Part One
First Year
Full year
subjects
- Computer Science I 7 Cred. E.C.T.S.
First term
subjects
- Real Analysis I 9 Cred. E.C.T.S.
- Geometry I 5.5
Cred. E.C.T.S.
- Introduction to Algebra
5.5 Cred. E.C.T.S.
Second term
subjects
- Calculus in Rn 5.5
Cred. E.C.T.S.
- Diferential Equations I
4 Cred. E.C.T.S.
- Elements of Differential Geometry and Topology 5.5 Cred. E.C.T.S.
- Introduction to Probability Calculus
5.5 Cred. E.C.T.S.
- Numerical Methods I 4.5
Cred. E.C.T.S.
[1]Free
configuration 8 Cred. E.C.T.S.
Second Year
First term
subjects
- Matrix Algebra: Canonic Forms 6 Cred. E.C.T.S.
- Real Analisis II 7 Cred. E.C.T.S.
- Probability Calculus I and Mathematical Statistics 7 Cred. E.C.T.S.
- Geometry II 5 Cred. E.C.T.S.
- Numerical Methods II 5 Cred. E.C.T.S.
Second
term subjects
- Real Analysis III 6 Cred. E.C.T.S.
- Differential Geometry and Topology
5 Cred. E.C.T.S.
- Numerical Methods III
3 Cred. E.C.T.S.
- Theory of Algebraic Equations 4.5 Cred. E.C.T.S.
Optional subjects 6.5 Cred. E.C.T.S.
Part One Optional Subjects
First term
subjects
- Comprehension of English Scientific Texts in
Mathematics 6.5 Cred. E.C.T.S.
- Computer Science II (Data Structure)
3.5 Cred. E.C.T.S.
- Introduction to Optimization 3.5 Cred. E.C.T.S.
- Statistical Methods: Use in Sciences
3.5 Cred. E.C.T.S.
Second
term subjects
- Algebraic Curves 3.5
Cred. E.C.T.S.
- Metric Geometry 3.5
Cred. E.C.T.S.
- Problems in Mathematics 3.5 Cred. E.C.T.S.
Second Part
Third Year
First term
subjects
- Complex Analysis I 7.5
Cred. E.C.T.S.
- Differential Equations II
3.5 Cred. E.C.T.S.
- Basics in Physics 4.5 Cred.
E.C.T.S.
Second
term subjects
- Probability Calculus 6.5 Cred.
E.C.T.S.
Optional subjects 11.5
Cred. E.C.T.S.
Free configuration 8 Cred.
E.C.T.S.
Fourth Year
First term
subjects
[2]Second
term subjects
Optionals 42 Cred. E.C.T.S.
Free configuration 10.5 Cred. E.C.T.S.
Part Two Optional Subjects
First term
subjects
- Mathematic Didactics in Baccalaureate
4.5 Cred. E.C.T.S.
- Experimental Design 3.5
Cred. E.C.T.S.
- Equations in Partial Derivatives 4.5 Cred.
E.C.T.S.
- Statistical Inference II
3.5 Cred. E.C.T.S.
- Stochastic Processes 3.5
Cred. E.C.T.S.
- Representation of Knowledge 4.5 Cred. E.C.T.S.
- Theory of
Algorithms 4.5
Cred. E.C.T.S.
- Theory of Measure 4.5
Cred. E.C.T.S.
- Algebraic Topology 4.5
Cred. E.C.T.S.
- Computational Topology
4.5 Cred. E.C.T.S.
- Image Digital Processing: Algorithms and
Applications 3.5 Cred. E.C.T.S.
Second
Term Subjects
- Complex Analysis II 3.5
Cred. E.C.T.S.
- Astrophysics 4.5
Cred. E.C.T.S.
- Astronomy of Position
4.5 Cred. E.C.T.S.
-
Numerical Calculus II 4.5
Cred. E.C.T.S.
- Computational Statistics
3.5 Cred. E.C.T.S.
- Geodesy 4.5
Cred. E.C.T.S.
- Algebraic Geometry 4.5
Cred. E.C.T.S.
- Computational Geometry 3.5 Cred. E.C.T.S.
- Differential Geometry
4.5 Cred. E.C.T.S.
- Mechanics 4.5
Cred. E.C.T.S.
- Practicals in Teaching
4.5 Cred. E.C.T.S.
- Numerical Solution of Partial Derivative Equations 4.5 Cred. E.C.T.S.
- Theory of Rings 4.5
Cred. E.C.T.S.
- Automata Theory and Formal Languages
4.5 Cred. E.C.T.S.
Computer Science I
Part One; Full year subject; 4 hours per week. 7 Cred.
E.C.T.S.
Algorithms. Data Structure. Programming Languages.
Uses in Mathematics.
Professor: Mercedes MARTÍNEZ DURBÁN.
Teaching Method:
Assesment Method:
Real Analysis I
Part One; First term subject; 8 hours per week. 9
Cred. E.C.T.S.
The real number set.
Successions of real numbers.
Continuity of real functions of real variable.
Functional limit.
Derivatives of real functions of real variable.
Professor: Enrique DE AMO ARTERO.
Teaching Method: Theoretical and practical lectures. Seminars
Assessment Method: Written.
Geometry I
Part One; First term subject; 5 hours per week. 5.5
Cred. E.C.T.S.
Basic elements in the solution of linear equations
systems.
Basic elements of
finite-dimension vector spaces as such as bases and generator systems,
linear maps and dual vector space. Study of alternate multilinear forms and
concept of determinant. Basic properties of determinants and their calculation.
Matrix diagonallization. Basic concepts of affine spaces with special mention
to dimensions 2 and 3. Definition of projective space and projectivization of an
affine space.
Professor: María Luz PUERTAS
GONZÁLEZ.
Teaching Method: Theoretical and practical lectures. Solution of
problems by the student in the lecture hall.
Assessment Method: Written examination. Positive assessment for problems
solved during lectures.
Part One; First term subject; 5 hours per week. 5.5
Cred. E.C.T.S.
