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We study Γ-convergence of nonconvex integrals of the calculus of variations in strongly connected sets when the integrands do not have polynomial growth and can take infinite values. Applications to homogenization of unbounded integrals in strongly perforated sets are also developed. [ABSTRACT FROM AUTHOR]

This research presents a first attempt to analytically determine the asymmetric dynamic transverse characteristics of thin to moderately thick viscoelastic functionally graded porous (VFGP) annular plates. Firstly, the material properties of the plate are assumed to have various nonlinear distributions in terms of porosity coefficient. Secondly, the motion equations are obtained through the first-order shear deformation theory (FSDT) of elasticity, the energy method, and the variations calculus. Thirdly, the standard linear solid (SLS) model is adopted to consider the viscoelastic behavior of the plate. Finally, the perturbation procedure together with Fourier series are utilized to solve the system of partial differential equations, and the asymmetrically dynamic response is found in a closed-form solution. To assess the veracity of the analytical findings, an algorithm based on the finite element (FE) method called the user-defined field (USDFLD) code is developed. In order to benchmark the present study, the dynamic response of VFGP annular plates is scrutinized under two types of excitations (impulsive and step), four different types of radial load profiles (such as constant, linear, parabolic, and sine distributions), and various asymmetric circumferential loads. Moreover, the influence of various geometrical and material characteristics on the dynamic response of VFGP annular plates is investigated. [ABSTRACT FROM AUTHOR]

We exploit the existence and non-existence of positive solutions to the eigenvalue problem driven by the nonhomogeneous fractional $ p\& q $ p &q Laplacian operator with indefinite weights \[ \left(-\Delta_p\right)^{\alpha}u + \left(-\Delta_q\right)^{\beta}u = \lambda\left[a \left|u\right|^{p-2}u + b \left|u\right|^{q-2}u \right]\quad{\rm in}\ \Omega, \] (− Δ p) α u + (− Δ q) β u = λ [ a | u | p − 2 u + b | u | q − 2 u ] in Ω , where $ \Omega \subseteq \mathbb {R}^N $ Ω ⊆ R N is a smooth bounded domain that has been extended by zero. We further show the existence of a continuous family of eigenvalues in the case $ \Omega =\mathbb {R}^N $ Ω = R N and $ b\equiv 0 $ b ≡ 0 a.e. Our approach relies strongly on variational Analysis, in which the Mountain pass theorem plays the key role. Due to the lack of spatial compactness and the embedding $ \mathcal {W}^{\alpha, p}\left (\mathbb {R}^N\right) \hookrightarrow \mathcal {W}^{\beta, q}\left (\mathbb {R}^N\right) $ W α , p (R N) ↪ W β , q (R N) in $ \mathbb {R}^N $ R N , we employ the concentration-compactness principle of P.L. Lions [The concentration-compactness principle in the calculus of variations. The limit case. II, Rev Mat Iberoamericana. 1985;1(2):45–121]. to overcome the difficulty. Our paper can be considered as a counterpart to the important works [Alves et al. Existence, multiplicity and concentration for a class of fractional $ p\& q $ p &q Laplacian problems in $ \Bbb R^N $ R N , Commun Pure Appl Anal, 2019;18(4):2009–2045], [Benci et al. An eigenvalue problem for a quasilinear elliptic field equation. J Differ Equ, 2002;184(2):299–320], [Bobkov et al. On positive solutions for $ (p,q) $ (p , q) -Laplace equations with two parameters, Calc Var Partial Differ Equ, 2015;54(3):3277–3301], [Colasuonno and Squassina. Eigenvalues for double phase variational integrals, Ann Mat Pura Appl (4), 2016;195(6):1917–1956], [Papageorgiou et al. Positive solutions for nonlinear Neumann problems with singular terms and convection, J Math Pures Appl (9), 2020;136:1–21], [Papageorgiou et al. Ground state and nodal solutions for a class of double phase problems, Z Angew Math Phys, 2020;71:1–15], and may have further applications to deal with other problems. [ABSTRACT FROM AUTHOR]

