damping characteristics of a fractional oscillator abstract a study of sinusoidally forced oscillations of a fractional oscillator shows that the system exhibits a rich variety of damping characteristics while some aspects of the damping mimic the characteristic features of a damped harmonic oscillator there are others which do not find any parallel in the damped harmonic oscillator system it is clearly demonstrated that the free and forced oscillations of a fractional oscillator are characterized by different damping parameters while both depend on the fractional index the free oscillation damping depends on the natural frequency 0 of the oscillator the forced oscillation damping depends in addition on the driving frequency furthermore there is a different power law tail associated with each of these cases

perceived ambiguity and relevant measures we axiomatize preferences that can be represented by a monotonic aggregation of subjective expected utilities generated by a utility function and some set of i i d probability measures over a product state space s for such preferences we define relevant measures show that they are treated as if they were the only marginals possibly governing the state space and connect them with the measures appearing in the aforementioned representation these results allow us to interpret relevant measures as reflecting part of perceived ambiguity meaning subjective uncertainty about probabilities over states under mild conditions we show that increases or decreases in ambiguity aversion cannot affect the relevant measures this property necessary for the conclusion that these measures reflect only perceived ambiguity distinguishes the set of relevant measures from the leading alternative in the literature we apply our findings to a number of well known models of ambiguity sensitive preferences for each model we identify the set of relevant measures and the implications of comparative ambiguity aversion

bifurcation phenomena in the optimal velocity model for traffic flow in the optimal velocity model with a time lag we show that there appear multiple exact solutions in some ranges of car density describing a metastable uniform flow a metastable congested flow and an unstable congested flow this establishes the presence of subcritical hopf bifurcations our analytical results have implications for continuum traffic flow such as hysteresis phenomena associated with discontinuous transitions between uniform and congested flow

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

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

map of discrete system into continuous continuous limits of discrete systems with long range interactions are considered the map of discrete models into continuous medium models is defined a wide class of long range interactions that give the fractional equations in the continuous limit is discussed the one dimensional systems of coupled oscillators for this type of long range interactions are considered the discrete equations of motion are mapped into the continuum equation with the riesz fractional derivative

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

refined boundary conditions on the free surface of an elastic half space taking into account non local effects the dynamic response of a homogeneous half space with a traction free surface is considered within the framework of non local elasticity the focus is on the dominant effect of the boundary layer on overall behaviour a typical wavelength is assumed to considerably exceed the associated internal lengthscale the leading order long wave approximation is shown to coincide formally with the local problem for a half space with a vertical inhomogeneity localized near the surface subsequent asymptotic analysis of the inhomogeneity results in an explicit correction to the classical boundary conditions on the surface the order of the correction is greater than the order of the better known correction to the governing differential equations the refined boundary conditions enable us to evaluate the interior solution outside a narrow boundary layer localized near the surface as an illustration the effect of non local elastic phenomena on the rayleigh wave speed is investigated

emd based refined composite multiscale entropy analysis of complex signals abstract multiscale entropy mse is an effective method to measure the complexity of signals from complex systems which has been applied to various fields successfully however mse may yield an inaccurate estimate of entropy and induce undefined entropy as the coarse graining procedure reduces the length of data considerably at large scales refined composite multiscale entropy rcmse is then developed to solve this problem however trends superimposed in signals may significantly affect the complexity measurement thus we introduce an empirical mode decomposition based rcmse called emd rcmse to first eliminate such effects of trends and then measure the complexity of signals it is validated from simulated signals and has good estimation in addition this method is also applied to study the complexity of traffic signals and we obtain some interesting results 1 traffic signals are more complex than the results showing from rcmse mse 2 weekday and weekend patterns different combination of trends greatly affect the results 3 complexity indices change with time at each day due to the degree of human activities

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

continuous limit of discrete systems with long range interaction discrete systems with long range interactions are considered continuous medium models as continuous limit of discrete chain system are defined long range interactions of chain elements that give the fractional equations for the medium model are discussed the chain equations of motion with long range interaction are mapped into the continuum equation with the riesz fractional derivative we formulate the consistent definition of continuous limit for the systems with long range interactions in this paper we consider a wide class of long range interactions that give fractional medium equations in the continuous limit the power law interaction is a special case of this class

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

robotic reactions delay induced patterns in autonomous vehicle systems fundamental design principles are presented for vehicle systems governed by autonomous cruise control devices by analyzing the corresponding delay differential equations it is shown that for any car following model short wavelength oscillations can appear due to robotic reaction times and that there are tradeoffs between the time delay and the control gains the analytical findings are demonstrated on an optimal velocity model using numerical continuation and numerical simulation

