# Course Descriptions

Below are the descriptions of each class typically included in the Masters of Science of Mathematics. As always, the official descriptions are in the Hardin-Simmons University Graduate Catalog.

### The First Summer

In the first summer, the program builds the graduate level foundation of Introduction of Topology and Advanced Topics in Calculus I. In these courses, students review and build on their abilities to write mathematical proofs and their knowledge of calculus.

### The Second School Year

In the fall, we will continue our study with Advanced Topics of Calculus II. Then in the spring, we turn our attention to Linear Algebra. After this first year foundation, we will be ready to study the three major foci of the program

### Three Major Focuses of the Program

**Real Analysis I and II:**
Real
Analysis is the typical foundation of advanced mathematics found in most
graduate programs across the United States.

**Mathematical Modeling with
Differential Equations I and II: **Much of mathematics was developed to solve a specific
problem, which is the focus of applied
mathematics. To apply mathematics
correctly, one must understand and develop the necessary theory. The goal
is not the development of mathematics, but the correct and appropriate
applications of mathematics.

**Mathematical Statistics I
and II:
**
To teach or use statistics, a strong understanding of the
theory of statistics is required. Mathematical
Statistics focuses on the mathematical foundations of statistics typically
taught at the undergraduate level.

**MATH 5301: Introduction to Topology**

**Catalog Description:**

MATH 5301 Introduction to Topology: Topological spaces, continuous functions, metric spaces, connectedness, compactness, separation axioms, fundamental group, covering spaces, metrication theorems. The focus of the course is on proving theorems.

**Goals of the Course:**

Introduction to Topology covers the mathematical background for Real Analysis I and II. By first focusing on a review of set theory and the structure of the real number line, the course provides the student with a mathematical foundation for understanding the proofs of Real Analysis. The course then extends this foundation in additional topics of topology. The course is focused on proofs, refreshing the student’s ability to read, understand, create and write proofs using a correct structure and mathematical language.

**MATH 5302: Topics Advanced Topics in Calculus I**

MATH 5303: Topics Advanced Topics in Calculus II

MATH 5303: Topics Advanced Topics in Calculus II

**Catalog Description**

MATH 5302 Topics Advanced Topics in Calculus I: Sequences and series of functions, multiple integrals, improper multiple integrals, functions of several variables, extreme value problems, and implicit function theorems. The focus of this course will be the proofs of calculus and advanced computational skills.

**MATH 5303 Topics Advanced Topics in Calculus II: A continuation of MATH 5302.**

**Goals of the Courses:**To teach calculus effectively at the college level+, an instructor should know the theorems and fundamental ideas of calculus. Topics in Advanced Calculus I and II extend the mathematical knowledge of calculus beyond that typically found in an introductory calculus text. Students will master the key theorems of calculus. Computational skills of multivariable calculus which many instructors find difficult will be reviewed and mastered by the student. The course will focus on the proofs of calculus while ensuring strong computational skills needed to do the ‘hard’ problems in calculus.

**MATH 5103: History of Mathematics I**

MATH 5104: History of Mathematics II

MATH 5104: History of Mathematics II

**Catalog Description**

MATH 5103: The historical development of mathematics in a cultural context. Emphasis will be on the developments of algebra and analytical geometry up to the discovery of calculus. MATH 5104: The historical development of mathematics in a cultural context. Emphasis will be on the developments of differential and integral calculus.

**Goals of the Courses**

Knowing the history of mathematics, doing problems in their original context and notation, leads to deeper appreciation of the beauty, power, and effectiveness of today’s mathematics. This course is primarily a one unit reading course with evaluation based on discussion and assignments.

**MATH 6305: Linear Algebra**

**Catalog Description:**

MATH 6305: Linear Algebra: Matrix algebra, eigenvalues and eigenvectors, canonical forms, orthogonal and unitary transformations, and quadratic forms. Proofs and applications of these concepts

**Goals of the Course:**

The theory and power of matrices and the theory of linear functionals have all the structure found in algebra. A strong understanding of Linear Algebra will increase the teacher’s abilities in multiple dimension calculus, Mathematical Statistics, Real Analysis, and Mathematical Modeling.

**MATH 6310: Mathematical Statistics I**

MATH 6312: Mathematical Statistics II

MATH 6312: Mathematical Statistics II

**Catalog Descriptions:**

MATH 6310: Mathematical Statistics I: Random Variables, Probability, Sampling Distributions, Limiting Distributions, Point Estimation, Interval Estimation. MATH 6312: Mathematical Statistics II: A continuation of MATH 6310. Hypothesis Testing, General Linear Models, Analysis of Variance, Empirical Methods

**Goals of the Courses:**

Statistics is increasingly important in today’s society. An instructor should know the theoretical background of statistics, and know the importance of satisfying the hypothesis before applying a statistical method.

**MATH 6320, Mathematical Modeling with Differential Equations I**

MATH 6321, Mathematical Modeling with Differential Equations II

MATH 6321, Mathematical Modeling with Differential Equations II

**Catalog Description**

MATH 6320: The course emphasizes applications of modeling and analysis for deterministic and stochastic biological systems. Analysis includes equilibria, stability, phase-plane methods, limit-cycles, bifurcations, separation of variables. Applications include but are not limited to continuous and discrete-time models for population growth, epidemiology, predator-prey, age-structure, Markov chains, and others. MATH 6321: A continuation of Mathematical Modeling with Differential Equations I.

**Goals of the Courses:**

Applying mathematics requires understanding, developing a strong understanding of the relevant theory and then using this mathematical knowledge to solve significant problems.This course uses theory of differential equations.

**MATH 6340, Real Analysis I**

MATH 6341, Real Analysis II

MATH 6341, Real Analysis II

**Catalog Descriptions:**

MATH 6340: Borel sets, measure and measurable sets, measurable functions, and the Lebesgue integral.

MATH 6341: Real Analysis II is a continuation of Real Analysis I. Function spaces, abstract measure, and differentiation.

**Goals of the Courses:**

Real Analysis is considered an extension of continuous mathematics beyond Calculus I, II and II and beyond Differential Equations (undergraduate level).Real Analysis is, in some sense, what comes after calculus and the related theory. One specific example: Real Analysis extends the Riemann integral found in Calculus I to the more general Lebesgue integration.

**MATH 6150: Comprehensive Review and Exam**

**Catalog Description:**

MATH 6150: Summary review of topics from MATH 6310, 6312, 6320, 6321, 6340, and 6341 with a comprehensive exam.

**(Official
Description is in the Graduate Catalog for the year you are admitted into the
program.)*