# Applied and Computational Mathematics and Statistics Courses

## ACMS 10140. Elements of Statistics

(3-0-3)
This course is intended for those students who may or may not plan to use statistics in their chosen careers, but wish nevertheless to become informed and astute consumers. Topics include statistical decision-making, sampling, data representation, random variables, elementary probability, conditional probabilities, independence, and Bayes' rule. The methodology will focus on a hands-on approach. Concepts and terminology will be introduced only after thorough exposure to situations that necessitate the concepts and terms. Care will be exercised to select a variety of situations from the many fields where statistics are used in modern society. Examples will be taken from biology and medicine (e.g. drug testing, wild animal counts), the social sciences, psychology, and economics. This course counts only as general elective credit for students in the College of Science.

## ACMS 10141. Statistical Reasoning in Politics

(3-0-3)
Essential concepts of statistical reasoning are explored through the analysis of politics, elections, and media. Topics covered include data exploration, measures of variability, inference, correlation, and linear regression. Real datasets are used to illustrate many concepts, and concepts are introduced by references to actual events. Calculations will be conducted in Microsoft Excel and/or with a graphing calculator.

## ACMS 10145. Statistics for Business and Economics I

(3-0-3)
A conceptual introduction to the science of data for students of business. Descriptive statistics: graphical methods, measures of central tendency, spread, and association. Basic probability theory and probability models for random variables. Introduction to statistical inference: confidence intervals and hypothesis tests. Many examples will be based on real, current business and economics datasets. Calculations will be illustrated in Microsoft Excel. Not eligible for science credit for students in the College of Science. Credit is not given if a student takes both ACMS 10145 and ACMS 10140 or ACMS 10145 and ACMS 10141. This course is proposed to satisfy one university mathematics requirement.

## ACMS 10150. Elements of Statistics II

(3-0-3)
Prerequisite: MATH 10140. The goal of this course is to give students an introduction to a variety of the most commonly used statistical tools. A hands-on approach with real data gathered from many disciplines will be followed. Topics include inferences based on two samples, analysis of variance, simple linear regression, categorical data analysis, and non-parametric statistics. This course counts only as general elective credit for students in the College of Science.

## ACMS 19999. ACMS Transfer Elective

This course provides a number and range of credits to assign to a transfer student's classes that do not have an ACMS equivalent at Notre Dame so that ACMS coursework being transferred from another university or college will transfer as a General elective only and not as a science (ACMS) credit. Students can transfer 1 to 4 credits.

## ACMS 20210. Scientific Computing

(3.5-0-3.5)
Prerequisite: MATH 10560 or MATH 10860 or MATH 10360 or MATH 14360. An introduction to solving mathematical problems using computer programming in high-level languages such as C.

## ACMS 20340. Statistics for the Life Sciences

(3-0-3)
Prerequisites: MATH 10360 or MATH 10460 or MATH 10560 or MATH 14360. An introduction to the principles of statistical inference following a brief introduction to probability theory. This course does not count as a science or mathematics elective for mathematics majors. NOTE: Students may not take more than one of ACMS 20340, BIOS 40411 and MATH 20340. Not open to students who have taken MATH 30540.

## ACMS 20550. Introduction to Applied Mathematics Methods I

(3.5-1-3)
An introduction to the methods of applied mathematics. Topics include: basic linear algebra, partial derivatives, Taylor and power series in multiple variables, Lagrange multipliers, multiple integrals, gradient and line integrals, Green's theorem, Stokes theorem and divergence, Fourier series and transforms, introduction to ordinary differential equations. Applications to real-world problems in science, engineering, the social sciences and business will be emphasized in this course and ACMS 20750. Computational methods will be taught. Credit is not given for both ACMS 20550 and PHYS 20451.

## ACMS 20620. Applied Linear Algebra

(3-0-3)
The objective of this class is to impart the fundamental knowledge in linear algebra and computational linear algebra that is needed to solve matrix algebra problems in application areas. Appropriate software packages will be used.

## ACMS 20750. Introduction to Applied Mathematics Methods II

(3.5-1-3.5)
Prerequisite: ACMS 20550 or PHYS 20451 or MATH 20550. The fundamental methods of applied mathematics are continued in this course. Topics include: variational calculus, special functions, series solutions of ordinary differential equations (ODE), orthogonal functions in the solution of ODE, basic partial differential equations and modeling heat flow, vibrating string, and steady-state temperature. Topics in complex function theory include contour integrals, Laurent series and residue calculus, and conformal mapping. The course concludes with a basic introduction to probability and statistics. Credit is not given for both ACMS 20750 and PHYS 20452.

