introduction to fuzzy partial differential equations this paper considers solutions to elementary fuzzy partial differential equations we follow the same strategy as in buckley and feuring fuzzy sets and systems to appear which is 1 first check to see if the buckley feuring method produces a solution and 2 if the buckley feuring method does not give a solution then see if the seikkala procedure generates a solution examples are presented showing the buckley feuring solution and the seikkala solution

instability in bacterial populations and the curvature tensor in the geometry associated with equilibrium thermodynamics the scalar curvature rs is a measure of the volume of correlation and therefore the singularities of rs indicates the system instabilities we explore the use of a similar approach to study instabilities in non equilibrium systems and we choose as a test example a colony of bacteria in this regard we follow the proposal made by obata et al of using the curvature tensor for studying system instabilities bacterial colonies are often found in nature in concentrated biofilms or other colony types which can grow into spectacular patterns visible under the microscope for instance it is known that a decrease of bacterial motility with density can promote separation into bulk phases of two coexisting densities this is opposed to the logistic law for birth and death that allows only a single uniform density to be stable although this homogeneous configuration is stable in the absence of bacterial interactions without logistic growth a density dependent swim speed v leads to phase separation via a spinodal instability thus we relate the singularities in the curvature tensor r to the spinodal instability that is the appearance of regions of different densities of bacteria

the measurement of surface cracks in metals by using a c electric fields this paper describes some recent studies associated with the development of an instrument the crack microgauge at university college london for the detection and measurement of surface cracks in metals by using a c electric currents in the metal surface the design of the instrument has enhanced the accuracy of measurement of surface voltages in the neighbourhood of surface breaking cracks to the point where it is now advantageous to consider the distribution of surface voltages by mathematical analysis this enables better use to be made of the instrument since measurements can then be more accurately interpreted in terms of crack size and geometry in contrast earlier applications of the method relied on the calibration of the instruments against test blocks the first generation of users of the crack microgauge interpreted their readings according to a simple one dimensional normal crack model this however is an oversimplification for cracks of finite aspect ratio and it leads to underestimates of centre line crack depth typically of the order of 30 40 when the aspect ratio is about 3 the first major contribution of the theory was therefore to provide a model for surface current flow around cracks of finite aspect ratio for this purpose a useful unfolding theorem was deduced in cases in which the current skin depth is small compared with crack dimensions this allows the surface field to be unfolded into a plane distribution of potential satisfying laplace s equation and it enables us to obtain solutions for many different crack shapes by using standard mathematical methods an account of these developments is given in xi 2 of this paper the unfolding of the surface field gives the global distribution of surface current around a crack that is on the length scale of the crack dimensions on the length scale of the skin depth however the solutions need modification in the neighbourhood of the surface and interior edges of the crack in developing the unfolding theorem it was argued that these modifications which we describe as corner solutions could be neglected the purpose of this paper is to construct these corner solutions to confirm that this is so the mathematical theory described in xi 2 was developed for the most important case in which cracks develop in planes perpendicular to the surface of the metal it is clear from the construction of that solution however that the unfolding may be applied with the same result to plane cracks inclined to the surface thus the global solution is not able to distinguish the inclination of the crack plane to the surface we shall find however that the corner solutions are sensitive to this inclination and they are given in s 3 for arbitrary angles of inclination these studies suggest therefore that information on crack inclination could be deduced from experimental investigation of these corner solutions since the solutions are found to possess a phase of frac 1 4 pi it might be possible to extract this information by measuring perturbation signals in quadrature with the main field or perhaps by increasing the skin depth so that the scale of the solutions is enhanced for a given specimen this could be done by reducing the frequency of the a c field but if the frequency is reduced significantly the field distributes itself through the interior of the specimen on the scale of the crack dimensions and may no longer be described as a surface distribution further work to describe such field distributions when the skin depth is of the order of the crack depths is in progress

generalized differentiability for n dimensional fuzzy number valued functions and fuzzy optimization in this paper we present a characteristic theorem for the generalized difference concept of n dimensional fuzzy numbers by means of support functions the concepts of differentiability for n dimensional fuzzy number valued functions are presented and discussed using the new generalized difference the relationships of g differentiability s differentiability and l differentiability are characterized furthermore optimization problems with s convex and s differentiable fuzzy number valued functions are described

