Search Results (49)

The Robertson–Walker minimum length (RWML) theory considers stochastically perturbed spacetime to describe an expanding universe governed by geometry and diffusion. We explore the possibility of static, torsionless universe eras with conserved energy density. We find that the RWML theory provides asymptotically static equations of state under positive curvature both far in the past and far into the future, with a Big Bang singularity in between. Full article

In this study, we explore the concept of symmetry as it applies to the dynamics of the Hepatitis B Virus (HBV) epidemic model. By incorporating symmetric principles in the stochastic model, we ensure that the control strategies derived are not only effective but also consistent across varying conditions, and ensure the reliability of our predictions. This paper presents a stochastic optimal control analysis of an HBV epidemic model, incorporating vaccination as a pivotal control measure. We formulate a stochastic model to capture the complex dynamics of HBV transmission and its progression to acute and chronic stages. By leveraging stochastic differential equations, we examine the model’s stationary distribution and asymptotic behavior, elucidating the impact of random perturbations on disease dynamics. Optimal control theory is employed to derive control strategies aimed at minimizing the disease burden and vaccination costs. Through rigorous numerical simulations using the fourth-order Runge–Kutta method, we demonstrate the efficacy of the proposed control measures. Our findings highlight the critical role of vaccination in controlling HBV spread and provide insights into the optimization of vaccination strategies under stochastic conditions. The symmetry within the proposed model equations allows for a balanced approach to analyzing both acute and chronic stages of HBV. Full article

We analyze truncated series generated as divergent formal solutions of non-linear ordinary differential equations. Motivating the study is a specific non-linear, first-order differential equation, which is the basis of the resurgent formulation of renormalized perturbation theory in quantum field theory. We use the Borel–Padé approximant and classical analysis to determine the analytic structure of the solution using the first few terms of its asymptotic series. Afterward, we build an approximant, consistent with the resurgent properties of the equation. The procedure gives an approximate expression for the Borel–Ecalle resummation of the solution useful for practical applications. Connections with other physical applications are also discussed. Full article

In this article, we study a predator–prey system, which includes impulsive stocking prey and a nonlinear harvesting predator at different moments. Firstly, we derive a sufficient condition of the global asymptotical stability of the predator–extinction periodic solution utilizing the comparison theorem of the impulsive differential equations and the Floquet theory. Secondly, the condition, which is to maintain the permanence of the system, is derived. Finally, some numerical simulations are displayed to examine our theoretical results and research the effect of several important parameters for the investigated system, which shows that the period of the impulse control and impulsive perturbations of the stocking prey and nonlinear harvesting predator have a significant impact on the behavioral dynamics of the system. The results of this paper give a reliable tactical basis for actual biological resource management. Full article

Three approaches for determining the thermodynamic stability of irreversible processes are described in generalized formulations. The simplest is the Gibbs–Duhem theory, specialized to irreversible trajectories, which uses the concept of virtual displacement in the reverse direction. Its only drawback is that even a trajectory leading to an explosion is identified as a thermodynamically stable motion. In the second approach, we use a thermodynamic Lyapunov function and its time rate from the Lyapunov thermodynamic stability theory (LTS, previously known as CTTSIP). In doing so, we demonstrate that the second differential of entropy, a frequently used Lyapunov function, is useful only for investigating the stability of equilibrium states. Nonequilibrium steady states do not qualify. Without using explicit perturbation coordinates, we further identify asymptotic thermodynamic stability and thermodynamic stability under constantly acting disturbances of unperturbed trajectories as well as of nonequilibrium steady states. The third approach is also based on the Lyapunov function from LTS, but here we additionally use the rates of perturbation coordinates, based on the Gibbs relations and without using their explicit expressions, to identify not only asymptotic thermodynamic stability but also thermodynamic stability under constantly acting disturbances. Only those trajectories leading to an infinite rate of entropy production (unstable states) are excluded from this conclusion. Finally, we use these findings to formulate the Fourth Law of thermodynamics based on the thermodynamic stability. It is a comprehensive statement covering all nonequilibrium trajectories, close to as well as far from equilibrium. Unlike previous suggested “fourth laws”, this one meets the same level of generality that is associated with the original zeroth to third laws. The above is illustrated using the Schlögl reaction with its multiple steady states in certain regions of operation. Full article

