A moment symbolic representation of Lévy processes with applications

By using a symbolic method, known in the literature as the classical umbral calculus, a symbolic representation of Lévy processes is given and a new family of time-space harmonic polynomials with respect to such processes, which includes and generalizes the exponential complete Bell polynomials, is introduced. The usefulness of time-space harmonic polynomials with respect to Lévy processes is that it is a martingale the stochastic process obtained by replacing the indeterminate x of the polynomials with a Lévy process, whereas the Lévy process does not necessarily have this property. Therefore to find such polynomials could be particularly meaningful for applications. This new family includes Hermite polynomials, time-space harmonic with respect to Brownian motion, Poisson-Charlier polynomials with respect to Poisson processes, Laguerre and actuarial polynomials with respect to Gamma processes , Meixner polynomials of the first kind with respect to Pascal processes, Euler, Bernoulli, Krawtchuk, and pseudo-Narumi polynomials with respect to suitable random walks. The role played by cumulants is stressed and brought to the light, either in the symbolic representation of Lévy processes and their infinite divisibility property, either in the generalization, via umbral Kailath-Segall formula, of the well-known formulae giving elementary symmetric polynomials in terms of power sum symmetric polynomials. The expression of the family of time-space harmonic polynomials here introduced has some connections with the so-called moment representation of various families of multivariate polynomials. Such moment representation has been studied here for the first time in connection with the time-space harmonic property with respect to suitable symbolic multivariate Lévy processes. In particular, multivariate Hermite polynomials and their properties have been studied in connection with a symbolic version of the multivariate Brownian motion, while multivariate Bernoulli and Euler polynomials are represented as powers of multivariate polynomials which are time-space harmonic with respect to suitable multivariate Lévy processes.

Non-Markovian stochastic processes and their applications: from anomalous diffusion to time series analysis

This work provides a forward step in the study and comprehension of the relationships between stochastic processes and a certain class of integral-partial differential equation, which can be used in order to model anomalous diffusion and transport in statistical physics. In the first part, we brought the reader through the fundamental notions of probability and stochastic processes, stochastic integration and stochastic differential equations as well. In particular, within the study of H-sssi processes, we focused on fractional Brownian motion (fBm) and its discrete-time increment process, the fractional Gaussian noise (fGn), which provide examples of non-Markovian Gaussian processes. The fGn, together with stationary FARIMA processes, is widely used in the modeling and estimation of long-memory, or long-range dependence (LRD). Time series manifesting long-range dependence, are often observed in nature especially in physics, meteorology, climatology, but also in hydrology, geophysics, economy and many others. We deepely studied LRD, giving many real data examples, providing statistical analysis and introducing parametric methods of estimation. Then, we introduced the theory of fractional integrals and derivatives, which indeed turns out to be very appropriate for studying and modeling systems with long-memory properties. After having introduced the basics concepts, we provided many examples and applications. For instance, we investigated the relaxation equation with distributed order time-fractional derivatives, which describes models characterized by a strong memory component and can be used to model relaxation in complex systems, which deviates from the classical exponential Debye pattern. Then, we focused in the study of generalizations of the standard diffusion equation, by passing through the preliminary study of the fractional forward drift equation. Such generalizations have been obtained by using fractional integrals and derivatives of distributed orders. In order to find a connection between the anomalous diffusion described by these equations and the long-range dependence, we introduced and studied the generalized grey Brownian motion (ggBm), which is actually a parametric class of H-sssi processes, which have indeed marginal probability density function evolving in time according to a partial integro-differential equation of fractional type. The ggBm is of course Non-Markovian. All around the work, we have remarked many times that, starting from a master equation of a probability density function f(x,t), it is always possible to define an equivalence class of stochastic processes with the same marginal density function f(x,t). All these processes provide suitable stochastic models for the starting equation. Studying the ggBm, we just focused on a subclass made up of processes with stationary increments. The ggBm has been defined canonically in the so called grey noise space. However, we have been able to provide a characterization notwithstanding the underline probability space. We also pointed out that that the generalized grey Brownian motion is a direct generalization of a Gaussian process and in particular it generalizes Brownain motion and fractional Brownain motion as well. Finally, we introduced and analyzed a more general class of diffusion type equations related to certain non-Markovian stochastic processes. We started from the forward drift equation, which have been made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation has been interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time-evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the same memory kernel K(t). We developed several applications and derived the exact solutions. Moreover, we considered different stochastic models for the given equations, providing path simulations.

