# MATE - Mathematics

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## MATE-1010 Seminario de Matemáticas

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.

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## MATE-1064B Thought through Numbers

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.

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## MATE-1102 Matemática Estructural

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.

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## MATE-1103 Teoría de Números

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.

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### Instructor

Caicedo Ferrer Xavier

## MATE-1105 Capitalism and Entrepreneurship in the Global Economy

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.

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### Instructor

Arteaga Bejarano Jose

## MATE-1106 Algebra Lineal 1 (Honores)

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.

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## MATE-1107 Algebra Lineal II

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.

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## MATE-1201 Precalculo

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.

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### Instructor

Bernal Pedraza Oscar

## MATE-1203 Calculo Diferencial

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.

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## MATE-1204 Calculo Diferencial (Honores)

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.

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## MATE-1207 Calculo Vectorial

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.

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## MATE-1208 Calculo Vectorial (Honores)

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.

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## MATE-1209 Calculo III (Economía y Administración)

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.

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## MATE-1212 Matemáticas I (Biología - Medicina)

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.

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Vanegas Gordillo Orlando

## MATE-1213 Matemáticas III (Biología - Medicina)

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.

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### Instructor

Casas Sanchez Oscar

## MATE-1214 Calculo Integral con Ecuaciones Diferenciales

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.

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Ramirez Vallejo Jorge

## MATE-1215 Calculo Integral con Ecuaciones Diferenciales (Honores)

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.

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Goodrick John

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## MATE-1407 Geometría Analítica

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.

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## MATE-1441B Escher: Geometry and Art

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.

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## MATE-1442B Geometries of the Imagination

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.

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## MATE-1501 Estadística I (Ciencias Sociales)

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.

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### Instructor

Rocuts Schweitzer

## MATE-1502 Estadística II (Ciencias Sociales)

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.

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### Instructor

Casas Monsegny Marta

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.

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### Instructor

Benitez Castro Ferney

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.

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### Instructor

Rodriguez Granobles Fredy

## MATE-1507 Matemáticas II (Biología - Medicina)

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.

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## MATE-2101 Algebra Abstracta I

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.

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## MATE-2201 Análisis I

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.

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## MATE-2210 Calculo de Variable Compleja (Ingeniería)

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.

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Lopez Garcia Jose

## MATE-2211 Calculo de Variable Compleja

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.

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Arteaga Bejarano Jose

## MATE-2230 Complex Variable and Numerical Analysis

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.

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De La Vega Ramiro

## MATE-2301 Ecuaciones Diferenciales

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.

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### Instructor

Giniatoulline Andrei

## MATE-2411 Geometría de Curvas y Superficies

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Rodriguez Granobles Fredy

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## MATE-2510 Developing the Creative Organization

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.

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## MATE-2604 Teoría de Análisis Numérico

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.

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Echeverry Navarro Luz

## MATE-2711 Métodos Matemáticos para Economistas

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.

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Ferri Stefano

## MATE-3101 Algebra Abstracta II

Rings, integral domains and fields. Algebraic and transcendental extensions. Unique factorization domains and Euclidean domains. Field automorphisms, Galois’ theory and solvability of equations.

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## MATE-3120 Developing the Creative Organization

This course is an introduction with mathematical emphasis to logic. The minimum content of the course includes the study of the calculation of propositions and predicates: symbolization, syntax, semantics, formal deduction, validity and completeness theorems for these calculations. It gives an introduction to calculations: recursive functions, Turing-calculable functions, equivalence between them. We study some links between calculation and formal properties of logical calculations studied.

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Corredor Londoño Luis

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## MATE-3401 Geometría Riemanniana

Introduce students to the basic tools provided by the differential geometry (Riemannian) for the study of differential varieties with metrics. Specially, study in depth fundamental notions as parallel transport, curvature and geodesic and its properties. Study the key results on Riemannian varieties and get to know the classic examples from which we can study the mentioned concepts.

