Courses in Mathematics
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Courses in Mathematics All courses have been taken at the Faculty of Mathematics and Statistics of the Technical University of Catalonia (UPC) Academic year 2000-2001 Algebraic Computation Description and contents -Introduction to Arithmetic -Groups -Rings and Ideals -Polynomials and finite fields Principal Textbooks Biggs, N.L. . Discrete Mathematics Vicens-Vives 1993 Childs, L. Springer-Verlag 1979 Concrete Introduction to Higher Algebra Gallian, J.A. Contemporary Abstract Algebra D.C. Heath and Company 1994 Rosen, K.H. Elementary Number Theory and its Applications Addison-Wesley Number of credits (ECTS): 6 Linear Algebra Description and contents -Structures: introduction to groups, rings and fields -Vector Spaces -Polynomials -Matrices and systems of equations -Determinants -Linear maps -Diagonalized form -Jordan form Principal Textbooks M. Castellet, I. Llerena Àlgebra Lineal i Geometria UAB F. Puerta Àlgebra Lineal 1988 Edicions UPC 1993 1993 Number of credits (ECTS): 6 Calculus 1 Description and contents -Introduction to real numbers and their axioms -Sequences and series of real numbers -Limit of a real-valued function with real variable -Continuous, derivable and integrable functions Principal Textbooks Bartle, G.B., Sherbert, D.R. Introducción al análisis matemático de 2a Limusa una variable 1996 Ortega, J.M. Introducció a l'anàlisi matemàtica. Publicacions UAB 1990 Burgos, J Cálculo infinitesimal de una variable McGraw-Hill 1994 Spivak, M Càlcul Infinitesimal Reverté 1995 Number of credits (ECTS): 6 Calculus 2 Description and contents -Topology in R^n -Continuity of a function of multiple variables -Differentiability of functions of multiple variables -Theorems on differentiable functions -Integration with multiple variables Principal Textbooks J.E. Marsden, M.J. Hoffman Análisis clásico elemental Addison Wesley Iberoamericana J.E. Marsden, A.J. Tromba Cálculo vectorial Addison Wesley Longman 1998 J. de Burgos Cálculo infinitesimal en varias McGraw-Hill variables F del Castillo Análisis Matemático II Number of credits (ECTS): 6 Alhambra Universidad 1998 1995 Academic year 2001-2002 Calculus 3 Description and contents - Curves and surfaces; scalar and vector fields, integrals -Vector analysis and classical theorems of integration: Green, Stokes, Gauss -Differential forms and tensor analysis -Integration and Stokes Theorem on forms -Holomorphic functions -Reminder Theorem Principal Textbooks Dubrovin,B.A, S.P.Novikov, A.T.Fomenko Modern Geometry. Methods and Applications, Vol. 1 Springer 1984 Marsden, J.E.; Tromba, A.J Cálculo vectorial. Addison-Wesley Iberoamericana, Argentina 1991 Spivak, M. Cálculo en variedades Reverté 1979 Ahfolrs, L.V. Complex Analysis Arnold, V.I. Mathematical Methods of Classical Mechanics 2 McGraw Hill Edition Springer-Verlag Number of credits (ECTS): 6 Geometry Description and contents -Classical geometry -Geometry in metric spaces -Geometry in the projective space -3-dimensional figures Principal Textbooks Xambó, S. Geometria 2a Edicions UPC 2001 Berger, M. Geometry (2 vols.) Springer-Verlag 1987 Santaló, Ll. Geometría Proyectiva EUDEBA 1966 1966 1989 Reventós, A. Geometria Projectiva de Guzmán, M. Mirar y ver Edicions UAB 2000 Alhambra 1977 Number of credits (ECTS): 6 Numerical methods 1 Description and contents -Error analysis -Function interpolation and its application -Linear systems -Eigenvectors and Eigenvalues Principal Textbooks Volum 7 de Manulas de la Univ. Autònoma de Barcelona; en castellà Labor, 1993 1991 Edicions UPC 1994 Introducción al análisis numérico Ed. Vicenc Vives 1977 Elementos de cálculo numérico Ed. Paraninfo, Madrid 1977 Aubanell, A., Benseny, A., Delshams, A. Eines bàsiques de càlcul numèric Bonet, C. i altres Càlcul Numèric Froberg, C. E. Demidovich, B, Maron, I. Grau, M., Noguera, Càlcul numèric M. Aula teòrica 23 Aula Ed. UPC teòrica 1 Number of credits (ECTS): 6 Probability and Statistics Description and contents -Probability space -Random variable -Moments and moment generating function of a random variable -Random vectors and sequences of random variables; Central limit theorem Principal Textbooks 1993 Ash, R.B. Basic Probability Theory Wiley 1970 Baldi, P. Calcolo delle Probabilita e Estatistica McGraw-Hill Libri 1993 Chung, K.L. Elementary Probability Theory with Stochastic Processes Springer-Verlag 1979 Rohatgi, V.K. An Introduction to Probability Theory and Mathematical Statistics Wiley 1976 Sanz, M. Probabilitats UB 1999 Number of credits (ECTS): 6 Introduction