PAPER – I

Differential Equations

CO 1.Solve linear differential equations

CO2.Convert non exact homogeneous equations to exact differential equations by using integrating factors.

CO 3.Know the methods of finding solutions of differential equations of the first order but not of the first degree.

CO 4. Solve higher-order linear differential equations, both homogeneous and non homogeneous, with constant coefficients.

CO 5. Understand the concept and apply appropriate methods for solving differential equations

PAPER – II

Solid Geometry

CO 1. Get the knowledge of planes.

CO 2. Basic idea of lines, sphere and cones.

CO 3. Understand the properties of planes,                 lines, spheres

CO 4. Express the problems geometrically and then to get the solution.

CO 5. Understand the properties of cones

PAPER – III

Abstract Algebra

CO 1: Acquire the basic knowledge and structure of groups, subgroups and cyclic groups.

CO 2 : Acquire the basic knowledge and structure of subgroups and normal groups

CO 3 :Get the significance of the notation of a normal subgroups. study the homomorphisms and isomorphism with applications

CO 4: Get the behavior of permutations and operations on them

CO5.Understand the ring theory concepts with the help of knowledge in group theory and to prove the theorems. understand the applications of ring theory in various fields.

PAPER – IV:

Real Analysis

CO 1 :Students will be able to recognize bounded, convergent, divergent and other features of real lines.

CO 2: Students will be able to apply the ratio test, root and alternating series tests as well as the limit comparison test, to determine the convergence and absolute convergence of an infinite series of real numbers.

CO 3: Test the continuity and differentiability

CO 4: Know the geometrical interpretation of mean value theorem

CO 5 : Conceptualize Upper Darboux sum

U(P, f) and lower Darboux sum L(P, f) and associated results. Upper integral and lower integral.

PAPER – V:

Linear Algebra

CO 1 : Understand the concepts of vector spaces, subspaces

CO 2 : Demonstrate understanding of linear independence, span, and basis

CO 3: Understand the concepts of linear transformations and their properties

CO 4 : Apply Cayley- Hamilton theorem to problems for finding the inverse of a matrix and higher powers of matrices without using routine methods CO 5 : Learn the properties of inner product spaces and determine orthogonality in inner product spaces

PAPER – VI(A)

Numerical Methods

CO 1 : Understand the subject of various numerical methods that are used to obtain approximate solutions

CO 2 : Understand various finite difference concepts and interpolation methods.

CO 3 : Work out numerical differentiation and integration whenever and wherever routine methods are not applicable.

CO 4 : Find numerical solutions of ordinary differential equations by using various numerical methods.

CO 5 : Analyze and evaluate the accuracy of numerical methods.

PAPER – VI(B)

Mathematical Special Functions

CO 1 : Understand the Beta and Gamma functions, their properties and relation between these two functions, understand the orthogonal properties of Chebyshev polynomials and recurrence relations.

CO 2 :.Find power series evolutions of ordinary differential equations.

CO 3 : solve Hermit equation and write

the Hermit Polynomial of order (degree) n, also find the generating function for Hermit  Polynomials, study the orthogonal properties of Hermit Polynomials and recurrence relations.

CO 4 : Solve Legendre equation and write the Legendre equation of first kind, also find the generating function for Legendre Polynomials, understand the orthogonal properties of Legendre Polynomials.

CO 5 : Solve Bessel equation and write the Bessel equation of first kind of order n, also find the generating function for Bessel function understand the orthogonal properties of Bessel unction

Outcomes

After completion of the course the student should be able to

CO 1:  Introduce the basic concepts of group theory and study the structure of groups.

CO 2:  Introduce the concepts of conjugacy and G sets and prove cayley theorem. To introduce explicitly the properties of permutation groups

CO 3:  Determine structure of any abelian groups. To determine structure of finite non abelian groups through Sylow theorems.

CO 4: Introduce concepts of ring theory. To introduce different types of ideals. To apply Zorn’s lemma on the set of ideals.

CO 5:Study on UFD as a generalization of fundamental theorem of arithmetic, PID based on ideals and ED is division algorithm applied on polynomials introduces to fundamental techniques adapted in advanced algebra

CO 1: Describe elementary concepts on metric spaces to get the general idea that is relevant to Euclidean spaces.

