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The matrix product state (MPS) Ansatz offers a promising approach for finding the ground state of molecular Hamiltonians and solving quantum chemistry problems. Building on this concept, the proposed technique of quantum circuit MPS (QCMPS) enables the simulation of chemical systems using a relatively small number of qubits. In this study, we enhance the optimization performance of the QCMPS Ansatz by employing the variational quantum imaginary time evolution (VarQITE) approach. Guided by McLachlan's variational principle, the VarQITE method provides analytical metrics and gradients, resulting in improved convergence efficiency and robustness of the QCMPS. We validate these improvements numerically through simulations of H2, H4, and LiH molecules. In addition, given that VarQITE is applicable to non-Hermitian Hamiltonians, we evaluate its effectiveness in preparing the ground state of transcorrelated Hamiltonians. This approach yields energy estimates comparable to the complete basis set (CBS) limit while using even fewer qubits. In particular, we perform simulations of the beryllium atom and LiH molecule using only three qubits, maintaining high fidelity with the CBS ground state energy of these systems. This qubit reduction is achieved through the combined advantages of both the QCMPS Ansatz and transcorrelation. Our findings demonstrate the potential practicality of this quantum chemistry algorithm on near-term quantum devices. [ABSTRACT FROM AUTHOR]

The MPSDynamics.jl package provides an easy-to-use interface for performing open quantum systems simulations at zero and finite temperatures. The package has been developed with the aim of studying non-Markovian open system dynamics using the state-of-the-art numerically exact Thermalized-Time Evolving Density operator with Orthonormal Polynomials Algorithm based on environment chain mapping. The simulations rely on a tensor network representation of the quantum states as matrix product states (MPS) and tree tensor network states. Written in the Julia programming language, MPSDynamics.jl is a versatile open-source package providing a choice of several variants of the Time-Dependent Variational Principle method for time evolution (including novel bond-adaptive one-site algorithms). The package also provides strong support for the measurement of single and multi-site observables, as well as the storing and logging of data, which makes it a useful tool for the study of many-body physics. It currently handles long-range interactions, time-dependent Hamiltonians, multiple environments, bosonic and fermionic environments, and joint system–environment observables. [ABSTRACT FROM AUTHOR]

Numerous variational methods have been proposed for solving quantum many-body systems, but they often face exponentially increasing computational complexity as the Hilbert space dimension grows. To address this, we introduce a novel approach using quantum neural networks to simulate the dissipative dynamics of many-body open quantum systems. This method combines neural-network quantum state representation with the time-dependent variational principle, both implemented via quantum algorithms. This results in accurate open quantum dynamics described by the Lindblad quantum master equation, exemplified by the spin-boson and transverse field Ising models. Our approach avoids the computational expense of classical algorithms and demonstrates the potential advantages of quantum computing for many-body simulations. To reduce measurement errors, we introduce a projection reset procedure, which could benefit other quantum simulations. In addition, our approach can be extended to simulate non-Markovian quantum dynamics. [ABSTRACT FROM AUTHOR]

In quantum chemical calculations of heterogeneous structures in solids, e.g., when an impurity is located on the surface, the conventional cluster model is insufficient to describe the electronic structure of substrates due to its finite size. The open-boundary cluster model (OCM) overcomes this problem by performing cluster calculations under the outgoing-wave boundary condition. In this method, a complex absorbing potential (CAP) is used to impose the boundary condition, but the CAP used in the previous studies required parameter optimization based on the complex variational principle. This study proposes and applies a parameter-free CAP to OCM calculations. This approach makes it possible to uniquely determine the band-specific CAP based on the surface Green's function theory. Using this CAP, we conducted OCM calculations of the tight-binding model of a one-dimensional semi-infinite chain, and we found that the calculated density of states agreed with the exact one. Surface states of the Newns–Anderson–Grimley model were also computed using the CAP, and the projected density of states on the adsorbed atom was successfully reproduced. [ABSTRACT FROM AUTHOR]

The vibration of a transformer core induced by magnetostriction is closely related to the environment and safety of devices. The DC bias excitation leads to the over saturation of the transformer core and causes more severe vibration. In this paper, a magnetic-mechanical strong coupling model based on a fixed-point time-periodic 2D finite element method with excitation voltage and DC bias current as the input is established in which the vector magnetic potential A, winding current I, and displacement in the core u are simultaneously taken as solution variables, and the fixed-point reluctivity is introduced to deal with the nonlinear problem of the core. In this model, the magnetic field and mechanical field are associated with the piezomagnetic equations and variational principle, and the circuit and magnetic fields are connected through the law of electromagnetic induction. The equivalent thickness of the core model and the determination of the initial value are discussed. A test transformer with a laminated core is made for experimental measurement. The accuracy and efficiency of the model proposed are verified by comparing the calculated results with the experimental results, and the influence of DC bias components on the magnetic characteristics and vibration of the transformer core is analyzed. [ABSTRACT FROM AUTHOR]

Equilibrium thermodynamics describes the energy exchange of a body with its environment. Here, we describe the global energy exchange of an ideal gas in the Coutte flow in a thermodynamic-like manner. We derive a fundamental relation between internal energy as a function of parameters of state. We analyze a non-equilibrium transition in the system and postulate the extremum principle, which determines stable steady states in the system. The steady-state thermodynamic framework resembles equilibrium thermodynamics. [ABSTRACT FROM AUTHOR]

