By using the generalization of the gamma function (p−gamma function: Γp(.)), we introduce a generalization of the fractal-fractional calculus which is called p−fractal-fractional calculus. Examples are illustrated including the basic power functions. As applications, we formulate the p−fractal-fractional difference operators. A class of maps, called gingerbread-man maps, is investigated. We present a new idea of a stability for continuous system, based on three parameters. Sufficient conditions are illustrated to obtain the stability of the system.
In this paper, the problem of a doubly nonlinear degenerate parabolic system with nonlinear sources and absorption terms not located in a homogeneous medium was considered. It obeys zero Dirichlet boundary conditions in a smooth bounded domain. The comparison principle and self-similar approach was used to study the problem. In this paper, the nonlinear splitting method was used to prove the existence of global and blow-up in finite time solutions. It is shown that the role of the nonlinear source and nonlinear absorption is important for the existence and non-existence of the solution. The results contribute to a broader understanding of nonlinear parabolic systems.
L.A. Alexeyeva, A.N. Dadayeva, D.A. Prikazchikov
et al.
In this study, heat conductivity boundary value problems on a star graph are considered, inspired by engineering applications, e.g., heat conduction phenomena in mesh-like structures. Based on the generalized function method, a unified technique for solving boundary value problems on such graphs is developed. Generalized solutions to transient and stationary boundary value problems are constructed for different conditions at the end edges, with the Kirchhoff conditions at the common node. Regular integral representations of solutions to boundary value problems are obtained using the properties and symmetry of the fundamental solution’s Fourier transform. The derived results allow the action of various heat sources to be simulated, including concentrated ones by using singular generalized functions. The generalized function method enables a wide variety of boundary value problems to be tackled, including those with local boundary conditions at the ends of the graph, and various transmission conditions at the common node. Based on the research, the authors propose an analytical solution method under the action of various heat sources to solve various boundary value problems on a star thermal graph.
This paper investigates the relationship between categorical entropy and von Neumann entropy of quantum lattices. We begin by studying the von Neumann entropy, proving that the average von Neumann entropy per site converges to the logarithm of an algebraic integer in the low-temperature and thermodynamic limits. Next, we turn to categorical entropy. Given an endofunctor of a saturated A-infinity-category, we construct a corresponding lattice model, through which the categorical entropy can be understood in terms of the information encoded in the model. Finally, by introducing a gauged lattice framework, we unify these two notions of entropy. This unification leads naturally to a sufficient condition for a conjectural algebraicity property of categorical entropy, suggesting a deeper structural connection between A-infinity-categories and statistical mechanics.
We study the global solvability and unsolvability of a nonlinear diffusion system with nonlinear boundary conditions in the case of slow diffusion. We obtain the critical exponent of the Fujita type and the critical global existence exponent, which plays a significant part in analyzing the qualitative characteristics of nonlinear models of reaction-diffusion, heat transfer, filtration, and other physical, chemical, and biological processes. In the global solvability case, the key components of the asymptotic solutions are obtained. Iterative methods, which quickly converge to the exact solution while maintaining the qualitative characteristics of the nonlinear processes under study, are known to require the presence of an appropriate initial approximation. This presents a significant challenge for the numerical solution of nonlinear problems. A successful selection of initial approximations allows for the resolution of this challenge, which depends on the value of the numerical parameters of the equation, which are primarily in the computations recommended using an asymptotic formula. Using the asymptotics of self-similar solutions as the initial approximation for the iterative process, numerical calculations and analysis of the results are carried out. The outcomes of numerical experiments demonstrate that the results are in excellent accord with the physics of the process under consideration of the nonlinear diffusion system.
This paper examines the initial value problem for a third-order delay partial differential equation with involution and Robin boundary condition. We construct a first-order accurate difference scheme to obtain the numerical solution for this equation. Illustrative numerical results are provided.
The scientific foundations of group analysis are explored by drawing analogies between the group analytic process and quantum ontology. The group-analytic situation is conceived as a wave function process leading to collapses, according to Schrödinger’s equation and the interpretation of the Copenhagen School, through the conductor’s interpretative interventions. Superposition, as expressed by the conductor’s three positions (as therapist/member, father/mother and leader/conductor) and those of the members (adults/infants), as linked with mirror/resonance phenomena, are investigated. Heisenberg’s uncertainty principle is related to the impossibility of simultaneously “measuring” the conductor/father’s paternal position (Signifier/Name-of-the-Father) and his/ her dynamic influence (“speed”) on the group/mother or Signified (the mother/ group’s desire) (Lacan). A reservation regarding the exact determination of the Signifier and/or Signified using the uncertainty principle leads to a balance of power between them, which favours the pre-eminence of the conductor/father as expressing an effective paternal function by recognizing the group/mother as the major therapeutic figure (Foulkes).
Pablo Díez-Valle, Fernando Martínez-García, Juan José García-Ripoll
et al.
