Desh Bhagat University



Ordinances

For

M.Sc. Mathematics

(Under Choice Based Credit System)

Session

(2019-20)

Ordinances

for the Master of Science in Mathematics

(Under Choice Based Credit System)

1. Duration of Course:

The duration of course shall be two academic years consisting of four (4) semesters i.e. two semesters in each year. The duration of each semester will be 18-20 weeks with ninety (90) teaching days.

2. Maximum period for passing M.Sc. Mathematics.

The candidate must pass all the subjects of all the semesters of M.Sc. Mathematics in four (4) years. If the candidate fails to pass all the subjects of the course within stipulated period, his/her registration will be cancelled.

3. Eligibility for admission

A candidate must have passed B.Sc. from any recognized University with 50% marks in aggregate. 5% relaxation in marks shall be given to Schedule Caste/ Schedule Tribe or any rural and under privileged candidates.

4. Medium of Instructions

The medium of instruction during the course and examinations shall be Punjabi/Hindi/English.

5. Examination Schedule, examination fee and examination forms:

1. The examination of Odd Semesters shall ordinarily be held in the month of December and that of Even Semesters in the month of May, or on such other dates as may be fixed by the competent authority.

2. The candidates will be required to pay examination fees as prescribed by the University from time to time.

3. The Examination Form must reach in the office of the Controller of Examinations as per the schedule notified, from time to time.

4. The Examination Forms must be countersigned by the Director/Head of the Department along with the following certificate :--

i) that he/she has been on the rolls of the University Teaching Department during the academic term preceding the end semester examination;

ii) that he/she has attended not less than 75% lectures delivered to that class in each paper; and

(iii) that he/she has a good moral character.

5. The shortage in the attendance of lectures of the candidate may be condoned by the Vice-Chancellor, on the recommendations of Head of the Department, as per rules.

6. Re-admission

In case name of a student is struck off from the rolls due to non-payment of fee or continued absence from classes in any subject for one month and he/she will be re-admitted after payment of re-admission fee as prescribed by the University from time to time. However, the student will be allowed to appear in the end semester examination of that paper (s) only after attending the required lectures/practical delivered to that paper(s). However, if a student falls short of attendance in all courses offered in a semester, he/she shall be required to repeat the semester, along with the next batch of students.

7. Scheme of Examinations

The examination in each semester shall be conducted according to the syllabus prescribed for the semester. The end semester examination for each paper shall be of three hours duration.

8. Minimum pass marks

The minimum number of marks required to pass in each semester shall be 40% marks in each in Theory and Practical/Laboratory/Seminar/Viva-Voce paper and in Internal Assessment, separately.

9. Grading of performances

1. Letter grades and grade points allocations:

Based on the performances, each student shall be awarded a final letter grade at the end of the semester for each course. The letter grades and their corresponding grade points are given here under: -

|Percentage of marks obtained |Letter Grade |Grade Point |Performance |

|90.00 – 100 |O |10 |Outstanding |

|80.00 – 89.99 |A+ |9 |Excellent |

|70.00 – 79.99 |A |8 |Very Good |

|60.00 – 69.99 |B+ |7 |Good |

|50.00 – 59.99 |B |6 |Average |

|40.00 – 49.99 |C |5 |Pass |

|Less than 40.00 |F |0 |Fail |

|Absent |AB |0 |Fail |

2. Grades from ‘O’ to ‘C’ are pass grades.

3. A student who fails in any end semester shall be assigned a letter grade ‘F’ and a corresponding grade point of zero. He/she should reappear for the said evaluation/examination in due course.

4. A student who remains absent for any end semester examination shall be assigned a letter grade of ‘AB’ and a corresponding grade point of zero.

Semester Grade Point Average (SGPA) = (∑ Ci Gi) ∕ (∑ Ci)

Where C i = No. of credits assigned to ith semester

G i = No. of Grade equivalent point assigned to ith semester.

∑ (SGPAj X Cj)

Cumulative Grade Point Average (CGPA) = ______________

∑ Cj

Where SGPAj = SGPA score of jth semester

Cj = Total no. of credits in the jth Semester

9.5 Percentage can be calculated as CGPA *10

10. Declaration of class and Division

The class shall be awarded on the basis of CGPA as follows:

|CGPA: ≥7.5 provided that the candidate must have passed all the Semester Examinations in |First Division with |

|the first available attempt. |Distinction |

|CGPA: 6.0 to 7.49 |First Division |

|CGPA: 5.0 to 5.99 |Second Division |

|CGPA: 4.0 to 4.99 |Third Division |

11. Internal Assessment of failed candidate

The internal assessment award of a candidate who fails in the external examination shall be carried forward to the next Examination, if passed in Internal Assessment.

12. Grace Marks

1. The grace marks of 1% of total marks of the semester shall be given to a candidate to his best advantage so as to enable him to pass in one or more written papers, to make up aggregate to pass the examination/paper or for changing the result from FAIL to COMPARTMENT/PASS. If a fraction works out to be half or more, it shall be counted as one mark and fraction less than half shall be ignored

2. If a candidate appears in an examination to clear re-appear/compartment paper, the grace marks of 1% will be given only on the total marks of that particular paper.

13. Re-evaluation

A candidate who is not satisfied with his result may apply to the Examination Branch for re-evaluation in a subject/paper within 15 days of declaration of result along with a fee as prescribed by the university from time to time.

14. Re-checking

A candidate who is not satisfied with his result may apply to the Examination Branch for re-evaluation in a subject/paper within 15 days of declaration of result along with a fee as prescribed by the university from time to time.

15. Special examination

A Special Examination will be conducted for those students who are passing out but having re-appear(s) in the last semester and/or in the lower semesters. The special examination will be conducted within one month of the declaration of final semester result. The student shall have to pay prescribed fee for Special Examination.

16. Re-appear/Supplementary examination

In case of re-appear examination, the University will adopt even/odd semester examination or open semester system. The student will be eligible to appear in the re-appear papers of odd semester along with the odd semester regular examinations of subsequent batches and re-appear of even semester’s paper of the even semester regular examinations in the case of even/odd semester examination. The student will be eligible to appear in the re-appear papers of all semesters (even/odd) along with regular examinations of open semester examinations. Controller of Examination will implement any of the above examination system with the approval of the Vice-Chancellor.

17. Mercy Chance

The candidate will be given maximum two chances to appear in the supplementary examinations. After that, mercy chance may be given by the Vice-Chancellor on the recommendations of the Director of the concerned school on payment of a special fee.

18. Syllabus for re-appear candidates

A student who obtains re-appear(s) in a subject will be examined from the same syllabus which he/she studied as a regular student.

19. Promotion Criteria

1. A candidate who joins First Semester of M.Sc. Mathematics may on completing attendance requirements appear in 1st semester examination. He/she shall be allowed to continue his/her studies in the 2nd Semester even if he/she does not clear any paper of the 1st semester and on completing attendance requirements may appear in the 2nd Semester examination.

2. A candidate shall not be eligible to join 3rd Semester of M.Sc. Mathematics if he/she has yet to clear more than 50% papers of First and Second Semesters taken together. A candidate who has cleared 50% or more papers of M.Sc. Mathematics 1st and 2nd Semesters taken together may join 3rd Semester and on completing attendance requirements may take 3rd Semester Examination. He/she shall be allowed to continue his/her studies in the 4th Semester even if he/she does not clear any paper of the 3rd Semester and on completing attendance requirements may appear in 4th Semester examination.

19. Division Improvement

A candidate who has passed M.Sc. Mathematics examination from this University may re-appear for improvement of division in one or more subjects in the succeeding semesters with regular candidates in order to increase the percentage for obtaining higher division. However, final year candidates who have passed an examination of the University may re-appear for improvement of performance under special examination as per rules of the university.

20. Migration to this University

1. Migration to this University will be allowed only after completion of the 1st year and is applicable only to those students who are eligible to register for 3rd semester.

