inseparable local uniformization abstract it is known since the works of zariski in the early 40s that desingularization of varieties along valuations called local uniformization of valuations can be considered as the local part of the desingularization problem it is still an open problem if local uniformization exists in positive characteristic and dimension larger than three in this paper we prove that zariski local uniformization of algebraic varieties is always possible after a purely inseparable extension of the field of rational functions and therefore any valuation can be uniformized by a purely inseparable alteration

product matrix msr codes with bandwidth adaptive exact repair in the distributed storage systems dsss with k systematic nodes robustness against node failure is commonly provided by storing redundancy in a number of other nodes and performing repair mechanism to reproduce the content of the failed nodes efficiency is then achieved by minimizing the storage overhead and the amount of data transmission required for data reconstruction and repair provided by coding solutions such as regenerating codes common explicit regenerating code constructions enable efficient repair through accessing a predefined number d of arbitrary chosen available nodes namely helpers in practice however the state of the system dynamically changes based on the request load the link traffic and so on and the parameters which optimize system s performance vary accordingly it is then desirable to have coding schemes which are able to operate optimally under a range of different parameters simultaneously specifically adaptivity in the number of helper nodes for repair is of interest while robustness requires capability of performing repair with small number of helpers it is desirable to use as many helpers as available to reduce the transmission delay and total repair traffic in this paper we focus on the minimum storage regenerating msr codes where each of the n nodes in the network is supposed to store alpha information units and the source data of size k alpha could be recovered from any arbitrary set of k nodes we introduce a class of msr codes that realize the optimal repair bandwidth simultaneously with a set of different choices for the number of helpers namely d d 1 ldots d delta our coding scheme follows the product matrix pm framework introduced by rashmi et al and could be considered as a generalization of the pm msr code presented by rashmi et al such that any d i i 1 k 1 helpers can perform an optimal repair as a result the coding rate in our construction is limited by k n leq 1 2 however similar to the original design of pm msr codes our solution can realize practical values of the parameter alpha recently ye and barg have presented another explicit msr coding scheme which is capable of performing optimal repair for various number of helpers the solution presented by ye and barg works for any arbitrary set of parameters k and d and can achieve high coding rates but the required alpha for this code is exponentially large we show that the required value for alpha in the coding scheme presented in this paper is exponentially smaller when compared with the work of ye and barg for the same set of other parameters particularly for a dss with n nodes and k systematic nodes the required value for alpha is reduced from s n to sk where s mathrm l cm d 1 k 1 cdots d delta k 1 we also show the required field size in the presented coding scheme is equal to n

applications of soft sets in ideal theory of bck bci algebras the notion of soft ideals and idealistic soft bck bci algebras is introduced and several examples are given relations between soft bck bci algebras and idealistic soft bck bci algebras are provided the intersection union and operation and or operation of soft ideals and idealistic soft bck bci algebras are established

shortening array codes and the perfect 1 factorization conjecture the existence of a perfect 1 factorization of the complete graph with n nodes namely kn for arbitrary even number n is a 40 year old open problem in graph theory so far two infinite families of perfect 1 factorizations have been shown to exist namely the factorizations of kp 1 and k2 p where p is an arbitrary prime number p 2 it was shown in previous work that finding a perfect 1 factorization of kn is related to a problem in coding specifically it can be reduced to constructing an mds minimum distance separable lowest density array code in this paper a new method for shortening arbitrary array codes is introduced it is then used to derive the kp 1 family of perfect 1 factorization from the k2p family namely techniques from coding theory are used to prove a new result in graph theory that the two factorization families are related

new array codes for multiple phased burst correction an optimal family of array codes over gf q for correcting multiple phased burst errors and erasures where each phased burst corresponds to an erroneous or erased column in a code array is introduced as for erasures these array codes have an efficient decoding algorithm which avoids multiplications or divisions over extension fields replacing these operations with cyclic shifts of vectors over gf q the erasure decoding algorithm can be adapted easily to handle single column errors as well the codes are characterized geometrically by means of parity constraints along certain diagonal lines in each code array thus generalizing a previously known construction for the special case of two erasures algebraically they can be interpreted as reed solomon codes when q is primitive in gf q the resulting codes become conventional reed solomon codes of length p over gf q sup p 1 in which case the new erasure decoding technique can be incorporated into the berlekamp massey algorithm yielding a faster way to compute the values of any prescribed number of errors

