New path integral simulation algorithms and their application to creep in the quantum sine-Gordon chain

A path integral simulation algorithm which includes a higher-order Trotter approximation (HOA)is analyzed and compared to an approach which includes the correct quantum mechanical pair interaction (effective Propagator (EPr)). It is found that the HOA algorithmconverges to the quantum limit with increasing Trotter number P as P^{-4}, while the EPr algorithm converges as P^{-2}.The convergence rate of the HOA algorithm is analyzed for various physical systemssuch as a harmonic chain,a particle in a double-well potential, gaseous argon, gaseous helium and crystalline argon. A new expression for the estimator for the pair correlation function in the HOA algorithm is derived. A new path integral algorithm, the hybrid algorithm, is developed.It combines an exact treatment of the quadratic part of the Hamiltonian and thehigher-order Trotter expansion techniques.For the discrete quantum sine-Gordon chain (DQSGC), it is shown that this algorithm works more efficiently than all other improved path integral algorithms discussed in this work. The new simulation techniques developed in this work allow the analysis of theDQSGC and disordered model systems in the highly quantum mechanical regime using path integral molecular dynamics (PIMD)and adiabatic centroid path integral molecular dynamics (ACPIMD).The ground state phonon dispersion relation is calculated for the DQSGC by the ACPIMD method.It is found that the excitation gap at zero wave vector is reduced by quantum fluctuations. Two different phases exist: One phase with a finite excitation gap at zero wave vector, and a gapless phase where the excitation gap vanishes.The reaction of the DQSGC to an external driving force is analyzed at T=0.In the gapless phase the system creeps if a small force is applied, and in the phase with a gap the system is pinned. At a critical force, the systems undergo a depinning transition in both phases and flow is induced. The analysis of the DQSGC is extended to models with disordered substrate potentials. Three different cases are analyzed: Disordered substrate potentials with roughness exponent H=0, H=1/2,and a model with disordered bond length. For all models, the ground state phonon dispersion relation is calculated.

Die Faktorisierungsmethode für die elektrische Impedanztomographie im Halbraum

In der vorliegenden Arbeit wird die Faktorisierungsmethode zur Erkennung von Inhomogenitäten der Leitfähigkeit in der elektrischen Impedanztomographie auf unbeschränkten Gebieten - speziell der Halbebene bzw. dem Halbraum - untersucht. Als Lösungsräume für das direkte Problem, d.h. die Bestimmung des elektrischen Potentials zu vorgegebener Leitfähigkeit und zu vorgegebenem Randstrom, führen wir gewichtete Sobolev-Räume ein. In diesen wird die Existenz von schwachen Lösungen des direkten Problems gezeigt und die Gültigkeit einer Integraldarstellung für die Lösung der Laplace-Gleichung, die man bei homogener Leitfähigkeit erhält, bewiesen. Mittels der Faktorisierungsmethode geben wir eine explizite Charakterisierung von Einschlüssen an, die gegenüber dem Hintergrund eine sprunghaft erhöhte oder erniedrigte Leitfähigkeit haben. Damit ist zugleich für diese Klasse von Leitfähigkeiten die eindeutige Rekonstruierbarkeit der Einschlüsse bei Kenntnis der lokalen Neumann-Dirichlet-Abbildung gezeigt. Die mittels der Faktorisierungsmethode erhaltene Charakterisierung der Einschlüsse haben wir in ein numerisches Verfahren umgesetzt und sowohl im zwei- als auch im dreidimensionalen Fall mit simulierten, teilweise gestörten Daten getestet. Im Gegensatz zu anderen bekannten Rekonstruktionsverfahren benötigt das hier vorgestellte keine Vorabinformation über Anzahl und Form der Einschlüsse und hat als nicht-iteratives Verfahren einen vergleichsweise geringen Rechenaufwand.

