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Recently, some of the authors of the present note provided a generalization of Bumby's theorem to the injectivity of modules over some classes of monomorphisms. The question of whether Bumby's criterion could be extended to more general classes of categories was left without an affirmative response, though. However, the present note provides an extension of that theorem to Grothendieck categories. More concretely, we establish that any two injective objects in a Grothendick category are isomorphic when they are isomorphic to subobjects of each other. As a corollary, we prove that two objects have isomorphic injective envelops (if they exist) whenever each of them is isomorphic to a subobject of the other. In the way, it will be indispensable to note that the main results of our previous work are still valid under weaker conditions.Keywords: Grothendiek categories, Bumby's theorem, isomorphism criterion, injective objects, injective envelops

A class of one-dimensional time-fractional parabolic differential equations with delay effects of functional type in the time component is numerically investigated in this work. To that end, a compact difference scheme is constructed for the numerical solution of those equations based on the idea of separating the current state and the prehistory function. In these terms, the prehistory function is approximated by means of an appropriate interpolation–extrapolation operator. A discrete form of the fractional Gronwall inequality is employed to provide an optimal error estimate. The existence and uniqueness of the numerical solutions, the order of approximation error for the constructed scheme, the stability and the order of convergence are mathematically investigated in this work. © 2019 Wiley Periodicals, Inc. Russian Foundation for Basic Research, RFBR: 19‐01‐00019 A1‐S‐45928 The authors want to thank the associate editor in charge of handling this manuscript and anonymous reviewers for all their comments and criticisms. Their suggestions contributed decisively to improve the overall quality of this work. The first two authors wish to acknowledge the support of RFBR Grant 19‐01‐00019. Meanwhile, the last author wishes to acknowledge the financial support of the National Council for Science and Technology of Mexico through grant A1‐S‐45928.

A discrete monotone iterative method is reported here to solve a space-fractional nonlinear diffusion–reaction equation. More precisely, we propose a Crank–Nicolson discretization of a reaction–diffusion system with fractional spatial derivative of the Riesz type. The finite-difference scheme is based on the use of fractional-order centered differences, and it is solved using a monotone iterative technique. The existence and uniqueness of solutions of the numerical model are analyzed using this approach, along with the technique of upper and lower solutions. This methodology is employed also to prove the main numerical properties of the technique, namely, the consistency, stability, and convergence. As an application, the particular case of the space-fractional Fisher’s equation is theoretically analyzed in full detail. In that case, the monotone iterative method guarantees the preservation of the positivity and the boundedness of the numerical approximations. Various numerical examples are provided to illustrate the validity of the numerical approximations. More precisely, we provide an extensive series of comparisons against other numerical methods available in the literature, we show detailed numerical analyses of convergence in time and in space against fractional and integer-order models, and we provide studies on the robustness and the numerical performance of the discrete monotone method. © 2019, The Author(s). Russian Foundation for Basic Research, RFBR: 19-01-00019 Consejo Nacional de Ciencia y Tecnología, CONACYT: A1-S-45928 The first author would like to acknowledge the financial support of the National Council for Science and Technology of Mexico (CONACYT). The second (and corresponding) author acknowledges financial support from CONACYT through grant A1-S-45928. ASH is financed by RFBR Grant 19-01-00019.

Departing from a two-dimensional hyperbolic system that describes the interaction between some activator and inhibitor substances in chemical reactions, we investigate a general form of that model using a finite-difference approach. The model under investigation is a nonlinear system consisting of two coupled partial differential equations with generalized reaction terms. The presence of two-dimensional diffusive terms consisting of fractional operators of the Riesz type is considered here, using spatial differentiation orders in the set (1, 2] . We impose initial conditions on a closed and bounded rectangle, and a finite-difference methodology based on the use of weighted-shifted Grunwald differences is proposed. Among the most important results of this work, we establish analytically the second-order consistency of our scheme. Moreover, the discrete energy method is employed to prove the stability and the quadratic convergence of the technique. Some numerical simulations obtained through our method show the appearance of Turing patterns and wave instabilities, in agreement with some reports found in the literature on superdiffusive hyperbolic activator-inhibitor systems.