a smooth model of decision making under ambiguity we propose and axiomatize a model of preferences over acts such that the decision maker prefers act f to act g if and only if e e u f e e u g where e is the expectation operator u is a vn m utility function is an increasing transformation and is a subjective probability over the set of probability measures that the decision maker thinks are relevant given his subjective information a key feature of our model is that it achieves a separation between ambiguity identified as a characteristic of the decision maker s subjective information and ambiguity attitude a characteristic of the decision maker s tastes we show that attitudes towards risk are characterized by the shape of u as usual while attitudes towards ambiguity are characterized by the shape of we also derive x 1 e x as the special case of constant ambiguity aversion ambiguity itself is defined behaviorally and is shown to be characterized by properties of the subjective set of measures this characterization of ambiguity is formally related to the definitions of subjective ambiguity advanced by epstein zhang 2001 and ghirardato marinacci 2002 one advantage of this model is that the welldeveloped machinery for dealing with risk attitudes can be applied as well to ambiguity attitudes the model is also distinct from many in the literature on ambiguity in that allows smooth rather than kinked indifference curves this leads to different behavior and improved tractability while still sharing the main features e g ellsberg s paradox etc the maxmin eu model e g gilboa and schmeidler 1989 with a given set of measures may be seen as an extreme case of our model with infinite ambiguity aversion two illustrative applications to portfolio choice are offered

uniqueness of initial boundary value problems in nonlocal elasticity abstract in this study we have dealt with the uniqueness of the solutions of a class of initial boundary value problems in linear isotropic homogeneous nonlocal elasticity the proof of uniqueness is based on the positive definiteness of total strain energy after giving the sufficient conditions under which total strain energy is positive definite it has been shown that the solutions of the initial boundary value problems which are considered in this study are unique existence of the solutions of these problems is an a priori assumption

vector expected utility and attitudes toward variation this paper proposes a model of decision under ambiguity deemed vector expected utility or veu in this model an uncertain prospect or savage act is assessed according to a a baseline expected utility evaluation a nd b anadjustment that reflects the individual s perception of ambiguity and her attitudes toward it the adjustment is itself a function of the act s exposure to distinct sources of ambiguity as well as its variability the key elements of the veu model are a baseline probability and a collection of random variables or adjustment factors which represent acts exposed to distinct ambiguity sources and also reflect complementarities among ambiguous events the adjustment to the baseline expected utility evaluation of an act is a function of the covariance of its utility profile with each adjustment factor which reflects exposure to the corresponding ambiguity source a behavioral characterization of the veu model is provided furthermore an updating rule for veu preferences is proposed and characterized the suggested updating rule facilitates the analysis of sophisticated dynamic choice with veu preferences

bifurcation analysis of a class of car following traffic models abstract we consider a follow the leader traffic model describing the dynamics of n cars on a circular road where each car driver chooses his acceleration according to a certain law the model is represented by a nonlinear system of ode s this model is known to have a solution with constant velocities and headways which in a certain parameter regime is stable varying the density of the cars we prove that the loss of stability is generally due to a hopf bifurcation also we investigate numerically the global bifurcation diagram for periodic solutions and obtain a complete picture of the dynamics of general optimal velocity models finally some analytical results on the stability of solutions in the case of non equal drivers are given

long range cohesive interactions of non local continuum faced by fractional calculus abstract a non local continuum model including long range forces between non adjacent volume elements has been studied in this paper the proposed continuum model has been obtained as limit case of two fully equivalent mechanical models i a volume element model including contact forces between adjacent volumes as well as long range interactions distance decaying between non adjacent elements ii a discrete point spring model with local springs between adjacent points and non local springs with distance decaying stiffness connecting non adjacent points under the assumption of fractional distance decaying interactions between non adjacent elements a fractional differential equation involving marchaud type fractional derivatives has been obtained for unbounded domains it is shown that for unbounded domains the two mechanical models revert to lazopoulos and eringen model with fractional distance decaying functions it has also been shown that for a confined bar the stress strain relation is substantially different from that obtained simply using the truncated marchaud derivatives since a double integral instead of convolution integral appears moreover in the analysis of bounded domains the governing equations turn out to an integro differential equation including only the integral part of marchaud fractional derivatives on finite interval the mechanical boundary condition for the proposed model has been introduced consistently on the basis of mechanical considerations and the constitutive law of the proposed continuum model has been reported by mathematical induction several numerical applications have been reported to show verify and assess the concepts listed in this paper

lattice with long range interaction of power law type for fractional non local elasticity abstract lattice models with long range interactions of power law type are suggested as a new type of microscopic model for fractional non local elasticity using the transform operation we map the lattice equations into continuum equation with riesz derivatives of non integer orders the continuum equations that are obtained from the lattice model describe fractional generalization of non local elasticity models particular solutions and correspondent asymptotic of the fractional differential equations for displacement fields are suggested for the static case

eigenvalue problems of the model from nonlocal continuum mechanics this article studies the eigenvalue problem of a fractional differential equation which is a foundation model of a bar of finite length with long range interactions arising from non local continuum mechanics we show that this problem has countable simple real eigenvalues and the corresponding eigenfunctions form a complete orthogonal system in the hilbert space l2 furthermore the asymptotic behavior of eigenvalues and the numbers of zeros of eigenfunctions are studied by using the analytic perturbation theory

