2016/2017

2017/2018

DM270

MAT/05 (MATHEMATICAL ANALYSIS)

DEPARTMENT OF ELECTRICAL,COMPUTER AND BIOMEDICAL ENGINEERING

BIOENGINEERING

PERCORSO COMUNE

2°

1st semester (02/10/2017 - 19/01/2018)

9

88 lesson hours

Italian

WRITTEN AND ORAL TEST

SAVARE' GIUSEPPE (titolare) - 6 ECTS
GUALANDI STEFANO - 3 ECTS

Differential and integral calculus for scalar and vector functions, matrices and linear transformations, sequences and series, power series in the real line, complex numbers, polar coordinates.

Learn how to work in the complex framework, evaluate integrals of holomorphic functions, manipulate power and Fourier series, adopt the point of view of signal theory, calculate and operate with Fourier, Laplace and Zeta transforms, solve simple ordinary differential equations with constant coefficients, understand convolutions.

The second part (Optimization and discrete transforms, 3CFU only for Bioengineering) will be devoted to the elementary notions of free and constraint optimization and to the basic techniques of the mathematical theory of discrete signals (DFT, FFT, convolutions) with simple applications to difference equations and numerical approximations.

Complex functions
Manipulation of complex numbers
Rational, exponential, and trigonometric functions, logarithms
Power series
Conplex derivatives, olomorphic functions, Cauchy-Riemann conditions
Line integrals, Cauchy theorem, , analyticity of olomorphic functions
Singularities, Laurent series, residue formula
Evaluation of integrals, Jordan lemma

The language of signals
Continuous and discrete signals.
Basic operations on signals: sum and linear combinations of signals, traslation and rescalings.
Scalar products and norms.

Z trasform
Definition and simple examples
Simple applications to difference equations


Fourier series
Periodic signals, trigonometric and exponential functions, Fourier series.
Pointwise and energy convergence, Gibbs phenomenon.
Parseval identity
Applications

Fourier Transform
Definition of Fourier transform, relationships with Fourier series, elementary properties
Riemann-Lebesgue lemma
Inversion theorem for piecewise regular functions
Plancherel identity, Fourier transform for L^2 functions

Laplace transform
Definition, links with the Fourier transform, main properties
Inversion of Laplace transform, residue and Heaviside formula
Application to simple ordinary differential equations
Convolution
Definition and simple example of convolutions
Links with Fourier and Laplace transform
Simple applications to differential equations

Second parte (only for Bioengineering)

Optimization
Unconstrained Optimization Problems
- Gradient methods and line-searches
- Newtonian methods: trust-regions, quasi-Newton and Gauss-Newton for least-squares problems
Constrained Optimization Problems
- Optimality conditions, penalization and SQP methods

Discrete transforms
Discrete Fourier transform (DFT)
The algorithm of Fast Fourier Transform (FFT)
Discrete convolution
Applications to difference and approximation problems, stability

Lectures and exercises classes with blackboard or powerpoint presentations, lab activities.

M. Codegone. Metodi Matematici per l'Ingegneria. Zanichelli.

M. Giaquinta, G. Modica. Note di Metodi Matematici per Ingegneria Informatica. Pitagora, Bologna..

F. Tomarelli. Metodi Matematici per l'Ingegneria. CLU.

Lecture notes available on the web page of the course.

Matlab Optimization and Signal Proccessing Toolbox. User's guide. The MathWorks Inc..

F.J. Bonnan, C.J. Gilbert, C. Lemarechal C, C.A. Sagastizabal. Numerical Optimization. Theoretical and practical aspects. Springer Verlag (Universitext), 2006. Second edition.

Written examination (with a computer lab test, for the part concerning Optimization and discrete transforms).
The written examination is divided in three parts: a simple test on eementary computations, a more theoretically oriented series o questions, and some problems, whose solutions require ad hoc strategies. Papers that fail all the questions of one section will not be considered sufficient. The final grade will result from the sum of the grades of each single part.

The course is divided in two parts. The first part, Mathematical methods (6CFU), is shared with the degree programs in Electronic and Computer Engineering and in Industrial Engineering. The second part, Optimization and Discrete Transforms (3CFU), is taught by Professor Stefano Gualandi.