what is the fractional laplacian a comparative review with new results abstract the fractional laplacian in r d which we write as 2 with 0 2 has multiple equivalent characterizations moreover in bounded domains boundary conditions must be incorporated in these characterizations in mathematically distinct ways and there is currently no consensus in the literature as to which definition of the fractional laplacian in bounded domains is most appropriate for a given application the riesz or integral definition for example admits a nonlocal boundary condition where the value of a function must be prescribed on the entire exterior of the domain in order to compute its fractional laplacian in contrast the spectral definition requires only the standard local boundary condition these differences among others lead us to ask the question what is the fractional laplacian beginning from first principles we compare several commonly used definitions of the fractional laplacian theoretically through their stochastic interpretations as well as their analytical properties then we present quantitative comparisons using a sample of state of the art methods we discuss recent advances on nonzero boundary conditions and present new methods to discretize such boundary value problems radial basis function collocation for the riesz fractional laplacian and nonharmonic lifting for the spectral fractional laplacian in our numerical studies we aim to compare different definitions on bounded domains using a collection of benchmark problems we consider the fractional poisson equation with both zero and nonzero boundary conditions where the fractional laplacian is defined according to the riesz definition the spectral definition the directional definition and the horizon based nonlocal definition we verify the accuracy of the numerical methods used in the approximations for each operator and we focus on identifying differences in the boundary behaviors of solutions to equations posed with these different definitions through our efforts we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications

free vibrations of rectangular cantilever plates part 1 out of plane motion an analysis of the free in plane vibrations of a cantilevered rectangular plate is performed by means of a variational approximation procedure the problem is treated by first obtaining the exact solution for waves in the plate satisfying the equations of plane stress including in plane inertia with two opposite edges traction free the solution results in a set of dispersion curves a number of the resulting waves are used in what remains of the variational equation in which all conditions occur as natural conditions roots of the resulting transcendental equation are calculated which yield the eigensolutions and associated eigenfrequencies the results are compared with results obtained using fem and good agreement is shown

a fast semi discrete kansa method to solve the two dimensional spatiotemporal fractional diffusion equation abstract fractional order diffusion equations fdes extend classical diffusion equations by quantifying anomalous diffusion frequently observed in heterogeneous media real world diffusion can be multi dimensional requiring efficient numerical solvers that can handle long term memory embedded in mass transport to address this challenge a semi discrete kansa method is developed to approximate the two dimensional spatiotemporal fde where the kansa approach first discretizes the fde then the gauss jacobi quadrature rule solves the corresponding matrix and finally the mittag leffler function provides an analytical solution for the resultant time fractional ordinary differential equation numerical experiments are then conducted to check how the accuracy and convergence rate of the numerical solution are affected by the distribution mode and number of spatial discretization nodes applications further show that the numerical method can efficiently solve two dimensional spatiotemporal fde models with either a continuous or discrete mixing measure hence this study provides an efficient and fast computational method for modeling super diffusive sub diffusive and mixed diffusive processes in large two dimensional domains with irregular shapes

fractional laplacian time space models for linear and nonlinear lossy media exhibiting arbitrary frequency power law dependency frequency dependent attenuation typically obeys an empirical power law with an exponent ranging from 0 to 2 the standard time domain partial differential equation models can describe merely two extreme cases of frequency independent and frequency squared dependent attenuations the otherwise nonzero and nonsquare frequency dependency occurring in many cases of practical interest is thus often called the anomalous attenuation in this study a linear integro differential equation wave model was developed for the anomalous attenuation by using the space fractional laplacian operation and the strategy is then extended to the nonlinear burgers equation a new definition of the fractional laplacian is also introduced which naturally includes the boundary conditions and has inherent regularization to ease the hypersingularity in the conventional fractional laplacian under the szabo s smallness approximation where attenuation is assumed to be much smaller than the wave number the linear model is found consistent with arbitrary frequency power law dependency

a fuzzy syllogistic reasoning schema for generalized quantifiers in this paper a new approximate syllogistic reasoning schema is described that expands some of the approaches expounded in the literature into two ways i a number of different types of quantifiers logical absolute proportional comparative and exception taken from theory of generalized quantifiers and similarity quantifiers taken from statistics are considered and ii any number of premises can be taken into account within the reasoning process furthermore a systematic reasoning procedure to solve the syllogism is also proposed interpreting it as an equivalent mathematical optimization problem where the premises constitute the constraints of the searching space for the quantifier in the conclusion

vibration analysis of a continuous system subject to generic forms of actuation forces and sensing devices this work provides a general formulation to solve vibration problems for continuous systems with damping effects including modal transient harmonic and spectrum response analyses in modal analysis the system eigenvalues and corresponding eigenfunctions can be determined the orthogonal relations of eigenfunctions are shown for transient harmonic and spectrum analyses the generic force actuator functions and response sensing operators are introduced respectively and used to derive the system response the time domain response is obtained for transient analysis the frequency response function is derived for harmonic analysis and statistical quantities of response variables due to random excitation are determined in spectrum analysis the solution for each type of analysis can be formulated and expressed in a concise format in terms of generic force actuator and response sensor mode shape functions in particular one dimensional beam and two dimensional plate vibration analyses are illustrated by following the developed generic formulation this work provides the complete analytical solutions of four types of vibration analyses for continuous systems and can be applied to other engineering structures as well

the fractional laplacian operator on bounded domains as a special case of the nonlocal diffusion operator we analyze a nonlocal diffusion operator having as special cases the fractional laplacian and fractional differential operators that arise in several applications in our analysis a nonlocal vector calculus is exploited to define a weak formulation of the nonlocal problem we demonstrate that when sufficient conditions on certain kernel functions hold the solution of the nonlocal equation converges to the solution of the fractional laplacian equation on bounded domains as the nonlocal interactions become infinite we also introduce a continuous galerkin finite element discretization of the nonlocal weak formulation and we derive a priori error estimates through several numerical examples we illustrate the theoretical results and we show that by solving the nonlocal problem it is possible to obtain accurate approximations of the solutions of fractional differential equations circumventing the problem of treating infinite volume constraints

