wavenumber based filtering of marine point source data in seismic processing plane wave decomposition has played a fundamental role serving as a basis for developing sophisticated processing techniques valid for depth dependent models by comparing analytical expressions for the decomposed wavefields we review several processing algorithms of interest for the geophysicist the algorithms may be applied to marine point source data acquired over a horizontally layered viscoelastic and anisotropic medium the plane wave decomposition is based on the fourier transform integral for line source data and the hankel transform integral for point source data the involved wavenumber integrals of the cosine or bessel function type are difficult to evaluate accurately and efficiently however a number of the processing techniques can easily be run as a filtering operation in the spatial domain without transforming to the wavenumber domain the mathematical expressions for the spatial filters are derived using plane wave analysis with numerical examples we demonstra

fuzzy equivalence relations and their equivalence classes in this paper we investigate various properties of equivalence classes of fuzzy equivalence relations over a complete residuated lattice we give certain characterizations of fuzzy semi partitions and fuzzy partitions over a complete residuated lattice as well as over a linearly ordered complete heyting algebra in the latter case for a fuzzy equivalence relation over a linearly ordered complete heyting algebra we construct an algorithm for calculation of a minimal family of its equivalence classes which generates it most of the presented results are new but some of them are generalizations of known results given in a way which simplifies and clarifies them

building a class of fuzzy equivalence relations in this paper we propose a practical method given a strict triangular norm with a convex additive generator for deriving a fuzzy equivalence relation whose reflexivity condition generalizes ruspini s definition of fuzzy partitions the properties of the relations their comparison their transitivity the construction of fuzzy equivalence relations on cartesian products are presented a large part of the paper is devoted to applications with fuzzy partitions defined on the real line several examples including the fairy tale problem from de cock and kerre on un suitable fuzzy relations to model approximate equality fuzzy sets and systems 133 2 2003 137 153 the comparison of colored objects and comfort situations are proposed

fuzzy congruences and compatible fuzzy partitions the main purpose of this paper is to prove that fuzzy congruence relations and compatible fuzzy partitions on a fuzzy algebra f can be described with the help of crisp congruence relations and compatible partitions on the cuts of f

on the imaging of reflectors in the earth in this paper i present a modification of the beylkin inversion operator this modification accounts for the band limited nature of the data and makes the role of discontinuities in the sound speed more precise the inversion presented here partially dispenses with the small parameter constraint of the born approximation this is shown by applying the proposed inversion operator to upward scattered data represented by the kirchhoff approximation using the angularly dependent geometrical optics reflection coefficient a fully nonlinear estimate of the jump in sound speed may be extracted from the output of this algorithm interpreted in the context of these kirchhoff approximate data for the forward problem the inversion of these data involves integration over the source receiver surface the reflecting surface and frequency the spatial integrals are computed by the method of stationary phase the output is asymptotically a scaled singular function of the reflecting surface the singular function of a

green s function representations for seismic interferometry the term seismic interferometry refers to the principle of generating new seismic responses by crosscorrelating seismic observations at different receiver locations the first version of this principle was derived by claerbout 1968 who showed that the reflection response of a horizontally layered medium can be synthesized from the autocorrelation of its transmission response for an arbitrary 3d inhomogeneous lossless medium it follows from rayleigh s reciprocity theorem and the principle of time reversal invariance that the acoustic green s function between any two points in the medium can be represented by an integral of crosscorrelations of wavefield observations at those two points the integral is along sources on an arbitrarily shaped surface enclosing these points no assumptions are made with respect to the diffusivity of the wavefield the rayleigh betti reciprocity theorem leads to a similar representation of the elastodynamic green s function when a part of the enclosing surface is the earth s free surface the integral needs only to be evaluated over the remaining part of the closed surface in practice not all sources are equally important the main contributions to the reconstructed green s function come from sources at stationary points when the sources emit transient signals a shaping filter can be applied to correct for the differences in source wavelets when the sources are uncorrelated noise sources the representation simplifies to a direct crosscorrelation of wavefield observations at two points similar as in methods that retrieve green s functions from diffuse wavefields in disordered media or in finite media with an irregular bounding surface

connectives for fuzzy partitions abstract the fuzzy intersection and union are defined by means of a certain pair t s of non strict archimedean operators dual with respect to a strict negation having the same additive generator as t we prove that the equivalence between the butnariu s definition and the ruspini s one concerning the fuzzy partitions holds if and only if the connectives for fuzzy operations are t and s

