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A comprehensive study is reported herein for the evaluation of Lagrangian functions for linear systems possessing symmetric or nonsymmetric coefficient matrices. Contrary to popular beliefs, it is shown that many coupled linear systems do not admit Lagrangian functions. In addition, Lagrangian functions generally cannot be determined by system decoupling unless further restriction such as classical damping is assumed. However, a scalar function that plays the role of a Lagrangian function can be determined for any linear system by decoupling. This generalized Lagrangian function produces the equations of motion and it contains information on system properties, yet it satisfies a modified version of the Euler-Lagrange equations. Subject to this interpretation, a solution to the inverse problem of linear Lagrangian dynamics is provided.

In the present paper, we discuss the existence and uniqueness of solution for higher-order calculus of variations problems, involving composition of functionals. Also, higher-order DuBois-Reymond conditions in the Sobolev space W m , p ([ t 1 , t 2 ] ; R) are proven, both in integral and differential form, and under additional constraints. We consider the higher-order Noether's theorem and discuss invariance conditions for the main problem. [ABSTRACT FROM AUTHOR]

In this paper, we study discrete elliptic Dirichlet problems. Applying a variational technique together with Morse theory, we establish several results on the existence and multiplicity of nontrivial solutions. Finally, two examples and numerical simulations are provided to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

Of paramount importance in both ecological systems and economic policies are the problems of harvesting of natural resources. A paradigmatic situation where this question is raised is that of fishing strategies. Indeed, overfishing is a well-known problem in the management of live-stocks, as being too greedy may lead to an overall dramatic depletion of the population we are harvesting. A closely related topic is that of Nash equilibria in the context of fishing policies. Namely, two players being in competition for the same pool of resources, is it possible for them to find an equilibrium situation? The goal of this paper is to provide a detailed analysis of these two queries (i.e optimal fishing strategies for single-player models and study of Nash equilibria for multiple players games) by using a basic yet instructive mathematical model, the logistic-diffusive equation. In this framework, the underlying model simply reads - μ Δ θ = θ (K (x) - α (x) - θ) where K accounts for natural resources, θ for the density of the population that is being harvested and α = α (x) encodes either the single player fishing strategy or, when dealing with Nash equilibria, a combination of the fishing strategies of both players. This article consists of two main parts. The first one gives a very fine characterisation of the optimisers for the single-player game where one aims at solving sup α ∫ Ω α θ , under L ∞ and L 1 constraints on the fishing strategies α . In particular, we show that, depending on the value of these constraints, this optimal control problem may behave like a convex or, conversely, concave problem. We also provide a detailed analysis of the large diffusivity limit of this problem. In the case where two players are involved, we rather write α as α 1 + α 2 where α i , the fishing strategy of the i-th player, also satisfies L ∞ and L 1 constraints. Defining I 1 : = ∫ Ω α i θ we aim at finding a Nash equilibrium. We prove the existence of Nash equilibria in several different regimes and investigate several related qualitative queries, for instance providing examples of the well-known tragedy of commons. Our study is completed by a variety of numerical simulations that illustrate our results and allow us to formulate open questions and conjectures. [ABSTRACT FROM AUTHOR]

This paper studies the design of Pareto-optimal insurance under the heterogeneous beliefs of the insured and insurer. To accommodate a wide range of belief heterogeneity, we allow the likelihood ratio function to be non-monotone. To prevent the ex post moral hazard issue, the incentive compatibility condition is exogenously imposed to restrict the indemnity function. An implicit characterization of the optimal indemnity function is presented first by using the calculus of variations. Based on the point-wise maximizer to the problem, we partition the domain of loss into disjoint pieces and derive the parametric form of the optimal indemnity function over each piece through its implicit characterization. The main result of this paper generalizes those in the literature and provides insights for related problems. [ABSTRACT FROM AUTHOR]