linguistic quantifiers modeled by sugeno integrals since quantifiers have the ability of summarizing the properties of a class of objects without enumerating them linguistic quantification is a very important topic in the field of high level knowledge representation and reasoning this paper introduces a new framework for modeling quantifiers in natural languages in which each linguistic quantifier is represented by a family of fuzzy measures and the truth value of a quantified proposition is evaluated by using sugeno s integral this framework allows us to have some elegant logical properties of linguistic quantifiers we compare carefully our new model of quantification and other approaches to linguistic quantifiers a set of criteria for linguistic quantification was proposed in the previous literature the relationship between these criteria and the results obtained in the present paper is clarified some simple applications of the sugeno s integral semantics of quantifiers are presented

computation of propagative waves in free rail using a finite element technique abstract a method is presented for the numerical computation of the wavenumbers and associated modes of the cross section of solid waveguides the method based on the finite element technique is well suited to the computation of both the propagative and the evanescent waves in a straight waveguide with an arbitrary cross section the solution is obtained by factorization with the cross section of the waveguide being modelled by numerical discretization the dispersion curves of propagative waves in a free rail are computed by using triangular and quadrilateral finite elements of the cross section the evolution of cross section modes as the frequency increases is evaluated and discussed

error estimation of perturbations under cri the analysis of stability and robustness of fuzzy reasoning is an important issue in areas like intelligent systems and fuzzy control an interesting aspect is to what extent the perturbation of input in a fuzzy reasoning scheme causes the oscillation of the output in particular when the error limits restrictions of the input values are given what the error limits of the output values are in this correspondence we estimate the upper and lower bounds of the output error affected by the perturbation parameters of the input and obtain the limits of the output values when the input values range over some interval in many fuzzy reasoning schemes under compositional rule of fuzzy inference cri

group decision making with a fuzzy linguistic majority abstract the determination of solutions in group decision making is considered the point of departure is a collection of individual fuzzy preference relations a solution is derived either directly from the individual fuzzy preference relations or by constructing first a social fuzzy preference relation as opposed to conventional approaches in which a crisp threshold type majority rule is used we employ a fuzzy majority rule specified by a fuzzy linguistic quantifier e g most much more than 50 etc a calculus of linguistically quantified propositions is applied using the fuzzy majority various solution concepts are derived mainly of the type of core minimax opposition set and consensus winner

finite difference methods for two dimensional fractional dispersion equation fractional order partial differential equations as generalizations of classical integer order partial differential equations are increasingly used to model problems in fluid flow finance and other areas of application in this paper we discuss a practical alternating directions implicit method to solve a class of two dimensional initial boundary value fractional partial differential equations with variable coefficients on a finite domain first order consistency unconditional stability and therefore first order convergence of the method are proven using a novel shifted version of the classical grunwald finite difference approximation for the fractional derivatives a numerical example with known exact solution is also presented and the behavior of the error is examined to verify the order of convergence

on fuzzy type theory this paper is a generalization of classical simple type theory we have developed a formal system of fuzzy type theory which differs from the classical one essentially in extension of truth values from two to infinitely many the structure of truth values is assumed to be an imtl algebra residuated lattice with prelinearity and double negation extended by the baaz delta operation various properties of fuzzy type theory are proved including its completeness

fuzzy association rules general model and applications the theory of fuzzy sets has been recognized as a suitable tool to model several kinds of patterns that can hold in data in this paper we are concerned with the development of a general model to discover association rules among items in a crisp set of fuzzy transactions this general model can be particularized in several ways each particular instance corresponds to a certain kind of pattern and or repository of data we describe some applications of this scheme paying special attention to the discovery of fuzzy association rules in relational databases

free vibrations of annular sector cantilever plates part 1 out of plane motion analysis of the free in plane vibrations of a cantilevered annular sector plate is performed by means of a variational approximation procedure the problem is treated by first obtaining the exact solution for waves in the plate satisfying the equations of plane stress including in plane inertia with the inner and outer circumferential edges traction free in this exact solution new radial vector functions are obtained from a frobenius type expansion the solution results in a set of dispersion curves a number of the resulting waves are used in what remains of the variational equation in which all conditions occur as natural conditions roots of the resulting transcendental equation are calculated which yield the eigensolutions and associated eigenfrequencies the results are compared with results obtained using fem and good agreement is shown

searching for rare growth factors using multicanonical monte carlo methods the growth factor of a matrix quantifies the amount of potential error growth possible when a linear system is solved using gaussian elimination with row pivoting while it is an easy matter n j higham and d j higham siam j matrix anal appl 10 1989 pp 155 164 to construct examples of n times n matrices having any growth factor up to the maximum of 2 n 1 the weight of experience and analysis n j higham accuracy and stability of numerical algorithms siam philadelphia 1996 l n trefethen and r s schreiber siam j matrix anal appl 11 1990 pp 335 360 l n trefethen and i d bau numerical linear algebra siam philadelphia 1997 suggest that matrices with exponentially large growth factors are exceedingly rare here we show how to conduct numerical experiments on random matrices using a multicanonical monte carlo method to explore the tails of growth factor probability distributions our results suggest for example that the occurrence of an 8 times 8 matrix with a growth factor of 40 is on the order of a once in the age of the universe event

cayley bacharach theorems and conjectures a theorem of pappus of alexandria proved in the fourth century a d began a long development in algebraic geometry in its changing expressions one can see reflected the changing concerns of the field from synthetic geometry to projective plane curves to riemann surfaces to the modern development of schemes and duality we survey this development historically and use it to motivate a brief treatment of a part of duality theory we then explain one of the modern developments arising from it a series of conjectures about the linear conditions imposed by a set of points in projective space on the forms that vanish on them we give a proof of the conjectures in a new

metric truth as a basis for fuzzy linguistic reasoning abstract this article presents some basic assumptions of an alternative approach to linguistic truth qualification and quantification in fuzzy logic the examination is based on distance function and the semantic differential technique these ideas have been applied to fuzzy reasoning

finite element prediction of wave motion in structural waveguides a method is presented by which the wavenumbers for a one dimensional waveguide can be predicted from a finite element fe model the method involves postprocessing a conventional but low order fe model the mass and stiffness matrices of which are typically found using a conventional fe package this is in contrast to the most popular previous waveguide fe approach sometimes termed the spectral finite element approach which requires new spectral element matrices to be developed in the approach described here a section of the waveguide is modeled using conventional fe software and the dynamic stiffness matrix formed a periodicity condition is applied the wavenumbers following from the eigensolution of the resulting transfer matrix the method is described estimation of wavenumbers energy and group velocity discussed and numerical examples presented these concern wave propagation in a beam and a simply supported plate strip for which analytical solutions exist and the more complex case of a viscoelastic laminate which involves postprocessing an ansys fe model the method is seen to yield accurate results for the wavenumbers and group velocities of both propagating and evanescent waves

