Mathematics finds applications in some of the most diverse and interdisciplinary research in the physical, engineering, and biological sciences. The Applied Mathematics program at FLAME is designed to expose students to those fields of Mathematics that are fundamental to the Modeling and Simulation of phenomena straddling diverse disciplines of the Physical and Natural sciences, the social sciences, and in business. The programs lays emphasis on taking a practical problem and turn it into a mathematical problem, also known as Mathematical Modeling, and to use analytical and computational techniques that help in the solution of such mathematical problems.

The Applied Mathematics major enables students to become adept in the use of mathematical techniques to solve problems in diverse fields where mathematics has a role to play. A sequence of courses will serve to provide the background for students’ understanding of the essential principles of mathematics. It will then introduce them to advanced methods and techniques that will provide the necessary skills for solving problems in diverse areas.

At the introductory level the program lays the foundation for study of subsequent higher level courses. At the intermediate level, a strong theoretical foundation is set by introducing subjects that form the backbone of Applied Mathematics subjects while introducing the students to some of the applied fields. At the advanced level students are introduced to advanced computational techniques.

The program prepares the student for graduate level courses with specialization in fields such as Applied Mathematics, Economics, Finance, Statistics, Optimization, and Environment sciences among others. Equipped with modeling and computational skills, a student can pursue a career in research not only in the conventional Applied Mathematics areas but also in any area that requires Mathematical modeling and computational proficiency such as Modeling in Biology, Ecology, Social Sciences, Actuarial Sciences and Finance.

#### Program Aims

- To provide the necessary mathematical skills required to solve problems of practical interest
- To provide an understanding of the process undertaken to arrive at a suitable mathematical model
- To teach the fundamental analytical techniques and computational methods used to develop solutions to mathematical models.
- To expose students to a range of problems from diverse areas along with their associated conceptual models, and the appropriate methods employed to solve them.

#### 17 MAJOR COURSES

1. Introduction to Programming |
7. Fundamental of Data Structure |
13. Mathematical Modelling |

2. Introduction to Discrete Mathematics |
8. Intermediate Multivariate Calculus |
14. Complex Analysis |

3. Linear Algebra |
9. Ordinary Differential Equations |
15. Applied Probability |

4. Calculus of One Variable |
10. Abstract Algebra |
16. Numerical Methods |

5. Introduction to Probability and Statistics |
11. Design and Analysis of Algorithms |
17. Applied Multivariate Statistics |

6. Introduction to Real Analysis |
12. Partial Differential Equations |

#### 1. Introduction to Programming

This is a first course in computer programming for those with little or no previous programming experience. It equips the student with basic tools to efficiently solve problems on the computer. Programming places considerable emphasis on algorithms and requires you to specify every step of the solution process. By doing so, it hones your analytical and logical skills and in the end makes you a better problem solver.

#### 2. Introduction to Discrete Mathematics

This course aims to cover the basics of discrete mathematics. Discrete mathematics is the sturdy of discrete mathematical structures which do not rely on the notion of continuity. It introduces fundamental mathematical structures and various proof techniques and methods for solving different kind of problems. This course prepares the student to do advanced courses in applied mathematics and computer science.

#### 3. Linear Algebra

This course emphasizes matrix and vector calculations and applications. It delves deeply into the theory of Matrices and other algebraic constructs such as Vector spaces, Determinants and Linear Transformations with particular emphasis on understanding the underlying theory and develops the analytical skills to prove theorems.

#### 4. Calculus of One Variable

Calculus forms the foundation for a variety of subjects and finds applications in fields like Physics, Engineering, Economics, and Finance among others. In this course students will learn the concepts and techniques of single variable Differential and Integral Calculus

#### 5. Introduction to Probability and Statistics

This course provides an elementary introduction to probability theory and its application to statistics with emphasis on the theorems and proofs of univariate statistics. Addressed to a beginning Mathematics Major, it provides a foundation for advanced courses in probability and statistics.

#### 6. Introduction to Real Analysis

This is a first course in mathematical analysis. In this course a rigorous analysis of the real numbers, and provides training in the methods of mathematical proof. It is the first course in the analysis sequence and is followed by the introduction to Functional Analysis course.

#### 7. Fundamental of Data Structure

This course will introduce students to the common data structures and their applications. It introduces the concepts and techniques of structuring and operating on Abstract Data Types in problem solving. It will equip them with a range of approaches and established algorithms for solving common classes of problems. Common sorting and searching algorithms will be discussed, and the complexity and comparisons among these various techniques will be studied. The course takes a practical approach, focusing on coded examples and applications.