Sets: Boolean Algebra of the set parts. Cartesian
product. Correspondences. Mappings. Relations. Equivalence and order relations.
Natural and Integer numbers. Euclidean algorithm and applications. Introduction
to groups: General concepts. Subgroup grating. Module classes of a subgroup. Normal subgroups. Quotient group. Direct product of groups. Homomorphisms of groups. Isomorphy
theorems. Cyclic groups. . Symmetrical groups. Arithmetics in euclidean
domains.; rings of polynomials: General Concepts of rings. Subrings and ideals.
Ring homomorphisms. Theorems of isomorphy. I.D.. Body of fractions. Divissibility.
U.F.D., P.I.D. and E.D. Rings of polynomials. K[x]. K[x] is a E.D.. Factorization
of polynomials. Irreducibility criteria. The ring of polynomials in some
uncertainities. Symmetrical polynomials
Professor: Antonio LIROLA TERREZ.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination.
Second
half part subjects
Calculus in RN
Part One; Second term subject; 5 hours per week. 5.5
Cred. E.C.T.S.
Series of real Numbers: convergent series. Absolute
and conmutative convergence. Convergence criteria for non-negative term series.
Direchlet and Abel criteria. The Real Plane: forms of expression of complex
numbers and calculation. Topology in Rn. Vector Space Rn.
Euclidean Norm. Open and closed sets. Interior, exterior and frontier points. Adherent
points and clustering points. Compact sets. Multivariable functions: Functional
limit. Continuity and uniform continuity. Partial derivatives and matrix.
Jacobian. Derivatives of superior order. Functions of class CN. The
Taylor´s theoreme. Calculus of extremi. Riemann Multiple Integral: double
integral of bound sets. Zero content
and zero measure. Integration on bound sets. Area of a plane set. Variable
replacement. Multiple integral. Mappings.
Professors: El Amin KAIDI LHACHMI y
Antonio MORALES CAMPOY.
Teaching Method: Formal and practical lectures, together with some
practicals
Assessment Method: Written examination and presentation of works.
Differential equations I
Part One; Second term subject; 3 hours per week. 4
Cred. E.C.T.S.
Basic concepts. Separate variables equations and
equations reducible to them. Homogeneous equations and equations reducible to
them. Linear equations of firt order. Exact differential equations. Integrating
factor. Lagrange´s and Quirant´s equations. Composition of differential
equations of superior order. Order reduction of equations. Homogeneous and
non-homogeneous linear equations with constant coefficients of n-order. Euler´s
equations. Linear differential equations of variable coefficients: Lagrange´s
constant variation method.
Professor: Antonio JIMÉNEZ VARGAS.
Método de Enseñanza:
Theoretical and practical lectures.
Método de examen: Written
Elements of Differential Geometry
and Topology
Part One; Second term subject; 5 hours per week. 5.5
Cred. E.C.T.S.
Metric spaces and topological spaces. Convergence,
continuity and homeomorphisms. Basis of environments and opens. Construction of
topological spaces: Elements of topology, topological subspaces, products and
quotients. Topological properties: Conection, compacity. Compactations.
Introduction to regular curves.
Professor: David LLENA CARRASCO.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination.
Introduction to the Calculus of
Probabilities.
Part One; Second term subject; 5 hours per week. 5.5 Cred. E.C.T.S.
An introductory subject which briefly tackles an
introduction to Descriptive Statistics, followed by some basic elements of
probability from the Kolgomorov Axiomacy. Following, the concept of random
variable is treated, basically the study of the Distribution Function, types of
variables, changes of variable and their characteristics. The subject ends with
an overview of the different types of distribution models.
Professor: Francisco HERRERA CUADRA.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination.
Numerical Method I
Part One;
Second term subject; 4 hours per week. 4.5 Cred. E.C.T.S.
In many scientific disciplines the presence of
mathematical problems requiring the solution of large linear equation systems,
mostly difficult to be explicitally solved, is frequently found.
This subject develops direct and indirect solution
methods for linear equation systems, analyzing the method´s convergence order. A
basic study of Matrix Algebra for its
use in some practical problems like population matrixes, constant coefficients
differential equation systems a.o. are also introduced.
Professor: Manuel GÁMEZ CÁMARA;
Antonio ANDÚJAR RODRÍGUEZ.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination. Task works.
First term subjects
Matrix Algebra: Canonic Forms.
Part One; First term subject; 6 hours per week. 6
Cred. E.C.T.S.
Rings and Modules: Factorization in a domain of
principal ideals. Basic concepts of
the theory of modules. Special classes of modules. Theorem of structures. Submodules
of free modules. Theorems of descomposition. Applications of the Structure
Theoreme: Finitelly generated abelian groups. Canonic forms of matrixes. Effective
calculus of canonic forms. Multilinear Algebra: Tensor product of modules and
algebras: tensor algebra of one module. External algebra of one module:
determinants.
Professor: María Jesús ASENSIO DEL
ÁGUILA.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written/oral examination.
Part One; First term subject; 7 hours per week. 7
Cred.E.C.T.S.
Analysis of some real variables: Lebesque´s Integral.
Vector functions and sets: continuity,
differentiability.
Extremi. Inverse functions and Implicits.
Conditioned extremi.
Professor: Antonio JIMÉNEZ VARGAS.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination.
Calculus of Probabilities and
Mathematical Statistics.
Part One; First term subject; 7 hours per week. 7
Cred.E.C.T.S.
This subject is a continuation of the subject
"Introduction to the Calculus of Probabilities". It starts with some
basic elements of the Measurement Theory in order to harshen the approach to
Probability. Following, the study of bidimensional random variables is tackled,
specifying the importance in the obtention of the relationship among
unidimensional variables. This is generalized to n-dimensional variables. The
subject ends with the study of Succesions of random variables and their boundary
theorems. An introduction to Sampling Statistics and their distributions is
given.
Professor: Francisco HERRERA CUADRA.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination.
