CSIR-MATH-U4 — Probability, Statistics, Linear Models, and Operations Research
CSIR-NET Unit 4: Probability theory, statistical inference, regression and linear models, multivariate methods, sampling theory, design of experiments, and operations research (LPP, queues, reliability).
📋 Course Learning Outcomes
On successful completion of this course, the learner will be able to:
- Apply axiomatic probability and Bayes' theorem to compute event probabilities [Apply]
- Analyze standard discrete and continuous distributions; compute moments and MGFs [Analyze]
- Apply CLT and laws of large numbers to derive asymptotic distributions [Apply]
- Construct MLE / MoM estimators and verify properties (unbiased, consistent, efficient) [Create]
- Evaluate hypotheses using NP lemma, likelihood-ratio, and chi-square tests [Evaluate]
- Solve LP problems via simplex/duality; analyze queues and reliability systems [Apply]
📚 Chapters
- Probability Fundamentals & Bayes
- Random Variables & Distributions
- Expectation, Moments & Generating Functions
- Inequalities, LLN, CLT & Convergence
- Markov Chains
- Sampling Distributions & Order Statistics
- Estimation Theory — Properties & Methods
- Hypothesis Testing — MP, UMP, LRT
- Nonparametric Tests, Chi-square & Bayesian Methods
- Linear Models, Gauss–Markov & ANOVA
- Regression — Linear, Logistic, Diagnostics
- Multivariate Methods — MVN, PCA, Discriminant
- Sampling Theory & Design of Experiments
- Operations Research — LPP, Queues, Reliability
TutorDA LMS
- المعلم: Student One
- المعلم: Student Two
CSIR-MATH-U3 — ODE, PDE, Numerical Analysis, Calculus of Variations
CSIR-NET Unit 3: Ordinary and Partial Differential Equations, Numerical Analysis, Calculus of Variations, Linear Integral Equations, and Classical Mechanics (Lagrangian and Hamiltonian formulations).
📋 Course Learning Outcomes
On successful completion of this course, the learner will be able to:
- Apply Picard's existence-uniqueness theorem to verify well-posedness of IVPs [Apply]
- Solve linear ODEs of higher order using variation of parameters and Green's functions [Apply]
- Classify second-order PDEs and solve via separation of variables [Analyze]
- Apply iterative methods (Newton, Gauss-Seidel, RK) with error analysis [Apply]
- Solve Euler-Lagrange equations and standard variational problems [Apply]
- Evaluate Fredholm and Volterra integral equations using resolvent kernels [Evaluate]
📚 Chapters
- First-order ODEs — Existence, Uniqueness
- Linear ODEs & Variation of Parameters
- Sturm–Liouville Problems & Green's Function
- First-order PDEs — Lagrange & Charpit
- Classification of Second-order PDEs
- Separation of Variables — Heat, Wave, Laplace
- Roots of Equations — Iteration & Newton
- Linear Systems — Gauss & Gauss–Seidel
- Interpolation — Lagrange, Hermite, Splines
- Numerical Differentiation & Integration
- Numerical ODEs — Picard, Euler, RK
- Calculus of Variations — Euler–Lagrange
- Linear Integral Equations — Fredholm & Volterra
- Classical Mechanics — Lagrangian & Hamiltonian
TutorDA LMS
- المعلم: Student One
- المعلم: Student Two
CSIR-MATH-U2 — Complex Analysis, Algebra, and Topology
CSIR-NET Unit 2: Complex Analysis (analytic functions, contour integration, residues, conformal maps), Abstract Algebra (groups, Sylow, rings, fields, Galois theory), Combinatorics & Number Theory, and General Topology.
📋 Course Learning Outcomes
On successful completion of this course, the learner will be able to:
- Apply Cauchy-Riemann equations to identify analytic functions and harmonic conjugates [Apply]
- Evaluate contour integrals using Cauchy's theorem, integral formula, and residues [Evaluate]
- Analyze ring structure (PID/UFD/ED) and field extensions including Galois groups [Analyze]
- Apply Sylow's theorems to classify finite groups of small order [Apply]
- Analyze topological properties (compactness, connectedness, separation) of standard spaces [Analyze]
- Apply combinatorial and number-theoretic identities to CSIR-NET style problems [Apply]
📚 Chapters
- Complex Numbers & the Complex Plane
- Analytic Functions & Cauchy–Riemann
- Power Series & Elementary Functions
- Contour Integrals & Cauchy's Theorem
- Cauchy Integral Formula & Consequences
- Taylor & Laurent Series
- Residue Calculus
- Conformal Mappings & Möbius Transformations
- Combinatorics & Number Theory
- Groups: Basics & Homomorphisms
- Sylow Theory & Permutation Groups
- Rings, Ideals, UFD / PID / Euclidean
- Polynomial Rings, Fields & Galois Theory
- Topology — Basis, Compactness, Connectedness
TutorDA LMS
- المعلم: Student One
- المعلم: Student Three
- المعلم: Student Two
CSIR-MATH-U1 — Analysis and Linear Algebra
CSIR-NET Unit 1: Real Analysis (sets, countability, sequences, series, continuity, differentiability, integration, multivariable calculus, metric spaces, bounded variation, Lebesgue) and Linear Algebra (vector spaces, matrices, rank, eigenvalues, canonical forms, inner products, quadratic forms).
📋 Course Learning Outcomes
On successful completion of this course, the learner will be able to:
- Apply the order-completeness of R and countability arguments to set-theoretic problems [Apply]
- Analyze convergence of sequences and series using standard tests [Analyze]
- Evaluate continuity, differentiability, and Riemann integrability of real functions [Evaluate]
- Apply uniform convergence to justify term-by-term differentiation and integration [Apply]
- Analyze metric-space properties of subsets of R and R^n [Analyze]
- Compute eigenvalues, canonical forms, and quadratic-form classifications [Apply]
📚 Chapters
- Real Numbers, Sets & Countability
- Sequences, Series & Convergence
- Continuity, Differentiation, MVT
- Sequences & Series of Functions
- Riemann Integration
- Bounded Variation & Lebesgue
- Functions of Several Variables
- Metric Spaces, Compactness, Connectedness
- Vector Spaces, Basis & Dimension
- Matrices, Rank & Linear Equations
- Eigenvalues & Cayley–Hamilton
- Canonical Forms — Diagonal, Triangular, Jordan
- Inner Product Spaces & Orthonormality
- Quadratic Forms — Reduction & Classification
TutorDA LMS
- المعلم: Student One
- المعلم: Student Three
- المعلم: Student Two