The Earth–Moon libration points no longer exhibit the dynamical characteristics of “equilibrium points” due to perturbation effects when applying the ephemeris model. By decoupling the forced motions within the ephemeris model and computing the dynamical substitute trajectories, we can reconstruct a dynamical system that recovers the “equilibrium points” feature. Diverging from the conventional analytical approach rooted in the framework of Newtonian mechanics, this paper presents a novel method for calculating dynamical substitute based on the Hamiltonian mechanics framework. First, the Hamiltonian equations for the ephemeris model are formulated. Subsequently, the problem of decoupling forced motions is reformulated as solving a nonautonomous differential equation through canonical transformations. Then, an iterative method based on frequency analysis is employed for the computation. Eventually, approximate analytical solutions for five libration points over a 360 yr period are provided. Simulation results demonstrate that the computed approximate analytical solutions are in excellent agreement with the numerical integration results derived from the ephemeris model, thereby validating the efficacy of the proposed method. The Hamiltonian dynamical system derived herein enables the analysis of nonlinear central manifold motions via canonical transformations, facilitating the construction of higher-order analytical solutions for libration point orbits. This framework also provides a robust foundation for exploring characterization parameters of libration point orbits within the real Earth–Moon system.
We present results concerning new notion connected with the study of Jonsson theories. The new notion is a Kaiser class of models for arbitrary Jonsson theories. All results are obtained within the framework of the normality of the considered Jonsson theory. Additionally, we describe the properties of lattices formed by perfect fragments of a fixed Jonsson theory and their relationship with the #-companion of these fragments. The results we obtained are the model-theoretic properties of the #-companion of a normal perfect Jonsson fragment. Furthermore, we establish necessary and sufficient conditions for a normal Jonsson theory to be perfect, expressed in terms of the lattices of existential formulas.
Accurately predicting the growth rates of stationary cross-flow instabilities is crucial for understanding transition mechanisms in swept-wing configurations, which can significantly impact aerodynamic performance. This paper introduces a model for the prediction of cross-flow instability growth rates, focusing on both accuracy and ease of implementation. The proposed model consists of straightforward expressions involving key boundary layer quantities. Validation against established methods demonstrates that the new model achieves comparable or superior accuracy in predicting growth rates. Additionally, tests conducted on a three-dimensional (3-D) prolate spheroid show strong alignment with transition lines computed by means of exact linear stability. Overall, this model provides a practical and efficient alternative for accurately predicting cross-flow transitions in complex 3-D geometries, contributing to improved aerodynamic design and analysis.
M.T. Sabirzhanov, B.D. Koshanov, N.M. Shynybayeva
et al.
This article consists of three sections. In the first section the concept of Vekua space is introduced, where for elliptic systems of the first order, the theorem on the representation of the solution of a homogeneous equation and the theorem on the continuity of the solution of an inhomogeneous equation are valid. In the second section the Vekua method for solving boundary value problems for a polyharmonic equation is described. In the third section the Otelbaev method describes the correct boundary value problems for a polyharmonic equation in a multidimensional sphere.
The notions of almost quasi-Urbanik structures and theories, and studied possibilities for the degrees of quasi-Urbanikness, both for existential and universal cases were introduced. Links of these characteristics and their possible values are described. These values for structures of unary predicates, equivalence relations, linearly ordered, preordered and spherically ordered structures and theories as well as for strongly minimal ones, and for some natural operations including disjoint unions and compositions of structures and theories were studied. A series of examples illustrates possibilities of these characteristics.
In this article, a question regarding the solvability of an inverse boundary value problem for the linearized equation of longitudinal waves in rods with an integral condition of the first kind was considered. For the considered inverse boundary value problem, the definition of a classical solution was introduced. Using the Fourier method, the problem was reduced to solving a system of integral equations. The method of contraction mappings is applied to prove the existence and uniqueness of a solution to the system of integral equations. The problem is to deduce the existence and uniqueness of the classical solution for the original problem.
This study is devoted to the study of the solution of a boundary value problem for an ordinary linear differential equation of half order with constant coefficients. Using of the fundamental solution of the main part of the considered equation, we obtained the principal relations, from which we obtain the necessary conditions for the Fredholm property of the original problem. Further, using the Mittag-Leffler function, a general solution of the homogeneous equation is obtained. Finally, the problem under consideration is reduced to an integral Fredholm equation of the second kind with a non-singular kernel, i.e., the Fredholm property of the stated problem is proved.
In this paper, we consider an optimal control problem with a «pure», integral boundary condition. The Green’s function is constructed. Using contracting Banach mappings, a sufficient condition for the existence and uniqueness of a solution to one class of integral boundary value problems for fixed admissible controls is established. Using the functional increment method, the Pontryagin‘s maximum principle is proved. The first and second variations of the functional are calculated. Further, various necessary conditions for optimality of the second order are obtained by using variations of controls.
Difference schemes of the finite difference method and the finite element method of high-order accuracy in time and space are proposed and investigated for a pseudo-parabolic Sobolev type equation. The order of accuracy in space is improved in two ways using the finite difference method and the finite element method. The order of accuracy of the scheme in time is improved by a special discretization of the time variable. The corresponding a priori estimates are determined and, on their basis, the accuracy estimates of the proposed difference schemes are obtained with sufficient smoothness of the solution to the original differential problem. Algorithms for the implementation of the constructed difference schemes are proposed.