a new design of binary mds array codes with asymptotically weak optimal repair binary maximum distance separable mds array codes are a special class of erasure codes for distributed storage that not only provides fault tolerance with minimum storage redundancy but also achieves low computational complexity they are constructed by encoding k information columns into r parity columns in which each element in a column is a bit such that any k out of the k r columns suffice to recover all information bits in addition to providing fault tolerance it is critical to improve repair performance in practical applications specifically if a single column fails then our goal is to minimize the repair bandwidth by downloading the least amount of bits from d healthy columns where k leq d leq k r 1 if one column of an mds code is failed it is known that we need to download at least 1 d k 1 fraction of the data stored in each of the d healthy columns if this lower bound is achieved for the repair of the failure column from accessing arbitrary d healthy columns we say that the mds code has optimal repair however if such lower bound is only achieved by d specific healthy columns then we say the mds code has weak optimal repair in this paper we propose two explicit constructions of binary mds array codes with more parity columns i e r geq 3 that achieve asymptotically weak optimal repair where k 1 leq d leq k lfloor r 1 2 rfloor and asymptotic means that the repair bandwidth achieves the minimum value asymptotically in d codes in the first construction have odd number of parity columns and asymptotically weak optimal repair for anyone information failure while codes in the second construction have even number of parity columns and asymptotically weak optimal repair for any one column failure

new constructions of cooperative msr codes reducing node size to exp o n we consider the problem of multiple node repair in distributed storage systems under the cooperative model where the repair bandwidth includes the amount of data exchanged between any two different storage nodes recently explicit constructions of mds codes with optimal cooperative repair bandwidth for all possible parameters were given by ye and barg ieee transactions on information theory 2019 the node size or sub packetization in this construction scales as exp theta text n text h where h is the number of failed nodes and n is the code length in this paper we give new explicit constructions of optimal mds codes for all possible parameters under the cooperative model and the node size of our new constructions only scales as exp text o text n for any number of failed nodes furthermore it is known that any optimal mds code under the cooperative model including in particular our new code construction also achieves optimal repair bandwidth under the centralized model where the amount of data exchanged between failed nodes is not included in the repair bandwidth we further show that the node size of our new construction is also much smaller than that of the best known mds code constructions for the centralized model

codes with local regeneration and erasure correction regenerating codes and codes with locality are two coding schemes that have recently been proposed which in addition to ensuring data collection and reliability also enable efficient node repair in a situation where one is attempting to repair a failed node regenerating codes seek to minimize the amount of data downloaded for node repair while codes with locality attempt to minimize the number of helper nodes accessed this paper presents results in two directions in one this paper extends the notion of codes with locality so as to permit local recovery of an erased code symbol even in the presence of multiple erasures by employing local codes having minimum distance 2 an upper bound on the minimum distance of such codes is presented and codes that are optimal with respect to this bound are constructed the second direction seeks to build codes that combine the advantages of both codes with locality as well as regenerating codes these codes termed here as codes with local regeneration are codes with locality over a vector alphabet in which the local codes themselves are regenerating codes we derive an upper bound on the minimum distance of vector alphabet codes with locality for the case when their constituent local codes have a certain uniform rank accumulation property this property is possessed by both minimum storage regeneration msr and minimum bandwidth regeneration mbr codes we provide several constructions of codes with local regeneration which achieve this bound where the local codes are either msr or mbr codes also included in this paper is an upper bound on the minimum distance of a general vector code with locality as well as the performance comparison of various code constructions of fixed block length and minimum distance

x code mds array codes with optimal encoding we present a new class of mds maximum distance separable array codes of size n spl times n n a prime number called x code the x codes are of minimum column distance 3 namely they can correct either one column error or two column erasures the key novelty in x code is that it has a simple geometrical construction which achieves encoding update optimal complexity i e a change of any single information bit affects exactly two parity bits the key idea in our constructions is that all parity symbols are placed in rows rather than columns

scalar mscr codes via the product matrix construction an n k d cooperative regenerating code provides the optimal bandwidth repair for any t t 1 node failures in a cooperative way in particular an mscr minimum storage cooperative regenerating code retains the same storage overhead as an n k mds code suppose each node stores alpha symbols which indicates the sub packetization level of the code a scalar mscr code attains the minimum sub packetization i e alpha d k t by now all existing constructions of scalar mscr codes restrict to very special parameters eg d k or k 2 etc in a recent work ye and barg construct mscr codes for all n k d t however their construction needs alpha approx text exp n t which is almost infeasible in practice in this paper we give an explicit construction of scalar mscr codes for all d geq max 2k 1 t k which covers all possible parameters except the case of k leq d leq 2k 2 t when k moreover as a complementary result for k we prove the nonexistence of linear scalar mscr codes that have invariant repair spaces our construction and most of the previous scalar mscr codes all have invariant repair spaces and this property is appealing in practice because of convenient repair in this sense this work presents an almost full description of usual scalar mscr codes

characterizing diophantine henselian valuation rings and valuation ideals we give a characterization in terms of the residue field of those henselian valuation rings and those henselian valuation ideals that are diophantine this characterization gives a common generalization of all the positive and negative results on diophantine henselian valuation rings and diophantine valuation ideals in the literature we also treat questions of uniformity and we apply the results to show that a given field can carry at most one diophantine nontrivial equicharacteristic henselian valuation ring or valuation ideal

