UC Merced's Applied Mathematics courses emphasize learning fundamental mathematics as well as its application to other fields of study such as biology, economics and physics. To that end, we emphasize learning mathematics conceptually and concretely by exploring topics analytically, graphically, computationally and verbally. Below we describe each of our undergraduate course offerings.

## Lower Division Courses

Lower division mathematics courses (e.g. pre-calculus and calculus) are required for many of the majors at UC Merced because mathematics is the foundation for many different areas of study. To help our students to learn mathematics as best as they can, our math instructors provide students a variety of experiences to learn mathematics through lectures, discussion sections, homework and various workshops. In addition, there is free tutoring available on campus through the Math Center. (Courses with * are currently not scheduled to be taught.)

#### MATH 005: Preparatory Calculus

Preparation for calculus. Analyzing data by means of functions (linear, quadradic, polynomial, logarithmic, exponential and trigonometric) and graphs with an emphasis on mathematical modeling of real-world applications.

#### MATH 011: Calculus I

Introduction to differential and integral calculus of functions of one variable, including exponential, logarithmic and trigonometric functions, emphasizing conceptual understanding and applying mathematical concepts to real-world problems (approximation, optimization). Course does not lead to MATH 023, MATH 024.

Prerequisite: MATH 005 or equivalent exam

#### MATH 012: Calculus II

Continuation of MATH 011. Introduction to integral calculus of functions of one variable and differential equations, emphasizing conceptual understanding and applying mathematical concepts to real-world problem. Course does not lead to MATH 023, MATH 024.

Prerequisite: MATH 011 or MATH 021 or equivalent exam

#### MATH 015: Introduction to Scientific Data Analysis

Fundamental analytical and computational skills to find, assemble and evaluate information, and to teach the basics of data analysis and modeling using spreadsheets, statistical tool, scripting languages, and high-level mathematical languages.

Concurrent Prerequisites: MATH 005 or MATH 011 or MATH 021, or equivalent exam

#### MATH 018: Statistics for Scientific Data Analysis*

Analytical and computational methods for statistical analysis of data. Descriptive statistics, graphical representations of data, correlation, regression, causation, experiment design, introductory probability, random variables, sampling distributions, inference and significance.

Prerequisite: (MATH 005 or MATH 011 or MATH 021 or equivalent exam) and (MATH 015 or CSE 020 or CSE 005 or ENVE 105 or equivalent exam)

#### MATH 021: Calculus I for Physical Sciences and Engineering

An introduction to differential and integral calculus of functions of one variable. Elementary functions such as the exponential and the natural logarithm, rates of change and the derivative with applications to natural sciences and engineering.

Prerequisite: MATH 005 or equivalent exam

#### MATH 022: Calculus II for Physical Sciences and Engineering

Continuation of MATH 021. Analytical and numerical techniques of integration with applications, infinite sequences and series, first order ordinary differential equations.

Prerequisite: MATH 021 or equivalent exam

#### MATH 023: Vector Calculus

Calculus of several variables. Topics include parametric equations and polar coordinates, algebra and geometry of vectors and matrices, partial derivatives, multiple integrals, and introduction to the theorems of Green, Gauss, and Stokes.

Prerequisite: MATH 022 or equivalent exam

#### MATH 023H: Honors Vector Calculus*

Honors version of MATH 023. Topics cover vectors, calculus of multi-variable functions, coordinate systems, parametric curves and surfaces, and theorems of Green, Gauss and Stokes. Small class size and innovative pedagogical methods are adopted to help students develop a deep understanding of theories and a mastery of skills.

Prerequisite: MATH 022 with A- or better, or equivalent exam

#### MATH 024: Linear Algebra and Differential Equations

Introduces ordinary differential equations, systems of linear equations, matrices, determinants, vector spaces, linear transformations and linear systems of differential equations.

Prerequisite: MATH 022 or equivalent exam

#### MATH 032: Probability and Statistics

Concepts of probability and statistics. Conditional probability, independence, random variables, distribution functions, descriptive statistics, transformations, sampling errors, confidence intervals, least squares and maximum likelihood. Exploratory data analysis and interactive computing.

Concurrent Prerequisites: MATH 023 or MATH 023H

#### MATH 050: Beginning MATLAB Programming*

This half-semester minicourse introduces students to the writing, implementing, and testing of MATLAB algorithms to solve mathematical problems. Topics include programming syntax, data visualization, debugging, and coding aesthetics.

Prerequisite: MATH 022 or equivalent exam

#### MATH 090X: Freshman Seminar*

Topics in mathematics.

#### MATH 091: General Topics in Applied Mathematics*

Introduction to a variety of concepts useful in applied mathematics. Topics covered included floating point arithmetic, methods of proofs, random walks, stereographic projections, transforms, etc. Students are exposed to advanced mathematical topics in preparation for their ongoing studies.

Concurrent Prerequisites: (MATH 023 or MATH 023H) and MATH 024

#### MATH 095: Lower Division Undergraduate Research

Supervised research in mathematics.

## Upper Division Courses

(Courses with * are currently not scheduled to be taught.)

#### MATH 101: Real Analysis

Introduction to rigorous mathematical proofs and concepts pertaining to real numbers. The class will cover the structure of real numbers, sequences, series and functions of real numbers, and, time permitting, concepts of abstract algebra.

