## M.A/M.Sc. (Mathematics)

Teaching Pattern for Semester I, II, III and IV

1. Four lectures per week per course. Each lecture is of 60 minutes duration.
2. In addition, there shall be tutorials, seminars as necessary for each of the five courses.

## PSMT401 / PAMT401: ALGEBRA IV

Course Outcomes:

1. Students will learn about algebraic extensions and their properties.
2. Splitting fields and their degrees can be computed. The notion of normal extension is introduced and its equivalent properties are discussed.
3. Finite fields as splitting fields are visualized and notion of algebraic closure is discussed in detail.
4. Galois extensions are studied and the fundamental theorem of Galois theory is established.
5. Cyclotomic extensions are studied in detail and order of its Galois group is computed.
6. Examples of fixed fields, field automorphisms and fundamental theorem are studied in special cases.

## PSMT402 / PAMT402: Fourier Analysis

Course Outcomes:

1. Students will be able to understand the Fourier series expansion of a periodic function and their convergence.
2. Students will be able to grasp properties of the Dirichlet kernel, Fejer kernel, Poisson kernel and the concept of a good kernel.
3. Students will aware about application of a Fourier series in the solution of the Dirichlet problem and heat equation.

## PSMT 403/PAMT 403: Calculus on Manifolds

Course Outcomes:

1. Students will be able to grasp the concept of tensor, alternating tensor, wedge product and differential forms.
2. Students will be able to understand fields and forms on manifolds.
3. Students will be able to understand the application of Classical theorems: Stoke’s theorem, Green’s theorem, Gauss divergence theorem.

## Skill Course I: Business Statistics

Course Outcomes: Students should know the following:

1. Classification of Data: Requisites of Ideal Classification, Basis of Classification.
2. Organizing Data Using Data Array: Frequency Distribution, Methods of Data Classification, Bivariate Frequency Distribution, Types of Frequency Distributions.
3. Tabulation of Data: Objectives of Tabulation, Parts of a Table, Types of Tables, General and Summary Tables, Original and Derived Tables.
4. Graphical Presentation of Data: Functions of a Graph, Advantages and Limitations of Diagrams(Graph), General Rules for Drawing Diagrams.
5. Types of Diagrams: One-Dimensional Diagrams, Two-Dimensional Diagrams, Three- Dimensional Diagrams, Pictograms or Ideographs, Cartograms or Statistical Maps.
6. Exploratory Data Analysis: Stem-and-Leaf Displays.
7. Objectives of Averaging, Requisites of a Measure of Central Tendency, Measures of Central Tendency.
8. Mathematical Averages: Arithmetic Mean of Ungrouped Data, Arithmetic Mean of Grouped (Or Classified) Data, Some Special Types of Problems and Their Solutions, Advantages and Disadvantages of Arithmetic Mean, Weighted Arithmetic Mean.
9. Geometric Mean: Combined Geometric Mean, Weighted Geometric Mean, Advan- tages, Disadvantages and Applications of G.m.
10. Harmonic Mean: Advantages, Disadvantages and Applications of H.M. Relationship Between A.M., G.M. and H.M.
12. Partition Values quartiles, Deciles and Percentiles: Graphical Method for Calculating Partition Values.
13. Mode: Graphical Method for Calculating Mode Value. Advantages and Disadvantages of Mode Value.
14. Relationship Between Mean, Median and Mode, Comparison Between Measures of Central Tendency.
15. Significance of Measuring Dispersion (Variation):Essential Requisites for a Measure of Variation.
16. Classification of Measures of Dispersion.
17. Distance Measures: Range, Interquartile Range or Deviation.
18. Average Deviation Measures: Mean Absolute Deviation, Variance and Standard Deviation.
19. Mathematical Properties of Standard Deviation, Chebyshev’s Theorem, Coefficient of Variation.
20. Measures of Skewness: Relative Measures of Skewness.
21. Moments: Moments About Mean, Moments About Arbitrary Point, Moments About Zero or Origin, Relationship Between Central Moments and Moments About Any Arbitrary Point, Moments in Standard Units, Sheppard’s Corrections for Moments.
22. Kurtosis: Measures of Kurtosis.

## Skill Course II: Statistical Methods

Course Outcomes:Students should know the following:

1. Measures of central tendencies: Mean, Median, Mode.
2. Measures of Dispersion: Range, Mean deviation, Standard deviation. Measures of skewness. Measures of relationship: Covariance, Karl Pearson’s coefficient of Correlation, Rank Correlation. Basics of Probability.
3. Sampling Distribution, Student’s t-Distribution, Chi-square (χ2) Distribution, Snedecor’s F-Distribution. Standard Error. Central Limit theorem. Type I and Type II Errors, Critical Regions. F-test, t-test, χ2 test, goodness of Fit test.
4. The Anova Technique. The basic Principle of Anova. One Way ANOVA, Two Way ANOVA. Latin square design. Analysis of Co-variance.
5. R as Statistical software and language, methods of Data input, Data accessing, usefull built-in functions, Graphics with R, Saving, storing and retrieving work.

## Skill Course III: Computer Science

Course Outcomes:Students should know the following:

1. Basics of object oriented programming principles, templates, reference operators NEW and delete in C++, the java innovation which avoids use of delete, classes polymorphism friend functions, inheritance, multiple inheritance operator overloading basics
2. Basic algorithms, selection sort, quick sort, heap sort, priory queses, radix sort, merge sort, dynamic programming, app pairs, shortest paths, image compression, topological sorting, single source shortest paths reference, hashing intuitive evaluation of running time.
3. Stacks queues, linked lists implementation and simple applications, trees implementa- tion and tree traversal ( stress on binary trees).
4. Concept of relational databases , normal forms BCNF and third normal forms. Arm- strongs axioms. Relational algebra and operations in it.

## Skill Course IV: Linear and Non-linear Programming

Course Outcomes:

1. Understand the concept of an objective function, a feasible region, and a solution set of an optimization problem.
2. Understand the broad classification of optimization problems, and where they arise in simple applications.
3. Use the simplex method to find an optimal vector for the standard linear program- ming problem and the corresponding dual problem.
4. Use Lagrange multipliers to solve nonlinear optimization problems.
5. Write down and apply Kuhn-Tucker conditions for constrained nonlinear optimiza- tion problems.
6. Apply approximate methods for constraint problems.
7. Understand the importance of convexity in nonlinear optimization problems.
8. Apply basic line search methods to one-dimensional optimization problems, gradient methods to optimization problems, ,conjugate gradient methods to optimization problems.

## Skill Course V: Computational Algebra

Course Outcomes:

1. Students will learn to balance between theory and practicals via the use of computers.
2. Previously learnt concepts can be strengthened by allowing them to explore topics using the mathematical softwares.