# Syllabus Dips Academy-Csir mathematics (1)

Distributions. Characteristic functions. Probability inequalities (Tchebyshef, Markov, Jensen). Modes of convergence, weak and strong laws of large numbers, Central Limit theorems (i.i.d. case). Markov chains with finite and countable state space, classification of states, limiting behaviour of n-step transition probabilities, stationary distribution. Standard discrete and continuous univariate distributions. Sampling distributions. Standard errors and 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, tests for linear hypotheses and confidence intervals. 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, randomized blocks and Latin-square designs. Connected, complete and orthogonal block designs, BIBD. 2K factorial experiments: confounding and construction. Series and parallel systems, hazard function and failure rates, censoring and life testing.

Linear programming problem. Simplex methods, duality. Elementary queuing and inventory models. Steady-state solutions of Markovian queuing models: M/M/1, M/M/l with limited waiting space, M/M/C, M/M/C with limited waiting space, M/G/1.

Linear Algebra : Finite dimensional vector spaces. Linear transformations and their matrix representations, rank; systems of linear equations, eigenvalues and eigenvectors, minimal polynomial, Cayley-Hamilton theorem, diagonalisa-tion, Hermitian, Skew –Hermitian and unitary matrices. Finite dimensional inner product spaces, selfadjoint and Normal linear operators, spectral theorem, Quadratic forms.

of linear equations; direct and itervative methods, (Jacobi Gauss- Seidel and SOR) with convergence; matrix eigenvalue problems; Jacobi and Given’s methods, numerical solution of ordinary differential equations; initial value problems. Taylor series method, Runge-Kutta methods, predictorcorrector methods; convergence and stability. Partial Differential Equations: Linear and quasilinear first order partial differential equations, method of characteristics; second order linear equations in two variables and their classification: Cauchy, Dirichlet and Neumann problems, Green’s function of Laplace, wave and diffusion equations in two variables Fourier series and transform methods of solutions of the above equations and applications to physical problems. Mechanics: Forces in three dimensions, Poinsot central axis, virtual work, Lagrange’s equations for holonomic systems, theory of small oscillations, Hamiltonian equations. Topology : Basic concepts of topology, product topology, connectedness, compactness, countability and separation axioms, Urysohn’s Lemma, Tietze extension theorem, metrization theorems, Tychonoff theorem on compactness of product spaces. Probability and Statistics : Probability space, conditional probability, Baye’s theorem, independence, Random variables, joint and conditional distributions, standard probability distributions and their properties, expectation, condition expectation, moments. Weak and strong law of large numbers, central limit theorem. Sampling distributions, UMVU estimators, sufficiency and consistency, maximum likelihood estimtors. Testing of hypothesis, Neymann-Pearson tests, monotone likelihood ratio, likelihood ratio tests, standard parametric tests based on normal, X2, t, F-distributions. Linear regression and test for linearity of regression, Interval estimation. Linear Programming: Linear programming problem and its formulation, convex sets their properties, graphical method, basic feasible solution, simplex method, big-M and two phase methods, infeasible and unbounded LPP’s alternate optima. Dual problem and duality theorems, dual simplex method and its application in post optimality analysis, interpretation of dual variables. Balanced and unbalanced transport-ation problems, unimodular property and u-v method for solving transportation problems. Hungarian method for solving assignment problems. Calculus of Variations and Integral Equations : Variational problems with fixed boundaries; sufficient conditions for extrremum, linear integral equations of Fredholm and Volterra type, their iterative solutions, Fredholm alternative.

