Syllabus for
Master of Philosophy (Mathematics)
Academic Year  (2018)

This course is intended to assist students in planning and carrying out research projects.  The students are exposed to the principles, procedures and techniques of implementing a research project.

• foster a clear understanding about research design that enables students in analyzing and evaluating the published research.
• acquire sound knowledge in theoretical and quantitative methods.
• analyze and interpret data for evaluating alternative perspectives.
• understand the components and techniques of effective report writing.
• obtain necessary skills in preparing scientific documents using LaTeX..
• employ computers in managing and planning research activities effectively.

An introduction–meaning of research-objectives of research- motivation in research –types of research- research approaches-significance of research-research methods versus methodology-research and scientific method-importance of knowing how research done-research processes-criteria of good research-defining research problem-selecting the problem-necessity of defining the problem-technique involved in defining a problem-Research design- meaning of research design-need for research design-features of good design-different research design-basic principles of experimental design

Measurement and Scaling Techniques- Methods of Data Collection, - processing and Analysis of Data,- Sampling Fundamentals, Testing of Hypotheses - I (Parametric or Standard Tests of Hypotheses), Chi-square Test, Analysis of Variance and Covariance, Testing of Hypotheses - II (Nonparametric or Distribution - Free Test),Multivariate Analysis Techniques.

Interpretation and report writing, technique of report writing-precaution in interpretation-significance- different steps of report writing- layout of research report-oral presentation- mechanics of writing- Exposure to writing tools like Latex/PDF, Camera Ready Preparation

Originality in research, resources for research, Research skills, Time management, Role of supervisor and Scholar, Interaction with subject expert,  The Computer: Its Role in Research, Case study interpretation: minimum 5 case studies.

1. C.R.Kothari, Research Methodology- Methods and Techniques, 2nd ed., Vishwa Prakashan Publications, New Delhi, 2006.

2. R. Pannerselvam, Research methodology, 3rd Printing, New Delhi, PHI 2006.

3. Santosh Gupta,  Methodology And Statistical Techniques, 1st ed., Deep and Deep Publications, 2004.

4. E. B. Wilson Jr., An Introduction to scientific research, 1st ed., (Reprint), New York: Dover publications Inc, 2000.

5. Ram Ahuja, Research Methods, 1st ed., New Delh: Rawat Publications, 2002.

6. Gopal Lal Jain, Research Methodology, 2nd ed., Jaipur: Mangal Deep Publications, 2003.

7. B. C. Nakra and K. K. Chaudhry: Instrumentation, measurement and analysis,2nd ed., New Delhi: Tata McGraw-Hill Education, 2004.

8. S. L. Mayers,  Data analysis for Scientists, Reprint,John Wiley & Sons, 2000.

9. L. Blaxter, C. Hughes, M. Tight, How to research, 4th ed., McGraw-Hill, 2010.

10. J. Bell, Doing your research project, 5th ed., McGraw-Hill, 2010.

11. A. Thomas, J. Chataway, M. Wuyts, Finding our fast-Investigative Skills for Policy and Development, Reprint, SAGE Publications Inc., 2000.

12. P.J.M. Costello,  Effective Action Research: Developing Reflective Thinking and Practice, 2nd ed., Continuum, 2005 (NIAS)

13. B. Gilham, Case study research methods,1st ed.,Continuum,  2011.

14. S. Kleinman, M.A.Copp, Emotions and fieldwork, Reprint, SAGE Publications Inc., 2000.

15. I. Gregory, Ethics in research, Continuum, 2005 (NIAS)

16. J. Bennet, Evaluation methods in research, Continuum, 2005 (NIAS)

17. D. L. Morgan, Focus groups as qualitative research, Reprint, Sage Pub., 2000 (NIAS)

18. Illingham,Jo., Giving presentations, OUP, 2003 (NIAS)

19. M. Denscombe, The good research guide, Reprint, Viva, 2000 (NIAS)

20. D. Ezzy, Qualitative analysis, Routledge, 2002 (NIAS)

21. M. Q. Patton, Qualitative evaluation and research methods, Reprint, Sage Pub, 2000 (NIAS)

22. J. Kirk, Reliability and validity in qualitative research, Rerpint, Sage Pub, 2000 (NIAS)

This course provides an opportunity to explore interlink with Mathematics and various philosophical areas, particularly metaphysics, epistemology and philosophy of science. 

