# Graduate Course Descriptions

- IPFW.edu
- College of Arts and Sciences
- Department of Mathematical Sciences
- Course Information
- Graduate Course Descriptions

MA 51000 - Vector Calculus

MA 51100 - Linear Algebra with Applications

MA 52100 - Introduction to Optimization Problems

MA 52300 - Introduction To Partial Differential Equations

MA 52500 - Introduction to Complex Analysis

MA 54000 - Analysis I

MA 54100 - Analysis II

MA 55300 - Introduction to Abstract Algebra

MA 55400 - Linear Algebra

MA 55600 - Introduction to the Theory of Numbers

MA 56000 - Fundamental Concepts of Geometry

MA 57100 - Elementary Topology

MA 57500 - Graph Theory

MA 58000 - History of Mathematics

MA 58100 - Introduction to Logic for Teachers

MA 59800 - Topics in Mathematics

STAT 51200 - Applied Regression Analysis

STAT 51400 - Design of Experiments

STAT 51600 - Basic Probability and Applications

STAT 51700 - Statistical Inference

STAT 51900 - Introduction to Probability

STAT 52000 - Time Series And Applications

STAT 52800 - Introduction to Mathematical Statistics

### MA 51000 - Vector Calculus 3 cr.

Calculus of functions of several variables and of vector fields in orthogonal coordinate systems; optimization problems; the implicit function theorem; Green’s, Stokes’, and the Divergence theorems; applications to engineering and the physical sciences.

Cannot be used toward a graduate degree or graduate certificate in Mathematical Sciences.

Prerequisite: MA 26100 or 26300

### MA 51100 - Linear Algebra with Applications 3 cr.

Real and complex vector spaces; linear transformations; Gram-Schmidt process and projections; least squares; QR and LU factorization; diagonalization, real and complex spectral theorem; Schur triangular form; Jordan canonical form; quadratic forms.

Prerequisite: MA 35100

### MA 52100 - Introduction to Optimization Problems 3 cr.

Necessary and sufficient conditions for local extrema in programming problems and in the calculus of variations. Control problems, statement of maximum principles, and applications. Discrete control problems.

Prerequisite: MA 51000, and 35100 or 51100

### MA 52300 - Introduction To Partial Differential Equations 3 cr.

First order quasi-linear equations and their application to physical and social sciences; the Cauchy-Kovalevsky theorem; characteristics, classification, and canonical form of linear equations; equations of mathematical physics; study of the Laplace, wave, and heat equations; methods of solution.

Prerequisite: MA 26100 or 26300, and 36300

### MA 52500 - Introduction to Complex Analysis 3 cr.

Complex numbers and complex-valued functions of one variable; differentiation and contour integration; Cauchy’s theorem; Taylor and Laurent series; residues; conformal mapping; applications.

Prerequisite: MA 26300 or 44100 or 51000

### MA 54000 - Analysis I 3 cr.

Metric spaces, compactness and connectedness, sequences and series, continuity and uniform continuity, differentiability, Taylor’s Theorem, Riemann-Stieltjes integrals.

Prerequisite: MA 44100

### MA 54100 - Analysis II 3 cr.

Sequences and series of functions, uniform convergence, equicontinuous families, the Stone-Weierstrass Theorem, Fourier series, introduction to Lebesgue measure and integration.

Prerequisite: MA 54000

### MA 55300 - Introduction to Abstract Algebra 3 cr.

Group theory: Sylow theorems, Jordan-Holder theorem, solvable groups. Ring theory: unique factorization in polynomial rings and principal ideal domains. Field theory: straightedge and compass constructions, roots of unity, finite fields, Galois theory, and solubility of equations by radicals.

Prerequisite: MA 45300

### MA 55400 - Linear Algebra 3 cr.

Review of basics: vector spaces, dimension, linear maps, matrices, determinants, linear equations. Bilinear forms; inner product spaces; spectral theory; eigenvalues. Modules over a principal ideal domain; finitely generated abelian groups; Jordan and rational canonical forms for a linear transformation.

Prerequisite: MA 45300

### MA 55600 - Introduction to the Theory of Numbers 3 cr.

Divisibility, congruences, quadratic residues, Diophantine equations, and the sequence of primes.

Prerequisite: MA 26300 or 26100

### MA 56000 - Fundamental Concepts of Geometry 3 cr.

Foundations of Euclidean geometry, including a critique of Euclid’s Elements and a detailed study of an axiom system such as that of Hilbert. Independence of the parallel axiom and introduction to non-Euclidean geometry.