Geometry II
Part One; First term subject; 5 hours per week. 5
Cred. E.C.T.S.
General vision of vector spaces, particularly
euclidean spaces. Study of projectivities among projective spaces and
classification of P1, P2,
P3 ´s. Study of the double ratio and armonic quatern in the projective line and of the dual
projective space, the principle of duality and dual projectivities. Study of
projective hyperquadrics, classified for conics and quadrics, tackling
polarity, tangency, cone of tangents and tangential hyperquadric. Study of the
hyperquadrics sheafs and classification of the conic and quadric sheafs, and
their reduced equations. At last, the study of euclidean hyperquadrics, metric
elements and unvariants for conics and quadratics.
Professor: Rosendo RUIZ SÁNCHEZ.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination.
Requirement: Theoretical/practical written examination.
Numerical Methods II
Part One; First term subject; 5 hours per week. 5 Cred. E.C.T.S.
The approximation to functions of one real variable is
one of the pillars of numerical methods. The first chapter considers Hermite´s
interpolation, together with the convergence and minimization of the
interpolation error. Splines interpolation is also studied. The second chapter
consists of an approximation in euclidean spaces and in the uniform norm
(theorems of existence and unicity, characterization of the best approximation,
constructive methods, ortogonal polynomials). Rational approximation is briefly
introduced. These topics support others
like: numerical derivation methods (based upon Taylor expansion, the
interpolation polynomial, together with the convergence acceleration by means
of the Richardson´s extrapolation) and numerical integration (simple and
compound Newton-Cotes quadratures, Romberg method and Gaussian quadratures).
Professor: Juan José MORENO BALCÁZAR; Andrei MARTÍNEZ
FINKELSHTEIN.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written
examination. Task works.
Second term
subjects
Advanced Mathematical Statistics.
Part One; Second term subject; 5 hours per week: 5
Cred. E.C.T.S.
The major techniques and methods of the Statistical
inference are developed, specially in parametrical models, studying in depth
the sufficiency, completion, invariance, unbiasness, efficiency and asintotic
properties of statistics. The Theory of Point Estimation, the Hypothesis Test
and the Regions of Confidence are studied in depth. The basic techniques of
non-parametrical Inference and linear models are presented, together with an
introduction to the Theory of Decision.
Professor: Carmelo RODRÍGUEZ
TORREBLANCA y Fernando RECHE LORITE.
Teaching Method: Theoretical and practical lectures.Practicals.
Seminars.
Assessment Method: Written examination. Task works. Work in groups.
Part One; Second term subject; 6 hours per week. 6
Cred. E.C.T.S.
Lebesgue´s integral in R: Introduction. Stepped
functions. First extension. Integral functions. Two theorems of limits. The
Rieman´s integral. Measurable Functions: Measurable Sets. Structure of the
measurable function. Integration on measurable sets. The Fundamental Theorem of
Calculus: Total variation. Absolute continuity. Differentiability in almost
every point. Theorem of Lebesgue´s differentiation.
The Fundamental Theorem of Calculus: Integration by
parts: Replacement integration. Theorems of the Mean Value. The Lp
spaces: Hölder´s and Minkowski´s unequalities. Integration on Rn . Fubini´s
theorem. Theorem of variable replacement.
Professor: Agripina RUBIO FLORES.
Teaching Method: Theoretical and practical lectures. Seminars.
Assessment Method: Written
Differential Geometry and Topology
Part One; Second term subject; 5 hours per week. 5
Cred. E.C.T.S.
Three differentiated parts. The first one completes
some of the topological concepts not studied in previous subjects (separation
axiomes and numberability, included in the Tietze´s extension theorem and the
Urysolin´s theorem. The second part studies the differential curves in plane
and space by means of the Frenet-Serre equations (thus, locally). It also uses
topology introduced in the first part to demonstrate the Jordan´s curve theorem
(by means of the Brouwer´s fixed point theorem)
and the Isopermetric theorem (the most important global theorem for plane
curves). In the third part, the followig concepts are introduced: smoothing of
maps among them, Gauss mapping, fundamental quadratic forms, the various types
of curvature and methods for their calculation, and the Gauss´s Egrgium
Theorem. This forms the starting point of the most classical topics of the
Intrinsec Geometry.
Professor: Francisco GARCÍA ARENAS.
Teaching Method: Theoretical and practical lectures.Seminars.
Discussions.
Assessment Method: Written examination. Presentations.
Numerical Methods III
Part One; Second term subject; 3 hours per
week. 3 Cred. E.C.T.S.
In many scientific disciplines the presence of
mathematical problems that require the solution of equations, in most of the
cases difficult or impossible to be explicitally solved, is frequently found. This
subject introduces numerical methods developed to obtain approximate solutions
for non-linear equations. The search of the approximate solution does not pose
any ambiguity because error bounding is also studied. An approximate solution
with a pre-determined margin of error can be equally found.
Professor: Antonio ANDÚJAR RODRÍGUEZ.
Teaching Method: Theoretical and practical lectures. Practicals.
Assessment Method: Written examination. Presentation of works.
Part One; second term; 4 hours per week. 4.5 Cred.
E.C.T.S.
The concept of the body extension. Simple extensions.
Construction of simple extensions. Classification of simple extensions.
Transitivity. Algebraic numbers. Constructions with rule and compass. General
idea of the Galois Theory. Scission bodies. Normality. Separability. Linear
independence of monomorphisms. Normal clausures. Fundamental theorem of the
Galois´s Theory; examples. Solvable groups. Simple groups. Radical extensions.
Quintic unsolvability. Trascendence degree. Solution of the cubic and quartic. The
structure of finite bodies.
Professor: Antonio LIROLA TERREZ.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination.
First Part Optional Subjects
First term
subjects
Comprehension of English Scientific
Texts in Mathematics
Part One; First term subject; 6 hours per week. 6.5
Cred. E.C.T.S.
