**M.Sc. MATHEMATICS**

**PROGRAMME SPECIFIC OUTCOMES:**

PSOs describe what students are expected to know or be able to do by the time of graduation. The Program Specific Outcomes of PG in Mathematics are:

At the end of the programme, the students will be able to:

**PO 1 : **Demonstrate in-depth knowledge of Mathematics, both in theory and application.

**PO 2 : **Innovate, invent and solve complex mathematical problems using the knowledge of pure and applied mathematics.

**PO 3 : **Have strong foundation in core areas of Mathematics and able to communicate Mathematics effectively .

**PO 4 : S**olve one dimensional Wave and Heat equations employing the methods in Partial Differential equations.

**PO 5 : **Create, select, and apply appropriate techniques, resources, and modern IT tools including prediction and modeling to complex activities with an understanding of the limitations.

**PO 6 : **Present papers in seminars and conferences in order to defend their mathematical skills on various topics in the curriculum.

**PO 7 :**Provide knowledge of a wide range of mathematical techniques and application of mathematical methods/tools in other scientific and engineering domains.

**PO 8 : **Equip the student with skills to analyze problems, formulate hypothesis, evaluate and validate results, so as to draw reasonable conclusions thereof.

**PO 9 : **Recognise the need to engage in lifelong learning through continuous education and research leading to higher degrees like Ph.D,D.Sc etc.

**PO 10 :** Present mathematics clearly and precisely, make vague ideas precise by formulating them in the language of mathematics describe mathematical ideas from multiple perspectives and explain fundamental concepts of mathematics to non-mathematics.

**M.Sc., Mathematics**

**Course Outcomes**

**Course Name : **ABSTRACT ALGEBRA ** Course Code: **21PMT01** **

S. NO. |
COURSE OUTCOME |
PSOs Addressed |
BLOOMS VERB |

CO1 | Know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations) | 1,2,3,7,8,10 | Understand |

CO2 | Provide the examples and non-examples of the various concepts. | 1,2,3,7,8,10 | Apply |

CO3 | Demonstrate their mastery by solving non-trivial problems related to theseconcepts | 1,2,3,7,8,10 | Apply |

CO4 | Proving simple (but non-trivial) theorems about the concepts, related to, but not identical to, statements proven by the text or instructor. | 1,2,3,7,8,10 | Apply |

CO5 | Differentiate applications | 1,2,3,7,8,10 | Analyze |

**Course Name : **REAL ANALYSIS ** Course Code: **21PMT02** **

S. NO. |
COURSE OUTCOME |
PSOs Addressed |
BLOOMS VERB |

CO1 | Identity the logic behind the execution of the Riemann Stieltjesintegral ,Existence of the integral, Properties of the integral, identity the Rectifiable curves. | 1,2,3,4,5,6,7,8,9,10 |
Knowledge |

CO2 | Analyze the field of sequences and series of functions, uniform convergence. | 1,2,3,4,5,6,7,8,9,10 | Apply |

CO3 | Develop the Equi-continuous families of functions , families of functions and Stone Weierstrass Theorem. | 1,2,3,4,5,6,7,8,9,10 | Analyze |

CO4 | Identify the special Functions, Power series , exponential , Logarithmic functions and trigonometric functions. | 1,2,3,4,5,6,7,8,9,10 | Knowledge |

CO5 | Develop the Fourier series and Gamma function | 1,2,3,4,5,6,7,8,9,10 | Analyze |

**Course Name : ****Ordinary Differential Equations** ** Course Code: **21PMT03** **

S. NO. |
COURSE OUTCOME |
PSOs Addressed |
BLOOMS VERB |

CO1 | Identify a system of linear equations (or linear systems) and describe its solution. Have to know the formula for Wronskian and solve the DE by Wronskian method. | 1,2,3,5,6,7,8,9,10 |
Understand |

CO2 | Find the general solution of second order linear homogeneous equation and know annihilator method to solve non homogeneous equation. | 1,2,3,5,6,7,8,9,10 | Understand |

CO3 | Have to solve the linear differential equation with variable coefficient. | 1,2,3,5,6,7,8,9,10 | Apply |

CO4 | Compute the solutions and properties of Legendre and Bessel’s equations. | 1,2,3,5,6,7,8,9,10 | Apply |

