1000

Introduction to the different areas that make up the Math program. Get students close to the Faculty through the various talks of the professors in their work areas. Approaching the life experiences of each Math professor through interviews on academic biography. Introduction to mathematical work by developing a brief monograph.

This course attempts to open a window for students from different disciplines to approach the theory of numbers and immerse themselves in interesting, demanding problems, thus developing their analytical skills.

Sets, set operations, demonstration by elements, set algebra. Theory of numbers, principle of proper order, principles of induction, applications on counting. Divisibility, division algorithm, Euclid’s’ algorithm, prime numbers, congruencies, Chinese theorem of residuals, Fermat’s little theorem. Relationships, orders, equivalence relations, functions, applications on counting. Cardinality of finite and infinite sets, Cantor-Schröder-Bernstein theorem. Mathematical structures, isomorphism of structures, rudiments of group theory.

The theory of numbers has determined and keeps on determining, to a large extent, the history of mathematics. The study requires the development and execution of rigorous reasoning. The problems tabled, many of them disconcerting by its apparent simplicity, have been a source of inspiration for mathematics creation at all levels. Topics: During the first three months, these topics addressed will include divisibility, primes, linear equations, congruence, quadratic residues, multiplicative functions, diofantinas nonlinear equations, continued fractions, approximation of irrational, distribution of primes. If time allows, the last month will be devoted to topics selected by the instructor.

At a first approximation (i.e. when considering small perturbations), problems in Engineering and Physics can be treated as linear problems. In this course the main objective is to introduce students to the main tools to model these linear situations: vector spaces and linear transformations. Vector spaces provide the structures of admissible objects, and linear transformation a way to relate these structures. In order to familiarize students with this very abstract branch of Mathematics, this course is focused on the study of finite dimen- sional vector spaces, matrices, scalar products and diagonalization, without leaving aside the computational aspects of this subject.

The content of this course is the same as MATE-1105 but in more depth. Vectors in the Euclidean space, scalar product and rule. Matrices and their algebra, linear equation systems. Inverse squared matrices, homogeneous systems, subspaces and bases. Independence and dimension, the range of a matrix. Linear transformations in Euclidean spaces, linear transformations of a plane. Vector spaces, basic concepts in vector spaces, vectors in coordinates. Determinants and linear transformations. Volume areas and cross product, the determinant of a squared matrix, calculation of determinants and Cramer’s rule. Values and vectors, diagonalization and applications. Projections, the Gram-Schmidt orthogonalization process, orthogonal matrices. Projection matrix and the minimum square method. Base change, matrix representations and similarity. Diagonalization of quadratic forms, applications on geometry.

Review of the previous course (Mate-1105) in more depth: Vector spaces, subspaces, linear combinations, bases and dimension, Linear transformations, core and image, matrix representation of a linear transformation, coordinate change matrix, dual space, elemental matrices and linear equation systems, determinants, their characterization as a multilinear form, values and vectors, diagonalizability, invariant subspaces, the Cayley-Hamilton Theorem, Spaces with an Internal Product: Adjoint operator, normal, self-adjoint, unitary and orthogonal operators, orthogonal projection and spectral theorem, bilinear and quadratic forms. Applications on the theory of relativity: Einstein’s principle of relativity, Lorentz’ transformations. Jordan Canonical Form: Jordan normal form, minimal polynomial. Multilinear algebra and tensors: Tensors on a vector space, examples and applications.

Algebra and arithmetic: Operations with fractions, real numbers, scientific notation, exponents and radicals, polynomials, factorization, rational expressions, equations, applications, inequalities. Functions: Definition of function, function graphs, linear functions, slope, operations between functions, compound function, inverse function, distance, midpoint, circles. Polynomial and rational functions: Complex numbers, quadratic functions, polynomial functions, roots and their graphs, rational functions and their graphs, inequalities of polynomial and rational functions, applications. Geometry and trigonometry: Angles, similar triangles, Pythagorean Theorem, trigonometry in right triangles, trigonometric functions, graphs of trigonometric functions, trigonometric identities, trigonometric equations.

