interpretation of image flow a spatio temporal approach a general formulation is described for the interpretation of image flow in the framework of this formulation computational methods have been derived for image flow interpretation for many important cases including simple cases of nonrigid and nonuniform motions it is possible to derive computational methods for other situations not considered explicitly the formulation is general in that it is applicable to any situation provided that the scene geometry the scene transformation and the image flow are all smooth or analytic

on some logical points of fuzzy conditional decision making abstract the paper deals with some properties of pl probability logic and fl fuzzy logic fl is derived from pl by means of choosing the connectives such that the logical system created is truth functional from many alternatives the usual connectives of fl emerged as is known pl is not truth functional conditionality is considered by means of implication and using a fuzzy counterpart of conditional probability in that way some basic tools are created for bayesian like decision making in a fuzzy environment the analysis shows that the implication and the conditional probability like counterpart are not necessarily the same thing a labelling system is introduced in fuzzy valued fl such that a finite set of non negative integers forms the set of truth values which actually are fuzzy sets of the unit interval 0 1 in this way new quantitative expressions are obtained for each qualitative linguistic variable in doing so the system preserves fuzziness some general rules of inference are also considered

interpretation of af c algebras in łukasiewicz sentential calculus on donne un critere pour la non simplicite d une af c algebre 7b u en termes de proprietes de la theorie de la recursivite du groupe dimension k 0 7b u

on fuzzy implication in de morgan algebras the behaviour of some fuzzy implications with respect to measure of fuzziness and their deductive power is studied in a lattice framework the point of view of bandler and kohout 1 and willmott 7 is followed

fuzzy sets and toposes towards higher order fuzzy logic abstract a series of paper which have appeared in fuzzy sets and systems over the last three years are reviewed in an expository manner the claims and counter claims made in them are dealt with some of the problems of representing knowledge in the first order predicate calculus are discussed briefly and a case is made for exposing the relationship between topoi and fuzzy mathematics with a view to making a true fuzzy logic both consistent complete and respectable the paper is an attempt to present the controversy to the non specialist and especially to practitioners in expert systems and artificial intelligence

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

applications of product space algebra of conditional events and one point random set representations of fuzzy sets to the development of conditional fuzzy sets abstract the field of conditional event algebra is a newly emerging approach to establishing algebraic bases for probabilistic conditioning this has been further enhanced more recently by the development of a conditional event algebra based upon a countable product space construction this paper considers a new rigorous approach to the development of conditional fuzzy sets based upon the further extension of conditional event algebra to a fuzzy setting using the previously established one point random set coverage probability representation of fuzzy sets

nonlinear galerkin methods based on the concept of determining modes for the magnetohydrodynamic equations a new nonlinear galerkin method based on an approximate inertial manifold and on a finite number of determining modes has been implemented for the three dimensional magnetohydrodynamic equations it is found that these active modes which regulate the dynamics of the system are not necessarily those with the smallest wavenumbers but those containing most of the time averaged enstrophy it is demonstrated that owing to the reduced number of equations the modified nonlinear galerkin method which includes only these determining modes in a truncation improves the computational efficiency in comparison with the traditional galerkin method

evaluation of combinations of conditioned information a history this paper considers the history of the problem of combining probabilistic related information in implicative or conditional form at present no satisfactory technique exists for handling the probability evaluation p a b c d or other compounds of conditional expressions where a b c d are all ordinary statements or events lying in some common boolean algebra and p is a probability measure compatible with the component evaluationsp a b p a b and p c d p c d where p a b p a b p b p c d p c d p d the ordinary conditional probability evaluations assuming in general p b p d 0 surprisingly relatively few individuals have ever considered this problem formally and the proposed solutions have been either mathematically unsatisfactory incomplete or ad hoc