Departing from a general stochastic differential equation with Brownian diffusion, we establish that the distribution of probability of the stopping time is governed by a parabolic partial differential equation. A particular form of the problem under investigation may be associated to a stochastic generalization of the well-known Paris’ law from structural mechanics, in which case, the solution of the boundary-value problem represents the probability distribution of the hitting time. An implicit, convergent and probability-based discretization to approximate the solution of the boundary-value problem is proposed in this work. Using a convenient vector representation of our scheme, we prove that the method preserves the most relevant properties of a probability distribution function, namely, the non-negativity, the boundedness from above by 1, and the monotonicity. In addition, we establish that our method is a convergent technique, and provide some illustrative comparisons against known exact solu...

We depart from the well-known one-dimensional Fisher’s equation from population dynamics and consider an extension of this model using Riesz fractional derivatives in space. Positive and bounded initial-boundary data are imposed on a closed and bounded domain, and a fully discrete form of this fractional initial-boundary-value problem is provided next using fractional centered differences. The fully discrete population model is implicit and linear, so a convenient vector representation is readily derived. Under suitable conditions, the matrix representing the implicit problem is an inverse-positive matrix. Using this fact, we establish that the discrete population model is capable of preserving the positivity and the boundedness of the discrete initial-boundary conditions. Moreover, the computational solubility of the discrete model is tackled in the closing remarks.

In this work, we propose a discrete mathematical system to model the evolution of the thickness of two-dimensional viscous thin films subject to a dewetting process. The continuous model under consideration is a degenerate partial differential equation that generalizes the classical thin film equation, and considers the inclusion of a singular potential. The analytical model is discretized using an exponential method that is capable of preserving the positive character of the approximations. In addition, the explicit nature of our approach results in an economic computer implementation which produces fast simulations. We provide some illustrative examples on the dynamics of the growth of thin films in the presence/absence of a dewetting process. The qualitative results exhibit the appearance of typical patterns obtained in experimental settings. The technique was validated against Bhattacharya’s method and a standard explicit discretization of the mathematical model.

In this work, we use a computational technique with multiple properties of consistency in order to approximate solutions of a bounded β-Fermi–Pasta–Ulam lattice in two space dimensions subject to harmonic driving in two adjacent boundaries. The processes of nonlinear supratransmission and infratransmission are employed in order to propagate binary signals in the medium by means of a suitable modulation of the driving amplitude.

In this work, for the first time in the literature, we provide solid computational evidence that the phenomenon of nonlinear supratransmission is present in time-fractional sine-Gordon equations with damping. The model under study considers a Caputo-type partial derivative of order α ∈ ( 1 , 2 ] with respect to the temporal variable, and it possesses an associated continuous energy density function which is dissipated throughout time. In the present work, we determine the dispersion relation of the physical model and employ a reliable numerical technique to approximate the solutions. Associated with the finite-difference scheme, we also propose a discrete energy density function. This methodology was implemented computationally to approximate the solution of the model on a closed and bounded interval, considering sinusoidal perturbations on one end, and homogeneous Neumann conditions on the other. As the most relevant physical conclusions of our experiments, we have confirmed the presence of nonlinear supratransmission in nonlinear wave equations with time-fractional derivatives. Additionally, we have found out that nonlinear supratransmission gradually ceases to be present in the system as α tends from 2 to 1. Moreover, we provide some bifurcation diagrams depicting the relation between the driving amplitude at which supratransmission is triggered versus driving frequency. To that end, we follow a systematic approach which is relatively standard nowadays. For the sake of convenience, we provide a Matlab implementation of our code as an appendix at the end of this manuscript.

The phenomenon of supratransmission in nonlinear systems with global interactions is predicted analytically for the first time. The model considered is a physically significant extension of the classical β-Fermi–Pasta–Ulam–Tsingou (β-FPUT) chain, with power-law interaction of degree α ⩾ 0. Using a relatively simple analytical theory, we derive the threshold that triggers the supratransmission process. The threshold for the short-range model is a particular case of our predictions. Our theoretical derivation shows that supratransmission is present for any power α > 1. We confirm numerically the validity of these results. Moreover, the numerical simulations also confirm the presence of supratransmission in the case of long-range interactions, in spite that the theoretical arguments are not valid for that case. It is worth pointing out that the present approach may be applied to other systems with global interactions.