Algorithms for on-line image registration from multiple views

Images of a scene, static or dynamic, are generally acquired at different epochs from different viewpoints. They potentially gather information about the whole scene and its relative motion with respect to the acquisition device. Data from different (in the spatial or temporal domain) visual sources can be fused together to provide a unique consistent representation of the whole scene, even recovering the third dimension, permitting a more complete understanding of the scene content. Moreover, the pose of the acquisition device can be achieved by estimating the relative motion parameters linking different views, thus providing localization information for automatic guidance purposes. Image registration is based on the use of pattern recognition techniques to match among corresponding parts of different views of the acquired scene. Depending on hypotheses or prior information about the sensor model, the motion model and/or the scene model, this information can be used to estimate global or local geometrical mapping functions between different images or different parts of them. These mapping functions contain relative motion parameters between the scene and the sensor(s) and can be used to integrate accordingly informations coming from the different sources to build a wider or even augmented representation of the scene. Accordingly, for their scene reconstruction and pose estimation capabilities, nowadays image registration techniques from multiple views are increasingly stirring up the interest of the scientific and industrial community. Depending on the applicative domain, accuracy, robustness, and computational payload of the algorithms represent important issues to be addressed and generally a trade-off among them has to be reached. Moreover, on-line performance is desirable in order to guarantee the direct interaction of the vision device with human actors or control systems. This thesis follows a general research approach to cope with these issues, almost independently from the scene content, under the constraint of rigid motions. This approach has been motivated by the portability to very different domains as a very desirable property to achieve. A general image registration approach suitable for on-line applications has been devised and assessed through two challenging case studies in different applicative domains. The first case study regards scene reconstruction through on-line mosaicing of optical microscopy cell images acquired with non automated equipment, while moving manually the microscope holder. By registering the images the field of view of the microscope can be widened, preserving the resolution while reconstructing the whole cell culture and permitting the microscopist to interactively explore the cell culture. In the second case study, the registration of terrestrial satellite images acquired by a camera integral with the satellite is utilized to estimate its three-dimensional orientation from visual data, for automatic guidance purposes. Critical aspects of these applications are emphasized and the choices adopted are motivated accordingly. Results are discussed in view of promising future developments.

Zweischleifenkorrekturen zum leptonischen Zerfall des Z-Bosons

Im Rahmen der vorliegenden Dissertation wurde, basierend auf der Parallel-/Orthogonalraum-Methode, eine neue Methode zur Berechnung von allgemeinen massiven Zweischleifen-Dreipunkt-Tensorintegralen mit planarer und gedrehter reduzierter planarer Topologie entwickelt. Die Ausarbeitung und Implementation einer Tensorreduktion fuer Integrale, welche eine allgemeine Tensorstruktur im Minkowski-Raum besitzen koennen, wurde durchgefuehrt. Die Entwicklung und Implementation eines Algorithmus zur semi-analytischen Berechnung der schwierigsten Integrale, die nach der Tensorreduktion verbleiben, konnte vollendet werden. (Fuer die anderen Basisintegrale koennen wohlbekannte Methoden verwendet werden.) Die Implementation ist bezueglich der UV-endlichen Anteile der Masterintegrale, die auch nach Tensorreduktion noch die zuvor erwaehnten Topologien besitzen, abgeschlossen. Die numerischen Integrationen haben sich als stabil erwiesen. Fuer die verbleibenden Teile des Projektes koennen wohlbekannte Methoden verwendet werden. In weiten Teilen muessen lediglich noch Links zu existierenden Programmen geschrieben werden. Fuer diejenigen wenigen verbleibenden speziellen Topologien, welche noch zu implementieren sind, sind (wohlbekannte) Methoden zu implementieren. Die Computerprogramme, die im Rahmen dieses Projektes entstanden, werden auch fuer allgemeinere Prozesse in das xloops-Projekt einfliessen. Deswegen wurde sie soweit moeglich fuer allgemeine Prozesse entwickelt und implementiert. Der oben erwaehnte Algorithmus wurde insbesondere fuer die Evaluation der fermionischen NNLO-Korrekturen zum leptonischen schwachen Mischungswinkel sowie zu aehnlichen Prozessen entwickelt. Im Rahmen der vorliegenden Dissertation wurde ein Grossteil der fuer die fermionischen NNLO-Korrekturen zu den effektiven Kopplungskonstanten des Z-Zerfalls (und damit fuer den schachen Mischungswinkel) notwendigen Arbeit durchgefuehrt.

