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Optimal Control Theory

The optimal control problem basic mathematical notion from the variational calculus - minimization of functionals - Euler-Lagrance equation - minimization of functional under constraints - optimal control of continuous or discrete time systems with or without state/input constraints - the minimum principle of Pontryagin - the linear quadratic (LQ) regulation and tracking problem - Riccati equation - bang-bang control - Hamilton-Jacobi-Bellman theory - dynamic programming - the linear quadratic Gaussian (LQG) problem - applications in Matlab.

Cryptography

Classical Cryptosystems - Perfect Security - Feedback Shift Register - Basic Computational Number Theory - RSA Cryptosystem - Rabin Cryptosystem - Primality Testing - Factorization Methods - Discrete Logarithm - Diffie-Hellman Protocol - ElGamal Cryptosystem - Okamoto-Uchiyama Cryptosystem - Digital Signatures - Cryptographic Protocols.
Remark: The basic concepts of Linear Algebra, Algebraic Structures and Number Theory are needed for the aforementioned course.

Special Topics ( Automata over Semirings )

Semirings, weighted automata over semirings. Recognizable series. Properties of recognizable series. The problem of determinization. Decidability problems. Applications: Fuzzy languages. Digital image compression.

Quantum Information and Computation

Introduction to Quantum Mechanics: Mathematical introduction (Hilbert spaces - Spectrum of autoadjoint and unitary operators-Lie group U(N) - control theory of groups) - Quantum states and observables - the state space of Hilbert space - The state space as a set of definite operators.

Dynamic Modeling

Phenomenological Laws, Scientific Method, Mathematical Modeling, The Prediction Problem. Differential Equations, Difference Equations and Dynamical Systems.Classification, Stability, Solutions (Analytic, Approximate, Numerical), Simulations.Selective Applications.Chaos, Random Number Generators, Population Dynamics and ChemicalReactions, Economics, Biology, Signals and Filters, Cellular Automata, Dynamics of Communication Νetworks. Τhe Objectives of the course are:

Statistics and Decision Making

The characteristic functions for the multivariate random variables - The multivariate normal distribution and related topics - Application in statistical analysis (Cochran’s theorem, ANOVA, regression, X2) - Statistical inference: The Neyman-Pearson lemma - Likelihood ratio test and related procedures - decision theory.

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