ramification of valuations abstract in this paper we develop a very explicit theory of ramification of general valuations in algebraic function fields in characteristic zero and arbitrary dimension we obtain the strongest possible generalization of the classical ramification theory of local dedekind domains we further develop a ramification theory of algebraic functions fields of dimension two in positive characteristic we prove that local monomialization and simultaneous resolution hold under very mild assumptions and give pathological examples

subfields of ample fields rational maps and definability abstract pop proved that a smooth curve c over an ample field k with c k has k many rational points we strengthen this result by showing that there are k many rational points that do not lie in a given proper subfield even after applying a rational map this has several consequences for example we gain insight into the structure of existentially definable i e diophantine subsets of ample fields

repairing reed solomon codes we study the performance of reed solomon rs codes for the exact repair problem in distributed storage our main result is that in some parameter regimes reed solomon codes are optimal regenerating codes among maximum distance separable mds codes with linear repair schemes moreover we give a characterization of mds codes with linear repair schemes which holds in any parameter regime and which can be used to give non trivial repair schemes for rs codes in other settings more precisely we show that for k dimensional rs codes whose evaluation points are a finite field of size n there are exact repair schemes with bandwidth n 1 log n 1 n k bits and that this is optimal for any mds code with a linear repair scheme in contrast the naive commonly implemented repair algorithm for this rs code has bandwidth k log n bits when the entire field is used as evaluation points the number of nodes n is much larger than the number of bits per node which is o log n and so this result holds only when the degree of sub packetization is small however our method applies in any parameter regime and to illustrate this for high levels of sub packetization we give an improved repair scheme for a specific 14 10 rs code used in the facebook hadoop analytics cluster

e e v q fuzzy normal quasinormal and maximal subgroups abstract the notions of and q fuzzy normality and quasinormality for a q fuzzy subgroup are introduced the notions of an q fuzzy core subgroup and an q fuzzy p sylow subgroup are also introduced and some fundamental properties are dealt with q fuzzy normality and q fuzzy quasinormality are found to be more generalised concepts than those introduced by others a detailed study with a fuzzy maximal subgroup is incorporated with the limitation of this concept

error correction based on partial information we consider the decoding of linear and array codes from errors when we are only allowed to download a part of the codeword more specifically suppose that we have encoded k data symbols using an n k code with code length n and dimension k during storage some of the codeword coordinates might be corrupted by errors we aim to recover the original data by reading the corrupted codeword with a limit on the transmission bandwidth namely we can only download an alpha proportion of the corrupted codeword for a given alpha our objective is to design a code and a decoding scheme such that we can recover the original data from the largest possible number of errors a naive scheme is to read alpha n coordinates of the codeword this method used in conjunction with mds codes guarantees recovery from any lfloor alpha n k 2 rfloor errors in this paper we show that we can instead download an alpha proportion from each of the codeword s coordinates for a well designed mds code this method can guarantee recovery from lfloor n k alpha 2 rfloor errors which is 1 alpha times more than the naive method and is also the maximum number of errors that an n k code can correct by downloading only an alpha proportion of the codeword we present two families of such optimal constructions and decoding schemes of which one is based on interleaved reed solomon codes and the other on folded reed solomon codes we further show that both code constructions attain asymptotically optimal list decoding radius when downloading only a part of the corrupted codeword we also construct an ensemble of random codes that with high probability approaches the upper bound on the number of correctable errors when the decoder downloads an alpha proportion of the corrupted codeword

generalizations of fuzzy subalgebras in bck bci algebras more general form of the notion of quasi coincidence of a fuzzy point with a fuzzy set is considered and generalizations of results in the papers y b jun on a b fuzzy subalgebras of bck bci algebras bull korean math soc 42 4 2005 703 711 y b jun fuzzy subalgebras of type a b in bck bci algebras kyungpook math j 47 2007 403 410 are discussed the notions of q k fuzzy subalgebras and email protected q k fuzzy subalgebras in a bck bci algebra x are introduced and several properties are investigated characterizations of email protected q k fuzzy subalgebra in a bck bci algebra x are discussed

rosenfeld s fuzzy subgroups with interval valued membership functions abstract in this paper we define interval valued fuzzy subgroups i e i v fuzzy subgroups of rosenfeld s nature a necessary and sufficient condition for an i v fuzzy subset to be an i v fuzzy subgroup is stated intersection of two i v fuzzy subgroups is also an i v fuzzy subgroup the images and inverse images of i v fuzzy subgroups are defined and how the images or inverse images of i v fuzzy subgroups become i v fuzzy subgroups is studied

