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Syllabus M.Sc.-Mathematics Applicable w.e.f. Academic Session 2012-13 Page 1

Study & Evaluation Scheme

Of

Master of Sciences

Mathematics

[Applicable w.e.f. Academic Year 2012-13]

TEERTHANKER MAHAVEER UNIVERSITY

N.H. 24, Delhi Road, Moradabad, Uttar Pradesh-244001

Website: www.tmu.ac.in

[With changes in MHM301 wide approval of V.C dated 26-10-2013]

http://www.tmu.ac.in/


TEERTHANKER MAHAVEER UNIVERSITY (Established under Govt. of U.P. Act No. 30,2008)

Delhi Road, Bagarpur, Moradabad, U.P.-244001

Study & Evaluation Scheme

Of

Master of Sciences - Mathematics

SUMMARY Programme : M.Sc. Mathematics

Duration : Two year full time (Four Semesters)

Medium : English

Minimum Required Attendance : 75 percent

Credit :

Maximum Credit : 90

Minimum Credit required for the degree : 86

Assessment- theory :

Internal Evaluation

(Theory papers)

:

Evaluation of Practical :

Evaluation of Project /Seminar

Reports :

Duration of Examination : To qualify the course a student is required to secure a minimum of 40 % marks in aggregate including the

semester end examination and teachers continuous evaluation.(i.e. both internal and external).

A candidate who secures less than 40% of marks in a course shall be deemed to have failed in that course.

The student should have at least 50% marks in aggregate to clear the semester. In case a student has more

than 40% in each course, but less than 50% overall in a semester, he/she shall re-appear in courses where

the marks are less than 50% to achieve the required aggregate percentage (of 50%) in the semester.

Question Paper Structure

1. The question paper shall consist of eight questions. Out of which first question shall be of short answer

type (not exceeding 50 words) and will be compulsory. Question No. 1 shall contain 8 parts representing

all units of the syllabus and students shall have to answer any five (weightage 4 marks each).

2. Out of the remaining seven questions, students shall be required to attempt any five questions. There will

be minimum one and maximum two questions from each unit of the syllabus. The weightage of Question

No. 2 to 8 shall be 10 marks each.

Internal External Total

30% 70% 100%

1st

Class

Test

2nd

Class

Test

3rd

Class

Test

Assignment(s)

Other

Activity

(including

attendance)

Total

Best two shall be

taken

10 10 10 5 05 30


50 50 100


50 50 100

External Internal

3 hrs 1 ½ hrs


Study and Evaluation Scheme

Course: M.Sc. - Mathematics

Semester I

Semester II

L – Lecture T- Tutorial P- Practical C-Credits

1L = 1Hr 1T= 1 Hr 1P=1Hr 1C=1Hr of theory

1C= 2 Hrs of Practical

S.

No.

Subject

Code Subject

Periods Credits

Evaluation Scheme

L T P Internal External Total

1 MAT 101 Linear Algebra 3 1 - 4 30 70 100

2 MAT 102 Real Analysis 3 1 - 4 30 70 100

3 MAT 103 Graph Theory 3 1 - 4 30 70 100

4 MAT 104 Fluid Dynamics 3 1 - 4 30 70 100

5 MAT 105

Programming in C & Data

Structures 3 1 - 4 30 70 100

6 MHM101 Industrial Management

2 - - 2 30 70 100

7 MAT 151

Programming in C & Data

Structures Lab - - 4 2 50 50 100

Total 17 5 4 24 230 470 700

S.

No.

Subject

Code Subject

Periods Credits

Evaluation Scheme


1 MAT 201 Complex Analysis 3 1 - 4 30 70 100

2 MAT 202 Advance Abstract Algebra 3 1 - 4 30 70 100

3 MAT 203 Functional Analysis 3 1 - 4 30 70 100

4 MAT 204 Numerical Techniques

(NT) 3 1 - 4 30 70 100

5 MAT 205 Differential Equations 3 1 - 4 30 70 100

6 MHM201 Organization Behavior

2 - - 2 30 70 100

7. MAT 251

Numerical Techniques

Lab. - - 4 2 50 50 100

Total 17 5 4 24 230 470 700


Study and Evaluation Scheme

Course: M.Sc. - Mathematics

Semester III

Semester IV

(Minor Research Project may be carried out from III semester to IV semester with some additional

applications.) L – Lecture T- Tutorial P- Practical C-Credits

1L = 1Hr 1T= 1 Hr 1P=1Hr 1C=1Hr of theory

1C= 2 Hrs of Practical

S.

No

.

Subject

Code Subject

Periods Credits

Evaluation Scheme


1 MAT 301 Topology 3 1 - 4 30 70 100

2 MAT 302 Partial Differential

Equations 3 1 - 4 30 70 100

3 MAT 303 Mathematical Statistics 3 1 - 4 30 70 100

4 MAT 304 Operation Research 3 1 - 4 30 70 100

5 MAT 305 Advanced Discrete

Mathematics 3 1 - 4 30 70 100

6 2 - - 2 30 70 100

7 MAT 391 Minor Research Project

and Seminar - - 4 2 50 50 100

Total 17 5 4 24 230 470 700

S.

No.

Subject

Code Subject

Periods Credits

Evaluation Scheme


1 MAT 401 Number Theory 3 1 - 4 30 70 100

2 MAT 402 Fuzzy Sets and its

Applications 3 1 - 4 30 70 100

3 MAT 491 Project work, Seminar &

Viva - - 20 10 50 50 100

Total 6 2 20 18 110 190 300

MHM302 Project Management


M.Sc. – Mathematics Ist Year: Semester I

Linear Algebra

Course code: MAT 101 L T P C

3 1 0 4

Unit I (Lectures 08)

Vector Spaces: Definition, General properties of vector spaces; Vector subspaces; Algebra of subspaces;

Linear Spans; Row space of Matrix; Linear dependence and independence of vectors; Finite-dimensional

vector spaces; Dimension of vector space and sub-spaces; Quotient spaces; Direct sum of spaces;

Coordinates; Disjoint subspaces. .

Unit II (Lectures 08)

Vectors in Rn; Curves in R

n ; Vectors in R

3; Vector in C

3; Matrices: Addition and scalar multiplication,

Transpose of matrix, Square matrices; Systems of linear equations; Diagonalisation; Eigen values and

Eigen vectors; Minimal polynomial; Cayley-Hamilton Theorem; Hermitian & Skew-Hermitian and

unitary matrices; Powers of Matrices; Polynomials in Matrices; Invertible Matrices; Special types of

Square Matrices; Complex and Block Matrices.

Unit III (Lectures 08)

Linear Transforms; Linear operator; Range and null space of a linear Transformation; Rank and nullity;

Product of linear Transformation; Singular Transformation; Representation of linear Transformation by

matrix; Dual spaces; Dual Bases; Projections.

Unit IV (Lectures 08)

Inner Product Spaces: Definition, Euclidean and unitary spaces; Norm and length of vector; Cauchy-

Schwarz’s inequality and Applications; Orthogonality, Orthogonal Sets and Basis, Gram-Schmidt

orthogonalization process; self-adjoint operators, Complex Inner Product Spaces; Unitary and Normal

operators; Projection theorem; Spectral theorem.

Unit V (Lectures 08) Bilinear Forms: Definition, Bilinear form as vectors; Matrix of a bilinear form; Symmetric & skew

Symmetric bilinear forms.

Text Books:

1. Vivek Sahai, Vikas Bist; Linear Algebra, Narosa Publishing House.

2. Sharma & Vashistha, Linear Algebra, Krishna Prakashan Media Ltd.

Reference Books:

1. Schaum’s series Linear Algebra, Tata McGraw- Hill.

2. Kenneth Hoffman & Ray Kunze, Linear Algebra, Pearson Education.



Real Analysis

Course code: MAT 102 L T P C 3 1 0 4


Definition and existence of Reimann- Stieltjes integral; Properties of the integral; Integration and

differentiation; Fundamental theorem of calculus; Integration of vector-valued functions.


Sequences and series of functions; Point wise and uniform convergence; Cauchy criterion for uniform

convergence; Uniform convergence and continuity; Uniform convergence and Riemann-Stieltjes

integration; Uniform convergence and differentiation; Weierstrass approximation theorem.


Power series; Algebra of power series; Uniqueness theorem for power series; Abel’s and Tauber’s

theorems.