finite element analysis of the vibrations of waveguides and periodic structures many structural components can be regarded as waveguides they are uniform in one direction so that the cross section of the waveguide has the same physical and geometric properties at all points along the axis of the waveguide in this paper a method is presented to calculate the forced response of such a structure using a combination of wave and finite element fe approaches the method involves post processing a conventional but low order fe model in which the mass and stiffness matrices are typically found using a conventional fe package a section of the waveguide is meshed and the eigenvalues and eigenvectors of the resulting transfer matrix found the eigenvectors form a set of basis functions for the analysis of the structure as a whole allowing the global dynamic stiffness matrix to be built easily and then the forced response to be calculated very efficiently the main advantage of the approach over the alternative waveguide fe approach often termed the spectral fe method is that conventional fe packages can be used to form the stiffness and mass matrices so that structures with complex geometries or material distributions can be analysed with relative ease to demonstrate the efficacy of the method examples of the forced response for a finite beam and plate strip are presented

generalized gregory s series abstract in this paper the countable family of generalized gregory s series all of which are the power series having the radius of convergence equal to one is defined sums s n x n 1 2 of these series are found first in the definite integral form and next in the form of finite linear combination of arcus tangents of some linear forms of x it enables to extend the definition of s n x for all x c 1 2 n for every n n moreover for new functions s n x obtained in result of this extension many fundamental and surprising integral and trigonometric identities are received sums of the series of differences of odd harmonic numbers are presented as well

a new definition of fractional laplacian with application to modeling three dimensional nonlocal heat conduction this paper proposes a new implicit definition of the fractional laplacian compared with the existing explicit definitions in literature this novel definition has clear physical significance and is mathematically simple and numerically easy to calculate for multidimensional problems in stark contrast to a quick increasing and extensive applications of time fractional derivative to diverse scientific and engineering problems little has been reported on space fractional derivative modeling this is largely because the existing definitions are only feasible for one dimensional case and become mathematically too complicated and computationally very expensive when applied to higher dimensional cases in this study we apply the newly defined fractional laplacian for modeling the power law behaviors of three dimensional nonlocal heat conduction the singular boundary method sbm a recent boundary only collocation discretization method is employed to numerically solve the proposed fractional laplacian heat equation and the computational costs are observed moderate owing to the proposed new definition of fractional laplacian and the boundary only discretization meshfree and integration free natures of the sbm technique numerical experiments show the validity of the proposed definition of fractional laplacian

linguistic characterization of time series the goal of this paper is to provide an overview of applications of special soft computing theories the fuzzy transform and fuzzy natural logic to analysis forecasting and mining information from time series the focus is especially placed on the ability of methods of fuzzy natural logic to provide information in sentences of natural language our approach is based on the decomposition of time series into three components the trend cycle seasonal component and noise the trend cycle is extracted using the f transform and its course is characterized by automatically generated linguistic description the latter is then used to forecast the trend cycle the trend cycle can be furthermore decomposed into trend general tendency and cycle the former is computed again using the f transform moreover the f1 transform makes it possible to estimate the direction of the trend which can then be characterized by expressions of natural language stagnating slightly increasing sharply decreasing etc finally we focus on selected problems of mining information from time series first we suggest an algorithm for finding intervals of monotonous behavior and then show how the theory of intermediate quantifiers a constituent of fuzzy natural logic and generalized aristotle s syllogisms can be applied to automatic summarization of information on time series

l fuzzy quantifiers of type determined by fuzzy measures the aim of this paper is first to introduce two new types of fuzzy integrals namely fuzzy integral and fuzzy integral the first integral is based on a fuzzy measure of l fuzzy sets and the second one on a complementary fuzzy measure of l fuzzy sets where l is a complete residuated lattice some of their properties and a relation to the fuzzy sugeno integral are investigated second using these integrals two classes of monadic l fuzzy quantifiers of type are defined these l fuzzy quantifiers can be used for modeling the semantics of natural language quantifiers like all some many none at most half etc several semantic properties of these l fuzzy quantifiers are studied

a local structural derivative pde model for ultraslow creep abstract in this short communication an ultraslow creep pde model is proposed via the local structural derivative the inverse mittag leffler ml function is selected as the structural function which is capable of characterizing ultraslow creep the data on the ultra high performance concrete uhpc are tested to examine the proposed pde model the results indicate that the proposed pde model has better usability and higher accuracy compared with the traditional kelvin model where the parameter reflects the pore content in different time scales the structural derivative model can be considered a potential modeling methodology to describe challenging ultraslow creep problem

monadic l fuzzy quantifiers of the type the aim of this paper is to introduce monadic l fuzzy quantifiers of the type and investigate some of their semantic properties that are well known in generalized quantifier theory relationships between the properties of generalized quantifiers are also studied for monadic l fuzzy quantifiers finally a property to be extensional with respect to a given l permutation equivalence is proposed and some properties are shown

a comprehensive theory of trichotomous evaluative linguistic expressions in this paper a logical theory of the so called trichotomous evaluative linguistic expressions tev expressions is presented these are frequent expressions of natural language such as small very small roughly medium extremely big etc the theory is developed using the formal system of higher order fuzzy logic namely the fuzzy type theory generalization of classical type theory first we discuss informally what are properties of the meaning of tev expressions then we construct step by step axioms of a formal logical theory t e v of tev expressions and prove various properties of t e v all the proofs are syntactical and so our theory is very general we also outline construction of a canonical model of t e v the main elegancy of our theory consists in the fact that semantics of all kinds of evaluative expressions is modeled in a unified way we also prove theorems demonstrating that essential properties of the vagueness phenomenon can be captured within our theory

fuzzy cardinality and the modeling of imprecise quantification abstract the concept of cardinality of a fuzzy set has received attention from several researchers and has been defined in several apparently independent manners a systematic investigation of this notion is performed which unifies and improves previous attempts the cardinality of a fuzzy set viewed as a fuzzy integer is related to scalar cardinality indices the closely related question of the probability of a fuzzy event is dealt with lastly the usefulness of fuzzy cardinality for meaning representation of statements or queries involving fuzzy linguistic quantifiers is emphasized