In this paper, a Euler wavelets method for solving the variational problems is presented. The operational matrices of integration and product of Euler wavelets are calculated. Then, by using Euler wavelets and the operational matrices, the variational problems are reduced into the system of algebraic equations. Furthermore, the convergence analysis and error bound of the Euler wavelets method are given. Some examples are included to demonstrate the applicability and validity of the schemes. [ABSTRACT FROM AUTHOR]

Motivated by some landslide models, and related estimation challenges, this paper presents an optimal estimation method for state and parameter in a special class of so-called ODE–PDE system based on the adjoint method for discrete-time asynchronous measurements. This system is described by a pair of coupled Ordinary Differential Equation (ODE) and Partial Differential Equation (PDE), with a mixed boundary condition for the PDE. The coupling appears both in the ODE and in the Neuman boundary condition of the PDE. For this system, initial conditions or state variables and some empirical parameters are assumed to be unknown and need to be estimated. The Lagrangian multiplier method is used to connect the dynamics of the system and the cost function defined as the least square error between the simulation values and the available measurements. The adjoint state method is applied to the objective functional to get the adjoint system and the gradients with respect to parameters and initial state. The cost functional is optimised, employing the steepest descent method to estimate parameters and initial state. This general approach is illustrated by two application examples corresponding to two different landslide models that validate the presented optimal estimation approach. The first one is about state and parameter estimation in an extended sliding-consolidation landslide model, and the second one is in the viscoplastic sliding-consolidation landslide model. [ABSTRACT FROM AUTHOR]

We study the dynamics of particles in cold electron plasma medium based on two dissimilar approaches: the fractional actionlike variational and fractal calculus approaches. In each case, the corresponding Boltzmann and Vlasov–Boltzmann equations were derived. Although the mathematical relationships between fractional calculus and fractal calculus were established in the literature, it was revealed throughout this study that the corresponding physics for its approach is quite different. Each model is characterized by its corresponding Boltzmann and Vlasov–Boltzmann equations which describe dissimilar dynamics and gives rise to unrelated Bohm–Gross formulas for electron plasma waves (dispersion relations) and different group velocities connected to the numerical ranges of the matching fractional parameter. Several consequences were obtained and discussed accordingly. [ABSTRACT FROM AUTHOR]

Non-zonal methods of hybrid Reynolds Averaged Navier–Stokes (RANS) equations-Large Eddy Simulation (LES) with subfilter transport equations enable to simulate turbulent flows out of spectral equilibrium on relatively coarse grid resolution. They can operate from RANS to LES depending on a control parameter linked to the grid step size. Variational analysis gives a useful framework to specify the functional dependence of this parameter to the grid size. This is the case for the partially integrated transport modelling method originally established in spectral space for homogeneous turbulence. The partitioning control function then monitors the ratio of the subfilter part to the total turbulent energy and the same process is developed in the present work for the thermal or transported scalar variance as well. So, we demonstrate here that this control function mechanism can be derived by variational calculus from a mathematical physics formalism developed both for first-order and second-order moment closures. We show that the result is entirely consistent with the spectral model derivation made in previous papers and that this approach can be transposed to almost any subfilter transport model developed in hybrid RANS-LES methodology in full generality. As a result, it will be evidenced also that resolved turbulent diffusion terms play a significant role in the acting mechanisms of turbulence and cannot be therefore neglected, even if these terms do not explicitly appear in LES subfilter closure because they are computed by the simulation itself and not modelled by the subfilter model. In addition, numerical simulations have been performed for illustrating the theoretical results of the variational analysis. [ABSTRACT FROM AUTHOR]