fuzzy logic of lukasiewicz logic a clarification pavelka 10 had shown in 1979 that the only natural way of formalizing fuzzy logic for truth value in the unit interval 0 1 is by using lukasiewicz s implication operator a b min 1 1 a b or some isomorphic form of it a considerable number of other papers around the same time had attempted to formulate alternative definitions for a b by giving intuitive justifications for them there continues to be some confusion however even today about the right notion of fuzzy logic much of this has its origin in the use of improper and or and the not operations and a misunderstanding of some of the key differences between proofs or inferencing in fuzzy logic and those in lukasiewicz s logic we point out the need for defining the strong conjunction operator in connection with fuzzy modus ponens rule and why we do not need the fuzzy syllogism rule we also point out the shortcomings of many of the alternative definitions of a b which indicate further support for pavelka s result we hope that these discussions help to clarify the misconceptions about fuzzy logic

an algebraic synthesis of the foundations of logic and probability abstract this paper develops a unified algebraic theory of logic and probability based upon an initial boolean algebra logic of propositions together with its extensionally associated boolean algebra of models probabilistic events that satisfy the axioms of the boolean algebra of propositions a proposition may be true in some models and false in others but each model must unambiguously assign true or false to each proposition thus the examples models in which different propositions are true may be all some or none according as those propositions are necessary possible or impossible a probability measure p on the collection of models events induces a probability measure on the propositions the crucial if then relation is defined so that p if p then q equals p q p the conditional probability of q given p this conditional probability is less than the probability of the material conditional q or not p unless p p 1 or p q p 1 at these two extremes the two propositional constructions coalesce thus an additional relation q p is needed in logic and probability namely q given p this is accomplished in the algebraic logic generated by the various sum ideals entailments of various propositions that may serve as the explicit condition of a conditional proposition q p the corresponding relation b a in the model realm defines the notion of a conditional event in this context of algebraic logic and model theory a new algebraic structure called the conditional closure l l of a boolean logic l is defined consisting of the ordered pairs q p of all propositions of l excluding those for which the condition p is equivalent to 0 that is impossible the conditional closure is not altogether a boolean algebra not altogether distributive for instance but it has many subalgebras that are boolean including l itself this paper contains partially assumed formulas for operating on the conditional propositions q p using and or not and of course if then this allows complex conditional expressions to be reduced to simple conditionals of boolean propositions which have a conditional probability an example calculation from the dice table illustrates how to simplify a compound conditional proposition and compute its probability the structure of l l is explored including the finite conditional closures of finite boolean logics formulas are given for the probabilistic difference between familiar conditional propositions that are equivalent in the two valued logic where all propositions are either necessary or impossible but not equivalent in the conditional closure of a logic with possible propositions the probability that a proposition p is a theorem true in all models is contrasted from the probability of the models in which the proposition is true the latter being p p while the former is p p 1 this leads to a natural connection with fuzzy set theory since p 1 can also be expressed as p 1 where 1 is the sum ideal filter of necessary propositions given the boolean axioms this paper ends with a representation of each conditional proposition q p as the p measurable two valued truth function of q restricted to those models that satisfy p

formal concept analysis and lattice valued chu systems this paper links formal concept analysis fca both to order theoretic developments in the theory of galois connections and to chu spaces or systems viewed as a common rubric for both topological systems and systems arising from predicate transformers in programming semantics 13 these links are constructed for each of traditional fca and l fca where l is a commutative residuated semiquantale surprising and important consequences include relationships between formal l contexts and l topological systems within the category of l chu systems relationships justifying the categorical study of formal l contexts and linking such study to l chu systems applications and potential applications are primary motivations including several example classes of formal l contexts induced from data mining notions throughout categorical frameworks are given for fca and lattice valued fca in which morphisms preserve the birkhoff operators on which all the structures of fca and lattice valued fca rest and further the results of this paper show that under very general conditions these categorical frameworks are both sufficient and necessary for the interchange or preservation of l concepts and l protoconcepts structures central to fca and lattice valued fca

