Central University of Rajasthan

Department of Mathematics

Revised syllabus for Pre-Ph.D. course work-2016-17

Course work for the doctoral program

The Department of Mathematics offers a variety of courses that enables the students to get a

deep and state-of-the-art understanding of their field of specialization, while at the same time

acquiring an excellent background in other relevant areas of applied and pure mathematics. On

joining the Ph.D. program, the student is assigned a faculty advisor, who will help the student

choose the appropriate course. The first year’s program aims to fill whatever gaps there are in the

student’s background, as well as to provide the foundations in analysis and in other basic

subjects which are necessary preparation for the more specialized courses in the following years.

The intellectual demands on applied mathematicians in university research as well as in

industrial research and development are continually changing. It is essential that a program be

chosen with both the `long view’ as well as the more immediate demands of impending research

in mind.

Scheme for the course work

The total credits in the Ph. D. coursework are 12. The credits are distributed as follows:

A. Research Methodology in Mathematics: 4 credits

B. Core Mathematics: 4 credits

C. Specialized course with the research supervisor: 4 credits.

• Research Methodology in Mathematics:

Course code: PMC-101

• Core Mathematics Course

Course code: PMC-102

Students must choose any one of the following:

• Advanced Complex Analysis

• Fractional Calculus and Geometric Function Theory

• Mathematical Programming

• Dynamical Systems

• Complex Dynamics

• Computational Fluid Dynamics

• Course with the research supervisor

Course code: PMC-103

(Students must choose any one of the following)

• Holomorphic dynamics

• Geometric Function Theory

• Applied Dynamical Systems

• Mathematical Biology

• Queuing theory

• Magnetohydrodynamics

• Thermal Instabilities and Methods

• Celestial Mechanics

Course code: PMC-101 LTP: 3+1+0

Research Methodology in Mathematics

UNIT-I:Introduction of research, Importance of research, Research methods and research

methodology, Importance of research methodology, Types of research, Various stages of

research, Selection of research topic, Identification of research problems, Definition and

formulation of the problem, Literature survey, Internet as a medium for research, Knowledge of

web search:, Elements of an article: Title, Abstract, Keyword, Introduction, Formulation, Result

and discussion, References, Evaluation of research: plagiarism, citation, impact factor etc.

Review of research papers, Working knowledge of Google Scholar, Research Gate, Web of

Science, MathSciNet, SCOPUS and/or other open-source/subscribed journals and books. (15L)

UNIT-II:Introduction:- Latex and open office, Writing of simple article, letters and applications,

Mathematical symbols and commands, arrays, formulas and equations, Spacing, Borders and

Colors,Creating different templets, Writing of research article, repots etc. Preparation of templets

of thesis and books.Preparation of ppt. poster, etc., Pictures and Graphics.(15L)

UNIT-III:MATLAB and Mathematica: Basic introduction: Arithmetic operations, functions,

plotting the graphs of different functions, Matrix operations, finding roots of an equation,

Finding roots of a system of equations, Solving differential equations. Basic 2-D plots and 3-D

plots, (15 Lectures)

Reference books:

1. R. Pratap: Getting started with MATLAB, Oxford University Press, 2010.

2. S. Lynch, Dynamical Systems with Applications using MATLAB, Birkhäuser, 2014.

3. M. L. Abell, J.P. Braselton, Differential Equations with Mathematica, Elsevier Academic

Press, 2004.

4. I. P. Stavroulakis, S.A. Tersian,An Introduction with Mathematica and MAPLE, World

Scientific, 2004.

5. L.W. Lamport, LaTeX: A document Preparation Systems, Addison-Wesley Publishing

Company, 1994.

6. H. Kopka, P.W. Daly, Guide to LATEX, Fourth Edition, Addison Wesley, 2004

Detail syllabus of core mathematics courses

a. Advanced Complex Analysis

Unit-I- Schwarz’s Lemma and its consequences, Zeros of certain polynomials, Meromorphic

functions, Essential Singularities and Picard’s Theorem, Analytic Continuation, Monodrmy

Theorem, Poisson integral formula, Analytic continuation via reflexion.