scaling laws and renormalization groups for strength and toughness of disordered materials abstract the abundant literature on tensile strength and fracture energy size effects as well as the newly introduced fractal and renormalization group theories would appear to indicate the need for a dramatic change in our conceptual framework if we want to consider and measure material constants in strength of materials as well as in fracture mechanics for disordered materials such as for example concrete and rocks the renormalized tensile strength is given by a force acting on a surface having a fractal dimension lower than 2 just as the renormalized fracture energy is represented by a dissipation over a surface with a dimension higher than 2 in the case of tensile strength the dimensional decrement represents self similar weakening of the reacting cross section or ligament due to pores voids defects cracks aggregates inclusions etc likewise in the case of fracture energy the dimensional increment represents self similar tortuosity of the fracture surface due to aggregates and inclusions as well as self similar microcrack overlapping and distribution also in the direction orthogonal to that of the forming macrocrack it can be demonstrated that the sum of the dimensional decrement for material ligament and the dimensional increment for fracture surface must be lower than unity

analysis of optimal velocity model with explicit delay we analyze the optimal velocity model ovm with explicit delay the properties of congestion and the delay time of car motion are investigated by analytical and numerical methods it is shown that the small explicit delay time has almost no effects in the case of the large explicit delay time a new phase of congestion pattern of ovm seems to appear

bifurcations and multiple traffic jams in a car following model with reaction time delay we investigate an optimal velocity car following model for n cars on a circular singlelane road where reaction time delay of drivers is taken into account the stability of the uniform ow equilibrium is studied analytically while bifurcating periodic solutions for difierent wave numbers are investigated with numerical continuation techniques this reveals that the periodic solution with the smallest wave number may be stable and all other periodic solutions are unstable as n is increased periodic solutions develop stop and go fronts that correspond to rapid deceleration and acceleration between regions of uniformly owing and stagnant tra c in terms of the positions of all cars on the ring these fronts are associated with tra c jams all tra c jams form a tra c pattern that evolves under time due to slow motion of the fronts the tra c pattern corresponding to the stable periodic motion of cars is the only stable one however we flnd that other periodic orbits may be unstable only so weakly that they give rise to transient tra c jams that may persist for long times eventually such tra c jams either merge with one another or disperse until the stable tra c pattern is reached

kink soliton characterizing traffic congestion we study traffic congestion by analyzing a one dimensional traffic flow model developing an asymptotic method to investigate the long time behavior near a critical point we derive the modified korteweg de vries mkdv equation as the lowest order model there is an infinite number of kink solitons to the mkdv equation while it has been found by numerical simulations that the kink pattern arising in traffic congestion is uniquely determined irrespective of initial conditions in order to resolve this selection problem we consider higher order corrections to the mkdv equation and find that there is a kink soliton that can deform continuously with the perturbation represented by the addition of these corrections with numerical confirmation we show that this continuously deformable kink soliton characterizes traffic congestion we also discuss the relationship between traffic congestion and the slugging phenomenon in granular flow

the finite element method for the mechanically based model of non local continuum summary in this paper the finite element method fem for the mechanically based non local elastic continuum model is proposed in such a model each volume element of the domain is considered mutually interacting with the others beside classical interactions involved by the cauchy stress field by means of central body forces that are monotonically decreasing with their inter distance and proportional to the product of the interacting volume elements the constitutive relations of the long range interactions involve the product of the relative displacement of the centroids of volume elements by a proper distance decaying function which accounts for the decrement of the long range interactions as long as distance increases in this study the elastic problem involving long range central interactions for isotropic elastic continuum will be solved with the aid of the fem the accuracy of the solution obtained with the proposed fem code is compared with other solutions obtained with galerkins approximation as well as with finite difference method moreover a parametric study regarding the effect of the material length scale in the mechanically based model and in the kroner eringen non local elasticity has been investigated for a plane elasticity problem copyright 2011 john wiley sons ltd

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

a symmetric nonlocal damage theory abstract the paper presents a thermodynamically consistent formulation for nonlocal damage models nonlocal models have been recognized as a theoretically clean and computationally efficient approach to overcome the shortcomings arising in continuum media with softening the main features of the presented formulation are i relations derived by the free energy potential fully complying with nonlocal thermodynamic principles ii nonlocal integral operator which is self adjoint at every point of the solid including zones near to the solid s boundary iii capacity of regularizing the softening ill posed continuum problem restoring a meaningful nonlocal boundary value problem in the present approach the nonlocal integral operator is applied consistently to the damage variable and to its thermodynamic conjugate force i e nonlocality is restricted to internal variables only the present model when associative nonlocal damage flow rules are assumed allows the derivation of the continuum tangent moduli tensor and the consistent tangent stiffness matrix which are symmetric the formulation has been compared with other available nonlocal damage theories finally the theory has been implemented in a finite element program and the numerical results obtained for 1 d and 2 d problems show its capability to reproduce in every circumstance a physical meaningful solution and fully mesh independent results

