Previous Scheme of Studies

Till Spring, 2012 Semester

Department of Mathematics and Statistics

Faculty of Basic and Applied Sciences

International Islamic University, Islamabad

Scheme of Studies for BS in Mathematics (4-years)

1st Semester

GC-101 Functional English – I 3GC-I02 Introduction to the Use of Computer 3GC-I03 Physics-I 3MATH 101 Fundamentals of Mathematics 3MATH 111 Calculus-I 3

15

2nd Semester

GC-I04 Functional English – II 3GC-105 Physics-II 3GC-106 Introduction to Economics 3MATH 112 Calculus-II 3MATH 121 Introduction to Linear Algebra 3

15

3rd Semester

GC-107 Basics of Academic Writing 3 GC-108 Islamic Worldview and Civilization-I 3

4th Semester

GC-111 Understanding of Quran -II 3 GC-112 Islamic Worldview and Civilization-II 3

1

GC-109 Psychology 3 GC-110 Understanding of Quran-I 3MATH 213 Calculus-III 3 MATH 231 Mechanics-I 3

18

GC-113 Introduction to Law 3 GC-114 Introduction to Management 3

MATH 232 Mechanics-II 3 MATH 241 Elementary Differential Equations with

Applications 3 18

5th Semester

GC-115 Computer Language-I 3MATH 314 Advanced Calculus 3MATH 322 Algebra-I 3MATH 342 Ordinary Differential Equations 3MATH 351 Differential Geometry-I 3MATH 361 Complex Analysis 3

18

6th Semester

GC-116 Computer Language-II 3GC-117 Pakistani Culture & Society 3MATH 323 Linear Algebra 3MATH 333 Analytical Mechanics 3 MATH 352 Introduction to Topology 3 MATH 362 Real Analysis 3

18

7th Semester

MATH 443 Partial Differential Equations 3MATH 463 Functional Analysis-I 3MATH 471 Numerical Analysis-I 3 MATH 472 Statistics and Probability-I 3 Elective-I 3

15

8th Semester GC-118 Software Tools 3 MATH 473 Numerical Analysis-II 3 Elective-II 3

MATH 491 Project 1 or two electives 6 ( IV & V )

15

Remarks:At present the codes allotted to the general courses (GC) are temporary. The exact codes will be given with an

approval of concerned faculty or committee later on.

M. Sc in Mathematics

It will be a two years (4 semesters) 72 credit hours degree programme.

Eligibility: BA/B.Sc (14 years of schooling) from a recognized university with at least 50%

aggregate of marks having Mathematics as a major subject.

Scheme of Studies for M. Sc Mathematics Ist Semester

GC-110 Understanding of Quran-I 3 MATH 314 Advanced Calculus 3

2nd Semester

GC-111 Understanding of Quran-II 3 GC-115 Computer Language-I 3

2

MATH 322 Algebra-I 3MATH 342 Ordinary Differential Equations 3MATH 351 Differential Geometry-I 3MATH 361 Complex Analysis 3

18

MATH 323 Linear Algebra 3MATH 333 Analytical Mechanics 3 MATH 352 Introduction to Topology 3 MATH 362 Real Analysis 3

18 3rd Semester GC-116 Computer Language-II 3MATH 443 Partial Differential Equations 3MATH 463 Functional Analysis-I 3MATH 471 Numerical Analysis-I 3MATH 472 Statistics and Probability-1 3 Elective-I 3

18

4th Semester

GC-118 Software Tools 3 MATH 473 Numerical Analysis-II 3

Elective-II 3 Elective-III 3 Elective-IV 3 Elective-V 3

18

List of Elective Courses BS/M.Sc. Mathematics

1. MATH 424 Elementary Number Theory

2. MATH 425 Algebra-II

3. MATH 434 Analytical Dynamics

4. MATH 435 Fluid Mechanics

5. MATH 444 Integral Equations

3

6. MATH 453 Advanced Topology

7. MATH 454 Algebraic Topology

8. MATH 455 Differential Geometry-II

9. MATH 456 Riemannian Geometry

10. MATH 457 Combinatorics and Graph Theory

11. MATH 464 Measure and Integration

12. MATH 465 Functional Analysis-II

13. MATH 474 Operations Research

14. MATH 475 Optimization Theory

15. MATH 476 Mathematical Modelling and Simulation

16. MATH 477 Statistics and Probability -II

17. MATH 481 Special Relativity

18. CS 111 Programming Fundamentals

19. CS 212 Data Structures and Algorithms

20. CS 314 Theory of Automata

21. CS 322 Computer Communications and Networks

22. CS 361 Computer Graphics

Note: Each Course-I shall be a pre-requisite for its Course-II.

MS/M.Phil in Mathematics

This will be a 2-4 years (4-8 semesters), 30 credit hour’s degree programme. Students will be

required to carry out 24 credit hours of course work in addition to 6 credit hours of thesis

work.

Eligibility: To be eligible for selection as a Junior Research Assistant, a candidate:-

4

(i) M.Sc. in Mathematics/BS in Mathematics/M.Sc Physics having 16 years

education with at least 60 % marks (annual system) or 2.5/4.0 GPA

(semester system) from a recognized institute / university.

(ii) Should not have obtained 3rd division in F.A/F.Sc. B.A/B.Sc. M.A. or

M.Sc. provided that any of the conditions (i) and (ii) above may by relaxed

by the President in case of the teachers of the Universities/

colleges/research organizations nominated by them.

Scheme of Studies for MS/M.Phil in Mathematics

1st Semester

4 courses 12 credit hours

2nd Semester

4 courses 12 credit hours

3rd Semester

MATH 692 Project -I 3 credit hours

4th Semester

MATH 692 Project -II 3 credit hours

5

Details of MS/M. Phil programme are given in Procedure for regulating post graduate studies in Department of Mathematics.

Ph.D. in Mathematics

This will be a 3 – 5 years research degree programme. Students having MS (18 years) and M.

Sc (16 years) education will be required to carry out 18 and 42 credit hours course work

respectively in addition to 9 credit hours of thesis work.

The student will have to complete all the requirement of HEC subject to the approval of

academic council of IIUI.

Eligibility: To be eligible for selection as a Senior Research Assistant, a candidate:-

(a) An M. Phil/MS degree or its equivalent to 18 years education in the relevant subject from a recognized University.

(b) M.Sc. in Mathematics/BS in Mathematics/M.Sc Physics with 16 years

education/MS or M. Phil with at least 65 % marks (annual system) or 3.0/4.0

CGPA (semester system) from a recognized institute.

(c) A college/university teacher or a member of the research staff of a Research organization who has shown undoubted promise for research and holds M.A./M.Sc. degree, may be recommended by the Admission Committee for admission to Ph.D. programme.

Details of PHD programme are given in Procedure for regulating post graduate studies in Department of Mathematics.

Electives Courses for MS/Ph. D. in Mathematics

MATH 526 Ring and Modules-I

MATH 527 Ring and Modules - II

MATH 528 Advanced Linear Algebra

MATH 529 Lie Algebras

6

MATH 536 Newtonian Fluids

MATH 537 Advanced Analytical Dynamics-I

MATH 538 Advanced Analytical Dynamics-II

MATH 539 Non-Newtonian Fluids

MATH 545 Advanced Integral Equations

MATH 546 Advanced Partial Differential Equations

MATH 547 Mathematical Techniques for Boundary Value Problems

MATH 548 Group Theoretic Methods

MATH 549 Perturbation Methods

MATH 558 Topological Vector Spaces

MATH 559 Fixed Point Theory and Applications

MATH 566 Advanced Functional Analysis

MATH 567 Advanced Real Analysis

MATH 568 Advanced Complex Analysis

MATH 569 Advanced Numerical Analysis

MATH 577 Numerical Solutions of Ordinary Differential Equations

MATH 578 Numerical Solutions of Partial Differential Equations

MATH 579 Advanced Optimization Theory

MATH 582 Magneto hydrodynamics-I

MATH 583 Magneto hydrodynamics-II

MATH 584 Electrodynamics-I

MATH 585 Electrodynamics-II

MATH 586 General Relativity

MATH 587 Spectrum Methods in Fluid Dynamics

MATH 588 Thermodynamics I

MATH 589 Thermodynamics II

MATH 692 Project I & II (MS dissertation: 03 credit hours +03 credit hours = 06 credit hours)

MATH 792 Topics in AlgebraMATH 793 Topics in Mechanics

MATH 794 Topics in Differential Equations

MATH 795 Topics in Topology

MATH 796 Topics in Analysis

MATH 897 Topics in Computational Math

MATH 898 Topics in Applied Mathematics

MATH 993 Project (PhD Seminar)

MATH 994 Project (PhD Thesis) (4 credit hours +5 credit hours)

7

8

Revised Scheme of Studies

From Fall, 2012 Semester to onward

Department of Mathematics and Statistics

Faculty of Basic and Applied Sciences

International Islamic University, Islamabad

9

Scheme of Studies of BS Mathematics (4 years)

1st Semester

GC-101 Understanding of Quran-I 3GC-102 Functional English – I 3GC-I03 Physics-I 3MATH 101 Fundamentals of Mathematics 3MATH 102 Calculus-I 3

15

2nd Semester

GC-I04 Functional English – II 3GC-105 Physics-II 3GC-106 Introduction to the Use of Computer 3MATH 103 Calculus-II 3MATH 104 Introduction to Linear Algebra 3

15 3rd Semester

GC-107 Basics of Academic Writing 3GC-108 Islamic Worldview and Civilization-I 3GC-109 Physics-III 3 GC-110 Understanding of Quran –II 3MATH 201 Calculus-III 3 MATH 202 Mechanics 3

18

4th Semester GC-111 Introduction to Economics 3GC-112 Islamic Worldview and Civilization-II 3GC-113 Introduction to Law

3GC-114 Psychology 3MATH 203 Elementary number theory and

Combinatorics 3 MATH 204 Elementary Differential Equations

with Applications 3

185th Semester

GC-115 Pakistani Culture & Society 3MATH 301 Advanced Calculus 3MATH 302 Linear Algebra 3MATH 303 Ordinary Differential Equations 3MATH 304 Set Topology 3MATH 305 Complex Analysis 3

18

6th Semester

GC-116 Software Tools 3GC-117 Computer Language-I 3MATH 306 Group Theory 3MATH 307 Analytical Mechanics 3 MATH 308 Partial Differential Equations 3 MATH 309 Real Analysis 3

18

7th Semester

MATH 401 Differential Geometry-I 3MATH 402 Functional Analysis-I 3MATH 403 Numerical Method s 3 MATH 404 Statistics & Probability-I 3

Elective-I 3

15

8th Semester GC-118 Discrete Structures 3

Elective-II 3 Elective-III 3

MATH 400 Project 1 or two electives 6( IV & V )

15

Scheme of Studies of M.Sc Mathematics (2 years)

1st Semester

GC-101 Understanding of Quran-I 3 MATH 301 Advanced Calculus 3MATH 302 Linear Algebra 3MATH 303 Ordinary Differential Equations 3MATH 304 Set Topology 3MATH 305 Complex Analysis 3

18

2nd Semester

GC-116 Software Tools 3GC-117 Computer Language-I 3MATH 306 Group Theory 3MATH 307 Analytical Mechanics 3 MATH 308 Partial Differential Equations 3 MATH 309 Real Analysis 3

18 3rd Semester

GC-110 Understanding of Quran-II 3MATH 401 Differential Geometry-I 3MATH 402 Functional Analysis-I 3 MATH 403 Numerical Methods 3 MATH 404 Statistics & Probability-I 3

4th Semester

GC-118 Discrete Structures 3 Elective-II 3 Elective-III 3 Elective-IV 3 Elective-V 3 Elective-VI 3

10

Elective-I 3

18 18

List of Elective Courses BS/M.Sc. Mathematics

1. MATH-406 Continuous Groups2. MATH-425 Fluid Mechanics-II3. MATH-435 Fuzzy Logics4. MATH-409 Advanced Group Theory5. MATH-410 Theory of Modules6. MATH-411 Decomposition of Modules7. MATH-412 Galois Theory8. MATH-416 Algebraic Geometry9. MATH-417 Algebraic Systems and Coding Theory10. MATH-421 Quantum Mechanics11. MATH 425 Rings and Fields12. MATH-426 Elasticity Theory13. MATH-427 Electromagnetism14. MATH-428 Theory of Manifolds15. MATH-434 Group Algorithms Programming16. MATH 434 Analytical Dynamics17. MATH 435 Fluid Mechanics-I18. MATH 444 Integral Equations19. MATH 453 Advanced Topology20. MATH 454 Algebraic Topology21. MATH 455 Differential Geometry-II22. MATH 456 Riemannian Geometry23. MATH 457 Combinatorics and Graph Theory24. MATH 464 Measure and Integration25. MATH 465 Functional Analysis-II26. MATH 474 Operations Research27. MATH 475 Optimization Theory28. MATH 476 Mathematical Modelling and Simulation29. MATH 477 Statistics and Probability -II30. MATH 481 Special Relativity31. CS 111 Programming Fundamentals32. CS 212 Data Structures and Algorithms33. CS 314 Theory of Automata34. CS 322 Computer Communications and Networks35. CS 361 Computer Graphics

11

Scheme of Studies for MS Mathematics Programs

1st Semester4 Courses 12 credit hours

2nd Semester 4 courses 12 credit hours

3rd SemesterMATH 600 Project –I 3 credit hours

4th Semester MATH 600 Project –II 3 credit hours

Eligibility:

M.Sc/BS-(4years) (Mathematics or Physics) with minimum CGPA 2.50/4.00 or 60% marks in annual system and appropriate NTS/GAT (General) with minimum 50% score.

List of Core Courses for MS Mathematics Programs

1. MATH 501 Advanced Mathematical Analysis2. MATH 502 Advanced Partial Differential Equations3. MATH 503 Advanced Linear Algebra4. MATH 504 Advanced Mathematical Methods

12

List of Elective Courses for MS Mathematics Programs

1. MATH 505 Semigroup Theory2. MATH 506 Theory of Group Actions3. MATH 507 Theory of Several Complex Variables4. MATH 508 Topological Vector Spaces5. MATH 509 Loop Groups6. MATH 510 Nilpotent and Soluble Groups7. MATH 511 Commutative Algebra8. MATH 512 Banach Algebras9. MATH 513 Lie Algebras10. MATH 514 Spectral Theory in Hilbert Spaces11. MATH 515 Heat and Mass Transfer 12. MATH 516 Introduction to Modeling and Simulation 13. MATH 551 Newtonian Fluids14. MATH 552 Advanced Integral Equation15. MATH 553 Numerical Solutions of Ordinary Differential Equations16. MATH 554 Electrodynamics17. MATH 555 General Relativity18. MATH 556 Elastodynamics19. MATH 557 Plasma Theory20. MATH 601 Variational Inequalities21. MATH 602 Theory of Complex Manifolds22. MATH 603 C *-Algebras23. MATH 604 Von Neumann Algebras24. MATH 651 Perturbation Methods25. MATH 652 Numerical Solutions of Partial Differential Equations26. MATH 653 Cosmology27. MATH 654 Solid Mechanics28. MATH 655 Numerical Optimization29. MATH 656 The Classical Theory of Fields

MATH 600 MS dissertation (06 credit hours)

13

Scheme of Studies for Ph.D Mathematics Programs

1st Semester3 Courses 9

2nd Semester3 Courses 9

3rd SemesterMATH 800 Ph.D. Thesis 9

4th Semester MATH 800 Ph.D. Thesis 9

5th SemesterMATH 800 Ph.D. Thesis 9

6th Semester MATH 800 Ph.D. Thesis 9

Eligibility:

18 years of education in Mathematics with minimum CGPA 3.00/4.00 or 65% marks in annual system. GRE/GAT (Subject) with minimum 60% score.

Details of PHD program are given in Procedure for regulating post graduate studies in Department of Mathematics.

List of Core Courses for Ph.D. Mathematics Programs

1. MATH 801 Advances in Analysis2. MATH 802 Advanced Perturbation Methods

14

List of Elective Courses for MS/PhD Mathematics Program

1. MATH 701 Near Rings2. MATH 702 Advanced Ring Theory-I3. MATH 703 Fixed Point Theory4. MATH 704 Commutative Semigroup Rings5. MATH 705 Homological Algebras6. MATH 706 Representation of Finite Algebra and Quivers7. MATH 707 Theory of Semirings8. MATH 708 Ordered Vector Spaces9. MATH 709 Banach Lattices10. MATH 710 Approximation Theory11. MATH 711 Topological Algebras12. MATH 712 Fuzzy Algebra13. MATH 713 Algebraic Number Theory14. MATH 714 Hopf Algebra and Quantum Groups15. MATH 751 Advanced Analytical Dynamics-I16. MATH 752 Non-Newtonian Fluids17. MATH 753 Mathematical Techniques for Boundary Value Problems18. MATH 754 Group Theoretic Methods19. MATH 755 Advanced Numerical Analysis20. MATH 756 Advanced Optimization Theory21. MATH 757 Magnetohydrodynamics22. MATH 758 Advanced Electrodynamics23. MATH 759 Stochastic Processes24. MATH 760 Multivariate Methods and Analysis25. MATH 761 Nonlinear Differential Equation 26. MATH 762 Advanced Plasma Theory27. MATH 763 Convective Heat Transfer: Viscous Fluids28. MATH 764 Finite Elements Analysis29. MATH 765 Momentum and Thermal Boundary Layer Theory30. MATH 766 Astrophysics31. MATH 767 Advanced Elastodynamics32. MATH 768 Statistical Mechanics33. MATH 769 Advanced Quantum Theory34. MATH 770 Nonlinear Waves

15

List of Elective Courses for PhD Mathematics Program

1. MATH 803 LA-Semigroups2. MATH 804 Advanced Analytical Dynamics-II3. MATH 805 Advanced Magnetohydrodynamics4. MATH 806 Spectral Methods in Fluid Dynamics5. MATH 807 Advanced Semigroup Theory6. MATH 808 Advanced Near Rings7. MATH 809 Theory of Group Graphs8. MATH 810 Advanced Ring Theory-II9. MATH 811 Non-Standard Analysis10. MATH 812 Numerical Ranges of Operators on Normal Spaces11. MATH 813 Strict Convexity12. MATH 814 Advanced Commutative Algebra13. MATH 815 Advanced Homological Algebra14. MATH 816 Advanced Theory of Semirings15. MATH 851 Advanced Heat Transfer16. MATH 852 Convective Heat Transfer: Porous Media17. MATH 853 Advanced Finite Elements Analysis18. MATH 854 Advanced Multivariate Methods and Analysis19. MATH 855 Robotics20. MATH 856 Group Analysis of Partial Differential Equations21. MATH 857 Advanced Nonlinear Differential Equations22. MATH 858 Modeling and Simulation of Dynamical Systems23. MATH 859 Topics in Fluid Mechanics24. MATH 860 Topics in Mechanics25. MATH 861 Topics in Differential Equations26. MATH 862 Topics in Computational Mathematics27. MATH 863 Topics in Applied Mathematics 28. MATH 864 Topics in Algebra29. MATH 865 Topics in Topology30. MATH 866 Topics in Analysis31. MATH 867 Topics in Complex Analysis

MATH 800 PhD Thesis (9 credit hours)

16

Scheme of Studies and Course Contents of BS/M.Sc Mathematics programs

GC-102 Functional English-ITo provide language support to mixed ability groups of students form different faculties. To cater to their deficiencies especially in advanced level grammar related to structure and style. To expose them to reading strategies of deal with a variety of reading texts, especially in their own area of studies. To help them improve their vocabulary to cope with specific texts in English. To equip them with good paragraph writing skills to form strong paragraphs and provide foundation for essay writing.

Course Contents. English has three main components:. Reading. 27 hours. Paragraph Writing and Introduction to essay writing 12 hours. Grammar, structure and style 09 hours

Total: 48 hours

Reading: Nine units have been allocated for reading. The units consist of a variety long and short text from different fields of studies. The rich vocabulary, from simple to complex concepts in the texts makes them good reading texts.

List of reading textsGetting to know your bookThe true studentSelf-discipline and studiesHow to help hard of hearingLearning to seeWhat do you know about nutritionSocial factors that shape up our livesBread and CireusDoodles

WritingIntroduction to paragraph writing.Writing descriptive paragraphsWriting narrative paragraphsWriting process paragraphsWriting expository paragraphs with emphasis on the following:

. Cause and effect

. Comparison and contrast

. Situation-problem-solutionGrammar, Structure and Style

. Any grammar deficiency found in students as a class

. Simple, compound and complex sentence structures

. Coordination

. Parallelism

. Fragments

. Style: choice of words, positioning of the subject, length of sentence repetition etc.

Recommended Books:

17

A course book with the above-mentioned objectives and contents has been developed by the syllabus designing committee especially constituted for this purpose by the Dean (FLL & II).

GC-103 Physics-I

Objectives:This course sets the foundation of undergraduate physics. The students will be able to know about the basic concepts of charge, matter, electric field and electric potential.

Course Outline:Vector and scalar field, differential and integral vector analysis. Electric charge, Electric field, Electric dipole, continuous charge distribution, line and surface integrals, Gauss's law and its application, conductors and insulators. Electrostatic force and electric potential: Potential due to discrete and continuous charge distribution. Poisson’s and Laplace equation, capacitors and dielectrics, DC circuits: Kirchhoff's laws, loop analysis and network theorems, RC circuit.

Recommended Books:1. Raymond A. Serway and John W. Jewett, Jr., Physics for Scientists and Engineers.

Vol. 2. 6th Ed Belmont, CA: Thomson-Brooks/Cole, c2004.2. Theodore F. Bogart, Jr., Electric Circuits, McGraw-Hill.3. Halliday, D., Resnick, R., and Krane, K. S., Physics Vol II, John Wiley and Sons. 5 th

ed 2004.4. Cutnell and Johnson, Physics. John Wiley and Sons 6th edition.5. Giancoli, D., Physics for Scientists and Engineers, Prentice Hall Inc., 1988.6. Serway, R. A., Physics for Scientists and Engineers, Saunders Golden Sunburst

Series.7. Young Hugh D. and Freedman, Roger A., University Physics, Addison Wesiey, 2003.8. Purcell, E. M. Electricity and Magnetism, Berkeley Physics Course. Vol. II 2nd ed.

New York, NY: McGraw-Hill Science/Engineering/Math, 1984.9. Griffiths, D. J. Introduction to Electrodynamics, 3rd ed. Upper Saddler River, NJ:

Prentice Hall, 1998.

GC-104 Functional English–IITo provide students reading skills and strategies to help them in reading their subject specified materials. To enhance their writing skills form simple paragraph to full y developed essay. To equip them with writing skills e.g. summary writing, job appellations, resume writing , which serve a practical purpose in professional life. To expose them to texts rich in vocabulary and varied in context.

Course ContentsEnglish II has four components:

. Reading. 27 hours

. Essay Writing 09 hours

. Summary writing 06 hours

. Job Application, resumes/C.Vs 06 hoursTotal: 48 hours

ReadingNine units have been allocated for reading . They are based on informative and interesting texts form different fields of life like education ,sports, health and society. The texts are exploited to help students learn reading strategies.

List of Reading Texts18

The Book of KnowledgeListening faultsSports and warThe miracle of Zam ZamNot just a parrot talkThe weather and how you feelHow to build a healthy response to stressSocial factors that shape our livesTo paint a portrait of a bird (poem)

Essay Writing.. Different parts of essay: Introduction, body, and conclusion, how each part is

developed and linked. Descriptive essays. Expository essays. Argumentative essays

Summary Writing. Writing summaries of descriptive, expository and argumentative texts. What’s summary writing. Locating main and important supporting details. Looking for irrelevant details. Structure and style wherever necessary

Job Applications. Types of job applications. Format. Language and style in job applications

Recommended BooksA course book with the above-mentioned objectives and contents has been developed syllabus designing committee especially constituted for this purpose by the dean (FLL & H).

GC-105 Physics-II:

Objectives:After the completion of the course, the students will be able to;

Apply the basic laws of electricity and magnetism. Solve problem concerning motion and distribution of charge. Learn fundamental principles of electromagnetism to continue to develop solid and

systematic problems skill.

Course Outline:Magnetic fields, moving charge in a magnetic field, magnetic force, Lorentz force,, Ampere’s law, vector potential, Biot Savart law, applications of Ampere and Biot Savart law, magnetic properties of matter, Gauss's Law for Magnetism, differential form, concepts of conservation of magnetic flux, Differential form of Gauss’s Law. Origin of Atomic and Nuclear magnetism, Basic ideas. Bohr Magneton. Magnetization, Defining M, B, ì. Magnetic Materials, Paramagnetism, Diamagnetism, Ferromagnetism- Discussion. Hysteresis in Ferromagnetic materials. Faraday’s law, Lenz’s law, mutual and self induction, LR Circuits, Undriven RLC circuit, phasor representation of driven AC Circuits, impedance, power and energy,filters, quality factor and resonance, displacement current, Maxwell's Equations, wave equations,electromagnetic radiation.

Recommended Books:19

1. Purcell, E. M., Electricity and Magnetism, Berkeley Physics Course, Vol. 2. 2nd ed. New York, NY: McGraw-Hill Science/Engineering/Math, august 1, 1984.

2. Halliday, D., Resnick, R., and Krane, K. S., Pysics Vol II, John wiley and Sons. 5th ed. 2004.

3. Cutnell and Johnson, Physics, John Wiley and Sons 6th edition.4. Giancoli, D., Physics for Scientists and Engineers, Prentice Hall Inc., 1988.5. Serway, R. A., Physics for Scientists and Engineers, Saunders golden Sunburst Series.6. Young Hugh D., and Freedman, Roger A., University Physics, Addison Wesley,

2003.7. Griffiths, D. J., Introduction to Electrodynamics 3rd ed. Upper Saddler River, NJ:

Prentice Hall, 1998.