UNIT-II- Infinite sums and infinite product of complex numbers, Infinite product of analytic

functions, Factorization of entire functions, The Gamma functions, The Zeta functions.

UNIT-III- Open mapping theorem and Hurwitz’ theorem, Basic results on univalent functions,

The Riemann mapping theorem (statement only), Area theorem, Biberbachtheorem, Biberbach

Conjecture.

Reference books:

1. S. Ponnusamy, Foundation of Complex Analysis, 2nd edition, Narosa Publishing House.

2. L. R. Ahlofrs, Complex Analysis, McGraw Hill

3. A. S. B. Holland, Introduction to the theory of entire functions, Academic Press.

b. Fractional Calculus and Geometric Function Theory

Fractional derivatives and Integrals,application of fractional calculus, Laplace transforms of

fractional integrals and fractional derivatives, fractional ordinary differential equations,

fractional integral equations, Initial value problem of fractional differential equations.

Univalence in Complex plane.Area theorem.Growth, covering and distortion results.Starlike and

Convex functions.Starlike and Convex functions of order α. Alpha convexity. Close to

convexity, spirallikeness and Φ-likeness in unit disk.

Subordination.Application of subordination principle.First and second order differential

subordination.Briot-Bouquet differential subordinations.Briot-Bouquet application in Univalent

function theory.

Reference books:

1. L. Debnath, D. Bhatta, Intgral Transforms and Special Functions, CRC press, 2010.

2. I. Graham, G. Kohr, Geometric function theory in one and higher dimensions, Marcel Dekker,

2003.

3. S. S. Miller, P. T. Mocanu, Differential Subordinations theory and Applications, Marcel

Dekker, 2000.

c. Mathematical Programming

Linear programming, duel Simplex method. Non-linear programming techniques:

classical optimization techniques: Lagrangian method and Kuhn-Tucker condition’s.

Quadratic Programming: Wolfe’s method and Beal’s method

Separable programming, Geometric programming, Fractional programming, Dynamic

programming.

Optimization by Direct Search and Gradient methods: one-dimensional search methods,

multi-dimensional search methods, constrained optimization methods.

Reference books:

1. S.D. Sharma: Non-linear and Dynamic programming, KedarNath Ram Nath& Co.

Meerut ;

2. S.M. Sinha: Mathematical programming: Theory and Methods, Elsevier-2006.

d. Dynamical Systems

Non-linear Systems- local analysis: the fundamental existence-uniqueness theorem, The flow

defined by a differential equation, Linearization, The stable manifold theorem, The Hartman-

Grobman theorem, Stability and Liapunov functions, Saddles, Nodes, Foci, and Centers.

Non-linear Systems- global analysis: Dynamical systems and global existence theorem, Limit

sets and Attractors, Periodic orbits, Limit Cycles, and Seperatrix cycles, the Poincare map, the

stable manifold theorem for periodic orbits, the Poincare-Bendixon theory in R2, Lineard

Systems, Bendixon’s Criteria.

Discrete dynamical systems: finite dimensional maps, limit sets, Stability, Invariant manifolds,

Runge-Kutta methods: the framework, linear decay, Lipschitz conditions, Dissipative systems,

Generalized dissipative systems, Gradient system.

Reference books:

1.L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, 2006.

2. A.M. Stuart and A.R. Humphries. Dynamical Systems and Numerical Analysis, Cambridge

University Press, 1998.

3. S. Lynch, Dynamical Systems with applications using MATLAB, Birkhause press, 2004.

e. Complex Dynamics

Iteration of a Mobius transformation, attracting, repelling and indifferent fixed points.Iterations

of R(z) = z2, z

2+c, z + . The extended complex plane, chordal metric, spherical metric, rational

maps, Lipschitz condition, conjugacy classes of rational maps, valency of a function, fixed

points, Critical points, Riemann Hurwitz relation.

Equicontinuous functions, normality sets ,Fatou sets and Julia sets, completely invariant sets,

Normal families and equicontinuity, Properties of Julia sets, exceptional points Backward orbit,

minimal property of Julia sets.

