### Master’s Degree

#### Areas of Concentration: Applied Mathematics

**Algebra** (4 Credit Units, 60 Credit Hours)

Syllabus: Groups and Subgroups; cyclic groups; Lagrange's Theorem; sub-normal groups and quotient groups; homomorphism and group isomorphism; groups of permutations; Sylow's theorem; finitely generated abelian groups; soluble groups; rings and bodies; maximal ideals and prime ideals; homomorphism and ring isomorphism; single factorization domains; Euclidean domains; rings of polynomials in one and in several indeterminate ones.

**Commutative Algebra **(4 Credit Units, 60 Credit Hours)

Syllabus: Rings and Ideals; modules; finitely generated modules; homeomorphism of modules; exact sequences; tensor product; rings and fractional modules; primary decomposition; integral dependence; Noetherian rings; dedekind domains; dimension of Krull.

**Linear Algebra** (4 Credit Units, 60 Credit Hours)

Syllabus: Linear transformations; dual and bidual spaces; spaces with internal product; primary decomposition theory; Spectral Theorem; quadratic forms; rational and Jordanian forms; bilinear forms.

**Analysis of Time Series** (4 Credit Units, 60 Credit Hours)

Syllabus: Basic concepts: stochastic processes and time series, stationarity, self-covariance and spectrum function; stationary ARMA processes: autoregressive, moving-ave and discrete-mixed models; ARIMA models, the general linear model and harmonic models; spectral analysis: Fourier series, analysis of periodic and non-periodic functions, spectral representation of stationary processes, mixed spectrum and linear filters; estimation in the time domain: estimation of mean and self-covariance function, identification, estimation and prediction of ARIMA model parameters; Frequency domain estimation: finite Fourier transform and gram period, smoothed estimators.

**Functional Analysis **(4 Credit Units, 60 Credit Hours)

Syllabus: Topological vector spaces; Banach space; Hahn-Banach theorems; category and Baire's Theorem; the Banach-Steinhauss Theorem; The Open Application Theorem and Closed Graph Theorem; weak and weak topologies; Alaoglu-Banach's Theorem; reflective spaces; Hilbert spaces; operators; compact operators; or Spectral Theorem for compact auto-adjunct operators.

**Multivariate Analysis** (4 Credit Units, 60 Credit Hours)

Syllabus: Multivariate normal distribution; hypothesis tests for the mean vector; multivariate analysis of variance to one and two factors; hypothesis testing on covariance matrices; principal component analysis; factor analysis; conglomerate analysis; discriminant analysis; correspondence analysis, canonical analysis; multidimensional scheduling.

**Rings and Modules** (4 Credit Units, 60 Credit Hours)

Syllabus: Rings. Sub-rings. Ideals. Ratios quotients. Homomorphism of rings. Modules. Submodules. Modules quotients. Homomorphism of modules. Direct product of modules. Direct external summation of modules. Direct internal sum of modules. Projections. Exact sequences. Direct sum and exact sequences. Finite modules generated. Free modules. Modules on main domains. Modules on integrity domain. Torsion modules. Structural theorem (elementary divisor theorem). Projective modules and injection modules.

**Advanced Calculus **(4 Credit Units, 60 Credit Hours)

Syllabus: RN topology; partial and directional derivatives; derived as linear transformation; chain rule; the classes of differentiability; the Taylor formula; Theorem of inverse function; Implicit Function Theorem; Lagrange multipliers; multiple integrals; zero measurement sets; iterated integrals; Fubini's theorem; change of variable in multiple integrals; integral of line; the Green Theorem.

**Computer Graphics **(4 Credit Units, 60 Credit Hours)

Syllabus: History and applications of computer graphics. Geometry and computer graphics. Color spaces. Trim. Package graphics. Algorithms of hidden lines and surfaces. Rasterization of images. Filling of polygons. Tonalization of images: Flat, Gouraud, Phong. Modeling techniques: introductory view of the area.