Practical density functional theory (DFT) owes its success to the groundbreaking work of Kohn and Sham that introduced the exact calculation of the non-interacting kinetic energy of the electrons using an auxiliary mean-field system. However, the full power of DFT will not be unleashed until the exact relationship between the electron density and the non-interacting kinetic energy is found. Various attempts have been made to approximate this functional, similar to the exchange–correlation functional, with much less success due to the larger contribution of kinetic energy and its more non-local nature. In this work, we propose a new and efficient regularization method to train density functionals based on deep neural networks, with particular interest in the kinetic-energy functional. The method is tested on (effectively) one-dimensional systems, including the hydrogen chain, non-interacting electrons, and atoms of the first two periods, with excellent results. For atomic systems, the generalizability of the regularization method is demonstrated by training also an exchange–correlation functional, and the contrasting nature of the two functionals is discussed from a machine-learning perspective. [ABSTRACT FROM AUTHOR]

In this paper, a thorough analysis is conducted to examine the propagation characteristics of SH surface waves in a layered medium with a nanoscale piezoelectric guiding layer deposited on an isotropic elastic substrate. Specifically, a two-dimensional analytical model is established, within which the effects of strain gradient, electric field gradient, inertia gradient, and flexoelectricity are considered, as well as interfacial imperfections at the interface between the piezoelectric guiding layer and the elastic substrate, which are characterized by spring models. Within the framework of the variational principle, the governing equations, boundary conditions, and continuity conditions at the interface are derived. Based on these equations, the dispersion relations for SH surface waves are deduced and numerically solved for both the electrically open-circuit and electrically short-circuit cases. A comprehensive investigation of the dispersion relations for the fundamental mode of SH surface waves is subsequently provided, with a detailed discussion on the influence of critical factors. The developed theoretical model, encompassing various size effects observed in nano-scale structures, enables a more precise prediction of surface wave propagation behavior, thereby enhancing the design and application of surface acoustic wave devices. [ABSTRACT FROM AUTHOR]

This study presents an innovative approach for developing a reduced-order model (ROM) tailored specifically for nearly incompressible materials at large deformations. The formulation relies on a three-field variational approach to capture the behavior of these materials. To construct the ROM, the full-scale model is initially solved using the finite element method (FEM), with snapshots of the displacement field being recorded and organized into a snapshot matrix. Subsequently, proper orthogonal decomposition is employed to extract dominant modes, forming a reduced basis for the ROM. Furthermore, we efficiently address the pressure and volumetric deformation fields by employing the k-means algorithm for clustering. A well-known three-field variational principle allows us to incorporate the clustered field variables into the ROM. To assess the performance of our proposed ROM, we conduct a comprehensive comparison of the ROM with and without clustering with the FEM solution. The results highlight the superiority of the ROM with pressure clustering, particularly when considering a limited number of modes, typically fewer than 10 displacement modes. Our findings are validated through two standard examples: one involving a block under compression and another featuring Cook's membrane. In both cases, we achieve substantial improvements based on the three-field mixed approach. These compelling results underscore the effectiveness of our ROM approach, which accurately captures nearly incompressible material behavior while significantly reducing computational expenses. [ABSTRACT FROM AUTHOR]

The objective of this research is to establish a representation of the general Hausdorff and packing dimensions of compact sets in Euclidean space. This representation is formulated in terms of the lower and upper local dimensions with respect to two functions, denoted as h and g, which are associated with a probability measure defined on R n . Additionally, we establish a relationship that enables us to determine the general fractal dimensions by utilizing corresponding versions of these measures. These dimensions are defined as the supremum over the lower and upper general dimensions across all Borel probability measures. As a practical application of these findings, we introduce a variational principle for these general fractal dimensions. This principle establishes an equality between a parameter that is inherently linked to a given space or mapping and the supremum of specific values associated with a class of probability measures supported by the set, denoted as A. Furthermore, we propose two additional dimensions: the general concentration dimension and the general thin dimension. The concentration dimension is defined through the utilization of the Lévy concentration function and offers a relatively straightforward calculation method. It is closely connected to the mass distribution principle and often simplifies the determination of the general Hausdorff dimension of sets. The thin dimension, on the other hand, is grounded in the concept of the thin function, which can be seen as an anti-concentration function in relation to the Lévy function. This thin dimension also exhibits connections with the general fractal dimension. [ABSTRACT FROM AUTHOR]

High-performance finite element research has always been a major focus of finite element method studies. This article introduces isogeometric analysis into the finite element method and proposes a new isogeometric finite element method. Firstly, the physical field is approximated by uniform B-spline interpolation, while geometry is represented by non-uniform rational B-spline interpolation. By introducing a transformation matrix, elements of types C0 and C1 are constructed in the isogeometric finite element method. Subsequently, the corresponding calculation formats for one-dimensional bars, beams, and two-dimensional linear elasticity in the isogeometric finite element method are derived through variational principles and parameter mapping. The proposed method combines element construction techniques of the finite element method with geometric construction techniques of isogeometric analysis, eliminating the need for mesh generation and maintaining flexibility in element construction. Two elements with interpolation characteristics are constructed in the method so that boundary conditions and connections between elements can be processed like the finite element method. Finally, the test results of several examples show that: (1) Under the same degree and element node numbers, the constructed elements are almost consistent with the results obtained by traditional finite element method; (2) For bar problems with large local field variations and beam problems with variable cross-sections, high-degree and multi-nodes elements constructed can achieve high computational accuracy with fewer degrees of freedom than finite element method; (3) The computational efficiency of isogeometric finite element method is higher than finite element method under similar degrees of freedom, while as degrees of freedom increase, the computational efficiency between the two is similar. [ABSTRACT FROM AUTHOR]

