element size and time step selection procedures for the numerical analysis of elasticity with higher order inertia continuum theories with additional higher order inertia terms have been suggested in the literature in order to be able to describe dispersive wave propagation and in order to include microstructural information in the macroscopic model in this short note we investigate two numerical discretisation aspects of such a model firstly the critical time step is derived which is relevant for conditionally stable time integrators secondly a discrete dispersion analysis is carried out by which the accuracy of the discretised model can be assessed a priori recommendations for the finite element size and the time step size can thus be made

dynamic analysis of continuous plane trusses with equidistant supports the natural and forced vibrations of a continuous warren truss resting on equidistant roller supports are studied in the analyses the longitudinal inertial forces are neglected in the equation of motion and the mass of the truss is replaced by two sets of lumped masses at the nodes in the forced vibration case the truss is subjected to transverse harmonic loads acting at the nodes the truss can be considered as an equivalent system with cyclic bi periodicity periodicities due to the repetitive units of the truss and the supports respectively therefore the governing equation for such a system can be uncoupled by applying u transformation twice to form a set of single degree of freedom equations such equations can be used to solve for the natural frequencies of the trusses or their responses due to external excitations typical warren trusses are taken as examples to demonstrate the advantages of the proposed method and the accuracy that can be achieved the response of a truss with six substructures and four supports with a harmonic load acting at the centre node is investigated

multi scale second order computational homogenization of multi phase materials a nested finite element solution strategy this paper presents the detailed implementation and computational aspects of a novel second order computational homogenization procedure which is suitable for a multi scale modelling of macroscopic localization and size effects the second order scheme is an extension of the classical first order computational homogenization framework and is based on a proper incorporation of the gradient of the macroscopic deformation gradient tensor into the kinematical macro micro scale transition from the microstructural analysis the macroscopic stress and higher order stress tensors are obtained thus delivering a microstructurally based constitutive response of the macroscopic second gradient continuum the higher order macroscopic constitutive tangents are derived through static condensation of the microscopic global tangent matrix for the solution of the second gradient equilibrium problem on the macrolevel a mixed finite element formulation is developed as an example the second order computational homogenization approach is applied for the multi scale analysis of simple shear of a constrained heterogeneous strip where a pronounced boundary size effect appears

on repeated games with complete information we consider n person repeated games with complete information and standard signalling we first prove several properties of the sets of feasible payoffs and nash equilibrium payoffs for the n stage game and for the discounted game in the second part we determine the set of equilibrium payoffs for the prisoner s dilemma corresponding to the critical value of the discount factor

subdivision surfaces a new paradigm for thin shell finite element analysis we develop a new paradigm for thin shell finite element analysis based on the use of subdivision surfaces for i describing the geometry of the shell in its undeformed configuration and ii generating smooth interpolated displacement fields possessing bounded energy within the strict framework of the kirchhoff love theory of thin shells the particular subdivision strategy adopted here is loop s scheme with extensions such as required to account for creases and displacement boundary conditions the displacement fields obtained by subdivision are h2 and consequently have a finite kirchhoff love energy the resulting finite elements contain three nodes and element integrals are computed by a one point quadrature the displacement field of the shell is interpolated from nodal displacements only in particular no nodal rotations are used in the interpolation the interpolation scheme induced by subdivision is non local i e the displacement field over one element depend on the nodal displacements of the element nodes and all nodes of immediately neighbouring elements however the use of subdivision surfaces ensures that all the local displacement fields thus constructed combine conformingly to define one single limit surface numerical tests including the belytschko et al 10 obstacle course of benchmark problems demonstrate the high accuracy and optimal convergence of the method

a boundary element method for solving 2 d and 3 d static gradient elastic problems part i integral formulation abstract a boundary element formulation is developed for the static analysis of two and three dimensional solids and structures characterized by a linear elastic material behavior taking into account microstructural effects the simple gradient elastic theory of aifantis expressed in the framework of mindlin s general theory is used to model this material behaviour a variational statement is established to determine all possible classical and non classical due to gradient terms boundary conditions of the general boundary value problem the gradient elastic fundamental solution for both two and three dimensional cases is explicitly derived and used to construct the boundary integral representation of the solution with the aid of the reciprocal integral identity especially established for the gradient elasticity considered here it is found that for a well posed boundary value problem in addition to a boundary integral representation for the displacement a second boundary integral representation for its normal derivative is also necessary explicit expressions for interior displacements and stresses in integral form are also presented all the kernels in the integral equations are explicitly provided

cooperation in repeated games when the number of stages is not commonly known it is shown that an exponentially small departure from the common knowledge assumption on the number t of repetitions of the prisoners dilemma already enables cooperation more generally with such a departure any feasible individually rational outcome of any one shot game can be approximated by a subgame perfect equilibrium of a finitely repeated version of that game the sense in which the departure from common knowledge is small is as follows i with probability one the players know t with precision k ii with probability 1 e the players know t precisely moreover this knowledge is mutual of order et iii the deviation of t from its finite expectation is exponentially small

obtaining eringen s length scale coefficient for vibrating nonlocal beams via continualization method abstract eringen s length scale coefficients e 0 are presented herein for initially stressed vibrating nonlocal beams with various boundary conditions the coefficients are obtained by applying the continualization method to the discrete equations of microstructured beam models when compared to another continualized approach that is based on the pade approximant the proposed continualization method which is based on assuming continuous exponential displacement functions for the discrete displacement field is a more straightforward approach for solution unlike the pade approximant approach this latter method needs no sophisticated mathematical manipulations for obtaining the continuous equation moreover the continualization method allows one to directly solve the discrete governing equations for vibration of microstructured beams in the discrete domain rather than solving an approximated continuous equation obtained via the pade approximant by using this method it is found that e 0 1 12 for the buckling case of beams and e 0 1 6 for the vibration case for any combination of end conditions however the coefficients are not the same in the presence of an initial stress for vibrating beams with different boundary conditions

a class of contractions in fuzzy metric spaces using the notion of geometrically convergent t norms a fixed point theorem in fuzzy metric spaces in the sense of kramosil and michalek for a class of contractions larger than the class of e l contraction mappings has been proved