GC-106 Introduction to EconomicsNature, scope and importance of Economics, Microeconomics versus Macroeconomics, Scarcity and choice, Commodities ( goods and services). In come and resources, Opportunity costs, Factors of production. Production possibility frontier.Demand, Supply and Equilibrium: Concepts of demand and supply. Laws of demand and supply, Market equilibrium . Shifts in demand and supply curves and price determination, Concept of elasticity: Own-price Income elasticity and Cross-price Elasticity of Demand, Price elasticity of supply. Importance and determinants of elasticity.Consumer’s Behavior: Utility Theory. Consumer’s preferences and utility function, Laws of increasing and diminishing marginal utility . Classification of goods ( Normal and Inferior goods). Theory of the Firm: Factors of production and their rewards, Total, average. And marginal products. Laws of returns. Cost of production. Total, average, and marginal costs, Revenues of a firm: Total , average, and marginal revenues, Concept of profit maximization and Cost minimization, Form and Industry.Market Structure: Theory of Exchange, Classification of markets, Competitive markets, Imperfect competition: Monopoly. Monopolistic . Price and Output determination under different markets.National Income: Concepts of national income, GDP & GNP, Real vs. Nominal income, Per capita income, Measurement of national income, Saving and Personal Consumption.Money and Banking: Definition of money, Demand for and supply of money. Commercial banking system, Role of the Central Bank. The exchange rate and need for foreign currency.Public Finance: The need for government, Provision of public goods, Public revenues. Forms and kinds of taxes, Heads of Public expenditure, Budget and deficit financing.Macroeconomic Issues: Concept of inflation, unemployment, Balance of payment, Exchange rate and Business cycles, Role of Public Policies in the economy ( Monetary, Fiscal , Commercial and Labour Policies): Brief discussion.

Recommended Books1. Michel Parkin-Economics-5th Ed. (2004). Addision Wesley.2. Samualson and Nordhaus- Economics-18th Ed. (2004)- McGraw Hill, Inc.3. Lipsey and Crystal –Economics- (1999) – Oxford University Press.

GC-107 Basics of Academic Writing Introduction to Basic Academic WritingStructure and CohesionDescription, process, procedure & physicalNarrative, Definitions, Exemplification, and classifications.Comparison and contrast, cause and effect.Interpretation of Data.Academic style.Proofreading. Surveys Questionnaires and projects.

20

PresentationsShort presentations using overhead projector and audio visual aidsIntroduction to public speakingSkills involved in presentingMaking a good presentationLonger presentations using multi-media based on deeper research project of Academic nature. 7- 10 mins. For final project.

Language difficulties and types of errorCorrection codeResearch Reports.

Recommended Books1. Academic Writing Course, R.R Jordan. ( Longman)

GC-108 Islamic Worldview and Civilization-I Introduction to the course, introduction to the Islamic culture and civilization, fundamental sources of Islamic civilization, role of civilization in human society, distinguished aspects of islamic civilization, impact of Islamic civilization upon the world, the world before the advent of the prophet.(s.a.w), outstanding aspects about the biography of prophet (p.b.u.h), status of women in the light of search of prophet (p.b.u.h), history of the Holy Quran, compilation of hadith, nature of the creator, nature of the creature , eschatology (end of the world).

Recommended Books1. The emergence of islam, lectures on the development of islamic worldview,

intellectual tradition and polity by Mohammad hamidullah.2. English and islam: a clash of civilization by Ratnawati mohd Asraf, international islamic

university Malaysia.3. What happened in history (penguin, 1942) and man makes himself by Gordon chide, V.4. Decline of the west perspectives of world history Spengler, oswald, (1919).5. Islam the natural way of life by Abdul Hamid Abdul Wahid.6. Islam and the world (S. Abul hasan Ali Nadvi)7. Islam between east and west (Alija Izat)8. Islamic resurgence (Pro. Khurshid Ahmad)9. Clash of civilization (Huntington)10. Internet (islamonline.net , sultan.org, soundvision.net, islamweb.net)

GC-109 Physics-III

Objectives:After the completion of the course, the students will be able to;

State the relationship between temperature and heat. Analyze thermodynamic charges using law of thermodynamics. Use equation of state and simple kinematic energy in solving problems. Apply the laws of thermodynamics to real world.

Course Outline:Heat transfer mechanisms, Zeroth Law of Thermodynamics. First law, Second law: Entropy, Third law of thermodynamics Ideal gas and temperature, Thermodynamic potentials and Internal energy, Free energies, Euler and Gibbs-Duhem, The Gibbs Equation, Entropy Changes, Maxwell relations, Response functions, Relations between partial derivatives, Conditions for equilibrium, Stability requirements on other fee energies, Phase transitions,

21

Phase diagrams, Clausius-Clapeyron relation, Multi-phase boundaries, Binary phase diagram, Van der Waals equation of state,

Recommended Books:1. Fundamentals of Classical Thermodynamics Richard E. Sonntag, Claus Borgnakke

and Gordon V. Van Wylen., 6th ed., John Wiley & Sons, 1988.2. Thermodynamics: Processes and Applications: Jr., Earl Logan fundamentals of

Thermodynamics, Richard E. Sunntag, Claus Borgnakke, Gordon J. Van Wylen.3. Thermodynamics, Henri J. F. Jansen, 2003.4. Michael J. Moran and Howard N. Shapiro. Fundamentals of Engineering

Thermodynamics. John wiley & Sons. (any edition)5. Yunus A. Cengel and Michael A. Boles. Thermodynamics, an Engineering Approach.

McGraw-Hill. (any edition)

GC-111 Introduction to Management Introduction to management, evolution of management theory, ethics, social Responsibility and external environment of organizations, planning and decision making,Strategic planning, problem solving and decision making, operations management and Productivity, organizing for stability and change, coordination and organizational design, Authority, delegation, and decentralization. Staffing and human resource management, Managing organizational change and development, managing organizational conflict andCreativity, motivation, performance and satisfaction. Leadership, groups and committees.Interpersonal and organizational communication. Effective controlling , financial control methods and international management, approaches to management, effective decision making the planning Process, organizing for action, organizing individuals, human resources, copying with Change, self management, group dynamics, communication, motivation, leadership, Concepts of control, managing information systems, presentations. What is entrepreneurship, financial plane, fund generation, legal aspects.

Recommended Books1. Management by Robins S. 5th ed. Prentice-Hall, 1998.2. Management by Harold Koontz and Heinz Weihrich M. by Mc Graw Hill 1998.3. Management by James A. F. Stoner and Charles Wankel by Prentice-Hill, 1986.4. Entrepreneurship and new venture formation by Thomas W. Zimmerer and Norman

M. Scarvocough by Printice-HAall International, 1996. GC-112 Islamic Worldview and Civilization-II Salient features of islam, functions of devine deen, introduction of muslim ummah purpose & qualities, decline of mulsim ummah, challenges of muslim ummah in contemporary world, rise of muslim ummah and its distinguished qualities, a plea for science in the muslim culture, achievements of muslims in the field of science, social and family system of islam, introduction of muslim minorities, statues of muslim women & issue of hijab, statues of Non muslims in islamic states, jihad introduction ,importance and relevant conditions, caliphate of man, duties of caliph (head of state),introduction of sharia & fundamental sources of islamic law, clash of civilization myth of reality, challenges and future of islamic civilization in contemporary world.

Recommended Books1. The emergence of islam, lectures on the development of islamic worldview,

intellectual tradition and polity by Mohammad hamidullah.2. English and islam: a clash of civilization by Ratnawati mohd Asraf, international

islamic university Malaysia.

22

3. What happened in history (penguin, 1942) and man makes himself by Gordon chide, V.

4. Decline of the west perspectives of world history Spengler, oswald, (1919).5. Islam the natural way of life by Abdul Hamid Abdul Wahid.6. Islam and the world (S. Abul hasan Ali Nadvi)7. Islam between east and west (Alija Izat)8. Islamic resurgence (Pro. Khurshid Ahmad)9. Clash of civilization (Huntington)10. Internet (islamonline.net , sultan.org, soundvision.net, islamweb.net)

GC-113 Introduction to LawA brief study of basic principles of Law and legal theory coupled with comparativeIslamic doctrines will enable students to have a grass root level comprehension of the discipline itself. It will develop legal acumen and in-depth capability to Comprehend basic rules of Law & Society with particular reference to theoretical perspective.

Introduction:I. Definitions:

(a) Scope(b) State & Law(c) Advantages & Disadvantages of Law

II. Source of Law:a. Western

(i) Legislation(ii) Precedent(iii) Custom

Islamic(i) Primary Sources Quran

Sunnah

(ii) Secondary SourcesIjthihadIjma-QiyasIstehsan Istidlal

III. Classification of Law

(i) Kinds of Law(ii) Public Law, Private Law

Recommended Books1. Jurisprudence by John Salmond.2. Early Development of Islamic Jurisprudence by Dr. Ahmad Hassan.3. Text Book of Jurisprudence by G.W. Paton.4. Islamic Jurisprudence by Imran Ahsan Nyazee5. Jurisprudence by W.N. Hibbert.

GC-114 Psychology 1. Introduction & Methods of Psychology Nature and Application of Psychology with special

reference to Pakist

23

2. Historical Background and Schools of Psychology (A Brief Survey)3. Observation4. Case History Method5. Experimental6. Survey Method7. Interviewing Techniques8. Sensation, Perception and Attention sensation

Characteristics and Major Functions of Different Sensation

Vision: Structure and functions of the Eye.Audition: Structure and functions of the Ear.

PerceptionNature of PerceptionFactors of Perception: Subjective, Objective and Social Kinds of Perception:

Spatial Perception (Perception of Depth and Distance)Temporal Perception; Auditory Perception.

AttentionFactors, Subjective and ObjectiveSpan of AttentionFluctuation of AttentionDistraction of Attention (Causes and Control)

3. MotivesDefinition and NatureClassification

Primary (Biogenic) Motives: Hunger, Thirst, Defection and Urination, Fatigue, Sleep, Pain, Temperature, Regulation, Maternal Behavior, Sex.

Secondary (Sociogenic) Motives: Play and Manipulation, Exploration and Curiosity, Affiliation ,Achievement and Power, Competition, Cooperation Social Approval and Self Actualization.

4. EmotionsDefinition and NaturePhysiological changes during Emotions (Neural, Cardial,Visceral, Glandular), Galvanic Skin Response; PupilliometricsTheories of EmotionJames Lange Theory; Cannon-Bard TheorySchachter- Singer Theory

5. Learning & MemoryDefinition of LearningTypes of Learning: Classical and Operant ConditioningMethods of Learning: Trial and Error: Learning by Insight;Observational LearningNature of MemoryMemory Processes: Retention, Recall and recognitionForgetting: Nature and Causes

24

6. ThinkingDefinition of NatureTools of Thinking: Imagery; Language; ConceptsKinds of ThinkingProblem Solving; Decision Making; Reasoning

7. PersonalityDefinitionDevelopment of Personality: Biological and Environmental Factors

Recommended Books1. Atkinson R.C., & Smith E. E. (2000) . Introduction to Psychology (13th ed),2. Harcourt Brace College Publishers.3. Hayes, N. (2000) Foundation of Psychology (3trd ed), Thomson Learning.4. Lahey, B.B. (2004) to Psychology: An introduction (8th ed). Mc Graw Hill Companies, Inc.]5. Leahey, T.H. (1992) A history of Psychology : Main Current in Psychological6. Myers, D.G. (1992) Psychology: (3trd ed), New York: Wadsworth Publishers.7. Glassman, W.E. (2000). Approaches to psychology. Open University Press.8. Lahey, B.B. (2004). Psychology: An introduction (8th ed.). McGraw-Hill Companies, Inc.9. Myers, D.G. (1992) Psychology: (3trd ed), New York: Wadsworth Publishers.10. Ormord, J E.. (1995). Educational psychology: Developing learners Prentice

GC-115 Pakistani Culture & Society Land and People : Physical Features of Pakistan (location, mountains and rivers, climate, mineral recourses). Language and culture. Religions of people and minorities. Relations with neighboring communities, with special reference to the Middle East and newly independent republics of Central Asia.Historical and Ideological perspective : Advent of Islam in South Asia. Role of the Muslim Empire and the intellectuals in spreading the message of Islam. A short review of the Muslim rule (from 712 to 1857). The decline of the Muslim rule and rise of British colonialism. Muslim’s efforts to maintain their socio-political identity with reference to the Aligarh Movement and other educational movement. Two-Nation Theory and its elaboration by the stalwarts like Sayyid Ahmad Khan and the leaders of All India Muslim League (Iqbal, Quaid-I-Azam Muhammad Ali Jinnah and others). Political and Constitutional developments : Pakistan at the time of independence. Political and constitutional developments. Comparative study of the constitution of 1956, 1962 and 1973. The amendments in the 1973 constitution. Achievements and failures : Education. Democratic traditions. Social welfare. Health care. Development of human resources. Economy.

Recommended Books1. Following books or their more recent equivalents or manuals, magazines and journals articles,

at the discretion of the instructor:2. Afzal Iqbal, Islamization of Pakistan.3. Govt. of Pakistan, Handbook about Pakistan.4. Kennedy, C., Islamization of Laws and Economy: Case Studies on Pakistan (Islamabad,

1967).5. Qureshi, I.H. (ed.), A Short History of Pakistan (Karachi, 1967).

GC-116 Software Tools Introduction to Matlab and Mathematica and how these can be used to solve problems in algebra, calculus, linear algebra, statistics and differential equations.

Recommended Books

25

1. Getting started with MATLAB7: A quick introduction for Scientists & Engineers, by Rudra Pratap.

2. A guide to MATLAB: for beginners and experienced users, by Hunt / Lipsman / Rosenberg.3. The Mathematica Book, Fourth Edition by Stephen Wolfram.4. Schaum’s outline of Mathematica, by Eugene Don.

GC-117 Computer language-I Historical background, IDE, character set, variables, constants, data types, input/output, arithmetic operators, logical operators, bit wise operators. comments, casting, preprocessor directives, decision control statements, loops, functions, external variables, storage classes, recursion, command line arguments, arrays, strings, and pointers, structures, array of structures, unions, files, character display memory, attribute byte, bit field, equipment list word, graphics, handling larger programs.

Recommended Books 1. Following books or their more recent equivalents, manuals, computer magazines and journals

articles, at the discretion of the instructor:2. Turbo C Programming for PC by Robert Lafore.

GC-118 Discrete Structures:Objectives:After successful completion, the students will be able to;

Understand clear thinking and creative problem solving. Thoroughly train in construction and understanding of mathematical proofs. Exercise common mathematical arguments and proof strategies. Cultivate sense of familiarity and ease in the working with mathematical notations

and common concepts in discrete mathematics.

Course Outline:Introduction to logic, quantifiers and conditional statements; proofs, valid and invalid arguments, Predicates and quantified statements; arguments with quantified statements, Direct proofs; counterexamples; quotient-remainder theorem; floor and ceiling functions; irrationality of some square roots; infinitude of primes, Mathematical Induction, Strong induction, Set Theory; set properties; partitions; power sets, Recursively defined sequences; solving recurrences by iteration, Big Oh notation; efficiency of algorithms; exponential and logarithmic functions, Relations; equivalence relations, finite state automata, partial order relations, Trees, Graphs and graph theory.Recommended Books:

1. Discrete Mathematics and Its Applications by (K. Rosen)2. Discrete Mathematics with Applications by Susanna S.3. Bernard Kolman; Discrete Mathematical structures, 4th ed.

MATH-101 Fundamentals of Mathematics

Overview of basic number theory and basic set theory, binary relations, functions, injective, surjective and bijective functions, inverse image of a function, inverse of a function, composition of functions, denumerable, non-denumerable, countable and uncountable sets, partial order relations and equivalence relations, counting the combinations or permutations of a set. The Inclusion-Exclusion Principle. Mathematical induction and the well-ordering principle. Complex numbers System, polar form of complex number, De Moiver’s Theorem. Exponentional, Trigonometric, Hyperbolic and Logarithmic functions of complex numbers, complex powers, summation of series. Metric space, open and closed balls, open and closed sets. Binary operations, groups, subgroups, cyclic groups and cosets.

Recommended Books26

1. Sets: Naive, Axiomatic and Applied by D. van Dalen, H.C. Doets and H. Deswart, Published by Pergamon Press, 1978

2. Finite Mathematics with Applications by A.W. Goodman and J. S. Ratli, Published by Macmillan Publishing Co.

3. Fraleigh J.B., A first course in Algebra (Addison-Wesley 1982). 4. James W.B., Churchill R.V., Complex variables and applications (Mc-Graw Hill).

MATH-102 Calculus-IReal numbers system, algebra of real valued functions and their graphs, Limits, continuity, differentiation, Applications of derivatives, mean value Theorems, Maxima and Minima, Concavity, singular points, Higher order derivatives and Leibniz Rule. Techniques of Integration. Properties of definite integral, Fundamental Theorem of integral Calculus, Improper Integrals, Reduction Formulas, Applications of definite integral, Quadrature, Area in polar coordinates, Lengths of arcs.

Recommended Books5. Calculus and Analytic Geometry by G.B. Thomas and R.L. Finney, published by Addison

Wesley.6. Calculus, A new horizon by H. Anton, published by John Wiley and Sons.7. Calculus by James Stewart, fifth edition, published by Brooks/Cole, 2002.

MATH-103 Calculus-IIConic sections and polar coordinates, General equation of conic, Ellipse, Parabola, Hyperbola, Cycloid, Area and Length in polar coordinates, conics in polar coordinates, Tangents and normal to the conics in rectangular and polar coordinates, pedal equation, Infinite sequences and series, convergence and divergence of infinite sequences and series, comparison tests, Ratio and Root tests, integral test, alternating and mixed series, absolute and conditional convergence of series, power series and its applications. Taylor and Maclaurin series.

Recommended Books1. Calculus and Analytic Geometry by G.B. Thomas and R.L. Finney, published by Addison-

Wesley.2. Calculus, A new horizon by H. Anton, published by John Wiley and sons3. Calculus by James Stewart, fifth edition, published by Brooks/Cole, 2002

MATH-104 Introduction to Linear AlgebraSystems of linear equations, echelon and reduced echelon forms, matrices, matrix algebra, invertible matrices, elementary matrices, finding the inverse of a matrix, vector spaces, subspaces, linear dependence and independence, basis and dimension, rank, coordinates and transition matrices, determinants, properties, formula for the inverse of a matrix, Cramer’s rule, inner product spaces, norm, Cauchy - Schwarz and triangle inequalities, orthonormal sets. Gram-Schmidt process, projections, least squares, linear transformations, kernel and range, similarity, eigenvalues, eigenvectors, diagonalization.

Recommended Books1. Linear Algebra and its Applications by David C. Lay, second edition, published by Addison –

Wesley, 2000.2. Linear Algebra with Applications by Gareth Williams, published by Jones and Bartlett

Publications.3. Introductory Linear Algebra with Applications by Bernard Kobman and David R. Hill.

MATH-201 Calculus-IIIVector functions, derivatives and integrals of vector functions, functions of several variables, limits and continuity, partial derivatives, tangent planes, linear approximations, chain rule,

27

directional derivatives, gradient vector, maximum and minimum values, Lagrange multipliers. Multiple integrals. Analytic Geometry of three dimensions. Volume of a solid of revolution, Area of a surface of revolution.

Recommended Books1. Calculus and analytic Geometry by G.B. Thomas and R.L. Finney, published by Addison-

Wesley.2. Calculus, A new horizon by H. Anton, published by John Wiley and sons.3. Calculus by James Stewart, fifth edition, published by Brooks/Cole, 2002.

MATH-202 Mechanics:

Objectives:The main objectives of this course are as follows;

To give a comprehensive exposure to students about the fundamentals of static and dynamics.

To develop analytical skills among the students in dealing with the problems. To develop the understanding of basic laws governing the static and dynamics of rigid

body and particles.

Course Outline:Composition of forces, friction, equilibrium, center of mass and gravity, friction, virtual work, Kinematics, rectilinear motion, projectile motion, constrained motion.

Recommended Books:1. Mechanics by Q. K. Ghori, published by West Pakistan Publ.2. Vector Analysis by Munawar Hussain, published by Caravan Book Publ., Lahore.

MATH-203 Elementary Number Theory and Combinatorics:

Objectives:Upon the completion the students will be able to;

Learn the role of number theory in mathematics and how to do number theoretic proofs.

Understand prime factorization, linear from of gcd, linear congruences and system of simultaneous linear congruences primitive roots module primes with important theorem.

Course Outline:Divisibility, greatest common divisor and least common multiple, Euclidean algorithm, Primes, fundamental theorem of arithmetic, linear Diophantine equations, congruence, linear convergences, Chinese’s remainder theorem, divisibility tests, Wilsons theorem, Fermat’s little theorem, Euler’s phi functions, Euler’s theorem, the sum and number of divisors, perfect numbers and Mersenne primesGraphs, Trees, Colorings of graphs and Ramsey's theorem, Turan's theorem (Statements and applications), and extremal graphs, Systems of distinct representatives, Elementary counting Stirling numbers , Recursions and generating functions, Partitions, (0,1)-matrices, Latin squares.

Recommended Books:1. Elementary Number Theory and its applications by Kenneth H. Rosen, fifth edition,

Published by Edison Wesley, 2005.

28

2. A Friendly Introduction to Number Theory by Joseph H. Silverman, third edition, Published by Prentice Hall 2006.

3. Hardy, G. H.; Wright, E. M. An introduction to the theory of numbers. Sixth edition. Revised by D. R. Heath-Brown and J. H. Silverman. Oxford University Press, Oxford, 2008.

4. Van Lint, J. H.; Wilson, R. M.,A Course in Combinatorics, Cambridge University Press, Cambridge, 1992.

MATH 204 Elementary Differential Equations with ApplicationsIntroduction, First order equations: Separable equations, homogeneous equations, linear equations, Bernoulli’s equation, exact equations, integrating factors, Ricatti equation, Clairaut’s equation, orthogonal trajectories.Second order equations: homogeneous linear equations with constant coefficients, non-homogeneous linear equations with constant coefficients, method of undetermined coefficients, variation of parameters, Euler-Cauchy equations, some special second order equations, mechanical and electrical vibrations and other applications to mathematical modeling, total differential equations, Laplace transforms, properties, application to initial value problems, power series solutions.

Recommended Books1. Elementary Differential Equations and Boundary Value Problems by William E. Boyce and

Richard C. Diprima, fifth edition, published by John Wiley and Sons, 1992.2. Differential Equations with Applications and Historical Notes by George F. Simmons, second

edition, published by McGraw Hill, 1991.3. Differential Equations by C. Ray Wylie, published by McGraw Hill.

MATH-301 Advanced Calculus The real numbers: algebraic and order properties of R; the completeness property; cluster points; open and closed sets in R. Sequences, the limit of a function, limit theorems. Continuous functions on intervals: boundedness theorem, maximum-minimum theorem and the intermediate value theorem; uniform continuity. The derivative: the mean value theorem; Taylor’s theorem. Functions of several variables: Limit and continuity of functions of two and three variables; partial derivatives; differentiable functions. Multiple Integrals: regions in the x-y plane, iterated integrals, double integrals, change in the order of integration, transformation of double integrals. Line and surface integrals: Jordan curve, regular region, line integral, Green’s theorem, independence of the path, surface integrals, Gauss theorem.

Recommended Books1. Bartle, R.G. and Sherbert, D.R. Introduction to Real Analysis, John Wiley & Sons 1994. 2. Widder, D.V. Advanced Calculus, Prentice-Hall, 1982. Rudin, W Principles of Real

Analysis, McGraw-Hill, 1995.

MATH-302 Linear AlgebraVector spaces, subspaces, basis and dimension, coordinates, linear transformations, rank and nullity, algebra of linear transformations, isomorphisms, matrix representation of a linear transformation, similarity, linear functionals, dual space, the double dual, transpose of a linear transformation, algebras, the algebra of polynomials, polynomial ideals, prime factorization of a polynomial, eigenvalues, characteristic polynomials, diagonalization, minimal polynomials, Cayley-Hamilton theorem, invariant subspaces, direct sum decompositions, invariant direct sums, primary decomposition theorem, rational and Jordan canonical forms, inner product spaces, linear functionals and adjoints, unitary and normal operators.

29

Recommended Books 1. Linear Algebra by Kenneth Hoffman and Ray Kunze, second edition, published by Prentice

Hall, 19712. Linear Algebra by Je Shilov, G.E. 1977, Dover Publication , Inc., New York.3. Linear Algebra with Applications by J.T. Scheick 1997, McGram Hill,

MATH-303 Ordinary Differential EquationsExistence and uniqueness of solutions, general theory of linear differential equations, method of undetermined coefficients, variation of parameters, power series solutions, ordinary points, regular singular points, Bessel, hypergeometric, Legendre and Hermite equations and the associated functions, Fourier series, boundary value problems, Sturm -Liouville theory.

Recommended Books 1. Elementary Differential Equations and Boundary Value Problems by William E. Boyce and

Richard C. Diprima, fifth edition, published by John Wiley and Sons, 1992.2. Differential Equations with Applications and Historical Notes by George F. Simmons, second

edition, published by McGraw Hill, 1991.3. Differential Equations by M. Morris and O. E. Brown, published by Prentice Hall, 1964.4. Applied Differential Equations by M. R. Spiegel, published by Prentice Hall 1967.5. Ordinary Differential Equations and Difference Groups by F. Chorlton, published by Van

Nostrand, 1965.6. Differential and Difference Equations by L. Brand, published by John Wiley 1966.7. Advanced Engineering Mathematics by D. G. Zill and M. R. Cullen, published by PWS

Publishing Co. 1992.8. Elementary Differential Equations by E. D. Rainville and P. E. Bedient, published by

Macmillan, 1963.

MATH-304 Set TopologyMetric spaces, open sets, closed sets, convergence and continuity in metric spaces, Topological spaces, bases and subbases, product topology, subspace topology, closed sets and limit points, closure, interior and boundary, Hausdorff spaces, continuous functions, homeomorphisms, metric topology, connectedness, path connectedness, components and local connectedness, compact spaces, compact subspaces of the real line, limit point compactness, sequential compactness, local compactness, first-countable and second countable spaces, regular and normal spaces.

Recommended Books 1. Topology by James R. Munkres, second edition, published by Prentice Hall, 20002. Elementary Topology by Michael C. Gemignani, second edition, published by Addison-

Wesley, 19723. Introduction to Topology Modern Analysis, by J.F. Simmons, lealest edition McGram Hill,

N.Y.4. An Introduction to General Topology by Paul E. Long, published by Charles E. Merill

Publishing Company, 1971

MATH-305 Complex Analysis Complex Numbers, basic properties, De- Moivre’s theorem, roots of complex numbers, regions in the complex plane, functions of a complex variable, limits, continuity, derivatives, Cauchy- Riemann equations, analytic functions, harmonic functions, exponential, trigonometric, hyperbolic and logarithmic functions, contour integrals, Cauchy-Goursat theorem, Cauchy- integral formula, derivatives of analytic functions, Liouville’s theorem, maximum modulus principle, sequences and series, Taylor and Laurent series, residues and poles, Cauchy’s residue theorem, application to evaluation of real definite integrals, argument

30

principle and Rouche’s theorem, mapping by elementary functions, linear fractional transformations.