2. Migration shall be allowed after completion of the second semester but before start of the 3rd semester.

3. The candidates shall not be allowed to change his/ her discipline of study in the process of migration.

4. Migration to an affiliated College /Institute of the University from other recognized universities will be allowed 15 days prior to of the start of the 3rd semester. The following conditions shall be applying: -

i) The candidate should have passed all the courses of the first year of the University from where he/she wants to migrate.

ii) The courses studied by the candidate in first year must be equivalent to the courses offered in this University. Deficiency, if any, should not be of more than two subjects. The candidate would be required to furnish an undertaking that he/she will attend classes and pass these courses (found deficient). The institute and the University where the student is studying and the Institute, to which migration is sought, have no objection to the migration.

iii) There is a vacant seat available in the discipline in the college in which migration is sought.

5. Power of Relaxation: Notwithstanding the existing Migration Rules, the Vice-Chancellor, after obtaining an undertaking/affidavit from the candidate, to his satisfaction, to be recorded in writing, shall be authorized to consider the migration for the cases that are not otherwise covered under the above Migration Rules, with the approval of the Chancellor.

21. Migration to any other University

1. Migration to any other University will be allowed 15 days prior to of the start of the 3rd semester.

2. The candidate seeking migration from this University shall be apply for the approval of his migration to the University within 15 working days after passing the 2nd Semester/First Year Examination.

3. The Director/Head of the department concerned of the University will issue “No Objection Certificate” after the candidate has paid all the fees due for the remaining period of the full session as well as the annual dues as per rules. In addition to the above, Migration fee as prescribed by the University shall be charged from such candidates.

4. If a candidate, on completion of any course, applies for Migration Certificate, the same shall be issued on receipt of fee prescribed for Migration Certificate and on completion of other formalities etc.

22. Award of Detail Marks Card

Each candidate of First Year M.Sc. Mathematics (i.e. Semester-I & Semester-II), Second Year (i.e. Semester-III & Semester-IV) and Third Year (i.e. Semester-V & Semester- VI), on successfully completion of course and passing all the papers of each semester, shall be supplied Detail of Marks Cards indicating CGPA score and Division obtained by him/her in the examination.

23. Award of Degree

The degree of Master of Science in Mathematics in the concerned stream stating the CGPA score and Division, will be awarded to the candidate who has successfully completed the course and passed all the papers of all the semesters. The degree will be awarded at the University Convocation. However, a degree in absentia can be issued before the convocation, on completion of required formalities and payment of prescribed fee.

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Scheme of Examinations

SEMESTER: I

|Sr. No. |

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1. |MSM-406 |Advanced Operation Research |EC |4 |0 |0 |40 |60 |100 |4 | | |MSM-407 |Graph Theory |EC |4 |0 |0 |40 |60 |100 |4 | |Total | |20 |0 |2 |300 |300 |600 |21 | |

Pattern of Question Papers w.e.f. 2019-20

The question paper covering the entire course shall be divided into three sections.

A) Multiple Choice Questions:

➢ This section will cover 20% marks of the total marks.

➢ All the questions will be compulsory.

➢ Each question will carry 1 (one) mark and there will be 4 (four) options in the answer to each question.

➢ There will be not negative marking

➢ OMR Sheets will be provided to students for attempting this section

B) Short Answer Type Questions:

➢ This section will cover 30% marks of the total marks.

➢ There will be choice in questions i.e. attempt 10 out of 12, 12 out of 15, 15 out of 18 etc.

➢ Each question will carry 2 (two) marks and the answer is to be given in about 150 words.

C) Essay Type Questions:

➢ This section will cover 50% marks of the total marks.

➢ To cover the whole syllabus the questions may be divided in two parts as follows

Q. a)

b)

OR

a)

b)

➢ Each part will carry weight-age of 5 (five) marks. The answer to each question is to be given in about 250 words.

******

Instructions to Paper Setters

Total marks 60

Section A: Multiple Choice Questions (12 marks)

Q.1 There will be 12 MCQs with four options of answers. All the questions will be compulsory and each question will carry 1 (one) mark. There will be no negative marking. OMR Sheets will be provided to students for attempting this section

Section B: Short Answer Type Questions (18 marks)

Q.2 There will be 12 questions out of which 9 will have to be attempted. Each question will carry 2 (two) marks. The answer is to be given in about 150 words.

Section C: Essay Type Questions (30 marks)

➢ This section will cover 50% marks of the total marks.

➢ To cover the whole syllabus the three questions may be divided in two parts as follows

Q.3 a) (5 marks)

b) (5 marks)

OR

a) (5 marks)

b) (5 marks)

Q.4 a) (5 marks)

b) (5 marks)

OR

a) (5 marks)

b) (5 marks)

Q.5 a) (5 marks)

b) (5 marks)

OR

a) (5 marks)

b) (5 marks)

➢ Each part will carry weight-age of 5 (five) marks. The answer to each question is to be given in about 250 words.

******

Semester-I

Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-I

Subject Code: MSM-101 Max. Marks: 60

Subject Title: Real Analysis Internal Assessment: 40 Time Allowed: 3 Hours Total: 100

SECTION-A

Metric Spaces: Definition and examples, The Euclidean space RK as a metric space, Neighborhoods, Open and Closed sets, Interiors, Closures and Relative open sets. Compactness, Compactness of k- cells in RK, Weierstrass Theorem, Perfect sets, Cantor set, Connected sets in R1.

SECTION-B

Riemann- Stieljes Integration: Definition and Existence of Riemann-Stieljes Integral, Properties of Integral, Integration and Differentiation, The Fundamental Theorem of Calculus, Change of Variables, Integration of vector valued functions, Rectifiable Curves, Uniform convergence and continuity, Uniform convergence and Riemann-Stieljes Integration, Uniform convergence and Differentiation, Stone-Weierstrass Theorem.

SECTION-C

Rearrangement of terms of a series, Riemann’s Theorem. Power series, uniqueness theorem for power series, Abel’s theorem and Tauber’s theorem. Exponential and Logarithmic functions. Trigonometric functions. Fourier series.

SECTION-D

Functions of several variables, Linear transformations, Derivatives in an open subset of Rn, Chain Rule, Partial derivatives, Interchange of the order of differentiation, Derivatives of higher orders, Taylor’s theorem, Inverse function theorem, Implicit function theorem.

Textbooks:

1. W. Rudin, Principles of Mathematical Analysis, Mc-Graw Hill, 1976.

2. E. Kreyszig,Introductory FunctionalAnalysis with Applications, John Wiley and Sons, 2010.

References Books:

1. H. L. Royden, Real Analysis, Macmillan Publishing Company, 1998.

2. T. Tao, Analysis I, Hindustan Book Agency, 2006.

3. T. Tao, Analysis II, Hindustan Book Agency, Springer, 2015.

4. T. M. Apostol, Mathematical Analysis, Addison-Wesley, 1974.

5. G. F. Simmons, Topology and Modern Analysis, Kreiger, 2003.

6. C. C. Pugh, Real Mathematical Analysis, Springer, 2002.

7. R. G. Bartle and D. R. Sherbert, Introduction to Real Analysis, Wiley, 2000.

E-book Links:

1. [1]

2. [2]

References Links:

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Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-I

Subject Code: MSM-102 Max. Marks: 60

Subject Title: ALGEBRA I Internal Assessment: 40

Time Allowed: 3 Hours Total: 100

SECTION-A

Normal and subnormal Series, Jordan Holder theorem for finite groups, Fundamental theorem of arithmetic, solvable groups, Nilpotent groups, Zassenhaus Lemma, Scherer's refinement theorem and Jorden Holder theorem for groups (not necessarily finite).