boundary particle method for laplace transformed time fractional diffusion equations highlights error analysis and numerical experiments demonstrate the efficiency of our method for time fractional diffusion equations our method avoids costly convolution integral calculation and dilemmatic selection of time step size a truly boundary only meshless method is applied to laplace transformed inhomogeneous problem our method effectively simulates 3d long time history fractional diffusion systems this paper develops a novel boundary meshless approach laplace transformed boundary particle method ltbpm for numerical modeling of time fractional diffusion equations it implements laplace transform technique to obtain the corresponding time independent inhomogeneous equation in laplace space and then employs a truly boundary only meshless boundary particle method bpm to solve this laplace transformed problem unlike the other boundary discretization methods the bpm does not require any inner nodes since the recursive composite multiple reciprocity technique rc mrm is used to convert the inhomogeneous problem into the higher order homogeneous problem finally the stehfest numerical inverse laplace transform nilt is implemented to retrieve the numerical solutions of time fractional diffusion equations from the corresponding bpm solutions in comparison with finite difference discretization the ltbpm introduces laplace transform and stehfest nilt algorithm to deal with time fractional derivative term which evades costly convolution integral calculation in time fractional derivation approximation and avoids the effect of time step on numerical accuracy and stability consequently it can effectively simulate long time history fractional diffusion systems error analysis and numerical experiments demonstrate that the present ltbpm is highly accurate and computationally efficient for 2d and 3d time fractional diffusion equations

accurate free vibration analysis of the orthotropic cantilever plate abstract utilizing the superposition method an accurate analytical type solution is described for resolving the problem of free vibration of cantilever plates with rectangular orthotropy a number of verification checks performed to verify the mathematical procedure are described convergence is found to be rapid and eigenvalues are tabulated for a wide range of plate aspect ratios and a number of combinations of values of orthotropic parameters full delineation is made between symmetric and antisymmetric modes it is believed the tabulated results will prove useful to designers and provide a reference against which other researchers can compare their results

fuzzy quantifiers defined over fuzzy domains abstract in this paper we introduce a generalization of the concept of fuzzy quantifier called fuzzy quantifiers over fuzzy domains the development of this generalization is motivated by the impossibility to satisfactorily define the important operation of relativization for standard fuzzy quantifiers we provide basic examples of these quantifiers and define and investigate also several semantic properties of them finally we prove a theorem saying among other things that all these two argument fuzzy quantifiers that possess properties of extension and conservativity are relativizations of one argument fuzzy quantifiers of this kind

linguistic summarization of time series using a fuzzy quantifier driven aggregation we propose new types of linguistic summaries of time series data that extend those proposed in our previous papers the proposed summaries of time series refer to the summaries of trends identified here with straight line segments of a piecewise linear approximation of time series we first show how to construct such an approximation then we employ a set of features attributes to characterize the trends such as the slope of the line segment the goodness of approximation and the length of the trend the derivation of a linguistic summary of a time series is then related to a linguistic quantifier driven aggregation of trends for this purpose we employ the classic zadeh s calculus of linguistically quantified propositions but extending our previous works with different t norms in addition to the basic minimum we show an application to the analysis of time series data on daily quotations of an investment fund over an eight year period present some interesting linguistic summaries obtained and show results for different t norms the results are very promising

a formal theory of generalized intermediate syllogisms in this paper we continue developing the formal theory of intermediate quantifiers expressions such as most few almost all a lot of many a great deal of a large part of a small part of the theory is a fuzzy logic formalization of the concept introduced by peterson in his book we will syntactically prove that 105 generalized aristotle s syllogisms introduced in this book are valid in our theory at the same time we will also prove that syllogisms listed there as invalid are invalid also in our theory therefore we believe that our theory provides a reasonable mathematical model of the generalized syllogistics

connectives and quantifiers in fuzzy sets abstract we discuss some of the issues involved in the selection of appropriate operators for implementing the union and intersection of fuzzy subsets we discuss the extension principles used in extending operators to fuzzy subsets we provide a structure for aggregating linguistic variables in the theory of approximate reasoning we look at mean operators on fuzzy subsets as well as the inclusion of importances in aggregation we discuss non monotone set operators we introduce the concept of linguistic quantifiers and look at some applications of these structures we are particularly concerned with their application to linguistic summarizers syllogistic reasoning and multi criteria decision making

multidimensional advection and fractional dispersion extension of the fractional diffusion equation to two or three dimensions is not as simple as extension of the second order equation this is revealed by the solutions of the equations unlike the gaussian the most general stable vector cannot be generated with an atomistic measure on the coordinate axes a random combination of maximally skewed stable variables on the unit sphere generates a stable vector that is a general model of a diffusing particle subsets are symmetric stable vectors that have previously appeared in the literature and the well known multidimensional brownian motion a multidimensional fractional differential operator is defined in the process

a fast finite difference method for three dimensional time dependent space fractional diffusion equations and its efficient implementation fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the non local property of fractional differential operators numerical methods for space fractional diffusion equations generate dense or even full coefficient matrices with complicated structures traditionally these methods were solved with gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory to store where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of these methods makes a numerical simulation of three dimensional space fractional diffusion equations computationally prohibitively expensive in this paper we develop an efficient and faithful solution method for the implicit finite difference discretization of time dependent space fractional diffusion equations in three space dimensions by carefully analyzing the structure of the coefficient matrix of the finite difference method and delicately decomposing the coefficient matrix into a combination of sparse and structured dense matrices the fast method has a computational work count of o nlogn per iteration and a memory requirement of o n while retaining the same accuracy as the underlying finite difference method solved with gaussian elimination numerical experiments of a three dimensional space fractional diffusion equation show the utility of the fast method

a graded approach to cardinal theory of finite fuzzy sets part i in this article we propose a fuzzy class relation assigning to each pair of finite fuzzy sets a degree to which they are equipollent which indicates that they have the same number of elements the concepts of fuzzy sets and fuzzy classes in the class of all sets in zfc are introduced and several standard relations and constructions such as the fuzzy power set and exponentiation are defined a functional approach to the cardinal theory of finite fuzzy sets based on graded equipollence is shown and a relation to generalized cardinals and wygralak s cardinal theory of finite fuzzy sets defined over triangular norms is demonstrated

