dimensionally dependent tensor identities by double antisymmetrization some years ago lovelock showed that a number of apparently unrelated familiar tensor identities had a common structure and could all be considered consequences in n dimensional space of a pair of fundamental identities involving trace free p p forms where 2p n we generalize lovelock s results and by using the fact that associated with any tensor in n dimensional space there is associated a fundamental tensor identity obtained by antisymmetrizing over n 1 indices we establish a very general master identity for all trace free k l forms we then show how various other special identities are direct and simple consequences of this master identity in particular we give direct application to maxwell lanczos ricci bel and bel robinson tensors and also demonstrate how relationships between scalar invariants of the riemann tensor can be investigated in a systematic manner

first order linear fuzzy differential equations under generalized differentiability first order linear fuzzy differential equations are investigated we interpret a fuzzy differential equation by using the strongly generalized differentiability concept because under this interpretation we may obtain solutions which have a decreasing length of their support which means a decreasing uncertainty in several applications the behaviour of these solutions better reflects the behaviour of some real world systems derivatives of the h difference and the product of two functions are obtained and we provide solutions of first order linear fuzzy differential equations using different versions of the variation of constants formula some examples show the rich behaviour of the solutions obtained

fuzzy differential equations and the extension principle we study the cauchy problem for differential equations considering its parameters and or initial conditions given by fuzzy sets these fuzzy differential equations are approached in two different ways a by using a family of differential inclusions and b the zadeh extension principle for the solution of the model we conclude that the solutions of the cauchy problem obtained by both are the same we also provide some illustrative examples

brief note on the variation of constants formula for fuzzy differential equations this note gives a theory of state transition matrices for linear systems of fuzzy differential equations this is used to give a fuzzy version of the classical variation of constants formula a simple example of a time independent control system is used to illustrate the methods while similar problems to the crisp case arise for time dependent systems in time independent cases the calculations are elementary solutions of eigenvalue eigenvector problems in particular for nonnegative or nonpositive matrices the problems at each level set can easily be solved in matlab to give the level sets of the fuzzy solution

periodic boundary value problems for first order linear differential equations with uncertainty under generalized differentiability we study the existence of solutions to a class of first order linear fuzzy differential equations subject to periodic boundary conditions from the point of view of generalized differentiability the objective of this paper is to show that fuzzy differential equations under generalized differentiability can be used in the study of periodic phenomena by considering a combination of two types of derivatives with a switching point we provide sufficient conditions which guarantee that the piecewise defined solutions match adequately and illustrate through some examples the process of construction and calculation of solutions

positivity properties of the bel robinson tensor the bel robinson tensor is a rank 4 tensor constructed from the weyl curvature tensor it is well known that when it is contracted with four future pointing vectors a non negative value is always obtained in this paper we investigate various cases in which the value zero is attained and show that it occurs in some situations where space time is of an algebraically special type this is of importance for expressions of gravitational energy related to the bel robinson tensor some new invariant ways of classifying the petrov type of space times using the bel robinson tensor are also presented

necessary and sufficient conditions for n dimensional conformal einstein spaces via dimensionally dependent identities listing has recently extended results of kozameh newman and tod for four dimensional space times and presented a set of necessary and sufficient conditions for a metric to be locally conformally equivalent to an einstein metric in all semi riemannian spaces of dimension n 4 subject to a nondegeneracy restriction on the weyl tensor by exploiting dimensionally dependent identities we demonstrate how to construct two alternative versions of these necessary and sufficient conditions which we believe will be useful in applications the four dimensional case is discussed in detail and examples are also given in five and six dimensions

on the solutions to first order linear fuzzy differential equations in this paper we study different formulations of first order linear fuzzy differential equations using the concept of generalized differentiability we present sufficient conditions for the existence of solutions and obtain the general expression of these solutions which exhibit different behavior some examples are given to illustrate our results

