Level I

Vector Algebra: Definition of a Vector, Addition and Subtraction, Components, Physical examples. Vector Products:Scalar and Vector products including a brief introduction to determinants, triple products, Geometrical applica-tions. Differentiation and Integration of a Vector functions. Vector Analysis:Scalar and Vector fields, grad, div, curl, Manipulation with combinations of these operators acting on combinations of fields. Integral transformations: Line, Surface and Volume integrals, the divergence theorem, conservative and solenoidal fields, Greens theorem, Stokes theorem (3-D) form. General Co-ordinates: Unit vectors in orthogonal curvilinear co-ordinates, elementary arc length and volume, curl, div, grad in curvilinear co-ordinates. Method of assessment: end of semester examination

Introduction, Equations of first order and first degree, Orthogonal trajectories, Clairant’s form, Linear equations, Theory of operators, Euler’s form, Simultaneous equations. Method of assessment: end of semester examination

Definition of Probability, Conditional Probability and the Independence of events, , The Law of Total Probability and Bayes’ Rule, Definition of random variables, Cumulative distribution function, Density functions for discrete random variables and continuous random variables, Expectations, Mean, Variance, standard deviation, Expected value of a function of a random variable, Moments, Central Moments, Moment Generating function, Bernoulli and Binomial Distributions, Hypergeometric Distribution, Poisson Distribution, Geometric Distribution, Uniform Distribution, Normal Distribution, Exponential and Gamma Distribution, Approximation: Binomial and Poisson by Normal. Method of assessment: end of semester examination

Basic Algebra (including Complex Numbers), Logarithms, Trigonometric functions, Limits, The principle of Dif- ferentiation, Differentiation of a Product, Quotient and a function of a function, Maxima and Minima, Partial Differentiation, Total Differentiation, Homogeneous Functions and Eulers Theorem on Homogeneous functions, Integration as the converse of Differentiation, Integration by parts, Exact Differential equations, Definite Integral, Vectors, Determinants, Matrices, Introduction to Group Theory, Statistics for Chemistry( permutations, Configurations and Microstates, Molecular Assemblies, The importance of , W=W!/na! nb! ,The Boltzman Distribution.) Method of assessment: end of semester examination

Elementary set theory, Relations, mappings and functions, theory of polynomial equations in one variable including the statement of the fundamental theory, Newton’s relations between roots, solution of cubic and biquadratic equations, determinants, solution of equations using determinants nth roots of unity, factors of x

Classical Logic, Set theory, Field axioms, Real number system as a field, Functions and its properties, Real sequences, Continuity and Limits of functions, Differentiability. Method of assessment: end of semester examination

Frame of reference, Inertial frames, Forces, Velocity, Acceleration, Linear momentum, Angular velocity, Angular acceleration, Angular momentum, motion of a particle (Newton laws), motion of a system particles, Rotating coordinate systems, moments and products of inertia. Parallel axes theorem, perpendicular axes theorem for moments and products of inertia. Principle axes and principle moments of inertia of a system of particles, Rotating coordinate systems, Infinitesimal rotation, Euler equations, Generalized coordinates, Lagrange’s formulations, Hamiltonian functions, Theory of small oscillation, Impulsive motion.

Logic Syllogisms, propositional logic, propositions, arguments, predicates and quantifiers, logic programming. Number Systems: Number Systems (decimal, Roman etc.), Binary number system, Octal system, Binary arithmetic (including complements methods) Boolean Algebra and Logic circuits: Boolean Algebra, Switching circuits, logic circuits.