The present paper is devoted to explore the variational principles for nonlinear topological pressure, which was introduced by Buzzi, Kloeckner and Leplaideur recently. We apply the Carathéodory structure to the theory of nonlinear topological pressure and introduce several types of nonlinear topological pressures for subsets. Based on this, we develop several variational principles in different forms for the nonlinear topological pressures, some of which are even new for the case of topological entropy. [ABSTRACT FROM AUTHOR]

By using the well-known mountain-pass theorem and Ekeland's variational principle, we prove that there exist at least two fully nontrivial solutions for a (p , q) -Kirchhoff elliptic system with the Dirichlet boundary conditions and perturbation terms on a locally weighted and connected finite graph G = (V , E) . We also present a necessary condition of the existence of semitrivial solutions for the system. Moreover, by using Ekeland's variational principle and Clark's Theorem, respectively, we prove that the system has at least one or multiple semitrivial solutions when the perturbation terms satisfy different assumptions. Finally, we present a nonexistence result of solutions. [ABSTRACT FROM AUTHOR]

We introduce the notion of Feldman–Katok metric mean dimensions in this note. We show metric mean dimensions defined by different dynamical metrics of phase space coincide under the (weak) tame growth of covering numbers, and establish variational principles for Feldman–Katok metric mean dimensions in terms of FK Katok ϵ-entropy and FK local ϵ-entropy function. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discover some new exact solutions for two kinds of nonlinear Schrödinger equation (NLSE), including the (2+1)-dimensional NLSE and the derivative NLSE, through the use of the variational principle method and amplitude ansatz method. We will find the solutions to these equations by selecting a trial function with a single nontrivial variational parameter, and it is continuous at all intervals in three cases from a region of a rectangular box, then using this trial function to obtain the functional integral and the Lagrangian of the system without any loss. After that, we approximate this trial function by quadratic polynomials with two free parameters rather than a piecewise linear ansatz function. Similarly, this trial function will be approximated by the tanh function. Additionally, we accept several new types of soliton solutions, including bright solitons, dark solitons, bright–dark solitary wave solutions, rational dark–bright solutions, and periodic solitary wave solutions. Also, conditions for the stability of the solutions will be proposed. These explanations are of great importance in the fields of applied science and engineering. The results will be shown through different types of graphs, including 2D, 3D, and contour plots. [ABSTRACT FROM AUTHOR]

The evolution of microstructures can be observed in various engineering and natural materials, such as special alloys, silica glasses or soils. These macroscopic and microscopic effects are often controlled by thermal and chemical changes or extern forces. In addition, several phases are usually involved in these processes, whose energy potentials result in a non‐convex total energy. A theory is introduced that can be applied to a wide range of elasto‐plastic materials with kinematic hardening. In order to obtain a general model, a relaxed free energy in small strain theory is defined using variational principles for inelastic materials. This energy includes the dissipation of the substance as so‐called dissipation distance to facilitate a time‐stepped incremental representation. Furthermore, the phase volume fractions are contained in the weighted sum of the individual energies and are expressed here using Young measures. In the centre of the model, transition rates with underlying ordinary differential equations (ODEs) are calculated, which allow access to the phase evolution of the system. The initial results and the behaviour of this theory are explained using numerical simulations created with the programming language Julia. [ABSTRACT FROM AUTHOR]

In this study, a layer-wise theory satisfying displacement continuity at each layer's interface and the Chebyshev collocation technique are adopted to analyze the vibration behavior of functionally graded composite nanobeams under non-uniform temperature variation in the thickness direction. The temperature-dependent mechanical properties vary according to the rule of mixture. Boundary conditions are modified by including the effect of size parameter and named as nonlocal boundary conditions. These nonlocal boundary conditions along with Eringen's nonlocal elasticity theory are developed to capture the effect of size-dependency for such a nanobeam based on the first-order shear deformation theory (FSDT) and physical neutral plane. The energy principle and the variational approach are used to obtain force-moment equilibrium equations, which lead to governing differential equations and local boundary conditions. Two types of sandwich constructions are considered, namely, Type 1 (functionally graded-homogeneous-functionally graded) and Type 2 (homogeneous-functionally graded-homogeneous), by maintaining the material continuity at the interfaces. Natural frequencies are analyzed in the fundamental mode for various values of volume fraction index, thickness of the layers, size scale parameter, and thermal environment. The convergence rate of the Chebyshev collocation technique is also shown over the differential quadrature method. [ABSTRACT FROM AUTHOR]

This paper investigated the periodic and quasi-periodic orbits in the symmetric collinear four-body problem through a variational approach. We analyze the conditions under which homographic solutions minimize the action functional. We compute the minimal value of the action functional for these solutions and show that, for four equal masses organized in a linear configuration, these solutions are the minimizers of the action functional. Additionally, we employ numerical experiments using Poincaré sections to explore the existence and stability of periodic and quasi-periodic solutions within this dynamical system. Our results provide a deeper understanding of the variational principles in celestial mechanics and reveal complex dynamical behaviors, crucial for further studies in multi-body problems. [ABSTRACT FROM AUTHOR]

