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In this paper, the author deals with a well-known problem of Celestial Mechanics, namely the three-body problem. A numerical analysis has been done in order to prove existence of chaotic motions of the full-averaged problem in particular configurations. Full because all the three bodies have non-negligible masses and averaged because the Hamiltonian describing the system has been averaged with respect to a fast angle. A reduction of degrees of freedom and of the phase-space is performed in order to apply the notion of covering relations and symbolic dynamics.

We discuss some perturbation results concerning certain pairs of sequences of vectors in a Hilbert space $\Hil$ and producing new sequences which share, with the original ones, { reconstruction formulas on a dense subspace of $\Hil$ or on the whole space}. We also propose some preliminary results on the same issue, but in a distributional settings., To appear in Journal of Mathematical Physics

This work focuses on the dynamics of vegetation stripes in sloped semi-arid environments in the presence of secondary seed dispersal and inertial effects. To this aim, a hyperbolic generalization of the Klausmeier model that encloses the advective downhill transport of plant biomass is taken into account. Analytical investigations were performed to deduce the wave and Turing instability loci at which oscillatory and stationary vegetation patterns arise, respectively. Additional information on the possibility of predicting a null-migrating behavior was extracted with suitable approximations of the dispersion relation. Numerical simulations were also carried out to corroborate theoretical predictions and to gain more insights into the dynamics of vegetation stripes at, close to, and far from the instability threshold.

In this paper we propose a dynamical approach based on the Gorini-Kossakowski-Sudarshan-Lindblad equation for a problem of decision making. More specifically, we consider what was recently called a quantum parliament, asked to approve or not a certain law, and we propose a model of the connections between the various members of the parliament, proposing in particular some special form of the interactions giving rise to a {\em collaborative} or non collaborative behaviour.

We consider a gas of bosonic particles confined in a box with Neumann boundary conditions. We prove Bose-Einstein condensation in the Gross-Pitaevskii regime, with an optimal bound on the condensate depletion. Our lower bound for the ground state energy in the box implies (via Neumann bracketing) a lower bound for the ground state energy of the Bose gas in the thermodynamic limit., Comment: 45 pages

The goal of my Ph.D. research is to analyze three species models in order to describe the behavior of an ecological community. In particular, two reaction-diffusion systems describing different local interactions between three species have been considered to obtain species coexistence, diversity, and distribution patterns. The first analyzed model describes intraguild predation: there are an IG-predator species, an IG-prey species, and a common resource species, which is shared by both of them. The IGP interaction is of Lotka-Volterra type, coupled with nonlinear diffusion, since we assume that the IG-prey moves towards lower density areas of the IG-predator. In this model, the extinction of species has been surveyed. Performing the linear stability analysis in the neighborhood of the coexistence point, the conditions for the occurrence of Hopf instability have been established. Cross-diffusion is able to induce Turing instability for this system, which would not admit this bifurcation in presence of only classical diffusion terms. Moreover, the effect of each parameter on Turing and Turing-Hopf instability has been detected. Numerical solutions of the system have been computed using spectral method, showing the rich dynamics of the model, including the Turing pattern, time oscillation pattern, Turing-Hopf pattern, and chaotic behavior. The weakly nonlinear analysis also has been employed to predict the amplitude of patterned solutions have been compared with numerical spectral solutions of the reaction-diffusion system. Furthermore, we have used multiscale methods to determine normal form of the model around Turing-Hopf codimension-2 points. Finally, by utilizing the fixed point argument and energy estimate, the existence of the global solution to the system has been established, assuming some conditions on initial data. The second three species model describes the dynamics of two predators competing with each other to feed on the same prey. The functional response of predators is the Holling type. This local dynamics has been coupled with linear cross-diffusion terms taking into account the movement of each species towards lower-density areas of the other species. We have applied linear analysis of the system with and without diffusion to obtain the necessary conditions of stability and the occurrence of Hopf and Turing instability. In particular, weakly nonlinear analysis, Turing regions, and maximum growth rate have been investigated. Using a numerical finite elements method, Turing patterns have also been displayed and compared with WNL solutions. Finally, to prove the existence of global in time of the solutions, a rectangular invariant method has been presented for a particular case.

