2000

Preliminary concepts: Sets and equivalency relations. Groups and subgroups: Binary operations, groups and subgroups, cyclic groups and generators. Groups and cosets: Permutation Groups, Orbits, Cycles and Alternating Groups, Introduction to Isomorphism and Cayley’s Theorem, Cosets and Lagrange’s Theorem, Direct Products and Finitely Generated Abelian Groups. Homomorphisms and Factor Groups: Homomorphisms, factor groups, simple groups, series of groups, action groups on sets, applications of G-sets in combinatorial analysis. Advanced Group Theory: Sylow’s Theorems, Free Abelian Groups, Free Groups. Rings and Fields: Rings, Fields and Integral Domains.

Real Numbers: Dedekind cuts. Metric Spaces. Successions and numeric series. Limits, continuity, differentiation and fundamental theorems. Riemann Stieltjes’ Integral. Successions and series of functions. Functions of several variables.

COMPLEX NUMBERS: Basic concepts and representations. ANALYTICAL FUNCTIONS: Cauchy-Riemann Equations, Harmonic Functions. ELEMENTARY COMPLEX FUNCTIONS: exponential, trigonometric, hyperbolic and logarithmic. Transformations with exponential functions. COMPLEX INTEGRATION: Path integrals, Cauchy-Goursat’s Theorem, Cauchy’s integral formula. SUCCESSIONS AND SERIES: Convergence, Taylor’s and Laurent’s Series. remainders: The Cauchy remainder theorem. APPLICATIONS OF REMAINDERS: Calculus of improper integrals, improper integrals in Fourier analysis, Jordan’s lemma.

Complex numbers, algebra of the complex, geometry of the complex. Compliant applications. Analytical functions. Elementary complex functions: exponential, trigonometric functions, logarithms. Complex integration: Cauchy Goursat theorem. Liouville theorem. Successions and series, power series, series by Taylor and Laurent. Calculation of waste. Agreed representation. Harmonic functions.

Complex Numbers. Analytic Functions. Elemental Functions. Integrals. Cauchy-Goursat Theorems. Numerical Solution of Equations in a Variable. Initial Value Problems in Ordinary Differential Equations. Solution methods for Linear, direct and iterative systems. Nonlinear Equation System Solutions. Solution of edge problems in Partial Differential Equations. Finite Differences.

General methods to solve equations of the first order. Linear equations of the second order or higher. Linear equations of the second order with variable coefficients. Applications in physics. Systems of equations of the first order. Homogenous and non-homogenous linear systems. Applications of series of powers in solving differential equations. Laplace transform. Fourier series. Orthogonal functions. Partial differential equations. Applications: waves, vibrations, heat conduction.

Probability Spaces. Counting, permutations, combinations, multinomial coefficients, sample space, events, probability axioms, equally likely events, probability as a continuous function, as a measure of credibility, conditional probability, Bayes’ formula, independent events, P(.|F) is a probability, random variables (r.v.), discrete variables, expected value, expectancy of a r.v. function, variance, Bernoulli and Binomial, Poisson, other discrete variables, accumulated distribution function, continuous random variables, expectancy and variance, uniform, normal, exponential, other continuous variables, distribution of a function of a random variable, joint distributions, independent random variables, sum of independent r.v., conditional distribution, order statistics, joint probability of the function of a r.v., sum expectancy, moments of the number of events, covariance, correlations, conditional expectancy and prediction, moment generating functions, normal multivariate, weak law of large numbers, central limit theorem, strong law of large numbers.

Many problems in real life are modeled using algebraic or differential equations. The people who solve these problems have to make sure the solution exists, but in many cases (almost always) it is impossible to find said solution. Numeric analysis consists of finding approaches to said solutions and, therefore, a fundamental part of the course is the analysis of different types of mistakes that are made by employing each method in accordance with the accuracy desired in the result. Very different sorts of problems discussed in class, such as: Finding the roots of a function, interpolating or predicting the behavior of a variable based on a series of observations, solving linear equation systems, solving problems with an initial value or boundary value for ordinary and partial differential equations. The computer is a fundamental tool in this course to implement the algorithms.

Several real-life problems are modeled using algebraic or differential equations. The mathematician who solves these problems should make sure that the solution exists. But in many cases (almost all) is impossible to find such a solution. Precisely the numerical analysis is to find approximations to these solutions. Contents: Interpolation. Numerical integration. Matrix calculation. Vector and matrix norms. Direct Resolution, Linear Systems. Iterative methods. Optimization-based methods. Equations with partial derivatives: finite differences and finite elements.

This course targets the modeling and management of uncertainty in design. Uncertainty primarily results from models used for analysis and from variable aspects such as loads, building material and the structural topology. One good practice in engineering consists of making economical designs that meet an acceptable safety standard. It is not an advanced course of probability, but requires basic probability principles knowledge.