CO5 | Solve the DE by the method of successive approximations. | 1,2,3,5,6,7,8,9,10 | Apply |

**Course Name : ****CLASSICAL MECHANICS** ** Course Code: **21PMT04** **

S. NO. |
COURSE OUTCOME |
PSOs Addressed |
BLOOMS VERB |

CO1 | Understand the fundamentals of mechanical systems, constraints, virtual work and energy and momentum. | 1,2,3,4,5,6,7,8,9,10 | Remember |

CO2 | Provide the Lagrange’s equation, derivative of Lagrange’s equation and examples of integral of motions. | 1,2,3,4,5,6,7,8,9,10 | Apply |

CO3 | Comprehend the concepts of Hamilton’s principle and Hamilton’s equation. | 1,2,3,4,5,6,7,8,9,10 | Understand |

CO4 | Identify the Applications of Hamilton’s-Jacobi theory and Hamilton’s principle in Hamilton Jacobi equation. | 1,2,3,4,5,6,7,8,9,10 |
Analyze |

CO5 | Explore the concepts of canonical transforms, differential forms and generating functions, especially Lagrange and Poisson Brackets. | 1,2,3,4,5,6,7,8,9,10 |
Evaluate |

**Course Name : ****CALCULUS OF VARIATIONS AND INTEGRAL EQUATIONS ****Course Code: **21PMTM1** **

S. NO. |
COURSE OUTCOME |
PSOs Addressed |
BLOOMS VERB |

CO1 | Define Variational problems with fixed boundaries. | 1,2,3,4,5,6,7,8,9,10 | Understand |

CO2 | Illustrate the Variational problems with moving boundaries. | 1,2,3,4,5,6,7,8,9,10 | Understand |

CO3 | Demonstrate Integral Equation. | 1,2,3,4,5,6,7,8,9,10 | Apply |

CO4 | Implement method for Solution of Fredholm integral equation. | 1,2,3,4,5,6,7,8,9,10 | Apply |

CO5 | Evaluate Hilbert – Schmidt Theory | 1,2,3,4,5,6,7,8,9,10 | Apply |

**Course Name : ****MATHEMATICAL METHODS ** ** Course Code: **21PMTM2** **

S. NO. |
COURSE OUTCOME |
PSOs Addressed |
BLOOMS VERB |

CO1 | Understand the concept of Fourier Sine and Cosine transforms, Fourier transforms of derivatives | 1,2,3,4,5,6,7,8,9,10 | Understand |

CO2 | Provide the Fourier transforms of simple functions, Solution of PDE by Fourier transform. | 1,2,3,4,5,6,7,8,9,10 | Understand |

CO3 | Concept of Mellin Transforms of derivatives and Integrals | 1,2,3,4,5,6,7,8,9,10 | Apply |

CO4 | Identify the Applications of Parseval Relation for Kontorovich-Lebedev Transforms | 1,2,3,4,5,6,7,8,9,10 | Apply |

CO5 | Explore the concepts of finite Fourier Transforms- Generalized Finite Fourier Transforms | 1,2,3,4,5,6,7,8,9,10 | Apply |

**Course Name : ****COMPLEX ANALYSIS ** ** Course Code: **21PMT05** **

S. NO. |
COURSE OUTCOME |
PSOs Addressed |
BLOOMS VERB |

CO1 | Identity the Analytic functions, power series, – Exponential and Trigonometric function. | 1,2,3,4,5,6,7,8,9,10 | Knowledge |

CO2 | Analyze the Cross ratio, conformal mapping and elementary Riemann surfaces. | 1,2,3,4,5,6,7,8,9,10 | Apply |

CO3 | Develop the Cauchy’s integral Formula, Taylor’s Theorem and Maximum Principle. | 1,2,3,4,5,6,7,8,9,10 | Analyze |

CO4 | Identity chains and Cycles, Simple Connectivity, Locally exact differentials , Multiply connected regions. | 1,2,3,4,5,6,7,8,9,10 | Knowledge |

CO5 | Develop the Harmonic Functions, Taylor’s series and Laurent series | 1,2,3,4,5,6,7,8,9,10 | Analyze |

**Course Name : ****Partial Differential Equations ** ** Course Code: **21PMT06** **

S. NO. |
COURSE OUTCOME |
PSOs Addressed |
BLOOMS VERB |

CO1 | Have a sound knowledge of second order PDE reducible to canonical forms and Riemann’s method. | 1,2,3,5,6,7,8,9,10 | Remember |

