# Connectedness and Necessary Conditions for an Extremum (Mathematics and Its Applications)

Simplex algorithm and duality, shortest paths, network flows, min-cost flows and circulations, out-of-kilter method, assignments and matchings. Using generating functions and Polya theory to do sophisticated counting. Permutations and combinations, inclusion-exclusion, partitions, recurrence relations, group actions, Polya theory with applications.

Algorithmic and theoretical aspects of graph theory: matchings, colorings, scheduling problems, Hamilton cycles. Euler tours, spanning trees, network reliability, connectivity, extremal graphs, planar graphs, disjoint paths. Latin squares, mutually orthogonal latin squares, orthogonal and perpendicular arrays, Steiner triple systems, block designs, difference sets and finite geometries. A development of the mathematical theory of life insurance and annuities.

Utility functions, mortality models, life tables, insurance plans, premiums. Divisibility, Diophantine equations, congruencies. The numerical solutions of selected problems arising in calculus and algebra along with the programming techniques. Computer familiarity. Options and spreads, pricing of such options in accordance with the Black-Scholes Equation, and the binomial pricing model.

Topics may vary as needed. Includes general principles involving continuous deterministic problems and a detailed, specific term-project. Fundamental concepts of extrema functions and functionals; first and second generalizations; sufficient conditions; constrained functionals; the general Legrange problem; optimal control.

Fourier transforms, Z-transforms, Function spaces, eigenfunction methods. Approximation of functions by polynomials, spline functions or trigonometric function, expansions in series. Introduction to the theoretical foundations of Linear Algebra Algebra including vector spaces, basis, dimension, linear transformations, fundamental subspaces matrix representations, eigenvalues, eigenspaces.

The symmetry group of a set, homotheties and similitudes, path, arcs and length of curves, and advanced theorems on the circle. Analytical solutions of nonlinear problems, ODES, PDEs, multiple scales, and transcendental equations in engineering, mathematics, and physics using both regular and singular perturbation methods. Lorenz map, Henonmap, toral automorphisms, stable and unstable manifolds, strange attractors, quadratic maps of the complex plane, Julia sets, Mandelbrot set.

Numerical solutions of systems of linear equations, numerical computation of eigenvalues and eigenvectors, error analysis. Central Limit Theorem, Laplace transforms, convolutions, simulation, renewal processes, Continuous-time Markov Chains, Markov renewal and semi- regenerative processes, Brownian motion and diffusion. Computer simulation and experimenting within Mathematica, supported by Internet resources. Linear spaces, matrices, eigenvalues, least squares solutions to linear systems, Hilbert spaces, orthogonal expansions, integral equations, compact operators, Green's functions for boundary value problems, eigenfunction expansions.

Calculus of variations, asymptotic expansions, Spectral theory, Fourier transform, Partial differential equations, transform methods and eigenfunction expansions, vibrations, diffusion processes, equilibrium states, Green's functions, boundary layer problems. Introduction and theory of some of the important methods of approximation.

## Welcome to LNMIIT, Jaipur

Includes uniform approximation, best approximation, best trigonometric approximation. Least square approximation and rational approximation, and advanced topics of current interest. Techniques of approximation by interpolation, rates of convergence and methods of estimating error. Simultaneous approximation of functions and their derivatives; spline function interpolation; curve and surface fitting. Timan's Theorem. Telyakovski and Gopengauz Theorems. Brudnyi's inequality, Polynomial based discrete cosine transforms, Piecewise polynomial interpolation, B-splines, Recurrence relations, Schoenberg-Whitney Theorem, Convergence properties of spline approximations.

## Meaning of "extremum" in the English dictionary

Special functions from classical complex analysis which play an important role in the mathematics of physics, chemistry, and engineering. Geometric objects and configurations with discrete symmetry groups. Regular polygons and polyhedra. Regular arrangements.

Plane tilings and patterns. Convexity and related geometric extremum problems. Packing and covering.

Arrangements of extreme density. Manifolds, differential structure, vector and tensor fields, vector and tensor bundles, differential forms, chains.

### Translation of «extremum» into 25 languages

First-order languages, Satisfaction. The completeness and compactness theorems, models constructed from constants. Elementary substructures and embeddings, Lowenheim-Skolem-Tarskj theorems. Ultraproducts and ultrapowers. Introduction to modern set theory. Introduction to forcing, independence results, iterated forcing, consistency of Martin's Axiom. Theory and practice of discrete algorithms: complexity class classes, reductions, approximate algorithms, greedy algorithms, search techniques, heuristics, randomized algorithms, and numeric algorithms.

Fundamental aspects of algorithmic algebra. Noetherian rings. Theory of Groebner bases. Hilbert Nullstellensatz. Elimination theory.

Applications to graph theory and algebraic geometry. Groebner bases for modules and syzygy computations. Improved Buchberger's algorithm. Computation of Ext. Groebner bases over rings. Primary decomposition of ideals. Dimension theory. Sigma algebras, measures, measurable functions, integratability, properties of Lebesgue measure, density, Lusin's theorem, Egeroff's theorem, product measures, Fubini's theorem.

Limit theorems involving pointwise convergence and integration. L-p spaces, completeness, duals. Weak convergence, norm convergence, pointwise convergence, convergence in measure.

## International Master's in Mathematics and its Applications - MSc

Signed and complex measures. Absolute continuity, Lebesgue decomposition. Measure theory, Lebesgue integration, introductory functional analysis. Complex numbers, analytic functions, derivatives, Cauchy integral theorem and formulae, Taylor and Laurent series, analytic continuation, residues, maximum principles, Riemann surfaces. Conformal mappings, families of analytic functions and harmonic analysis.

Existence and continuation theorems for ordinary differential equations, continuity and differentiability with respect to initial conditions, linear systems, differential inequalities, Sturm theory. Stability theory, periodic solutions, boundary value problems, disconjugacy of linear equations, Green's functions, upper and lower solutions, a priori bounds methods, current research.

Groups, Lagrange's Theorem, normal subgroups, factor groups, Isomorphism and Correspondence Theorems. Symmetric groups, alternating groups, free groups, torsion groups. Introduction to rings, correspondence theorems. Rings, modules, vector spaces, and semi-simple modules. Maschke's Theorem, characters, orthogonality relations, induced modules, Frobenius reciprocity, Clifford's Theorem, Mackey's Subgroup Theorem, Burnside's theorem on solvability. Topics on: commutative rings Cohen-Seidenberg theorems, Krull Intersection Theorem, Dedekind domains , or noncommutative rings projective modules over Artinian algebras, representation type, Noether-Skolem Theorem, division algebras.

Torsion groups: Decompositions, Ulm's theorem, uniqueness theorem for Axion 3 groups, Torsion-free groups: Completely decomposable groups, Butler groups, p-local groups, Warfield groups, splitting criteria.