In this article, a new approach based on fuzzy systems is used for solving time delay fractional order optimal control problems. The fractional derivatives are considered in the Atangana–Baleanu sense that is a new derivative with the nonsingular and nonlocal kernel. By means of the calculus of variations and the formula for fractional integration by parts, the necessary optimality conditions associated to the time delay problem is derived. In order to solve the obtained optimality system, the solution of the system is first approximated by fuzzy solutions with adjustable parameters. The optimality system is then reduced to an unconstrained optimization problem by using appropriate error function. A learning algorithm is also presented to achieve the parameters of these fuzzy solutions. The efficiency and accuracy of the proposed approach are assessed through some illustrative examples of the time delay fractional optimal control problems. [ABSTRACT FROM AUTHOR]

This paper studies a class of superlinear damped vibration equations with nonlocal boundary conditions on time scales by using the calculus of variations. We consider the Cerami condition, while the nonlinear term does not satisfy Ambrosetti-Rabinowitz condition such that the critical point theory could be applied. Then we establish the variational structure in an appropriate Sobolev's space, obtain the existence of infinitely many large energy solutions. Finally, two examples are given to prove our results. [ABSTRACT FROM AUTHOR]

Minimal set, 49Q20, Plateau problem, Reifenberg problem, calculus of variation, geometric measure theory Keywords: Plateau problem; Reifenberg problem; minimal set; calculus of variation; geometric measure theory; 49Q20 EN Plateau problem Reifenberg problem minimal set calculus of variation geometric measure theory 49Q20 913 927 15 10/03/22 20221001 NES 221001 1 Definitions and main results 1.1 Introduction We present the Reifenberg approach to the Plateau problem ([[19]], 1960). [Extracted from the article]

26 F. Seuffert, An extension of the Bianchi-Egnell stability estimate to Bakry, Gentil, and Ledoux's generalization of the Sobolev inequality to continuous dimensions, J. Funct. Morrey's inequality, calculus of variations, extremal, Euler-Lagrange, p-Laplacian, functional inequality, p-Laplace, 49J40, 31C45, nonlinear potential theory, stability estimate, 35J62 Keywords: Morrey's inequality; nonlinear potential theory; calculus of variations; extremal; Euler-Lagrange; p-Laplacian; functional inequality; stability estimate; p-Laplace; 35J62; 49J40; 31C45 EN Morrey's inequality nonlinear potential theory calculus of variations extremal Euler-Lagrange p-Laplacian functional inequality stability estimate p-Laplace 35J62 49J40 31C45 863 877 15 10/03/22 20221001 NES 221001 1 Introduction xtremals, Trans. Amer. Math. Soc. 360 (2008), no. 8, 4335-4347. [Extracted from the article]

8 V. Caselles, On the entropy conditions for some flux limited diffusion equations, J. Differential Equations 250 (2011), no. 8, 3311-3348. 18 (2018), no. 1, 1-28. 30 G. P. Leonardi and G. Saracco, The prescribed mean curvature equation in weakly regular domains, NoDEA Nonlinear Differential Equations Appl. 25 (2018), no. 2, Paper No. 9. 31 G. P. Leonardi and G. Saracco, Rigidity and trace properties of divergence-measure vector fields, Adv. Calc. Keywords: Divergence-measure vector fields; functions of bounded variation; coarea formula; Gauss-Green formula; semicontinuity; 26B30; 49Q15; 49J45 EN Divergence-measure vector fields functions of bounded variation coarea formula Gauss-Green formula semicontinuity 26B30 49Q15 49J45 787 810 24 10/03/22 20221001 NES 221001 1 cations, Ann. [Extracted from the article]

Differential games are a class of game theory problems governed by differential equations. Differential games are often defined in the continuous domain and solved by the calculus of variations. However, modelling and solving these games are not straightforward tasks. Differential games, like game theory, are often involved with social dilemmas and social behaviours. Modelling these social phenomena with mathematical tools is often problematic. In this paper, we modelled deception to increase the pay-off in differential games. Deception is modelled as a bi-level policy system, and each level is modelled with a fuzzy controller. Fuzzy controllers are trained using a novel hierarchical fuzzy actor-critic learning algorithm. A deceitful player plays against multiple opponents. Although there is one ultimate goal for the player, it can choose multiple fake goals as well. The intention is to find a strategy to switch between the fake goals and the true goal to fool the opponents. The simulation platform is the game of guarding territories, a specific form of the pursuit–evasion games. We propose a method to easily increase the number of defenders with minimum changes in the policies. We create a universal structure that is not affected by the curse of dimensionality. We show that a discerning invader capable of using deception can improve its performance against the defenders by increasing the chance of invasion. We investigate the single-invader single-defender game and the single-invader multi-defender game. We study the superior invader and agents with the same speed. In all mentioned situations, the invader increases its pay-off by using deception versus being honest. A two-level policy system is used in this paper to model deception. The lower-level policy controls each goal's invasion actions, while the higher-level policy controls deception where a successful game is not initially possible. [ABSTRACT FROM AUTHOR]