The article considers a high-order optimal control problem and its dual problems described by high-order differential inclusions. In this regard, the established Euler–Lagrange type inclusion, containing the Euler–Poisson equation of the calculus of variations, is a sufficient optimality condition for a differential inclusion of a higher order. It is shown that the adjoint inclusion for the first-order differential inclusions, defined in terms of a locally adjoint mapping, coincides with the classical Euler–Lagrange inclusion. Then the duality theorems are proved. [ABSTRACT FROM AUTHOR]

In this paper, an optimal switch-time control problem is solved for a class of impulsive switched autonomous systems. The considered systems jump at the switching times, and the sequence of active subsystems is pre-specified. The control variables consist of the impulse times and a set of scalars which determine the jump amplitudes. Moreover, the subsystems do not require a refractory period, which can bring more generality. Using the calculus of variation, the partial derivatives of the cost with respect to the control variables are derived, based on which the optimality conditions are given. Meanwhile, the obtained formulas can be used in some gradient descent algorithms to locate the optimal control variables. Finally, the viability of the proposed method is illustrated through two numerical examples. [ABSTRACT FROM AUTHOR]

We study a variational approach introduced by S.D. Fisher and the author in the 1970s in the context of norm minimization for differentiable mappings occurring in nonlinear elliptic boundary value problems. It may be viewed as an abstract version of the calculus of variations. A strong hypothesis, initially limiting the scope of this approach, is the assumption of a bounded minimizing sequence in the least squares formulation. In this article, we employ regularization and invariant regions to overcome this obstacle. A consequence of the framework is the convergence of approximations for regularized problems to a desired solution. The variational method is closely associated with the implicit function theorem, and it can be jointly studied, so that continuous parameter stability is naturally deduced. A significant aspect of the theory is that the reaction term in a reaction-diffusion equation can be selected to act globally as in the steady Schrödinger-Hartree equation. Local action, as in the non-equilibrium Poisson-Boltzmann equation, is also included. Both cases are studied at length prior to the development of a general theory. [ABSTRACT FROM AUTHOR]

This paper investigates a Pareto-optimal insurance problem, where the insured maximizes her rank-dependent utility preference and the insurer is risk-neutral and employs the mean-variance premium principle. To eliminate potential moral hazard issues, we only consider the so-called moral-hazard-free insurance contracts that obey the incentive compatibility constraint. The insurance problem is first formulated as a non-concave maximization problem involving Choquet expectation, then turned into a concave quantile optimization problem and finally solved by the calculus of variations method. The optimal contract is expressed by a semi-linear second-order double-obstacle ordinary differential equation with nonlocal operator. An effective numerical method is proposed to compute the optimal contract assuming the probability weighting function has a density. Also, we provide an example that is analytically solved. [ABSTRACT FROM AUTHOR]

This approach present improved higher-order layer-wise theory for the investigation of flawlessly wedged sandwich-composite beams with general laminate configurations. Our analysis incorporates the continuity assumption of interlaminar shear stresses and in-plane and flexural displacements between interfaces. In addition, interlaminar shear stresses are constrained using the Lagrange multiplier technique by introducing new unknown variables. These unknown variables are expressed with interlaminar strain energy, assuming that the strain energy is continuous throughout the overall thickness of the beam. To govern the newly introduced and other unknown variables, the total potential energy (TPE) is minimised using variational calculus. The numerical analysis results show that our approach provides enhanced accuracy to examine sandwich-composite beams. [ABSTRACT FROM AUTHOR]

A web (continuum) traveling in a flow of ideal fluid between two fixed supports with a constant velocity and performing transverse vibrations is considered by modeling the web as an isotropic type elastic panel. The fluid flow is considered as a potential flow. The web transverse vibration process is analyzed taking into account both material elasticity and the fluid-structure interaction between the traveling material and the surrounding flowing fluid. It is supposed that the actuator system is used for suppression of occurring transverse vibrations. An optimization approach based on the modern control theory and calculus of variations is proposed for determining the optimal suppressive actions at a given time interval. The quality of suppression vibration process is evaluated by means of the functional depending on displacements and velocities at the terminal time. The example (carried out in an analytical way) illustrating all basic stages of finding of optimal vibration suppression is presented. [ABSTRACT FROM AUTHOR]