The modeling of instantaneous phenomena is very important in the study of Natural and Social sciences. The description of instantaneous phenomena is engulfed in the notion of limit, and it is through its celebrated offspring, the derivative and the integral of a function, that it finds its main applications. This course provides an introduction to the modeling of instantaneous phenomena, mainly those concerned with rates of change, i.e., the study of limits and of derivatives. As it ends up being unavoidable, and being so intimately related to the concept of differentiation, a short introduction on the subject of integration is also given in this course. This course is a first step towards making students aware of the power of the language of Mathematics when used as a tool for describing and analyzing natural and man-made processes.

This course is recommended for students with a better mathematical background from high school, students most interested in this discipline and, in particular, students of mathematics. In the University’s different programs of study, it is equivalent to the course MATE-1203 and its content is the same but in more depth, as it is offered to students who are more prepared and more competitive. Functions: Function algebra, trigonometric functions, exponential functions, inverse functions, logarithms, trigonometric inverses, problem solving principles. Limits and derivatives: Velocity and tangents, limit of a function, calculating limits, continuity, limits at infinity, rates of change, derivatives, derived function, derivation rules, derivatives of trigonometric functions, chain rule, implicit derivation, derivatives of logarithms, derivatives of a higher order, hyperbolic functions. Applications of derivatives: Related rates, maximums and minimums, Mean Value Theorem, derivatives and graphs, L’Hopital Rule, curve plotting, optimization. Integrals: antiderivatives, areas and distance, defined integral, Fundamental Theorem of Calculus, integration by substitution, areas between curves, volumes of rotation solids using the slice and cylindrical shell methods.

The aim of this course is to provide the student with the basic tools of multidimensional calculus and to give a framework to hone his/her abilities at modeling and solving problems that require a basic knowledge of the geometry of 2 and 3 dimensional space. Also, it is of particular interest in this course to enhance the relation between the lan- guage provided by Mathematics and physical phenomena, through the study of the three Fundamental Theorems of Vector CalcuIus, namely, Green’s, Stokes’ and Gauss’ Theorem.

Curves on a plane and in space. Cylindrical and quadric surfaces. Straight lines and planes. Partial derivatives, chain rule. Differentiation in scalar and vector fields: definitions, techniques and applications. Notion of gradient. Maximums and minimums of functions with more than one variable. Lagrange multipliers. Line, multiple, surface and volume integrals. Theorems of Green, Stokes and Gauss. Physical and geometric applications. The difference between this course and MATE-1207 is not the content, but rather the depth of the topics discussed.

This course introduces topics of calculus of several variables – differentiation and integration and, mainly, the topic of optimization with or without restrictions. One of the objectives is for students to see the application of these topics on Economics. With emphasis on the use of mathematics, the techniques have applications not only in the economic context, but also in other areas such as Administration, Engineering, Physics or Biology. Students also become familiar with mathematical rigor, as many of the results and theorems are formally demonstrated. Functions of several variables. Partial derivatives, quadratic forms. Chain rule. Derivatives of implicitly defined functions. Partial elasticity. Homogenous functions. Equation systems. Implicit Function Theorem. Optimization. Maximums and minimums. Extreme Value Theorems. Local extremes. Convex sets. Concave and convex functions. Second derivative tests. Lagrange multiplier methods.

Function. Function graphs. Quadratic functions. Operations in functions. Inverse functions. Polynomials and rational functions. Exponential, logarithmic and trigonometric functions. Logarithmic scales. Graph transformations. Vertical and horizontal translating. Tangent and velocity problems. Limit of a function. Limit. Continuity. Limits at infinity. Tangents, velocities and other change rates. Derivatives. Derived function. Isolating Power Product and quotient rules Derivatives in natural and social sciences. Derivatives of trigonometric functions. Chain rule. Derivatives of a higher order. Implicit differentiation. Mean value theorem. Antiderivatives. Areas and distances. Definite integral. Fundamental theorem of calculus. Indefinite integral. Rule of substitution. Logarithm as an integral. Areas between curves. Mean value of a function. Integration by parts.