Hamiltonian variational principles have provided, since the 1960s, the means of developing very successful wave theories for nonlinear free-surface flows, under the assumption of irrotationality. This success, in conjunction with the recognition that almost all flows in the sea are not irrotational, raises the question of extending Hamilton's principle to rotational free-surface flows. The Euler equations governing the bulk fluid motion have been derived by means of Hamilton's principle since the late 1950s. Nevertheless, a complete variational formulation of the rotational water-wave problem, including the derivation of the free-surface boundary conditions, seems to be lacking until now. The purpose of the present work is to construct such a missing variational formulation. The appropriate functional is the usual Hamilton's action, constrained by the conservation of mass and the conservation of fluid parcels' identity. The differential equations governing the bulk fluid motion are derived as usually, applying standard methods of the calculus of variations. However, the standard methodology does not provide enough structure to obtain the free-surface boundary conditions. To overcome this difficulty, differential-variational forms of the aforementioned constraints are introduced and applied to the boundary variations of the Eulerian fields. Under this transformation, both kinematic and dynamic free-surface conditions are naturally derived, ensuring the Hamiltonian variational formulation of the complete problem. An interesting feature, appearing in the present variational derivation, is a dual possibility concerning the tangential velocity on the boundary; it may be either the same as in irrotational flow (no condition) or zero, corresponding to the small-viscosity limit. The deeper meaning and the significance of these findings seem to deserve further analysis. [ABSTRACT FROM AUTHOR]

The aim of this work is to study several problems of the calculus of variations, where the dynamics of the state function is given by a generalized fractional derivative. This derivative combines two well-known concepts: fractional derivative with respect to another function and fractional derivative of variable order. We present the Euler-Lagrange equation, which is a necessary condition that every optimal solution of the problem must satisfy. Other problems are also studied: with integral and holonomic constraints, with higher order derivatives, and the Herglotz variational problem. [ABSTRACT FROM AUTHOR]

We present a mixed finite element method for approximating a fourth-order elliptic partial differential equation (PDE), the Kirchhoff plate equation, on a surface embedded in |${\mathbb {R}}^{3}$|⁠ , with or without boundary. Error estimates are given in mesh-dependent norms that account for the surface approximation and the approximation of the surface PDE. The method is built on the classic Hellan–Herrmann–Johnson method (for flat domains), and convergence is established for |$C^{k+1}$| surfaces, with degree |$k$| (Lagrangian, parametrically curved) approximation of the surface, for any |$k \geqslant 1$|⁠. Mixed boundary conditions are allowed, including clamped, simply-supported and free conditions; if free conditions are present then the surface must be at least |$C^{2,1}$|⁠. The framework uses tools from differential geometry and is directly related to the seminal work of Dziuk, G. (1988) Finite elements for the Beltrami operator on arbitrary surfaces. Partial Differential Equations and Calculus of Variations , vol. 1357 (S. Hildebrandt & R. Leis eds). Berlin, Heidelberg: Springer, pp. 142–155. for approximating the Laplace–Beltrami equation. The analysis here is the first to handle the full surface Hessian operator directly. Numerical examples are given on nontrivial surfaces that demonstrate our convergence estimates. In addition, we show how the surface biharmonic equation can be solved with this method. [ABSTRACT FROM AUTHOR]

We consider a family of three-dimensional stiffened plates whose dimensions are scaled through different powers of a small parameter ε . The plate and the stiffener are assumed to be linearly elastic, isotropic, and homogeneous. By means of Γ -convergence, we study the asymptotic behavior of the three-dimensional problems as the parameter ε tends to zero. For different relative values of the powers of the parameter ε , we show how the interplay between the plate and the stiffener affects the limit energy. We derive twenty-three limit problems. [ABSTRACT FROM AUTHOR]