a new view of fuzzy hypernear rings in this survey article we first introduce the concept of quasi coincidence of a fuzzy interval value with an interval valued fuzzy set this concept is a generalized concept of quasi coincidence of a fuzzy point within a fuzzy set by using this new idea we consider the interval valued q fuzzy sub hypernear rings hyperideals of a hypernear ring and hence a generalization of a fuzzy sub near ring ideal is given some related properties of fuzzy hypernear rings are described finally we consider the concept of implication based interval valued fuzzy sub hypernear rings hyperideals in a hypernear ring in particular the implication operators in lukasiewicz system of continuous valued logic are discussed

fuzzy hyperrings hv rings based on fuzzy universal sets the aim of this paper is to define and study the concept of a fuzzy hyperring which depends on the concept of a fuzzy universal set a correspondence relation between a fuzzy hyperring based on a fuzzy universal set and both ordinary and fuzzy hyperideal in the sense of davvaz is obtained

explicit constructions of optimal access mds codes with nearly optimal sub packetization an n k l maximum distance separable mds array code of length n dimension k n r and sub packetization l is formed of l times n matrices over a finite field f with every column of the matrix stored on a separate node in the distributed storage system and viewed as a coordinate of the codeword repair of a failed node recovery of one erased column can be performed by accessing a set of d le n 1 surviving helper nodes the code is said to have the optimal access property if the amount of data accessed at each of the helper nodes meets a lower bound on this quantity for optimal access mds codes with d n 1 the sub packetization l satisfies the bound l ge r k 1 r in our previous work ieee trans inf theory vol 63 no 4 2017 for any n and r we presented an explicit construction of optimal access mds codes with sub packetization l r n 1 in this paper we take up the question of reducing the sub packetization value l to make it to approach the lower bound we construct an explicit family of optimal access codes with l r lceil n r rceil which differs from the optimal value by at most a factor of r 2 these codes can be constructed over any finite field f as long as f ge r lceil n r rceil and afford low complexity encoding and decoding procedures we also define a version of the repair problem that bridges the context of regenerating codes and codes with locality constraints lrc codes which we call group repair with optimal access in this variation we assume that the set of n sm nodes is partitioned into m repair groups of size s and require that the amount of accessed data for repair is the smallest possible whenever the d s k 1 helper nodes include all the other s 1 nodes from the same group as the failed node for this problem we construct a family of codes with the group optimal access property these codes can be constructed over any field f of size f ge n and also afford low complexity encoding and decoding procedures

an encoding algorithm of triply extended reed solomon codes with asymptotically optimal complexities in this paper we devise a fast encoding algorithm for triply extended reed solomon codes the proposed approach requires approximately two xors per bit which improves the prior result of three xors per bit established by certain maximum distance separable mds array codes we also prove that for mds codes with two and three parities the scheduling algorithms require at least two xors per bit to the best of our knowledge this is the first provable scheduling algorithm for the triple parity mds codes to approach the theoretical lower bounds the implementation with simd instructions is provided the simulations show that the proposed approach is competitive as compared with other cutting edge implementations

redefined fuzzy hv submodules and many valued implications this paper considers a relationship between fuzzy sets and algebraic hyperstructures a generalization of ideas presented by davvaz b davvaz fuzzy h v submodules fuzzy sets syst 117 2001 477 484 and bhakat and das s k bhakat p das email protected q fuzzy subgroup fuzzy sets syst 80 1996 359 368 is presented in fact the main purpose of this paper is to introduce and study a new sort of fuzzy h v submodule of an h v module called email protected q fuzzy h v submodules these fuzzy h v submodules are characterized by their level h v submodules the concept of a fuzzy h v submodule with thresholds is introduced and some interesting properties are investigated

new types of fuzzy ideals in bck bci algebras a generalization of an email protected q fuzzy ideal of a bck bci algebra is discussed characterizations of an email protected q k fuzzy ideal and an email protected email protected email protected q email protected fuzzy ideal are provided conditions for an email protected q k fuzzy ideal resp email protected email protected email protected q email protected fuzzy ideal to be a fuzzy ideal are provided using the notion of a fuzzy ideal with thresholds characterizations of a fuzzy ideal an email protected q k fuzzy ideal and an email protected email protected email protected q email protected fuzzy ideal are discussed

embedding problems over large fields in this paper we study galois theoretic properties of a large class of fields a class which includes all fields satisfying a universal local global principle for the existence of rational points on varieties as applications we give a positive answer to a long standing conjecture which originates in an unpublished note of roquette and asserts that the absolute galois group of a countable hilbertian pac field is profinite free see f j problem 24 41 secondly we give new evidence for the conjecture of shafarevich which asserts that the absolute galois group of qab is profinite free finally one of the most interesting applications of the theory we develop here is the insight in the galois structure of the totally eg adic numbers

