Sharp estimates for eigenvalues of integral operators generated by dot product kernels on the sphere

We obtain explicit formulas for the eigenvalues of integral operators generated by continuous dot product kernels defined on the sphere via the usual gamma function. Using them, we present both, a procedure to describe sharp bounds for the eigenvalues and their asymptotic behavior near 0. We illustrate our results with examples, among them the integral operator generated by a Gaussian kernel. Finally, we sketch complex versions of our results to cover the cases when the sphere sits in a Hermitian space.

Maximum principle, mean value operators and quasi boundedness in non-euclidean settings

This work deals with some classes of linear second order partial differential operators with non-negative characteristic form and underlying non- Euclidean structures. These structures are determined by families of locally Lipschitz-continuous vector fields in RN, generating metric spaces of Carnot- Carath´eodory type. The Carnot-Carath´eodory metric related to a family {Xj}j=1,...,m is the control distance obtained by minimizing the time needed to go from two points along piecewise trajectories of vector fields. We are mainly interested in the causes in which a Sobolev-type inequality holds with respect to the X-gradient, and/or the X-control distance is Doubling with respect to the Lebesgue measure in RN. This study is divided into three parts (each corresponding to a chapter), and the subject of each one is a class of operators that includes the class of the subsequent one. In the first chapter, after recalling “X-ellipticity” and related concepts introduced by Kogoj and Lanconelli in [KL00], we show a Maximum Principle for linear second order differential operators for which we only assume a Sobolev-type inequality together with a lower terms summability. Adding some crucial hypotheses on measure and on vector fields (Doubling property and Poincar´e inequality), we will be able to obtain some Liouville-type results. This chapter is based on the paper [GL03] by Guti´errez and Lanconelli. In the second chapter we treat some ultraparabolic equations on Lie groups. In this case RN is the support of a Lie group, and moreover we require that vector fields satisfy left invariance. After recalling some results of Cinti [Cin07] about this class of operators and associated potential theory, we prove a scalar convexity for mean-value operators of L-subharmonic functions, where L is our differential operator. In the third chapter we prove a necessary and sufficient condition of regularity, for boundary points, for Dirichlet problem on an open subset of RN related to sub-Laplacian. On a Carnot group we give the essential background for this type of operator, and introduce the notion of “quasi-boundedness”. Then we show the strict relationship between this notion, the fundamental solution of the given operator, and the regularity of the boundary points.

On differential operators of Calabi-Yau type

This thesis is devoted to the study of Picard-Fuchs operators associated to one-parameter families of $n$-dimensional Calabi-Yau manifolds whose solutions are integrals of $(n,0)$-forms over locally constant $n$-cycles. Assuming additional conditions on these families, we describe algebraic properties of these operators which leads to the purely algebraic notion of operators of CY-type. rnMoreover, we present an explicit way to construct CY-type operators which have a linearly rigid monodromy tuple. Therefore, we first usernthe translation of the existence algorithm by N. Katz for rigid local systems to the level of tuples of matrices which was established by M. Dettweiler and S. Reiter. An appropriate translation to the level of differential operators yields families which contain operators of CY-type. rnConsidering additional operations, we are also able to construct special CY-type operators of degree four which have a non-linearly rigid monodromy tuple. This provides both previously known and new examples.

Data sanitization in a clearing system for public transport operators

In this work we will discuss about a project started by the Emilia-Romagna Regional Government regarding the manage of the public transport. In particular we will perform a data mining analysis on the data-set of this project. After introducing the Weka software used to make our analysis, we will discover the most useful data mining techniques and algorithms; and we will show how these results can be used to violate the privacy of the same public transport operators. At the end, despite is off topic of this work, we will spend also a few words about how it's possible to prevent this kind of attack.

