### Mathematics Course Description

#### Level III – Semester I

**MAT311β: Group Theory (30 lecture hrs + 15 tutorial hrs) -( Credit Value 2.5)**

Groups and subgroups: Groups, subgroups/normal subgroups, quotient groups, cyclic groups, Cayley diagrams. Permutations, cosets and direct products: groups of permutation, orbits, cycles and alternating groups, cosets and the theorem of Lagrange, direct product, finite groups. Homomorphism and factor groups: Homomorphism factor group, factor group computation and simple groups.

*Method of assessment: end of semester examination*

**MAT312β: Real Analysis-III (30 lecture hrs + 15 tutorial hrs) -Prerequisite MAT221β or MAT223β (Credit Value 2.5)**

Non Linear Analysis: Functions from Rn to Rm , Open balls and Open sites, limit points. Limit and continuity, The derivative of scalar field with respect to a vector. Directional derivatives and partial derivatives, Partial derivatives of higher order, Directional derivatives and continuity, The Total derivative, The gradient of Scalar field, A sufficient condition for differentiability, A chain rule for derivatives of scalar fields. Derivatives of vector fields, Differentiability implies continuity, The chain rule for derivatives of vector fields, Sufficient conditions for the equality of mixed partial derivatives. Derivatives of functions defined implicitly, Maximum, Minimum and saddle points, Extremes with constraints, Lagrange’s Multipliers, Double integrals.

*Method of assessment: end of semester examination*

**MAT313β: Mathematical Statistics-II (30 lecture hrs + 15 tutorial hrs) -( Credit Value 2.5) Op. for students following Mathematics, Prerequisite: MAT225β**

Point estimation: The method of moments, The method of Maximum Likelihood, Properties of point estimation: Unbiasedness, Efficiency, Consistency, Sufficiency, Minimal Sufficient Statistics, Exponential family, Cramer – Rao

Inequality, Completeness.

Interval Estimation: Confidence Interval for the mean and variance. Tests of Hypotheses: Simple Hypothesis, Composite Hypothesis, Critical Region, Types of Error, Power Function, Size of Test, Simple Likelihood-ratio Test, Most powerful Test, Neyman-Pearson lemma, Generalized Likelihood ratio Test, Uniformly Most Powerful Test, Tests of Hypotheses – Sampling from the Normal Distribution.

*Method of assessment: end of semester examination*

**Level III – Semester II**

**Refer the Optional course units offered by the department of Mathematics for Level III- Semester II, for details.**