constructions of high rate minimum storage regenerating codes over small fields a novel technique for construction of minimum storage regenerating msr codes is presented based on this technique three explicit constructions of msr codes are given the first two constructions provide access optimal msr codes with two and three parities respectively which attain the sub packetization bound for access optimal codes the third construction provides longer msr codes with three parities i e codes with larger number of systematic nodes this improvement is achieved at the expense of the access optimality and the field size in addition to a minimum storage in a node all three constructions allow the entire data to be recovered from a minimal number of storage nodes that is given storage ell in each node the entire stored data can be recovered from any 2 log 2 ell for two parity nodes and either 3 log 3 ell or 4 log 3 ell for three parities second in the first two constructions a helper node accesses the minimum number of its symbols for repair of a failed node access optimality the goal of this paper is to provide a construction of such optimal codes over the smallest possible finite fields the generator matrix of these codes is based on perfect matchings of complete graphs and hypergraphs and on a rational canonical form of matrices for two parities the field size is reduced by a factor of two for access optimal codes compared to previous constructions for three parities in the first construction a field size of at least 6 log 3 ell 1 or 3 log 3 ell 1 for fields with characteristic 2 is sufficient and in the second construction the field size is larger yet linear in log 3 ell both constructions with three parities provide a significant improvement over previous works due to either decreased field size or lower subpacketization

on fuzzy ideals in bck bci algebras in the present paper we give a procedure by which we generate a fuzzy ideal resp closed fuzzy ideal by a fuzzy set in a bci algebra as applications we prove the set of all the fuzzy ideals in a bci algebra forms a complete lattice called fuzzy ideal lattice the set of all the closed fuzzy ideals in a bci algebra is a modular sublattice of the fuzzy ideal lattice but it is not distributive in general for commutative bck algebras we establish the prime fuzzy ideal theorem in particular we give some characterizations of noether bck bci algebras by fuzzy ideals

pliable fractional repetition codes for distributed storage systems design and analysis a distributed storage system dss is one of the most vital components of a cloud computing system used for storing and sharing big data among authorized users a typical dss consists of n storage nodes each with a storage capacity of alpha units of data such that the entire file stored on the dss can be recovered by accessing any k nodes the objective of this paper is to construct fractional repetition fr codes with applications in dsss fr codes are the key to constructing a class of distributed storage codes with exact repair by transfer i e upon failure a failed storage node is exactly regenerated through simple downloading a major drawback of existing fr codes is that they are insufficiently flexible to adapt adequately to system changes in dsss to address this problem this paper introduces a new type of fr codes called pliable fr codes in which both the per node storage alpha and repetition degree rho can easily and simultaneously be adjusted in addition this paper presents several constructions of pliable fr codes and provides a relatively comprehensive analysis of the constructed pliable fr codes which reveals the following versatile properties of the codes 1 the transposed codes of the constructed codes are also pliable fr codes 2 the constructed codes attain a tight upper bound on the file size with equality at least for 1 le k where k is the smallest value of k such that k alpha k 1 frac n 1 rho 1 3 the constructed codes also meet a singleton like bound on the minimum distance at least for 1 le k which demonstrates their optimality 4 the computational complexity necessary for determining the file size or the minimum distance of the constructed codes can be greatly reduced when it is hard to exactly determine them and 5 the constructed codes can be used as fractional repetition batch codes to provide load balancing in dsss for which the batch size i e the number of symbols that can be read in parallel can be exactly determined

explicit constructions of high rate mds array codes with optimal repair bandwidth maximum distance separable mds codes are optimal error correcting codes in the sense that they provide the maximum failure tolerance for a given number of parity nodes suppose that an mds code with k information nodes and r n k parity nodes is used to encode data in a distributed storage system it is known that if h out of the n nodes are inaccessible and d surviving helper nodes are used to recover the lost data then we need to download at least h d h k fraction of the data stored in each of the helper nodes dimakis et al 2010 and cadambe et al 2013 if this lower bound is achieved for the repair of any h erased nodes from any d helper nodes we say that the mds code has the h d optimal repair property we study high rate mds array codes with the optimal repair property also known as minimum storage regenerating codes or msr codes explicit constructions of such codes in the literature are only available for the cases where there are at most three parity nodes and these existing constructions can only optimally repair a single node failure by accessing all the surviving nodes in this paper given any r and n we present two explicit constructions of mds array codes with the h d optimal repair property for all h le r and k le d le n h simultaneously codes in the first family can be constructed over any base field f as long as f ge sn where s text lcm 1 2 dots r the encoding decoding repair of failed nodes and update procedures of these codes all have low complexity codes in the second family have the optimal access property and can be constructed over any base field f as long as f ge n 1 moreover both code families have the optimal error resilience capability when repairing failed nodes we also construct several other related families of mds codes with the optimal repair property