dispersion and wave propagation in discrete and continuous models for granular materials abstract a generalised continuum model for granular media is derived by direct homogenisation of the discrete equations of motion in contrast to previous works on this topic continuum concepts such as stress and moment stress are introduced after homogenisation first a very simple one dimensional model is considered and the continuum version for this model is derived by replacing the difference quotients of the discrete model by differential quotients the dispersion relations of the discrete and the continuous model are derived and compared variational boundary conditions for the continuous model are deduced from the stationarity of the corresponding lagrangian the three dimensional case is treated in an essentially similar fashion the resulting continuum theory is a combination of a cosserat continuum and a higher order deformation gradient continuum the salient features of the theory are illustrated by means of the dispersion relations for planar wave propagation

nonlocal and nonlinear dispersion in a nonlinear schrodinger type equation exotic solitons and short wavelength instabilities we study the continuum limit of a nonlinear schrodinger lattice model with both on site and inter site nonlinearities describing weakly coupled optical waveguides or bose einstein condensates the resulting continuum nonlinear schrodinger type equation includes both nonlocal and nonlinear dispersion looking for stationary solutions the equation is reduced to an ordinary differential equation with a rescaled spectral parameter and a single parameter interpolating between the nonlocality and the nonlinear dispersion it is seen that these two effects give a similar behaviour for the solutions we find smooth solitons and beyond a critical value of the spectral parameter also nonanalytic solitons in the form of peakons and capons the existence of the exotic solitons is connected to the special properties of the phase space of the equation stability is investigated numerically by calculating eigenvalues and eigenfunctions of the linearized problem and we particularly find that with both nonlocal and nonlinear dispersion simultaneously present all solutions are unstable with respect to a break up into short wavelength oscillations

a note on fixed point theorems of hadzci c abstract in this paper we give a fixed point theorem for multi valued probabilistic contraction mappings on a menger space as a special case a fixed point theorem for multi valued contraction mappings on a fuzzy metric space is obtained the results presented improve and generalize in a sense some recent fixed point theorems proved by had ic

the response of two dimensional periodic structures to impulsive point loading abstract this work is concerned with the response of an infinite two dimensional periodic structure to an impulsive point load or equivalently the response of a finite system at times before the disturbance reaches the system boundaries initially a model approach is employed to yield an expression for the response of a finite system under born von karman boundary conditions by allowing the system size to become large the modal response summation is converted to an integral and the method of stationary phase is employed with de allowance being made for the occurrence of caustics with this approach the response is expressed in terms of the properties of the phase constant surfaces which arise in the analysis of plane wave motion through the system the method is applied to an example lumped mass system and a comparison is made with results yield by direct numerical simulation the method is found to be efficient and accurate and a number of observations are made regarding the physical nature of the system response

accuracy of the finite difference method in stochastic setting in this paper we study the accuracy of the finite difference method when the finite difference method is applied to approximately analyze the structure

multiscale coupling in function space weak coupling between molecular dynamics and continuum mechanics we present a function space oriented coupling approach for the multiscale simulation of non linear processes in mechanics using finite elements and molecular dynamics concurrently the key idea is to construct a transfer operator between the different scales on the basis of weighted local averaging instead of using point wise taken values the local weight functions are constructed by assigning a partition of unity to the molecular degrees of freedom shepard s approach this allows for decomposing the micro scale displacements into a low frequency and a high frequency part by means of a weighted l2 projection numerical experiments illustrating the stabilizing effect of our coupling approach are given copyright 2009 john wiley sons ltd

elastic solutions for periodically layered strip with perfect bonding or with an interface crack a non homogenization approach to the analysis of a strip consisting of a finite number of isotropic elastic layers arranged periodically is presented the result is obtained by the use of the representative cell method based on the discrete fourier transform two types of problems are addressed in the problems of the first type the bonding between the layers is perfect the solution for this case is found in a closed form in terms of the laplace integrals the numerical results for different boundary conditions are given and a comparison with plate theory is carried out in the problems of the second type there is a flaw crack at one of the interfaces the solution in this case hinges on the analytic expression for the green function corresponding to a single interface dislocation in the uncracked strip by using this expression the crack problem is reduced to a singular integral equation the influence of the elastic mismatch and other problem parameters on the fracture characteristics is examined it is found that similar to the case of the periodic plane when the thinner layeres in the strip are stiffer further increase of their stiffness may lead to the increase or to the decrease of the absolute value of the stress intensity factor depending upon the elastic mismatch between the composite constituents

comparative review of applied linear algebra 3rd ed by b noble and j w daniel and linear algebra and its applications 3rd ed by g strang

on the theory of infinitely repeated games with discounting presentation d un cadre systematique pour etudier des jeux repetes indefiniment avec actualisation en s attachant plus particulierement aux equilibres parfaits de strategie pure

a new barrier to the existence of moving kinks in frenkel kontorova lattices an explanation is offered for an observed lower bound on the wave speed of travelling kinks in frenkel kontorova lattices kinks exist at discrete wavespeeds within a parameter regime where there is a resonance with linear waves phonons however they fail to exist even in this codimension one sense whenever there is more than one phonon branch in the dispersion relation inside such bands only quasi kinks with nondecaying oscillatory tails are possible the results are presented for a discrete sine gordon lattice with an onsite potential that has a tunable amount of anharmonicity novel numerical methods are used to trace kinks with topological charge q 1 and 2 in three parameters representing the propagation speed lattice discreteness and anharmonicity although none of the analysis is presented as rigorous mathematics numerical results suggest that the bound on allowable wavespeeds is sharp the results also explain why the vanishing at discrete values of anharmonicity of the peierls nabarro barrier between stationary kinks as discovered by savin et al does not lead to bifurcation of kinks with small wave speed 2003 elsevier b v all rights reserved