In this work, the phenomenon of supratransmission in nonlinear sine-Gordon chains with global interactions will be predicted analytically for the first time in the literature. The model considered is an extension of the classical sine-Gordon chain which describes the dynamics of pendula attached through strings. Using a relatively simple analytical theory, we derive the threshold that triggers the supratransmission process for relatively high degrees of interaction. The threshold for the classical model is a particular case of our predictions when the degree of interaction tends to infinity. We confirm numerically the validity of our results using a systematic computational approach. We obtain numerically bifurcation diagrams for the supratransmission threshold and compare them with the theoretical predictions, obtaining a good agreement.

In this work, we provide a generalization of the β -Fermi–Pasta–Ulam–Tsingou (FPUT) chains which considers the absolute value of a power-law in a semi-infinite interval of the real line. It is well known that the β -FPUT chain shows the presence of nonlinear supratransmission, but not the α -FPUT model. The purpose of this work is to provide a generalization of the β -FPUT chain which considers potentials with arbitrary real power-laws, in such way that supratransmission is present independently of the value of the power. The systems considered in this work are all Hamiltonian, and we will use a Hamiltonian numerical discretization of the system to produce simulations. In our computational experiments, we will consider finite FPUT arrays which are perturbed sinusoidally on one end, and have a free boundary on the other. The system will be initially at rest, and the particles will be in resting position. For illustration purposes, we will consider various particular cases in order to exhibit the existence of supratransmission in those models, both in the undamped and the damped scenarios. Bifurcation diagrams will be obtained for various values of the power-law orders, and we will elucidate the behavior of the supratransmission threshold as a function of the driving amplitude and the power-law order.

In this note, we provide a generalization of a well-known result of module theory which states that two injective modules are isomorphic when they are isomorphic to submodules of each other. More precisely, we show here that two RD-injective (respectively, pure-injective) modules over an integral domain are isomorphic if they are isomorphic to relatively divisible (respectively, pure) sub-modules of each other. Keywords: Relatively divisible submodules, pure submodules, injective modules, RD- injective modules, pure-injective modules

In this work, we propose an efficient implementation of a finite-difference method employed to approximate the solutions of a system of partial differential equations that appears in the investigation of the growth of biological films. The associated homogeneous Dirichlet problem is discretized using a linear approach. This discretization yields a positivity- and boundedness-preserving implicit technique which is represented in vector form through the multiplication by a sparse matrix. A straightforward implementation of this methodology would require a substantial amount of computer memory and time, but the problem is conveniently coded using a continual reduction of the zero sub-matrices of the representing matrix. In addition to the conditions that guarantee the positivity and the boundedness of the numerical approximations, we establish some parametric constraints that assure that the same properties for the discrete total mass at each point of the mesh-grid and each discrete time are actually satisfi...

We present a finite element scheme capable of preserving the nonnegative and bounded solutions of the generalized Burgers–Huxley equation. Proofs of existence and uniqueness of a solution to the continuous problem together with some results concerning the boundedness and the nonnegativity of the solution are given. Under appropriate conditions on the mesh and the initial and boundary data, boundedness and nonnegativity of the finite element approximation are established. An a priori error estimate for the approximation is also derived. Numerical experiments are presented which support the derived theoretical results.

We investigate analytically and numerically the conditions for wave instabilities in a hyperbolic activator-inhibitor system with species undergoing anomalous superdiffusion. In the present work, anomalous superdiffusion is modeled using the two-dimensional Weyl fractional operator, with derivative orders $\ensuremath{\alpha}\ensuremath{\in}$ [1,2]. We perform a linear stability analysis and derive the conditions for diffusion-driven wave instabilities. Emphasis is placed on the effect of the superdiffusion exponent $\ensuremath{\alpha}$, the diffusion ratio $d$, and the inertial time $\ensuremath{\tau}$. As the superdiffusive exponent increases, so does the wave number of the Turing instability. Opposite to the requirement for Turing instability, the activator needs to diffuse sufficiently faster than the inhibitor in order for the wave instability to occur. The critical wave number for wave instability decreases with the superdiffusive exponent and increases with the inertial time. The maximum value of the inertial time for a wave instability to occur in the system is ${\ensuremath{\tau}}_{max}=3.6$. As one of the main results of this work, we conclude that both anomalous diffusion and inertial time influence strongly the conditions for wave instabilities in hyperbolic fractional reaction-diffusion systems. Some numerical simulations are conducted as evidence of the analytical predictions derived in this work.