Theory of Forces and Couples: Force acting at a point, Resultant of a system of Forces acting at a point, Condition for equilibrium of a system of Forces acting at a point, Vector moment of a Force, Couple, Moment of a Couple, Resultant of a system of Forces in 3D, Invariants, Wrench, Coplanar Systems of Forces, Varignon’s Theorem of Moments, Parallel Systems, Conjugate forces. Bending of Beams: Shear and Bending moment in a beam, Relations among Load, Shear and Bending Moment, Thin Elastic Beams, Bernoulli-Euler Law, Macaulay’s Notation, Clapeyron’s equation for three moments. The Catenary: Flexibility, The common catenary, Parabolic chain, suspension bridge, Catenary of uniform strength, General equations of equilibrium of a string in one plane under given forces, Strings on rough curves, Variable chain hanging under gravity.

Introduction: General Introduction to Modelling, concepts of system identification, Deterministic vs Stochastic, classification of models. Modelling via First Order Differential Equations: Modelling Through First Order and Simple Higher Order Differential Equations, Linear Differential Equations (LDEs), systems of Ordinary Differential Equations (ODEs). Analysis of Solutions: Existence and uniqueness of solutions, continuation of solutions, dependence on initial conditions and parameters, linear systems of equations with constant and variable coefficients, autonomous systems, phase space, and stability, Interpretation of solutions in modelling. Applications: Population ecology, chemical kinetics, traffic dynamics, Mechanics, Biology and Medicine, Pharmokinetics, Economics, Engineering, Special topics in modelling.

Frame of reference, Inertial frames, Forces, Velocity, Acceleration, Linear momentum, Angular velocity, Angular acceleration, Angular momentum, motion of a particle (Newton laws), motion of a system particles, Rotating coordinate systems, moments and products of inertia. Parallel axes theorem, perpendicular axes theorem for moments and products of inertia. Principle axes and principle moments of inertia of a system of particles, Rotating coordinate systems, Infinitesimal rotation, Euler equations, Generalized coordinates, Lagrange’s formulations, Hamiltonian functions, Theory of small oscillation, Impulsive motion.

Introduction to computer systems and its historical development, contribution from mathematicians (e.g. Libnietz, Boole, Pascal, Babage, Turing, von Neumann),numerical computation and mechanical computing devices; General overview of: computer architecture, hardware, software and liveware, programming languages, application packages, the present day use of computers and its future; Introduction to Linux Operating System: UNIX commands, directory structure, text editors, user accounts and file permissions, text editors, virtual terminals in text mode. programming with C on Unix system – editing (with emacs), compilation, debugging etc,. ; Formatted input-output, control structures, loops, C-functions, pointers, File input/output, command-line arguments (the above shall be discussed with mathematical applications); Introduction to X-Windows system.

Semester II

Theory of Forces and Couples: Force acting at a point, Resultant of a system of Forces acting at a point, Condition for equilibrium of a system of Forces acting at a point, Vector moment of a Force, Couple, Moment of a Couple, Resultant of a system of Forces in 3D, Invariants, Wrench, Coplanar Systems of Forces, Varignon’s Theorem of Moments, Parallel Systems, Conjugate forces. Bending of Beams: Shear and Bending moment in a beam, Relations among Load, Shear and Bending Moment, Thin Elastic Beams, Bernoulli-Euler Law, Macaulay’s Notation, Clapeyron’s equation for three moments. The Catenary: Flexibility, The common catenary, Parabolic chain, suspension bridge, Catenary of uniform strength, General equations of equilibrium of a string in one plane under given forces, Strings on rough curves, Variable chain hanging under gravity.

Introduction: General Introduction to Modelling, concepts of system identification, Deterministic vs Stochastic, classification of models. Modelling via First Order Differential Equations: Modelling Through First Order and Simple Higher Order Differential Equations, Linear Differential Equations (LDEs), systems of Ordinary Differential Equations (ODEs). Analysis of Solutions: Existence and uniqueness of solutions, continuation of solutions, dependence on initial conditions and parameters, linear systems of equations with constant and variable coefficients, autonomous systems, phase space, and stability, Interpretation of solutions in modelling. Applications: Population ecology, chemical kinetics, traffic dynamics, Mechanics, Biology and Medicine, Pharmokinetics, Economics, Engineering, Special topics in modelling.