Julia sets of commuting rational functions, structure of Fatou set, Topology of the Sphere,

Completely invariant components of the Fatouset , The Euler characteristic, Riemann Hurwitz

formula for covering maps, maps between components of the Fatou sets, the number of

components of Fatou sets, components of Julia sets.

Reference books:

1. A. F. Beardon, Iteration of rational functions, Springer Verlag , New York, 1991.

2. L. Carleson and T . W. Gamelin, Complex dynamics, Springer Verlag, 1993.

3. S. Morosawa, Y. Nishimura, M. Taniguchi, T. Ueda, Holomorphic dynamics,

Cambridge University Press, 2000.

4. X. H. Hua, C. C. Yang, Dynamics of transcendental functions, Gordan and Breach Science

Pub. 1998.

f. Computational Fluid Dynamics (Note:- Course comprise of 60% theory and 40% practical)

Theory: Finite difference method: Implicit method for pipe and starting flow in a channel,

Solution of Bihormonic equation.

Finite volume method: Solution of first and second order ordinary differential equations and

Laplace equation.

Finite element method: Linear and quadratic interpolation, Two-dimensional interpolation,

Application to diffusion equation and viscous flow in a rectangular duct.

Grid generation: Grid generation by conformal mapping, algebraic mapping and solution of

elliptic partial differential equations, application to transport equations in non-rectangular cavity.

Practical: Practical on numerical computation of the problems based on the above methods

using a programming language or software.

Reference books:

1. Chuen-Yen Chow, An Introduction to Computational Fluid Mechanics, John Wiley & Sons,

New York, 1979.

2. Clive A. J. Fletcher, Computational Techniques for Fluid dynamics, Vol. 1 and 2, Springer-

Verlag Berlin Heidelberg New York, 2nd

Edition, 2003.

3. T. K. Bose, Numerical Fluid Dynamics, Narosa Publishing House, New Delhi, 1997.

Frederick S. Sherman, Viscous Flow, McGraw-Hill Publishing Company, New York, 1990.

Detail syllabus of specialized mathematics courses

a. Holomorphic Dynamics

Escaping points of polynomials, Basic properties of filled Julia set, local behavior near fixed

point, Quadratic polynomials and the Mandelbrot sets, Hausdorff dimension and capacity,

Polynomial like mappings.

Hyperbolic domains, contraction principle, Normal families, Julia set and the Fatou set,

classification theorem, Snail Lemma of Fatou set, Value distribution theory, The second main

theorem and direct consequence, Nevanlinna inequality, basic properties of the Julia set,

The first fundamental theorem, Rescaling Lemma of Zalcman, singular values and Fatou

components, the second fundamental theorem, topological properties of Fatou set and the Julia

set, special classes of entire functions, Escaping points of transcendental entire and meromorphic

functions.

Reference books:

1. S. Morosawa, Y. Nishimura, M. Tanuguchi, T. Ueda, Holomorphic Dynamics, Cambridge

studies in Advanced mathematics, Cambridge University Press, 2000.

2. X. H. Hua, C. C. Yang, Dynamics of transcendental functions, Gordon and Breach Science

Pub. 1998.

3. W. Bergweiler, Iteration of meromorphic functions, Bull.Amer. Math. Soc. 29(1993), 151-

188.

4. J. Milnor, Dynamics in one complex variable, (Third Ed. )Annals of Mathematics studies, No.

160, Princeton University Press, 2006.

b. Geometric Function Theory

Univalence in Complex plane, Schwarz Lemma, Normal families & applications, Riemann

mapping theorem, Area theorem, Growth, covering and distortion theorems, Biberbach

Conjecture, Starlike and Convex functions, Close-to-Convex functions.

Radius of starlikeness, the rotation theorem, Subordination, Majorization, Convolutation of

convex functions, Criteria of univalence functions.

Subordination.Application of subordination principle.First and second order differential

subordination.Briot-Bouquet differential subordinations.Briot-Bouquet application in Univalent

function theory.

Reference books:

• P. L Duren, Univalent functions, Springer-Verlag, New York 1983.

• I. Graham and G. Kohr, Geometric function theory in one and higher dimensions, Marcel

Dekker, 2003.