t partitions of the real line generated by idempotent shapes abstract the idea of generating fuzzy numbers as equivalence classes of particular t equivalences on the real line r is fully exploited scales or generators are used to define certain pseudo metrics on r by means of a shape function these pseudo metrics are then transformed into binary fuzzy relations on r shapes leading to t equivalences and hence to a class of fuzzy numbers forming a t partition of r are completely characterized in the case of a continuous generator this characterization problem is shown to be closely related to determining the idempotents w r t the t addition of fuzzy numbers

comparison of fuzzy partitions based on their a cuts abstract this paper proposes a measure of degree of equivalence between two fuzzy partitions a method which allows one to find values of 0 1 at which two or more partitions are considered to be equivalent is introduced partial order of fuzzy partitions with respect to their sharpness is discussed a procedure for constructing a lower bound and an upper bound of a family of fuzzy partitions is suggested this procedure can be used in aggregation of fuzzy partitions

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

approximate solutions of fuzzy relational equations and a characterization of t norms that define metrics for fuzzy sets abstract if one considers the membership degrees of fuzzy sets as truth values of a many valued logic there is a canonical way to define a fuzzified equality for fuzzy sets this generalized equality relation can be used to define a degree to which some value of the unknown variable is a solution of the fuzzy equation as well as a degree of solvability of the whole equation furthermore there is an intimate relationship between such a degree to which some fuzzy set is a solution and the quality to which it is an approximate solution the background is that the negation of the fuzzified equality measures a kind of distinctness of fuzzy sets formally the definition of this fuzzified equality contains as parameter some t norm which acts as truth functions of a conjunction connective for some such t norms this negation of the fuzzified equality really has the properties of a metric in the mathematical sense of the word the paper gives a necessary and sufficient condition which characterizes all those t norms which in that way yield a metric for fuzzy sets

crossline wavefield reconstruction from multicomponent streamer data part 2 joint interpolation and 3d up down separation by generalized matching pursuit computation of the 3d upgoing downgoing separated wavefield at any desired position within a marine streamer spread is enabled by multicomponent streamers that can measure the crossline and vertical components of water particle motion in addition to the pressure we introduce the concept of simultaneous interpolation and deghosting and describe a new technique generalized matching pursuit gmp to achieve this this method is based on the matching pursuit technique and iteratively reconstructs the signal as a combination of optimal basis functions in the gmp method the basis functions describing the unknown 3d upgoing wavefield are filtered by appropriate forward ghost operators before being matched to the multicomponent measurements as a data dependent method gmp can operate on data samples that are highly aliased in the crossline direction without relying on assumptions about seismic events such as linearity the technique is naturally suitable for data with only a small number of samples that may be irregularly spaced we demonstrate the efficacy and robustness of the gmp method on several synthetic data sets of increasing complexity and in the presence of noise

fuzzy equivalence relations abstract in this note we define and discuss properties of fuzzy equivalence relations on a set in particular we show that there exists a correspondence between fuzzy equivalence relations and certain classes of fuzzy subsets we also study the cuts of fuzzy equivalence relations finally certain lattice theoretic properties of fuzzy equivalence relations are discussed

metrics and t equalities abstract the relationship between metrics and t equalities is investigated the latter are a special case of t equivalences a natural generalization of the classical concept of an equivalence relation it is shown that in the construction of metrics from t equalities triangular norms with an additive generator play a key role conversely in the construction of t equalities from metrics this role is played by triangular norms with a continuous additive generator or equivalently by continuous archimedean triangular norms these results are then applied to the biresidual operator e t of a triangular norm t it is shown that e t is a t equality on 0 1 if and only if t is left continuous furthermore it is shown that to any left continuous triangular norm t there correspond two particular t equalities on f x the class of fuzzy sets in a given universe x one of these t equalities is obtained from the biresidual operator e t t by means of a natural extension procedure these t equalities then give rise to interesting metrics on f x