## ACMS 22550. Introduction to Applied Mathematics Methods I Tutorial

Tutorial for Introduction to Applied Mathematics Methods I.

## ACMS 22750. Introduction to Applied Mathematics Methods II Tutorial

(0-1-0)

Tutorial for Introduction to Applied Mathematics Methods II.

## ACMS 29999. ACMS Transfer Elective

This course provides a number and range of credits to assign to a transfer student's classes that do not have an ACMS equivalent at Notre Dame so that ACMS coursework being transferred from another university or college will transfer as science (ACMS) credit. As all ACMS courses that currently may be prerequisites for other ACMS courses would not be in this category, ACMS 20999 would not represent a prerequisite, but rather an elective, negating any registration problem(s). Students can apply 1 to 4 credits.

## ACMS 30440. Probability and Statistics

(3-0-3)
An introduction to the theory of probability and statistics, with applications to the computer sciences and engineering. Topics include discrete and continuous random variables, joint probability distributions, the central limit theorem, point and interval estimation, and hypothesis testing.

## ACMS 30530. Introduction to Probability

(3-0-3)
Prerequisites: MATH 20550 or ACMS 20550 or MATH 20850. Cross-listed with MATH 30530. An introduction to the theory of probability, with applications to the physical sciences and engineering. Topics include discrete and continuous random variables, conditional probability and independent events, generating functions, special discrete and continuous random variables, laws of large numbers and the central limit theorem. The course emphasizes computations with the standard distributions of probability theory and classical applications of them.

## ACMS 30540. Mathematical Statistics

(3-0-3)
Prerequisites: ACMS 30530 or MATH 30530. An introduction to mathematical statistics. Topics include distributions involved in random sampling, estimators and their properties, confidence intervals, hypothesis testing including the goodness-of-fit test and contingency tables, the general linear model and analysis of variance.

## ACMS 30600. Statistical Methods & Data Analysis I

(3-0-3)
Prerequisite: ACMS 30440 OR ACMS 30530 OR MATH 30530. Introduction to statistical methods with an emphasis on analysis of data. Estimation of central values. Parametric and nonparametric hypothesis tests. Categorical data analysis. Simple and multiple regression. Introduction to time series. The SOA has approved this course for VEE credit in Applied Statistics.

## ACMS 30610. Introduction to Financial Mathematics

(3 -0- 3)
Prerequisite: ACMS 20550 or ACMS 20620 or ACMS 20750 or ACMS 30530 and MATH 30610
The course serves as a preparation for the first actuarial exam in financial mathematics, known as Exam FM or Exam 2. The first part of the course deals with pricing of fixed income securities, such as bonds and annuities. The second part of the course can serve as an introduction to derivative securities, such as options and futures. Although the amount of material for both parts is almost the same, Exam FM devotes usually about 2/3 of its questions to Part 1. Therefore, about 2/3 of the course is devoted to Part 1. Topics covered: interest rates, annuities, loans and bonds, forwards, options, hedging, and swaps.

## ACMS 31600. Statistical Methods & Data Analysis I Lab

Lab for Statistical Methods & Data Analysis I

## ACMS 34540. Inferential Statistics

(3-0-3)

Taught at a host institution. STAT 20100 Inferential Statistics at UCD; Continuous bivariate and multivariate distributions. Covariance and correlation. Chebyshev inequality. Law of Large Numbers Theory of Estimation. Method of moments and maximum likelihood. Point and interval estimation. Hypothesis Testing. Simple and Composite Hypotheses. Neyman Pearson Lemma and applications. Likelihood ratio tests. Bayesian statistical inference. Loss functions Normal/Normal, Binomial/Beta and Exponential/Gamma models. Probability generating functions.

## ACMS 40212. Advanced Scientific Computing

(3-0-3)
This course covers fundamental material necessary for using high performance computing in science and engineering. There is a special emphasis on algorithm development, computer implementation, and the application of these methods to specific problems in science and engineering.

