Ph. D StatisticsNorth Eastern Hill University
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Recap of elements of set theory; Introduction to real numbers. Introduction to ndimensional Euclidian space; open and closed intervals (rectangles), compact sets,
Bolzano – Weirstrass theorem, Heine – Borel theorem. Sequences and series; their convergence.
UNIT – II : Real valued function; continuous functions, uniform continuity. Differentiation; maxima
– minima of functions, functions of several variables, constrained maxima – minima of functions, mean value theorem, Taylor’s theorem, differentiation under the sign of
integral – Leibnitz rule.
UNIT – III : Fields, vector spaces, subspaces, linear dependence, basis and dimension of a vector space, completion theorem, linear equations. Vector spaces with an inner
product, Gram-Schmidt ortghogonalization process, orthonormal basis and
orthogonal projection of a vector.
UNIT – IV : Linear transformations, algebra of matrices, row an column spaces of a matrix, elementary matrices, determinants, rank and inverse of a matrix, partitioned matrices, Kronecker product. Real quadratic forms, positive definite matrix. Characteristic roots and vectors, Cayley – Hamilton theorem, minimal polynomial, similar matrices.
UNIT – V : Difference table. Methods of interpolation; Lagrange’s interpolation formula, Newton’s
divided difference formula. Numerical differentiation based on Newton’s and
Lagrange’s formula. Numerical integration; Trapezoidal, Simpson’s one-third and
three-eighth formula for numerical integration.