**Reading Course** (2 Credit Units, 30 Credit Hours)

**Deconvolution and Inversion** (4 Credit Units, 60 Credit Hours)

Syllabus: The HWB Method. The direct and inverse problems: reflection, refraction, AVO, tomography. Data description and the Decomposition Theorem. Deterministic processes. Stochastic processes. The Bayes Theorem in inversion. The linear inverse problem. The inverse nonlinear problem. Parameterization. Minimizing function and hyperspace. Linearization. Regularization. Methods of the first order. Methods of the second order. Quality measures and matrixes resolution. Deconvolution by the Wiener and Kalman methods. The minimum-phase pulse and determination of the source signal. Historical cases.

**Dissertation Development** (6 Credit Units, 90 Credit Hours)

Syllabus: Writing about scientific work.

**Differential Equations **(4 Credit Units, 60 Credit Hours)

Syllabus: Theorems of Existence; Theorems of Existence and Oneness; The Carathéodory Theorem; continuous and differentiable dependence on initial data; linear equations; exponential matrix; classification of linear fields; singularities; linearization around singularities; stability of Liapunov; Poincaré-Bendixon Theorem.

**Partial Differential Equations** (4 Credit Units, 60 Credit Hours)

Syllabus: Classification of second order equations in two independent variables; problems of boundary conditions and initials. The method of separating variables; Fourier series; convergence of the Fourier series; application of heat conduction problems in a bar and vibrating string; double Fourier series; Dirichlet problems in a rectangle; the Fourier transform; the Schwartz space, temperate distributions.

**Internship Teaching **(2 Credit Units, 30 Credit Hours)

Syllabus: Aid in undergraduate courses offered by the Faculties of Mathematics and Statistics. Assistance to students with learning difficulties. Resolution of exercises and related activities.

**Mathematical Statistics **(4 Credit Units, 60 Credit Hours)

Syllabus: Random sample; statistical models; exponential family of distributions; statistics and estimators; sufficient statistics; sample distributions; efficient estimators; maximum likelihood estimators; asymptotic properties; confidence intervals; hypothesis testing; even more powerful tests; likelihood ratio test.

**Transportation Phenomena** (4 Credit Units, 60 Credit Hours)

Syllabus: Fundamentals of transport phenomena and thermodynamics. Equations of conservation of mass, energy and momentum. Diffusive and convective transport of heat and mass. Flow in permeable media.

**Applied Geostatistics** (4 Credit Units, 60 Credit Hours)

Syllabus: Spatial variability. Kriging, Co-Kriging. Space uncertainty. Geostatistical simulations.

**Computational Geometry** (4 Credit Units, 60 Credit Hours)

Syllabus: Computational complexity. Basic geometric algorithms (angles, distances, relative positions, orientation). Convex closure in two and three dimensions. Triangulations. Voronoi diagram and Delaunay triangulation. Search and localization problems. Intersection and visibility.

**Differential Geometry** (4 Credit Units, 60 Credit Hours)

Syllabus: Curves in space: local theory of curves parameterized by arc length; Frenet formulas; Fundamental Theorem of Curves in Space; the local canonical form; global properties of flat curves; regular surfaces of R3, the normal application of Gauss and its fundamental properties; the main curvatures. Gaussian and mean; controlled surfaces and minimum surfaces; Gauss's Theorem Egregium; the exponential application; complete surfaces and Hopf-Rinow's Theorem, first and second bow compliance variations; the Gauss-Bonet Theorem.

**Bayesian Inference**

Syllabus: A priori distributions. Subsequent distributions. Point estimators. Credibility intervals. Computational methods.

**Introduction to Geometric Modeling **(4 Credit Units, 60 Credit Hours)

Syllabus: Overview of the modeling area. Basic concepts and tools. Representation of curves. Representation of surfaces. Solid representation.

**Introduction to Matroid Theory** (4 Credit Units, 60 Credit Hours)

Syllabus: Definition of Matroid; independent sets; circuits; base; station; duality; children; connectivity; graphite matroids; representable matroids.