Level II

Integers: Prime and irreducible, division algorithm, Euclid’s algorithm, Fundamental Theorem of Arithmetic, Integers mod n, Chinese Remainder Theorem, Euler’s function Prime integers: Sieve of Eratosthenes, perfect numbers, Mersenne numbers, Fermat numbers, infinite number of primes, the prime number theorem. Gaussian integers Modular calculations: Fermat’s Little Theorem, Wilson’s theorem. Sums of squares, Fermat’s Last Theorem, Sums of 4 squares. Primitive elements: Roots of unity, factors of Fermat primes, roots of polynomial equations, the number of n roots of unity, the Primitive Element theorem. Integer polynomials: Hensel’s Lemma, primitive elements mod n. Special Topics in Number Theory.

Sequences and series of functions, Point-wise convergence of sequence of functions, Uniform convergence of sequence of functions, Convergence and Uniform convergence of series of functions, Integration and differentiation of series of functions.

Sets, Relations on sets, Equivalence relations, Equipotent sets, Finite and infinite sets, Countability and uncount- ability, Topology of line and plane, Bolzano Weierstrass theorem , Heine Borel Theorem, Metric spaces, Complete metric spaces, Compact metric spaces, Connected metric spaces, Continuous functions on metric spaces.

Plane: Various forms of the equation of a plane. Straight Line, Various forms of the equation of a line. Sphere: Various forms of the equation of a sphere, Tangent line to a sphere, Tangent plane to a sphere, Condition of Tangency, Intersection of two spheres. The Central Conicoids: Ellipsoid, Hyperboloid of one sheet, Hyperboloid of two sheets, Intersection of a conicoid and a line, Tangent Line to a conicoid, Tangent Plane to a conicoid, Normal to a conicoid, Number of Normals from a given point.

Joint Density Functions, Joint Cumulative Distribution Function, Conditional Distribution Function, Independence, Covariance and correlation coefficient, Conditional Expectations, Joint Moment Generating Function and Moments, Independence and Expectation, Bivariate Normal Distribution, Expectations of Functions of Random Variables. Distribution of Function of Random variables: Cumulative Distribution Function Technique, Moment Generating Function Technique, Transformation Technique. Population and Samples, Random Sample, Statistic, and Sample Moments, Sample Mean, Law of Large Numbers, Central limit Theorem. Sampling from the normal distribution: Sample mean, chi-square distribution, F distribution, Student t Distribution.

Equations of stream lines, Equations of vortex lines, Differentiation following the motion of a fluid. Equations of continuity, Euler’s and Bernoulli’s equations, Irrotational motion, uniqueness theorem, Kinetic energy, Sources and sinks, Images, Potential flow, Complex potential.

Numerical computing and computers: Introduction, Using a computer to do numerical analysis, Computer arithmetic and errors. Solving Non Linear equations: Bisection Method, Newton’s Method, Fixed point Iteration x = g(x) Method, Secant Method, Regular-Falsi Method. Interpolation and Curve Fitting: Interpolation, Lagrange polynomials, Divided Differences, Interpolating with a Cubic Spline, Least Square Approximation. Numerical Differentiation and numerical Integration: Getting derivatives and integrals numerically, Trapezoidal rule (composite formula), Simpson’s rules, Applications of cubic splines.

Introductory Numerical Solutions of Differential Equations, Mathematical Modelling through Difference Equations, Further Study on Systems of Differential Equations with Matrices. Modelling with Partial Differential Equations (PDEs): The concept of a PDE, Method of separation of variables, Mass-Balance equation (The first method of obtaining PDE Models), Momentum-Balance Equation (The second method of obtaining PDE Models), Variational Principles (The third method of obtaining PDE Models), Probability Generating functions (The fourth method of obtaining PDE Models), Nature of PDEs Initial and Boundary Conditions.

