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The validity of the inviscid limit for the incompressible Navier-Stokes equations is one of the most important and challenging problems in the mathematical theory of fluid dynamics: the motion of inviscid fluids is described by the Euler equations, so, when the viscosity goes to zero, one would expect the convergence of NS solutions to the Euler solutions. However, NS equations are a singular perturbation of the Euler equations: the change of order of the equation implies that fewer boundary conditions can be imposed on the inviscid flows. Therefore, the no-slip boundary conditions, imposed on the NS solutions, are not satisfied by the Euler flow, for which a tangential slip is allowed. This mismatch between the behaviour at the boundary of the NS solutions and the same behaviour of their supposed limit creates a boundary layer, with large gradients of velocity in the normal direction, which make the diffusive effects comparable to the inertial ones: this situation is classically described by Prandtl's equation. The ill posedness of Prandtl's equation in Sobolev settings require the use of more regular functional spaces: in a holomorphic setting, Prandtl's equation is well posed, and the inviscid limit holds. In this thesis, we extend this result for incompatible initial data, which satisfy the no-penetration boundary condition, but allow a tangential slip, a kind of data of both numerical and theoretical interest: this extension is not trivial, since the singularity formed by this kind of initial data forces us to use different function spaces, where some of the properties used in the proof of the compatible case do not hold. In Sobolev settings, we see that, for the linearization around an inviscid flow of the NS equations, the inviscid limit actually holds: in this case, Prandtl's asymptotic expansion is not necessary, and convergence can be proved through energy methods in conormal Sobolev spaces. The linearization can be limited to the tangential part of the flow: indeed, this is enough to avoid the interaction between the diffusive effects on the boundary and strong inertial terms, which is believed to cause boundary layer separation.

In the recent papers [5, 18], respectively, the existence of motions where the perihelions afford periodic oscillations about certain equilibria and the onset of a topological horseshoe have been proved. Such results have been obtained using, as neighbouring integrable system, the so-called two-centre (or Euler) problem and a suitable canonical setting proposed in [16, 17]. Here we review such results.

Wave front propagation with non-trivial bottom topography is studied within the formalism of hyperbolic long wave models. Evolution of non-smooth initial data is examined, and in particular the splitting of singular points and their short time behaviour is described. In the opposite limit of longer times, the local analysis of wavefronts is used to estimate the gradient catastrophe formation and how this is influenced by the topography. The limiting cases when the free surface intersects the bottom boundary, belonging to the so-called "physical" and "non-physical" vacuum classes, are examined. Solutions expressed by power series in the spatial variable lead to a hierarchy of ordinary differential equations for the time-dependent series coefficients, which are shown to reveal basic differences between the two vacuum cases: for non-physical vacuums, the equations of the hierarchy are recursive and linear past the first two pairs, while for physical vacuums the hierarchy is non-recursive, fully coupled and nonlinear. The former case may admit solutions that are free of singularities for nonzero time intervals, while the latter is shown to develop non-standard velocity shocks instantaneously. We show that truncation to finite dimensional systems and polynomial solutions is in general only possible for the case of a quadratic bottom profile. In this case the system's evolution can reduce to, and is completely described by, a low dimensional dynamical system for the time-dependent coefficients. This system encapsulates all the nonlinear properties of the solution for general power series initial data, and in particular governs the loss of regularity in finite times at the dry point. For the special case of parabolic bottom topographies, an exact, self-similar solution class is introduced and studied to illustrate via closed form expressions the general results., 32 pages, 15 figures

In this paper we address, from a purely numerical point of view, the question, raised in [20, 21], and partly considered in [22, 9, 3], whether a certain function, referred to as "Euler Integral", is a quasi-integral along the trajectories of the three-body problem. Differently from our previous investigations, here we focus on the region of the "unperturbed separatrix", which turns to be complicated by a collision singularity. Concretely, we reduce the Hamiltonian to two degrees of freedom and, after fixing some energy level, we discuss in detail the resulting three-dimensional phase space around an elliptic and an hyperbolic periodic orbit. After measuring the strength of variation of the Euler Integral (which are in fact small), we detect the existence of chaos closely to the unperturbed separatrix. The latter result is obtained through a careful use of the machinery of covering relations, developed in [13, 24, 23]., This paper is supported by the the ERC project 677793 Stable and Chaotic Motions in the Planetary Problem (2016-2022)

We minimize elastic energies on framed curves which penalize both curvature and torsion. We also discuss critical points using the infinite dimensional version of the Lagrange multipliers’ method. Finally, some examples arising from the applications are discussed and also numerical experiments are presented.