• S. S. Miller and P. T. Mocanu, Differential Subordinations theory and Applications,

Marcel Dekker, 2000.

c. Applied Dynamical Systems

Dynamical systems: continuous and discrete. Poincare surface of section, limit sets, attractors,

basin of attraction, sensitive dependence on parameters and initial conditions.

One-dimensional maps, fixed points, periodic orbits and their stability, logistic map, bifurcations,

Universal scaling of period doubling bifurcations in quadratic maps, other types of bifurcations

in one-dimensional maps, Lyapunov exponents.

Quasi-periodicity, circle map, Hopf bifurcations, Hamiltonian systems, Symplectic structure,

Canonical transformations, Integrable systems, Pertubation of integrable systems, Kolmogorov-

Arnold-Moser theorem, Resonant tori, Chaotic transitions, intermittency, crises.

Time series of discrete dynamical system: system evolution and attractor, correlation integral,

estimators of correlation integral, Takens theorem.

Reference books:

• E. Ott, Chaos in dynamical systems. Cambridge University Press, 2002.

• K. Alligood, T.D. Sauer, J.A. Yorke, Chaos: An introduction to Dynamical Systems,

Springer Verlag, 1996.

• S.H. Strogatz, Nonlinear dynamics and chaos. Perseus books, 2000.

• R.L. Devaney, An introduction to Chaotic dynamical systems. Addison-Wesley, 2003.

• Wiggins, Introduction to applied nonlinear dynamical systems. Springer Verlag, 1997.

• A.M. Stuart and A.R. Humphries. Dynamical Systems and Numerical Analysis,

Cambridge University Press, 1998.

d. Mathematical Biology

Introduction: Goals and Challenges of mathematical modeling in biology and ecology.

Idealization and general principle of model building, deterministic and stochastic models,

different types of mathematical models and differential and difference equations as relevant

mathematical techniques, complex network dynamics, biological and ecological examples,

spatial and non-spatial models, time-discrete and time-continuous models.

Population, community and ecosystem dynamics: single species and multi-species

models, competition and facilitation, predator-prey systems, spatially structured models,

metapopulation and metacommunity dynamics, Individual-based models. Modeling applications:

biological invasions, biological pattern formations, species succession, biodiversity maintenance

mechanisms.

Molecular interactions, dynamics and evolution, enzyme kinetics, genome structure and

evolution, biological stoichiometry, bio-informatics tools for dealing genomic sequences.

Reference books:

• J.D. Murray, Mathematical biology: An introduction, Springer, 2007.

• N.F. Britton, Essential mathematical biology, Springer, 2004.

• M. Kot, Elements of mathematical ecology, Cambridge University Press, 2001.

• A. Okubo, S.A. Levin, Diffusion and ecological problems, Springer, 2002.

• S.V. Petrovskii, B.L. Li, Exactly solvable models of biological invasions, CRC Press/

Chapman and Hall, 2005.

• R.W. Sterner, J.J. Elser, Ecological stoichiometry: the biology of elements from

molecules to the biosphere, Princeton University Press, 2002.

e. Queuing Theory

Description of the Queuing of the Queuing problem, characteristics of Queuing process, Poisson

process and the exponential distribution, Markovian property of the exponential distribution,

stochastic process and Markovian chains. Simple Markovian Queuing models: Birth-death

process, M/M/1, M/M/C, M/M/C/K, M/M/∞ etc.

Advanced Markovian Queuing models: Mx/M/1, M/M

y/1, Erlang models, Priority Queue

discipline, Retrial Queues.

General Arrival or Service Patterns: M/G/1, M/G/C, M/G/∞, G/M/1, G/M/C General models:

G/EK/I, G(K)

/M/I and G/PHK/I and G/G/I

Reference books:

1. D. Gross, J.F. Shortle, J.M. Thompson, G.M. Harries, Fundamental of Queuing Theory,

John Wiley & Sons. Inc. Publication

f. Magnetohydrodynamics (MHD) 4cr.(3-1-0)

Qualitative Overview of MHD: Basic concepts of Magnetohydrodynamics and its applications,

Maxwell’s equations, Frame of reference, Ampere’s Law, Faraday’s Law in differential form, Lorentz

force, Electromagnetic body force, Ohm’s law for a moving conductor, Hall current, Conduction

current, Kinematic aspect of MHD, Magnetic Reynolds number.