CO2 | Have a sufficient exposure to get the solution of Laplace equation and Poisson equation, Dirichlet, Neumann’s problem for sphere and circle. | 1,2,3,5,6,7,8,9,10 |
Apply |

CO3 | Have an idea of Boundary conditions, separation of variables method and diffusion equation in cylindrical and spherical coordination. | 1,2,3,4,5,6,7,8,9,10 |
Apply |

CO4 | Have an idea of D’Alemberts solution, Duhamel’s principle and also wave equation its periodic solution in cylindrical and spherical polar coordinates. | 1,2,3,4,5,6,7,8,9,10 |
Apply |

CO5 | Have knowledge of Laplace transform and Fourier transform and their applications to PDE such as Laplace, diffusion and wave equation. | 1,2,3,5,6,7,8,9,10 |
Apply |

**Course Name : ****MEASURE THEORY AND INTEGRATION** ** Course Code: **21PMT07** **

S. NO. |
COURSE OUTCOME |
PSOs Addressed |
BLOOMS VERB |

CO1 | Understand the measurable sets and lebegue measure. | 1,2,3,5,6,7,8,9,10 | Understand |

CO2 | Understand the Riemann integral ,Lebesgue integral. | 1,2,3,5,6,7,8,9,10 | Understand |

CO3 | Have knowledge on differentiation of monotone functions | 1,2,3,5,6,7,8,9,10 | Remember |

CO4 | Understand lebegue integral and measurable functions | 1,2,3,5,6,7,8,9,10 | Apply |

CO5 | Apply these spaces in space theory. | 1,2,3,5,6,7,8,9,10 | Apply |

**Course Name : ****OPTIMIZATION TECHNIQUES ** ** Course Code: **21PMT08** **

S. NO. |
COURSE OUTCOME |
PSOs Addressed |
BLOOMS VERB |

CO1 | An ability to solve integer programming, some applications of integer programming, cutting plane algorithm and Dynamic Programming, DP models, problem of dimensionality in dynamic programming. | 1,2,3,4,5,6,7,8,9,10 |
Remember |

CO2 | Knowledge of inventory models like ABC Inventory System, A Generalized Inventory Model-Deterministic models, Single Item static model with price breaks and Single Item N-period dynamic model. | 1,2,3,4,5,6,7,8,9,10 |
Understand |

CO3 | Scope for Markovian decision process in the finite stage and infinite stage decision model. Policy iteration method with and without discounting. | 1,2,3,4,5,6,7,8,9,10 |
Apply |

CO4 | An ability to use the basic elements of the queuing model, Roles of the Poisson and exponential distributions and queues with combined arrivalsand departures. | 1,2,3,4,5,6,7,8,9,10 |
Analyze |

CO5 | An exposure to classical optimization theory and non linear programming. | 1,2,3,4,5,6,7,8,9,10 | Evaluate |

**Course Name : ****DIFFERENTIAL GEOMETRY ** ** Course Code: **21PMTM3** **

S. NO. |
COURSE OUTCOME |
PSOs Addressed |
BLOOMS VERB |

CO1 | Define the theory of Space curves for building differential calculus based applications. | 1,2,3,4,5,6,7,8,9,10 |
Understand |

CO2 | Illustrate the theory of Space curves and its related results. | 1,2,3,4,5,6,7,8,9,10 | Understand |

CO3 | Demonstrate the Local Intrinsic properties of surface. | 1,2,3,4,5,6,7,8,9,10 | Apply |

CO4 | Implement method for Local Intrinsic properties of surface. | 1,2,3,4,5,6,7,8,9,10 | Apply |

CO5 | Apply Geodesic on a surface and its applications. | 1,2,3,4,5,6,7,8,9,10 | Apply |

**Course Name : ****COMMUTATIVE ALGEBRA ** ** Course Code: **21PMTM4

S. NO. |
COURSE OUTCOME |
PSOs Addressed |
BLOOMS VERB |

CO1 | Know operations on ideals, extension and contraction of ideals | 1,2,3,5,6,7,8,9,10 | Apply |

CO2 | Understand modules, sub modules, homomorphism, direct sum and product of modules | 1,2,3,5,6,7,8,9,10 | Understand |