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]

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]

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]

The Landau-de Gennes elastic free energy of liquid crystals is a functional expressed in terms of a symmetric traceless tensor order parameter. We derive convexity criteria for SO(3)-invariant expansions of the elastic energy that are quadratic in the gradient and up to fourth-order overall in the tensor order parameter. To do so we write the integrand as the sum of squares of linearly independent spherical tensors from which definiteness criteria can be ascertained. In particular, transformations between Cartesian and spherical tensors representations are computed based on irreducible tensor representations of the rotation group. [ABSTRACT FROM AUTHOR]

The paper provides objective motivation for the class of non–local moment prior, introduced for the purposes of Bayesian hypothesis testing using Bayes factors. The motivation is that it minimizes Fisher information among a large class of density. Being a minimizer of some measure of information is an important feature so that the test, even within a Bayesian framework, has an objective criterion. [ABSTRACT FROM AUTHOR]

This paper proposed a new land cover mapping algorithm for remote sensing images under the presence of clouds. Here, we modelled the observed image as the weighted sum between the reflectances from the cloud-free image and clouds where weights depend on the degree of cloud contamination called the 'cloud thickness' map. Next, we represented the cloud thickness map using a level set function. Similarly, we also represented the land cover map as functions of different level set functions whose values indicate the class label. Next, the maximum a posteriori (MAP) criterion is employed where the most likely land cover and cloud thickness maps are chosen. Under the MAP criterion, the corresponding energy function whose values depend on the level set functions can be derived where the minimum energy point corresponds to the optimum land cover and cloud thickness maps. Since the level set functions are continuous, the optimum solution can be obtained by applying the calculus of variation where the gradient descent algorithm is employed. From the observation model, the cloud-free data can only be reconstructed only if thin to medium clouds are present since thick clouds can completely block the reflectance from the land cover materials. Thus, our algorithm is more suitable for thin to medium than thick cloud covers. Our synthesis and real cloud-contamination examples support our statement since our algorithm achieves significantly higher overall accuracies over other classification techniques, especially, in the thin to medium cloud contamination, and fails to recover the true land cover classes over the thick clouds. [ABSTRACT FROM AUTHOR]

In this paper, we give an overview of some recent applications of a minimax theorem. These applications deal with: uniquely remotal sets in normed spaces; multiple global minima for the integral functional of the Calculus of Variations; multiple periodic solutions for Lagrangian systems of relativistic oscillators; variational inequalities in balls of Hilbert spaces. [ABSTRACT FROM AUTHOR]

In this article, we give an in-depth analysis of the problem of optimising the total population size for a standard logistic-diffusive model. This optimisation problem stems from the study of spatial ecology and amounts to the following question: assuming a species evolves in a domain, what is the best way to spread resources in order to ensure a maximal population size at equilibrium? In recent years, many authors contributed to this topic. We settle here the proof of two fundamental properties of optimisers: the bang-bang one, which had so far only been proved under several strong assumptions, and the other one is the fragmentation of maximisers. We prove the bang-bang property in all generality using a new spectral method. The technique introduced to demonstrate the bang-bang character of optimisers can be adapted and generalised to many optimisation problems with other classes of bilinear optimal control problems where the state equation is semilinear and elliptic. We comment on it in a conclusion section. Regarding the geometry of maximisers, we exhibit a blow-up rate for the BV-norm of maximisers as the diffusivity gets smaller: if Ω is an orthotope and if m μ is an optimal control, then | | m μ | | B V ≳ 1 / μ. The proof of this results relies on a very fine energy argument. [ABSTRACT FROM AUTHOR]

