Variational Principles

We i n vestigate the dynamics of one dimensional mass-spring chain with non-monotone dependence of the spring force vs. spring elongation. For this strongly nonlinear system we nd a family of exact solutions that represent the nonlinear waves. We h a ve found numerically that this system displays a dynamical phase transition from the stationary phase when all masses are at rest to the twinkling phase when the masses oscillate in a wave motion. This transition has two fronts, which propagate with di erent speeds. We study this phase transition analytically and derive the relations between its quantitative c haracteristics.

In this paper we address the adequacy of various approximate methods of including Coulomb distortion effects in (e, e ′ ) reactions by comparing to an exact treatment using Dirac-Coulomb distorted waves. In particular, we examine approximate methods and analyses of (e, e ′ ) reactions developed by Traini et al. using a high energy approximation of the distorted waves and phase shifts due to Lenz and Rosenfelder. This approximation has been used in the separation of longitudinal and transverse structure functions in a number of (e, e ′ ) experiments including the newly published 208 P b(e, e ′ ) data from Saclay. We find that the assumptions used by Traini and others are not valid for typical (e, e ′ ) experiments on medium and heavy nuclei, and hence the extracted structure functions based on this formalism are not reliable. We describe an improved approximation which is also based on the high energy approximation of Lenz and Rosenfelder and the analyses of Knoll and compare our results to the Saclay data. At each step of our analyses we compare our approximate results to the exact distorted wave results and can therefore quantify the errors made by our approximations. We find that for light 2 sin 4 θ 2 , and S L and S T are the longitudinal and transverse structure functions which depend only on the momentum transfer q and the energy transfer ω. By keeping the momentum and energy transfers fixed while varying the electron energy E and scattering angle θ e , it is possible to extract the two structure functions with two measurements. However, when the electron wavefunctions are not Dirac plane waves, but rather are distorted by the static Coulomb field of the target nucleus, such a simple formulation as given in Eq. ( ) is no longer possible and, in general, the cross section does not separate into the sum of longitudinal and transverse structure functions with coefficients which only depend on the electron kinematics. Clearly the size of the Coulomb distortion effects depends on the charge of the target nucleus and on the energy of the electrons. In general, Coulomb effects are not too large near the peaks of cross sections, but can have greatly magnified effects when one extracts "structure functions" by subtracting one cross section from another. For example, in a recent distorted wave calculation [1] of 16 O(e, e ′ p) in the quasielastic region, we found Coulomb effects on the extracted spectroscopic factors to be approximately 3%, while the effects on the extracted fourth structure function was approximately 15%. However, some sort of extraction of structure functions, albeit approximate, is still very appealing since structure functions are sensitive to different aspects of the underlying knockout process or the final state interaction. Furthermore, (e, e ′ ) reactions in the quasielastic

Abstract: The smallest enclosing circle problem asks for the circle of smallest radius enclosing a given set of finite points on the plane. This problem was introduced in the 19th century by Sylvester [17]. After more than a century, the problem remains very active. This ...

The dissertation is devoted to the development of variational analysis and generalized differentiation in infinite dimensions. We derive new calculus rules for both first-order partial subdifferentials and second-order partial subdifferentials in the framework of general Banach spaces as well as more developed rules in the framework of Asplund spaces. This calculus is applied in the study of sensitivity analysis for

We generate conservation laws for the one dimensional nonconservative Fokker-Planck (FP) equation, also known as the Kolmogorov forward equation, which describes the time evolution of the probability density function of position and velocity of a particle, and associate these, where possible, with Lie symmetry group generators. We determine the conserved vectors by a composite variational principle and then check if the condition for which symmetries associate with the conservation law is satisfied. As the Fokker-Planck equation is evolution type, no recourse to a Lagrangian formulation is made. Moreover, we obtain invariant solutions for the FP equation via potential symmetries.