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

algebraic properties and topological properties of the quotient space of fuzzy numbers based on mares equivalence relation abstract in this paper we obtain some algebraic properties and topological properties of the quotient space of fuzzy numbers with respect to the equivalence relation defined by mares every fuzzy number has only one mares core and equivalent fuzzy numbers have the same mares core in addition equivalence classes of fuzzy numbers can be only expressed as the sum of its mares core and the set of symmetric fuzzy numbers which shows the notable difference between the equivalence classes of fuzzy numbers and the cosets of the normal subgroup in a group based on these results we introduce a new concept of convergence under which the quotient space is complete as an application of the main results it is shown that if we identify every fuzzy number with the corresponding equivalence class there would be more differentiable fuzzy functions than what is found in the literature

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

a note on fuzzy differential equations and the extension principle in this note we show by an example that the main result in a paper by bassanezi et al m t mizukoshi l c barros y chalco cano h roman flores r c bassanezi fuzzy differential equations and the extension principle information sciences 177 2007 3627 3635 is incorrect

the lanczos potential for the weyl curvature tensor existence wave equation and algorithms in the last few years renewed interest in the 3 tensor potential l abc proposed by lanczos for the weyl curvature tensor has not only clarified and corrected lanczos9s original work but generalized the concept in a number of ways in this paper we first of all carefully summarize and extend some aspects of these results make some minor corrections and clarify some misunderstandings in the literature the following new results are also presented the computer checked complicated second order partial differential equation for the 3 potential in arbitrary gauge for weyl candidates satisfying bianchi type equations is given in those n dimensional spaces with arbitrary signature for which the potential exists this is easily specialized to lanczos potentials for the weyl curvature tensor it is found that it is only in four dimensional spaces with arbitrary signature that the nonlinear terms disappear and that certain awkward second order derivative terms cancel for four dimensional spacetimes with lorentz signature this remarkably simple form was originally found by illge using spinor methods it is also shown that for most four dimensional vacuum spacetimes any 3 potential in the lanczos gauges which satisfies a simple homogeneous wave equation must be a lanczos potential for the non zero weyl curvature tensor of the background vacuum spacetime this result is used to prove that the form of a possible lanczos potential recently proposed by dolan and kim for a class of vacuum spacetimes is in fact a genuine lanczos potential for these spacetimes

positivity of general superenergy tensors several of the most important results in general relativity require or assume positivity properties of certain tensors the positive energy theorem and the singularity theorems make assumptions about the energy momentum tensor and ricci tensor respectively positivity of the bel robinson tensor is needed in the proof of the global stability of minkowski spacetime senovilla has recently presented a procedure of how to construct a superenergy tensor from any tensor for a maxwell field or a scalar field the procedure yields the usual energy momentum tensor for the weyl tensor and the riemann tensor one obtains the bel robinson tensor and bel tensor respectively in general by considering any tensor as an r fold n 1 n r form one constructs a rank 2r superenergy tensor from it by using spinor methods we prove that the contraction of any such superenergy tensor with 2r future pointing vectors is non negative we refer to this as the dominant superenergy property and it generalizes several previous positivity results obtained for certain tensors as well as it provides a unified way of treating them some more examples are given and applications discussed

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

comparison results for fuzzy differential equations in this paper we present several comparison results for the solutions of fuzzy differential equations by using the hukuhara derivative these results constitute the extension to the fuzzy context of some comparison results for the solutions of linear differential equations besides some side results about the hukuhara difference and the partial orderings are given we illustrate the applicability of the new results by showing several examples

linear independence of renormalization counterterms in curved space times of arbitrary dimensionality the counterterms in the lagrangian of a renormalizable quantum field theory that involve the riemann curvature tensor are considered it is proved that for six or fewer dimensions the counterterms not containing derivatives of the riemann tensor are linearly independent it is shown that there appears to be a maximum space time dimension for which identities can exist among invariants involving the product of n riemann tensors without derivatives acting on them space times that would require these products as counterterms for a renormalizable theory have a dimensionality which is one higher than this maximum dimension this makes it plausible that the required set of counterterms not involving derivatives of the riemann tensor is linearly independent in arbitrary dimensions in the appendix a relation cubic in the weyl tensor which relates invariants cubic in the riemann tensor in five dimensions is proved