Fermentative processes for environmental remediation

The growing interest in environmental protection has led to the development of emerging biotechnologies for environmental remediation also introducing the biorefinery concept. This work mainly aimed to evaluate the applicability of innovative biotechnologies for environmental remediation and bioenergy production, throught fermentative processes. The investigated biotechnologies for waste and wastewater treatment and for the valorisation of specific feedstocks and energy recovery, were mainly focused on four research lines. 1. Biotechnology for textile wastewater treatment and water reuse that involving anaerobic and aerobic processes in combination with membrane technologies. Combinations of different treatments were also implemented for water reuse in a textile company. 2. Biotechnology for the treatment of solid waste and leachate in landfill and for biogas production. Landfill operated as Bioreactor with recirculation of the generated leachate was proposed for organic matter biostabilisation and for ammonia removal from leachate by favouring the Anammox process. 3. An innovative two-stage anaerobic process for effective codigestion of waste from the dairy industry, as cheese whey and dairy manure, was studied by combining conventional fermentative processes with a simplified system design for enhancing biomethanisation. 4) The valorisation of the glycerol waste as surplus by-product of the biodiesel industry was investigated via microbial conversion to value-added chemicals, as 1,3-propanediol. The investigated fermentative processes have been successfully implemented and reached high yields of the produced bio-chemical. The studied biotechnological systems proved to be feasible for environmental remediation and bioenergy and chemicals production.

Studio di fenomeni d'instabilità gravitativa sui fondali marini, con particolare riferimento all'isola di Stromboli