Review of Integrals. Integration techniques. Differential equations. Balance and stability. Points and vectors. The norm of a vector. Vector product. Lines on a plane. Scalar product. Parametric equation of the straight line. Functions of several variables. Partial derivatives. Tangent planes, derivable functions and linearization. The chain rule. Directional derivative and gradient. Maximums and minimums. Regression line. Multiple integrals. Linear systems. Autonomous non-linear systems and applications in biology.

This course is in part a continuation of the study of one-dimensional Calculus: one of the main concepts introduced in this course is that of integration. Here the student enlarges his/her toolkit to model instantaneous phenomena by the use of integration techniques. On the other hand, an Introduction to the theory of linear differential equations is also given: differential equations are the butter and bread of scientists and engineers, and, without any hint of doubt, they enormously expand the realm of possible applications of Cal- culus. Finally, as a first encounter with infinity and its paradoxical properties, series is a topic of study covered in this course; and again, as we want students to notice that mathematical concepts are not that far from real world applications, the study of series and its convergence properties derives in the study of Taylor series.

In content, this course is similar to MATE-1214, but topics are discussed more in depth. The content covers integration by parts, trigonometric integrals, trigonometric substitution, partial fractions, integration strategies, improper integrals, arc length, surface area of revolution, applications in other disciplines, modeling with differential equations of the first order, separable equations, exponential growth and decay, logistic equations, linear equation of the first order, parametric equations, calculus with parametric equations, polar coordinates, areas and length in polar coordinates, successions, series, criteria of integrals, comparison criteria, alternating series, rate and nth root criteria, series of powers, representation in series of powers, Taylor and Maclaurin series, complex numbers, linear differential equations of the second order of constant coefficients.

The main objective of this course is for students to expand their imagination and exercise their logic by analyzing different geometries from the axiomatic standpoint. Preliminary Notions: Euclidean and Cartesian geometry, affine planes, projective plane, duality, theorems of Desargues and Pappus, finite planes. Affine Planes: Addition and multiplication on lines, properties of operations, reciprocal of Desargues’ theorem. Co-ordinate affine planes on D division rings: coordinates, linear equations. Projective planes: Projective points, homogeneous 3D equations. Co-ordinate projective planes: Coordination, projective conics, Pascal’s theorem. Affine Space: Axiomatization, sub-geometries of an affine space, closure operator, Desargues’ Theorem, coordination. Projective Space: Axiomatization, planes in a projective space, dimension, consequences of Desargues´ theorem, coordination. Reticules of sub-geometries: Closure spaces, reticule properties. Collineations: Automorphism of planes, perspectivities in projective spaces. Fundamental theorem of projective geometry. Comparison with other non-Euclidean geometries: Spherical geometry, neutral geometry, hyperbolic geometry.

By the end of the course, students are expected to have an understanding of the relationship between geometry and art, and have a command of techniques to construct particular decorative designs, with a geometric background and the support of freely accessible software. The course covers important moments in the development of Geometry and its corresponding artistic manifestation: Basic concepts of geometry. Intuitive topology. Rigid body movements. Mosaics. Rates, proportions and similarity. Polyhedrons. Fractal geometry.

It is intended for students to discover by themselves the beauty of mathematics hidden in one of its most representative areas, that is to say, Geometry. Upon the previous study (informal) of the Euclidean geometry and some non - Euclidean, as well as its applications in art, architecture and physics, it is intended for students to understands (in a non-technical manner) the notion of truth in both Mathematics and science in general, added to its evolution through History.