In this note, we establish analytically the convergence of a nonlinear finite-difference discretization of the generalized Burgers–Fisher equation. The existence and uniqueness of positive, bounded and monotone solutions for this scheme was recently established in [J. Diff. Eq. Appl. 19 (2014), pp. 1907–1920]. In the present work, we prove additionally that the method is convergent of order one in time, and of order two in space. Some numerical experiments are conducted in order to assess the validity of the analytical results. We conclude that the methodology under investigation is a fast, nonlinear, explicit, stable, convergent numerical technique that preserves the positivity, the boundedness and the monotonicity of approximations, making it an ideal tool in the study of some travelling-wave solutions of the mathematical model of interest. This note closes proposing new avenues of future research.

In this manuscript, we follow a nonlinear approach in the problem of numerically estimating solutions of a generic, diffusive partial differential equation with nonpolynomial advection/convection, and reaction laws. The nonlocal, finite-difference methodology presented in this manuscript has the advantage of conditionally preserving the properties of non-negativity, boundedness, and temporal and spatial monotonicity of approximations, which are characteristics that are inherent to some kink-like solutions of particular forms of the mathematical model under investigation. We establish theorems on the stability of the technique, on the existence and uniqueness of non-negative and bounded solutions of our computational model, and on the property of preservation of monotonicity. As corollaries, we show that the method is capable of preserving the spatial and temporal monotonicity of numerical approximations. We provide some illustrative, numerical simulations that support our analytical results. Our simulations show that the method provides good approximations to the exact solutions of the particular problems considered and that the properties of non-negativity, boundedness, and monotonicity (both temporal and spatial) are conditionally preserved at each iteration. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 652–669, 2015

In this note, we establish the property of convergence for a finite-difference discretization of a diffusive partial differential equation with generalized Burgers convective law and generalized Hodgkin–Huxley reaction. The numerical method was previously investigated in the literature and, amongst other features of interest, it is a fast and nonlinear technique that is capable of preserving positivity, boundedness and monotonicity. In the present work, we establish that the method is convergent with linear order of convergence in time and quadratic order in space. Some numerical experiments are provided in order to support the analytical results.

In this work, we investigate numerically a system of partial differential equations that describes the interactions between populations of predators and preys. The system considers the effects of anomalous diffusion and generalized Michaelis&ndash, Menten-type reactions. For the sake of generality, we consider an extended form of that system in various spatial dimensions and propose two finite-difference methods to approximate its solutions. Both methodologies are presented in alternative forms to facilitate their analyses and computer implementations. We show that both schemes are structure-preserving techniques, in the sense that they can keep the positive and bounded character of the computational approximations. This is in agreement with the relevant solutions of the original population model. Moreover, we prove rigorously that the schemes are consistent discretizations of the generalized continuous model and that they are stable and convergent. The methodologies were implemented efficiently using MATLAB. Some computer simulations are provided for illustration purposes. In particular, we use our schemes in the investigation of complex patterns in some two- and three-dimensional predator&ndash, prey systems with anomalous diffusion.

In this work, we numerically investigate a three-dimensional nonlinear reaction-diffusion susceptible-infected-recovered hepatitis B epidemic model. To that end, the stability and bifurcation analyses of the mathematical model are rigorously discussed using the Routh&ndash, Hurwitz condition. Numerically, an efficient structure-preserving nonstandard finite-difference time-splitting method is proposed to approximate the solutions of the hepatitis B model. The dynamical consistency of the splitting method is verified mathematically and graphically. Moreover, we perform a mathematical study of the stability of the proposed scheme. The properties of consistency, stability and convergence of our technique are thoroughly analyzed in this work. Some comparisons are provided against existing standard techniques in order to validate the efficacy of our scheme. Our computational results show a superior performance of the present approach when compared against existing methods available in the literature.

