Complex singularities and PDEs

In this paper we give a review on the computational methods used to characterize the complex singularities developed by some relevant PDEs. We begin by reviewing the singularity tracking method based on the analysis of the Fourier spectrum. We then introduce other methods generally used to detect the hidden singularities. In particular we show some applications of the Pad\'e approximation, of the Kida method, and of Borel-Polya method. We apply these techniques to the study of the singularity formation of some nonlinear dispersive and dissipative one dimensional PDE of the 2D Prandtl equation, of the 2D KP equation, and to Navier-Stokes equation for high Reynolds number incompressible flows in the case of interaction with rigid boundaries

Algebras for Tree Decomposable Graphs

Complex problems can be sometimes solved efficiently via recursive decomposition strategies. In this line, the tree decomposition approach equips problems modelled as graphs with tree-like parsing structures. Following Milner’s flowgraph algebra, in a previous paper two of the authors introduced a strong network algebra to represent open graphs (up to isomorphism), so that homomorphic properties of open graphs can be computed via structural recursion. This paper extends this graphical-algebraic foundation to tree decomposable graphs. The correspondence is shown: (i) on the algebraic side by a loose network algebra, which relaxes the restriction reordering and scope extension axioms of the strong one; and (ii) on the graphical side by Milner’s binding bigraphs, and elementary tree decompositions. Conveniently, an interpreted loose algebra gives the evaluation complexity of each graph decomposition. As a key contribution, we apply our results to dynamic programming (DP). The initial statement of the problem is transformed into a term (this is the secondary optimisation problem of DP). Noting that when the scope extension axiom is applied to reduce the scope of the restriction, then also the complexity is reduced (or not changed), only so-called canonical terms (in the loose algebra) are considered. Then, the canonical term is evaluated obtaining a solution which is locally optimal for complexity. Finding a global optimum remains an NP-hard problem

Asymptotic analysis of the linearized Navier-Stokes equation on an exterior circular domain: Explicit solution and the zero viscosity limit

In this paper we study and derive explicit formulas for the linearized Navier-Stokes equations on an exterior circular domain in space dimension two. Through an explicit construction, the solution is decomposed into an inviscid solution, a boundary layer solution and a corrector. Bounds on these solutions are given, in the appropriate Sobolev spaces, in terms of the norms of the initial and boundary data. The correction term is shown to be of the same order of magnitude as the square root of the viscosity. Copyright © 2001 by Marcel Dekker, Inc

Excitable FitzHugh-Nagumo model with cross-diffusion: close and far-from-equilibrium coherent structures

In this paper, we shall study the formation of stationary patterns for a reaction-diffusion system in which the FitzHugh-Nagumo (FHN) kinetics, in its excitable regime, is coupled to linear cross-diffusion terms. In (Gambino et al. in Excitable Fitzhugh-Nagumo model with cross-diffusion: long-range activation instabilities, 2023), we proved that the model supports the emergence of cross-Turing patterns, i.e., close-to-equilibrium structures occurring as an effect of cross-diffusion. Here, we shall construct the crossTuring patterns close to equilibrium on 1-D and 2-D rectangular domains. Through this analysis, we shall show that the species are out-of-phase spatially distributed and derivethe amplitude equations that govern the pattern dynamics close to criticality. Moreover, we shall classify the bifurcation in the parameter space, distinguishing between super-and sub-critical transitions. In the final part of the paper, we shall numerically investigate the impact of the cross-diffusion terms on large-amplitude pulse-like solutions existing outside the cross-Turing regime, showing their emergence also in the case of lateral activation and short-range inhibition

Excitable FitzHugh-Nagumo model with cross-diffusion: long-range activation instabilities

In this paper, we shall study a spatially extended version of the FitzHugh-Nagumo model, where one describes the motion of the species through cross-diffusion. The motivation comes from modeling biological species where reciprocal interaction influences spatial movement. We shall focus our analysis on the excitable regime of the system. In this case, we shall see how cross-diffusion terms can destabilize uniform equilibrium, allowing for the formation of close-to-equilibrium patterns; the species are out-of-phase spatially distributed, namely high concentration areas of one species correspond to a low density of the other (cross-Turing patterns). Moreover, depending on the magnitude of the inhibitor’s cross-diffusion, the pattern’s development can proceed in either case of the inhibitor/activator diffusivity ratio being higher or smaller than unity. This allows for spatial segregation of the species in both cases of short-range activation/long-range inhibition or long-range activation/short-range inhibition