on lowest density mds codes let f sub q denote the finite field gf q and let h be a positive integer mds maximum distance separable codes over the symbol alphabet f sub q sup b are considered that are linear over f sub q and have sparse low density parity check and generator matrices over f sub q that are systematic over f sub q sup b lower bounds are presented on the number of nonzero elements in any systematic parity check or generator matrix of an f sub q linear mds code over f sub q sup b along with upper bounds on the length of any mds code that attains those lower bounds a construction is presented that achieves those bounds for certain redundancy values the building block of the construction is a set of sparse nonsingular matrices over f sub q whose pairwise differences are also nonsingular bounds and constructions are presented also for the case where the systematic condition on the parity check and generator matrices is relaxed to be over f sub q rather than over f sub q sup b

fuzzy strong implicative hyper bck ideals of hyper bck algebras the fuzzy setting of strong implicative hyper bck ideals in hyper bck algebras is considered some of their properties are investigated relations among fuzzy strong hyper bck ideals fuzzy implicative hyper bck ideals and fuzzy strong implicative hyper bck ideals are given a characterization of a fuzzy strong implicative hyper bck ideal is provided the hyper homomorphic preimage of a fuzzy strong implicative hyper bck ideal is discussed

maximizing burst erasure correction capability of mds codes in general an n k maximum distance separable mds code over gf pm can not correct all burst erasures of length n k m over gf p in this letter we constructively show that such a linear mds code over gf pm can be modified to correct all burst erasures of length up to n k m over gf p while maintaining its mds property over gf pm

some kinds of c ccq interval valued fuzzy ideals of bci algebras in this paper we first introduce the notions of positive implicative implicative and commutative interval valued fuzzy ideals of bci algebras which are generalizations of positive implicative implicative and commutative fuzzy ideals respectively and investigate some of their related properties the concept of quasi coincidence of an interval valued fuzzy point within an interval valued fuzzy set is introduced in fact the concept is a generalized concept of quasi coincidence of a fuzzy point within a fuzzy set by using this new idea we further introduce the notions of positive implicative implicative and commutative q interval valued fuzzy ideals of bci algebras and investigate some of their related properties some characterization theorems of these generalized interval valued fuzzy ideals are derived the relationship among these generalized interval valued fuzzy ideals of bci algebras is also considered

new efficient mds array codes for raid part ii rabin like codes for tolerating multiple spl ges 4 disk failures for pt 1 see ibid vol 54 no 9 p 1071 1080 2005 a new class of binary maximum distance separable mds array codes which are based on circular permutation matrices are introduced in this paper these array codes are used for tolerating multiple spl ges 4 disk failures in redundant arrays of inexpensive disks raid architecture the size of the information part is m spl times n where n is the number of information disks and m 1 is a prime integer the size of the parity check part is m spl times r the minimum distance is r 1 and the number of parity check disks is r in practical applications m can be very large and n ranges from 20 to 50 the code rate is r n n r these codes can be used for tolerating up to r disk failures with very fast encoding and decoding the complexities of encoding and decoding algorithms are o rmn and o m sup 3 r sup 4 respectively when r 4 there need to be 9mn xor operations for encoding and 9n 95 m 1 xor operations for decoding

fuzzy ideals in bci algebras abstract the aim of this paper is to introduce the notions of fuzzy bci positive implicative ideals and fuzzy bci implicative ideals of bci algebras and to investigate their properties we give several characterizations of fuzzy bci positive implicative ideals and fuzzy bci implicative ideals the extension theorems of fuzzy bci positive implicative ideals and fuzzy bci implicative ideals are obtained we discuss the relations among various fuzzy ideals and show that a fuzzy set of a bci algebra is fuzzy bci implicative ideal if and only if is both fuzzy bci positive implicative ideal and fuzzy bci commutative ideal finally we characterize implicative positive implicative bci algebras via fuzzy bci implicative fuzzy bci positive implicative ideals

new efficient mds array codes for raid part i reed solomon like codes for tolerating three disk failures this paper presents a class of binary maximum distance separable mds array codes for tolerating disk failures in redundant arrays of inexpensive disks raid architecture based on circular permutation matrices the size of the information part is m spl times n the size of the parity check part is m spl times 3 and the minimum distance is 4 where n is the number of information disks the number of parity check disks is 3 and m 1 is a prime integer in practical applications m can be very large and n is from 20 to 50 the code rate is r n n 3 these codes can be used for tolerating three disk failures the encoding and decoding of the reed solomon like codes are very fast there need to be 3mn xor operations for encoding and 3mn 9 m 1 xor operations for decoding