embedding problem of fuzzy number space part ii abstract using a concrete structure into which we embed the fuzzy number space e1 several necessary and sufficient conditions of fuzzy set valued functions are given by means of abstract function theory

on fuzzy solutions for partial differential equations in this study we investigate heat wave and poisson equations as classical models of partial differential equations pdes with uncertain parameters considering the parameters as fuzzy numbers the fuzzy solution is built from fuzzification of the deterministic solution the continuity of the zadeh extension is used to obtain qualitative properties on regular a cuts of the fuzzy solution we prove the stability with respect to the initial boundary data and show that as time goes to zero the diameter of the fuzzy solution converges to zero and as a consequence to the cylindrical surface determined by the curve of the degree of membership numerical simulations are used to obtain a graphical representation of the fuzzy solution and a defuzzification of this solution is obtained using the center of gravity method we theoretically show that the surface obtained by defuzzification with the plane determined by fixing time is indeed the solution of the same initial boundary problem for this time point for the heat and poisson equations and in a particular case for the wave equation the deterministic solution and the defuzzified surface intercept are numerically compared using the euclidean distance

characterization of compact subsets of fuzzy sets a characterization of compact subsets is presented for the metric space of normal fuzzy convex fuzzy sets on the base space rn the metric for which is the supremum over the hausdorff distances between corresponding level sets it is shown that a closed subset is compact if and only if it is uniformly support bounded and the corresponding set of support functions is equileftcontinuous in the membership grade variable

riemannian geometry and stability of thermodynamical equilibrium systems a geometrical approach to statistical thermodynamics is proposed it is shown that any r parameter generalised gibbs distribution leads to a riemannian metric of parameter space the components of the metric tensor are represented by second moments of stochastic variables the scalar curvature r as a geometrical invariant is a function of the second and third moments so is strictly connected with fluctuations of the system in the case of a real gas r is positive and tends to infinity as the system approaches the critical point in the case of an ideal gas r 0 the obtained results and the results of the authors previous work suggests that for a wide class of models r tends to infinity near the critical point they treat r as a measure of the stability of the system they propose some sort of statistical principle only such models may be accepted for which r tends to infinity if the system is approaching the critical point it is shown that if this criterion is adopted for a class of models for which the scaling hypothesis holds then they obtain the new inequalities for the critical indices these inequalities are in good agreement with model calculations and experiment

generalized differentiability of fuzzy valued functions in the present paper using novel generalizations of the hukuhara difference for fuzzy sets we introduce and study new generalized differentiability concepts for fuzzy valued functions several properties of the new concepts are investigated and they are compared to similar fuzzy differentiabilities finding connections between them characterization and relatively simple expressions are provided for the new derivatives

refined nine parameter triangular thin plate bending element by using refined direct stiffness method based on a new generalized variational principle a refined direct stiffness method rdsm which can be directly used to improve non conforming elements is proposed the formulation is similar to that of the direct stiffness method dsm but the constraint condition of interelement continuity is satisfied in an average sense and as a result convergence and high accuracy are insured the well known bciz nine parameter triangular thin plate bending element is refined by the rdsm to yield a new nine parameter thin plate bending element rt 9 numerical examples are presented to show that the present model passes the patch test and possesses high accuracy

ill posed fuzzy initial boundary value problems based on generalized differentiability and regularization in this study based on a new generalized differentiability and parametric representation for fuzzy valued functions we discuss the existence of solutions for fuzzy partial differential equations fpdes and their ill posedness a regularization method is required to recover the numerical stability a stable approximation of the exact solution is obtained as an example of a cauchy problem for the fuzzy laplace equation

generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations the usual concept of differentiability of fuzzy number valued functions has the following shortcoming if c is a fuzzy number and g a b r is an usual real valued function differentiable on x 0 a b with g x 0 0 then f x c g x is not differentiable on x 0 in this paper we introduce and study generalized concepts of differentiability of any order n n which solves this shortcoming newton leibnitz type formula is obtained and existence of the solutions of fuzzy differential equations involving generalized differentiability is studied also some concrete applications to partial and ordinary fuzzy differential equations with fuzzy input data of the form c g x are given

