Relay interlayer synchronisation: invariance and stability conditions

In this paper, the existence (invariance) and stability (locally and globally) of relay interlayer synchronisation (RIS) are investigated in a chain of multiplex networks. The local dynamics of the nodes in the symmetric positions layers on both sides of the non-identical middlemost layer(s) are identical. The local and global stability conditions for this synchronisation state are analytically derived based on the master stability function approach and by constructing a suitable Lyapunov function, respectively. We propose an appropriate demultiplexing process for the existence of the RIS state. Then the variational equation transverse to the RIS manifold for demultiplexed networks is derived. In numerical simulations, the impact of interlayer and intralayer coupling strengths, variations of the system parameter in the relay layers and demultiplexing on the emergence of RIS in triplex and pentaplex networks are explored. Interestingly, in this multiplex network, enhancement of RIS is observed when a type of impurity via parameter mismatch in the local dynamics of the nodes is introduced in the middlemost layer. A common time-lag with small amplitude shift between the symmetric positions and central layers plays an important role for the enhancing of relay interlayer synchrony. This analysis improves our understanding of synchronisation states in multiplex networks with nonidentical layers.

Assessment of stress state and dynamic characteristics of plane and spatial structure

This study is devoted to the assessment of the stress state and dynamic characteristics of various structures. The actual task at the design stage is to determine the parameters of a structure. In this article, a mathematical model was developed for assessing the stress state and dynamic characteristics of plane and spatial structures based on the Lagrange variational equation using the d’Alembert principle. The variational problem for the structures considered by the finite element method leads to the solution of nonhomogeneous algebraic equations or to the solution of algebraic eigenvalue problems. To assess the adequacy of the model and the accuracy of the numerical results obtained, a plane and spatial test problem with an exact solution was solved. Using the proposed model, the eigenfrequencies and modes of oscillations of the gravitational and earth dams (296 m high) of the Nurek reservoir were investigated. At that, it was revealed that in the natural modes of vibration of earth dams, the greatest displacements under low frequencies are observed at the crest part or at the middle of the slopes.

Contact interactions II; Gross-Pitaewskii equation, Bose-Einstein condensate, Fermi sea

In Dell’Antonio (Eur Phys J Plus 13:1–20, 2021), we explored the possibility to analyse contact interaction in Quantum Mechanics using a variational tool, Gamma Convergence. Here, we extend the analysis in Dell’Antonio (Eur Phys J Plus 13:1–20, 2021) of joint weak contact of three particles to the non-relativistic case in which the free one particle Hamiltonian is $$ H_0 = - \frac{\Delta }{2M} $$ H 0 = - Δ 2 M . We derive the Gross–Pitaevskii equation for a system of three particles in joint weak contact. We then define and study strong contact and show that the Gross–Pitaevskii equation is also the variational equation for the energy of the Bose–Einstein condensate (strong contact in a four-particle system). We add some comments on Bogoliubov’s theory. In the second part, we use the non-relativistic Pauli equation and weak contact to derive the spectrum of the conduction electrons in an infinite crystal. We prove that the spectrum is pure point with multiplicity two and eigenvalues that scale as $$ \frac{1}{log {n}}$$ 1 logn .

Interior Elastic Scattering by a Non-Penetrable Partially Coated Obstacle and Its Shape Recovering

In this paper, the interior elastic direct and inverse scattering problem of time-harmonic waves for a non-penetrable partially coated obstacle placed in a homogeneous and isotropic medium is studied. The scattering problem is formulated via the Navier equation, considering incident circular waves due to point-source fields, where the corresponding scattered data are measured on a closed curve inside the obstacle. Our model, from the mathematical point of view, is described by a mixed boundary value problem in which the scattered field satisfies mixed Dirichlet-Robin boundary conditions on the Lipschitz boundary of the obstacle. Using a variational equation method in an appropriate Sobolev space setting, uniqueness and existence results as well as stability ones are established. The corresponding inverse problem is also studied, and using some specific auxiliary integral operators an appropriate modified factorisation method is given. In addition, an inversion algorithm for shape recovering of the partially coated obstacle is presented and proved. Last but not least, useful remarks and conclusions concerning the direct scattering problem and its linchpin with the corresponding inverse one are given.

OPTIMIZATION PROBLEMS FOR A THERMOELASTIC FRICTIONAL CONTACT PROBLEM

In the present paper, we analyze and study the control of a static thermoelastic contact problem. We consider a model which describes a frictional contact problem between a thermoelastic body and a deformable heat conductor obstacle. We derive a variational formulation of the model which is in the form of a coupled system of the quasi-variational inequality of elliptic type for the displacement and the nonlinear variational equation for the temperature. Then, under a smallness assumption, we prove the existence of a unique weak solution to the problem. Moreover, we establish the dependence of the solution with respect to the data and prove a convergence result. Finally, we introduce an optimization problem related to the contact model for which we prove the existence of a minimizer and provide a convergence result.

