8th Semester AMPS

Group actions, Burnside s theorem and applications, p-groups, the Sylow theorems, the class equation, applications. Introduction to Ring Theory. Polynomial rings, integral domains. Prime and maximal ideals, ring homomorphisms, isomorphism theorems. Field extensions, geometric constructions, finite fields. Automophisms, Galois theory, applications.This course is useful to those who have selected the stream «Mathematics for Information Sciences».

Introductory Example: The Dirichlet problem. Weak form. Numerical solution using the Finite Element Method. Boundary Value Problems and Galerkin Method: General weak form. Lax-Milgram Theorem. Galerkin Method. Error estimation. Variational form. Rayleigh-Ritz-Galerkin Method. Generalized derivatives and Sobolev spaces. Green s types. Elliptic boundary value problems. Existence and uniqueness. Mixed boundary conditions. Applications. Finite Element Methods for Elliptic Boundary Value Problems: One-dimensional finite elements. Piecewise polynomial functions. Cubic Hermite functions and splines. Two-dimensional and three-dimensional finite elements. Element-wise polynomial functions. Tensor product functions. Error estimates. Applications: Fluid flow, Heat flow, Various electrical potentials, Loaded beam, Loaded plate. Finite Difference Methods: Sturm-Liouville and Dirichlet problems. Heat equation. Wave equation. Compatibility, stability and convergence.

Propagation of EM waves in infinite space with emphasis on attenuation and dispersion issues. Propagation in transmission lines, composite resistance, load matching, scattering matrices. Propagation in metallic waveguides. Transverse Electric (TE) and Transverse Magnetic (TM) normal modes. Parallel plate, rectangular and circular waveguides. Waveguiding in dielectric waveguides. Propagation in single and multi-mode optical waveguides, dispersion phenomena in optical fibers.

Database Management Systems (DBMS) and their architecture. Data Structures for Databases. Modeling - The E-R model. Reference to classical Database models (Hierarchical, Network). The Relational Model. Database Languages - The SQL language. File Systems and Physical Database Design. Logical Design and Normalization. Management and Operation Issues (integrity, optimization, reorganization, security, functionality, etc.). Current Topics (object-oriented systems, multi-systems, personal computer systems, etc.).

Introduction to Calculus of Variations: Necessary and sufficient conditions for extrema. Euler-Lagrange equations. Extrema with constraints, Lagrange multipliers. Optimal Control: Control systems, Attainable sets, Topological properties, Controllability. The minimum time problem in the linear case, Extremal control, Maximum Principle. Minimization of quadratic cost in the linear case without input set constraints, Riccati equation. Nonlinear systems: Topological properties of attainable sets, extremal control, the general Maximum Principle (Pontryagin s Maximum Principle), Necessary conditions in optimal control problems with and without control constraints. Sufficient conditions and existence theorems. The Hamilton-Jacobi-Bellman equation. Applications.

This course provides the essential mathematical foundation for understanding General Relativity (GR). It covers: Special Relativity: Geometry, relativistic particle dynamics, and the classical relativistic field. Differential Geometry: Manifolds, tensor fields, connections, geodesics, metric, curvature. Einsteins Equation: Interaction of geometry and matter, symmetries, and conserved quantities, Lagrangian formalism. Schwarzschild Solution: Black holes, free particle trajectories, and causal structure. The course also introduces Cosmology and the physics of the expanding and accelerating Universe, including: Friedman-Lemaitre-Robertson-Walker (FLRW) model: Cosmological principle, Hubbles law, phenomenology. Cosmological Constant: Examining de Sitter spacetime and relevant cosmological data. Accelerating expansion, inflation, cosmic epochs, dark energy.

Review of regression and analysis of variance. Multiple comparisons. Fixed and random effects models. Tests for homogeneity of variances. Nonparametric one-way ANOVA. Two-way ANOVA with and without interaction; random and mixed models. Orthogonal contrasts. ANOVA with 3 or more factors; random and mixed effects models. Graeco–Latin square designs. Balanced incomplete block (BIB) designs and their statistical analysis. Recovery of interblock information in BIB designs. Partially BIB designs and their statistical analysis. Youden squares. Lattice designs. 2k factorial designs. Design projection. The addition of centre points to the 2k design. Yates’ algorithm for the 2k design. Confounding in the 2k factorial design. Partial confounding. Two level fractional factorial designs. The notion of resolution. 3k factorial designs. Confounding in the 3k factorial design. Factorial designs with mixed levels. Response curve methodology. Nested and split-plot designs. Applications using statistical packages.