towards fuzzy differential calculus part 1 integration of fuzzy mappings abstract this paper deals with fuzzy set valued mappings of the real line and more particularly focuses on mappings from the real line to the set of convex normal fuzzy sets of the real line these mappings can also be viewed as fuzzy relations using zadeh s extension principle the integral of such fuzzy mappings over a crisp interval is defined provided a special analytical representation of the fuzzy mapping the practical computation of such an integral is shown to be easy practically speaking it yields the fuzzy surface of a fuzzily bounded area

numerical analysis of thick plates bending behaviour of a rectangular plate is analysed with the help of a refined higher order theory the theory is based on a higher order displacement model and the three dimensional hooke s laws for plate material giving rise to a more realistic quadratic variation of the transverse shearing strains and linear variation of the transverse normal strain through the plate thickness it is shown that the segmentation method for the numerical analysis of plates simply supported on two opposite edges can be considered to be the most competitive method in terms of efficiency economy reliability and accuracy in such applications

dynamical behaviour of a statistical manifold associated with correlated walks a riemann space associated with a correlated walk cw model is investigated by a numerical method which is applicable to many other stochastic processes the behaviour of the riemann curvature is shown to have a close relation to the stability of the cw system the riemann space is spanned by two parameters representing jump probabilities of the cw model the cw manifold evolves through some characteristic eras it starts at a point and changes to a line and then to a homogeneous spherical surface of the scalar curvature r 1 2 the homogeneous space immediately transforms to an inhomogeneous space the inhomogeneity gradually grows a region of the inhomogeneous space violently oscillates in time and another region expands fast as a whole the curvature of the space decreases and soon becomes negative the oscillations have already started to fade away the negative curvature goes on decreasing and finally the space converges to a homogeneous saddle surface of r 1 such a behaviour of the curvature is shown to be well understood by the terms of stability and order parameter of stochastic processes in other words the inhomogeneity the oscillation and the decrease behaviour of the curvature are closely related to stability and orderliness of the correlated system

generalized metric phase space for quantum systems and the uncertainty principle abstract we demonstrate that when the gibbs entropy is an invariant of motion following information theory procedures it is possible to define a generalized metric phase space for the temporal evolution of the mean values of a given hamiltonian the metric is positive definite and this fact leads to a metric tensor k t whose properties are well defined working with these properties we shown that a the generalized uncertainty principle gup is always the summation over the principal minors of order 2 belonging to k t b several invariants of motion can be derived from the metric tensor and c particularly under certain conditions the gup itself is also a motion invariant

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

a generalization of hukuhara difference and division for interval and fuzzy arithmetic we propose a generalization of the hukuhara difference first the case of compact convex sets is examined then the results are applied to generalize the hukuhara difference of fuzzy numbers using their compact and convex level cuts finally a similar approach is suggested to attempt a generalization of division for real intervals and fuzzy numbers applications to solving interval and fuzzy linear equations and fuzzy differential equations are shown

analysis of thick sandwich construction by a 3 2 order theory abstract this study is an extension of the 1 2 order plate theory to a higher order 3 2 theory based on the equivalent single layer assumptions the in plane and transverse displacement components are expressed as cubic and quadratic expansions through the thickness of the sandwich construction also the transverse stress component is assumed to vary as a cubic function through the thickness utilizing reissner s definitions for kinematics of thick plates the displacement components at any point on the plate are approximated in terms of weighted average quantities displacements and rotations that are functions of the in plane coordinates the undetermined coefficients defining the in plane and transverse displacement fields are then expressed in terms of the weighted average displacements and rotations and their derivatives by directly employing reissner s definitions and enforcing the zero transverse shear stress conditions on the upper and lower surfaces of the sandwich panel the coefficients defining the transverse stress component are obtained by requiring the transverse strain component which is expressed in terms of the unknown coefficients of the transverse stress component from a mixed constitutive relation to be the least squares equivalent of the kinematic definition of the transverse strain component the resulting expressions for the unknown coefficients of the transverse stress component are related to resultant strains and curvatures defined from kinematic relations the equations of equilibrium and boundary conditions of the sandwich plate based on the 3 2 higher order theory are derived by employing the principles of virtual displacements the robustness and accuracy of this 3 2 order plate theory are established through comparisons with exact solutions available in the literature the finite element implementation of the present 3 2 order plate theory is also discussed