to the mathematics of engineering sciences Description and contents -Diffential equations and applications -Fourier analysis -Probability and stochastic processes Principal Textbooks M.Braun Differential Equations and their application J.Marsden,M.Hoffman Analisis Clasico Elemental SpingerVerlag Addison Wesley 1998 1994 Leon Garcia,A. Probability and random processes for electrical engineering Addison Wiley Papoulis,A. Probability, random variables and stochastic processes McGraw-Hill 1991 Number of credits (ECTS): 6 Real Analysis Description and contents -Sequences and series of functions -Space of continuous functions; Theorem of Ascoli-Arzelá; Theorem of Stone-Weierstrass -Lebesgue integration -Trigonometric Fourier series Principal Textbooks Bartle, R. The elements of integration and Lebesgue measure John Wiley 1995 Batlle, C. i Fossas, E. Anàlisi Real -Apunts Ahlens, S.L. 2002 Folland, G.B. Fourier Analysis and its applications Brooks/Cole Publ. 1992 Marsden, J. i Hoffmann, M. Análisis Clásico Elemental Addison-Wesley 1998 Sprecher, D.A. Elements of Real Analysis Dover 1987 Number of credits (ECTS): 6 Topology Description and contents -Metric spaces -Topological spaces -Construction of topological spaces -Compact and connected spaces -Separation in topological spaces: spaces of Fréchet and Hausdorff, Urysohn lemma;Theorems of Urysohn and Tietze -Homotopy of continuous functions -Topology of the plane Principal Textbooks Armstrong, M.A. Topología Básica Reverté 1987 Kosniowski, C. Topología Algebraica Reverté 1986 Munkres, J.R. Topología PrenticeHall 1975 UPC 2004 PWSKENT 1992 Pascual, P.; Roig, Topologia A. Sieradski, A. An introduction to topology and homotopy Number of credits (ECTS): 6 Academic year 2002-2003 Ed. Virtuals Differential Equations 1 Description and contents -Types of ordinary differential equations -Fundamental theorems -Linear systems and equations -Qualitative theory Principal Textbooks Arnold, V.I. Ordinary Differential Equations M.I.T. Press, Cambridge, MA 1973 Braun, M. Differential Equations and Their Applications Springer-Verlag 1993 Coddington, E.A. and Levinson, N. Theory of Ordinary Differential Equations McGraw-Hill 1955 Guzman, M. De. Ecuaciones diferenciales ordinarias: Teoría de estabilidad y control Alhambra, Madrid 1975 Martínez Carracedo, C. Introducción a las ecuaciones y Sanz Alix, M.A. diferenciales ordinarias Reverté, Barcelona 1991 Number of credits (ECTS): 6 Numerical Methods 2 Description and contents -Basic concepts on approximation of functions -Approximation of functions and techniques of minimum quadratic error -Solving problems of minimum quadratic error -Techniques of interpolation -Numerical integration -Solving non-linear equations -Iterative methods for systems of equations Principal Textbooks Axelsson, 0 Iterative solution methods . Cambridge University Press 1983 Dahlquist, G.; Björck, A Numerical methods. Prentice Hall 1975 Hamming, R.W Numerical methods for scientists and engineers Dover 1986 Hildebrand, F.B Introduction to numerical analysis Dover 1987 Ralston, A.; Rabinowitz, P A first course in numerical analysis Mc Graw-Hill 1978 Number of credits (ECTS): 6 Statistical Inference Description and contents -Convergence of sequences of random variables and limit theorems -Ways of exploring data -Statistical structures -Estimation of parameters -Proofs of hypothesis -Non-parametrical methods -Introduction to the multiple linear model Principal Textbooks Bickel, P.J., Doksum, K.A. Mathematical Statistics: Basic Ideas and Holden-Day Selected Topics Breiman, L. Statistics Houghton and Mifflin 1973 Casella, G., Berger, R.L. Statistical Inference Duxbury Press 1990 DeGroot, M.H. Probabilidad y Estadística Addison-Wesley 1988 Kalbfleisch, J.G. Probability and Statistical Inference Springer Number of credits (ECTS): 6 Operative Investigation Description and contents -Introduction: mathematical modelation -Linear models -Network flows -Models of integer programming -Non-linear optimization -Game theory 1977 1985 Principal Textbooks R.K. Ahuja; T.L. Magnanti, J.B. Orlin Network Flows: Theory, Algorithms, Applications. M. Padberg Linear Optimization and Extensions. Bazaraa, M.S., Sherali, and Shetti, C.M. Prent.