particle lattices and fuzzy particle lattices abstract this paper introduces several new concepts a particle in a lattice which is a generalization of an atom in a distributive lattice by weakening the minimality property of atom a particle lattice which is a generalization of distributive complete atomic lattices by replacing atoms by particles a fuzzy particle lattice a particle lattice with an order reversing involution a particulate boolean algebra a fuzzy particle lattice whose involution is a complementation operator and other related concepts points fuzzy points boundary points and boundary quasi points the paper establishes representation theorems for fomse of these concepts establishes connections between others and some well established lattice theoretic concepts and provides some elementary properties for some of the new concepts

q convergence of nets and weak separation axioms in fuzzy lattices abstract in this paper we introduce the concept of q remote neighborhood and establish the q convergence theory of nets in fuzzy lattices next we introduce the st i i 1 0 1 2 separation axioms in fuzzy lattices and discuss some of their characterizations finally we introduce and study s irresolute order homomorphism and s homeomorphism discuss the application of q convergence of nets and show that the st i i 1 0 1 2 separability is preserved under s homeomorphism

on two weakers forms of continuous order homomorphisms the aim of this paper is to find further connections between two different current researches on fuzzy topology the one based on the concepts of neighborhood filter and zadeh function and the other based on the concepts of remote neighborhood ideal and order homomorphism on fuzzy lattices to approach this the paper studies two weaker forms of continuous order homomorphisms the semi continuous order homomorphism and the weakly semi continuous order homomorphism between l fuzzy topological spaces these order homomorphisms can be characterized in each of the notions remote neighborhood neighborhood ideal filter and molecular net in addition some relations connecting weakly semi continuous order homomorphisms semi hausdorff l fts and urysohn l fts have also been established

lattice valued mappings completely distributive law and induced spaces the lattice valued mappings whose value domain is a completely distributive lattice are investigated we describe the analytic condition the semicontinuity of lattice valued mappings on a topological space x via the algebraic property the completely distributive law and conversely we use the semicontinuity to depict the completely distributive law the family of all the upper semicontinuous lattice valued mappings on a topological space x or a part of them constitutes a l fuzzy topological space on x a series of global properties of these lattice valued mappings is presented in particular we investigate the impact of n compactness on the semicontinuity using these results we solve some problems on fuzzy stone cech compactification

multiple burst correcting array codes two families of binary linear multiple burst correcting array codes are presented the codes consist of all possible n sub 1 n sub 2 arrays over gf 2 where the columns have even parity and the rows belong to any given code of length n sub 2 and minimum distance 2t it is shown that if the bits are read out diagonally instead of horizontally each diagonal followed by the preceding one viewed cyclically then the code can correct up to t bursts of length or tn sub 1 1 if each diagonal is followed by the next one the code can correct up to t bursts of length or 2t n 2 1 for t 1 some of these results are already known decoding algorithms are presented and the case t 1 is discussed in more detail

interferometry by deconvolution part 1 theory for acoustic waves and numerical examples interferometry allows for synthesis of data recorded at any two receivers into waves that propagate between these receivers as if one of them behaves as a source this is accomplished typically by crosscorrelations based on perturbation theory and representation theorems we show that interferometry also can be done by deconvolutions for arbitrary media and multidimensional experiments this is important for interferometry applications in which 1 excitation is a complicated source time function and or 2 when wavefield separation methods are used along with interferometry to retrieve specific arrivals unlike using crosscorrelations this method yields only causal scattered waves that propagate between the receivers we offer a physical interpretation of deconvolution interferometry based on scattering theory here we show that deconvolution interferometry in acoustic media imposes an extra boundary condition which we refer to as the free point or clamped point boundary condition depending on the measured field quantity this boundary condition generates so called free point scattering interactions which are described in detail the extra boundary condition and its associated artifacts can be circumvented by separating the reference waves from scattered wavefields prior to interferometry three wavefield separation methods that can be used in interferometry are direct wave interferometry dual field interferometry and shot domain separation each has different objectives and requirements