In this work, we investigate numerically a one-dimensional wave equation in generalized form. The system considers the presence of constant damping and functional anomalous diffusion of the Riesz type. Reaction terms are also considered, in such way that the mathematical model can be presented in variational form when damping is not present. As opposed to previous efforts available in the literature, the reaction terms are not only functions of the solution. Instead, we consider the presence of smooth functions that depend on fractional derivatives of the solution function. Using a finite-difference approach, we propose a numerical scheme to approximate the solutions of the fractional wave equation. Along with this integrator, we propose discrete forms of the local and the total energy operators. In a first stage, we show rigorously that the energy properties of the continuous system are mimicked by our discrete methodology. In particular, we prove that the discrete system is dissipative (respectively, conservative) when damping is present (respectively, absent), in agreement with the continuous model. The theoretical numerical analysis of this system is more complicated in light of the presence of the functional form of the anomalous diffusion. To solve this problem, some novel technical lemmas are proved and used to establish the stability and the quadratic convergence of the scheme. Finally, we provide some computer simulations to show the capability of the scheme to conserve/dissipate the energy. Various fractional problems with functional forms of the anomalous diffusion of the solution are considered to that effect.

In this work, we investigate a numerical diffusion equation with nonlinear reaction, defined spatially over a closed and bounded interval of the real line. The partial differential equation is expressed in an equivalent logarithmic form, and initial and Dirichlet boundary data are imposed upon the problem. An implicit finite-difference discretization of this logarithmic model is proposed then. We show that the numerical scheme is capable of preserving the constant solutions of the continuous model. Moreover, we establish the existence of positive and bounded numerical solutions using analytical arguments. Finally, an extensive set of numerical simulations is provided in order to illustrate the performance of the scheme. The results verify that the logarithmic scheme converges to the exact solution of the continuous problem, with first order of convergence in time and second order in space. Moreover, we provide some comparisons on the efficiency of the implicit method against an explicit logarithmic scheme of the literature. The results show that the present discretization is a more efficient technique.

Departing from a generalized Burgers–Huxley partial differential equation, we provide a Mickens-type, nonlinear, finite-difference discretization of this model. The continuous system is a nonlinear regime for which the existence of travelling-wave solutions has been established previously in the literature. We prove that the method proposed also preserves many of the relevant characteristics of these solutions, such as the positivity, the boundedness and the spatial and the temporal monotonicity. The main results provide conditions that guarantee the existence and the uniqueness of monotone and bounded solutions of our scheme. The technique was implemented and tested computationally, and the results confirm both a good agreement with respect to the travelling-wave solutions reported in the literature and the preservation of the mathematical features of interest.

In the present work, we develop a computational technique to approximate the changes of phase in temporal series associated to electric signals of muscles which perform activities at three different levels of intensity. We suppose that the temporal series are samples of independent, normally distributed random variables with mean equal to zero, and variance equal to one of three possible values, each of them associated to a certain degree of electric intensity. For example, these intensity levels may represent a leg muscle at rest, or active during a light activity (walking), or active during a highly demanding performance (jogging). The model is presented as a maximum likelihood problem involving discrete variables. In turn, this problem is transformed into a continuous one via the introduction of continuous variables with penalization parameters, and it is solved recursively through an iterative numerical method. An a posteriori treatment of the results is used in order to avoid the detection of relatively short periods of silence or activity. We perform simulations with synthetic data in order to assess the validity of our technique. Our computational results show that the method approximates well the occurrence of the change points in synthetic temporal series, even in the presence of autocorrelated sequences. In the way, we show that a generalization of a computational technique for the change-point detection of electric signals with two phases of activity (Esquivel-Frausto et al., 2010 [40]), may be inapplicable in cases of temporal series with three levels of intensity. In this sense, the method proposed in the present manuscript improves previous efforts of the authors.