The systematic development of Coarse-Grained (CG) models via the Mori-Zwanzig projector operator formalism requires the explicit description of several terms, including a deterministic drift term, a dissipative memory term and a random fluctuation term. In many applications, the memory and fluctuation terms are related by the fluctuation-dissipation relation and are, in general, more challenging to derive than the drift term. In this work we analyse an approximation of the memory term and propose a rational basis for a data-driven approach to an approximation of the memory and fluctuating terms which can be considered included in the class of the Markovian ones., 39 pages, 2 Figures

When prices reflect all available information, the price dynamics is a martingale and the market is said to be ef-ficient. However, much empirical evidence supports the conclusion about the inefficiency of financial markets, especially at high-frequency timescales. We investigate the sources and dynamics of the inefficiency of the ETF market at a 1 min timescale by proposing a computational methodology for a genuine estimation of the Shannon entropy. Since several sources of regularity lead to the detection of apparent inefficiencies, we build a multi-step filtering method, which allows (i) to remove the seasonality of volatility and heteroskedasticity, (ii) to detect and remove spurious effects due to price staleness, and (iii) to filter out microstructure noise. We corroborate our findings with an extensive analysis of the ETF market. We conclude that, after removing all known patterns of regularity, the market is not efficient at a one-minute time scale and on a weekly basis; however, the signal is weak.(c) 2022 Elsevier Ltd.

The motion of the lymph has a very important role in the immune system, and it is influenced by the porosity of the lymph nodes: more than 90% takes the peripheral path without entering the lymphoid compartment. In this paper, we construct a mathematical model of a lymph node assumed to have a spherical geometry, where the subcapsular sinus is a thin spherical shell near the external wall of the lymph node and the core is a porous material describing the lymphoid compartment. For the mathematical formulation, we assume incompressibility and we use Stokes together with Darcy–Brinkman equation for the flow of the lymph. Thanks to the hypothesis of axisymmetric flow with respect to the azimuthal angle and the use of the stream function approach, we find an explicit solution for the fully developed pulsatile flow in terms of Gegenbauer polynomials. A selected set of plots is provided to show the trend of motion in the case of physiological parameters. Then, a finite element simulation is performed and it is compared with the explicit solution.

A Smoluchowski type model of coagulation in a turbulent fluid is given, first expressed by means of a stochastic model, then in a suitable scaling limit as a deterministic model with enhanced diffusion in the velocity component. A precise link between mean intensity of the turbulent velocity field and coagulation enhancement is obtained by numerical simulations, and a formula for the mean velocity difference, in agreement with the gas-kinetic model, is proved by a new method., Comment: 18 pages, double column, 20 figures. Added appendix, more details on presented methods. Added derived formulas on kernel, added new notations

In this thesis we develop a mathematically rigorous framework of the so-called ''classical limit'' of quantum systems and their semi-classical properties. Our methods are based on the theory of strict, also called C*- algebraic deformation quantization. Since this C*-algebraic approach encapsulates both quantum as classical theory in one single framework, it provides, in particular, an excellent setting for studying natural emergent phenomena like spontaneous symmetry breaking (SSB) and phase transitions typically showing up in the classical limit of quantum theories. To this end, several techniques from functional analysis and operator algebras have been exploited and specialised to the context of Schr���dinger operators and quantum spin systems. Their semi-classical properties including the possible occurrence of SSB have been investigated and illustrated with various physical models. Furthermore, it has been shown that the application of perturbation theory sheds new light on symmetry breaking in Nature, i.e. in real, hence finite materials. A large number of physically relevant results have been obtained and presented by means of diverse research papers.

In this paper we extend some previous results on weak pseudo-bosons and on their related bi-coherent states. The role of {\em compatible} functions is discussed in details, and some examples are considered. The pseudo-bosonic ladder operators analysed in this paper generalize significantly those considered so far, and a class of new diagonalizable manifestly non self-adjoint Hamiltonians are deduced., Comment: in press in JMAA

This manuscript tackles the study of vegetation pattern dynamics driven by inertial effects and secondary seed dispersal. To achieve this goal, an hyperbolic extension of the classical parabolic Klausmeier model of vegetation, generally used to predict the formation of banded vegetation along the slopes of semiarid environments, has been here considered together with an additional advective term mimicking the downslope motion of seeds. Linear stability analyses have been carried out to inspect the dependence of the wave instability locus on the model parameters, with particular emphasis on the role played by inertial time and seed advection speed. Moreover, periodic travelling wave solutions are taken into account to better characterize modulus and direction of the migration speed of striped vegetation patterns. Theoretical predictions are corroborated by numerical Investigations and ecological implications are also discussed. In particular, it is highlighted how the hyperbolic nature of the model may provide possible justifications about some controversial field observations.

This Ph.D. project was initially born from the motivation to contribute to the depth-averaged and 3D modeling of lava flows. Still, we can frame the work done in a broader and more generalist vision. We developed two models that may be used for generic viscous fluids, and we applied efficient numerical schemes for both cases, as explained in the following. The new solvers simulate free-surface viscous fluids whose temperature changes are due to radiative, convective, and conductive heat exchanges. A temperature-dependent viscoplastic model is used for the final application to lava flows. Both the models behind the solvers were derived from mass, momentum, and energy conservation laws. Still, one was obtained by following the depth-averaged model approach and the other by the 3D model approach. The numerical schemes adopted in both our models belong to the family of finite volume methods, based on the integral form of the conservation laws. This choice of methods family is fundamental because it allows the creation and propagation of discontinuities in the solutions and enforces the conservation properties of the equations. We propose a depth-averaged model for a viscous fluid in an incompressible and laminar regime with an additional transport equation for a scalar quantity varying horizontally and a variable density that depends on such transported quantity. Viscosity and non-constant vertical profiles for the velocity and the transported quantity are assumed, overtaking the classic shallow-water formulation. The classic formulation bases on several assumptions, such as the fact that the vertical pressure distribution is hydrostatic, that the vertical component of the velocity can be neglected, and that the horizontal velocity field can be considered constant with depth because the classic formulation accounts for non-viscous fluids. When the vertical shear is essential, the last assumption is too restrictive, so it must relax, producing a modified momentum equation in which a coefficient, known as the Boussinesq factor, appears in the advective term. The spatial discretization method we employed is a modified version of the central-upwind scheme introduced by Kurganov and Petrova in 2007 for the classical shallow water equations. This method is based on a semi-discretization of the computational domain, is stable, and, being a high-order method, has a low numerical diffusion. For the temporal discretization, we used an implicit-explicit Runge-Kutta technique discussed by Russo in 2005 that permits an implicit treatment of the stiff terms. The whole scheme is proved to preserve the positivity of flow thickness and the stationary steady-states. Several numerical experiments validate the proposed method, show the incidence on the numerical solutions of shape coefficients introduced in the model and present the effects of the viscosity-related parameters on the final emplacement of a lava flow. Our 3D model describes the dynamics of two incompressible, viscous, and immiscible fluids, possibly belonging to different phases. Being interested in the final application of lava flows, we also have an equation for energy that models the thermal exchanges between the fluid and the environment. We implemented this model in OpenFOAM, which employs a segregated strategy and the Finite Volume Methods to solve the equations. The Volume of Fluid (VoF) technique introduced by Hirt and Nichols in 1981 is used to deal with the multiphase dynamics (based on the Interphase Capturing strategy), and hence a new transport equation for the volume fraction of one phase is added. The challenging effort of maintaining an accurate description of the interphase between fluids is solved by using the Multidimensional Universal Limiter for Explicit Solution (MULES) method (described by Marquez Damian in 2013) that implements the Flux-Corrected Transport (FCT) technique introduced by Boris and Book in 1973, proposing a mix of high and low order schemes. The choice of the framework to use for any new numerical code is crucial. Our contribution consists of creating a new solver called interThermalRadConvFoam in the OpenFOAM framework by modifying the already existing solver interFoam (described by Deshpande et al. in 2012). Finally, we compared the results of our simulations with some benchmarks to evaluate the performances of our model.

We investigate a reaction-diffusion-chemotaxis system that describes the immune response during an inflammatory attack. The model is a modification of the system proposed in Penner, Ermentrout, and Swigon [SIAM J. Appl. Dyn. Syst., 11 (2012), pp. 629-660]. We introduce a logistic term in the immune cell dynamics to reproduce the macrophages' activation, allowing us to describe the disease evolution from the early stages to the acute phase. We focus on the appearance of pattern solutions and their stability. We discover steady-state (Turing) and wave instabilities and classify the bifurcations deriving the corresponding amplitude equations. We study stationary radially symmetric solutions and show that they reproduce various inflammatory aggregates observed in the clinical practice. Moreover, the model supports oscillating-in-time spatial patterns, thus giving a theoretical explanation of the periodic appearance of inflammatory eruptions typical of recurrent erythema multiforme. A detailed numerical bifurcation analysis indicates that the inclusion of the logistic growth term is crucial for the occurrence of a sequence of bifurcations leading to spatio-temporal chaos. In the parameter space, there are large regions where the model system displays critical behavior.

Section 7 of the original paper contained several errors which are corrected here. Equations (54) and (55) are incorrect. In the following, the corrected versions of these equations are given and the subsequent results of Section 7 are also revised.

We consider the edge transport properties of a generic class of interacting quantum Hall systems on a cylinder, in the infinite volume and zero temperature limit. We prove that the large-scale behavior of the edge correlation functions is effectively described by the multi-channel Luttinger model. In particular, we prove that the edge conductance is universal, and equal to the sum of the chiralities of the non-interacting edge modes. The proof is based on rigorous renormalization group methods, that allow to fully take into account the effect of backscattering at the edge. Universality arises as a consequence of the integrability of the emergent multi-channel Luttinger liquid combined with lattice Ward identities for the microscopic $2d$ theory., Extended introduction, minor corrections. 77 pages

We provide an analytical approximation to the dynamics in each of the three most important low order secondary resonances (1:1, 2:1, and 3:1) bifurcating from the synchronous primary resonance in the gravitational spin-orbit problem. To this end we extend the perturbative approach introduced in Gkolias et. al. (2016), based on normal form series computations. This allows to recover analytically all non-trivial features of the phase space topology and bifurcations associated with these resonances. Applications include the characterization of spin states of irregular planetary satellites or double systems of minor bodies with irregular shapes. The key ingredients of our method are: i) the use of a detuning parameter measuring the distance from the exact resonance, and ii) an efficient scheme to `book-keep' the series terms, which allows to simultaneously treat all small parameters entering the problem. Explicit formulas are provided for each secondary resonance, yielding i) the time evolution of the spin state, ii) the form of phase portraits, iii) initial conditions and stability for periodic solutions, and iv) bifurcation diagrams associated with the periodic orbits. We give also error estimates of the method, based on analyzing the asymptotic behavior of the remainder of the normal form series., Comment: Accepted for publication in Communications in Nonlinear Science and Numerical Simulation

The behavior of a class of solutions of the shallow water Airy system originating from initial data with discontinuous derivatives is considered. Initial data are obtained by splicing together self-similar parabolae with a constant background state. These solutions are shown to develop velocity and surface gradient catastrophes in finite time and the inherent persistence of dry spots is shown to be terminated by the collapse of the parabolic core. All details of the evolution can be obtained in closed form until the collapse time, thanks to formation of simple waves that sandwich the evolving self-similar core. The continuation of solutions asymptotically for short times beyond the collapse is then investigated analytically, in its weak form, with an approach using stretched coordinates inspired by singular perturbation theory. This approach allows to follow the evolution after collapse by implementing a spectrally accurate numerical code, which is developed alongside a classical shock-capturing scheme for accuracy comparison. The codes are validated on special classes of initial data, in increasing order of complexity, to illustrate the evolution of the dry spot initial conditions on longer time scales past collapse., 28 pages 14 figures

We have recently shown that pseudo-bosonic operators realize concrete examples of finite dimensional nilpotent Lie algebras over the complex field. It has been the first time that such operators were analyzed in terms of nilpotent Lie algebras (under prescribed conditions of physical character). On the other hand, the general classification of a finite dimensional nilpotent Lie algebra $\mathfrak{l}$ may be given via the size of its Schur multiplier involving the so-called corank $t(\mathfrak{l})$ of $\mathfrak{l}$. We represent $\mathfrak{l}$ by pseudo-bosonic ladder operators for $t(\mathfrak{l}) \le 6$ and this allows us to represent $\mathfrak{l}$ when its dimension is $\le 5$., Comment: Accepted in Journal of Geometry and Physics

We consider the quantum hydrodynamic system on a $d$-dimensional irrational torus with $d=2,3$. We discuss the behaviour, over a "non trivial" time interval, of the $H^s$-Sobolev norms of solutions. More precisely we prove that, for generic irrational tori, the solutions, evolving from $\varepsilon$-small initial conditions, remain bounded in $H^s$ for a time scale of order $O(\varepsilon^{-1-1/(d-1)+})$, which is strictly larger with respect to the time-scale provided by local theory. We exploit a Madelung transformation to rewrite the system as a nonlinear Schr\"odinger equation. We therefore implement a Birkhoff normal form procedure involving small divisors arising from three waves interactions. The main difficulty is to control the loss of derivatives coming from the exchange of energy between high Fourier modes.This is due to the irrationality of the torus which prevent to have "good separation" properties of the eigenvalues of the linearized operator at zero. The main steps of the proof are: (i) to prove precise lower bounds on small divisors; (ii) to construct a modified energy by means of a suitable \emph{high/low} frequencies analysis, which gives an \emph{a priori} estimate on the solutions., Comment: 21 pag

We give explicit expressions for the finite frequency greybody factor, quasinormal modes and Love numbers of Kerr black holes by computing the exact connection coefficients of the radial and angular parts of the Teukolsky equation. This is obtained by solving the connection problem of the confluent Heun equation in terms of the explicit expression of irregular Virasoro conformal blocks as sums over partitions via the AGT correspondence. In the relevant approximation limits our results are in agreement with existing literature. The method we use can be extended to solve the linearized Einstein equation in other interesting gravitational backgrounds., 37 pages, 7 figures, comments welcome

In this paper, we consider a member of an integrable family of generalized Camassa–Holm (GCH) equations. We make an analysis of the point Lie symmetries of these equations by using the Lie method of infinitesimals. We derive nonclassical symmetries and we find new symmetries via the nonclassical method, which cannot be obtained by Lie symmetry method. We employ the multiplier method to construct conservation laws for this family of GCH equations. Using the conservation laws of the underlying equation, double reduction is also constructed. Finally, we investigate traveling waves of the GCH equations. We derive convergent series solutions both for the homoclinic and heteroclinic orbits of the traveling-wave equations, which correspond to pulse and front solutions of the original GCH equations, respectively.

We analyze the top Lyapunov exponent of the product of sequences of two by two matrices that appears in the analysis of several statistical mechanics models with disorder: for example these matrices are the transfer matrices for the nearest neighbor Ising chain with random external field, and the free energy density of this Ising chain is the Lyapunov exponent we consider. We obtain the sharp behavior of this exponent in the large interaction limit when the external field is centered: this balanced case turns out to be critical in many respects. From a mathematical standpoint we precisely identify the behavior of the top Lyapunov exponent of a product of two dimensional random matrices close to a diagonal random matrix for which top and bottom Lyapunov exponents coincide. In particular, the Lyapunov exponent is only $\log$-H\"older continuous., Comment: 29 pages, 2 figures. Introduction is restructured, a few misprints corrected. Accepted for publication on Alea

Today we know that neutrinos are massive, but we ignore both the origin of their masses and the reason why those masses are so tiny. A natural mechanism that can explain the smallness of neutrino masses is the See-saw Mechanism, that in this work is studied in its Type III variant. However, even in the scope of the See-saw Mechanism, the origin of the huge neutrino Majorana mass (which is a fundamental prerequisite for any see-saw model) remains to be explained. A possible explanation can be found in Twisted Non-Commutative Geometry (NCG): for particular twists of the so-called Connes Model (which is the NCG formulation of the Standard Model) a new scalar field appears naturally, whose vacuum expectation value generates a Majorana mass for the neutrinos, and the order of magnitude of said v.e.v. is precisely the natural scale of the Majorana mass of See-saw Mechanisms (10^15 GeV). In this work, three different twists of the Connes Model are studied, paying particular attention to the new boson field content, their gauge transformations, as well as the fermionic actions of each model.

In this paper, we describe our study of the mixed convection of a Boussinesquian Bingham fluid in a vertical channel in the absence and presence of an external uniform magnetic field normal to the walls. The velocity, the induced magnetic field, and the temperature are analytically obtained. A detailed analysis is conducted to determine the plug regions in relation to the values of the Bingham number, the buoyancy parameter, and the Hartmann number. In particular, the velocity decreases as the Bingham number increases. Detailed considerations are drawn for the occurrence of the reverse flow phenomenon. Moreover, a selected set of diagrams illustrating the influence of various parameters involved in the problem is presented and discussed.

La modellizzazione matematica è diventata ormai uno strumento essenziale nell’avanzamento delle conoscenze in qualsiasi campo della scienza, dall’econometria alla psicologia, dalla chimica all’epidemiologia, come vissuto di recente nella pandemia da SARS-coV-2. Anche quelle che erano una volta discipline “soft” come la biomedicina o l’ecologia possono ora usufruire di tecniche di rappresentazione ed esplicazione dei processi estremamente utili alla comprensione delle dinamiche in gioco. Questa evoluzione, dovuta in non piccola parte all’attuale disponibilità di abbondante potenza di calcolo elettronico a costi ridotti, non è però opera ed appannaggio soltanto della nostra generazione. Lo stesso Consiglio nazionale delle ricerche (CNR) fondato nel 1923 ha avuto come primo presidente il Prof. Vito Volterra, uno dei fondatori della biomatematica, il cui nome è storicamente legato alle equazioni di dinamica di popolazione da lui introdotte insieme a Lotka. Nel solco di tale tradizione oggi la biomatematica ha un ruolo primario nello studio delle dinamiche di inquinamento, diffusione di sostanze tossiche, bioaccumulo nelle specie di rilevanza alimentare, tossicocinetica, tossicodinamica ed epidemiologia delle malattie legate all’inquinamento. In tutte queste discipline viene affidato alla biomatematica il ruolo di svolgere o supportare interpretazioni quantitative dei dati e facilitare l’inquadramento delle esperienze specifiche degli specialisti di campo (esperti ambientali, biologi, epidemiologi e clinici), in una cornice protesa a collegare dati disparati in una logica di insieme coerente. Si tratta di un compito non facile, che richiede non solo lo sforzo di molti ricercatori con competenze complementari (biologi, medici, epidemiologi, analisti numerici, sistemisti, statistici), ma che deve anche fare i conti con le limitazioni dovute alla mancata disponibilità di dati talvolta cruciali, un fenomeno purtroppo frequente nelle indagini su sistemi complessi. Le pagine che seguono si propongono di illustrare, in modo comprensibile anche per non specialisti, alcuni di questi temi. Il capitolo comprende una panoramica su cosa sono i modelli statici e dinamici, ed in particolare quelli sviluppati in ambito “ambiente e salute”. Inoltre viene offerta al lettore una scheda riassuntiva del sistema formalizzato BioMatLab (MoSpec) dedicato all’implementazione dei modelli matematici in diversi linguaggi di programmazione, corredato da un link alla piattaforma online nella quale sono rese disponibili alla comunità scientifica alcuni modelli prodotti/implementati per lo studio delle dinamiche di interazione ambiente e salute sviluppate durante le attività del progetto CISAS (capitolo 14).

The brain’s structural connectivity plays a fundamental role in determining how neuron networks generate, process, and transfer information within and between brain regions. The underlying mechanisms are extremely difficult to study experimentally and, in many cases, large-scale model networks are of great help. However, the implementation of these models relies on experimental findings that are often sparse and limited. Their predicting power ultimately depends on how closely a model’s connectivity represents the real system. Here we argue that the data-driven probabilistic rules, widely used to build neuronal network models, may not be appropriate to represent the dynamics of the corresponding biological system. To solve this problem, we propose to use a new mathematical framework able to use sparse and limited experimental data to quantitatively reproduce the structural connectivity of biological brain networks at cellular level.

It is known that self-adjoint Hamiltonians with purely discrete eigenvalues can be written as (infinite) linear combination of mutually orthogonal projectors with eigenvalues as coefficients of the expansion. The projectors are defined by the eigenvectors of the Hamiltonians. In some recent papers, this expansion has been extended to the case in which these eigenvectors form a Riesz basis or, more recently, a ${\mathcal{D}}$ D -quasi basis (Bagarello and Bellomonte in J. Phys. A 50:145203, 2017, Bagarello et al. in J. Math. Phys. 59:033506, 2018), rather than an orthonormal basis. Here we discuss what can be done when these sets are replaced by Parseval frames. This interest is motivated by physical reasons, and in particular by the fact that the mathematical Hilbert space where the physical system is originally defined, contains sometimes also states which cannot really be occupied by the physical system itself. In particular, we show what changes in the spectrum of the observables, when going from orthonormal bases to Parseval frames. In this perspective we propose the notion of $E$ E -connection for observables. Several examples are discussed.

Brain-predicted age difference scores are calculated by subtracting chronological age from ‘brain’ age. Positive scores reflect accelerated ageing and are associated with increased mortality risk and poorer physical function. To date, however, the relationship between brain-predicted age difference scores and specific cognitive functions has not been systematically examined. First, applying machine learning to 1,359 T1-weighted MRI scans, we predicted the relationship between chronological age and voxel-wise grey matter data. This model was then applied to MRI data from three independent datasets, significantly predicting chronological age: Dokuz Eylul University (n=175), the Cognitive Reserve/Reference Ability Neural Network study (n=380), and The Irish Longitudinal Study on Ageing (n=487). Each independent dataset had rich neuropsychological data. Brain-predicted age difference scores were significantly negatively correlated with general cognitive status (two datasets); processing speed, visual attention, cognitive flexibility (three datasets); visual attention and cognitive flexibility (two datasets); and semantic verbal fluency (two datasets). They were not significantly correlated with processing speed, cognitive flexibility, response inhibition and selective attention, sustained attention, verbal episodic memory or working memory in any dataset. As such, there is firm evidence of correlations between increased brain-predicted age differences and reduced cognitive function only in some domains that are implicated in cognitive ageing.

The general relativistic Poynting-Robertson effect is a dissipative and non-linear dynamical system obtained by perturbing through radiation processes the geodesic motion of test particles orbiting around a spinning compact object, described by the Kerr metric. Using the Melnikov method we find that, in a suitable range of parameters, chaotic behavior is present in the motion of a test particle driven by the Poynting-Robertson effect in the Kerr equatorial plane., Comment: 10 pages, 8 figures; published on Phys. Rev. D

We study the Boussinesq hierarchy in the geometric context of the theory of bi-Hamiltonian manifolds. First, we recall how its bi-Hamiltonian structure can be obtained by means of a process called bi-Hamiltonian reduction, choosing a specific symplectic leaf S of one of the two Poisson structures. Then, we introduce the notion of a bi-Hamiltonian S-hierarchy, that is, a bi-Hamiltonian hierarchy that is defined only at the points of the symplectic leaf S, and we show that the Boussinesq hierarchy can be interpreted as the reduction of a bi-Hamiltonian S-hierarchy.

We give an explicit arithmetical condition which guarantees the existence of the unstable manifold of the MacKay approximate renormalisation scheme for the breakup of invariant tori in one and a half degrees of freedom Hamiltonian systems, correcting earlier results. Furthermore, when our condition is violated, we give an example of points on which the unstable manifold does not converge. (c) 2022 Elsevier B.V. All rights reserved.

The reliability of deep learning algorithms is fundamentally challenged by the existence of adversarial examples, which are incorrectly classified inputs that are extremely close to a correctly classified input. We explore the properties of adversarial examples for deep neural networks with random weights and biases, and prove that for any $p\ge1$, the $\ell^p$ distance of any given input from the classification boundary scales as one over the square root of the dimension of the input times the $\ell^p$ norm of the input. The results are based on the recently proved equivalence between Gaussian processes and deep neural networks in the limit of infinite width of the hidden layers, and are validated with experiments on both random deep neural networks and deep neural networks trained on the MNIST and CIFAR10 datasets. The results constitute a fundamental advance in the theoretical understanding of adversarial examples, and open the way to a thorough theoretical characterization of the relation between network architecture and robustness to adversarial perturbations.