Algorithms for UAS insertion in civil air space

One of the most interesting challenge of the next years will be the Air Space Systems automation. This process will involve different aspects as the Air Traffic Management, the Aircrafts and Airport Operations and the Guidance and Navigation Systems. The use of UAS (Uninhabited Aerial System) for civil mission will be one of the most important steps in this automation process. In civil air space, Air Traffic Controllers (ATC) manage the air traffic ensuring that a minimum separation between the controlled aircrafts is always provided. For this purpose ATCs use several operative avoidance techniques like holding patterns or rerouting. The use of UAS in these context will require the definition of strategies for a common management of piloted and piloted air traffic that allow the UAS to self separate. As a first employment in civil air space we consider a UAS surveillance mission that consists in departing from a ground base, taking pictures over a set of mission targets and coming back to the same ground base. During all mission a set of piloted aircrafts fly in the same airspace and thus the UAS has to self separate using the ATC avoidance as anticipated. We consider two objective, the first consists in the minimization of the air traffic impact over the mission, the second consists in the minimization of the impact of the mission over the air traffic. A particular version of the well known Travelling Salesman Problem (TSP) called Time-Dependant-TSP has been studied to deal with traffic problems in big urban areas. Its basic idea consists in a cost of the route between two clients depending on the period of the day in which it is crossed. Our thesis supports that such idea can be applied to the air traffic too using a convenient time horizon compatible with aircrafts operations. The cost of a UAS sub-route will depend on the air traffic that it will meet starting such route in a specific moment and consequently on the avoidance maneuver that it will use to avoid that conflict. The conflict avoidance is a topic that has been hardly developed in past years using different approaches. In this thesis we purpose a new approach based on the use of ATC operative techniques that makes it possible both to model the UAS problem using a TDTSP framework both to use an Air Traffic Management perspective. Starting from this kind of mission, the problem of the UAS insertion in civil air space is formalized as the UAS Routing Problem (URP). For this reason we introduce a new structure called Conflict Graph that makes it possible to model the avoidance maneuvers and to define the arc cost function of the departing time. Two Integer Linear Programming formulations of the problem are proposed. The first is based on a TDTSP formulation that, unfortunately, is weaker then the TSP formulation. Thus a new formulation based on a TSP variation that uses specific penalty to model the holdings is proposed. Different algorithms are presented: exact algorithms, simple heuristics used as Upper Bounds on the number of time steps used, and metaheuristic algorithms as Genetic Algorithm and Simulated Annealing. Finally an air traffic scenario has been simulated using real air traffic data in order to test our algorithms. Graphic Tools have been used to represent the Milano Linate air space and its air traffic during different days. Such data have been provided by ENAV S.p.A (Italian Agency for Air Navigation Services).

Enabling Techniques and Algorithms for Integrated Communication and Navigation Satellite Systems

This thesis presents the outcomes of my Ph.D. course in telecommunications engineering. The focus of my research has been on Global Navigation Satellite Systems (GNSS) and in particular on the design of aiding schemes operating both at position and physical level and the evaluation of their feasibility and advantages. Assistance techniques at the position level are considered to enhance receiver availability in challenging scenarios where satellite visibility is limited. Novel positioning techniques relying on peer-to-peer interaction and exchange of information are thus introduced. More specifically two different techniques are proposed: the Pseudorange Sharing Algorithm (PSA), based on the exchange of GNSS data, that allows to obtain coarse positioning where the user has scarce satellite visibility, and the Hybrid approach, which also permits to improve the accuracy of the positioning solution. At the physical level, aiding schemes are investigated to improve the receiver’s ability to synchronize with satellite signals. An innovative code acquisition strategy for dual-band receivers, the Cross-Band Aiding (CBA) technique, is introduced to speed-up initial synchronization by exploiting the exchange of time references between the two bands. In addition vector configurations for code tracking are analyzed and their feedback generation process thoroughly investigated.