## ACMS 40390. Numerical Analysis

(3-0-3)
Prerequisites: (MATH 20750 or MATH 20860 or MATH 30650 or ACMS 20750 or PHYS 20452) and (ACMS 20620 or MATH 20610) and ACMS 20210. An introduction to the numerical solution of ordinary and partial differential equations. Topics include the finite difference method, projection methods, cubic splines, interpolation, numerical integration methods, analysis of numerical errors, numerical linear algebra and eigenvalue problems, and continuation methods.

## ACMS 40395. Numerical Linear Algebra

(3 -0- 3) Zhang
Prerequisite: (MATH 20610 or ACMS 20620) and (ACMS 40390 or MATH 40390)
The course will cover numerical linear algebra algorithms which are useful for solving problems in science and engineering. Algorithm design, analysis, and computer implementation will be discussed.

## ACMS 40485. Applied Complex Analysis

(3 -0- 3)
Complex analysis is a core part of applied and computational mathematics. Asymptotic methods for evaluation of functions and integrals, special functions (Gamma, elliptic, Bessel, ...), and conformal mappings arise naturally in applications, e.g., in the solution of physical models from electromagnetism, optics, tumor growth, fluid flow... In this course, an introduction to complex analysis will be given with special regard to those topics occurring in modeling and computation.

## ACMS 40570. Mathematical Methods in Financial Economics

(3-0-3)
Prerequisites: (ACMS/MATH 30530) AND (MATH 20750 OR MATH 30650 OR ACMS 20750) AND (MATH 30750 OR MATH 30850) OR (FIN 30600) OR (FIN 70670). Cross-listed with MATH 40570. An introduction to financial economic problems using mathematical methods, including the portfolio decision of an investor and the determination of the equilibrium price of stocks in both discrete and continuous time, will be discussed. The pricing of derivative securities in continuous time, including various stock and interest rate options, will also be included. Projects reflecting students’ interests and background are an integral part of this course.

## ACMS 40630. Nonlinear Dynamical Systems

(3-0-3)

Theory of nonlinear dynamical systems has applications to a wide variety of fields, from physics, biology, and chemistry, to engineering, economics, and medicine. This is one of its most exciting aspects - that it brings researchers from many disciplines together with a common language. A dynamical system consists of an abstract phase space or state space, whose coordinates describe the dynamical state at any instant; and a dynamical rule which specifies the immediate future trend of all state variables, given only the present values of those same state variables. Dynamical systems are "deterministic" if there is a unique consequent to every state, and "stochastic" or "random" if there is more than one consequent chosen from some probability distribution. A dynamical system can have discrete or continuous time. The discrete case is defined by a map and the continuous case is defined by a "flow. Nonlinear dynamical systems have been shown to exhibit surprising and complex effects that would never be anticipated by a scientist trained only in linear techniques. Prominent examples of these include bifurcation, chaos, and solitons. This course will be self-contained.

## ACMS 40730. Mathematical and Computational Modeling

(3-0-3)
Prerequisites: MATH 20750 or MATH 30650 or ACMS 20750. Introductory course on applied mathematics methods with emphasis on modeling of biological problems in terms of differential equations and stochastic dynamical systems. Students will be working in groups on several projects and will present them in class at the end of the course.

## ACMS 40740. Mathematical and Computational Modeling in Neuroscience

(3-0-3)

This course will introduce students to some of the most common computational and mathematical models used in neuroscience. The models will progress gradually from the scale of ion channels to the systems scale. In addition to developing a deeper understanding of the biological processes modeled in the course, the students will learn numerical and analytical approaches to studying dynamical systems and random processes. The course is appropriate as an elective for Neuroscience or ACMS majors. After completing the course, students will be able to create models of neural systems, simulate these models in MATLAB and use basic techniques from dynamical systems theory and probability theory to analyze these models. Students are expected to have some exposure to ordinary differential equations and probability before taking the course.

## ACMS 40750. Partial Differential Equations

(3-0-3)
Prerequisites: MATH 20750 or MATH 30650 or MATH 30850 or ACMS 20750. An introduction to partial differential equations. Topics include Fourier series, solutions of boundary value problems for the heat equation, wave equation and Laplace’s equation, Fourier transforms, and applications to solving heat, wave, and Laplace’s equations in unbounded domains.