discrete dispersion relation for hp version finite element approximation at high wave number the dispersive properties of high order finite element schemes are analyzed in the setting of the helmholtz equation and an explicit form of the discrete dispersion relation is obtained for elements of arbitrary order it is shown that the numerical dispersion displays three different types of behavior depending on the size of the order of the method relative to the mesh size and the wave number quantitative estimates are obtained for the behavior and rates of decay of the dispersion error in the differing regimes all estimates are fully explicit and are shown to be sharp limits are obtained where transitions between the different regimes occur and used to provide guidelines for the selection of the mesh size and the polynomial order in terms of the wave number so that the dispersion error is controlled

softer optimization and control models via fuzzy linguistic quantifiers a softening of a basic formulation of multicriterion optimization and control multistage decision making is presented for optimization instead of seeking an optimal solution that best satisfies all the fuzzy objectives as has been done so far we seek an optimal solution that best satisfies most much more than 50 etc a linguistic quantifier in general of the fuzzy objectives for control we seek in turn an optimal sequence of controls that best satisfies the fuzzy constraints and fuzzy goals at most much more than 50 etc of the control stages a calculus of linguistically quantified statements based upon fuzzy sets and possibility theory is used some applications to softer evidence aggregation in expert systems are also indicated

relaxing constraints in enhanced entity relationship models using fuzzy quantifiers while various articles about fuzzy entity relationship er and enhanced entity relationship eer models have recently been published not all examine how the constraints expressed in the model may be relaxed in this paper our aim is to relax the constraints which can be expressed in a conceptual model using the modeling tool so that these constraints can be made more flexible we will also study new constraints that are not considered in classic eer models we use the fuzzy quantifiers which have been widely studied in the context of fuzzy sets and fuzzy query systems for databases in addition we shall examine the representation of these constraints in an eer model and their practical repercussions the following constraints are studied the fuzzy participation constraint the fuzzy cardinality constraint the fuzzy completeness constraint to represent classes and subclasses the fuzzy cardinality constraint on overlapping specializations fuzzy disjoint and fuzzy overlapping constraints on specializations fuzzy attribute defined specializations fuzzy constraints in union types or categories and fuzzy constraints in shared subclasses we shall also demonstrate how fuzzy min max notation can substitute the fuzzy cardinality constraint but not the fuzzy participation constraint all these fuzzy constraints have a new meaning they offer greater expressiveness in conceptual design and are included in the so called fuzzy eer model

a galerkin finite element method for time fractional stochastic heat equation abstract in this study a galerkin finite element method is presented for time fractional stochastic heat equation driven by multiplicative noise which arises from the consideration of heat transport in porous media with thermal memory with random effects the spatial and temporal regularity properties of mild solution to the given problem under certain sufficient conditions are obtained numerical techniques are developed by the standard galerkin finite element method in spatial direction and gorenflo mainardi moretti paradisi scheme is applied in temporal direction the convergence error estimates for both semi discrete and fully discrete schemes are established finally numerical example is provided to verify the theoretical results

the structure of generalized intermediate syllogisms abstract in this paper we continue development of the formal theory of intermediate quantifiers which are expressions of the natural language most many few etc in the previous paper we demonstrated that 105 generalized syllogisms are valid in our theory we introduce the generalization of all the figures and we show that for the proof of validity of all the generalized syllogisms we need to prove the validity of only few of them so that the validity of the other ones immediately follows

the theory of intermediate quantifiers in fuzzy natural logic revisited and the model of many abstract the fuzzy natural logic fnl is a formal theory whose goal is to provide a mathematical model of natural human reasoning whose typical feature is the use of natural language one of its tasks is to develop a mathematical model of a special class of linguistic quantifiers called intermediate ones in our previous papers we introduced a general principle of how intermediate quantifiers can be formed and suggested generalized definitions of the relations of contrary contradictory subcontrary and subalterns between these quantifiers based on these relations and following peterson s book 62 we were able to suggest a formal model of the 5 square of opposition with intermediate quantifiers this paper picks up the threads of this work and has the following goals it contributes to the development of the theory of fuzzy natural logic first it gets back to some concepts of the theory of evaluative linguistic expressions introduces slight modifications and proves some not yet presented properties furthermore it improves the formal theory of intermediate quantifiers some definitions are made more precise and some new theorems are proved finally it contributes to the theory of the intermediate quantifier many and the analysis of its position in the mentioned 5 square we will show that many manifests an ambiguous relation towards the other quantifiers

exact enclosures of roots of interval quadratic equations by sridhara s and fagnano s or modified fagnano s formulas in this study we investigate the question of accurate determination of the root enclosures of quadratic equations whose coefficients constitute interval variables we treat several special cases where either sridhara s classical formula or fagnano s alternative expression provide exact interval enclosures for the roots of the quadratic equation in the case of a single coefficient serving as an interval it is shown that the classical interval analysis by either sridhara s fagnano s or specifically modified fagnano s formulas provide lower and upper bounds of roots then we follow with the case of two coefficients being intervals whereas the third coefficient is a deterministic quantity numerous examples are provided in which the solutions are compared with the direct numerical evaluation of the roots enclosures in three arising cases either sridhara s or fagnano s expressions suffice to obtain exact enclosures

cardinality quantifiers and the aggregation of fuzzy criteria abstract in this paper we are interested in counting the number of elements of a fuzzy set we study two concepts of fuzzy cardinality one which gives the answer as a fuzzy set or fuzzy number and the other which gives an ordinary integer relationships between these concepts and others which have been studied in the literature e g sigma count are also investigated some properties of the new concepts are studied such as fuzzy convexity monotonicity and additivity the concepts of fuzzy cardinality are important for a better understanding of fuzzy quantified rules of the form qx s are a or qa s are b where q is a fuzzy quantifier and a b are fuzzy sets we study such rules as well as aggregation rules of the form q of a1 ap contain x

computational aspects of fem approximation of fractional advection dispersion equations on bounded domains in r 2 in this paper we investigate the numerical approximation of the variational solution to the fractional advection dispersion equation fade on bounded domains in r2 more specifically we investigate the computational aspects of the galerkin approximation using continuous piecewise polynomial basis functions on a regular triangulation of the domain the computational challenges of approximating the solution to fractional differential equations using the finite element method stem from the fact that a fractional differential operator is a non local operator several numerical examples are given which demonstrate approximations to fades