CO 2: Study the continuity and its properties of real valued functions in metric spaces.

CO 3: Describe the derivatives of real valued functions defined on intervals or segments, and study its properties.

CO 4: Introduce Riemann-Stieltjes integral as a generalization of Riemann integral and discuss the existence of this integral.

CO 5: Study differentiation of integrals and further the extension of integration to vector valued functions

CO 1 : Will be able to handle operations on sets and functions and their properties

CO 2: Understand the concepts of Metric spaces, open sets, closed sets, convergence, some important theorems like Cantor’s intersection theorem and Baire’s theorem

CO 3: Be familiar with the concept of Topological spaces, continuous functions in more general and characterize continuous functions in terms of open sets, closed sets etc.

CO 4: Explain the concept of compactness in topological spaces

CO 5: Characterise compactness in metric spaces and their properties

CO 1: Familiarize with essential concepts of real function theory that help to grasp the theory of ordinary differential equations

CO 2: Introduce basic theorems in theory of ordinary differential equations pertaining to existence, uniqueness, continuation of solutions.

CO 3: Understand dependence of solutions on initial conditions and parameters

CO 4: Transform nth order differential equations in to differential systems and extend the theory to differential systems.

CO 5: Study the qualitative behaviour of solutions of homogeneous and non homogeneous linear equations and systems

CO-1:Bridge the relation between matrix theory and vector spaces.

CO-2:Understand the applications of Cayley-Hamilton Theorem.

CO-3:Find an inverse of a linear transformation(a matrix) using Cayley-Hamilton Theorem.

CO-4:Find the Jordan forms of a complex matrix with a given characteristic polynomial.

CO-5: Understand the relation between semi-simple operators and diagonalizable operators.

CO 1:Understand the concept of extensions of a field, based on the study of irreducible polynomials.

CO 2:Understand the concept of normal extensions and separable extensions based on the study multiplicity of roots of a polynomial

CO 3:Introduce the concept of group of automorphisms on a field. To introduce fixed fields. To prove the fundamental theorem of Galois theory.

CO 4:Apply Galois theory and prove the fundamental theorem of algebra. To study the properties of nth cyclotomic polynomial.

CO 5:Understand Galois theory and study its applications

CO 1: Discuss the most important aspects of the problems that arise when limit processes are interchanged.

CO 2: Study the approximation of continuous complex function and its generalization and an introduction of power series.

CO 3: Study of exponential and logarithmic functions, the trigonometric functions and Fourier series and their properties.

CO 4: Discuss linear transformations on finite-dimensional vector spaces over any field of scalars and derivative of functions of several variables.

CO 5: Study the method of solving implicit functions. Interesting illustration of the general principle that the local behaviour of a continuously differentiable mapping near a point. Further study of derivatives of higher order and differentiation of integrals.

CO 1:Understand various toplogical spaces like T 1 spaces, Hausdorff spaces, Completely regular spaces, normal spaces

CO 2:Prove the existence of continuous functions on normal spaces

CO 3: Characterize connected subsets of Real number system , understand local connectedness and totally disconnected spaces

CO 4:Prove various approximation theorems for continuous functions

CO 5: Locally compact spaces and generalise Stone – Weirstrass theorems

CO 1:Solve problems using the properties of analytic functions like power series expansion, Cauchy-Riemann equations etc.

CO2: Analyze the properties of power series and apply them to understand properties of analytic functions.

CO 3: Apply the Cauchy integral formula to solve problems.

CO4: Analyze the zeros of analytic functions.

CO5: Identify and analyze the nature of singularities and behaviour of functions near the singularities.

CO 1: Understand The Four Colour Theorem and applications in chemistry and physics.

CO 2: Familiarize the basic concepts of graphs and different types of graphs.

CO 3: Learn the modelling of Konigsberg Bridge Problem and Hamilton’s Game by graphs.

CO 4: Characterize graphs which are both Eulerain and Hamiltonian.

CO 5: Understand specific difference between modular and distributive lattices.

CO 1: The concept of Banach space through which it helps to consider the combination of algebraic and metric structures opens up the possibility of studying linear transformations of one Banach space into another with the additional property of being continuous.

CO 2: To understand the algebraic and topological aspects of the continuous linear functionals.