In the last decade the interest for submarine instability grew up, driven by the increasing exploitation of natural resources (primary hydrocarbons), the emplacement of bottom-lying structures (cables and pipelines) and by the development of coastal areas, whose infrastructures increasingly protrude to the sea. The great interest for this topic promoted a number of international projects such as: STEAM (Sediment Transport on European Atlantic Margins, 93-96), ENAM II (European North Atlantic Margin, 96-99), GITEC (Genesis and Impact of Tsunamis on the European Coast 92-95), STRATAFORM (STRATA FORmation on Margins, 95-01), Seabed Slope Process in Deep Water Continental Margin (Northwest Gulf of Mexico, 96-04), COSTA (Continental slope Stability, 00-05), EUROMARGINS (Slope Stability on Europe’s Passive Continental Margin), SPACOMA (04-07), EUROSTRATAFORM (European Margin Strata Formation), NGI's internal project SIP-8 (Offshore Geohazards), IGCP-511: Submarine Mass Movements and Their Consequences (05-09) and projects indirectly related to instability processes, such as TRANSFER (Tsunami Risk ANd Strategies For the European region, 06-09) or NEAREST (integrated observations from NEAR shore sourcES of Tsunamis: towards an early warning system, 06-09). In Italy, apart from a national project realized within the activities of the National Group of Volcanology during the framework 2000-2003 “Conoscenza delle parti sommerse dei vulcani italiani e valutazione del potenziale rischio vulcanico”, the study of submarine mass-movement has been underestimated until the occurrence of the landslide-tsunami events that affected Stromboli on December 30, 2002. This event made the Italian Institutions and the scientific community more aware of the hazard related to submarine landslides, mainly in light of the growing anthropization of coastal sectors, that increases the vulnerability of these areas to the consequences of such processes. In this regard, two important national projects have been recently funded in order to study coastal instabilities (PRIN 24, 06-08) and to map the main submarine hazard features on continental shelves and upper slopes around the most part of Italian coast (MaGIC Project). The study realized in this Thesis is addressed to the understanding of these processes, with particular reference to Stromboli submerged flanks. These latter represent a natural laboratory in this regard, as several kind of instability phenomena are present on the submerged flanks, affecting about 90% of the entire submerged areal and often (strongly) influencing the morphological evolution of subaerial slopes, as witnessed by the event occurred on 30 December 2002. Furthermore, each phenomenon is characterized by different pre-failure, failure and post-failure mechanisms, ranging from rock-falls, to turbidity currents up to catastrophic sector collapses. The Thesis is divided into three introductive chapters, regarding a brief review of submarine instability phenomena and related hazard (cap. 1), a “bird’s-eye” view on methodologies and available dataset (cap. 2) and a short introduction on the evolution and the morpho-structural setting of the Stromboli edifice (cap. 3). This latter seems to play a major role in the development of largescale sector collapses at Stromboli, as they occurred perpendicular to the orientation of the main volcanic rift axis (oriented in NE-SW direction). The characterization of these events and their relationships with successive erosive-depositional processes represents the main focus of cap.4 (Offshore evidence of large-scale lateral collapses on the eastern flank of Stromboli, Italy, due to structurally-controlled, bilateral flank instability) and cap. 5 (Lateral collapses and active sedimentary processes on the North-western flank of Stromboli Volcano), represented by articles accepted for publication on international papers (Marine Geology). Moreover, these studies highlight the hazard related to these catastrophic events; several calamities (with more than 40000 casualties only in the last two century) have been, in fact, the direct or indirect result of landslides affecting volcanic flanks, as observed at Oshima-Oshima (1741) and Unzen Volcano (1792) in Japan (Satake&Kato, 2001; Brantley&Scott, 1993), Krakatau (1883) in Indonesia (Self&Rampino, 1981), Ritter Island (1888), Sissano in Papua New Guinea (Ward& Day, 2003; Johnson, 1987; Tappin et al., 2001) and Mt St. Augustine (1883) in Alaska (Beget& Kienle, 1992). Flank landslide are also recognized as the most important and efficient mass-wasting process on volcanoes, contributing to the development of the edifices by widening their base and to the growth of a volcaniclastic apron at the foot of a volcano; a number of small and medium-scale erosive processes are also responsible for the carving of Stromboli submarine flanks and the transport of debris towards the deeper areas. The characterization of features associated to these processes is the main focus of cap. 6; it is also important to highlight that some small-scale events are able to create damage to coastal areas, as also witnessed by recent events of Gioia Tauro 1978, Nizza, 1979 and Stromboli 2002. The hazard potential related to these phenomena is, in fact, very high, as they commonly occur at higher frequency with respect to large-scale collapses, therefore being more significant in terms of human timescales. In the last chapter (cap. 7), a brief review and discussion of instability processes identified on Stromboli submerged flanks is presented; they are also compared with respect to analogous processes recognized in other submerged areas in order to shed lights on the main factors involved in their development. Finally, some applications of multibeam data to assess the hazard related to these phenomena are also discussed.

Second law analysis and simulation techniques for the energy optimization of buildings