Introductory course whose objective is to provide descriptive and inference tools to handle data in a social experiment to reach conclusions on the behavior of an individual as regards his/her social, political, economic environment, etc. Content: basic terms, Descriptive analysis, histograms, ogives, measures of central tendency, dispersion, graph interpretation. Introduction to probability: Definition of event, probability density function and its rules, mutually exclusive events, independent events, conditional probability. Discrete random variables, binomial distribution, standard deviation and mean of binomial distribution, standard normal distribution. Central limit theorem and applications. Point and interval estimation of one mean, two means, one and two proportions. Hypothesis testing of one and two means, and one and two proportions. Independence testing.

Point and interval estimation, hypothesis testing, hypothesis testing and confidence intervals for small samples, dependent and independent samples, testing for the difference of two means of independent populations, standard deviation), testing for the standard deviation of a population, testing for the standard deviation of two independent populations, estimating a proportion, testing relating to one proportion, testing relating to two proportions, contingency tables, goodness of fit testing, simple linear regression, regression analysis, correlation analysis, multiple linear regression, F test and relationship with linear regression, introduction to variance analysis, variance decomposition, analysis in a factor classification problem, a priori comparisons, post-hoc testing. Non-parametric testing: Sign test, range test, non-parametric testing, Wilcoxon signed rank test, Kruskal Wallis test.

The objective of this course is to familiarize students with the basic concepts of probability and the most common distributions. This knowledge will be useful not only for future courses of Statistics of Stochastic Processes, but is also directly applicable in many situations where chance or randomness prevail. Combinatory Methods. Binomial coefficients. Sample Spaces. Probability, rules. Conditional probability, independence. Bayes’ Theorem. Probability distributions. . Continuous random variables, density functions. Multivariate distributions. Marginal distributions. Conditional distributions. Expected value. Moments, Chebyshev’s Theorem. Moment generating functions. Product moments. Comb moments. Linear moments, conditional expectation. Uniform, Bernoulli, Binomial. Negative binomial, geometric, hyper-geometric. Poisson. Multinomial, multivariate hyper-geometric. Uniform, gamma, exponential, j-I squared. Beta distribution. Normal distribution. Normal to binomial approximation. Normal bivariate. Functions of random variables. Transformation technique: one variable. Transformation technique: several variables. Moment generating function technique. Sampling distributions. Mean distribution.

The objective of this course is to familiarize students with statistical inference, estimation and hypothesis testing concerning the parameters of a population and multiple linear regression models. The theory is explained with practical examples and exercises with statistical software such as SPSS, SAS or STATA. Mean distribution. Ji squared distribution. t distribution. F distribution. Order Statistics. Unbiased estimators. Efficiency. Consistency. Sufficiency. The moment method. The maximum likelihood method. Estimating means. Estimating the difference between means. Estimating proportions. Estimating the difference between proportions. Estimating variances and quotient. Hypothesis testing. Neyman Pearson Lemma. Power function, likelihood ratio. Mean testing. Testing the difference between means. Variance testing. Proportion testing. Analysis of an rXc table. Goodness of fit. Minimum square method. Analysis of normal regression. Analysis of normal correlation. Multiple linear regression. Matrix notation.

Linear equation systems. Matrices. Addition and multiplication of matrices. Inverse of a matrix. Determinant. Descriptive Statistics: basic terms, measures of central tendency and dispersion. Descriptive analysis, Pareto graphs, estograms, graph interpretation, bivariate data. Discrete mathematics: Sets, set operations, counting. Basic principles of counting. Permutations. Combinations. Relations, equivalence relation, partitions, binomial coefficient. Functions: pigeonhole principle, composition, symmetry. Probability: Introduction to probability, definition of event, probability function, probability function rules, mutually exclusive events, independent events, conditional probability, Bayes´ rule, discrete random variables, binomial distributions, geometrics and Poisson. Expected value, variance and standard deviation of discrete distributions, continuous distributions: normal, uniform and exponential. Tools of Statistics. Confidence intervals. Linear regression.