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

true amplitude vector acoustic imaging application of gaussian beams abstractacoustic gaussian beam migration is an attractive imaging method because it is flexible with input geometry efficient and accurate in imaging multipath arrivals however one of the hurdles that this method must overcome in production processing is its extension to use multimeasurement data as recently allowed by novel acquisition technologies this is inevitable when the compensation of the ghost effect is best corrected within a true amplitude imaging process a necessity for amplitude variation with offset work for this purpose i introduced a novel formalism for vector acoustic imaging based on green s function theory which can remove the ghost effect and produce amplitudes on reflectors that are proportional to the reflection coefficients i established a theoretical framework with gaussian beam representations of green s functions including the weighted beam stacking approach that reduced the cost of computation i extended my formulas to use the steep descent i e stationary phase

crossline wavefield reconstruction from multicomponent streamer data part 1 multichannel interpolation by matching pursuit mimap using pressure and its crossline gradient we introduce a technique that uses multicomponent seismic measurements to reconstruct the seismic wavefield at any desired crossline position between towed streamers this method called multichannel interpolation by matching pursuit mimap operates on pressure and crossline particle motion measurements it is based on the matching pursuit technique and iteratively reconstructs the signal as a combination of optimal basis functions being a data dependent technique mimap can interpolate severely aliased data without assumptions about seismic events such as linearity or the model related to the seismic wavefield mimap has the capability to perform well in the presence of irregular sampling and is robust when only a small number of samples are available using synthetic data examples we show that the new method has the potential to interpolate signals that are sampled at realistic crossline streamer spacing and in the presence of noise

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

quasistatic crack growth in nonlinear elasticity in this paper we prove a new existence result for a variational model of crack growth in brittle materials proposed in 19 we consider the case of n dimensional nonlinear elasticity for an arbitrary n 1 with a quasiconvex bulk energy and with prescribed boundary deformations and applied loads both depending on time

fuzzy partitions and fuzzy quotient rings abstract the cosets of a fuzzy ideal m in a ring a form another ring a m called the fuzzy quotient ring of a and m we show that a m is always a fuzzy partition of a and a fuzzy partition of a will be a fuzzy quotient ring with respect to some fuzzy ideal if and only if its associated fuzzy similarity relation is compatible with the ring operations we also discuss relationships between the two operations of the fuzzy quotient ring a m and the extensions over ia of the two operations of a by zadeh s extension principle we study relationships between the level ring structures of a fuzzy quotient ring and the level relations of its associated fuzzy similarity relations we introduce and study a new notion of the factorization of a ring homomorphism through fuzzy quotients finally a relationship between fuzzy prime ideals and fuzzy similarity relation has been established

on the structure of f indistinguishability operators abstract some properties concerning the structure of f indistinguishability operators are analyzed it is shown that any of such operators is generated by a family of fuzzy subsets this result since it gives the way to construct f indistinguishabilities facilitates new applications of fuzzy relations the links between f indistinguishability operators and a kind of generalized metrics in the unit interval which are also explored are used to define the canonical generators of an f indistinguishability operator that is the fuzzy partition associated with the operator

depth migration by the gaussian beam summation method seismic depth migration aims to produce an image of seismic reflection interfaces ray methods are suitable for subsurface target oriented imaging and are less costly compared to two way wave equation based migration but break down in cases when a complex velocity structure gives rise to the appearance of caustics ray methods also have difficulties in correctly handling the different branches of the wavefront that result from wave propagation through a caustic on the other hand migration methods based on the two way wave equation referred to as reverse time migration are known to be capable of dealing with these problems however they are very expensive especially in the 3d case it can be prohibitive if many iterations are needed such as for velocity model building our method relies on the calculation of the green functions for the classical wave equation by per forming a summation of gaussian beams for the direct and back propagated wavefields the subsurface image is obtained by cal culating

fuzzy equivalence relation redefined abstract in this note a weak fuzzy equivalence relation is defined generally superseding most of the established results the concept is discussed in the setting where the min transitivity of a fuzzy relation is replaced by the general t norm transitivity

migration inversion for incident waves synthesized from common shot data gathers wavefield synthesis is a process for producing reflection responses from more general sources or from prescribed incident waves by combining common shot data gathers synthesis can provide surveywide data sets similar in that regard to common offset data gathers but with the added advantage that each synthesized data set is a solution to a single wave equation a common offset data set does not have this last feature thus synthesized data sets can be processed by true amplitude wave equation migration the output is then known to be true amplitude in the same sense as is the output of kirchhoff inversion that is the peak amplitude is proportional to the ray theoretic reflection coefficient at a determinable specular incidence angle multiplied bythe area under the frequency domain source signature and scaled by 1 2 alternatively the kirchhoff inversion of synthesized data has a beylkin determinant that is expressed in terms of the ray theoretic green s function amplitude this is in contrast to 3

similarity relations the calculation of minimal generating families abstract the structure of similarity relations is studied from the point of view of the representation theorem of t indistinguishabilities the generators are characterized the concept of dimension is introduced and algoriths in oder to calculate a minimal generating family in the finite case are developed

quotients with respect to similarity relations abstract the problem of the quotient w r t a similarity relation is closely related to poincare s paradox and central problems in cluster analysis amazingly this problem remains open for more than 30 years as a generalization of strong m categories we introduce the concept of m valued sets which leads to the definition of the category m set in m set we give a complete solution of the quotient problem