We introduce a variational algorithm based on Matrix Product States that is trained by minimizing a generalized free energy defined using Tsallis entropy instead of the standard Gibbs entropy. As a result, our model can generate the probability distributions associated with generalized statistical mechanics. The resulting model can be efficiently trained, since the resulting free energy and its gradient can be calculated exactly through tensor network contractions, as opposed to standard methods which require estimating the Gibbs entropy by sampling. We devise a variational annealing scheme by ramping up the inverse temperature, which allows us to train the model while avoiding getting trapped in local minima. We show the validity of our approach in Ising spin-glass problems by comparing it to exact numerical results and quasi-exact analytical approximations. Our work opens up new possibilities for studying generalized statistical physics and solving combinatorial optimization problems with tensor networks.
The paper deals with the construction of minimizing sequences for the problem of minimizing a weakly nonlinearly perturbed quadratic performance index on trajectories of a weakly nonlinear system with threetempo state variables. For this purpose, the so-called direct scheme for constructing an asymptotic solution is used, which consists in immediate substituting the postulated asymptotic expansion of the solution into the problem conditions and constructing a series of optimal control problems (in the case under consideration, linear-quadratic ones), the solutions of which are terms of the asymptotic expansion of the solution of the original nonlinear control problem. An estimate is obtained for the proximity of the optimal trajectory to the trajectory of the equation of state when some asymptotic approximation to the optimal control is used as a control. An example is given that illustrates in detail the proposed scheme for constructing minimizing sequences.
Ismail Gad Ameen, Rasha Osman Ahmed Taie, Hegagi Mohamed Ali
The objective of this work is to implement two efficient techniques, namely, the Laplace Adomian decomposition method (LADM) and the modified generalized Mittag–Leffler function method (MGMLFM) on a system of nonlinear fractional partial differential equations (NFPDEs) to get an analytic-approximate solution. The nonlinear time-fractional Schrödinger equation (TFSE) and coupled fractional order Schrödinger-Korteweg-de Vries (Sch-KdV) equation are found in various areas such as quantum mechanics and physics. These equations describe different types of wave propagation like dust-acoustic waves, Langmuir and electromagnetic waves in plasma physics. Using the proposed methods, a convenient solution is established for the considered nonlinear fractional order models. The obtained analytic-approximate travelling-waves solutions and the effect of the fractional order α on the behaviour of these projected solutions are presented in some figures and tables along with the exact solution. We compare the approximate values with their corresponding values of the known exact solution and compute the absolute error. Consequently, we can deduce that the used methods are very efficient, reliable and simple to construct a series form that rapidly convergent to the exact solution, which indicates the advantages of the methods.
S.M. Lutsak, A.O. Basheyeva, A.M. Asanbekov
et al.
The questions of the standardness of quasivarieties have been investigated by many authors. The problem "Which finite lattices generate a standard topological prevariety?" was suggested by D.M. Clark, B.A. Davey, M.G. Jackson and J.G. Pitkethly in 2008. We continue to study the standardness problem for one specific finite modular lattice which does not satisfy all Tumanov’s conditions. We investigate the topological quasivariety generated by this lattice and we prove that the researched quasivariety is not standard, as well as is not finitely axiomatizable. We also show that there is an infinite number of lattices similar to the lattice mentioned above.
V. P. Golubyatnikov, A. A. Akinshin, N. B. Ayupova
et al.
Periodic processes of gene network functioning are described with good precision by periodic trajectories (limit cycles) of multidimensional systems of kinetic-type differential equations. In the literature, such systems are often called dynamical, they are composed according to schemes of positive and negative feedback between components of these networks. The variables in these equations describe concentrations of these components as functions of time. In the preparation of numerical experiments with such mathematical models, it is useful to start with studies of qualitative behavior of ensembles of trajectories of the corresponding dynamical systems, in particular, to estimate the highest likelihood domain of the initial data, to solve inverse problems of parameter identification, to list the equilibrium points and their characteristics, to localize cycles in the phase portraits, to construct stratification of the phase portraits to subdomains with different qualities of trajectory behavior, etc. Such an à priori geometric analysis of the dynamical systems is quite analogous to the basic section “Investigation of functions and plot of their graphs” of Calculus, where the methods of qualitative studies of shapes of curves determined by equations are exposed. In the present paper, we construct ensembles of trajectories in phase portraits of some dynamical systems. These ensembles are 2-dimensional surfaces invariant with respect to shifts along the trajectories. This is analogous to classical construction in analytic mechanics, i. e. the level surfaces of motion integrals (energy, kinetic moment, etc.). Such surfaces compose foliations in phase portraits of dynamical systems of Hamiltonian mechanics. In contrast with this classical mechanical case, the foliations considered in this paper have singularities: all their leaves have a non-empty intersection, they contain limit cycles on their boundaries. Description of the phase portraits of these systems at the level of their stratifications, and that of ensembles of trajectories allows one to construct more realistic gene network models on the basis of methods of statistical physics and the theory of stochastic differential equations.