the wave equation for the lanczos potential i the non local part of the gravitational field in general relativity is described by the 10 component conformal curvature tensor c abcd of weyl for this field lanczos found a tensor potential l abc with 16 independent components we can make l abc have only 10 effective degrees of freedom by imposing the 6 gauge conditions l smallmatrix ab s s endsmallmatrix 0 both fields c abcd l abc satisfy wave equations the wave equation satisfied by l abc is nonlinear even in vacuo however a linear spinor wave equation for the lanczos potential has been found by illge but no correct tensor wave equation for l abc has yet been published here we derive a correct tensor wave equation for l abc and when it is simplified with the aid of some four dimensional identities it is equivalent to illge s wave equation we also show that the nonlinear spinor wave equation of penrose for the weyl field can be derived from illge s spinor wave equation a set of analogues of well known results of classical electromagentic radiation theory can now be given we indicate how a green s function approach to gravitational radiation could be based on our tensor wave equation when a global study of space time is attempted

on linear fuzzy differential equations by differential inclusions approach abstract in this paper we study first order linear fuzzy differential equations under differential inclusions and strongly generalized differentiability approaches we present some new results on the relation between their solutions finally some examples are given to illustrate our results

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

some remarks on fuzzy differential equations via differential inclusions abstract in this paper we discuss the formulation and procedure for solving fuzzy differential equations via differential inclusions we give several examples showing the correct and incorrect procedure for solving fuzzy differential equations we show the connection between fuzzy differential equations and fuzzy differential inclusions finally we give some remarks on numerical algorithms for solving fuzzy differential equations via differential inclusions

fuzzy differential equations we will concentrate on the second order linear constant coefficient ordinary differential equation for x in interval i i can be 0 t for t 0 or i can be 0 the initial conditions are y 0 0 y 0 1 we assume g is continuous on i

the cauchy problem of fuzzy differential equations under generalized differentiability the generalization of the concept of h differentiability can be of great help in the dynamic study of fuzzy differential equations in this paper the concept of generalized differentiability is described from a new perspective on the basis of this concept the class of differentiable fuzzy set valued mappings is enlarged the cauchy problem for fuzzy differential equations is investigated in this enlarged setting as a result some new solutions are obtained the lengths of the support sets of these solutions may be non monotonic several examples are also shown

comparation between some approaches to solve fuzzy differential equations in this paper we study the class of fuzzy differential equations where the dynamics is given by a continuous fuzzy mapping which is obtained via zadeh s extension principle we get a fuzzy solution for this class of fuzzy differential equations and several illustrative examples are presented we also give some properties and we show the relationships with others interpretation finally we propose a numerical procedure for obtaining the fuzzy solution

almost periodic fuzzy number valued functions in this paper we develop a theory of almost periodic fuzzy functions i e of the almost periodic functions of real variable and with values fuzzy real numbers although the class of fuzzy real numbers does not form a linear normed space the majority of main properties of almost periodic functions with values in banach spaces are extended to this case applications to fuzzy differential equations and to fuzzy dynamical systems are given

old and new results for superenergy tensors from dimensionally dependent tensor identities it is known that some results for spinors and in particular for superenergy spinors are much less transparent and require a lot more effort to establish when considered from the tensor viewpoint in this paper we demonstrate how the use of dimensionally dependent tensor identities enables us to derive a number of 4 dimensional identities by straightforward tensor methods in a signature independent manner in particular we consider the quadratic identity for the bel robinson tensor tabcxtabcy xy tabcdtabcd 4 and also the new conservation law for the chevreton tensor both of which have been obtained by spinor means both of these results are rederived by tensor means for 4 dimensional spaces of any signature using dimensionally dependent identities and moreover we are able to conclude that there are no direct higher dimensional analogs in addition we demonstrate a simple way to show the nonexistense of such identities via counter examples in particular we show that there is no nontrivial bel ten

on fuzzy differential equations in the quotient space of fuzzy numbers since addition possesses an inverse in the quotient space of fuzzy numbers based on mares equivalence relation in this paper we introduce a metric on the quotient space of fuzzy numbers and then deal with fuzzy mappings of a real variable whose values are equivalence classes of fuzzy numbers we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation finally some examples are given to illustrate the main theorems