Exact Algorithms for Different Classes of Vehicle Routing Problems

We deal with five problems arising in the field of logistics: the Asymmetric TSP (ATSP), the TSP with Time Windows (TSPTW), the VRP with Time Windows (VRPTW), the Multi-Trip VRP (MTVRP), and the Two-Echelon Capacitated VRP (2E-CVRP). The ATSP requires finding a lest-cost Hamiltonian tour in a digraph. We survey models and classical relaxations, and describe the most effective exact algorithms from the literature. A survey and analysis of the polynomial formulations is provided. The considered algorithms and formulations are experimentally compared on benchmark instances. The TSPTW requires finding, in a weighted digraph, a least-cost Hamiltonian tour visiting each vertex within a given time window. We propose a new exact method, based on new tour relaxations and dynamic programming. Computational results on benchmark instances show that the proposed algorithm outperforms the state-of-the-art exact methods. In the VRPTW, a fleet of identical capacitated vehicles located at a depot must be optimally routed to supply customers with known demands and time window constraints. Different column generation bounding procedures and an exact algorithm are developed. The new exact method closed four of the five open Solomon instances. The MTVRP is the problem of optimally routing capacitated vehicles located at a depot to supply customers without exceeding maximum driving time constraints. Two set-partitioning-like formulations of the problem are introduced. Lower bounds are derived and embedded into an exact solution method, that can solve benchmark instances with up to 120 customers. The 2E-CVRP requires designing the optimal routing plan to deliver goods from a depot to customers by using intermediate depots. The objective is to minimize the sum of routing and handling costs. A new mathematical formulation is introduced. Valid lower bounds and an exact method are derived. Computational results on benchmark instances show that the new exact algorithm outperforms the state-of-the-art exact methods.

Messung der B s-Oszillation mit dem semileptonischen Zerfall B 0 s D - s (phi pi -) my + ny my mit dem D0-Detektor

Das Standardmodell (SM) der Teilchenphysik beschreibt sehr präzise die fundamentalen Bausteine und deren Wechselwirkungen (WW). Trotz des Erfolges gibt es noch offene Fragen, die vom SM nicht beantwortet werden können. Ein noch noch nicht abgeschlossener Test besteht aus der Messung der Stärke der schwachen Kopplung zwischen Quarks. Neutrale B- bzw. $bar{B}$-Mesonen können sich innerhalb ihrer Lebensdauer über einen Prozeß der schwachen WW in ihr Antiteilchen transformieren. Durch die Messung der Bs-Oszillation kann die Kopplung Vtd zwischen den Quarksorten Top (t) und Down (d) bestimmt werden. Alle bis Ende 2005 durchgeführten Experimente lieferten lediglich eine untere Grenze für die Oszillationsfrequenz von ms>14,4ps-1. Die vorliegenden Arbeit beschreibt die Messung der Bs-Oszillationsfrequenz ms mit dem semileptonischen Kanal BsD(-)+. Die verwendeten Daten stammen aus Proton-Antiproton-Kollisionen, die im Zeitraum von April 2002 bis März 2006 mit dem DØ-Detektor am Tevatron-Beschleuniger des Fermi National Accelerator Laboratory bei einer Schwerpunktsenergie von $sqrt{s}$=1,96TeV aufgezeichnet wurden. Die verwendeten Datensätze entsprechen einer integrierten Luminosität von 1,3fb-1 (620 millionen Ereignisse). Für diese Oszillationsmessung wurde der Quarkinhalt des Bs-Mesons zur Zeit der Produktion sowie des Zerfalls bestimmt und die Zerfallszeit wurde gemessen. Nach der Rekonstruktion und Selektion der Signalereignisse legt die Ladung des Myons den Quarkinhalt des Bs-Mesons zur Zeit des Zerfalls fest. Zusätzlich wurde der Quarkinhalt des Bs-Mesons zur Zeit der Produktion markiert. b-Quarks werden in $pbar{p}$-Kollisionen paarweise produziert. Die Zerfallsprodukte des zweiten b-Hadrons legen den Quarkinhalt des Bs-Mesons zur Zeit der Produktion fest. Bei einer Sensitivität von msenss=14,5ps-1 wurde eine untere Grenze für die Oszillationsfrequenz ms>15,5ps-1 bestimmt. Die Maximum-Likelihood-Methode lieferte eine Oszillationsfrequenz ms>(20+2,5-3,0(stat+syst)0,8(syst,k))ps-1 bei einem Vertrauensniveau von 90%. Der nicht nachgewiesene Neutrinoimpuls führt zu dem systematischen Fehler (sys,k). Dieses Resultat ergibt zusammen mit der entsprechenden Oszillation des Bd-Mesons eine signifikante Messung der Kopplung Vtd, in Übereinstimmung mit weiteren Experimenten über die schwachen Quarkkopplungen.

