COURSE DESCRIPTION


Ph.D. Mathematics or Doctor of Philosophy in Mathematics is Doctorate Mathematics course. Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns and formulate new conjectures. On the completion of doctorate degree scholars should submit their ‘thesis’ and then they deserve the respective degree. The duration of the program is minimum two years from the date of registration (3 years for external candidates) and maximum 5-6 years. The minimum qualifications and other eligibility criteria for admission are the same as for regular full time students. Ph.D. is a broad-based course involving a minimum course credit requirement and research thesis.


Ph.D. Mathematicians Eligibility

  • A Master’s degree is required to gain admission to a doctoral program. In some subjects, doing a Masters in Philosophy (M.Phil.) is a prerequisite to start Ph.D.
  • In cases, where the admission at the M.Phil. has been conducted through an entrance examination and course work has been prescribed at the M.Phil. level, such M.Phil. candidates when admitted to the Ph.D. programs shall not be required to undertake entrance examination or course work and it shall be considered to have complied with the UGC (Minimum standards and procedure for award M.Phil./Ph.D. Degree).
  • In other cases, where a candidate has done M.Phil. from one university and moves to another university for Ph.D., the new university may give credit and exempt for the course work done in the previous university. However, such a candidate will have to appear in the entrance test as applicable to a fresh candidate directly joining Ph.D. This procedure will apply in case of those candidates who have also obtained Ph.D. degree from abroad.
  • For some prestigious Universities, a candidate is required to qualify the all India level examination such as ‘National Eligibility Test’ (NET) for Lectureship conducted by University Grants Commission. Candidates appearing in the final year of qualifying degree examination are eligible to apply. However, they must submit attested copies of qualifying degree certificates/final transcripts.
  • Admission is offered based on an interview held usually a month before the commencement of the semester for which admission is sought. The interview may be supplemented by a written test, if necessary.
  • Normally all candidates in Ph.D. program are required to be resident on campus. Candidates sponsored from reputed research organizations, who are conducting their Ph.D. work co-advised by a reputed guide at their organization, may be eligible to complete their Ph.D. off campus. Only for such sponsored candidates, the minimum residency requirement is 1 semester, during which they will need to complete the course requirements. However, frequent visit to the Institution where admission has been taken is still normally required.

Ph.D. Mathematics Course Suitability

  • They should have the ability to approach problems in an analytical and rigorous way and to formulate theories and apply them to solve problems.
  • Other essential skills are ability to present mathematical arguments and conclusions from them with accuracy and clarity.
  • Advanced numeracy and the ability skills to handle and analyse large quantities of data, clear, logical thinking also is necessary for it.
  • If one who needs a Ph.D. for promotion in academic career, hike in salary, or stable job in academics then they can join this course. Such candidates should take admission in State-run Universities and part time Ph.D.'s courses.

How is Ph.D. Mathematics Course Beneficial?

  • Doctorate Degree enables one to keep pace with the expanding frontiers of knowledge and provides research training relevant to the present social and economic objectives of the country.
  • It learns to write a good research report and acquires the skill of presenting data in graphical form.
  • They can also go for the accountancy and business services, banking, investment and insurance, government and public administration.
  • They can become teacher and lecturer in schools and colleges/universities respectively; for becoming lecturer in colleges they should pass NET exam and the UGC guidelines.

COURSE ELIGIBILITY


A Master’s degree is required to gain admission to a doctoral program. In some subjects, doing a Masters in Philosophy (M.Phil.) is a prerequisite to start Ph.D.
  • Ph.D. Mathematicians Syllabus

    Syllabus of Mathematicians as prescribed by various Universities and Colleges.

       Paper I (Research Methodology)

    Sections

    Subjects of Study

    A

    Note: Common to all faculty

     