Functions of several variables; Linear transformation; Derivatives in an open subset of Rn; Chain rule;

Partial derivatives; Interchange of the order of differentiation; Derivatives of higher orders; Taylor’s

theorem.

Unit V (Lectures 08)

Inverse function theorem and Implicit function theorem (without proof); Jacobians; Extremum problems

with constraints; Lagrange’s multiplier method; Differentiation of integrals.

Text Books:

1. Walter Rudin, Principles of Mathematical Analysis, McGraw-Hill.

Reference Books:

1. T. M. Apostol, Mathematical Analysis, Narosa Publishing.

2. J. White, Real Analysis, An Introduction, Addison-Wesley Publishing.

3. H. L. Royden, Real Analysis, Macmillan Publishing Co. Inc.



Graph Theory


3 1 0 4


Graph; Applications of Graph; Finite and Infinite Graphs; Null Graph; Incidence and Degree; Isolated

Vertex; Pendant Vertex; Isomorphism; Sub graphs; Walks; Paths; Circuits; Connected Graphs,

Disconnected Graphs and Components.

Unit II (Lectures 08) Euler’s Graph; Operations On Graphs; Hamiltonian Paths and Circuits; Shortest Path Algorithms; The

Traveling Salesman Problem; Dijkastra’s Algorithm; Fleury’s Algorithm


Trees; Properties of Trees; Pendant Vertices in a Tree; Distance and Centers in a Tree; Rooted and Binary

Trees, On Counting Trees; Spanning Trees; Fundamental Circuits; Finding All Spanning Trees of a

Graph; Spanning Trees in a Weighted Graph; Cut-Sets; Some Properties of a Cut-Set; Fundamental

Circuits and Cut-Sets, Connectivity and Separability; Network Flows.


Combinatorial and Geometric Graphs; Planar Graphs; Kuratowski's Two Graphs; Different, Detection of

Planarity; Geometric Dual; Combinatorial Dual; Thickness and Crossings; Vectors and Vector Space;

Associated with a Graph.


Matrix representation of graphs; Incidence matrix; Sub matrix of ( )A G ; Circuit matrix, Fundamental

circuit matrix and Rank of B ; Cut-set matrix; Path matrix; Adjacency Matrix; Adjacency Matrix;

Chromatic Number; Chromatic Partitioning; Chromatic Polynomial; Matching Coverings, The Four Color

Problem.

Text Books:

1. Narsingh Deo; Graph Theory; Prentice-Hall, Inc.

2. Douglas B. West; Introduction to Graph Theory; Pearson Education Pvt. Ltd.

3. Gary Chartrand; Chromatic Graph Theory; CRC Press.

.

Reference Books:

1. J.A. Bondy U.S.R. Murty; Graph Theory, Springer.

2. Reinhard Diestel ; Graph Theory, Springer.



Fluid Dynamics



Classification of fluid and its physical properties, Continuum hypothesis; Kinematics of fluids Methods of

describing fluid motion, Translation, Rotation and deformation of fluid elements, Stream Lines, Path lines

and Streak lines, concepts of Vortices.


General theory of stress and rate of strain in a real fluid–Symmetry of stress tensor, Principal axes and

Principle values of stress tensor, Constitutive equation for Newtonian fluid; Conservation laws-

Conservation of mass, momentum and energy.


One and two dimensional in-viscid incompressible flow-Equation of continuity and motion using stream

tube, Circulation, Velocity potential, Irrotational flow; Some theorems about rotational and irrotational

flows Stokes theorem, Kelvin’s minimum energy theorem, Gauss theorem, Kelvin’s circulation theorem.


Vortex motion and its elementary properties; Integration of Euler’s equation under different conditions;

Bernoulli’s equation; Stream function in two dimensional motion; Complex variable technique; Flow past

a circular cylinder; Blasius theorem; Milne’s circle theorem; Sources, Sinks and Doublets; Dynamical

similarity; Buckingham’s π theorem; Non-dimensional numbers and their physical significance.


Incompressible viscous fluid flows-Steady flow between two parallel plates (non-porous and porous)-

Plane couette flow; Plane poiseuille flow, Generalized plane couette flow, Steady flow of two immiscible

fluids between two rigid parallel plates; Steady flow through tube of uniform circular cross section,

Steady flow through annulus under constant pressure gradient.

Text Books:

1. S. W. Yuan, Foundations of fluid mechanics, Prentice Hall of India Prt. Limited.

2. R. K. Rathy, An introduction of fluid dynamics, Oxford and IBH Publishing Company.

Reference Books:

1. G. K. Betchelor, An introduction of fluid dynamics, Oxford University Books.

2. F. Charlton, Text book of fluid dynamics, C.B.S. Publishers.



Programming in C & Data Structures


3 1 0 4


Computer system introduction; Characteristics and classification of computers, CPU, ALU, Control unit,

data & instruction flow, primary, secondary and cache memories; RAM, ROM, PROM, EPROM;

Programming language classifications.


C-Programming : Representation of integers, real, characters, constants, variables; Operators: Precedence

& associative, Arithmetic, Relation and Logical operators, Bitwise operators, increment and decrement

operators, comma operator, Arithmetic & Logical expression.


Assignment statement, Looping, Nested loops, Break and continue statements, Switch statement,

goto statement; Arrays, String processing, functions, Recursion, Structures & unions.


Simple Data Structures: Stacks, queues, single and double linked lists, circular lists, trees, binary search

tree. C-implementation of stacks, queues and linked lists.


Algorithms for searching, sorting and merging e.g., sequential search, binary search, insertion sort, bubble

sort, selection sort, merge sort, quick sort, heap sort.

Text Books:

1. Balaguruswami, Programming in C, Tata McGraw- Hill. 2. Y.P. Kanetkar, Let us C, BPB, India.

Reference Books:

1. Brian Kernighan and Dennis Ritchie, The C- programming Language, Prentice Hall of India Pvt.

Ltd.


M.Sc. – Mathematics Ist Year: Semester I INDUSTRIAL MANAGEMENT

Course code: MHM 101 L T P C 2 0 0 2


General Management: Principles of scientific management; Brief description of managerial functions.

Business Organizations: Salient features of sole Proprietorship, Partnership, Joint stock Company– rivate

and public limited.


Financial Management: Concept of interest; Compound interest; Present worth method, Future worth

method. Depreciation–purpose, Types of Depreciation; Common methods of depreciation-Straight line

method, Declining balance method, Sum of the years digits method.


Personnel Management: Leadership and motivation; Staff role of the personnel department; Personnel

functions; Organizational structure.

Human Resource Planning: Reasons for human resource planning; Planning process; Goals and plans of

the organizations; Implementation programs; Brief description of recruitment, selection, placement,

performance appraisal, career development, promotion, transfer, retirement, training and development,

motivation and compensation.


Material Management: Importance; Definition; Source selection, Vendor rating and Value analysis; Scope

of MRP.

Inventory Control: Definition, objectives, reasons, and requirements for inventory management;

Inventory methods - ABC Analysis, VED.

Economic Order Quantity models - Basic EOQ, Economic production run size and Quantity discounts.


Marketing Management: Product life cycle; Channels of distribution; Advertising & sales promotion;

Market Research.

Managing Marketing Effort: Marketing implementation and evaluation; Appraisal and prospects.

Text books:

1. K. K. Ahuja, Industrial Management, Vol. I & II, Khanna Publisher.

2. E.Paul Degarmo, John R.Chanda, William G.Sullivan, Engineering Economy, Mac Millan

Publishing Co.

Reference Books:

1. Philip Kotler, Principles of Marketing Management, Prentice Hall.

2. P. Gopalakrishnan, M. Sundaresan, Materials Management, Prentice Hall of India Ltd.

3. Koontz & Weirich, Management, McGraw-Hill.

http://www.google.co.in/search?tbo=p&tbm=bks&q=inauthor:%22P.+Gopalakrishnan%22

http://www.google.co.in/search?tbo=p&tbm=bks&q=inauthor:%22M.+Sundaresan%22



Programming in C & Data Structures Lab

Course code: MAT 151 L T P C 0 0 4 2 Write programs in C:

1. To search an element in array using Linear search.

2. To search an element in the 2-diamensional array using Linear search.

3. To merge two sorted array into one sorted array.

4. To perform the following operation in Matrix-

a. 1. Addition 2. Subtraction 3. Multiplication 4. Transpose.

5. To perform the swappingof two numbers using call by value and cell by reference.

6. To perform the following operation on strings using strings functions-

b. 1. Addition 2.Copying 4.Reverse 5.Lenght of string.

7. To search an element in the array using Iterative Binary search.

8. To search an element in the array using Recursive Binary search.

9. To implement Bubble sort.

10. To implement selection sort.

11. To implement Insertion sort.

12. To implement Quick sort.

13. To implement Merge sort.

14. To implement Stack using array.

15. To implement Queue using array.

16. To implement Linked List.

Evaluation of Practical Examination:

Internal Evaluation (50 marks) Each experiment would be evaluated by the faculty

concerned on the date of the experiment on a scale of 5 which would include the practical

conducted by the students and a Viva taken by the faculty concerned. The marks shall be entered

on the index sheet of the practical file.

Evaluation Scheme: Evaluation scheme:

PRACTICAL PERFORMANCE & VIVA

DURING THE SEMESTER (30 MARKS)

ATTENDANCE

(5 MARKS)

QUIZ

(5 MARKS)

VIVA

(10 MARKS)

TOTAL

INTERNAL

(50 MARKS) EXPERIMENT

(10 MARKS)

FILE WORK

(10 MARKS)

VIVA

(10 MARKS)

External Evaluation (50 marks)

The external evaluation would also be done by the external Examiner based on the experiment

conducted during the examination.

EXPERIMENT

(20 MARKS)

FILE WORK

(10 MARKS) VIVA

(20 MARKS) TOTAL EXTERNAL

(50 MARKS)


M.Sc. – Mathematics Ist Year: Semester II

Complex Analysis

Course code MAT 201 L T P C

3 1 0 4


Functions of complex variables; Limit and continuity, Differentiability; Power Series as an

analytic function, Exponential and Trigonometric functions, Complex Logarithms, Zeros of

analytic functions.

Unit II (Lectures 08) Complex integration, curves in the complex plane , basic properties of complex integrals

winding number of a curve; Cauchy–Goursat Theorem, Cauchy’s Integral formula, Morera’s

Theorem, Laurent’s series Maximum modulus principle, Schwarz lemma, Liouville’s theorem.


Isolated singularities, removable singularity, poles, Singularity at infinity calculus of residue at

finite point, residue at the point at in finite residue theorem, Number of zeros, Poles, Rouche’s

Theorem.


Bilinear transformations, their properties and classifications, Definitions and examples of

conformal mappings; spaces of analytic functions, Hurwitz’s theorem, Montel’s theorem,

Riemann mapping theorem; Mobius transformations.

Unit V (Lectures 08) Hyper-geometric Series, Generalized Hyper-geometric functions, Gamma function and its

properties, Riemann Zeta function, Riemann’s functional equation.

Text Books:

1. J.B. Conway, Narosa Complex Analysis, Publishing House.

2. Ruel V. Churchill, Complex Variables and Applications, Tata McGraw-Hill.

3. Foundation of Complex Analysis, S. Ponnusamy, Narosa Publishing House.

Reference Books:

1. H.A. Priestly, Introduction to Complex Analysis, Clarendon Press, Oxford.

2. J.B. Conway, Function of one Complex Variable, Springer-Verlag.

3. L.V. Ahlfors, Complex Analysis, McGraw-Hill.

4. Walter Rudin, Real and Complex Analysis, McGraw-Hill.



Advance Abstract Algebra


3 1 0 4


Groups–Properties, Examples; subgroups, cyclic groups, homomorphism of groups and

Lagrange’s theorem; permutation groups, permutations as products of cycles, even and odd

permutations, normal subgroups, quotient groups, isomorphism theorems, correspondence

theorem.

Unit II (Lectures 08) Group action; Cayley's theorem, group of symmetries, dihedral groups and their elementary

properties; orbit decomposition; counting formula; class equation, consequences for p-groups;

Sylow’s theorems.

Unit III (Lectures 08) Applications of Sylow’s theorems, conjugacy classes in Sn and An, simplicity of An. Direct

product; structure theorem for finite abelian groups; invariants of a finite Abelian group.


Basic properties and examples of ring, domain, division ring and field; direct products of rings,

characteristic of a domain, field of fractions of an integral domain; ring homomorphisms (always

unitary); ideals, factor rings, prime and maximal ideals, principal ideal domain; Euclidean

domain, unique factorization domain.

Unit V (Lectures 08) A brief review of polynomial rings over a field; reducible and irreducible polynomials, Gauss’

theorem for reducibility of f(x) ∈ Z[x]; Eisenstein’s criterion for irreducibility of f(x) ∈ Z[x]

over Q, roots of polynomials; finite fields of orders 4, 8, 9 and 27 using irreducible polynomials

over Z2 and Z

3.

Text Books:

1. I.N. Herstein, Topics in Algebra, Wiley Eastern Ltd.

2. M. Artin, Algebra, Prentice-Hall of India.

3. N. Jacobson, Basic Algebra, Hindustan Publishing Corporation.

Reference Books:

1. Maclane and Birkhoff, Algebra, Macmillan Company.

2. S.Lang Addision, Linear Algebra, Wesley.

3. Hofmann and Kunz, Linear Algebra, Prentice Hall.



Function Analysis


3 1 0 4


Normed linear spaces, Banach spaces, Examples and counter examples, Quotient space of

normed linear spaces and its completeness; Equivalent norms.


Reisz Lemma, Basic properties of finite dimensional normed linear spaces; Bounded linear

transformations and normed linear spaces of bounded linear transformations; Uniform

boundedness theorem and some of its applications.


Dual spaces, weak convergence, open mapping and closed graph theorems; Hahn Banch theorem

for real and complex linear spaces.


Inner product spaces, Hilbert spaces–Orthonormal sets; Bessel’s inequality, complete

orthonormal sets and Perseval ’s identity.


Structure of Hilbert spaces, Projection theorem, Riesz representation theorem, Adjoint of and

operator on Hilbert space, Self adjoint operators, Normal and Unitary operators. Projections

Text Books:

1. E. Kreyszig, Functional Analysis and its application, John Wiley and sons.

2. J.N. Sharma & A. R. Vashistha, Functional Analysis, Krishana Publication.

Reference Books:

1. G. Bachman & L.Narici, Functional Analysis Academic Press.

2. H.C. Goffman and G.Fedrick, First course in Functional Analysis, Prentice Hall of India.

3. B.V. Limaye, Functional Analysis, New Age International Limited.



Numerical Technique (NT)


Unit I (Lectures 08) Errors in numerical calculations: Absolute, Relative and percentage errors, A general error formula, Error

in a series approximation; Solutions of algebraic & transcendental equations: The Bisection method, The

iteration method, Regula-Falsi method, Secant method, Newton- Raphson method

Unit II (Lectures 08) Interpolation: Errors in Polynomial interpolation; Finite differences: Forward, Backward and Central

differences, Symbolic relations, Difference of polynomial, Newton’s formulae of interpolation, Central

difference interpolation formulae: Gauss’s , Bessel’s & Stirling’s formulae, Interpolation with unevenly

spaced points: Lagrange’s interpolation formula, Interpolation with cubic splines, Divided differences and

their properties, Newton’s general interpolation formula, Inverse interpolation, Method of successive

approximations.

Unit III (Lectures 08) Numerical differentiation and integration: Forward, Backward and Central difference formulae for first

and second order derivatives; Errors in numerical differentiation; Numerical integration, Trapezoidal rule;

Simpson’s 1/3 rule, Simpson’s 3/8 rule; Boole’s and Weddle’s rules; Newton’s-Cotes integration

formulae.

Unit IV (Lectures 08) Numerical solution of ordinary differential equations: Taylor’s series, Picard’s successive

approximations, Euler’s, Modified Euler’s, Runge-Kutta & Milne’s Predictor-Corrector methods;

Simultaneous and higher order equations: Taylor’s series method and Runge-Kutta method, Boundary

value problems: Finite differences method.

Unit V (Lectures 08) Numerical solution of partial differential equations: Finite difference approximations to derivatives;

Laplace’s equation: Jacobi’s method, Gauss Seidel method, The ADI method; Parabolic equations:

Explicit scheme, C-N scheme; Hyperbolic equations: Explicit scheme, Implicit scheme.