Fingerprint Recognition: Enhancement, Feature Extraction and Automatic Evaluation of Algorithms

The identification of people by measuring some traits of individual anatomy or physiology has led to a specific research area called biometric recognition. This thesis is focused on improving fingerprint recognition systems considering three important problems: fingerprint enhancement, fingerprint orientation extraction and automatic evaluation of fingerprint algorithms. An effective extraction of salient fingerprint features depends on the quality of the input fingerprint. If the fingerprint is very noisy, we are not able to detect a reliable set of features. A new fingerprint enhancement method, which is both iterative and contextual, is proposed. This approach detects high-quality regions in fingerprints, selectively applies contextual filtering and iteratively expands like wildfire toward low-quality ones. A precise estimation of the orientation field would greatly simplify the estimation of other fingerprint features (singular points, minutiae) and improve the performance of a fingerprint recognition system. The fingerprint orientation extraction is improved following two directions. First, after the introduction of a new taxonomy of fingerprint orientation extraction methods, several variants of baseline methods are implemented and, pointing out the role of pre- and post- processing, we show how to improve the extraction. Second, the introduction of a new hybrid orientation extraction method, which follows an adaptive scheme, allows to improve significantly the orientation extraction in noisy fingerprints. Scientific papers typically propose recognition systems that integrate many modules and therefore an automatic evaluation of fingerprint algorithms is needed to isolate the contributions that determine an actual progress in the state-of-the-art. The lack of a publicly available framework to compare fingerprint orientation extraction algorithms, motivates the introduction of a new benchmark area called FOE (including fingerprints and manually-marked orientation ground-truth) along with fingerprint matching benchmarks in the FVC-onGoing framework. The success of such framework is discussed by providing relevant statistics: more than 1450 algorithms submitted and two international competitions.

Metaheuristics for Search Problems in Genomics - New Algorithms and Applications

In this thesis we made the first steps towards the systematic application of a methodology for automatically building formal models of complex biological systems. Such a methodology could be useful also to design artificial systems possessing desirable properties such as robustness and evolvability. The approach we follow in this thesis is to manipulate formal models by means of adaptive search methods called metaheuristics. In the first part of the thesis we develop state-of-the-art hybrid metaheuristic algorithms to tackle two important problems in genomics, namely, the Haplotype Inference by parsimony and the Founder Sequence Reconstruction Problem. We compare our algorithms with other effective techniques in the literature, we show strength and limitations of our approaches to various problem formulations and, finally, we propose further enhancements that could possibly improve the performance of our algorithms and widen their applicability. In the second part, we concentrate on Boolean network (BN) models of gene regulatory networks (GRNs). We detail our automatic design methodology and apply it to four use cases which correspond to different design criteria and address some limitations of GRN modeling by BNs. Finally, we tackle the Density Classification Problem with the aim of showing the learning capabilities of BNs. Experimental evaluation of this methodology shows its efficacy in producing network that meet our design criteria. Our results, coherently to what has been found in other works, also suggest that networks manipulated by a search process exhibit a mixture of characteristics typical of different dynamical regimes.

Testing future weak lensing surveys through simulations of observations

Weak lensing experiments such as the future ESA-accepted mission Euclid aim to measure cosmological parameters with unprecedented accuracy. It is important to assess the precision that can be obtained in these measurements by applying analysis software on mock images that contain many sources of noise present in the real data. In this Thesis, we show a method to perform simulations of observations, that produce realistic images of the sky according to characteristics of the instrument and of the survey. We then use these images to test the performances of the Euclid mission. In particular, we concentrate on the precision of the photometric redshift measurements, which are key data to perform cosmic shear tomography. We calculate the fraction of the total observed sample that must be discarded to reach the required level of precision, that is equal to 0.05(1+z) for a galaxy with measured redshift z, with different ancillary ground-based observations. The results highlight the importance of u-band observations, especially to discriminate between low (z < 0.5) and high (z ~ 3) redshifts, and the need for good observing sites, with seeing FWHM < 1. arcsec. We then construct an optimal filter to detect galaxy clusters through photometric catalogues of galaxies, and we test it on the COSMOS field, obtaining 27 lensing-confirmed detections. Applying this algorithm on mock Euclid data, we verify the possibility to detect clusters with mass above 10^14.2 solar masses with a low rate of false detections.