Spectral theory of differential operators on finite metric graphs and on bounded domains

Die vorliegende Arbeit widmet sich der Spektraltheorie von Differentialoperatoren auf metrischen Graphen und von indefiniten Differentialoperatoren auf beschränkten Gebieten. Sie besteht aus zwei Teilen. Im Ersten werden endliche, nicht notwendigerweise kompakte, metrische Graphen und die Hilberträume von quadratintegrierbaren Funktionen auf diesen betrachtet. Alle quasi-m-akkretiven Laplaceoperatoren auf solchen Graphen werden charakterisiert, und Abschätzungen an die negativen Eigenwerte selbstadjungierter Laplaceoperatoren werden hergeleitet. Weiterhin wird die Wohlgestelltheit eines gemischten Diffusions- und Transportproblems auf kompakten Graphen durch die Anwendung von Halbgruppenmethoden untersucht. Eine Verallgemeinerung des indefiniten Operators $-tfrac{d}{dx}sgn(x)tfrac{d}{dx}$ von Intervallen auf metrische Graphen wird eingeführt. Die Spektral- und Streutheorie der selbstadjungierten Realisierungen wird detailliert besprochen. Im zweiten Teil der Arbeit werden Operatoren untersucht, die mit indefiniten Formen der Art $langlegrad v, A(cdot)grad urangle$ mit $u,vin H_0^1(Omega)subset L^2(Omega)$ und $OmegasubsetR^d$ beschränkt, assoziiert sind. Das Eigenwertverhalten entspricht in Dimension $d=1$ einer verallgemeinerten Weylschen Asymptotik und für $dgeq 2$ werden Abschätzungen an die Eigenwerte bewiesen. Die Frage, wann indefinite Formmethoden für Dimensionen $dgeq 2$ anwendbar sind, bleibt offen und wird diskutiert.

Efficient implementation of discrete Pascal transform using difference operators

An alternative way is provided to define the discrete Pascal transform using difference operators to reveal the fundamental concept of the transform, in both one- and two-dimensional cases, which is extended to cover non-square two-dimensional applications. Efficient modularised implementations are proposed.

TV Signal Delivery to Cable Operators and DTH Platform Operators

The paper discusses new business models of transmission of television programs in the context of definitions of broadcasting and retransmission. Typically the whole process of supplying content to the end user has two stages: a media service provider supplies a signal assigned to a given TV channel to the cable operators and satellite DTH platform operators (dedicated transmission), and cable operators and satellite DTH platform operators transmit this signal to end users. In each stage the signals are encoded and are not available for the general public without the intervention of cable/platform operators. The services relating to the supply and transmission of the content are operated by different business entities: each earns money separately and each uses the content protected by copyright. We should determine how to define the actions of the entity supplying the signal with the TV program directly to the cable/digital platform operator and the actions of the entity providing the end user with the signal. The author criticizes the approach presented in the Chellomedia and Norma rulings, arguing that they lead to a significant level of legal uncertainty, and poses the basic questions concerning the notion of “public” in copyright.

Lepton Flavor Violation in the Standard Model with general Dimension-Six Operators

We study lepton flavor observables in the Standard Model (SM) extended with all dimension-6 operators which are invariant under the SM gauge group. We calculate the complete one-loop predictions to the radiative lepton decays μ → eγ, τ → μγ and τ → eγ as well as to the closely related anomalous magnetic moments and electric dipole moments of charged leptons, taking into account all dimension-6 operators which can generate lepton flavor violation. Also the 3-body flavor violating charged lepton decays τ ± → μ ± μ + μ −, τ ± → e ± e + e −, τ ± → e ± μ + μ −, τ ± → μ ± e + e −, τ ± → e ∓ μ ± μ ±, τ ± → μ ∓ e ± e ± and μ ± → e ± e + e − and the Z 0 decays Z 0 → ℓ+iℓ−j are considered, taking into account all tree-level contributions.

Eigenvalue estimates for non-selfadjoint Dirac operators on the real line

We show that the non-embedded eigenvalues of the Dirac operator on the real line with complex mass and non-Hermitian potential V lie in the disjoint union of two disks, provided that the L1-norm of V is bounded from above by the speed of light times the reduced Planck constant. The result is sharp; moreover, the analogous sharp result for the Schrödinger operator, originally proved by Abramov, Aslanyan and Davies, emerges in the nonrelativistic limit. For massless Dirac operators, the condition on V implies the absence of non-real eigenvalues. Our results are further generalized to potentials with slower decay at infinity. As an application, we determine bounds on resonances and embedded eigenvalues of Dirac operators with Hermitian dilation-analytic potentials.