**Introduction to Graph Theory** (4 Credit Units, 60 Credit Hours)

Syllabus: Definition of graphs; ways; cycles; circuits; connectivity; trees; forests; planar graphs; duality; coloring of graphs; color number; flows; ambitious algorithm.

**Measurement and Integration **(4 Credit Units, 60 Credit Hours)

Syllabus: Measure of Lebesque in RN; Fatou's lemma; Monotonic Convergence Theorem; Dominated Convergence Theorem; Space LP. The L2 Space. Riesz-Fischer's Theorem; bases; absolutely continuous functions; differentiation in R; duality between LP spaces.

**Numerical Methods of Finite Differences** (4 Credit Units, 60 Credit Hours)

Syllabus: Approximation of derivatives by finite differences. Finite difference methods for ordinary equations. Finite difference methods for parabolic, elliptic and hyperbolic partial differential equations. Convergence, consistency and stability.

**Regression Models **(4 Credit Units, 60 Credit Hours)

Syllabus: General linear model; minimum squares method; inference; exponential family of distributions; generalized linear models; estimation by the maximum likelihood method; hypothesis testing; deviation analysis; models for binary responses; models for contingency tables; models for counting.

**Generalized Linear Models** (4 Credit Units, 60 Credit Hours)

Syllabus: Exponential family with one parameter. Algorithm of estimation. Deviance and Pearson X2 statistics generalized. Deviance analysis and model selection. Hypothesis testing. Confidence Intervals. Techniques for verifying model fit.

**Probability** (4 Credit Units, 60 Credit Hours)

Syllabus: Random experiment; probability space, events; conditional probability; random variable; main probability distributions; generating moment function; characteristic function; weak and strong laws of large numbers; Central Limit Theorem.

**Processing, Signs and Images** (4 Credit Units, 60 Credit Hours)

Syllabus: Signal aspects and discrete time systems. Discrete Fourier series and discrete Fourier transform. Discrete Hilbert transform. Flow diagram of digital filters. Matrix representation of digital filters. Techniques of design of digital filters.

**Stochastic Processes **(4 Credit Units, 60 Credit Hours)

Syllabus: Introduction and fundamentals. Construction of Markov chains. Invariant measures. Loss of memory and convergence to balance. Study of some special processes; Poisson, Birth and Death, branching, renewal, Markovian processes and Salto, processes of diffusion.

**Scientific Programming** (4 Credit Units, 60 Credit Hours)

Syllabus: Introduction to MatLab. Basic concepts. Control flow. Statistical measures. Solutions of linear equations. Interpolation and curve fitting. Polynomial analysis. Initial value problem for ordinary differential equations. Numerical solutions of systems of nonlinear equations. Limit value problems for ordinary differential equations. Numerical solutions for partial differential equations. Finite differences. Finite elements. Signal processing.

**Seminary **(2 Credit Units, 30 Credit Hours)

Syllabus:

**Item Response Theory**

Syllabus: Classical Theory of Measurement. Introduction to latent trait. Main cumulative models. Examples of TRI applications in different areas of knowledge. Estimation of the parameters of the items (calibration). Dichotomous model and a single group. Estimation of proficiencies (latent trait). Construction and interpretation of the measurement scale. Applications in practical problem solving and scientific modeling based on the analysis of scientific articles from journals and congress annals. Data Simulation and Computational Implementation.

**Theory of Galois **(4 Credit Units, 60 Credit Hours)

Syllabus: Extensions of bodies; Finite extensions and algebraic extensions; Normal extensions and separable extensions; bodies of decomposition; Galois groups; Galois Fundamental Theorem; cyclotomic bodies; finite bodies; solubility by radicals; buildings with ruler and compass; transcendent extensions.

**Critical Point Theory** (4 Credit Units, 60 Credit Hours)

Syllabus: Introduction to distributions theory. Notions about Sobolev spaces. Sobolev Immersion Theorems. Definition of weak solutions for elliptical problems. Existence of a weak solution for linear problems using the Lax-Milgran theorem. Spectral theory of the Laplacian operator. Theorems of the Mini-max type.