equation free modelling of evolving diseases coarse grained computations with individual based models we demonstrate how direct simulation of stochastic individualbased models can be combined with continuum numericalanalysis techniques to study the dynamics of evolving diseases sidestepping the necessity of obtaining explicit populationlevel models the approach analyses the unavailable in closed form coarse macroscopic equations estimating the necessary quantities through appropriately initialized short bursts of individualbased dynamic simulation we illustrate this approach by analysing a stochastic and discrete model for the evolution of disease agents caused by point mutations within individual hosts building up from classical susceptibleinfectedrecovered and susceptibleinfectedrecoveredsusceptible models our example uses a onedimensional lattice for variant space and assumes a finite number of individuals macroscopic computational tasks enabled through this approach include stationarystate computation coarse projective integration parametric continuation and stability analysis

second gradient constitutive theory for granular material with random packing structure abstract a second gradient constitutive law for granular media is derived in which stress is a function of the second order of strain gradient the constitutive coefficients for granular materials with an isotropic packing structure are derived in explicit terms of inter particle stiffness and particle size in the present constitutive theory the total number of elastic constants for the isotropic granular material is four two of them are the usual lame constants the other two are associated with the second gradient of strain the derived constitutive relationships illustrate the important role of micro scale properties in the macro scale behavior the influence of inter particle stiffness and particle size on the response of material due to the second gradient of strain is discussed this paper also discusses the connection between the second gradient theory and non local theory the second gradient theory can be regarded as the first order approximation of non local theory

orthogonality condition for a multi span beam and its application to transient vibration of a two span beam abstract this paper treats the orthogonality condition for a multi span beam and its application to forced transient vibration of a two span beam the beam is modeled as a bernoulli euler beam the boundary conditions for the particular case of two span beam are clamped pinned pinned an exact closed form solution is obtained for this problem even though there has been an enormous amount of work on beam vibration most of the studies are conducted on a single span beam there are some studies on the multi span beam vibration however their treatment is rather specialized in terms of the applied loading and the initial conditions none of the studies in the past treats an exact solution for a forced transient vibration of a general two span beam with arbitrary initial conditions and arbitrary forcing functions therefore the solution obtained in this paper is new the key development in the solution is the orthogonality condition for a multi span beam the method of solution developed in this paper establishes a general methodology for the forced transient vibration of a multi span beam the closed form solution obtained in this paper can be used as a benchmark solution for the transient vibration of a two span beam

gradient elasticity in statics and dynamics an overview of formulations length scale identification procedures finite element implementations and new results in this paper we discuss various formats of gradient elasticity and their performance in static and dynamic applications gradient elasticity theories provide extensions of the classical equations of elasticity with additional higher order spatial derivatives of strains stresses and or accelerations we focus on the versatile class of gradient elasticity theories whereby the higher order terms are the laplacian of the corresponding lower order terms one of the challenges of formulating gradient elasticity theories is to keep the number of additional constitutive parameters to a minimum we start with discussing the general mindlin theory that in its most general form has 903 constitutive elastic parameters but which were reduced by mindlin to three independent material length scales further simplifications are often possible in particular the aifantis theory has only one additional parameter in statics and opens up a whole new field of analytical and numerical solution procedures we also address how this can be extended to dynamics an overview of length scale identification and quantification procedures is given finite element implementations of the most commonly used versions of gradient elasticity are discussed together with the variationally consistent boundary conditions details are provided for particular formats of gradient elasticity that can be implemented with simple linear finite element shape functions new numerical results show the removal of singularities in statics and dynamics as well as the size dependent mechanical response predicted by gradient elasticity

transfer matrix analysis of the elastostatics of one dimensional repetitive structures transfer matrices are used widely for the dynamic analysis of engineering structures increasingly so for static analysis and are particularly useful in the treatment of repetitive structures for which in general the behaviour of a complete structure can be determined through the analysis of a single repeating cell together with boundary conditions if the structure is not of infinite extent for elastostatic analyses non unity eigenvalues of the transfer matrix of a repeating cell are the rates of decay of selfequilibrated loading as anticipated by saint venant s principle multiple unity eigenvalues pertain to the transmission of load e g tension or bending moment and equivalent homogenized continuum properties such as cross sectional area second moment of area and poisson s ratio can be determined from the associated eigen and principal vectors various disparate results the majority new others drawn from diverse sources are presented these include calculation of principal vectors using the moore penrose inverse bi and symplectic orthogonality and relationship with the reciprocal theorem restrictions on complex unity eigenvalues effect of cell left to right symmetry on both the stiffness and transfer matrices eigenvalue veering in the absence of translational symmetry and limitations on possible jordan canonical forms it is shown that only a repeating unity eigenvalue can lead to a non trivial jordan block form so degenerate decay modes cannot exist the present elastostatic analysis complements langley s langley 1996 proc r soc a 452 1631 1648 transfer matrix analysis of wave motion energetics

large time behavior of solutions to the nonlinear pseudo parabolic equation abstract in this paper we investigate the initial value problem for the nonlinear pseudo parabolic equation global existence and optimal decay estimate of solution are established provided that the initial value is suitably small moreover when n 2 and the nonlinear term f u disappears we prove that the global solutions can be approximated by the linear solution as time tends to infinity when n 1 and the nonlinear term f u disappears we show that as time tends to infinity the global solution approaches the nonlinear diffusion wave described by the self similar solution of the viscous burgers equation

initial tension in randomly disordered periodic lattices this paper is concerned with probabilistic analysis of initial member stress in geometrically imperfect regular lattice structures with periodic boundary conditions spatial invariance of the corresponding statistical parameters is shown to arise on the born von karman domains this allows analytical treatment of the problem where the parameters of stress distribution are obtained in a closed form several benchmark problems with beam and plate like lattices are considered and the results are verified by the direct monte carlo simulations behaviour of the standard deviation as a function of lattice repetitive cell number is investigated and dependence on the lattice structural redundancy is pointed out