Curves: General concepts, parametric curves in space, implicit form. Arc length, curvature, torsion, Frenet-Serret formulas, spherical indicatrices. Local form of curve. Osculating circles and spheres, involutes and evolutes. Fundamental Theorem of curves. Plane curves, envelopes, convexity. Closed curves and global theorems: Jordan Curve Theorem, Schonflies Theorem, Riemann-Hopf Theorem, Four Vertex Theorem, Isoperimetric Inequality. Surfaces: General concepts, definition of a surface (surface patch, simple surface, smooth surface), tangent space, atlases, orientability. First and second fundamental forms. Gauss-Rodrigues map, Weingarten map, shape operator. Normal curvature, principal directions, Gaussian curvature, mean curvature, geodesics, lines of curvature, asymptotic curves. Classification of surface points, local form of a surface. Theorema Egregium, Gauss-Bonnet Theorem. Surface mappings: isometric, conformal, equiareal, minimal surfaces.

Introduction to controlled thermonuclear fusion: Basic nuclear reactions, effective cross-section, and power produced. Energy balance and Lawsons criterion. Plasma definition: Ionization, macroscopic charge neutrality, shielding, Debye length, oscillations, and plasma frequency. Properties of ideal plasmas. Methodological approaches to studying plasma. Charged particle motion in electromagnetic fields: Homogeneous magnetic field, cyclotronic frequency, Larmor radius, magnetic moment. Homogeneous static electric and magnetic fields, the ExB drift. Weakly inhomogeneous magnetic field, gradient, and curvature drifts. Magnetic mirrors. Time-varying fields and polarization drift. Magnetic confinement of particles in toroidal devices. Collisions: Coulomb collisions, Rutherford scattering, energy and momentum transfer through collisions, collision frequencies. Plasma as a multiple fluid: Material derivative, Lagrange derivative, conservation of mass and momentum, Reynolds transport theorem. Kinetic pressure and energy conservation. Ideal fluid equations, plasma resistance. Plasma as a single fluid (Magnetohydrodynamics): Single fluid equations, ideal and resistive magnetohydrodynamics. Magnetic flux, magnetic viscosity, Reynolds number, magnetic pressure, Virial theorem. Magnetohydrodynamic Equilibrium: Equilibrium equations, theta pinch, straight-line screw pinch, axisymmetric toroidal magnetic configuration. Stability analysis: Linear analysis and normal modes, energy method, Rayleigh-Taylor instability, waves in ideal magnetohydrodynamics.

Principles of the Physics of ionizing radiation. Basic and applied radiobiological theories and models (LQ model, etc.). Ionizing radiation characteristics as properties of the atomic nucleus. Theory of the interaction of ionizing radiation with biological and living matter. Nuclear reactions and isotope production. Ionizing radiation effects on biological organisms. Biological effects of ionizing radiations and their use in clinical medicine (proton therapy). Advanced techniques for clinical applications and the use of accelerating devices. Introduction to dosimetry and radiation protection.

First-order predicate calculus, models, Herbrand models, clauses, normal forms, prenex, Skolem normal form, resolution. Correctness and completeness of Robinson resolution. Theory of Logical Programming: Horn clauses, negation as failure and its semantics, non-monotonic reasoning, 3-valued models. Functional programming: untyped, typed, proofs as programs, Curry-Howard isomorphism. Second-order logical systems, polymorphism. Semantics: programming languages, fixed point. Algebraic specifications. Introduction to category theory.