Text Books:

1. S.S. Sastry, Introductory Methods of Numerical Analysis, Prentice Hall of India.

2. Grewal B. S, Numerical Methods in Engineering and Science, Khanna Publishers.

Reference Books:

1. . M.K. Jain, S.R.K Iyengar & R.K.Jain, Numerical methods of Scientific and Engineering

Computation, New Age Pub.



Differential Equations


3 1 0 4


Linear equations with constant coefficients; the second and higher order homogeneous equation; initial

value problems for second order equations; existence theorem; uniqueness theorem; linear dependence

and independence of solutions; the Wronskian and linear independence; a formula for the Wronskian; the

non-homogeneous equation of order two.

Unit II (Lectures 08) Linear equations with variable coefficients, initial value problems for the homogeneous equations;

existence theorem; uniqueness theorem; solutions of homogeneous equations; the theorem on n linearly

independent solutions; the Wronskian and linear independence.


Linear equations with regular singular points – introduction; Euler equation; second order equations with

regular singular points – example and the general case, convergence proof, exceptional cases; Bessel

equation; regular singular points at infinity.


Initial value problems for the homogeneous equations; solutions of homogeneous equations; Wronskian

and linear independence; non-homogeneous equations; homogeneous equations with analytic coefficients;

Legendre equation, justification of power series method; Legendre polynomials and Rodrigues’ formulae.

Unit V (Lectures 08) Existence and uniqueness of solutions – introduction; equations with variable separated; exact equations,

Lipschitz condition; non-local existence of solutions; uniqueness of solutions; existence and uniqueness

theorem for first order equations; statement of existence and uniqueness theorem for the solutions of

ordinary differential equation of order n.

Text Books:

1. E. A. Coddington, An Introduction to Ordinary Differential Equations, Prentice-Hall of India.

2. G.F. Simmons, Differential equations with applications and historical note, Tata McGraw Hill.

3. M. Rama Mohana Rao, Ordinary Differential Equations, East-West Press.

Reference Books:

1. G. Birkhoff and G.C. Rota, Ordinary differential equations, John Wiley and Sons.

2. S. G. Deo, V. Lakshmikantham, V. Raghvendra. Text book of ordinary Differential Equations

Tata Mc-Graw Hill.


M.Sc. – Mathematics Ist Year: Semester II ORGANISATIONAL BEHAVIOUR

Course Code: MHM 201 L T P C

2 0 0 2 Course Content

Unit – I (Lecture 06)

Concept, Nature, Characteristics, Models of Organizational Behaviour, Management Challenge,

Organizational Goal. Global challenges and Impact of culture.

Unit – II (Lecture 06)

Perception: Concept, Nature, Process, Importance. Attitudes and Workforce Diversity.

Personality: Concept, Nature, Types and Theories of Personality Shaping, Learning: Concept

and Theories of Learning.

Unit – III (Lecture 06)

Motivation: Concepts and Their Application, Principles, Theories, Motivating a Diverse

Workforce.

Leadership: Concept, Function, Style and Theories of Leadership-Trait, Behavioural and

Situational Theories. Analysis of Interpersonal Relationship.

Unit – IV (Lecture 06)

Organizational Power and Politics: Concept, Sources of Power, Approaches to Power,

Political Implications of Power. Knowledge Management & Emotional Intelligence in

Contemporary Business Organization.

Organizational Change: Concept, Nature, Resistance to change, Managing resistance to

change, Implementing Change.

Unit –V (Lecture 06)

Conflict: Concept, Sources, Types, Functionality and Dysfunctional of Conflict, Classification

of Conflict Intra, Individual, Interpersonal, Intergroup and Organizational, Resolution of

Conflict, Stress: Understanding Stress and Its Consequences, Causes of Stress, Managing Stress.

Text Books:

1. Dwivedi, D. N, Managerial Economics, Vikas Publishing House.

2. Varshney & Maheshwari, Managerial Economics, Sultan Chand & Sons.

Reference Books:

1. Robbins Stephen P., Organizational Behavior Pearson Education

2. Hersey Paul, “Management of Organsational Behavior: Leading Human Resources”

Blanchard, Kenneth H and Johnson Dewey E., Pearson Education

3. Khanka S. S. “Organizational Behavior



Numerical Technique Lab

Course code MAT 251 L T P C 0 0 4 2 Write programs in C:

1. To implement floating point arithmetic operations i.e., addition, subtraction,

multiplication and division.

2. To deduce errors involved in polynomial interpolation.

3. Algebraic and transcendental equations using Bisection, Newton Raphson, Iterative

method of false position, rate of conversions of roots in tabular form for each of these

methods.

4. To implement formulae by Bessel’s and Stirling etc.

5. Gauss Interpolation, flowchart C program and output.

6. Implement numerical differentiation.

7. Implement numerical integration using Simpson's 1/3 and 3/8 rules.

8. Implement numerical integration using trapezoidal rule.

9. Solution of differential equations using 4th

order Runge-Kutta method.

10. Numerical solution of ordinary first order differential equation -Euler’s method with

algorithm, flowchart C Program and output.

11. Newton’s and Lagrange’s interpolation with algorithm, flowchart C Program and output.

12. Iteration method, flowchart C program and output.

Evaluation of Practical Examination:

Internal Evaluation (50 marks)

Each experiment would be evaluated by the faculty concerned on the date of the experiment on a 5

point scale which would include the practical conducted by the students and a Viva voce taken by the

faculty concerned. The marks shall be entered on the index sheet of the practical file.

Evaluation scheme:

PRACTICAL PERFORMANCE & VIVA

DURING THE SEMESTER (30 MARKS)

ATTENDANCE

(5 MARKS)

QUIZ

(5 MARKS)

VIVA

(10 MARKS)

TOTAL

INTERNAL

(50 MARKS) EXPERIMENT

(10 MARKS)

FILE WORK

(10 MARKS)

VIVA

(10 MARKS)

External Evaluation (50 marks)

The external evaluation would also be done by the external Examiner based on the experiment

conducted during the examination.

EXPERIMENT

(20 MARKS)

FILE WORK

(10 MARKS) VIVA

(20 MARKS) TOTAL EXTERNAL

(50 MARKS)


M.Sc. – Mathematics IInd Year: Semester III

Topology


3 1 0 4

UNIT I (Lectures 08) Metric space, Open sets, closed sets, Convergence, Completeness, Continuity in metric space,

Cantor intersection theorem.

UNIT II (Lectures 08) Topological space, Elementary concept, Basis for a topology, Open and closed sets, Interior and

closure of sets, Neighborhood of a point, Limits points, Boundary of a set, Subspace topology ,

Weak topology, Product topology, Quotient topology.

UNIT III (Lectures 08) Continuous maps, Continuity theorems for Open and closed sets, Homeomorphism, Connected

spaces, Continuity and connectedness, Components, Totally disconnected space, Locally

connected space, Compact space, Limit point compact, Sequentially compact space, Local

compactness, Continuity and compactness, Tychonoff theorem.

UNIT IV (Lectures 08) First and second countable space, T

1 spaces, Hausdorff spaces, Regular spaces, Normal spaces,

Completely normal space, Completely regular space, Tietz- Extention theorem, Metrizability,

Uryshon Lemma, Uryshon metrization theorem.

UNIT V (Lectures 08) Fundamental group function, Homotopy of maps between topological spaces, Homotopy

equivalence, Contractible and simple connected spaces, Fundamental groups of S1, and S1x S1

etc., Calculation of fundamental groups of Sn , n>1 using Van Kampen’s theorem , Fundamental

groups of a topological group.

Text Books:

1. James R. Munkres, Topology, Pearson Education Pvt. Ltd.

2. J. R Munkres, Topology A First Course, Prentice- Hall.

3. J.L. Kelly, General Topology, Van Nostrand, Reinhold Co.

Reference Books:

1. G.F. Simmons, Introduction to Topology and Mordern Analysis.

2. K. D. Joshi, Introduction to General Topology, Wiley Eastern Limited.

3. L. A. Steen and J. A. Seebach Jr, Counterexamples in Topology, Holt Rinehart and

Winston.



Partial Differential Equations


3 1 0 4


Examples of PDE. Classification, Transport Equation: Initial value Problem, Non-homogeneous

Equation. Laplace's Equation: Fundamental Solution, Mean Value Formulas, Properties of Harmonic

Functions, Energy Methods.