after the fuzzy wave reached europe abstract we briefly describe the situation of the fuzzy community in europe before the fuzzy wave we show that there was a lot of expertise in almost all subareas of fuzzy logic then we describe some aspects of the fuzzy wave itself in particular fuzzy control and first generation fuzzy systems that played a role during the time the fuzzy wave reached europe we also make some remarks about european companies and their activities during this time then we discuss the situation after the fuzzy wave some aspects of second generation fuzzy systems the combination of fuzzy logic with other theories and techniques and some new application areas finally we will discuss the chances of fuzzy systems in europe in the future

on standard models of fuzzy modal logics abstract we improve the valuations of modal sentences of schotch 7 and build a more reasonable kind of standard models so some important results about d t b 4 5 are generalized into fuzzy situation

motion field and optical flow qualitative properties it is shown that the motion field the 2 d vector field which is the perspective projection on the image plane of the 3 d velocity field of a moving scene and the optical flow defined as the estimate of the motion field which can be derived from the first order variation of the image brightness pattern are in general different unless special conditions are satisfied therefore dense optical flow is often ill suited for computing structure from motion and for reconstructing the 3 d velocity field by algorithms which require a locally accurate estimate of the motion field a different use of the optical flow is suggested it is shown that the smoothed optical flow and the motion field can be interpreted as vector fields tangent to flows of planar dynamical systems stable qualitative properties of the motion field which give useful informations about the 3 d velocity field and the 3 d structure of the scene usually can be obtained from the optical flow the idea is supported by results from the theory of structural stability of dynamical systems

a resolution like strategy based on a lattice valued logic as the use of nonclassical logics becomes increasingly important in computer science artificial intelligence and logic programming the development of efficient automated theorem proving based on nonclassical logic is currently an active area of research this paper aims at the resolution principle for the pavelka type fuzzy logic 1979 pavelka showed that the only natural way of formalizing fuzzy logic for truth values in the unit interval 0 1 is by using the lukasiewicz s implication operator a spl rarr b min 1 1 a b or some isomorphic forms of it hence we first focus on the resolution principle for the lukasiewicz logic l sub spl aleph with 0 1 as the truth valued set some limitations of classical resolution and resolution procedures for fuzzy logic with kleene implication are analyzed then some preliminary ideals about combining resolution procedure with the implication connectives in l sub spl aleph are given moreover a resolution like principle in l sub spl aleph is proposed and the soundness theorem of this resolution procedure is also proved second we use this resolution like principle to horn clauses with truth values in an enriched residuated lattice and consider the l type fuzzy prolog

computerized flow field analysis oriented texture fields an approach to the solution of signal to symbol transformation in the domain of flow fields such as oriented texture fields and velocity vector fields is discussed the authors use the geometric theory of differential equations to derive a symbol set based on the visual appearance of phase portraits which are a geometric representation of the solution curves of a system of differential equations they also provide the computational framework to start with a given flow field and derive its symbolic representation specifically they segment the given texture derive its symbolic representation and perform a quantitative reconstruction of the salient features of the original texture based on the symbolic descriptors results of applying this technique to several real texture images are presented this technique is useful in describing complex flow visualization pictures defects in lumber processing defects in semiconductor wafer inspection and optical flow fields

the logic of tied implications part 1 properties applications and representation a conjunction t ties an implication operator a if the identity a a a b z a t a b z holds a a abdel hamid n n morsi associatively tied implications fuzzy sets and systems 136 2003 291 311 we study the class of tied adjointness algebras which are five connective algebras on two partially ordered sets in which the implications are tied by triangular norms this class contains besides residuated implications several other implications employed in fuzzy logic nevertheless we show that the algebraic inequalities of residuated algebras remain true for our tied implications but in forms that distribute roles over the five connectives of the algebra we apply the properties of tied implications to a generalized modus ponens inference scheme with two successive rules we prove its equivalence to a scheme with one compound rule when both schemata are interpreted by the compositional rule of inference and all connectives are taken from one tied adjointness algebra then we quote another application of this rich theory a notion of many valued rough sets which exhibit the basic mathematical behaviour of the rough sets of pawlak a comparator h is said to be prelinear if it satisfies h y z h z y 1 for all y z hajek we introduce prelinear tied adjointness algebras in which two comparators are prelinear we provide a representation of those algebras as subdirect products of tied adjointness chains on the lines of hajek s representation of bl algebras but our representations are more economical because we employ minimal prime filters on residuated lattices only rather than all prime filters