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

on fuzzy solutions for heat equation based on generalized hukuhara differentiability abstract in this paper a fuzzy heat equation with fuzzy initial values is considered the concept of generalized hukuhara differentiation is interpreted thoroughly in the univariate and multivariate cases and also several properties for generalized hukuhara differentiability are obtained on the topics such as switching point the univariate and multivariate fuzzy chain rules fuzzy mean value theorem among others the objective of this paper is to prove the uniqueness of a solution for a fuzzy heat equation and show that a fuzzy heat equation can be modeled as two systems of fuzzy differential equations by considering the type of differentiability of solutions finally some examples show the behavior of the solutions obtained

refined global local higher order theory and finite element for laminated plates based on completely three dimensional elasticity theory a refined global local higher order theory is presented as enhanced version of the classical global local theory proposed by li and liu int j numer meth engng 1997 40 1197 1212 in which the effect of transverse normal deformation is enhanced compared with the previous higher order theory the refined theory offers some valuable improvements these are able to predict accurately response of laminated plates subjected to thermal loading of uniform temperature however the previous higher order theory will encounter difficulty for this problem a refined three noded triangular element satisfied the requirement of c1 weak continuity conditions in the inter element is also presented the results of numerical examples of moderately thick laminated plates and even thick plates with span thickness ratios l h 2 are given to show that in plane stresses and transverse shear stresses can be reasonably predicted by the direct constitutive equation approach without smooth technique in order to accurately obtain transverse normal stresses the local equilibrium equation approach in one element is employed herein copyright 2006 john wiley sons ltd

thermodynamics of boson systems related to dunkl differential difference operators abstract we study the thermodynamics of systems based on a fock space representation inspired by the differential difference operators proposed in dunkl 1989 we calculate thermodynamic functions as the entropy and heat capacity and compare them with the standard boson case a calculation of the second virial coefficient and the scalar curvature in two and three dimensions show that these systems become repulsive within an interval of negative values of the reflection operator parameter in addition the stability of this system is examined as a function of

on retarded fuzzy functional differential equations and nonabsolute fuzzy integrals abstract we prove a generalized convergence theorem for fuzzy henstock integrals based on the sequence of fuzzy number valued functions which are weak generalized uniformly absolute continuous u a c g to widen the applications of this convergence theorem we provide some existence theorems for the generalized solution for retarded fuzzy functional differential equations and the dependence of the solutions on a parameter under the assumption of the strong generalized derivative of the solution our results generalize existence results in a kaleva integral setting to a fuzzy henstock integral setting

geometrical aspects of a generalized statistical mechanics abstract we discuss here the use of generalized forms of entropy taken as information measures to characterize phase transitions and critical behavior in thermodynamic systems our study is based on geometric considerations pertaining to the space of parameters that describe statistical mechanics models the thermodynamics of the system is the focus of attention in this geometric context

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

stability and anyonic behavior of systems with m statistics abstract starting with the partition function z for systems with m statistics as proposed in chen et al 1996 1 we calculate from the metric g ln z the scalar curvature r in two and three dimensions our results exhibit the details of the anyonic behavior as a function of the fugacity z and the identical particle maximum occupancy number m we also compare the stability of systems for m 1 with the fermionic m 1 and bosonic m cases

a study of transverse normal stress effect on vibration of multilayered plates and shells this paper evaluates transverse normal stress zzeffect on vibration of multilayered structures to this purpose a mixed plate model initially introduced by toledano and murakami has been extended to dynamics analysis of double curved shells these models allow both continuous interlaminar transverse shear and normal stresses as well as the zigzag form of the displacement distribution in the shell thickness directions to be modelled governing equations have been derived by employing a reissner s mixed theorem classical models on the basis of standard displacement formulations have been considered for comparison purposes the evaluations of transverse stress effects have been conducted by comparing constant linear and higher order distributions of transverse displacement components in the plate thickness directions free vibrational response of layered simply supported plates cylindrical and spherical shells made of isotropic as well as orthotropic layers has been analyzed the numerical investigation carried out and comparison with earlier results has concluded that 1 the possibility of describing interlaminar continuous transverse normal stress makes the mixed theories more attractive with respect to other available modelling 2 any refinements of classical models are meaningless unless the effects of interlaminar continuous transverse shear and normal stresses are both taken into account in a multilayered shell theory