The fundamental notion of periodicity and bifurcation. The concepts of chaos and strong chaos for functions of one variables Fractals, Fractal dimensions, Julia sets and Manderbolt sets.

Event Spaces, probability measure, probability space, sample space, continuity of a probability measure, Defining random variables on probability spaces and their functions, partition theorem, conditional probabilities, Distribution Functions, The law of large numbers, Introduction to Information theory and Claude Shannon’s remarkable work on mathematical formulation of the central problem in telecommunication channels, Error correcting codes for binary symmetric channel and their performances, Shannon’s noisy channel coding theorem, probability and entropy, entropy and mutual information, convex functions and Jensen’s inequality, the data processing theorem, Discrete memoryless channels and their capacity-cost functions, measuring the information content of an ensemble,the Source-Channel Coding Theorem for the Binary Symmetric Channel.

Collecting and Summarizing data: Constructing tables and graphs, Measures of center of a set of observations, Median, Arithmetic Mean, Mode. Samples and Populations: Methods of choosing a sample, Measures of variability: Range, Mean deviation, Variance and Standard deviation, Semi-interquartile range, five number summaries, Box and Whisker plots, stem and leaf plots. Joint distributions of data: The Scatter diagram, the concept of a statistical relation, Quantitative description of a statistical relation, Covariance, Correlation coefficient Linear regression: Regression equation, Prediction and error, Interpreting regression. Statistical Applications with probability models: Bernoulli, Binomial, Poisson, Normal approximations, Statistical software packages.

Equations of stream lines, Equations of vortex lines, Differentiation following the motion of a fluid. Equations of continuity, Euler’s and Bernoulli’s equations, Irrotational motion, uniqueness theorem, Kinetic energy, Sources and sinks, Images, Potential flow, Complex potential.

Introduction to the computer package Mathematica, how to type mathematics, special characters, basic constructions, Numerical computations, Standard functions, Accuracy, The use of variables, Working with whole numbers, Finding prime numbers, Handling algebraic expressions and doing symbolic computations, Graphics in Mathematica, Calculus in Mathematica, Solving equations, Introduction to programming (procedural vs functional) using various numerical and algebraic methods to solve equations and sets of equations, Defining new functions, Writing new commands which perform more complicated tasks.

Introductory Numerical Solutions of Differential Equations, Mathematical Modelling through Difference Equations, Further Study on Systems of Differential Equations with Matrices. Modelling with Partial Differential Equations (PDEs): The concept of a PDE, Method of separation of variables, Mass-Balance equation (The first method of obtaining PDE Models), Momentum-Balance Equation (The second method of obtaining PDE Models), Variational Principles (The third method of obtaining PDE Models), Probability Generating functions (The fourth method of obtaining PDE Models), Nature of PDEs Initial and Boundary Conditions.

The fundamental notion of periodicity and bifurcation. The concepts of chaos and strong chaos for functions of one variables Fractals, Fractal dimensions, Julia sets and Manderbolt sets.

Event Spaces, probability measure, probability space, sample space, continuity of a probability measure, Defining random variables on probability spaces and their functions, partition theorem, conditional probabilities, Distribution Functions, The law of large numbers, Introduction to Information theory and Claude Shannon’s remarkable work on mathematical formulation of the central problem in telecommunication channels, Error correcting codes for binary symmetric channel and their performances, Shannon’s noisy channel coding theorem, probability and entropy, entropy and mutual information, convex functions and Jensen’s inequality, the data processing theorem, Discrete memoryless channels and their capacity-cost functions, measuring the information content of an ensemble,the Source-Channel Coding Theorem for the Binary Symmetric Channel.

Collecting and Summarizing data: Constructing tables and graphs, Measures of center of a set of observations, Median, Arithmetic Mean, Mode. Samples and Populations: Methods of choosing a sample, Measures of variability: Range, Mean deviation, Variance and Standard deviation, Semi-interquartile range, five number summaries, Box and Whisker plots, stem and leaf plots. Joint distributions of data: The Scatter diagram, the concept of a statistical relation, Quantitative description of a statistical relation, Covariance, Correlation coefficient Linear regression: Regression equation, Prediction and error, Interpreting regression. Statistical Applications with probability models: Bernoulli, Binomial, Poisson, Normal approximations, Statistical software packages.