Recommended Books 1. Complex Variables and Applications by James Ward Brown and Ruel V. Churchill, seventh

edition, published by Mc-Graw Hill.2. Complex Variables by Mark J. Ablowitz and A.S. Fokas, published by Cambridge

University Press.

MATH-306 Group TheoryIntroduction to Sets and Structures. Motivation for groups. Finite groups. Subgroups. Permutations and cyclic groups. Isomorphisms and homomorphisms, with separate reference to Abelian groups. Cosets, normal groups, factor groups and simple groups. Series of groups. The Sylow theorems. Group actions, free groups and group presentations. Geometric, Analytic and Dynamical, applications. A brief introduction to continuous groups and to group representations.

Recommended Books 1. Fraleigh, J.B., A First Course in Algebra, Addison-Wesley 1982. 2. Hamermesh, M., Group Theory, Addison-Wesley 1972. 3. Herstein, I.N., Topics in Algebra, John Wiley 1975.

MATH-307 Analytical MechanicsReview of basic principles: Kinematics of particle and rigid body in three dimension; Euler’s theorem. Work, Power, Energy, Conservative field of force. Motion in a resisting medium. Variable mass problem. Moving coordinate systems, Rate of change of a vector, Motion relative to the rotating Earth. The motion of a system of particles, Conservation laws. Generalized coordinates, Lagrange’s equations, Hamilton’s equations, Simple applications. Motion of a rigid body, Moments and products of inertia, Angular momentum, kinetic energy about a fixed point; Principal axes; Momental ellipsoid; Equimomental systems. Gyroscopic motion, Euler’s dynamical equations, Properties of a rigid body motion under no forces. Review of material. Recommended Books

1. Principles of Mechanics by F. Chorlton, published by McGraw Hill, N.Y, 1983. 2. Mechanics by K. R. Symon, published by Addison Wesley, 1964.3. Classical Mechanics by H. Goldstein, published by Addison Wesley, 2nd Edition,

1980.4. Principles of Mechanics by J. I. Synge and B. A. Griffith, published by McGraw-Hill,

N.Y., 1986. 5. Mechanics for Engineers by F. P. Beer and E. R. Johnston, Vols.I&II, published by

McGraw- Hill, N.Y, 1975.

MATH-308 Partial Differential EquationsPartial differential equations of the first order and applications, mathematical modeling of heat, Laplace and wave equations, classification of second order partial differential equations, reduction to canonical form and the solution of second order partial differential equations, technique of separation of variables with emphasis on heat, Laplace and wave equations, Laplace, Fourier and Hankel transforms for the solution of partial differential equations and their application to boundary value problems.

Recommended Books 1. Elements of Partial Differential Equations by I. N. Sneddon, published by Mc-Graw Hill, 19872. Introduction to Partial Differential Equations and Boundary Value Problems by R. Dennemyer,

published by McGraw Hill, 196831

3. Boundary Value Problems and Partial Differential Equations by M. Humi and W. B. Miller, published by PWS-Kent publishing company, 1992

4. Techniques in Partial Differential Equations by C. R. Chester, published by McGraw Hill, 19715. Elementary Applied Partial Differential Equations byR. Haberman, published by Prentice Hall

1983.6. Partial Differential Equations of Applied Mathematics by E. Zauderer, published by John

Wiley, 1983.

MATH-309 Real Analysis The Riemann Integral: Upper and lower sums, definition of a Riemann integral, integrability criterion, classes of integrable functions, properties of the Riemann integral. Infinite Series: Review of sequences, the geometric series, tests for convergence, conditional and absolute convergence. Regrouping and rearrangement of series. Power series, radius of convergence. Uniform Convergence: Uniform convergence of a sequence and a series, the M-test, properties of uniformly convergent series. Weierstrass approximation theorem. Improper Integrals: Classification, tests for convergence, absolute and conditional convergence, convergence of f(x) sinx dx, the gamma function. Uniform convergence of integrals, the M-text, properties of uniformly convergent integrals. Fourier Series: Orthogonal functions, Legendre, Hermite and Laguerre polynomials, convergence in the mean. Fourier-Legendre and Fourier-Bessel series, Bessel inequality, Parseval equality. Convergence of the trigonometric Fourier series.

Recommended Books 1. Bartle, R.G. and Sherbert, D.R., Introduction to Real Analysis, John Wile Sons 1994. 2. Widder, D.V., Advanced Calculus, Prentice Hall 1982. 3. Rudin, W., Principles of Real Analysis, McGraw-Hill 1995. 4. Rabenstein, R.L., Elements of Ordinary Differential Equations, Academic Press, 1984.

MATH-401 Differential Geometry -IHistorical background, Curve theory: Regular curve, reparametrization, arc length, tangent, normal and binormal lines, tangent, normal and rectifying planes, curvature and torsion, Frenet-Serret formulas and their applications, spherical images, involutes and evolutes, Bertrand curves.Surface theory: tensors, Christoffel symbols and their applications.Local surface theory: First and second fundamental forms and their applications, Gaussian and mean curvatures.

Recommended Books 1. Elements of Differential Geometry by R. S. Millman and G. D. Parker, published by Prentice

Hall, 19772. Lectures on Classical Differential Geometry by D. J. Struik, published by Addison-Wesley,

19773. Differential Geometry of Curves and Surfaces by M. P. Do Carmo, published by Prentice

Hall, 19854. Elementary Differential Geometry by B. O. Neil, published by Academic Press, 1966 5. Introduction to Differential Geometry by A. Goetz, published by Addison Wesley, 19706. Vector and Tensor Methods by F. Charlton, published by Ellis Horwood, 1976

MATH-402 Functional Analysis-IMetric spaces, open sets, closed sets, convergence, completeness, normed spaces, Banach spaces, linear operators, bounded linear operators, linear functionals, dual space, inner product spaces, Hilbert spaces, orthogonal complements and direct sums, Riesz representation theorem, Hilbert adjoint operator, self adjoint, unitary and normal operators,

32

Hahn-Banach theorem, reflexive spaces, Baire Category theorem, Uniform boundedness theorem, open mapping and closed graph theorems.

Recommended Books 1. Introductory Functional Analysis and Applications by E. Kreyszig, published by John Wiley

and Sons 2. Elements of Functional Analysis by L. Maddox, published by Cambridge University Press3. Introduction to Topology and Modern Analysis by G. F. Simmons, published by McGraw

Hill

MATH-403 Numerical MethodsNumber systems and errors: Loss of significance and error propagation, condition and instability, error estimation, floating point arithmetic. Interpolation by polynomials: existence and uniqueness of the interpolating polynomial, Lagrangian interpolation, the divided difference table, error of the interpolating polynomial, interpolation with equally spaced data, Newton’s forward and backward difference formulas, Bessel’s interpolation formula, Hermite interpolation. Solution of non-linear equations: Bisection method, iterative methods, secant method, fixed point iteration, Newton -Raphson method, order of convergence of Newton-Raphson and secant methods. System of linear equations: Gauss elimination method, triangular factorization, Crout method. Iterative methods: Jacobi method, Gauss-Seidel method, SOR method, convergence of iterative methods.Numerical Differentiation: Numerical differentiation formulae based on interpolation polynomials, error estimates. Numerical Integration: Newton-Cotes formulae, trapezoidal rule, Simpson’s formulas, composite rules, Romberg improvement, Richardson extrapolation, error estimates of integration formulas, Gaussian quadrature.

Recommended Books 1. Elementary Numerical Analysis by S. D. Conte and C. Boor, published by McGraw Hill,

19722. Elements of Numerical Analysis by F. Ahmad and M. A. Rana, published by National Book

Foundation, Islamabad, 19953. Numerical Analysis for Engineers and Physicists by R. Zurmuhl, published by Springer

Verlag, 1976

MATH-404 Statistics and Probability-IIntroduction to statistics: Frequency distribution and graphical presentation of data, Measures of central tendency, Arithmetic Mean, Median, Mode, Geometric Mean, Harmonic Mean, Measures of Dispersion, Absolute measures and relative measures, Range, Quartile deviation, Mean deviation, Standard deviation, Variance, Coefficient of variation.Probability: Random experiment, Mutually exclusive and Not-mutually exclusive events, Conditional probability, Independent and dependent events, Laws of probability and their applications, Independent repeated trials with only two outcomes, Baye’s theorem (derivation and application).Random Variable: Discrete and continuous random variables, Distribution function, Probability density function, Joint distributions, Bivariate distribution function and bivariate probability functions, Marginal probability functions, Conditional probability functions. Independence of random variables, Mathematical expectation of a random variable, laws of expectations, Variance, covariance and correlation defined by expectation, Moment generating function and Characteristic function. Central limit theorem, limiting distributions and stochastic convergence. Regression and correlation analysis, Probabilistic and deterministic models, Simple and multiple linear regressions, Method of least squares, intercept and slope, Standard error of estimate, Correlation coefficient and its properties. Partial and multiple correlations.

Recommended Books33

1. Mood, A. M. Graybill, F. A. Boes, D.C introduction to the theory of Statistics 3rd Ed., (McGraw-Hill Book Company New york, 1974)

2. Degroot, M. H. Probability and Statistics, 2nd Ed., (Addison-Wesley Publishing Company, USA, 1986)

MATH-406 Continuous Groups:

Objectives:Upon successful completion of this course students will be able to

Demonstrate knowledge of the syllabus material; Use the definitions continuous groups, Lie groups and related topics to identify and

construct examples and to distinguish examples from non-examples; Classify groups of low dimensions; Apply the techniques of finding curvature of invariant metrices on Lie groups and

homogeneous spaces.

Course Outline:Continuous Groups; Gl(n,R), Gl(n,C), So(p,q), Sp(2n); generalities on continuous groups; groups of isometries, classification of two and three dimensional Euclidean space accoding to their isometries; introduction to Lie groups with special emphasis on matrix Lie groups; relationship of isometries and Lie group; theorem of Cartan; correspondence of continuous groupswith Lie algebras; classification of groups of low dimensions; homogeneous spaces and orbit types; curvature of invariant metrics on Lie groups and homogeneous spaces.

Recommended Books: 1. Bredon, G.E., Introduction to compact transformation groups, Academic Press, 1972.1. Eisenhart, L.P., Continuous groups of transformations, Priceton U.P., 1933.2. Pontrjagin, L.S., Topological groups, Princeton University Press, 1939.3. Husain Taqdir., Introduction to Topological Groups, W.B. Saunder’s Company, 1966.4. Miller Willard, Jr., Symmetry groups and their application, Academic Press New

York and London 1972.

MATH-425 Fluid Mechanics-II:

Objectives:This course is designed;

To give the students clear concepts equation of motion of viscous fluids. To develops skill of applying equation of motion to real problems. To familiarize the students about the flow of fluid in rotating frame. To introduce the concept of boundary layer flows.

Course Outline:Constitutive equations; Navier-Stokes’ equations; Exact solutions of Navier-Stoke’s equations; Steady unidirectional low; Poiseuille flow; Couette flow; Unsteady unidirectional low; sudden motion of a plane boundary in a fluid at rest; Flow due to an oscillatory boundary; Equations of motion relative to a rotating system; Ekman flow; Dynamical similarity and the Reynold’s number; Flow over a flat plate (Blasius’ solution); Reynold’s equations of turbulent motion, Backingum Pi Theorem, Similarity Transformations.

Recommended Books:1. L.D. Landau and E.M. Lifshitz., Fluid Mechanics, Pergamon Press, 1966.2. Batchelor, G.K. , An Introduction to Fluid Dynamics, Cambidge University

Press,1969.3. Walter Jaunzemis, Continuum Mechanics, MacMillan Company, 1967.

34

4. Milne-Thomson, Theoretical Hydrodynamics, MacMillan Company, 1967.

MATH-435 Fuzzy Logics:

Objectives:Upon successful completion, the students will be able to;

To provide machinery for carrying out approximate reasoning when available is uncertain, incomplete or vague.

Demonstrate the basics of fuzzy logic and fuzzy set theory. Apply in the field of engineering, psychology, economics and sociology.

Course Outline:Examples of fuzziness, Modeling of fuzziness, Operations on fuzzy Sets, Fuzziness as uncertainty, Boolean algebra and lattices, Equivalence relations and partitions, Composing mappings, Isomorphism and homomorphisms, Alpha cuts, Images of alpha level sets, Fuzzy quantities, Fuzzy numbers, Fuzzy intervals, t – norms, Generators of t – norms, Isomorphisms of t – norms, Negations, t – conorms, Strict De Morgan Systems, Nilpotent De Morgan Systems, Nonuniqueness of negations in strict De Morgan Systems, Fuzzy implications, Averaging operators and negations, Averaging operators and nilpotent t-norms, De Morgan systems with averaging operators, Power of t-norms, Sensitivity of connectives, Binary fuzzy relations, Operations on fuzzy relations

Recommended Books:1. Fuzzy Logic by H.T. Nguyen and E. A. Walker2. Introduction to the Basic Principles of Fuzzy Set Theory and some of its Applications

by E. E. Kerre3. Fuzzy Set Theory and its Applications by H. J. Zimmermann.4. Fundamentals of Fuzzy Sets by D. Dubois and H. Prade

MATH-409 Advance Group Theory:

Objectives:Upon successful completion of this course students will be able to

Demonstrate knowledge of the syllabus material; Use the definitions studied in this course to identify and construct examples and to

distinguish examples from non-examples; Know about use of group actions and the related topics in application point of view. Write about group theory in a coherent, grammatically correct and technically

accurate manner.

Course Outline:Actions of Groups, Permutation representation, Equivalence of actions, Regular representation, Cosets spaces, Linear groups and vector spaces, Affine group and affine spaces, Transitivity and orbits, Partition of G-spaces into orbits, Orbits as conjugacy class Computation of orbits, The classification of transitive G-spaces Catalogue of all transitive G-spaces up to G-isomorphism, One-one correspondence between the right coset of Ga and the G-orbit, G-isomorphism between coset spaces and conjugation in G, Simplicity of A5, Frobenius-Burnside lemma, Examples of morphisms, G-invariance, Relationship between morphisms and congruences, Order preserving one-one correspondences between congruences on Ω and subrroups H of G that contain the stabilizer Gα, The alternating

35

groups, Linear groups, Projective groups, Mobius groups, Orthogonal groups, unitary groups, Cauchy’s theorem, P-groups, Sylow P-subgroups, Sylow theorems, Simplicity of An when n>5.

Recommended Books:1. J.S. Rose, A Course on Group Theory, Cambridge University Press, 1978.2. H. Wielandt, Finite Permutation Groups. Academic Press, 1964.3. J.B. Fraleigh, A Course in Algebra, Addison-Wesley 1982.

MATH-410 Theory of Modules:

Objectives:Upon successful completion of this module students will be able to

Write precise and accurate mathematical definitions of objects in module theory; Use definitions to identify and construct examples of modules, submodules, quotient

modules and all related topics and to distinguish examples from non-examples; Validate and critically assess mathematical proofs of the topics covered and

discussed; Use a combination of theoretical knowledge and independent mathematical thinking

to investigate questions in module theory and to use them for practical applications;

Course Outline:Motivations to modules. Submodules, quotient modules, finitely generated and cyclic modules, exact sequences and elementary notions of homological algebra, Noetherian and Artinian rings and modules, radicals, semisimple rings and modules.

Recommended Books:1. Adamson, J., Rings and modules.2. Blyth, T. S., Module theory, Oxford University Press, 1977.3. Fraleigh J.B., A first Course in Algebra (Addison-Wesley 1982).4. Herstein, I.N., Topics in Algebra (John Wiley 1975).5. Hartley, B., and Hawkes, T.O., Ring, Modules and Linear Algebra, Chapman and

Hall, 1980.

MATH-411 Decomposition of Modules:

Objectives:After completion of the course, the students will be able to;

Decompose a module as a direct sum of its sub modules. Relate linear algebra and group theory to modules. Learn invariance and canonical forms and conjugacy classes in general linear group.

Course Outline:Rings and modules, decomposition of modules, decomposition theorem, the primary decomposition theorem, The primary decomposition, Abelian groups as Z-modules, Abelian groups, Sylow’s theorem, linear transformation and matries, invariants and the Jordan canonical form, the rational canonical form theorem - (linear transformation version), The Jordan canonical form theorem, conjugacy classes in general linear groups.

Recommended Books:1. Blyth, T., Module theory, O.U.P., Oxford, 1977.

36

2. Hartley, B. and Hawkes, T., Rings, modules and linear algebra, Chapman, G., Lecture Nortes on Modules, Michigan University Press.

MATH-412 Galois Theory:

Objectives:The main objectives of this course are;

To make a connection between field theory and group theory. To describe how the various roots of a given polynomial equation are related to each

other.

Course Outline:Basics:Integral domains and Fields, Homorphisms and ideals, Quotient Rings, Polynomial rings in one indeterminate over Fields, Prime ideals and Maximal ideals, irreducible Polynomials.Field Extensions:Algebraic and Transcendental field extensions, Simple Extensions, Composite Extensions, Splitting Fields, The Degree of and Extension, Ruler and Compass Constructions. Normality and Separability.Finite Field Extensions:Circle Division, The Galois Group, Toots of Unity, Solvability by Radicals, Galois Extensions, The Fundamental Theorem of Galois Theory, Galois’s Great Theorem, Algebraically Closed Fields.

Recommended Books:1. Joseph Rotman, “Galois Theory”, Springer-Veriog, New York, Inc. (2005)2. Lan Steward, “Galois Theory”, Chapman & Hall, New York (2004)3. David S. Dummit and Richard M. Foote, “Abstract Algebra”, John Wiley & Sons,

Inc, New York (2002).

MATH-416 Algebraic Geometry:

Objectives:After the completion of the this course, the students will be in position to;

Combine techniques of abstract algebras especially commutative algebra with language and the problem of geometry.

Gain an understanding of algebraic varieties, geometric manifestations of solutions of systems of polynomial equations.

Explain dimensions, tangent spaces, smoothness and completeness.

Course Outline:Algebraic varieties: Affine algebraic varieities, Hibert basis Theorem, Decomposition of variety into irreducible components, Hibert’s Nulttstellensatz, The Sectrum of a Ring, Projective variety and the homogeneous Spectrum.Functions and Morphisms: Some properties of Zariski topology, Rings and modules of franctions and their properties, Coordinate ring and polynomial functions, Polynomial maps, Regular and rational functions, Morphisms, Rational maps.Dimension: The Krull dimension of Topological Spaces and Rings, Prime Ideal Chain and Integral Extensions, The Dimension of Affine Algebras and Affine Algebraic Varieties, The Dimension of Projective Varieties.Applications: The product of varieties, On dimension, Tangent space and smoothness, Completeness.

Recommended Books:

37

1. O. Zariski and P. Samual, Commutative Algebra, Vol. 1, Van Nostrand, Princeton, N. J., 1958.

2. M.F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison Wesley Pub. Co., 1969.

3. I.R. Shafarevich, Basic Algebraic Geometry, Springer Verlag, 1974.4. R. Hartshorne, Algebraic Geometry, Springer Verlag, 1977.5. E, Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Boston;

Basel; Stuttgrat: Birkhauser, 1985.

MATH-417 Algebraic Systems and Coding Theory:

Objectives:After the completion of the this course, the students will be in position to;

To relate algebra to coding theory. To make applications of algebraic structures, groups, rings and fields and vector

spaces.

Course Outline:An introduction to the use of abstract methods in mathematics, using algebraic systems that play an important role in many application so mathematics. Abelian groups, Commutative rings with identity, fields, Ideals, Polynonial rings, Principal Ideal domains, arithmetic of integers mod n and finite fields. Vector spaces over arbitraty fields, Examples of Algebra of Polynomial rings over an arbitrary field, subspaces, bass, linear transformations. Eigenvalues, eigenvectors, eigenspaces, Characteristies, Polynomial, Minimal Polynomial, Linear Transformation as a matrix operator, geometric and algebraic multiplicity and diagonalisation. Groups: subgroups, cosets, Lagrange’s theorem, homomorphisms.Applications to coding theory will be chosen from: linear codes, encoding and decoding, the dual code, the parity check matrix, syndrome decoding, Hamming codes, perfect codes, cyclic codes, BCH codes.

Recommended Books:1. John B Fraleigh, A First Course in Abstract Algebra, 5th edition, Addison-Wesley,

1994.2. Richard Laatsch, An Introduction to Abstract Algebra, McGraw-Hill, 1968.3. Max D Larsen, Introduction to Modern Algebraic Concepts, Addison-Wesley, 1969.4. F.J. Budden, The Fascination of Groups, Cambridge University Press, 1972.5. Joel G Broida and S Gill Williamson, A comprehensive Introduction to Linear

Algebra, Addison-Wesley, 1989.6. Hill, Raymond, 1942, A first course in coding theory, Oxford University Press, 1986.7. McEliece, Robert J, The theory of information and coding, Cambridge, U.K; New

York: Cambridge University Press, 2002.8. Roman, Steven, Introduction to coding and information theory, New York: Springer,

1997.9. Assmus, E.F, Designs and their codes, Cambridge: Cambridge University Press, 1992.10. Hamming R. W. (Richard Wesley), 1915-Coding and information theory / Richard W.

Hamming, Englewood Cliffs N.J: Prentice-hall, 1986.

MATH-421 Quantum Mechanics:

Objectives:On completing the course, the students will be able to;

Know about quantum mechanical operator, Eigen states, potential barrier, potential well, angular momentum and spin.

38

To explain the behavior of matter and its interaction with energy on the scale of atoms and atomic scales at basic level.

Understand the wave and particle nature of entities.

Course Outline:Basic postulates of quantum mechanics. State vectors. Formal properties of quantum mechanical operators. Eigenvalues and eigenstates, simple harmonic oscillator. Schrodinger representation. Heisenberg equation of motion Schrodinger equation. Potential step, potential barrier, potential well. Orbital angular momentum. Motion in a centrally symmetric field. Hydrogen atom. Matrix representation of angular momentum and spin. Time independent perturbation theory, degeneracy. The Stark effect. Introduction to relativistic Quantum Mechanics.

Recommended Books:1. Fayyazuddin and Riazuddin, Quantum Mechanics, World Scientific 1990. 2. Merzbacher, E., Quantum Mechanics, John Wiley 2nd Ed. 1970.3. Liboff, R.L., Introductory Quantum Mechanics, Addision-Wesley 2nd Ed. 1991. 4. Dirac, P.M.A., Principles of Quantum Mechanics, (Latest Edition), Oxford University

Press.

MATH 425 Rings and Fields

Difinitions and basic concepts, homomorphisms, homomorphism theorems, polynamical rings, unique factorization domain, factorization theory, Euclidean domains, arithemtic in Eclidean domains, extension fields, algebraic and transeendental elements, simple extension, introduction to Galois theory.

RECOMMENDED BOOKS: 1) Fraleigh, J.A., A First Course in Abstract Algebra, Addision Wesley Publishing

Company, 1982.2) Herstein, I.N., Topies in Algebra, John Wiley & Sons 1975.3) Lang, S., Algebra, Addison Wesley, 1965.4) Hartley, B., and Hawkes, T.O., Ring, Modules and Linear Algebra, Chapman and

Hall, 1980.

MATH-426 Elasticity Theory:

Objectives:Upon the completion of the course, the students will be able to;

Analyze linear elastic under mechanical and thermal behavior. Understand wave propagation, tensor analysis of stress and strain and solid

mechanics.

Course Outline:Cartesian tensors; analysis of stress and strain, generalized Hooke’s law; crystalline structure, point groups of crystals, reduction in the number of elastic moduli due to crystal symmetry; equations of equilibrium; boundary conditions, compatibility equations; plane stress and plane strain problems; two dimensional problems in rectangular and polar co-ordinates; torsion of rods and beams.

Recommended Books:1. Sokolinikoff., Mathematical theory of Elasticity, McGraw-Hill, New York.2. Dieulesaint, E. and Royer, D., Elastic Waves in Solids, John Wiley and Sons, New

York, 1980.39

3. Funk, Y.C., Foundations of Solid Mechanics, Prentice-Hall, Englewood Cliffs, 1965.

MATH-427 Electromagnetism:

Objectives:After the successful completion of this course, the students will be able to;

Gain the knowledge of electrostatic energy, magnetic energy, maxwell’s equations, electromagnetic waves, radiations and motion of electric charges.

Learn fundamental principle of electromagnetism to continue to develop solid and systematic problems solving skills and to lay the foundations for further studies in physical science.

Apply basic law of electricity and magnetism.

Course Outline:Electrostatics and the solution of electrostatic problems in vacuum and in media, Electrostatic energy, Electric currents, The magnetic field of steady currents, Magnetic properties of matter. Magnetic energy, Electromagnetic Induction, Maxwell’s equations, Boundary Value Potential Problems in two dimensions, Electromagnetic Waves, Radiation, Motion of electric charges.

Recommended Books:1. Reitz, J.R. and Milford, F.J., Foundation of electromagnetic theory, Addision-Wesley,

1969.2. Panofsky, K.H. and Philips, M., Classical Electricity and Magnetism, Addision-

Wesley, 1962. 3. Corson, D. and Lerrain, P., Introduction to Electromagnetic fields and waves,

Freeman, 1962.4. Jackson, D.W., Classical Electrodynamics, John-Wiley. 5. Ferraro, V.C.A., Electromagnetic theory, The Athlone Press, 1968.

MATH-428 Theory of Manifolds:

Objectives:After completing the course, the students will be in position;

To introduce a notion of abstract smooth manifold. To develop skill in manipulation of differential objects. To generalize the concept of curves and surfaces. To apply the subject to general relativity, modern physics and partial differential

equations.

Course Outline:Manifolds and smooth maps; Derivatives and Tangents; The inverse function theorem and Immersions; Submersions; Transversality, homotopy and stability; Embedding manifolds in Euclidean space; Manifolds with boundary; One manifolds and some consequences; Exterior algebra; Differential forms; Partition of unity; Integration on manifolds; Exterior derivative; Cohomology with forms; Stoke’s theorem; Integration and mappings; The Gauss-Bonnet --theorem; Lie groups as examples of manifolds; Their Lie algebras; Examples of matrix Lie groups and their Lie algebras.

Recommended Books:1. Guillemin, V. and Pollock, A., Differential Topology, Prentice-Hall, Inc., Englewood

Cliffs, New Jersey, 1974.

