Khinchin's theorem in Teichmüller dynamics

This thesis is concerned with two themes which are strictly linked with each other, and therefore will be developed in parallel. The first one is dynamics in Teichmueller space. The Teichmueller space of a (topological, closed and orientable) surface S is defined as the set of the complex structures one can endow S with, up to isotopies. Such a space can be given a structure of geodesic metric space. The description of this structure requires the notion of flat structures on the underlying surface, i.e. flat Riemannian metrics with conical singularities, such that a foliation in straight lines in each direction is defined. The space of all flat structures is a sort of tangent bundle to the Teichmueller space, and the geodesic flow, knows as Teichmueller flow, has a simple description in these terms. It becomes interesting from a dynamical viewpoint when projected onto the moduli space, namely the set of the complex structures up to diffeomorphisms. Invariant subspaces under the flow are called strata; we are concerned in particular with dynamics in the strata made up by translation structures, a subspecies of the flat structures. The second theme treated in this work are interval exchange maps (i.e.m.s)i.e. injective maps of an interval which are locally a translation except at finitely many singularities. They are completely determined by providing some combinatorial data as well as the length data of the sub-intervals. If one considers an adequate leftmost portion of the considered interval, the first return map of the i.e.m. on this portion is a new i.e.m.. This yields a dynamics on the parameter space for i.e.m.s, called Rauzy dynamics. The themes above are linked on two levels. First of all, if one fixes a translation surface, the first return map induced by the flow in vertical direction on a horizontal segment is an i.e.m.; and a `generic' i.e.m. can always be obtained this way. But a link at a higher level is possible, too: the Teichmueller flow admits a transverse section such that the return map can be interpreted as Rauzy dynamics. Chapter 0 of the thesis is an introduction: it includes the preliminary material from the theory of dynamical systems which will be used in this work, as well as a description of the simplest case of the theory, represented by flat tori and rotations of the circumference. In Chapter 1 Teichmueller dynamics is formally, but rapidly, introduced; whereas Chapter 2 is concerned with the formalism related to interval exchange maps and Rauzy dynamics. Moreover, it is explained how it is possible to switch from this setting to the one of translation structures, and conversely. The first half of Chapter 3 treats, still in an extremely concise manner, classical questions related to the themes above. In particular it deals with ergodicity of i.e.m.s and of the Teichmueller and Rauzy dynamics and briefly introduces the Kontsevich-Zorich cocycle. The chapter ends with a technical discussion needed for the results tackled in the following chapters: its protagonists will be the reduced triples for an i.e.m. T, namely triples (b,a;n) where b is a singular point for $T^{-1}$, a is a singular point for T, and n is a positive integer, such that no singularities for $T^{-1},...,T^-n$ appears between $T^n(b)$ and a. Chapter 4 thus deals with a first generalisation of a theorem of A.Ya. Khinchin, found by Luca Marchese (2010). The Khinchin theorem in its classical formulation states a condition for a Diophantine inequality to have finitely, or infinitely many, solutions. Its generalisation to i.e.m.s states: Let f(n) be a positive, decreasing sequence. We are concerned with the quantity of solutions (b,a;n) to the condition $|T^n(a)-b|<f(n)$ for a fixed i.e.m. T, where b is a singular point of $T^-1$, and a is a singular point for T. If the sequence f(n) has a finite sum, then solutions are finitely many for almost any T; if nf(n) is still a decreasing sequence, with infinite sum, then solutions are infinitely many for almost any T. This result will be partially proved. It is interesting not only as a property of singularities of an i.e.m., but also because it yields a weaker version of a theorem of Jon Chaika, which states a similar property for generic points. Chapter 5 is again about translation surfaces. The theorem above is restated as a property of lengths of connections, namely segments connecting two singular points on a translation surface. Hence it is possible to gain another result of Chaika, which gives a property of 'strong recurrence' of foliations. And, eventually, this restatement of the generalised Khinchin theorem yields a logarithm law for the Teichmueller flow: Let X be a translation surface, and let Sys(X) be the minimum length of a connection of X. Denote $g^t$ the Teichmueller flow. Then, for almost any X, it holds that $\limsup [-\log (Sys(g^t(X))]/(\log t)=1/2$

A direct elimination algorithm for quasi-static and dynamic contact problems

This paper deals with the computational modeling and numerical simulation of contact problems at Unite deformations using the Finite element method. Quasi-static and dynamic problems are considered and two particular frictional conditions, full stick friction and frictionless cases, are addressed. Lagrange multipliers and regularized formulations of the contact problem, such as penalty or augmented Lagrangian methods, are avoided and a new direct elimination method is proposed. Conserving algorithms are also introduced for the proposed formulation for dynamic contact problems. An assessment of he performance of the resulting formulation is shown in a number of selected benchmark tests and numerical examples, including both quasi-static and dynamic contact problems under full stick friction and frictionless contact conditions. Conservation of key discrete properties exhibited by the time stepping algorithm used for dynamic contact problems is also shown in an example. (C) 2014 Elsevier B.V. All rights reserved.Peer ReviewedPostprint (author’s final draft

A study on exponential-size neighborhoods for the bin packing problem with conflicts

We propose an iterated local search based on several classes of local and large neighborhoods for the bin packing problem with conflicts. This problem, which combines the characteristics of both bin packing and vertex coloring, arises in various application contexts such as logistics and transportation, timetabling, and resource allocation for cloud computing. We introduce $O(1)$ evaluation procedures for classical local-search moves, polynomial variants of ejection chains and assignment neighborhoods, an adaptive set covering-based neighborhood, and finally a controlled use of 0-cost moves to further diversify the search. The overall method produces solutions of good quality on the classical benchmark instances and scales very well with an increase of problem size. Extensive computational experiments are conducted to measure the respective contribution of each proposed neighborhood. In particular, the 0-cost moves and the large neighborhood based on set covering contribute very significantly to the search. Several research perspectives are open in relation to possible hybridizations with other state-of-the-art mathematical programming heuristics for this problem.Comment: 26 pages, 8 figure

Large mass dimuon detection in the LHCb experiment

The structure of this thesis consists of two main parts: in the first part (Detector studies), the work is focused on the performances of the Multiwire Proportional Chambers, adopted for the detection of muons in LHCb, while in the second part (Physics studies) it is focused on the study of the LHCb potentialities to improve the knowledges of the proton Parton Distribution Functions with the physical channel Z0 -> mu+ mu-. The work described in the first part is concentrated on the cosmic rays test station developed in Rome2 in order to carry out the study of the detectors performances. In particular, the cosmic rays stand allows to perform a detailed study of the detector tracking capabilities and to obtain precise measurements of the efficiency and gas gain uniformity of the produced chambers. In the second part of the thesis is reported a study of the process pp -> Z0 -> mu+ mu-. The aim of the study is to demonstrate that, in spite of the limited angular acceptance and the optimization for a different kind of physics, the number of Z0 detected at LHCb is sufficient to make profitable physics. Moreover, the foward design of the spectrometer allows to study the proton structure in a kinematic region not probed by the present experiments. A particular focus has been put on the effect of the LHCb geometrical acceptance on the cross section sensitivity to the various set of partons, simulating the process with two event generators, PYTHIA and MC@NLO