i v fuzzy relations and sanchez s approach for medical diagnosis abstract this paper introduces the concept of i v fuzzy relations i v fuzzy tolerance relations composition of i v fuzzy relations etc several useful propositions are proved finally sanchez s approach for medical diagnosis is considered and discussed using the idea of i v fuzzy relations

new constructions of mds codes with complementary duals linear complementary dual lcd for short codes are linear codes that intersect with their duals trivially lcd codes have been used in certain communication systems it is recently found that lcd codes can be applied in cryptography this application of lcd codes renewed the interest in the construction of lcd codes having a large minimum distance maximum distance separable mds codes are optimal in the sense that the minimum distance cannot be improved for given length and code size constructing lcd mds codes is thus of significance in theory and practice recently jin constructed several classes of lcd mds codes through generalized reed solomon codes in this paper a different approach is proposed to obtain new lcd mds codes from generalized reed solomon codes consequently new code constructions are provided and certain previously known results by jin are extended

necessary and sufficient conditions for generalized uniform fuzzy partitions abstract the fundamental concept in the theory of fuzzy transform f transform is the fuzzy partition which is a generalization of the classical concept of partition the original definition assumes that every two normal fuzzy subsets in a partition overlap in such a way that the sum of the membership degrees at each point is equal to 1 this condition can be generalized by relaxing the assumption of normality for fuzzy sets the result is a denser fuzzy partition that may improve the approximation properties and the smoothness of the inverse f transform a fuzzy partition with this property will be referred to as general the problem is how a general fuzzy partition can be effectively constructed if we use a generating function with special properties then it is not immediately clear whether it defines a general fuzzy partition in this paper we find a necessary and sufficient condition that will enable the optimal generalized fuzzy partition to be designed more easily which is important in various practical applications of the f transform for example image processing time series analysis and solving differential equations with boundary conditions

on local entropy of fuzzy partitions in this paper we will introduce a new kind of fuzzy version of the local entropy of a continuous dynamical system on a compact metric space and will prove some of its properties then we will define an average entropy of fuzzy partitions by integrating the local fuzzy entropy we will also state the relationship between this entropy and the fuzzy entropy of dynamical systems based on the entropy of fuzzy partitions in the sense of dumitrescu

i fuzzy equivalence relations and i fuzzy partitions a t fuzzy equivalence relation is a fuzzy binary relation on a set x which is reflexive symmetric and t transitive for a t norm t recently mesiar et al r mesiar b reusch h thiele fuzzy equivalence relations and fuzzy partitions j multi valued logic soft comput 12 2006 167 181 have generalised the t norm t to any general conjunctor c and investigated the minimal assumptions required on such operations called duality fitting conjunctors such that the fuzzification of the equivalence relation admits any value from the unit interval and also the one one correspondence between the fuzzy equivalence relations and fuzzy partitions is preserved in this work we conduct a similar study by employing a related form of c transitivity viz i transitivity where i is an implicator we show that although every i fuzzy equivalence relation can be shown to be a c fuzzy equivalence relation there exist c fuzzy equivalence relations that are not i fuzzy equivalence relations and hence these concepts are not equivalent most importantly we show that the class of duality fitting implicators i is much richer than the residuals of the duality fitting conjunctors in the study of mesiar et al we also show that the i fuzzy partitions have a constant wise structure

fuzzy equivalence relation and its multigranulation spaces multigranulation rough sets focus on describing uncertain sets in different granularity spaces different approximation accuracy of an uncertain set can be obtained through changing knowledge granularities according to user s need however it is a bottleneck for establishing a multigranulation space to better describe an uncertain set in a proper granulation space and the typical way of a multigranulation space is a hierarchical quotient space determined by a fuzzy equivalence relation in this paper a new concept granularity of binary relation gb is put forward firstly and knowledge granulation gk is proved to be a special case of gb the changing rules of granularity series for describing a hierarchical quotient space structure are proposed and approximation accuracy series of an uncertain set is presented for automatically searching optimal knowledge space according to user s requirement a utility function with both knowledge space granularity and approximation accuracy is proposed then classification isomorphism and granularity isomorphism between two hierarchical quotient space structures are defined and discussed through the hierarchical quotient space structure of a fuzzy equivalence relation the essence of classification ability of a fuzzy equivalence relation is also revealed finally a new algorithm for establishing isomorphic fuzzy equivalence relations based on a hierarchical quotient space structure is presented and this algorithm can further explain the nature of those isomorphic fuzzy equivalence relations from a new perspective

finding the permutation between equivalent linear codes the support splitting algorithm two linear codes are permutation equivalent if they are equal up to a fixed permutation on the codeword coordinates we present here an algorithm able to compute this permutation it operates by determining a set of properties invariant by permutation one for each coordinate called a signature if this signature is fully discriminant i e different for all coordinates the support of the code splits into singletons and the same signature computed for any permutation equivalent code will allow the reconstruction of the permutation a procedure is described to obtain a fully discriminant signature for most linear codes the total complexity of the support splitting algorithm is polynomial in the length of the code and exponential in the dimension of its hull i e the intersection of the code with its dual

on fuzzy equivalence i fuzzy reflexive symmetric and transitive relations on fuzzy subsets are studied various types of reflexivities and corresponding equivalences are considered the possible fuzzy partitioning of a fuzzy subset is investigated