**Topics in Statistics - Statistical Quality Control** (4 Credit Units, 60 Credit Hours)

Syllabus: Basic Statistical Quality Control Tools. Stratification. Check sheets. Check sheet for the distribution of the production process. Check sheet for item. Check sheet for troubleshooting. Causes check sheet. Customer satisfaction check sheet. Diagram of Ishikawa. Construction of the Ishikawa Diagram. Exercises - Ishikawa diagram. Pareto's chart. Construction of the Pareto Chart. Analysis and use of the Pareto Chart. Histogram. Histogram construction. Comparison of the histogram with specification limits. Correlation diagram. Positive correlation. Negative correlation. Absence of correlation. Construction of the correlation diagram. Calculation of the linear correlation coefficient. Overview of control charts. Introduction. Some of the top searches involving control charts. Principles of control charts. Control chart in practice. Planning a control chart. Benefits of using control charts. Types of control charts. Control charts for variables. Introduction. Control charts to monitor process dispersion. Standard Deviation Chart (Chart S). Graph of Variance (Chart S2). Amplitude Chart (Graph R). Control Charts to monitor the process level. Average Chart. Median Chart. Process capacity indices. Introduction. Cp index. Hypothesis test and the Cp index. Cpu, Cpl and Cpk indexes. Principles of estimation in statistical quality control. Introduction. Properties of the estimators. Estimating the dispersion of the process. Estimating the process level. Quarte estimator based on Quartiles applied to CEQ. Introduction. Qbar Applied to the Control Charts for variables to monitor process dispersion. Qbar applied to the standard deviation graph. Qbar applied to the variance plot. Qbar applied to the amplitude plot. Qbar applied to the control charts for variables to monitor the process level. Qbar applied to the average graph. Qbar applied to the Median Chart. Qbar applied to the capacity indexes. Applied to the development of new tools for CEQ. Chart Qbar. Graph Q2. CCO and ARL for Shewhart Control Charts. Error Type I and Error Type II. CCO and ARL for Level and Dispersion Control Charts. CCO and ARL for Average Control Chart. CCO and ARL for Control Chart of the Median CCO and ARL for Variance Control Chart. CCO and ARL for Standard Deviation Control Chart. The Regression Control Chart. Introduction. Elements of the Regression Control Graph. Building the Regression Control Graph. Analyzing the Regression Control Graph. Multivariate Control Charts (GCM). Hotelling T2 pictures. The method of decomposition. Comparison with other methods of decomposition. Control Charts for attributes. Graph for nonconforming fraction or Graph. Chart for the number of non-conforming items or Chart np. Control Graphs p and np with variable sample size. Control graph p and np with mean sample size. Standardized Control Chart p. Graph for Number of Defects or Graph c. Graph for the Average Number of Defects per Unit or Graph u. Control chart u with mean sample size. Standardized Control Chart u. Control Charts for individual measurements. Control Chart for Mobile Range or MR Chart. Control Chart for individual observations or Chart x 12 Control Charts for small process changes. CUSUM control chart. Chart of CUSUM Tabular. EWMA Control Chart.

**Special Topics: variational calculation** (4 Credit Units, 60 Credit Hours)

Syllabus: n-th variation. Required condition for local extreme. Euler-Lagrange equation. Sufficient condition for extreme. Semicontinuity and results of existence. Regularity of critical points. Weierstrass generalized theory of existence. Applications.

**Special Topics in Algebra **(4 Credit Units, 60 Credit Hours)

Syllabus: Algebras on a body, structure constants, ideals, non-associative algebras, multiplication algebras, derivation algebras, Bernstein algebras, algebra character, Peirce decomposition, invariant subspaces, subspaces of invariant dimension.