SECTION-B

Review of permutation groups, Alternating group An, simplicity of An, Structure theory of groups, Direct products, Fundamental theorem of finite abelian groups. Invariants of finite abelian groups, Sylow theorems, Groups of order P2, pq

SECTION-C

Ideals, Ring homomorphiseas, algebra of ideals: sum and direct sum of ideals, maximal and prime ideals, Nilpotent and nil ideals, Statement of Zorn's Lemma, Field of Quotients of integral domain.

SECTION-D

Review of vector spaces, Equal spaces, dual Bases, second dual space, Reflexivity, Annihilators, inner product spaces, Schwarz inequality, Gram-Schmidt orthogonalisation process.

Text Books:

1. S. Singh and Q. Zameruddin, Modern Algebra, Vikash Publishing House, 2006

2. K. Hoffman and R. Kunz, Linear Algebra, Prentice Hall, 1965

Reference Books:

1. Seymour Lipschutz, Schaum's Outline ofLinear Algebra, The McGraw-Hill Companies, Inc., 2013

2. I. N.Herstein, Topics in Algebra, John Wiley& Sons, 1975.

3. C. Musili, Introduction to Rings and Modules, Narosa Publishing House, 1994

4. D. S. Malik, J.M. Mordeson and M. K. Sen, Fundamentals of Abstract Algebra, McGrawHill Company, 1997

5. K. B. Datta, Matrix and linear algebra, PHI Pvt. Limited, 2004

E-book Links:

1. (2009)Lipschutz-Lipson.pdf [1]

References Links:

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Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-I

Subject Code: MSM-103 Max. Marks: 60

Subject Title: DIFFERENTIAL GEOMETRY Internal Assessment: 40

Time Allowed: 3 Hours Total: 100

SECTION-A

A simple are, Curves and their parametric representation, are length and natural parameter, contact of curves, Tangent to a curve, osculating plane, Frenet trihedron, Curvature and Torsion, Serret Frenet formulae, fundamental theorem for spaces curves, helices, contact between curves and surfaces.

SECTION-B

Evolute and involute, Bertrand Curves, spherical indicatrix, implicit equation of the surface, Tangent plane, the first fundamental form of a surface, length of tangent vector and angle between two tangent vectors, area of a surface.

SECTION-C

The second fundamental form, Gaussian map and Gaussian curvature, Gauss and Weingarten formulae, the Christoffel symbols, Codazzi equation and Gauss theorem, curvature of a curve on a surface, geodesic curvature.

SECTION-D

Geodesic, normal curvature, principal curvature, Mean Curvature, principal directions, lines curvature, Rodrigue formula, asymtotic Lines, conjugate directions, envelopes, developable surfaces associated with spaces curves, minimal surfaces, ruled surfaces.

Text books:

1. Kuhnel, Wolfgang. Differential Geometry: Curves – Surfaces – Manifolds. Student mathematical library, vol. 16. Providence, RI: American Mathematical Society, 2002.

2. Pressley, Andrew. Elementary Differential Geometry. Springer undergraduate mathematics series. London, UK: Springer, 2002.

References Books:

1. Weatherburn, C.E., Differential Geometry of Three Dimensions, Cambridge University Press, 2016.

2. Willmore, T.J., Introduction to Differential Geometry, Dover Publications Inc., United States, 2012.

E-book Links:

1. [1]

References Links:

1.

Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-I

Subject Code: MSM-104 Max. Marks: 60

Subject Title: DIFFERENTIAL EQUATIONS Internal Assessment: 40

Time Allowed: 3 Hours Total: 100

SECTION-A

Existence of solution of ODE of first order, initial value problem, Ascoli’s Lemma, Gronwall’s inequality, Cauchy Peano Existence Theorem, Uniqueness of Solutions. Method of successive approximations, Existence and Uniqueness Theorem.

SECTION-B

System of differential equations, nth order differential equation, Existence and Uniqueness of solutions, dependence of solutions on initial conditions and parameters.

SECTION-C

Linear system of equations (homogeneous & non homogeneous). Superposition principle, Fundamental set of solutions, Fundamental Matrix, Wronskian, Abel Liouville formula, Reduction of order, Adjoint systems and self adjoint systems of second order, Floquet Theory.

SECTION-D

Linear 2nd order equations, preliminaries, Sturm’s separation theorem, Sturm’s fundamental comparison theorem, Sturm Liouville boundary value problem, Characteristic values & Characteristic functions, Orthogonality of Characteristic functions, Expansion of a function in a series of orthonormal functions.

Textbooks:

1. E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955.

2. G.F. Simmons, Differential Equations with Applications and Historical Notes, 2ndEd,McGraw- Hill, 1991.

3. R.P. Agarwal and D. O'Regan, An Introduction to Ordinary Differential Equations, Springer- Verlag, 2008.

References Books:

1. G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, John Willey and Sons, 4th Ed., 1989.

2. R.P. Agarwal and R.C.Gupta, Essentials of Ordinary Differential Equations, McGraw-Hill, 1993.

3. E.A. Coddington an, An Introduction to Ordinary Differential Equations, PHI Learning 1999.

4. M. Braun, Differential Equations and Their Applications, 3rdEd., Springer-Verlag, 1983.

5. S. G. Deo, V. Raghavendra, R. Kar and V. Lakshmikantham, Textbook of Ordinary Differential Equations, McGraw Hill Education, 3rd Ed., 2015.

6. G.F. Simmons and S.G. Kantz, Differential Equations: Theory, Technique and Practice, Tata McGraw-Hill, 2007.

E-Book Links:

1. [2]

References Links:

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Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-I

Subject Code: CSE-101 Max. Marks: 60

Subject Title: Fundamentals of Computer and IT Internal Assessment: 40

Time Allowed: 3 Hours Total: 100

Section-A

Computer basics Computer basics, hardware and software, flowchart, flowchart symbols, computer languages, low level languages, high level languages, FORTRAN language, implicit, constants and variables, declaration of reals and integers, arithmetic expressions, real and integer expressions, some problems due to rounding of real numbers, mixed mode expressions, special functions.

Section-B

Computer programming in FORTRAN Program preparation preliminaries, Input/output statements, list directed input/output statements, PRINT statement, Control statements, relational operators, logical IF statements, nested IF statements, arithmetic IF statement, DO statement, rules to be followed in utilizing DO loops, REPEAT WHILE structure, subscripted variable, use of multiple subscripts, subscript expressions, DIMENSION statement, FORMAT description for PRINT statement, WRITE statement, multi record For Mats, Logical expressions and decision tables.

Section-C

Functions and subroutines in FORTRAN Functions, statement functions, function subprograms, syntax rules for function subprograms, subroutines, COMMON declaration, processing files in FORTRAN, creating a sequential file, updating a sequential file, merging two sequential files, direct access files, CHARACTER manipulations in FORTRAN, string expressions, substrings, double precision facility in FORTRAN, use of complex quantities, DATA statement, EQUIVALENCE declaration.

Section-D

Graphical plotting of data Graphical plotting of data, 2D plotting, 3D plotting, contour plotting, data plotting softwares such as Excel, Origin, Sigma etc.

References Books:

1. V Rajaramanm, Computer Programming in Fortran 77, PHI Learning Pvt. Ltd., 1997.

2. Ian D Shivers and J Sleight, Interactive Fortran 77, A hands on Approach, Ellis Horwood Ltd; 1990.

E-Book Links:

1. [2]

References Links:

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Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-I

Subject Code: CSE-101P Max. Marks: 20

Subject Title: Fundamentals of Computer and IT Lab Internal Assessment: 30

Time Allowed: 3 Hours Total: 50

Programming in FORTRAN 77

Write an algorithm, flowchart and computer program for the following problems.

1. Picking greatest number of given five numbers.

2. Sum of n consecutive integers, sum of n consecutive odd integers and sum of n consecutive even integers.

3. Compute the total resistance of four resistances connected in parallel and in series.

4. Compute the roots of a quadratic equation.

5. Compute the determinant and inverse of a matrix.

6. Solution of n linear equations.

7. Compute the value of energy and radius of first Bohr´s orbit of Hydrogen atom.

8. Compute and plot the trajectory of a projectile thrown with some velocity u at an angle θ. Compute the horizontal range for a projectile fired with velocity 10 m/s and θ=35.