multiary α resolution principle for a lattice valued logic this paper focuses on resolution based automated reasoning theory in a lattice valued logic system with truth values that are defined in a lattice valued logical algebraic structure lattice implication algebras lias which essentially aims to extend the classical logic to handle automated deduction under an uncertain environment concretely we investigate a generalization of the known conjunctive normal form cnf in the classical logic representation that we call the generalized conjunctive normal form gcnf which aims to characterize the constants and implication connectives that are essentially different from the ones in classical logic we then extend the established resolution principle at a certain truth value level called resolution in this lattice valued logic to a more general form i e from binary resolution to multiary resolution the extension to multiary resolution starts from lattice valued propositional logic lp x while its theorems of both soundness and completeness are proved multiary resolution principle is then further established in the corresponding lattice valued first order logic lf x along with its soundness theorem lifting lemma and completeness theorem meanwhile an important result that multiary resolution principle in lf x can be equivalently transformed into that in lp x to some extent is obtained all these works will theoretically support the establishment of an automated reasoning algorithm and its implementation with further applications into automated deduction and decision making problems under uncertainty

large volume limit of the distribution of characteristic exponents in turbulence for spatially extended conservative or dissipative physical systems it appears natural that a density of characteristic exponents per unit volume should exist when the volume tends to infinity in the case of a turbulent viscous fluid however this simple idea is complicated by the phenomenon of intermittency in the present paper we obtain rigorous upper bounds on the distribution of characteristic exponents in terms of dissipation these bounds have a reasonable large volume behavior for two dimensional fluids a particularly striking result is obtained the total information creation is bounded above by a fixed multiple of the total energy dissipation at fixed viscosity the distribution of characteristic exponents is estimated in an intermittent model of turbulence see 7 and it is found that a change of behavior occurs at the valued 2 6 of the self similarity dimension

a representation of exponential form on fuzzy logic abstract this paper proposes a representation of exponential form ef in fuzzy logic by ef any fuzzy value a in fuzzy valued logic or fuzzy linguistic valued logic can be represented as b c where b is called the truth base and c the confidence exponent the basic idea of this form and its properties are discussed by using a different truth base the exponential form can be applied to represent positive and negative logic in fuzzy valued logic as well as fuzzy linguistic valued logic some applications of ef for approximate reasoning and fuzzy neural networks are also introduced in this paper

mapping abelian l groups with strong unit one one into mv algebras abstract we exhibit a functor from lattice ordered abelian groups with strong unit l fuchs partially ordered algebraic systems pergamon new york 1963 to c c chang s mv algebras trans amer math soc 88 1958 467 490 in such a way that every mv algebra is the image of at most one such group

do inertial manifolds apply to turbulence abstract in this article we survey some recent results concerning attractors inertial manifolds and approximate inertial manifolds for dissipative evolution equations and in particular for the two dimensional navier stokes equation the role of inertial manifolds as a modelization of the interaction of small and large eddies in turbulent flows is described and discussed

a small set of axioms for residuated logic we show that a set of three axioms together with modus ponens suffices to build up a formal system for the class r of all quadruples l t j in which l is a partially ordered set with a top element and t is a triangular norm on l that has a residuated implication j we show that this r is not a variety

a fuzzy inverse relation based on godelian logic and its applications abstract fuzzy logic applicable to decision making processes such as diagnostic expert systems and fuzzy controllers is the prime concern of this paper in these applications an inverse operation or backward inference plays an important role in the current paper a new backward inference technique based on godelian logic is proposed the relation between the forward and backward inferences is established by introducing fuzzy joint and conditional relations

