f quantum spaces and their dynamics abstract f quantum space has been suggested by riecan as an alternative model for quantum mechanics in this paper we study the dynamics of f quantum spaces

exponential error bounds for algebraic soft decision decoding of reed solomon codes algebraic soft decision decoding of reed solomon codes is a promising technique for exploiting reliability information in the decoding process while the algorithmic aspects of the decoding algorithm are reasonably well understood and in particular complexity is polynomially bounded in the length of the code the performance analysis has relied almost entirely on simulation results analytical exponential error bounds that can be used to tightly bound the performance of reed solomon codes under algebraic soft decision decoding are presented in this paper the analysis is used in a number of examples and several extensions and consequences of the results are presented

hierarchical pattern classification abstract the aim of this paper is to present a divisive hierarchical clustering method a chain of fuzzy partitions ordered by the refinement relation and the corresponding binary fuzzy hierarchy are obtained by this method since no a priori knowledge concerning the number of real clusters is needed the method represents a solution for the cluster validity problem an algorithm based on this method is given an adaptive distance is used to make the algorithm efficient also for unequal population clusters

applications of list decoding to tracing traitors we apply results from algebraic coding theory to solve problems in cryptography by using recent results on list decoding of error correcting codes to efficiently find traitors who collude to create pirates we produce schemes for which the traceability ta traitor tracing algorithm is very fast we compare the ta and identifiable parent property ipp traitor tracing algorithms and give evidence that when using an algebraic structure the ability to trace traitors with the ipp algorithm implies the ability to trace with the ta algorithm we also demonstrate that list decoding techniques can be used to find all possible pirate coalitions finally we raise some related open questions about linear codes and suggest uses for other decoding techniques in the presence of additional information about traitor behavior

shuffle compatible permutation statistics abstract since the early work of richard stanley it has been observed that several permutation statistics have a remarkable property with respect to shuffles of permutations we formalize this notion of a shuffle compatible permutation statistic and introduce the shuffle algebra of a shuffle compatible permutation statistic which encodes the distribution of the statistic over shuffles of permutations this paper develops a theory of shuffle compatibility for descent statistics statistics that depend only on the descent set and length which has close connections to the theory of p partitions quasisymmetric functions and noncommutative symmetric functions we use our framework to prove that many descent statistics are shuffle compatible and to give explicit descriptions of their shuffle algebras thus unifying past results of stanley gessel stembridge aguiar bergeron nyman and petersen

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

fuzzy conditional logic abstract a certainty or realization degree of a fuzzy formula is defined the conditional certainty degree is also studied fuzzy bayes theorem for exclusive and exhaustive sequences of fuzzy formulas is obtained

the entropy of fuzzy dynamical systems general scheme and generators in this paper some fuzzy modifications of the kolmogorov sinaj entropy are studied we introduce the general scheme which includes the mentioned entropies finally the kolmogorov sinaj theorem on generators is proved for special cases

entropy of a fuzzy process abstract using an additive fuzzy measure 1 in 5 a general concept of entropy of a fuzzy partition has been considered this concept is used to define the entropy of a fuzzy process in this paper some properties of this entropy are summarized these results could be used to build a fuzzy ergodic theory in a further paper the entropy of a measure preserving transformation of a fuzzy measure space will be studied

correction to list decoding of generalized reed solomon codes over commutative rings in the above titled paper ibid vol 51 no 1 pp 411 419 jan 05 lemma 6 is incorrect due to an error on the proof consequently proposition 2 is restated here

grobner basis solutions of constrained interpolation problems this paper extends the previous work of the authors on recursive grobner basis techniques in coding theory pade approximation partial realization interpolation and modelling discrete time behaviours we present a general algorithm applicable to a wide range of constrained interpolation problems in coding theory and systems theory including list decoding and m pade approximation

entropy of fuzzy dynamical systems abstract entropy of a finite fuzzy partition had been defined dumitrescu 1983 1993 using an additive fuzzy measure butnariu 1983 in this paper we define the independence of fuzzy partitions with respect to a fuzzy measure some properties concerning the entropy of independent fuzzy partitions are proved this concept of independence is also relevant for the study of the entropy of a fuzzy process dumitrescu 1993 we prove some new results concerning the entropy of a fuzzy process the fuzzy measure preserving transformations dumitrescu 1993 may be related with the fuzzy ergodic theory the main problem in ergodic theory is to build the isomorphism invariants kolmogorov s 1958 transformation entropy is such an invariant in this paper we define the entropy of a fuzzy dynamical system we prove that this entropy is an isomorphism invariant