wave propagation in continuous periodic structures research contributions from southampton 1964 1995 abstract after brief reference to some early studies by other investigators this paper focuses mainly on methods developed at the university of southampton since 1964 to analyze and predict the free and forced wave motion in continuous periodic engineering structures beginning with receptance methods which have been applied to periodic beams and rib skin structures it continues with a method of direct solution of the wave equation this uses floquet s principle and has been applied to beams and quasi one dimensional periodic plates and cylindrical shells sample curves of the propagation and attenuation constants pertaining to these structures are presented a limited discussion of the transfer matrix then follows after which the method of space harmonics is introduced as the method best suited to the prediction of sound radiated from a vibrating periodic structure reviewed next are some theorems and variational principles relating to periodic structures which have been developed at southampton and which form a basis for finding natural frequencies of finite structures or for computing free and forced wave notion by energy methods this has led to the finite element method in its standard and hierarchical forms being used to study wave motion in genuine two dimensional and three dimensional structures examples of this work are shown the method of phased array receptance functions is then introduced as possibly the easiest way of setting up exact equations for the propagation constants of uniform quasi one dimensional periodic structures a summary is finally presented of the limited and early work performed at southampton on simple disordered periodic structures

continuum approach to discreteness we study analytically and numerically continuum models derived on the basis of pade approximations and their effectiveness in modeling spatially discrete systems we not only analyze features of the temporal dynamics that can be captured through these continuum approaches e g shape oscillations radiation effects and trapping but also point out ones that cannot be captured such as peierls nabarro barriers and bloch oscillations we analyze the role of such methods in providing an effective homogenization of spatially discrete as well as of heterogeneous continuum equations finally we develop numerical methods for solving such equations and use them to establish the range of validity of these continuum approximations as well as to compare them with other semicontinuum approximations

boundary element analysis of mode i and mixed mode i and ii crack problems of 2 d gradient elasticity abstract a boundary element method is developed for fracture analysis of gradient elastic planar solids under static loading a simple version of mindlin s general theory of gradient elastic materials is employed and the two required boundary integral equations one for the displacement vector and the other for its normal derivative are presented use is made of the fundamental solution of the problem and this leads to a formulation that requires only a boundary discretization two representative numerical examples are presented to illustrate the method demonstrate its accuracy and efficiency and assess the gradient effect on the response the first deals with a mode i crack while the second with a mixed mode i ii crack for the second case the proposed method is used in conjunction with the method of subregions the method is employed with regular and special near the crack tip boundary elements the gradient effect consists of modifying both the displacement and the stress field around the crack tip and resulting in a response which is more physically acceptable than the one coming from the classical theory of elasticity

on static analysis of finite repetitive structures by discrete fourier transform functional solutions for the static response of beam and plate like repetitive lattice structures are obtained by discrete fourier transform the governing equation is set up as a single operator form with the physical stiffness operator acting as a convolution sum and containing a matrix kernel which relates to the mechanical properties of the lattice boundary conditions do not affect the equation form and are taken into account at a subsequent stage of the analysis the technique of virtual load and substructure is proposed to formally close the repetitive lattice into a cyclic structure and to assure the equivalence of responses of the modified cyclic and original repetitive lattices a discrete periodic green s function is introduced for the modified structure and the final displacement solutions are written as convolution sums over the green s function and the actual external and virtual loads several example problems illustrate the approach

domain reduction method for atomistic simulations in this paper a quasi static formulation of the method of multi scale boundary conditions msbcs is derived and applied to atomistic simulations of carbon nano structures namely single graphene sheets and multi layered graphite this domain reduction method allows for the simulation of deformable boundaries in periodic atomic lattice structures reduces the effective size of the computational domain and consequently decreases the cost of computations the size of the reduced domain is determined by the value of the domain reduction parameter this parameter is related to the distance between the boundary of the reduced domain where msbcs are applied and the boundary of the full domain where the standard displacement boundary conditions are prescribed two types of multi scale boundary conditions are derived one for simulating in layer multi scale boundaries in a single graphene sheet and the other for simulating inter layer multi scale boundaries in multi layered graphite the method is tested on benchmark nano indentation problems and the results are consistent with the full domain solutions

non linear analysis of beam like structures on unilateral foundations a lattice spring model abstract a computationally efficient method is presented for static and dynamic analysis of a beam like structure on a viscoelastic foundation with the unilateral contact constraint at their interface the non smooth dynamics of the system is modelled using the euler bernoulli theory for the beam and the bilinear winkler model for the substrate thus the unilateral contact is the only source of non linearity the proposed approach relies on reducing the non smooth continuous system to a piecewise smooth multi degree of freedom model the beam is represented as a chain of discrete units through the use of the lattice spring model lsm and consequently the phase space is divided into a number of subdomains hence the system smoothness is lost when the trajectory of lattice nodes cross the boundaries between these subdomains an effective algorithm has been developed to handle the unilateral constraints by tracking the trajectories and successively capturing the intervals that the nodes spend in each of the smooth regions unlike more commonly used methods it neither relies on any prior information such as the number and location of the lift off areas nor on the non linear solvers the accuracy and robustness of this numerical technique have been demonstrated in several application examples furthermore the developed algorithm is utilised in combination with the shooting method and continuation of the periodic motions to obtain the non linear normal modes nnms of the system we show that frequency stiffness i u k w plots can be used to describe the salient features of the nnms for the beam on the full foundation the first mode in the configuration space has a linear shape and thus the conventional bilinear formulation can accurately approximate the non linear frequency for higher modes the modal lines are open curves and the bilinear approximation loses its accuracy in addition internal resonances may occur once the frequencies are commensurate they are characterised as new multimodal motions emanating from the backbone in the frequency stiffness plot

an improved finite difference analysis of uncoupled vibrations of cantilevered beams abstract natural frequencies and mode shapes of uniform cantilever beams are obtained with use of the first and second order central difference schemes it is observed that the improved finite difference scheme with second order central differences produces the natural frequencies and characteristic functions with a rapid convergence as compared to the conventional approach of using the first order central differences further the present approach facilitates a direct determination of the dynamic characteristics of beams without any necessity of extrapolations of the results or application of iterative procedures for improving the accuracy satisfaction of the boundary conditions and elimination of the fictitious stations lying outside the beam domain has been a serious problem in the previous investigations in which higher order central differences have been used in the present study some recursive relations have been implemented which can be derived by using the first order central differences and application of these simple and elegant relations to higher order difference schemes has been demonstrated to improve the accuracy over conventional approaches

