Bøger i Publications of the Scuola Nor serien

In this thesis, we study the regularity of optimal transport maps and its applications to the semi-geostrophic system. The first two chapters survey the known theory, in particular there is a self-contained proof of Brenier' theorem on existence of optimal transport maps and of Caffarelli's Theorem on Holder continuity of optimal maps. In the third and fourth chapter we start investigating Sobolev regularity of optimal transport maps, while in Chapter 5 we show how the above mentioned results allows to prove the existence of Eulerian solution to the semi-geostrophic equation. In Chapter 6 we prove partial regularity of optimal maps with respect to a generic cost functions (it is well known that in this case global regularity can not be expected). More precisely we show that if the target and source measure have smooth densities the optimal map is always smooth outside a closed set of measure zero.

This book is taken from the lectures given at the ENSAE in March 1998 and at the Scuola Normale Superiore in May 1998. At the beginning of the 70's, the set of mathematical methods of finance were reduced to actuarial calculus. The modern approach uses the stochastic calculus theory and evidences the duality between problems of arbitrage and valorisation and a set of martingale probabilities.

These notes contain my lectures on light scattering by matter presented at the Scuola Normale Superiore, Pisa during May and June 1995. I have deleted some of the topics discussed then and added a few related to my recent work. The notes are not to be regarded as exhaustive but rather as a selection of topics. In particular I have discussed, as examples, the theoretical basis for the interpretation of experiment on light scattering by photons in alpha-quartz and by electronic excitations in boron-doped diamond.

The subject of this thesis is the Mu-calculus, which nowadays represents a very active research area in both theoretical and practical Computer Science. The Mu-calculus is a logic capable of expressing least and greatest solutions of fixpoint equations x=f(x), where f is a monotone function mapping some complete ordered set into itself. The thesis has two goals, one expository and one technical. The expository goal is to present some results and problems about the Mu-calculus. On the technical respect, this thesis addresses a specific expressiveness problem of the Mu-calculus, the alternation depth hierarchy problem.

The aim of this book is to give an elementary and essential introduction to the stochastic integration with respect to a semi-martingale. It is supposed to be useful to the beginner as a starting point for additional studies and readings in the subject. The exposed theory is only about real-valued processes. The prerequisites for this book are of an average level; basically the reader is supposed to be familiar with the language of probability and with the basilar notions of measure theory.

Cp20 is a neuronal protein involved in the molecular mechanisms of synaptic plasticity. Cp20 was first identified in the CNS of the marine snail Hermissenda, which can be trained to learn an association between visual and vestibular stimuli in a Pavlovian conditioning procedure. Upon learning, cp20 is phosphorylated in a few defined neurons of Hermissenda, and inhibits the Ca2+-dependent K+ channels. This results in an increase in membrane resistence and therefore a hyperexcitability of the neuronal pathway. Similar biochemical mechanisms have been proposed to underlie memory in higher species, such as rabbit and rat. Cp20 was isolated from squid optic lobes by a 5-step purification. The protein was cloned, fused with an oligohistidine tail, and expressed in Esterichia coli. The activation of PKC, the enzyme responsible for cp20 phosphorylation, was also studied by an in vivo imaging system.

The subject of the present thesis is a study on bipolaron formation in heavily doped and polar materials. The study is applied to the physics of superconducting compounds with high transition temperature. In particular this thesis addresses the problem of the formation and superconducting properties of real space electron (hole) pairs in these systems and their dependence on the carrier density. The pairing mechanism results from the dynamical cooperation of photonic and electronic degrees of freedom which characterize the heavily doped polar material. We call these pairs biplasmapolarons (BBP) because they can be thought as quasi-particles composed by two electrons (holes) and their cloud of virtual photons and plasmons. In the present work we extend the theory to the calculation of both BPP binding energy and effective mass and we discuss the features of the effective electron-electron interaction.

Spin glass theory is really one of the most fascinating and appealing topics in theoretical and statistical physics. Since the oldest paper in 1959, such systems, which appearently are neither ferromagnetic nor antiferromagnetic, have been studied far and wide, but a full and complete understanding of the transition of disordered systems is still missing. The physical reason of their complexity is that the free energy at low temperature occurs as a rough landscape with a huge number of metastable states and the conventional statistical techniques fail.

Electron gas theory is one of the broadest fields in theoretical condensed matter physics, and even its most elementary application to the study of collective excitations and screening in the simpe metals poses interesting questions. Recent Electron Energy Loss and Inelastic X-Ray Scattering experiments have shown that traditional electron gas theories are unable to account for the measured plasmon dispersion relation. While it has become clear that neither correlation nor band structure alone can explain those results, the recently developed Time Dependent Density Functional Theory provides a general framework which can account for both.

Sensory experience can dynamically modify both the anatomy and function of the brain. In particular, neurons can modify their connections in response to enviromental stimuli. This process, generally referred to as "neuronal plasticity", is the basis of learning and memory processes in the brain. Which are the bases of neuronal plasticity? That is, how can environmental factors and sensory experience modify the connections between neurons? Neurotrophic factors of the Nerve Growth Factor (NGF) family are known to finely regulate the development and maintenance of synaptic connections in the PNS. The results obtained by the group of Maffei firstly suggested the hypothesis that neurotrophic factors could play a key role in the plasticity of the mammalian visual system. The purpose of this thesis is to give a general overview about the recent advances in the field of neurotrophine-regulated plasticity of the mammalian visual system.

This set of lecture notes is the outcome of a lecture series, given in April 2000 by the author while holding the "Cattedra Galileiana" at Scuola Normale Superiore in Pisa. The purpose of the lectures was to give an overview of some recent work concerning structural properties of the evolution of the forward rate curve in an arbitrage free bond market.