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

structures of fuzzy order homomorphisms abstract first the concept of fuzzy order homomorphism is introduced which is a variety of the generalized order homomorphism given in 7 then we show that the fuzzy order homomorphism possesses a nicely local structure furthermore we establish a necessary and sufficient condition under which the fuzz function is a zadeh type function the present condition is simpler than that given in 6 finally for each fuzzy order homomorphism f we yield some characterizations about f 1 1 f which sharpen the corresponding theorem in 6

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

the uniform boundedness principle in l topological vector spaces in this paper we introduce and study the equicontinuity of a family of l fuzzy linear order homomorphisms we prove a generalization of the uniform boundedness principle in the classical topological vector spaces to l topological vector spaces

retrieving the exact green s function by wavefield crosscorrelation recent development on the green s function retrieval by wavefield crosscorrelation has substantially advanced the physical research in a multidisciplinary and unprecedented fashion however the underlying assumption of the theory that the sources are in the far field limits the technology to extracting only the high frequency part of the green s function in an open system this critical approximation can be eliminated using the exact boundary integral equation method a scheme involving the crosscorrelation kernel is proposed to recover the exact green s function including all frequency content symmetric difference kernels are analytically constructed for sources on a plane or on a circle and can be reduced to the known dirac delta kernel under the far field approximation

equivalence of the virtual source method and wave field deconvolution in seismic interferometry seismic interferometry and the virtual source method are related approaches for extracting the green s function that accounts for wave propagation between receivers by making suitable combinations of the waves recorded at these two receivers these waves can either be excited by active controlled sources or by natural incoherent sources we compare this technique with the deconvolution of the wave field recorded at different receivers we show that the deconvolved wave field is a solution of the same wave equation as that for the physical system but that the deconvolved wave forms may satisfy different boundary conditions than those of the original system we apply this deconvolution approach to the wave motion recorded at various levels in a building after an earthquake and show how to extract the building response for different boundary conditions extracting the response of the system with different boundary conditions can be used to enhance or suppress the normal mode response in seismic exploration this principle can be used for the suppression of surface related multiples

connectedness on fuzzy topological spaces in this paper properties of connectedness on completely distributive topological lattices definition 2 1 and fuzzy topological spaces 1 are studied some equivalent propositions on the concepts of functions in completely distributive lattices and connected elements in completely distributive lattices and fuzzy topological spaces are proved specially connectedness is characterized both by the r neighborhoods of molecules and c separations for any family li i i i of fuzzy topological spaces the product 1 is connected iff every li i is connected a continuous function maps connected elements onto connected ones the connectedness on the fuzzy real line r l is discussed in the last section as an important example the intervals defined here definition 4 1 must be connected in the fuzzy real line r l and so the intervals defined by rodabaugh in 13 are too any connected element is connected in the fuzzy real line r l if and only if r l r

fuzzy boundary and characteristic properties of order homomorphisms abstract in this paper first the concept of the boundary of an element in a fuzzy topological space is defined and studied then the concepts of boundary preserving interior preserving and closure preserving order homomorphisms etc are introduced and discussed finally the relations among these are obtained

h λ completely hausdorff axiom on l topological spaces this paper defines the new concept of completely hausdorff axiom of an l topological space by means of l continuous mappings from an l topological space to the refined hutton s unit l interval by wang some characterizations of the completely hausdorff axiom defined in this paper are given and many nice properties of this kind of completely hausdorff axiom are proved for example it is hereditary and product invariant the refined hutton s unit l interval satisfy this kind of completely hausdorff axiom and when an l topological space satisfy this kind of completely hausdorff axiom every f convergent ideal does not have f limit points with different supports etc the relation between the completely hausdorff axiom defined in the paper and other separation axioms is discussed also