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

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

on the redundancy of fuzzy partitions abstract we introduce a concept of redundancy for families of fuzzy subsets based on a left continuous t norm and its corresponding operator and study its basic properties including a characterization of families with redundancy zero in some important special cases examples illustrate that in the case of t norms with zero divisors several definitions for fuzzy partitions from the literature are insensitive to the phenomenon of redundancy finally a geometric interpretation of the redundancy based on the ukasiewicz t norm and an upper bound for its value are given

crack growth in polyconvex materials we discuss a model for crack propagation in an elastic body where the crack path is described a priori in particular we develop in the framework of finite strain elasticity a rate independent model for crack evolution which is based on the griffith fracture criterion due to the nonuniqueness of minimizing deformations the energy release rate is no longer continuous with respect to time and the position of the crack tip thus the model is formulated in terms of the clarke differential of the energy generalizing the classical crack evolution models for elasticity with strictly convex energies we prove the existence of solutions for our model and also the existence of special solutions where only certain extremal points of the clarke differential are allowed

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

non uniqueness of minimizers for strictly polyconvex functionals in this note we solve a problem posed by ball in philos trans r soc lond ser a 306 1496 557 611 1982 about the uniqueness of smooth equilibrium solutions to boundary value problems for strictly polyconvex functionals mathcal f u int omega f nabla u x rm d x quad rm and quad u vert partial omega u 0 where is homeomorphic to a ball

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

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

prestack gaussian beam depth migration kirchhoff migration is the most popular method of three dimensional prestack depth migration because of its flexibility and efficiency its effectiveness can become limited however when complex velocity structure causes multipathing of seismic energy an alternative is gaussian beam migration which is an extension of kirchhoff migration that overcomes many of the problems caused by multipathing unlike first arrival and most energetic arrival methods which retain only one traveltime this alternative method retains most arrivals by the superposition of gaussian beams this paper presents a prestack gaussian beam migration method that operates on common offset gathers the method is efficient because the computation of beam superposition isolates summations that do not depend on the seismic data and evaluates these integrals by considering their saddle points gaussian beam migration of the two dimensional marmousi test data set demonstrates the method s effectiveness for structural imaging in a case w

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

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

a new approach to the similarity relations in the fuzzy set theory abstract the subtle theory of fuzzy sets has long been proposed by zadeh as an extension of the classical theory of hard sets in this paper we show how the properties of reflexivity and transitivity of a fuzzy relation may be reconsidered if they are used in view of another intuitive support and new mathematical developments as a consequence we propose a new modified system of leading definitions for the fuzzy equivalence relationship that could be used profitably to support the concept of fuzzy partition the role of an internal basis is also emphasized corollary properties are derived from this new approach

global existence results for viscoplasticity at finite strain we study a model for rate dependent gradient plasticity at finite strain based on the multiplicative decomposition of the strain tensor and investigate the existence of global in time solutions to the related pde system we reveal its underlying structure as a generalized gradient system where the driving energy functional is highly nonconvex and features the geometric nonlinearities related to finite strain elasticity as well as the multiplicative decomposition of finite strain plasticity moreover the dissipation potential depends on the left invariant plastic rate and thus depends on the plastic state variable the existence theory is developed for a class of abstract nonsmooth and nonconvex gradient systems for which we introduce suitable notions of solutions namely energy dissipation balance and energy dissipation inequality solutions hence we resort to the toolbox of the direct method of the calculus of variations to check that the specific energy and dissipation functionals for our viscoplastic models comply with the conditions of the general theory

degenerate and non degenerate convex decomposition of finite fuzzy partitions ii abstract in bezdek and harris 1979 an algorithm called minimax shortly mm algorithm for the convex decomposition of a fuzzy partition has been proposed in this paper another decomposition algorithm called miniminimax shortly mmm algorithm is considered a comparative study of these algorithms is done from this study we may conclude that 1 i the mm convex decomposition sequence is not lexicographically larger than any other convex decomposition 2 ii the conjecture from bezdek and harris 1979 concerning the length of the mm decomposition fails some properties of the spaces of fuzzy partitions are also given in part ii a theorem which states a necessary and sufficient condition for the existence of non degenerate convex decompositions is proved an algorithm for non degenerate convex decomposition of a fuzzy partition inspired by the constructive proof of this theorem is proposed this algorithm builds a positive path through the matrix representing a fuzzy partition the convergence of the algorithm and the monotony of the coefficients sequence in the convex decomposition are proved other two results the limited cardinality and the heredity property concerning this algorithm are also given