We address the properties of the catenary in its role of generating a minimal rotational area around the abscissa. We first correct a small flaw in an excellent, highly respected text on the calculus of variations. We then highlight the fact that, for numerous boundary conditions, there is an infinitely large number of continuously differentiable functions that outperform the catenary. We will show this thanks to beautiful parametric expressions found and provided to us by Ernst Hairer. [ABSTRACT FROM AUTHOR]

In the article 'How long is my toilet roll-a simple exercise in mathematical modelling' several models of increasing complexity are introduced and solved to calculate indirectly the length of paper on a toilet-roll. All these results are presented without errors. The authors of this comment believe the error analysis of measurements made in a laboratory is an important part of the experiment. The calculation of the absolute errors in indirect measurements provides us with information about the confidence that we must have in our experimental results. For that reason, the error calculation for all approximations has been made. [ABSTRACT FROM AUTHOR]

In this note, we propose an elementary method to study the existence and uniqueness of solutions to a type of variational problems which arise naturally in the theory of large deviations. This type of problems involves a movable boundary and may not have the coercivity condition in general. Our method is elementarily based on direct analysis over the space of absolutely continuous functions and specific properties of the underlying functional. [ABSTRACT FROM AUTHOR]

In this paper, the fuzzy variational problem and fuzzy optimal control problem are considered. The granular Euler–Lagrange condition for the fuzzy variational problem and necessary conditions of Pontryagin type for fixed and free final state fuzzy optimal control problem are derived based on the concepts of horizontal membership function (HMF) and granular differentiability with the calculus of variations. Further, based on the proposed solution method, the solutions of fuzzy optimal control problem, i.e., optimal fuzzy control, and corresponding optimal fuzzy state are always fuzzy functions. Finally, the proposed algorithm used to summarize the main steps of solving the fuzzy variational problem and fuzzy optimal control problem numerically using He's variational iteration method (VIM). [ABSTRACT FROM AUTHOR]

In the present work, by taking advantage of a so-called practical limitation of fractional derivatives, namely, the absence of a simple chain and Leibniz's rules, we proposed a generalized fractional calculus of variation where the Lagrangian function depends on fractional derivatives of differentiable functions. The Euler–Lagrange equation we obtained generalizes previously results and enables us to construct simple Lagrangians for nonlinear systems. Furthermore, in our main result, we formulate a Noether-type theorem for these problems that provides us with a means to obtain conservative quantities for nonlinear systems. In order to illustrate the potential of the applications of our results, we obtain Lagrangians for some nonlinear chaotic dynamical systems, and we analyze the conservation laws related to time translations and internal symmetries. [ABSTRACT FROM AUTHOR]

We study some variational problems related to the controllability of the transition between two quantum states of the driven quantum harmonic oscillator. We focus on the existence of minimum points of the total energy functional which is obtained by adding to the cost of the control a term penalizing the distance from the maximum transition probability between given energy states. [ABSTRACT FROM AUTHOR]

We define the concept of good trade execution and we construct explicit adapted good trade execution strategies in the framework of linear temporary market impact. Good trade execution strategies are dynamic, in the sense that they react to the actual realisation of the traded asset price path over the trading period; this is paramount in volatile regimes, where price trajectories can considerably deviate from their expected value. Remarkably, however, the implementation of our strategies does not require the full specification of an SDE evolution for the traded asset price, making them robust across different models. Moreover, rather than minimising the expected trading cost, good trade execution strategies minimise trading costs in a pathwise sense, a point of view not yet considered in the literature. The mathematical apparatus for such a pathwise minimisation hinges on certain random Young differential equations that correspond to the Euler–Lagrange equations of the classical Calculus of Variations. These Young differential equations characterise our good trade execution strategies in terms of an initial value problem that allows for easy implementations. [ABSTRACT FROM AUTHOR]