    1. Introduction to Research Methodology: Meaning of Research, Objectives of Research, Motivations in Research, Types of Research, Research Approaches, Significance of Research, Research Methods v/s Methodology, Research and Scientific Methods, Research Process, Criteria of Good Research.
    2. Defining the Research Problem: What is Research Problem? Selecting the Problem, Necessity of and Techniques in defining the problem.
    3. Research Design: Meaning, Need, Features of Good Design, Concepts, Types. Basic Principles of Experimental Design, Developing a Research Plan.
    4. Sample Design: Implication, Steps. Criteria for selecting a sample procedure, Characteristics of Good Sampling Procedure, Types of Sample Design, Selecting Random Samples, Complex random sampling Design.
    5. Measurement and Scaling Techniques: Measurement in Research, Measurement Scales, Sources of Errors in measurement, Tests of Second measurement, Technique of developing Measurement Tools, Meaning of Scaling, Scale Classification Bases, Important Scaling Techniques, Scale Construction Techniques.
    6. Methods of Data Collection: Collection of Primary Data, Observation Method, Interview method, Collection of Data through questionnaire and Schedules, Other methods. Collection of Secondary Data, Selection of appropriate method for data collection, Case Study Method, Guidelines for developing questionnaire, successful interviewing. Survey v/s experiment.
    7. Processing and Analysis of Data: Processing Operations (Meaning, Problems), Data Analysis (Elements), Statistics in Research, Measures of Central Tendency, Dispersion, Asymmetry, and Relationship. Regression Analysis, Multiple correlation and Regression, Partial Correlation, Association in case of Attributes.
    8. Sampling Fundamentals: Definition, Need, Important sampling Distribution, Central limit theorem Sampling Theory, Sandler’s A-test, Concept of Standard Error, Estimation, estimating population mean, proportion. Sample size and its determination, Determination of sample size Based on i) Precision Rate and Confidence level ii) Bayesian Statistics.
    9. Testing of Hypothesis: Meaning, Basic concepts, Flow diagram, Power of a hypothesis test, Important parametric tests, Hypothesis Testing of Means, Differences between Means, Comparing Two related samples, Testing of Proportion, Difference between proportions, comparing variance to hypothesized population variance, Equality of variances of two normal populations, hypothesis testing of Correlation coefficients, Limitations of Tests of hypothesis.
    10. Chi-square test: Applications, Steps, characteristics, limitations.
    11. Analysis of Variance and Covariance: Basic Principles, techniques, applications, Assumptions, limitations.
    12. Analysis of Non-parametric or distribution-free Tests: Sign Test, Fisher-Irwin Test, McNemer Test, Wilcoxon Matched Pair Test (Signed Rank Test), Rank.
    13. Sum Tests: a) Wilcoxon-Mann-Whitney Test b) Kruskal-Wallis Test, One sample Runs Test, Spearman’s Rank Correlation, Kendall’s Coefficient of Concordance, Multivariate Analysis Techniques: Characteristics, Application, Classification, Variables, Techniques, Factor Analysis (Methods, Rotation), Path Analysis

    B

    Faculty of Mathematics

     

    1. Select the area of research
    2. Study of research papers in the relevant area
    3. Analysis of studied research papers
    4. Formulate Model / problem
    5. Components of the model
    6. Critical parameters of each component
    7. Solution methodology of proposed model
    8. Evaluation
    9. Future scope and limitation of model

       Paper II (Scientific Communication)

    Sections

    Subjects of Study

    A

    Note: Common to all faculty

     

    1. Basics of Communication skill
    2. Types of Scientific Communications
    3. Importance of publishing research papers
    4. Publishing Research paper
    5. Writing Review Articles
    6. Preparing and Delivering of Oral and Poster Presentations
    7. Avoiding Plagiarism
    8. Preparing documents for MoUs, Confidentiality Agreements
    9. IUPAC symbols and Terminology for physicochemical quantities and Units, SI prefixes, Fundamental Constants, Standard Abbreviations and Symbols

    B

    Faculty of Mathematics

     

    Exposure on:
    1. Study of general guidelines for authors in journals
    2. Compilation of manuscript
    3. Preparation of Hardcopy and Softcopy version of manuscript
    4. Selection of Journal
    5. Submission of manuscript
    6. Final Submission of paper after review comments
    7. Select an area from emerging methodologies
    8. Plan for an innovative project
    9. Plan for project proposal
    10. Compilation of proposal with data
    11. Selection of funding agency (UGC, AICTE, GUJCOST, DST, IT Ministry, CSIR, etc.
    12. Submission proposal to the agency.
    13. Use of MS-OFFICE, MATLAB, MAPLE, MATHEMATICA for scientific visualization of data.

       Paper III (Faculty of Mathematics)

     

    Subjects of Study

     

    1. Basic Concepts of Real and Complex Analysis: Limits, Continuity, Uniform Continuity, Differentiability, Riemann Integral, Metric space, Sequence and series, Algebra of complex numbers, Analytic functions, Power series, Taylor’s and Laurent’s series, Conformal mapping.
    2. Basic Concepts of Linear Algebra: Vector space, Subspace, Linear dependence, Basis, Linear transformation, Algebra of matrices, Rank of matrix, Determinants, Linear equations, eigen values and eigen vectors, Quadratic forms.
    3. Discrete Mathematics: Partially ordered sets, Lattices, Complete Lattices, Distributive lattices, Complements, Boolean algebra, Elements of Graph Theory, Eulerian and Hamiltonian graphs, Planar Graphs, directed graphs, Trees, Spanning trees, Fuzzy set theory
    4. Differential Equations: First order ODE, singular solutions, initial value problem of first order ODE, and general theory of homogeneous and non-homogeneous linear ODE, variation of parameters.
    5. Basic concepts of probability: Sample space, discrete probability, simple theorems on probability, independence of events, Bayes Theorem. Discrete and continuous random variables, Binomial, Poisson, Uniform, Exponential, Weibull and Normal distributions; Expectation and moments, independence of random variables.
    6. Linear/Non-Linear Programming Basic Concepts: Convex sets. Linear Programming Problem (LPP). Examples of LPP, Hyperplane, open and closed half – spaces. Feasible, basic feasible and optimal solutions, Extreme point and graphical method, K-T conditions.
    7. Operational Research Modelling: Definition and scope of Operational Research, Different types of models, Replacement models and sequencing theory, inventory problems and their analytical structure. Simple deterministic and stochastic models of inventory control, Basic characteristics of queuing system, different performance measures. Steady state solution of Mark ovian queuing models: M/M/1, WW1 with limited waiting space MWC, M/M/C with limited waiting space.

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