finitely repeated games etude systematique des equilibres parfaits de sous jeux pour des jeux repetes de facon finie avec information complete

on causality of the gradient elasticity models abstract this paper is concerned with causality of the gradient elasticity models of heterogeneous materials as a rule these models are not strictly causal since they allow an infinite speed of energy transfer by means of either propagating or transient evanescent waves a discussion is presented in this paper of both physical and mathematical implications of this fact this discussion is carried out employing one dimensional 1d second order gradient models a phenomenological enhancement is proposed which makes these models causal the main idea behind this enhancement is that a partial differential equation that governs dynamic behaviour of a causal gradient elasticity model must be of the same order with respect to spatial coordinate and with respect to time the validity of this idea is confirmed in this paper by deriving a second order 1d continuum model for concrete a brief comparison is provided in conclusion of the equations of motion of the 1d second gradient elasticity models and those used in dynamics of thin bars it is shown that the proposed causal model corresponds to the most advanced dynamic models of such bars namely to the timoshenko model for the bending motion and the mindlin model for the longitudinal motion of a bar

the folk theorem for repeated games a neu condition we are concerned here with perfect folk theorems for infinitely repeated games with complete information folk theorems assert that any feasible and individually rational payoff vector of the stage game is a subgame perfect equilibrium payoff in the associated infinitely repeated game with little or no discounting where payoff streams are evaluated as average discounted or average values respectively it is obvious that feasibility and individual rationality are necessary conditions for a payoff vector to be an equilibrium payoff the surprising content of the folk theorems is that these conditions are also almost sufficient perhaps the first folk theorem type result is due to friedman 1971 who showed that any feasible payoff which pareto dominates a nash equilibrium payoff of the stage game will be an equilibrium payoff in the associated repeated game with sufficiently patient players this kind of result is sometimes termed a nash threats folk theorem a reference to its method of proof for the more permissive kinds of folk theorems considered here the seminal results are those of aumann and shapley 1976 and rubinstein 1977 1979 these authors assume that payoff streams are undiscounted 2 fudenberg and maskin 1986 establish an analogous result for discounted repeated games as the discount factor goes to 1 their result uses techniques of proof rather different from those used by aumann shapley and rubinstein respectively see their paper for an insightful discussion of this point and quite generally for more by way of background it is a key reference for subsequent work in this area including our own for the two player case the result of fudenberg and maskin 1986 is a complete if and only if characterization modulo the requirement of strict rather than weak individual rationality which we retain in this note and does not employ additional conditions for three or more players fudenberg and maskin introduced a full dimensionality condition the convex hull f of the set of feasible payoff vectors of the stage game must have dimension n where n is the number of players or equivalently a nonempty interior this condition has been widely adopted in proving folk theorems for related environments such as finitely repeated games benoit and krishna 1985 and overlapping generations games kandori 1992 smith 1992 full dimensionality is a sufficient condition fudenberg and maskin present an example of a three player stage game in which the conclusion of the folk theorem is false in this example all players receive the same payoffs in all contingencies the convex hull of the set of feasible payoffs is one dimensional this example violates full dimensionality in a rather extreme way less extreme violations may also lead to

higher order continua derived from discrete media continualisation aspects and boundary conditions abstract in this paper the derivation of higher order continuum models from a discrete medium is addressed with the following aims i for a given discrete model and a given coupling of discrete and continuum degrees of freedom the continuum should be defined uniquely ii the continuum is isotropic and iii boundary conditions are derived consistently with the energy functional and the equations of motion of the continuum firstly a comparison is made between two continualisation methods namely based on the equations of motion and on the energy functional they are shown to give identical results secondly the issue of isotropy is addressed a new approach is developed in which two rather than one layers of neighbouring particles are considered finally the formulation and interpretation of boundary conditions is treated by means of the hamilton ostrogradsky principle boundary conditions are derived that are consistent with the energy functional and the equations of motion a relation between standard stresses and higher order stresses is derived and used to make an interpretation of the higher order stresses an additional result of this study is the non uniqueness of the higher order contributions to the energy

large deformations of planar extensible beams and pantographic lattices heuristic homogenization experimental and numerical examples of equilibrium the aim of this paper is to find a computationally efficient and predictive model for the class of systems that we call pantographic structures the interest in these materials was increased by the possibilities opened by the diffusion of technology of three dimensional printing they can be regarded once choosing a suitable length scale as families of beams also called fibres interconnected to each other by pivots and undergoing large displacements and large deformations there are however relatively few ready to use results in the literature of nonlinear beam theory in this paper we consider a discrete spring model for extensible beams and propose a heuristic homogenization technique of the kind first used by piola to formulate a continuum fully nonlinear beam model the homogenized energy which we obtain has some peculiar and interesting features which we start to describe by solving numerically some exemplary deformation problems furthermore we consider pantographic structures find the corresponding homogenized second gradient deformation energies and study some planar problems numerical solutions for these two dimensional problems are obtained via minimization of energy and are compared with some experimental measurements in which elongation phenomena cannot be neglected

dynamic response of infinite continuous beams subjected to a moving force an exact method this paper describes an exact method for investigating the dynamic response of an infinite uniform beam resting on periodic rolle supports and subjected to a moving force at first an exact solution for a rotationally periodic continous beam subjected to a moving force is obtained by the u transformation and mode method then by a limiting process with the number of spans approaching infinity the result is generalized to give a solution for an infinite continuous beam subjected to a moving force this method will be applicable to the forced vibration analysis of an infinite continuous beam under arbitrary excitation

common fixed point theorems under strict contractive conditions in menger spaces abstract the main purpose of this paper is to establish some common fixed point theorems under strict contractive conditions for mappings satisfying the property e a in menger probabilistic metric spaces as applications we obtain the corresponding common fixed point theorems under strict contractive in metric spaces