Introduction: Basic terminology. Vertex Degrees: Properties. Vertex-degree sequences (the Havel-Hakimi and the Erdos-Gallai theorems). Vertex-degree sets (the Kapoor-Polimeni-Wall theorem). Paths, Circles, Distances: Definitions and properties. Eccentricity. Distance decomposition. Graph circumference and graph girth. Graph Connectivity: Vertex connectivity, minimum separators, properties of connected graphs. Biconnectivity, k-connectivity (Menger theorem, Whitney theorem, Halin theorem). Edge connectivity. Trees: Definitions and basic properties. Spanning tree. Labeled tree. Prufer sequence. Cayley’s theorem. Eulerian Graphs: Euler tour and Euler path. Characterization of Eulerian graphs. Fleury’s algorithm. Euler’s theorem. The Chinese-postman problem. Hamiltonian Graphs: Definitions and properties. Dirac’s theorem. Ore’s theorem. Bondy-Chvatal theorem. Vertex Coloring: k-coloring, chromatic number. Brook’s theorem. Matchings: Maximum and perfect matchings, alternating and augmenting paths. Berge’s theorem. Bipartite matchings. Hall’s theorem. Edge Coloring: k-edge coloring, chromatic index. Properties. Planar Graphs: Definitions and properties. Euler’s theorem. Kuratowski’s theorem. Outerplanar graphs. Coloring of planar graphs.

Hilbert Spaces: Introduction, basic concepts. Bounded Operators: Operators in Hilbert spaces. Definitions, properties. Bilinear forms. Operator norm. Adjoint of an operator. Self-adjoint, normal, unitary operators. Projections: Orthogonal projections, properties. Invariant subspaces. Compact Operators: Finite rank operators, compact and self-adjoint compact operators. Spectral theory: Spectral theorem for self-adjoint compact operators. Applications to integral operators and Sturm-Liouville systems. Green s functions. Operators in Banach spaces: The adjoint operator, compact operators. Fredholm Operators: Definitions, properties, Fredholm index. Unbounded Operators: Closed operators, symmetric and self-adjoint operators.

Propositional Logic: Language, Unique readability, Logical connectives, Truth assignments, Semantic concepts, Completeness of connectives, Disjunctive and conjunctive normal form, Compactness theorem of propositional logic, Applications. First-Order Predicate Logic: Language, Variables, Concepts of free and bound variables, Substitution, Analogy with programming, The concept of structure, Interpretation of language, Tarski s definition of truth. Axiomatization of First-Order Logic: The concept of an axiomatic system, Analogies with algorithmic concepts, The concept of consistency, Gödel s completeness theorems, and Gödel-Church decidability. Proof Theory of Propositional and Predicate Logic: The Gentzen system, Propositional resolution, Elimination of cuts, Tableau systems, Completeness via tableau systems.

General on models: Types, reliability, construction. Mechanical models. Population dynamics (single-species, multi-species, Lotka-Volterra competitive models). Lanchester combat models. Ecological – Biological models. Dimensional Analysis: Buckingham s pi theorem. Normalization. Perturbation Methods: Model construction, regular and singular perturbation, boundary layer analysis. Calculus of Variations: Model construction, variational problems (brachistochrone), Euler – Lagrange equation, Hamilton s principle, isoperimetric problems, geodesics. Traffic models. Elliptic problems: Gravitational field. Electromagnetism. Acoustics. Electrochemical plating. Hyperbolic problems: Traveling waves. Telegraph equation, pantograph. Scattering. Parabolic problems: Electromagnetism. Heat and mass transfer. Probabilistic heat model. Economic model. Wave phenomena in continuous media: Linear and nonlinear waves. Burger s, KdV equations, mathematical models of continuous media. Stochastic models. Prerequisite knowledge: Mathematical Analysis, Differential Equations, Mathematica, Matlab. Convex sets and convex functions. Fréchet derivatives and directional derivatives. Extrema. Existence and uniqueness theorems. Basic necessary and sufficient conditions for optimality. Lagrange and Kuhn-Tucker-Lagrange multiplier theorems. Quadratic functions. Least Squares Methods and applications. Golden Section, Gradient, Conjugate Gradient, Newton, Frank-Wolfe, Projected Gradient, Penalty, Gradient-Penalty Methods. Applications in Optimal Control.