Re-activation of the language knowledge by means of
grammar exercises and syntax structurization. Reading and comprehension of
scientific texts. Study of the retorical functions of the scientific speech
pursuying cohesion, organization and coherence of the information to be
transmited. Specific lexicon and word formation. Arithmetical operations in
spoken english. Mathematical definitions. The use of mathematical simbols
taking notes in English-spoken Congresses. Listening
exercises.
Professor: María Enriqueta CORTÉS DE
LOS RÍOS.
Teaching Method: Theoretical/practical lectures.
Assessment Method: Written examination.
Computer Science II (Structure of
Data)
Part One; First term subject; 3 hours per week. 3.5
Cred. E.C.T.S.
Abstract data types.
Testing and efficiency of the representation of data
types.
Structures of linear data: Lists, stacks and queues.
Elemental set structures. Advanced representations of
sets: tables, hash, binary trees.
Professor: Irene MARTÍNEZ MASEGOSA.
Teaching Method:
Assessment Method:
Introduction to Optimizatión
Part One; First term subject; 3 hours per week. 3.5
Cred. E.C.T.S.
Concept. Existence and unicity of the Solution.
Optimization without constraints and with constraints.
Introduction to non-linear programming.
Introduction to linear programming: the Simplex
method. Uses in Economy and Engineering.
Professor: Ramón CARREÑO CARREÑO.
Teaching Method:
Assessment Method:
Statistical Methods: Uses in
Sciences
Part One; First term subject; 3 hours per week. 3.5
Cred. E.C.T.S.
Variance analysis of I and II ways.
Regression analysis. Covariance analysis.
Statistical Unidimensional varieties.
Statistical bidimensional varieties. Regression and
correlation. Index numbers. Time series. Practicals with Statgraphics Plus.
Professor: Inmaculada OÑA CASADO.
Antonio SALMERÓN CERDÁN.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written and oral examinations.
Second Term
Subjects
Operational Calculus
Part One; Second term subject; 3 hours per week. 3.5
Cred. E.C.T.S.
The linear finite differences equation with constant
coefficients. The space of possible strings. The obtaining of the Riemann
integrals. The factorial function (Gamma) as a solution for a equation in
linear difference, of first degree and variable coefficients. Linear SDO and
complete linears. Improper integrals. Convergence criteria in improper integral
of first and second specie. The Cauchy Criteria. Circular elemental functions.
Solutions of the EDO. The Flee´s transform. Transform of EDO and linear EDP of
constant coefficients. Series of Fourier integrals. The theorem of convolution.
Fourier inverse transform.
Professor: José Juan RODRÍGUEZ CANO.
Teaching Method: Theoretical and practical lectures. Practical works.
Seminars. Discussion. Literature discussion.
Assessment Method: Written examination. Task works.
Part One; Second term subject; 3 hours per week. 3.5
Cred. E.C.T.S.
Plane Algebraic Curves. Curves in the affine space. Irreducible
and reducible curves. Saddle Points: Line-curve interception. Tangent lines. Saddle
points. Examples. Curves in the projective plane: Rational curves. Curve linear
systems. Multiplicity of interception. Bezout´s Theoreme. Cubics. Point
addition on cubic.
Professor: Blas TORRECILLAS JOVER.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Task works.
Metric Geometry
Part One; Second term subject; 3 hours per week. 3,5
Cred. E.C.T.S.
The objective of this optional subject is to complete
the geometry background of the student in aspects not tackled in the core
subjects of the programme.
The name of the subject comes from the study of the
vector metric Geometry, that is to say quadratic forms (symmetrical or not) and
the type of Geometry they determine. The student is also introduced in the
study of convex bodies in Rn, the simplest objects after vector
spaces. Other classical topics of elementary geometry like tessellates,
simplicees and polyhedrons are also treated.
Professor: Francisco GARCíA ARENAS.
Teaching Method: Theoretical and practical lectures. Seminars.
Assessment Method: Written examination. Presentations. Practical reports.
Problems in Mathematics
Part One; Second term subject; 3 hours per week. 3.5
Cred. E.C.T.S.
This subject tackles the classification and solution
of problems. Different classifications of mathematical problems are analyzed,
classroom problems, classification of arithmetical problems. Phases in the
problem-solving process. Teaching models of the problem-solving process. Heuristic
techniques and strategies. Factors and requirements of the problem-solving
process. Obstacles and blockings in the problem-solving process. Elaboration
and analysis of protocols. Practicals about the theoretical contents by means
of individual and small-groups
problem-solving.
Professor: Antonio FRÍAS ZORRILLA.
Met. Enseñanza: Formal lectures.
Met. Examen: Theoretical/practical written examination on the
subject contents. Practicals Report (group problems). Attendance and
participation.
Part Two; First term subject; 6 hours per week. 6.5
Cred. E.C.T.S.
Artin-Wenddeburn´s Theoreme, Theorem of density.
Jacobson´s Radical. Rings with chain conditions. Representation
of finite groups.
Maschke´s Theoreme. Irreducible representations.
Theory of characters. Forbinius´s Theorem of reciprocity.
Burnside´s theorem.
Professor: Juan Ramón GARCÍA ROZAS.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written.
Requisito: Basic linear algebra, polynomials, groups of
permutations.
Complex Analysis I
Part Two; First term subject; 7 Hours per week. 7.5
Cred. E.C.T.S.
Holomorphic functions: Basic Theory: the concept of
derivative, Cauchy-Riemann´s equations. Holomorphic functions. Series of
powers. Analytical functions. Exponential function, trigonometric functions, multiform
functions. The Local Cauchy´s theory: curvilinear integral. Cauchy´s theorem
for star-shaped domains. Cauchy´s integral formula. Taylor´s expansion in
serie. Equivalence between analyzicity and holomorphy: Riemann´s theory of
avoidable singularities. The Principle of identity. The principle of maximal
module. Theorems of the open map and the inverse function. Singularities:
Laurent´s expansion in serie. Classification of singularities. The Theory of
residues. Mappings. The Principle of the argument. The Theorems of Rouché and
Hurwitz.