40

2. Boecker, T. and Dieck, T., Representations of Compact Lie groups, Springer Verlag,1985.

3. Bredon, G.E., Introduction to Compact Transformation Groups, Academic Press, 1972.

MATH-434 Group Algorithm Programming:

Objectives:On completing the course, the students will gain the ability to;

Use GAP in research and teaching for studying groups and their representations, rings, vector, spaces, algebras and combinatorial structures.

Understand parallel computing, number theoretic algorithms. Apply GAP in computer science for software engineering applications regarding

binary trees, flavor graph, graph coloring and spanning trees.

Course Outline:Algorithms and its Analysis – Basic concepts and its applications. Mathematical Foundations: Growth of functions, Asymptotic functions, Summations, Recurrences, Counting and probability.Divide-and-Conquer algorithms; General method and its analysis, Binary search and its analysis, Merge sort and its analysis, Quick sort and its analysis, Insertion sort and its analysis.Advanced Design and Analysis Techniques: Dynamic Programming, Greedy algorithms and its applications in scheduling, Generating functions and its application in Recurrences, Permutation Algorithms and its application in sorting, Amortized analysis, Worst-case analysis, Average case analysis.Graph algorithms: Basic search techniques, Algorithmic binary tees and its application, breadth-first search, Depth-first search, Planner graphs, Graph colouring, Minimum Spanning Trees, Single source shortest paths.Special Topics:Algorithms for parallel computers. Matrix Operations. Polynomials and the FFT. Number-Theoretic algorithms. NP-completeness. Approximations algorithms.Encyption/Decryption algorithms.

Recommended Books:1. Thomas H. Cormen and Charles E, Leiserson, Introduction to Algorithms, MIT Press,

McGraw-Hill (2nd Edition) 1990.2. H. Sedgwick Analysis of Algorithms, Addison Wesley, (1st Edition) 1995.3. K. Rosen., Discrete Mathematics and its Applications, McGraw Hill, (5th Edition)

1999.

MATH 434 Analytical Dynamics (Cr.3)Constraints, generalized co-ordinates, generalized forces, general equation of dynamics, Lagrange’s equations, conservation laws, ignorable co-ordinates, Explicit form of Lagrange’s equation in terms of tensors. Hamilton’ principle of least action. Hamilton’s equations of motion, Hamilton-Jacobi Method. Poisson Brackets (P.B’s); Poisson’s theorem; Solution of mechanical problems by algebraic technique based on (P.B’s) Small oscillations and normal modes, vibrations of strings, transverse vibrations normal modes, forced vibrations and damping, reflection and transmission at a discontinuity, longitudinal vibrations, Rayleigh’s principle.

Recommended Books 1) Textbook of Dynamics by F. Chorlton, published by Van Nostrand, 1963.

41

2) Mechanics by W. Chester, published by George Allen and Unwin Ltd. London, 1979.3) Classical Mechanics by H. Goldstein, published by Cambridge University, 19804) Methods of Analytical Dynamics by G. Meirovitch, published by McGraw-Hill,

1970.

MATH 435 Fluid Mechanics-I (Cr. 3)Real fluids and ideal fluids, velocity of a fluid at a point, streamlines and pathlines, steady and unsteady flows, velocity potential, vorticity vector, local and particle rates of change, equation of continuity, acceleration of a fluid, conditions at a rigid boundary, general analysis of fluid motion, Euler’s equations of motion, Bernoulli’s equation, steady motion under conservative body forces, some potential theorems, impulsive motion, sources, sinks and doublets, images in rigid infinite plane and solid spheres, axi-symmetric flows, Stokes’ stream function, complex potential for two dimensional irrotational, incompressible flow, complex velocity potential for uniform stream, line sources and line sinks, line doublets and line vortices, image systems, Miline –Thomson circle theorem, Blasius theorem, use of conformal transformation and Schwartz-Christoffel transformation in solving problems, vortex rows, Kelvin’s minimum energy theorem, uniqueness theorem, fluid streaming past a circular cylinder, irrotational motion produced by a vortex filament, Helmholtz vorticity equation, Karman’s vortex-street.

Recommended Books1) Textbook of Fluid Dynamics by F. Chorlton, published by Van Nostrand, 19672) Theoretical Hydrodynamics by M. Thomson, published by Macmillan Press, 19793) Continuum Mechanics by W. Jaunzemics, published by Macmillan Company, 19674) An Introduction to Fluid Dynamics by G. K. Batchelor, published by Cambridge

University Press, 19695) Fluid Mechanics by L. D. Landau and E. M. Lifshitz, published by Pergamon Press,

1966.

MATH 444 Integral Equations (Cr. 3)Integral equations, formulation of boundary value problems, classification of integral equations, method of successive approximation, Hilbert-Schmidt theory, Schmidt’s solution of non-homogeneous integral equations, Fredholm theory, case of multiple roots of characteristic equation, degenerate kernels, introduction to Wiener-Hopf technique.

Recommended Books1) Linear Integral Equations by W. V. Lovitt, published by Dover, 19502) Integral Equations by F. Smith, published by Cambridge University Press3) Integral Equations by F. G. Tricomi, published by Interscience, 19574) Methods based on the Weiner-Hopf technique by B. Noble, published by Pergamon

Press, 1958 5) Introduction to Integral Equations with Applications by J. Jerri Abdul, published by

Marcel Dekker, 1985 MATH 453 Advanced Topology (Cr.3)

Urysohn lemma, Urysohn metrization theorem, Tietze extension theorem, Tychonoff theorem, Stone-Cech compactification, further metrization theorems, paracompactness, complete metric spaces, Baire spaces.Recommended Books

1) Topology by James R. Munkres, second edition, published by Prentice Hall, 20002) A First Course in Algebraic Topology by C. Kosniowski, published by Cambridge

University Press 3) Algebraic topology, A First Course by M. J. Greenberg, published by

Benjamin/Commings42

MATH 454 Algebraic Topology (Cr.3)Quotient topology, homotopy of paths, the fundamental group, covering spaces, fundamental group of the circle, retractions and fixed points, Borsuk-Ulam theorem, deformation retracts and homotopy type, fundamental groups of various surfaces, direct sums of Abelian groups, free products of groups, free groups, Seifert-van Kampen theorem and applications.

1) Kosniowski, C., A first course in algebraic topology, Cambridge University Press, 1980.2) Greenberg, M.J., Algebraic topology, A first course, Benjamin/Commings, 1967. 3) Wallace, A.H., Algebraic Topology, Homology and Cohomology, Benjamine, 1968.

MATH 455 Differential Geometry-II (Cr.3)Definition and examples of manifolds, differential maps, submanifolds, tangents, coordinate vector fields, tangent spaces, dual spaces, multilinear functions, algebra of tensors, vector fields, tensor fields, integral curves, flows, Lie derivatives, brackets, differential forms, introduction to integration theory on manifolds, Riemannian and semi-Riemannian metrics, flat spaces, affine connections, parallel translations, covariant differentiation of tensor fields, curvature and torsion tensors, connection of a semi-Riemannian tensor, Killing equations and Killing vector fields, geodesics, sectional curvature.

Recommended Books1) Tensor Analysis on Manifolds by R. L. Bishop and S. I. Goldberg, published by

Dover, 1980 2) Riemannian Geometry by M. P. Do Carmo, published by Birkhauser, 1992 3) Differential Forms and Variational Principles by D. Lovelock and H. Rund,

published by John Wiley, 19754) Differential and Riemannian Geometry by D. Langwitz, published by Academic

Press, 1970 5) Manifolds, Tensor Analysis and Applications by R. Abraham, J. E. Marsden and T.

Ratiu, published by Addison Wesley, 1983

MATH 456 Riemannian Geometry (Cr.3)

Geodesics and their length minimizing properties, Jacobi fields, equation of geodesic deviation, geodesic completeness, theorem of Hopf-Rinow, curvature and its influence on topology, theorem of Cartan-Myers and Hadamard geometry of submanifolds, second fundamental form, curvature and convexity, minimal surfaces, mean curvature of minimal surfaces, calculus of differential forms and integration on manifolds, Theorem of Stokes, elementary applications of differential forms to algebraic topology.

Recommended Books1) M.P., Riemannian Geometry by M. P. Do Carmo, published by Birkhauser, 19922) Riemannian Geometry by S. Gallot and J. Lafontaine, published by Springer-Verlag,

19903) Differential forms in algebraic topology by R. Bott and M. tu, published by Springer-

Verlag, 1987

MATH 457 Combinatorics and Graph Theory (Cr. 3)Counting, pigeonhole principle, permutations and combinations, probability, permutations and combinations with repetition, recurrence relations, generating functions, principle of inclusion-exclusion, graphs, adjacency matrices, incidence matrices , isomorphism of graphs, paths, connectivity , Euler and Hamilton paths, Dijkstra’s shortest path algorithm, planar graphs, Euler’s formula, graph coloring, applications, trees, applications, spanning trees.

43

Recommended Books1) Discrete Mathematics and its Applications by Kenneth H. Rosen, fifth edition, published by

McGraw Hill, 2005.2) Discrete and Combinatorial Mathematics by Ralph P. Grimaldi.

MATH 464 Measure and Integration (Cr.3)Measure Spaces: Definition and examples of algebras and σ algebras, basic properties of measurable spaces, definition and examples of measure spaces, outer measure, Lebesgue measure, measurable sets, complete measure spaces.Measurable Functions: Some equivalent formulations of measurable functions, examples of measurable functions, various characterization of measurable functions, property that holds almost everywhere, Egorov’s theorem.Lebesgue Integration: Definition of Lebesgue integral, basic properties of Lebesgue integrals, comparison between Riemann integration and Lebesgue integration, L2-space. The Riesz-Fischer theorem.

Recommended Books1) Real Analysis by H. L. Royden, third edition, published by Prentice Hall, 1988 2) Measure Theory by D. L. Cohn, published by Birkhauser, 19803) Measure Theory by P. R. Halmos, published by D.Van Nostrand, 1950

MATH 465 Functional Analysis-II (Cr.3)The Hahn-Banach theorem, principle of uniform boundedness, open mapping theorem, closed graph theorem; Weak topologics and the Banach-Alouglu theorem, extreme points and the Krein-Milman theorem.The dual and bidual spaces, reflexive spaces, compact operators, Spectrum and eigenvalue, of an operator, elementary spectral theory.

Recommended Books1) Introductory Functional Analysis and Applications by E. Kreyszig, published by John

Wiley, 19732) Introduction to Functional Analysis by A. E. Taylor and D. C. Lay, published by

John Wiley1) Functional Analysis by H. G. Heuser, published by John Wiley, 19822) Elements of Applied Functional Analysis by C. W. Groetsch, published by Marcel

Dekker, 1980

MATH 474 Operations Research(Cr.3):To provide an appreciation of a number of different techniques used in operations research and their application. In particular the formulation and solution of linear programming problems, queuing systems reliability analysis, probalistic risk analysis. To provide an understanding of their probabilistic and theoretical backgrounds. This course includes the formulation of a linear programming problem; solving a problem using the appropriate solution method i.e. either the primal or dual simplex algorithm;

Recommended Books 1) Operations Research, an Introduction, 6th edition, published by Prentice Hall2) Introduction to Operations Research by F. Hillier, 6th edition, published by McGraw

Hill, 1995

MATH 475 Optimization Theory (Cr. 3)Introduction to optimization, relative and absolute extrema, concave and unimodal Functions, constraints, mathematical programming problems, optimization of one, two and several variables functions and necessary and sufficient conditions of their optima, optimization by

44

equality constraints, direct substitution method and Lagrange multiplier method, necessary and sufficient conditions for an equality constrained optimum with bounded independent variables, inequality constraints and Lagrange multipliers, Kuhn-Tucker Theorem, multidimensional optimization by gradient method, convex and concave programming, Calculus of variation and Euler-Lagrange equations, functionals depending on several independent variables, variation problems in parametric form, generalized mathematical formulation of dynamics programming, non-linear continuous models, dynamics programming and variational calculus, Control theory.

Recommended Books1) Introduction to Optimization Theory by B. S. Gotfried and J. Weisman, published by Prentice Inc. New Jersey, 19732) Differential Equations and the Calculus of Variations by L. Elsgolts, published by

Mir Publishers Moscow, 19703) Introduction to Nonlinear Optimization by D. A. Wismer and R. Chattergy,

published by North Holland New York, 19784) Mathematical Optimization and Economic Theory by M. D. Intriligator,

published by Prentice-Hall Inc. New Jersey, 1971

MATH 476 Mathematical Modeling & Simulation (Cr. 3)Basic concepts of computer modeling in science and engineering using discrete particle systems and continuum fields. Techniques and software for statistical sampling, simulation, data analysis and visualization. Use of statistical, quantum chemical, molecular dynamics, Monte Carlo, mesoscale and continuum methods to study fundamental physical phenomena encountered in the fields of computational physics, chemistry, mechanics, materials science, biology, and applied mathematics. Applications drawn from a range of disciplines to build a broad-based understanding of complex structures and interactions in problems where simulation is on equal-footing with theory and experiment.

Recommended Books3) Graph Models and Finite Mathematics by Malkevitich, published by Prentice Hall4) Analytical and Computational Methods of Advanced Engineering by G. B. Gustafson,

published by SV, 1998

MATH 481 Special Relativity (Cr.3)Historical background and fundamental concepts of special theory of relativity, Lorentz transformations (one dimensional), length contraction, time dilation and simultaneity, velocity addition formulae, 3- dimensional Lorentz transformations, introduction to 4-vector formalism, Lorentz transformations in the 4-vector formalism, the Lorentz and Poincare groups, introduction to classical mechanics, Minkowski spacetime and null cone, 4-velocity, 4-momentum and 4-force, application of special relativity to Doppler shift and Compton effect, particle scattering, binding energy, particle production and decay, electromagnetism in relativity, electric current, Maxwell’s equations and electromagnetic waves, the 4-vector formulation of Maxwell’s equations, special relativity with small acceleration.

Recommended Books1) Relativity: An Introduction to the Special Theory by Abdul Qadir, published by

World Scientific, 19892) Introducing Einstein’s Relativity by R. D. Inverno, published by Oxford University

Press, 19923) Classical Mechanics by H. Goldstein, published by Addison Wesley, 19624) Classical Electrodynamics by J. D. Jackson, published by John Wiley 19625) Essential Relativity by W. Rindler, published by Springer Verlag, 1977

45

MATH 491 Project I & II (Cr. 9)

List of Computer Science Courses

CS 111 Programming Fundamentals (Cr.3)Introduction to algorithms, different ways of representing algorithms, pseudo code, pseudo code languages, how to write a computer program, problem solving using algorithms, flowcharts, tracing and timing, series and sequence manipulation, matrix manipulation, searching and sorting algorithms, recursion and related algorithms, trees and related algorithms, graphs and related algorithmsRecommended Books

1) Computer Science, an overview by Glenn Brook shears, 6th edition2) Discrete Mathematics and its applications by Kenneth H. Rosen.

CS 212 Data Structures and Algorithms Introduction to data structures, importance and role of data structures in software development, space and time complexity, big 0 notation, data abstraction, ADT, storage and retrieval properties and techniques for various data structures, strings, arrays, linked lists, stacks, queues, trees, heaps, hash tables, and graphs and itsapplications, when to use what, sorting, searching, lab assignments and projects.

Recommended Books1) Data Structures Using C and C++ by Augenstein and Tenenbaum, published by Prentice Hall.2) Data Structures and Algorithms by Aho, Hopcroft and Ullman, published by Addison

Wesley.

3) C++ An Introduction to Data Structures by Larry Nayhotf, International Edition.4) Fundamentals of Data Structures in C++ by Horowitz, Sahni, and Mehta, published

by Computer Science Press.5) Data Structures and Algorithm Analysis in C++ by Weiss, Mark Allen, published by

Addison-Wesley.

CS 314 Theory of Automata (Cr.3)Preliminaries; Introduction, graphs, trees, inductive proofs, set notation. Finite Automata; Finite state automaton, deterministic finite automaton, simulating a finite state automaton using software, non deterministic finite automata, finite automata with E-'moves, two way finite automata, finite automata with output, regular expression; equivalence of FA and regular expression, the pumping lemma for regular sets, closure properties of regular sets, decision algorithms for regular sets, the Myhill-Nerode theorem, minimizing finite automata, context-free grammars and languages, grammars, derivations and languages, properties of context free languages; kinds of properties, Greibach normal form, eliminating A-productions from a CFG, unit productions from a CFG, useless variables from a CFG, Chomsky normal form, pushdown automata, pushdown automata and context free grammar, deterministic pushdown automata, parsing.

Recommended Books 1) Introduction to Automata Theory, Languages, and Computation by J. E. Hopcroft

and J. D. Ullman, published by Nayosa Publishing House, New Delhi, India, 1987/1994.

2) Elements of the Theory of Computations by H. R. Lewis and C. S. Papadimitrious, published by Prentice Hall Inc. Englewood Chirrs NJ. USA. 1981.

3) Introduction to Computer Theory by D. I. A. Cohen, published by John Wiley & Sons Inc., NY, USA,1941.

46

4) Theory of Automata and Computation by M. Sikander Hayat Khiyal, published by National Book Foundation, Pakistan.

CS 322 Computer Communication & Networks (Cr. 3)Communication model, communication tasks, transmission system utilization, interfacing, signal generation, exchange management, error detection and correction, flow control, addressing, routing, recovery, message formatting, security, network management protocol and protocol architecture, OSI standard, TCP/IP suite, bus, tree, ring, star LANs, circuit switching, packet switching, frame relay, ATM, ISDN and broadband ISDN, point to point and multipoint, simplex, half-duplex and full-duplex transmission, analog and digital data transmission. attenuation, delay distortion, noise, channel capacity, transmission media, data encoding.Digital Data & Analog Signals, –Modem Encoding Techniques, CODEC Encoding Techniques, Modulation Techniques, Data communication interface, data link control, error detection techniques, error control techniques, high level data link control protocols(HDLC), multiplexing.network models, network operating systems, network adapter cards, network services, network printing, network applications, LAN technology, LAN system & network standards, connectivity devices, connection services, managing and securing a network, disaster recovery.

Recommended Books Following books or their more recent equivalents, manuals, computer magazines and journals articles, at the discretion of the instructor:

1) Data And Computer Communication by William Stalling, 5th edition, published by Prentice Hall.

2) Data Communication, Computer Networks And Open Systems by Fred Halsall, published by Addison-wesley.

3) Computer Networks by A. S. Tanen , published by Prentice Hall.4) Data Communications, Networks And Systems by Thomas C.Bartee, Editor-in-Chief

BPB publications5) Networking Essentials, MCSE Training Guide by Joe Casad & Dan Newland,

published by Techmedia.

CS 361 Computer Graphics

Details of Core and Elective Courses for MS Mathematics Programs

MATH-501 Advanced Mathematical AnalysisTheory of distance function (Real valued, Vector valued and multivalued), Continuous functions (Single valued, Vector valued and multivalued), Random variables (Measureable sets, Measurable and Lebesgue measurable

Functions (linear and non non-linear operators), spaces, Axiom of Coice, Zoarn's lemma.

Objectives: It is advanced level course in Mathematical Analysis. The students will understand distance between two objects. After studying the student should be able to apply the above mentioned concept in different related areas.

Recommended Books:1. Real Analysis by H. L. Royden, Third Edition, Published Prentice Hall, 1988.2. Measure Theory by D. L. Cohn, Published by Birkhaser, 1980.3. Principals of Real Analysis, by W. Rudin, Published by McGraw Hill, 1995.

MATH 502 Advanced Partial Differential Equations Cauchy’s problems for linear second order equations in n-independent variables, Cauchy Kowalewski theorem, characteristic surfaces, adjoint operations, bi-characteristics spherical

47

and cylindrical waves, heat equation, wave equation, Laplace equation, maximum-minimum principle, integral transforms.

Objectives: This course is designed to give the students a rigorous treatment of the basic nomenclature for partial differential equations, the three basic types of partial differential equations and the fundamental theory for existence of solutions including the use of functional analysis, Hilbert and Sobolov Spaces. The student will learn to apply Fourier series and Transforms for solution to partial differential equations, Green's function to the solution of boundary value problems. Both analytic and numerical methods will be explained to obtain the solution of hyperbolic, parabolic and elliptic partial differential equations.

Recommended Books:1. Introduction to Partial Differential Equations and Boundary Value problems by R.

Dennemyer, published by McGraw-Hill Book Company, 1968.2. Techniques in Partial Differential Equations by C. R. Chester, published by McGraw-

Hill Book Company, 1971.3. Advanced Topics in Computational Partial Differential Equations by H. P.

Lengtangen and A. Treito, 2003.

MATH 503 Advanced Linear Algebra Basic properties of vector spaces and linear transformations, algebra of polynomials, characteristic values and diagonalizable operators, invariant subspaces and triangulable operators, the primary decomposition theorem, cyclic decompositions and generalized Cayley-Hamilton theorem, rational and Jordan forms, inner product spaces, the spectral theorem, bilinear forms, symmetric and skew-symmetric bilinear forms.

Objectives: Upon successful completion of this course students will be able to demonstrate knowledge of the syllabus material. Students will also be able to apply eigen value problems and related material in the linear algebra to different areas of applied and pure Mathematics.

Recommended Books 1. Linear Algebra by Kenneth Hoffman and Ray Kunze, second edition, published by

Prentice Hall, 19712. Advanced Linear Algebra by S. Roman, 2nd edition, 2005.3. Advanced Linear Algebra by B. Cooperstein, 2010.

MATH 504 Advanced Mathematical MethodsContents: Analytical Solutions concept, Analytical methods. Variational Iteration method. Modified Adomain decomposition method. Perturbation and Asymptotic methods, Homotopy perturbation method, Homotopy analysis method, Newton’s method, Householder method, Decomposition method, Modified HPM, Halley method, Single step method, Taylor series and convergence, Explicit and Implicit multistep methods.

Objectives: The focus of this course is on the derivation and application advanced analytical methods used for solution for algebraic and differential equations. A proper framework will be presented for the derivation of methods based on variational approach. The stability and convergence of explicit as well implicit methods will be discussed.

Recommended Books:1. Shijun Liao, Beyond Perturbation, Chapman and Hall/CRC London New York

Washington, D.C (2003).2. Abdul Majid Wazwaz, A first course in integral equation, World Scientific publishing

Co (1997).

48

3. Andrei D. Polynin and Alexander V. Manzhirov, A note book on Integral equation, Chapman and Hall/CRC London New York Washington, D.C (2010).

MATH-505 Semigroup TheoryIntroductory ideas: Basic definitions, Cyclic semigroups; Ordered sets, semi lattices and lattices. Binary relations; Equivalences; Congruences; Free semigroups; Green’s Equivalences; L,R,H,J and D; Regular semigroups, O-Simple semigroups; Simple and O-Simple semigroups; Isomorphism and normalization, Rees’s theorem; Primative idempotents; Completely O-Simple semigroups; Finite congruence-free semigroups, Union of groups; Bands; Free bands; varieties of bands.

Objectives: To introduce students to semigroup theory, which is the study of sets with one associative binary operation and comparisons between semigroups, groups and rings. Further to be familiar with the most important examples of semigroups and be able to perform calculations in them. Understand the basic structure theory of semigroups.

Recommended Books:1. The Algebraic Theory of Semigroups; A.H. Clifford and G.B. Preston, Vol. I & II.

AMS Math. Surveys, 1961 and 1967.2. An Introduction to Semigroup Theory by J.M. Howie, Academic Press 1967. 3. Semigroup Theory and Applications by P. Clement, 2001.

MATH-506 Theory of Group ActionsSurvey of theory of group actions, Applications of group actions, Transitivity and k-transitivity, Primitivity, Finite fields and their extensions, Projective line over finite fields, Finite geometries, Projective spaces and their groups, Actions of PGL (n,q) and PSL (n,q) on PG (n-I,q), Simplicity of projective special linear groups over finite fields, Modular group, Parameterization of action of the extended modular group on projective lines over finite fields. Projective and linear groups through actions.

Objectives: The basic objective of this course is to make a better understanding of finite geometries , Projective spaces and their groups, finite fields and their extensions Projective lines over finite fields, transitivity of action, K- transitive and Primitive groups . And then to teach a student about actions of different kind of linear, Modular and projective groups over finite fields on Projective lines over finite fields.

Recommended Books:1. Generators and Relations for Discrete Groups by Coxeter, H.S.M. and Moser, W.O.,

Springer-Verlag, 1980.2. A Course in Group Theory by J.S. Rose, Cambridge University Press. 1978.3. Presentation of Groupsby Johnson, D.L., Cambridge Lecture Notes, 1976. 4. Finite Group Theory by I. M. Isaacs, 2008.

MATH-507 Theory of Several Complex VariablesReview of 1-variable theory, Real and complex differentiability, Power series, Complex differentiable functions, Cauchy integral formula for a polydisc, Cauchy inequalities, The maximum principle. Hartogs figures, Hartogs theorem, Domains of holomorphy, Holomorphic convexity, Theorem of Cartan Thullen. The Levi form, Geometric interpretation of its signature, E.E. Levi’s theorem, Connections with Kahlerian geometry, Elementary properties of plurisubharmonic functions. Definition and examples of complex manifolds. The d-operators, The Poincare Lemma and the Dolbeaut Lemma, The Cousin problems, Introduction to Sheaf theory.

Objectives: After the successful completion of this course students will be able to demonstrate knowledge of the syllabus material. Also they will be able to know about

49

continuity and differentiability of the functions of several complex variables and they will be able to apply the Theory of several complex variables to solve the problems Chomology.

Recommended Books:1. Complex Manifolds by J. Morrow and K. Kodaira, Holt, Rinehart and Winston, New

York, 1971.2. An Introduction to Complex Analysis in Several Variables by L. Hormander, D. Van

Nostrand, New York, 1966.3. Several Complex Variables by H. Grauert and K. Fritsche, Springer Verlag, 1976.4. Several Complex Variables and Complex Manifolds by M. Field, Cambridge

University Press, 1982.5. Function Theory of Several Complex Variables by S. G. Krantz, 2001.6. Complex Analysis: Fundamentals of the Classical Theory of Functions by John

Stalker, Birkhauser Verlag, 2003.7. Complex Analysis (Princeton Lectures in Analysis Series Vol. II) by Elias M. Stein,

Rami Shakarchi, Princeton University Press, 2003.

