Number Theory Lecture 1(Modern Algebra)By Dr Neha Arora Anand institute of Mathematics

Number Theory Lecture 1(Modern Algebra) #Anand institute of Mathematics#RealAnalysis #Elementary_settheory, #finite_set, #countable_set,#uncountable_sets,#Real_number_system as complete ordered field, #Archimedean_property, #supremum, #infimum.# Sequences_and_series,#convergence,# limsup,# liminf. #Bolzano_Weierstrass-theorem, #Heine_Borel_theorem. #Continuity, #uniform-continuity, #differentiability,#mean-valu- theorem. #Sequences_and_series_of_functions, #uniform convergence. #Rieman sums #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, linear, #transformation,# inverse_function , #implicit_function #Metric_spaces, #compactness, #connectedness. #Normed_linear_Spaces. #Spaces_of_continuous_functions #Linear_Algebra #Complex_Analysis# Algebra_of_complex_ numbers, #the_complex_plane, #polynomials, #power_series, #transcendental_functions #exponential_functions,# trigonometri_functions #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, #divisibilit in_Z, #congruences, #Chines-Remainde_Theorem, #Euler’s Ø-function, #primitive_roots. #Groups, #subgroups,#normal_subgroups, #quotient_groups, #homomorphisms, #cycli groups, #permutation_groups, Cayley’stheorem, class_equations, Sylow_theorems. #Rings, #ideals, #prime_and_maximal_ideals, #quotient_rings, #uniqu_ 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 #Ordinary_Differential_Equations #ODE #Existence_and_uniqueness of solutions,#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 #PDE #Lagrange_method,#Charpit_methods_for_solving_first_order_PDEs, #Cauch_ problem #Classification_of_second_order,#General_solution_of_higher_order_PDEs_with_constant_coefficients, #Method_of_separation_of_variables_for_Laplace, #Heat_and_Wave_equations. #Numerical_Analysis ,#Numerical solutions of algebraic equations, #Method_of_iteration ,#Newton-Raphson_method, #Rate_of_convergence,