Prerequisite: MA 30500

### MA 57100 - Elementary Topology 3 cr.

Fundamentals of point-set topology with a brief introduction to the fundamental group and related topics; topological and metric spaces; compactness and connectedness; separation properties; local compactness; introduction to function spaces; basic notions involving deformations of continuous paths.

Prerequisite: MA 44100

### MA 57500 - Graph Theory 3 cr.

Introduction to graph theory with applications.

Prerequisite: MA 30500 or 35100

### MA 58000 - History of Mathematics 3 cr.

The origins of mathematical ideas and their evolution over time, from early number systems and the evolution of algebra, geometry, and calculus to 20th-century results in the foundations of mathematics. Connections between mathematics and society, including the role of applications in the development of mathematical concepts.

Prerequisite: Two semesters of calculus and MA 30500

### MA 58100 - Introduction to Logic for Teachers 3 cr.

Sentential and general theory of inference and nature of proof, elementary axiom systems.

Prerequisite: MA 35100

### MA 59800 - Topics in Mathematics 1-5 cr.

Supervised reading courses as well as dual-level special topics courses are given under this number.

### STAT 51200 - Applied Regression Analysis 3 cr.

Inference in simple and multiple linear regression, residual analysis, transformations, polynomial regression, model building with real data, nonlinear regression. One-way and two-way analysis of variance, multiple comparisons, fixed and random factors, analysis of covariance. Use of existing statistical computer programs.

Prerequisite: STAT 51100 or STAT 51700 or STAT 52800 with a grade of C- or higher.

### STAT 51400 - Design of Experiments 3 cr.

Fundamentals, completely randomized design; randomized complete blocks; latin square; multi-classification; factorial; nested factorial; incomplete block and fractional replications for 2n, 3n, 2m x 3n; confounding; lattice designs; general mixed factorials; split plot; analysis of variance in regression models; optimum design. Use of existing statistical programs.

Prerequisite: STAT 51200 with a grade of C- or higher.

### STAT 51600 - Basic Probability and Applications 3 cr.

A first course in probability intended to serve as a background for statistics and other applications. Sample spaces and axioms of probability, discrete and continuous random variables, conditional probability and Bayes’ theorem, joint and conditional probability distributions, expectations, moments and moment generating functions, law of large numbers and central limit theorem. (The probability material in Course 1 of the Society of Actuaries and the Casualty Actuarial Society is covered by this course.)

Cannot be used toward a graduate degree or graduate certificate in Mathematical Sciences

Prerequisite: MA 26100 or MA 26300 with a grade of C- or higher.

### STAT 51700 - Statistical Inference 3 cr.

A basic course in statistical theory covering standard statistical methods and their application. Estimation including unbiased, maximum likelihood and moment estimation; testing hypotheses for standard distributions and contingency tables; confidence intervals and regions; introduction to nonparametric tests and linear regression.

Cannot be used toward a graduate degree or graduate certificate in Mathematical Sciences

Prerequisite: STAT 51600 with a grade of C- or higher.

### STAT 51900 - Introduction to Probability 3 cr.

Algebra of sets, sample spaces, combinatorial problems, independence, random variables, distribution functions, moment generating functions, special continuous and discrete distributions, distribution of a function of a random variable, limit theorems.

Prerequisite: MA 26100 or MA 26300 with a grade of C- or higher.

### STAT 52000 - Time Series And Applications 3 cr.

A first course in stationary time series with applications in engineering, economics, and physical sciences. Stationarity, autocovariance function and spectrum; integral representation of a stationary time series and interpretation; linear filtering, transfer functions; estimation of spectrum; multivariate time series. Use of computer programs for covariance and spectral estimation.

Prerequisite: STAT 51200 with a grade of C- or higher.

### STAT 52800 - Introduction to Mathematical Statistics 3 cr.

Distribution of mean and variance in normal samples, sampling distributions derived from the normal distribution, Chi square, t and F. Distribution of statistics based on ordered samples. Asymptotic sampling distributions. Introduction to multivariate normal distribution and linear models. Sufficient statistics, maximum likelihood, least squares, linear estimation, other methods of point estimation, and discussion of their properties. Cramer-Rao inequality and Rao-Blackwell theorem. Tests of statistical hypotheses, simple and composite hypotheses, likelihood ratio tests, power of tests.

Prerequisite: STAT 51900 with a grade of C- or higher.