variational formulations and a consistent finite element procedure for a class of nonlocal elastic continua the structural boundary value problem in the context of nonlocal integral elasticity and quasi static loads is considered in a geometrically linear range the nonlocal elastic behaviour is described by the so called eringen model in which the nonlocality lies in the constitutive relation the diffusion processes of the nonlocality are governed by an integral relation containing a recently proposed symmetric spatial weight function expressed in terms of an attenuation function a firm variational basis to the nonlocal model is given by providing the complete set of variational formulations composed by ten functionals with different combinations of the state variables in particular the nonlocal counterpart of the classical principles of the total potential energy complementary energy and mixed hu washizu principle and hellinger reissner functional are recovered some examples concerning a piecewise bar in tension are provided by using the fredholm integral equation and the proposed nonlocal fem

beams on a two dimensional pasternak base subjected to loads that cause lift off abstract first two closely related problems shown in fig 1 published by chernigovskaya 1961 in issledovaniya po dinamike sooruzhenii i raschetu konstruktsii na uprugoni osnovanii edited by b g korenev pp 113 141 gosstroiizdat moscow and ting 1973 j franklin inst 296 2 77 89 are discussed it is shown that neither of their formulations is correct the aim of this paper is to show how to correctly formulate and solve problems of this type utilizing the variational approach for variable matching points derived by kerr 1976 int j solids structures 12 1 11 a formulation lor the problem analyzed by ting is presented that is mechanically reasonable and mathematically well posed the analytical solution obtained is evaluated numerically and then compared with related test results by durelli et al 1969 j struct div asce 95 1713 1725 this paper concludes with a discussion of the results obtained

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

derivation of continuum traffic flow models from microscopic follow the leader models in this paper we establish a connection between a microscopic follow the leader model based on ordinary differential equations and a semidiscretization of a macroscopic continuum model based on a c

space time fractional zener wave equation the space time fractional zener wave equation describing viscoelastic materials obeying the time fractional zener model and the space fractional strain measure is derived and analysed this model includes waves with finite speed as well as non propagating disturbances the existence and the uniqueness of the solution to the generalized cauchy problem are proved special cases are investigated and numerical examples are presented

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

lattice fractional calculus integration and differentiation of non integer orders for n dimensional physical lattices with long range particle interactions are suggested the proposed lattice fractional derivatives and integrals are represented by kernels of lattice long range interactions such that their fourier series transformations have a power law form with respect to components of wave vector continuous limits for these lattice fractional derivatives and integrals give the continuum derivatives and integrals of non integer orders with respect to coordinates lattice analogs of fractional differential equations that include suggested lattice differential and integral operators can serve as an important element of microscopic approach to nonlocal continuum models in mechanics and physics

ambiguity aversion robustness and the variational representation of preferences we characterize in the anscombe aumann framework the preferences for which there are a utility functionu on outcomes and an ambiguity indexc on the set of probabilities on the states of the world such that for all acts f and g the function u represents the decision maker s risk attitudes while the index c captures his ambiguity attitudes these preferences include the multiple priors preferences of gilboa and schmeidler and the multiplier preferences of hansen and sargent this provides a rigorous decision theoretic foundation for the latter model which has been widely used in macroeconomics and finance

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

a generalized fractional order elastodynamic theory for non local attenuating media this study presents a generalized elastodynamic theory based on fractional order operators capable of modelling the propagation of elastic waves in non local attenuating solids and across complex

on a functional differential equation arising from a traffic flow model we provide a derivation in the context of a traffic flow model as well as both analytical and numerical studies of the functional differential equation z s alpha 2 z s beta z s z s z s here alpha and beta are positive parameters and we are in particular investigating the existence and properties of nonconstant traveling wave type solutions

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

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

uncertainty aversion with second order utilities and probabilities subjective expected utility theory does not distinguish between attitudes toward uncertainty ambiguous probabilities and attitudes toward risk unambiguous probabilities both are explained in terms of nonlinear utility for money rather than properties of events per se hence the decision maker displays the same attitude toward all sources of risk and uncertainty there is ample evidence that real decision makers do not always behave or even wish to behave in this way and instead they often distinguish between risk and uncertainty as in ellsbergs 1961 paradox this paper presents a simple axiomatic model of nonneutral attitudes toward uncertainty and a behavioral test for uncertainty aversion that is applicable even if utility is state dependent the decision maker may display different degrees of aversion toward gambles on different kinds of events e g being systematically more averse toward gambles on events whose probabilities are more ambiguous for such a decision maker the elicitation of preferences among objective gambles may not yield the correct measure of risk aversion for modeling real world decisions

static kinematic duality and the principle of virtual work in the mechanics of fractal media abstract the framework for the mechanics of solids deformable over fractal subsets is outlined while displacements and total energy maintain their canonical physical dimensions renormalization group theory permits to define anomalous mechanical quantities with fractal dimensions i e the fractal stress and the fractal strain e a fundamental relation among the dimensions of these quantities and the hausdorff dimension of the deformable subset is obtained new mathematical operators are introduced to handle these quantities in particular classical fractional calculus fails to this purpose whereas the recently proposed local fractional operators appear particularly suitable the static and kinematic equations for fractal bodies are obtained and the duality principle is shown to hold finally an extension of the gauss green theorem to fractional operators is proposed which permits to demonstrate the principle of virtual work for fractal media