pointwise uniformities in fuzzy set theory a theory of pointwise quasi uniformities is based on fuzzy sets many theorems corresponding to the usual theorems in general topology are generalized on fuzzy sets in particular each fuzzy topological space is pointwise quasi uniformizable a pointwise uniformity is characterized in terms of a kind of complete regularity a uniformity on the fuzzy unit interval is constructed

pointwise topology on completely distributive lattices abstract in this paper the author presents a kind of pointwise topology theory on complete completely distributive lattices which is both a generalization of fuzzy topology and a further generalization of the author s papers 14 15

another look at order homomorphisms on fuzzes abstract order homomorphisms between fuzzy spaces are considered a general result is established which leads to an extension and simplification of earlier work by g j wang fuzzy sets and systems 12 1984 281 288

srn continuous order homomorphism and its characterizations in this paper the concept of semi regular continuity of order homomorphism is introduced and characterized by molecular nets and ideals in l topological spaces furthermore several properties of this continuity including hereditability and multiplicable properties are studied

t convergence of nets of l fuzzy sets and its applications abstract in this paper the theory of convergence on nets of l fuzzy sets which is a generalization of the convergence theory of fuzzy nets introduced by sarma and ajmal is established some important and interesting results in l fuzzy topology are obtained by means of the theory

nonlinear extended images via image domain interferometry wave equation finite frequency imaging and inversion still face many challenges in addressing the inversion of highly complexvelocitymodelsaswellasindealingwithnonlinearimaging e g migration of multiples amplitude preserving migration extended images eis are particularly important for designing image domain objective functions aimed at addressing standing issues in seismic imaging such as two way migration velocity inversion or imaging inversion using multiples general oneand two way representations for scattered wavefields can describe and analyze eis obtained in wave equation imaging we have developed a formulation that explicitly connects the wavefield correlations done in seismic imaging with the theory and practiceofseismicinterferometry inlightofthisconnection we define eis as locally scattered fields reconstructed by model dependent image domain interferometry because they incorporate the same one and two way scattering representations used for seismic interferometry the reciprocity based eis can in principle account for all possible nonlinear effects in the imaging process i e migrationofmultiplesandamplitudecorrections in this case the practice of two way imaging departs considerably from the one way approach we have studied the differences betweentheseapproachesinthecontextofnonlinearimaging analyzingthedifferencesinthewavefieldextrapolationstepsaswell as in imposing the extended imaging conditions when invoking single scatteringeffectsandignoringamplitudeeffectsingenerating eis the one and two way approaches become essentially the same as those used in today s migration practice with the straightforwardadditionofspaceandtimelagsinthecorrelationbased imaging condition our formal description of the eis and theinsightthattheyarescatteredfieldsintheimagedomainmay beusefulinfurtherdevelopmentofimagingandinversionmethodsinthecontextoflinear migration basedvelocityinversionor inmoresophisticatedimage domainnonlinearinversescattering approaches

nonreciprocal green s function retrieval by cross correlation the cross correlation of two recordings of a diffuse acoustic wave field at different receivers yields the green s function between these receivers in nearly all cases considered so far the wave equation obeys time reversal invariance and the green s function obeys source receiver reciprocity here the theory is extended for nonreciprocal green s function retrieval in a moving medium it appears that the cross correlation result is asymmetric in time the causal part represents the green s function from one receiver to the other whereas the acausal part represents the time reversed version of the green s function along the reverse path