MATH-508 Topological Vector SpacesBalanced sets, absorbent sets, convex sets, linear functional, linear manifolds, sublinear functionals and extension of linear functional.Definitions and general properties, product spaces and guohent spaces, bounded and totally bounded sets, convex sets and compact sets in toplogical vector spaces, closed hyperplanes and separation of convex sets, complete topological vector spaces, meterizable topological vector spaces, normed vector spaces, normal toplogical vector spaces and finite dimensional spaces.General properties, subspaces, product spaces, quotient spaces, convex and compact sets in locally convex spaces, hornological spaces, barreled spaces, spaces of continuous function, spaces of indefinitely differentiable function, the notion of distribution, unclear spaces, montal spaces, Sehwartz spaces, (DF)-spaces and Silva spaces.

Objectives: The goal of this course is to study vector spaces in light of different topological concepts. A student is given knowledge about linear functional linear manifold and about balanced, Absorbent and convex sets in a vector space. A topological vector space, product spaces bounded and totally bounded sets, Convex sets and locally convex spaces, closed hyper planes and separation of convex sets, and some basic knowledge of meterizable normal and complete topological vector spaces is given.

Recommended Books:1. Toplogical Vector Spaces by Robertoson, A.P. and Robertson, W., Cambridge

University Press, 1966.2. Toplogical Vector Spaces by Cristescu, R., Noordhoff International Publishing,

Netherlands, 1977.3. Toplogical Vector Spaces by Treves, F., Distributions and Kernels Academic Press

New York, 1967.4. Toplogical Vector Spaces by Horvath, J., Addison-Wesley, 1966.5. Topological Vector Spaces by Schaefer, H., Springer-Verlage, 1966.6. Topological Vector Spaces by L. Narici and E. Becktnstein, 2010.

MATH-509 Loop GroupsComplex Groups, Compact Groups, Root Systems, Weyl Groups, Complex Homogeneous Spaces, Borel-Weil theorem. Infinite dimensional manifolds, Groups of maps as infinite dimensional Lie groups, The Loop group L(G) = Maps (S1 ,G) and its basic properties. Lie algebra extensions, the Co-adjoint action of the loop group on its Lie algebra, Kirillov method of orbits, group extension of simply connected Lie groups, Circle bundles,

50

Connections and curvature. The affine Weyl group and its root system, Generators and relations.

Objectives: Upon the successful completion of this course students will be able to demonstrate knowledge of the finite dimensional Lie groups, Groups of smooth maps and Kac-Moody Lie Algebras. Also they will be able to know about the affine Weyl group and its root systems.

Recommended Books:1. Loop Groups by A. Pressley and G. Segal., Oxford University Press, 1986.2. Infinite Dimensional Lie Algebras by V.G. Kac, Birkhauser, 1983.3. Lie Algebra; Theory and Algorithms by W. A. D. Graff, 2000.4. Langland Correspondence for Loop Groups by Edward Frenkel, Cambridge Studies in

Advanced Mathematics, 2007.

MATH-510 Nilpotent and Soluble GroupsNormal and Subnormal Series, Abelian and Central Series, Direct Products, Finitely Generated Abelian Groups, Splitting Theorems, Solube and Nilpotent Groups, Commutators Subgroup, Derived Series, The Lower and Upper Central Series, Characterization of Finite Nilpotent Groups, Fitting Subgroup, Frattini Subgroup, Dedekind Groups, Supersoluble Groups, Soluble Groups with Minimal Condition. Subnormal Subgroups, Minimal Condition on Subnormal Subgroups, The Subnormal Socle, the Wielandt Subgroup and Wielandt Series, T-Groups, Power Automorphisms, Structure and Construction of Finite Soluble T-Groups.

Objectives: The objective of this course is to make a student understand about some important series of groups, direct products, finitely generated Abelian groups, Soluble and Nilpotent groups. In this course students learn to characterize Finite Nilpotent groups, Fitting groups, Frattini subgroup, Dedikind groups, subnormal subgroups, Weilandt subgroup and Weilandt series and then T-groups.

Recommended Books:1. A Course in the Theory of Groups by Robinson, D.J.S., Graduate Textes in

Mathematics 80, Springer, New York, 1982. 2. Finite Soluble Groups by Doerk, K. and Hawkes, T., De Gruyter Expositions in

Mathematics 4, Walter De Gruyter, Berlin, 1992.3. The Theory of Infinite soluble Groups by J. Carson and D. J. S. Robinson, 2004.

MATH-511 Commutative AlgebraCommutative Rings: Definition and examples, Integral domains, unit, irreducible and prime elements in ring, Types of ideals, quotient rings, Rings of fractions, Ring homomorphism, Euclidean Domains, Principal ideal domains and Unique Factorization domains. Polynomial and Formal Power series Rings, Factorization in polynomial rings, Irreducibility Criteria. Noetherian Rings, Polynomial extension of Noetherian domains, Quotient ring of Noetherian rings, Ring of Fractions of Noetherian rings. Dimension of Rings: Chain of prime ideals in a domain, Length of chain of prime ideals, Dimension of ring, Dimension of Polynomial rings.Integral Dependence: Ring extension, Integral element, Almost integral element, Integral closure of a domain, Complete integral closure of domain, integrally closed domain. Completely integrally closed domain. Valuation Rings: Valuation map and value group, Rank of a valuation, Discrete Valuation Rings and Dedekind domains: Fractional ideals, finitely generated fractional ideals, invertible fractional ideals, Dedekind domain.

Objectives: After the successful completion of this course students will be able to demonstrate knowledge of the commutative rings, polynomials and formal power series rings, Noetherian rings and valuation rings. Also they will be able to know about the

51

definitions of different types of Rings and related topics to identify and construct examples and to distinguish examples from non-examples.

Recommended Books:1. Commutative. Algebra by O. Zariski and P. Samual, Vol. l, Springer-Verlag, New

York, 1958. 2. Introduction to Commutative Algebra by M. F. Anayah and L. G. Macdonald,

Addison Wesley Pub. Co., 1969.3. Multipllcative Ideal Theory by R. Gilmer, Marcell Dekker, New York, 1972. 4. Commutative Ring theory by H. Matsumura, Cambridge University Press, 1986.5. Combinatorial Commutative Algebra by E. Miller and B. Sturmfuls, 2005.6. Introduction to Ring Theory by P.M. Cohn, Springer undergraduate mathematics

series, Springer, 2000.7. Exercises in Classical Ring Theory by T. Y. Lam, 2nd ed., Springer, 2003.

MATH-512 Banach AlgebrasBanach Algebra: Ideals, Homomorphisms, Quotient algebra, Wiener’s lemma. Gelfand’s Theory of Commutative Banach Algebras: The notions of Gelfand’s Topology, Radicals, Gelfand’s Transforms. Basic properties of spectra. Gelfand-Mazur Theorem, Symbolic calculus: differentiation, analytic functions, integration of A-Valued functions. Normed rings. Gelfand-Naimark theorem.

Objectives: The basic goal of this course is to make understanding of what Banach Algebras are what are ideals, Homomorphisms and Quotient Algebras of the Banach Algebras. Next to teach a student the Gelfand theory of commutative Banach algebras and basic properties of Spectra, Symbolic Calculus, Normed rings and in the end Gelfand-NAinmark Theorem.

Recommended Books:1. Functional Analysis by Rudin, W., McGraw Hill Publishing Company Inc. New

York.2. Normed Algebras by M.A. Naimark, M., Wolters Noordhoff Publishing Groningen.

The Netherlands 1972.3. Banach Algebras by Zelazko, W., American Elsevier Publishing Company Inc.New

York, 1973.4. Banach Algebras by Rickart, C.E., D. Van Nostrand Company Inc. New York 1960.5. Banach Algebras and General Theory of Algebras by T. W. Palmer, 2001.

MATH-513 Lie AlgebrasDefinitions and Examples of Lie algebras ideals and quotients simple, solvable and nilpotent Lie algebras radical of a Lie algebra, Semisimple Lie algebras, Engel’s milpotency criterion, Lie’s and Cartan theorems Jordan-Chevalley decomposition. Killing forms dimension 4; Applications of Lie algebras.Objectives: After the successful completion of this course the students will be able to know about Lie algebras, nilpotent and solvable Lie algebras and related topics. They will also be able to know about Jordan-Chevalley decomposition and its applications.

Recommended Books:1. Introcution to Lie Algebras and Representation Theory by Humphreys, J.E. , Springer

Verlag, 1972.2. Elementary Lie algebra theory by Lepowsky, J. and Mecollum, G.W., Yale

University, 1974.3. Lie algebras by Jacobson, N., Intersciences, New York, 1962.4. Introduction to Lie Algebras by K. Erdmann and M. J. Wildon, 2007.5. Lie Algebras; Theory and Algorithms by W. A. D. Graff, North Holland, 2000.6. Introduction to Lie Algebras, Karin Erdmann, Mark J. Wildon, Springer, 2006.

52

MATH-514 Spectral Theory in Hilbert SpacesSpectral analysis of unitary and self-adjoint operators: resultion of the indentity, integral representations. The Caley transform. Spectral types, commutative operators. Rings of bounded self-adjoint operators and their examples.

Objectives: This course presents the basic tools of modern analysis within the context of the fundamental problem of operator theory: to calculate spectra of specific operators on infinite dimensional spaces, especially operators on Hilbert spaces. The tools are diverse, and they provide the basis for more refined methods that allow one to approach problems that go well beyond the computation of spectra: the mathematical foundations of quantum physics, noncommutative k-theory, and the classification of simple C*-algebras being three areas of current research activity which require mastery of the material presented here. The book is based on a fifteen-week course which the author offered to first or second year graduate students with a foundation in measure theory and elementary functional analysis.

Recommended Books:1. Theory of linear operators: Vol. II by Akhiezer and Clazman., Frederick Ungar

Publishing Co., 1963.2. Theory of Differential Operators by Naimark, M., George Harrapand Co., 1967.3. Introduction to Spectral Theory in Hilbert Spaces by G. Helmberg, Dover

Publications, 2008.

MATH 515 Heat and Mass TransferLaws of thermodynamics, Momentum transfer, Relation between heat and momentum transfer, Modes of heat transfer, Fourier’s law of heat conduction, Law of conservation of energy, Equation for temperature field, Concept of velocity and thermal boundary layers, Forced and free convection, Radiation and its applications, Heat transfer by conduction and convection-heat exchanger, Mass transfer and its modes, Fick’s law of diffusion, Equation for concentration field, Steady mass diffusion through a wall, Diffusion of a mass in a moving medium, Convective mass transfer, Mass transfer equation with chemical reaction.

Objectives: This course enables the student to understand the terminology and principal use in heat and Mass transfer analysis and solve the energy and momentum balance equation by the computer or hand generated solutions.

Recommended Books:1. Advances in Heat Transfer by Young I. Cho, George A. Green, Academic press,

2011.2. Heat and Mass transfer by Sawheny G. S., Second Edition, I. K. International Pvt.

Ltd., 2010.3. Heat and Mass transfer by Hans Dieter Baehr, Karl Stephan, Springer, 2006.4. Heat Convection by Latif M. Jiji, Springer-Verlag Berlin Heidelberg, 2006.

MATH 516 Introduction to Modeling and SimulationThis course surveys the basic concepts of computer modeling in science and engineering using discrete particle systems and continuum fields. It convers techniques and software for statistical sampling, simulation, data analysis and visualization and use statistical, quantum chemical, molecular dynamics, Monte Carlo, mesoscale and continuum methods to study fundamental physical phenomena encountered in the fields of computational physics, chemistry, mechanics, materials science, biology and applied mathematics. Applications are drawn from a range of disciplines to build a broad-based understanding of complex structures and interactions in problems where simulation is on equal footing with theory and

53

experiment. A term project allows development of individual interests. Students are mentored by a coordinated team of participating faculty from across the Institute.

Objectives: The course will give brief and concise discussion on models of complex system in engineering and control systems. Modeling of layers of society’s critical infrastructure networks will be performed. Building tools to view and control simulation and their results is an essential part of the course. This course establishes an unambiguous, common vocabulary to discuss modeling and simulation. Identify, general characteristics of simulations determine under which circumstances simulations are useful in engineering.

Recommended Books:1. Applied numerical methods with softwares by Schoichito Nakamura, Prentice Hall

1991.2. Numerical methods using Matlab by J. H. Methews and K. K. Fink, 4th Edition,

Prentice Hall 2004.3. Numerical Analysis by R. L. Burden, 9 Edition, Brooks Cole, 2010.

MATH 551 Newtonian Fluids Some examples of viscous flow phenomena, properties of fluids, boundary conditions, equation of continuity, the Navier-Stokes equations, the energy equation, boundary condition, orthogonal coordinate system, dimensionless parameters, velocity considerations, two dimensional considerations, and the stream functions. Couette flows, Poiseuille flows, unsteady duct flows, similarity solutions, some exact analytical solutions from the papers. Introduction; laminar boundary layer equations, similarity solutions, two-dimensional solutions, thermal boundary layer, Some exposure will also be given from the recent literature appearing in the journal.

Objectives: The aim of the course is to use the Reynolds Transport Theorem to derive the Continuity Equation, Energy Equation and the Momentum Equation. Use the Energy Equation to evaluate the work done by a fluid in motion. Use the Energy Equation to determine the energy loss through any fluid system and to evaluate flow through a multi-pipe system. Students will learn about the presence of the planetary boundary layer within the troposphere. Then using real life data, observe the changing levels of the boundary layer.

Recommended Books:1. F.M.White, Viscous fluid flow, McGraw Hill inc., 1991.2. H.Schlichting and K.Gertsen, Boundary layer theory, Springer, 1991.3. P.A.Davidson, An introduction to magnetohydrodynamics, Cambridge University

Press, 2001.

MATH 552 Advanced Integral EquationsExistence Theorems, Integral Equations with L2 Kernels. Applications to partial differential equations. Integral transforms, Wiener-Hopf Techniques.

Objectives: The purpose of this course is to transform the BVP and IVP into integral transform and then solve the integral transform by different analytic and numerical methods.

Recommended Books:1. Harry Hoch Stadl, Integral Equations, John Wiley, 1973.2. Stakgold, I., Boundary Value Problems of Mathematical Physics, Macmillan, New

York, 1968.3. Multidimensional Integral Equations and inequalities by B. G. Pschpatte, 2011.

54

MATH 553 Numerical Solutions of Ordinary Differential Equations Introduction: Ordinary differential equations: IVP’s, BVP’s and applications, Numerical integration of IVP’s, Method of superposition, method of Chasing, The Adjoint Operator Method, Iterative Methods – The Shooting Methods, Iterative methods, The Finite-Difference Method, Method of Transformation- Direct Transformation,

Objectives: The objective of this course is to introduce students to the methods, tools and ideas of numerical computation. The emphasis of the course will be on problem solving using computer to give students an opportunity to sharpen their skills in programming. A final objective is to familiarize students with the intelligent use of powerful and versatile systems such Matlab, Maple, Mathematica, Mathcad, Fortron and others in attacking numerical problems and obtaining not only numerical but also graphical results.

Recommended Books:1. Computer Methods for Ordinary Differential Equations and Differential-Algebraic

Equations by Uri M. Ascher and Linda R. Petzold, published by SIAM, 19982. Computational Methods in Engineering Boundary Value Problems by T.Y. Na.3. Numerical Analysis by R. L. Burden and J. D. Faires, Seventh Edition, PWS

Publishing Company, Boston, USA.4. Numerical Solutions of Ordinary Differential Equations by K. Atkinson, W. Han and

D. E. Stewart, 2011.

MATH 554 Electrodynamics Maxwell’s equations, electromagnetic wave equation, boundary conditions, waves in conducting and non-conducting media, reflection and polarization, energy density and energy flux, Lorentz formula, wave guides and cavity resonators, spherical and cylindrical waves, inhomogeneous wave equation, retarded potentials, Lenard-Wiechart potentials, field of uniformly moving point charge, radiation from a group of moving charges, field of oscillating dipole, field of an accelerated point charge.

Objectives: The aim in this course is to present the basic subject matter with emphasis on the unity of electric and magnetic phenomena. Moreover a number of topics will be developed and utilized in mathematical physics which are useful in electromagnetic theory and wave mechanics. The effects of radiation and fields of uniform and accelerated point charges will be discussed.

Recommended Books:1. Foundations of Electromagnetic Theory by J. R. Reitz and F. J. Milford, published by

Addison Wesley, 1969.2. Classical Electricity and Magnetism by K. H. Panofsky and M. Phillips, published by

Addison Wesley, 1962.3. Introduction to Electromagnetic Fields and Waves by D. Corson and P. Lorrain,

published by Freeman, 1962.4. Classical Electrodynamics by D. W. Jackson, published by John Wiley.5. Electrodynamics by F. Melia, 2001.

MATH 555 General RelativityThe Einstein field equations, the principles of general relativity, the stress-energy momentum tensor, the vacuum Einstein equations and the Schwarzschild solution, the three classical tests of general relativity, the homogeneous sphere and the interior Schwarzschild solution, Birkhoff’s theorem, the Reissner-Nordstrom solution and the generalized Birkhoff’s theorem, the Kerr and Kerr-Newman solution, essential and coordinate singularities, event horizon and black holes, Eddington-Finkelstein, Kruskal-Szekres coordinates, Penrose diagrams for Schwarzschild, Reissner-Nordstrom solutions.

55

Objectives: The objectives of this course are to give students a basic understanding of Einstein's theory of General Relativity. It will provide understanding for how space-time geometry underlies the structure of our universe and how this leads to theory of gravity that reproduces Newton's force of gravity in the appropriate limit, but also loads to new phenomena.

Recommended Books:1. Introduction to General Relativity by R. M. Wald, published by University of Chicago

Press, Chicago, 19842. Inroduction to General Relativity by R. Alder, M. Bazine, and M. Schiffer, published

by McGraw-Hill Inc., 19653. Essential Relativity by W. Rindler, published by Springer Verlag, 1977.4. General Relativity by N. M. J. Woodhouse, 2007.

MATH 556 ElastodynamicsWaves in infinite media. Half-space problems; Surface waves. Dispensive media. Diffraction and scattering due to irregular structures.

Objectives: The main objective is to study certain general consequences of the equations governing classical elastodynamics with limitation mechanically homogeneous and isotropic solids.

Recommended Books:1. Introduction to Elastodynamics by Zaman, F.D., U.G.C. Monograph.2. Wave Propagations in Elastic Solids by Achenbach.3. Reciprocity in Elastodynamics by J. D. Achenbach, 2003.

MATH 557 Plasma TheoryDefinition of plasma, temperature, Debye shielding, the plasma parameter, criteria for plasmas, introduction to controlled fusion.Wave propagation in plasma, derivation of dispersion relations for simple electrostatic and electromagnetic modes. Equilibrium and stability (with fluid model), Hydromagnetic equilibrium/diffusion of magnetic field into a plasma, classification of instabilities, two-stream instability, the gravitational instability, resistive drift waves. Atomospheric source of magnetospheric plasma and its temperature, plasma from Jupiter.

Objectives: The objectives of this course are to educate students in the fundamentals of plasma physics, to teach students to become proficient in using fluid equations to study wave, stability and transport phenomena, to provide an introduction to plasma kinetic theory and waves in plasmas and to provide an introduction to controlled fusion research.

Recommended Books:1. Introduction to Plasma Physics by Chen, F.F., Plenum Press, New York, 1974.2. Principles of Plasma Physics by Krall, N.A. and Trivelpiece, A.W., McGraw-Hill

Book Company, 1973.3. Controlled thermonuclear reactions by Glasstone, S., and Lovberg, R.H., Van

Nartrand Company, 1960.4. Magnetospheric Plasma Physics by Nishida, A., D. Reidel Publishing Compnay,

1982.5. Plasma Astrophysics by Melrose, D.B., Gordon and Breach Science Publishers, 1980.6. High Informative Plasma Theory by V. Erofeer, 2011.

MATH-601 Variational InequalitiesVariational Problems, Existence results for the general implicit variational problems, Implicit Ky Fan’s inequality for monotone functions, Jartman Stampacchia theorem for monotone

56

compact operators, Selection of fixed points by monotone functions, Variational and quasivariational inequalities for monotone operators.

Objectives: At the end of this course the students will be able to understand the basic concept of variational inequalities, convex function, minimum of convex functions, formulation of variational inequalities, approximation and projection theorems. Study the existence of unique solution of variational inequalities and related problems. Preparing students to be self independent and enhancing their mathematical ability by giving them home work and projects.

Recommended Books:1. Variational Inequalities by J.L. Liions., and G. StamPacchia, Comm. Pure Appl. Math

20, 1967.2. Implicit Variational Problems and Quasi Variational Inequalities by V. Mosco.,

Lecture Notes in Mathematics-543, Springer-Verlag, Berlin, 1976.3. Variational and Quasi-variational Inequalities by C. Baiocchi and A. Capelo, Wiley,

1984.4. An Introduction to Variational Inequalities and their applications by D. Kinderlehrer

and G. Stampacchia, 2000.

MATH-602 Theory of Complex ManifoldsAlgebraic preliminaries; Almost complex manifolds and complex manifolds; connections in almost complex manifolds; Hermitian metrics and Kaehler matrics; Kaehler metric in local coordinate systems; Examples of Kaehler manifolds; Holomorphic sectional curvature; De Rham decomposition of Kaehler manifolds; Curvature of Kaehler submanifolds; Topology of Kaehler manifolds with positive curvature. Hermitian connections in Hermitian vector bundles. Homogeneous spaces: Structure theorems on homogeneous complex manifolds; Invariant connections on homogeneous spaces. Invariant connections on reductive homogeneous spaces; invariant indefinite Riemannian metrics; holonomy groups of invariant connections; the deRham decomposition and irreducibility; Invariant almost complex structures.

Objectives: In this course the student will study holomorphic functions in several variables in some ways, which are similar to the familiar theory of functions in one complex variable, but there are also many interesting differences. At the end of the, the students will be able to apply basic theory complex manifolds

Recommended Books:1. Shoshichi Kobayashi and Katsumi Nomizu, Foundations of Differential Geometry,

Vol.II, Interscience Publishers, John Wiley & Sons, 1969.2. Shabat, B.V., Introduction to Complex Analysis, Part II, American Mathematical

Society, 1992.3. Griffiths and Harris, Principles of Algebraic Geometry, Wiley and Sons, Inc., 1994.

MATH-603 C *-AlgebrasInvolutive Algebras, Normed Involutive algebra, C*-Algebras, Gelfand-Naimark theorem, Positive functions, A characterization of C*-Algebras, Positive forms and representations, Applications of C*-Algebras to differential operators.

Objectives: It is an advanced course of functional analysis and it will mainly focus on normed related characterizations. At the end of the course the students will be able to apply C* -algebra to differential operators.

Recommended Books:1. C*-Algebras by Dixmier, J., North Holland Publishing Company 1977.

57

2. Functional Analysis by Rudin, W., McGraw Hill Publishing Company Inc. New York, 2006.

3. Normed Algebras by Naimark, M.A., Wolters. Noordhoff Publishing Groningen. The Netherlands 1972.

4. C*-Algebras Vol. 3; General Theory of C*-Algebras by C. Constantinescu, North Holland, 2001.

MATH-604 Von Neumann AlgebrasThe weak - and strong topologies, Elementary properties of Von Neumann Algebras, Commutant and bicommutant, the density theorems, comparison of projections, introduction to the classification of factors, Normal states and the predual, Gelfand-Naimark-Siegal construction (GNS-constructions).

Objectives: Upon successful completion of this course students will be able to demonstrate knowledge of the syllabus material. Students will also be able to use the definitions of Von-Neumann algebras and associated topics to identify and construct examples and to distinguish examples from non-examples. This course will extremely help the students for research in algebra and analysis.

Recommended Books:1. Von Neumann Algebras by Dixmier, J., North Holland, 1977.2. C*-Algebras by Dixmier, J., North Holland, 1977.3. W*-Algebras and Breach by Schwartz, J., Gordon, New York, 1967.4. C*-Algebras and W*-Algebras by Sakai, S., Springer-Verlag, 2005.

MATH 651 Perturbation Methods Parameter perturbations, coordinate perturbations, order symbols and gauge functions, asymptotic series and expansions. Asymptotic expansion of intergrals, intergration by parts, Laplace’s method and Watson’s lemma, method of stationary phase and method of steepest descent. Straightforward expansions and sources of nonuniformity, the Duffing equation, small Reynolds number flow past a phere, small parameter multiplying the highest derivative. The method of strained coordinates, the Lindstedt – Poincare’ methods, renormalization method. Variation of parameters and method of averaging examples. Method of Multiple scales with examples.Objectives: Perturbation methods underlie almost all applications of physical applied mathematics for example in boundary layer theory of viscous flow, celestial mechanics, optics, shock waves, reaction-diffusion equations and nonlinear oscillations. The aims of the course are to give a clear and systematic account of modern perturbation theory and how it can be applied to differential equations.

Recommended Books:1. Perturbation methods by Nayfeh, A.H., John Wiley & Sons, 2000.2. Problems in Perturbation by Nayfeh, A.H., John Wiley & Sons, 1985.

MATH 652 Numerical Solutions of Partial Differential Equations Boundary and initial conditions, polynomial approximations in higher dimensions. Finite element method: The Galarkin method in one and more dimensions, error bound on the Galarkin method, the method of collocation, error bounds on the collocation method, comparison of efficiency of the finite difference and finite element method. Finite difference method: Finite difference approximations. Application to solution of linear and non-linear partial differential equations appearing in physical problems.

Objectives: The objective of this course is to introduce students to numerical methods for partial differential equations, especially those of physical importance. It will be shown that many obvious methods are unsuccessful, and that the majority of the successful methods are

58

guided by the physics and mathematics of the problem at hand. Simple model problems representing several major classifications are studied for the sake of the general messages that they convey. Relevant numerical techniques are discussed.

Recommended Books:1. An Analysis of Finite Element Method by G. Strang and G. fix, published by Prentice

Hall, 1973.2. Finite Element Analysis from Concepts to Applications by David S. Burnett,

published by Addison Wesley, 1987.3. Numerical Modelling in Science and Engineering by Myron B. Allen, Ismael Herrera

and George F. Pinder.4. Elementary Finite Element Method by G. S. Desai, published by Prentice Hall, 1988.5. Numerical Solutions of Partial Differential Equations by K. W. Morton and B. F.

Mayers, 2005.

MATH 653 CosmologyReview of Relativity. Historical background, Astronomy, Astrophysics, Cosmology. The cosmological principle and its strong form. The Einstein and DeSitter universe models. Measurement of cosmic distances. The Hubble law and the Friedmann models. Steady state models, The hot big bang model, The microwave background. Discussion of significance of a start of time. Fundamentals of high energy physics. The chronology and composition of the Universe. Non-baryonic dark matter, Problems of the standard model of cosmology. Bianchi spacetimes. Mixmaster models. Inflationary cosmology. Further developments of inflationary models. Kaluza-Klein cosmologies. Review of material.