## ACMS 40760. Introduction to Stochastic Modeling.

(3-0-3)

Stochastic modeling is a technique of presenting data or predicting outcomes that takes into account a certain degree of randomness, or unpredictability. Topics include (i) Short Review of Probability - Major discrete and continuous distributions, properties of random variables. (ii) Conditional probability and conditional expectation, sums of random variables, martingales. (iii) Introduction to Discrete Markov Chains - Transition probability matrix of a Markov chain, some Markov chain models, first step analysis, the absorbing Markov chains, various types and classifications of Markov chains. (iv) Long Run (asymptotic) Behavior of Markov Chains: Limiting distribution, the classification of states, irreducible Markov chains, periodicity of Markov chains, recurrent and transient states, the basic limit theorem of Markov chains. (v) Poisson Processes - The Poisson distribution and the Poisson process, the law of rare events, distributions associated with the Poisson process, the Uniform distribution and Poisson processes. (vi) Continuous Time Markov Chains - Pure birth and death processes and it's limiting behavior. (vii) Introduction to Brownian Motion, Drift and Diffusion, Geometric Brownian motion, Ornstein-Uhlenbeck process and its long run behavior. (viii) Monte Carlo Simulations for Diffusion.

## ACMS 40790. Topics in Applied Mathematics

(3-0-3)
Topics in Applied and Computational Mathematics.

## ACMS 40842. Time Series Analysis

(3-0-3)
This is an introductory and applied course in time series analysis. Popular time series models and computational techniques for model estimation, diagnostics and forecasting will be discussed. Although the book focuses on financial data sets, other data sets such as climate data, earthquake data, and biological data will also be included and discussed within the same theoretical framework.

## ACMS 40852. Statistical Methods in the Biological and Health Sciences

(3-0-3)

This course surveys the statistical methods used in biological and biomedical research. Topics include study designs commonly used in health research including case-control, cross-sectional, prospective and retrospective studies; statistical analysis of different types of data arising from biological and health research including categorical data analysis, count data analysis, survival analysis, linear mixed models, lab data, and diagnostic tests. Design and analysis of clinical trials, relative risk assessment, statistical power and sample size calculations will also be covered by the class. Additional topics of introduction to statistical genetics and bioinformatics might also be covered.

## ACMS 40860. Statistical Methods in Molecular Biology

(3-0-3)

Prerequisite: ACMS 30600. This is an introductory and applied course in statistical genetics and bioinformatics. Problems and statistical techniques in various fields of genetics, genomics and bioinformatics will be discussed. Since knowledge in these areas is evolving rapidly, novel and prevailing methods, such as next generation sequencing data analysis and network models, will also be introduced. Moreover, guest lectures may be given by visiting speakers.

## ACMS 40870. Statistical Methods in Social Sciences

(3-0-3)
Prerequisite: ACMS 30600. This is an introductory and applied course in the statistical methods used in social science research.

## ACMS 40875. Statistical Methods in Data Mining and Prediction

(3-0-3)

Data mining is widely used to discover useful patterns and relationships in data. We will emphasize on large complex datasets such as those in very large databases or web-based mining. The topics will include data visualization, decision trees, association rules, clustering, case based methods, etc.

## ACMS 40880. Statistical Methods in Pattern Recognition and Prediction

(3-0-3)
Prerequisite: ACMS 30600. Statistical theories and computational techniques for extracting information from large data sets. Building and testing predictive models.

## ACMS 40890. Statistical Methods for Financial Risk Management

(3-0-3)

This course is an introduction to some of the models and statistical methodology used in the practice of managing market risk for portfolios of financial assets. Throughout the course, the emphasis will be on the so-called loss distribution approach, a mapping from the individual asset returns to portfolio losses. Methodology presented will include both univariate and multivariate statistical modeling, Monte Carlo simulation, and statistical inference. This course will make heavy use of the R statistical computing environment.

## ACMS 40900. Topics in Applied and Computational Mathematics and Methods

(3 -0- 3)
This course will include the study of topics related to the instructor’s research interests in applied and computational mathematics and methods.