The goal of this paper is to investigate the Theta invariant -an invariant of framed 3manifolds associated with the lowest order contribution to the Chern-Simons partition function -in the context of the quantum BV-BFV formalism. Namely, we compute the state on the solid torus to low degree in , and apply the gluing procedure to compute the Theta invariant of lens spaces. We use a distributional propagator which does not extend to a compactified configuration space, so to compute loop diagrams we have to define a regularization of the product of the distributional propagators, which is done in an ad hoc fashion. Also, a polarization has to be chosen for the quantization process. Our results agree with results in the literature for one type of polarization, but for another type of polarization there are extra terms.

We consider a Neumann problem of the type −ε∆u + F (u(x)) = 0 in an open bounded subset Ω of R n , where F is a real function which has exactly k maximum points. Using Morse theory we find that, for ε suitably small, there are at least 2k nontrivial solutions of the problem and we give some qualitative information about them.

The dissertation is devoted to the development of variational analysis and generalized differentiation in infinite dimensions. We derive new calculus rules for both first-order partial subdifferentials and second-order partial subdifferentials in the framework of general Banach spaces as well as more developed rules in the framework of Asplund spaces. This calculus is applied in the study of sensitivity analysis for

On the basis of the implicit standard materials that introduces a function, called bipotential, depending on both the stress and plastic strain rate, this paper is devoted to present a new approach of shakedown analysis for non standard elastoplastic materials. The bipotential theory was successfully applied to geomaterials with non-associated flow rule and Coulomb's dry friction law. The present analysis is different to the Melan's potential and it is based on a corner stone inequality satisfied by the bipotential and the existence of time-independent residual stress field. The deduction of bound theorems, static and kinematic, is detailed in the present article.

The metriplectic formalism couples Poisson brackets of the Hamiltonian description with metric brackets for describing systems with both Hamiltonian and dissipative components. The construction builds in asymptotic convergence to a preselected equilibrium state. Phenomena such as friction, electric resistivity, thermal conductivity and collisions in kinetic theories are well represented in this framework. In this paper we present an application of the metriplectic formalism of interest for the theory of control: a suitable torque is applied to a free rigid body, which is expressed through a metriplectic extension of its "natural" Poisson algebra. On practical grounds, the effect is to drive the body to align its angular velocity to rotation about a stable principal axis of inertia, while conserving its kinetic energy in the process. On theoretical grounds, this example shows how the non-Hamiltonian part of a metriplectic system may include convergence to a limit cycle, the first example of a non-zero dimensional attractor in this formalism. The method suggests a way to extend metriplectic dynamics to systems with general attractors, e.g. chaotic ones, with the hope of representing biophysical, geophysical and ecological models.

We introduce a new class of algebras that we call Lie antialgebras. The subject is situated in between commutative algebra, symplectic/contact geometry and Lie superalgebra theory. We define the odd Lie-Poisson structure on the space dual to a Lie antialgebra and study the notion of central extensions. We classify simple finite-dimensional Lie antialgebras.

A method is proposed for deriving dynamical equations for systems with both rigid and flexible components. During the derivation, each flexible component of the system is represented by a "surrogate element" which captures the response characteristics of that component and is easy to mathematically manipulate. The derivation proceeds essentially as if each surrogate element were a rigid body. Application of an extended form of Lagrange's equation yields a set of simultaneous differential equations which can then be transformed to be the exact, partial differential equations for the original flexible system. This method's use facilitates equation generation either by an analyst or through application of software-based symbolic manipulation.

Clarke has given a robust definition of subgradients of arbitrary Lipschitz continuous functions f on R", but for purposes of minimization algorithms it seems essential that the subgradient multifunction af have additional properties, such as certain special kinds of semicontinuity, which are not automatic consequences of f being Lipschitz continuous. This paper explores properties of 3 f that correspond to f being subdifferentially regular, another concept of Clarke's, and to f being a pointwise supremum of functions that are k times continuously differentiable.

In order to provide a vision of power electrical engineering for future energy-supply crossroads and challenges, variational principles are used to express local laws of electromagnetism : they are derived from thermodynamic principles and by assuming a reversible transfer of the global power exchanged throughout the electrical network. This approach is suitable for consolidating energy processes involved in electromagnetic and electromechanical conversions, from deep within the structure of the materials to the power network operations. Two kinds of devices are identified: (i) those dedicated to power conversion; and (ii) their couplings involved in transmission lines, distribution busways, cablings and connections. Furthermore, the approach is shown to provide a unique description framework for power management, energy efficiency and sustainable long-term planning exercises.

