The aim of the course is, on one hand, to broaden and consolidate the acquisition of the mathematical method as a fundamental investigation tool for economic, financial and business disciplines, and, on the other hand, to provide the necessary notions for understanding the financial markets and the main financial instruments. In particular, we will provide tools in the field of risk analysis and management in the financial markets.
scheda docente
materiale didattico
Eigenvalues, eigenvectors, eigenspace, diagonalizable matrices, diagonalization of matrices, algebraic and geometric multiplicity, eigenvalues of symmetric matrices, properties of eigenvalues and eigenvectors, definitions and related theorems.
Sets in \( \mathbb{R}^2 \) and \( \mathbb{R}^n \).
Topology in \( \mathbb{R}^2 \): Open, closed, compact, bounded and unbounded sets, connected, disconnected, star-shaped, and convex sets.
Metric spaces, normed spaces.
Real functions of several real variables.
Functions defined between Euclidean spaces, graphs, domain of a function, bounded functions, continuous functions, concave and convex functions.
Level curves.
Level curves for a function, upper and lower level curves.
Sign of a matrix.
Definition, sign of a matrix through the determinant and through the eigenvalues.
Multivariable differential calculus.
Partial derivatives, gradient, equation of the tangent plane and linear approximation of a function, higher-order partial derivatives, Hessian matrix, Schwartz's theorem, functions of class \( C^2 \), second-order Taylor polynomial, second-order approximation of a function.
Unconstrained optimization.
Definitions of local and absolute maximum, local and absolute minimum, saddle point, first-order conditions, sufficient and necessary conditions. Second-order conditions, optimization for convex functions.
Section on Graphs
Directed graph. Successor of a vertex. Undirected graph; examples: Medici family graph, Erdős number. Incident edges and vertices, empty graph, order and size of a graph, neighborhood, multigraph. Königsberg bridges problem.
Degree of a vertex in an undirected graph. Isolated vertices. Handshaking lemma and its corollary. Outdegree and indegree for a directed graph. Complete graph of order \( n \). Weighted graph.
Mathematical representation of a graph: unweighted and undirected; unweighted and directed; weighted and undirected; weighted and directed; multigraph. Adjacency matrix. "Sparse" representation of a graph.
Cucker-Smale model. Galam model and binomial distribution.
Combinatorial calculus: permutations and arrangements. Combinatorial calculus: arrangements with repetition, combinations, binomial coefficient, first and second properties of binomial coefficients.
Isomorphic graphs. Isomorphic graphs and permutation matrices (theorem). Isomorphic graphs and eigenvalues (theorem). Necessary conditions for isomorphism.
Definition of path, length of the path, closed path, simple paths. kth power theorem of the adjacency matrix. Subgraph, connected graph, bridge, minimum path, distance matrix, graph diameter, underlying graph, weakly connected graph, strong connection, minimum path for weighted graphs.
Minimum weight path. Number of simple paths in a complete graph. Dijkstra's algorithm.
Eulerian paths on undirected graphs, solution of the Seven Bridges of Königsberg problem. Theorem on Eulerian paths.
Centrality measures, degree centrality. Closeness centrality and betweenness centrality. Clustering coefficient, star graph and clique. Average clustering coefficient of a graph, graph resilience.
Mastroeni - Mazzoccoli. Mathematics for economic applications PEARSON
Notes and other material downloadable online from the course on the Moodle platform at: https://economia.el.uniroma3.it/
Programma
Section on FunctionsEigenvalues, eigenvectors, eigenspace, diagonalizable matrices, diagonalization of matrices, algebraic and geometric multiplicity, eigenvalues of symmetric matrices, properties of eigenvalues and eigenvectors, definitions and related theorems.
Sets in \( \mathbb{R}^2 \) and \( \mathbb{R}^n \).
Topology in \( \mathbb{R}^2 \): Open, closed, compact, bounded and unbounded sets, connected, disconnected, star-shaped, and convex sets.
Metric spaces, normed spaces.
Real functions of several real variables.
Functions defined between Euclidean spaces, graphs, domain of a function, bounded functions, continuous functions, concave and convex functions.
Level curves.
Level curves for a function, upper and lower level curves.
Sign of a matrix.
Definition, sign of a matrix through the determinant and through the eigenvalues.
Multivariable differential calculus.
Partial derivatives, gradient, equation of the tangent plane and linear approximation of a function, higher-order partial derivatives, Hessian matrix, Schwartz's theorem, functions of class \( C^2 \), second-order Taylor polynomial, second-order approximation of a function.
Unconstrained optimization.
Definitions of local and absolute maximum, local and absolute minimum, saddle point, first-order conditions, sufficient and necessary conditions. Second-order conditions, optimization for convex functions.
Section on Graphs
Directed graph. Successor of a vertex. Undirected graph; examples: Medici family graph, Erdős number. Incident edges and vertices, empty graph, order and size of a graph, neighborhood, multigraph. Königsberg bridges problem.
Degree of a vertex in an undirected graph. Isolated vertices. Handshaking lemma and its corollary. Outdegree and indegree for a directed graph. Complete graph of order \( n \). Weighted graph.
Mathematical representation of a graph: unweighted and undirected; unweighted and directed; weighted and undirected; weighted and directed; multigraph. Adjacency matrix. "Sparse" representation of a graph.
Cucker-Smale model. Galam model and binomial distribution.
Combinatorial calculus: permutations and arrangements. Combinatorial calculus: arrangements with repetition, combinations, binomial coefficient, first and second properties of binomial coefficients.
Isomorphic graphs. Isomorphic graphs and permutation matrices (theorem). Isomorphic graphs and eigenvalues (theorem). Necessary conditions for isomorphism.
Definition of path, length of the path, closed path, simple paths. kth power theorem of the adjacency matrix. Subgraph, connected graph, bridge, minimum path, distance matrix, graph diameter, underlying graph, weakly connected graph, strong connection, minimum path for weighted graphs.
Minimum weight path. Number of simple paths in a complete graph. Dijkstra's algorithm.
Eulerian paths on undirected graphs, solution of the Seven Bridges of Königsberg problem. Theorem on Eulerian paths.
Centrality measures, degree centrality. Closeness centrality and betweenness centrality. Clustering coefficient, star graph and clique. Average clustering coefficient of a graph, graph resilience.
Testi Adottati
Recommended: Mastroeni - Mazzoccoli. Mathematics for economic applications PEARSON
Notes and other material downloadable online from the course on the Moodle platform at: https://economia.el.uniroma3.it/
Modalità Valutazione
The exam consists of a written and an oral test. The written test consists of exercises, theoretical questions regarding the whole program of the course. The oral test is recommended only to those who pass the written test with at least 18/30 and will consist of one or more questions on the whole program including proofs of theorems indicated in the program