Power series. Existence and uniqueness theory. Picard's theorem. Method of undetermined coefficients, reduction of order, variation of parameters. Theory of linear differential equations. Formulate definitions, basic results and their proofs in relation to uniform convergence of sequences and series of functions, the corresponding Cauchy criteria, and some of their applications;?

Formulate and prove basic results about absolute and uniform convergence of power series and determine the radius of convergence. Determine existence and uniqueness of solutions of an ordinary differential equation;? Solve ordinary differential equations by elementary methods such as separation of variables, undetermined coefficients, or variation of parameters;? Apply reduction of order to differential equations;? Solve constant-coefficient linear ordinary differential equations of all orders;?

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Solve inhomogeneous linear ordinary differential equations, either by Laplace transform or by other elementary methods;? Apply Picard iteration to approximate solutions of ordinary differential equations. MA Linear Algebra. Module Objective: To provide an introduction to the concepts of the theory of linear algebra. Module Content: Linear equations and matrices; vector spaces; determinants; linear transformations and eigenvalues; norms and inner products; linear operators and normal forms. Verify the linearity of mappings on real and complex vector spaces,? Evaluate bases, transition matrices and matrices representing linear transformations;?

Compute eigenvalues and eigenvectors of linear operators;? Construct orthonormal bases for vector spaces;? Verify properties of projection mappings, adjoint mappings, self-adjoint operators and isometries. Module Objective: To provide an introduction to business-related applications of linear algebra. Module Content: Applications of linear analysis to game theory, Markov chains, linear programming. Solve linear optimization problems subject to linear inequalities;?

Use the simplex method;? Find optimal strategies and calculate expected values of games;? Employ Markov matrices in business models. MA Financial Mathematics. Module Objective: To develop facility with quantitative techniques for finance and investment. Module Content: An introduction to the theory of options, the time value of money, rate of return of an investment cash-flow sequence and the arbitrage theorem. Calculate probabilities and expectations of events and random variables associated to finite probability spaces and to standard variants of Brownian motion, using conditioning and independence techniques;?

Carry out calculations based on present-value analysis and arbitrage arguments;? Calculate the price of European call and put options using the multiperiod model;? Derive and apply the Black-Scholes formula for option pricing;? Estimate volatility of shares from price history data. The mark for Continous Assessment is carried forward.

## Mathematical Analysis : Foundations and Advanced Techniques for Functions of Several Variables

MA Multivariable Calculus. Module Objective: To provide a foundation in multivariable calculus. Module Content: Calculus of several variables, including continuity, differentiability and constrained and unconstrained optimisation. Line, surface and volume integrals. Use the definitions to verify continuity and differentiability for simple functions of two or more variables;?

Compute partial derivatives, mixed partial derivatives and higher-order partial derivatives using various methods including the chain rules;? Find the equations of tangent planes and normal lines to surfaces that are the graphs of functions of several variables;? Set up and calculate line integrals including examples arising in physical and geometrical problems involving curvature, mass, and work done by forces;?

Calculate double and triple integrals of continuous functions, defined over closed, bounded regions, by means of iterated integrals and the fundamental theorem of computation. Module Objective: To complete the introduction to fundamental mathematical techniques for business. Module Content: Least squares approximation and curve fitting. Matrices and simultaneous equations. Further methods and applications of calculus.

Module Objective: To present elementary linear analysis in a concrete setting, emphasizing specific techniques important to analysis and its applications. Module Content: Normed linear spaces, Banach spaces; Hilbert spaces, convexity, orthogonal expansions and their applications, Riesz representation theorem for functionals. Verify the axioms of normed vector spaces, Banach spaces and Hilbert spaces in specific examples, applying relevant tests for completeness or the existence of an inner product;?

Use completeness arguments to produce existence proofs for operators with desired properties;?

## Mathematical Analysis Foundations And Advanced Techniques For Functions Of Several Variables

Compute orthogonal expansions with respect to an orthonormal basis of a Hilbert space, and apply such expansions to solve problems;? Prove the Riesz Representation Theorem for bounded linear functionals on Hilbert spaces. Assessment: Total Marks Formal Written Examination 75 marks; Continuous Assessment 25 marks 2 assignments 5 marks each , 1 in-class test 15 marks.

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MA Ring and Field Theory. Module Content: Divisibility, irreducibles and primes in Z and Q[X], rings and ideals and unique factorization, construction of fields, Galois theory, straight edge and compass constructions. Accurately manipulate computations in a given ring or field, and in ring quotients by appropriate ideals;?

Solve arithmetic problems like finding the largest common divisor in a Euclidean ring;? Master basic techniques for investigating when a polynomial over rational numbers is irreducible;? Solve problems in an integral domain via the induced problem in its field of fractions;? Recognize when a ring is principal ideal domain or unique factorization domain;?

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Solve problems in field extensions of a field F by translating them into linear problems over F;? Compute the Galois group of a given extension. MA Complex Analysis. Module Objective: To provide an introduction to the theory of functions of a complex variable. Module Content: Bilinear mappings, complex differentiable functions, power series, complex contour integrals, Cauchy's theorem and integral formula, Taylor's theorem, zeros of analytic functions and Rouche's theorem, maximum modulus principle, singularities and Laurent series, poles and residues, residue calculus. Analyse the mapping properties of fundamental functions bilinear, exponential, trigonometric of a complex variable;?

Define the derivative of a complex function and derive and apply the Cauchy-Riemann equations;?

## Mathematical Analysis : Giuseppe Modica :

Prove and apply Cauchy's integral formula, and Taylor's theorem on the power series expansion of analytic functions;? Derive Liouville's theorem on entire functions and establish the fundamental theorem of algebra;? Compute residues and apply Cauchy's theorem to the evaluation of integrals and the summation of series;? Apply Rouche's theorem on the zeros of analytic functions.

- Foundations and Advanced Techniques for Functions of Several Variables.
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MA Metric Spaces and Topology. Module Objective: To provide an introduction to the theory and concepts of metric and topological spaces. Module Content: Topological spaces, compactness, connectedness, product spaces, continuity, countability, homotopy. State the basic concepts of topological and metric spaces. Construct examples and counterexamples of topological spaces with certain properties. Perform set theoretic computations. Formulate the basic properties of continuous functions. Apply the theory of various notions of convergence. Determine if a set is simply-connected.

MA Introduction to Modern Algebra. Module Objective: To provide an introduction to major concepts in Modern Algebra with applications. Module Content: Basic algebraic concepts and techniques such as rings and modules, computational algebra techniques and optimization, group representations, fields and field extensions, Galois theory. Identify generators for given ideals of polynomials;?

Apply division algorithms for multivariable polynomials;? Construct free resolutions of modules;? Define the homology group of a complex;?