a generalized model of elastic foundation based on long range interactions integral and fractional model abstract the common models of elastic foundations are provided by supposing that they are composed by elastic columns with some interactions between them such as contact forces that yield a differential equation involving gradients of the displacement field in this paper a new model of elastic foundation is proposed introducing into the constitutive equation of the foundation body forces depending on the relative vertical displacements and on a distance decaying function ruling the amount of interactions different choices of the distance decaying function correspond to different kind of interactions and foundation behavior the use of an exponential distance decaying function yields an integro differential model while a fractional power law decay of the distance decaying function yields a fractional model of elastic foundation ruled by a fractional differential equation it is shown that in the case of exponential decaying function the integral equation represents a model in which all the gradients of the displacement function appear while the fractional model is an intermediate model between integral and gradient approaches a fully equivalent discrete point spring model of long range interactions that may be used for the numerical solution of both integral and fractional differential equation is also introduced some green s functions of the proposed model have been included in the paper and several numerical results have been also reported to highlight the effects of long range forces and the governing parameters of the linear elastic foundation proposed

bifurcation analysis of traffic flow through an improved car following model considering the time delayed velocity difference abstract in this paper a new car following model with consideration of time delayed velocity difference is proposed to investigate the effect of time delayed velocity on traffic flow via bifurcation analysis based on this model the linear stability analysis is carried out and the corresponding stability condition is obtained then the hopf bifurcation analysis is performed in term of theoretical analysis and numerical simulation the result shows that there exist hopf bifurcation critical points and the position of the critical points is found to be located on the boundary of the instability region finally the influence of the delay time and sensitivity of velocity difference on hopf bifurcation is discussed respectively

nonlocal elasticity and related variational principles abstract the eringen model of nonlocal elasticity is considered and its implications in solid mechanics studied the model is refined by assuming an attenuation function depending on the geodetical distance between material particles such that in the diffusion processes of the nonlocality effects certain obstacles as holes or cracks existing in the domain can be circumvented a suitable thermodynamic framework with nonlocality is also envisaged as a firm basis of the model the nonlocal elasticity boundary value problem for infinitesimal displacements and quasi static loads is addressed and the conditions for the solution uniqueness are established three variational principles nonlocal counterparts of classical ones i e the total potential energy the complementary energy and the mixed hu washizu principles are provided the former is used to derive a nonlocal type fem nl fem in which the symmetric global stiffness matrix reflects the nonlocality features of the problem an alternative standard fem based solution method is also provided which consists in an iterative procedure of the type local prediction nonlocal correction in which the nonlocality is simulated by an imposed like correction strain the potentialities of these analysis methods are pointed out their numerical implementations being the object of an ongoing research work

fractional differential equations and related exact mechanical models the aim of the paper is the description of fractional order differential equations in terms of exact mechanical models this result will be archived in the paper for the case of linear multiphase fractional hereditariness involving linear combinations of power laws in relaxation creep functions the mechanical model corresponding to fractional order differential equations is the extension of a recently introduced exact mechanical representation di paola and zingales 2012 33 and di paola et al 2012 34 of fractional order integrals and derivatives some numerical applications have been reported in the paper to assess the capabilities of the model in terms of a peculiar arrangement of linear springs and dashpots

on pattern formation in a class of traffic models abstract we consider a simple microscopic follow the leader model of n identical cars on a circular road in the classical setting a car is not allowed to reach its leader if it happens at a particular time t e we propose to extend the model beyond this time t e the idea is to define a physically reasonable leader for each car mathematically the resulting model is a kind of filippov system the objective is to study a long time behavior of the model in case n 3 we observed several invariant patterns which apparently exhibit spatial temporal symmetries we managed to classify and identify two kinds of rotating waves the classification exploits symbolic dynamical notions

derivation of mindlin s first and second strain gradient elastic theory via simple lattice and continuum models abstract mindlin in his celebrated papers of arch rat mech anal 16 51 78 1964 and int j solids struct 1 417 438 1965 proposed two enhanced strain gradient elastic theories to describe linear elastic behavior of isotropic materials with micro structural effects since then many works dealing with strain gradient elastic theories derived either from lattice models or homogenization approaches have appeared in the literature although elegant none of them reproduces entirely the equation of motion as well as the classical and non classical boundary conditions appearing in mindlin theory in terms of the considered lattice or continuum unit cell furthermore no lattice or continuum models that confirm the second gradient elastic theory of mindlin have been reported in the literature the present work demonstrates two simple one dimensional models that conclude to first and second strain gradient elastic theories being identical to the corresponding ones proposed by mindlin the first is based on the standard continualization of the equation of motion taken for a sequence of mass spring lattices while the second one exploits average processes valid in continuum mechanics furthermore mindlin developed his theory by adding new terms in the expressions of potential and kinetic energy and introducing intrinsic micro structural parameter without however providing explicit expressions that correlate micro structure with macro structure this is accomplished in the present work where in both models the derived internal length scale parameters are correlated to the size of the considered unit cell