Copyright of Problems of Friction & Wear is the property of National Aviation University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In the setting of the Euclidean space equipped with an arbitrary Radon measure, we prove the equivalence between several notions of function of bounded variation present in the literature. We also study the relation between various definitions of 1-Sobolev function. [ABSTRACT FROM AUTHOR]

The long-time average behavior of the value function in the calculus of variations is known to be connected to the existence of the limit of the corresponding Abel means. Still in the Tonelli case, such a limit is in turn related to the existence of solutions of the critical (or ergodic) Hamilton-Jacobi equation. The goal of this paper is to address similar issues when set on the whole Euclidean space and the Hamiltonian fails to be Tonelli. We first study the convergence of the time-averaged value function as the time horizon goes to infinity, proving the existence of the critical constant (Mañé critical value) for a general control system. Then, we show that the ergodic equation admits solutions for systems associated with a family of vector fields which satisfies the Lie Algebra rank condition. Finally, we construct a solution to the critical HJB equation on the whole space which coincides with its Lax-Oleinik evolution. [ABSTRACT FROM AUTHOR]

The evaluation of the roof collapse in tunnels or cavities remains one of the most complex issues in geotechnical engineering. Taking the detaching surface of the tunnel roof collapse as an orthotropic weak interlayer, an analytical approach for determining the limit collapse range considering the arch effect of the tunnel is presented in this paper by the variation calculus. A discontinuity criterion moving from the anisotropic criterion proposed by the present authors is applied to the orthotropic interlayer. The phenomenon of sharp points in collapse blocks is further analyzed. Based on the proposed approach, illustrated examples are analyzed to investigate the effect of the strength parameters and the consideration of the collapse cusp, which show different influence laws on the range of collapse blocks. Those interesting conclusions can provide guidance for the prediction of the collapse mechanism of the tunnel. [ABSTRACT FROM AUTHOR]

In this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from inherent misuse of objects and for finding natural formulations of various objects. A tensor bundle formalism, crucially relying on the notion of pullback bundle, will be used to create a rich type system with which to distinguish objects. The type system and relevant notation is designed to "telescope" to accommodate a level of detail appropriate to a set of calculations. Various techniques using this formalism will be developed and demonstrated with the goal of providing a relatively complete and uniform method of coordinate-free computation. The calculus of variations pertaining to maps between Riemannian manifolds will be formulated using the strongly typed tensor formalism and associated techniques. Energy functionals defined in terms of first-order Lagrangians are the focus of the second half of this paper, in which the first variation, the Euler–Lagrange equations, and the second variation of such functionals will be derived. [ABSTRACT FROM AUTHOR]

As a continuation of Part I, dedicated to the ground state of He‐like and Li‐like isoelectronic sequences for nuclear charges Z≤20, and Part II, dedicated to two excited states of He‐like sequence, two ultra‐compact wave functions in the form of generalized Guevara–Harris–Turbiner functions are constructed for Li‐like sequence. They describe accurately the domain of applicability of the non‐relativistic Quantum Mechanics of Coulomb Charges (QMCC) for energies (2–3 significant digits [s.d.]) of the spin‐quartet state 140+ of Li‐like ions (in static approximation with point‐like, infinitely heavy nuclei). Variational parameters are fitted in Z by 2nd degree polynomials. The most accurate ultra‐compact function leads to the absolute accuracy ∼10−3 a.u. for energy, and ∼10−4 for the normalized electron‐nuclear cusp parameter for Z≤20. Critical charge Z=ZB, where the ultra‐compact trial function for the 140+ state loses its square‐integrability, is estimated, ZB140+∼1.26−1.30. As a complement to Part I, square integrability for the compact functions constructed for the ground, spin‐doublet state120+ of the Li‐like sequence is discussed. The critical charge, for which these functions stop to be normalizable, is estimated as ZB120+=1.62−1.65. It implies that at Z=2—the negative helium ion He−—both states, 120+ and 140+, exist as states embedded in the continuum. [ABSTRACT FROM AUTHOR]