an axiomatizable logical foundation for lattice ordered qualitative linguistic approach for reasoning with words it is commonsense that people do express think reason and make decision directly using linguistic terms in natural language rather than using numerical quantification how to establish the formalized approach imitating the common way of human being s to manipulate directly linguistic terms without numerical approximation to provide the rational decision is still an open research area investigations on the algebraic structure of linguistic term set for varied decision making applications especially in social science still lack a formalism for development of strict linguistic valued logic system as a theoretical foundation and its approximate reasoning scheme in practice to attain this goal we characterize and construct a typical structure of commonly used linguistic term sets in natural language by a lattice ordered algebra structure lattice implication algebra lia where lukasiewicz implication algebra as a special case of lia plays a substantial role by using lukasiewicz logic s axiomatizability in terms of pavelka type fuzzy logic we propose a new axiomatizable lattice ordered qualitative linguistic truth valued logic system based on lia to place an important foundation for further establishing formal linguistic truth valued logic based approximate reasoning and decision making with applications this proposed logic system has a distinct feature of handling comparable or incomparable linguistic terms directly without numerical quantification will be especially beneficial for perception based decision making processes it attempts to enhance the quantitative theory of decision science with qualitative algebraic and logic oriented approaches to achieve reasoning with words

issues on adjointness in multiple valued logics we contribute to the theory of implications and conjunctions related by adjointness in multiple valued logics we suggest their use in zadeh s compositional rule of inference to interpret generalized modus ponens inference schemata we provide new complete characterizations of implications that distinguish left arguments implications that satisfy the exchange principle divisible conjunctions commutative conjunctions associative conjunctions and triangular norms we also introduce and characterize pseudo strict and pseudo continuous implications and conjunctions and we explore the close relationship between these two notions

direct estimation of range flow on deformable shape from a video rate range camera a method of estimating range flow space displacement vector field on nonrigid as well as rigid objects from a sequence of range images is described this method can directly estimate the deformable motion parameters by solving a system of linear equations that are obtained from substituting a linear transformation of nonrigid objects expressed by the jacobian matrix into motion constraints based on an extension of the conventional scheme used in intensity image sequences the range flow is directly computed by substituting these estimated motion parameters into the linear transformation the algorithm is supported by experimental estimations of range flow on a sheet of paper a piece of cloth human skin and a rubber balloon being inflated using real range image sequences acquired by a video rate range camera

inertial manifolds for the 3d modified leray α model abstract in this paper we investigate the critical modified leray model in t 3 and prove the existence of an n dimensional inertial manifold for this problem

an example of absence of turbulence for any reynolds number ii we study a viscous incompressible fluid moving in a two dimensional flat torus 0 l 0 2 l 2 we show a set of external forces for which the stationary state is attractive for any reynolds numberr moreover the size of this set and the basin of attraction are independent ofr

theory of expertons and fuzzy logic abstract since 1965 when l a zadeh presented the concept of fuzzy sets various types of extensions have been proposed the treatment of subjectivity knowledge and uncertain data from one or several experts is now an important field in computer science in this paper a general concept the experton is described it is especially adapted for the case where several experts and or users are concerned the properties of expertons give also a monoid and a distributive lattice with the norms min and max then every process used with fuzzy sets stays available for expertons with a constraint of monotonicity expertons are based on probabilistic sets when intervals of probabilities are considered

the accuracy of the computation of optical flow and of the recovery of motion parameters the accuracy and the dependence on parameters of a general scheme for the analysis of time varying image sequences are discussed the approach is able to produce vector fields from which it is possible to recover 3 d motion parameters such as time to collision and angular velocity the numerical stability of the computed optical flow and the dependence of the recovery of 3 d motion parameters on spatial and temporal filtering is investigated by considering optical flows computed on subsampled images or along single scanlines it is also possible to recover 3 d motion parameters from reduced optical flows an adequate estimate of time to collision can be obtained from sequences of images with spatial resolution reduced to 128 128 pixels or from sequences of single scanlines passing near the focus of expansion the use of kalman filtering increases the accuracy and the robustness of the estimation of motion parameters the proposed approach seems to be able to provide not only a theoretical background but also practical tools that are adequate for the analysis of time varying image sequences