At first sight, aiming for energy efficiency appears to penalize electricity carriers. A close look at electrical energy flow, from primary to final energy, reveals abysmally low efficiency, resulting from the generation mix, transmission and distribution losses, and obsolete equipment. As a consequence, the share of primary electricity has to be reduced in order to achieve low carbon society scenarios in line with energy saving policies. Since it relies on a partial description of the electrical workflow, this viewpoint reveals certain weaknesses that need to be qualified with a long-term planning assessment. We propose circumventing this drawback using a thermodynamic analysis based on the Gibbs free energy of the electromagnetic field. In particular, the so-called Maxwell-Faraday law of induction is the local result of an optimal path towards reversibility of the entire power system, described as a monotherm engine. This approach supports a multi-scale analysis that naturally addresses those levels that involve huge losses, namely the functional materials, conversion devices and transmission network. As a result, the role played by magnetic energy is clarified: although it is required locally for conversion purposes, magnetic energy's overall value governs the transmission of electrical power through the grid. In addition to indicating kinetic reserve, an evaluation of the power system's reliability should include an assessment of magnetic energy from the very first moment of fluctuation (typically a few ms).

At first sight, aiming for energy efficiency appears to penalize electricity carriers. A close look at electrical energy flow, from primary to final energy, reveals abysmally low efficiency, resulting from the generation mix, transmission and distribution losses, and obsolete equipment. As a consequence, the share of primary electricity has to be reduced in order to achieve low carbon society scenarios in line with energy saving policies. Since it relies on a partial description of the electrical workflow, this viewpoint reveals certain weaknesses that need to be qualified with a long-term planning assessment. We propose circumventing this drawback using a thermodynamic analysis based on the Gibbs free energy of the electromagnetic field. In particular, the so-called Maxwell-Faraday law of induction is the local result of an optimal path towards reversibility of the entire power system, described as a monotherm engine. This approach supports a multi-scale analysis that naturally addresses those levels that involve huge losses, namely the functional materials, conversion devices and transmission network. As a result, the role played by magnetic energy is clarified: although it is required locally for conversion purposes, magnetic energy's overall value governs the transmission of electrical power through the grid. In addition to indicating kinetic reserve, an evaluation of the power system's reliability should include an assessment of magnetic energy from the very first moment of fluctuation (typically a few ms).

In order to provide a vision of power electrical engineering for future energy-supply crossroads and challenges, variational principles are used to express local laws of electromagnetism : they are derived from thermodynamic principles and by assuming a reversible transfer of the global power exchanged throughout the electrical network. This approach is suitable for consolidating energy processes involved in electromagnetic and electromechanical conversions, from deep within the structure of the materials to the power network operations. Two kinds of devices are identified: (i) those dedicated to power conversion; and (ii) their couplings involved in transmission lines, distribution busways, cablings and connections. Furthermore, the approach is shown to provide a unique description framework for power management, energy efficiency and sustainable long-term planning exercises.

Variational principles for magnetohydrodynamics were introduced by previous authors both in Lagrangian and Eulerian form. In this paper we introduce simpler Eulerian variational principles from which all the relevant equations of barotropic magnetohydrodynamics can be derived. The variational principle is given in terms of six independent functions for non-stationary barotropic flows and three independent functions for stationary barotropic flows. This is less then the seven variables which appear in the standard equations of barotropic magnetohydrodynamics which are the magnetic field B the velocity field v and the density ρ. The equations obtained for non-stationary barotropic magnetohydrodynamics resemble the equations of Frenkel, Levich & Stilman [1]. The connection between the Hamiltonian formalism introduced in [1] and the present Lagrangian formalism (with Eulerian variables) will be discussed. Finally the relations between barotropic magnetohydrodynamics topological constants and the functions of the present formalism will be elucidated.