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

fuzzy subrings and ideals redefined abstract v q fuzzy subrings and ideals of a ring are defined the concepts of v q fuzzy semiprime prime semi primary primary and maximal ideals are introduced and characterization of such fuzzy ideals are obtained v q fuzzy cosets determined by v q fuzzy ideals and v q fuzzy radicals of v q fuzzy ideals are discussed

computing automorphism groups of error correcting codes an algorithm is described for computing the automorphism group of an error correcting code the algorithm determines the order of the automorphism group and produces a set of monomial permutations which generate the group it has been implemented on a computer and has been used successfully on a great number of codes of moderate length

preconditioned iterative methods for space time fractional advection diffusion equations in this paper we propose practical numerical methods for solving a class of initial boundary value problems of space time fractional advection diffusion equations first we propose an implicit method based on two sided grunwald formulae and discuss its stability and consistency then we develop the preconditioned generalized minimal residual preconditioned gmres method and preconditioned conjugate gradient normal residual preconditioned cgnr method with easily constructed preconditioners importantly because resulting systems are toeplitz like fast fourier transform can be applied to significantly reduce the computational cost we perform numerical experiments to demonstrate the efficiency of our preconditioners even in cases with variable coefficients

constructions of linear codes with one dimensional hull the hull of a linear code is defined to be the intersection of the code and its dual and was originally introduced to classify finite projective planes the hull plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code it has been shown that these algorithms are very effective in general if the size of the hull is small the objective of this paper is to present some sufficient and necessary conditions that linear codes and cyclic codes have one dimensional hull it is shown that there are no such binary or ternary cyclic codes based on these characterizations some constructions of linear codes with one dimensional hull were given by employing quadratic number fields partial difference sets and difference sets we also construct cyclic codes with one dimensional hull some optimal codes with one dimensional hull are obtained

three way fuzzy partitions defined by shadowed sets abstract pedrycz introduces the notion of a shadowed set as a three way approximation of a fuzzy set given a pair of thresholds a shadowed set approximates a fuzzy set by three regions an core of the fuzzy set the difference between a support and the core and the complement of the support a fuzzy partition is a family of nonempty fuzzy sets satisfying certain conditions when applying ideas of shadowed sets to a family of nonempty fuzzy sets the result is a three way approximation of the family that is a family of shadowed sets this enables us to study a new type of three way fuzzy partitions we introduce and examine three pairs of properties for defining a three way fuzzy partition disjointness property overlap property and coverage property of cores and supports we propose a general definition of a three way fuzzy partition by using two sets of properties by considering all possible non equivalent subsets of properties we obtain 21 classes of three way fuzzy partition we study three classes of three way fuzzy partition and establish their connections to partitions fuzzy partitions and interval set clusters

novel numerical methods for solving the time space fractional diffusion equation in two dimensions in this paper a time space fractional diffusion equation in two dimensions tsfde 2d with homogeneous dirichlet boundary conditions is considered the tsfde 2d is obtained from the standard diffusion equation by replacing the first order time derivative with the caputo fractional derivative t d gamma gamma in 0 1 and the second order space derivatives with the fractional laplacian delta alpha 2 alpha in 1 2 using the matrix transfer technique proposed by ilic et al m ilic f liu i turner and v anh fract calc appl anal 9 2006 pp 333 349 the tsfde 2d is transformed into a time fractional differential system as t d gamma mathbf u k alpha mathbf a alpha 2 mathbf u where mathbf a is the approximate matrix representation of delta traditional approximation of mathbf a alpha 2 requires diagonalization of mathbf a which is very time consuming for large sparse matrices the novelty of our proposed numerical schemes is that using either the finite difference method or the laplace transform to handle the caputo time fractional derivative the solution of the tsfde 2d is written in terms of a matrix function vector product f mathbf a mathbf b at each time step where mathbf b is a suitably defined vector depending on the method used to generate the matrix mathbf a the product f mathbf a mathbf b can be approximated using either the preconditioned lanczos method when mathbf a is symmetric or the mathbf m lanzcos method when mathbf a is nonsymmetric which are powerful techniques for solving large linear systems we give error bounds for the new methods and illustrate their roles in solving the tsfde 2d we also derive the analytical solution of the tsfde 2d in terms of the mittag leffler function finally numerical results are presented to verify the proposed numerical solution strategies