Professor: Juan Carlos NAVARRO
PASCUAL.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination. Practical reports.
Part Two; First term subject; 3 hours per week. 3.5
Cred. E.C.T.S.
Cauchy´s problem in Ordinary Differential Equations
(ODE) and Ordinary Differential Systems (ODS). Existence. Existence and units.
Picard-Lindelof method. Caeciopoli-Banach
method. Cauchi-Jacobi´s theorem in Eu of solutions the implicit case. Conditions
of existence. The fundamental theorem of calculus. Solution dependence on
initial conditions. Igrowall theorem. Peano theorem. G. Bendixon theorem. Pomearé
theorem. Wintner´s prolongability theorem. Linear ODS and complex linears. Feoquet
theorem. Lyapunov´s theorem. Stability. The Fundamental Theorem of Linear
Differential Algebra. Exponential of a matrix. Linear ODS of constant
coefficients. Linear EDO and boundary conditions. Fourier expansions in serie.
Autoattached problems of autovalues in linear quadratic ODE. The Green
function. Fourier expansion in series and completion for a quadratic (a,b)
function in terms of proper functions.
Professor: José Juan RODRÍGUEZ CANO.
Teaching Method: Theoretical and practical lectures. Practicals.
Seminars. Debates. Literature retrieval.
Assessment Method: Written examination. Task works. Group works. Presentations.
Practical reports.
Requirement: Every 1st and 2nd year
Mathematic subjects should have been passed.
Basics in Physics
Part Two; First term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
In this subject some of the most fundamental Nature
processes are studied. The subject has been structured in three blocks tackling
different aspects of the Physical sciences: Mechanics, Thermodynamics and
Electricity and Magnetism. The first block tackles Classical mechanics
(Kinematics and Dynamics of the particle; the gravitational Field; Dynamics of
the Rigid Solid; Elasticity) with an introduction to the Quantum Mechanics. The
second block tackles concepts like Heat and his propagation, ideal and real
gases, and the Principles of Thermodynamics. The last block shows and studies
the electric and magnetic fields in the vacuum.
Professor: Francisco LUZÓN
MARTÍNEZ.
Teaching Method: Theoretical and practical lectures.
Mét. Exámen: Written examination.
Second Term
Functional Analysis I
Part Two; Second term subject; 5 hours per week. 5.5
Cred. E.C.T.S.
Basic theory of normed spaces: linear and continuous
mappings among normed spaces, finite dimension normed spaces, topological dual.
Hilbert´s spaces: The theorems of the optimal approximation, of the orthogonal
projection and Riesz-Fréchet´s. Orthonormal bases. Operators in Hilbert´s
spaces, spectral theorem for a compact normal operator. Fundamental Principles
of the Functional Analysis and Duality in Banach´s Spaces: the Hahn-Banach´s
Theorems (the extension and separation theorems). Banach´s reflexive spaces,
weak topologies, the Helly´s, Goldstine and Milman-Pettis Theorems. The
Banach-Alaoglú Theoreme. Consequences of the Baire Theoreme: Theorems of the
open map, of the Banach´s isomorphisms and of the closed graphic. The
Steinhaus-Banach Theoreme.
Professor: Juan Carlos NAVARRO
PASCUAL.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination. Practical reports.
Numerical Calculus I
Part Two; Second term subject; 6 hours per week. 6.5
Cred. E.C.T.S.
The mathematical modelization of the real phenomena
around us requires the solution of ordinary differential equations or equations
in partial derivatives. In most of the cases these equations do not show
analytically explicit solutions, so the use of numeric methods is necessary in
order to obtain a solution. The mathematical fundaments of numeric methods for
the solution of problems of initial
values are studied: 1-step methods (special attention paid to the Runge-Kutta
methods), multi-step methods (the Adams methods), and methods for special
problems (stiff problems, and so on). Numeric methods to solve contour problems
are also studied: shots, resolution in differences, variational methods, and so
on. Finally solving methods for Fredholm and Volterra´s integral equations are
introduced.
Professor: Juan José MORENO BALCÁZAR.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination.
Calculus of Probabilities
Part Two; Second term subject; 6 hours per week. 6.5
Cred. E.C.T.S.
Structures of set classes. Measurable space. Product
of measurable spaces. The Borel´s space. Set functions. Measure. Probability
measure. Complete, regular, tight measures. Extent of a measure. Lebesgue-Stieltjes
measures and distribution functions. Measurable functions. Lebesgue´s integral.
Indefinite integral: Characterization. Product measure. Successions of
measurable functions. Types of convergence. Distribution functions:
descomposition and types of convergences. Convergence of integral strings. Probability
and independence: Kolmogorov zero-one rule. Characteristic function: The
Inversion, Unicity, Convolution and Continuity Theorems.Criteria of
characteristic functions. Multi-dimensional case. Infinitely and stable
divisible distributions. Canonic representations. The Laws of the Big Numbers. Central
Theorem of the Limit. Dependence: conditioned expectation. Stochastic process:
Definition and description.
Professor: Alicia JUAN GONZÁLEZ.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination.
Fourth Year
First Term
Geometry and Topology
Part Two; First term subject; 6 hours per week. 7.5
Cred. E.C.T.S.
This subject consists of two parts: 1st.- Advanced
Geometry of regular surfaces. Intrinsic Geometry of surfaces. Global Theorems
of the Theory of surfaces. Need of abstraction and generalization of the
surface concept. 2 nd .- Theory of the differentable varieties.
Definition of variety, differentable structures,
fibrous tangent spaces and connections.
Professor: Juan T. LÓPEZ RAYA.
Teaching Method: Theoretical and practical lectures.
Assessment
Method: Written examination.
*Second Term
Part One; First term subject; 5 Hours per week. 6
Cred. E.C.T.S.