Spatio-temporal variability of micro-, nano- and pico-phytoplankton in the Mediterranean Sea from satellite ocean colour data of SeaWiFS

Abstract. The seasonal and year-to-year variability of the phytoplankton size class (PSC) spatial distribution has been examined in the Mediterranean Sea by using the entire time series of Sea-viewing Wide Field-of-view Sensor (SeaWiFS) space observations (1998–2010). Daily maps of PSCs have been determined using an empirical model based on a synoptic relationship between surface chlorophyll a and diagnostic pigments referred to different taxonomic groups. The analysis of micro-, nano- and pico-phytoplankton satellite time series (1998–2010) describes, quantitatively, the algal assemblage structure over the basin and reveals that the main contribution to chlorophyll a in most of the Mediterranean Sea comes from the pico-phytoplankton component, especially in nutrient-poor environments. Regions with different and peculiar features are the Northwestern Mediterranean Sea, the Alborán Sea and several coastal areas, such as the North Adriatic Sea. In these areas, local interactions between physical and biological components modulate the composition of the three phytoplankton size classes. It results that, during the spring bloom season, micro-phytoplankton dominates in areas of intense vertical winter mixing and deep/intermediate water formation, while in coastal areas micro-phytoplankton dominates in all seasons because of the nutrient supply from the terrestrial inputs. In the Alborán Sea, where the Atlantic inflow modulates the nutrient availability, any predominance of one class over the other two has been observed. The nano-phytoplankton component instead remains widespread over the entire basin along the year, and its contribution to chlorophyll a is of the order of 30–40 %. The largest inter-annual signal occurs in the Northwestern Mediterranean Sea, driven by the year-to-year variation in intensity and extension of the spring bloom, followed by the Alborán Sea, in which the inter-annual variability is strongly modulated by the Atlantic inflow. In absence of sufficient in situ data of community composition, the satellite-based analysis demonstrated that pico-, nano- and micro-phytoplankton classes often coexist. The predominance of one group over the other ones is strongly dependent on the physical and biological processes occurring at the mesoscale. These processes directly influence the nutrient and light availability, which are the principal forcing for the algae growth

The platform switching approach to optimize split crest technique.

The split crest technique is a reliable procedure used simultaneously in the implant positioning. In the literature some authors describe a secondary bone resorption as postoperative complication. The authors show how platform switching can be able to avoid secondary resorption as complication of split crest technique

Multidisciplinary Approach to Fused Maxillary central Incisors: a Case Report

Introduction: The fusion of permanent teeth is a development anomaly of dental hard tissue. It may require a hard multidisciplinary approach with orthodontics, endodontics, surgery and prosthetics to solve aesthetic and functional problems. Case presentation: A 20-year-old Caucasian man presented to our Department to solve a dental anomaly of his upper central incisors. An oral investigation revealed the fusion of his maxillary central incisors and dyschromia of right central incisor. Vitality pulp tests were negative for lateral upper incisors and left central incisor. Radiographic examinations showed a fused tooth with two separate pulp chambers, two distinct roots and two separate root canals. There were also periapical lesions of central incisors and right lateral incisor, so he underwent endodontic treatment. Six months later, OPT examination revealed persistence of the periapical radiolucency, so endodontic surgery was performed, which included exeresis of the lesion, an apicoectomy and retrograde obturation with a reinforced zinc oxide-eugenol cement (SuperEBA) Complete healing of the lesion was obtained six months postoperatively. Fused teeth crowns were separated and orthodontic appliances were put in place. When correct teeth position was achieved (after nine months), the anterior teeth were prosthetically rehabilitated. Conclusion: Many treatment options have been proposed in the literature to solve cases of dental fusion. The best treatment plan depends on the nature of the anomaly, its location, the morphology of the pulp chamber and root canal system, the subgingival extent of the separation line, and the patient compliance. Following an analysis of radiographical and clinical data, it was possible to solve our patient’s dental anomaly with a multidisciplinary approach