We present a result which allows us to deform a Poisson-Nijenhuis manifold into a Poisson quasi-Nijenhuis manifold by means of a closed 2-form. Under an additional assumption, the deformed structure is also Poisson-Nijenhuis. We apply this result to show that the canonical Poisson-Nijenhuis structure on R^2n gives rise to both the Poisson-Nijenhuis structure of the open (or non periodic) n-particle Toda lattice, introduced by Das and Okubo [6], and the Poisson quasi-Nijenhuis structure of the closed (or periodic) n-particle Toda lattice, described in our recent work [7].

We perform a detailed study of a class of irregular correlators in Liouville Conformal Field Theory, of the related Virasoro conformal blocks with irregular singularities and of their connection formulae. Upon considering their semi-classical limit, we provide explicit expressions of the connection matrices for the Heun function and a class of its confluences. Their calculation is reduced to concrete combinatorial formulae from conformal block expansions., Comment: 61 pages, many diagrams, 2 figures, huge list of symbols, comments welcome

Mathematical models for the collective dynamics of interacting and spatially distributed populations find applications in several contexts (biology, ecology, social sciences). Their formulation depends primarily on the (continuous or discrete) description of the space. Reaction-diffusion equations have been widely used in bioecology (morphogenesis, migration of biological species, tumor growth, neuro-degenerative diseases) and in the social sciences (diffusion of opinions or decisionmaking processes), and exhibit complex behaviors (propagation of oscillatory phenomena, pattern formation caused by instability). A reaction–diffusion system exhibits diffusion-driven instability, sometimes called Turing instability, if the homogeneous steady state is stable to small perturbations in the absence of diffusion but unstable to small spatial perturbations when diffusion is present. In this thesis, we move from this classical approach, considering a so called crimo-taxis model (Epstein, 1997), and proposing two variants (Inferrera et al., 2022) enabling us to study the formation of some patterns due to instability driven by self- and cross-diffusion terms, to operatorial models built by means of some techniques typical of quantum mechanics (see Bagarello, 2012; Bagarello, 2019). The leading idea in this approach relies on the evidence, shown in the last fifteen years in several applications, that the operatorial framework provides useful tools for describing the interactions occurring within macroscopic systems. Therefore, three applications of the operatorial formalism are discussed: 1)an operatorial version of crimo-taxis model; 2)a model where two populations spatially distributed in a one–dimensional domain compete both locally and nonlocally and are able to migrate (Inferrera and Oliveri, 2022); 3) a model of a finite number of agents subjected both to cooperative and competitive interactions (Gorgone, Inferrera, and Oliveri, 2022).