Heat Equation: Fundamental Solution; Mean Value Formula, Properties of Solutions, Energy Methods.

Wave Equation: Solution by Spherical Means. Non-homogeneous Equations, Energy Methods.


Nonlinear First Order PDE-Complete Integrals, Envelopes, Characteristics; Hamilton –Jacobi Equations

(Calculus of Variations, Hamilton's ODE, Legendre Transform, Hopf-Lax Formula, Weak Solutions,

Uniqueness), Conservation Laws ( Rankine-Hugoniot condition, Lax-Oleinik formula, Weak Solutions,

Uniqueness).

Unit IV (Lectures 08) Representation of Solutions-Separation of Variables, Similarity Solutions (Plane and Traveling Waves,

Solitons, Similarity Linder Scaling), Fourier and Laplace Transform, Hopf-Cole Transform, Hodograph

and Legendre Transforms. Potential Functions.


Deriving Difference Equations, Elliptic Equations: Solution of Laplace’s equation, Leibmann,s method,

relaxation method, solution of Poisson’s equation, Parabolic equation : solution of heat equation, Bender-

Schmidt method. The Crank-Nicholson method, Hyperbolic equations: solution of hyperbolic equation.

Text Books:

1. L.C. Evans, Partial Differential equations, Graduate Studies in Mathematics. Volume 19, AMS,

1998.

2. P. Prasad and R. Ravindran, Partial Differential equations, New Age International Pub. F. John,

Partial Differential equations, Springer- Verlag.

Reference Books:

1. W. E. William, Partial Differential equations, Clarendon press-oxford.

2. E. T. Copson, Partial differential equations, Cambridge university press.

3. I.N. Sneddon, Elements of partial differential equations, Mc-Graw Hill book company.



Mathematical Statistics


3 1 0 4


Random variable and sample space, notion of probability, axioms of probability, empirical approach to

probability, conditional probability, independent events, Bayes’ Theorem; probability distributions with

discrete and continuous random variables, joint probability mass function, marginal distribution function,

joint density function.


Mathematical expectation, moment generating function; Chebyshev’s inequality, weak law of large

numbers, Bernoullian trials; the Binomial, negative binomial, geometric, Poisson, normal, rectangular,

exponential, Gaussian, beta and gamma distributions and their moment generating functions; fit of a given

theoretical model to an empirical data.

Unit III (Lectures 08) Sampling and large sample tests, Introduction to testing of hypothesis, tests of significance for large

samples, chi-square test, SQC, analysis of variance,T and F tests; Theory of estimation, characteristics of

estimation, minimum variance unbiased estimator, method of maximum likelihood estimation.


Scatter diagram, linear and polynomial fitting by the method of least squares; linear correlation and

linear regression, rank correlation, correlation of bivariate frequency distribution.

Unit V (Lectures 08) Limit Theorems: Stochastic convergence, Bernoulli law of large numbers, Conv ergence of sequence of

distribution functions, Levy-Cramer Theorems, de-Moivre Theorem, Laplace Theorem, Poisson,

Chebyshev, Khintchine Weak law of large numbers, Lindberg Theorem, Lyapunov Theroem, Borel

Cantelli Lemma, Kolmogorov Inequality and Kolmogorov Strong Law of large numbers.

Text Books:

1. Robert V. Hogg and Allen T. Craig, Introduction to Mathematical Statistics, Macmillan

Publishing Co. Inc.

2. Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, American Mathematical

Society.

Reference Books:

1. Feller, W: Introduction to Probability and its Applications, Wiley Eastem Pvt. Ltd.

2. K.L.Chung, A course in Probability, Academic Press.

3. R.Durrett, Probability Theory and Examples, Duxbury Press.



Operation Research


3 1 0 4

UNIT I (Lectures 08) Introduction: Definition and scope of O.R., Different O.R. models, General methods for solving O.R.

models, Main characterization and phases of O.R., Linear programming and Simplex method with simple

problems, Two-phase and Big-M methods.

UNIT II (Lectures 08)

Inventory Management: Inventory control, Types of inventories, Cost associated with inventories, Factors

affecting inventory control, Single item deterministic problems with and without shortages, Inventory

control with price breaks, Inventory control for one period without setup cost with uncertain demands

(News paper boy type problem).

UNIT III (Lectures 08) Sequencing Theory: Introduction, Processing with n-jobs and two machines, n-jobs and three machines,

n-jobs and m- machines, Concept of jobs blocks; Non-linear Programming: Convex sets and convex

functions, Quadratic programming, Wolfe’s complementary pivot method and Beale’s methods.

UNIT IV (Lectures 08) Queuing Theory: Introduction, Characteristics of queuing systems, Poisson process and Exponential

distribution; Classification of queues, Transient and steady states; Poisson queues (M/M/1, M/M/C).

UNIT V (Lectures 08) Non-Poisson Queuing systems: (M/Ek/1) queuing systems; Replacement Problems: Replacement of items

that deteriorate gradually and value of money does not change with time; Replacement of items that fail

suddenly, Individual and group replacement policies.

Text Books:

1. S.D.Sharma Operation Research, Kedar Nath Ram Nath.

2. H.A. Taha, Operation Research An introduction, Macmillan Publishing Company.

Reference Books:

1. P.K.Gupta, Kanti Swarup & Man Mohan, Operation Research, Sultan Chand & Co.

2. R.L.Ackoff and N.W. Sasieni, Fundamental of Operations Research, John Willy.



Advanced Discrete Mathematics


3 1 0 4

UNIT I (Lectures 08) Propositional Calculus: Proposition and logical operations; Proposition and truth tables; Tautologies &

contradiction; Logical equivalence; Conditional & Bi-conditional statements; Logical implication;

Propositional functions & quantifiers. Semi groups & Monoids: Definitions and examples of semi groups

and monoids; Isomorphism & homomorphism of semi groups and monoids.

UNIT II (Lectures 08) Partially ordered set; Hasse diagram; External element of poset; Lattices as algebraic system; Sub lattices,

Isomorphic lattices, Bounded lattices; Complete, Compliment, Complemented lattices, Modular lattices,

Penthogonal lattices, Pentagonal.

UNIT III (Lectures 08) Boolean algebra: Definition, Principle of duality, Basic Theorems, Sub algebra, Isomorphic; Boolean

algebra as lattices; Boolean functions and min-terms; Disjunctive normal form; Complete disjunctive

normal form; Conjugate normal form.

UNIT IV (Lectures 08) Language & Grammar and their types: Regular expressions and Regular sets; Regular language, Finite

state Automata.

UNIT V (Lectures 08) Finite state Machine, Semi-Machines and languages; Application of Logic Circuit: Sum-of products form

for Boolean algebra; Minimal Boolean expressions, Prime implicants, Logic and Circuits; Boolean

functions, Karnaugh map.

Text Books:

1. B.Colman, R.C. Busby & S.Ross, Discrete Mathematical Structures, Prentice Hall of India Pvt.

Ltd.

2. Schaum’s series Discrete Mathematics, Tata Mc-Graw- Hill Edition.

Reference Books:

1. Susanna S. Epp, Discrete Mathematics with Applications, Thomson Learning TM.


M.Sc. – Mathematics IInd Year: Semester III PROJECT MANAGEMENT

Objective: To impart knowledge of project management to the students.


Fundamental Topics: Introduction to Project Management and Corporate Planning Process; Corporate

Financial Objectives; Time Value of Money; Future Value and Present Value of Multi-period Cash Flow;

Interest Rate.


Concept Stage: Strategic Investment Decisions and Project Ideas; Project Feasibility Study; Demand

Forecasting Techniques; Project Financing; Forms of Business Organization.

Unit III (Lectures06)

Analysis Stage: Cost-Benefit Analysis; Financial Analysis; Required Rate of Return from Projects;

Economic and Social Cost-Benefit Analysis; Project Portfolio Risk; Project Risk Analysis; Framework of

Project Risk Management.

Unit IV (Lectures06)

Planning, Execution and Completion Stage: Introduction to PERT & CPM; Allocation of Limited Capital

- Capital Rationing; Project Planning and Control Project Execution and Control; Post Completion Audit

(PCA).


Special Topics: Inflation and Project Investment; Economic Life of Projects and Replacement Policy;

Infrastructure Projects; International Capital Budgeting.