Level III

Groups and subgroups: Groups, subgroups/normal subgroups, quotient groups, cyclic groups, Cayley diagrams. Permutations, cosets and direct products: groups of permutation, orbits, cycles and alternating groups, cosets and the theorem of Lagrange, direct product, finite groups. Homomorphism and factor groups: Homomorphism factor group, factor group computation and simple groups.

Non Linear Analysis: Functions from Rn to Rm , Open balls and Open sites, limit points. Limit and continuity, The derivative of scalar field with respect to a vector. Directional derivatives and partial derivatives, Partial derivatives of higher order, Directional derivatives and continuity, The Total derivative, The gradient of Scalar field, A sufficient condition for differentiability, A chain rule for derivatives of scalar fields. Derivatives of vector fields, Differentiability implies continuity, The chain rule for derivatives of vector fields, Sufficient conditions for the equality of mixed partial derivatives. Derivatives of functions defined implicitly, Maximum, Minimum and saddle points, Extremes with constraints, Lagrange’s Multipliers, Double integrals.

Point estimation: The method of moments, The method of Maximum Likelihood, Properties of point estimation: Unbiasedness, Efficiency, Consistency, Sufficiency, Minimal Sufficient Statistics, Exponential family, Cramer – Rao Inequality, Completeness. Interval Estimation: Confidence Interval for the mean and variance. Tests of Hypotheses: Simple Hypothesis, Composite Hypothesis, Critical Region, Types of Error, Power Function, Size of Test, Simple Likelihood-ratio Test, Most powerful Test, Neyman-Pearson lemma, Generalized Likelihood ratio Test, Uniformly Most Powerful Test, Tests of Hypotheses – Sampling from the Normal Distribution.

Solving Linear Systems: Matrix Notation, Direct Methods Gauss, Jordan, Aitken Method, etc. Iterative Methods Jacobi, Gauss – seidel, S. O. R Method, etc. Numerical solution of Ordinary differential equations: Euler and Modified Euler methods, Runge – Kutta method, Convergence criteria, Errors and error propagation. Numerical solution of partial differential equations: Parabolic type, Elliptic type, Hyperbolic type.

Laplace transformations, Inverse Laplace Transformations, Gamma, Beta and Bessel functions, Applications in Solving the wave equation and the heat equation, Fourier series.

Testing hypotheses about many population means: Introduction to analysis of variance, Linear model for analysis of variance, variability as sum of squares, Test statistics and rejection rules. The population regression: Formulating hypotheses about regression, Analysis of Variance for regression Nonparametric tests: Chi-square test,Contingency tables (test for independence), Kolmogorov-Smirnov test,The sign test, The Rank test (Mann-Whitney U-test), Runs test (one sample runs test, two sample runs test), Kruskal- Walis, H-test

Refer the Optional course units offered by the Department of Mathematics for Level III- Semester II, for details.

This includes a real world problem solving task, using mathematical (modelling) techniques and computational tools that the student has learnt in Level-I, II and III.

Solution of Linear Differential Equations by Laplace Transforms, Mathematical Modelling through Graphs, Mathematical Modelling Through Calculus of Variations and Dynamic Programming or Special Topics and/or Project,Stochastic Modelling, A survey on Ancient Sri Lankan Science and Technological Methods, Topics in Mathematical Modelling of Life-Environmental relationships.