Spectral properties of the Hamiltonian function which characterizes a trapped ion are investigated. In order to study semiclassical dynamics of trapped ions, coherent state orbits are introduced as sub-manifolds of the quantum state space, with the Kähler structure induced by the transition probability. The time dependent variational principle is applied on coherent states' orbits. The Hamilton equations of motion on Kähler manifolds of the type of classical phase spaces naturally arise. The associated classical Hamiltonian is obtained from the expected values on symplectic coherent states of the quantum Hamiltonian. Spectral information is thus coded within the phase portrait. We deal with the bosonic realization of the Lie algebra of the SU(1,1) group, which we particularize for the case of an ion confined in a combined, Paul and Penning trap. This formalism can be applied to Hamiltonians which are nonlinear in the infinitesimal generators of a dynamical symmetry group, such as the case of ions confined in electrodynamic traps. Discrete quasienergy spectra are obtained and the corresponding quasienergy states are explicitly realized as coherent states parameterized by the stable solutions of the corresponding classical equations of motion. A correspondence between quantum and classical stability domains is thus established, using the Husimi representation.

Collective many-body dynamics for time-dependent quantum Hamiltonian functions is investigated, for a dynamical system that exhibits multiple degrees of liberty, that is a combined (Paul and Penning) trap. We introduce the generators of the Lie algebra of the symplectic group SL(2,R), which we use to build the coherent states (CS) associated to the system under investigation. The trapped ion is treated as a harmonic oscillator (HO) to which we associate the quantum Hamilton function. We obtain the kinetic and potential energy operators as functions of the Lie algebra generators, and supply the expressions for the classical coordinate, momentum, kinetic and potential energy, as well as the total energy. We also infer the dispersions for the coordinate and momentum, the asymmetry and the flatness parametre for the distribution. The system interaction with the laser radiation is also examined for a system of identical two-level atoms. The Hamilton function for the Dicke model is obtain...

The variational principle of statistical mechanics accurately predicts behavior of materials and their mixtures, phase transitions, systems at extreme conditions, and intermolecular interactions. We have used it to develop variational calculation schemes to predict thermodynamic properties and phase transitions, including analytic equations of state.
This includes the chronological list of our selected publications:

Integrated and differential cross sections for the excitation of the b '2"+ state of H" from the ground state, by electron impact, are presented. The energy range of interest is 10.5 to 40.0 eV. The close-coupling equations for this process are solved in five different exchange approximations: exact two-electron exchange terms, no orthogonality; exact two-electron exchange terms, orthogonality; semiclassical exchange terms, noorthogonality, with barrier term; semiclassical exchange terms, no orthogonality, no barrier term; and semiclassical exchange terms, orthogonality, with barrier term. The purpose is to judge the reliability of the semiclassical exchange approximation for electronic excitation of molecules by electron impact at intermediate energies. The results are compared with recent theoretical estimates for the same process.

The variational principle is an approximation method that allows one to obtain accurate estimates of a quantity using relatively crude trial functions for the physical behavior. This principle is applied to transducer analysis by coupling a variational principle developed for the driving element (including piezoelectric effects) to one for the shell. The motion of the transducer is, in turn, coupled to a variational principle for the pressure in a fluid medium. The modeling of a class V flextensional ring-shell projector will be presented in detail. The in-vacuo resonance frequencies for the piezoelectric ring and spherical shell, as well as the in-vacuo mode shapes for the driver–shell combination will be compared with existing theory and finite element analyses. Results for a free-flooded piezoelectric ring and immersed spherical shell will be presented as tests of the variational fluid loading formulation. Admittance calculations and beam patterns for a single element will be com...

We derive a family of ideal (nondissipative) 3D soundproof fluid models that includes both the Lipps-Hemler anelastic approximation (AA) and the Durran pseudo-incompressible approximation (PIA). This family of models arises in the Euler-Poincaré framework involving a constrained Hamilton's principle expressed in the Eulerian fluid description. The derivation in this framework establishes the following properties of each member of the entire family: the Kelvin-Noether circulation theorem, conservation of potential vorticity on fluid parcels, a Lie-Poisson Hamiltonian formulation possessing conserved Casimirs, a conserved domain integrated energy and an associated variational principle satisfied by the equilibrium solutions. Having set the stage with the derivations of 3D models using the constrained Hamilton's principle, we then derive the corresponding 2D vertical slice models for these soundproof theories.