fuzzy quantification a state of the art abstract quantified sentences are a very powerful notion for modelling statements in natural language nl but in practice they have been used to solve several problems this paper is intended to offer a global view of the development on this branch until now focusing in the different approaches dealing with quantification specially those involving imprecision called fuzzy quantification we put attention to the different mechanisms for defining them the evaluation methods for measuring their fulfilment as well as the properties they should satisfy

a logic for default reasoning the need to make default assumptions is frequently encountered in reasoning about incompletely specified worlds inferences sanctioned by default are best viewed as beliefs which may well be modified or rejected by subsequent observations it is this property which leads to the non monotonicity of any logic of defaults in this paper we propose a logic for default reasoning we then specialize our treatment to a very large class of commonly occuring defaults for this class we develop a complete proof theory and show how to interface it with a top down resolution theorem prover finally we provide criteria under which the revision of derived beliefs must be effected

space fractional advection dispersion equations by the kansa method the paper makes the first attempt at applying the kansa method a radial basis function meshless collocation method to the space fractional advection dispersion equations which have recently been observed to accurately describe solute transport in a variety of field and lab experiments characterized by occasional large jumps with fewer parameters than the classical models of integer order derivative however because of non local property of integro differential operator of space fractional derivative numerical solution of these novel models is very challenging and little has been reported in literature it is stressed that local approximation techniques such as the finite element and finite difference methods lose their sparse discretization matrix due to this non local property thus the global methods appear to have certain advantages in numerical simulation of these non local models because of their high accuracy and smaller size resultant matrix equation compared with the finite difference method popular in the solution of fractional equations the kansa method is a recent meshless global technique and is promising for high dimensional irregular domain problems in this study the resultant matrix of the kansa method is accurately calculated by the gauss jacobi quadrature rule numerical results show that the kansa method is highly accurate and computationally efficient for space fractional advection dispersion problems

a generalization of discounted multistage decision making andcontrol through fuzzy linguistic quantifiers an attempt to introduce commonsense knowledge abstract this paper is an extension of work published by kacprzyk in 1983 in which by using a fuzzy logic based calculus of linguistically quantified propositions some generalizations of multistage decision making and control have been proposed namely a new class of models has been developed that make it possible roughly speaking to find an optimal sequence of controls best satisfying fuzzy constraints and goals at say most or any other suitable linguistic quantifier for example almost all at least a few of the control stages in this paper we further extend the above mentioned approach by introducing discounting a variable importance of particular control stages in principle the earlier the more important we propose a new model that can find an optimal sequence of controls best satisfying the fuzzy constraints and goals at say most of the earlier control stages we employ a fuzzy logic based calculus of linguistically quantified propositions and develop a branch and bound algo

fuzzy sets with triangular norms and their cardinality theory the most advanced and adequate approach to the question of cardinality of a fuzzy set seems to be that offering a fuzzy perception of cardinality the resulting convex fuzzy sets of usual cardinals of nonnegative integers in the finite case are then called generalized cardinal numbers three types of them are of special interest and importance namely fgcounts flcounts and fecounts in this paper first we show that their original forms are suitable only for fuzzy sets with the classical min and max operations second we propose an appropriate generalization to fuzzy sets with triangular norms and conorms further we investigate the resulting generalized fgcounts flcounts and fecounts from the viewpoint of the corresponding equipotency relations inequalities and arithmetical operations

type 1 1 fuzzy quantifiers determined by fuzzy measures on residuated lattices part iii extension conservativity and extensionality we study the properties of extension conservativity and extensionality of fuzzy quantifiers of type 1 1 defined using fuzzy measures and integrals the property of extension states that truth values of quantifier applications are invariant with respect to possible extensions of the universe conservativity expresses the property that quantifiers are sensitive in their second argument only to objects that lie in the intersection of their arguments extensionality represents a form of the smoothness of quantifiers we characterize these properties by the corresponding properties of functionals used in the definition of fuzzy quantifiers

a framework for fuzzy quantification models analysis a framework for description of fuzzy quantification models is presented within this framework the fuzzy quantified statements evaluation problem is described as the compatibility between the fuzzy quantifier and a fuzzy cardinality or a fuzzy aggregation measure a list of desirable properties for quantification models is presented and those models that fit the framework are confronted with it

on the relation between fuzzy and generalized quantifiers fuzzy quantified sentences have been widely applied for modeling statements in natural language since they allow to represent a broad type of intrinsically imprecise quantified statements that are usually linguistically expressed two different frameworks arise in order to model this imprecision in quantification zadeh s fuzzy quantifiers and the fuzzification of generalized quantification for both models several approaches have been developed for assessing the degree of satisfiability of fuzzy quantified sentences we present and compare both models concluding that although the concept of quantifier in each model has not the same semantics we can establish a connection between them and between the evaluation methods in both theories

an analysis of a vibratory angular rate gyroscope using polarized piezoceramic bimorph plates part 1 derivation of variational equations in the absence of angular velocity a variational equation is derived for the in plane and out of plane motions of a configuration composed of two rectangular and annular sector piezoceramic bimorph plates arranged in the shape of one half of a single tuning fork the required variational equation for the structure is obtained by means of a low order expansion in the three dimensional variational equation the elements that connect any of the two bimorph plates which are at right angles with respect to each other are modelled as rigid bodies in the continuity conditions at the interface edges of the connected bimorphs

simplified description of out of plane waves in thin annular elastic plates abstract dispersion relations are derived for the out of plane wave propagation in planar elastic plates with constant curvature using the classical kirchhoff thin plate theory the dispersion diagrams and the mode shapes are compared with their counterparts for a straight plate strip and the role of curvature is assessed for plates with unconstrained edges elementary bernoulli euler theory for a beam of rectangular cross section with the circular shape of its axis is also employed to analyze the wave guide properties of this structure in its out of plane deformation the applicability range of the elementary beam theory is validated the wave finite element method in the formulation of the three dimensional elasticity theory is used to ensure that the comparison of dispersion diagrams is performed in the frequency range where the classical thin plate theory is valid thus the paper summarizes the effects brought to the propagation of out of plane waves in thin elastic plates by their constant curvature and the models of these plates