A mathematical justification of some basic structural properties of stochastically perturbed gene regulatory networks, including those with autoregulation and delay, is offered in this paper. By using the theory of stochastic differential equations, it is, in particular, shown how to control the asymptotic behavior of the diffusion terms in order to not destroy certain qualitative features of the networks, for instance, their sliding modes. The results also confirm that the level of randomness is gradually reduced if the gene activation times become much smaller than the time of interaction of genes. Finally, the suggested analysis explains why the deterministic numerical schemes based on replacing smooth, steep response functions by the simpler yet discontinuous Heaviside function, the well-known simplification algorithm, are robust with respect to uncertainties in data. The main technical difficulties of the analysis are handled by applying the uniform version of the stochastic Tikhonov theorem in singular perturbation analysis suggested by Yu. Kabanov and S. Pergamentshchikov. Full article

We explore the dependence of vacuum energy on the boundary conditions for massive scalar fields in (2 + 1)-dimensional spacetimes. We consider the simplest geometrical setup given by a two-dimensional space bounded by two homogeneous parallel wires in order to compare it with the non-perturbative behaviour of the Casimir energy for non-Abelian gauge theories in (2 + 1) dimensions. Our results show the existence of two types of boundary conditions which give rise to two different asymptotic exponential decay regimes of the Casimir energy at large distances. The two families are distinguished by the feature that the boundary conditions involve or not interrelations between the behaviour of the fields at the two boundaries. Non-perturbative numerical simulations and analytical arguments show such an exponential decay for Dirichlet boundary conditions of SU(2) gauge theories. The verification that this behaviour is modified for other types of boundary conditions requires further numerical work. Subdominant corrections in the low-temperature regime are very relevant for numerical simulations, and they are also analysed in this paper. Full article

A class of fractional viscoelastic Kirchhoff equations involving two nonlinear source terms of different signs are studied. Under suitable assumptions on the exponents of nonlinear source terms and the memory kernel, the existence of global solutions in an appropriate functional space is established by a combination of the theory of potential wells and the Galerkin approximations. Furthermore, the asymptotic behavior of global solutions is obtained by a combination of the theory of potential wells and the perturbed energy method. Full article

We continue our work on the study of spherically symmetric loop quantum gravity coupled to two spherically symmetric scalar fields, with one that acts as a clock. As a consequence of the presence of the latter, we can define a true Hamiltonian for the theory. In previous papers, we studied the theory for large values of the radial coordinate, i.e., far away from any black hole or star that may be present. This makes the calculations considerably more tractable. We have shown that in the asymptotic region, the theory admits a large family of quantum vacua for quantum matter fields coupled to quantum gravity, as is expected from the well-known results of quantum field theory on classical curved space-time. Here, we study perturbative corrections involving terms that we neglected in our previous work. Using the time-dependent perturbation theory, we show that the states that represent different possible vacua are essentially stable. This ensures that one recovers from a totally quantized gravitational theory coupled to matter the standard behavior of a matter quantum field theory plus low probability transitions due to gravity between particles that differ at most by a small amount of energy. Full article

The method of Fractional Borel Summation is suggested in conjunction with self-similar factor approximants. The method used for extrapolating asymptotic expansions at small variables to large variables, including the variables tending to infinity, is described. The method is based on the combination of optimized perturbation theory, self-similar approximation theory, and Borel-type transformations. General Borel Fractional transformation of the original series is employed. The transformed series is resummed in order to adhere to the asymptotic power laws. The starting point is the formulation of dynamics in the approximations space by employing the notion of self-similarity. The flow in the approximation space is controlled, and “deep” control is incorporated into the definitions of the self-similar approximants. The class of self-similar approximations, satisfying, by design, the power law behavior, such as the use of self-similar factor approximants, is chosen for the reasons of transparency, explicitness, and convenience. A detailed comparison of different methods is performed on a rather large set of examples, employing self-similar factor approximants, self-similar iterated root approximants, as well as the approximation technique of self-similarly modified Padé–Borel approximations. Full article

We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension $d>2$. This solution family is equivalent to a fractal curve in complex space ${\mathbb{C}}^d$ with random steps parametrized by N Ising variables ${\sigma }_i=\pm 1$, in addition to a rational number $\frac{p}{q}$ and an integer winding number r, related by $\sum {\sigma }_i=qr$. This equivalence provides a dual theory describing a strong turbulent phase of the Navier-Stokes flow in ${\mathbb{R}}_d$ space as a random geometry in a different space, like ADS/CFT correspondence in gauge theory. From a mathematical point of view, this theory implements a stochastic solution of the unforced Navier-Stokes equations. For a theoretical physicist, this is a quantum statistical system with integer-valued parameters, satisfying some number theory constraints. Its long-range interaction leads to critical phenomena when its size $N\to \infty$ or its chemical potential $\mu \to 0$. The system with fixed N has different asymptotics at odd and even $N\to \infty$, but the limit $\mu \to 0$ is well defined. The energy dissipation rate is analytically calculated as a function of $\mu$ using methods of number theory. It grows as $\nu /{\mu }^2$ in the continuum limit $\mu \to 0$, leading to anomalous dissipation at $\mu \propto \sqrt{\nu }\to 0$. The same method is used to compute all the local vorticity distribution, which has no continuum limit but is renormalizable in the sense that infinities can be absorbed into the redefinition of the parameters. The small perturbation of the fixed manifold satisfies the linear equation we solved in a general form. This perturbation decays as $t^{-\lambda }$, with a continuous spectrum of indexes $\lambda$ in the local limit $\mu \to 0$. The spectrum is determined by a resolvent, which is represented as an infinite product of $3\otimes 3$ matrices depending of the element of the Euler ensemble. Full article