A variational framework is defined for vertical slice models with three-dimensional velocity depending only on x and z. The models that result from this framework are Hamiltonian, and have a Kelvin-Noether circulation theorem that results in a conserved potential vorticity in the slice geometry. These results are demonstrated for the incompressible Euler-Boussinesq equations with a constant temperature gradient in the y-direction (the Eady-Boussinesq model), which is an idealized problem used to study the formation and subsequent evolution of weather fronts. We then introduce a new compressible extension of this model. Unlike the incompressible model, the compressible model does not produce solutions that are also solutions of the threedimensional equations, but it does reduce to the Eady-Boussinesq model in the low Mach number limit. Hence, the new model could be used in asymptotic limit error testing for compressible weather models running in a vertical slice configuration.

In this article, we consider the rolling (or development) of two Riemannian connected manifolds $(M,g)$ and $(\hat{M},\hat{g})$ of dimensions $2$ and $3$ respectively, with the constraints of no-spinning and no-slipping. The present work is a continuation of \cite{MortadaKokkonenChitour}, which modelled the general setting of the rolling of two Riemannian connected manifolds with different dimensions as a driftless control affine system on a fibered space $Q$, with an emphasis on understanding the local structure of the rolling orbits, i.e., the reachable sets in $Q$. In this paper, the state space $Q$ has dimension eight and we show that the possible dimensions of non open rolling orbits belong to the set $\{2,5,6,7\}$. We describe the structures of orbits of dimension $2$, the possible local structures of rolling orbits of dimension $5$ and some of dimension $7$.

A variational principle for Lagrangian densities containing derivatives of real order is formulated and the invariance of this principle is studied in two characteristic cases. Necessary and sufficient conditions for an infinitesimal transformation group (basic Nöther's identity) are obtained. These conditions extend the classical results, valid for integer order derivatives. A generalization of Nöther's theorem leading to conservation laws for fractional Euler-Lagrangian equation is obtained as well. Results are illustrated by several concrete examples. Finally, an approximation of a fractional Euler-Lagrangian equation by a system of integer order equations is used for the formulation of an approximated invariance condition and corresponding conservation laws.

The deterministic KPZ equation has been recently formulated as a gradient flow. Its non-equilibrium analog of a free energy-the "non-equilibrium potential" [h], providing at each time the landscape where the stochastic dynamics of h(x,t) takes place-is however unbounded, and its exact evaluation involves all the detailed histories leading to h(x,t) from some initial configuration h 0 (x,0). After pinpointing some implications of these facts, we study the time behavior of [h] t (the average of [h] over noise realizations at time t) and show the interesting consequences of its structure when an external driving force F is included. The asymptotic form of the time derivative ˙[ h] is shown to be valid for any substrate dimensionality d, thus providing a valuable tool for studies in d > 1.

Maximal monotone relations and the second derivatives of nonsmooth functions Annales de l'I. H. P., section C, tome 2, n o 3 (1985), p. 167-184 <http://www.numdam.org/item?id=AIHPC_1985__2_3_167_0> © Gauthier-Villars, 1985, tous droits réservés. L'accès aux archives de la revue « Annales de l'I. H. P., section C » (http://www.elsevier.com/locate/anihpc) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Maximal monotone relations and the second derivatives of nonsmooth functions R. T. ROCKAFELLAR

READ INSTRUCTIONS •__BEFORE COMPLETING FORM 1. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER optimization, network flows, nonlinear programming, Lagrange multipliers 20. ABSTRACT (Continue on reverse side it necessary and identify by block number) This report describes the research under this grant in 1979./ monograph on network optimizatiqn and monotropic programming was completed. New results were obtained on'subradient methods and open horizon control. DDFORM173 DD I JAN 73 EDITION OF I NOV 65 IS ORSOLETE UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered

The theory of second-order epi-derivatives of extended-real-valued functions is applied to convex functions on R n {\mathbb {R}^n} and shown to be closely tied to proto-differentiation of the corresponding subgradient multifunctions, as well as to second-order epi-differentiation of conjugate functions. An extension is then made to saddle functions, which by definition are convex in one argument and concave in another. For this case a concept of epi-hypo-differentiability is introduced. The saddle function results provide a foundation for the sensitivity analysis of primal and dual optimal solutions to general finite-dimensional problems in convex optimization, since such solutions are characterized as saddlepoints of a convex-concave Lagrangian function, or equivalently as subgradients of the saddle function conjugate to the Lagrangian.

The dissertation is devoted to the development of variational analysis and generalized differentiation in infinite dimensions. We derive new calculus rules for both first-order partial subdifferentials and second-order partial subdifferentials in the framework of general Banach spaces as well as more developed rules in the framework of Asplund spaces. This calculus is applied in the study of sensitivity analysis for

It is well known that k-contact geometry is a suitable framework to deal with non-conservative field theories. In this paper, we study some relations between solutions of the k-contact Euler-Lagrange equations, symmetries, dissipation laws and Newtonoid vector fields. We review the k-contact Euler-Lagrange equations written in terms of k-vector fields and sections and provide new results relating the solutions in both approaches. We also study different kind of symmetries depending on the structures they preserve: natural (preserving the Lagrangian function), dynamical (preserving the solutions), and k-contact (preserving the underlying geometric structures) symmetries. For some of these symmetries, we provide Noether-like theorems relating symmetries and dissipation laws. We also analyse the relation between k-contact symmetries and Newtonoid vector fields. Throughout the paper, we will use the damped vibrating string as our main illustrative example.

Solution stability of a class of linear generalized equations in finite dimensional Euclidean spaces is investigated by means of generalized differentiation. Exact formulas for the Fréchet and the Mordukhovich coderivatives of the normal cone mappings of perturbed Euclidean balls are obtained. Necessary and sufficient conditions for the local Lipschitz-like property of the solution maps of such linear generalized equations are derived from these coderivative formulas. Since the trust-region subproblems in nonlinear programming can be regarded as linear generalized equations, these conditions lead to new results on stability of the parametric trust-region subproblems.

This study focuses on determining the engineering characteristics of Hot Mix Asphalt (HMA) using waste roofing shingle, PVC and plastic glass scraps. These waste materials were added 1%, 2%, 3%, and 5% mixing ratio in the asphalt concrete which designed with predetermined bitumen content. The asphalt concretes which made of waste roofing shingle, PVC, plastic glass and conventional asphalt concrete were evaluated through their fundamental engineering properties such as Marshall Stability and flow. The results indicate that stability of Marshall samples using with waste roofing shingle and waste PVC scraps decrease slightly depending on the ratio. Besides, Marshall flows of the specimens increase depending on the these waste material contents. However, these waste materials can be used in HMA in accordance with Turkey General Directorate of Highways (TCK) wearing course within specification limits. Finally, there is a high possibility for waste roofing shingle, PVC and plastic glass ...

Within the framework of MindlinÕs dipolar gradient elasticity, general energy theorems are proved in this work. These are the theorem of minimum potential energy, the theorem of minimum complementary potential energy, a variational principle analogous to that of the Hellinger-Reissner principle in classical theory, two theorems analogous to those of Castigliano and Engesser in classical theory, a uniqueness theorem of the Kirchhoff-Neumann type, and a reciprocal theorem. These results can be of importance to computational methods for analyzing practical problems. In addition, the J-integral of fracture mechanics is derived within the same framework. The new form of the J-integral is identified with the energy release rate at the tip of a growing crack and its path-independence is proved. The theory of dipolar gradient elasticity derives from considerations of microstructure in elastic continua [Mindlin, R.D., 1964. Microstructure in linear elasticity. Arch. Rational Mech. Anal. 16, 51-78] and is appropriate to model materials with periodic structure. According to this theory, the strain-energy density assumes the form of a positive-definite function of the strain (as in classical elasticity) and the second gradient of the displacement (additional term). Specific cases of the general theory considered here are the well-known theory of couple-stress elasticity and the recently popularized theory of strain-gradient elasticity. The latter case is also treated in the present study.

