# Alpha Plus - Best Coaching For CSIR-NET Maths Preparation

Continuing Education and Certification / by tarun bhalla / 59 views

Alpha Plus got incepted in Delhi in the year 2003 at various locations to cater the needs of Mathematics and Actuarial aspirants. Today Alpha Plus become a name synonym to academic excellence and is a hallmark of success for Mathematics and Actuarial aspirant. Also It has produced excellent results year after year since 2003 and till this year trend is continued. Our commitment towards the success of student is achieved due to the unique academic system & teaching methodologies which completely takes care of Graduation syllabus, other features includes practice test, Special Doubt Clearing Session, Micro & Macro Level analysis to give Regular feedback to students etc. Alpha Plus has a huge pool of good faculties, which ensures that our system is immune to any faculty movement and also we are able to incorporate any change in system immediately. Thus, undisrupted quality teaching has become inherent feature of our academic system. This has already been proven through our unmatched results year after year.

Incorporate the years of experience in formulating a comprehensive learning environment that solidifies the strengths of students and makes them ready for the tough challenge of examinations. Discover and rejuvenate the unique qualities of each student, motivating them to respect their own special abilities and those of others. Honour sensitivity, ethical behaviour, trust and compassion and nurture these qualities in the Alpha Plus Family. Embrace diversity through instructions and examples; foster understanding of the full spectrum of human social and cultural differences. Commit to the principle of serving, teaching the grace of giving, the warmth of community, and the dignity of a cooperative spirit. To maintain high academic standards, teaching quality, promote the analytical thinking and independent judgement necessary to function responsibility.

CSIR-UGC National Eligibility Test (NET)

UNIT–1:

Analysis:Elementary set theory, finite, countable and uncountable sets, Real number system as a complete ordered field, Archimedean property, supremum, infimum. Sequences and series, convergence, limsup, liminf. Bolzano Weierstrass theorem, Heine Borel theorem. Continuity, uniform continuity, differentiability, mean value theorem. Sequences and series of functions, uniform convergence. Riemann sums and Riemann integral, Improper Integrals. Monotonic functions, types of discontinuity, functions of bounded variation, Lebesgue measure, Lebesgue integral. Functions of several variables, directional derivative, partial derivative, derivative as a linear transformation, inverse and implicit function theorems. Metric spaces, compactness, connectedness. Normed linear Spaces. Spaces of continuous functions as examples.

Linear Algebra:Vector spaces, subspaces, linear dependence, basis, dimension, algebra of linear transformations. Algebra of matrices, rank and determinant of matrices, linear equations. Eigenvalues and eigenvectors, Cayley-Hamilton theorem. Matrix representation of linear transformations. Change of basis, canonical forms, diagonal forms, triangular forms, Jordan forms. Inner product spaces, orthonormal basis. Quadratic forms, reduction and classification of quadratic forms.

UNIT – 2:Complex Analysis:Algebra of complex numbers, the complex plane, polynomials, power series, transcendental functions such as exponential, trigonometric and hyperbolic functions. Analytic functions, Cauchy-Riemann equations. Contour integral, Cauchy’s theorem, Cauchy’s integral formula, Liouville’s theorem, Maximum modulus principle, Schwarz lemma, Open mapping theorem. Taylor series, Laurent series, calculus of residues. Conformal mappings, Mobius transformations.

Algebra:Permutations, combinations, pigeon-hole principle, inclusion-exclusion principle, derangements. Fundamental theorem of arithmetic, divisibility in Z, congruences, Chinese Remainder Theorem, Euler’s Ø-function, primitive roots. Groups, subgroups, normal subgroups, quotient groups, homomorphisms, cyclic groups, permutation groups, Cayley’s theorem, class equations, Sylow theorems. Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domain, principal ideal domain, Euclidean domain. Polynomial rings and irreducibility criteria. Fields, finite fields, field extensions, Galois Theory.

Topology:Basis, dense sets, subspace and product topology, separation axioms, connectedness and compactness.

UNIT – 3:Ordinary Differential Equations (ODEs):Existence and uniqueness of solutions of initial value problems for first order ordinary differential equations, singular solutions of first order ODEs, system of first order ODEs. General theory of homogenous and non-homogeneous linear ODEs, variation of parameters, Sturm-Liouville boundary value problem, Green’s function.

Partial Differential Equations (PDEs): Lagrange and Charpit methods for solving first order PDEs, Cauchy problem for first order PDEs. Classification of second order PDEs, General solution of higher order PDEs with asymptotic distributions, distribution of order statistics and range. Methods of estimation, properties of estimators, confidence intervals. Tests of hypotheses: most powerful and uniformly most powerful tests, likelihood ratio tests. Analysis of discrete data and chi-square test of goodness of fit. Large sample tests. Simple nonparametric tests for one and two sample problems, rank correlation and test for independence. Elementary Bayesian inference. Gauss-Markov models, estimability of parameters, best linear unbiased estimators, confidence intervals, tests for linear hypotheses. Analysis of variance and covariance. Fixed, random and mixed effects models. Simple and multiple linear regression. Elementary regression diagnostics. Logistic regression. Multivariate normal distribution, Wishart distribution and their properties. Distribution of quadratic forms. Inference for parameters, partial and multiple correlation coefficients and related tests. Data reduction techniques: Principle component analysis, Discriminant analysis, Cluster analysis, Canonical correlation. Simple random sampling, stratified sampling and systematic sampling. Probability proportional to size sampling. Ratio and regression methods. Completely randomized designs, randomized block designs and Latin-square designs. Connectedness and orthogonality of block designs, BIBD. 2K factorial experiments: confounding.

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