Applicazione ai gruppi di Lie della prolungabilità per Equazioni Differenziali Ordinarie

Lo scopo di questa tesi è studiare in dettaglio l'articolo "A Completeness Result for Time-Dependent Vector Fields and Applications" di Stefano Biagi e Andrea Bonfiglioli, dove si ottiene una condizione sufficiente per la completezza di un campo vettoriale (dipendente dal tempo) in RN, che generalizza la ben nota condizione di invarianza a sinistra per i gruppi di Lie.

Sigma models for genuinely non-geometric backgrounds

The existence of genuinely non-geometric backgrounds, i.e. ones without geometric dual, is an important question in string theory. In this paper we examine this question from a sigma model perspective. First we construct a particular class of Courant algebroids as protobialgebroids with all types of geometric and non-geometric fluxes. For such structures we apply the mathematical result that any Courant algebroid gives rise to a 3D topological sigma model of the AKSZ type and we discuss the corresponding 2D field theories. It is found that these models are always geometric, even when both 2-form and 2-vector fields are neither vanishing nor inverse of one another. Taking a further step, we suggest an extended class of 3D sigma models, whose world volume is embedded in phase space, which allow for genuinely non-geometric backgrounds. Adopting the doubled formalism such models can be related to double field theory, albeit from a world sheet perspective.

T-duality without isometry via extended gauge symmetries of 2D sigma models

Target space duality is one of the most profound properties of string theory. However it customarily requires that the background fields satisfy certain invariance conditions in order to perform it consistently; for instance the vector fields along the directions that T-duality is performed have to generate isometries. In the present paper we examine in detail the possibility to perform T-duality along non-isometric directions. In particular, based on a recent work of Kotov and Strobl, we study gauged 2D sigma models where gauge invariance for an extended set of gauge transformations imposes weaker constraints than in the standard case, notably the corresponding vector fields are not Killing. This formulation enables us to follow a procedure analogous to the derivation of the Buscher rules and obtain two dual models, by integrating out once the Lagrange multipliers and once the gauge fields. We show that this construction indeed works in non-trivial cases by examining an explicit class of examples based on step 2 nilmanifolds.

Lie symmetries in cortical inspired CNNs

Our scope in this thesis is to propose architectures of CNNs in such a way to model the early visual pathway, including the Lateral Geniculate Nucleus and the Horizontal Connectivity of the primary visual cortex. Moreover, we will show how cortically inspired architectures allow to perform contrast perceptual invariance as well as grouping and the emergence of visual percepts. Particularly, the LGN is modeled with a first layer l0 containing a single filter Ψ0 that pre-filters the image I. Since the RPs of the LGN cells can be modeled as a LoG, we expect to obtain a radially symmetric filter with a similar shape; to this end, we prove the rotational invariance of Ψ0 and we study the influence of this filter to the subsequent layer. Indeed, we compare the statistic distribution of the filters in the second layer l1 of our architecture with the statistic distribution of the RPs of V1 cells of a macaque. Then, we model the horizontal connectivity of V1 implementing a transition kernel K1 to the layer l1. In this setting, we study the vector fields and the association fields induced by the connectivity kernel K1. To this end, we first approximate the filters bank in l1 with a Gabor function and use the parameters just found to re-parameterize the kernel. Thanks to this step, the kernel is now re-parameterized into a sub-Riemmanian space R2 × S1. Now we are able to compare the vector and association fields induced by K1 with the models of the horizontal connectivity.

Heat kernel and worldline path integrals for the Proca theory

In this thesis we study the heat kernel, a useful tool to analyze various properties of different quantum field theories. In particular, we focus on the study of the one-loop effective action and the application of worldline path integrals to derive perturbatively the heat kernel coefficients for the Proca theory of massive vector fields. It turns out that the worldline path integral method encounters some difficulties if the differential operator of the heat kernel is of non-minimal kind. More precisely, a direct recasting of the differential operator in terms of worldline path integrals, produces in the classical action a non-perturbative vertex and the path integral cannot be solved. In this work we wish to find ways to circumvent this issue and to give a suggestion to solve similar problems in other contexts.

Lipschitz regularity for weak solutions of parabolic p-Laplacian type equations in certain subriemannian structures

The main aim of the thesis is to prove the local Lipschitz regularity of the weak solutions to a class of parabolic PDEs modeled on the parabolic p-Laplacian. This result is well known in the Euclidean case and recently has been extended in the Heisenberg group, while higher regularity results are not known in subriemannian parabolic setting. In this thesis we will consider vector fields more general than those in the Heisenberg setting, introducing some technical difficulties. To obtain our main result we will use a Moser-like iteration. Due to the non linearity of the equation, we replace the usual parabolic cylinders with new ones, whose dimension also depends on the L^p norm of the solution. In addition, we deeply simplify the iterative procedure, using the standard Sobolev inequality, instead of the parabolic one.