This paper addresses the study and characterizations of variational convexity of extendedreal-valued functions on Banach spaces. This notion has been recently introduced by Rockafellar, and its importance has been already realized and applied to continuous optimization problems in finitedimensional spaces. Variational convexity in infinite-dimensional spaces, which is studied here for the first time, is significantly more involved and requires the usage of powerful tools of geometric functional analysis together with variational analysis and generalized differentiation in Banach spaces.

SUMMARYThe dynamic response of an end bearing pile embedded in a linear visco‐elastic soil layer with hysteretic type damping is theoretically investigated when the pile is subjected to a time‐harmonic vertical loading at the pile top. The soil is modeled as a three‐dimensional axisymmetric continuum in which both its radial and vertical displacements are taken into account. The pile is assumed to be vertical, elastic and of uniform circular cross section. By using two potential functions to decompose the displacements of the soil layer and utilizing the separation of variables technique, the dynamic equilibrium equation is uncoupled and solved. At the interface of soil‐pile system, the boundary conditions of displacement continuity and force equilibrium are invoked to derive a closed‐form solution of the vertical dynamic response of the pile in frequency domain. The corresponding inverted solutions in time domain for the velocity response of a pile subjected to a semi‐sine excitati...

The Plateau-Bézier problem consists in finding the Bézier surface with minimal area from among all Bézier surfaces with prescribed border. An approximation to the solution of the Plateau-Bézier problem is obtained by replacing the area functional with the Dirichlet functional. Some comparisons between Dirichlet extremals and Bézier surfaces obtained by the use of masks related with minimal surfaces are studied.

In this paper we present a method for generating Bézier surfaces from the boundary information based on a general 4th-order PDE. This is a generalisation of our previous work on harmonic and biharmonic Bézier surfaces whereby we studied the Bézier solutions for Laplace and the standard biharmonic equation, respectively. Here we study the Bézier solutions of the Euler-Lagrange equation associated with the most general quadratic functional. We show that there is a large class of fourth-order operators for which Bézier solutions exist and hence we show that such operators can be utilised to generate Bézier surfaces from the boundary information. As part of this work we present a general method for generating these Bézier surfaces. Furthermore, we show that some of the existing techniques for boundary based surface design, such as Coons patches and Bloor-Wilson PDE method, are indeed particular cases of the generalised framework we present here.

We provide some necessary and sufficient conditions for a proper lower semicontinuous convex function, defined on a real Banach space, to be locally or globally Lipschitz continuous. Our criteria rely on the existence of a bounded selection of the subdifferential mapping and the intersections of the subdifferential mapping and the normal cone operator to the domain of the given function. Moreover, we also point out that the Lipschitz continuity of the given function on an open and bounded (not necessarily convex) set can be characterized via the existence of a bounded selection of the subdifferential mapping on the boundary of the given set and as a consequence it is equivalent to the local Lipschitz continuity at every point on the boundary of that set. Our results are applied to extend a Lipschitz and convex function to the whole space and to study the Lipschitz continuity of its Moreau envelope functions.

The concept of the lower limit for vector-valued mappings is the main focus of this work. We first introduce a new definition of adequate lower and upper level sets for vector-valued mappings and establish some of their topological and geometrical properties. Characterization of semicontinuity for vector-valued mappings is thereafter presented. Then, we define the concept of vector lower limit, proving its lower semicontinuity, and furnishing in this way a concept of lower semicontinuous regularization for mappings taking their values in a complete lattice. The results obtained in the present work subsume the standard ones when the target space is finite dimensional. In particular, we recapture the scalar case with a new flexible proof. In addition, extensions of usual operations of lower and upper limits for vector-valued mappings are explored. The main result is finally applied to obtain a continuous D.C. decomposition of continuous D.C. mappings.