polynomials, diagonalizability, Jordan canonical form. Abstract Algebra : Groups: subgroups, Lagrange’s theorem, normal subgroup, quotient group, homomorphism, permutation groups, Cayley’s theorem, Sylow theorems, Rings, Ideals Fields. Ordinary Differential Equations : First order ODEs and their solutions, singular solutions, experience and uniqueness of initial value problems for first order ODE. Gewneral theory of homogeneous and homomor-geneous linear differential

equations. Variation of parameters. Types of singular points in the phase plane of an autonomous system of two equations. Partial Differential Equations: Elements of first order PDE. Second order linear PDE: Classification, wave Laplace and Heat equations. Basic properties and important solutions of classical initial and boundary value problems. Elements of Numerical Analysis: Interpolation : Lagrange and Newton’s forms, error in interpolation. Solution of nonlinear equations by iteration, various iterative methods including Newton. Raphson method, fixed point iteration. Convergence, integration: trapezoidal rule, Simpson’s rule, Gaussian rule, expressions for the error terms. Solution of ordinary differential equations: simple difference equations, series method, Euler’s method, Runge Kutta methods, predictor- corrector methods, error estimates.

Fundamental operations in Algebra, Expansions, Factorization, simultaneous linear and quadratic equations, indices, logarithms, arithmetic, geometric and harmonic progressions, binomial theorem, permutations and combinations, surds, determinants, matrices and application to solution of simultaneous linear equations, Set Theory, Group Theory. Coordinate Geometry: Rectangular Cartesian coordinates, equations of a line, midpoint, intersections etc., equations of a circle, distance formulae, pair of straight lines, parabola, ellipse and hyperbola, simple geometric transformations such as translation, rotation, scaling. Calculus: Limit of functions, continuous functions, differentiation of functions tangents and normals, simple examples of maxima and minima, Integration of function by parts, by substitution and by partial fraction, definite integrals, and applications of Definite Integrals to areas. Differential Equations: Differential equations of first order and their solutions, linear differential

equations with constant coefficients, homogenous

linear differential equations.

Vector: Position Vector,

additions and subtraction of

vectors, scalar and vector

products and their

applications to simple

geometrical problems and

mechanics.

Absolute convergence, Uniform convergence, properties of continuous functions, Rolle’s theorem, Mean value theorem, Taylor’s and Maclaurian’s series, Maxima and Minima, Indeterminate forms.

Statistics & Linear Programming: Frequency distribution and measure of dispersion, skewness and Kurtosis, Permutations and Combinations, Probability, Random variables and distribution function, Mathematical expectation and generating function, Binomial, Poisson normal distribution curve fitting and principle of least squares, Correlation and Regression, Sampling and large sample tests, Test of significance base on t, x2 and f distribution, Formulation of simple linear programming problems, basic concepts of graphical and simple methods.

Analytical Ability and Logical Reasoning

The questions in this section will cover logical reasoning, quantitative reasoning.

ComputerAwareness

Data Representation: Representation of characters, integers and fractions, binary and hexadecimal representations, Binary Arithmetic: Addition, subtraction, division, multiplication, single arithmetic and two’s complement arithmetic, floating point representation of numbers, normalized floating point representation, Boolean algebra, truth tables, Venn diagrams.

Elements of Data Structures Computer Organization C Language

**Operation Research (O.R)**Linear programming problem. Simplex methods, duality. Elementary queuing and inventory models. Steady-state solutions of Markovian queuing models: M/M/1, M/M/l with limited waiting space, M/M/C, M/M/C with limited waiting space, M/G/1.

**GATE**Linear Algebra : Finite dimensional vector spaces. Linear transformations and their matrix representations, rank; systems of linear equations, eigenvalues and eigenvectors, minimal polynomial, Cayley-Hamilton theorem, diagonalisa-tion, Hermitian, Skew –Hermitian and unitary matrices. Finite dimensional inner product spaces, selfadjoint and Normal linear operators, spectral theorem, Quadratic forms.