9. Compute and plot the trajectory of Earth around the Sun.

10. Compute the root of equation x=sin (x) using the Newton-Raphson´s method.

11. Integrate the equation F(x)=x7/3+x2+x in the limits 0 and 1 for N=10 using Simpson and Trapezoidal rule and compare the results.

12. Solve Harmonic oscillator problem using Runge-Kutta method.

Semester –II

Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-II

Subject Code: MSM 201 Max. Marks: 60

Subject Title: Real Analysis II Internal Assessment: 40

Time Allowed: 3 Hours Total: 100

SECTION-A

Lebesgue measure: Introduction, outer measure, Lebesgue measure, Measurable sets, properties of measurable sets, Borel sets and their measurability, Non measurability sets.

SECTION-B

Measurable functions: Definitions and properties of measurable function, step function, characteristics function, Simple function.

Lebesgue integral: Lebesgue integral of bounded function, Properties of Lebesgue integral, Integration of non-negative measurable functions, Fatou’s lemma, Monotone convergence theorem, Lebesgue convergence theorem, General Lebesgue integral, Integration of series, Comparison of Riemann and Lebesgue integrals.

SECTION-C

Differentiation and integration; Dini derivatives. Functions of bounded variation, Vitalis Cover, Vitalis Lemma, Lebesgue differentiation theorem, Differentiation of an Integral, Absolute Continuity.

SECTION-D

LP spaces: LP spaces, Convex Functions and Jensens inequality, Holder and Minkowski's inequalities, Riesz Fisher Theorem, Approximation in LP, Bounded Linear Functionals on the LP spaces; The Riesz Representation Theorem

Text Books:

1. S.C. Malik and SavitaArora, Mathematical Analysis, New Age International Private Limited, 2017

2. H.L. Royden, Real Analysis, Prentice Hall of India, 2011

3. P.K. Jain and V.P. Gupta, Lebesgue Measure and Integration, Anshan Ltd., 2012

Reference Books:

1. R.R. Goldberg, Methods of Real Analysis, Oxford and IBH Publishing, 2012.

2. W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Education, 1976

E-Book Links:

1. [1]

Reference Links:

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Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-II

Subject Code: MSM-202 Max. Marks: 60

Subject Title: COMPLEX ANALYSIS I Internal Assessment: 40

Time Allowed: 3 Hours Total: 100

SECTION-A

Analytic function: Function of complex variables, limit, continuity, derivatives, Cauchy -Riemann equations, Analytic function, Power series, Exponential functions, Trigonometric functions, Logarithmic functions, hyperbolic and inverse hyperbolic functions, Branches of multivalued functions with reference to arg z, log z and [pic].

SECTION-B

Complex Integration, Cauchy Goursat theorem, Cauchy integral formula, Cauchy integral formula for higher order derivatives, Liouville's theorem, Moreara's theorem, Power series, Taylors Theorem, Taylor series, Laurent series, Singularities, Types of singularities, Residue theorem , Zero, Poles.

SECTION-C

Maximum Modules Principle, Schwarz Lemma, Argument principle, Rouche's theorem, Fundamental theorem of algebra. Cauchy's residue theorem, Definite integral using residue theorem.

SECTION-D

Conformal Mappings: Elementary conformal mapping, Bilinear transformation, Schwartz-christoffel transformation, analytic continuation, Method of analytic continuation by Power series.

Text Books:

1. M.R. Spiegel, Complex Variables. Schaum’s Outlines series, McGraw Hill Education, 2017

2. E. G. Philips, Functions of a complex variables with applications, Oliver and Boyd, 1957

Reference Books:

1. Walter Rudin, Real and Complex Analysis, McGraw Hill Education, 2017

2. L. V. Ahlfors, Complex Analysis, McGraw Hill., 2000

3. H. A. Priestly, Introduction to Complex Analysis, Clarendon Press Oxford, 1990

4. Mark J. Ablowitz and A.S. Fokas, Complex Variables, Introduction and Application, CUP, 1998.

5. John B Conway, Functions of Complex Variable, Springer, 1872.M. J. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications, Cambridge University Press, NY, 2003.

E-Book Links:

1. [1]

Reference Links:

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Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-II

Subject Code: MSM-203 Max. Marks: 60

Subject Title: ALGEBRA II Internal Assessment: 40

Time Allowed: 3 Hours Total: 100

SECTION-A

Group Theory: Review of group theory, subgroup, Homomorphism, normal subgroup, Normal Quotient group, Composition series, Jordan Holder theorem, Solvable group, Nilpotent group, Conjugate element, class equation for a finite group, Sylow p-subgroup, Sylows theorem and applications.

SECTION-B

Field Theory: Algebraic and transcendental extensions, Separable and inseparable extensions, Normal extensions, Perfect fields, Finite field, Primitive elements, Algebraically closed fields, Automorphism of extensions, Gralosis extensions, Fundamental theorem of Gralosis theory.

SECTION-C

Modules: Review of vector space, modules, submodules, quotient modules, free modules, difference modules and vector, Homomorphism of modules, simple module, Modules of PID.

SECTION-D

Modules with chain conditions: Artinian Modules, Noetherian Modules, composition series of a module, Length of a module, Hilbert Basis Theorem

Textbooks:

1. J.A. Gallian, Contemporary Abstract Algebra, Narosa, 4th Ed., 1999.

2. I. N. Herstein,Topics in Algebra, John-Wiley, 1995.

References Books:

1. M.Artin, Algebra, Prentice Hall Inc., 1994.

2. T. A. Hungerford, Algebra, Graduate Texts in Mathematics, Vol. 73, Springer-Verlag, 1980.

3. D. S. Dummit and R. M. Foote, Abstract Algebra, John-Wiley, 2nd Ed., 1999.

4. S. Lang, Algebra, Addison-Wesley, 3rd Ed., 1999. 5. J. B.Fraleigh,A First Course in Abstract Algebra, Pearson, 7thEd., 2003.

E-Book Links:

1. [1]

Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-II

Subject Code: MSM-206 Max. Marks: 60

Subject Title: Discrete Mathematics Internal Assessment: 40

Time Allowed: 3 Hours Total: 100

SECTION-A

Lattices:Lattices as Partially Ordered Sets. Their Properties, Lattices as algebraic Systems, Sublattices, Direct Product and homomorphisms. Some Special Lattices e.g. Complete, Complemented and Distributive Lattices, Isomorphic Lattices. Join Irreducible elements. Atoms.

SECTION-B

Boolean Algebra : As Lattices, Various Boolean identiies, The switching Algebra Example, Sub Algebras, Direct Production and Homomorphisms. Boolean Forms and their Equivalence, Minterm Boolean forms, Sum of Products Cannonical Forms, Minimization of Boolean Functions, Application to Switching Theory, The Karnuagh Map Method.

SECTION-C

Graph Theory: Definition of (undirected, Multigraphs, Subgraphs, Paths, Circuits, cycles, Induced Subgraphs, degree of vertex, Connectivity, Planar graphs, Euler's theorem, Directed Graphs, Warshall's Algorithm of shortest paths, Reular and Bipartite Graphs, Kuratowski's Theorem.

SECTION-D

Trees, Spanning Trees, Cut Sets, Fundamental Cut sets and Cycles, Minimal Spanning Trees and Kruskal's Algorithm, Matrix representation of graphs, Euler's Theorem on the existence of Eulerian paths and circuits. Directed Graphs, indegree and Outdegree of vertex. Directed trees, Search trees, Tree Traversals.