This paper investigates the (semi)group action of |$\textrm{SL}_{n}({\mathbb R})$| on |${\mathbb P}({\mathbb R}^{n})$|⁠ , a primary example of non-conformal, non-linear, and non-strictly contracting action. We establish variational principles of the affinity exponent for two main examples: the Borel Anosov representations and the Rauzy gasket. In [ 32 ], they obtain a dimension formula for the stationary measures on |${\mathbb P}({\mathbb R}^{3})$|⁠. Combined with our result, it allows us to study the Hausdorff dimension of limit sets of Anosov representations in |$\textrm{SL}_{3}({\mathbb R})$| and the Rauzy gasket. It yields the equality between the Hausdorff dimensions and the affinity exponents in both settings, generalizing the classical Patterson–Sullivan formula. In the appendix, we improve the numerical lower bound of the Hausdorff dimension of Rauzy gasket to |$1.5$|⁠. [ABSTRACT FROM AUTHOR]

A technique for developing convex dual variational principles for the governing PDE of nonlinear elastostatics and elastodynamics is presented. This allows the definition of notions of a variational dual solution and a dual solution corresponding to the PDEs of nonlinear elasticity, even when the latter arise as formal Euler–Lagrange equations corresponding to non-quasiconvex elastic energy functionals whose energy minimizers do not exist. This is demonstrated rigorously in the case of elastostatics for the Saint-Venant Kirchhoff material (in all dimensions), where the existence of variational dual solutions is also proven. The existence of a variational dual solution for the incompressible neo-Hookean material in 2-d is also shown. Stressed and unstressed elastostatic and elastodynamic solutions in 1 space dimension corresponding to a non-convex, double-well energy are computed using the dual methodology. In particular, we show the stability of a dual elastodynamic equilibrium solution for which there are regions of non-vanishing length with negative elastic stiffness, i.e. non-hyperbolic regions, for which the corresponding primal problem is ill-posed and demonstrates an explosive 'Hadamard instability;' this appears to have implications for the modeling of physically observed softening behavior in macroscopic mechanical response. [ABSTRACT FROM AUTHOR]

In this paper, the thin-film ferroelectric material equation (TFFME) which enables the propagation of solitary polarization in thin-film ferroelectric materials is investigated, and also illustrated through the nonlinear evolution equations. Ferroelectrics are dielectric materials that exhibit nonlinear behaviors in wave propagation. Thin films constructed from the ferroelectric materials are utilized in different modern electronic devices. To investigate the characteristics of new waves, the solitary wave dynamics of the mentioned equation is used in the generalized trial equation scheme. The bright and periodic solutions are obtained by semi-inverse variational principle scheme. Many alternative responses may be obtained through different formulae; each of these solutions offers a distinct graph. The validity of such methods and solutions may be demonstrated by assessing how well the relevant techniques and solutions match up. The effects of free variables on the behavior of few achieved solutions for nonlinear rational exact cases are also plotted and explored depending upon the nature of nonlinearities. The dynamic properties of the obtained results are shown and analyzed by some density, two- and three-dimensional images. The results provide a way for future research on generating optical memories based on the nonlinear solitons. [ABSTRACT FROM AUTHOR]

Background: The variational method, a quantum mechanical approach, estimates effective charge distributions and ground-state energy by minimizing the Hamiltonian's expectation value using trial wave functions with adjustable parameters. This method provides valuable insights into system behavior and is widely used in theoretical chemistry and physics. This paper aims to investigate ground-state energies and isoelectronic sequences using the variational method, introducing a novel approach for analyzing multi-electron systems. This technique allows for determining effective charge values and ground-state energies for 2–5 electrons sequence up to Z ≤ 20. Hydrogenic wave functions are used as a trial wave function to calculate effective charge in 1 s, 2 s, and 2p states. Two varying parameters were used to calculate an approximate wave function for the system. These values are then used in non-relativistic Hamiltonian with electron–electron interaction terms to calculate the ground-state energy of an atom. Result: The results align with the reported experimental values, showing a marginal 1% error. Conclusion: A Python algorithm is established based on the variational principle. It was found that, based on a few selected parameters in scripting the program, a very promising result was obtained. Furthermore, adding more variational parameters can minimize the difference between experimental and theoretical values, and this technique can be extended to elements with higher atomic numbers. [ABSTRACT FROM AUTHOR]

The pull-down instability is a newly found phenomenon for oscillators with even nonlinearities. It is a seemingly periodic motion, which will suddenly pull down after some cycles. The phenomenon is similar to the pull-in instability in the micro-electromechanical systems. This paper restudies the well-known Toda oscillator, which is considered to be in periodic motion. Its pseudo-periodic property is studied and the pull-down instability is found for the first time ever. To this end, this paper establishes a variational principle for the Toda oscillator, and the criterion for prediction of the pull-down motion is obtained. A simple formulation to calculate the pull-down instability is proved and verified through examples. Furthermore, an approximate frequency formula is obtained by Li–He modified homotopy perturbation method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence and multiplicity of solutions for the critical Neumann problem 0.1 −▵u−12x·∇u=λ|u|2∗−2u+μ|u|p−2uinR+N,∂u∂n=λ|u|2∗−2uon∂R+N,$$\begin{equation} \hspace*{40pt}\left\{ \def\eqcellsep{&}\begin{array}{rcll} -\triangle u-\displaystyle\frac{1}{2}\left(x\cdot \nabla u\right)& =& \lambda |u{|}^{{2}^{*}-2}u+\mu |u{|}^{p-2}u & \text{in }{\mathbb{R}}_{+}^{N},\\ [12pt] \displaystyle\frac{\partial u}{\partial n}& =& \sqrt{\lambda}|u{|}^{{2}_{*}-2}u & \text{on }\partial {\mathbb{R}}_{+}^{N}, \end{array} \right. \end{equation}$$where R+N={(x′,xN):x′∈RN−1,xN>0}$ \mathbb {R}^{N}_{+}=\lbrace (x{^{\prime }}, x_{N}): x{^{\prime }}\in {\mathbb {R}}^{N-1}, x_{N}>0\rbrace$, N≥3$N\ge 3$, λ>0$\lambda >0$, μ∈R$\mu \in \mathbb {R}$, 20$\lambda >0$ and the different values of the parameters p$p$ and μ>0${\mu }>0$. Particularly, for λ>0$\lambda >0$, N≥4$N\ge 4$, 20$\mu >0$. Moreover, the existence of multiple solutions for (0.1) is also obtained by dual variational principle for all μ>0$\mu >0$ and suitable λ$\lambda$. [ABSTRACT FROM AUTHOR]