**Complex Analysis :**Analytic functions, conformal mappings, bilinear transformations, complex integration; Cauchy’s integral theorem and formula, Liouville’s theorem, maximum modulus principle, Taylor and Laurent’s series, residue theorem and applications for evaluating real integrals.**Real Analysis :**Sequences and series of functions , uniform convergence, power series, Fourier series, functions of several variables, maxima, minima, multiple integrals, line, surface and volume integrals, theorems of green, Stokes and Gauss; metric spaces, completeness, Weiestrass approxi-mation theorem, compactness, Lebesgue measure, measurable functions; Lebesgue integral, Fatou’s lemma, dominated convergence theorem. Ordinary Differential equations: First order ordinary differential equations, existence and uniqueness theorems, systems of linear first order ordinary differential equations, linear ordinary differential equations of higher order with constant coefficients; linear second order ordinary differential equations with variable coefficients, method of Laplace transforms for solving ordinary differential equations, series solutions; Legendre and Bessel functions and their orthogonality, Sturm Liouville system, Green’s functions. Algebra: Normal subgroups and homomorphisms theorems, automorphisms. Group actions, sylow’s theorems and their applications groups of order less than or equal to 20, Finite p-groups. Euclidean domains, principal, Principal ideal domains and unique factorizations domains. Prime ideals and maximal ideals in commutative rings. Functional Analysis: Banach spaces, Hahn-Banach theorems, open mapping and closed graph theorems, principle uniform boundedness; Hilbert spaces, orthonormal sets, Riesz representation theorem, self-adjoint, unitary and normal linear operators on Hilbert Spaces. Numerical Analysis: Numerical solution of algebraic and transcendental equations; bisection, secant method, Newton-Raphson method, fixed point iteration, interpolation: existence and error of polynomial interpolation. Lagrange, Newton, Hermite (osculatory) interpolations; numerical differenti-ation and integration, Trapezoidal and Simpson rules; Gaussian quadrature; (Gauss-Legendre and Gauss- Chebyshev), method of undetermined parameters, least square and orthonormal polynomial approximation; numerical solution of systems If you miss an opportunity, do not cloud your eyes with tears; keep your vision clear so that you will not miss the next one DIPS Academy /11**ISI Kolkata**of linear equations; direct and itervative methods, (Jacobi Gauss- Seidel and SOR) with convergence; matrix eigenvalue problems; Jacobi and Given’s methods, numerical solution of ordinary differential equations; initial value problems. Taylor series method, Runge-Kutta methods, predictorcorrector methods; convergence and stability. Partial Differential Equations: Linear and quasilinear first order partial differential equations, method of characteristics; second order linear equations in two variables and their classification: Cauchy, Dirichlet and Neumann problems, Green’s function of Laplace, wave and diffusion equations in two variables Fourier series and transform methods of solutions of the above equations and applications to physical problems. Mechanics: Forces in three dimensions, Poinsot central axis, virtual work, Lagrange’s equations for holonomic systems, theory of small oscillations, Hamiltonian equations. Topology : Basic concepts of topology, product topology, connectedness, compactness, countability and separation axioms, Urysohn’s Lemma, Tietze extension theorem, metrization theorems, Tychonoff theorem on compactness of product spaces. Probability and Statistics : Probability space, conditional probability, Baye’s theorem, independence, Random variables, joint and conditional distributions, standard probability distributions and their properties, expectation, condition expectation, moments. Weak and strong law of large numbers, central limit theorem. Sampling distributions, UMVU estimators, sufficiency and consistency, maximum likelihood estimtors. Testing of hypothesis, Neymann-Pearson tests, monotone likelihood ratio, likelihood ratio tests, standard parametric tests based on normal, X2, t, F-distributions. Linear regression and test for linearity of regression, Interval estimation. Linear Programming: Linear programming problem and its formulation, convex sets their properties, graphical method, basic feasible solution, simplex method, big-M and two phase methods, infeasible and unbounded LPP’s alternate optima. Dual problem and duality theorems, dual simplex method and its application in post optimality analysis, interpretation of dual variables. Balanced and unbalanced transport-ation problems, unimodular property and u-v method for solving transportation problems. Hungarian method for solving assignment problems. Calculus of Variations and Integral Equations : Variational problems with fixed boundaries; sufficient conditions for extrremum, linear integral equations of Fredholm and Volterra type, their iterative solutions, Fredholm alternative.