computing supergame equilibria we present a general method for computing the set of supergame equilibria in infinitely repeated games with perfect monitoring and public randomization we present a three stage algorithm that constructs a convex set containing the set of equilibrium values constructs another convex set contained in the set of equilibrium values and produces strategies that support them we explore the properties of this algorithm by applying it to familiar games

transverse vibration analysis of plane trusses by analytical method abstract the type of plane truss considered in this paper is the warren truss which is built up with basic units consisting of four sets of bars pin jointed at the nodes and subjected to transverse nodal loads the natural vibration and dynamic response are studied for the simply supported truss with lumped masses at the nodes closed form solutions are deduced by using the u transformation technique which leads to the uncoupling of the governing equations in order to explain the procedure of the present method the response functions for the nodal displacements of a truss subjected to the action of a suddenly applied load at a node are derived it is obvious that the present method is also applicable to the forced vibration analysis of trusses subjected to arbitrary transverse loads applied at the nodes

recursive methods in discounted stochastic games an algorithm for δ 1 and a folk theorem we present an algorithm to compute the set of perfect public equilibrium payoffs as the discount factor tends to 1 for stochastic games with observable states and public but not necessarily perfect monitoring when the limiting set of long run players equilibrium payoffs is independent of the initial state this is the case for instance if the markov chain induced by any markov strategy profile is irreducible we then provide conditions under which a folk theorem obtains if in each state the joint distribution over the public signal and next period s state satisfies some rank condition every feasible payoff vector above the minmax payoff is sustained by a perfect public equilibrium with low discounting

gradient elasticity and nonstandard boundary conditions abstract gradient elasticity for a second gradient model is addressed within a suitable thermodynamic framework apt to account for nonlocality the pertinent thermodynamic restrictions upon the gradient constitutive equations are derived which are shown to include besides the field differential stress strain laws a set of nonstandard boundary conditions consistently with the latter thermodynamic requirements a surface layer with membrane stresses is envisioned in the strained body which together with the above nonstandard boundary conditions make the body constitutively insulated i e no long distance energy flows out of the boundary surface due to nonlocality the total strain energy is shown to include a bulk and surface strain energy a minimum total potential energy principle is provided for the related structural boundary value problem the toupin mindlin polar type strain gradient material model is also addressed and compared with the above one their substantial differences are pointed out particularly for what regards the constitutive equations and the boundary conditions accompanying the solving displacement equilibrium equations a gradient one dimensional bar sample in tension is considered for a few applications of the proposed theory

a family of higher order mixed finite element methods for plane elasticity the dirichler problem for the equations of plane elasticity is approximated by a mixed finite element method using a new family of composite finite elements having properties analogous to those possessed by the raviart thomas mixed finite elements for a scalar second order elliptic equation estimates of optimal order and minimal regularity are derived for the errors in the displacement vector and the stress tensor inl 2 and optimal order negative norm estimates are obtained inh s for a range ofs depending on the index of the finite element space an optimal order estimate inl for the displacement error is given also a quasioptimal estimate is derived in an appropriate space all estimates are valid uniformly with respect to the compressibility and apply in the incompressible case the formulation of the elements is presented in detail

a regularized model equation for discrete breathers in anharmonic lattices with symmetric nearest neighbor potentials abstract we propose a regularized continuum model equation for describing discrete breathers or intrinsic localized modes in one dimensional anharmonic lattices with symmetric nearest neighbor potentials exact stationary breather solutions with purely hard quartic anharmonicity as well as approximate stationary breather solutions in the general case are found the application of the multiple scales analysis indicates the movability of the small amplitude breather solutions the results of numerical simulations for the model equation fully support the analytical solutions as regards the breather breather collisions the continuum model shares many common features with its discrete counterpart which provides an opportunity to clarify the energy exchange mechanism for collisions between discrete breathers in lattices

multivalued generalizations of probabilistic contractions we generalize some well known fixed point theorems for probabilistic contractions to multivalued contractions the contraction maps are considered in generalized menger spaces

subgame perfect equilibria in discounted stochastic games abstract this paper considers policies and payoffs corresponding to subgame perfect equilibrium strategies in discounted stochastic games with finitely many states it is shown that a policy is induced by an equilibrium strategy if and only if it can be supported with the threat of reverting to the induced policy that gives the least equilibrium payoff for the deviator it follows that the correspondence of subgame perfect equilibrium payoffs is the largest fixed point of a correspondence valued operator defined by the players incentive compatibility conditions moreover the fixed point iteration converges to the equilibrium payoff correspondence

three dimensional bridging scale analysis of dynamic fracture this paper presents a three dimensional generalization of the bridging scale concurrent method a finite temperature multiple scale method that couples molecular dynamics md to finite elements fe the generalizations include the numerical calculation of the boundary condition acting upon the reduced md region as such boundary conditions are analytically intractable for realistic three dimensional crystal structures the formulation retains key advantages emphasized in previous papers particularly the compact size of the resulting time history kernel matrix the coupled fe and reduced md equations of motion are used to analyze dynamic fracture in a three dimensional fcc lattice where interesting physical phenomena such as crack branching are seen the multiple scale results are further compared to benchmark md simulations for verification purposes

probabilistic fractals and attractors in menger spaces abstract probabilistic fractals are considered as self similar sets for contraction mappings in hyperspaces of the menger probabilistic metric spaces the completeness of hyperspaces of all closed or closed and bounded or compact nonempty subsets with respect to the probabilistic hausdorff metric is studied in some general conditions of triangular norm using the properties of hyperspaces and triangular norms of h type several existence results of attractors are obtained for a finite family and a countable family of probabilistic nonlinear contractions respectively our results are generalizations in many aspects

a fixed point theorem in probabilistic metric spaces and an application abstract the notion of a c contraction type multivalued mapping is introduced this notion is a generalization of the notion of c contraction introduced by t l hicks univ u novom sadu zb rad prirod mat fak ser mat 13 1983 63 72 a fixed point theorem for c contraction is proved an application on the existence of a random fixed point for random operator f m m where m d is a separable metric space and a m a measure space with a decomposable measure of nsa type is given