One of the most important parameters in the fluidization process is to determine the minimum fluidization velocity. For this purpose, the conservation equation of momentum in the vertical transport of fluids and solids were used. In this study, the pressure drop term was replaced with Ergun's equation and the drag force was considered a linear function. The drag coefficient was extremized using the calculus of variations and the relationship between fluidization velocity and voidage was determined. Two test cases with their experimental data were used for validating the presented drag model. Drag functions obtained in previous studies did not match with the empirical data in the bed volume fraction range of 0.45–0.59. The present study reveals that the reason for this difference lies in the use of the Darcy pressure-drop and neglecting the importance of the linear velocity term in the drag model. For vessels of small diameter, the wall effect becomes important. The frictional pressure drop proposed by Ergun is appropriate because this correlation consists of a viscous term. Also, it is expected that by increasing the beds' voidage, the energy drop decreases. Contrary to Darcy's model, presented drag model predicts this behavior perfectly. [ABSTRACT FROM AUTHOR]

In this paper, the analysis is focused on the optimization of some simple, multiple or curvilinear integral functionals (governed by second-order Lagrangians) subject to ODEs, PDEs or isoperimetric constraints. In this context, using appropriate techniques of calculus of variations and some geometric tools, necessary conditions of optimality are formulated. As well, the theory developed in this paper is illustrated with applications. [ABSTRACT FROM AUTHOR]

The theory of calculus of variations is a mathematical tool which is widely used in different scientific areas in particular in physics and chemistry. This theory is strongly related with optimisation. In fact the former seeks to optimise an integral related with some physical magnitude over some space to an extremum by varying a function of the coordinates. On the other hand, reaction paths and potential energy surfaces, in particular their stationary points, are the basis of many chemical theories, in particular reactions rate theories. We present a review where it is gathered together the variational nature of many types of reaction paths: steepest descent, Newton trajectories, artificial force induced reaction (AFIR) paths, gradient extremals, and gentlest ascent dynamics (GAD) curves. The variational basis permits to select the best optimisation technique in order to locate important theoretical objects on a potential energy surface. [ABSTRACT FROM AUTHOR]

A sonography image processing system for tumour segmentation is implemented by using the active contour model based on level set formulation. To achieve extraction of weak edges in speckle noisy images, the level set model called Edge Attraction Force on Chan and Vese model is developed. The cell competition algorithm is modified by adding the information of the cells' edges. Calculus of variation is taken to derive the evolution equation of the active contour. Experiments demonstrate that the boundaries of tumours extracted by the proposed active contour are more accurate for conquering the drawbacks coming from weak edges. Experiments demonstrate that the boundaries of tumors extracted by the proposed active contour are accurate and the results avoid the drawbacks of the weak edges. [ABSTRACT FROM AUTHOR]

Takahasi and Wakimoto (1968) derived a sharp upper bound on the efficiency of the balanced ranked-set sampling (RSS) sample mean relative to the simple random sampling (SRS) sample mean under perfect rankings. The bound depends on the set size and is achieved for uniform distributions. Here we generalize the Takahasi and Wakimoto (1968) result by finding a sharp upper bound in the case of unbalanced RSS. The bound depends on the particular unbalanced design, and the distributions where the bound is achieved can be highly nonuniform. The bound under perfect rankings can be exceeded under imperfect rankings. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with questions of the existence of solution for a class of nonlocal and nonhomogeneous Neumann boundary value problems involving the -Laplacian in which the nonlinear terms assume both critical growth. The main tools used are the Lions' Concentration-Compactness Principle [Lions PL. The concentration-compactness principle in the calculus of variations. The limit case I, part 1. Rev Mat Iberoam. 1985;1(1):145–201.] for variable exponent spaces and the Mountain Pass Theorem. [ABSTRACT FROM AUTHOR]

