Heuristic algorithms for the travelling deliveryman problem

Algoritmi euristici per la risoluzione del Travelling DEliveryman Problem

Rational algorithms for the decomposition of Feynman integrals via intersection theory

The decomposition of Feynman integrals into a basis of independent master integrals is an essential ingredient of high-precision theoretical predictions, that often represents a major bottleneck when processes with a high number of loops and legs are involved. In this thesis we present a new algorithm for the decomposition of Feynman integrals into master integrals with the formalism of intersection theory. Intersection theory is a novel approach that allows to decompose Feynman integrals into master integrals via projections, based on a scalar product between Feynman integrals called intersection number. We propose a new purely rational algorithm for the calculation of intersection numbers of differential $n-$forms that avoids the presence of algebraic extensions. We show how expansions around non-rational poles, which are a bottleneck of existing algorithms for intersection numbers, can be avoided by performing an expansion in series around a rational polynomial irreducible over $\mathbb{Q}$, that we refer to as $p(z)-$adic expansion. The algorithm we developed has been implemented and tested on several diagrams, both at one and two loops.

The Feynman-Kac formula and applications to PDEs

The thesis presents a probabilistic approach to the theory of semigroups of operators, with particular attention to the Markov and Feller semigroups. The first goal of this work is the proof of the fundamental Feynman-Kac formula, which gives the solution of certain parabolic Cauchy problems, in terms of the expected value of the initial condition computed at the associated stochastic diffusion processes. The second target is the characterization of the principal eigenvalue of the generator of a semigroup with Markov transition probability function and of second order elliptic operators with real coefficients not necessarily self-adjoint. The thesis is divided into three chapters. In the first chapter we study the Brownian motion and some of its main properties, the stochastic processes, the stochastic integral and the Itô formula in order to finally arrive, in the last section, at the proof of the Feynman-Kac formula. The second chapter is devoted to the probabilistic approach to the semigroups theory and it is here that we introduce Markov and Feller semigroups. Special emphasis is given to the Feller semigroup associated with the Brownian motion. The third and last chapter is divided into two sections. In the first one we present the abstract characterization of the principal eigenvalue of the infinitesimal generator of a semigroup of operators acting on continuous functions over a compact metric space. In the second section this approach is used to study the principal eigenvalue of elliptic partial differential operators with real coefficients. At the end, in the appendix, we gather some of the technical results used in the thesis in more details. Appendix A is devoted to the Sion minimax theorem, while in appendix B we prove the Chernoff product formula for not necessarily self-adjoint operators.

Numerical modeling on the short-term response of fiber-reinforced concrete round panels

Fiber-reinforced concrete is a composite material consisting of discrete, discontinuous, and uniformly distributed fibers in plain concrete primarily used to enhance the tensile properties of the concrete. FRC performance depends upon the fiber, interface, and matrix properties. The use of fiber-reinforced concrete has been increasing substantially in the past few years in different fields of the construction industry such as ground-level application in sidewalks and building floors, tunnel lining, aircraft parking, runways, slope stabilization, etc. Many experiments have been performed to observe the short-term and long-term mechanical behavior of fiber-reinforced concrete in the last decade and numerous numerical models have been formulated to accurately capture the response of fiber-reinforced concrete. The main purpose of this dissertation is to numerically calibrate the short-term response of the concrete and fiber parameters in mesoscale for the three-point bending test and cube compression test in the MARS framework which is based on the lattice discrete particle model (LDPM) and later validate the same parameters for the round panels. LDPM is the most validated theory in mesoscale theories for concrete. Different seeds representing the different orientations of concrete and fiber particles are simulated to produce the mean numerical response. The result of numerical simulation shows that the lattice discrete particle model for fiber-reinforced concrete can capture results of experimental tests on the behavior of fiber-reinforced concrete to a great extent.