## ACMS 40950. Selected Topics in Statistics

(3-0-3)

Selected advanced topics in Statistics. Possible topics include, but are not limited to, applied logistic and ordinal regression modeling including fitting, building, and interpreting regression models for binary and ordinal response variables, various modeling strategies addressing different sampling and experimental designs such as case-control studies and longitudinal data, advanced experimental designs, survey research, big data analysis, Bayesian analysis, survival analysis, spatial and longitudinal analysis, commonly-used nonparametric statistics, basics of robust statistics, tests of association in contingency tables, permutation tests, the bootstrap, introduction to data mining techniques, etc. Applications in a variety of fields such as medical biology, psychology, global health, psychiatry, etc will be introduced. The topic of the course could vary from one semester to another depending on the interests of the faculty member and the students. The course could potentially involve a student project in the area of the interests of the faculty member and could change from one semester to another. The course will count for science credit, ACMS elective credit as well as STAT major elective credit.

(V -0- V)
Readings not covered in the curriculum which relate to the student’s area of interest.

(V -0- V)
Research in collaboration with members of the faculty. Evaluation of performance will be accomplished through regular discussions with the faculty member in charge of the course.

To produce a thesis that describes work of an undergraduate research project. The undergraduate thesis must go beyond what is found in an undergraduate course, and present a novel approach to a subject.

## ACMS 50051. Numerical PDE Techniques for Scientists and Engineers

(3 -0- 3)
Prerequisite: MATH 20670 or MATH 20750 or MATH 20860 or MATH 30650 or PHYS 20452

Partial Differential Equations (PDEs) are ubiquitous in science and engineering and are usually discussed in classes as analytic solutions for specialized cases. This course will teach the students the basic methods for their numerical solution. The course starts with an overview of PDEs, then moves on to discuss finite difference approximations. Hyperbolic systems are introduced by the scalar advection and scalar non-linear conservation laws, followed by the Riemann problem for hyperbolic systems and approximate Riemann solvers. Multidimensional schemes for non-linear hyperbolic systems are then presented. Elliptic and parabolic systems and their solution methodologies are then discussed including Krylov subspace methods and Multigrid methods. The course explains the theory underlying the numerical solution of PDEs and also provides hands-on experience with computer codes. A recommended prerequisite for this course is programming courses or a programming background.

## ACMS 50052. Numerical PDE Techniques for Scientists and Engineers II

(3 -0- 3)
Partial Differential Equations (PDEs) are ubiquitous in science and engineering and are usually discussed in courses as analytic solutions for specialized cases. In PHYS 50051, students saw an overview of PDEs and were taught basic methods for their solution. Emphasis was on the theory underlying the numerical solution of PDEs and providing hands-on experience with computer codes. For background, students were expected to have some computer literacy at the level of familiarity with the Linux operating system and Fortran. The text was the first half of a book in development, written by the instructor. This second course will cover the second half of the text, with topics ranging from stiff source terms in hyperbolic PDEs to multigrid methods and Krylov subspace methods for elliptic and parabolic PDEs and adaptive mesh refinement. Interested advanced undergraduates and graduate students from applied mathematics, engineering, and the sciences may take the second course without having taken the first one.

## ACMS 50550. Functional Analysis

(3 -0- 3)
This one semester course will cover selected topics in Functional Analysis. The theory will be built on Banach and Hilbert spaces and will be applied to selected examples from application including Laplace equations, heat equations, and wave equations. Tools and methods such as fixed point theorems, Dirichlet principle, Semi-group, etc. will be covered in the course.

## ACMS 50730. Mathematical and Computational Modeling in Biology and Physics

(3 -0- 3)
Introductory course on applied mathematics and computational modeling with emphasis on modeling of biological problems in terms of differential equations and stochastic dynamical systems. Students will be working in groups on several projects and will present them in class in the end of the course.

## ACMS 50850. Numerical PDE Techniques for Scientists and Engineers

(3-0-3)

Partial Differential Equations (PDEs) are ubiquitous in science and engineering and are usually discussed in classes as analytic solutions for specialized cases. This course will teach the students the basic methods for their numerical solution. The course starts with an overview of PDEs, then moves on to discuss finite difference approximations. Hyperbolic systems are introduced by the scalar advection and scalar non-linear conservation laws, followed by the Riemann problem for hyperbolic systems and approximate Riemann solvers. Multidimensional schemes for non-linear hyperbolic systems are then presented. Elliptic and parabolic systems and their solution methodologies are then discussed including Krylov subspace methods and Multigrid methods. The course explains the theory underlying the numerical solution of PDEs and also provides hands-on experience with computer codes. A recommended prerequisite for this course is programming courses or a programming background.