ST.PIOUS
X DEGREE & PG COLLEGE WOMEN
DEPARTMENT
OF MATHEMATICS
M.Sc.
COURSE OUTCOMES
SEMESTER-I
PAPER-I: ABSTRACT ALGEBRA
1. Define group and subgroups
2. Understand and prove
fundamental results
3. Demonstrate knowledge and
understanding of rings, fields and their properties
4. Apply algebraic ways of
thinking
5. Discuss Sylow’s theorems
6. Extend group structure to
finite permutation groups
PAPER-II MATHEMATICAL
ANALYSIS
After
completion of this course, students will be able
1.
Describe
fundamental properties of the real numbers that lead to the formal development
of Real Analysis.
2.
Demonstrated
an understanding of limits and how that are used in sequences series and
differentiation
3.
The
course previous the basic for further studies with in function Analysis ,
topology,& function theory.
4.
Give
the definition of concepts related to metric spaces such as continuity,
compactness convergent ,etc.,
5.
Appreciate
how abstract ideas regions methods in mathematical analysis can be applied to
important practical problems.
6.
Construct
a definite integrals as the limit of a Riemann sum.
PAPER-III: ORDINARY DIFFERENTIAL
EQUATION &PARTIALDIFFERENTIAL EQUATION
On successful completion of the course, students will be
able to:
1.
Student will be
able to solve first order differential equations utilizing the standard
Techniques for separable, exact, linear, homogeneous, or Bernoulli
cases.
2. Student will be able to find the complete solution of a nonhomogeneous
differential
Equation as a linear combination of
the complementary function and a particular
Solution.
3. Student will be introduced to the complete solution of a nonhomogeneous
differential
Equation with constant
coefficients by the method of undetermined coefficients.
4. Student will be able to find the complete solution of a differential
equation with constant coefficients by variation of parameters.
5. Student will have a working knowledge of basic application problems
described by
Second order linear
differential equations with constant coefficients.
PAPER-IV: ELEMENTARY NUMBER THEORY
1. Define
and interpret the concept of Divisibility, Congruence, Greatest common Divisor,
Prime and Prime Factorization.
2. Learn
the methods and techniques used in Number Theory.
3. Determine
Multiplicative Inverses, modulo n and use to solve linear congruence.
4. Apply
Euclid’s Algorithm and backwards substitution.
5. Understand
the definitions of congruence’s, residue classes and Least residues.
PAPER-V: DISCRETE MATHEMATICS
1. Understands
the basic principles of sets and operations in sets
2. Apply
counting principles to determine probabilities
3. Demonstrate
different traversal methods for trees and graph
4. Write
model problems in Computer science using
trees and graphs
5. Determine
when a function is one-one and onto
6. Prove
basic set equalities
7. Demonstrate
the ability to write and evaluate a proof.
SEMESTER-II
PAPER-I: GALIOS THEORY
1.
Solving polynomial
equations using formulas for roots.
2.
How to test if a polynomial is irreducible
Finite Field (Galois Fields).
3.
Understanding which equations can be solved
using radials using the concepts. Ability to understand/obtain the roots of a
polynomial equation if the same has (or can be reduced to) degree less than
five.
4.
Facility in working with finite fields.
5.
Applying the concept of
a field extension to various mathematical problems including geometric
constructions and perfect division of a circle into n parts Facility in working
with mathematical problems that involve polynomial equations.
6.
Facility in handling problems involving
polynomial equations.
PAPER-II:
LEBESUGE MEASURE AND INTEGRATIONS
After completion of this course, students will be
able
1. Solving
the problems using real Analysis techniques applied to diverse situations in
physics, engineering & other mathematical contents.
2. Demonstrate capacity for mathematical resoning
through analysing proving and explaining concepts form real analysis.
3. Demon stare a competence in formulating analyzing
and solving problem in several core areas of mathematics at a detailed level
,including analysis
4. Fundamental objects, techniques & theorems
in the mathematical sciences including
the field of analysis.
PAPER-III: COMPLEX ANALYSIS
On successful completion of the course, students will be
able to:
1. carry out
computations with the complex exponential, logarithm and root functions and
know their domains of definition;
2. calculate the image
of circles and lines under Mobius transformations;
3. find the harmonic
conjugate to a harmonic function;
4. express analytic
functions in terms of power series and Laurent series;
5. calculate complex
line integrals and some infinite real integrals using Cauchy's integral theorem
or residue calculus;
6. find the number of
zeroes and poles within a given curve using the argument principle or Rouche's
theorem;
7. Calculate the flow
lines of an irrational and incompressible fluid.
PAPER-IV: TOPOLOGY
1.
Understand terms, definitions and
theorems related to topology.
2.
Demonstrate knowledge and understanding
of concepts such as open and closed sets, interior, closure and boundary.
3.
Create new topological spaces by using
subspace, product and quotient topologies.
4.
Use continuous functions and homeomorphisms to
understand structure of topological spaces.
5.
Demonstrate knowledge and understanding
of metric spaces.
6.
Apply theoretical concepts in topology to
understand real world applications.
PAPER-V: THEORY
OF ORDINARY DIFFERENTIAL EQUATION
1.
Define ordinary differential equations
2.
Apply the fundamental concepts of ordinary differential
equations and partial differential equations for the resolution
3.