We present a nonlinear method to approximate solutions of a Burgers–Huxley equation with generalized advection factor and logistic reaction. The equation under investigation possesses travelling-wave solutions that are temporally and spatially monotone functions; the travelling-wave fronts considered are bounded and connect asymptotically the stationary solutions of the model. For the linear regime, the method is consistent of first order in time and second order in space. In the nonlinear scenario, we investigate conditions under which bounded initial profiles evolve into bounded new approximations. The main results report on parametric conditions that guarantee the boundedness, the positivity and the monotonicity preservation of the method. As a consequence, our recursive method is capable of preserving the temporal and the spatial monotonicity of the solutions. We provide simulations that show that, indeed, our technique preserves the positivity, the boundedness and the temporal and spatial monotonicit...

In this work, we present a computational method to approximate the occurrence of the change-points in a temporal series consisting of independent and normally distributed observations, with equal mean and two possible variance values. This type of temporal series occurs in the investigation of electric signals associated to rhythmic activity patterns of nerves and muscles of animals, in which the change-points represent the actual moments when the electrical activity passes from a phase of silence to one of activity, or vice versa. We confront the hypothesis that there is no change-point in the temporal series, against the alternative hypothesis that there exists at least one change-point, employing the corresponding likelihood ratio as the test statistic; a computational implementation of the technique of quadratic penalization is employed in order to approximate the quotient of the logarithmic likelihood associated to the set of hypotheses. When the null hypothesis is rejected, the method provides estim...

In this manuscript, we propose a simple, two-step, finite-difference scheme to approximate the solutions of an advective Fisher's equation. The method proposed is nonlinear, explicit and, in the linear regime, it approximates the solutions of the equation of interest with a consistency of first order in time and second order in space. We prove that the technique is capable of preserving the positive, the bounded, and the temporally and spatially monotone characters of initial approximations; moreover, we establish that the method is conditionally stable under suitable constraints on the model and numerical parameters. Some simulations are provided to evince the validity of our analytical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

In the present manuscript, we introduce a finite-difference scheme to approximate solutions of the two-dimensional version of Fisher's equation from population dynamics, which is a model for which the existence of traveling-wave fronts bounded within (0,1) is a well-known fact. The method presented here is a nonstandard technique which, in the linear regime, approximates the solutions of the original model with a consistency of second order in space and first order in time. The theory of M-matrices is employed here in order to elucidate conditions under which the method is able to preserve the positivity and the boundedness of solutions. In fact, our main result establishes relatively flexible conditions under which the preservation of the positivity and the boundedness of new approximations is guaranteed. Some simulations of the propagation of a traveling-wave solution confirm the analytical results derived in this work; moreover, the experiments evince a good agreement between the numerical result and the analytical solutions.

Motivated by Hill’s criterion of freeness for abelian groups, we establish a generalization of that result to categories \({\mathcal{C}}\) of torsion-free modules over integral domains, which are closed with respect to the formation of direct sums, and in which every object can be decomposed into direct sums of objects of \({\mathcal{C}}\) of rank at most a fixed limit cardinal number κ. Our main result states that a module belongs to \({\mathcal{C}}\) if it is the union of a continuous, well-ordered, ascending chain of length κ, consisting of pure submodules which are objects of \({\mathcal{C}}\) . As corollaries, we derive versions of Hill’s theorem for some classes of torsion-free modules over domains, and a generalization of a well-known result by Kaplansky.

In this work, we propose a finite-difference scheme to approximate the solutions of a generalization of the classical, one-dimensional, Newell-Whitehead-Segel equation from fluid mechanics, which is an equation for which the existence of bounded solutions is a well-known fact. The numerical method preserves the skew-symmetry of the problem of interest, and it is a non-standard technique which consistently approximates the solutions of the equation under investigation, with a consistency of the first order in time and of the second order in space. We prove that, under relatively flexible conditions on the computational parameters of the method, our technique yields bounded numerical approximations for every set of bounded initial estimates. Some simulations are provided in order to verify the validity of our analytical results. In turn, the validity of the computational constraints under which the method guarantees the preservation of the boundedness of the approximations, is successfully tested by means of computational experiments in some particular instances.