discounted stochastic games we give an alternative proof to a result of mertens and parthasarathy stating that every n player discounted stochastic game with general setup and with a norm continuous transition has a subgame perfect equilibrium

uncoupling of dynamic equations for periodic structures abstract this paper is aimed at providing some explanation of the physical meaning and mathematical formulation of the u transformation method which has found many applications in obtaining solutions for structures with periodicity properties the u transformation was first derived from the mode method for rotational periodic structures the dynamic equation for cyclic periodic structures can be uncoupled in the domain of single substructure by u transformation it is then extended to the double u transformation method for structures with cyclic periodicity in two directions however it should be noted that the method may also be applied to some one way or two way linear periodic structures

a strain gradient generalized continuum approach for modelling elastic scale effects this work follows a generalized continuum framework c sansour a unified concept of elastic viscoplastic cosserat and micromorphic continua j phys iv proc 8 1998 341 348 to derive a first strain gradient formulation which features a generalized deformation description new strain and stress measures as a consequence of these new quantities a corresponding generalized variational principle is formulated and its underlying equilibrium equations are derived the approach is completed by dirichlet boundary conditions for the displacement field and its derivatives the basic idea behind this generalized continuum theory is the consideration of a micro and a macro space which span together the generalized space the approach is appealing from theoretical as well as numerical point of view as it allows for the consideration of classical material laws and circumvents the use of otherwise cumbersome representation theorems the resulting expressions for first and second order stresses are obtained by numerical integration over the micro space in this way material information of the micro space which is here only the geometrical specifications of the micro continuum can naturally enter the constitutive law moreover non linear material behaviour can be considered in a straightforward manner in this work conventional hyperelasticity will be considered four applications in the context of linear and non linear hyperelasticity demonstrate the potential of the proposed method in particular the use of a moving least square based meshfree method facilitates a pure displacement based approximation scheme as it can provide c 1 continuity which is required for

damage gradient of damage and principle of virtual power a theory of continuum damage mechanics is developed within the framework of the principle of virtual power because the damage in a solid results from microscopic movements we decided to include the power of these microscopic movements in the power of the internal forces the microscopic velocities are related to the damage rate the power of the internal forces we choose depends on the damage velocity and its gradient to take into account the interactions models issued from this theory are presented they are coherent from the mechanical and mathematical points of view the numerical computations show no mesh sensitivity they describe with good agreement the main experimental properties concrete is chosen as an example to illustrate the theory a model using two damage quantities is also presented it yields the correct description of the unilateral phenomenon observed in concrete finally an extension of the existing models is proposed to describe the fatigue damage behaviour

global existence of solution of cauchy problem for nonlinear pseudo parabolic equation abstract in this paper we prove that the cauchy problem for the nonlinear pseudo parabolic equation v t v x x t v x x v x f v x v x x g v g v x x admits a unique global generalized solution in c 1 0 h s r a unique global classical solution and asymptotic behavior of the solution we also prove that the cauchy problem for the equation v t v x x t v x x g v g v x x has a unique global generalized solution in c 1 0 w m p r l r a unique global classical solution and asymptotic behavior of the solution

dynamic analysis of a hollow cylinder subject to a dual traveling force imposed on its inner surface abstract the dynamic behavior of a hollow cylinder under a dual traveling force applied to the inner surface is investigated in this study the cylinder is constrained at both the top and bottom surfaces not to move in the length direction but free in other directions and a dual force travels at a constant velocity along the length direction on the inner surface of the hollow cylinder the resulting governing field equations and the associated boundary conditions are ruled by the general hooke s law due to the nature of the field equations proper adjoint system of equations and biorthogonality conditions were derived in a precise and detailed manner to solve these field equations in this study the method of separation of variable is used and the method of fr o benius is employed for the differential equations in the radial direction using the field equations the eigenanalyses on both the original and its adjoint system were performed with great care which results in the eigenfunction sets of both systems the biorthogonality conditions were applied to the field equations to obtain the discretized equation for each mode using the solutions of the discretized equations that account for the boundary forcing terms the critical speed for a dual traveling force for each mode could be computed

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

gradient dependent plasticity formulation and algorithmic aspects a plasticity theory is proposed in which the yield strength not only depends on an equivalent plastic strain measure hardening parameter but also on the laplacian thereof the consistency condition now results in a differential equation instead of an algebraic equation as in conventional plasticity to properly solve the set of non linear differential equations the plastic multiplier is discretized in addition to the usual discretization of the displacements for appropriate boundary conditions this formulation can also be derived from a variational principle accordingly the theory is complete

bridging multi scale method for localization problems abstract a bridging multi scale method is proposed for the analysis of a class of localization problems in which the micropolar continuum model is used to describe the localized deformation and the dynamic failure occurs only in a small number of localized regions it starts with a concurrent discretization of the entire domain with both coarse and fine scale finite element meshes the coarse scale mesh is employed to capture the nonlinear response with long wavelength outside the localized regions whereas the fine scale mesh captures the detailed physics of the localized deformation for both the coarse and fine scale meshes to coexist a bridging scale term is constructed so that the information common to both scales is correctly subtracted to achieve computational efficiency the localized regions are first identified by a preliminary calculation and the fine scale degrees of freedom dofs outside the localized regions are mathematically represented by the construction of dynamic interface conditions applied to the edges of these regions hence a large portion of the fine scale dofs are eliminated and the fine scale equations are reduced to a much smaller set with the added dynamic interface conditions the two way coupled coarse scale and reduced fine scale equations are then solved by a mixed time integration procedure applications to one dimensional and two dimensional dynamic shear localization problems are presented

molecular dynamics boundary conditions for regular crystal lattices we present a method for deriving molecular dynamics boundary conditions for use in multiple scale simulations that can be applied at a planar boundary for any solid that has a periodically repeating crystal lattice the method is based on a linearization in the vicinity of the boundary and utilizes a fourier and laplace transforms in space and time to eliminate the degrees of freedom associated with atoms outside the boundary this method is straightforward to implement numerically and thus can be automated for a general crystal lattice we show that this method reproduces the known kernel for a 1d linear chain and apply the approach to obtain the damping kernel matrices for two real crystal lattices the graphene and diamond structures of carbon

