00674 - Mathematics

Learning outcomes

The student will learn the basic concepts and techiques of multivariate differential calculus

More specificfally, the student will be able to calculate partial derivatives,  to find maxima and minima a nd to use the Lagrange multiplicator method.

Course contents

 

PART I: introduction to the Differential Calculus of multivariate functions.

Introduction to metric topology. Complete spaces. Connected spaces.

Spaces of limited functions. pointwise and uniform convergence for sequences of limited functions. Normed and Banach spaces. Compact spaces.

Total convergence for series in Banach spaces. Spces with inner product.

Partial derivatives and differentiable functions. The total differential theorem.

Multivariate Taylor expansion. Hessian matrices, local extrema.

Implicit functions: Dini's theorem.

Simple, open and regular curves in R^n and tangent vectors. Differentiable manifolds in R^n. Jacobian matrices. Tangent spaces and normal spaces.  The method of Lagrange multipliers for optimality in constrained problems.

PART II: Introduction to Enumerative Combinatorics and applications to Discrete Probability.

FUNCTIONS BETWEEN FINITE SETS.The occupancy model. The word model.The number of functions from an n-set to an m-set. Some general principles. Injective functions. Increasing words. Increasing functions.

MULTISETS. Enumeration of queues. Increasing factorial. Multisets and multiset coefficients. Non-decreasing words. Non-decreasing fonctions.

Equations with natural integer solutions. The Bose-Einstein statistics and the Fermi-Dirac statistics. The generalized Gergonne problem.

MULTINOMIAL COEFFICIENTS AND COMPOSITIONS OF A FINITE SET.

PARTITIONS. Partitions and equivalence relatitions. Stirling nubers of the II kind. Bell numbers. Faa' di Bruno coefficients.

PERMUTATIONS. Directed graphs. Permutation graphs. Cycles and factorization. Cauchy coefficients. Stirling numbers of the I kind.

SIEVE METHODS. The Moebius inversion formula (set-theoretic case).

The inclusion|exclusion principle: the formulae of Sylvester and C.Jordan.

Applications: the Euler function, enumeration of surjective functions, the "menages" problem.

Readings/Bibliography

*) Part I:

- E.Giusti, Analisi Matematica, vol.2, Boringhieri  (Part I)

**) Part II:

- CERASOLI, EUGENI. PROTASI, Introduzione alla Matematica Discreta, Zanichelli, Bologna

- Graham R., Knuth D., Patashnik O., Concrete Mathematics. A foundation for Computer Science, Addison-Wesley, 1989

Teaching methods

We will  introduce  general concept and methods pertaining to the Differentil Calculus for functions in several variables (Part I), and to Enumerative Combinatorics with applications to Discrete Probability (Part II).

We also analyze some concrete problems, in order to stimulate the student to find solutions in an autonomous way.

Assessment methods

Final oral examination.

Office hours

See the website of Andrea Brini