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

separation axioms in l fuzzy topological spaces i t0 and t1 abstract using the notion of remote neighborhood we define the separation axioms t 0 and t 1 in l fuzzy topological spaces l fts the relations between our definitions hutton and reilly s and wang s are discussed and the separations of hutton s fuzzy unit interval and gantner s fuzzy real line are examined characterizations of these l fts are proved and a series of properties of them are investigated moreover some results on minimal t 0 l fts and t 1 l fts are established which will be used immediately in our subsequent discussion

sr convergence theory in fuzzy lattices abstract convergence theory and order homomorphisms theory are two important objects of study in fuzzy lattices in this paper sr convergence theory for molecular nets and ideals is established and the concepts of r irresolute sr continuous and almost irresolute order homomorphisms in topological molecular lattices are introduced and systematically characterized some mutual relations between these concepts and others are obtained

imaging of multiple reflections current multiple removal algorithms in seismic processing use either differential moveout or predictability if the differential moveout between primaries and multiples is small prediction is the only option available in the last decade multidimensional prediction error filtering by weighted convolution such as surface related multiple elimination srme have proved to be very successful in practice so far multiples have been considered as noise and have been discarded after the removal process in this paper we argue that multiple reflections contain a wealth of information that can be used in seismic processing to improve the resolution of reservoir images beyond current capability in the near future one may expect that the so called weighted crosscorrelation wcc concept may offer an attractive alternative in approaching the multiple problem wcc creates an option to avoid the adaptive subtraction process as applied in prediction error algorithms moreover it allows the transformation of multiples into primaries the latter means that seismic imaging with primaries and multiples nonlinear process can be implemented by a sequence of linear processes including the transformation of multiples into primaries and the imaging of primaries

generalized zadeh function in 1986 he ming introduced the concept of biinduced mapping between two arbitrary l fuzzes in this paper we consider the order homomorphisms between two lattices on the basis of biinduced mapping and give the concept of generalized zadeh function the generalized zadeh function has almost all the fine properties of the zadeh function furthermore it has overcome the limitations of the zadeh function in applications and the shortcomings of order homomorphisms

a simple and exact acoustic wavefield modeling code for data processing imaging and interferometry applications improvements in industrial seismic seismological acoustic or interferometric theory and applications often result in quite subtle changes in sound quality seismic images or information which are nevertheless crucial for improved interpretation or experience when evaluating new theories and algorithms using synthetic data an important aspect of related researchisthereforethat numerical errors due towavefield modeling are reduced to a minimum we present a new matlab code based on the foldy method that models theoretically exact direct and scattered parts of a wavefield its main advantage lies in the fact that while all multiple scattering interactions aretakenintoaccount unlikefinite difference or finite element methods numerical dispersion errors are avoided the method is therefore ideal for testing new theory in industrial seismics seismology acoustics and in wavefield interferometry in particular because the latter is particularly sensitive to the dynamics of scattering interactions we present the theory behind the foldy acoustic modeling method and provide examples of its implementation we also benchmark the code against a good finite difference code because our foldy code was written and optimized to test new theory in seismic interferometry examples of its application to seismicinterferometryarealsopresented showing its validity and importance when exact modeling results are needed

fuzzy sets and de morgan algebras abstract this paper examines some relations between on the one hand the concept of fuzzy sets and fuzzy topologies and on the other hand the concept of complete de morgan algebras and the topologies associated herewith

representation theorems and green s function retrieval for scattering in acoustic media reciprocity theorems for perturbed acoustic media are provided in the form of convolution and correlation type theorems these reciprocity relations are particularly useful in the general treatment of both forward and inverse scattering problems using green s functions to describe perturbed and unperturbed waves in two distinct wave states representation theorems for scattered waves are derived from the reciprocity relations while the convolution type theorems can be manipulated to obtain scattering integrals that are analogous to the lippmann schwinger equation the correlation type theorems can be used to retrieve the scattering response of the medium by cross correlations unlike previous formulations of green s function retrieval the extraction of scattered wave responses by cross correlations does not require energy equipartitioning allowing for uneven energy radiation brings experimental advantages to the retrieval of fields scattered by remote lossless and or attenuative scatterers these concepts are illustrated with a number of examples including analytic solutions to a one dimensional scattering problem and a numerical example in the context of seismic waves recorded on the ocean bottom

minimal hausdorff l fuzzy topological spaces abstract in this paper the author introduced the concept of minimal hausdorff l fuzzy topological spaces investigated basic properties of it by means of fuzzy ideals and fuzzy nets and compared it with the concepts of compactness and h closedness

theory of topological molecular lattices abstract the aim of this paper is to bring the theories of point set topology fuzzy topology and l fuzzy topology into a unified theory the theory of topological molecular lattices for this purpose we first introduce the concepts of molecule for completely distributive complete lattice molecular lattice for short the cotopology on it and the concept of remote neighborhoods of molecules therefrom we develop the theory of pointwise topology on molecular lattices