Rail irregularity presents a major threat to the high-speed train (HST) safety on a high-speed rail (HSR) bridge. Lateral bridge deformation has a significant impact on geometric parameters of the rail. In this paper, the influence of lateral bridge deformation on the train operation on the HSR bridge is investigated. Variational calculus is employed to establish the analytical mapping between the lateral bridge and rail deformation. By superimposing the mapped rail deformation and initial random rail irregularity, the composed rail unevenness under lateral bridge deformation is obtained. Thereafter, by adopting the dynamical model of the train-track-bridge coupling system (TTBCS), the dynamic behaviour of the train moving across the HSR track-bridge system is analysed. Based on safety and comfortability control indicators of the HST operation, effects of different types of lateral bridge deformations with various amplitudes on the train's operation performances are investigated. The results indicate that the bilateral girder's mutual translation has the most significant impact on the train's operation. When the HST is moving with the velocity of 350 km/h, lateral bridge deformation amplitude thresholds are suggested as 17.43 and 8.12 mm to ensure the safety and comfortability of the HST operation, respectively. The obtained results can provide beneficial pointers for HSR bridge design. [ABSTRACT FROM AUTHOR]

The throughput of a belt conveyor system is the primary design parameter when considering a new installation, determined by the cross-section of the material on the belt, coupled with the belt speed. Optimizing this area not only improves efficiency, but also minimizes capital costs through the optimal selection of equipment. Whilst speed-related optimization has seen considerable attention, the cross-sectional area has largely been neglected due to existing design constraints of the system. Conventional belt conveyors typically utilize a 3-idler troughing configuration, which forms a trapezoidal cross-section with a parabolic surcharge. The rigidity of this support directly limits the geometry of the cross-section that may be considered. A new conveyor system developed at the University of Newcastle supports the conveyor belt by a rail-based carriage, with no relative movement between the belt and carriage. This configuration allows the cross-section of the belt to be freely optimized in order to maximize the material throughput for a given belt width, or alternatively to minimize the belt width for a given throughput. This article utilizes the calculus of variations to optimize the form of this cross section, and demonstrates that an increase in throughput of up to 30% is possible, compared to troughed installations. [ABSTRACT FROM AUTHOR]

In this paper we apply a smoothing technique for the maximum function, based on the compensated convex transforms, originally proposed by Zhang in [1] to construct some computable multiwell non-negative quasiconvex functions in the calculus of variations. Let with a finite set in a linear subspace without rank-one matrices of the space of real matrices. Our main aim is to construct computable quasiconvex lower bounds for the following two multiwell models with possibly uneven wells: i) Let be an -Lipschitz mapping with and , where is a control parameter, and ii) , where with invertible linear transforms for . If the control paramenter is sufficiently large, our quasiconvex lower bounds are 'tight' in the sense that near each 'well' the lower bound agrees with the original function, and our lower bound are of . We also consider generalisations of our constructions to other simple geometrical multiwell models and discuss the implications of our constructions to the corresponding variational problems. [ABSTRACT FROM AUTHOR]

In this work, we propose to succinctly review, in non technical terms, the current mathematical state of brittle fracture first put forth by A. A. Griffith in his seminal work Griffith (Philos Trans R Soc Lond CCXXI-A:163–198, 1920), then re-interpreted under the label "variational fracture" in Francfort and Marigo (J. Mech Phys Solids 46(8):1319–1342, 1998). and subsequent works. We will only address the sharp theory and will limit ourselves to a theoretical exposition, leaving phase-field approximations and other possible implementations to other articles within this volume. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of the steady-state solution of a nonlinear parabolic equation modeling the capillary formation in tumor angiogenesis. The analysis is based on the Lax-Milgram Theorem in variational calculus. Proving the existence and uniqueness of this steady-state shows that there is only one way for endothelial cells to follow the trail of transition probability density function. [ABSTRACT FROM AUTHOR]