Evaluation of Pendulum NGGM Scenarios by Full Closed-Loop Simulation

<p>The GRACE mission (2002-2017) delivered temporal gravity field solutions of the Earth for 15 years. It's successor, GRACE follow-on (GRACE-FO) is continuing it's legacy since May 2018. The time series of monthly gravity fields revealed global mass redistribution in in the near surface layer of the Earth with unprecedented accuracy. This assessed a completely new observable in geoscience disciplines and has become a crucial data product for climate research.<br>Despite the groundbreaking success and relevance of the GRACE mission(s) for Earth observation and climate science, no further successor gravity mission is planned, yet. Summarized by the name Next Generation Gravity Mission (NGGM) concepts for future gravimetry missions have been proposed and analyzed for a while. As an outcome of these studies the so called Bender-configuration (two GRACE-like satellite pairs, one in a polar orbit and a second in an inclined orbit around 60° to 70°) is the concept currently favored by the scientific community for a candidate of the next gravity mission to be realized.</p><p><br>However, an other concept still remains interesting due to specific advantages that might contribute to future improvements of gravity missions. In order to emphasize this, we present results of a full closed loop-simulation for a different ll-SST approach, the so called pendulum. It offers a quite similar overall performance with just two satellites. For this configuration the satellites are following each other in orbits with slightly different longitudes of the ascending nodes, thus the inter-satellite measurement direction is varying between along-track and cross-track. This configuration makes an interferometric laser ranging (LRI) quite challenging on the technical level. Nevertheless, the LRI accuracy is not necessarily needed. The relevance of the pendulum configuration has also been shifted into the focus of the French MARVEL mission proposal.</p><p><br>In this contribution we analyze in detail the performance of the pendulum formation with the main parameters being the angle between along-track and cross-track component of the ranging direction at the equator, and the mean distance between the satellites. We conduct the angle variation for different mean ranges and assumed ranging accuracies. As reference, the GRACE and Bender concepts are simulated, as well. The orbit simulations are performed using a derivative of the ZARM/DLR XHPS mission simulator including high precision implementations of non-gravitational accelerations.<br>The different concepts and configurations include complete GRACE-FO like attitude control and realistic environment models. State-of-the-art instrument noise models based on GRACE/-FO are used to generate observation data for accelerometer (ACC), range dependent inter satellite ranging (KBR/LRI), kinematic orbit solution (KOS) and star camera (SCA). For the gravity recovery process we use the classical variational equation approach. As for real GRACE processing, ACC calibration parameter are estimated and KOS and KBR range-rate observations are weighted by VCE.</p>

Natural Vibrations of the Reinforced Tapered Shell with Taking Into Account the Viscoelastic Properties of the Material

In this paper, the integro-differential equations of natural oscillations of a viscoelastic ribbed truncated conical shell are obtained based on the Lagrange variational equation. The general research methodology is based on the variational principles of mechanics and variational methods. Geometrically nonlinear mathematical models of the deformation of ribbed conical shells are obtained, considering such factors as the discrete introduction of edges. Based on the finite element method, a method for solving and an algorithm for the equations of natural oscillations of a viscoelastic ribbed truncated conical shell with articulated and freely supported edges is developed. The problem is reduced to solving homogeneous algebraic equations with complex coefficients of large order. For a solution to exist, the main determinant of a system of algebraic equations must be zero. From this condition, we obtain a frequency equation with complex output parameters. The study of natural vibrations of viscoelastic panels of truncated conical shells is carried out, and some characteristic features are revealed. The complex roots of the frequency equation are determined by the Muller method. At each iteration of the Muller method, the Gauss method is used with the main element selection. As the number of edges increases, the real and imaginary parts of the eigenfrequencies increase, respectively.

Chaos Detection of the Chen System with Caputo–Hadamard Fractional Derivative

In this paper, we investigate the chaotic behaviors of the Chen system with Caputo–Hadamard derivative. First, we construct some practical numerical schemes for the Chen system with Caputo–Hadamard derivative. Then, by means of the variational equation, we estimate the bounds of the Lyapunov exponents for the considered system. Furthermore, we analyze the dynamical evolution of the Chen system with Caputo–Hadamard derivative based on the Lyapunov exponents, and we found that chaos does exist in the considered system. Some phase diagrams and Lyapunov exponent spectra are displayed to verify our analysis.

Microscopic derivation of density functional theory for superfluid systems based on effective action formalism

Density-functional theory for superfluid systems is developed in the framework of the functional renormalization group based on the effective action formalism. We introduce the effective action for the particle-number and nonlocal pairing densities and demonstrate that the Hohenberg–Kohn theorem for superfluid systems is established in terms of the effective action. The flow equation for the effective action is then derived, where the flow parameter runs from 0 to 1, corresponding to the non-interacting and interacting systems. From the flow equation and the variational equation that the equilibrium density satisfies, we obtain the exact expression for the Kohn–Sham potential generalized to including the pairing potentials. The resultant Kohn–Sham potential has a nice feature that it expresses the microscopic formulae of the external, Hartree, pairing, and exchange-correlation terms, separately. It is shown that our Kohn–Sham potential gives the ground-state energy of the Hartree–Fock–Bogoliubov theory by neglecting the correlations. An advantage of our exact formalism lies in the fact that it provides ways to systematically improve the correlation part.