These are the informal notes of two seminars held at the Università di Roma "La Sapienza", and at the Scuola Normale Superiore in Pisa in spring and autumn 1997. We discuss in detail the content of the parts of a paper by Givental dealing with mirror symmetry for projective complete intersections.

During the academic year 1995/96, I was invited by the Scuola Normale Superiore to give a series of lectures. The purpose of these notes is to make the underlying economic problems and the mathematical theory of exterior differential systems accessible to a larger number of people. It is the purpose of these notes to go over these results at a more leisurely pace, keeping in mind that mathematicians are not familiar with economic theory and that very few people have read Elie Cartan.

The main aim of the work was initially to study the possible relevance of the cross-over from BCS superconductivity to Bose-Einstein condensation to explain some unifying features found in experiments on high temperature superconductors. During our work we have gradually recognized that many of the technical and physical difficulties that we had to face were present already in the purely Bose system. We passed then to study more closely the Bose system itself. The second part of the thesis is dedicated to renormalization group approach for the zero-temperature interacting bosons; in this way we have healed the divergences of the theory and found the exact infrared behavior of the Bose system.

The achievement of Bose-Einstein condensation in ultra-cold vapours of alkali atoms has given enormous impulse to the study of dilute atomic gases in condensed quantum states inside magnetic traps and optical lattices. High purity and easy optical access make them ideal candidates to investigate fundamental issues on interacting quantum systems. This review presents some theoretical issues which have been addressed in this area and the numerical techniques which have been developed and used to describe them, from mean-field models to classical and quantum simulations for equilibrium and dynamical properties. The attention given in this article to methods beyond standard mean-field approaches should make it a useful reference point for future advances in these areas.

The classical theory of braids is deeply connected with the theory of reflection groups and there are many relations between Artin groups and Coxeter groups. It turns out that the classifying spaces of Artin groups of finite type are affine varieties, the complement of the singularities associated to Coxeter groups.In order to study the topology of the Milnor fiber of these non-isolated singularities together with the monodromy action it is useful to compute the cohomology of the Artin groups with coefficients in an abelian representation.In this book a description of this cohomology for Artin groups of type A and B and for affine Artin groups of the same type is given.

Little more than one hundred years have gone by since the birth of Ettore Majorana, a highly renowned theoretical physicist. His career was brief and irregular but very intense, and he disappeared in March 1938 in circumstances that are still not completely clear. This volume is a contribution to a better understanding of the scientific, academic and human personality of Ettore Majorana, beyond the layers of legendary aspects which have accumulated over the years. Based on primary sources alone - scientific literature of the period and numerous archival documents - the figure of Ettore Majorana emerges in a completely new light. The young scientist is intensely involved in scientific research, completely independent, always striving to offer innovative contributions of the highest level according to the most advanced international standards, and very determined to make his results known by following a shrewd publication strategy. He is profoundly interested in his academic career, very scrupulous with institutional relationships, and attentive to his students. Moreover, his documented scholarly activity is much wider than previously reported. This historical analysis shows also that Majorana had a very important role in orienting research in Rome, especially in the sector of the statistical model of the atom and in Nuclear Physics. It is also clear that there are aspects of Ettore Majorana's life which transmit a solid historical legacy from the cultural and human points of view, besides the scientific legacy which is universally recognized to him. A rich reproduction of original documents completes the volume.

The experimental achievement of Bose-Einstein condensation (1995) and of Fermi degeneracy (1999) in ultra-cold, dilute gases has opened a new field in atomic physics and condensed matter physics. This thesis presents an overview of theoretical and experimental facts on ultra-cold atomic gases. A Green's function scheme is examined, and the book also applies a novel spin-density-functional approach to the study of Fermi gases inside one-dimensional optical lattices.

In this thesis I study the untwisted affine quantum algebras and their specialization at a primitive l-th root of unity. In particular my goal is the complete description of the center, when l is odd. The center of the specialization of the quantum algebra at odd roots of unity is already known in the finite case, that is for finite dimensional semisimple Lie algebras. In this thesis I prove that analogous results hold in the affine untwisted case.

Since neutrinos interact so weakly with matter, most of their basic properties are still largely unknown. One of the most important issues to be settled concerns their rest mass. We have no idea why neutrinos are so much lighter than their charged lepton partners; no fundamental symmetry in nature requires massless neutrinos. Massive neutrinos are demanded to explain the anomalous counting rate of experiments measuring the solar and the atmospheric neutrino fluxes. The discrepancy between experimental data and theoretical predictions can be accounted for in terms of neutrino oscillations, which would take place only in the case of massive neutrinos. The subject of this thesis is the search for neutrino oscillations in CHOOZ, the first long baseline experiment to explore a neutrino mass region where hints at neutrino oscillations came from the atmospheric neutrino anomaly.

This thesis is devoted to investigating some aspects of the geometry and function theory on domains in complex vector spaces. The link between geometry and function theory stems from the standard approach whereby, if D is a domain in Cn, S is a semigroup of holomorphic maps of D into itself, and Hol(D) is the space of all scalar valued holomorphic functions on D, then S induces a semigroup of linear operators on Hol(D) whose invariant subspaces give information on the structure of S. In the one-dimensional case, the theory of Riemann surfaces leads to precise results. Passing from one to several complex variables the situation changes radically. In order to obtain further information on Aut D, one has to drastically restrict the class of bounded domains D under consideration, focusing the attention on bounded homogeneous domains or even on the narrower class of bounded symmetric domains.