The research activity described in this thesis is focused mainly on the study of finite-element techniques applied to thermo-fluid dynamic problems of plant components and on the study of dynamic simulation techniques applied to integrated building design in order to enhance the energy performance of the building. The first part of this doctorate thesis is a broad dissertation on second law analysis of thermodynamic processes with the purpose of including the issue of the energy efficiency of buildings within a wider cultural context which is usually not considered by professionals in the energy sector. In particular, the first chapter includes, a rigorous scheme for the deduction of the expressions for molar exergy and molar flow exergy of pure chemical fuels. The study shows that molar exergy and molar flow exergy coincide when the temperature and pressure of the fuel are equal to those of the environment in which the combustion reaction takes place. A simple method to determine the Gibbs free energy for non-standard values of the temperature and pressure of the environment is then clarified. For hydrogen, carbon dioxide, and several hydrocarbons, the dependence of the molar exergy on the temperature and relative humidity of the environment is reported, together with an evaluation of molar exergy and molar flow exergy when the temperature and pressure of the fuel are different from those of the environment. As an application of second law analysis, a comparison of the thermodynamic efficiency of a condensing boiler and of a heat pump is also reported. The second chapter presents a study of borehole heat exchangers, that is, a polyethylene piping network buried in the soil which allows a ground-coupled heat pump to exchange heat with the ground. After a brief overview of low-enthalpy geothermal plants, an apparatus designed and assembled by the author to carry out thermal response tests is presented. Data obtained by means of in situ thermal response tests are reported and evaluated by means of a finite-element simulation method, implemented through the software package COMSOL Multyphysics. The simulation method allows the determination of the precise value of the effective thermal properties of the ground and of the grout, which are essential for the design of borehole heat exchangers. In addition to the study of a single plant component, namely the borehole heat exchanger, in the third chapter is presented a thorough process for the plant design of a zero carbon building complex. The plant is composed of: 1) a ground-coupled heat pump system for space heating and cooling, with electricity supplied by photovoltaic solar collectors; 2) air dehumidifiers; 3) thermal solar collectors to match 70% of domestic hot water energy use, and a wood pellet boiler for the remaining domestic hot water energy use and for exceptional winter peaks. This chapter includes the design methodology adopted: 1) dynamic simulation of the building complex with the software package TRNSYS for evaluating the energy requirements of the building complex; 2) ground-coupled heat pumps modelled by means of TRNSYS; and 3) evaluation of the total length of the borehole heat exchanger by an iterative method developed by the author. An economic feasibility and an exergy analysis of the proposed plant, compared with two other plants, are reported. The exergy analysis was performed by considering the embodied energy of the components of each plant and the exergy loss during the functioning of the plants.

Interaction between APOE4 genotype and environmental risk factors in Alzheimer's disease

Alzheimer's disease (AD) is probably caused by both genetic and environmental risk factors. The major genetic risk factor is the E4 variant of apolipoprotein E gene called apoE4. Several risk factors for developing AD have been identified including lifestyle, such as dietary habits. The mechanisms behind the AD pathogenesis and the onset of cognitive decline in the AD brain are presently unknown. In this study we wanted to characterize the effects of the interaction between environmental risk factors and apoE genotype on neurodegeneration processes, with particular focus on behavioural studies and neurodegenerative processes at molecular level. Towards this aim, we used 6 months-old apoE4 and apoE3 Target Replacement (TR) mice fed on different diets (high intake of cholesterol and high intake of carbohydrates). These mice were evaluated for learning and memory deficits in spatial reference (Morris Water Maze (MWM)) and contextual learning (Passive Avoidance) tasks, which involve the hippocampus and the amygdala, respectively. From these behavioural studies we found that the initial cognitive impairments manifested as a retention deficit in apoE4 mice fed on high carbohydrate diet. Thus, the genetic risk factor apoE4 genotype associated with a high carbohydrate diet seems to affect cognitive functions in young mice, corroborating the theory that the combination of genetic and environmental risk factors greatly increases the risk of developing AD and leads to an earlier onset of cognitive deficits. The cellular and molecular bases of the cognitive decline in AD are largely unknown. In order to determine the molecular changes for the onset of the early cognitive impairment observed in the behavioural studies, we performed molecular studies, with particular focus on synaptic integrity and Tau phosphorylation. The most relevant finding of our molecular studies showed a significant decrease of Brain-derived Neurotrophic Factor (BDNF) in apoE4 mice fed on high carbohydrate diet. Our results may suggest that BDNF decrease found in apoE4 HS mice could be involved in the earliest impairment in long-term reference memory observed in behavioural studies. The second aim of this thesis was to study possible involvement of leptin in AD. There is growing evidence that leptin has neuroprotective properties in the Central Nervous System (CNS). Recent evidence has shown that leptin and its receptors are widespread in the CNS and may provide neuronal survival signals. However, there are still numerous questions, regarding the molecular mechanism by which leptin acts, that remain unanswered. Thus, given to the importance of the involvement of leptin in AD, we wanted to clarify the function of leptin in the pathogenesis of AD and to investigate if apoE genotype affect leptin levels through studies in vitro, in mice and in human. Our findings suggest that apoE4 TR mice showed an increase of leptin in the brain. Leptin levels are also increased in the cerebral spinal fluid of AD patients and apoE4 carriers with AD have higher levels of leptin than apoE3 carriers. Moreover, leptin seems to be expressed by reactive glial cells in AD brains. In vitro, ApoE4 together with Amyloid beta increases leptin production by microglia and astrocytes. Taken together, all these findings suggest that leptin replacement might not be a good strategy for AD therapy. Our results show that high leptin levels were found in AD brains. These findings suggest that, as high leptin levels do not promote satiety in obese individuals, it might be possible that they do not promote neuroprotection in AD patients. Therefore, we hypothesized that AD brain could suffer from leptin resistance. Further studies will be critical to determine whether or not the central leptin resistance in SNC could affect its potential neuroprotective effects.