algebraic structures in fuzzy logic abstract common features in various fuzzy logical systems are investigated it turns out that adjoint couples and residuated lattices are very often present though not always explicitly expressed some minor new results are also proved

some representable de morgan algebras the stone theorems give us a representation of boolean algebras the structure associated to classical logic through set algebras the present paper deals with the representation of de morgan algebras one of the most important structures associated to multiple valued logic through fuzzy set algebras let us recall that a de morgan algebra 9 a v n is a structure such that pi a v is a distributive lattice with supremum u and infimum 0 and n is a decreasing involution in 9 such that n 0 u and n u 0 and consequently satisfies the de morgan laws we call these mappings n 9 9 strong negations or involutions of 40 see for instance 8 on the other hand let us recall that zadeh 3 defined the fuzzy subset of a universe x being characterized by a mapping a x 0 11 the generalized characteristic function which sends each x e x into the membership grade of x belonging to a as usual we denote by p x the set of fuzzy subsets of x f x o 11 and by q x n u the lattice obtained by defining f7 and u between fuzzy sets through a 7 b x min a x b x and a u b x max a x b x this lattice is distributive with supremum x defined by x x 1 for all x e x infimum 0 defined by 0 x 0 for all x e x and such that the boolean algebra of the boolean elements of f x is the boolean algebra of the classical subsets of x p x n u c the concept of fuzzy subset of a universe x is generalized goguen 4 by defining fuzzy subsets taking values in a lattice l a v it can still be more widely generalized by defining fuzzy subsets that for all x e x take values in different lattices thus if 9 r 1 x e x is a family of lattices r a v with supremum 1 and inlimum o we can define a fuzzy subset of x taking values in 5 as any mapping a x 1 u r such hat a x e r for all x e x if we denote by p x the set of fuzzy subsets of x taking values in r and define n and u through a n b x a x a b x and a u b x a x v b x then p x n u happens to be a lattice 463 0022 247 84 3 00

an lp bound for the riesz and bessel potentials of orthonormal functions let psi 1 psi n be orthonormal functions in rd and let u 1 delta 1 2 psi i or u 1 delta 1 1 2 psi i and let p x sum left u i x right 2 lp bounds are proved for p an example being for left p right p le a d n 1 p for rm d ge rm 3 with p d d 2 rm 1 the unusual feature of these bounds is that the orthogonality of the i yields a factor n 1 p instead of n as would be the case without orthogonality these bounds prove some conjectures of battle and federbush a phase cell cluster expansion for euclidean field theories i 1982 preprint and of conlon comm math phys in press

some new results concerning random sets and fuzzy sets abstract this paper is a continuation of past work showing direct connections between fuzzy set theory and classical probability theory through the use of random sets this includes characterizations of random intervals one point coverage equivalent to fuzzy sets determination of all nested random sets equivalent to fuzzy sets the solution of the one and multiple point coverage problems for random sets in finite spaces and the use of entropy for ordering random sets within the class of one point coverage equivalent ones for any given fuzzy set finally a connection between random variables random sets and fuzzy sets is pointed out

the measure of the degree of truth and the grade of membership abstract a method is presented to define the meaning of a fuzzy proposition and of a fuzzy set through the construction of fuzzy propositions with non ambiguous intermediate truth values

on characteristic exponents in turbulence ruelle has found upper bounds to the magnitude and to the number of non negative characteristic exponents for the navier stokes flow of an incompressible fluid in a domain the latter is particularly important because it yields an upper bound to the hausdorff dimension of attracting sets however ruelle s bound on the number has three deficiences i it relies on some unproved conjectures about certain constants ii it is valid only in dimensions 3 and not 2 iii it is valid only in the limit in this paper these deficiences are remedied and in addition the final constants in the inequality are improved