#### 8. Intermediate Multivariate Calculus

This course deals with functions and calculus of several variables. It follows the course on Single Variable calculus. Topics covered include geometry of 2 and 3 dimensions, Partial differentiation, scalar and vector fields and multiple integration. His course aims to provide students with working knowledge of functions in two or more variables, their partial derivatives, geometric interpretation of the derivatives, demonstrate the applications of these concepts in problems of finding extrema with or without constraints, and introduce the tools and techniques of evaluating Multiple Integrals.

#### 9. Ordinary Differential Equations

This course aims to introduce students to the basic theory of ordinary differential equations and the modelling of diverse practical phenomena by ordinary differential equations by a variety of examples. Students will learn both quantitative and qualitative methods for solving these equations.

#### 10. Abstract Algebra

Algebra is language of Mathematics. This course aims to introduce abstract objects such as Groups, Rings and Fields and its applications in other disciplines. This course covers some properties of groups, rings and fields. Permutation groups and polynomial rings are included also included.

#### 11. Design and Analysis of Algorithms

This course will introduce students to the common data structures and their applications. It introduces the concepts and techniques of structuring and operating on Abstract Data Types in problem solving. It will equip them with a range of approaches and established algorithms for solving common classes of problems. Common sorting and searching algorithms will be discussed, and the complexity and comparisons among these various techniques will be studied. The course takes a practical approach, focusing on coded examples and applications.

#### 12. Partial Differential Equations

Partial Differential Equations are differential equations with more than one independent variable. They arise naturally in the modelling of physical and natural phenomena such as waves, diffusion of heat and fluid flow. This course is an introduction to partial differential equations.

#### 13. Mathematical Modelling

This course equips the student with some of the methods and techniques of modelling continuous systems. It builds on knowledge gained through previous courses on Calculus and introduces new techniques for analysing and solving problems that arise in the application of mathematics in various disciplines.

#### 14. Complex Analysis

This course provides an introduction to the theory of function of a complex variable. Residue Theorem and its applications to the integrals and sums and also conformal mappings and their applications will be discussed. This course aims to introduce students to the principal techniques and methods of analytic function theory. This is quite different from real analysis and has much more geometric emphasis. It tries to show how complex analysis can be used to evaluate real integrals, investigate the location of roots of polynomial equations and also introduce students to some applications of complex analysis, for example in fluid flow

#### 15. Applied Probability

This course introduces probabilistic distributions and stochastic processes. It builds on knowledge acquired from elementary courses in probability and equips them to understand and apply advanced concepts to relatively more complex problems arising in diverse fields where uncertainty is a decisive factor.

#### 16. Numerical Methods

This course gives an introduction to the basic techniques for solving problems in science and engineering using numerical methods. It provides students with an understanding of the concepts and knowledge of the theory and practical application of numerical methods.

#### 17. Applied Multivariate Statistics

This course will introduce the theory and applications of multivariate statistical methods. It while emphasis will be on conceptual knowledge of the statistical tools and techniques used to analyse multivariate data, it also focusses on applying these techniques to real world using statistical packages. A background in calculus, probability and statistics is desirable.

#### 17 MINOR COURSES

1. Introduction to Programming |
7. Fundamental of Data Structure |
13. Mathematical Modelling |

2. Introduction to Discrete Mathematics |
8. Intermediate Multivariate Calculus |
14. Complex Analysis |

3. Linear Algebra |
9. Ordinary Differential Equations |
15. Applied Probability |

4. Calculus of One Variable |
10. Abstract Algebra |
16. Numerical Methods |

5. Introduction to Probability and Statistics |
11. Design and Analysis of Algorithms |
17. Applied Multivariate Statistics |

6. Introduction to Real Analysis |
12. Partial Differential Equations |

#### 1. Introduction to Programming

This is a first course in computer programming for those with little or no previous programming experience. It equips the student with basic tools to efficiently solve problems on the computer. Programming places considerable emphasis on algorithms and requires you to specify every step of the solution process. By doing so, it hones your analytical and logical skills and in the end makes you a better problem solver.

#### 2. Introduction to Discrete Mathematics

This course aims to cover the basics of discrete mathematics. Discrete mathematics is the sturdy of discrete mathematical structures which do not rely on the notion of continuity. It introduces fundamental mathematical structures and various proof techniques and methods for solving different kind of problems. This course prepares the student to do advanced courses in applied mathematics and computer science.