Effetti sulle funzioni autonomiche e sugli stati di veglia e di sonno della manipolazione centrale farmacologica del sistema ipocretinergico nel ratto

Obiettivo della tesi è stato quello di studiare il ruolo svolto dall’ipotalamo laterale (LH) nella regolazione dei processi di integrazione dell’attività autonomica e termoregolatoria con quella degli stati di veglia e sonno. A questo scopo, l’attività dell’LH è stata inibita per 6 ore (Esperimento A) mediante microiniezioni locali dell’agonista GABAA muscimolo nel ratto libero di muoversi, nel quale sono stati monitorati in continuo l’elelttroencefalogramma, l’elettromiogramma nucale, la pressione arteriosa (PA) e la temperatura ipotalamica (Thy) e cutanea. Gli animali sono stati studiati a temperatura ambientale (Ta) di 24°C e 10°C. I risultati hanno mostrato che l’inibizione acuta dell’LH riduce l’attività di veglia e sopprime la comparsa del sonno REM. Ciò avviene attraverso l’induzione di uno stato di sonno NREM caratterizzato da ipersincronizzazione corticale, con scomparsa degli stati transizionali al sonno REM. Quando l’animale è esposto a bassa Ta, tali alterazioni si associano a un ampio calo della Thy, che viene compensato da meccanismi vicarianti solo dopo un paio d’ore dall’iniezione. Sulla base di tali risultati, si è proceduto ad un ulteriore studio (Esperimento B) volto ad indagare il ruolo del neuropeptide ipocretina (prodotto in modo esclusivo a livello dell’LH) nei processi termoregolatori, mediante microiniezioni del medesimo nel bulbo rostrale ventromediale (RVMM), stazione cruciale della rete nervosa preposta all’attivazione dei processi termogenetici. La somministrazione di ipocretina è stata in grado di attivare la termogenesi e di potenziare la comparsa della veglia, con concomitante lieve incremento della PA e della frequenza cardiaca, quando effettuata alle Ta di 24°C o di 10°C, ma non alla Ta di 32°C. In conclusione, i risultati indicano che l’LH svolge un ruolo cruciale nella promozione degli stati di veglia e di sonno REM e, per tramite dell’ipocretina, interviene in modo coplesso a livello del RVMM nella regolazione dei processi di coordinamento dell'attività di veglia con quella termoregolatoria.