reduction of fuzzy strict order relations abstract fuzzy strict order relations and the notion of their reduction are defined a necessary and sufficient condition is obtained for the transitive closure of the reduction to coincide with the strict order itself some possible graph theoretic significances of these results are discussed in the conclusion

fourth order finite difference schemes for time space fractional sub diffusion equations in this paper we devote to the study of high order finite difference schemes for one and two dimensional time space fractional sub diffusion equations a fourth order finite difference scheme is invoked for the spatial fractional derivatives and the l 1 approximation is applied to the temporal fractional parts for the two dimensional case an alternating direction implicit scheme based on l 1 approximation is proposed the stability and convergence of the proposed methods are studied numerical experiments are performed to verify the effectiveness and accuracy of the proposed difference schemes

one to one correspondences between ɛ partitions 1 ɛ equivalences and ɛ pseudometrics the generalization of the concept of a classical or crisp partition to that of a fuzzy partition requires in our researches the choice of a family of fuzzy subsets which preserves the conditions characterizing crisp partitions however the usual constraints for the union and intersection of subsets are very limiting and seem to be far from the fuzzy spirit so that the union of subsets in the partitions is forced to be close to the universe of discourse and the intersection of any two subsets in the partition is forced to be close to the empty set furthermore since the classical definition of partition is very related to the union and intersection the generalization of this concept has to be applicable for any extension of these operations for fuzzy sets apart from this we need to redefine the concept of fuzzy partition to establish a one to one correspondence between fuzzy partitions and some kind of fuzzy equivalence relations the different definitions that we found in the literature did not exactly verify these requirements although some of them were more appropriate than others finally we have given a definition of fuzzy partition of any fuzzy subset of the universe which is a generalization of crisp partitions and of most of the different definitions of fuzzy partitions which preserves the classical ideas of covering and disjointness and such that it is appropriate in our researches classification problems fuzzy questionnaires fuzzy information theory once our fuzzy partitions epartitions are defined it is necessary to find the fuzzy binary relation which generates and characterizes the fuzzy partitions this fuzzy relation has to be by analogy with the crisp case a fuzzy equivalence relation two points in the same fuzzy subset of the partition will be related and two points in different subsets will not in relation with this equivalence relation it is also possible to obtain a way to measure the distance between two points by means of the degree of relation that there exists between them and therefore by means of the membership degree to a same subset in the fuzzy partition the definitions of these concepts and the relationships between them are the purpose of this paper

a computationally effective alternating direction method for the space and time fractional bloch torrey equation in 3 d the space and time fractional bloch torrey equation st fbte has been used to study anomalous diffusion in the human brain numerical methods for solving st fbte in three dimensions are computationally demanding in this paper we propose a computationally effective fractional alternating direction method fadm to overcome this problem we consider st fbte on a finite domain where the time and space derivatives are replaced by the caputo djrbashian and the sequential riesz fractional derivatives respectively the stability and convergence properties of the fadm are discussed finally some numerical results for st fbte are given to confirm our theoretical findings

an approach to group choice with fuzzy preference relations abstract this paper presents an approach to group choice with fuzzy preference relations the experts judgements are presented as fuzzy preorder relations as group choice we assume the nearest relation to these relations in the sense of the hamming distance in this paper an algorithm for finding a preorder relation as an expression of group choice is given and a way is pointed out to find the best alternative on the basis of this group choice furthermore the connection between the impossibility theorem of arrow and this group choice is considered

compatibility of t norms with the concept of e partition in this paper we study the behaviour of a kind of partitions formed by fuzzy sets the e partitions with respect to three important operations refinement union and product of partitions in the crisp set theory the previous operations lead to new partitions every refinement of a partition is also a partition the union of partitions of disjoint sets is a partition of the union set the product of two partitions of two sets is a partition of the intersection of the partitioned sets it has been proven that e partitions extend the three previous properties when the intersection of fuzzy sets is defined by the minimum t norm and the union by the maximum t conorm in this paper we consider any t norm defining the intersection of fuzzy sets and we characterize those t norms for which refinements unions and products of e partitions are e partitions we pay special attention to these characterizations in the case of continuous t norms

fuzzy antisymmetry and order abstract fuzzy antisymmetry and various types of ordering are defined and theorems relating to them proved a few constants associated with a fuzzy relation are introduced and some of their properties investigated

trapezoidal scheme for time space fractional diffusion equation with riesz derivative abstract in this paper a finite difference scheme is proposed to solve time space fractional diffusion equation which has second order accuracy in both time and space direction the time and space fractional derivatives are considered in the senses of caputo and riesz respectively first the centered difference approach is used to approximate the riesz fractional derivative in space then the obtained fractional ordinary differential equations are transformed into equivalent volterra integral equations and then the trapezoidal rule is utilized to approximate the volterra integral equations the stability and convergence of our scheme are proved via mathematical induction method finally numerical experiments are performed to confirm the high accuracy and efficiency of our scheme