Objectives: This course is a graduate-level introduction to astrophysical cosmology, with emphasis on the standard big bang theory of the universe and in the later part of the course, its extension to a more detailed theory (the inflation + cold dark matter + cosmological constant model) that is presently the leading scenario for explaining the origin of structure in the universe.

Recommended Books:1. Peebles, P.J.E., Principles of Physical Cosmology, Princeton University Press

1993. 2. Ryan, M.P.Jr. and Shepley, L.C., Homogeneous Relativistic Cosmologies, Princeton

University Press 1975.3. Kolb, E.W. and Turner, M.S., The Early Universe, Addison Wesley 1990.4. Abbott, L.F. and Pi, S.Y., Inflationary Cosmology, World Scientific 1986.5. Cosmology by M. R. Robinson, 2004.

MATH 654 Solid MechanicsFluid Mechanics Lagrangian and Eulerain descriptions of motion, analysis of strain, Balance laws of continuum mechanics, Frame-indifference, Constitutive equations for a nonlinear elastic material. Linear elasticity as a linearization of nonlinear elasticity. Incompressibility and models of rubber. Exact solutions for incompressible materials, phase transformations, shape-memory effect.

Objectives: Solid Mechanics is a collection of physical laws, mathematical techniques and computer algorithms that can be used to predict the behavior of a solid material that is subjected to mechanical or thermal loading. The field has a wide range of applications, including.

Recommended Books:1. Applied Mechanics of Solids by A. F. Bower, Ist Edition, CRC Press, 2009.2. Viscous Fluid Flow by F.M. White, McGraw Hill Inc., 1991.

59

MATH 655 Numerical OptimizationIntroduction, Fundamental of unconstrained optimization, Line search methods, Trust-Region methods, Conjugate gradient methods, Quasi-Newton methods, Large scale unconstrained optimization, Calculating Derivatives, Derivative free optimization, Least square problems, nonlinear equations, theory of constrained optimization, linear programming, the Simplex method, interior point methods, fundamental of algorithms for nonlinear constrained optimization, quadratic programming, penalty and augmented Lagrangian methods, sequential quadratic programming, interior point methods for nonlinear programming.

Objectives: In this course the students will learn numerical techniques used in problems dealing with constrained and unconstrained optimization. After learning these techniques they will use them in variety of problems arising in technological and industrial world.

Recommended Books:1. Numerical optimization by J. Nocedal and Stephen J. Wright, Second ed. Springer

2006.

MATH 656 The Classical Theory of FieldsReview of continuum mechanics; solid and fluid media; constitutive equations and conservation equations. The concept of a field. The four dimensional formulation of fields and the stress-energy momentum tensor. The scalar field. Linear scalar fields and the Klein-Gordon equation. Non-linear scalar fields and fluids. The vector field. Linear massless scalar fields and the Maxwell field equations. The electromagnetic energy-momentum tensor. Electromagnetic waves. Diffraction of waves. Advanced and retarded potentials. Multipole expansion of the radiation field. The massive vector (Proca) field. The tensor field. The massless tensor field and Einstein field equations. Gravitational waves. The massive tensor field. Coupled field equations.

Objectives: This course develops among students the understanding of concept of field and its applications in fluids, relativity and quantum mechanics. In particular linear and nonlinear scalar fields, vector fields will be addressed. Students will be given sound knowledge of some well known field equations like KDV equation, Maxwell equation, Einstein field equations and their implications.

Recommended Books:1. Principles of Continua with Applications by Scipio, L.A., John Wiley, New York,

1969.2. The Classical Theory of Fields by Landau, L.D., and Lifshitz, M., Pergamon Press,

1980.3. Classical Electrodynamics by Jackson, J.D., John Wiley, New York, 1975.4. Cravitation by Misner, C.W., Thorne, K.S., and Wheeler, J.A., W.H. Freeman, 1973.5. Introduction to quantum field theory by Romen, P., John Wiley, New York, 1969.6. The Classical Theory of Fields by C. S. Helrich, 2012.

60

Details of Core and Elective Courses for Ph. D. Mathematics Programs

MATH-701 Near RingsNear Rings, Ideals of Near-rings, Isomorphism Theorems, Near Rings on finite groups, Near-ring modules. Isomorphism theorem for R-modules, R-series of modules, Jorden-Holder- Schrier Theorem, Type of Representations, Primitive near-rings R-centralizers, Density theorem, Radicals of near-rings.

Objectives: Upon successful completion of this course students will be able to demonstrate knowledge of the syllabus material. Students will also be able to use the definitions of Near Rings and related topics to identify and construct examples and to distinguish examples from non-examples. Also they will be able to apply the Isomorphism Theorems, Jorden-Holder-Schrier Theorem and Density Theorem to solve the problems in Near Rings.

Recommended Books:1. Near Rings by Pilz, G., North Holland, 1977. 2. Near Rings by M. Alchediak, 1967.3. Near Rings; Some developments to link to Semigroups and Groups by G. Serreroro,

2000.

MATH-702 Advanced Ring Theory-IRadical classes, semisimple classes, the upper radical, semisimple images, the lower radical, hereditariness of the lower radical class and the upper radical class. Partitions of simple rings.

Objectives: After studying this course the students will be able to demonstrate knowledge of the Ring Theory of advanced level. Also they will be able to know about Use definitions to identify and construct examples of Radical classes, semisimple classes and to distinguish examples from non-examples.

Recommended Books:1. Radical and Semisimple classes of Rings by Wiegandt, R., Queen’s papers in Pure

and Applied Mathematics No. 37, queen’s University, Kingston, Ontario, 1974. 2. Advances in Ring Theory by D. V. Huynh, 2010.3. Introduction to Ring Theory by P.M. Cohn, Springer undergraduate mathematics

series, Springer, 2000.4. Exercises in Classical Ring Theory by T. Y. Lam, 2nd ed., Springer, 2003.

MATH-703 Fixed Point TheoryBanach’s contraction principle, Nonexpansive mappings, Sequential approximation techniques for nonexpansive mappings, Properties of fixed point sets and minimal set, Multivalued mappings, Brouwer’s fixed point theorem.

Objectives: In this course the student will learn the approximation of fixed points of contractive mappings. Mainly the focus of the course will be on the existence side. At the end of the course the student must be able to handle the fixed point solutions of non- linear functional equations both single and multi valued mappings.

Recommended Books:1. Topics in Metric Fixed Point Theory by K. Goebel and W.A. Kirk, Cambridge

University Press, 1990.2. Fixed Point Theory by J. Dugundji and A.Granas, Polish Scientific Publishers,

Warszawa, 1982.3. Fixed Point Theory by V.I. Istratescu, D. Reidel Publication Company, 1981.4. Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mapping by K. Goebel

and S. Reich, Marcel Dekker Inc. 1984.61

5. Fixed Point Theory by A. Grenas and J. Dugeundji, 2003.MATH-704 Commutative Semigroup RingsCommutative Rings: Definition and examples, Integral domains, unit, irreducible and prime elements in ring, Types of ideals, Quotient rings, Rings of fractions, Ring homomorphism, Definitions and examples of Euclidean Domains, Principal ideal domains and Unique Factorization domains. Dedekind and Krull Domains. Commutative Semigroups: Basic notions, Cyclic Semigroups, Numerical Monoids,Ordered Semigroups, Congruences, Noetherian Semigroups, Factorization in Commutative Monoids. Semigroup Ring and its Distinguished Elements: Introduction of Polynomial Rings in one indeterminate including its elements of distinct behaviours, Structure of Semigroup ring, Zero Divisors, Nilpotent Elements, Idempotents Units. Ring Theoretic Properties of Monoid Domains: Integral Dependence for Domains and Monoid Domains, Monoid Domains as Factorial Domains, Monoid Domains as Krull Domains, Divisor Class Group of a Krull Monoid Domain.

Objectives: After the successful completion of this course students will be able to demonstrate knowledge of the commutative semigroups, Ring Theoretic properties of Monoid Domains and Krull Domains. Also they will be able to know about the definitions of different related topics to identify and construct examples and to distinguish examples from non-examples.

Recommended Books:1. Introduction to Commutative Algebra by M. F. Atiyah and I. G. Macdonald, Addison

Wesley Pub. Co., 1969. 2. Multiplicative Ideal Theory by R. Gilmer, Marcell Dekker, New York, 1972. 3. Commutative Ring Theory by H. Matsumura, Cambridge University Press, 1986. 4. Commutative Semigroup Rings by R. Gilmer, The University of Chicago Press,

Chicago, 1984.5. Commutative Semigroups by P. A. Grillet, 2001.

MATH-705 Homological AlgebraRevision of basic concepts of Ring Theory. Modules, Homomorphisms and Exact Sequences, Product and co-product of Modules. Comparison of Free Modules and Vector Spaces Projective and Injective Modules. Hom and Duality Modules over Principal ideal Domain Notherian and Artinian Module and Rings Radical of Rings and Modules Semisimple Modules.

Objectives: The basic Goal of this course is to relate ring theory with module theory and then to compare free Modules and Vector spaces, Projective and Injective Modules. Next to give a student concepts of Hom and Duality Modules over Principal ideal Domain , Notherian and Artinian Module and Rings Radical of Rings and Module with semi simple modules.

Recommended Books:1. Rings and Categories of Modules by K. R. Fuller and F.W. Anderson: Stringer Verlag

1973. 2. lectures on Rings and Modules by J. Lambek: New York, 1966. 3. Modules and Rings by F. Kasch: Academic Press, 1982. 4. Algebra, Holt, Rinehart and Winston by T.W. Hungerford: Inc. New York, 1974. 5. An Introduction to Homological Algebra by J. J. Rotman, Academic Press, New

York, 1979. 6. Commutative Algebra by O. Zariski and P. Samual, Vol. I, Springer-Verlag, New

York, 1958. 7. Commutative Algebra by O. Zariski and P. Samual, Vol. II, Springer-Verlag, New

York, 19608. Introduction to Commutative Algebra by M. F. Atiyah and I. G. Macdonald, Addison

Wesley Pub. Co. 1969. 62

9. An Introduction to Homological Algebra by J. J. Rotman, 2009.

MATH 706 Representation of Finite-Algebra and QuiversFinite length modules, right and left modules, minimal morphisms, radicals of rings and modules, projective modules. Homological facts.Artin algebras and categories. Projectivization, duality, structures of injective modules. Blocks, quivers and their representations. Dynkin and Euclidean diagrams. Triangular matrix rings, group algebras and skew group algebras, the transpose. Nakayama algebra, self-injective algebras.Defect and exact sequences, almost split sequences and morphisms, projective and injective middle terms. Irreducible morphisms.Grothendieck groups, Asulander algebras. The Auslander-Reite-quiver and finite-type. Cartan matrix. Translation quivers.

Objectives: The purpose of the course is primarily to address the graduate students starting research in the representation theory of algebras. The representation of algebra involves the classification of indecomposable, irreducible and completely reducible modules over the algebras and the homomorphisms between them. The theory also involves the quiver-theoretical techniques (by P. Gabriel) or the representation of algebras. It also involves knowledge of the almost split sequences (by M. Auslander, I. Reiten)

Recommended Books:1. Representation theory of Artin Algebras by M. Auslander, I. Reiten, S. Smalo,

Cambridgestudies in advanced Mathematics Vol. 36, Cambridge, 1995.

2. Daniel Simson and Andrzej Skowronski by Ibrahim Assem, Elements of representation Theory of Associative Algebras, Cambridge University Press, 2006.

3. Lectures on Representations of Quivers by Willium Crawley-Boevey, Mathematical Institute, Oxford University, 1992.

4. Blocks of Tame Representation Type and related Algebras by K. Erdman, lecture Notice in Maths., Spring-Verlag, Berlin-Heidelberg 1990.

5. Auslander-Reiten sequences and representation-finite algebras by P. Gabriel, Lecture Notes in Maths., Spring-Verlag, 831, Berlin-Heidelberg 1980, Volume 831/1980, 1-71, DOI: 10.1007/BFb0089778.

MATH-707 Theory of SemiringsHemirings and Semirings: definitions and examples. Building new semirings from old. Complemented elements in semrings. Ideals in semirings. Prime and semiprime ideals in semirings. Factor semirings. Homomorphisms of semirings. Regular semirings. Hemiregular and Intra-hemiregular hemirings.

Objectives: The aim of this course is to discuss a more general structure called theory of Semirings. The different aspect of Semirings will be discussed. At the end of the course the student must be awarded of the detailed characterization of Semirings.

Recommended Books:1. The Theory of Semirings and Applications in Mathematics and Theoretical Computer

Science by J. S. Golan, Longman Scientific & Technical John Wiley & sons New York, 1992.

2. Semirings Algebraic Theory and Applications in Computer Science by U Hebisch and H. J. Weinert, Word Scientific Singapore, New Jersey London Hong Kong, 1998.

3. Advances in Ring Theory by D. V. Huynh, 2010.

63

MATH-708 Ordered Vector SpacesGeneral facts about ordered sets, lattices, convergence, with respect to the order relation. Topological vector spaces, locally convex spaces, uniform convergence, topologies in spaces of linear continuous operators, Duality between vector spaces.Ordered vector spaces, Directed spaces and Arehimedean spaces, Vector Lattice, Decomposition of a vector lattice, Concrete spaces, Topological ordered vector spaces.Objectives: The ordered will be considered on the set which would be most of the times vector space and convergence will be studied in the ordered sense. At the coerce students will be to apply their concepts in ordered spaces with topological sense.

Recommended Books:1. Ordered Topological Vector Spaces by Peressini, A.L., Harper and Row, 1967. 2. Ordered Vector Spaces and Linear Operators by Cristescu, R., Taylor and Francis,

1976.

MATH-709 Banach LatticesVector lattices over the real field, ideals, bands and projections, maximal and minimal ideals vector lattices of finite dimension, duality of vector lattices, normed vector lattices, abstract M-spaces, abstract L-spaces, duality of AL- and AM-spaces.

Objectives: After the successful completion of this course the students will be able to know about vector lattices over the real field and the related topics. They will also be able to know about M-spaces, L-spaces and duality of AL- and AM-spaces. This course will help the students in their research in advanced analysis.

Recommended Books:1. Banch lattices and positive operators by Schaeff, H.H., 1971.2. Banch lattices and positive operators by Schaeff, H.H.,, 1984.3. Introduction to Banach Algebras, Operators, and Harmonic Analysis by H. Garth

Dales, London Mathematical Society Student Texts, 2003.

MATH-710 Approximation TheoryBest approximation in metric and normed spaces, Least square approximation, Rational approximation, Haar condition and best approximation in function spaces, Interpolation, Stone-Weierstrass theorem for scalar-and vector-valued functions, Spline approximation.

Objectives: The purpose of this course is to guide the students in learning about new developments in approximation theory that have come up over the last 20 years. The emphasis is on multivariate approximation theory. Most of the topics appear here are still current areas of research. The instructor will cover positive definite functions, radial basis interpolation, thin-plate splines, neural networks, ridge functions, box splines, approximation on spheres, and wavelets. The students will learn practical problems from current research in areas of science, engineering, geophysics, business, and economics.

Recommended Books:1. Introduction to Approximation Theory by E.W. Cheney., McGraw-Hill, 1996.2. Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces by I.

Singer., Springer-Verlag, 2003.3. The Approximation of Functions I, II by J.R. Rice., Addison-Wesley, 1964.4. A Course on Optimization and Best Approximation by R.B. Holmes., Lecture Notes

in Mathematics No.257, Springer-Verlag, 1971.5. Approximation Theory and Methods by M.D. Powell., Cambridge University Press,

1981.

64

MATH-711 Topological AlgebrasDefinition of a Topological algebra and its Examples. Adjunction of Unity, Locally Convex Algebras, Idempotent and m-convex sets, Locally Multicatively convex (l.m.c) algebras, Q-algebras, Frechet algebras, Spectrum of an element, Spectral radius, Basic theorems on Spectrum, Gelfand-Mazur Theorem. Maximal ideals, Quotient algebras, Multiplicative linear functionals and their continuity, Gelfand transformations, Radical of an algebra, Semi-simple algebras, Involutive algebras, Gelfand-Naimark theorem l.m.c. algebras.

Objectives: After studying this course the students will be able to know about Topological Algebra and related topics. Also they will be able to know about Use of definitions to identify and construct examples of different types of algebras. Further this course is very useful for the students to do research in Topology, algebra and Topological Algebra.

Recommended Books:1. Topological Algebras by E. Beckenstein, L. Narici and C. Suffel, North-Holland

Company, 1977.2. Topological Algebras by A. Mallios, Selected Topics, North-Holland Compnay, 1993.3. Multiplicative Functions on Topological Algebras by T. Husain, Pitman Advanced

Publishing Program, 2001.4. Locally Multiplicatively-convex Topological Algebras by E. Michael, Memoirs

Amer. Math. Soc. No.11, 1951.5. Metric Generalization of Banach Algebras by W. Zelazko, Rozprawy

Matematyczne,1965. 6. Introduction to Banach Algebras, Operators, and Harmonic Analysis by H. Garth

Dales, London Mathematical Society Student Texts, 2003.

MATH-712 Fuzzy AlgebraIntroduction, The Concept of Fuzziness Examples, Mathematical Modeling, Operations of fuzzy sets, Fuzziness as uncertainty, Boolean Algebra and lattices, Equivalence relations and partions, Composing mappings, Alpha-cuts, Images of alpha-level sets, Operations on fuzzy sets, Definition and examples of Fuzzy Relations, Binary Fuzzy relations Operations on Fuzzy relations, fuzzy partitions, Fuzzy ideals of semigroups, Fuzzy quasi-ideals, Fuzzy bi-ideals of Semigroups, Characterization of different classes of semigroups by the properties of their fuzzy ideals fuzzy quasi-ideals and fuzzy bi-ideals, Fuzzy Rings, Fuzzy ideals of rings, Prime, semiprime fuzzy ideals, Characterization of rings using the properties of fuzzy ideals Objectives: We study the subject of Fuzzy Algebra. Originally, the revolutionary theory of Smarandache notions was born as a paradoxist movement that challenged the status quo of existing mathematics. The genesis of Smarandache Notions, a field founded by Florentine Smarandache, is alike to that of Fuzzy Theory: both the fields imperatively questioned the dogmas of classical mathematics.

Recommended Books:1. A First course in Fuzzy Logic by Hung T. Nguyen, Chapman and Hall/CRC Elbert A.

Walker 1999.2. Introduction to Fuzzy Sets and Fuzzy Logic by M. Ganesh, Prentice-Hall of India,

2006.3. Fuzzy Commutative algebra by John N. Mordeson, World Scientific, 1998.D.S.

Malik, 4. Fuzzy Semigroups, Springer-Verlage, 2003 by John N. Mordeson, D.S. Malik and

Nobuki Kuroki.

65

MATH-713 Algebraic Number Theory Algebraic Numbers: Algebraic Numbers and Number Fields, Discriminant, Norms and Traces, Algebraic integers and Integral Bases, Factorization and Divisibility, Applications of UFD. Arithmetic’s Number Fields: Quadratic Dields, Cyclotomic Fields, Units in Number rings.Ideals Theory: Properties of Ideals, PIDs and UFDs, Dedekind rings, Norms of ideals, Class group and Class Numbers of Quadratic Fields. Valuations: Definitions and First properties of valuations, Valuation rings, DVRs, P-adic valuation.

Objectives: The aims of this unit are to enable students to gain an understanding and appreciation of algebraic number theory and familiarity with the basic objects of study, namely number fields and their rings of integers. In particular, it should enable them to become comfortable working with the basic algebraic concepts involved, to appreciate the failure of unique factorisation in general, and to see applications of the theory to Diophantine equations.

Recommended Books:1. Algebraic Number Theory by Richard A. Molin, Chapman & Hall, Washington D. C.,

(2005).2. Number Theory I, Fundamental Problems, Ideas and Theories by A.N. Parshin and

I.R. Shafarevich, Springer-Varlag, Berlin Heidelbers, (1995).3. Algebraic Number Fields by G.J. Janusz, Academic Press, New York and London

(1973).

MATH 714 Hopf Algebra and Quantum Groups

Algebra and modules, affine line and plane, graded and filtered algebra. Tensor products of vector spaces, tensor products of algebras, tensor products of linear maps, dualities and traces. Tensor and symmetric algebras, The Hopf algebra GL(2) and SL(2). Modules and comodules over the Hopf algebras. Actions of finite-dimensional Hopf algebras and smash products. Coradical and filtration, pointed Hopf algebras. Inner actions, crossed products. Cleft extensions and existence of crossed products. Twisted H-comodule algebras, quantum plane, Gauss polynomials and the q-binomial formula. The algebra and bialgebra structures on Mq (2). The Hopf algebras GLq(2) and SLq(2). Coactions on quantum planes. Lie algebras, enveloping algebras. The Lie algebras sl(2), the enveloping algebra of sl(2). Hopf algebra structures on Uq(sl(2)). The Yang-Baxter equation. Braided bialgebra, R-Matrix. Antipode in braided Hopf algebra. Cobraided bialgebra. Bicrossed product of groups, Variation on the adjoint representation. Drinfeld’s quantum double.

Objectives: The aim of the course is to provide an introduction to the algebra behind the quantum groups. Its main objective is to provide an opportunity for researchers and students to Groups, Rings, Lie and Hopf Algebras. The quantum groups are the Hopf algebras which arose in mathematical physics and have connections to various areas of mathematics. Hopf algebra also came up in the representation theory of Lie groups and algebraic groups.

Recommended Books:1. S. DǎSčalescu, C. Nǎstǎsescu and S. Raianu, Hopf Algebras, An introduction, Marcel

Dekker, Inc, New York, Basel, 2001. 2. C. Kassel, Quantum Groups, Grad. Text Math., Vol. 155, Springer Verlag, New York,

1995.3. S. Montgomery, Hopf algebras and their actions on rings, CBMS Reg. Conference

Series 82, Providence R. I, 1993.4. S. Shnider and S. Sternberg, Quantum groups: From coalgebras to Drinfeld algebras,

(a guided tour), International Press, Inc., 1997.

66

5. M. E. Sweedler, Hopf Algebras, Benjamin, New York, 1969.

MATH 751 Advanced Analytical Dynamics-I Equations of dynamics and their various forms, equations of Lagrange and Euler, Jacobi’s elliptic functions and the qualitative and quantitative solutions of the problem of Euler and Poisson, the problems of Lagrange and Poisson, dynamical systems, equations of Hamilton and Appel, Hamilton-Jacobi theorem, separable systems, Holder’s variational principle and its consequences.

Objectives: The objectives of this course are to present the principles and methods of analytical mechanics based on Lagrange´s, Poisson, Jacobi and Hamilton´s formulations of the laws of classical mechanics and to give a theoretical basis for further studies in classical and quantum mechanics.

Recommended Books:1. Introduction to Dynamics by L. A. Pars, published by Heinmann, 2008.2. A Treatise on Dynamics of Rigid bodies and Particles by E. T. Whittaker, published

by Cambridge University Press.

MATH 752 Non-Newtonian Fluids Newtonian versus non-Newtonian behavior. Review of Newtonian fluid dynamics. Elementary constitutive equations and their use in solving fluid dynamics problems. Nonlinear viscoelastic constitutive equations and their use in solving fluid dynamics problems. Modelling and solution of flow problems using different constitutive equations.

Objectives: The aim of the study is to model different types of fluids and then apply the basic assumptions used in non-Newtonian fluids for ocean engineering and use scale models to predict the behavior of a real system.

Recommended Books:1. Dynamics of Polymeric Liquids by R.D.Bird, R.C.Armstrong, and O.Hassager, Vol.

1, Fluid Mechanics, 2nd ed., John Wiley & Sons, New York, 2087.2. Rheology and non-Newtonian flow by J.Harris, Longman, London.

MATH 753 Mathematical Techniques for Boundary Value Problems Green’s function method with applications to wave-propagation. Perturbation methods: regular and singular perturbation techniques with applications, variational methods, a study of transform techniques, Wiener-Hopf technique with application to diffraction problems.

Objectives: This course develops mathematical techniques which are useful in solving `real-world' problems involving linear and non-linear differential equations and aims to show in a practical way how equations `work' and what kinds of solution behaviors can occur.

Recommended Books:1. Perturbation Methods by A. Nayfeh, 1998.2. Boundary Value Problems of Mathematical Physics by I. Stakgold.3. Methods based on the Wiener-Hopf technique for the solution of Partial Differential

Equations by B. Noble.4. Analytical Techniques in the Theory of Guided Waves by R. Mitra and S. W. Lee.5. Mixed Boundary Value Problems by D. G. Duffy, 2008.

MATH 754 Group Theoretic MethodsBasic concepts of groups of transformation; parameter lie group of transformation (LGT); Infinitesimal transformation (I.T); Infinitesimal generators; Lie’s first fundamental theorem; Invariance; Canonical coordinates; Prolongations; Multi-parameter lie group of

67

transformations (MLGT); Lie algebra; Solvable lie algebra; Lie’s second and third fundamental theorems. Invariance of ODE’s under (LGT) and (MLGT); Mappings of solutions to other solutions from invariance of an ODE and PDE; Determining equations for (I.T) of an n-th order ODE and a system of PDE’s. Determination of n-th order ODE invariant under a given group; Reduction of order by canonical coordinates and differential invariants; Invariant solutions of ODE’s and PDE’s; Separatrices and envelops. Noether’s theorem and Lie-Backlund symmetries, Potential symmetries; Mappings of differential equations.

Objectives: The use of algebraic methods specifically group theory, representation theory, and even some concepts from algebraic geometry is an emerging new direction. The purpose of course is to give an entertaining but informative introduction to the background to these developments and sketch some of the many possible applications. The course is intended to be palatable by a non-specialist audience with no prior background in abstract algebra.

Recommended Books: 1. Symmetries and differential equations by G.W., Bluman and Sokeyuki Kumei.,

Springer-Verlag, N.Y. 1989.2. Differential equations and group methods by James M. Hill., CRC Press, Inc. N.Y.

1992.3. Continuous groups of transformations by I.P., Eisenhart., Dover Publications, Inc.

N.Y. 1961.4. Group Theoretical Methods in Physics by G. H. Pogosyan and L. E. Vicent, 2005.