Electromyograms are biomedical signals which are detected through electrodes. These signals represent measurements of the electric potentials associated to muscles contractions, and they are physically important in view that they provide information on the health of individuals. The computational modeling of electromyograms has been mainly carried out through general-purpose commercial software, which is typically provided with the purchase of computer equipment. In view of the well-known limitations inherent in the commercial software, a lot of research and efforts are devoted continuously to upgrade existing processing methods. However, as expected, these innovations take a considerable amount of time in order to be implemented in commercial software. In order to alleviate this situation, the package has been developed to provide a free and fully open-source package for both applied and theoretical researchers, which includes some fundamental tools to computationally process electromyograms. The purpose of the present paper is to provide an overview of the most important features of this package, and to supply codes and examples to illustrate its use.

The nonlinear processes of supratransmission and infratransmission are employed here in order to show numerically that binary information may be transmitted into (2 + 1)-dimensional, continuous Frenkel–Kontorova media spatially defined on a square, by perturbing harmonically two adjacent boundaries by means of Neumann conditions. The presence of these nonlinear phenomena is established numerically through a computational method that preserves the positivity of the energy operators, and that consistently approximates the solution of the model, the local energy density, the total energy and its dissipation; the existence of a region of bistability where two regimes coexist (conducting and insulating) is established as a corollary. The transmission of binary information is accomplished by fixing a frequency in the forbidden band-gap of the system, and modulating the amplitude of the signals as the sum of a seed (whose amplitude oscillates sinusoidally between the supratransmission and infratransmission thresholds) and small, positive, constant perturbations associated to nonzero bits. Our simulations show that a reliable transmission of information is indeed feasible.

In this work, we consider bounded one-dimensional systems consisting of particles with relativistic potentials and nonlocal interactions. The systems are initially at rest and at equilibrium, and are perturbed sinusoidally at one end. Meanwhile, we consider an absorbing boundary at the other end. A Hamiltonian computational method is employed in this work to simulate the dynamics of the continuous-time system. We show here that the driving boundary propagates energy into the system in the form of nonlinear modes. For each driving frequency at the forbidden band-gap, this process occurs at a critical amplitude: the supratransmission threshold. We find that supratransmission occurs for all frequencies in the forbidden band-gap independently of whether nearest-neighbor or global interactions are considered. Several qualitative conclusions are obtained in this work based on our computational results.

In this work, we present a non-standard finite-difference scheme to approximate solutions of a damped, hyperbolic partial differential equation with nonlinear reaction which follows a generalized logistic form. Our mathematical model is a hyperbolic generalization of the classical Fisher-Kolmogorov-Petrovsky-Piscounov equation from population dynamics, where the damping term has a constant coefficient; both zero Dirichlet and Neumann boundary conditions are considered. Under certain conditions of the model and the computational parameters, the method is able to preserve the positivity and the boundedness of the solutions of the mathematical model, as evidenced both theoretically (via techniques of matrix algebra which provide sufficient conditions on the positivity and the boundedness of the technique) and computationally (via numerical simulations). The validity of the method is tested against known exact solutions of the parabolic and the hyperbolic regimes, obtaining an excellent agreement between the analytical and numerical results.

In this work, we investigate computationally the dynamics of a nonlinear partial differential equation with anomalous diffusion that extends the well-known double sine–Gordon equation from relativistic quantum mechanics. The problem under consideration includes the presence of constant damping along with anomalous spatial derivatives. The model is defined on a close and bounded interval of the real line, and it is at rest at the initial time. One end of the interval is subject to sinusoidal driving, and the other considers the presence of an absorbing boundary in order to simulate a semi-infinite medium. The simulation of this system is carried out using a numerical method that resembles the energy properties of the continuous medium. The computational results shown in this work establish the presence of the nonlinear phenomenon of bistability in the system considered. We obtain hysteresis cycles for some particular scenarios, and employ the bistability of the system to simulate the transmission of binary signals from the driving boundary to the opposite end.