In this study, we present a general form of nonlinear two-time-scale systems, where singular perturbation analysis is used to separate the dynamics of the slow and fast subsystems. Machine learning techniques are utilized to approximate the dynamics of both subsystems. Specifically, a recurrent neural network (RNN) and a feedforward neural network (FNN) are used to predict the slow and fast state vectors, respectively. Moreover, we investigate the generalization error bounds for these machine learning models approximating the dynamics of two-time-scale systems. Next, under the assumption that the fast states are asymptotically stable, our focus shifts toward designing a Lyapunov-based model predictive control (LMPC) scheme that exclusively employs the RNN to predict the dynamics of the slow states. Additionally, we derive sufficient conditions to guarantee the closed-loop stability of the system under the sample-and-hold implementation of the controller. A nonlinear chemical process example is used to demonstrate the theory. In particular, two RNN models are constructed: one to model the full two-time-scale system and the other to predict solely the slow state vector. Both models are integrated within the LMPC scheme, and we compare their closed-loop performance while assessing the computational time required to execute the LMPC optimization problem. Full article

Baroclinic instability is one of the main mechanisms for the formation of synoptic scale systems. Previous studies examined the exponential growth of small perturbations for a stably stratified troposphere in the case of a constant meridional temperature gradient ignoring the stratosphere (Eady’s model). However, since stratospheric flow also affects to some extent the motions in the troposphere, in this work we investigate the effect of stratospheric wind shear on baroclinic instability using the tools of Generalized Stability Theory (GST). GST is a linear stability theory that addresses both the exponential growth of perturbations that is pertinent in the large time asymptotic limit and the transient growth of perturbations at finite time. The optimal initial perturbations leading to the largest growth over a specified time interval are calculated for three main cases of stratospheric shear: positive, zero and negative shear over the stratosphere. It is found that the inclusion of stratospheric shear in all three cases decreases perturbation growth and influences the scale of the structures that will dominate the flow. For optimizing times of the order of a week, the development of systems with larger spatial scale compared to the prediction of the Eady model is expected, while for optimizing times of the order of a day, smaller scale systems are found to develop. Full article

Based on the five-dimensional Einstein–Maxwell theory, Bah et al. constructed a singularity-free topology star/black hole [Phys. Rev. Lett. 126, 151101 (2021)]. After performing the Kaluza–Klein reduction, i.e., integrating the extra space dimension, it can obtain an effective four-dimensional spherically static charged black hole with scalar hair. In this paper, we study the quasinormal modes (QNMs) of the scalar, electromagnetic, and gravitational fields in the background of this effective four-dimensional charged black hole. The radial parts of the perturbed fields all satisfy a Schrödinger-like equation. Using the asymptotic iteration method, we obtain the QNM frequencies semianalytically. For low-overtone QNMs, the results obtained using both the asymptotic iteration method and the Wentzel–Kramers–Brillouin approximation method agree well. In the null coordinates, the evolution of a Gaussian package is also studied. The QNM frequencies obtained by fitting the evolution data also agree well with the results obtained using the asymptotic iteration method. Full article

The quantum-field renormalization group method is one of the most efficient and powerful tools for studying critical and scaling phenomena in interacting many-particle systems. The multiloop Feynman diagrams underpin the specific implementation of the renormalization group program. In recent years, multiloop computation has had a significant breakthrough in both static and dynamic models of critical behavior. In the paper, we focus on the state-of-the-art computational techniques for critical dynamic diagrams and the results obtained with their help. The generic nature of the evaluated physical observables in a wide class of field models is manifested in the asymptotic character of perturbation expansions. Thus, the Borel resummation of series is required to process multiloop results. Such a procedure also enables one to take high-order contributions into consideration properly. The paper outlines the resummation framework in dynamic models and the circumstances in which it can be useful. An important resummation criterion is the properties of the higher-order asymptotics of the perturbation theory. In static theories, these properties are determined by the method of instanton analysis. A similar approach is applicable in critical dynamics models. We describe the calculation of these asymptotics in dynamical models and present the results of the corresponding resummation. Full article