Basic relations and constitutive equations of the linear theory of thermoelasticity. Basic relations and constitutive equations of the linear theory of electroelasticity. Elements of crystallography and crystal physics. Interaction of physical fields in piezoelectric media. Waves in piezoelectric media. Fracture mechanics of piezoelectric materials. Basic relations and constitutive equations of magnetothermoelasticity.

For the description, please refer to the course with code 9186.

Basic concepts of reliability. Data truncation. Reliability or survival function, hazard function. Lifetime distributions: Gamma, Weibull, Gumbel, Log-logistic, etc. Non-parametric estimation: Kaplan-Meier estimator, Nelson-Aalen estimator. Log-rank test. Graphical tests. Model fitting using the maximum likelihood method. Goodness-of-fit tests. Regression models for lifetime data: proportional hazards models, accelerated failure time models and Cox s semi-parametric model. Model development and diagnostic methods, Cox-Snell residuals, Schoenfeld residuals. System reliability. Repairable systems. Computer applications.

The course is laboratory-based and takes place in a continuous 4-hour session per week. 12-13 laboratory exercises are conducted per academic year (during the spring semester), out of a total of 15 mentioned below (depending on the availability of the corresponding setups). During the 4-hour session, the presentation of the experimental method and the experimental setup will precede (1.5 hours), followed by the execution of the experiment and the acquisition of measurements (2.5 hours). Discussion and suggestions for result analysis will follow. For each exercise, each student submits a full report, no later than one week after the exercise is performed. The exercises are conducted in the Physics Section (PS) of the School of Applied Mathematical and Physical Sciences (SEMFE) of the National Technical University of Athens (NTUA) and at the National Center for Scientific Research Demokritos (NCSR D), within the framework of educational cooperation between the Physics Section and the Institute of Materials Science (IMS) of NCSR D, and are as follows: 1. Differential Scanning Calorimetry (PS) 2. Dielectric Spectroscopy (PS) 3. Dynamic Mechanical Analysis (PS) 4. Stress-Strain Measurement (PS) 5. Raman Spectroscopy (PS) 6. Infrared Spectroscopy (PS) 7. X-Ray Diffraction (PS) 8. Atomic Force Microscopy (PS) 9. Electrical measurements in semiconductor systems (PS) 10. Modulated Photoreflectance (PS) 11. Ellipsometry (PS) 12. Nuclear Magnetic Resonance (IMS) 13. Transmission Electron Microscopy (IMS) 14. X-Ray Crystallography (IMS) 15. Magnetic Measurements (IMS).

Computability: Logic as a Foundation for C.S. Historical overview of decidability for mathematical sentences, of solvability and computability of problems in an effective, i.e. algorithmic way. Simple equivalent models of computation: Turing Machines, WHILE programs. Induction and Recursion, encodings and Semantics. Fix-point theory. Arithmetical Hierarchy. Complexity: Inclusions among Complexity Classes. Reductions and Completeness. Oracles. Polynomial Hierarchy. Probabilistic,Interactive and Counting classes. Advanced topics of the theory of Formal grammars. Applications on the Syntax of programming languages.

Introduction. Classification, Polymer classification. Polymer molecular weight, exercises. Brief presentation of polymerization techniques. Macromolecule conformations, Introduction to crystallinity. Kinetics and Thermodynamics of crystallinity, Polymer transitions, WLF equation, DSC method. Reminder of Elasticity (briefly), Introduction to elastomers. Statistical molecular theory of elastomers. Viscoelasticity, exercises, dynamic mechanical analysis. Mechanical failure of polymers. Introduction to Polymer Rheology, exercises. Electrical properties of polymers – conductive polymers. General about composite materials. Classical reinforcement models. Introduction to nanocomposite materials. Properties of nanocomposite materials (mechanical, thermal, electrical, magnetic, optical).

Nuclear decay law, γ-ray emission. Electric and magnetic multipoles. Bound states of nucleons, deuterium nucleon exchange forces. Nuclear models (liquid drop, shell, collective). Nuclear deformation. Rutherford scattering. Nuclear reactions, nuclear reactions cross-section. Neutron physics. Applications of Nuclear Physics in: Study of materials (RBS, ERDA, PIXE, etc.) Medicine (diagnosis and therapy) Environment Archaeometry Industry Nuclear fission and nuclear fusion.