Basic elements of the Theory of the Vector Topological
Spaces: definitions and general results. The Tihonov and Riesz Theorems. Metrisable
topological vector spaces. Birkhoff- Katletani Theoreme. Introduction to the
Theory of the Locally Convex Spaces: locally convex typologies. Different
characterizations. L.C.S.. Spaces of distributions. Hahn-banach´s theorems in
L.C.S. Extreme points of a convex. The Krein-Milman´s Theoreme. Mappings.Weak
Topologies and Duality: Vector Spaces in duality. Weak and *-weak topologies. Polarity.
Theorem of the bipolar. Banach-Alaoglu Theoreme. Banach´s reflexive spaces. Some
theorems of the fixed point. Elemental Theory of the Banach Algebras. Espectrum.
Functional Calculus. Banach Conmutative Algebras. Elements of C*-algebra.
Professor: El Amin KAIDI LHACHMI.
Teaching Method: Theoretical and practical lectures. Practicals.
Seminars.
Assessment Method: Written/oral examination. Work presentation. Work in
groups. Presentations.
Mathematic Didactics in
Baccalaureate
Second Part; First term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
Curriculum of Mathematics. Teaching. Learning.
Elements of curriculum design. The process in class. Mathematics programmes in
second-grade education. Design and development of the curriculum. The community
of mathematic lecturers. Comparative vision of the Mathematics Curriculum.
Professor: María Francisca MORENO
CARRETERO.
Teaching Method: Theoretical and practical lectures. Practicals.
Seminars. Discussion. Literature Retrieval.
Assessment Method: Written/oral examination. Task works. Presentations.
Experimental Design
Part Two; First term subject; 3 hours per week. 3.5
Cred. E.C.T.S.
The objective of this subject is to introduce and
analyze the major models of design of statistical experiments. After tackling
the Analysis of Variance technique we develope the completely randomized,
randomized, factorial, nested and mixed
models. The analysis of models is carried out by using statistical packages
such as Statgraphics. The methodology of response surfaces is also introduced.
Professor: Carmelo RODRÍGUEZ
TORREBLANCA.
Teaching Method: Theoretical and practical lectures. Practicals.
Seminars.
Assessment Method: Written examination. Task works. Work in groups. Practical
reports.
Requirement: Basic knowledge in Statistical Inference.
Equations in Partial Derivatives
Second Part; First term subjects; 4 hours per week. 4.5
Cred. E.C.T.S.
First order partial derivative equations. The general
Cauchy´s problem. The Cauchy-Kowalewsky Theorem. The Unicity Theorem. Quadratic
equations. Classification. The Divergence Theorem. The potential equation. The
waves equation. An introduction to the Theory of Partial Derivative Equations
modern expansion.
Professor: Bernardo LAFUERZA GUILLÉN.
Teaching Method:
Assessment Method:
Statistical Inference II
Part Two; First term subject; 3 hours per week. 3.5 Cred. E.C.T.S.
The purpose of this subject is to tackle statistics
from different points of view: Bayes model and theory of decision. The subject
is divided in three different parts: 1,. Interpretation of probability:
Classical method, frequentist model and subjective model. 2.- Approach to
statistical problems under the bayes perspective. 3.- Introduction and study of
statistical problems by means of Theory of Decission´s tools.
Professor: Fernando RECHE LORITE.
Teaching Method: Theoretical and practical lectures. Seminars.
Assessment Method: Written/Oral examination. Task works.
Requirement: Proficiency in probability and classical statistics.
Stochastic Processes
Part Two; First term subject; 3 hours per week. 3.5
Cred. E.C.T.S.
The stochastic process: definition, description. Some
types of processes: process of independent and stationary increments; Markov,
stationary; renovation. Some elementary processes: Bernoulli, random course,
Poisson and Gaussian. Markov chains: transition matrixes, Chapman-Kolmogorov
equations, homogeneous chains and classification of the states. Markov finite
chains: chains with recurrent and transitory states, irreducible ergodic
chains. Analysis of a Markov chain with two states. Multiple, stacked and
inverted chains. Markov infinite chains: ergodic, null recurrents, and
transitories. Markov notable processes: the Poisson process, birth and death
processes. General properties of the Markov processes. Renovation processes.
Stationary processes.
Professor: Alicia JUAN GONZÁLEZ.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Reports. Presentations.
Representation of Knowledge
Part Two; First term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
Theory of first order. Propotitional and Predicates
Calculus.
Uses in computation.
Logic programmation.
Expert systems.
Collection and knowledge structuring methods.
Professor: Manuel Francisco CRUZ
MARTÍNEZ.
Teaching Method: Practicals. Seminars.
Assessment Method: Reports. Work in groups. Presentations.
Theory of Algorithms
Part Two, First term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
The objective of this subject is to instruct the
students in the analysis of efficient algorithms, the different techniques in
the conception of algorithms, and let them know the basic tools for the
development of their own algorithms applied to mathematics. The following
topics will be developed: 1- Analysis of the algorithm´s efficiency. 2.-
Algorithms "divide and rule". 4.- Voracious algorithms. 4.- Dynamic
programmation. 5.- Graph exploration. 6.- Elements of calculation complexity.
Professor: Mercedes MARTÍNEZ DURBÁN.
Teaching Method: Theoretical and practical lectures. Practicals.
Debates.
Assessment Method: Written examination or presentations. Practical
report.
Requirement: The student is recommended to follow Computer
Science II (first part).
Second Part; First term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
Families of sets. Measures and measures.
The theorem of convergence. The Lebesgue spaces.
The Fubini´s theorem. The Radon-Nikodym theorem.
Riesz´s theorem of Representation. Duality. The Haar
measure.
Professor: Enrique DE AMO ARTERO.
Teaching Method: Theoretical lectures. Practicals. Literature
retrieval.
Assessment Method: Written and oral, with reports.
