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.

teacher profile teaching materials

Eigenvalues, eigenvectors, eigenspace, diagonalization of matrices, eigenvalues of symmetric matrices, properties of eigenvalues.

Sets in R*2 and in R*n.

Metric spaces, normed spaces. Topology in R*n.

Real functions of several real variables.

Functions defined between Euclidean spaces, graphs, contour lines, continuous functions, concave functions and convex functions.

Differential calculus in several variables.

Partial derivatives, gradient, higher order derivatives, Hessian matrix, Schwartz theorem, Taylor polynomial.

Bilinear and quadratic forms.

Definitions, sign of quadratic forms, minor principal of a matrix, sign of a matrix.

Free optimization.

Definitions, first-order conditions, second-order conditions; optimization for convex functions.

Graphs.

Oriented graph, vertices and arcs, successor function, undirected graph, vertices and incident arcs, outgoing and incoming arcs, empty graph, order and size of a graph, adjacent vertices, neighborhood, multi-graph, loop, simple graph, degree of a vertex, isolated vertex.

Lemma of handshakes and its corollary. Incoming degree and outgoing degree, graphs with maximum dimension, complete graphs, weighted graph, weighted incoming degree and weighted outgoing degree.

Mathematical representation of an unweighted graph (oriented and unoriented). Adjacency matrix of an undirected graph, sum over rows and columns, degree vector; adjacency matrix of an oriented graph, sum over rows and columns; adjacency matrix of a multigraph. Mathematical representation of a weighted graph: adjacency matrix. Cucker-Smale model.

Adjacency lists, isomorphism between graphs, permutation matrices, Theorem on isomorphic graphs and permutation matrices, eigenvalues of adjacency matrices, Theorem on isomorphic graphs and eigenvalues. Isomorphism between graphs and degree distribution of vertices. Walks on undirected graphs: walk, length of a walk, simple walk, closed walk (cycle), simple cycle, acyclic graph. Walks on oriented graphs. Power k-th of adjacency matrix.

Power k-th theorem of adjacency matrix, subgraph, connected vertices, connected graph, components, maximal connected subgraph, bridge, minimal path, distance between vertices, distance matrix.

Subjacent graph, weakly connected graph, strongly connected graph, weight of a path, minimum path for weighted graphs.

Minimum weight path: properties of subpaths of a minimum path. Dijkstra algorithm for oriented and nonoriented graphs. Definition of centrality, degree centrality, betweenness and closeness.

Programme

Linear algebra.Eigenvalues, eigenvectors, eigenspace, diagonalization of matrices, eigenvalues of symmetric matrices, properties of eigenvalues.

Sets in R*2 and in R*n.

Metric spaces, normed spaces. Topology in R*n.

Real functions of several real variables.

Functions defined between Euclidean spaces, graphs, contour lines, continuous functions, concave functions and convex functions.

Differential calculus in several variables.

Partial derivatives, gradient, higher order derivatives, Hessian matrix, Schwartz theorem, Taylor polynomial.

Bilinear and quadratic forms.

Definitions, sign of quadratic forms, minor principal of a matrix, sign of a matrix.

Free optimization.

Definitions, first-order conditions, second-order conditions; optimization for convex functions.

Graphs.

Oriented graph, vertices and arcs, successor function, undirected graph, vertices and incident arcs, outgoing and incoming arcs, empty graph, order and size of a graph, adjacent vertices, neighborhood, multi-graph, loop, simple graph, degree of a vertex, isolated vertex.

Lemma of handshakes and its corollary. Incoming degree and outgoing degree, graphs with maximum dimension, complete graphs, weighted graph, weighted incoming degree and weighted outgoing degree.

Mathematical representation of an unweighted graph (oriented and unoriented). Adjacency matrix of an undirected graph, sum over rows and columns, degree vector; adjacency matrix of an oriented graph, sum over rows and columns; adjacency matrix of a multigraph. Mathematical representation of a weighted graph: adjacency matrix. Cucker-Smale model.

Adjacency lists, isomorphism between graphs, permutation matrices, Theorem on isomorphic graphs and permutation matrices, eigenvalues of adjacency matrices, Theorem on isomorphic graphs and eigenvalues. Isomorphism between graphs and degree distribution of vertices. Walks on undirected graphs: walk, length of a walk, simple walk, closed walk (cycle), simple cycle, acyclic graph. Walks on oriented graphs. Power k-th of adjacency matrix.

Power k-th theorem of adjacency matrix, subgraph, connected vertices, connected graph, components, maximal connected subgraph, bridge, minimal path, distance between vertices, distance matrix.

Subjacent graph, weakly connected graph, strongly connected graph, weight of a path, minimum path for weighted graphs.

Minimum weight path: properties of subpaths of a minimum path. Dijkstra algorithm for oriented and nonoriented graphs. Definition of centrality, degree centrality, betweenness and closeness.

Core Documentation

Mastroeni - Mazzoccoli. Matematica per le applicazioni economiche PEARSONReference Bibliography

Mastroeni - Mazzoccoli. Matematica per le applicazioni economiche PEARSONType of delivery of the course

Lectures will be held in presenceAttendance

Attendance is not compulsoryType of evaluation

The examination will consist of a written and an oral test. The written test will consist of exercises, theoretical questions covering the entire course program. The oral test is recommended only for those who pass the written with at least 18/30 and will consist of one or more questions on the entire program including demonstrations of the results given in the program. teacher profile teaching materials

Eigenvalues, eigenvectors, eigenspace, diagonalization of matrices, eigenvalues of symmetric matrices, properties of eigenvalues. Permutation matrices.