fuzzy solutions to partial differential equations adaptive approach a new technique using an adaptive fuzzy algorithm to obtain the solutions to a class of partial differential equations pdes is presented the design objective is to find a fuzzy solution to satisfy precisely the pdes with boundary conditions according to the adaptive scheme of fuzzy logic systems a fuzzy solution with adjustable parameters for the pde is first described then a set of adaptive laws for tuning the free parameters in the consequent part is derived from minimizing an appropriate error function in addition an elegant approximated error bound between the exact solution and the proposed fuzzy solution with respect to the number of fuzzy rules and solution errors has also been derived furthermore the convergence of error equations in mesh points is also discussed from the energy perspective in this paper we show that the proposed method can solve a variety of pdes encountered in engineering comparisons are also made with solutions obtained by the finite element method

generalized hukuhara differentiability of interval valued functions and interval differential equations abstract in the present paper we use a recently proposed generalization of the hukuhara difference for compact convex sets to introduce and study a generalization of the hukuhara differentiability to the case of interval valued functions we also consider several possible definitions for the derivative of an interval valued function recently proposed in the literature and we study connections between them and their properties using these concepts we study interval differential equations local existence and uniqueness of two solutions is obtained together with characterizations of the solutions of an interval differential equation by ode systems several examples are also shown

towards fuzzy differential calculus part 3 differentiation this paper deals successively with differentiation of ordinary functions at a fuzzy point and with differentiation of fuzzy valued functions at a nonfuzzy point in both cases the concept of differentiation is extended and the problem of the linearity is addressed in the second case practical methods of computation are presented in practical situations the available knowledge about mappings may be fuzzy it is the motivation of this work

a refined nonlinear theory of plates with transverse shear deformation abstract a higher order shear deformation theory of plates accounting for the von karman strains is presented the theory contains the same dependent unknowns as in the hencky mindlin type first order shear deformation theory and accounts for parabolic distribution of the transverse shear strains through the thickness of the plate exact solutions of simply supported plates are obtained using the linear theory and the results are compared with the exact solutions of 3 d elasticity theory the first order shear deformation theory and the classical plate theory the present theory predicts the deflections stresses and frequencies more accurately when compared to the first order theory and the classical plate theory

fuzzy continuous function and its properties abstract in this paper we introduce the concepts of fuzzy distance of fuzzy numbers fuzzy limit of fuzzy numbers fuzzy function and fuzzy limit of a fuzzy function and fuzzy continuous function on the set of fuzzy numbers and give some of their elementary properties we also discuss some important theorems of fuzzy numbers and the fuzzy continuous function on the m closed interval

on henstock integral of fuzzy number valued functions i abstract in this paper we define and discuss the fh integral for fuzzy number valued functions using a concrete structure into which we embed the fuzzy number space e 1 several necessary and sufficient conditions of integrability for fuzzy number valued functions are given by means of abstract function theory

riemann scalar curvature of ideal quantum gases obeying gentile s statistics the scalar curvature r of ideal quantum gases obeying gentile s statistics is investigated by the method of information geometrical theory the r value is specified by the fugacity and the maximum number p of particles in a state the lowest case p 1 corresponds to fermi dirac statistics and the unbounded case p to bose einstein statistics in contrast to r 0 for ideal classical gases obeying boltzmann statistics we find r 2 1 2 32 for p2 and r 2 1 2 32 for p 1 in 0 which is the classical limit this means that a quantum statistical character is left in r in the classical limit also a correlation between the sign of r and a quantum mechanical exchange effect is recognized for 0 and 1 furthermore we obtain results that support the instability interpretation of r proposed by janyszek and mrugala