Pattern formation and modulation is an active branch of mathematics, not only from the perspective of fundamental theory but also for its huge applications in many fields of physics, ecology, chemistry, biology, and other sciences. In this thesis, the occurrence of Turing and wave instabilities, giving rise to stationary and oscillatory patterns, respectively, is theoretically investigated by means of two-compartment reaction-transport hyperbolic systems. The goal is to elucidate the role of inertial times, which are introduced in hyperbolic models to account for the finite-time propagation of disturbances, in stationary and transient dynamics, in supercritical and subcritical regimes. In particular, starting from a quite general framework of reaction-transport model, three particular cases are derived. In detail, in the first case, the occurrence of stationary patterns is investigated in one-dimensional domains by looking for the inertial dependence of the main features that characterize the formation and stability process of the emerging patterns. In particular, the phenomenon of Eckhaus instability, in both supercritical and subcritical regimes, is studied by adopting linear and multiple-scale weakly-nonlinear analysis and the role played by inertia during the transient regime, where an unstable patterned state evolves towards a more favorable stable configuration through sequences of phase-slips, is elucidated. Then, in the second topic, the focus is moved to oscillatory periodic patterns generated by wave (or oscillatory Turing) instability. This phenomenon is studied by considering 1D two-compartment hyperbolic reaction-transport systems where different transport mechanisms of the species here involved are taken into account. In these cases, by using linear and weakly nonlinear stability analysis techniques, the dependence of the non-stationary patterns on hyperbolicity is underlined at and close to the criticality. In particular, it is proven that inertial effects play a role, not only during transient regimes from the spatially-homogeneous steady state toward the patterned state but also in altering the amplitude, the wavelength, the migration speed, and even the stability of the travelling waves. Finally, in the last case, the formation and stability of stationary patterns are investigated in bi-dimensional domains. To this aim, a general class of two-species hyperbolic reaction-transport systems is deduced following the guidelines of Extended Thermodynamics theory. To characterize the emerging Turing patterns, linear and weakly nonlinear stability analysis on the uniform steady states are addressed for rhombic and hexagonal planform solutions. In order to gain some insight into the above-mentioned dynamics, the previous theoretical predictions are corroborated by numerical simulations carried out in the context of dryland ecology. In this context, patterns become a relevant tool to identify early warning signals toward desertification and to provide a measure of resilience of ecosystems under climate change. Such ecological implications are discussed in the context of the Klausmeier model, one of the easiest two-compartment (vegetation biomass and water) models able to describe the formation of patterns in semi-arid environments. Therefore, it will be also here discussed how the experimentally-observed inertia of vegetation affects the formation and stability of stationary and oscillatory periodic vegetation patterns. Pattern formation and modulation is an active branch of mathematics, not only from the perspective of fundamental theory but also for its huge applications in many fields of physics, ecology, chemistry, biology, and other sciences. In this thesis, the occurrence of Turing and wave instabilities, giving rise to stationary and oscillatory patterns, respectively, is theoretically investigated by means of two-compartment reaction-transport hyperbolic systems. The goal is to elucidate the role of inertial times, which are introduced in hyperbolic models to account for the finite-time propagation of disturbances, in stationary and transient dynamics, in supercritical and subcritical regimes. In particular, starting from a quite general framework of reaction-transport model, three particular cases are derived. In detail, in the first case, the occurrence of stationary patterns is investigated in one-dimensional domains by looking for the inertial dependence of the main features that characterize the formation and stability process of the emerging patterns. In particular, the phenomenon of Eckhaus instability, in both supercritical and subcritical regimes, is studied by adopting linear and multiple-scale weakly-nonlinear analysis and the role played by inertia during the transient regime, where an unstable patterned state evolves towards a more favorable stable configuration through sequences of phase-slips, is elucidated. Then, in the second topic, the focus is moved to oscillatory periodic patterns generated by wave (or oscillatory Turing) instability. This phenomenon is studied by considering 1D two-compartment hyperbolic reaction-transport systems where different transport mechanisms of the species here involved are taken into account. In these cases, by using linear and weakly nonlinear stability analysis techniques, the dependence of the non-stationary patterns on hyperbolicity is underlined at and close to the criticality. In particular, it is proven that inertial effects play a role, not only during transient regimes from the spatially-homogeneous steady state toward the patterned state but also in altering the amplitude, the wavelength, the migration speed, and even the stability of the travelling waves. Finally, in the last case, the formation and stability of stationary patterns are investigated in bi-dimensional domains. To this aim, a general class of two-species hyperbolic reaction-transport systems is deduced following the guidelines of Extended Thermodynamics theory. To characterize the emerging Turing patterns, linear and weakly nonlinear stability analysis on the uniform steady states are addressed for rhombic and hexagonal planform solutions. In order to gain some insight into the above-mentioned dynamics, the previous theoretical predictions are corroborated by numerical simulations carried out in the context of dryland ecology. In this context, patterns become a relevant tool to identify early warning signals toward desertification and to provide a measure of resilience of ecosystems under climate change. Such ecological implications are discussed in the context of the Klausmeier model, one of the easiest two-compartment (vegetation biomass and water) models able to describe the formation of patterns in semi-arid environments. Therefore, it will be also here discussed how the experimentally-observed inertia of vegetation affects the formation and stability of stationary and oscillatory periodic vegetation patterns.