computing equilibria in discounted dynamic games game theory gt is an essential formal tool for interacting entities however computing equilibria in gt is a hard problem when the same game can be played repeatedly over time the problem becomes even more complicated the existence of multiple game states makes the problem of computing equilibria in such games extremely difficult in this paper we approach this problem by first proposing a method to compute a nonempty subset of approximate up to any precision subgame perfect equilibria in repeated games we then demonstrate how to extend this method to approximate all subgame perfect equilibria in a repeated game and also to solve more complex games such as markov chain games and stochastic games we observe that in stochastic games our algorithm requires additional strong assumptions to become tractable while in repeated and markov chain games it allows approximating all subgame perfect equilibria reasonably fast and under considerably weaker assumptions than previous methods

markov games with frequent actions and incomplete information the limit case we study the asymptotics of a class of two player zero sum stochastic game with incomplete information on one side when the time span between two consecutive stages vanishes the informed player observes the realization of a markov chain on which the payoffs depend whereas the noninformed player only observes his opponent s actions we show the existence of a limit value this value is characterized through an auxiliary optimization problem and as the solution of a hamilton jacobi equation

coincidence point theorems in probabilistic and fuzzy metric spaces in this paper we prove some new coincidence point theorems for a hybrid pair of single and multi valued mappings under more generalized contractive conditions in menger spaces and generalized menger spaces our results can be applied to the theory of the fuzzy metric spaces by relaxing some conditions several recent results given by zikic dos enovic mihet and ghaemi and razani are extended

a three dimensional c1 finite element for gradient elasticity in gradient elasticity strain gradient terms appear in the expression of virtual work leading to the need for c1 continuous interpolation in finite element discretizations of the displacement field only employing such interpolation is generally avoided in favour of the alternative methods that interpolate other quantities as well as displacement due to the scarcity of c1 finite elements and their perceived computational cost in this context the lack of three dimensional c1 elements is of particular concern in this paper we present a new c1 hexahedral element which to the best of our knowledge is the first three dimensional c1 element ever constructed it is shown to pass the single element and patch tests and to give excellent rates of convergence in benchmark boundary value problems of gradient elasticity it is further shown that c1 elements are not necessarily more computationally expensive than alternative approaches and it is argued that they may be more efficient in providing good quality solutions copyright 2008 john wiley sons ltd

a folk theorem for asynchronously repeated games we prove a folk theorem for asynchronously repeated games in which the set of players who can move in period t denoted by i i is a random variable whose distribution is a function of the past action choices of the players and the past realizations of i s 1 2 t 1 we impose a condition the finite periods of inaction fpi condition which requires that the number of periods in which every player has at least one opportunity to move is bounded given the fpi condition together with the standard nonequivalent utilities neu condition we show that every feasible and strictly individually rational payoff vector can be supported as a subgame perfect equilibrium outcome of an asynchronously repeated game

a boundary element method for solving 2 d and 3 d static gradient elastic problems part ii numerical implementation the boundary element formulation for the static analysis of two dimensional 2 d and three dimensional 3 d solids and structures characterized by a gradient elastic material behavior developed in the first part of this work is treated numerically in this second part for the creation of a highly accurate and efficient boundary element solution tool the discretization of the body is restricted only to its boundary and is accomplished by the use of quadratic isoparametric three noded line and eight noded quadrilateral boundary elements for the 2 d and 3 d cases respectively advanced algorithms are presented for the accurate and efficient numerical computation of the singular integrals involved numerical examples involving a cylindrical bar in tension and a cylinder and a sphere in radial deformation are solved by the proposed boundary element method and the results are found in excellent agreement with the derived by the authors analytical solutions the bar and sphere problems are solved in a 3 d context while the cylinder problem is solved in a 2 d context plane strain both the exterior and interior versions of the cylinder and sphere problems are considered

a fixed point theorem for multivalued mappings in probabilistic metric spaces and an application in fuzzy metric spaces in this paper we shall generalize a fixed point theorem for multivalued mappings in probabilistic metric spaces proved in j math anal appl 202 1996 433 449 as a corollary a fixed point result in menger spaces s f t where t is a strict t norm is obtained some special cases where t is a member of dombi schweizer sklar and aczel alsina families of t norms are investigated an application in fuzzy metric spaces is presented

mixed finite elements for elasticity there have been many efforts dating back four decades to develop stable mixed finite elements for the stress displacement formulation of the plane elasticity system this requires the development of a compatible pair of finite element spaces one to discretize the space of symmetric tensors in which the stress field is sought and one to discretize the space of vector fields in which the displacement is sought although there are number of well known mixed finite element pairs known for the analogous problem involving vector fields and scalar fields the symmetry of the stress field is a substantial additional difficulty and the elements presented here are the first ones using polynomial shape functions which are known to be stable we present a family of such pairs of finite element spaces one for each polynomial degree beginning with degree two for the stress and degree one for the displacement and show stability and optimal order approximation we also analyze some obstructions to the construction of such finite element spaces which account for the paucity of elements available

semi analytic solution in the time domain for non uniform multi span bernoulli euler beams traversed by moving loads abstract this paper presents a semi analytic solution of the moving load problem which is of great interest for the analysis of multi span uniform and non uniform beams subjected to moving forces such as high speed trains the solution is based on the response of the structure to a unit load circulating at a constant speed of the train viscous modal damping is considered using bernoulli euler beam elements with variable cross sectional properties the structure is discretized and the mode shapes are computed using standard procedure the moving load is represented by a unitary dirac delta function and the modal loads are obtained in terms of cubic hermitian polynomials this leads in a straightforward manner to the closed form solution for the unit load in the time domain the solution is expressed in terms of 10 coefficients per element and per mode the values of which are independent of the speed of the moving load finally the response to a series of loads is built simply by adding the contribution of each the overall procedure is fast and accurate depending only on the spatial discretization and the time step selected for evaluating the solution without the need of any integration step numerical tests have been included in order to show the efficiency of this technique