MATH 755 Advanced Numerical Analysis Ordinary differential equations: IVP’s, BVP’s and DAE’s and applications. Initial Value Problems: (IVP) On problem stability: test equation and general definitions, linear constant coefficient systems, linear variable coefficients systems, nonlinear problems and Hamiltonian systems. (IVP) Basic methods and Basic concepts: A simple method (forward Euler), convergence, accuracy, consistency, and 0-stability, absolute stability, stiffness, A-stability, P-stability and symmetry. (IVP) one-step methods: The First RK methods, general RK methods, convergence, 0-stability, order for RK methods, explicit RK methods, implicit RK methods and collocation methods. (IVP) linear multi-step methods: Adams methods, BDF methods, initial values for multi-step methods, order, 0-stability, convergence, absolute stability, predictor-corrector methods, modified Newton methods, variable step-size formulae, estimating and controlling the local error and approximating the solution at off-step points. linear BVP’s and Green’s function, stability of BVP’s BVP stiffness and some reformulation tricks. Boundary Value Problems (BVP) shooting: simple method and multiple shooting. Boundary Value Problems (BVP) Finite difference methods for BVP’s: midpoint and Trapezoidal methods, solving linear equations, higher order methods, error estimations and mesh selection, stiff problems and decoupling.

Objectives: The purpose of this course is to discuss the methods for IVP, BVP and DAE’s. The use of multistep methods along with error estimation will be presented. The student will thoroughly go through stability, convergence, accuracy, efficiency and reliability of numerical algorithms. They will also analyze and solve problems like mesh selection, stiffness and decoupling.

Recommended Books: 1. Computer Methods for Ordinary Differential Equations and Differential-Algebraic

Equations by Uri M. Ascher and Linda R. Petzold, published by SIAM, 19982. Numerical Analysis, Seventh Edition by R. L. Burden and J. D. Faires, PWS

Publishing Company, Boston, USA.3. An Introduction to Numerical Analysis by K. E. Atkinson, J. Wiley and Sons, 1989.

68

4. Numerical Solutions of Partial Differential Equations by W. Ames, Academic Press, New York, USA, 1992.

5. Numerical Analysis by W. Gautschi, 2011.

MATH 756 Advanced Optimization Theory Simplex method, bracketing method, Fibonacci method, golden section method, quadratic and cubic search methods, multivariate methods, gradient and conjugate methods, constrained optimization, penalty function approach and methods based on calculus of variation. Objectives: This course is designed to provide a good footing in well known single search methods such as bracketing, Fibonacci, golden section, quadratic and cubic search, as majority of the multivariate methods are ultimately reduced to a single search exercise. Live problems would be identified and their solutions would be suggested through some of the widely used concepts based on gradient and conjugate methods. The constrained optimization would be discussed with emphasis on penalty function approach along with topics related to calculus of variations.

Recommended Books:1. Introduction to Optimization Theory by B. S. Gotfried and J. Weisman, published by

Prentice Inc., 1973.2. Differential Equations and the Calculus of Variations by L. Elsgolts, published by Mir

Publishers, 1970.3. Introduction to Non-Linear Optimization by D. A. Wismer and R. Chattergy,

published by North Holland, 1978.4. Mathematical Optimization and Economic Theory by M. D. Intriligator, published by

Prentice Hall, 1971.5. Optimization Theory by H. T. Jongen, K. Meer and E. Triesch, 2004.

MATH 757 Magnetohydrodynamics Equations of electrodynamics, equations of fluid dynamics, Ohm’s law, equations of magneto hydrodynamics. Motion of a viscous electrically conducting fluid with linear current flow, steady state motion along a magnetic field, wave motion of an ideal fluid. Magneto-sonic waves, Alfve’s waves, damping and excitation of MHD waves, characteristic lines and surfaces. of simple waves, distortion of the profile of a simple wave, discontinuities, simple and shock waves in relativistic magneto hydrodynamics, stability and structure of shock waves, discontinuities in various quantities, piston problem, oblique shock waves.

Objectives: In this course the objective is to stress on basic formulation of MHD flow with modification of Maxwell’s equations in fluid flow. Students will learn about the motion of incompressible conducting fluid in presence of magnetic field. They will also develop understanding of small amplitude waves and shock waves in MHD.

Recommended Books:1. Magneto Hydrodynamics by T. G. Cowling, published by Interscience Publishers,

1963.2. Magneto Hydrodynamics by A. G. Kulikowshy and A. G. Lyabimov, published by

Addison Wesley, 1965.3. Cosmical Electrodynamics by H. Alfven and C. Falthammar, published by Clarendon

Press, 1965.4. Plasma Electrodynamics by A. I. Akhiezer, published by Pergamon Press, 1975.5. Magneto Hydrodynamics by P. C. Kendall and C. Plumption.6. An Introduction to Magnetohydrodynamics by P. A. Davidson, 2001.

69

MATH 758 Advanced Electrodynamics General angular and frequency distributions of radiation from accelerated charges, Thomson scattering, Cherenkov radiation, fields and radiation of localized oscillating sources, electric dipole fields and radiation, magnetic dipole and electric quadruple fields, multipole fields, multipole expansion of the electromagnetic fields, angular distributions, sources of multipole radiation, spherical wave expansion of a vector plane wave, scattering of electromagnetic wave by a conducting sphere.

Objectives: Our aim in this course is to focus on later developments in electromagnetic radiations, multiple fields and multipole expansions, scattering of electromagnetic waves by conductors and relation of electromagnetic theory with special relativity.

Recommended Books:1. Classical Electrodynamics by D. W. Jackson, published by John Wiley.2. Electromagnetic Theory by J. A. Straton, published by McGraw Hill.3. Classical and Quantum Electrodynamics by M. W. Evans and L. B. Crowell, 2001.

MATH 759 Stochastic ProcessesStochastic processes, Markov processes, Queuing theory. Markov chains, discrete and continuous time Morkov chains, transition matrix and probabilities, spatial Poisson processes, compound and marked Poisson processes. Renewal phenomenon, discrete renewal theory. Branching processes and population growth, queuing systems. Brownian motion and martingales.

Objective: This course emphasis on model building and probabilistic reasoning. The approach will be non-measure theoretic but otherwise rigorous. Though the subject is rich in mathematical theory, the learning objective of many students exposed to stochastic processes will be to develop knowledge in the subject for application to their own areas of interests. The contents are mathematically rigorous fashion, with exposure to its use as a modeling and analysis tool.

Books Recommended1. Ross. S. M., Stochastic Processes, (PB) (2006).2. Grimmet. G. and Stirzaker, D., Probability and Random Processes, 3rd Ed. Oxford

University Press, (2001).3. Koehler. U Soresen, M., Exponential Families of Stochastic Processes, Springer-

Verlag, New York, (1997).4. Suddhendu. B., Applied Stochastic Processes, A Bio statistical and Population

Oriented Approach, New Age International Publishers Limited, Wiley Eastern Limited. UK: London, (1995).

5. Srinivasan.S.K and Mehata. K.M., Stochastic Process, 2nd Ed., National Book Foundation, Islamabad.

MATH 760 Multivariate Methods and AnalysisIntroduction: Some multivariate problems and techniques. The data matrix. Summary statistics. Normal distribution theory: Characterization and properties. Linear Forms. The Wishart distribution. The Hotelling T2-dustribution. Distributions related to the multionormal. Estimation and Hypothesis testing: Maximum likelihood estimation and other techniques. The Behrens-Fisher problem. Simultaneous confidence intervals. Multivariate hypothesis testing. Design matrices of degenerate rank. Multiple correlation. Least squares estimation. Discarding of variables.

Objectives: For multivariate analysis, the necessary statistical and mathematical background is necessary. A course in multivariate methods and analysis traditionally tends to focus on techniques that can be described as being primarily exploratory and descriptive. The main

70

objective of this course is to understand multivariate distributions, all of their properties including for large sample sizes. Emphasis is also on the application and interpretation of these methods in practice.

Recommended Books:1. Multivariate Analysis by Mardia, K.V., Kent, J.T., and Bibby, J.M., Academic Press,

London, 1982.2. Multivariate Analysis by Kshirsagar, A.M., Marcell Dekker, New York, 1972.3. Methods of Multivariate Analysis by A. C. Renchar, 2002.

MATH 761 Nonlinear Differential EquationsGeneral classifications of nonlinear differential equations. Basic theory of logistic equations. Nonlinear ordinary differential equations and their logistic construction. Phase space analysis, euquilibrium points and stability conditions. Poincare-Lindstedt method, Asymptotic conditions, limit cycles and bifurcation, Krylov-Bogoliubov techniques for solution construction. Order reduction. System of Nonlinear ordinary differential equations. Method of averaging. Scaling techniques. Iterative maps and chaotic behaviour. Painleve analysis and Painleve analytical iteration-procedure. Applications to nonlinear harmonic oscillators, van der Pol oscillators, and nonlinear damping problem. Introduction to variational principles. Euler-Lagrange partial differential equations. First order quasi-linear partial differential equations. Method of characteristics. Exactly soluble problem. Geometrical interpretation of the general solutions.

Objectives: This course aims to apply mathematical modelling, differential equations, existence and stability theory and apply numerical methods, theory of invariant manifolds, different iteration procedure apply bifurcation theory and the implicit function theorem.

Recommended Books:1. Introduction to Nonlinear Differential and Integral Equations by Davis, H.T., Dover

Publications, New York, 1962.2. Ncnlinear Ordinary Differential Equations by Grimshaw, R., CRC Press, Baca Raton,

1991.3. Painleve Differential Equations in complex Plane Walter de Gruyter by Gromak, V.I.,

Berlin 2002.4. Asymptotic Expansions for Ordinary Differential Equation by Wasov, W., John

Wiley & Sons, New York, 1965.5. Nonlinear Partial Differential Equations by Debnth, L., (2nd Edition) Birkhauser,

Boston, 2005.

MATH 762 Advanced Plasma TheorySolution of localized Vlasov equation Vlasov theory of small amplitude waves in field free uniform/nonuniform magnetized cold/hot plasmas, the theory of instability, Conservation of particles, momentum and energy in quasilinear theory, Landau damping; the gentle-bump and two-stream instability in quasilinear theory, plasma wave echoes, nonlinear wave-particle interaction.Shielding of a moving test charge, electric field fluctuations in maxwellian and nonmaxwellian plasmas, emission of electrostatic waves, electromagnetic fluctuations, emission of radiation from plasma, black body radiation; cyclotron radiation.

Objectives: The main objective of this course is to use the governing equations of plasma physics to discuss the plasma theory of waves especially nonlinear Vlasov theory of waves. Moreover, students are exposed to fluctuations, correlation and radiations in plasmas.

71

Recommended Books:1. Principles of Plasma Physics by Krall, N.A., and Trivelpiece, A.W., McGraw-Hill

Book Company, 1973.2. Plasma instabilities and nonlinear effects by Hasegawa, A., Springer Verlag, 1975.

MATH 763 Convective Heat Transfer: Viscous FluidsFree convection boundary layer flow over a vertical flat plate, mixed convection boundary layer flow along a vertical flat plate, free and mixed convection boundary layer flow past inclined at horizontal plates, double-diffusion convection, convection flow in buoyant plumes and jets, conjugate heat transfer over vertical and horizontal flat plates, free and mixed convection from cylinders, free and mixed convection boundary layer flow over moving surfaces, unsteady free and mixed convection, free and mixed convection boundary layer flow of non-Newtonian fluids.

Objectives: The objective of this course is to discuss the convective flow of viscous fluid over flat horizontal and inclined plates under boundary layer assumption. The emphasis will be given on cylindrical geometries and mixed convection boundary layer flow of non-Newtonian fluids.

Recommended Books:1. Convective heat transfer: Mathematical and Computational modeling of viscous fluids

in porous media by I.Pop and D.B. Ingham, Elsevier, 20012. Convection in porous media by D.A.Nield and A.Bejan, Springer, Third Edition 2006.

MATH 764 Finite Element AnalysisRational Bezier curves, properties of rational Bezier curves, Marsden identity, construction of FEM basis function, the de Boor algorithm, dual functional, error approximation by orthogonal functional, cubic Hermite interpolation, natural spline interpolation, quasi interpolant, Schoenberg scheme, error of quasi interpolation, Lagrangian function for interpolation, interpolation error, curves on uniform grid and their properties, interpolation with curves on uniform grid, geometric Hermite interpolation, non-uniform rational B-splines, construction of finite element basis on multidimensional space, Box splines, recursion for Box splines, approximation on multidimensional space, ellipticity of approximation, Cea’s lemma, approximation theorems for FEM.

Objectives: The objective of finite element method is to discretize the domain into finite element for which the governing equations are algebraic equations. Solution of these algebraic equations gives the approximate solution of the non linear differential equations. The convergence is judged by the refinement of mesh.

Recommended Books:1. Introduction to the Mathematics of Subdivision Surfaces by Lars-Erik Andersson,

SIAM, 2010.2. Numerical Models for Differential Problems by Quarteroni A., Springer, 2009.3. Finite Element Method by Klaus-Jürgen Bathe, John Wiley & Sons, 2007.4. Splines and Variational Methods by Prenter, P. M., A Wiley-Interscience Publication,

2006.

MATH 765 Momentum and Thermal Boundary-Layer TheorySteady two-dimensional flows: Approximate integral methods, numerical, Flow along a vertical plates (cylinder/sphere): Approximate integral method, numerical method, Forced convection flows: similarity solution of thermal boundary-layer equations, Approximate integral method, numerical method, Natural convection flows: similarity solution of thermal boundary-layer equation, Approximate integral method, numerical method, Boundary-layer flow past a cylinder:

72

Three-dimensional boundary-layer flow at a cylinder, Similar and semi-similar solutions: Unsteady motion of a body at rest, Oscillation of bodies in a fluid at rest, Boundary-layer flow with periodic potential flow

Objectives: It visualize the development of velocity and thermal boundary layer during the flow over the surface and derive the differential equation that governs the convection over the basis of momentum, mass and energy balances and solve the equation for laminar and turbulent flow.

Recommended Books:1. Boundary layer theory by H.Schlichting and K.Gersten, 8th Edition 2000, Springer-

Verleg Berlin.2. Viscous Fluid Flow by F.M. White, McGraw Hill Inc., 1991.

MATH 766 AstrophysicsStatic stellar structure and the equilibrium conditions, Introduction to stellar modeling, The Hertzprung-Russell diagram and stellar evolution, Gravitational collapse and degenerate stars, White dwarfs, neutron stars and black holes, Systems of stars, irregular and globular clusters, galaxies superclusters and filaments, astrophysical dark matter and galactic haloes.

Objectives: Students will be taught about astrophysics and its research methods on selected examples. Aim is creating integrated actual astrophysical picture of the Universe. At the end of this course students should be able to: solve basic problems of selected themes of astrophysics and to understand the physical substance of cosmical bodies and astrophysical phenomena.

Recommended Books:1. An Introduction to the study of Stellar Structure by Chandrasekhar, S., Dover

Publications, Inc. 1967.2. Astrophysics by Richard, L., and Deeming, T., Vol.I and II, Jones and Bartlett

Publishers, Inc., 1984.3. Structure and Evolution of Stars by Schwarzschild, M., Dover Publications, New

York, 1965.4. Gravitation by Misner, C.w., Thorne, K.S., and Wheeler, J.A., W.H., Freeman & Co.

1973.5. Astrophysics; A new approach by W. Kundt, 2005.

MATH 767 Advanced ElastodynamicsStrain potential, Galerkin vector, vertical load on the horizontal surface of a half space, Love’s strain function, Biharmonic functions, Lamb’s problem, Cagniard-de Hoop transformation.Transient waves in a layer, forced shear motion of a layer.Thermoelasticity: thermal stresses Chadwick’s solution of thermoelastic solutions.Piezoelectricity. Tensor formultion of piezoelectricity, elastic waves in a piezoelectric solid, Bleustein-Gnlayev waves.

Objectives: This is graduate level course. In this course the student will learn about love’s strain function, biharmonic function, Lamb problem and transient waves in a layer. The material on elastic waves in a piezoelectric solid and Bleustein-Gnlayev waves will also be covered. The students will also gain insight in the important area of thermoelasticity.

Recommended Books:1. Elastic Waves in Solids by Dieulesant D. and Royer, F., John Wiley and Sons, New

York, 1980.

73

2. Foundations of Solid Mechanics by Fung, Y.C., Prentice-Hall, Englewood Cliffs, 1995.

3. Waves Propagation in Elastics Solids by Achenbach, North-Holland, Amsterdam, 1990.

MATH 768 Statistical MechanicsMicrostates; Macrostates; Multiplicity; The second law of thermodynamics; Microcanonical Ensemble; Indistinguishability; Free Energy and Chemical Potential; Gibbs free energy; Chemical Potential Dilute Solutions and Chemical Equilibrium. Boltzmann Factor; Averages; Canonical Ensemble; Equipartion Theorem; Maxwell Speed Distribution; Partition Functions, Free Energy; Composite Systems; Ideal Gas. Review of Quantum Mechanics (Schoedinger Equation; Angular Momentum; Systems of Many particles); The Gibbs Factor; Grand Canon9ical Ensemble; Bosons and Fermions; The Distribution Functions; Degenerate Fermi Gas. Weakly Interacting Gases; Partition function; configuration integral; Cluster Expansion; Second Virial Coefficient. Blackbody Radiation; Debye Theory of Solids; Bose-Einstein Condensation; Non-Equilibrium Systems and Chaos; Application of Degeneracy to White Dwarfs and Neutron Stars.

Objectives: This course aims to give students a deep understanding of the principles of statistical mechanics and how to apply them to a wide variety of problems. At the end of the course the student will have firm grasp of the fundamental principles of statistical mechanics. Students should be able to identify situations where the methods of statistical mechanics may be applied, simplify and model the situation in a physically reasonable and tractable fashion and then utilize the formal and mathematical techniques learnt in the course to predict various properties of the system at hand and be able to then verbally communicate what their predictions mean in a natural setting.

Recommended Books:1. Introduction to Modern Statistical Mechanics by D. Chandler , Oxford University

Press, 1987. 2. Mathematical Foundations of Statistical Mechanics by A. I. Khinchin, Dover

Publications, 1960.3. Introductory Statistical Mechanics by R. Bowley and M. Sanchez , Oxford University

Press, 1999.4. Statistical Physics by L. D. Landau, and E. M. Lifshitz, Butterworth-Heinemann,

1984. 5. Quantum Statistical Mechanics by Leo Kadanoff, Gordon Baym, Westview Press,

2001. 6. Computational Statistical Mechanics by William G. Hoover, Elsevier Science

Publishers, 1991.7. An Introduction to Chaos in Non equilibrium Statistical Mechanics by J. R. Dorfman,

Cambridge University Press, 2003.

MATH 769 Advanced Quantum TheoryLagrangian and Hamiltonian Formalisms; Hamilton-Jacobi Equations; Noether Theorem; Symmetries and Conservation laws; Lorentz Invariance and Relativistic Mechanics. Operators in Banach Space and Operator Calculus; Applications to Quantum Computing and Information Theory; Representation Theory (including Heisenberg; Schroedinger and Holomorphic Representations); Deformation Quantization. Classical Field Theory; Examples of Quantized Field Theories; Dirac Equation and Spinor Formulation; Electron Spin; Field Theoretic Methods in Quantum Statistics. Free Particle Scattering Problems; General Theory of Free Particle Scattering; Scattering by a Static Potential; Scattering Problems and Born Approximation. Feynman Path Integral Formalism and Related Wiener theory of Functional Integration; Perturbation theory and Feynman Diagrams; Regularized Determinants of Elliptic Operators Supersymmetry and Path Integral Formalism for Fermions.

74

Objectives: Main objectives are to learn more on some new tools for the solution of the quantum mechanical problems. Starting form the quantum mechanics in Hilbert space the student will be exposed to the Dirac theory and relativistic scattering theory. The theory of functional integration will also be discussed in detail. At the end of the course the student should be able to understand more of the formalism and interpretation of quantum mechanics and apply the formalism to the analysis of various quantum mechanical systems.

Recommended Books:1. Advanced Quantum Mechanics by J. J. Sakurai, Addison-Wesley, 2006. 2. Quantum Mechanics by A. Messiah, John Wiley & Sons Inc., 1961. 3. The Principles of Quantum Mechanics by P.A. M. Dirac, Oxford at the Clarendon

Press, 1958. 4. Mathematical Foundations of Quantum Mechanics by J. von Neumann , Princeton

University Press, 1955.5. Quantum Mechanics Non-Relativistic Theory by L. D. Landau and E. M. Lifshitz,

Pergamon Press, 1977. 6. Relativistic Quantum Theory by L. D. Landau, E. M. Lifshitz and L.P. Pitaevskii,

Pergamon Press, 1977.7. Quantum Mechanics for Mathematicians and Physicists by Ikenberry , Oxford

University Press, 1962.8. The Quantum Theory of Fields by S. Weinberg, Vol. 1. Cambridge University Press,

1995.

MATH 770 Nonlinear WavesFundamental of wave propagation. General classifications of dispersive and hyperbolic waves. Advection equation and characteristic curves. Nonlinear advection equation. Traveling wave solutions. Conservation laws. Quasi-linear wave equations. Age-structure models. Cauchy problem for nonlinear wave equations. Inverse-scattering methods. Shock dynamics in one, two and three dimensions. Non-linearization and weak shock solutions. Solutions using wave-front expansion and N wave expansions. Nonlinear water waves equation. Exact solutions by variational techniques. Korteweg-de Veries equation. Shape preserving nonlinear waves. Solution waves. Asymptotic analysis, Solution solutions using inverse scattering method. Miura transforms and applications to cubic Schroedinger wave equation, sine-Gordon waves, Toda chain problems, nonlinear Born-Infeld wave equations.

Objectives: The aim of the course is to give the students an introduction to the characterization of nonlinear wave PDEs, methods for finding wave-like solutions an the nature of nonlinear waves. It is desired in this course that the students should understand basic theory for modeling of linear and non-linear continuum systems, list and describe fundamental principles and basic methods of nonlinear dynamics and apply this knowledge to solve particular problems.

Recommended Books:1. Linear and Nonlinear Waves by Whitham, G.B., Wiley-Interscience, New York 1974.2. Nonlinear Diffasive Waves by Sachdev, P.L., Cambridge University Press,

Cambridge. 1987.3. Theoretical Foundations of Nonlinear Acoustics by Rudenko, O. V., Soluyan and S.I.,

Plenum Press, New York 1977.4. Nonlinear Waves by Leibovich, S. and Seebass, A.R., Cornell University Press, Ithaca

1972.5. Asymptotic Methods in Nonlinear Waves Theory by J. A. Kawahara and T. Pitman

Advance Publishing program, Boston 1982.6. A Modern Introduction to the mathematical Theory of Water Waves by Johnson, R.

S., Combridge University Press, Cambridge 1997.7. Nonlinear Waves by P. Popivanov and A. Savova, 2010.

75

MATH-801 Advances in AnalysisFundamental Theorems of Functional Analysis (Open mapping Theorem, Closed graph Theorem, Hahan Banach Theoren Analytic and Geometric forms), Bounded linear

functionals, Dual, Bidual and reflexive spaces, Weak and Weak Topologies. Adjoint Operators Riez representation Theorem.

Objectives: The course consists of the most celebrated results of analysis. The student should be able to understand the concept of extension and theory of open and closed balls. The idea of Topological Structure will be exploited.

Recommended Books:1. Introduction to Functional Analysis and Applications by E. Kreyszig, Published by

John Wiley and Sons2. Introduction to Topology and Modern Analysis by G.F. Simmons,Published by

McGraw Hill3. Elements of Functional analysis by I. Maddox, Published by Cambridge University

Press.

MATH 802 Advanced Perturbation MethodsApproximate Solution of Linear differential Equations Approximate Solution of Nonlinear Differential Equations Perturbation Series Regular and Singular Prturbation Theory Perturbation methods for Linear Eigenvalue problems Asymptotic Matching Boundary Layer Theory Mathematical Structure of Boundary Layer: Inner, Outer, and Intermediate Limits Higher-Order Boundary Layer Theory distinguished Limits and Boundary Layers of Thickness WKB Theory Exponential Approximation for Dissipative and Dispersive Phenomena Conditions for Validity of the WKB approximation Patched Asymptotic Approximation: WKB Solution of Inhomogeneous Linear quations. Matched Asymptotic Approximation: Solution of the One-Turning-Point Problem.

Objectives: The focus of this course is to prepare the students so that they can apply advanced asymptotic methods in various areas of science and engineering. Further they will be able to understand the theoretical background implementation and limitation of each method.

Recommended Books:1. Advanced mathematical Methods for Scientists and Engineers by Carl M. Bender,

Steven A. Orszag McGraw-Hill, Inc. 1978.2. Perturbation Methods for Differential Equations by B. K. Shivamoggi, 2003.

MATH-803 LA-SemigroupsLA-semigroups and basic results, Connection with other algebraic structures, Medial and exponential properties, LA-semigroups defined by commutative inverse semigroups, Homomorphism theorems for LA-semigroups, Abelian groups defined by LA-semigroups, Embedding theorem for LA-semigroups, Structural properties of LA-semigroups, LA-semigroups as a semilattice of LA-subsemigroups, Locally associative LA-semigroups, Relations on locally associative LA-semigroups, Maximal separative homomorphic images of locally associative LA-semigroups, Decomposition of locally associative LA-semigroups.

Objectives: To introduce the students to basic concepts of LA-Semigroup and its connection with other algebraic structures. Also students will learn Structural properties of LA-Semigroups, Locally associative LA-Semigroups, Decomposition of locally associative LA-Semigroups.

Recommended Books:

76

1. Clifford, A.H. and G.B. Preston., The Algebraic Theory of Semigroups, Vols. I & II, Amer. Math. Soc. Surveys, 7, Providence, R.I, 2000.

MATH 804 Advanced Analytical Dynamics-II Groups of continuous transformations and Poincare’s equations, systems with one degree of freedom, singular points, cyclic characteristics of systems with a degree of freedom, ergodic theorem, metric indecompossability, stability of motion.

Objectives: This course is designed to link the continuous groups and dynamics. The focus of the course is to develop the Poincare’s dynamical equations using group theoretic approach. Students will also go through ergodic theorem and thoroughly study stability of motion.

Recommended Books:1. Introduction to Dynamics Analytical Dynamics by L. A. Pars, published by

Heinmann, 2008.2. A Treatise on Dynamics of Rigid bodies and Particles by E. T. Whittaker, published

by Cambridge University Press.

MATH 805 Advanced Magnetohydrodynamics Flow of an ideal fluid past magnetized bodies, fluid of finite electrical conductivity flow past a magnetized body. Elasser’s theory, Bullard’s theory, Earth’s field turbulent motion and dissipation, vorticity analogy. Effects of molecular structure, currents in a fully ionized gas, partially ionized gases, interstellar fields, dissipation in hot and cool clouds.