Text and ReferenceBook:

1. Bhavesh Patel, PROJECT MANAGEMENT: Financial Evaluation with Strategic Planning,

Networking and Control, Vikas Publishing House Pvt Ltd.

Course Code MHM 302 L T P C

2 0 0 2


M.Sc. Mathematics IInd Year : IIIrd Semester

Minor Research Project and Seminar

Course code MAT- 391 L T P C

0 0 4 2

For students to enter into preliminary research field both in theory and experiment the concept of

Project has been introduced in the final Semester. In the Project the student will explore new

developments from the books and journals, collecting literature / data and write a Dissertation

based on his / her work and studies. The Project Work can also be based on experimental work in

industries / research laboratories.

Selection of Topic:

1. Students will make project which should be preferably a working of third thoughts based on

their subject.

2. The student will be assigned a faculty guide who good the supervisor of the students. The

faculty would be identified before the end of the III semester.

3. The assessment of performance of the students should be made at least twice in the

semester. Internal assessment shall be for 250 marks. The students shall present the final

project live using overhead projector PowerPoint presentation on LCD to the internal

committee and the external examiner.

4. The evaluation committee shall consists of faculty members constituted by the college

which would be comprised of at least three members comprising of the department

Coordinator’s Class Coordinator and a nominee of the Dirtier. The students guide would be

special in bitted to the presentation. The seminar session shall be an open house session. The

internal marks would be the average of the marks given by each members of the committee

separately to the director in a sealed envelope.

The Marking shall be as follows.

Internal : 50 marks

By the Faculty Guide – 25 marks

By Committee appointed by the Director – 25 marks

External : 50 marks

By External examiner by the University – 50


Seminar

Selection of Topic:

1. All students pursuing M.Sc. shall select and propose a topic of the seminar in the first

week of the semester. Care should be taken that the topic selected is not directly related

to the subjects of the course being pursued or thesis work, if any. The proposed topic

should be submitted to the course coordinator.

2. The course coordinator shall forward the list of the topics to the coordinator of concerned

department, who will consolidate the list including some more topics, in consultation

with the faculty of the department. The topics will then be allocated to the students along

with the name of the faculty guide and also forwarded to the director for approval.

3. On approval by the Director, the list shall be displayed on the notice board and the

students will also be accordingly informed by the course coordinator within three weeks

of the commencement of the semester.

Preparation of the Seminar:

1. The student shall meet the guide for the necessary guidance for their preparation for the

seminar.

2. During the next two to four weeks the student will read the primary literature related to

the topic under the guidance of supervisor.

3. After necessary collection of data and literature survey, the students must prepare a

report. The report shall be arranged in the sequence as per following format & lay out

plan :-

a. Top Sheet of transparent plastic.

b. Top cover.

c. Preliminary pages.

(i) Title page

(ii) Certification page.

(iii) Acknowledgment.

(iv) Abstract.

(v) Table of Content.

(vi) List of Figures and Tables.

(vii) Nomenclature.

d. Chapters (Main Material).

e. Appendices, If any.

f. Bibliography/ References.

g. Evaluation Form.

h. Back Cover (Blank sheet).

i. Back Sheet of Plastic (May be opaque or transparent).

a. Top Cover- The sample top cover shall be as Under:


TITLE OF THE SEMINAR

NAME OF THE STUDENT WITH COURSE, STREAM, SEMESTER & SECTION.

Department of Applied Science

College of Engineering

Teerthanker Mahaveer University

Moradabad-244001

MONTH AND YEAR


b. Title Page:- The Title Page cover shall be as Under:

Title of the seminar

(Submitted in Partial fulfillment of the requirement for the degree of

BACHELOR OF SCIENCE

in

Mathematics

by

Name of Student in capital Letters

(Roll No.)



Moradabad-244001

MONTH AND YEAR


c. Certification page:- This shall be as under

Department of Applied Science


Teerthankar Mahaveer University

Moradabad-244001

The seminar Report and Title “Name of the Topic of the Seminar.” Submitted by Mr./Ms.

(Name of the student) (Roll No.) may be accepted for being evaluated-

Date Signature

Place (Name of guide)

Note:

For Guide: If you choose not to sign the acceptance certificate above, please indicate reasons

for the same from amongst those given below:

i) The amount of time and effort put in by the student is not sufficient;

ii) The amount of work put in by the student is not adequate;

iii) The report does not represent the actual work that was done / expected to be done;

iv) Any other objection (Please elaborate)

d. Abstract:- A portion of the seminar grade will be based on the abstract. The abstract will be

graded according to the adherence to accepted principles of English grammar and according to

the adherence to the format described below.

The seminar abstract is an important record of the coverage of your topic and provides a valuable

source of leading references for students and faculty alike. Accordingly, the abstract must serve

as an introduction to your seminar topic. It will include the key hypotheses, the major scientific

findings and a brief conclusion. The abstract will be limited to 500 words, excluding figures,

tables and references. The abstract will include references to the research articles upon which

the seminar is based as well as research articles that have served as key background material.


(e). Table of Content:- This shall be as under

SAMPLE SHEET FOR TABLE OF CONTENTS

TABLE OF CONTENTS

Chapter No Title Page No.

Certificate ii

Abstract iii

Acknowledgement iv

List of Figures v

List of Table vi

1 Introduction 1

1.1

1.2

1.3

2 …………………..

3 …………………..

4 References/Bibliography

5 Evaluation sheets ……..


f. List of Figures and Tables :- This will be as under

List of Figures and Tables - sample entries are given below:

List of Figures

Figure No. Caption / Title Page

No.

2.1 Schematic representation of a double layered droplet . . . 21

…………..

3.2 Variation in rate versus concentration . . . 32

List of Tables - sample entries are given below:

List of Tables

Table No. Caption / Title Page

No.

2.1 Thickness of a double layered droplet . . . 22

…………….



(g). Main Pages- The Main report should be divided in chapters (1, 2, 3 ….. etc.) and

structured into sections (1.1, 1.2 ……..etc) and subsections (1.2.1, 1.2.2, ….. etc).

Suitable title should be given for sections and subsections, where necessary.

Referencing style- wherever reference is given in the main pages it should have the

following format.

The values of thermal conductivities for a variety of substances have been reported by

Varma (1982). For polymers, however, the information is more limited and some recent

reviews have attempted to fill the gaps (Batchelor and Shah, 1985).

For two authors - (Batchelor and Kapur, 1985)

For more than two authors - (Batchelor et al., 1986)

By same author/combination of authors in the same year -

(Batchelor, 1978a; Batchelor, 1978b; Batchelor et al., 1978)

(h) Bibliography/References- In the bibliography/ references list standard formats must

be used. The typical formats are given blow-

Journal articles: -

David, A.B., Pandit, M.M. and Sinha, B.K., 1991, "Measurement of surface viscosity by

tensiometric methods", Chem. Engng Sci.47, 931-945.

Books: -

Doraiswamy, L.K. and Sharma, M.M., 1984, "Heterogeneous Reactions-Vol 1", Wiley,

New

York, pp 89-90.

Edited books/Compilations/Handbooks: -

Patel, A.B., 1989, "Liquid -liquid dispersions", in Dispersed Systems

Handbook, Hardy, L.C. and Jameson, P.B. (Eds.), McGraw Hill, Tokyo, pp 165-178.

Lynch, A.B. (Ed.), 1972, "Technical Writing", Prentice Hall, London.


Theses/Dissertations: -

Pradhan, S.S., 1992, "Hydrodynamic and mass transfer characteristics of packed

extraction columns", Ph.D. Thesis, University of Manchester, Manchester, U.K..

Citations from abstracts: -

Lee, S. and Demlow, B.X., 1985, US Patent 5,657,543, Cf C.A. 56, 845674.

Personal Communications: -

Reddy, A.R., 1993, personal communication at private meeting on 22 October 1992 at

Physics Department, Indian Institute of Technology, Delhi.

Electronic sources (web material and the like)- For citing web pages and electronic

documents, use the APA style given at:

http://www.apastyle.org/elecsource.html

(I) Evaluation Form:- Three sheets of evaluation form should be attached in the

report as under.

a. Evaluation form for guide and other Internal Examiner.

b. Evaluation form for external examiners.

c. Summary Sheet.