Demonstrate understanding of the meaning of ODE, its order, its
general solution and its particular solution
4.
Apply the method of undermined coefficient to solve non
homogeneous linear differential equations with constant coefficients
5.
Determine solutions to first and second order linear
differential equations
6.
Use the method of Laplace transforms to solve initial value
problem for linear differential equations with constant coefficients
SEMESTER-III
PAPER-I: FUNCTIONAL ANALYSIS
1. Understand how functional analysis uses and unifies ideas from vector
spaces, the theory of metrics, and complex analysis.
- Understand and apply
fundamental theorems from the theory of normed and Banach spaces,
including the Hahn-Banach theorem, the open mapping theorem, the closed
graph theorem, and the Stone-Weierstrass theorem.
- Understand the role of
Zorn's lemma.
- Understand and apply ideas
from the theory of Hilbert spaces to other areas, including Fourier
series, the theory of Fredholm operators, and wavelet analysis.
- Understand the fundamentals
of spectral theory, and some of its power.
PAPER-II: GENERAL MEASURE &
INTEGRATION
On successful completion of the course, Students
will be able to:
1. Learn
the basic elements of measure theory.
2. Understanding
basics concept of measure and Integration theory
3. Apply
the general principles of measure theory and integration in such concrete
subject as the theory of probability or Financial Mathematics.
4. Deal
with theoretical problems of the area and to write complete and rigorous
proofs.
5. Understand
the Lebesgue integration can solve certain problems for which Reimann
integration does not provide adequate answers.
PAPER-III: LINEAR ALGEBRA
1. Explains
the concepts of base and dimension of Vector space
2. Express
Vector space in different dimension
3. Express
some of the algebra operation between linear transformations
4. Explain
concepts of Eigenvalues and Eigen vectors of a Matrix
5. Express
that a set is Orthogonal and orthonormal
6. Explains
the concepts of Inner products on vector space
7. Find
the length of a Vector in some Vector space and the angle between two vectors.
PAPER-IV: OPERATION RESEARCH
8. It
is used in data envelopment
9. Operation
Research is used for defence capability acquisition decision making
10. It
is used to find optimal or near optimal solution to complex decision making
problems
11. It
is used in finding maximum of profit in real world objective
12. It
is used in finding minimum of loss in real world objective
13. It
has strong ties in computer science and analytics
PAPER-V: NUMERICAL ANALYSIS
1. Wide
variety of numerical techniques to solve mathematical problems arising in
diverse scientific contexts.
2. Implementation of stable algorithms for
finding roots of nonlinear equations, solving linear system of equations, and
solution for ODEs, etc.
3. Influence of data representation on computers
on numerical algorithms. Implementing numerical algorithms through computer
programs.
4. Analysis of errors of numerical algorithms.
5. Obtain
approximate stable solution to mathematical problems making use of numerical
algorithms.
SEMESTER-IV
PAPER-I: INTEGRAL EQUATIONS
&CALCULUS OF VARIATION
On successful completion of the course, students will be able
to:
1. Recognize difference between Volterra and Fredholm Integral
Equations, First kind and Second kind, homogeneous and inhomogeneous etc.
2. Find the extreme values of functional.
3. Solve Boundary value problems through integral equations using Green’s Function
4. Evaluate the area of surfaces of Revolution
5. Find extreme values of Functional.
PAPER-II: ELEMENTARY OPERATOR
THEORY
1. Estimate
difficulties resulting in different approaches and knows basic sources of
specialized results.
2. Use
different methods of estimatimating action of a given operator T, depending on
the chosen model.
3. Understands
general principles governing the behaviour of linear equations.
4. Knows
basic results from spectral theory of bounded (as well as unbounded) linear
operators on Hilbert spaces and examples of their applications.
5. Recognize
structures related to systems of commuting operators, by using Banach and
Hilbert space techniques.
6. Knows
basic methods of investigating linear operators in various situations and knows
their relations to other branches of mathematics.
PAPER-III: ANALYTIC NUMBER THEORY
1.
Define fundamental objects appearing in the course such as
gamma function, theta functions and Riemann zeta functions
2.
State and prove many
of the fundamental theorems in the analytic theory of numbers
3.
Understand the
arithmetic functions and their utility in the analytic theory of numbers
4.
Understand dirichlet
characters and analytic properties of dirichlet L-functions
5.
To know dirichlet’s theorem on primes in arithmetic
progressions
6.
Analyze the
connections between zeros of the Riemann zeta function and properties of prime
numbers
PAPER-IV: GRAPH THEORY
1. Solve problems using basic graph theory
2. Identify induced sub graphs, cliques, matching’s, covers in graphs
3. Determine whether graphs are Hamiltonian and/or Eulerian
4. Solve problems involving vertex and edge connectivity, planarity and
crossing numbers
5. Solve problems involving vertex and edge coloring
6. Model real world problems using graph theory
PAPER-V: ADVANCED OPERATION
RESEARCH
1. Give
an appreciation of strategic importance of operations and supply chain
management in a global business environment
2. Understand
how an operation relates to other business function
3. Develop
a working knowledge of concepts and methods related to designing and managing
operations and supply chains
4. Develop
a skill set for equality and process
improvement
5. Develops
how to manage and control the resource allocation.