We build a general theory of microlocal (homogeneous) Fourier Integral Operators in real-analytic regularity, following the general construction in the smooth case by Hörmander and Duistermaat. In particular, we prove that the Boutet-Sjöstrand parametrix for the Szegő projector at the boundary of a strongly pseudo-convex real-analytic domain can be realised by an analytic Fourier Integral Operator. We then study some applications, such as FBI-type transforms on compact, real-analytic Riemannian manifolds and propagators of one-homogeneous (pseudo)differential operators.
T. F. A. Alves, J. F. da Silva Neto, F. W. S. Lima
et al.
In this Letter we introduce some field-theoretic approach for computing the critical properties of systems undergoing continuous phase transitions governed by the $κ$-generalized statistics, namely $κ$-generalized statistical field theory. In particular, we show, by computations through analytic and simulation results, that the $κ$-generalized Ising-like systems are not capable of describing the nonconventional critical properties of real imperfect crystals, \emph{e. g.} of manganites, as some alternative generalized theory is, namely nonextensive statistical field theory, as shown recently in literature. Although $κ$-Ising-like systems do not depend on $κ$, we show that a few distinct systems do. Thus the $κ$-generalized statistical field theory is not general, \emph{i. e.} it fails to generalize Ising-like systems for describing the critical behavior of imperfect crystals, and must be discarded as one generalizing statistical mechanics. For the latter systems we present the physical interpretation of the theory by furnishing the general physical interpretation of the deformation $κ$-parameter.
Travis Leadbetter, Prashant K. Purohit, Celia Reina
Far-from-equilibrium phenomena are critical to all natural and engineered systems, and essential to biological processes responsible for life. For over a century and a half, since Carnot, Clausius, Maxwell, Boltzmann, and Gibbs, among many others, laid the foundation for our understanding of equilibrium processes, scientists and engineers have dreamed of an analogous treatment of non-equilibrium systems. But despite tremendous efforts, a universal theory of non-equilibrium behavior akin to equilibrium statistical mechanics and thermodynamics has evaded description. Several methodologies have proved their ability to accurately describe complex non-equilibrium systems at the macroscopic scale, but their accuracy and predictive capacity is predicated on either phenomenological kinetic equations fit to microscopic data, or on running concurrent simulations at the particle level. Instead, we provide a framework for deriving stand-alone macroscopic thermodynamics models directly from microscopic physics without fitting in overdamped Langevin systems. The only necessary ingredient is a functional form for a parameterized, approximate density of states, in analogy to the assumption of a uniform density of states in the equilibrium microcanonical ensemble. We highlight this framework's effectiveness by deriving analytical approximations for evolving mechanical and thermodynamic quantities in a model of coiled-coil proteins and double stranded DNA, thus producing, to the authors' knowledge, the first derivation of the governing equations for a phase propagating system under general loading conditions without appeal to phenomenology. The generality of our treatment allows for application to any system described by Langevin dynamics with arbitrary interaction energies and external driving, including colloidal macromolecules, hydrogels, and biopolymers.
The paper studies a two-point boundary value problem with unlimited boundary conditions for a linear singularly perturbed differential equation. Asymptotic estimates are given for a linearly independent system of solutions of a homogeneous perturbed equation. Auxiliary, so-called boundary functions, the Cauchy function are defined. For sufficiently small values of the parameter, estimates for the Cauchy function and boundary functions are found. An algorithm for constructing the desired solution of the boundary value problem has been developed. A theorem on the solvability of a solution to a boundary value problem is proved. For sufficiently small values of the parameter, an asymptotic estimate for the solution of the inhomogeneous boundary value problem is established. The initial conditions for the degenerate equation are determined. The formula is determined; the phenomena of the initial jump are studied.
This paper studies a nonlocal boundary value problem with Steklov’s conditions of the first type for a linear ordinary delay differential equation of a fractional order with constant coefficients. The Green’s function of the problem with its properties is found. The solution to the problem is obtained explicitly in terms of the Green’s function. A condition for the unique solvability of the problem is found, as well as the conditions under which the solvability condition is satisfied. The existence and uniqueness theorem is proved using the representation of the Green’s function and its properties, as well as the representation of the fundamental solution to the equation and its properties. The question of eigenvalues is investigated. The theorem on the finiteness of the number of eigenvalues is proved using the notation of the solution in terms of the generalized Wright function, as well as the asymptotic properties of the generalized Wright function as λ →∞ and λ →-∞.
The paper is devoted to the study of the solvability of inverse coefficient problems for degenerate parabolic equations of the second order. We study both linear inverse problems – the problems of determining an unknown right-hand side (external influence), and nonlinear problems of determining an unknown coefficient of the equation itself. The peculiarity of the studied work is that its unknown coefficients are functions of a time variable only. The work aims to prove the existence and uniqueness of regular solutions to the studied problems (having all the generalized in the sense of S.L. Sobolev derivatives entering the equation).