representation by levels an alternative to fuzzy sets for fuzzy data mining abstract in this paper we describe and discuss the main contributions of the representation by levels approach to fuzzy data mining representation by levels is an alternative representation of fuzziness in information and data which is complementary to fuzzy sets in the sense that it provides tools and algebraic structures beyond the capabilities of fuzzy set theories based on t norms t conorms and fuzzy negations our approach allows to extend any crisp mining technique to the fuzzy case in a simple way keeping all the properties of the crisp technique we illustrate our discussion with examples and existing approaches based on representation by levels to fuzzy association rules and the related issues of mining exception anomalous rules and mining fuzzy bag databases

on usual values in commonsense reasoning abstract in many cases commonsense knowledge consists of knowledge of what is usual in this paper we develop a system for reasoning with usual information this system is based upon the fact that these pieces of commonsense information involve both a probabilistic aspect and a granular aspect we implement this system with the aid of probabilistic possibilistic granules

extensions of fuzzy relational compositions based on generalized quantifiers abstract the aim of this paper is to extend the fuzzy relational compositions motivated by their applications to medical diagnosis problem by bandler and kohout these compositions employ two quantifiers the universal quantifier and the existential quantifier as there exists a big gap between these two quantifiers that may be appropriately filled in by intermediate generalized quantifiers we take this as a motivation for our study in particular we introduce the concept of fuzzy relational compositions based on generalized quantifiers furthermore we show that the properties that are typically valid for standard fuzzy relational compositions are also valid for the compositions based on generalized quantifiers yet sometimes in a weaken form apart from fuzzy relational compositions the use of generalized quantifiers is also applied to images and preimages of fuzzy sets under fuzzy relations

on the direct solution of wave propagation for repetitive structures abstract the direct method of solving the wave constants for a repetitive structure with given frequency is developed in this paper the analogy between structural mechanics and optimal control theory is applied the dynamic stiffness matrix approach is used to solve for the eigensolutions of wave propagation the weighted adjoint symplectic orthonormality relationship and the eigenvector expansion theorem are also established for this approach the symplectic eigenproblem is derived for skew symmetric matrices a cell triangular decomposition for skew symmetric matrices is introduced to reduce the generalized eigenproblem and the symplectic householder transformation is introduced to cell tri diagonalize the skew symmetric matrix to solve the eigenequation the problem of a wave impinging on the boundary is considered by the eigenvector expansion method which could also be applied to a wave scattering when passing over an abnormal part numerical examples show that resonance will occur for the localized vibration case

gauss jacobi type quadrature rules for fractional directional integrals fractional directional integrals are the extensions of the riemann liouville fractional integrals from one to multi dimensional spaces and play an important role in extending the fractional differentiation to diverse applications in numerical evaluation of these integrals the weakly singular kernels often fail the conventional quadrature rules such as newton cotes and gauss legendre rules it is noted that these kernels after simple transforms can be taken as the jacobi weight functions which are related to the weight factors of gauss jacobi and gauss jacobi lobatto rules these rules can evaluate the fractional integrals at high accuracy comparisons with the three typical adaptive quadrature rules are presented to illustrate the efficacy of the gauss jacobi type rules in handling weakly singular kernels of different strengths potential applications of the proposed rules in formulating and benchmarking new numerical schemes for generalized fractional diffusion problems are briefly discussed in the final remarking section

a meshless integration free and boundary only rbf technique based on the radial basis function rbf nonsingular general solution and dual reciprocity method drm this paper presents an inherently meshless integration free boundary only rbf collocation technique for numerical solution of various partial differential equation systems the basic ideas behind this methodology are very mathematically simple in this study the rbfs are employed to approximate the inhomogeneous terms via the drm while nonsingular general solution leads to a boundary only rbf formulation for homogenous solution the present scheme is named as the boundary knot method bkm to differentiate it from the other numerical techniques in particular due to the use of nonsingular general solutions rather than singular fundamental solutions the bkm is different from the method of fundamental solution in that the former does not require the artificial boundary and results in the symmetric system equations under certain conditions the efficiency and utility of this new technique are validated through a number of typical numerical examples completeness concern of the bkm due to the sole use of the nonsingular part of complete fundamental solution is also discussed

spectral finite element analysis of the vibration of straight fluid filled pipes with flanges spectral finite element analysis of the vibration of straight fluid filled pipes with flanges

the evaluation of service quality using generalized choquet integral in view of the compatibility between psychology and linguistic terms this study generalizes the standard choquet integral whose measurable evidence and fuzzy measures are real numbers the proposed generalization can deal with fuzzy number types of measurable evidence and fuzzy measures in comparison with the previous research on service quality and fuzzy sets this study integrates the three column format servqual into the generalized choquet integral a numerical example evaluating the overall service quality of e stores is provided to illustrate how the generalized choquet integral evaluates service quality in the three column format servqual this integral is also compared with the possibility necessity model when measuring service quality

robustness of fuzzy reasoning and spl delta equalities of fuzzy sets fuzzy reasoning methods or approximate reasoning methods are extensively used in intelligent systems and fuzzy control in this paper the author discusses how errors in premises affect conclusions in fuzzy reasoning that is he discusses the robustness of fuzzy reasoning after reviewing his previous work 1996 he presents robustness results for various implication operators and inference rules all the robustness results are formulated in terms of spl delta equalities of fuzzy sets two fuzzy sets are said to be spl delta equal if they are equal to an extent of spl delta

an overview of fuzzy quantifiers i interpretations quantification is an important topic in fuzzy theory and its applications an overview is presented for quantification in fuzzy theory after a brief review of quantifiers in first order logic two approaches of generalizing quantifiers are given the algebraic method and the substitution method by distinguishing the fuzziness of predicates and quantifiers various approaches to quantification in fuzzy logic can be organized quantifiers in first order logic can be generalized in crisp sense and these generalized quantifiers can also be applied to fuzzy sets moreover quantifiers themselves can be fuzzy i e they can only be represented by a fuzzy set these different kinds of quantifications are identified quantifiers relate close to the concept of the cardinality of a fuzzy set which is summarized before investigating fuzzy quantifications different to classical logic various semantics of propositions in fuzzy logic fall into different frameworks which are known as the possibility distribution based reasoning system and the many valued fuzzy logics accordingly numerical and possibilistic interpretation explored in literature are reviewed conforming to these two frameworks

