Challenges for first-principles methods in theoretical and computational physics: multiple excitations in many-electrons systems and the Aharonov-Bohm effect in carbon nanotubes

In the first part of the thesis we will describe double excitations in the absorption spectrum. Double excitations are a peculiar effect of interacting systems which does not have a counterpart in non-interacting ones. The optical absorption spectrum of a system is obtained by shining light on it. At the microscopic level photons hit the electrons which sit in the ground state and change their configuration. If the light source is not too intense this can be described in linear response; that is only "one photon" processes are involved, only one electron per time can be influenced. Here is where the interaction comes in. The hit electron is linked to the others and so other process take place, one of these is the appearance of multiple excitations. In the second part of the thesis we focus on the application of more standard techniques to the description of carbon nanotubes (CNTs). In particular we focus on the effects of magnetic fields on CNTs. CNTs are quasi 1D-systems composed by carbon atoms which have been discovered in 1952. Under the effect of a magnetic field electrons delocalized on a cylindrical surface display a peculiar behaviour, known as Aharonov-Bohm effect. The Aharonov-Bohm is a pure quantum mechanical effect which does not have any counterpart in classical physics. In CNTs the Aharonov-Bohm modify the electronic gap and so can be used to tune the electronic properties. Though a model able to account for such process is available in the literature, in the present work we will describe the effect of magnetic fields "ab-initio". In the description of CNTs we will use standard approximations which are by far much more accurate and general than any approximation introduced in phenomenological descriptions based on model systems.Comment: PhD thesis. http://phd.fisica.unimi.i

Hilbert Spaces Without Countable AC

This article examines Hilbert spaces constructed from sets whose existence is incompatible with the Countable Axiom of Choice (CC). Our point of view is twofold: (1) We examine what can and cannot be said about Hilbert spaces and operators on them in ZF set theory without any assumptions of Choice axioms, even the CC. (2) We view Hilbert spaces as ``quantized'' sets and obtain some set-theoretic results from associated Hilbert spaces.Comment: 51 page

Foundations of Applied Mathematics I

This paper aims to study the foundations of applied mathematics, using a formalized base theory for applied mathematics: \ZFCA_{\sigma} (Zermelo-Fraenkel set theory (with Choice) with atoms, where the subscript used refers to a signature specific to the application. Examples are given, illustrating the following five features of applied mathematics: comprehension principles, application conditionals, representation hypotheses, transfer principles and abstract equivalents

Foundations of Applied Mathematics I

This paper aims to study the foundations of applied mathematics, using a formalized base theory for applied mathematics: \ZFCA_{\sigma} (Zermelo-Fraenkel set theory (with Choice) with atoms, where the subscript used refers to a signature specific to the application. Examples are given, illustrating the following five features of applied mathematics: comprehension principles, application conditionals, representation hypotheses, transfer principles and abstract equivalents

Lorentzian approach to noncommutative geometry

This thesis concerns the research on a Lorentzian generalization of Alain Connes' noncommutative geometry. In the first chapter, we present an introduction to noncommutative geometry within the context of unification theories. The second chapter is dedicated to the basic elements of noncommutative geometry as the noncommutative integral, the Riemannian distance function and spectral triples. In the last chapter, we investigate the problem of the generalization to Lorentzian manifolds. We present a first step of generalization of the distance function with the use of a global timelike eikonal condition. Then we set the first axioms of a temporal Lorentzian spectral triple as a generalization of a pseudo-Riemannian spectral triple together with a notion of global time in noncommutative geometry.Comment: PhD thesis, 200 pages, 9 figures, University of Namur FUNDP, Belgium, August 201

Approximation and Classification in the Ergodic Theory of Nonamenable Groups

This thesis is a contribution to the theory of measurable actions of discrete groups on standard probability spaces. The focus is on nonamenable acting groups. It is organized into two parts. The first part deals with a notion called weak equivalence, which describes a sense in which such actions can approximate each other. The second part deals with the concept of entropy for measure preserving actions of sofic groups.</p

Artículos

ber that a closed subspace M of a real Banach space X is said to be an L2- summand subspace if there exists another closed subspace N of X verifying X = (M ® N)2 (that is, ¡m + n||2 = ||m||2 + ||n||2 for every m e M and every n g N.) The linear projection t?m of X onto M that fixes the elements of M and maps the elements of N to {0} is called the L2-summand projection of X onto M. For a wider perspective about L2-summand subspaces, see [1], [2], and [3]. A vector e of a real Banach space X is an L2-summand vector if Re is an L2- summand subspace. Furthermore, if e A 0 then there exists a functional e* in X*, which is called the L2-summand functional of e, such that ||e*|| = ||e||—1, e* (e) = 1, and 7TRe G) = e* (a:) e for every x £ X. The set of all L2-summand vectors of A' will be denoted by Lx- For a wider perspective about L2-summand vectors, see [1]. Let us recall two relevant results about L2-summand vectors402 pág

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic