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We take up the investigation we had to put in the future-work stack at the end of Sec.VB2 of Ref.2, in which we pointed out the obvious necessity to inquire about existence or absence of values of the characteristic numbers $\alpha$ and $\beta$ in correspondence to which the perfect-gas model's self gravitational effects, namely upper boundedness of the gravitational number, spiraling behavior of peripheral density, oscillating behavior of central density, and existence of multiple solutions corresponding to the same value of the gravitational number, appear also for the van der Waals' model. The development of our investigation brings to the conversion of our M$_{2}$ scheme based on a second-order differential equation into an equivalent system of two first-order differential equations that incorporates Milne's homology invariant variables. The converted scheme 1oM$_2$ turns out to be much more efficacious than the M$_{2}$ scheme in terms of numerical calculations' easiness and richness of results. We use the perfect-gas model as benchmark to test the 1oM$_2$ scheme; we re-derive familiar results and put them in a more general and rational perspective that paves the way to deal with the van der Waals' gas model. We introduce variable transformations that turn out to be the key to study (almost) analytically the monotonicity of the peripheral density with respect to variations of the gravitational number. The study brings to the proof that the gravitational number is not constrained by upper boundedness, the peripheral density does not spiral, and the central density does not oscillate for any couple of values assumed by the characteristic numbers $\alpha$ and $\beta$; however, multiple solutions corresponding to the same value of the gravitational number can exist but their genesis is completely different from that of the perfect-gas model's multiple solutions. We provide the boundary in the $\alpha,\beta$ plane between the two regions of solution's uniqueness and multiplicity. Finally, by the application of the 1oM$_2$ scheme, we resolve the mystery of the missing steep curve corresponding to Aronson and Hansen's 60km case [29] that we unsuccessfully chased in Ref.2 and we even detect another solution that had somehow escaped the attention of those authors., Comment: 21 pages, 26 figures, 1 table

In this paper we describe the efficient numerical implementation of Fractional HBVMs, a class of methods recently introduced for solving systems of fractional differential equations. The reported arguments are implemented in the Matlab code fhbvm, which is made available on the web. An extensive experimentation of the code is reported, to give evidence of its effectiveness., Comment: 26 pages, 6 figures

In this paper we consider the numerical solution of fractional terminal value problems (FDE-TVPs). In particular, the proposed procedure uses a Newton-type iteration which is particularly efficient when coupled with a recently-introduced step-by-step procedure for solving fractional initial value problems (FDE-IVPs), able to produce spectrally accurate solutions of FDE problems. Some numerical tests are reported to make evidence of its effectiveness., Comment: 23 pages, 4 figures, 7 tables, one typo fixed

We subject to scrutiny the physical consistency of adopting the perfect-gas thermodynamic model within self-gravitation circumstances by studying the fluid statics of a self-gravitating isothermal sphere with the van der Waals' thermodynamic model, whose equation of state features well-known terms that account for molecular attraction and size. The governing equations are formulated for any thermodynamic model with two intensive degrees of freedom, applied with the van der Waals' model and solved numerically in nondimensional form by finite-difference algorithms. After a brief summary of thermodynamic characteristics possessed by the van der Waals' model, and relevant to the present study, we proceed to the description of the results in terms of comparative graphs illustrating radial profiles of density, pressure and gravitational field. We complement them with graphs that compare the dependence of central and wall densities on gravitational number for both perfect-gas and van der Waals' models and that attest dramatically and unequivocally how the presence of molecular-attraction and -size terms removes questionable fluid-statics results systematically found accompanying the perfect-gas model in standard treatments. We also describe, within a very brief and preliminary digression, how the sanitising action of the mentioned terms affects the thermodynamics of the isothermal sphere by providing evidence of how the gravitational correction to entropy corresponding to the van der Waals' model makes sure that there is no risk of gravothermal catastrophes, negative specific heats, and thermal instabilities. Furthermore, we investigate the phenomenology related to self-gravitationally induced both liquid-gas phase equilibria and metastable-gas states and we describe how they arise naturally and self-consistently from the governing equations. We conclude with a summary of the main results ..., Comment: Minor revision with correcte typos. 27 pages, 29 figures, 5 tables, 64 bibliographic references

We consider the weighted least squares spline approximation of a noisy dataset. By interpreting the weights as a probability distribution, we maximize the associated entropy subject to the constraint that the mean squared error is prescribed to a desired (small) value. Acting on this error yields a robust regression method that automatically detects and removes outliers from the data during the fitting procedure, by assigning them a very small weight. We discuss the use of both spline functions and spline curves. A number of numerical illustrations have been included to disclose the potentialities of the maximal-entropy approach in different application fields., Comment: 11 color pictures

We address the question of the infinitude of twin and cousin prime pairs from a probabilistic perspective. Our approach partitions the set of integer numbers greater than $2$ in finite intervals of the form $[p_{n-1}^2,p_n^2)$, $p_{n-1}$ and $p_n$ being two consecutive primes, and evaluates the probability $q_n$ that such an interval contains a twin prime and a cousin prime. Combining Merten's third theorem with the properties of the binomial distribution, we show that $q_n$ approaches $1$ as $n \to \infty$. A study of the convergence properties of the sequence $\{q_n\}$ allows us to propose a new, more stringent conjecture concerning the existence of infinitely many twin and cousin primes. In accord with the Hardy-Littlewood conjecture, it is also shown that twin and cousin primes share the same asymptotic distribution., Comment: 20 pages, 4 figures

We address two questions regarding square-well potentials from a didactic perspective. The first question concerns whether or not the justification of the standard a priori omission of the potential's vertical segments in the analysis of the eigenvalue problem is licit. The detour we follow to find out the answer considers a trapezoidal potential, includes the solution, analytical and numerical, of the corresponding eigenvalue problem and then analyzes the behavior of that solution in the limit when the slope of the trapezoidal potential's ramps becomes vertical. The second question, obviously linked to the first one, pertains whether or not eigenfunction's and its first derivative's continuity at the potential's jump points is justified as a priori assumption to kick-off the solution process, as it is standardly accepted in textbook approaches to the potential's eigenvalue problem., Comment: 45 pages, 14 figures, 3 tables

Recently, the efficient numerical solution of Hamiltonian problems has been tackled by defining the class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Their derivation relies on the expansion of the vector field along the Legendre orthonormal basis. Interestingly, this approach can be extended to cope with other orthonormal bases and, in particular, we here consider the case of the Chebyshev polynomial basis. The corresponding Runge-Kutta methods were previously obtained by Costabile and Napoli [33]. In this paper, the use of a different framework allows us to carry out a novel analysis of the methods also when they are used as spectral formulae in time, along with some generalizations of the methods., Comment: 25 pages, 2 figures, 2 tables

In recent years, the efficient numerical solution of Hamiltonian problems has led to the definition of a class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Such methods admit an interesting interpretation in terms of continuous-stage Runge-Kutta methods, which is here recalled and revisited for general differential problems., Comment: 20 pages

In this paper we are concerned with numerical methods for the one-sided event location in discontinuous differential problems, whose event function is nonlinear (in particular, of polynomial type). The original problem is transformed into an equivalent Poisson problem, which is effectively solved by suitably adapting a recently devised class of energy-conserving methods for Poisson systems. The actual implementation of the methods is fully discussed, with a particular emphasis to the problem at hand. Some numerical tests are reported, to assess the theoretical findings., Comment: 14 pages, 1 figure, 2 tables

In this paper we are concerned with energy-conserving methods for Poisson problems, which are effectively solved by defining a suitable generalization of HBVMs, a class of energy-conserving methods for Hamiltonian problems. The actual implementation of the methods is fully discussed, with a particular emphasis on the conservation of Casimirs. Some numerical tests are reported, in order to assess the theoretical findings., Comment: 30 pages, 5 figures, 3 tables

In this paper we discuss a framework for the polynomial approximation to the solution of initial value problems for differential equations. The framework, initially devised for the approximation of ordinary differential equations, is further extended to cope with constant delay differential equations. Relevant classes of Runge-Kutta methods can be derived within this framework., Comment: 35 pages, 4 figures, 2 tables

We exploit the idea to use the maximal-entropy method, successfully tested in information theory and statistical thermodynamics, to determine approximating function's coefficients and squared errors' weights simultaneously as output of one single problem in least-square approximation. We provide evidence of the method's capabilities and performance through its application to representative test cases by working with polynomials as a first step. We conclude by formulating suggestions for future work to improve the version of the method we present in this paper., Comment: 15 pages, 1 table, 10 figures. Accepted in Mathematical Methods in the Applied Sciences. Changes: fixed typo in Eq. (27)

Background. The paper concerns the SARS-CoV2 (COVID-19) pandemic that, starting from the end of February 2020, began spreading along the Italian peninsula, by first attacking small communities in north regions, and then extending to the center and south of Italy, including the two main islands. Objective. The creation of a forecast model that manages to alert the decision-making bodies and, in particular, the healthcare system, to hinder the emergence of any other pandemic outbreaks, or the arrival of subsequent pandemic waves. Methods. A new mathematical model to describe the pandemic is given. The model includes the class of undiagnosed infected people, and has a multi-region extension, to cope with the in-time and in-space heterogeneity of the epidemic. Results. We obtain a robust and reliable tool for the forecast of the total and active cases, which can be also used to simulate different scenarios. Conclusions. We are able to address a number of issues, such as assessing the adoption of the lockdown in Italy, started from 11 March 2020, and how to employ a rapid screening test campaign for containing the epidemic., Comment: 21 pages, 10 figures

In this note, we describe simple generalizations of the basic SIR model for epidemic, in case of a multi-region scenario, to be used for predicting the COVID-19 epidemic spread in Italy., Comment: 7 pages (1 typo fixed)

We devise a variable precision floating-point arithmetic by exploiting the framework provided by the Infinity Computer. This is a computational platform implementing the Infinity Arithmetic system, a positional numeral system which can handle both infinite and infinitesimal quantities symbolized by the positive and negative finite powers of the radix grossone. The computational features offered by the Infinity Computer allows us to dynamically change the accuracy of representation and floating-point operations during the flow of a computation. When suitably implemented, this possibility turns out to be particularly advantageous when solving ill-conditioned problems. In fact, compared with a standard multi-precision arithmetic, here the accuracy is improved only when needed, thus not affecting that much the overall computational effort. An illustrative example about the solution of a nonlinear equation is also presented., Comment: 11 pages, 2 figures, 6 tables

Gyrocenter dynamics of charged particles plays a fundamental role in plasma physics. In particular, accuracy and conservation of energy are important features for correctly performing long-time simulations. For this purpose, we here propose arbitrarily high-order energy conserving methods for its simulation. The analysis and the efficient implementation of the methods are fully described, and some numerical tests are reported., Comment: 23 pages, 4 figures

In the present paper we analyze and discuss some mathematical aspects of the fluid-static configurations of a self-gravitating perfect gas enclosed in a spherical solid shell. The mathematical model we consider is based on the well-known Lane-Emden equation, albeit under boundary conditions that differ from those usually assumed in the astrophysical literature. The existence of multiple solutions requires particular attention in devising appropriate numerical schemes apt to deal with and catch the solution multiplicity as efficiently and accurately as possible. In sequence, we describe some analytical properties of the model, the two algorithms used to obtain numerical solutions, and the numerical results for two selected cases., Comment: Revision accepted for publication in "Communications in Computational Physics"; 23 pages, 9 figures

In recent years, the class of energy-conserving methods named Hamiltonian Boundary Value Methods (HBVMs) has been devised for numerically solving Hamiltonian problems. In this short note, we study their natural formulation as continuous-stage Runge-Kutta(-Nystr\"om) methods, which allows a deeper insight in the methods., Comment: 12 pages, accepted manuscript

In this paper, we report about recent findings in the numerical solution of Hamiltonian Partial Differential Equations (PDEs), by using energy-conserving line integral methods in the Hamiltonian Boundary Value Methods (HBVMs) class. In particular, we consider the semilinear wave equation, the nonlinear Schr\"odinger equation, and the Korteweg-de Vries equation, to illustrate the main features of this novel approach., Comment: 33 pages, 3 figures, 3 tables

We open the paper with introductory considerations describing the motivations of our long-term research plan targeting gravitomagnetism, illustrating the fluid-dynamics numerical test case selected for that purpose, that is, a perfect-gas sphere contained in a solid shell located in empty space sufficiently away from other masses, and defining the main objective of this study: the determination of the gravitofluid-static field required as initial field ($t=0$) in forthcoming fluid-dynamics calculations. The determination of the gravitofluid-static field requires the solution of the isothermal-sphere Lane-Emden equation. We do not follow the habitual approach of the literature based on the prescription of the central density as boundary condition; we impose the gravitational field at the solid-shell internal wall. As the discourse develops, we point out differences and similarities between the literature's and our approach. We show that the nondimensional formulation of the problem hinges on a unique physical characteristic number that we call gravitational number because it gauges the self-gravity effects on the gas' fluid statics. We illustrate and discuss numerical results; some peculiarities, such as gravitational-number upper bound and multiple solutions, lead us to investigate the thermodynamics of the physical system, particularly entropy and energy, and preliminarily explore whether or not thermodynamic-stability reasons could provide justification for either selection or exclusion of multiple solutions. We close the paper with a summary of the present study in which we draw conclusions and describe future work., Comment: 32 pages, 26 figures

Recently, the numerical solution of stiffly/highly-oscillatory Hamiltonian problems has been attacked by using Hamiltonian Boundary Value Methods (HBVMs) as spectral methods in time. While a theoretical analysis of this spectral approach has been only partially addressed, there is enough numerical evidence that it turns out to be very effective even when applied to a wider range of problems. Here we fill this gap by providing a thorough convergence analysis of the methods and confirm the theoretical results with the aid of a few numerical tests., Comment: 18 pages, 1 figure

Recently, the numerical solution of multi-frequency, highly-oscillatory Hamiltonian problems has been attacked by using Hamiltonian Boundary Value Methods (HBVMs) as spectral methods in time. When the problem derives from the space semi- discretization of (possibly Hamiltonian) partial differential equations (PDEs), the resulting problem may be stiffly-oscillatory, rather than highly-oscillatory. In such a case, a different implementation of the methods is needed, in order to gain the maximum efficiency., Comment: 17 pages, 3 figures

In this paper we are concerned with the analysis of a class of geometric integrators, at first devised in [14, 18], which can be regarded as an energy-conserving variant of Gauss collocation methods. With these latter they share the property of conserving quadratic first integrals but, in addition, they also conserve the Hamiltonian function itself. We here reformulate the methods in a more convenient way, and propose a more refined analysis than that given in [18] also providing, as a by-product, a practical procedure for their implementation. A thorough comparison with the original Gauss methods is carried out by means of a few numerical tests solving Hamiltonian and Poisson problems., Comment: 28 pages, 2 figures, 4 tables

In this paper, we propose a second-order energy-conserving approximation procedure for Hamiltonian systems with holonomic constraints. The derivation of the procedure relies on the use of the so-called line integral framework. We provide numerical experiments to illustrate theoretical findings., Comment: 30 pages, 3 figures, 4 tables

In this paper we discuss energy conservation issues related to the numerical solution of the nonlinear wave equation, when a Fourier expansion is considered for the space discretization. The obtained semi-discrete problem is then solved in time by means of energy-conserving Runge-Kutta methods in the HBVMs class., Comment: 25 pages, 7 figure

In this paper we discuss energy conservation issues related to the numerical solution of the nonlinear wave equation. As is well known, this problem can be cast as a Hamiltonian system that may be autonomous or not, depending on the specific boundary conditions at hand. We relate the conservation properties of the original problem to those of its semi-discrete version obtained by the method of lines. Subsequently, we show that the very same properties can be transferred to the solutions of the fully discretized problem, obtained by using energy-conserving methods in the HBVMs (Hamiltonian Boundary Value Methods) class. Similar arguments hold true for different types of Hamiltonian Partial Differential Equations, e.g., the nonlinear Schr\"odinger equation., Comment: 41 pages, 11 figure

We here investigate the efficient implementation of the energy-conserving methods named Hamiltonian Boundary Value Methods (HBVMs) recently introduced for the numerical solution of Hamiltonian problems. In this note, we describe an iterative procedure, based on a triangular splitting, for solving the generated discrete problems, when the problem at hand is separable., Comment: 4 pages

In this paper we define an efficient implementation for the family of low-rank energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs), recently defined in the last years. The proposed implementation relies on the particular structure of the Butcher matrix defining such methods, for which we can derive an efficient splitting procedure. The very same procedure turns out to be automatically suited for the efficient implementation of Gauss-Legendre collocation methods, since these methods are a special instance of HBVMs. The linear convergence analysis of the splitting procedure exhibits excellent properties, which are confirmed by a few numerical tests., Comment: 19 pages, 7 tables, 2 figures

These are the lecture notes of a course given by the first author on December 27, 2012 - January 4, 2013, held at the Academy of Mathematics and Systems Science Chinese Academy of Sciences in Beijing., Comment: 68 pages

We introduce a family of fourth order two-step methods that preserve the energy function of canonical polynomial Hamiltonian systems. Each method in the family may be viewed as a correction of a linear two-step method, where the correction term is O(h^5) (h is the stepsize of integration). The key tools the new methods are based upon are the line integral associated with a conservative vector field (such as the one defined by a Hamiltonian dynamical system) and its discretization obtained by the aid of a quadrature formula. Energy conservation is equivalent to the requirement that the quadrature is exact, which turns out to be always the case in the event that the Hamiltonian function is a polynomial and the degree of precision of the quadrature formula is high enough. The non-polynomial case is also discussed and a number of test problems are finally presented in order to compare the behavior of the new methods to the theoretical results., Comment: 14 pages, 4 figures, 2 tables

We discuss the efficient implementation of Hamiltonian BVMs (HBVMs), a recently introduced class of energy preserving methods for canonical Hamiltonian systems, via their blended formulation. We also discuss the case of separable problems, for which the structure of the problem can be exploited to gain efficiency., Comment: 10 pages, 4 figures

When numerically integrating canonical Hamiltonian systems, the long-term conservation of some of its invariants, among which the Hamiltonian function itself, assumes a central role. The classical approach to this problem has led to the definition of symplectic methods, among which we mention Gauss-Legendre collocation formulae. Indeed, in the continuous setting, energy conservation is derived from symplecticity via an infinite number of infinitesimal contact transformations. However, this infinite process cannot be directly transferred to the discrete setting. By following a different approach, in this paper we describe a sequence of methods, sharing the same essential spectrum (and, then, the same essential properties), which are energy preserving starting from a certain element of the sequence on, i.e., after a finite number of steps., Comment: 15 pages

In this paper, we provide a simple framework to derive and analyse several classes of effective one-step methods. The framework consists in the discretization of a local Fourier expansion of the continuous problem. Different choices of the basis lead to different classes of methods, even though we shall here consider only the case of an orthonormal polynomial basis, from which a large subclass of Runge-Kutta methods is derived. The obtained results are then applied to prove, in a simplified way, the order and stability properties of Hamiltonian BVMs (HBVMs), a recently introduced class of energy preserving methods for canonical Hamiltonian systems. A few numerical tests with such methods are also included, in order to confirm the effectiveness of the methods., Comment: 11 pages, 2 figures (proofs of Thms. 2.2 and 3.1 simplified)