Textbooks:

1. C. L. Liu, Elements of Discrete Mathematics, Second Edition, McGraw Hill 1985.

2. J. L. Mott, A. Kandel and T. P. Baker,Discrete Mathematics for Computer Scientists and Mathematicians, Prentice Hall India, 2nd Ed., 1986.

References Books:

1. F. Harary, Graph Theory, Narosa, 1969.

2. H. C. Thomas, C. E. Leiserson, R. L. Rivest and C.Stein,An Introduction to Algorithms, MIT Press and McGrawHill, 2nd Ed.,2001.

E-Book Links:

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2. [1]

References Links:

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2.

Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-II

Subject Code: MSM-204 Max. Marks: 60

Subject Title: Numerical Analysis Internal Assessment: 40

Time Allowed: 3 Hours Total: 100

Section-A

Errors, Error propagation, Order of approximation. Solution of non-linear equations: Bisection, Regula-falsi, Secant, Newton-Raphson, Generalized Newton’s method, Chebyshev method, Halley’s methods, General iteration method, Muller’s method. Rate of convergence. Newton’s method for complex roots and multiple roots, Simultaneous non-linear equations by Newton-Raphson method.

Section-B

Operators: Forward, Backward and Shift (Definitions and some relations among them). Interpolation: Finite differences, divided differences, Newton’s formulae for interpolation, Lagrange and Hermite interpolation, Cubic Spline interpolation. Numerical integration-Trapezoidal, Simpson’s 1/3rd rule, Simpson’s 3/8th rule, Boole’s rule, Weddle’s rule, Errors in Integration formulae.

Section-C

Curve fitting: Linear and non-linear curve fitting, curve fitting by sum of exponentials, fitting of exponential Solution of Linear system of equations: Matrix inversion, Gauss-elimination and Gauss-Jordan method, LU decomposition method, Gauss Jacobi and Gauss Seidal method.

Section-D

Solution of differential equations: Taylor series method, Euler’s method, Modified Euler’s method, Runge - Kutta methods of order two, three and four, Predictor –Corrector methods, Finite Difference Method for ODE and PDE (Boundary value problem).

Textbooks:

1. D. Kincaid and W. Cheney, Numerical Analysis and Mathematics of Scientific Computing, Brooks/Cole,1999.

2. M. K. Jain, S. R. K. Iyengar and R. K. Jain, Numerical Methods for Scientific and Engineering Computation, New age International Publishers, 2012.

References Books:

1. K. Atkinson, Elementary Numerical Analysis, John Wiley, 3rd Ed., 2003.

2. S. D. Conte and C. de Boor, Elementary Numerical Analysis, McGraw-Hill, 1980.

3. A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 1996.

4. J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations, John Wiley, 1987.

5. H. R. Schwarz, Numerical Analysis: A Comprehensive Introduction, Wiley, 1st Ed., 1989.

6. R. L. Burden and J. D. Faires, Numerical Analysis, Brooks/Cole, 9th Ed., 2011.

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1. [1]

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1.

Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-II

Subject Code: MSM-204P Max. Marks: 20

Subject Title: Numerical Analysis Lab Internal Assessment: 30

Time Allowed: 3 Hours Total: 50

List of Practicals:

(i) To find the absolute value of an integer.

(ii) Bisection Method and error analysis.

(iii) Newton-Raphson Method and error analysis

(iv) Secant Method and error analysis

(v) RegulaFalsi Method and error analysis

(vi) Matrix-inversion method

(vii) Gauss-Jacobi Method.

(viii) Gauss-Seidal Method.

(ix) Calculation of derivatives using Newton interpolation.

(x) Lagrange interpolation.

(xi) Bulle’s rule.

(xii) Weddle’s rule.

(xiii) Taylor series method.

(xiv) Euler’s method.

(xv) Predictor –Corrector methods

Textbooks:

1. W. J. Palm III, Introduction to Matlab for Engineers, Tata McGraw Hill, New Delhi, 2011.

2. E. Kreyszig, Advanced Engineering Mathematics, Wiley, 10th Ed., 2011.

References Books:

1. S. C. Chopra, Applied Numerical Methods with Matlab For Engineers and Scientists, Tata McGraw Hill Company Limited, New Delhi, 2ndEd., 2007.

2. D. F. Griffith, Introduction to Matlab, 2005.

Semester-III

Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-III

Subject Code: MSM-301 Max. Marks: 60

Subject Title: Topology Internal Assessment: 40

Time Allowed: 3 Hours Total:100

Section-A

Cardinal numbers and their arithmetic, Cantor’s theorem and the continuum hypothesis, Zorn’s Lemma, Well-ordering theorem, Topological spaces: Definition and examples, Euclidean spaces as topological spaces, Basis for a given topology, Sub-basis, Equivalent basis, Elementary concepts: Closure, Interior, Frontier and Dense sets, Tautologizing with pre-assigned elementary operations, Relativization, Subspaces.

Section-B

Continuous functions, Characterization of continuity, Open maps and Closed maps, Homeomorphisms and embedding, Cartesian product topology, Elementary concepts in product spaces, Continuity of maps in product spaces and slices in Cartesian products.

Section-C

Connected spaces, Connected subspaces of the real line, Components and path components, Local connectedness, Compact spaces, Sequentially compact spaces, Heine-Borel theorem, compact subspaces of the real line, Local-compactness and one-point compactification.

Section-D

Countability Axioms: Separable spaces, Lindelo-f spaces, Separation axioms: T0, T1 and T2 spaces, Regular space, Completely regular and Normal spaces, Urysohn lemma, Urysohn metrization theorem, Tietze extension theorem, Tychnoff theorem.

Text Books:

1. J. R. Munkres, Topology: A first course, Prentice Hall of India, 1974

2. G. F. Simmons,Introduction to Topology and Modern Analysis, McGraw Hill, 2017

Reference Books:

1. K.D. Joshi,Introduction to General Topology, New Age International Private Limited, 2017

2. J. Dugundji,Topology, Allyn and Bacon, 1966 (Reprinted in India By PHI)

3. J. Hocking and G. Young,Topology, Addison Wiley Reading, 1961

4. L. A. Steen and J. A. Seebach,Counter Examples in Topology, Dover Publications, 1995.

E-Book Links:

1. [3]

Reference Links:

1.

Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-III

Subject Code: MSM-302 Max. Marks: 60

Subject Title: Operations Research Internal Assessment: 40

Time Allowed: 3 Hours Total: 100

Section-A

Introduction, Definition of operation research, Models in operation research. Formulation of linear programming problem (LPP): Graphical method, Basic Feasible Solution, optimal solution of LPP using Simplex, Big-M and Two phase methods, Exceptional cases in LPP i.e. Infeasible, unbounded, alternate and degenerate solutions, Extreme Points, Convex set, Convex linear combination.

Section-B

Duality in Linear Programming: General Primal-Dual pair, formulating a dual problem, duality theorems, Complementary slackness theorem, Duality & simplex method, Dual simplex method, Sensitivity analysis: change in right hand side of constraints, change in the objective function and coefficient matrix addition and deletion of constraint and variables.

Section-C

Transportation Problem: Initial basic Feasible solution, Balanced and unbalanced transportation problems, Optimal solutions of transportation problem using U-V/MODI methods, Assignment problems: Mathematical formulation of assignment problem, typical assignment problem, the traveling salesman problem, Test for optimality, degeneracy, Project management with critical path method.

Section-D

Concept of convexity and concavity, Maxima and minima of convex functions, Single and multivariate unconstrained problems, constrained programming problems, Kuhn-Tucker conditions for constrained programming problems, Quadratic programming, Wolfe's method.