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

fuzzy partitions with the connectives t s abstract the notion of a fuzzy partition obtained when the set operations are defined using the operators t and s and an appropriate concept of disjointness is according to butnariu 1983 equivalent with ruspini s definition we prove that such an equivalence holds if and only if the connectives for the fuzzy set operations are t and s

a matrix based list decoding algorithm for linear codes over integer residue rings abstract in this paper we address the problem of list decoding of linear codes over an integer residue ring z q where q is a power of a prime p the proposed procedure exploits a particular matrix representation of the linear code over z p r called the standard form and the p adic expansion of the to be decoded vector in particular we focus on the erasure channel in which the location of the errors is known this problem then boils down to solving a system of linear equations with coefficients in z p r from the parity check matrix representations of the code we recursively select certain equations that a codeword must satisfy and have coefficients only in the field p r 1 z p r this yields a step by step procedure obtaining a list of the closest codewords to a given received vector with some of its coordinates erased we show that such an algorithm actually computes all possible erased coordinates that is the provided list is minimal

conditional fuzzy entropy of fuzzy dynamical systems abstract entropy of fuzzy dynamical systems has been investigated as an isomorphism invariant in dumitrescu 1995 13 in this paper we will introduce a new invariant called the conditional fuzzy entropy of fuzzy dynamical systems with respect to a finite fuzzy partition and an invariant sub algebra which is an extension of fuzzy entropy on fuzzy dynamical systems this new invariant possesses some basic properties such as power rule fuzzy version of abramov rokhlin entropy additive formula and generating sequence

improved decoding of reed solomon and algebraic geometry codes given an error correcting code over strings of length n and an arbitrary input string also of length n the list decoding problem is that of finding all codewords within a specified hamming distance from the input string we present an improved list decoding algorithm for decoding reed solomon codes the list decoding problem for reed solomon codes reduces to the following curve fitting problem over a field f given n points x sub i spl middot y sub i sub i 1 sup n x sub i y sub i spl isin f and a degree parameter k and error parameter e find all univariate polynomials p of degree at most k such that y sub i p x sub i for all but at most e values of i spl isin 1 n we give an algorithm that solves this problem for e 1 3 where the result yields the first asymptotic improvement in four decades the algorithm generalizes to solve the list decoding problem for other algebraic codes specifically alternant codes a class of codes including bch codes and algebraic geometry codes in both cases we obtain a list decoding algorithm that corrects up to n spl radic n n d errors where n is the block length and d is the designed distance of the code the improvement for the case of algebraic geometry codes extends the methods of shokrollahi and wasserman see in proc 29th annu acm symp theory of computing p 241 48 1998 and improves upon their bound for every choice of n and d we also present some other consequences of our algorithm including a solution to a weighted curve fitting problem which may be of use in soft decision decoding algorithms for reed solomon codes

uncertainty measure on fuzzy partitions in this paper we face the problem of constructing the uncertainty or perhaps expected information of a fuzzy partition in terms of the crisp measures of its cuts we solve the problem in the cases in which the measure is either compositive or branching

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

improved list decoding of generalized reed solomon and alternant codes over galois rings we present a two stage list decoder comprising an errors only guruswami sudan gs decoder and an errors and erasures gs decoder as component decoders in the first and second stage respectively the two stages are coupled via a post processor which selects a codeword from the output list of the first component decoder from which erasure locations are obtained for the second stage when applied to a generalized reed solomon rs code over a galois ring r that maps into a generalized rs code of the same length n and minimum hamming distance d over the corresponding residue field the proposed decoder exploits the presence of zero divisors in r to correct s errors where w lceiln radic n n d 1rceil sleslceiln radic n w n d 1rceil with a probability determined by s w and the ratio of the number of nontrivial zero divisors to the number of units in the code alphabet focusing primarily on alternant codes over zopf 2l an important class of subring subcodes of generalized rs codes over gr 2l a we demonstrate that the gs decoding radius w can be exceeded by a substantial margin with significant probability

on the hamming distance of linear codes over a finite chain ring let r be a finite chain ring e g a galois ring k its residue field and c a linear code over r we prove that d c the hamming distance of c is d c spl alpha where c spl alpha is a submodule quotient spl alpha is a certain element of r and denotes the canonical projection to k these two codes also have the same set of minimal codeword supports we explicitly construct a generator matrix polynomial of c spl alpha from the generator matrix polynomials of c we show that in general d c spl les d c with equality for free codes i e for free r submodules of r sup n and in particular for hensel lifts of cyclic codes over k most of the codes over rings described in the literature fall into this class we characterize minimum distance separable mds codes over r and prove several analogs of properties of mds codes over finite fields we compute the hamming weight enumerator of a free mds code over r

additive fuzzy measures and integrals iii abstract a point of view concerning fuzzy measures is explained to this end a new concept of disjointness for fuzzy is introduced and studied also a concept of an additive class of fuzzy sets is defined to be a class of fuzzy sets closed under some additive operations the fuzzy measures are defined to be sum preserving real functions over such additive classes some basic properties of the fuzzy measures are derived in contrast with other homonymous concepts studied in literature our fuzzy measures lead to an additive fuzzy integral see the part ii of the paper

on a family of connectives for fuzzy sets abstract in this paper we find the general solution of the functional equation s t x y t x n y x where s is a t conorm t is a t norm and n is a strong negation on the unit interval in particular the result yields a family of connectives for fuzzy sets

entropy of fuzzy partitions a general model abstract the entropy of fuzzy partitions with regard to a fuzzy measure is analyzed a general model based on pseudo arithmetical operations and g functions will be created the entropy of fuzzy partitions introduced by ruspini inform and control 15 1969 22 and dumitrescu j math anal appl 176 1993 359 is shown to be reasonable only up to an isomorphism