In this work, we present a numerical method to approximate solutions of a continuous β -Fermi–Pasta–Ulam-like medium which, amongst other features, includes the presence of damping and a fourth-order partial derivative, obtained when approximating the continuous limit of the classical β -Fermi–Pasta–Ulam chain. The method includes a discrete energy scheme which consistently approximates the continuous energy formula; moreover, it is shown that the discrete rate of change of energy consistently approximates its continuous counterpart. Computational experiments are conducted in order to show that the method preserves energy in the case of conservative systems, and it loses energy in the case of dissipative media. An application is conducted in order to approximate the occurrence of the process of nonlinear supratransmission in the classical β -Fermi–Pasta–Ulam chain and in discrete sine-Gordon systems, obtaining results which are in good agreement with the theory available in the literature.

In this work, we consider the propagation speed of the superenergy flux associated to the Einstein–Rosen cylindrical waves propagating in vacuum and over the background of the gravitational field of an infinitely long mass line distribution. The velocity of the flux is determined considering the reference frame in which the super-Poynting vector vanishes. This reference frame is then considered as comoving with the flux. The explicit expressions for the velocities are given with respect to a reference frame at rest with the symmetry axis.

In this work, we establish computationally the existence of the process of nonlinear supratransmission in the simplest generalization of the classical β-Fermi–Pasta–Ulam chain to two-space dimensions, subject to harmonic boundary data in one end of the Dirichlet form. Our simulations employ a finite-difference scheme with multiple properties of consistency in the domains of the solutions, the local energy density and the total energy of the system. Moreover, our results establish the presence of the phenomenon of nonlinear infratransmission or lower-transmission, thus proving the existence of a bistable region where a conducting and an insulating regimes coexist.

The nonlinear phenomena of supratransmission and bistability in harmonically forced arrays of anharmonic oscillators are studied in the present article. Employing a finite-difference scheme with multiple properties of consistency, we show that these processes are actually present in such nonlinear systems. We show that, for every frequency in the forbidden band-gap of the medium, there exists a critical forcing amplitude above which the propagation of energy into the system from the boundary is inevitable. We provide diagrams of total energy vs amplitude for several values of the parameters of the model studied in this work, as well as a graph establishing the relationship between the critical amplitude at which supratransmission starts vs frequency. Likewise, we establish the existence of a bistable region, where a conducting and an insulating regimes coexist.

We present an implicit finite-difference scheme to approximate solutions of generalized α -Fermi–Pasta–Ulam systems defined on bounded domains which, amongst other features, include the presence of external and internal damping. Both continuous and semi-discrete media are considered in this paper, and several other scalar parameters are considered in the mathematical model. The numerical method is consistent with the problems under study, and it has a discrete energy scheme associated with it. It is shown that the method consistently approximates the continuous rate of change of energy of the mathematical problem with respect to time and, as a corollary, we obtain that the method is conservative when the damping coefficients are equal to zero, and the boundary points either are fixed or satisfy null Neumann conditions. We briefly state the computational details of the implementation, and simulations showing the validity of our method are provided in this work. As a result, we observe that our method preserves the energy of conservative systems at a high degree of accuracy. Finally, we present numerical experiments that evidence the effects of the presence of the damping coefficients in the problem that originated the investigation of α -Fermi–Pasta–Ulam chains more than 50 years ago.

The problem of controlling the propagation of moving breathers in semi-infinite discrete mechanical lattices in two dimensions is tackled here. Nonlinear supratransmission is employed to show that an adequate modulation of the driving amplitude on two adjacent boundaries of the lattice yields an efficient transmission of binary information.

In this work, we use supratransmission and infratransmission in the mathematical modeling of the propagation of digital signals in weakly damped, discrete Josephson-junction arrays, using energy-based detection criteria. Our results show an efficient and reliable transmission of binary information., Comment: Communicated by R. Wu

Es un tratado de algebra moderna que abarca desde el mismo preambulo de las estructuras algebraicas (monoides, semigrupos) hasta las teorias de grupos y anillos, las extensiones de campos y la Teoria de Galois. Los temas propuestos pueden ser dictados holgadamente en un par de semestres, lo que da una oportunidad interesante para resolver ejercicios dentro del aula de clase. Por su temario, por la seriacion de los contenidos y el nivel de profundidad de los mismos, este libro es un auxiliar idoneo para cursos introductorios de algebra abstracta en cualquier licenciatura en matematicas del pais.

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