Monitored quantum dynamics reveal quantum state trajectories, which exhibit a rich phenomenology of entanglement structures, including a transition from a weakly monitored volume-law-entangled phase to a strongly monitored area-law phase. For one-dimensional hybrid circuits with both random unitary dynamics and interspersed measurements, we combine analytic mappings to an effective statistical mechanics model with extensive numerical simulations on hybrid Clifford circuits to demonstrate that the universal entanglement properties of the volume-law phase can be quantitatively described by a fluctuating entanglement domain wall that is equivalent to a “directed polymer in a random environment” (DPRE). This relationship improves upon a qualitative “mean-field” statistical mechanics of the volume-law-entangled phase [Ruihua Fan, Sagar Vijay, Ashvin Vishwanath, and Yi-Zhuang You, Phys. Rev. B 103, 174309 (2021), Yaodong Li and Matthew P. A. Fisher, Phys. Rev. B 103, 104306 (2021)]. For the Clifford circuit in various geometries, we obtain agreement between the subleading entanglement entropies and error-correcting properties of the volume-law phase (which quantify its stability to projective measurements) with predictions of the DPRE. We further demonstrate that depolarizing noise in the hybrid dynamics near the final circuit time can drive a continuous phase transition to a non-error-correcting volume-law phase that is not immune to the disentangling action of projective measurements. We observe this transition in hybrid Clifford dynamics, and obtain quantitative agreement with critical exponents for a “pinning” phase transition of the DPRE in the presence of an attractive interface.
We study a (relativistic) Wiener process on a complexified (pseudo-)Riemannian manifold. Using Nelson’s stochastic quantization procedure, we derive three equivalent descriptions for this problem. If the process has a purely real quadratic variation, we obtain the one-sided Wiener process that is encountered in the theory of Brownian motion. In this case, the result coincides with the Feyman–Kac formula. On the other hand, for a purely imaginary quadratic variation, we obtain the two-sided Wiener process that is encountered in stochastic mechanics, which provides a stochastic description of a quantum particle on a curved spacetime.
This paper investigates the unique solvability of the boundary control problem for a one-dimensional wave equation loaded along one of its characteristic curves in terms of a regular solution. The solution method is based on an analogue of the d’Alembert formula constructed for this equation. We point out that the domain of definition for the solution of DE, when the initial and final Cauchy data given on intervals of the same length is a square. The side of the squire is equal to the interval length. The boundary controls are established by the components of an analogue of the d’Alembert formula, which, in turn, are uniquely established by the initial and final Cauchy data. It should be noted that the normalized distribution and centering are employed in the final formulas of sought boundary controls, which is not typical for initial and boundary value problems initiated by equations of hyperbolic type.
N. Auffray, Francesco dell'Isola, V. Eremeyev
et al.
In this paper a stationary action principle is proved to hold for capillary fluids, i.e. fluids for which the deformation energy has the form suggested, starting from molecular arguments. We remark that these fluids are sometimes also called Korteweg–de Vries or Cahn–Allen fluids. In general, continua whose deformation energy depends on the second gradient of placement are called second gradient (or Piola–Toupin, Mindlin, Green–Rivlin, Germain or second grade) continua. In the present paper, a material description for second gradient continua is formulated. A Lagrangian action is introduced in both the material and spatial descriptions and the corresponding Euler–Lagrange equations and boundary conditions are found. These conditions are formulated in terms of an objective deformation energy volume density in two cases: when this energy is assumed to depend on either C and ∇C or on C−1 and ∇C−1, where C is the Cauchy–Green deformation tensor. When particularized to energies which characterize fluid materials, the capillary fluid evolution conditions are recovered. A version of Bernoulli’s law valid for capillary fluids is found and useful kinematic formulas for the present variational formulation are proposed. Historical comments about Gabrio Piola’s contribution to analytical continuum mechanics are also presented.
Convolutions are a classical hallmark of most mammalian brains. Brain surface morphology is often associated with intelligence and closely correlated to neurological dysfunction. Yet, we know surprisingly little about the underlying mechanisms of cortical folding. Here we identify the role of the key anatomic players during the folding process: cortical thickness, stiffness, and growth. To establish estimates for the critical time, pressure, and the wavelength at the onset of folding, we derive an analytical model using the Föppl-von-Kármán theory. Analytical modeling provides a quick first insight into the critical conditions at the onset of folding, yet it fails to predict the evolution of complex instability patterns in the post-critical regime. To predict realistic surface morphologies, we establish a computational model using the continuum theory of finite growth. Computational modeling not only confirms our analytical estimates, but is also capable of predicting the formation of complex surface morphologies with asymmetric patterns and secondary folds. Taken together, our analytical and computational models explain why larger mammalian brains tend to be more convoluted than smaller brains. Both models provide mechanistic interpretations of the classical malformations of lissencephaly and polymicrogyria. Understanding the process of cortical folding in the mammalian brain has direct implications on the diagnostics of neurological disorders including severe retardation, epilepsy, schizophrenia, and autism.