Fractional fermion charges induced by axial-vector and vector gauge potentials and parity anomaly in planar graphenelike structures

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Calculating current densities and fields due to shielded bi-planar radio-frequency coils

A method for the accurate computation of the current densities produced in a wide-runged bi-planar radio-frequency coil is presented. The device has applications in magnetic resonance imaging. There is a set of opposing primary rungs, symmetrically placed on parallel planes and a similar arrangement of rungs on two parallel planes surrounding the primary serves as a shield. Current densities induced in these primary and shielding rungs are calculated to a high degree of accuracy using an integral-equation approach, combined with the inverse finite Hilbert transform. Once these densities are known, accurate electrical and magnetic fields are then computed without difficulty. Some test results are shown. The method is so rapid that it can be incorporated into optimization software. Some preliminary fields produced from optimized coils are presented.

Heat flow for Yang-Mills-Higgs fields, part I

The Yang-Mills-Higgs field generalizes the Yang-Mills field. The authors establish the local existence and uniqueness of the weak solution to the heat flow for the Yang-Mills-Higgs field in a vector bundle over a compact Riemannian 4-manifold, and show that the weak solution is gauge-equivalent to a smooth solution and there are at most finite singularities at the maximum existing time.

Monocular-based 3-D seafloor reconstruction and ortho-mosaicing by piecewise planar representation

Photo-mosaicing techniques have become popular for seafloor mapping in various marine science applications. However, the common methods cannot accurately map regions with high relief and topographical variations. Ortho-mosaicing borrowed from photogrammetry is an alternative technique that enables taking into account the 3-D shape of the terrain. A serious bottleneck is the volume of elevation information that needs to be estimated from the video data, fused, and processed for the generation of a composite ortho-photo that covers a relatively large seafloor area. We present a framework that combines the advantages of dense depth-map and 3-D feature estimation techniques based on visual motion cues. The main goal is to identify and reconstruct certain key terrain feature points that adequately represent the surface with minimal complexity in the form of piecewise planar patches. The proposed implementation utilizes local depth maps for feature selection, while tracking over several views enables 3-D reconstruction by bundle adjustment. Experimental results with synthetic and real data validate the effectiveness of the proposed approach

Minimization of Nyquist ghosting for echo-planar imaging at ultra-high fields based on a "negative readout gradient" strategy.

PURPOSE: To improve the traditional Nyquist ghost correction approach in echo planar imaging (EPI) at high fields, via schemes based on the reversal of the EPI readout gradient polarity for every other volume throughout a functional magnetic resonance imaging (fMRI) acquisition train. MATERIALS AND METHODS: An EPI sequence in which the readout gradient was inverted every other volume was implemented on two ultrahigh-field systems. Phantom images and fMRI data were acquired to evaluate ghost intensities and the presence of false-positive blood oxygenation level-dependent (BOLD) signal with and without ghost correction. Three different algorithms for ghost correction of alternating readout EPI were compared. RESULTS: Irrespective of the chosen processing approach, ghosting was significantly reduced (up to 70% lower intensity) in both rat brain images acquired on a 9.4T animal scanner and human brain images acquired at 7T, resulting in a reduction of sources of false-positive activation in fMRI data. CONCLUSION: It is concluded that at high B(0) fields, substantial gains in Nyquist ghost correction of echo planar time series are possible by alternating the readout gradient every other volume.

Monocular-based 3-D seafloor reconstruction and ortho-mosaicing by piecewise planar representation

Photo-mosaicing techniques have become popular for seafloor mapping in various marine science applications. However, the common methods cannot accurately map regions with high relief and topographical variations. Ortho-mosaicing borrowed from photogrammetry is an alternative technique that enables taking into account the 3-D shape of the terrain. A serious bottleneck is the volume of elevation information that needs to be estimated from the video data, fused, and processed for the generation of a composite ortho-photo that covers a relatively large seafloor area. We present a framework that combines the advantages of dense depth-map and 3-D feature estimation techniques based on visual motion cues. The main goal is to identify and reconstruct certain key terrain feature points that adequately represent the surface with minimal complexity in the form of piecewise planar patches. The proposed implementation utilizes local depth maps for feature selection, while tracking over several views enables 3-D reconstruction by bundle adjustment. Experimental results with synthetic and real data validate the effectiveness of the proposed approach

Indecomposable and noncrossed product division algebras over function fields of smooth p-adic curves

We construct indecomposable and noncrossed product division algebras over function fields of connected smooth curves X over Z(p). This is done by defining an index preserving morphism s: Br(<(K(X))over cap>)` --> Br(K(X))` which splits res : Br(K (X)) --> Br(<(K(X))over cap>), where <(K(X))over cap> is the completion of K (X) at the special fiber, and using it to lift indecomposable and noncrossed product division algebras over <(K(X))over cap>. (C) 2010 Elsevier Inc. All rights reserved.

On the scattering of massive spinless bosons by a nonminimal vector smooth step potential

The Duffin-Kemmer-Petiau (DKP) equation for massive spinless bosons in the presence of a nonminimal vector smooth step potential is revised. The problem is mapped into a Sturm-Lionville equation. The reflection and transmission coefficients are obtained and discussed in detail. Furthermore; we show that Klein's paradox does not show its face in this sort of interaction.

Scattering and bound states of spinless particles in a mixed vector-scalar smooth step potential

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)