Spacetime geometry is described by two–a priori independent–geometric structures: the symmetric connection Γ and the metric tensor g. Metricity condition of Γ (i.e. ∇g = 0) is implied by the Palatini variational principle, but only when the matter fields belong to an exceptional class. In case of a generic matter field, Palatini implies non-metricity of Γ. Traditionally, instead of the (first order) Palatini principle, we use in this case the (second order) Hilbert principle, assuming metricity condition a priori. Unfortunately, the resulting right-hand side of the Einstein equations does not coincide with the matter energy-momentum tensor. We propose to treat seriously the Palatini-implied non-metric connection. The conventional Einstein's theory, rewritten in terms of this object, acquires a much simpler and universal structure. This approach opens a room for the description of the large scale effects in General Relativity (dark matter?, dark energy?), without resorting to purely phenomenological terms in the Lagrangian of gravitational field. All theories discussed in this paper belong to the standard General Relativity Theory, the only non-standard element being their (much simpler) mathematical formulation. As a mathematical bonus, we propose a new formalism in the calculus of variations, because in case of hyperbolic field theories the standard approach leads to nonsense conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we first characterize generalized convex functions introduced by Linh and Penot Optimization (62: 943–959, 2013) by using generalized monotonicity of the generalized subdifferentials. We use vector variational inequalities in terms of generalized subdifferentials to identify efficient solutions of a multiobjective optimization problem involving quasiconvex functions. We also establish the Minty variational principle by utilizing the mean value theorem established by Kabgani and Soleimani-damaneh (Numer. Funct. Anal. Optim 38: 1548–1563, 2017) for quasiconvex functions in terms of Greenberg–Pierskalla subdifferentials. [ABSTRACT FROM AUTHOR]

We study an optimal control problem for a stochastic model of tumour growth with drug application. This model consists of three stochastic hyperbolic equations describing the evolution of tumour cells. It also includes two stochastic parabolic equations describing the diffusions of nutrient and drug concentrations. Since all systems are subject to many uncertainties, we have added stochastic terms to the deterministic model to consider the random perturbations. Then, we have added control variables to the model according to the medical concepts to control the concentrations of drug and nutrient. In the optimal control problem, we have defined the stochastic and deterministic cost functions and we have proved the problems have unique optimal controls. For deriving the necessary conditions for optimal control variables, the stochastic adjoint equations are derived. We have proved the stochastic model of tumour growth and the stochastic adjoint equations have unique solutions. For proving the theoretical results, we have used a change of variable which changes the stochastic model and adjoint equations (a.s.) to deterministic equations. Then we have employed the techniques used for deterministic ones to prove the existence and uniqueness of optimal control. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of the topological r-pressure and measure-theoretic r-pressure for countable discrete amenable group actions. In addition, we establish a variational principle between topological r-pressure and measure-theoretic r-pressure. [ABSTRACT FROM AUTHOR]

In this article, we are first concerned with the existence of multiple normalized solutions to the following fractional p-Laplace problem: { (- Δ) p s ⁢ v + 풱 ⁢ (ξ ⁢ x) ⁢ | v | p - 2 ⁢ v = λ ⁢ | v | p - 2 ⁢ v + f ⁢ (v) in ⁢ ℝ N , ∫ ℝ N | v | p ⁢ 푑 x = a p , where a , ξ > 0 , p ≥ 2 , λ ∈ ℝ is an unknown parameter that appears as a Lagrange multiplier, 풱 : ℝ N → [ 0 , ∞) is a continuous function, and f is a continuous function with L p -subcritical growth. We prove that there exists the multiplicity of solutions by using the Lusternik–Schnirelmann category. Next, we show that the number of normalized solutions is at least the number of global minimum points of 풱 , as ξ is small enough via Ekeland's variational principle. [ABSTRACT FROM AUTHOR]

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This article aims to challenge the prevailing dichotomous approaches to goal-directedness across diverse scientific disciplines. To that effect, it focuses on establishing the interconnectedness of diverse notions of purposiveness by examining the relationship between variational principles in physics and life sciences. Thus, the article aims to uncover a unifying aspect of goal-directedness that transcends the varying levels of complexity. [ABSTRACT FROM AUTHOR]