Recent studies have extended the classical space-charge limited current (SCLC) solution in a non-magnetic, planar diode with zero injection velocity to other geometries using variational calculus (VC). We further extend VC to solve for SCLC with a non-relativistic, monoenergetic injection velocity from first principles for nonplanar diodes. By extremizing either the current or a functional of the electric field (and not its derivative), we demonstrate that VC can find either the bifurcation or the SCLC solution, respectively. The bifurcation solution is characterized by the onset of particle reflection, resulting in a singularity in the derivative of the electric field at the virtual cathode, physically analogous to the singularity at the cathode in SCLC for zero injection velocity. Alternatively, using VC to extremize a functional of the potential and its gradient (electric field) yields the maximum current SCLC result. We then derive the SCLC solutions in cylindrical and spherical diodes; additionally, we develop a method to determine SCLC numerically and the bifurcation solution exactly for any orthogonal geometry. Implications for the potential profile and virtual cathode are discussed, especially the behavior for other geometries. [ABSTRACT FROM AUTHOR]

Λ-fractional differential equations are discussed since they exhibit non-locality and accuracy. Fractional derivatives form fractional differential equations, considered as describing better various physical phenomena. Nevertheless, fractional derivatives fail to satisfy the prerequisites of differential topology for generating differentials. Hence, all the sources of generating fractional differential equations, such as fractional differential geometry, the fractional calculus of variations, and the fractional field theory, are not mathematically accurate. Nevertheless, the Λ-fractional derivative conforms to all prerequisites demanded by differential topology. Hence, the various mathematical forms, including those derivatives, do not lack the mathematical accuracy or defects of the well-known fractional derivatives. A summary of the Λ-fractional analysis is presented with its influence on the sources of differential equations, such as fractional differential geometry, field theorems, and calculus of variations. Λ-fractional ordinary and partial differential equations will be discussed. [ABSTRACT FROM AUTHOR]

The augmented Young–Laplace (AYL) equation has been widely used to describe the liquid–vapor interfaces affected by capillarity and adsorption, particularly in nanoporous or clay minerals where strong adsorptive interaction prevails. Still, how adsorption ultimately shapes the liquid–vapor interface in pores remains elusive. All current forms of the AYL equation only consider the attractive van der Waals interaction for complete wetting surface and are mostly an extension of the general equation based on thermodynamic principles without rigorous derivation. In this paper, closed forms of the AYL equations are derived by variational calculus method for two typical pore models considering effects of capillarity and adsorption, based on the minimization of free energy for the equilibrium system. The van der Waals, electrical, and structural forces are incorporated as disjoining pressure into the proposed AYL equations to quantify effects of both attractive and repulsive interactions (hence complete and partial wetting conditions) on interfacial geometry and equilibrium matric potential. Numerical calculation algorithms are established to solve the interfacial geometric profiles and characteristics. The proposed model illustrates a transition region on the interface where geometry and physical characteristics are drastically altered under adsorption compared with sole capillarity. Different surface wettabilities are examined for interface profiles and curvature distributions. An apparent contact angle on the partial wetting surface is identified that is approaching complete wetting as the adsorption dominates when pore size reduces. Water retention and interparticle force demonstrate clear distinctions in their evolution, with different particle sizes and varying half‐filling angles under adsorptive conditions as water volume changes. Core Ideas: Adsorption is considered to derive the updated augmented Young–Laplace equations for pore liquid interface.A transition from capillary to adsorptive regime is identified on interface by vast pressure and curvature change.The intermolecular mechanism reveals apparent contact angle receding to zero due to adsorption on partial wetting. [ABSTRACT FROM AUTHOR]

t-SNE is a commonly used force-based nonlinear dimensionality reduction method. This paper has two contributions: the first is forceful colorings, an idea that is also applicable to other force-based methods (UMAP, ForceAtlas2,...). In every equilibrium, the attractive and repulsive forces acting on a particle cancel out: however, both the size and the direction of the attractive (or repulsive) forces acting on a particle are related to its properties: the force vector can serve as an additional feature. Secondly, we analyze the case of t-SNE acting on a single homogeneous cluster (modeled by affinities coming from the adjacency matrix of a random k-regular graph); we derive a mean-field model that leads to interesting questions in classical calculus of variations. The model predicts that, in the limit, the t-SNE embedding of a single perfectly homogeneous cluster is not a point but a thin annulus of diameter ∼ k - 1 / 4 n - 1 / 4 . This is supported by numerical results. The mean field ansatz extends to other force-based dimensionality reduction methods. query Please check the edit made in the article title. [ABSTRACT FROM AUTHOR]