Università Roma Tre

LINEAR ALGEBRA.
MULTIVARIABLE CALCULUS
IMPLICIT FUNCTIONS.
QUADRATIC FORMS.
OPTIMIZATION THEORY
CONVEX FUNCTIONS

L. Mastroeni, A. Mazzoccoli, Mathemathics for economic applications, Ed Pearson

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, 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.
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.
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.

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/