• understand Mathematics as the foundation of Philosophy
• discuss the theories of Realism, Structuralism, Formalism, Constuctivism etc
• have ability to philosophize Mathematical learning
• understand the particular philosophical issues that arise in the context of specific branches of Mathematics

Mathematical Realism, Mathematical Anti-Realism, Critique of Platonism, Critique of Anti-Platonism, Okham's Critique of Platonism, The Strong Epistemic Conclusion, The Metaphysical Conclusion

Mathematics: Science of Forms, Mathematics is not mere Logic, Formal Rules, Rules and Forms of Representation, Axiomatization and Structures, Philosophy of Sets, Ordinal, Cardinal and Two Kinds of Infinite, Intuition and the Theory of Pure Manifolds, Totalities and Quantifiers.

Varieties of Constructivism, Constructivism in the 19th Century, Intuitionism, Formal Intuitionistic Logic, Russian Constructivism, Predicativism, Finitism

1. A. D. Irvine, ed. Philosophy of Mathematics. Handbook of the Philosophy of Science series, Amsterdam: North Holland, 2009.

2. D. Bostock, Philosophy of Mathematics: An Introduction. Oxford: Wiley-Blackwell, 2009.

3. J. Azzouni, Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences. Reprint, Cambridge, UK: Cambridge University Press, 2000.

4. L. Horsten, Philosophy of Mathematics  in The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2008.

5. M. Balaguer, Platonism and Anti-Platonism in Mathematics. Reprint, New York: Oxford University Press, 2000.

6. M. Colyvan, The Indispensability of Mathematics. New York: Oxford University Press, 2001.

7. S. Shanker, Philosophy of Science, Logic and Mathematics in the Twentieth Century, Reprint, Psychology Press, 2003.

8. M. Schirn, ed. The Philosophy of Mathematics Today. Reprint, Oxford: Clarendon, 2000.

9. P. Kitcher, The Nature of Mathematical Knowledge. Reprint, New York: Oxford University Press, 2000.

10. P. Maddy, Realism in Mathematics. Reprint, Oxford: Clarendon, 2000.

11. S. Shapiro, Thinking about Mathematics: The Philosophy of Mathematics. Oxford: Oxford University Press, 2000.

This course emphasize the mathematics and analysis methods used in the study of linear and nonlinear natural convective instability problems in fluid mechanics, boundary layer problems and basic concepts of mathematical modeling of nano-liquids.

• Have a high level of technical competence in fluid mechanics.

• Analytically and numerically solve Navier-Stokes equations for a couple of simple fluid flow problems.

• Analyze and understand the nonlinear flows.

• Understand the principles underlying sustainable boundary layer flow.

• Apply fluid mechanics principles to the model the nano-liquids.

Linear and non-linear natural convective instabilities:  Governing equations for Rayleigh – Bénard and Bénard – Marangoni convections in Newtonian and non-Newtonian liquids for different velocity and thermal boundary combinations.  Local and Global nonlinear instability analysis – Lorenz and Ginzburg – Landau models for stress-free isothermal boundary comobinations in the case of Newtonian and non-Newtonial liquids.  Critical points of the linear autonomous system in the case of linearized Lorenz model.  Illustration of Energy Method:  The diffusion equation, Navier-Stokes equations and Bénard Problem in Newtonian liquids for stress-free isothermal boundary.

Boundary Layer:  Boundary layer approximation, governing equations for Blasius boundary layer equation, Stretching sheet problems (horizontal, Vertical and inclined) and Falkner-Skan family of equations, flow past a wedge and a flat plate, liquid thin film flow. Dispersions: Molecular Dispersion, Convective Dispersion, Different approaches to study dispersion – Taylor Approach, Aris Approach, Barton Approach, Gill – SankaraSubramanian Approach, Lighhill Approach, Random Walk Approach, Smith’ Delay-Diffusion Approach. Diffusion Equation ( conservation of Species).