q similitudes and q fuzzy partitions abstract the concept of q reflexive q transitive relation q similitude and q fuzzy partition of a fuzzy set are introduced and it is proved that there exists a one to one correspondence between q similitudes and q fuzzy partitions some results on compositions of q reflexive relations and q similitudes are also obtained

fuzzy δ ɛ partitions in this paper the concept of a fuzzy e partitions of any fuzzy subset of the universe is introduced as a generalization of that of a classical partition the approach is based on four observations the first one is that for any two members of a classical partition their intersection is the empty set if they are not equal the second one is that the union of all members of a classical partition is equal to the partitioned set the third one is that this definition has to be applicable for any extension of the union and intersection for fuzzy subsets and the fourth one is that this definition has to allow us to establish a one to one correspondence between fuzzy partitions and some kind of fuzzy equivalence relations once this new concept is introduced by considering that the constrains for union and intersection are too limiting and that they have to be slightly modified some equivalence definitions of e partitions are proposed in particular two very important cases are studied they are the case when is equal to e because this is very useful in practical applications and the case when is equal to 1 e that is when the degree of fuzziness for the union and the intersection are equal this case is very useful in theoretical applications and it will be related to the concept of e equal fuzzy subsets apart from the fact that they are a generalization of that of a classical partition the new concept of e partition is also a generalization of that of the best known definitions of fuzzy partitions proposed in the literature

θ schemes for finite element discretization of the space time fractional diffusion equations the numerical solution of a space time fractional diffusion equation used to model the anomalous diffusion is considered spatial discretization is effected using a finite element method whereas the scheme is used for temporal discretization the fully discrete scheme is analyzed for all 0 1 to determine conditional and unconditional stability regimes for the scheme and also to obtain error estimates for the approximate solution the analysis is facilitated by making use of a variational formulation of the equations that is based on a recently developed nonlocal calculus one dimensional numerical examples are provided that illustrate the theoretical stability and convergence results

studies in fuzzy relations over fuzzy subsets in this paper a generalisation of fuzzy relations is introduced fuzzy relations are defined on fuzzy subsets properties like ordered reflexivity symmetry transitity transitive closures of such generalised relations and operations on them are discussed

a fast solver for multidimensional time space fractional diffusion equation with variable coefficients abstract in this paper we study a discretization scheme and the corresponding fast solver for multi dimensional time space fractional diffusion equation with variable coefficients in which l 1 formula and shifted grunwald formula are employed to discretize the temporal and spatial derivatives respectively a divide and conquer strategy is applied to the large linear system assembling discrete equations of all time levels which in turn requires to solve a series of multidimensional linear systems related to the spatial discretization preconditioned generalized minimal residual method is employed to solve the spatial linear systems resulting from the spatial discretization the discretization is proven to be unconditionally stable and convergent in the sense of infinity norm for general nonnegative coefficients numerical results are reported to show the efficiency of the proposed method

some high order difference schemes for the space and time fractional bloch torrey equations in this paper several difference schemes are proposed for both one dimensional and two dimensional space and time fractional bloch torrey equations the spatial second order scheme and the spatial fourth order compact scheme are established respectively the obtained schemes can achieve the global second order numerical accuracy in time the unique solvability unconditional stability and convergence of the proposed schemes are proved by the energy method two adi schemes are also discussed for the two dimensional problem numerical examples are given to verify the numerical accuracy and efficiency of the difference schemes

construction of mds codes with complementary duals a linear complementary dual lcd code is a linear code with complimentary dual lcd codes have been extensively studied in literature on the other hand maximum distance separable mds codes are an important class of linear codes that have found wide applications in both theory and practice however little is known about mds codes with complimentary duals the main purpose of this paper is to construct several classes of mds codes with complimentary duals i e lcd mds codes through generalized reed solomon codes

a fast direct method for block triangular toeplitz like with tri diagonal block systems from time fractional partial differential equations in this paper we study the block lower triangular toeplitz like with tri diagonal blocks system which arises from the time fractional partial differential equation existing fast numerical solver e g fast approximate inversion method cannot handle such linear system as the main diagonal blocks are different the main contribution of this paper is to propose a fast direct method for solving this linear system and to illustrate that the proposed method is much faster than the classical block forward substitution method for solving this linear system our idea is based on the divide and conquer strategy and together with the fast fourier transforms for calculating toeplitz matrix vector multiplication the complexity needs o m n log 2 m arithmetic operations where m is the number of blocks the number of time steps in the system and n is the size number of spatial grid points of each block numerical examples from the finite difference discretization of time fractional partial differential equations are also given to demonstrate the efficiency of the proposed method