Heuristic algorithms for the Capacitated Location-Routing Problem and the Multi-Depot Vehicle Routing Problem

The Capacitated Location-Routing Problem (CLRP) is a NP-hard problem since it generalizes two well known NP-hard problems: the Capacitated Facility Location Problem (CFLP) and the Capacitated Vehicle Routing Problem (CVRP). The Multi-Depot Vehicle Routing Problem (MDVRP) is known to be a NP-hard since it is a generalization of the well known Vehicle Routing Problem (VRP), arising with one depot. This thesis addresses heuristics algorithms based on the well-know granular search idea introduced by Toth and Vigo (2003) to solve the CLRP and the MDVRP. Extensive computational experiments on benchmark instances for both problems have been performed to determine the effectiveness of the proposed algorithms. This work is organized as follows: Chapter 1 describes a detailed overview and a methodological review of the literature for the the Capacitated Location-Routing Problem (CLRP) and the Multi-Depot Vehicle Routing Problem (MDVRP). Chapter 2 describes a two-phase hybrid heuristic algorithm to solve the CLRP. Chapter 3 shows a computational comparison of heuristic algorithms for the CLRP. Chapter 4 presents a hybrid granular tabu search approach for solving the MDVRP.

Low Correlated Sequences & Architectures and Algorithms for Analog-to-Information Converters: Theory, Design, Implementation and Applications.

Most electronic systems can be described in a very simplified way as an assemblage of analog and digital components put all together in order to perform a certain function. Nowadays, there is an increasing tendency to reduce the analog components, and to replace them by operations performed in the digital domain. This tendency has led to the emergence of new electronic systems that are more flexible, cheaper and robust. However, no matter the amount of digital process implemented, there will be always an analog part to be sorted out and thus, the step of converting digital signals into analog signals and vice versa cannot be avoided. This conversion can be more or less complex depending on the characteristics of the signals. Thus, even if it is desirable to replace functions carried out by analog components by digital processes, it is equally important to do so in a way that simplifies the conversion from digital to analog signals and vice versa. In the present thesis, we have study strategies based on increasing the amount of processing in the digital domain in such a way that the implementation of analog hardware stages can be simplified. To this aim, we have proposed the use of very low quantized signals, i.e. 1-bit, for the acquisition and for the generation of particular classes of signals.

Spectrum Sensing Algorithms for Cognitive Radio Applications

Future wireless communications systems are expected to be extremely dynamic, smart and capable to interact with the surrounding radio environment. To implement such advanced devices, cognitive radio (CR) is a promising paradigm, focusing on strategies for acquiring information and learning. The first task of a cognitive systems is spectrum sensing, that has been mainly studied in the context of opportunistic spectrum access, in which cognitive nodes must implement signal detection techniques to identify unused bands for transmission. In the present work, we study different spectrum sensing algorithms, focusing on their statistical description and evaluation of the detection performance. Moving from traditional sensing approaches we consider the presence of practical impairments, and analyze algorithm design. Far from the ambition of cover the broad spectrum of spectrum sensing, we aim at providing contributions to the main classes of sensing techniques. In particular, in the context of energy detection we studied the practical design of the test, considering the case in which the noise power is estimated at the receiver. This analysis allows to deepen the phenomenon of the SNR wall, providing the conditions for its existence and showing that presence of the SNR wall is determined by the accuracy of the noise power estimation process. In the context of the eigenvalue based detectors, that can be adopted by multiple sensors systems, we studied the practical situation in presence of unbalances in the noise power at the receivers. Then, we shift the focus from single band detectors to wideband sensing, proposing a new approach based on information theoretic criteria. This technique is blind and, requiring no threshold setting, can be adopted even if the statistical distribution of the observed data in not known exactly. In the last part of the thesis we analyze some simple cooperative localization techniques based on weighted centroid strategies.

Implementation of network entropy algorithms on hpc machines, with application to high-dimensional experimental data

Network Theory is a prolific and lively field, especially when it approaches Biology. New concepts from this theory find application in areas where extensive datasets are already available for analysis, without the need to invest money to collect them. The only tools that are necessary to accomplish an analysis are easily accessible: a computing machine and a good algorithm. As these two tools progress, thanks to technology advancement and human efforts, wider and wider datasets can be analysed. The aim of this paper is twofold. Firstly, to provide an overview of one of these concepts, which originates at the meeting point between Network Theory and Statistical Mechanics: the entropy of a network ensemble. This quantity has been described from different angles in the literature. Our approach tries to be a synthesis of the different points of view. The second part of the work is devoted to presenting a parallel algorithm that can evaluate this quantity over an extensive dataset. Eventually, the algorithm will also be used to analyse high-throughput data coming from biology.