Mathematical Modelling of Nanoliquids:  Buongiorno Model (Two phase model) : Nanoparticle/Fluid Slip : Inertia, Brownian Diffusion, Thermophoresis, Diffusiophoresis, Magnus Effect, Fluid Drainage, Gravity, Relative importance of the Nanoparticle Transport Mechanisms.  Conservation Equation for two phase Nanoliquids : The Continuity equation, The Momentum equation and The energy equation.

This course concerns with the principal concepts of graph operations such as Cartesian product, Normal product, Lexicographic product and so on. Algebraic properties of graphs such as factorization and cancellation are studied in detail and important classes of subgraphs are explored. This course is aimed at providing a sound training in proof techniques and to lay a strong foundation for research in Graph Theory.

• understand dichotomy between the structure of products and their subgraphs.
• be aware of the relationship between graph parameters of the product and factors.
• gain mastery in employing proof techniques. 

Automorphisms, Vertex Transitivity, Graph Invariants, Hypercubes, Isometric Subgraphs, Median Graphs.

Introduction to Product of Graphs, Commutativity, Associativity and Multiple Factors, Projections and Layers, Classification of Factors.

The Cartesian Product, Prime Factor Decompositions, Cartesian Product and Its Group, Transitive Group Action on Products, Cancellation, S-Prime Graphs.

The Strong Product, Basic Properties and S-Thin Graphs, Cliques and the Extraction of Complete Factors, Unique Prime Factorization for Connected Graphs, Automorphisms.

The Direct Product, Nonuniqueness of Prime Factorization, R-Thin Graphs, The Cartesian Skeleton, Factoring Connected, Nonbipartite, R-Thin Graphs, Factoring Connected Nonbipartite Graphs, Automorphisms, Applications to the Strong Product.

The Lexicographic Product, Basic Properties, Self-Complementarity and Cancellation Properties, Commutativety, Factorizations and Nonuniqueness, Automorphisms.

1. B. Bollobás, Modern Graph Theory, Rerpint, New York: Springer-Verlag, 2001.
2. C. Berge,Graphs. Reprint, Oxford: North-Holland, 2000.
3. G. Chartrand, Introductory Graph Theory. Reprint, New York: Dover Publications, 2000.
4. J.A. Bondy, and U.S.R. Murty, Graph Theory with Applications. Reprint, Oxford: North-Holland, 2000.
5. O.  Øre, Theory of Graphs. New York: Reprint, AMS Coloquium Publications 38, 2000.
6. R. Diestel, Graph Theory. Reprint, New York: Springer-Verlag, 2000.
7. R.J.Gould, Graph Theory. Reprint, San Francisco: Benjamin/Cummings, 2000.
8. R.J. Trudeau, Introduction to Graph Theory. Reprint, New York: Dover Publications, 2000.
9. R.J. Wilson, and J.J. Watkins, Graphs: An Introductory Approach. Reprint, John Wiley & Sons, 2000.

.

Each candidate shall work under the supervision of a guide. Specific guiding for the research program/Dissertation may commence from the middle of first semester. The HOD/Coordinator will allot guides to the candidates approved by the Dean in the middle of the first semester depending upon the area of specialization.   

Submission of Dissertation

The title page of dissertation, contents etc. should strictly conform to the format as prescribed by the university and the dissertation (all copies) should carry a declaration by the candidate and certificate duly signed and issued by the guide.  The dissertation should be hard bound. The candidate shall submit five soft bound copies and a soft copy (CD) of his/her dissertation work for assessment.

Adjudication of the M. Phil Dissertation

The dissertation submitted by the candidate under the guidance of the guide will be assessed by two experts (one internal and one external).

The candidates also have to appear for final viva-voce. Assessment based on the viva-voce and the dissertation, along with the assessment of theory papers of both I & II semesters will be considered to declare the results.