Objectives: In advanced MHD student will analyze flow of conducting fluid past magnetized bodies, with a detailed discussion on dynamo theories with vorticity analogy. Motion of compressible ionized gas through magnetic field will also be discussed.

Recommended Books:1. Magneto Hydrodynamics by T. G. Cowling, published by Interscience Publishers,

1963.2. Magneto Hydrodynamics by A. G. Kulikowshy and A. G. Lyabimov, published by

Addison Wesley, 1965.3. Cosmical Electrodynamics by H. Alfven and C. Falthammar, published by Clarendon

Press, 1965.4. Plasma Electrodynamics by A. I. Akhiezer, published by Pergamon Press, 1975.5. Magneto Hydrodynamics by P. C. Kendall and C. Plumption.6. An Introduction to Magnetohydrodynamics by P. A. Davidson, 2001.

MATH 806 Spectral Methods in Fluid Dynamics Introduction to computational fluid dynamics, its geometry and equation structure.Partial differential equations: Hyperbolic PDE, parabolic PDE, elliptic PDE and PDE traditional solution methods. Preliminary computational techniques: discretisation, approximation to derivatives, wave representation, finite difference method.Theoretical background: convergence, consistency, stability, solution accuracy and computational efficiency. Weighted residual methods: finite volume method, finite element method and interpolation, finite element method and the Sturm-Liouville equation, spectral method. Steady problem: nonlinear steady problems, direct method for linear systems, iterative methods, pseudotransient methods. One-dimensional diffusion equation: explicit methods, implicit methods, boundary and initial condition. Multi-dimensional diffusion equation: two-dimensional diffusion equations, multi-dimensional splitting methods, splitting scheme and the finite element method, Neumann boundary condition and method of fractional steps. Linear convection dominated problems: one dimensional linear convection equation, numerical dissipation and dispersion, steady convection-diffusion equation, one-

77

dimensional transport equation, two-dimensional transport equation. Nonlinear convection-dominated problems: one-dimensional Burger’s equation, system of equations, group finite element method, two-dimensional Burger’s equation. Fluid dynamics: equation of motion, incompressible and compressible flow.Generalized curvilinear coordinates: transformation relationships, evaluation of the transformation parameters. Grid generation:Inviscid flow.Boundary layer flow.Incompressible viscous flow.Compressible viscous flow.

Objectives: This course is designed to equip the students with the advanced computational techniques based finite difference and finite element methods for the solution of fluid dynamics equations. Each technique will be discussed form theoretical as well as application point of view. At the end of the course student will be able to apply these techniques in fluid dynamics and related areas.

Recommended Books:1. Computational techniques for fluid dynamics by C.A.J. Fletcher, published by

Springer-Verlag.2. Computational fluid dynamics by P.J. Roache, published by Hermosa.3. Chebyschev and Fourier Spectral Methods by J. P. Boyd, 2001.

MATH-807 Advanced Semigroup TheoryBasic definitions, Inverse semigroups, The natural order relation, Congruences on inverse semigroups, Anti uniform semilattices, Fundamental inverse semigroups; Bisimple and simple inverse semigroups. Orthodox semigroups; Basic properties; The structure of orthodox semigroups.

Objectives: The main goal of the course is to point at the ubiquity and the versatility of semigroups, showing notions and results which link semigroups with various kinds of mathematical structure. The students will learn some important constructions and techniques used in semigroup, Commutative semigroups, semigroup of fractions, archimedean decomposition, Free inverse semigroups, solution of the word problem in free inverse semigroups.

Recommended Books:1. The Algebraic Theory of Semigroups; Vol.I & II by A.H. Clifford and G.B. Preston,,

AMS Math. Surveys, 1961 and 1967.2. An Introduction to Semigroup Theory by J.M. Howie, Academic Press 1976.3. Fundamental of Semigroup Theory by J.M. Howie, Oxford University Press, 1996.4. Semigroup Theory and Application (Lecturer Notes on Pure and Applied

Mathematics) by Clement, CRC Press, 1989.

MATH-808 Advanced Near RingsDistributively generated near-rings, ideals isomorphism theorems, Free d.g. near rings, Representations of d.g. near-rings, Types of representations, upper and lower faithful d.g. near rings, Endomorphism near-rings of groups.

Objectives: Upon successful completion of this course students will be able to know about distributively generated Near Rings, Free d.g. Near Rings and endomorphisms Near Rings of Groups. Also they will be able to apply the Isomorphism Theorems, to solve the problems in Near Rings. Further this course will also help the students while doing research work in Near rings.

Recommended Books:1. Near-Rings by Pilz, G., North Holland, 1976.

78

2. A course in Ring Theory by D. S. Passman, Chelsea Pub. Co., 2004.3. An Introduction to Ring Theory by P. M. Cohn, Springer 2002.

MATH-809 Theory of Group GraphsGenerators and relations, Factor groups, Direct Products, Automorphisms, Finite Presentations of Groups, Tiezte transformations, Coset enumerations, Graphs, Cayley diagram, Schrier’s cost diagrams, Coset diagrams for the modular group, Action of the modular group on finite sets, Symmetry in the diagrams, Composition of coset diagrams, Action of the modular group on real projective line, Action of the modular group on finite projective lines over finite fields.

Objectives: This course is designed to make understanding of representing a group by its costs in a diagram. In this course students learn about Factor geoups, direct products, Automorphisms Tietz transformations , presentations of finite groups Caley diagrams, Schriers coset diagrams , Modular group, its action on different sets and their corresponding coset diagrams.

Recommended Books:1. Generators and relations for discrete Groups by Coxeter, H.S.M. and Moser, W.O.,

Springer-Verlag.1965.2. A course in group theory by Rose, S., Cambridge University Press. 1980. 3. Combinatorial group theory by Magnus, W., Karrass, A and Solitar, D., Dover

Publications, 1976. 4. Groups, Graphs and Trees; An Introduction to the Geometry of Infinite Group by J.

Meier, Cambridge University Press, 2008.

MATH-810 Advanced Ring Theory-IIMinimal left ideals, Wedderburn-Artin strueture theorem, The Brown-McCoy radical, the Jacobson radical, Connections among radical classes, Homomorphically closed semisimple classes.

Objectives: After studying this course the students will be able to know about Wedderburn-Artin Structure Theorem and its applications. Also they will be able to know about Use definitions to identify and construct examples of Brown-McCoy Radical classes, Jacobson radical classes. Further this course is very useful for the students to do research in Ring Theory.

Recommended Books:1. Radical and Semisimple classes of Rings by Wiegandt, R., Queen’s papers in Pure

and Applied Mathematics No. 37, queen’s University, Kingston, Ontario, 1974.2. Introduction to Ring Theory by P.M. Cohn, Springer undergraduate mathematics

series, Springer, 2000.3. Exercises in Classical Ring Theory by T. Y. Lam, 2nd ed., Springer, 2003.

MATH-811 Non-Standard AnalysisUniverse and Languages: Set relations, Filters, Individuals and super structures, Universes, Languages, Semantics, Los Theorem, Concurrence, Infinite Integers, Internal sets. Ordered Fields, Non-standard Theory of Archimedean Fields, The hyperreal numbers, Real sequences and Functions. Prolongation Theorems. Non-standard Differential calculus, Additivity, The existence of Non-measurable sets. Topological spaces, Mapping and products, Topological Groups, The existence of Haar Measure, Metric Spaces, Uniform continuity and Equicontinuity, Compact mapping.

Objectives: Upon successful completion of this course students will be able to demonstrate knowledge of the syllabus material. Students will also be able to use the definitions of

79

Ordered Fields, non-standard Theory of Archimedean Field and related topics to identify and construct examples and to distinguish examples from non-examples. This course will also help the students for research in algebra and analysis.Recommended Books:

1. Lectures on Non-standard Analysis by Machover, M and Hirschfled, J., Springer-Verlag.

2. Applied Non-Standard Analysis by M. Davis, Dover Publications, 2005.3. Non. Standard Analysis by A. Robinson, Princeton University Press, 1996.

MATH-812 Numerical Ranges of Operators on Normal SpacesNumerical range in normed algebras, Numerical radius, Vidav’s theorem and applications to C*-algebras, The spatial numerical range, spectral properties, second dual of a Banach algebra, spectral states.

Objectives: After the successful completion of this course the students will be able to know about numerical range in normed algebras, numerical radius. They will also be able to know about Vidav’s Theorem and its application to C*-algebras. This course will help the students in their research in advanced analysis.

Recommended Books:1. Numerical ranges of operators on normed spaces and of elements of normed algebras

by Bonsall, F.F., and Duncan, J., LMS lecture note series 2, Cambridge University Press,1971.

MATH-813 Strict ConvexityLocally covex spaces, Banach spaces, basic theorems of linear functional analysis, strict convex spaces, product and quotient spaces and strict convexity, interpolation and strict convexity, modulus of convexity, strict convexity and approximation theory, strict convexity and fixed point theory.

Objectives: Convex functions play an important role in many areas of mathematics. They are especially important in the study of optimization problems. For instance, a (strictly) convex function on an open set has no more than one minimum. Even in infinite-dimensional spaces, under suitable additional hypotheses, convex functions continue to satisfy such properties and, as a result, they are the most well-understood functionals in the calculus of variations. In probability theory, a convex function applied to the expected value of a random variable is always less or equal to the expected value of the convex function of the random variable.

Recommended Books:1. Strict convexity and complex strict convexity by Istratescue, V.I., 1984.2. Normed linear spaces by Day, M.M., 2007.3. Geometry of Banach spaces by Diestel, J., 1975.6. Linear operators-I by Dunford, N., and Schwartz, J.T., 1958.

MATH-814 Advanced Commutative AlgebraUnique Factorization Domains: Basics and examples, Guass Theorem, Quotient of a UFD, Nagata Theorem. Class Groups: Divisor Cllasses, Divisor Class monoid, Divisor Class group. Krull Rings and Factorial Ring: Divisorial ideals, Divisors, Krull rings, Stability properties, Two classes of Krull rings, Divisor class groups, Application of a Theorem of Nagata, Examples of Factorial Rings. Atomic Domains: Definition and examples, Polynomial extension of Atomic domains. Domains Satisfying ACCP: Definition and examples, Polynomial extension of domains satisfying ACCP. Connection of domains satisfying ACCP and Atomic domains. Bounded Factorization Domains: Definition and examples Length function, Charecterization of BFD through length function. Polynomial extension of BFDs, Noetherian and Krull domains are

80

BFDs. Half Factorial Domains: Class number of a Field, Carlitz Theorem, Examples and basic results, Dedekind and krull examples, Integrability and HFD, On polynomial and polynomial like extensions. Finite Factorization Domains: Group of Divisibility G(D) of a domain D, G(D) and FFD, Atomic idf-domain is FFD.

Objectives: After the successful completion of this course students will be able to demonstrate knowledge of the commutative rings, polynomials and formal power series rings, Noetherian rings and valuation rings of advanced level. Also they will be able to know about the definitions of different types of Rings and related topics to identify and construct examples and to distinguish examples from non-examples. This course is very useful for the students to do research in Ring Theory.

Recommended Books:1. Lecture Notes on Unique Factorization Domains by P. Samuel, Tata Institute of

Fundamental Research, Bombay, 1964. 2. Multiplicative ideal Theory by R. Gilmer, Marcel Dekker, New York, 1972. 3. Divisor Class group of a krull Domain by R. M. Fossum, Spriger Verlag, 1973. 4. Factorization in Integral Domains by D. D. Anderson, Lecture Notes in Pure &

Applied Mathematics, Marcel Dekker, New York, Vol. 189, 1997. 5. Non Noetherian Commutative Ring Theory by S. T. Chapman and Sara Glaz,

Mathematics & its Applications series Vol. 520, Kluwar Academic Publishers, 2000. 6. Introduction to Ring Theory by P.M. Cohn, Springer undergraduate mathematics

series, Springer, 2000.7. Exercises in Classical Ring Theory by T. Y. Lam, 2nd ed., Springer, 2003.

MATH-815 Advanced Homological AlgebraTensor Products of Modules, Singular Homology Flat Modules. Categories and Functors Cogenerator. Finitely related (finitely presented) Modules. Ure Ideals of a ring Pure submodules and Pure Exact sequences. Hereditary and Semihereditary Rings. Ext. and extensions, Axioms Tor and Torsion, Universal co-efficient Theorems. Hilbert Syzygy Theorem, Serre’s Theorem, Mixed identities.

Objectives: After studying this course the students will be able to know about Homological Algebra and related topics. Also they will be able to know about Use of definitions to identify and construct examples of different types of modules. Further this course is very useful for the students to do research in Algebra.

Recommended Books:1. Rings and Categories of Modules by J. Fuller and F.W. Anderson, Stringer Verlag,

2004. 2. Lectures on Rings and Academic Modules by J. Lambek, New York, 1966. 3. Modules and Rings by F. Kasch, Academic Press, 1997. 4. Algebra by T. W. Hungerford, Holt, Rinehart and Winston, Inc. New York, 1974. 5. An Introduction to Homological Algebra by J. J. Rotman, Academic Press, New

York, 1979. 6. Commutative Algebra by O. Zariski and P. Samual, Vol. I, Springer-Verlag, New

York, 1958. 7. Commutative Algebra by O. Zariski and P. Samual, Vol. II, Springer-Verlag, New

York, 19608. Introduction to Commutative Algebra by M. F. Atiyah and I. G. Macdonald, Addison

Wesley Pub. Co. 1969. 9. An Introduction to Homological Algebra by J. J. Rotman, 2009.

MATH-816 Advanced Theory of Semirings

81

Basic Definitions, Semirings of fractions, Euclidean Semirings, Semimodules over semirings. Factor semimodules, Some constructions of semimodules, Morphisms of semimodules. Factor semimodules. Free, projective, and injective semimodules.

Objectives: After studying this course the students will be able to know about Theory of Semirings and related topics. Also they will be able to know about Use of definitions to identify and construct examples of different types of semimodules over semirings. Further this course is very useful for the students to do research in the field of Algebra and Theoretical Computer Science.

Recommended Books:1. The Theory of Semirings and Applications in Mathematics and Theoretical Computer

Science by J. S. Golan, Longman Scientific & Technical John Wiley & sons New York, 1992.

2. Semirings Algebraic Theory and Applications in Computer Science by U Hebisch and H. J. Weinert, Word Scientific Singapore, New Jersey London Hong Kong, 1998.

MATH 851 Advanced Heat TransferReview of heat transfer modes, thermal boundary layer without coupling of the velocity field to the temperature field: Boundary layer equations for the temperature field. Forced convection for constant properties, effects of the Prandtl number, similar solutions of the thermal boundary layer. Integral methods for computing the heat transfer, effects of dissipation, thermal boundary layer with coupling of the velocity field to the temperature field, Boundary layer equations, boundary layer with moderate wall heat transfer, natural convection, indirect natural convection, mixed convection, Radiation fin of trapezoidal profile, conduction through fins, natural convection of Powell-Eyring Fliud between vertical flat plates, Natural convection boundary layer flow, Natural convection over a semi-infinite vertical plate.

Objectives: Apply scientific and engineering principles to analyze thermo fluid aspects of engineering system and use the analytic and computational tools to investigate the heat transfer of fluid flow at micro and macroscopic level.

Recommeded Books:1. Heat transfer by Yunus A. Cengel, Second Edition 2003, Tata McGraw-Hill,

Publishing company limited, New Delhi.2. Boundary layer theory by H.Schlichting and K.Gersten, 8th Edition 2000, Springer-

Verleg Berlin.3. Introduction to convective heat transfer analysis by P.H.Oosthuizen and D.Naylor,

McGraw-Hill International, New York, 19994. Heat and Mass transfer by F.M.White, Addison-Wesley series 1988.5. Computational Methods in Engineering Boundary value problems by T.Y.Na,

Academic Press, 1979.

MATH 852 Convective Heat Transfer: Porous MediaFree and mixed convection boundary layer flow over vertical surface in porous media, free and mixed convection past horizontal inclined surfaces in porous media, conjugate free and mixed convection over vertical surfaces in porous media, free and mixed convection from cylinder and spheres in porous media, unsteady free and mixed convection in porous media, non-Darcy free and mixed convection boundary layer flow in porous media,

Objectives: The objective of the course is to formulate realistic analysis of the behavior of multiphase porous media, composed of solid, miscible and immiscible fluids, subjected to multiphysics mechanical and hydraulic phenomena. The course involves thermodynamics and constitutive formulation for single and multi-phase material, derivation of the

82

conservation and field equations, and developing the weak and matrix forms for finite element implementation and are to offer advanced testing capabilities that meet the requirements and demands of the geoenvironmental engineering industry.Recommended Books:

1. Convective heat transfer: Mathematical and Computational modeling of viscous fluids in porous media by Ioan I.Pop and D.B. Ingham, Elsevier, 2001

3. Convection in porous media by D.A.Nield and A.Bejan, Springer, Third Edition 2006.

MATH 853 Advanced Finite Element AnalysisIntroduction to Sobolev spaces, Ritz-Galerkin approximation of Poisson’s equation, weak form of Poisson’s equation, variational form of Poisson’s equation, Ritz-Galerkin approximation of Poisson’s equation with hat functions, elliptic bilinear form, elliptic variational form, Ritz-Galerkin approximation of an elliptic variational problem, construction of FE basis, properties of basis function, basis function of multidimensional space, linear independence of basis function, basis function on uniform grid, condition number of Galerkin matrix, uniform Lagrange polynomial, extension of basis function, coefficients of extended basis, weight functions, R-functions, partial weight function, WEB-splines, stability and approximation with WEB-spline, Ritz-Galerkin system, applications of WEB-approximation.

Objectives: The goal of this course is to model the microscopic and macroscopic structure of many physical and mechanical problems. We can analyze the biomechanics of the head and figure prints by the simulation.

Recommended Books:1. Introduction to the Mathematics of Subdivision Surfaces by Lars-Erik Andersson

SIAM, 2010.2. Numerical Models for Differential Problems by Quarteroni A., Springer, 2009.3. Finite Element Method by Klaus-Jürgen Bathe, John Wiley & Sons, 2007.4. Splines and Variational Methods by Prenter, P. M., A Wiley-Interscience Publication,

2006.

MATH 854 Advanced Multivariate Methods and AnalysisPrincipal component analysis: Definition and properties of principal components. Testing hypotheses about principal components. Correspondence analysis. Discarding of variables. Principal component analysis in regression. Factor analysis: The factor model. Relationships between factor analysis and principal component analysis. Canonical correlation analysis: Dummy variables and qualittive data. Qualitative and quantitative data. Discriminant analysis: Discrimination when the populations are known. Fisher’s linear discriminant function. Discrimination under estimation. Multivariate analysis of variance: Formulation of multivariate one-way classification. Testing fixed contrasts. Canonical variables and test of dimensionality. Two-way classification.

Objectives: For this course it will be of great helpful to study the basic knowledge of multivariate methods. The main objective is to derive mathematical results on advanced techniques of multivariate analysis such as Principal component analysis, Factor Analysis etc and then applying these results on data sets from different sectors of life.

Recommended Books:1. Multivariate Analysis by Mardia, K.V., Kent, J.T., and Bibby, J.M., Academic Press,

London, 1982.2. Multivariate Analysis by Kshirsagar, A.M., Marcell Dekker, New York, 1972.3. An Introduction to Multivariate Analysis by T. Raykov and G. A. Marcoulides, 2008.

MATH 855 Robotics 83

Introduction to Robot (Fundamental notions and Definitions), Transformations and Jacobians, Manipulator. Kinematics (Forward and Inverse) of manipulator, Manipulator Dynamics, Trajectory Generation, Manipulator Mechanism, Manipulator Design. Linear Control of Minipulator, Non-linear Control of Manipulator, Forced Control of Manipulator, Multivariable control, Feedback linearization, Variable structure and Adaptive Control.

Objectives: The objectives of this course are to introduce students to the concept of robotics, its applications and to teach students the mechanics of mechanical manipulators. It is also desired that the students also learn about the control of mechanical manipulators and how to program robots to perform certain tasks.

Recommended Books:1. Introduction to Robotics by John, J. Craig, Addison-Wesley Publishing Company

Inc., 1999.2. Robot Dynamics and Control by Mark, W. Sponge and M. Vidyasagar, John Wiley

and Sons Inc., 2004.3. Control of Dynamic Systems by Gene Franklin, J. David Powell, Abbas Emami-

Naeini, Addison-Wesley Publishing Company Inc., 1989.4. Modern Control System Theory and Applications by Stainley M. Shinners, Addison-

Wesley Publishing Company Inc., 1987.5. Adaptive Control of Mechanical Manipulators by John, J. Craig, Addison-Wesley

Publishing Company Inc., 1997.

MATH 856 Group Analysics of Partial Differential EquationsIntroduction, Mathematical idea of symmetry; Local solvability for systems; Maximal rank condition, Symmetry transformations, Lie group transformations in IRn+m. Canonical parameter for a group; Infinitesimal transformations in IRn+m. Lie equations; Exponential map; Symmetry groups of differential equation systems; Prolongation formulas; Invariant points; Invariant functions; Canonical variables; Infinitesimal criteria for invariance for systems; Lie algebras. Multi-parameter groups; Symmetries of partial differential equations; Construction of exact solutions; Group classification; Other topics.

Objectives: upon successful completion of this course student will have sufficient grasp on lie group transformations, infinitesimal transformations, symmetry group of differential equations and different criteria for invariance of system, multiparameter groups and symmetry of partial differential equations.

Recommended Books:1. Lie-Backlund and Noether symmetries by Ibragimov, NH, Kara AH and

Mahomed FM 1998.applications, Nonlinear Dynamics, 15.2. Handbook of Lie Group Analysis of Differential Equations by Ibragimov, vols I to

III, CRC Press, Boca Raton, 1994-1996.

MATH 857 Advanced Nonlinear Differential EquationGenallzed Method of characteristics, Complete integrals of first order nonlinear partial differential equations, Discontinuous solutions and shock conditions, Weak and generalized solutions, Higher order nonlinear partial equations, Nonlinear hyperbotic systems and Riemann invariants, Initial-value problems and asymptotic solutions, Teavelling wave solutions, Stability analysis, Inverse scattering techniques, Nonlinear transforms, Backlund transformations and nonlinear superposition principle, Method of multiple scales, Applications to nonlinear reaction-diffusion problems, strongly dispersive nonlinear equations, nonlinear Schroedinger equation, and Korteweg-de Vriss equation, Special techniques for solution generation, Transformation methods using fractional Fourier

84

transforms, Nonlinear Hankel transforms, Painleve test and truncated expansions, Optimization techniques, Geometrical construction of solution surfaces.

Objectives: This course reflects the nonlinear problems and to solve them by the asymptotic, numerical, perturbation and symmetry reduction technique.Recommended Books:

1. Nonlinear Partial Differential Equations by Debnth, I., (2nd Edition) Birkhauser, Boston, 2005.

2. An Introduction to Nonlinear Partial Differential Equations by Logan, J. D., Wiley-Interscience, New York, 1994.

3. Fractional Differential Equation by Prodlubny, I., Academic Press, Boston 1999.4. Similarity Solutions of Nonlinear Partial Differential Equations by Dresner, L.,

Pitrnan Books, London 1983.5. Nonlinear Elliptic and Parabolic Equations by Poa, C. V., Plenum Press, New York

1992.6. Methods of Mathaematical Physics by Courant, R. and Hilbert, D., Vol.2. Wiley-

Interscience, New York 1962.7. Nonlinear Partial differential Equations in Engineering by Ames, W. F., Vol. 2.

Academic Press, New York 1972.

MATH 858 Modeling and Simulation of Dynamical SystemsThis course is about modeling multidomain engineering systems at a level of detail suitable for design and control system implementation. It also describes Network representation, state-space models, Multiport energy storage and dissipation, Legendre transforms, Nonlinear mechanics, transformation theory, Lagrangian and Hamiltionian forms, Control-relevant properties. The application examples may include electro-mechanical transducers, mechanisms, electronics, fluid and thermal systems, compressible flow, chemical processes, diffusion and wave transmission.

Objectives: after successful completion of this course student will be able to model multidomain engineering system of different designs and control. They will have sufficient knowledge and the modeling of space model, multiport energy storage and dissipation, fluids and thermal system, and wave system. They will be in good position to analyse the simulations output of these systems.

Recommended Books:1. Modelling and simulation by Giuseppe Petrone and Giuliano Cammarata, InTech ,

2008.2. Applied numerical methods with softwares by Schoichito Nakamura, Prentice Hall

1991.

MATH 859 Topics in Fluid MechanicsObjectives: The aim of this course is to study the latest developments in the field. The instructor will discuss the latest research trends with the students. At the end the of the course the students must have a good understanding of applications in the other related areas.

MATH 860 Topics in MechanicsObjectives: The aim of this course is to study the latest developments in the field. The instructor will discuss the latest research trends with the students. At the end the of the course the students must have a good understanding of applications in the other related areas.

MATH 861 Topics in Differential EquationsObjectives: The aim of this course is to study the latest developments in the field. The instructor will discuss the latest research trends with the students. At the end the of the course the students must have a good understanding of applications in the other related areas.

85

MATH 862 Topics in Computational Mathematics

Objectives: The aim of this course is to study the latest developments in the field. The instructor will discuss the latest research trends with the students. At the end the of the course the students must have a good understanding of applications in the other related areas.

MATH 863 Topics in Applied Mathematics

Objectives: The aim of this course is to study the latest developments in the field. The instructor will discuss the latest research trends with the students. At the end the of the course the students must have a good understanding of applications in the other related areas.

MATH 864 Topics in Algebra

Objectives: The aim of this course is to study the latest developments in the field. The instructor will discuss the latest research trends with the students. At the end the of the course the students must have a good understanding of applications in the other related areas.

MATH 865 Topics in Topology

Objectives: The aim of this course is to study the latest developments in the field. The instructor will discuss the latest research trends with the students. At the end the of the course the students must have a good understanding of applications in the other related areas.

MATH 866 Topics in Analysis

Objectives: The aim of this course is to study the latest developments in the field. The instructor will discuss the latest research trends with the students. At the end the of the course the students must have a good understanding of applications in the other related areas.

MATH 867 Topics in Complex Analysis

Objectives: The aim of this course is to study the latest developments in the field. The instructor will discuss the latest research trends with the students. At the end the of the course the students must have a good understanding of applications in the other related areas.

86