes, and signal processing. Topics include single-neuron excitability and oscillations; network structure and synchrony; and stochastic and statistical dynamics of large cell populations.

## A List of Undergraduate Math Courses for Math Majors at UH

Prerequisite: either familiarity with dynamical systems and probability, or permission of instructor. Topics include the effects of density dependence, delays, demographic stochasticity, and age structure on population growth; population interactions predation, competition, and mutualism ; and application of optimal control theory to the management of renewable resources.

Particular emphasis on branching process models of cancer initiation, progression and response to therapy, and their relationship to clinical, epidemiological and sequencing data. The course introduces both analytic and computational approaches for modeling cancer, and gets students acquainted with the current research in the field.

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Prerequisite: Previous experience with calculus, probability, ODEs and programming or permission of instructor. AMATH Introduction to Probability and Random Processes 5 This course, aimed at scientists and engineers without background in measure theory, introduces concepts in probability and stochastic dynamics needed for mathematical modeling. It includes, in addition to the basics of probability, Markov chain, Q-process, Chapman-Kolmogorov equations, and discrete-time martingales. Emphasis is on presenting theories with examples and variety of computational methods.

Exposure to graduate level PDEs expected. Prerequisite: either undergraduate course in probability and statistics, or permission of instructor. Introduces basic concepts in continuous stochastic processes including Brownian motion, stochastic differential equations, Levy processes, Kolmogorov forward and backward equations, and Hamilton-Jacobi-Bellman PDEs.

Course presents theories with applications from physics, biology, and finance. Exposure to graduate level linear PDEs expected. Merges concepts from model selection, information theory, statistical inference, neural networks, deep learning, and machine learning for building reduced order models of dynamical systems using sparse sampling of high-dimensional data. Branch cuts, series and product expansions. Contour integration, numerical implications.

Harmonic functions. Complex potential and singularities in physical problems. Discrete mathematics is the study of mathematical structures that are isolated and discrete, rather than varying in a smooth or continuous way. In contrast to subjects such as calculus or real analysis, where the continuum of real numbers or smooth i. While discrete mathematics is a subject studied by many mathematicians, this particular course at UH is primarily a service course for Computer Science students, and there is significant overlap with Math Transition to Advanced Mathematics.

Math majors should not take this course, and should instead take Math Math majors interested in learning more about discrete mathematics can take Math Graph Theory with Applications. This is a course in mathematical probability theory, and it also teaches applications to real-world problems in subjects such as Finance, Insurance, and Engineering. This is a course in mathematical statistics, and it also teaches applications to real-world problems in subjects such as Finance, Engineering, and the Sciences.

This is a course in the mathematical modeling of finance. It shows how calculus-level mathematics may be used to determine costs, prices, and returns in various standard "fixed income problems" including the basic analysis of loans, bonds, and portfolios.

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This is a useful course for anyone interested in Financial Mathematics. This course in an introduction to PDEs and boundary value problems, including such topics as Fourier series, the heat equation, vibrations of continuous systems, the potential equation, and spectral methods. This is a good course for students interested in engineering, physical or biological sciences, economics, finance, or careers in industry. PDEs are used in many aspects of mathematical modeling that arise in real-world problems. Math majors interested in a more serious treatment of PDEs may wish to skip this course and instead take the Math Math sequence or only Math , if desired.

In mathematics, the subject of Analysis is often divided into three main areas: Real Analysis, Complex Analysis, and Functional Analysis. This course is one of the only opportunities for math majors at UH to gain exposure to an area of analysis outside of real analysis. Math , Math , and Math cover Real Analysis, and Functional Analysis is typically not covered until graduate school.

All math majors who want to go to graduate school in mathematics or a related subject should take this course. In addition, complex analysis has application in physics, particularly to some aspects of hydrodynamics and thermodynamics as well as in nuclear, aerospace, mechanical and electrical engineering. Therefore, certain physics or engineering students may also be interested in this course. Topics include synthetic and algebraic geometry, harmonic division, cross ratio, and groups of projective transformations.

This is essentially an independent study course in which you work on a project under the supervision of a professor. This is the course number assigned to a class that a professor wishes to teach, but which does not currently exist in our curriculum. The topic of the course depends on who is teaching it, and no two Math courses are the same. This is course in which you can work on a research project with under the supervision of a professor.

Mathematical Biology also sometimes called Theoretical Biology is an interdisciplinary subject where mathematics is used to study biological processes. Mathematical biology aims at the mathematical representation, treatment, and modeling of biological processes, using a variety of applied mathematical techniques and tools. It has both theoretical and practical applications in biological, biomedical, and biotechnology research. This course introduces a variety of discrete and continuous ordinary and partial differential equation models of biological systems.

Mathematical methods taught include phase plane analysis, bifurcation methods, separation of timescales, and some scientific computing in MATLAB. Biological topics include population dynamics, epidemiology, gene networks, neuroscience, and biological transport. Biostatistics also called biometrics is the application of statistics to a wide range of topics in biology.

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The subject of biostatistics encompasses the design of biological experiments, especially in medicine, pharmacy, agriculture, and fishery; the collection, summarization, and analysis of data from those experiments; and the interpretation of, and inference from, the results. A major branch of biostatistics is medical biostatistics, which is exclusively concerned with medicine and health. A graph is a mathematical structure used to model pairwise relations between objects. The definition of a graph is very simple: A graph consists of "dots" formally called vertices and "lines" formally called edges drawn between them.

A graph may be undirected, meaning that there is no distinction between the two vertices on each edge, or it may be directed, with its edges written as an arrow pointing from one vertex to another. Graphs are ubiquitous and arise in numerous subjects where discrete relations between objects are found. Graphs are particularly useful in computer science, and they are the main object of study in the subject of Discrete Mathematics.