**Level II – Semester II**

**MAT221β: Number Theory (30 lecture hrs + 15 tutorial hrs) -( Credit Value 2.5)**

Integers: Prime and irreducible, division algorithm, Euclid’s algorithm, Fundamental Theorem of Arithmetic, Integers mod n, Chinese Remainder Theorem, Euler’s function

Prime integers: Sieve of Eratosthenes, perfect numbers, Mersenne numbers, Fermat numbers, infinite number of primes, the prime number theorem. Gaussian integers Modular calculations: Fermat’s Little Theorem, Wilson’s theorem. Sums of squares, Fermat’s Last Theorem, Sums of 4 squares.

Primitive elements: Roots of unity, factors of Fermat primes, roots of polynomial equations, the number of n roots of unity, the Primitive Element theorem.

Integer polynomials: Hensel’s Lemma, primitive elements mod n. Special Topics in Number Theory.

*Method of assessment: end of semester examination*

**MAT222δ: Real Analysis II (15 lecture hrs + 7 tutorial hrs) -( Credit Value 1.25)**

Sequences and series of functions, Point-wise convergence of sequence of functions, Uniform convergence of sequence of functions, Convergence and Uniform convergence of series of functions, Integration and differentiation of series of functions.

*Method of assessment: end of semester examination*

MAT223β Elementary Topology (30 lecture hrs + 15 tutorial hrs) -( Credit Value 2.5)

Sets, Relations on sets, Equivalence relations, Equipotent sets, Finite and infinite sets, Countability and uncount- ability, Topology of line and plane, Bolzano Weierstrass theorem , Heine Borel Theorem, Metric spaces, Complete metric spaces, Compact metric spaces, Connected metric spaces, Continuous functions on metric spaces.

*Method of assessment: end of semester examination*

**MAT224δ: Geometry (15 lecture hrs + 8 tutorial hrs) -( Credit Value 1.25)**

Plane: Various forms of the equation of a plane. Straight Line, Various forms of the equation of a line.

Sphere: Various forms of the equation of a sphere, Tangent line to a sphere, Tangent plane to a sphere, Condition of Tangency, Intersection of two spheres. The Central Conicoids: Ellipsoid, Hyperboloid of one sheet, Hyperboloid of two sheets, Intersection of a conicoid and a line, Tangent Line to a conicoid, Tangent Plane to a conicoid, Normal to a conicoid, Number of Normals from a given point.

*Method of assessment: end of semester examination*

MAT225β : Mathematical Statistics-I (30 lecture hrs + 15 tutorial hrs) -( Credit Value 2.5)

Joint Density Functions, Joint Cumulative Distribution Function, Conditional Distribution Function, Independence, Covariance and correlation coefficient, Conditional Expectations, Joint Moment Generating Function and Moments, Independence and Expectation, Bivariate Normal Distribution, Expectations of Functions of Random Variables.

Distribution of Function of Random variables: Cumulative Distribution Function Technique, Moment Generating Function Technique, Transformation Technique. Population and Samples, Random Sample, Statistic, and Sample Moments, Sample Mean, Law of Large Numbers, Central limit Theorem. Sampling from the normal distribution: Sample mean, chi-square distribution, F distribution, Student t Distribution.

*Method of assessment: end of semester examination*