-Hall 1993 Springer-Verlag 1999 Nonlinear Programming: Theory and Algorithms John Wiley and Sons 1993 Vanderbei, R.J Linear Programming, Foundations and Extesions. Kluwer Academic Publishers 1996 Williams, H.P. Model Building in 3rd Mathematical Programming edition Wiley 1993 2nd edition. Number of credits (ECTS): 6 Academic year 2003-2004 Differential Geometry 1 Description and contents -Regular differentiable manifolds -Curves in the plane and in the 3-d space -Surfaces 1: parametrizations -Surfaces 2: maps of Gauss and Weingarten, symbols of Christoffel, Theorema egregium,etc -Surfaces 3: geodesics, geodesic curvature, etc. -Surfaces 4: geodesic coordinates, theorem of Gauss-Bonet,etc. Principal Textbooks Carmo, M.P. do Geometría diferencial de curvas y superficies Cordero,L; Geometría diferencial de Fernández, M.; Gray, curvas y superficies A. Alianza Universidad 1990 Addison-Wesley Iberoamericana 1995 Girbau, J. Geometria diferencial i relativitat Publicacions de la Universitat de Barcelona 1993 Lipschutz, M. Geometria diferencial McGraw-Hill 1991 Novikov, S.P.; Fomenko, A.T. Basics elements of differential geometry and topology Kluwer 1990 Number of credits (ECTS): 6 Complex Analysis Description and contents -Holomorphic functions -Local Cauchy Theory -Global Cauchy Theory -Theorem of Riemann, principle of Schwarz, Problem of Dirichlet, etc -Representation of fields; flows Principal Textbooks · Ahlfors, L.V. Complex Analysis ·Conway, J.B. Functions of one Complex Variable 3a ed. McGraw-Hill 1979 SpringerVerlag 1978 Churchill, Ruel V. y J. Warm Variable Compleja y Brown Aplicaciones 5a ed. McGraw-Hill 1992 Marsden, J.E. y M.J. Hoffman Basic Complex Analysis 3a ed. W.H. Freeman Rudin, W. Análisis Real y Complejo 1999 McGraw-Hill 1987 Number of credits (ECTS): 6 Differential Geometry 2 Description and contents -Differentiable manifolds -Tangent vectors -Submanifolds -Differential equations -Tensor fields -Some applications: Lie groups, Riemaniann geometry, etc Principal Textbooks J. M. Lee Introduction to smooth manifolds Springer 2003 L. Conlon Differential manifolds: a first course Birkhäuser 1993 W. M. Boothby An introduction to differentiable manifolds and riemannian geometry Academic Press 1986 F. W. Foundations of differentiable manifolds and Lie Springer 1983 Warner groups M. Spivak A comprehensive introduction to differential geometry, vol. I Publish or Perish 1979 Number of credits (ECTS): 6 Numerical Methods 3 Description and contents -Programming languages and mathematical software: Fortran 90/95 , Matlab , Maple -Qualitative study of ordinary differential equations -Equations by differences -Numerical resolution of problems with initial conditions: methods of Adams, Bashfort and Moulton, methods of Runge-Kutta and Runge-Kutta-Fehlberg, etc -Numerical resolution of boundary-value problems -Simplectic Integration Principal Textbooks Butcher, J. The Numerical Analysis of Ordinary Differential Equations John Wiley Grau, M.; Noguera, M. Càlcul Numèric Edicions UPC 1993 Keller, H.B. Numerical Methods for two-point Boundary-Value Problems Dover 1992 Lambert, J.D Numerical Methods for Ordinary Differential Systems John Wiley 1991 Sanz-Serna, J.M.; Calvo, M.P. Numerical Hamiltonian Problems Chapman & Hall 1994 Number of credits (ECTS): 6 Academic year 2004-2005 Abstract Algebra Description and contents -Groups -Rings -Field extensions -Galois Theory 1987 Principal Textbooks Fenrick, M. H. Introduction to the Galois correspondence Birkhäuser Rotman, J. An introduction to the theory of groups Springer-Verlag 1994 Rotman, J. Galois Theory Springer-Verlag 1990 Stewart, I. Galois Theory Chapman and Hall 1989 Xambó, S.; Delgado, F.; Fuertes, C. Introducción al Álgebra (2 vols) Complutense 1993 Lang,S. Algebra Springer 2002 1992 Number of credits (ECTS): 6 Differential Equations 2 Description and contents -Partial differential equations ( not more information available yet) Principal Textbooks Number of credits (ECTS): 6 Functional Analysis Description and contents -Normed spaces -Hilbert spaces -Duality -Compact operators -Applications: Sobolev spaces, etc Principal Textbooks H. Brézis Análisis Funcional Alianza Editorial 1984 W. Rudin Functional Analysis McGraw-Hill S. Lang Real and Functional Analysis 1973 3ª Springer-Verlag 1993 F. Hirsch & G. Lacombe Elements of Functional Analysis Springer-Verlag 1999 Number of credits (ECTS): 6 Algebraic Topology Description and contents -Simplicial homology - Homologic algebra -Singular homology -Sequence of Mayer-Vietoris -Applications to the topology of the ordinary space -Classification of surfaces Principal Textbooks Greenberg,M.; Harper,J. Algebraic Topology Benjamin 1981 Hatcher,A. Algebraic Topology Cambridge UP 2002 Munkres,J. Elements of Algebraic Topology Addison-Wesley 1984 Navarro,V; Pascual.P. Topologia Algebraica Edicions UB Vick,J. Homology theory Springer Verlag 1994 Number of credits (ECTS): 6 1999