CO3 | Knowledge on modules of fractions, Primary decomposition | 1,2,5,6,7,8,9,10 | Apply |

CO4 | Know different kinds of modules based on chain conditions | 1,2,3,5,6,7,8,9,10 | Remember |

CO5 | Know the notions of Noetherian and Artinian rings | 1,2,3,5,6,7,8,9,10 | Understand |

**Course Name : ****GENERAL TOPOLOGY ** ** Course Code: **21PMT09** **

S. NO. |
COURSE OUTCOME |
PSOs Addressed |
BLOOMS VERB |

CO1 | learn the concepts of topological spaces, connected and compact spaces, continuous functions, countability and separation axioms. | 1,2,3,4,5,6,7,8,9,10 | Recall |

CO2 | understand the attributes of continuous functions and inspect their applications in connected and compact spaces, countability and separation axioms | 1,2,3,4,5,6,7,8,9,10 | Understand , Analyse |

CO3 | apply the notions of different topological spaces and solve real world problems | 1,2,3,4,5,6,7,8,9,10 | Apply |

CO4 | interpret various forms of tological spaces and assess their attributes | 1,2,3,4,5,6,7,8,9,10 | Understand , Evaluate |

CO5 | prove extreme value theorem, lebesgue number lemma, uniform continuity theorem, countability and separation axioms and inspect their applications | 1,2,3,4,5,6,7,8,9,10 | Analyse,
Evaluate |

**Course Name : ****PROBABILITY THEORY ** ** Course Code: **21PMT10** **

S. NO. |
COURSE OUTCOME |
PSOs Addressed |
BLOOMS VERB |

CO1 | Define and apply basic concepts and methods of probability theory Use common probability distributions and analyze their properties (exponential distribution, multivariate normal distribution, etc.) | 1,2,3,5,6,7,8,9,10 | Recall ,Analyze |

CO2 | Compute conditional probability distributions and conditional expectations | 1,2,3,5,6,7,8,9,10 | Understand , Analyze |

CO3 | Solve problems and compute limits of distributions by use of transforms (characteristic functions, generating functions) | 1,2,3,5,6,7,8,9,10 | Apply |

CO4 | Explain the concept of measurability and define and work with sigma algebras and construct probability measures on sample spaces | 1,2,3,5,6,7,8,9,10 | Understand , Evaluate |

CO5 | Define and use the properties of Stochastic processes, especially random walks, branching processes, the Poisson and Wiener process, applied to real problems | 1,2,3,5,6,7,8,9,10 | Analyse,
Evaluate |

**Course Name : ****GRAPH THEORY ** ** Course Code: **21PMT11

S. NO. |
COURSE OUTCOME |
PSOs Addressed |
BLOOMS VERB |

CO1 | Know the basic definitions and concepts of Graphs and Subgraphs. | 1,2,3,4,5,6,7,8,9,10 | Recall |

CO2 | Getting acquainted with the concepts of Trees and connectivity study its applications. | 1,2,3,4,5,6,7,8,9,10 | Understand |

CO3 | Recognize the concepts and properties of Euler Tours and Matchings and study its applications. | 1,2,3,4,5,6,7,8,9,10 | Apply |

CO4 | Assimilate the knowledge about edge coloring of Graphs, its applications and to understand the notations of independent Sets. | 1,2,3,4,5,6,7,8,9,10 | Apply |

CO5 | Acquire the knowledge about the concepts of Vertex colorings colourings and model in the real life problem. | 1,2,3,4,5,6,7,8,9,10 | Analyze |

**Course Name : ****PROGRAMMING IN R ** ** Course Code: **21PMT12** **

S. NO. |
COURSE OUTCOME |
PSOs Addressed |
BLOOMS VERB |

CO1 | Understand the operations on vectors using R | 1,2,3,4,5,6,8,9,10 | Understand |

CO2 | Understand basic operations on matrices using R | 1,2,3,4,5,6,8,9,10 | Understand |

CO3 | Apply commands on data frames to understand various operations on it using R | 1,2,3,4,5,6,7,8,9,10 | Apply |