Prerequisite: MATH 023 or MATH 023H

#### MATH 122: Complex Variables and Applications

Introduction to complex variables, analytic functions, contour integration and theory of residues. Mappings of the complex plane. Introduction to mathematical analysis.

Prerequisite: (MATH 023 or MATH 023H) and MATH 024

#### MATH 125: Intermediate Differential Equations

An introduction of advanced solution techniques for ordinary differential equations (ODE) and elementary solution techniques for partial differential equations (PDE). Specific topics include higher-order linear ODE, power series methods, boundary value problems, Fourier series, Sturm-Liouville theory, Laplace transforms, Fourier transforms, and applications to one-dimensional PDE.

Prerequisite: (MATH 023 or MATH 023H) and MATH 024

#### MATH 126: Partial Differential Equations

An introduction to the theory of boundary value and initial value problems for partial differential equations with emphasis on linear equations. Topics covered include Laplace’s equation, heat equation, wave equation, application of Sturm-Liouville’s theory, Green’s functions, Bessel functions, Laplace transform, method of characteristics.

Prerequisite: MATH 125

#### MATH 130: Numerical Analysis

Introduction to numerical methods with emphasis on the analysis and implementation of numerical methods. Topics covered: computer arithmetic, solution of nonlinear equations in one variable, interpolation and polynomial approximation, elements of approximation theory, numerical differentiation and integration, and introduction to initial-value problems for ordinary differential equations.

Prerequisite: ME 021 or (CSE 020 and CSE 021 or equivalent exam), and MATH 024

#### MATH 131: Numerical Methods for Scientists and Engineers

Introduction to numerical methods with emphasis on algorithm construction, analysis and implementation. Programming, round-off error, solutions of equations in one variable, interpolation and polynomial approximation, approximation theory, direct solvers for linear systems, numerical differentiation and integration, initial-value problems for ordinary differential equations.

Prerequisite: MATH 024 and (ME 021 or (CSE 020 and CSE 021 or equivalent exam))

#### MATH 132: Numerical Methods for Differential Equations

Introduction to numerical methods with emphasis on the analysis and implementation of numerical methods. Topics covered: Initial- and boundary-value problems for ordinary differential equations, methods to solve linear systems, eigenvalue problems, and numerical solutions to partial differential equations.

Prerequisite: MATH 125 and (MATH 130 or MATH 131)

#### MATH 140: Mathematical Methods for Optimization

Linear programming and a selection of topics from among the following: matrix games, integer programming, semidefinite programming, nonlinear programming, convex analysis and geometry, polyhedral geometry, the calculus of variations and control theory. Matlab implementation of several algorithms.

Prerequisite: (MATH 023 or MATH 023H) and MATH 024 and (CSE 021 or ME 021 or equivalent exam)
Concurrent Prerequisites: MATH 130 or MATH 131

#### MATH 141: Linear Analysis I

Applied linear analysis of finite dimensional vector spaces. Review of matrix algebra, vector spaces, orthogonality, least-squares approximations, eigenvalue problems, positive definite matrices, singular value decomposition with applications in science and engineering.

Prerequisite: (MATH 023 or MATH 023H) and MATH 024

#### MATH 146: Numerical Linear Algebra

Matrix factorization and iterative methods for solving systems of linear equations. Topics include floating point arithmetic, eigenvalue problems, conditioning and stability, LU factorization, QR factorization, and SVD with applications in science and engineering.

Prerequisite: ME 021 or (CSE 020 and CSE 021 or equivalent exam)
Concurrent Prerequisites: MATH 141 or MATH 130 or MATH 131

#### MATH 150: Mathematical Modeling

Introduction to the basics of mathematical modeling emphasizing model construction, analysis and application. Using examples from a variety of fields such as physics, biology, chemistry and economics, students will learn how to develop and use mathematical models of real-world systems.

Prerequisite: (MATH 131 or MATH 130) and (MATH 125 or MATH 141)

#### MATH 160: Mathematical Logic*

Introduction to the meta-theory of first-order logic. Topics include the consistency, compactness, completeness and soundness proofs for propositional and first-order logic; model theory; the axiomatization of number theory; Gödel’s incompleteness theorems and related results.

Prerequisite: PHIL 005

#### MATH 170: Mathematical Biology

Introduces the design and analysis of mathematical models of biological phenomena. The course focuses on three different classes of mathematical models: difference equations, ordinary differential equations and partial differential equations. Biological topics covered are expected to vary but likely include population dynamics, enzyme kinetics, biochemical networks, cellular processes, epidemiology and pattern formation.

Prerequisite: MATH 024

#### MATH 180: Modern Applied Statistics

Introduction to modern applied statistics emphasizing computational methods to deal with high-dimensional data. Multivariate linear and nonlinear regression, model selection, overfitting, cross-validation, bootstrapping and quantification of uncertainty in model parameters and predictions, principal component analysis, and classification.

Prerequisite: MATH 024 and MATH 032

#### MATH 181: Stochastic Processes

Introduction to stochastic processes with emphasis on problem-solving using both analytical and computational techniques. Markov chains in discrete and continuous time, martingales, branching processes, renewal processes, and Brownian motion.

Prerequisite: MATH 024 and MATH 032

#### MATH 195: Upper Division Undergraduate Research

Supervised research.