Algebraic Topology
Part Two; First term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
Definition of the first group of homotopy in a
topological space. Calculation of the first group of homotopy of the unit
circunference. The Seifert-van Kampen
theorem and his application to the calculation of the first homotopy group in
different vector spaces. Definition of the singular homotopy groups of a
topological space. Contruction of the Mayer-Vietoris string of a topological
pair. The Scission theorem and construction of the Mayer-Vietoris string and
its use for the calculation of the sphere´s singular homology groups. Singular
homology techniques used for the demonstration of classical theorems in
Topology: the theorem of dimension invariance. The Brouwer´s theorem of the
fixed point and the Jordan-Brouwer´s
separation theoreme.
Professor: María Luz PUERTAS GONZÁLEZ
. Juan Torcuato LÓPEZ RAYA.
Teaching Method: Theoretical and practical lectures. Practicals.
Assessment Method: Written examination. Presentation of works.
Computational Topology
Part Two; First term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
Classical Digital Topology, associated problems. Study
of spaces partially ordered, geometrical realization. Study of CW-regular
complexes and polyhedrals. Spaces partially ordered and polyhedrals
relationship, digitalization module. Model of abstract images and operations on
them, slimmings. Introduction to cellular automata.
Professor: Juan T. LÓPEZ RAYA.
Teaching Method: Theoretical and practical lectures. Practicals.
Seminars. Debates. Literature retrieval.
Assessment
Method: Written/oral examination. Task
works. Work in groups. Presentations. Practical reports.
Part Two; First term subject; 3 hours per week. 3.5
Cred. E.C.T.S.
Introduction.
Elements of Image Digital Processing. Fourier
Transforms, Hotelling and others.
Image segmentation.
Representation and description.
Survey and interpretation.
Professor: Manuel CANTÓN GARBÍN.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination. Practical report.
Requirement: Proficiency in Advanced Mathematics First Part.
Part Two; Second term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
Representation of algebraic data. Factorization of
polinomials (computational methods). Practical methods. Reduce. Arithmetics in
euclidean domains. Algorithm complexity. Calculus by homomorphic images. The
Fourier´s quick transform. Series of potentials. Algorithms on premises and
matrixes.
Professor: Antonio LIROLA TERREZ.
Teaching Method: Theoretical and practical lectures. Practicals.
Assessment Method: Written examination. Practical reports.
Complex Analysis II
Part Two; Second term subject; 3 hours per week. 3.5
Cred. E.C.T.S.
Aditive expressions for analytical functions. The
Taylor series. The Cauchy series. The Fourier series. Characterization of
analytical indefinitely derivative functions. Multiplicative expression of
analytical functions. The Weiertrass theorem. Hadamard and Borel. The Preadr´s
small theoreme. Normal families of holomorphic functions.
Professor: José Juan RODRÍGUEZ CANO.
Teaching Method: Theoretical and practical lectures. Practicals.
Seminars . Debates. Literature retrieval.
Assessment Method: Written examination. Presentation of works. Discussion.
Practical reports.
Astrophysics
Part Two; Second term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
Global description of the Universe; a description
surging from interpretation of observational aspects through fundamental
physical theories: the theory of radiation, classical and relativity dynamics
and nuclear physics.
The two-bodies problem. Motion of the Solar System
bodies. Earth motion. Double-stars.
Professor: F. Javier BARBERO.
Teaching Method: Theoretical and practical lectures. Practicals.
Seminars.
Mét. Exámen: Presentations. Practical report. Written
examination.
Astronomy of Position
Part Two; Second term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
Brief introduction to spheric trigonometry. Introduction
to co-ordinates systems and co-ordinate change systems. Problems associated to
the daily movement (rising and setting of stars, maximal disgressions and first
vertical). Correction in the astronomic co-ordinates: refraction and light
aberration, equinox precession, parallax. The problem of time: Sidereal, true,
mean, civil, and official. The solar system: the sun, the moon, the planets. Eclipses.
Professor: David LLENA CARRASCO.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination.
Numerical Calculus II
Part Two; Second term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
The theoretical fundaments and mostly used algorithms
in interpolation and approximation of functions of some real variables are
studied. The general analysis of the problems of existence and unisolvency, is
followed by the interpolation in regular grids. The interpolation of scattered
data is discussed: unisolvency, error, algorithms, together with different
approximation methods (uniform, square minimums, a.o.) The local methods are
analyzed; a central place occupied by multivariated splines and their use for
approximation and interpolation. In this sense, triangulation and surface
partitioning methods are studied. As a Bezier´s application, the Coon patches
and the method of finite elements.
Professor: Andrei MARTÍNEZ FINKELSHTEIN.
Teaching Method: Theoretical and practical lectures. Practicals.
Assessment Method: Written examination. Presentation of works. Practical
reports. Presentations.
Computational Statistics
Part Two; Second term subject; 3 hours per week. 3.5
Cred. E.C.T.S.
This subject tackles the bonds between Statistics and
Computer Science. We start studying the generation of randomized numbers and
variables (in general), followed by the Monte Carlo´s simulation and its use
for integrals estimation. The use of statistical techniques in the construction
of expert systems is also studied. Different statistical packages are used
during practicals.
Professor: Antonio SALMERÓN CERDÁN.
Teaching Method: Theoretical and practical lectures. Practicals.
Assessment Method: Written examination. Presentation of works. Practical
report.
Part Two; Second term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
Geodesic measures. Theoretical/practical programmes on
geometric Geodesy. Physical Geodesy. Spatial Geodesy.
Seminars: The Earth motions. Determination of
latitude, longitude and astronomical azimuts. Levelling of altitudes. Error analysis and processing of
laboratory data.
Professor: Víctor CORCHETE FERNÁNDEZ.
Teaching Method: Formal lectures. Practical reports. Practicals. Seminars. Literature retrieval.
Assessment Method: Theoretical/practical written examinations. The
students can choose an oral examination. To do so, the lecturer will suggest a
practical work to be done and presented individually.
Part Two; Second term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
Affine and projective varieties. Morphisms among
varieties. Dimension of rational maps. Non-saddle
points. Non-saddle curves. Bezout´s
theorem of curve interception. Sheafs. Cohomology of sheafs. Sheafs and
varieties. Arithmetic genus of curves, the Riemann-Roch theoreme, weak form. Geometrical
genus, unicurval curves.