Governing Equations of MHD:Navier-Stokes equation incorporating of Lorentz forces in

different co-ordinate systems, MHD heat transfer, magnetic field equations, induced

magnetic field equations with high and low magnetic Reynolds number.

Boundary Layers: Hartmann flow, Hartmann boundary layer, Hartmann flow between

two planes, Couette flow, Couette boundary layer, Couette flow between two planes, MHD

Stoke’s flow, MHD Rayleigh’s flow, Aligned flow in two dimensional MHD flow, MHD flow in

rotating medium.

Reference books :

1. T.G.Cowling, Magnetohydrodynamics, Interscience Publishers New York, 1957.

2. J.A. Shercliff, A Text Book of Magnetohydrodynamics, Pergamon Press, Oxford, 1965.

3. S.I. Pai, Magnetohydrodynamics and Plasma Dynamics, Springer Verlag, New York, 1962.

4. K. R. Cramer and S. I. Pai, Magnetofluid Dynamics for Engineers and Applied Physicists,

McGraw Hill, New York, 1973.

g. Thermal Instabilities and Methods 4cr.(3-1-0)

Mechanism of instability, various types of convection instabilities; Rayleigh-Benard

convection, oberbeck convection, magnetoconvection, Marangoni convection, magneto-

Marangoni convection, magnetic fluid convection, electroconvection, double diffusive

convection, cross diffusion convection, biconvection. Boundary conditions.

Techniques to solve linear and nonlinear instability problems; Galerkin technique,

perturbation techniques involving regular and singular perturbations, truncated

representation of Fourier series (finite amplitude technique), numerical techniques,

moment method, energy method, power integral technique, spectral method.

1. P.G. Drazin, W.H. Reid, Hydrodynamic Stability, Ed. 2nd, Cambridge University Press, 2004. 2. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford University Press,

1961. 3. D.A. Nield, A. Bejan, Convection in Porous Medium, Springer, 2006. 4. I.S. Shivakumara, M. Venkatachalappa, Advances in Fluid Mechanics, Vol 4, Tata McGraw-

Hill, 2004.

h. Celestial Mechanics LTP: 3+1+0

UNIT-I: Introduction, Kepler’s Laws of Planetary Motion, Newton’s law of gravitation, Central

force motion, Integral of energy, Differential equation of orbit, Inverse square force, Geometry

of orbits, Two body problem, Motion of center of mass, Relative motion, Earth bound satellite

circular orbit, Classical orbital elements, Position in elliptic orbit, Position in parabolic orbit and

Position in hyperbolic orbit. (15 Lectures)

UNIT-II: N-body problem, Mathematical formulation of N-body problem, Integrals of motion,

The Virial theorem, Equation of relative motion, Three body problem, Stationary solution of

three body problem, Restricted three body problem- formulation and its solution, Restricted three

body problem, Stability of motion near Lagrangian points.(15 Lectures)

UNIT-III: Theory of perturbations, Variation of parameter, Properties of Lagrange’s brackets,

Evaluation of Lagrange’s brackets, Solution of the perturbation equations, Perturbation function,

Earth-Moon system, Potential due to an oblate spheroid, Perturbations due to oblate planet,

Perturbation due atmospheric drag, Perturbation due to solar radiation. (15 Lectures)

Recommended Reading:

1. Introduction to Celestial Mechanics by S. W. McCuskey,Addison-Wesley Publishing

Company, 1963.

2.Solar System Dynamics by C. D. Murray and S. F. Dermott, Cambridge University

Press, 2000.

3.An Introduction to Celestial Mechanics by F. R. Moulton, the MacMillan Company,

1914.

4. Theory of orbits. The Restricted problem of three bodies by V. Szebehely, New York

Academic Press, 1967.

5. Classical Mechanics by K. Sankara Rao, PHI Learning Pvt. Ltd., 2009