objective and subjective rationality in a multiple prior model a decision maker dm is characterized by two binary relations the first reflects choices that are rational in an objective sense the dm can convince others that she is right in making them the second relation models choices that are rational in a subjective sense the dm cannot be convinced that she is wrong in making them in the context of decision under uncertainty we propose axioms that the two notions of rationality might satisfy these axioms allow a joint representation by a single set of prior probabilities and a single utility index it is objectively rational to choose f in the presence of g if and only if the expected utility of f is at least as high as that of g given each and every prior in the set it is subjectively rational to choose f rather than g if and only if the minimal expected utility of f with respect to all priors in the set is at least as high as that of g in other words the objective and subjective rationality relations admit respectively a representation a la bewley 2002 and a la gilboa and schmeidler 1989 our results thus provide a bridge between these two classic models as well as a novel foundation for the latter

inclusions and inhomogeneities in strain gradient elasticity with couple stresses and related problems abstract at small length scales and or in presence of large field gradients the implicit long wavelength assumption of classical elasticity breaks down postulating a form of second gradient elasticity with couple stresses as a suitable phenomenological model for small scale elastic phenomena we herein extend eshelby s classical formulation for inclusions and inhomogeneities while the modified size dependent eshelby s tensor and hence the complete elastic state of inclusions containing transformation strains or eigenstrains is explicitly derived the corresponding inhomogeneity problem leads to integrals equations which do not appear to have closed form solutions to that end eshelby s equivalent inclusion method is extended to the present framework in form of a perturbation series that then can be used to approximate the elastic state of inhomogeneities the approximate scheme for inhomogeneities also serves as the basis for establishing expressions for the effective properties of composites in second gradient elasticity with couple stresses the present work is expected to find application towards nano inclusions and certain types of composites in addition to being the basis for subsequent non linear homogenization schemes

stochastic master equation approach to aggregation in freeway traffic in analogy to the usual aggregation phenomena such as the formation of liquid droplets in a supersaturated vapor the nucleation growth and condensation of car clusters are considered in a one lane freeway traffic flow model the clustering behavior known as congestion in an initially homogeneous traffic stream is described by a master equation the construction of the stochastic equation is given as well as its relationship to other dynamical models numerical experiments in heavy traffic with well explained transition probabilities show the transition from the initial free particle situation free jet of vehicles to the final congested cluster state where one big aggregate of cars is formed the results are presented analytically in dependence of the stable cluster size from car concentration and numerically as stochastic trajectories

improved stability bound for steady state flow in a car following model of road traffic on a circular route this note revisits a car following model of road traffic on a circular route that was studied in recent literature and improves a stability result for steady state flows that was obtained in this literature it will be shown through a counter example that the stability bound obtained in the literature only gives a sufficient condition for stability which only becomes necessary when the number of cars on the route tends to infinity we will further present a result that gives a necessary and sufficient condition for stability

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

closed form solution for a nonlocal elastic bar in tension a simple mechanical one dimensional problem in the context of nonlocal integral elasticity is solved analytically the nonlocal elastic material behaviour is described by the eringen model whose nonlocality features all reside in the constitutive relation this relation of integral type contains an attenuation function usually assumed symmetric aimed to capture the diffusion process of the nonlocality effects it also exhibits a convolution format the governing equation is a fredholm integral equation of second kind whose analytical treatment even for the usual choice of a symmetric kernel is not easy to develop in the present paper assuming a specific shape for the attenuation function a closed form solution in terms of strains is alternatively obtained by solving a volterra integral equation of second kind the latter can be easily solved with standard techniques at least for the adopted kernel taking also advantage from the symmetry of the solution such a closed form solution is an essential result to validate the effectiveness of numerical procedures aimed to solve more complex mechanical problems in the context of nonlocal elasticity 2002 elsevier science ltd all rights reserved

on a theory of nonlocal elasticity of bi helmholtz type and some applications abstract a theory of nonlocal elasticity of bi helmholtz type is studied we employ eringen s model of nonlocal elasticity with bi helmholtz type kernels to study dispersion relations screw and edge dislocations the nonlocal kernels are derived analytically as green functions of partial differential equations of fourth order this continuum model of nonlocal elasticity involves two material length scales which may be derived from atomistics the new nonlocal kernels are nonsingular in one two and three dimensions furthermore the nonlocal elasticity of bi helmholtz type improves the one of helmholtz type by predicting a dispersion relationship with zero group velocity at the end of the first brillouin zone new solutions for the stresses and strain energy of screw and edge dislocations are found

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

toward lattice fractional vector calculus an analog of fractional vector calculus for physical lattice models is suggested we use an approach based on the models of three dimensional lattices with long range inter particle interactions the lattice analogs of fractional partial derivatives are represented by kernels of lattice long range interactions where the fourier series transformations of these kernels have a power law form with respect to wave vector components in the continuum limit these lattice partial derivatives give derivatives of non integer order with respect to coordinates in the three dimensional description of the non local continuum the fractional differential operators have the form of fractional partial derivatives of the riesz type as examples of the applications of the suggested lattice fractional vector calculus we give lattice models with long range interactions for the fractional maxwell equations of non local continuous media and for the fractional generalization of the mindlin and aifantis continuum models of gradient elasticity