CO 3: To study elementary theory of Hilbert spaces and their operators to provide an adequate foundation for the higher studies.

CO 4: To understand a natural correspondence between H and its conjugate space H* , and the ad joint of an operator on a Hilbert space.

CO 5: To study the spectral resolution of an operator T on a Hilbert space H..

CO 1: To learn about method of variations with fixed boundaries

CO 2: To learn about method of variations with moving boundaries

CO 3: To gain knowledge on some specific variational problems such as those involving extremals with corners and one sided variations

CO 4: To understand about sufficient conditions for an extremum for variational problems.

CO 5: To learn about variational problems involving a conditional extremum

Electives:

Number theory-I

CO 1 :     To introduce arithmetical functions and explore their role in the study of distribution of primes.

CO 2 :     To study the averages of arithmetical functions and some related asymptotic formulas.

CO 3 :     To introduce the foundations of congruence’s and study the polynomial congruence’s.

CO 4 :     To understand the prime number theorem on distribution of primes and develop some equivalent forms.

CO 5:To introduce the characters of a group and apply to the Dirichlet Theorem on primes in a progression..

CO 1 :     To familiarize the concepts of poset, chain conditions.

CO 2 :     To learn the lattice theoretic duality principle.

CO 3 :     To study complements, relative complements and semi-complements of elements of a bounded lattices.

CO 4 :     To learn the properties of compact elements and compactly generated lattices.

CO 5 :     To study the posets as topological spaces.

CO 1 :     To familiarize the essential concepts of ideals, quotient rings and homomorphisms.

CO 2 :     To understand the difference between zero divisors, nilpotent elements and units.

CO 3 :      To study the properties of finitely generated modulus.

CO 4 :     To introduce tensor product of modulus and its exactness properties.

CO 5 :     To learn the concepts of extended and contracted ideals in ring of fractions

CO 1 :     Introduce a special theory on sets, called outer measure of a set and measurable sets, which are useful to get an idea on real number system.

CO 2 :     To understand measurable functions through the certain construction of measurable sets and their properties.

CO 3 :     To introduce and understand the Lebesgue integral of various measurable functions and their properties.

CO 4 :     To study differentiation of Lebesgue integral and convex functions.

CO 5 :     To study some spaces of functions of a real variable, the Lp spaces

CO 1 :     To be introduced to categorization of partial differential equations such as linear, quasi linear and nonlinear equations.

CO 2 :     To learn a few methods of solving linear, semi linear and quasi linear equations and construction of Cauchy problem for first order partial differential equations

CO 3 :     To understand the classification pertaining to second order equation and learn the procedure of reducing equations to their cannonical forms.

CO 4 :     To understand the structure of hyperbolic equation, know its properties and solve related problems

CO 5 :     To understand the structure of elliptic equation, know its properties and solve related problems

ELECTIVES:

Number theory-II

CO 1 :      To introduce the concept of Quadratic residues. To define Legendre symbol and evaluate Quadratic residue. To generalize Legendre symbol to Jacobi symbol and to study applications of Quadratic residues

CO 2 :      To introduce the concept of primitive roots. To understand the study on existence of primitive roots.

CO 3 :      To define Dirichlet Series and identify the plane of absolute convergence and convergence of Dirichlet series. To establish Euler products to Dirichlet series.

CO 4 :      To derive some analytic properties of Dirichlet series. To develop some expressions as exponential and integral form for Dirichlet series.

CO 5 :      To understand the analytic proof of prime number theorem based on the analytic properties of the particular Dirichlet series, Riemann Zeta function.

CO 1 :       To study equivalent conditions for a lattice to become modular and distributive.

CO 2 :      To learn meet-representations of modular and distributive lattices.

CO 3 :      To understand the equivalent conditions for a complete Boolean algebra to become atomic.

CO 4 :      To study the properties of valuations of Boolean algebras.

CO 5 :       To learn the properties of rings of sets.

CO 1 :  To learn the decomposition of ideals into primary ideals.

CO 2 :  To learn Going-Up and Going-Down theorems concerning prime ideals in an integral extensions.

CO 3 :   To study valuation rings of a given field of fractions.

CO 4 :  To characterize Noetherian rings and Art in rings.

CO 5 :  To study primary decomposition in Noetherian rings and to learn The Structure Theorem for Artin rings.