crack initiation in brittle materials in this paper we study the crack initiation in a hyper elastic body governed by a griffith type energy we prove that during a load process through a time dependent boundary datum of the type t tg x and in the absence of strong singularities e g this is the case of homogeneous isotropic materials the crack initiation is brutal that is a big crack appears after a positive time ti 0 conversely in the presence of a point x of strong singularity a crack will depart from x at the initial time of loading and with zero velocity we prove these facts for admissible cracks belonging to the large class of closed one dimensional sets with a finite number of connected components the main tool we employ to address the problem is a local minimality result for the functional epsilon nu gamma int omega f x nabla v rm d x k mathcal h 1 gamma where omega subseteq mathbb r 2 k 0 and f is a suitable caratheodory function we prove that if the uncracked configuration u of relative to a boundary displacement has at most uniformly weak singularities then configurations u with mathcal h 1 gamma small enough are such that epsilon u emptyset epsilon u gamma gamma

complement of fuzzy k partitions abstract we define the unary operation of complementation on the set of all fuzzy k partitions of a finite set of objects we show that for k 2 the complement of fuzzy k partition results in zadeh s complementation of fuzzy sets some further properties of the complement of fuzzy k partitions are investigated

inverse extrapolation of primary seismic waves forward wave field extrapolation operators simulate propagation effects from one depth level to another inverse wave field extrapolation operators eliminate those propagation effects since forward wave field extrapolation can be described in terms of spatial convolution inverse wave field extrapolation can be described in terms of spatial deconvolution a simple approximation to a stable spatially band limited deconvolution operator is obtained by taking the complex conjugate of the convolution operator a one way version of the kirchhoff integral that contains the conjugate complex green s function is derived unlike the situation with respect to the forward problem the modification of the closed surface integral into an open boundary integral involves an approximation that is identical to the approximation in the conjugate complex deconvolution approach this approximation neglects the evanescent field and causes a second order amplitude error for a plane acquisition surface the one way kirchhoff

equality relations as a basis for fuzzy control abstract the aim of this paper is to introduce a fuzzy control model with well founded semantics in order to explain the concepts applied in fuzzy control assuming that the domains of the input and output variables for the process are endowed with equality relations that reflect the indistinguishability of values lying closely together the use of triangular and trapezoidal membership functions can be justified and max inference where is a t norm turns out to be a consequence of our model distinguishing between a functional and a relational view of the control rules it is possible to explain when defuzzification strategies like mom or coa are appropriate or lead to undesired results

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

maximal smoothness of solutions to certain euler lagrange equations from nonlinear elasticity we investigate the maximal smoothness of stationary states for the multiple integral such variational problems are motivated by the study of nonlinear elasticity assuming certain structure conditions for and given a stationary state we derive an a priori l p estimate for for any p and where as a consequence we show that a c 1 stationary state necessarily satisfies det and is of class c 2 in nevertheless singular stationary states do exist we construct a nonsmooth c 1 solution for a particular in two dimensions such that det in and det vanishes at precisely one point in

source receiver reverse time imaging of dual source vector acoustic seismic data abstractnovel technologies in seismic data acquisition allow for recording full vector acoustic va data pointwise recordings of pressure and its multicomponent gradient excited by pressure only as well as dipole gradient sources building on recent connections between imaging and seismic interferometry we present a wave equation based nonlinear reverse time imaging approach that takes full advantage of dual source multicomponent data the method s formulation relies on source receiver scattering reciprocity thus making proper use of va fields in the wavefield extrapolation and imaging condition steps in a self consistent manner the va imaging method is capable of simultaneously focusing energy from all in and outgoing waves the receiver side up and downgoing receiver ghosts fields are handled by the va receiver extrapolation whereas source side in and outgoing source ghosts arrivals are accounted for when combining dual source data at the imaging condition additionally va imaging handl

equivalent finite fuzzy sets and stirling numbers in this paper we study the number of k level equivalence classes of fuzzy subsets of a finite set of n elements under a natural equivalence this number is related to stirling numbers viewing fuzzy subsets as functions from a set into the unit interval we also associate a kernel partition with every equivalence class of fuzzy subsets after some elementary properties of the equivalence we provide a recurrence relation and a generating function concerning the number of k level fuzzy subsets using stirling numbers