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

coupling of nonlocal and local continuum models by the arlequin approach summary the objective of this work is to develop and apply the arlequin framework to couple nonlocal and local continuum mechanical models a mechanically based model of nonlocal elasticity which involves both contact and long range forces is used for the fine scale description in which nonlocal interactions are considered to have non negligible effects classical continuum mechanics only involving local contact forces is introduced for the rest of the structure where these nonlocal effects can be neglected both models overlap in a coupling subdomain called the gluing area in which the total energy is separated into nonlocal and local contributions by complementary weight functions a weak compatibility is ensured between kinematics of both models using lagrange multipliers over the gluing area the discrete formulation of this specific arlequin coupling framework is derived and fully described the validity and limits of the technique are demonstrated through two dimensional numerical applications and results are compared against those of the fully nonlocal elasticity method copyright 2011 john wiley sons ltd

a critical comparison of nonlocal and gradient enhanced softening continua continuous models of material degradation may cease to produced meaningful results in the presence of high strain gradients these gradients may occur for instance in the propagation of waves with high wave numbers and at stress concentrators adding nonlocal or gradient terms to the constitutive modelling may enhance the ability of the models to describe such situations the effect of adding nonlocal or gradient terms and the relation between these enhancements are examined in a continuum damage setting a nonlocal damage model and two different gradient damage models are considered in one of the gradient models higher order deformation gradients enter the equilibrium equations explicitly while in the other model the gradient influence follows in a more implicit way from an additional partial differential equation the latter implicit gradient formulation can be rewritten in the integral format of the nonlocal model and can therefore be regarded as truly nonlocal this is not true for the explicit formulation in which the nonlocality is limited to an infinitesimal volume this fundamental difference between the formulations results in quite different behaviour in wave propagation localisation and at crack tips this is shown for the propagation of waves in the models their localisation properties and the behaviour at a crack tip the responses of the nonlocal model and the implicit gradient model agree remarkably well in these situations while the explicit gradient formulation shows an entirely different and sometimes nonphysical response

a traffic flow model with overtaking as a filippov system abstract we investigate microscopic models of road traffic in particular we consider a car following model for a single line traffic flow of n identical cars on a circular road the classical differentiable model breaks down at the time instant when two cars collide however the natural action of a driver would be to overtake a slower car in our previous work we proposed a model which simulates the overtaking we observed a large variety of oscillatory solutions oscillatory patterns of the model in the present contribution we formulate our model as a particular filippov system i e odes with discontinuous right hand sides hence we can identify our problem with a well defined solution class we define oscillatory patterns as invariant objects of this filippov system

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

mechanically based approach to non local elasticity variational principles abstract the mechanically based approach to non local elastic continuum will be captured through variational calculus based on the assumptions that non adjacent elements of the solid may exchange central body forces monotonically decreasing with their interdistance depending on the relative displacement and on the volume products such a mechanical model is investigated introducing primarily the dual state variables by means of the virtual work principle the constitutive relations between dual variables are introduced defining a proper convex potential energy it is proved that the solution of the elastic problem corresponds to a global minimum of the potential energy functional moreover the euler lagrange equations together with the natural boundary conditions associated to the total potential energy functional are established with variational calculus and they coincide with analogous relations already obtained by means of mechanical considerations numerical analysis of a tensile specimen has been introduced to show the capabilities of the proposed approach

wave propagation in 1d elastic solids in presence of long range central interactions abstract in this paper wave propagation in non local elastic solids is examined in the framework of the mechanically based non local elasticity theory established by the author in previous papers it is shown that such a model coincides with the well known kroner eringen integral model of non local elasticity in unbounded domains the appeal of the proposed model is that the mechanical boundary conditions may easily be imposed because the applied pressure at the boundaries of the solid must be equilibrated by the cauchy stress in fact the long range forces between different volume elements are modelled in the body domain as central body forces applied to the interacting elements it is shown that the shape change of travelling disturbances coalesces with those predicted by the non local integral theory of elasticity in unbounded domains but several differences arise in the case of bounded domains the wave propagation problem has been formulated by means of the hamiltonian functional of the proposed mechanically based model of non local elasticity introducing an additional term to the elastic potential energy that accounts for elastic long range interactions in this way the wave equation may be obtained in a weak formulation and be further used to provide approximate analytical solutions to the governing equation in the context of standing wave analysis an equivalent discrete point spring model similar to lattice type networks has also been introduced to show the mechanical equivalence of the non local elastic model as well as to provide a mechanical scheme suitable for the numerical treatment of pressure waves travelling in non local bounded domains

subcritical hopf bifurcations in a car following model with reaction time delay a nonlinear car following model of highway traffic is considered which includes the reaction time delay of drivers linear stability analysis shows that the uniform flow equilibrium of the system loses its stability via hopf bifurcations and thus oscillations can appear the stability and amplitudes of the oscillations are determined with the help of normal form calculations of the hopf bifurcation that also handles the essential translational symmetry of the system we show that the subcritical case of the hopf bifurcation occurs robustly which indicates the possibility of bistability we also show how these oscillations lead to spatial wave formation as can be observed in real world traffic flows