We consider the minimal average action (Mather's β function) for area preserving twist maps of the annulus. The regularity properties of this function share interesting relations with the dynamics of the system. We prove that the β-function associated to a standard-like twist map admits a unique C 1 -holomorphic (canonical) complex extension, which coincides with this function on the set of real diophantine frequencies. In particular, we deduce a uniqueness result for Mather's β function.

The aim of the paper is to propose a mathematical model for the flow of the interstitial fluid in a lymph node, which main feature is the presence of a porous bulk region with a very low permeability, surrounded by a thin channel where the fluid can flow freely. The flow is driven by a pulsatile pressure gradient. An explicit solution is found in the case of a laminar flow in a simplified situation, while some finite element simulations are presented in a more realistic geometry.

We analyze the relevance of the generalized Kronheimer construction for the gauge-gravity correspondence. We study the general structure of IIB supergravity D3-brane solutions on crepant resolutions $Y$ of singularities $\mathbb{C}^3/\Gamma$ with $\Gamma$ a finite subgroup of $SU(3)$. Next we concentrate on another essential item for the D3-brane construction, i.e., the existence of a Ricci-flat metric on $Y$, with particular attention to the case $\Gamma=\mathbb{Z}_4$. We conjecture that on the exceptional divisor the Kronheimer K\"ahler metric and the Ricci-flat one, that is locally flat at infinity, coincide. The conjecture is shown to be true in the case of the Ricci-flat metric on ${\rm tot} K_{{\mathbb WP}[112]}$ that we construct, which is a partial resolution of $\mathbb{C}^3/\mathbb{Z}_4$. For the full resolution we have $Y=\operatorname{tot} K_{\mathbb{F}_{2}}$, where $\mathbb{F}_2$ is the second Hizebruch surface. We try to extend the proof of the conjecture to this case using the one-parameter K\"ahler metric on $\mathbb{F}_2$ produced by the Kronheimer construction as initial datum in a Monge-Amp\`{e}re (MA) equation. We exhibit three formulations of this MA equation, one in terms of the K\"ahler potential, the other two in terms of the symplectic potential; in all cases one can establish a series solution in powers of the fiber variable of the canonical bundle. The main property of the MA equation is that it does not impose any condition on the initial geometry of the exceptional divisor, but uniquely determines all the subsequent terms as local functionals of the initial datum. While a formal proof is still missing, numerical and analytical results support the conjecture. As a by-product of our investigation we have identified some new properties of this type of MA equations that we believe to be so far unknown., Comment: 63 pages, 8 figures. To appear in the Lett. Math. Phys. special issue in memory of Boris A. Dubrovin

This book includes the joint proceedings of the International Conference on Particle Systems and PDEs VI, VII and VIII. Particle Systems and PDEs VI was held in Nice, France, in November/December 2017, Particle Systems and PDEs VII was held in Palermo, Italy, in November 2018, and Particle Systems and PDEs VIII was held in Lisbon, Portugal, in December 2019. Most of the papers are dealing with mathematical problems motivated by different applications in physics, engineering, economics, chemistry and biology. They illustrate methods and topics in the study of particle systems and PDEs and their relation. The book is recommended to probabilists, analysts and to those mathematicians in general, whose work focuses on topics in mathematical physics, stochastic processes and differential equations, as well as to those physicists who work in statistical mechanics and kinetic theory.

It has been proved that the general relativistic Poynting-Robertson effect in the equatorial plane of Kerr metric shows a chaotic behavior for a suitable range of parameters. As a further step, we calculate the timescale for the onset of chaos through the Lyapunov exponents, estimating how this trend impacts on the observational dynamics. We conclude our analyses with a discussion on the possibility to observe this phenomenon in neutron star and black hole astrophysical sources., Comment: 11 pages; 4 figures; 3 tables; accepted for publication on PRD