two classes of binary mds array codes with asymptotically optimal repair for any single column an m times k r binary maximum distance separable mds array code contains k information columns and r parity columns with each entry being a bit where any k out of k r columns can recover the k information columns when there is a failed column it is critical to minimize the repair bandwidth that is the total number of bits downloaded from d out of k r 1 surviving columns in repairing the failed column in this article we first propose two explicit constructions of binary mds array codes that have asymptotically optimal repair bandwidth for any information column where r geq 2 and d k r 1 for the first construction and r geq 4 is an even number and d k frac r 2 1 for the second construction by applying a generic transformation for the proposed two classes of binary mds array codes we then obtain two classes of new binary mds array codes that also have optimal repair bandwidth for any parity column and asymptotically optimal repair bandwidth for any information column

redefined fuzzy implicative filters using the belongs to relation and quasi coincidence with relation q between fuzzy points and fuzzy sets the concept of q j fuzzy implicative filters where q j are any two of q email protected q email protected q with q email protected q is introduced and related properties are discussed relations between email protected q email protected q fuzzy implicative filters and email protected q fuzzy implicative filters are investigated and conditions for an email protected q fuzzy implicative filter to be an fuzzy implicative filter are provided characterizations of email protected q fuzzy implicative filters are given and conditions for a fuzzy set to be a q email protected q fuzzy implicative filter are provided

some kinds of fuzzy ideals of bci algebras the concepts of c c q d fuzzy positive implicative implicative and commutative ideals and c c q d fuzzy positive implicative implicative and commutative ideals in bci algebras are introduced some new characterizations are investigated in particular we prove that a fuzzy set m of a bci algebra x is an c c q d fuzzy implicative ideal of x if and only if it is both an c c q d fuzzy positive implicative ideal and an c c q d fuzzy commutative ideal finally we give some characterizations of three particular classes of bci algebras by these generalized fuzzy ideals

bci implicative ideals of bci algebras abstract this paper introduces the notion of bci implicative ideals and characterizes bci implicative ideals and closed bci implicative ideals using these characterizations the connections between bci implicative ideals and other ideals in bci bck algebras are investigated additionally the extension property of bci implicative ideals is established finally the implicative bci algebras are completely described using bci implicative ideals the above work generalizes the corresponding results in bck algebras

xi code a family of practical lowest density mds array codes of distance 4 designing the lowest density maximum distance separable mds array codes has gained much attention in recent years due to the optimal redundancy and minimum update penalty lowest density of such codes however the existing lowest density mds array codes of distance 4 have extremely strict constraints on the code length which makes them impractical in particular most of them require the code length to be p or p 1 where p is a prime that satisfies i 3 p 1 and ii 2 is primitive in mathrm gf p in this paper we propose a new family of the lowest density mds array codes of distance 4 called xi code it has the properties of 1 being capable of correcting both triple erasures and a single error combined with one erasure 2 having code length of either p or p 1 with p being an odd prime number 3 achieving optimality in encoding and update and 4 achieving optimality in erasure decoding for certain erasure patterns and near optimality for other erasure patterns it is worth mentioning that xi code is the first discovered family of the lowest density mds array codes of distance 4 that supports the code length of p or p 1 with p being an odd prime

value groups residue fields and bad places of rational function fields we classify all possible extensions of a valuation from a ground field k to a rational function field in one or several variables over k we determine which value groups and residue fields can appear and we show how to construct extensions having these value groups and residue fields in particular we give several constructions of extensions whose corresponding value group and residue field extensions are not finitely generated in the case of a rational function field k x in one variable we consider the relative algebraic closure of k in the henselization of k x with respect to the given extension and we show that this can be any countably generated separable algebraic extension of k in the tame case we show how to determine this relative algebraic closure finally we apply our methods to power series fields and the p adics

cyclic lowest density mds array codes three new families of lowest density maximum distance separable mds array codes are constructed which are cyclic or quasi cyclic in addition to their optimal redundancy mds and optimal update complexity lowest density the symmetry offered by the new codes can be utilized for simplified implementation in storage applications the proof of the code properties has an indirect structure first mds codes that are not cyclic are constructed and then transformed to cyclic codes by a minimum distance preserving transformation

the repair problem for reed solomon codes optimal repair of single and multiple erasures with almost optimal node size the repair problem in distributed storage addresses recovery of the data encoded using an erasure code for instance a reed solomon rs code we consider the problem of repairing a single node or multiple nodes in rs coded storage systems using the smallest possible amount of inter nodal communication according to the cut set bound communication cost of repairing h geqslant 1 failed nodes for an n k n r maximum distance separable mds code using d helper nodes is at least dhl d h k where l is the size of the node guruswami and wootters 2016 initiated the study of efficient repair of rs codes showing that they can be repaired using a smaller bandwidth than under the trivial approach at the same time their work as well as follow up papers stopped short of constructing rs codes or any scalar mds codes that meet the cut set bound with equality in this paper we construct the families of rs codes that achieve the cut set bound for repair of one or several nodes in the single node case we present the rs codes of length n over the field mathbb f q l l exp 1 o 1 n log n that meet the cut set bound we also prove an almost matching lower bound on l showing that super exponential scaling is both necessary and sufficient for scalar mds codes to achieve the cut set bound using linear repair schemes for the case of multiple nodes we construct a family of rs codes that achieve the cut set bound universally for the repair of any h 1 2 dots r failed nodes from any subset of d helper nodes k leqslant d leqslant n h for a fixed number of parities r the node size of the constructed codes is close to the smallest possible node size for codes with such properties