Understanding the Combined Role of Internal and External Factors in New Service Development Process within the Professional Knowledge Intensive Business Service Firms

Our research asked the following main questions: how the characteristics of professionals service firms allow them to successfully innovate in exploiting through exploring by combining internal and external factors of innovation and how these ambidextrous organisations perceive these factors; and how do successful innovators in professional service firms use corporate entrepreneurship models in their new service development processes? With a goal to shed light on innovation in professional knowledge intensive business service firms’ (PKIBS), we concluded a qualitative analysis of ten globally acting law firms, providing business legal services. We analyse the internal and factors of innovation that are critical for PKIBS’ innovation. We suggest how these firms become ambidextrous in changing environment. Our findings show that this kind of firms has particular type of ambidexterity due to their specific characteristics. As PKIBS are very dependant on its human capital, governance structure, and the high expectations of their clients, their ambidexterity is structural, but also contextual at the same time. In addition, we suggest 3 types of corporate entrepreneurship models that international PKIBS use to enhance innovation in turbulent environments. We looked at how law firms going through turbulent environments were using corporate entrepreneurship activities as a part of their strategies to be more innovative. Using visual mapping methodology, we developed three types of innovation patterns in the law firms. We suggest that corporate entrepreneurship models depend on successful application of mainly three elements: who participates in corporate entrepreneurship initiatives; what are the formal processes that enhances these initiatives; and what are the policies applied to this type of behaviour.

Visual-somatosensory interactions in mental representations of the body and the face

The body is represented in the brain at levels that incorporate multisensory information. This thesis focused on interactions between vision and cutaneous sensations (i.e., touch and pain). Experiment 1 revealed that there are partially dissociable pathways for visual enhancement of touch (VET) depending upon whether one sees one’s own body or the body of another person. This indicates that VET, a seeming low-level effect on spatial tactile acuity, is actually sensitive to body identity. Experiments 2-4 explored the effect of viewing one’s own body on pain perception. They demonstrated that viewing the body biases pain intensity judgments irrespective of actual stimulus intensity, and, more importantly, reduces the discriminative capacities of the nociceptive pathway encoding noxious stimulus intensity. The latter effect only occurs if the pain-inducing event itself is not visible, suggesting that viewing the body alone and viewing a stimulus event on the body have distinct effects on cutaneous sensations. Experiment 5 replicated an enhancement of visual remapping of touch (VRT) when viewing fearful human faces being touched, and further demonstrated that VRT does not occur for observed touch on non-human faces, even fearful ones. This suggests that the facial expressions of non-human animals may not be simulated within the somatosensory system of the human observer in the same way that the facial expressions of other humans are. Finally, Experiment 6 examined the enfacement illusion, in which synchronous visuo-tactile inputs cause another’s face to be assimilated into the mental self-face representation. The strength of enfacement was not affected by the other’s facial expression, supporting an asymmetric relationship between processing of facial identity and facial expressions. Together, these studies indicate that multisensory representations of the body in the brain link low-level perceptual processes with the perception of emotional cues and body/face identity, and interact in complex ways depending upon contextual factors.

Synthetic and mechanistic investigation of new radical processes: reaction of organic azides with group-XIII Lewis acids

Dichloroindium hydride revealed to be a valid alternative to tributyltin hydride for radical reduction of organic (alkyl, aryl, acyl, solfonyl) azides. The new approach entails mild reaction conditions and provides high yields of the corresponding amines and amides, also showing high degrees of selectivity. The system dichloroindium hydride / azides can be utilised in fivemembered ring closures of g-azidonitriles, as a new source of aminyl radicals for the attractive synthesis of interesting amidine compounds in the absence of both toxic reagents and tedious purification procedures. Allylindium dichloride seems a good substitute for dichloroindium hydride for generation of indium centred radicals under photolytic conditions, since it allows allylation of electrophilic azides (e.g. phenylsulfonyl azide) and halogen or ester δ-substituted azides, the latter through a 1,5-H transfer rearrangement mechanism. Evidences of the radical nature of the reactions mechanism were provided by ESR spectroscopy, furthermore the same technique, allowed to discover that the reaction of azides with indium trichloride and other group XIII Lewis acids, in particular gallium trichloride, gives rise to strongly coloured, persistent paramagnetic species, whose structure is consistent with the radical cation of the head-to-tail dimer of the aniline corresponding to the starting azide.