(J). Evaluation form for Guide & Internal Examiners:-


EVALUATION SHEET

(To be filled by the GUIDE & Internal Examiners only)

Name of Candidate:

Roll No:

Class and Section:

Please evaluate out of Five marks each.

S. No. Details Marks (5) Marks (5) Marks (5)

Guide Int. Exam.

1

Int. Exam.

2

1. OBJECTIVE IDENTIFIED &

UNDERSTOOD

2. LITERATURE REVIEW /

BACKGROUND WORK

(Coverage, Organization, Critical

review)

3. DISCUSSION/CONCLUSIONS

(Clarity, Exhaustive)

4. SLIDES/PRESENTATION

SUBMITTED

(Readable, Adequate)

5. FREQUENCY OF

INTERACTION ( Timely

submission, Interest shown, Depth,

Attitude)

Total (Out of 25)

Average out of 50

Signature: Signature: Signature:

Date: Date: Date:


EVALUATION SHEET FOR EXTERNAL EXAMINER

(To be filled by the External Examiner only)

Name of Candidate:

Roll No:

I. For use by External Examiner ONLY

S.No. Details Marks (5)

1. OBJECTIVE IDENTIFIED & UNDERSTOOD

2. LITERATURE REVIEW / BACKGROUND

WORK

(Coverage, Organization, Critical review)



4. POWER POINT PRESENTATION

(Clear, Structured)

5. SLIDES


Total (Out of 50)

Signature:

Date:


EVALUATION SUMMARY SHEET

(To be filled by External Examiner)

Name and Roll

No.

Internal

Examiners

(50)

External

Examiner

(50)

Total (100)

Result

(Pass/Fail)

Note:- The summary sheet is to be completed for all students and the same shall also be

Compiled for all students examined by External Examiner. The Format shall be provided

by the course coordinator.

(K). General Points for the Seminar

1. The report should be typed on A4 sheet. The Paper should be of 70-90 GSM.

2. Each page should have minimum margins as under-

(i) Left 1.5 inches

(ii) Right 0.5 Inches

(iii) Top 1 Inch

(iv) Bottom 1 Inch (Excluding Footer, If any)

3. The printing should be only on one side of the paper

4. The font for normal text should Times New Roman, 14 size for text and 16 size for

heading

and should be typed in double space. The references may be printed in Italics or in a

different

fonts.

5. The Total Report should not exceed 50 pages including top cover and blank pages.

6. A CD of the report should be pasted/ attached on the bottom page of the report.

7. Similarly a hard copy of the presentation (Two slides per page) should be attached along

with the report and a soft copy be included in the CD.

8. Three copies completed in all respect as given above is to be submitted to the guide.

One copy will be kept in departmental/University Library, One will be return to the

student and third copy will be for the guide.

9. The power point presentation should not exceed 30 minutes which include 10 minutes

for discussion/Viva.


M.Sc. – Mathematics IInd Year: Semester IV

Number Theory


3 1 0 4


The Division Algorithm, the gcd, The Euclidean Algorithm, Diophantine equation ax + by = c;

The fundamental theorem of arithmetic; The Sieve of Eratosthenes; The Goldbach conjecture.


Theory of Congruences – Basic properties of Consequence; Linear Congruences, Chinese

remainder theorem, Fermat’s Theorem, Wilson’s Theorem. Statement of Prime number

theorem. Some primality testing.


Number-Theoretic Functions – The functions T and Sigma; The mobius inversion formula; The

Greatest integer function, Euler’s Phi function – Euler Theorem, Properties of the Phi-function,

Applications to Cryptography.

Unit IV

(Lectures 08)

The order of an integer modulo n, Primitive roots for primes; The theory of indices, Euler’s

criterion, Legendre’s symbol and its properties; Quadratic reciprocity, Quadratic congruences

with composite moduli.


Perfect Numbers; Representation of integers as sum of two squares and sum of more than two

squares.

Text Books:

1. Davis M. Burton, Elementary Number Theory, USB.

Reference Books:

1. U. Dudley, Elementary Number Theory, Freeman & Co.

2. George Andrews, Number Theory, Courier Dover Publications.


M.Sc. – Mathematics IInd Year: Semester IV

Fuzzy Sets and its Applications


3 1 0 4

UNIT I (Lectures 08)

Crisp Sets, Fuzzy Sets (basic types), Fuzzy Sets (basic concepts); Representation of fuzzy sets;

Decompositions theorems; Extension principle for fuzzy sets.

UNIT II (Lectures 08) Operations on fuzzy sets (Fuzzy compliment, Intersection and union); Combinations of

operations.

UNIT III (Lectures 08) Fuzzy numbers, Linguistic variables; Arithmetic operations on fuzzy numbers; Lattice of fuzzy

numbers, Fuzzy equations.

UNIT IV (Lectures 08) Crisp and fuzzy relations; Projections; Binary fuzzy relations; Binary relations on a single set;

Fuzzy equivalence relations; Fuzzy compatibility relations; Fuzzy ordering relations; Fuzzy

morphism; Sup-i compositions of binary fuzzy relations; Inf-wi compositions of fuzzy relations.

UNIT V (Lectures 08) Fuzzy relation equations; Fuzzy logic; Fuzzy decision making; Fuzzy linear programming;

Linear Regression with fuzzy parameters; Fuzzy regression with fuzzy data.

Text Books:

1. George J. Klier and Bo Yuan, Fuzzy Sets and Fuzzy Logic, Prentice Hall of India.

2. H.J. Zimmerman, Fuzzy Set Theory and Its Applications, Kluwer Academic Publishers.

Reference Books:

1. Kaufmann, A. and Gupta, M.M. Fuzzy Mathematical Models in Engineering and

Management Science, Elsevier Science Inc.


M.Sc. Mathematics IInd Year : Semester – IV

Project Work, Seminar & Viva


0 0 20 10

For students to enter into preliminary research field both in theory and experiment the concept of

Project has been introduced in the final Semester. In the Project the student will explore new

developments from the books and journals, collecting literature / data and write a Dissertation

based on his / her work and studies. The Project Work can also be based on experimental work in

industries / research laboratories.

Selection of Topic:

5. Students will make project which should be preferably a working of third thoughts based on

their subject.

6. The student will be assigned a faculty guide who good the supervisor of the students. The

faculty would be identified before the end of the III semester.

7. The assessment of performance of the students should be made at least twice in the

semester. The students shall present the final project live using overhead projector

PowerPoint presentation on LCD to the internal committee and the external examiner in the

form of seminar.

8. The evaluation committee shall consists of faculty members constituted by the college

which would be comprised of at least three members comprising of the department

Coordinator’s Class Coordinator and a nominee of the Dirtier. The students guide would be

special in bitted to the presentation. The seminar session shall be an open house session. The

internal marks would be the average of the marks given by each members of the committee

separately to the director in a sealed envelope.

The Marking shall be as follows.

Internal : 50 marks

By the Faculty Guide – 25 marks

By Committee appointed by the Director – 25 marks

External : 50 marks

By External examiner by the University –50


Seminar

Selection of Topic: 4. All students pursuing M.Sc. shall select and propose a topic of the seminar in the first

week of the semester. Care should be taken that the topic selected is not directly related

to the subjects of the course being pursued or thesis work, if any. The proposed topic

should be submitted to the course coordinator.

5. The course coordinator shall forward the list of the topics to the coordinator of concerned

department, who will consolidate the list including some more topics, in consultation

with the faculty of the department. The topics will then be allocated to the students along

with the name of the faculty guide and also forwarded to the director for approval.

6. On approval by the Director, the list shall be displayed on the notice board and the

students will also be accordingly informed by the course coordinator within three weeks

of the commencement of the semester.

Preparation of the Seminar: 4. The student shall meet the guide for the necessary guidance for their preparation for the

seminar.

5. During the next two to four weeks the student will read the primary literature related to

the topic under the guidance of supervisor.

6. After necessary collection of data and literature survey, the students must prepare a

report. The report shall be arranged in the sequence as per following format & lay out

plan :-

a. Top Sheet of transparent plastic.

b. Top cover.

c. Preliminary pages.

(i) Title page

(ii) Certification page.

(iii) Acknowledgment.

(iv) Abstract.

(v) Table of Content.

(vi) List of Figures and Tables.

