Research Methodology (MT-PHD-01)

Credit -4/100 Marks Duration: 4 Hrs/ Week

UNIT – I: Foundations of Research

Meaning and Objective of Research, Types of Research, Data Sources: Primary and Secondary, Data Collection, Generation, Arrangement, and Processing

UNIT-II: Basic Computer Applications

Basic computer knowledge, Features and applications related to presentation of text in suitable format and saving the data for future applications. Use of word processing, Practical knowledge of MS Word to type the script, insert tables, figures and graphs, plotting of graphs in excel, Preparation of power point presentations based on the topic of research. Insertion of figures, graphs, charts in presentation. Use of spreadsheet and database software, Preparation of scientific posters for presentations, Internet and its application: Email, WWW, Web browsing, acquiring technical skills, drawing Inferences from data, Cloud computing. Introduction to LaTeX programming for scientific writing, document structuring, mathematical typesetting, insertion of images, tables, references and preparation of research articles, theses and presentations using Beamer.

UNIT-III: Quantitative methods, Statistics and application of Computer in statistics

Measures of Central tendency and Dispersion. Probability distribution- Normal, Binomial and Poisson distribution. Parametric and non-parametric statistics. Confidence interval, Errors. Quantitative Techniques: Levels of significance, Regression and Correlation coefficient. Statistical analysis and fitting of data; Chi‐Square Test, Association of Attributes t‐Test ANOVA Standard deviation, Co‐efficient of variations. Open-source software for quantitative and statistical analysis.

UNIT-IV: Documentation and scientific writing:

Results and Conclusions, Preparation of manuscript for Publication of Research paper, Presenting a paper in scientific seminar, Thesis writing. Structure and Components of Research Report, Types of Report: research papers, thesis, Research proposal, Research Project Reports, Pictures and Graphs, citation styles, writing a review of paper, Bibliography.

Research and Publication Ethics (MT-PHD-02)

Credits-2/100 Marks Duration: 2 Hrs/ Week

UNIT-I: Philosophy, Ethics, and Scientific Conduct

1. Introduction to Philosophy: Definition, nature, and scope; concept, branches.

2. Ethics: Definition, moral philosophy, nature of moral judgments and reactions. Ethics with respect to science and research; intellectual honesty and research integrity.

3. Scientific Misconduct: Falsification, fabrication, and plagiarism (FFP).

4. Redundant Publications: Duplicate and overlapping publications, salami slicing. Selective reporting and misrepresentation of data.

UNIT-II: Publication Ethics

Publication Ethics: Definition, introduction, and importance. Violation of publication ethics, authorship, and contributor ship.

2. Basic Practices and Standards: Setting initiatives and guidelines (COPE, WAME, etc.), conflict of interest.

3. Publication Misconduct: Definition, concept, problems leading to unethical behaviour, identification of publication misconduct, complaints and appeals.

4. Predatory Publishers and Journals.

Practice

1. Open Access Publishing:

A. Publications and initiatives; SHERPA/RoMEO for copyright policies; tools to identify predatory publications (e.g., SPPU software);

B. Journal suggestion tools (JANE, Elsevier Journal Finder, Springer Suggester).

2. Publication Misconduct:

A. Group discussion: Subject-specific ethical issues, FFP, authorship, conflict of interest, complaints and appeals; examples and fraud cases (India and abroad).

B. Software tools: Use of plagiarism checkers (Turnitin, Unkind, etc.).

3. Database and Research Metrics:

A. Databases: Indexing databases, citation databases (Web of Science, Scopus, etc.).

B. Research Metrics: Journal impact factor (JCR), SNIP, SJR, IPP, Cite Score, h-index, g-index, i10 index, altimetric.

Major Paper

Bicomplex Numbers (MT-PHD-03)

Credit-4/100 Marks Duration: 4 Hrs/ Week

UNIT I – Foundations of Complex and Multicomplex Numbers (Compact Overview): Concise revision of classical complex numbers, covering basic algebra, geometry, analytic functions, and Cauchy–Riemann equations. It then introduces multicomplex systems, explaining how Bicomplex numbers arise as a natural extension of the complex field.

UNIT II – Algebraic & Topological Structure of Bicomplex Space: Introducing the idempotent basis and the decomposition of BI complex numbers into two auxiliary complex components. It discusses zero divisors, the null cone, conjugations, and BI complex modulus, and then introduces the induced topology of BI complex space, including open and closed discus regions.

UNIT III – Bicomplex Holomorphic Functions and Analytic Theory: The theory of Bicomplex differentiability and holomorphy. Students learn BI complex Cauchy–Riemann equations, the structure of Bicomplex regular functions, and the formulation of power series in idempotent form. Emphasis is placed on convergence behaviour, analytic regions, BI complex entire functions, and the relationship between bicomplex holomorphy and two classical holomorphic functions.

UNIT IV – Bicomplex Functional Analysis: Spaces and Operators: Introduces Bicomplex normed, Banach and Hilbert spaces, with special attention to hyperbolic-valued norms and bicomplex inner products. It covers the bicomplex Schwarz inequality, orthogonality, projections, and the Riesz representation theorem. The theory of bicomplex linear and bounded operators is developed, including spectrum, eigenvalues, and dual spaces. This forms the analytical foundation for advanced Bicomplex research.

UNIT V – Bicomplex Sequence Spaces and Generalized Convergence: Sequence spaces defined over Bicomplex numbers, including Bicomplex versions of classical spaces such as ∫p, C, and C,C₀

Students study entire Bicomplex sequences, Kothe–Toeplitz duals, and the sixteen dual structures emerging from idempotent decomposition. It also introduces advanced convergence concepts such as bicomplex statistical convergence, λ-statistical convergence, I-convergence, and deferred statistical convergence.

Minor/Elective Sequence Spaces, Summability Theory & Statistical Convergence (MT-PHD-04)

Credits: 4/100 Marks, Duration: 4 Hrs/ Week

UNIT I – Basics of Sequences and Classical Convergence: Introduces sequences, subsequences, boundedness, monotonicity, and ordinary limits. It revisits essential theorems such as Bolzano–Weierstrass and Monotone Convergence, forming the analytical foundation needed for modern convergence concepts.

UNIT II – Statistical Convergence: Definitions and Core Ideas: Natural density and the notion of statistical convergence, explaining how it generalizes ordinary convergence. Statistical Cauchy sequences, regularity and examples highlighting the difference between statistical and classical convergence are developed.

UNIT III – Statistical Limit Points and Completeness Properties: Studies statistical limit points and cluster points, exploring their relationship with ordinary limits. It includes statistical analogues of classical completeness theorems and describes how non-thin subsequences determine statistical behaviour.

UNIT IV – Statistical Monotonicity, Boundedness and Behaviour: Examines statistically monotone and statistically bounded sequences, showing how monotonicity “almost everywhere” shapes convergence. It includes characterizations, decomposition results, and counterexamples important for understanding non-classical behaviour.

UNIT V – Sequence Spaces and Summability Methods: Introduces classical and generalized sequence spaces such as C,C₀ and l∞ along with summability methods like Cesàro and matrix transformations. Students study dual spaces and how statistical convergence fits into summability theory.