Textbooks:

1. H. A. Taha,Operations Research: An Introduction, MacMillan Pub Co., NY, 9th Ed., 2013.

2. Kanti Swarup, P.K. Gupta and Man Mohan, Opeartions Research, S. Chand and Co., 2010

3. Richard Bronson, Govindasami Naadimuthu,Operations Research, Schaum Outlines Series, McGraw Hill Education, 2017.

Reference Books:

1. F. S. Hillier and G. J. Lieberman, B. Nag and P. Basu, Introduction to Operations Research, McGraw Hill Education; Tenth edition (5 July 2017)

2. P. K. Gupta and D. S. Hira,Introduction to Operations Research, S. Chand Publishing, 2012

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1. [1]

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Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-III

Subject Code: MSM-303 Max. Marks: 60

Subject Title: Mathematical Statistics Internal Assessment: 40

Time Allowed: 3 Hours Total: 100

Section-A

Concept of random variables and probability distributions: Two dimensional random variables, Joint, Marginal and conditional distributions, Independence of random variables, Expectation, Conditional expectation, Moments, Product moments, Probability generating functions, Moment generating function and its properties, Moment inequalities, Techebyshey's, inequalities, Characteristic function and its elementary properties.

Section-B

Study of various discrete and continuous distributions: Binomial, Poison, Negative binomial, Geometric, Hyper geometric, Rectangular, Normal, Exponential, Beta and gamma distributions.

Section-C

Concept of sampling distribution and its standard error, Derivation of sampling distributions of Chi-square, t and F (null case only) distribution of sample mean and sample variance and their in random sampling from a normal distribution.

Section-D

Elementary concepts in testing of statistical hypotheses, Tests of significance: tests based on normal distribution, Chi-square, t and F statistic and transformation of correlation coefficient, tests for regression coefficients and partial and multiple correlation coefficients.

Analysis of variance: One-way classification, two-way classification with one observation per cell.

Text books:

1. Goon, Gupta and Das Gupta, ‘Fundamentals of Statistics’, 5th Edn., World Press, 1975.

2. V.K. Rohatgi, ‘Introduction to Probability Theory & Mathematical Statistics’, 2009.

References Books:

1. Feller W., An Introduction to Probability Theory and Its Applications (Vol-I), 3 rd Edition. John Wiley & Sons, 2003.

2. Gun A.M., Gupta, M.K. and Dasgupta B., Fundamentals of Statistics (Vol-I), World Press, 2013

3. R.V. Hogg & Craige, ‘Introduction to Mathematical Statistics’, 7th Edn., 2005.

4. J.W. Mckean and A.T. Craig, P. Mukhopadhyay, ‘Mathematical Statistics’, 2000.

5. Gupta S. C. and Kapoor V. K., Fundamentals of Mathematical Statistics, 11th Edition. Sultan Chand & Sons, 2014

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Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-III

Subject Code: MSM-304 Max. Marks: 60

Subject Title: Mathematical Methods Internal Assessment: 40

Time Allowed: 3 Hours Total: 100

Section-A

Linear integral equations of first and second kind, Abel’s problem, Relation between linear differential equation and Volterra’s equation, Nonlinear and Singular equations, Solution by successive substitutions, Volterra’s equation, Iterated and reciprocal functions, Volterra’s solution of Fredholm’s equation.

Section-B

Fredholm’s equation as limit of finite system of linear equations, Hadamard’s theorem, Convergence proof, Fredholm’s two fundamental relations, Fredholm’s solution of integral equation when D(λ)≠0, Fredholm’s solution of dirichet’s problem and neumann’s problem, Lemmas on iterations of symmetric kernel, Schwarz’s inequality and its applications.

Section-C

Simple variational problems, Necessary condition for an extremum, Euler’s equation, End point problem, Variational derivative, Invariance of Euler’s equation, Fixed end point problem for n-unknown functions, Variational problem in parametric form, Functional depending on higher order derivatives.

Section-D

Euler -Lagrange equation, First integral of Euler-Lagrange equation, Geodesics, The Brachistochrone, Minimum surface of revolution, Brachistochrone from a given curve to a fixed point, Snell’s law, Fermat’s principle and calculus of variations.

Text Books:

1. R. P. Kanwal, Linear Integral Equations, Theory and Techniques, Academic Press, New York, 1971

2. M. R. Spiegel, Schaum's Outline Series: Theory and Problems of Laplace Transforms, McGrawHill Book Company, 1965

3. A. S. Gupta, Calculus of variation with Applications : Prentice Hall of India, 1999

References Books:

1. M.D. Raisinghania, ‘Integral Equations and Boundary Value Problems’, 6th Edn., S. Chand,2015.

2. W.W. Lovitt, ‘Linear Integral Equations’, 2nd Edn., Dover, India, 2005.

3. S. G. Mikhlin, Linear Integral Equations (Translated from Russia), Hindustan Book Agency, 1960

4. F. B. Hilderbrand, Methods of Applied Mathematics, Dover Publications, 1992

5. R. Courant and D. Hilbert, Methods of Mathematical Physics- Vol- I, Wiley Interscience, New York 1953.

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1. [1]

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Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-III

Subject Code: MSM-305 Internal Assessment:100

Subject Title: SEMINAR-I Total: 100

1. Each of these Courses of Seminar will consist of 100 marks (internal only) having L T P C as 0 0 2 1.

2. In the beginning of the semester, a teacher will be allocated maximum 30 students. The latter will guide/teach them how to prepare/present 15 minutes Power Point Presentation for the Seminar.

3. If there are more than 30 students in the class, then class will be divided into two groups having equal students. Each group may be allocated to a different teacher.

4. Each student will be allotted a topic by the teacher at least one week in advance for the presentation. The topic for presentation may be from the syllabus or relevant to the syllabus of the programme.

5. During the presentation being given by a student, all the other students of his/her group will attend the Seminar. The assessment/evaluation will be done by the teacher. However, Head of Department and other faculty members may also attend the Seminar, ask questions and give their suggestions.

6. This is a turn wise continuous process during the semester and a student will give minimum two presentations in a Semester.

7. For the evaluation, the following criteria will be adopted,

a) Attendance in Seminar: 25 Marks

b) Knowledge of Subject along with Q/A handling during the Seminar: 25 Marks

c) Presentation and Communication Skills: 25 Marks

d) Contents of the Presentation: 25 Marks.

Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-III

Subject Code: MSM-306 Max. Marks: 60

Subject Title: Fourier anylysis & application Internal Assessment: 40

Time Allowed: 3 Hours Total: 100

Section-A

Fourier Series: Fourier series, Theorems, Dirichlet’s conditions, Fourier series for even and odd functions, Half range Fourier series, Other forms of Fourier series.

Section-B

Convergence and Uniform convergence of Fourier series, Cesaro and Abel Summability of

Fourier series, The Dirichlet Kernal, The Fejer kernel, 2-theory: Orthogonality, Completeness.

Section-C

Fourier Transforms: Dirichlet’s conditions, Fourier integral formula (without proof), Fourier transform, Inverse Theorem for Fourier transform, Fourier sine and cosine transforms and their inversion formulae. Properties of Fourier transform, Convolution theorem of Fourier transforms, Parseval’s identity, Finite Fourier sine and cosine transform, Inversion formula for sine transform,

Application of Fourier Transforms: Simultaneous ordinary differential equations, second order Partial differential equations (Heat, Wave and Laplace).

Section-D

The Discrete Fourier Transform (DFT): Definition, Theorems, Properties: Periodic and Linear Convolution by DFT, The Fast Fourier Transform, FFT convolutions, Two dimensional FFT Analysis.

Text books:

1. B.S. Grewal, ‘Higher Engineering Mathematics’, Khanna Publisher, 2014.

2. Duraisamy Sundararajan, ‘The Discrete Fourier Transform: Theory, Algorithms and Applications’, World Scientific Publishing Co. Pte Ltd., 2001.

References Books:

1. Javier Duoandikoetxe, ‘Fourier Analysis’, University Press, 2012.

2. Gerald B. Folland, ‘Fourier Analysis and its Applications’, American Mathematical Society, 2010.

3. N.K. Bary, ‘A Treatise on Trigonometric Series’ Vol. 1, Pergamon, 2014.

E-Book Links:

1. [2]

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1.

Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-III

Subject Code: MSM-307 Max. Marks: 60

Subject Title: Advanced Numerical Analysis Internal Assessment: 40

Time Allowed: 3 Hours Total: 100

Section-A

Iterative Methods for Linear Systems: The classical iterative methods (Jacobi, Gauss-seidel, Muller method and successive over relaxation (SOR) methods), Krylov subspace methods, Conjugate gradient, Bi-conjugate-gradient (BiCG), BiCG stability methods, Preconditioning techniques, parallel implementations.

Section-B

Finite Difference Methods: Explicit and implicit schemes, consistency, stability and convergence, Lax equivalence theorem, Numerical solutions to elliptic, parabolic and hyperbolic partial differential equations.

Section-C

Approximate Methods of Solution: Rayleigh-Ritz, Collocation and Galerkin methods, properties of Galerkin approximations, Petrov-Galerkin method, Generalized method, Spline (Quadratic, Cubic) Theory.

Section-D

Finite Element Method (FEM): FEM for second order problems, one and two dimensional problems, the finite elements (elements with a triangular mesh and a rectangular mesh and three dimensional finite elements), Fourth-order problems, Hermite families of elements, Iso-parametric elements, Numerical integration.

Text books:

1. R.S. Gupta, ‘Elements of Numerical Analysis’, McMillan India, 2009

2. P. Seshu, ‘Textbook of Finite Element Analysis’, Prentice Hall India, 2003.

References Books:

1. M.K. Jain, S.R.K. Iyengar, and R.K. Jain, ‘Numerical Methods for Scientific and Engineering Computation’, 5th Edn., New Age international, 2008.

2. Joe D. Hoffman, ‘Numerical methods for Engineers and Scientists’, McGraw Hill, 1993.

3. K.E Atkinson, ‘An Introduction to Numerical Analysis’, 2nd Edn., John Wiley, 2004.

E-Book Links:

1. [2]

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SEM-IV

Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-IV

Subject Code: MSM-401 Max. Marks: 60

Subject Title: Number Theory Internal Assessment: 40

Time Allowed: 3 Hours Total: 100

Section-A

Arithmetical Functions: Mobius function, Euler’s totient function, Mangoldt function, Liouville’s function, the divisor function, Relation connecting ф and µ Product formula for ф(n), Dirichlet product of arithmetical functions, Dirichlet inverse and Mobius invertion formula, Multiplicative function, Dirichlet multiplication, the inverse of a completely multiplicative function, Generalized convolutions.

Section-B

Averages of Arithmetical Function: The Big oh notation, Asymptotic equality of functions, Euler’s summation formula, Elementary asymptotic formulas, Average order of d(n), ф(n), a(n), µ(n), Ʌ(n), The partial sums of a Dirichlet product, application to µ(n) and Ʌ(n), Legendre’s identity.

Section-C

Some elementary theorems on the Distribution of prime numbers Chebyshev’s functions

(X) & ( ), Relation Connecting ( ) (X), Abel’s identity, equivalent forms of prime number theorem, Inequalities for (n) and Pn Shapiro’s Tauberian theorem, Application of Shapiro’s theorem.

Section-D

Elementary properties of groups, characters of finite abelian groups, the character group, Orthogonality relation for characters, Dirichlet character, Dirichlet theorem for prime of the form 4n-1 and 4n+1, Dirichlet theorem in primes on Arithmetical progression, Distribution of primes in arithmetical progression.

Text Books:

1. David M. Burton,Elementary Number Theory (Unit 1, 2, 3), McGraw Hill Education, 2017

2. G.E. Andrews,Number Theory (Unit 4), Dover Publications, 2012

3. Richard A Molin,Algebraic number theory,Chapman and Hall/CRC, 2011

Reference Books:

1. Niven, H. S. Zuckerman and H. L. Montgomery,Introduction to Theory of Numbers, Wiley, 2008.

2. H. Rosen Kenneth, ‘Elementary Number Theory’, 6th Edn.

3. G.H. Hardy, ‘An Introduction to the Theory of Numbers’, 6th Edn.

E-Book Links:

1. [2]

Reference Links:

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Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-IV

Subject Code: MSM-402 Max. Marks: 60

Subject Title: Functional analysis Internal Assessment: 40

Time Allowed: 3 Hours Total: 100

Section-A

Normed linear space, Banach Spaces, Properties of normed spaces, Finite dimensional normed spaces and subspaces, Equivalent norms, Linear operator, Bounded and Continuous linear operators, Linear functional, Normed space of operators.

Section-B

Uniform boundedness Theorem, Open mapping theorem, Closed graph theorem, Projections on Banach spaces, Projection theorem.

Section-C

Conjugate spaces, Reflexivity, Hahn-Banach theorems for real/complex vector spaces and normed spaces, Application to bounded linear functional on C [a, b], Hilbert spaces.

Section-D

Inner product spaces, Properties of inner product spaces, Orthogonal complements,

Orthonormal sets, Riesz representation thm. Bessel’s inequality, Hilbert – adjoint operator,

Self-adjoint, Unitary and normal operators.

Textbooks:

1. G. F. Simmons, Introduction to Topology and Modern Analysis, Tata McGraw-Hill International Ed.2004, Fourteenth reprint 2010.

2. M. T. Nair, Functional Analysis: A First Course, PHI-Learning (Formerly: Prentice-Hall of India), New Delhi, 2002.

3. B.V. Limaye, Functional Analysis, New Age International, 2014

4. E. Kreyszig, Introductory Functional Analysis with applications, Wiley Classics Lib. Ed. 2007.

5. W. Rudin- Functional Analysis, McGraw Hill Education, 2017

References Books:

1. E.Kreyszig,Introductory Function Analysis with Applications, John Wiley and Sons, 2010.

2. G.F. Simmons, ‘Introduction to Topology and Modern Analysis’, 2008.

3. Walter Rudin, ‘Functional Analysis: International Series in Pure and Applied Mathematics’,McGraw Hill, inc., 1991.

4. Erwin Kreyszig, ‘Introductory Functional Analysis with Applications’, John Wiley and Sons (Asia), Pvt. Ltd., 2006.

5. George Bachman and Lawrence Narici, ‘Functional Analysis’, Dover, 2000.

6. John B. Conway, ‘A course in Functional Analysis’, second Edn., Springer-Verlag, 2006

E-Book Links:

1. [2]

2. [1]

References Links:

1.

Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-IV

Subject Code: MSM-403 Max. Marks: 60

Subject Title: Partial Differential Equation Internal Assessment: 40

Time Allowed: 3 Hours Total: 100

Section-A

Non-linear PDE of First Order: Complete Integrals, Envelopes, Characteristics, Hamilton-

Jacobi equations, Hamilton’s ODE, Legengre transform, Hopf – Lax formula, Cauchy’s

method of characteristic; Compatible system of first order PDE, Charpit’s method of solution,

Solutions satisfying given conditions, Jacobi’s method of solution.

Section-B

Second Order PDE: Partial Differential equations of 2nd and Higher order, Classification, Examples of PDE, Solutions of Elliptic, Hyperbolic and Parabolic equations, Canonical Form, Initial and Boundary Value Problems, Lagrange-Green's identity and uniqueness by energy methods, Stability theory, energy conservation and dispersion.

Section-C

Method of Solution: Separation of variables in a PDE, Laplace equation: mean value property,Weak and strong maximum principle, Green's function, Poisson's formula, Dirichlet's principle,Existence of solution using Perron's method (without proof).

Section-D

Heat Equation: Initial value problem, Fundamental solution, Weak and strong maximumprinciple and uniqueness results, wave equation: uniqueness, D’Alembert’s method, Method of spherical means and Duhamel’s principle.

Textbooks:

1. I. P. Stavroulakis and S. A. Tersian, Partial Differential Equations- An Introduction with Mathematica and Maple, world - Scientific, Singapore, 1999.