Recent coarse-grained (CG) models have often supplemented conventional pair potentials with potentials that depend upon the local density around each particle. In this work, we investigate the temperature-dependence of these local density (LD) potentials. Specifically, we employ the multiscale coarse-graining (MS-CG) force-matching variational principle to parameterize pair and LD potentials for one-site CG models of molecular liquids at ambient pressure. The accuracy of these MS-CG LD potentials quite sensitively depends upon the length-scale, rc, that is employed to define the local density. When the local density is defined by the optimal length-scale, r c * , the MS-CG potential often accurately describes the reference state point and can provide reasonable transferability across a rather wide range of temperatures. At ambient pressure, the optimal LD length-scale varies linearly with temperature over a very wide range of temperatures. Moreover, if one adopts this temperature-dependent LD length-scale, then the MS-CG LD potential appears independent of temperature, while the MS-CG pair potential varies linearly across this temperature range. This provides a simple means for predicting pair and LD potentials that accurately model new state points without performing additional atomistic simulations. Surprisingly, at certain state points, the predicted potentials provide greater accuracy than MS-CG potentials that were optimized for the state point. [ABSTRACT FROM AUTHOR]

The hierarchical equations of motion (HEOM) method is a numerically exact open quantum system dynamics approach. The method is rooted in an exponential expansion of the bath correlation function, which in essence strategically reshapes a continuous environment into a set of effective bath modes that allow for more efficient cutoff at finite temperatures. Based on this understanding, one can map the HEOM method into a Schrödinger-like equation, with a non-Hermitian super-Hamiltonian for an extended wave function being the tensor product of the central system wave function and the Fock state of these effective bath modes. In this work, we explore the possibility of representing the extended wave function as a tree tensor network state (TTNS) and the super-Hamiltonian as a tree tensor network operator of the same structure as the TTNS, as well as the application of a time propagation algorithm using the time-dependent variational principle. Our benchmark calculations based on the spin-boson model with a slow-relaxing bath show that the proposed HEOM+TTNS approach yields consistent results with those of the conventional HEOM method, while the computation is considerably sped up. In addition, the simulation with a genuine TTNS is four times faster than a one-dimensional matrix product state decomposition scheme. [ABSTRACT FROM AUTHOR]

We discuss the connection between the nonlinear thermodynamic formalism recently introduced by Buzzi, Kloeckner, and Leplaideur and the theory of generalized rotation sets which goes back to Poincaré. [ABSTRACT FROM AUTHOR]

We investigate the dynamics of Landau–Zener (LZ) transitions in an anisotropic, dissipative three-level LZ model (3-LZM) using the numerically accurate multiple Davydov D2Ansatz in the framework of the time-dependent variational principle. It is demonstrated that a non-monotonic relationship exists between the Landau–Zener transition probability and the phonon coupling strength when the 3-LZM is driven by a linear external field. Under the influence of a periodic driving field, phonon coupling may induce peaks in contour plots of the transition probability when the magnitude of the system anisotropy matches the phonon frequency. The 3-LZM coupled to a super-Ohmic phonon bath and driven by a periodic external field exhibits periodic population dynamics in which the period and amplitude of the oscillations decrease with the bath coupling strength. [ABSTRACT FROM AUTHOR]

The theory of diffusion processes in solids has achieved significant results in recent decades, but the development of methods for calculating diffusion in a multicomponent thermodynamic system is still an urgent task. Problems of diffusion in solid and liquid solutions with small deviations from the equilibrium state, or fluctuations, are of significant interest. The work develops a general methodology for calculating diffusion flows in a multicomponent thermodynamic system for small deviations from the equilibrium state. A connection has been established between the mechanical approach to the analysis of generalized systems and the phenomenological equations of nonequilibrium thermodynamics. Examples are given of the use of the developed methodology for the analysis of carbide transformations in chromium steel.

This article is about the following Kirchhoff–Schrödinger–Poisson system with steep potential well 1.1 −(a+b∫ℝ3|∇u|2dx)△u+λV(x)u+μϕ(x)u=f(x,u)+h(x)|u|αinℝ3,−△ϕ=u2,inℝ3,$$ \left\{\begin{array}{ll}-\left(a&#x0002B;b{\int}_{{\mathrm{\mathbb{R}}}&#x0005E;3}{\left&#x0007C;\nabla u\right&#x0007C;}&#x0005E;2 dx\right)\triangle u&#x0002B;\lambda V(x)u&#x0002B;\mu \phi (x)u&#x0003D;f\left(x,u\right)&#x0002B;h(x){\left&#x0007C;u\right&#x0007C;}&#x0005E;{\alpha }& \mathrm{in}\kern0.4em {\mathrm{\mathbb{R}}}&#x0005E;3,\\ {}-\triangle \phi &#x0003D;{u}&#x0005E;2,& \mathrm{in}\kern0.4em {\mathrm{\mathbb{R}}}&#x0005E;3,\end{array}\right. $$ where a,b,λ>0$$ a,b,\lambda >0 $$ are constants, μ>0$$ \mu >0 $$, and 0<α<1,f∈C(ℝN×ℝ,ℝ)$$ 0<\alpha <1,f\in C\left({\mathrm{\mathbb{R}}}&#x0005E;N\times \mathrm{\mathbb{R}},\mathrm{\mathbb{R}}\right) $$. By using the variational principle, we overcome the difficulties caused by Poisson's term and obtain system (1.1) that has two nontrivial solutions under certain assumptions. Moreover, we study the concentration of solutions and obtain new conclusions of system (1.1). Finally, we present the case where the solution to system (1.1) does not exist. [ABSTRACT FROM AUTHOR]