We study the quantum evolution of many-body Fermi gases in three dimensions, in arbitrarily large domains. We consider both particles with non-relativistic and with relativistic dispersion. We focus on the high-density regime, in the semiclassical scaling, and we consider a class of initial data describing zero-temperature states. In the non-relativistic case we prove that, as the density goes to infinity, the many-body evolution of the reduced one-particle density matrix converges to the solution of the time-dependent Hartree equation, for short macroscopic times. In the case of relativistic dispersion, we show convergence of the many-body evolution to the relativistic Hartree equation for all macroscopic times. With respect to previous work, the rate of convergence does not depend on the total number of particles, but only on the density: in particular, our result allows us to study the quantum dynamics of extensive many-body Fermi gases., 44 pages, no figures. The proof of the main result has been simplified and generalized. Furthermore, we included the derivation of the time-dependent relativistic Hartree equation, for all macroscopic times

Given a self-adjoint operator A in a Hilbert space H, we analyze its spectral behavior when it is expressed in terms of generalized eigenvectors. Using the formalism of Gel’fand distribution bases, we explore the conditions for the generalized eigenspaces to be one-dimensional, i.e., for A to have a simple spectrum.

In this paper, meant as a companion to arXiv:2006.04458, we consider a class of non-integrable $2D$ Ising models in cylindrical domains, and we discuss two key aspects of the multiscale construction of their scaling limit. In particular, we provide a detailed derivation of the Grassmann representation of the model, including a self-contained presentation of the exact solution of the nearest neighbor model in the cylinder. Moreover, we prove precise asymptotic estimates of the fermionic Green's function in the cylinder, required for the multiscale analysis of the model. We also review the multiscale construction of the effective potentials in the infinite volume limit, in a form suitable for the generalization to finite cylinders. Compared to previous works, we introduce a few important simplifications in the localization procedure and in the iterative bounds on the kernels of the effective potentials, which are crucial for the adaptation of the construction to domains with boundaries., 69 pages. Minor revisions. Version accepted by Annales Henri Poincar\'e. arXiv admin note: substantial text overlap with arXiv:2006.04458

In nature, many fluid-like materials exhibit a yield stress below which they behave like a solid. The Bingham model aims to describe such materials. This paper draws some mathematical considerations on the flow of a Bingham fluid in a vertical channel. The situation due to the presence of an external magnetic field and natural convection is analyzed: the external magnetic field, which is orthogonal to the walls of the channel, generates the Lorentz forces that influence the motion through the Hartmann number. The behavior of the velocity, the induced magnetic field and the thickness of the plug regions are discussed and presented graphically. We find that the velocity is a decreasing function of the Bingham and Hartmann numbers. In particular, the presence of the external magnetic field increases the thickness of the plug region. The modulus of the induced magnetic field is not monotone when the Hartmann number changes, but it is a decreasing function of the Bingham number.

Although mathematical modelling of pressure-flow dynamics in the cardiocirculatory system has a lengthy history, readily finding the appropriate model for the experimental situation at hand is often a challenge in and of itself. An ideal model would be relatively easy to use and reliable, besides being ethically acceptable. Furthermore, it would address the pathogenic features of the cardiovascular disease that one seeks to investigate. No universally valid model has been identified, even though a host of models have been developed. The object of this review is to describe several of the most relevant mathematical models of the cardiovascular system: the physiological features of circulatory dynamics are explained, and their mathematical formulations are compared. The focus is on the whole-body scale mathematical models that portray the subject’s responses to hypovolemic shock. The models contained in this review differ from one another, both in the mathematical methodology adopted and in the physiological or pathological aspects described. Each model, in fact, mimics different aspects of cardiocirculatory physiology and pathophysiology to varying degrees: some of these models are geared to better understand the mechanisms of vascular hemodynamics, whereas others focus more on disease states so as to develop therapeutic standards of care or to test novel approaches. We will elucidate key issues involved in the modeling of cardiovascular system and its control by reviewing seven of these models developed to address these specific purposes.