Testing hypotheses about many population means: Introduction to analysis of variance, Linear model for analysis of variance, variability as sum of squares, Test statistics and rejection rules. The population regression: Formulating hypotheses about regression, Analysis of Variance for regression Nonparametric tests: Chi-square test,Contingency tables (test for independence), Kolmogorov-Smirnov test,The sign test, The Rank test (Mann-Whitney U-test), Runs test (one sample runs test, two sample runs test), Kruskal- Walis, H-test

Optional course units

Metric spaces: Definition and examples, Open set, Closed set, neighbourhood, Convergence, Cauchy Sequence, Complete Linear, Completion of metric spaces, Banach’s fixed point theorem. Normed spaces: Linear space, Normed space, Banach space, Finite dimensional normed spaces and sub spaces, Compactness and finite dimensions, Linear operators, Bounded and Continuous linear operators, Linear operators and functional, on finite dimensional spaces, Normed spaces of operators, Dual space, Inner product space, Hilbert spaces. Fundamental Theorems for Normed and Banach spaces: Zorn’s Lemma, Hann-Banach Theorems, Reflexive spaces, Strong and weak convergence, Numerical integration and weak convergence.

Theory of Complex Variables: Complex Functions, Complex differentiability, the Cauchy-Riemann equations, Analytic functions, Cauchy’s Theorem, Cauchy’s Integral Formula, Taylor’s and Laurent’s Theorem, Classification of singularities, Laurent expansions, Contour Integration, The cauchy’s residue Theorem, Integration of rational and trigonometric functions using residue theorem

Differential Geometry: Unit tangent vector, principal normal, binomial vector and curvature of a curve, surfaces, parametric curves, surfaces of revolution, metric, directional ratios and coefficients, Gauss and mean curvature, orthogonal trajectories, families of dual curves, Geodesics.Tensor Analysis: Transformation of coordinates, summation convention, the Kronecker-delta, contravariant and covariant vectors, contravariant, covariant and mixed tensors, symmetric and skew-symmetric tensors, tensor algebra, metric tensor, conjugate metric tensor, Christoffel’s symbols of first and second kind, covariant derivatives.

Basic description of mathematical modelling, Introduction of models in Ecology, Analysis of Dynamical systems. Non linear Dynamical systems, Web analysis, population Dynamics. Logistic model, Graphical and Analytical Approaches to Harvesting, Economics of Harvesting. Breeding Season and age structure, Predator-prey system with age structure, Analysis based on competition with aid of logistic equation, Stability and Complexity, The statistical mechanics of population .

Matrix algebra: Definition of matrices; rules of matrix algebra; determinants; ranks, inverses and solutions; Cramer’s Rule; quadratic forms; matrix definacy. Applications: solutions of multi-equation models; input-output analysis. Optimization: Unconstrained optimization in the n-variable case; second order conditions and Hessian matrices. Constrained optimization in the n-variable case; multiple constraint cases and bordered Hessian matrices. Applications: Maximization and minimization of various economic magnitudes in multi-variable settings. An Introduction to inequality-constrained optimization: profit maximization; non-negativity constraints. Difference equations: Introduction to dynamics; applications: the cobweb pricing model; macroeconomic trade cycles.

Logic: Syllogisms, propositional logic, propositions, arguments, predicates and quantifiers, logic programming. Number Systems: Number Systems (decimal, Roman etc.), Binary number system, Octal system, Binary arithmetic (including complements methods) Boolean Algebra and Logic circuits: Boolean Algebra, Switching circuits, logic circuits.

Introduction to the RSA Encryption Scheme: Raising integers to large powers to a given modulus, ’Egyptian exponentiation’, Discussion of primality testing, The Little Fermat and Rabin tests, Implications for the RSA system, Verifying authenticity Topics in Rings and Fields: GF(p), Polynomials over a ring, The Primitive Element Theorem, Recurrent Sequences, shift registers, The ideal and minimal polynomial of a sequence, Indexing polynomials. Congruence modulo a polynomial, Construction of finite fields, Construction of indexing polynomials, Cyclotomic polynomials, Factorizing polynomials over finite fields Error detection and correction in telecommunication: ISBN codes, The Hamming metric, The minimum distance of a code, Elementary bounds on the minimum distance of a code, Equivalence of codes, Parity checks, The sphere-packing bound, Reed-Muller codes, Linear Codes, Dual codes, The parity check matrix of a linear code Syndrome decoding, The Hamming codes, Cyclic Codes, Generator polynomials and check polynomials, Construction of binary Hamming codes as cyclic codes, The BCH codes, the Golay code.