(vii) Nomenclature.

d. Chapters (Main Material).

e. Appendices, If any.

f. Bibliography/ References.

g. Evaluation Form.

h. Back Cover (Blank sheet).

i. Back Sheet of Plastic (May be opaque or transparent).


a. Top Cover- The sample top cover shall be as under:

TITLE OF THE SEMINAR

NAME OF THE STUDENT WITH COURSE, STREAM, SEMESTER & SECTION.

Department of Applied Sciences



Moradabad-244001

MONTH AND YEAR


b. Title Page:- The Title Page cover shall be as Under:

Title of the seminar

(Submitted in Partial fulfillment of the requirement for the degree of

BACHELOR OF SCIENCE

in

Mathematics

by

Name of Student in capital Letters

(Roll No.)



Moradabad-244001

MONTH AND YEAR


c. Certification page:- This shall be as under

Department of Applied Sciences


Teerthankar Mahaveer University

Moradabad-244001

The seminar Report and Title “Name of the Topic of the Seminar.” Submitted by Mr./Ms.

(Name of the student) (Roll No.) may be accepted for being evaluated-

Date Signature

Place (Name of guide)

Note:

For Guide: If you choose not to sign the acceptance certificate above, please indicate reasons

for the same from amongst those given below:

i) The amount of time and effort put in by the student is not sufficient;

ii) The amount of work put in by the student is not adequate;

iii) The report does not represent the actual work that was done / expected

to be done;

iv) Any other objection (Please elaborate)

d. Abstract:- A portion of the seminar grade will be based on the abstract. The abstract will be

graded according to the adherence to accepted principles of English grammar and according to

the adherence to the format described below.

The seminar abstract is an important record of the coverage of your topic and provides a valuable

source of leading references for students and faculty alike. Accordingly, the abstract must serve

as an introduction to your seminar topic. It will include the key hypotheses, the major scientific

findings and a brief conclusion. The abstract will be limited to 500 words, excluding figures,

tables and references. The abstract will include references to the research articles upon which

the seminar is based as well as research articles that have served as key background material.


(e). Table of Content:- This shall be as under

SAMPLE SHEET FOR TABLE OF CONTENTS

TABLE OF CONTENTS

Chapter No Title Page No.

Certificate ii

Abstract iii

Acknowledgement iv

List of Figures v

List of Table vi

1 Introduction 1

1.1

1.2

1.3

2 …………………..

3 …………………..

4 References/Bibliography

5 Evaluation sheets ……..


f. List of Figures and Tables :- This will be as under

List of Figures and Tables - sample entries are given below:

List of Figures

Figure No. Caption / Title Page No.

2.1 Schematic representation of a double layered droplet . . . 21

…………..


List of Tables - sample entries are given below:

List of Tables

Table No. Caption / Title Page No.

2.1 Thickness of a double layered droplet . . . 22

…………….



(g). Main Pages- The Main report should be divided in chapters (1, 2, 3 ….. etc.) and

structured into sections (1.1, 1.2 ……..etc) and subsections (1.2.1, 1.2.2, ….. etc).

Suitable title should be given for sections and subsections, where necessary.

Referencing style- wherever reference is given in the main pages it should have the

following format.

The values of thermal conductivities for a variety of substances have been reported by

Varma (1982). For polymers, however, the information is more limited and some recent

reviews have attempted to fill the gaps (Batchelor and Shah, 1985).

For two authors - (Batchelor and Kapur, 1985)

For more than two authors - (Batchelor et al., 1986)

By same author/combination of authors in the same year -

(Batchelor, 1978a; Batchelor, 1978b; Batchelor et al., 1978)

(h) Bibliography/References- In the bibliography/ references list standard formats must

be used. The typical formats are given blow-

Journal articles: -

David, A.B., Pandit, M.M. and Sinha, B.K., 1991, "Measurement of surface viscosity by

tensiometric methods", Chem. Engng Sci.47, 931-945.

Books: -

Doraiswamy, L.K. and Sharma, M.M., 1984, "Heterogeneous Reactions-Vol 1", Wiley, New

York, pp 89-90.

Edited books/Compilations/Handbooks: -

Patel, A.B., 1989, "Liquid -liquid dispersions", in Dispersed Systems

Handbook, Hardy, L.C. and Jameson, P.B. (Eds.), McGraw Hill, Tokyo, pp 165-178.

Lynch, A.B. (Ed.), 1972, "Technical Writing", Prentice Hall, London.


Theses/Dissertations: -

Pradhan, S.S., 1992, "Hydrodynamic and mass transfer characteristics of packed

extraction columns", Ph.D. Thesis, University of Manchester, Manchester, U.K..

Citations from abstracts: -

Lee, S. and Demlow, B.X., 1985, US Patent 5,657,543, Cf C.A. 56, 845674.

Personal Communications: -

Reddy, A.R., 1993, personal communication at private meeting on 22 October 1992 at

Physics Department, Indian Institute of Technology, Delhi.

Electronic sources (web material and the like)- For citing web pages and electronic

documents, use the APA style given at:

http://www.apastyle.org/elecsource.html

(II) Evaluation Form:- Three sheets of evaluation form should be attached in the

report as under.

a. Evaluation form for guide and other Internal Examiner.

b. Evaluation form for external examiners.

c. Summary Sheet.

(J). Evaluation form for Guide & Internal Examiners:-


EVALUATION SHEET

(To be filled by the GUIDE & Internal Examiners only)

Name of Candidate:

Roll No:

Class and Section:

S. No. Details Marks (5) Marks (5) Marks (5)

Guide Int. Exam.

1

Int. Exam.

2

1. OBJECTIVE IDENTIFIED &

UNDERSTOOD

2. LITERATURE REVIEW /

BACKGROUND WORK

(Coverage, Organization, Critical

review)



4. SLIDES/PRESENTATION

SUBMITTED


5. FREQUENCY OF

INTERACTION ( Timely

submission, Interest shown,

Depth, Attitude)

Total (Out of 25)

Average out of 50

Signature: Signature: Signature:

Date: Date: Date:


EVALUATION SHEET FOR EXTERNAL EXAMINER

(To be filled by the External Examiner only)

Name of Candidate:

Roll No:

I. For use by External Examiner ONLY

S.No. Details Marks (5)

1. OBJECTIVE IDENTIFIED & UNDERSTOOD

2. LITERATURE REVIEW / BACKGROUND

WORK

(Coverage, Organization, Critical review)



4. POWER POINT PRESENTATION

(Clear, Structured)

5. SLIDES


Total (Out of 50)

Signature:

Date:


EVALUATION SUMMARY SHEET

(To be filled by External Examiner)

Name and Roll No. Internal

Examiners

(50)

External

Examiner

(50)

Total (100) Result

(Pass/Fail)

Note:- The summary sheet is to be completed for all students and the same shall also be

Compiled for all students examined by External Examiner. The Format shall be provided

by the course coordinator.

(K). General Points for the Seminar

1. The report should be typed on A4 sheet. The Paper should be of 70-90 GSM.

2. Each page should have minimum margins as under-

(i) Left 1.5 inches

(ii) Right 0.5 Inches

(iii) Top 1 Inch

(iv) Bottom 1 Inch (Excluding Footer, If any)

3. The printing should be only on one side of the paper

4. The font for normal text should Times New Roman, 14 size for text and 16 size for heading

and should be typed in double space. The references may be printed in Italics or in a different

fonts.

5. The Total Report should not exceed 50 pages including top cover and blank pages.

6. A CD of the report should be pasted/ attached on the bottom page of the report.

7. Similarly a hard copy of the presentation (Two slides per page) should be attached along

with the report and a soft copy be included in the CD.

8. Three copies completed in all respect as given above is to be submitted to the guide.

One copy will be kept in departmental/University Library, One will be return to the

student and third copy will be for the guide.

9. The power point presentation should not exceed 30 minutes which include 10 minutes

for discussion/Viva.


Viva- voce

Students will prepare the viva, which should be based on their subject.

The student will be assigned a faculty guide who good the supervisor of the students. The

faculty would be identified before the end of the III semester. The faculty will take the full

responsibility for preparing the viva to the students.

The evaluation committee shall consists of faculty members constituted by the college which

would be comprised of at least three members comprising of the department Coordinator’s

Class Coordinator and a nominee of the Director. The students guide would be special invitee to

the viva. The viva session shall be an open house session. The internal marks would be the

average of the marks given by each members of the committee in a sealed envelope.

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