an overview of fuzzy quantifiers ii reasoning and applications abstract in the second part of the overview reasoning with fuzzily defined quantifiers and applications are presented possibility based reasoning is summarized in detail the widely used quantifier extension principle and various reasoning schemas derived from it are discussed some disadvantages of the quantifier extension principle are pointed out although intervalvalued quantifiers are only a special case of general quantifiers some specific properties and results arise from them furthermore they suggest another approach to interence with fuzzy quantifiers reasoning with quantification and several kinds of qualification is introduced this kind of reasoning requires a seamless synthesis of probability theory possibility theory fuzzy logic and belief theory at last the applications of fuzzily defined quantifiers in decision making and fuzzy database systems are introduced

monoidal t norm based logic towards a logic for left continuous t norms hajek s bl logic is the fuzzy logic capturing the tautologies of continuous t norms and their residua in this paper we investigate a weaker logic mtl which is intended to cope with the tautologies of left continuous t norms and their residua the corresponding algebraic structures mtl algebras are defined and completeness of mtl with respect to linearly ordered mtl algebras is proved besides several schematic extensions of mtl are also considered as well as their corresponding predicate calculi

on the analysis of set based fuzzy quantified reasoning using classical syllogistics syllogism is a type of deductive reasoning involving quantified statements the syllogistic reasoning scheme in the classical aristotelian framework involves three crisp term sets and four linguistic quantifiers for which the main support is the linguistic properties of the quantifiers a number of fuzzy approaches for defining an approximate syllogism have been proposed for which the main support is cardinality calculus in this paper we analyze fuzzy syllogistic models previously described by zadeh and dubois et al and compare their behavior with that of the classical aristotelian framework to check which of the 24 classical valid syllogistic reasoning patterns or moods are particular crisp cases of these fuzzy approaches this allows us to assess to what extent these approaches can be considered as either plausible extensions of the classical crisp syllogism or a basis for a general approach to the problem of approximate syllogism

non monotonic logic i abstract non monotonic logical systems are logics in which the introduction of new axioms can invalidate old theorems such logics are very important in modeling the beliefs of active processes which acting in the presence of incomplete information must make and subsequently revise assumptions in light of new observations we present the motivation and history of such logics we develop model and proof theories a proof procedure and applications for one non monotonic logic in particular we prove the completeness of the non monotonic predicate calculus and the decidability of the non monotonic sentential calculus we also discuss characteristic properties of this logic and its relationship to stronger logics logics of incomplete information and truth maintenance systems

type fuzzy quantifiers determined by fuzzy measures on residuated lattices part i basic definitions and examples abstract we study fuzzy quantifiers of type 1 1 defined using fuzzy measures and integrals residuated lattices are considered as structures of truth values we present basic notions on fuzzy measures over algebras of fuzzy subsets of a given fuzzy set and the definition and necessary properties of so called fuzzy integrals residuated lattice operations are established to derive operations between fuzzy sets describing degrees in which formulas expressing relations between fuzzy sets are true finally a general definition of type 1 1 fuzzy quantifiers determined by fuzzy measures and integrals is introduced and several examples of important natural language quantifiers are modeled using this approach

level based fuzzy generalized quantification abstract we propose a new type of generalized quantification using a recently developed representation of fuzziness by levels this proposal integrates the existent models allowing the extension of crisp quantification to the fuzzy case keeping all the algebraic boolean properties besides the model covers new types of quantification not covered by the existent approaches the advantages of our proposal are clear from the computational and practical perspective since it allows a parallelization of the algorithm by taking into account a fixed number of levels some authors have proposed several properties that a good evaluation method should satisfy we also prove that our proposal fulfills all the desired properties to be considered a good evaluation method

a probabilistic quantifier fuzzification mechanism the model and its evaluation for information retrieval in this paper we propose a new quantifier fuzzification mechanism which is deeply rooted in the theory of probability this quantifier fuzzification mechanism skips the nested assumption which is inherent to other probabilistic quantification methods the new quantification approach complies with the properties required for determiner fuzzification schemes dfs with finite sets and hence its good behavior is assured moreover this new approach is suitable for some application domains in particular the use of fuzzy quantifiers for implementing query quantified statements for information retrieval exemplifies the adequacy of the new proposal the new quantifier fuzzification mechanism has been efficiently implemented and empirically tested for a retrieval task this practical evaluation followed the standard methodology in the field of information retrieval and was conducted against a popular benchmark consisting of a large collection of documents the retrieval performance evaluation made evident that 1 the new method can work in realistic scenarios and 2 it can overcome recent proposals for applying fuzzy quantifiers in information retrieval

fractional spectral collocation method we develop an exponentially accurate fractional spectral collocation method for solving steady state and time dependent fractional pdes fpdes we first introduce a new family of interpolants called fractional lagrange interpolants which satisfy the kronecker delta property at collocation points we perform such a construction following a spectral theory recently developed in m zayernouri and g e karniadakis j comput phys 47 2013 pp 2108 2131 for fractional sturm liouville eigenproblems subsequently we obtain the corresponding fractional differentiation matrices and we solve a number of linear fodes in addition to linear and nonlinear fpdes to investigate the numerical performance of the fractional collocation method we first examine space fractional advection diffusion problem and generalized space fractional multiterm fodes next we solve fpdes including the time and space fractional advection diffusion equation time and space fractional multiterm fpdes and finally the space

on fuzzification mechanisms for unary quantification abstract we address a salient issue arising in glockner s two step methodology for modeling vague quantifiers the design of quantifier fuzzification mechanisms qfms these are mechanisms to turn formal models of vague quantifiers operating only on crisp arguments semi fuzzy quantifiers into vague quantifiers accepting vague arguments as well fully fuzzy quantifiers we critically examine desiderata formulated by glockner for qfms and also point out that previous approaches to quantifier fuzzification largely ignored the question whether the resulting quantifiers can be expressed in suitable extensions of t norm based fuzzy logics in particular in ukasiewicz logic we also introduce a new family of qfms and assess it fares with respect to the mentioned desiderata we exclusively focus on unary quantifiers in order to circumvent interference with vagueness related problems arising for all truth functional accounts of quantifiers that refer to more than one argument formula