Sets in R*2 and in R*n.

Metric spaces, regulated spaces. Topology in R*n.

Real functions of several real variables.

Functions defined between Euclidean spaces, graphs, contour lines, continuous functions, concave functions and convex functions.

Differential calculus in several variables.

Partial derivatives, gradient, higher order derivatives, Hessian matrix, Schwartz theorem, Taylor polynomial.

Bilinear and quadratic forms.

Definitions, sign of quadratic forms, principal minors of a matrix, sign of a matrix.

Optimization.

Definitions, first order conditions, second order conditions; optimization for convex functions. Least squares method, regression line.

Implicit functions.

Dini's theorem, geometric interpretation of the theorem, regular points, gradient theorem.

Graphs.

Recalls of combinatorial calculus. Directed graph, vertices and edges, successor function, undirected graph, incident vertices and edges, outgoing and incoming edges, empty graph, order and dimension of a graph, adjacent vertices, neighborhood, multi-graph, loop, simple graph, degree of a vertex, isolated vertex.

Lemma of handshakes and its corollary. Incoming degree and outgoing degree, graphs with maximum size, complete graphs, weighted graph, weighted incoming degree and weighted outgoing degree.

Mathematical representation of an unweighted graph (directed and not). Adjacency matrix of an undirected graph, sum by rows and by columns, degree vector; adjacency matrix of a directed graph, sum by rows and by columns; adjacency matrix of a multigraph. Mathematical representation of a weighted graph: adjacency matrix. Cucker-Smale model. Galam's model.

Adjacency lists, isomorphism between graphs, permutation matrices, Theorem on isomorphic graphs and permutation matrices, eigenvalues of adjacency matrices, Theorem on isomorphic graphs and eigenvalues. Isomorphism between graphs and vertex degree distribution. Paths on undirected graphs: path, length of a path, simple path, closed path (cycle), simple cycle, acyclic graph. Walks on directed graphs. kth power of the adjacency matrix.

Theorem of the k-th power of the adjacency matrix, subgraph, connected vertices, connected graph, components, maximal connected subgraph, bridge, shortest path, distance between vertices, distance matrix.

Underlying graph, weakly connected graph, strongly connected graph, weight of a path, shortest path for weighted graphs.

Shortest weight path: properties of subpaths of a shortest path. Dijkstra's algorithm for directed and undirected graphs. Definition of centrality, degree centrality, betweenness and closeness.

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/

Programme

Linear algebra.Eigenvalues, eigenvectors, eigenspace, diagonalization of matrices, eigenvalues of symmetric matrices, properties of eigenvalues. Permutation matrices.

Sets in R*2 and in R*n.

Metric spaces, regulated spaces. Topology in R*n.

Real functions of several real variables.

Functions defined between Euclidean spaces, graphs, contour lines, continuous functions, concave functions and convex functions.

Differential calculus in several variables.

Partial derivatives, gradient, higher order derivatives, Hessian matrix, Schwartz theorem, Taylor polynomial.

Bilinear and quadratic forms.

Definitions, sign of quadratic forms, principal minors of a matrix, sign of a matrix.

Optimization.

Definitions, first order conditions, second order conditions; optimization for convex functions. Least squares method, regression line.

Implicit functions.

Dini's theorem, geometric interpretation of the theorem, regular points, gradient theorem.

Graphs.

Recalls of combinatorial calculus. Directed graph, vertices and edges, successor function, undirected graph, incident vertices and edges, outgoing and incoming edges, empty graph, order and dimension of a graph, adjacent vertices, neighborhood, multi-graph, loop, simple graph, degree of a vertex, isolated vertex.

Lemma of handshakes and its corollary. Incoming degree and outgoing degree, graphs with maximum size, complete graphs, weighted graph, weighted incoming degree and weighted outgoing degree.

Mathematical representation of an unweighted graph (directed and not). Adjacency matrix of an undirected graph, sum by rows and by columns, degree vector; adjacency matrix of a directed graph, sum by rows and by columns; adjacency matrix of a multigraph. Mathematical representation of a weighted graph: adjacency matrix. Cucker-Smale model. Galam's model.

Adjacency lists, isomorphism between graphs, permutation matrices, Theorem on isomorphic graphs and permutation matrices, eigenvalues of adjacency matrices, Theorem on isomorphic graphs and eigenvalues. Isomorphism between graphs and vertex degree distribution. Paths on undirected graphs: path, length of a path, simple path, closed path (cycle), simple cycle, acyclic graph. Walks on directed graphs. kth power of the adjacency matrix.

Theorem of the k-th power of the adjacency matrix, subgraph, connected vertices, connected graph, components, maximal connected subgraph, bridge, shortest path, distance between vertices, distance matrix.

Underlying graph, weakly connected graph, strongly connected graph, weight of a path, shortest path for weighted graphs.

Shortest weight path: properties of subpaths of a shortest path. Dijkstra's algorithm for directed and undirected graphs. Definition of centrality, degree centrality, betweenness and closeness.

Core Documentation

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/

Type of evaluation

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