**IISc**polynomials, diagonalizability, Jordan canonical form. Abstract Algebra : Groups: subgroups, Lagrange’s theorem, normal subgroup, quotient group, homomorphism, permutation groups, Cayley’s theorem, Sylow theorems, Rings, Ideals Fields. Ordinary Differential Equations : First order ODEs and their solutions, singular solutions, experience and uniqueness of initial value problems for first order ODE. Gewneral theory of homogeneous and homomor-geneous linear differential

equations. Variation of parameters. Types of singular points in the phase plane of an autonomous system of two equations. Partial Differential Equations: Elements of first order PDE. Second order linear PDE: Classification, wave Laplace and Heat equations. Basic properties and important solutions of classical initial and boundary value problems. Elements of Numerical Analysis: Interpolation : Lagrange and Newton’s forms, error in interpolation. Solution of nonlinear equations by iteration, various iterative methods including Newton. Raphson method, fixed point iteration. Convergence, integration: trapezoidal rule, Simpson’s rule, Gaussian rule, expressions for the error terms. Solution of ordinary differential equations: simple difference equations, series method, Euler’s method, Runge Kutta methods, predictor- corrector methods, error estimates.

**MCA Entrance Exam****Algebra:**Fundamental operations in Algebra, Expansions, Factorization, simultaneous linear and quadratic equations, indices, logarithms, arithmetic, geometric and harmonic progressions, binomial theorem, permutations and combinations, surds, determinants, matrices and application to solution of simultaneous linear equations, Set Theory, Group Theory. Coordinate Geometry: Rectangular Cartesian coordinates, equations of a line, midpoint, intersections etc., equations of a circle, distance formulae, pair of straight lines, parabola, ellipse and hyperbola, simple geometric transformations such as translation, rotation, scaling. Calculus: Limit of functions, continuous functions, differentiation of functions tangents and normals, simple examples of maxima and minima, Integration of function by parts, by substitution and by partial fraction, definite integrals, and applications of Definite Integrals to areas. Differential Equations: Differential equations of first order and their solutions, linear differential

equations with constant coefficients, homogenous

linear differential equations.

Vector: Position Vector,

additions and subtraction of

vectors, scalar and vector

products and their

applications to simple

geometrical problems and

mechanics.

**Trigonometry:**Simple identities, trigonometric equations, properties of triangles, solution of triangles, height and distance, inverse function, Inverse Trigonometric functions, General solutions of trigonometric equations, Complex numbers. Real Analysis: Sequence of real numbers, Convergent Sequences, Cauchy’s Sequences, Monotonic Sequences, Infinite series and their different tests of convergence, The only way to gain respect is, firstly to give it DIPS Academy /14

**JNU**Absolute convergence, Uniform convergence, properties of continuous functions, Rolle’s theorem, Mean value theorem, Taylor’s and Maclaurian’s series, Maxima and Minima, Indeterminate forms.

Statistics & Linear Programming: Frequency distribution and measure of dispersion, skewness and Kurtosis, Permutations and Combinations, Probability, Random variables and distribution function, Mathematical expectation and generating function, Binomial, Poisson normal distribution curve fitting and principle of least squares, Correlation and Regression, Sampling and large sample tests, Test of significance base on t, x2 and f distribution, Formulation of simple linear programming problems, basic concepts of graphical and simple methods.

Analytical Ability and Logical Reasoning

The questions in this section will cover logical reasoning, quantitative reasoning.

ComputerAwareness

**Computer Basics:**Organization of a Computer, Central Processing Unit (CPU), Structure of instructions in CPU, input/output devices, computer memory, memory organization, back- up devices.Data Representation: Representation of characters, integers and fractions, binary and hexadecimal representations, Binary Arithmetic: Addition, subtraction, division, multiplication, single arithmetic and two’s complement arithmetic, floating point representation of numbers, normalized floating point representation, Boolean algebra, truth tables, Venn diagrams.

Elements of Data Structures Computer Organization C Language

**Resource:- net-mathematics****Category**:
CSIR-NET

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