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Vargas Dominguez Andres

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## MATE-3420 Topologia 1

Topological spaces, bases and sub-bases, sub-spaces, continuous functions. Order topology, product topology, quotient topology and metric topology. Connected spaces, arc-connected spaces and local connectedness. Compact and locally compact spaces, compactifications, Tychonoff’s Theorem. Axioms of enumerability. Regular and completely regular spaces. Normal spaces, Urysohn’s Lemma, Tietze’s Theorem of extension. Metrizability.

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## MATE-3510 Procesos Estocásticos

Introduction. Stochastic processes specifications. Some important classes of processes such as stationary processes. Processes with stationary increments and processes with independent increments. Markov processes. Martingales. MARKOV CHAINS: definitions and examples. Features. Finite Markov chains. Classification of states and chains. Markov chains accounting. Limit theorems. Stationary distribution. POISSON PROCESSES: generalizations of Poisson processes. not homogeneous process. Compound Poisson processes. Conditional processes. Processes of birth and death. MARTINGALA in discrete time: conditional expected value. Definition and examples. Time to Stop. Optional stopping theorem. Inequalities of the Doob Martingale. Convergence Theorem of Martingala. RENEWAL PROCESSES : Renewal Equation. Laws of large numbers. Age and residual life. Applications to the theory of the tail. Brownian motion: Preliminaries. Simple Features of standard Brownian motion. Variations in the Brownian motion. Brownian motion with drift. Kolomogorov equations. Ornstein-Uhlenbeck process.

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Arunachalam Viswanatham

## MATE-3520 Estadística Matemática I

Estimation methods: time for confidence intervals. Methods of moments, least squares, maximum likeliness. Theory of optimality criteria estimate UMVU, the information. Consistent estimators, asymptotic distribution, efficient estimators, non-biased. Confidence intervals and hypothesis tests. Theme of Neyman-Pearson. Likeliness reason. Tests of fitting, contingency tables. Linear models, Gauss-Markov theorem, Testing in linear models.

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Garrido Lopeña Liliana

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## MATE-3801 Practica Enseñanza I

Training in teaching methodology. Micro-teaching practice on how to handle questions, the use of the blackboard and diagnostic sessions. Instructions prior to each class, observations on development, creation of exams. Students teach one section of problems relating to a lecture under the guidance of a professor from the Department.

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Paramo Fonseca Aquiles

## MATE-3802 Practica Enseñanza II

Instructions on how to teach a class, reading and discussion of articles in Mathematical Education, creation of exams, observations. Students teach a course under the guidance of a professor from the Department.

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Botero De Meza Maria

## MATE-3902 Proyecto de Grado

Students write a paper in a specific area of mathematics in which they show their capacity as regards research and presenting a topic with all the requirements of clarity, correctness and proper style.

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Ferri Stefano

## MATE-3904 Seminario de Proyecto de Grado I

Students start out in the current events of research in mathematics.

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Ferri Stefano

## MATE-3905 Seminario de Proyecto de Grado II

Students start out in research as an independent activity, as well as the proper oral and written communication of mathematics by reading articles and solving problems. This course requires students to choose a topic for their graduation thesis.

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## MATE-4001 Curso Tutorial de Maestria

The objective of this course is for students to develop a study and research plan proposed in advance under the guidance of a professor from the Department.

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## MATE-4101 Algebra Conmutativa

Rings and Ideals. Modules. Rings and fraction modules. Primary decomposition. Whole dependence and valuations. Chain conditions. Noetherian rings. Artin rings. Discrete valuation rings and Dedeking rings. Completions. Theory of dimension and, if there is time left, other themes that the professor may deem convenient.