We have theoretically investigated the phenomenon of Eckhaus instability of stationary patterns arising in hyperbolic reaction–diffusion models on large finite domains, in both supercritical and subcritical regime. Adopting multiple-scale weakly-nonlinear analysis, we have deduced the cubic and cubic–quintic real Ginzburg–Landau equations ruling the evolution of pattern amplitude close to criticality. Starting from these envelope equations, we have provided the explicit expressions of the most relevant dynamical features characterizing primary and secondary quantized branches of any order: stationary amplitude, existence and stability thresholds and linear growth rate. Particular emphasis is given on the subcritical regime, where cubic and cubic–quintic Ginzburg–Landau equations predict qualitatively different dynamical pictures. As an illustrative example, we have compared the above-mentioned analytical predictions to numerical simulations carried out on the hyperbolic modified Klausmeier model, a conceptual tool used to describe the generation of stationary vegetation stripes over flat arid environments. Our analysis has also allowed to elucidate the role played by inertia during the transient regime, where an unstable patterned state evolves towards a more favorable stable configuration through sequences of phase-slips. In particular, we have inspected the functional dependence of time and location at which wavelength adjustment takes place as well as the possibility to control these quantities, independently of each other.

Onboard ships, fire is one of the most dangerous events that can occur. For both military and commercial ships, fire risks are the most worrying; for this reason they have an important impact on the design of the vessel. The intumescent coatings react when heated or in contact with a living flame, and a multi-layered insulating structure grows up, protecting the underlying structure. In this concern, the aim of the paper is to evaluate the intumescent capacity of different composite coatings coupling synergistically modeling and experimental tests. In particular, the experiments have been carried out on a new paint formulation, developed by Colorificio Atria S.r.l., in which the active components are ammonium polyphosphate or pentaerythritol. The specimens were exposed to a gas-torch flame for about 70 s. The degree of thermal insulation of the coating was monitored by means of a thermocouple placed on the back of the sample. In order to get insights into the intumescent mechanism, experimental data was compared with the results of a mathematical model and a good agreement is detected. Furthermore, a predictive model on the swelling rate is addressed. The results highlight that all coatings exhibit a clear intumescent and barrier capacity. The best results were observed for coating enhanced with NH4PO3 where a regular and thick, porous char was formed during exposure to direct flame.

Starting from a recent proposal of a nonlinear Maxwell-Cattaneo equation for the heat transport with relaxational effects at nanoscale, in a special case of thermal-wave propagation we derive a nonlinear Schrodinger equation for the amplitudes of the heatflux perturbation. The complete integrability of the obtained equation is investigated in order to prove the existence of infinite conservation laws, as well as the existence of infinite exact solutions. In this regards, we have considered the simplest nontrivial solutions, namely, the bright and dark (thermal) solitons, which may be interesting for energy transport and for information transmission in phononic circuits. (c) 2022 Elsevier B.V. All rights reserved.