#### 3. Linear Algebra

This course emphasizes matrix and vector calculations and applications. It delves deeply into the theory of Matrices and other algebraic constructs such as Vector spaces, Determinants and Linear Transformations with particular emphasis on understanding the underlying theory and develops the analytical skills to prove theorems.

#### 4. Calculus of One Variable

Calculus forms the foundation for a variety of subjects and finds applications in fields like Physics, Engineering, Economics, and Finance among others. In this course students will learn the concepts and techniques of single variable Differential and Integral Calculus

#### 5. Introduction to Probability and Statistics

This course provides an elementary introduction to probability theory and its application to statistics with emphasis on the theorems and proofs of univariate statistics. Addressed to a beginning Mathematics Major, it provides a foundation for advanced courses in probability and statistics.

#### 6. Introduction to Real Analysis

This is a first course in mathematical analysis. In this course a rigorous analysis of the real numbers, and provides training in the methods of mathematical proof. It is the first course in the analysis sequence and is followed by the introduction to Functional Analysis course.

#### 7. Fundamental of Data Structure

This course will introduce students to the common data structures and their applications. It introduces the concepts and techniques of structuring and operating on Abstract Data Types in problem solving. It will equip them with a range of approaches and established algorithms for solving common classes of problems. Common sorting and searching algorithms will be discussed, and the complexity and comparisons among these various techniques will be studied. The course takes a practical approach, focusing on coded examples and applications.

#### 8. Intermediate Multivariate Calculus

This course deals with functions and calculus of several variables. It follows the course on Single Variable calculus. Topics covered include geometry of 2 and 3 dimensions, Partial differentiation, scalar and vector fields and multiple integration. His course aims to provide students with working knowledge of functions in two or more variables, their partial derivatives, geometric interpretation of the derivatives, demonstrate the applications of these concepts in problems of finding extrema with or without constraints, and introduce the tools and techniques of evaluating Multiple Integrals.

#### 9. Ordinary Differential Equations

This course aims to introduce students to the basic theory of ordinary differential equations and the modelling of diverse practical phenomena by ordinary differential equations by a variety of examples. Students will learn both quantitative and qualitative methods for solving these equations.

#### 10. Abstract Algebra

Algebra is language of Mathematics. This course aims to introduce abstract objects such as Groups, Rings and Fields and its applications in other disciplines. This course covers some properties of groups, rings and fields. Permutation groups and polynomial rings are included also included.

#### 11. Design and Analysis of Algorithms

This course explores the methods for the design of efficient and reliable algorithms. It introduces common techniques to decrease computational resources to find solutions to problems. It will also introduce the mathematical framework for evaluating the correctness, running time and space requirements of algorithms. It will review common data structures and their applications and introduce a wide range of approaches and established algorithms for solving common classes of problems. It will cover common programming paradigms like Divide and Conquer, Greedy algorithms, Dynamic Programming to solve a wide variety of problems. Some common Graph algorithms will also be covered.

#### 12. Partial Differential Equations

Partial Differential Equations are differential equations with more than one independent variable. They arise naturally in the modelling of physical and natural phenomena such as waves, diffusion of heat and fluid flow. This course is an introduction to partial differential equations.

#### 13. Mathematical Modelling

This course equips the student with some of the methods and techniques of modelling continuous systems. It builds on knowledge gained through previous courses on Calculus and introduces new techniques for analysing and solving problems that arise in the application of mathematics in various disciplines.

#### 14. Complex Analysis

This course provides an introduction to the theory of function of a complex variable. Residue Theorem and its applications to the integrals and sums and also conformal mappings and their applications will be discussed. This course aims to introduce students to the principal techniques and methods of analytic function theory. This is quite different from real analysis and has much more geometric emphasis. It tries to show how complex analysis can be used to evaluate real integrals, investigate the location of roots of polynomial equations and also introduce students to some applications of complex analysis, for example in fluid flow

#### 15. Applied Probability

This course introduces probabilistic distributions and stochastic processes. It builds on knowledge acquired from elementary courses in probability and equips them to understand and apply advanced concepts to relatively more complex problems arising in diverse fields where uncertainty is a decisive factor.

#### 16. Numerical Methods

This course gives an introduction to the basic techniques for solving problems in science and engineering using numerical methods. It provides students with an understanding of the concepts and knowledge of the theory and practical application of numerical methods.

#### 17. Applied Multivariate Statistics

This course will introduce the theory and applications of multivariate statistical methods. It while emphasis will be on conceptual knowledge of the statistical tools and techniques used to analyse multivariate data, it also focusses on applying these techniques to real world using statistical packages. A background in calculus, probability and statistics is desirable.