- Unit Coordinator: Roman Ger
- ECTS Credits: 6
- Semester: 2
- Year: 2
- Campus: University of Silesia in Katowice
- Language: English
- Content:
- Applications in Geometry:
1. Joint characterization of Euclidean, hyperbolic and elliptic geometries.
2. Characterizations of the cross ratio.
3. A description of certain subsemigroups of some Lie groups.
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Applications in Functional Analysis:
1. Analytic form of linear-multiplicative functionals in the Banach algebra of integrable functions on the real line.
2. A characterization of strictly convex spaces.
3. Some new characterizations of inner product spaces.
4. Birkhoff-James orthogonality.
5. Addition theorems in Banach algebras; operator semigroups.
- Reading list:
1. J. Aczel & J. Dhombres, Functional equations in several variables, Cambridge University Press, Cambridge, 1989.
2. J. Aczel & S. Gołąb, Funktionalgleichungen der Theorie der Geometrischen Objekte, PWN, Warszawa, 1960.
3. J. Dhombres, Some aspects of functional equations, Chulalongkorn Univ., Bangkok, 1979.
4. D. Ilse, I. Lehman and W. Schulz, Gruppoide und Funktionalgleichungen, VEB Deutscher Verlag der Wissenschaften, Berlin, 1984.
5. M. Kuczma, An introduction to the theory of functional equations and inequalities, Polish ScientiSc Publishers & Silesian University, Warszawa-Kraków-Katowice, 1985.
- Unit Coordinator: Ekaterina Shulman
- ECTS Credits: 6
- Semester: 2
- Year: 2
- Campus: University of Silesia in Katowice
- Language: English
- Content:
The course establishes the fundamental concepts of the graph theory and shows several applications in various topics.
In particular, the famous problems of the graph theory will be discussed: Minimum Connector Problem, Hall's Marriage Theorem, the Assignment Problem, the Network Flow Problem, the Committee Scheduling Problem, the Four Color Problem, the Traveling Salesman Problem.
- Reading list:
1. Bollobas B., Modern Graph Theory, Springer-Verlag, 2001
2. Diestel G. T., Graph Theory, Springer-Verlag, 1997, 2000
3. Foulds L. R., Graph Theory Applications, Springer-Verlag, 1992
4. Hartland G., Zhang P., A First Course in Graph Theory (Dover Books on Mathematics), 2012
5. Matousek J., Nesetril J., An invitation to discrete mathematics, Oxford, 2008
- Code: I0183
- Unit Coordinator: Amadori Debora
- ECTS Credits: 6
- Semester: 1
- Year: 1
- Campus: University of L'Aquila
- Language: English
- Aims:
LEARNING OBJECTIVES.
The course aims at providing basic properties and main techniques to solve basic partial differential equations.
Those objectives contribute to the learning goals of the entire course of studies, as the inner coherence of the master degree in Mathematical Modelling was verified at the time of the planning of the master program.LEARNING OUTCOMES.
At the end of the course, the student should:1. know basic properties (existence, uniqueness, etc.) and main techniques (characteristics, separation of variables, Fourier methods, Green's functions, similarity solutions, etc.) to solve basic partial differential equations (semilinear first order PDEs, heat, Laplace, wave equations);
2. understand and be able to explain thesis and proofs in the field of basic partial differential equations;
3. have strengthened the logic and computational skills;
4. be able to read and understand other mathematical texts on related topics. - Content:
First order partial differential equations. Definition of characteristic vectors and characteristic surfaces. Characteristics for (semi)linear partial differential equations of first order in two independent variables. Existence and uniqueness to initial value problems for first order semilinear partial differential equations in two independent variables Duhamel’s principle for non homogeneous first order partial differential equations.
Second order partial differential equations. Classification of second order semilinear partial differential equations in two independent variables. Canonical form for second order semilinear partial differential equations in two independent variables. Classification for second order semilinear partial differential equations in many independent variables.
Heat equation. Derivation of heat equation and well–posed problems in one space dimension. Solution of Cauchy–Dirichlet problem for one dimensional heat equation by means of Fourier method of separation of variables. Energy method and uniqueness. Maximum principle. Fundamental solution. Solution of global Cauchy problem. Non homogeneous problem: Duhamel’s principle.
Laplace equation. Laplace and Poisson equation: well-posed problems; uniqueness by means of energy method. Mean value property and maximum principles. Laplace equation in a disk by means of separation of variables. Poisson’s formula. Harnack’s inequality and Liouville’s Theorem. Fundamental solution of Laplace operator. Solution of Poisson’s equation in the whole space. Green’s functions and Green’s representation formula.
Wave equation. Transversal vibrations of a string. Well–posed problems in one space dimension. D’Alembert formula. Characteristic parallelogram. Domain of dependence and range of influence. Fundamental solution for one dimensional wave equation. Duhamel’s principle for non homogeneous one dimensional wave equations. Special solutions of multi–d wave equation: planar and spherical waves. Well–posedness for initial, boundary value problems: uniqueness by means of energy estimates. Separation of variables. Domain of dependence and range of influence in several space variables. Fundamental solution for multi–dimensional wave equation. Solution of 3–d wave equation: Kirchhoff’s formula and strong Huygens’ principle. Wave equation in two dimensions: method of descent. Fundamental solution in 2–d. Duhamel’s principle for non homogeneous wave equation in 3–d: delayed potentials. - Pre-requisites:
Students must know the basic notions of mathematical analysis, including Fourier series and ordinary differential equations, and the basic notions of continuum mechanics.
- Reading list:
- L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, AMS, 2010.
- S. Salsa. Partial Differential Equations in Actions: from Modelling to Theory. Springer–Verlag Italia, 2008.
- S. Salsa, G. Verzini. Equazioni a derivate parziali: complementi ed esercizi. Springer–Verlag Italia. 2005.
- W.A. Strauss. Partial Differential Equations: An Introduction. John Wiley & Sons Inc., 2008.
- E.C. Zachmanoglou, D.W. Thoe. lntroduction to Partial Differential Equations with Applications. Dover Publications, Inc., 1986.
- Unit Coordinator: Federica Di Michele
- Programme: InterMaths
- ECTS Credits: 6
- Semester: 1
- Year: 2
- Campus: University of L'Aquila
- Language: English
- Delivery: In-class
- Aims:
The course aims to introduce students to the study and modelling of natural disasters using artificial intelligence techniques. After an initial introduction to the programming language used (Python), the student learns how to prepare a dataset and how to build, validate and optimise machine learning based models. The concepts of supervised, semi-supervised and unsupervised machine learning are introduced and the basics of building neural network-based models for image recognition (convolution neural network (CNN)) and signal analysis (recurrent neural network (RNN) ) are provided.
All datasets used during the laboratory phase are related to seismic and environmental risk assessment and mitigation.