determination of α resolution in lattice valued first order logic lf x key issues for resolution based automated reasoning in lattice valued first order logic lf x are investigated with truth values in a lattice valued logical algebraic structure lattice implication algebra lia the determination of resolution at a certain truth value level called a resolution in lf x is proved to be equivalently transformed into the determination of a resolution in lattice valued propositional logic lp x based on lia the determination of a resolution of any quasi regular generalized literals and constants under various cases in lp x is further analyzed specified and subsequently verified hence the determination of a resolution in lf x can be accordingly solved to a very broad extent which not only lays a foundation for the practical implementation of automated reasoning algorithms in lf x but also provides a key support for a resolution based automated reasoning approaches and algorithms in lia based linguistic truth valued logics

asymptotic analysis of the navier stokes equations abstract new bounds are established on the number of modes which determine the solutions of the navier stokes equations in two dimensions the best bound available at present is nearly proportional to the generalized grashof number defined in the paper and less than logarithmically dependent on the spatial structure or the shape of the force driving the flow to the extent than for the case of 2 dimensional rayleigh benard convection the generalized grashof number may be identified with the usual grashof number the resulting bound on the number of modes is found to differ only slightly from a bound obtained earlier on heuristic grounds

propositional calculus under adjointness we develop a formal system for the class of all implications a and conjunctions k on partially ordered sets l with top elements 1 such that a and k are related by adjointness and they satisfy the neutrality principle that is 1 is their left identity element we call the resulting logic propositional calculus under adjointness abbreviated adjpc most algebraic theorems on those l a and k are inequalities in the posets l of truth values in consequence we have to find means for abstracting inequalities within syntax which must be free from partial truth values this is achieved by employing a further implication like adjoint h of a and k whereby the partial order of l coincides with the binary relation h 1 accordingly the semantical domain for adjpc should become the class of all such quintuples l a k h which we call adjointness algebras our axiom scheme for adjpc features seven axioms however it may be the case that no finite set of axioms can complete adjpc if inference is carried out by means of modus ponens mp alone this is because adjpc is too general it lacks some basic theorems of the more restricted logics such as residuated logic and intuitionistic logic as a result we have to adopt four inference rules for adjpc namely mp and three bits of the substitution theorem we deduce enough theorems and inferences in adjpc to delineate its basic structure this enables us to establish the completeness of adjpc for the semantical domain of adjointness algebras by means of a quotient algebra structure a lindenbaum type of algebra we also show how certain models can help disprove some incorrect inferences in adjpc we end by developing complete syntax with fewer axioms and inference rules for the smaller semantical domain of all adjointness algebras whose implications satisfy the exchange principle

on the dimension of the attractors in two dimensional turbulence abstract using a new version of the sobolev lieb thirring inequality we derive an upper bound for the dimension of the universal attractor for two dimensional space periodic navier stokes equations this estimate is optimal up to a logarithmic correction the relevance of this estimate to turbulence and related results are also briefly discussed

the use of optical flow for the analysis of non rigid motions this paper analysis the 2d motion field on the image plane produced by the 3d motion of a plane undergoing simple deformations when the deformation can be represented by a planar linear vector field the projected vector field i e the 2d motion field of the deformation is at most quadratic this 2d motion field has one singular point with eigenvalues identical to those of the singular point describing the deformation as a consequence the nature of the singular point of the deformation is a projective invariant when the plane moves and experiences a linear deformation at the same time the associated 2d motion field is at most quadratic with at most 3 singular points in the case of a normal rototranslation i e when the angular velocity is normal to the plane and of a linear deformation the 2d motion field has one singular point and substantial information on the rigid motion and on the deformation can be recovered from it experiments with image sequences of planes moving and undergoing linear deformations show that the proposed analysis can provide accurate results in addition experiments with deformable objects such as water oil textiles and rubber show that the proposed approach can provide information on more general 3d deformations