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

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

a sufficient and necessary condition for fuzzy functions to be zadeh type functions abstract in this paper for an arbitrary fuzz l a sufficient and necessary condition for fuzz functions to be zadeh type functions is given

emergence rate of the time domain green s function from the ambient noise cross correlation function it has been demonstrated experimentally and theoretically that an estimate of the green s function between two receivers can be obtained from the time derivative of the long time average cross correlation of ambient noise between these two receivers the emergence rate of these deterministic coherent arrival times of the cross correlation function i e the green s function estimate from the recordings of an isotropic distribution of random noise sources is derived by evaluating the amplitude of the variance of the cross correlation function the knowledge of the emergence rate of ncf is essential for practical applications to the first order the variance is equal to the ratio of the product of the recorded energy by both receivers and the time bandwidth product of the recordings the variance of the time derivative of the correlation function is shown to have a similar dependency these simple analytic formulas show a good agreement with the variance determined experimentally for the correlation of ocean ambient noise recorded in shallow water at a depth of 21 m in the frequency band 300 530 hz for receiver separation up to 28 m and averaging time from 1 to 33 min

some results on i l and r l abstract this paper gives a partial solution to problem 1 in wang and lu 1992 and a suggestion about how to construct a better fuzzy unit interval it defines a new kind of fuzzy real line and investigates a series of its properties

generalization of inductive topologies to l topological vector spaces in this paper using the theory of l topological vector spaces we study some properties of inductive topologies determined by the family of l fuzzy linear order homomorphisms defined by fang jin xuan j fuzzy math 4 1 1996 93 and give a characterization of inductive topologies determined by a single l fuzzy linear order homomorphism as an example we point out the quotient l topology of l tvs is an inductive topology

two dimensional half rate codes using two dimensional finite field filter banks this correspondence introduces two dimensional 2 d filter bank codes tdfbcs the synthesis bank of a two channel two dimensional filter bank over a finite field is used to design the 2 d code and the corresponding analysis bank is used to generate the syndrome of the code first we study the encoder of half rate tdfbcs and show that these linear codes are lattice cyclic it is proven that any 2 d lattice cyclic code can also be generated by a 2 d filter bank second we introduce a methodology to design tdfbcs over binary erasure channels these codes have a simple and efficient maximum likelihood ml decoding algorithm for burst erasures we show that half rate tdfbcs of dimensions n1 x n2 can recover burst erasures of size up to n1 x n2 2 and n1 2 x n2 using the proposed decoding technique finally we present examples of tdfbcs that satisfy the reiger bound with equality i e they are capable of correcting any burst of size n1 x n2 2

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

q convergence of ideals in fuzzy lattices and its applications abstract in this paper we study q convergence and q convergence of ideals in fuzzy lattices by the concept of q remote neighborhood then we introduce the strong s irresoluteness s irresoluteness and s strong semicontinuity on fuzzy lattices we also study some properties of the notions above and strong semicontinuity 2 3 and s irresoluteness 3 4 with the aid of the q convergence q convergence of ideals

acoustic daylight imaging via spectral factorization helioseismology and reservoir monitoring the acoustic time history of the sun s surface is a stochastic t x y cube of information helioseismologists cross correlate these noise traces to produce impulse response seismograms providing the proof of concept for a long standing geophysical conjecture we pack the x y mesh of time series into a single super long one dimensional time series we apply kolmogoroff spectral factorization to the super trace unpack and findthe multidimensional acoustic impulse response of the sun state of the art seismic exploration recording equipment offers tens of thousands of channels and permanent recording installations are becoming economically realistic helioseismology therefore provides a conceptual prototype for using natural noises for continuous reservoir monitoring

h l completely regular l fuzzy sets and their applications first we introduce the concept of h completely regular l fuzzy set in l fts and give some characterizations of it then we propose the concept of h complete regularization of l fts and investigate some properties of this complete regularization we also establish the relations between h completely regular l fts completely regular l fts and weakly induced l fts based on these results we obtain several characterizations of h completely regular l fts

antialiasing condition and filter for the reciprocity equation of correlation type an antialiasing formula has been derived for interferometric redatuming of seismic data more generally this formula is valid for numerical implementation of the reciprocity equation of correlation type which is used for redatuming extrapolation interpolation and migration the antialiasing condition can be surprisingly more tolerant of a coarser trace sampling compared to the standard antialiasing condition numerical results with synthetic vertical seismic profile vsp data show that interferometry artifacts are effectively reduced when the antialiasing condition is used as a constraint with interferometric redatuming

representation theorem of fuzzy lattices and its applications in fuzzy topology abstract a fuzzy lattice is expressed as a direct product of all its minimal fuzzy sublattices and a pointwise characterization of fuzzy complete regularity is proved

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

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

a class of array codes correcting multiple column erasures a family of binary array codes of size p 1 spl times n with p a prime correcting multiple column erasures is proposed the codes coincide with a subclass of shortened reed solomon codes and achieve the maximum possible correcting capability complexity of encoding and decoding is proportional to rnp where r is the number of correctable erasures i e is simpler than the forney decoding algorithm the length n of the codes is at most 2p 1 that is twice as big as the length of the blaum roth codes having comparable decoding complexity

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

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

source receiver wave field interferometry correlation or convolution of recordings of diffuse fields at a pair of locations have been shown to result in estimates of the green s function between the two locations variously referred to as wave field or seismic interferometry in different fields of research green s functions can thus be constructed between either pairs of receivers or pairs of energy sources proofs of these results rely on representation theorems we show how to derive three acoustic and elastic representation theorems that unify existing correlational and convolutional approaches we thus derive three forms of interferometry that provide green s functions on source to receiver paths using only energy that has propagated from surrounding sources or to surrounding receivers the three forms correspond to three possible canonical geometries we thus allow interferometric theory and methods to be applied to commonly used source receiver configurations

the characterizations of semi continuous and irresolute order homomorphisms on fuzzes abstract the concepts of semi continuous and irresolute order homomorphisms on fuzzes were introduced in j jianghan petroleum institute 2 1988 96 97 in this paper we first present the theory with respect to s convergence s convergence of molecular nets and ideals on fuzzes secondly we introduce the notions of strong semicontinuity strong irresoluteness and s irresoluteness on fuzzes then we systematically study the characterizations of these notions with the aid of the s convergence s convergence of molecular nets and ideals and give the relations of them

order homomorphisms on fuzzes abstract in this paper basic properties of the concept of order homomorphisms on fuzzes 4 which is a proper generalization of the concept of fuzz functions 1 are studied conditions for fuzz functions to be functions of zadeh s type are obtained moreover the concepts of continuous closed and open order homomorphisms are introduced and studied

codes correcting phased burst erasures we introduce a family of binary array codes of size t spl times n correcting multiple phased burst erasures of size t the codes achieve maximal correcting capability i e being considered as codes over gf 2 sup t they are mds the length of the codes is n spl sigma sub l 1 sup l sub l sup t where l is a constant or is slowly growing in t the complexity of encoding and decoding is proportional to rnml where r is the number of correctable erasures and m is the smallest number such that 2 sup t 1 modulo m this compares favorably with the complexity of decoding codes obtained from the shortened reed solomon codes having the same parameters

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