CO4 | Multiple Lines in a Line Chart using R | 1,2,3,4,5,6,7,8,9,10 | Analyse |

CO5 | Applying Trim Option using R | 1,2,3,4,5,6,7,8,9,10 | Evaluate |

**Course Name : ****MATHEMATICAL ECONOMICS** ** Course Code: **21PECMT** **

S. NO. |
COURSE OUTCOME |
PSOs Addressed |
BLOOMS VERB |

CO1 | Apply simple derivatives to find Elasticity of Demand, Marginal Utility, Marginal Revenue etc. | 1,2,3,5,6,7,8,9,10 | Understand |

CO2 | Find Total, Average and Marginal cost and the relation between Average and Marginal Cost curves. | 1,2,3,5,6,7,8,9,10 | Understand |

CO3 | Use LPP in many industrial and Economic Problems. | 1,2,3,5,6,7,8,9,10 | Apply |

CO4 | Apply Transportation Model to determine the amount to be shifted from each destination such that the total transportation cost is minimum. | 1,2,3,5,6,7,8,9,10 | Apply |

CO5 | Apply Assignment Problem in which the objective is to assign a number of tasks (Jobs or origin or sources) to an equal number of facilities (machines or persons or destinations) at a minimum cost (or maximum profit). | 1,2,3,5,6,7,8,9,10 | Apply |

**Course Name : **FUNCTIONAL ANALYSIS ** Course Code: **21PMT13** **

S. NO. |
COURSE OUTCOME |
PSOs Addressed |
BLOOMS VERB |

CO1 | learn the central concepts of Banach Space, Hilbert spaces | 1,2,3,4,5,6,7,8,9,10 | Recall |

CO2 | understand the notions of continuous linear transformations, Natural imbedding ,orthogonal complements, various operators, Banach algebra | 1,2,3,4,5,6,7,8,9,10 | Understand |

CO3 | recognize and analyze conjugate of an operator, axiomatic knowledge of the properties of a Hilbert space, including orthogonal complements, orthonormal sets | 1,2,3,4,5,6,7,8,9,10 | Understand , Analyze |

CO4 | apply the properties of various operators to the resolution of integral equations and projection | 1,2,3,4,5,6,7,8,9,10 | Apply |

CO5 | Prove Hahn BanachTheorm, open mapping theorem, properties of Hilbert spaces, the spectral theorem. | 1,2,3,4,5,6,7,8,9,10 | Evaluate |

**Course Name : ****Fuzzy Sets & Fuzzy Logic** ** Course Code: **21PMT14** **

S. NO. |
COURSE OUTCOME |
PSOs Addressed |
BLOOMS VERB |

CO1 | Identity the logic behind the execution and solving of the Crisp sets and Fuzzy sets | 1,2,3,4,5,6,7,8,9,10 |
Knowledge |

CO2 | Analyze the field of Operations on Fuzzy sets,Union, intersection and complement and the Properties of De-Morgan’s Laws. | 1,2,3,4,5,6,7,8,9,10 | Apply |

CO3 | Develop the solution methods of solving Binary fuzzy relation, Binary relations on a single set, Fuzzy equivalence relations, Fuzzy compatibility relation and Fuzzy ordering relations. | 1,2,3,4,5,6,7,8,9,10 | Analyze |

CO4 | Identify the difference between Evidence Theory and Possibility theory. | 1,2,3,4,5,6,7,8,9,10 | Knowledge |

CO5 | Develop the concept of Fuzzy clustering and Fuzzy image processing. | 1,2,3,4,5,6,7,8,9,10 | Analyze |

**Course Name : **NUMBER THEORY ** Course Code: **21PMT15** **

S. NO. |
COURSE OUTCOME |
PSOs Addressed |
BLOOMS VERB |

CO1 | To know what is meant by Divisibility – Primes – Congruence’s and finding solutions | 1,2,3,4,5,6,7,8,9,10 | Understand |

CO2 | To analyze the function – Congruence of higher degree – Prime power moduli & modulus | 1,2,3,4,5,6,7,8,9,10 | Analyze |

CO3 | To Know Quadratic residues – Quadratic reciprocity – The Jacobi symbol – Greatest Integer function | 1,2,3,4,5,6,7,8,9,10 | Understand |

CO4 | To apply the Mobius inverse formula and its functions | 1,2,3,4,5,6,7,8,9,10 | Apply |

CO5 | To know the application of Sums of four and five squares – Waring’s problem | 1,2,3,4,5,6,7,8,9,10 | Understand |