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Onshuus Niño Alf

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## MATE-4140 Teoría de Modelos 1

Initiate studies of Model Theory of First-Order Logics. Completitude, Compacity, Lowenheim-Skolem Theorems, Categorical K-Theory, Complete Theories, Decidable and Undecidable Theory, Elementary Equivalence and Summersion. Characterization of Universal Theories, Universal-Existential. Existentially Closed Models, Complete Model Theories, Elimination of Quantifiers, Partial Isomorphisms, Feferman-Vaugth Theorems. Interpolation and Definibility Theorems. Automorphisms, Indiscernible, Ehrenfeucht-Mostowski Theorem. Fraissé Generic Models. Boolean Algebra, Filters, Ultra filters. Ultra products, Ultra product Saturation. Types of Elements, Types Realization and Omission, Saturation, Homogeneity, Universality. Atomic and Prime Models, Omega-Categoric Theories. Type Spaces, Stability, Stable Omega Theories. Keisler-Shelah Theorem, Characterization of Elementary Classes. Morley Categoricity Theorem. Baldwin-Lachlan Theorem. After all of the foregoing, the instructor may concentrate further on themes such as: Laws 0-1 in Finite Models. Finite Models Spectrum. Relations with Complexity.

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Corredor Londoño Luis

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## MATE-4220 Medida e Integración

Algebras and sigma-algebras, measures, Lebesgue Measure, Constitution and regularity. integrated functions: Roles measurable properties almost everywhere, the definition of integral, theorems of boundaries, Riemann integral, complex functions and extent of image. Convergence: Modes of convergence, regulated spaces, Lp spaces, dual spaces. Real and complex measures: Absolute continuity, singularity, functions of bounded variation, dual spaces Lp. Product measures: construction, Fubini Theorem, applications.

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Berenstein Opscholtens Alexander

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## MATE-4330 Analisis Funcional

Banach Spaces: Definitions and examples. Subspaces, linear transformations, quotient spaces. Duality: Hahn-Banach’s theorem. Banach-Steinhaus’ theorems, the Open Mapping and the Closed Graph theorems. Applications: Adjoint operators. Hilbert Spaces: Definitions and examples, orthogonality. Continuous operators: Operator convergence. Hermetian, normal and unitary operations. Orthogonal projections. Compact operators: Introduction to spectral theory.

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## MATE-4421 Topología Algebraica

Introduce the basic concepts of algebraic topology, and the classical algebraic tools used in the calculation of topological invariants of spaces simplicials. Content: Variety topological. Homotopy group fundamental properties. Building simplicials, homology, Ulama-Borsuk theorem, fixed point theorem Lefschetz, Cohomology, Poincare duality. Theory of beams, Prebeams, Resolutions. Fiber vector.

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Malakhaltsev Mikhail

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## MATE-4901 Seminario de Postgrado I

The objective of the course is to place students in contact with a wide range of advanced mathematical topics and teach them to synthesize and orally explain these topics with clarity and accuracy.

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Ould Ahmed

## MATE-4902 Seminario de Postgrado II

The objective of the course is to place students in contact with a wide range of advanced mathematical topics and teach them to synthesize and orally explain these topics with clarity and accuracy. In this seminar, students decide on the topic in the area they would like to do their graduation thesis and prepare a presentation on this topic with the professor who will possibly be the director of the graduation thesis.

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Ould Ahmed

## MATE-4903 Seminario de Trabajo de Grado

The objective of this course is to introduce students to research activities, through the direct study of specialized mathematical literature and train them, not only to solve problems but also to formulate them properly. Students will present their theses to the Graduate and Research Committee of the Department before the last week of withdrawals for the semester. Students are expected to move forward in their research during the following period.

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## MATE-4904 Trabajo de Grado

Students will carry out a research project in one of the mathematical areas offered by the Master’s Program. This is to demonstrate that the author has assimilated and systematized or carried out thorough exploration of a particular topic, showing evidence of a certain degree of creativity and great familiarity with recent information on the topic. The Graduation Thesis must be written in Spanish or English and have the formal organization expected of a scientific paper.

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### Instructor

Uribe Jongbloed Bernardo

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Ould Ahmed

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Ould Ahmed

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