2. I. N. Sneddon, Elements of Partial Differential Equations, Mcgraw Hill 2006

References Books:

Lawrence C. Evans, Partial DifferentialEquations, Second Edition, American Mathematical Society, 2014.

Erich Zauderer, Partial Differential Equations of Applied Mathematics, A Wiley- Interscience Publication, John Wiley and Sons, 1983.

H.F. Weinberger, A first course in partial differential equations, Blaisdell, 1965.

4. C.R. Chester, Techniques in partial differential equations, McGraw Hill, New York, 1971.

R. Courant and D. Hilbert :Methods of Mathematical Physics: Partial differential equations, Vol – II, Wiley-VCH, 1989

E-Book Links:

1. [1]

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2.

Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-IV

Subject Code: MSM-404 Max. Marks: 60

Subject Title: Advanced Complex Analysis Internal Assessment: 40

Time Allowed: 3 Hours Total: 100

Sectiom-A

Fundamental theorems connected with zeros of analytic functions, the argument (counting) principle, Rouche's theorem, Fundamental theorem of algebra, Morera’s theorem, Normal limits of analytic functions, Hurwitz’s theorem, Normal limits of univalent functions, Open mapping theorem, Inverse function theorem.

Section-B

Implicit function theorem, Analyticity of the explicit function, Riemann surfaces for multivalued functions, Direct and indirect analytic continuation, Lipschitz nature of the radius of convergence, Analytic continuation along paths via power series.

Section-C

Monodromy theorem (first version and second version), The Mean value property, Harmonic functions, Maximum principle (with proof), Schwarz’s lemma (with proof), Differential or infinitesimal schwarz’s lemma.

Section-D

Pick’s lemma, Hyperbolic geometry on the unit disc, Arzela-ascoli theorem (with proof), Montel’s theorem (with proof), Riemann mapping theorem (with proof).

Text books:

1. M.R. Spiegel, Complex Variables. Schaum’s Outlines series, McGraw Hill Education, 2017

2. E. G. Philips, Functions of a complex variables with applications, Oliver and Boyd, 1957

References Books:

1. Walter Rudin, Real and Complex Analysis, McGraw Hill Education, 2017

2. L. V. Ahlfors, Complex Analysis, McGraw Hill., 2000

3. L.V. Ahlfors, ‘Complex Analysis’, 2nd Edn., McGraw Hill International Student Edn.,

1990.

4. E.T. Capson, ‘An Introduction to the Theory of Functions of a Complex Variable’, Oxford University Press, 1995.

5. Theodore Gamelin, ‘Complex Analysis (UTM)’, Springer, 2003.

E-Book Links:

1.

2. [2]

References Links:

1.

2.

Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-III

Subject Code: MSM-405 Internal Assessment:100

Subject Title: SEMINAR-II Total: 100

1. Each of these Courses of Seminar will consist of 100 marks (internal only) having L T P C as 0 0 2 1.

2. In the beginning of the semester, a teacher will be allocated maximum 30 students. The latter will guide/teach them how to prepare/present 15 minutes Power Point Presentation for the Seminar.

3. If there are more than 30 students in the class, then class will be divided into two groups having equal students. Each group may be allocated to a different teacher.

4. Each student will be allotted a topic by the teacher at least one week in advance for the presentation. The topic for presentation may be from the syllabus or relevant to the syllabus of the programme.

5. During the presentation being given by a student, all the other students of his/her group will attend the Seminar. The assessment/evaluation will be done by the teacher. However, Head of Department and other faculty members may also attend the Seminar, ask questions and give their suggestions.

6. This is a turn wise continuous process during the semester and a student will give minimum two presentations in a Semester.

7. For the evaluation, the following criteria will be adopted,

a) Attendance in Seminar: 25 Marks

b) Knowledge of Subject along with Q/A handling during the Seminar: 25 Marks

c) Presentation and Communication Skills: 25 Marks

d) Contents of the Presentation: 25 Marks.

Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-IV

Subject Code: MSM-406 Max. Marks: 60

Subject Title: Advanced Operation Resarch Internal Assessment: 40

Time Allowed: 3 Hours Total: 100

Section-A

Queueing Problems: Characteristics of queueing system, Distributions in queueing systems, Poisson arrivals and exponential service times, the M/M/I, M/M/S queueing systems, Steady state solutions and their measure of effectiveness.

Section-B

Inventory problems, definition, the nature and structure of inventory system, Deterministic models and their solution, multi item inventory problems, stochastic inventory models.

Section-C

Replacement and Maintenance Problems: replacement of capital equipment, discounting cost, replacement in anticipation of failure, preventive maintenance, the general renewal process

Section-D

Network Analysis: Introduction to Networks, Minimal spanning tree problem, Shortest path

problem: Dijkstra’s algorithm, Floyd’s algorithm, Maximum flow problem, Project

management: Critical path method, Critical path computations, Optimal scheduling by CPM, Review techniques (PERT).

Text books:

1. Introduction to Operations Research 10th edition by Hillier and Lieberman Solution Manual

2. Operations Research: Applications and Algorithms by Wayne L. Winston, 2003

References Books:

1. Friderick S. Hillier and Gerald J. Lieberman, ‘Operations Research’, 2nd Edn., Holden-Day Inc., USA, 1974.

2. M.S. Bazaraa, H.D. Sherali, C.M. Shetty, ‘Nonlinear Programming: Theory and

Algorithms’, John Wiley and Sons, 1993.

3. S. Chandra, Jayadeva, A. Mehra, ‘Numerical Optimization and Applications’, Narosa Publishing House, 2013.

E-Book Links:

1. [1]

References Links:

1.

Desh Bhagat University, Mandi Gobindgarh

MSc. (Mathematics)

Syllabus

Semester-IV

Subject Code: MSM-407 Max. Marks: 60

Subject Title: Graph Theory Internal Assessment: 40

Time Allowed: 3 Hours Total: 100

Section-A

Fundamental Concepts: Graph- Definitions an examples, graphs as models, Matrices and isomorphism, paths, Connected graphs, Bipartite graphs, Externality vertex degree, Pigeonhole principal, Turan’s theorem, Degree sequences, Graphic sequences, Degree and digraphs.

Section-B

Tree and distances: Properties of tree, Distance in graphs, Stronger results, Disjoint spanning trees, Shortest paths, Tress in computer science, Eulerian circuits.

Section-C

Matching and Factors: Matching in bipartite graphs, Maximum matching, Hall’s matching conditions, mismatching in bipartite graphs, sets, applications and algorithms, maximum bipartite matching, weighted bipartite matching, in general graphs, Tutte’s 1- factor theorem, f-factors of graphs.

Section-D

Connectivity and Paths: Cuts connectivity, Edge-connectivity, Blocks, 2-connected graphs, Connectivity of digraphs, k connected and k-edge connected graphs, Applications of merger’s theorem, Network flow problems, Maximum network flow, Integral flows.

Edges and cycles: Line graph and edge coloring, Hamiltonian cycles: Necessary and sufficient conditions.

Textbooks:

1. J.A.Bondy and U.S.R. Murty, Graph Theory, Springer-Verlag London, 2008.

2. J. Clark and D. A. Holton, A First look at Graph Theory, Allied Publishers, New Delhi, 1995

References Books:

1. 1. D. B. West, Introduction to Graph Theory, Pearson Education India, 2015

2. R. Gould, Graph Theory, Dover Publications, Reprint Edition, 2012.

3. A.Gibbons, Algorithmic Graph Theory, Cambridge University Press, Cambridge, 1999.

4. R.J. Wilson, Introduction to Graph Theory, Pearson Education, 4th Ed., 2004(Indian Print).

5. S.A.Choudum, A First Course in Graph Theory, MacMillan India Ltd., 2000.

E-Book Links:

1. [4]

References Links:

1.

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