In this article, a nonlinear geometrically exact (GeX) hybrid-mixed four-node solid-shell element under non-conservative loading using the sampling surfaces (SaS) method is developed. The term "geometrically exact" means that the parametrization of the middle surface is known and, therefore, the coefficients of the first and second fundamental forms and the Christoffel symbols are taken exactly at element nodes. The SaS formulation is based on the choice of N SaS parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the displacements of these surfaces as fundamental shell unknowns. Such a choice of unknowns with the use of Lagrange polynomials of degree N–1 in the through-thickness distributions of displacements, strains, and stresses leads to an efficient higher-order shell formulation. To develop the hybrid-mixed four-node solid-shell element, which is free of shear and membrane locking, the Hu–Washizu variational principle is utilized. The proposed GeX hybrid-mixed solid-shell element subjected to follower loads shows superior performance in the case of coarse meshes and allows the use of only one load step in most of the considered numerical examples. It is established that the difference between the second Piola-Kirchhoff and Cauchy stresses in some non-conservative problems for isotropic and composite shells undergoing finite rotations can be significant. [ABSTRACT FROM AUTHOR]

We present a rigorous mathematical analysis of a non‐convex optimal control problem for mosquito populations. The nonlinear model for the dynamics of the mosquito population takes in consideration the iterations among the immature (aquatic) subpopulation, the adult winged subpopulation, and the environment resources; the immature subpopulation is assumed to be age‐structured. Moreover, the action of certain control mechanisms on these subpopulations is also taken in account. The cost functional to be minimized is non‐convex. The proof of the existence of an optimal control is done by using fixed point arguments and a special minimizing sequence obtained with the help of Ekeland's variational principle. [ABSTRACT FROM AUTHOR]

In this work, the variational direct method (VDM) based on the variational theory and Ritz-like method, combined with the fractal two-scale transformation, is employed to construct the semi-domain solutions of the (3+1)-dimensional fractal Jimbo–Miwa (JM) equation, which is derived by He's fractal derivative. The applicability and effectiveness of the acquired solutions on the Cantor sets are presented through the numerical results in the form of 3-dimensional graphs. In addition, as a comparison, the exact solutions of the classic (3+1)-dimensional JM equation for γ = 1 are also illustrated. The obtained results in this work are expected to open new perspectives for the traveling wave theory. [ABSTRACT FROM AUTHOR]

When the structure size is reduced to the micro- and nano-scale, a new mechanoelectric coupling effect, the flexoelectric effect, cannot be ignored. The governing equations and boundary conditions for the sound insulation problem of thin plate acoustic metamaterial structure with the flexoelectric effects were derived by means of the variational principle. The sound insulation curves of thin plate mass blocks were predicted based on the Kirchhoff theory. The effects of flexural effects, geometric sizes and mass densities on the sound insulation performances of the structure were systematically discussed. The results show that, for the micro- and nano-scale structure sizes, the flexoelectric effect significantly increases the sound insulation valley values and peak frequencies of the sound insulation curves, so it is necessary to consider the flexoelectric effects. This work provides a theoretical basis for the research of noise control in MEMS. [ABSTRACT FROM AUTHOR]

A recent mixed formulation of the Virtual Element Method in 2D elastostatics, based on the Hu-Washizu variational principle, is here extended to 2D elastodynamics. The independent modeling of the strain field, allowed by the mixed formulation, is exploited to derive first order quadrilateral Virtual Elements (VEs) not requiring a stabilization (namely, self-stabilized VEs), in contrast to the standard VEs, where an artificial stabilization is always required for first order quads. Lumped mass matrices are derived using a novel approach, based on an integration scheme that makes use of nodal values only, preserving the correct mass in the case of rigid-body modes. In the case of implicit time integration, it is shown how the combination of a self-stabilized stiffness matrix with a self-stabilized lumped mass matrix can produce excellent performances both in the compressible and quasi-incompressible regimes with almost negligible sensitivity to element distortion. Finally, in the case of explicit dynamics, the performances of the different types of derived VEs are analyzed in terms of their critical time-step size. [ABSTRACT FROM AUTHOR]

A mathematical model for time-nonlocal six-phase-lag generalized thermoelastic diffusion with two-temperature is proposed for a linear, isotropic and homogeneous thermoelastic diffusive continuum by considering modified Fourier's law of heat conduction together with modified Fick's law of mass diffusion. The modified Fourier's law includes temperature gradient and thermal displacement gradient among the constitutive variables whereas the modified Fick's law includes chemical potential gradient and the chemical potential displacement gradient among the constitutive variables. The Fourier's law of heat conduction is replaced by a fractional-order approximation to a modification of the Fourier's law with three different phase lags for the heat flux vector, the temperature gradient and the thermal displacement gradient whereas the Fick's law of mass diffusion is replaced by a fractional-order approximation to a modification of the Fick's law with three different phase lags for the mass flux vector, the chemical potential gradient and the chemical potential displacement gradient. The proposed model includes some of the existing thermoelastic diffusion models as special cases. A variational principle is derived and a uniqueness theorem is proved. Finally, a dynamic reciprocity theorem is established for the proposed generalized thermoelastic diffusion model. [ABSTRACT FROM AUTHOR]