**Special Topics of Analysis **(4 Credit Units, 60 Credit Hours)

Syllabus: Uniqueness Theorem. Power method. Tricomi problem study. Introduction to the Schwartz distribution and the Sobolev spaces. Free solution. Variational formulation of the Dirichlet and Neumann problems. Notions about regularity of weak solutions. Study of the weak solutions of the wave equations and the heat transfer, in the linear case. The Galerkin Approach Method.

**Special Topics of Statistics** (4 Credit Units, 60 Credit Hours)

Syllabus: Introduction and main concepts in Item Response Theory; parameters associated with the models; classical and Bayesian estimators for the parameters; computational aspects in the estimation.

**Special Topics of Differential Geometry **(4 Credit Units, 60 Credit Hours)

Syllabus: Advanced topics and recent development of Differential Geometry.

**Special Topics in Applied Mathematics** (4 Credit Units, 60 Credit Hours)

Syllabus: Basic concepts. Initial value problem for ordinary differential equations. Numerical solutions of systems of nonlinear equations. Limit value problems for Ordinary differential equations. Numerical solutions for partial differential equations. Finite differences. Finite elements.

**Special Topics of Probability **(4 Credit Units, 60 Credit Hours)

Syllabus: Introduction and main concepts in Percolation Theory; critical density in regular percolation; oriented percolation of sites; computational aspects in the implementation of a percolation simulator.

**Special Topics in Algebra** (4 Credit Units, 60 Credit Hours)

Syllabus: Algebras on a body. Constants of structure. Ideals. Non-associative algebras. Algebra of multiplications. Lead algebra. Algebras of Bernstein. Character of an algebra. Peirce's decomposition. Subspaces invariant. Subspaces of invariant dimension.

**Special Topics in Differential Equations** (4 Credit Units, 60 Credit Hours)

Syllabus: Notion of Schwartz distribution and Sobolev spaces. Duality. Equivalent standards. Sobolev spaces immersions. Faedo-Galerkin method. Parabolic equations. Hyperbolic equations.

**Special Topics in Partial Differential Equations I** (4 Credit Units, 60 Credit Hours)

Syllabus: Parabolic equations: resolution techniques. Hyperbolic equations: Resolution techniques.

**Special Topics in Partial Differential Equations II** (4 Credit Units, 60 Credit Hours)

Syllabus: System of Navier-Stokes equations. Method of monotony. Compactness method. Nonlinear telegraph equation.

**Special Topics in Statistics - Statistical Quality Control** (4 Credit Units, 60 Credit Hours)

Syllabus: Basic Statistical Quality Control Tools. Overview of control charts. Control charts for variables. Process capacity indices. Principles of estimation in statistical quality control. Sigma estimator based on Quartiles Applied to CEQ. CCO and ARL for Shewhart Type Control Charts. Regression Control Chart. Multivariate Control Charts (GCM). Control Charts for attributes. Control Charts for individual measurements and Control Charts for small process changes.

**Special Topics in Bayesian Inference**

Syllabus: A priori distributions. Subsequent distributions. Point estimators. Credibility intervals. Computational methods.

**Special Topics in Neural Networks** (4 Credit Units, 60 Credit Hours)

Syllabus: Introduction. Principles of neurocomputation and learning process. Perceptron single layer. Perceptrons of multiple layers. Radial Base Function Networks (RBF). Self-organizing maps.

**General Topology **(4 Credit Units, 60 Credit Hours)

Syllabus: Complete metric spaces; completing a metric space; Baire's Theorem; successive approximations; topological spaces; bases of a topology; function spaces; compact spaces; Tychonov's theorem; Ascoli's theorem; Stone-Weierstrass's Theorem; Topology Quotient; normal spaces; Urysohn Metering Theorem; homotopy; the fundamental group; induced homeomorphism; the fundamental group of the circle; index of a closed curve; overlapping spaces.

**Computer Vision and Digital Imaging **(4 Credit Units, 60 Credit Hours)

Syllabus: Digital images. Display and image acquisition devices. Color. Spatial transformations and interpolation in intensity. Transforming images. Image enhancement. Filters. Segmentation of images. Descriptors. Pattern Recognition. Applications.