Modern Graph Theory is a newly emerging field that involves all of those approaches that study graphs differently from Classic Graph Theory. The main difference between Classic and Modern Graph Theory regards the analysis and the use of graph's structures (micro/macro). The former aims to solve tasks hosted on graph nodes, most of the time with no insight into the global graph structure, the latter aims to analyze and discover the most salient features characterizing a whole network of each graph, like degree distributions, hubs, clustering coefficient and network motifs. The activities carried out during the PhD period concerned, after a careful preliminary study on the applications of the Modern Graph Theory, the development of an innovative Convolutional Model to model brain connections at the cellular level capable of combining exponential models and power law models. This new theoretical framework has been introduced in the first instance with an aspatial graph formulation and then proposed a spatial graph model with Convolutive connectivity able to fit the degree distributions of data driven Connectome reconstructions. In order to evaluate the qualities of the Convolutional Model, theoretical graphical models capable of characterizing brain activity were taken into consideration. In the specific case, the model examined characterizes the epileptic activity through a simple Hindmarsh-Rose model system of point neurons and reproduces the functional characteristics observed in the data driven model. Such a model provides insight into the deep impact of micro connectivity in macro-scale brain activity. Other evaluations have been done in different applications, in the field of image cell segmentation with Explainable Artificial Intelligence's neuronal agents in which has been used a methodology that is not only explainable but also resistant to adversarial noise and also in the field of modelling Covid-19 outbreak in gaining insight on vaccines and role of our habits as individuals in the pandemic spread. Therefore, the core of the thesis is to introduce Modern Graph Theory with a new competitive Convolutive Model and then expose some applications to real-world problems like a characterization of Brain networks, simulation and analysis of Brain dynamics with a particular focus on Epilepsy, Immunofluorescence images segmentation with neuronal based agents and modelling of Covid-19 Epidemic spread with a specific interest in human social networks. All this takes continuously into account the whole dialogue between Graph Theory and its applications.

In this paper, we show that the position and the derivative operators, $${{\hat{q}}}$$ q ^ and $${{\hat{D}}}$$ D ^ , can be treated as ladder operators connecting various vectors of two biorthonormal families, $${{{\mathcal {F}}}}_\varphi $$ F φ and $${{{\mathcal {F}}}}_\psi $$ F ψ . In particular, the vectors in $${{{\mathcal {F}}}}_\varphi $$ F φ are essentially monomials in x, $$x^k$$ x k , while those in $${{{\mathcal {F}}}}_\psi $$ F ψ are weak derivatives of the Dirac delta distribution, $$\delta ^{(m)}(x)$$ δ ( m ) ( x ) , times some normalization factor. We also show how bi-coherent states can be constructed for these $${{\hat{q}}}$$ q ^ and $${{\hat{D}}}$$ D ^ , both as convergent series of elements of $${{{\mathcal {F}}}}_\varphi $$ F φ and $${{{\mathcal {F}}}}_\psi $$ F ψ , or using two different displacement-like operators acting on the two vacua of the framework. Our approach generalizes well- known results for ordinary coherent states.

This thesis deals with classification of special classes of 1-variable holomorphic rational functions. In the first part we focus on the class of rational maps with bounded orbits of post-critical points – so-called Thurston maps. These maps can be viewed as a class of topological objects – branching coverings. There exists a natural equivalence relation on the class of Thurston maps, such that different rational functions are almost never equivalent. We provide an algorithm which allows to represent these algebraic (or topological) objects by means of completely combinatorial objects – invariant graphs with marked vertices. We also introduce a computational procedure for finding such graphs. Then we look at the cubic polynomials with connected Julia sets with a fixed multiplier, thus we get a slice of such polynomials. Such slices can be considered as parameter spaces. We introduce a parametrization of these cubic slices via reglued Julia set of the quadratic polynomial with the same multiplier. We also show that the parametrizing map is continuous. 2

In this work we consider a quite general class of two-species hyperbolic reaction-advection-diffusion system with the main aim of elucidating the role played by inertial effects in the dynamics of oscillatory periodic patterns. To this aim, first, we use linear stability analysis techniques to deduce the conditions under which wave (or oscillatory Turing) instability takes place. Then, we apply multiple-scale weakly nonlinear analysis to determine the equation which rules the spatiotemporal evolution of pattern amplitude close to criticality. This investigation leads to a cubic complex Ginzburg-Landau (CCGL) equation which, owing to the functional dependence of the coefficients here involved on the inertial times, reveals some intriguing consequences. To show in detail the richness of such a scenario, we present, as an illustrative example, the pattern dynamics occurring in the hyperbolic generalization of the extended Klausmeier model. This is a simple two-species model used to describe the migration of vegetation stripes along the hillslope of semiarid environments. By means of a thorough comparison between analytical predictions and numerical simulations, we show that inertia, apart from enlarging the region of the parameter plane where wave instability occurs, may also modulate the key features of the coherent structures, solution of the CCGL equation. In particular, it is proven that inertial effects play a role, not only during transient regime from the spatially-homogeneous steady state toward the patterned state, but also in altering the amplitude, the wavelength, the angular frequency, and even the stability of the phase-winding solutions.