generalized rational interpolation over commutative rings and remainder decoding we propose a new decoding procedure for bose chaudhuri hocquenghem bch and reed solomon rs codes over z sub m where m is a product of prime powers our method generalizes the remainder decoding technique for rs codes originally introduced by welch and berlekamp and retains its key feature of not requiring the prior evaluation of syndromes it thus represents a significant departure from other algorithms that have been proposed for decoding linear block codes over integer residue rings our decoding procedure involves a welch berlekamp wb type algorithm for solving a generalized rational interpolation problem over a commutative ring r with identity the solution to this problem includes as a special case the solution to the wb key equation over r which is central to our decoding procedure a remainder decoding approach for decoding cyclic codes over z sub m up to the hartmann tzeng bound is also presented

the entropy of fuzzy dynamical systems and generators abstract in this paper the entropy and conditional entropy of stochastical complete repartitions are defined the represented measure satisfies all properties analogous to the properties of entropy of measurable repartitions in the crisp case further the notion of entropy of fuzzy dynamical systems is defined the above notion generalizes the notion of entropy of dynamical systems from the classical probability theory finally the kolmogorov sinaj theorem on generators is proved for the fuzzy case

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

a bijective proof of stanley s shuffling theorem for two permutations a and w on disjoint sets of integers consider forming a permutation on the combined sets by shuffling a and w i e a and w appear as subsequences stanley 10 by considering p partitions and a q analogue of saalschutz s 3f2 summation obtained the generating function for shuffles of a and w with a given number of falls an element larger than its successor with respect to greater index sum of positions of falls it is a product of two q binomial coefficients and depends only on remarkably simple parameters namely the lengths numbers of falls and greater indexes of a and w a combinatorial proof of this result is obtained by finding bijections for lattice path representations of shuffles which reduce a and w to canonical permutations for which a direct evaluation of the generating function is given

on hudetz entropy localization abstract in rahimi and riazi 2014 22 a type of entropy of continuous maps on compact metric spaces is defined via integrating a local entropy map this average entropy is represented in terms of the entropy in the sense of dumitrescu but the equality of these two types of entropy is left as an open problem in 22 in this paper we modify the problem stated in 22 by replacing the dumitrescu entropy by the hudetz entropy since in our case the two previous types of entropy take an infinite value and the problem stated in 22 reduces to a trivial equality then we conclude the paper by proving the equality of the modified average entropy and the hudetz entropy the key tool is a version of jaccob s theorem for the hudetz entropy

fuzzy measures and the entropy of fuzzy partitions abstract using an additive fuzzy measure a notion of the entropy of a finite fuzzy partition has been defined d dumitrescu on fuzzy partitions in itinerant seminar on functional equations approximation and convexity cluj napoca 1983 pp 57 60 d dumitrescu and m barbu fuzzy entropy and processes in itinerant seminar on functional equations approximation and convexity cluj napoca 1985 pp 71 74 this notion contains as a particular case a well known concept of entropy m a gil et al fuzzy sets and systems15 1985 65 78 h tanaka et al kybernetes5 1976 25 30 i j taneja bivariate entropies of finite partitions of fuzzy sets in preprints of second ifsa congress tokyo 1987 vol 2 pp 696 699 the aim of this paper is to review some basic properties of the considered entropy

list decoding of generalized reed solomon codes over commutative rings we show that the list decoding procedure of guruswami and sudan may be used to decode generalized reed solomon rs codes defined over commutative rings with identity an algorithm for performing the first of the two phases of this decoding procedure is also given in particular we show that for generalized rs codes over chain rings the probability of picking an incorrect codeword from a given list can be reduced by increasing the size of the code alphabet while keeping the length and minimum hamming distance of the code constant

triangular norm based measures and their markov kernel representation abstract we approach the problem whether left continuous triangular norm based valuations called t measures or t probability measures defined on triangular normbased tribes of the unit cube can be disintegrated by markov kernels we prove that each t measure based on a fundamental triangular norm these triangular norms t together with their corresponding triangular conorms s satisfy the functional equation t x y s x y x y can be uniquely represented as a sum of a disintegrable t measure and a hard core which is either identically zero or which is monotonically irreducible i e cannot be disintegrated

on generalized reed solomon codes over commutative and noncommutative rings in this paper we study generalized reed solomon codes grs codes over commutative and noncommutative rings we show that the classical welch berlekamp and guruswami sudan decoding algorithms still hold in this context and we investigate their complexities under some hypothesis the study of noncommutative grs codes over finite rings leads to the fact that grs codes over commutative rings have better parameters than their noncommutative counterparts also grs codes over finite fields have better parameters than their commutative rings counterparts but we also show that given a unique decoding algorithm for a grs code over a finite field there exists a unique decoding algorithm for a grs code over a truncated power series ring with a better asymptotic complexity moreover we generalize a lifting decoding scheme to obtain new unique and list decoding algorithms designed to work when the base ring is for example a galois ring or a truncated power series ring or the ring of square matrices over the latter ring