Basic Concepts of Fluid Flow: Introduction, Conservation Principles, Dimensionless form of a flow equation Simplified Mathematical Models for fluid flows: Incompressible Flow, Inviscid (Euler), Stokes (Creeping) Flow Mathematical Classification of Flows: Hyperbolic Flows, Parabolic Flows, Elliptic Flows, Introduction to the Navier- tokes Equation Introduction to Numerical Methods: Approaches to Fluid Dynamical Problems, What is CFD? , Possibilities and Limitations of Numerical Methods. Components of a Numerical Solution Process: Mathematical Model, Discretization Method, Numerical Grid, Finite Approximation, Solution process, Convergence Criteria, Properties of Numerical Schemes Discretization Approaches: Finite Difference Methods, Application of Finite Difference Methods to Different types of Models, Idea of Finite volume and Finite Element Methods with motivating examples.

Biological computers and their capabilities over digital computers, problem of classification and recognition, biological neurons, artificial neural networks, Mathematics of single-layer neural networks – the Perceptron, learning and training, learning rate, Perceptron training algorithm, Introducing Mathematica, methods to adjust the learning rate, convergence of solutions, basins of attractions, Baysian inference methods. Types of neural networks (feed-forward, back-propagation etc.) and algorithms for implementation. Monte- Carlo Methods, Hopfield network for optimization problems, e.g., traveling salesman problem, Applications in forecasting problems in finance, meteorology, particle physics.

Introduction to Statistical Packages, Data Analysis using a computer package, Descriptive Statistics, Graphical representation of data, Estimation, Hypothesis Testing, Regression, Analysis of Variance, non-parametric methods.

Electrostatic field equations, electrostatic potential, boundary value problems, magnetostatic field equations, boundary value problems, vector potential, Maxwell’s equations, Lorentz condition and gauge transformations, electromagnetic waves in non-conducting media, Electromagnetic waves in conductors.

Introduction (Inadequacy of Newtonian mechanics and the need of a new mechanics), The Space-time continuum and separation between events, Events and particles, Space-time, world lines and space-time diagrams, the motion of a material particle, the light-cone, the fundamental quadratic form, space-time as a Riemanian space, proper time and speed of light, Minkowskian coordinates, The Lorentz Transformations, Length contraction, the time dialation, composition of velocities, the velocity 4-vector and acceleration 4-vector, the expanding universe in S.R., The red-shift. Particles and mass, equation of motion, motion under a constant relative force.

The failure of Newtonian Mechanics to explain phenomena at microscopic level, problem of separation of observable from the observer. Quantum states, representation of quantum states by state (column) vectors, Observables as Hermitian Matrices, mean values and correspondence principle, the angular momentum of a photon, Uncertainty. Equations of motion, quantum particles in one-dimension and three dimension. The Spin of the electron, quantum particle in a spherically symmetric potential. The bound states of the hydrogen atom, The Dirac notation. Fourier transform, Applications to wave-packets, Basic Ideas of Hilbert space theory, theory of linear operators in Hilbert Spaces, Cauchy-Schwarz and Bessel inequalities, Completeness. Special Topics in Quantum Mechanics and applications: The EPR Paradox and Entanglement, Quantum effects in the computer-chip, Introduction to Quantum Computer.

Fundamental concepts in probability, Random variables, Mean, variance and expected values, Classification and Description of Sample Data, Sampling Distributions, Estimations, Hypothesis Testing, Regression Analysis, Analysis of Variance and Scientific Applications.

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