We consider a system of $N\gg 1$ interacting fermionic particles in three dimensions, confined in a periodic box of volume $1$, in the mean-field scaling. We assume that the interaction potential is bounded and small enough. We prove upper and lower bounds for the correlation energy, which are optimal in their $N$-dependence. Moreover, we compute the correlation energy at leading order in the interaction potential, recovering the prediction of second order perturbation theory. The proof is based on the combination of methods recently introduced for the study of fermionic many-body quantum dynamics together with a rigorous version of second-order perturbation theory, developed in the context of non-relativistic QED., Comment: 40 pages; introduction rewritten, proof of Lemma 4.7 corrected. To appear in Comm. Math. Phys

We obtain the minimal energy solution of the Plateau problem with elastic boundary as a variational limit of the minima of the Kirchhoff-Plateau problems with a rod boundary when the cross-section of the rod vanishes. The limit boundary is a framed curve that can sustain bending and twisting.

We study the evolution of a 2D vortex layer at high Reynolds number. Vortex layer flows are characterized by intense vorticity concentrated around a curve. In addition to their intrinsic interest, vortex layers are relevant configurations because they are regularizations of vortex sheets. In this paper, we consider vortex layers whose thickness is proportional to the square-root of the viscosity. We investigate the typical roll-up process, showing that crucial phases in the initial flow evolution are the formation of stagnation points and recirculation regions. Stretching and folding characterizes the following stage of the dynamics, and we relate these events to the growth of the palinstrophy. The formation of an inner vorticity core, with vorticity intensity growing to infinity for larger Reynolds number, is the final phase of the dynamics. We display the inner core's self-similar structure, with the scale factor depending on the Reynolds number. We reveal the presence of complex singularities in the solutions of Navier–Stokes equations; these singularities approach the real axis with increasing Reynolds number. The comparison between these singularities and the Birkhoff–Rott singularity seems to suggest that vortex layers, in the limit $Re\rightarrow \infty$ , behave differently from vortex sheets.

The n-body problem is one of the most important issue in Celestial Mechanics. This article aims to retrace the historical and scientific events that led the Paduan mathematician, Tullio Levi-Civita, to deal with the problem first from a classic and then a relativistic point of view. We describe Levi-Civita's contributions to the theory of relativity focusing on his epistolary exchanges with Einstein, on the problem of secular acceleration and on the proof of Brillouin's cancellation principle. We also point out that the themes treated by Levi-Civita are very topical. Specifically, we analyse how the mathematical formalism used nowadays to test General Relativity can be found in Levi-Civita's texts and evolves over the years up to the current Parametrized version of the Post-Newtonian approximation (PPN) which is used in high precision contexts such as important space missions designed also to test General Relativity and which aim to estimate with very high accuracy the PPN parameters., 18 pages, 2 figures

We prove that skew systems with a sufficiently expanding base have approximate exponential decay of correlations, meaning that the exponential rate is observed modulo an error. The fiber maps are only assumed to be Lipschitz regular and to depend on the base in a way that guarantees diffusive behaviour on the vertical component. The assumptions do not imply an hyperbolic picture and one cannot rely on the spectral properties of the transfer operators involved. The approximate nature of the result is the inevitable price one pays for having so mild assumptions on the dynamics on the vertical component. However, the error in the approximation goes to zero when the expansion of the base tends to infinity. The result can be applied beyond the original setup when combined with acceleration or conjugation arguments, as our examples show.

We present some works of Corinna Ulcigrai closely related to Diophantine approximations and generalizing classical notions to the context of interval exchange maps, translation surfaces and Teichmüller dynamics.