phased burst error correcting array codes various aspects of single phased burst error correcting array codes are explored these codes are composed of two dimensional arrays with row and column parities with a diagonally cyclic readout order they are capable of correcting a single burst error along one diagonal optimal codeword sizes are found to have dimensions n sub 1 n sub 2 such that n sub 2 is the smallest prime number larger than n sub 1 these codes are capable of reaching the singleton bound a new type of error approximate errors is defined in q ary applications these errors cause data to be slightly corrupted and therefore still close to the true data level phased burst array codes can be tailored to correct these codes with even higher rates than before

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

characterizations of regular ordered semigroups in terms of α β fuzzy generalized bi ideals in this paper the concept of an a b fuzzy generalized bi ideal in an ordered semigroup is introduced which is a generalization of the concept of a fuzzy generalized bi ideal in an ordered semigroup using this concept some characterization theorems are provided the upper lower parts of an q fuzzy generalized bi ideal are introduced and some characterizations of regular ordered semigroups are given also we consider the concept of implication based fuzzy generalized bi ideals in an ordered semigroup in particular the implication operators in lukasiewicz system of continuous valued logic are discussed

fuzzy h ideals of hemirings a characterization of an h hemiregular hemiring in terms of a fuzzy h ideal is provided some properties of prime fuzzy h ideals of h hemiregular hemirings are investigated it is proved that a fuzzy subset z of a hemiring s is a prime fuzzy left right h ideal of s if and only if z is two valued z 0 1 and the set of all x in s such that z x 1 is a prime left right h ideal of s finally the similar properties for maximal fuzzy left right h ideals of hemirings are considered

tame fields and tame extensions abstract letvbe a henselian valuation of any rank of a fieldkand let v be the extension ofvto a fixed algebraic closure k ofk in this paper it is proved that k v is a tame field i e every finite extension of k v is tamely ramified if and only if to each k k there correspondsa kfor which v a k where k min v runs over allk conjugates of a special case of the previous result whenkis a perfect field of nonzero characteristic was proved in 1995 with the purpose of completing a result of james ax s k khanduja j algebra172 1995 147 151

on diophantine definability and decidability in some infinite totally real extensions of ℚ let m be a number field and w m a set of its non archimedean primes then let o m wm x m ord t x 0 t w m let p 1 p r be a finite set of prime numbers let f inf be the field generated by all the p j i th roots of unity as j and i 1 r let k inf be the largest totally real subfield of f inf then for any e 0 there exist a number field m k inf and a set w m of non archimedean primes of m such that w m has density greater than 1 e and z has a diophantine definition over the integral closure of o m w m in k inf

access versus bandwidth in codes for storage maximum distance separable mds codes are widely used in storage systems to protect against disk node failures a node is said to have capacity l over some field f if it can store that amount of symbols of the field an n k l mds code uses n nodes of capacity l to store k information nodes the mds property guarantees the resiliency to any n k node failures an optimal bandwidth respectively optimal access mds code communicates respectively accesses the minimum amount of data during the repair process of a single failed node it was shown that this amount equals a fraction of 1 n k of data stored in each node in previous optimal bandwidth constructions l scaled polynomially with k in codes when the asymptotic rate is less than 1 moreover in constructions with a constant number of parities i e when the rate approaches 1 l is scaled exponentially with k in this paper we focus on the case of linear codes with linear repair operations and constant number of parities n k r and ask the following question given the capacity of a node l what is the largest number of information disks k in an optimal bandwidth respectively access k r k l mds code we give an upper bound for the general case and two tight bounds in the special cases of two important families of codes the first is a family of codes with optimal update property and the second is a family with optimal access property moreover the bounds show that in some cases optimal bandwidth codes have larger k than optimal access codes and therefore these two measures are not equivalent

pseudo d algebras we introduce the notion of a pseudo d algebra as a generalization of the idea of a d algebra the class of pseudo d algebras can be included in the class of coupled d algebras we show that the class of pseudo d algebras behaves well with respect to standard mappings such as direct products and standard mappings such as homomorphisms indeed we are able to prove a fundamental theorem of homomorphisms and we are able to identify kernel objects as d ideals in this theory we also discuss several large classes of examples of pseudo d algebras which occur in different settings and we identify properties sufficient for or equivalent to membership in important subclasses of the class of pseudo d algebras in these settings