Professor: Blas TORRECILLAS JOVER.
Teaching Method: Theoretical and practical lectures. Practicals.
Assessment Method: Reports.
Computational Geometry.
Part Two; Second term subject; 3 hours per week. 3.5
Cred. E.C.T.S.
The objective of Computational Geometry is to tackle
geometry problems with computational methods. The focus of the subject lies on
the discovery of effective algorithms (necessary to introduce first the
concepts of algorithm and efficiency) for rather simple problems (due to the
impossibility for the student to solve complex problems which are in some cases
still object of research). An example
of the treated topics would be: Voronoi´s diagrams, "guarded
vigilance" (the Chevall´s art gallery theoreme), uses in visibility and
robotics.
Professor: Mª Luz PUERTAS GONZÁLEZ y
M. A. SÁNCHEZ GRANERO.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination.
Differential Geometry
Part Two; Second term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
In the solving-process of many mathematical problems
lies the idea of reasonably disturbing the given hypothesis in order to
simplify the situation; this is the idea of "general position" in
Geometry, or "non-degenerated case" in Analysis. The most fruitful
expression of this argument is the transversality notion, which, created by
René Thorn, introduced the Differential Topology. In order to tackle this notion, it was necessary to introduce the
fundaments of the study of varieties with boundary (fibrated, inmersions and
summersions, orientations) and from there, reach the Sand-Brown theorem and the
Whitney´s theorem. On the other hand the construction of tubular environments
in the normal fibrate gives us some approximation theorems that, if combined
with transversality, constitute a powerful tool to classify curves, demonstrate
the Brouwer´s fixed point theorem and introduce the concept of degree.
Professor: Francisco GARCÍA ARENAS.
Teaching Method: Theoretical and practical lectures. Seminars. Debates.
Assessment Method: Written examination. Presentations.
Mechanics
Part Two; Second term; 4 hours per week. 4.5 Cred.
E.C.T.S.
Brief overview of the Newton Mechanics and
methodology. General approach of the Analytical Mechanics. Concept of
constraints and types. Generalized co-ordinates, degrees of freedom. Configuration
space. Study of the transforming relationships between cartesian and orthogonal
co-ordinates (plane-polar, cilindrical,
and spherical) and their properties. Velocities, moments and generalized
forces. The kinetic energy in generalized co-ordinates. Virtual motion.
Principle of kinetics in generalized co-ordinates. Virtual motion. Dálembert
Principle. Lagrange´s equations. Basic elements of variational calculus. Conditions
of Extreme. Hamilton´s principle: Lagrange´s equations for motion. Noether
theorem. Hamiltonian Dynamics. Hamilton´s equations. Hamilton-Jacobi equation.
Seminars: Action angular variables. Theory of the canonic perturbations. Eometric
and algebraic interpretations of motion.
Professor: María Dolores ROMACHO
ROMERO.
Teaching Method: Theoretical and practical lectures. Practicals.
Seminars. Discussion on specific topics which in relation with the subject,
have been collected in recent journals. Literature retrieval.
Assessment Method: Written examination about basic theoretics and
problems. Presentation of complementary exercises. Work in groups (max. two students).
Presentation of complementary advanced topics.
Teaching Practicals
Part Two; Second term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
Planning of teaching. Secondary
education programme. Other education programmes. Planning in Mathematics. Observation techniques. Practicals. Class
observation. Programme, impart, and evaluate Mathematic topics. Fill the
lecture´s diary. Evolution, patterns to evaluate a programme of Mathematics. Analysis
of the evaluation results. Elaboration of a practicals memory.
Professor: Francisco GIL CUADRA.
Teaching Method: Practical lectures. Seminars.
Assessment Method: Written/oral examination. Works in group. Presentations.
Practical reports.
Requirement: In order to follow this subject it is necessary to be
proficient in Mathematics Didactics.
Numerical Solution of Partial
Derivative Equations
Part Two; Second term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
The subject starts introducing physical situations
modelled with P.D.E. The classical equations of Mathematical Physics are
studied, together with the maximum-minimum concepts for armoric and parabolic
functions. By means of the variable separation method mixed problems like the
heat and waves equations are solved, together with problems of the Dirichlet
type in rectangle and disc. The core part of the programme is dedicated to the
study of explicit and implicit methods in finite differences, together with the
finite elements method and the Galerkin semi-discrete methods. Mathematica and
Ansys will be used as software packages.
Professor: Florencio CASTAÑO
IGLESIAS.
Teaching Method: Theoretical and practical lectures. Practicals.
Assessment Method: Written examination. Reports.
Theory of Rings
Part Two; Second term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
Functions among Module Categories. 1.- The Hom
functors and accuracy. Projectivity and injectivity. 2. Projective modules and
generators. 3. Injective modules and co-generators. 4. The tensor functors and
plane modules. 5. Natural transformations. Equivalence and Duality for Module
Categories. 1. Equivalent Rings. 2. Morita equivalence characterizations. 4. Dualities. 4. Morita Dualities.
Professor: María Jesús ASENSIO DEL
ÁGUILA.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Presentation of works.
Part Two; Second term subject; 4 hours per week. 4.5 Cred. E.C.T.S.
Finite
automata. Regular expressions. Context
free grammar. Battery automata. Turing´s machines. Computability. Chomsky´s
hierarchy. This subject deals with some of the principles or fundaments of
Computer Science supporting the global theoretical and practical frame of this
science, for example, the automata theory, the computation theory and the
formal languages theory.
Professor: Manuel CANTÓN GARBÍN.
Teaching Method: Theoretical/practical (problem-solving) lectures.
Assessment Method: Two examinations during this term.
Requirement: Basic knowledge of Mathematics.
[1] Subjects that been of a different
degree, the student can choose between the ones that have been offered by the
University in order to complete the number of credits needed.
[2] All subjects in this term are Optional or Free Configuration subjects.
* Every subject of this term period are Optionals or Free Configuration.