We consider a general calculus of variations problem with lower semicontinuous data which includes an optimal control problem with a dynamic differential inclusion constraint as a particular case. We provide a new approach to the derivation of necessary optimality conditions for such a problem, based on a new variant of the multidirectional mean-value inequality for smooth Banach spaces. The multidirectional mean-value inequality is a result in nonlinear and nonsmooth analysis which provides estimates for the minimum difference in function values between a closed convex subset of a Banach space E and a point. [ABSTRACT FROM AUTHOR]

We extend some results of the Itô calculus to functionals of the current path of a process to reflect the fact that often the impact of randomness is cumulative and depends on the history of the process, not merely on its current value. We express the differential of the functional in terms of adequately defined partial derivatives to obtain an Itô formula. We develop an extension of the Feynman-Kac formula to the functional case and an explicit expression of the integrand in the Martingale Representation Theorem. We establish that under certain conditions, even path dependent options prices satisfy a partial differential equation in a local sense. We exploit this fact to find an expression of the price difference between two models and compute variational derivatives with respect to the volatility surface. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of at least three distinct solutions for a perturbed anisotropic discrete Dirichlet problem. Our approach is based on recent variational methods for smooth functionals defined on reflexive Banach spaces. Some examples are presented to demonstrate the application of our main results. [ABSTRACT FROM AUTHOR]

Real world problems are embedded with uncertainties. Therefore, to tackle these problems, one must consider probabilistic nature of the problems both in modeling and solution. In this work, concepts of convergence of the solution of variational inequality in classical functional analysis are extended to a stochastic domain for a random Mann-type iterative and Ishikawa-type iterative schemes in a Banach space. A mean square convergence result is proved for this extension. [ABSTRACT FROM AUTHOR]

Cone-beam X-ray luminescence computed tomography (CB-XLCT) is an attractive hybrid imaging modality, and it has the potential of monitoring the metabolic processes of nanophosphors-based drugs in vivo. However, the XLCT imaging suffers from a severe ill-posed problem. In this work, a sparse nonconvex Lp (0 < p < 1) regularization was utilized for the efficient reconstruction for early detection of small tumour in CB-XLCT imaging. Specifically, we transformed the non-convex optimization problem into an iteratively reweighted scheme based on the L1 regularization. Further, an iteratively reweighted split augmented lagrangian shrinkage algorithm (IRW_SALSA-Lp) was proposed to efficiently solve the non-convex Lp (0 < p < 1) model. We studied eight different non-convex p-values (1/16, 1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8) in both 3D digital mouse experiments and in vivo experiments. The results demonstrate that the proposed non-convex methods outperform L2 and L1 regularization in accurately recovering sparse targets in CB-XLCT. And among all the non-convex p-values, our Lp(1/4 < p < 1/2) methods give the best performance. [ABSTRACT FROM AUTHOR]

We provide a maximum entropy derivation of a new family of BFGS-like methods. Similar results are then derived for block BFGS methods. This also yields an independent proof of a result of Fletcher 1991 and its generalization to the block case. [ABSTRACT FROM AUTHOR]

In this paper, a scheduling and routing based on AGVs processing and waiting times and the existing paths in a jobshop manufacturing system is proposed. A mathematical formulation is developed to schedule and route multiple AGVs handling jobs to shops so that the total delay of AGVs including the earliness and tardiness is minimized. A Lagrangian relaxation method is developed and a sub-gradient algorithm is composed to update the iterations in the searching process of method. The results show that the method is efficient in larger sizes problems while exact method cannot obtain the solutions in reasonable time. Two different problems are solved using two algorithms of Lagrangian relaxation and linear relaxation. Statistical comparisons showed much better performance of Lagrangian relaxation approach in a negligible run time. For larger sizes that exact method cannot be obtained even in long run times, Lagrangian relaxation approach is useful and efficient. [ABSTRACT FROM AUTHOR]