A novel method for determining the optimized parameter of the four-node incompatible generalized mixed element is presented based on the equilibrium between strain energy and complementary energy. The presented energy formulations are derived from the generalized mixed variational principle, which contains an arbitrary additional parameter. The initial solutions expressed by the displacement field are firstly assumed for the description of the energy of each generalized mixed element. Then, the identical relationship between strain energy and complementary energy is subsequently expressed at element level, which includes the arbitrary parameter. At the same time, a formulation for determining the optimized parameter at element level is proposed. Several representative examples with varying geometrical parameters, boundary and loading conditions are used to validate this method. By contrasting with the results of generalized mixed elements with different parameter values and other traditional finite elements. The effectiveness of the presented method has been demonstrated. On one hand, by ensuring the strain energy and complementary energy remain consistent under both coarse and fine meshes, the optimized parameter can adjust the stiffness of the generalized mixed element, thereby enhancing its resemblance to the real elastic body. On the other hand, the generalized mixed element has the additional advantage of conveniently introducing stress boundary conditions, thereby satisfying the requirement for zero-conditions of shear stresses on the exterior surfaces of beams. The numerical results obtained by the proposed method are accurate and stable. [ABSTRACT FROM AUTHOR]

The use of global displacement basis functions to solve boundary-value problems in linear elasticity is well established. No prior work uses a global stress tensor basis for such solutions. We present two such methods for solving stress problems in linear elasticity. In both methods, we split the sought stress σ into two parts, where neither part is required to satisfy strain compatibility. The first part, σ p , is any stress in equilibrium with the loading. The second part, σ h is a self-equilibrated stress field on the unloaded body. In both methods, σ h is expanded using tensor-valued global stress basis functions developed elsewhere. In the first method, the coefficients in the expansion are found by minimizing the strain energy based on the well-known complementary energy principle. For the second method, which is restricted to planar homogeneous isotropic bodies, we show that we merely need to minimize the squared L 2 norm of the trace of stress. For demonstration, we solve nine stress problems involving sharp corners, multiple-connectedness, non-zero net force and/or moment on an internal hole, body force, discontinuous surface traction, material inhomogeneity, and anisotropy. The first method presents a new application of a known principle. The second method presents a hitherto unreported principle, to the best of our knowledge. [ABSTRACT FROM AUTHOR]

Influenza A is a highly contagious respiratory illness that spreads globally and results in millions of cases worldwide. The transmission rate and mortality rate of influenza in the population are determined by the load of the influenza A virus. In this study, we have developed a multi-scale immuno-influenza model considering incomplete vaccine immunity. We have calculated the basic reproduction numbers for both immunological and epidemiological models. Additionally, we have examined the stability of the disease-free equilibrium and the existence of the endemic equilibrium at the population level. To assess the cost-effectiveness of antiviral therapy at an individual level and vaccination at a population level, we have formulated an optimal control problem. The objective of this problem is to minimize both the number of influenza cases and the cost of vaccination. To solve this optimal control problem, we have employed the Ekeland variational principle. Furthermore, we have designed a global forward and backward sweep algorithm to estimate the impact of antiviral treatment on reducing influenza transmission among humans. Our findings indicate that the use of antiviral treatment significantly reduces viral loads in individuals. However, it may also increase the potential risk at the population level. Therefore, vaccination against influenza is the most effective option from a public health perspective. • A novel multi-scale influenza transmission model was proposed by the nested law. • A global sweep algorithm was established for solving an optimal control problem. • The real-world application was used to evaluate cost-effectiveness of strategies. • Vaccination was the best choice for curtailing an influenza spread in a long term. [ABSTRACT FROM AUTHOR]

For $ \bar \alpha >1 $ and $ \alpha \in (0, \bar \alpha] $, we study weighted transfer operators associated to a piecewise expanding $ {\mathcal{C}}^{\bar \alpha} $ map $ T $ on a compact manifold of dimension $ d\ge 1 $, and a piecewise $ {\mathcal{C}}^\alpha $ weight $ g $, acting on Sobolev spaces. We bound the essential spectral radius in terms of a topological pressure for a subadditive potential. Under a new small boundary pressure condition, we improve the estimate by establishing a variational principle for piecewise expanding maps and subadditive potentials. [ABSTRACT FROM AUTHOR]

In this work, we introduce a new principle that provides a tool for searching for odd periodic responses of a general nonlinear oscillator with symmetries. This result arises as the dual version of the variational principle first introduced in Ortega (2016), because in this case the nonlinearity $ xD(t, x) $ verifies the reversed inequality, i.e., $ D(t, 0)

We investigate the existence and stability of traveling waves for the sixth‐order Boussinesq equation, considering a broad class of nonlinearities adhering to power‐like scaling relations. Employing the Nehari manifold method, we establish the existence of traveling waves and derive variational criteria for assessing their stability or instability. Subsequently, we develop a numerical approach based on these variational principles to delineate regions of stability and instability. Finally, for homogeneous nonlinearities, we establish a sufficient condition for strong instability. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the following fractional p & q -Laplacian problem: { (− Δ) p s u + (− Δ) q s u = λ a (x) | u | θ − 2 u + μ b (x) | u | r − 2 u in Ω , u (x) = 0 in R N ∖ Ω , where Ω ⊂ R N is a bounded domain with smooth boundary, s ∈ (0 , 1) , (− Δ) m s (m ∈ { p , q }) is the fractional m-Laplacian operator, p , q , r , θ ∈ (1 , p s ∗ ] , p s ∗ = N p N − s p , λ , μ > 0 , and the weights a (x) and b (x) are possibly sign changing. Using the concentration compactness principle for fractional Sobolev spaces and the Ekeland variational principle, we prove that the problem admits a nonnegative solution for the critical case r = p s ∗ . Moreover, for the subcritical case r < p s ∗ , we obtain two existence results by applying the Ekeland variational principle and the mountain pass theorem. [ABSTRACT FROM AUTHOR]