The idea behind this course is to train students on how to make presentations at the scientific level and prepare on a topic they choose or are given from approved lists. Each student presents to the class a topic related to the Physics courses taught in the Applied Physics concentration.

Recall of the Schroedinger equation. Probability density and current. Lorentz transformations and relativistic formalism. Klein-Gordon equation. Probability density and current. Negative energies and negative probability density. Quantum electrodynamics of spinless particles. Electron-muon scattering. Effective cross-section and invariant amplitude. Decay rate. Dirac equation. Gamma matrices, spinors, probability density. Solutions of the Dirac equation, spin=1/2 particles, antiparticles, helicity, massless fermions. Transformations of the Dirac spinor. Bilinear quantities. The Lagrange-Hamilton formalism. Euler-Lagrange equations. Classical field mechanics. Noether s Theorem. Invariance of the Lagrangian under internal transformations. Invariance in non-Abelian global symmetries. Non-Abelian local symmetries. Quantum Chromodynamics (QCD). Spontaneous symmetry breaking. Spontaneous symmetry breaking in gauge theories. Abelian and non-Abelian cases. The Standard Model. The Higgs particle. The masses of gauge bosons. The masses of fermions. Generation mixing. CKM matrix. Neutrino masses. The physics of the Higgs particle.

Continuous-time Stochastic Processes. Filtrations. Stopping times. Basic properties of continuous-time Martingales. Markov processes. Definition and basic properties of Brownian motion. Ito stochastic integral. Ito processes and diffusion processes. The quadratic variation of Brownian motion and Ito s formula. Girsanov s Theorem. Stochastic differential equations. Feynman-Kac Theorem. Examples and applications. Continuous-time market models. The Black-Scholes model. Valuation of options. European options. Special topics.

Estimation of probability density function with applications. Randomization tests and Monte Carlo. Stochastic simulation: generating uniform random deviates, inversion method, rejection sampling, variance reduction techniques. MCMC algorithms: Metropolis-Hastings algorithm. Resampling methods: Jackknife, bootstrap, cross-validation. EM algorithm. Applications of these methods using the R program.

This course introduces Monte Carlo simulation techniques, focusing on their applications in Statistical Physics. Students will explore simplified models of ferromagnetic materials, such as Ising and Potts models, gaining a foundational understanding of how Monte Carlo methods are applied across diverse scientific and technological fields, including biology, medicine, finance, condensed matter physics, and particle physics. Key topics covered include: Importance sampling principles and efficient algorithms. Calculating estimators of observable quantities and their associated errors. Analyzing autocorrelations in statistical samples generated by Markov chains. From a physics perspective, the course emphasizes the study of critical properties in thermodynamic systems undergoing continuous phase transitions. Students will explore the concept of universality, where diverse microscopic systems exhibit similar macroscopic behavior. This concept has significant implications for various thermodynamic systems and quantum field theories in condensed matter and particle physics. The course also covers: Critical point approximation techniques. Calculating critical quantities using finite-size scaling methods.

Introduction: Semiconductor Physics and Transport Phenomena. Planar Technology of Microelectronic devices. Bipolar Devices: p-n junctions, Space-charge region, Ideal diode characteristic, Recombination currents, transient phenomena, solar cells. Strong electric fields and breakdown phenomena. Bipolar transistors: operating principle, static current-voltage characteristics, transient phenomena, equivalent circuits. Ohmic and rectifying contacts in semiconductors. Tunneling phenomenon. Metal-semiconductor contacts (Schottky). Current-voltage characteristics. MOS devices (metal-oxide-semiconductor). MOS capacitance: energy band bending, surface states, capacitance and conductance as a function of bias voltage and frequency. Techniques for characterizing surface states. MOS Transistors: Static and dynamic response, simulation, surface carrier mobility. High-field and small-dimension phenomena, Scaling theory: Device types [JFET, MESFET, C-MOS etc.]. Scaling theory. Device types. JFET – MESFET – C-MOS. Small-dimension systems: Quantum wells. Quantum wires and dots. Introduction to nano-electronic devices.