Multistage ranked-set sampling (MRSS) is a generalization of ranked-set sampling in which multiple stages of ranking are used. It is known that for a fixed distribution under perfect rankings, each additional stage provides a gain in efficiency when estimating the population mean. However, the maximum possible efficiency for the MRSS sample mean relative to the simple random sampling sample mean has not previously been determined. In this paper, we provide a method for computing this maximum possible efficiency under perfect rankings for any choice of the set size and the number of stages. The maximum efficiency tends to infinity as the number of stages increases, and, for large numbers of stages, the efficiency-maximizing distributions are symmetric multi-modal distributions where the number of modes matches the set size. The results in this paper correct earlier assertions in the literature that the maximum efficiency is bounded and that it is achieved when the distribution is uniform. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a contraction algorithm for solving monotone variational inclusion problem. To reach this goal, our main iterative algorithm combine Dong’s projection and contraction algorithm with resolvent operator. Under suitable assumptions, we prove that the sequence generated by our main iterative algorithm converges weakly to the solution of the considered problem. Finally, we give two numerical examples to verify the feasibility of our main algorithm. [ABSTRACT FROM AUTHOR]

We consider a deformable body in frictionless unilateral contact with a moving rigid obstacle. The material is described by a viscoelastic law with short memory, and the contact is modeled by a Signorini condition with a time-dependent gap. The existence and uniqueness results for a weak formulation based on a Lagrange multipliers approach are provided. Furthermore, we discuss an efficient algorithm approximating the weak solution for the more general case of a two-body contact problem including friction. In order to illustrate the theory we present two numerical examples in 3D. [ABSTRACT FROM AUTHOR]

A mathematical model describing the contact between a viscoplastic body and a deformable foundation is analyzed under small deformation hypotheses. The process is quasistatic and in normal direction the contact is with adhesion, normal compliance, memory effects and unilateral constraint. We derive a mixed-variational formulation of the problem using Lagrange multipliers. Finally, we prove the unique weak solvability of the contact problem. [ABSTRACT FROM AUTHOR]

Numerous types of carbon nanostructure have been found experimentally, including nanotubes, fullerenes and nanocones. These structures have applications in various nanoscale devices and the joining of these structures may lead to further new configurations with more remarkable properties and applications. The join profile between different carbon nanostructures in a symmetric configuration may be modelled using the calculus of variations. In previous studies, carbon nanostructures were assumed to deform according to perfect elasticity, thus the elastic energy, depending only on the axial curvature, was used to determine the join profile consisting of a finite number of discrete bonds. However, one could argue that the relevant energy should also involve the rotational curvature, especially when its size is comparable to the axial curvature. In this paper, we use the Willmore energy, a natural generalisation of the elastic energy that depends on both the axial and rotational curvatures. Catenoids are absolute minimisers of this energy and pieces of these may be used to join various nanostructures. We focus on the cases of joining a fullerene to a nanotube and joining two fullerenes along a common axis. By comparing our results with the earlier work, we find that both energies give similar joining profiles. Further work on other configurations may reveal which energy provides a better model. [ABSTRACT FROM AUTHOR]

This paper investigates causal optimal transport problems. Within this framework, primal attainments and dual formulations are obtained under standard hypothesis, for the related variational problems. Causal transport plans are intrinsically related to martingales by a preserving property. Specific concretizations yield primal problems equivalent to several classical problems of stochastic control, and of stochastic calculus; trivial filtrations yield usual problems of optimal transport. [ABSTRACT FROM AUTHOR]

In this work, we consider the optimal control problem of linear quadratic regulation for discrete time-variant systems with single input and multiple input delays. An innovative and simple method to derive the optimal controller is given. The studied problem is first equivalently converted into a problem subject to a constraint condition. Last, with the established duality, the problem is transformed into a static mathematical optimisation problem without input delays. The optimal control input solution to minimise performance index function is derived by solving this optimisation problem with two methods. A numerical simulation example is carried out and its results show that our two approaches are both feasible and very effective. [ABSTRACT FROM AUTHOR]