ambiguity in the small and in the large this paper considers local and global multiple prior representations of ambiguity for preferences that are i monotonic ii bernoullian that is admit an affine utility representation when restricted to constant acts and iii locally lipschitz continuous we do not require either certainty independence or uncertainty aversion we show that the set of priors identified by ghirardato maccheroni and marinacci s 2004 unambiguous preference relation can be characterized as a union of clarke differentials we then introduce a behavioral notion of locally better deviation at an act and show that it characterizes the clarke differential of the preference representation at that act these results suggest that the priors identified by these preference statements are directly related to local optimizing behavior

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

time series forecasting using a system of ordinary differential equations this paper presents a hybrid evolutionary method for identifying a system of ordinary differential equations odes to predict the small time scale traffic measurements data we used the tree structure based evolutionary algorithm to evolve the architecture and a particle swarm optimization pso algorithm to fine tune the parameters of the additive tree models for the system of ordinary differential equations we also illustrate some experimental comparisons with genetic programming gene expression programming and a feedforward neural network optimized using pso algorithm experimental results reveal that the proposed method is feasible and efficient for forecasting the small scale traffic measurements data

representing preferences with a unique subjective state space we extend kreps 1979 analysis of preference for flexibility reinterpreted by kreps 1992 as a model of unforeseen contingencies we enrich the choice set consequently obtaining uniqueness results that were not possible in kreps model we consider several representations and allow the agent to prefer commitment in some contingencies in the representations the agent acts as if she had coherent beliefs about a set of possible future ex post preferences each of which is an expected utility preference we show that this set of ex post preferences called the subjectie state space is essentially unique given the restriction that all ex post preferences are expected utility preferences and is minimal even without this restriction because the subjective state space is identified the way ex post utilities are aggregated into an ex ante ranking is also essentially unique hence when a representation that is additive across states exists the additivity is meaningful in the sense that all representations are intrinsically additive uniqueness enables us to show that the size of the subjective state space provides a measure of the agent s uncertainty about future contingencies and that the way the states are aggregated indicates whether these contingencies lead to a desire for flexibility or commitment

fractional liouville equation on lattice phase space abstract in this paper we propose a lattice analog of phase space fractional liouville equation the liouville equation for phase space lattice with long range jumps of power law types is suggested we prove that the continuum limit transforms this lattice equation into liouville equation with conjugate riesz fractional derivatives of non integer orders with respect to coordinates of continuum phase space an application of the fractional liouville equation with these riesz fractional derivatives to describe properties of plasma like nonlocal media is considered

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

fractional order difference equations for physical lattices and some applications fractional order operators for physical lattice models based on the grunwald letnikov fractional differences are suggested we use an approach based on the models of lattices with long range particle interactions the fractional order operators of differentiation and integration on physical lattices are represented by kernels of lattice long range interactions in continuum limit these discrete operators of non integer orders give the fractional order derivatives and integrals with respect to coordinates of the grunwald letnikov types as examples of the fractional order difference equations for physical lattices we give difference analogs of the fractional nonlocal navier stokes equations and the fractional nonlocal maxwell equations for lattices with long range interactions continuum limits of these fractional order difference equations are also suggested

subjective probability and expected utility without additivity comonotic acts are defined and an axiom of comonotomic independence 19 introduced that weakens the von newmann morgenstern axiom of independence if a nondegenerate continuous and monotonic state independent weak order over acts satisfies comonotonic independence then it induces a unique non necessarily additive probability and a von neumann morgenstern utility an extension of theexpected utility theory covers situations as the ellsberg paradox which are inconsistent with additive expected utility the concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included

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

the mechanically based approach to 3d non local linear elasticity theory long range central interactions abstract this paper presents the generalization to a three dimensional 3d case of a mechanically based approach to non local elasticity theory recently proposed by the authors in a one dimensional 1d case the proposed model assumes that the equilibrium of a volume element is attained by contact forces between adjacent elements and by long range forces exerted by non adjacent elements specifically the long range forces are modelled as central body forces depending on the relative displacement between the centroids of the volume elements measured along the line connecting the centroids further the long range forces are assumed to be proportional to a proper material dependent distance decaying function and to the products of the interacting volumes consistently with the modelling of the long range forces as central body forces the static boundary conditions enforced on the free surface of the solid involve only local stress due to contact forces the proposed 3d formulation is developed both in a mechanical and in a variational context for this the elastic energy functionals of the solid with long range interactions are introduced based on the principle of virtual work to set the proper correspondence between the mechanical and the kinematic variables of the model numerical applications are reported for 2d solids under plane stress conditions

ambiguity and second order belief anscombe and aumann 1963 wrote a classic characterization of subjective expected utility theory this paper employs the same domain for preference and a closely related but weaker set of axioms to characterize preferences that use second order beliefs beliefs over probability measures such preferences are of interest because they accommodate ellsberg type behavior