Crustal heterogeneities and complexities of fault processes

In this work we study the relation between crustal heterogeneities and complexities in fault processes. The first kind of heterogeneity considered involves the concept of asperity. The presence of an asperity in the hypocentral region of the M = 6.5 earthquake of June 17-th, 2000 in the South Iceland Seismic Zone was invoked to explain the change of seismicity pattern before and after the mainshock: in particular, the spatial distribution of foreshock epicentres trends NW while the strike of the main fault is N 7◦ E and aftershocks trend accordingly; the foreshock depths were typically deeper than average aftershock depths. A model is devised which simulates the presence of an asperity in terms of a spherical inclusion, within a softer elastic medium in a transform domain with a deviatoric stress field imposed at remote distances (compressive NE − SW, tensile NW − SE). An isotropic compressive stress component is induced outside the asperity, in the direction of the compressive stress axis, and a tensile component in the direction of the tensile axis; as a consequence, fluid flow is inhibited in the compressive quadrants while it is favoured in tensile quadrants. Within the asperity the isotropic stress vanishes but the deviatoric stress increases substantially, without any significant change in the principal stress directions. Hydrofracture processes in the tensile quadrants and viscoelastic relaxation at depth may contribute to lower the effective rigidity of the medium surrounding the asperity. According to the present model, foreshocks may be interpreted as induced, close to the brittle-ductile transition, by high pressure fluids migrating upwards within the tensile quadrants; this process increases the deviatoric stress within the asperity which eventually fails, becoming the hypocenter of the mainshock, on the optimally oriented fault plane. In the second part of our work we study the complexities induced in fault processes by the layered structure of the crust. In the first model proposed we study the case in which fault bending takes place in a shallow layer. The problem can be addressed in terms of a deep vertical planar crack, interacting with a shallower inclined planar crack. An asymptotic study of the singular behaviour of the dislocation density at the interface reveals that the density distribution has an algebraic singularity at the interface of degree ω between -1 and 0, depending on the dip angle of the upper crack section and on the rigidity contrast between the two media. From the welded boundary condition at the interface between medium 1 and 2, a stress drop discontinuity condition is obtained which can be fulfilled if the stress drop in the upper medium is lower than required for a planar trough-going surface: as a corollary, a vertically dipping strike-slip fault at depth may cross the interface with a sedimentary layer, provided that the shallower section is suitably inclined (fault "refraction"); this results has important implications for our understanding of the complexity of the fault system in the SISZ; in particular, we may understand the observed offset of secondary surface fractures with respect to the strike direction of the seismic fault. The results of this model also suggest that further fractures can develop in the opposite quadrant and so a second model describing fault branching in the upper layer is proposed. As the previous model, this model can be applied only when the stress drop in the shallow layer is lower than the value prescribed for a vertical planar crack surface. Alternative solutions must be considered if the stress drop in the upper layer is higher than in the other layer, which may be the case when anelastic processes relax deviatoric stress in layer 2. In such a case one through-going crack cannot fulfil the welded boundary conditions and unwelding of the interface may take place. We have solved this problem within the theory of fracture mechanics, employing the boundary element method. The fault terminates against the interface in a T-shaped configuration, whose segments interact among each other: the lateral extent of the unwelded surface can be computed in terms of the main fault parameters and the stress field resulting in the shallower layer can be modelled. A wide stripe of high and nearly uniform shear stress develops above the unwelded surface, whose width is controlled by the lateral extension of unwelding. Secondary shear fractures may then open within this stripe, according to the Coulomb failure criterion, and the depth of open fractures opening in mixed mode may be computed and compared with the well studied fault complexities observed in the field. In absence of the T-shaped decollement structure, stress concentration above the seismic fault would be difficult to reconcile with observations, being much higher and narrower.