on the equilibrium of shell membranes under normal loading hidden integrability it is established that a well known system of classical shell theory descriptive of membranes in equilibrium is in fact integrable the membranes are shown to have geometries within the integrable class of so called o surfaces the membrane o surfaces include inter alia minimal constant mean curvature constant gaussian curvature and more generally linear weingarten surfaces as well as canal surfaces and dupin cyclides

superposition principles associated with the moutard transformation an integrable discretization of a 2 1 dimensional sine gordon system superposition principles both linear and nonlinear associated with the moutard transformation are found on suitable reinterpretation these constitute an integrable discrete nonlinear system and its associated linear system further it is shown that in a particular form this system is an integrable discretization of a 2 1 dimensional sine gordon system solutions of the discrete nonlinear system are constructed by means of a discrete analogue of the moutard transformation included in these solutions are discrete analogues of the kink solutions of the continuous system

reasoning about noisy sensors and effectors in the situation calculus abstract agents interacting with an incompletely known world need to be able to reason about the effects of their actions and to gain further information about that world they need to use sensors of some sort unfortunately both the effects of actions and the information returned from sensors are subject to error to cope with such uncertainties the agent can maintain probabilistic beliefs about the state of the world with probabilistic beliefs the agent will be able to quantify the likelihood of the various outcomes of its actions and is better able to utilize the information gathered from its error prone actions and sensors in this paper we present a model in which we can reason about an agent s probabilistic degrees of belief and the manner in which these beliefs change as various actions are executed we build on a general logical theory of action developed by reiter and others formalized in the situation calculus we propose a simple axiomatization that captures an agent s state of belief and the manner in which these beliefs change when actions are executed our model displays a number of intuitively reasonable properties

integrable discrete differential geometry of plated membranes in equilibrium recently it has been established that a well known system of classical shell theory descriptive of membranes in equilibrium is in fact amenable to the techniques of soliton theory here it is d

orthogonal polyanalytic polynomials and normal matrices the hermitian lanczos method for hermitian matrices has a well known connection with a 3 term recurrence for polynomials orthogonal on a discrete subset of r this connection disappears for normal matrices with the arnoldi method in this paper we consider an iterative method that is more faithful to the normality than the arnoldi iteration the approach is based on enlarging the set of polynomials to the set of polyanalytic polynomials denoting by d the number of elements computed so far the arising scheme yields a recurrence of length bounded by 8d for polyanalytic polynomials orthogonal on a discrete subset of c like this slowly growing length of the recurrence the method preserves at least partially the properties of the hermitian lanczos method we employ the algorithm in least squares approximation and bivariate lagrange interpolation

uniform bivariate hermite interpolation i coordinate degree on considere des schemas d interpolation d hermite uniforme bivariable a partir d espaces de polynomes de degre coordonne m n on se pose la question de la presque regularite de tels schemas c est a dire celle de l existence d ensembles de n uds pour lesquels avec une constellation de derivees donnees l interpolation est possible on montre que dans le cas affirmatif le schema en question peut etre utilise pour presque tous les ensembles de n uds l etude est basee sur des techniques de reduction

a curvature theory for discrete surfaces based on mesh parallelity we consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel gauss images and where mean and gaussian curvatures of faces are derived from the faces areas and mixed areas remarkably these notions are capable of unifying notable previously defined classes of surfaces such as discrete isothermic minimal surfaces and surfaces of constant mean curvature we discuss various types of natural gauss images the existence of principal curvatures constant curvature surfaces christoffel duality koenigs nets contact element nets s isothermic nets and interesting special cases such as discrete delaunay surfaces derived from elliptic billiards

pseudospherical surfaces on time scales a geometric definition and the spectral approach we define and discuss the notion of pseudospherical surfaces in asymptotic coordinates on time scales thus we extend well known notions of discrete pseudospherical surfaces and smooth pseudosperical surfaces on more exotic domains e g the cantor set in particular we present a new expression for the discrete gaussian curvature which turns out to be valid for asymptotic nets on any time scale we show that asymptotic chebyshev nets on an arbitrary time scale have constant negative gaussian curvature we present also the quaternion valued spectral problem the lax pair and the darboux backlund transformation for pseudospherical surfaces in asymptotic coordinates on arbitrary time scales

on the geometry of an integrable 2 1 dimensional sine gordon system it is recorded that darboux9s method of linking the classical lame system governing triply orthogonal systems of surfaces with an integrable 2 1 dimensional sine gordon equation may be extended and applied to the integrable two component generalization of the latter introduced by konopelchenko and rogers thus in a reinterpretation this 2 1 dimensional sine gordon system is shown to define particular integrable motions of surfaces

vectorial ribaucour transformations for the lame equations the vectorial extension of the ribaucour transformation for the lame equations of orthogonal conjugate nets in multidimensions is given we show that the composition of two vectorial ribaucour transformations with appropriate transformation data is again a vectorial ribaucour transformation from which follows the permutability of the vectorial ribaucour transformations finally as an example we apply the vectorial ribaucour transformation to the cartesian background

three dimensional integrable lattices in euclidean spaces conjugacy and orthogonality it is shown that the discrete darboux system descriptive of conjugate lattices in euclidean spaces admits constraints on the adjoint eigenfunctions which may be interpreted as discrete orthogonality conditions on the lattices thus it turns out that the elementary quadrilaterals of orthogonal lattices are cyclic orthogonal lattices on lines planes and spheres are discussed and the underlying integrable systems in one two and three dimensions are derived explicitly a discrete analogue of bianchi s ribaucour transformation is set down and particular orthogonal lattices are given as a by product discrete dini surfaces are obtained

semantical considerations on nonmonotonic logic abstract commonsense reasoning is nonmonotonic in the sense that we often draw on the basis of partial information conclusions that we later retract when we are given more complete information some of the most interesting products of recent attempts to formalize nonmonotonic reasoning are the nonmonotonic logics of mcdermott and doyle 1 2 these logics however all have peculiarities that suggest they do not quite succeed in capturing the intuitions that prompted their development in this paper we reconstruct nonmonotonic logic as a model of an ideally rational agent s reasoning about his own beliefs for the resulting system called autoepistemic logic we define an intuitively based semantics for which we can show autoepistemic logic to be both sound and complete we then compare autoepistemic logic with the approach of mcdermott and doyle showing how it avoids the peculiarities of their nonmonotonic logic

knowledge action and the frame problem this paper proposes a method for handling the frame problem for knowledge producing actions an example of a knowledge producing action is a sensing operation performed by a robot to determine whether or not there is an object of a particular shape within its grasp the work is an extension of reiter s approach to the frame problem for ordinary actions and moore s work on knowledge and action the properties of our specification are that knowledge producing actions do not affect fluents other than the knowledge fluent and actions that are not knowledge producing only affect the knowledge fluent as appropriate in addition memory emerges as a side effect if something is known in a certain situation it remains known at successor situations unless something relevant has changed also it will be shown that a form of regression examined by reiter for reducing reasoning about future situations to reasoning about the initial situation now also applies to knowledge producing actions

reductions of the multidimensional quadrilateral lattice the multidimensional circular lattice we apply a recently introduced 21 15 reduction method based on the dressing to construct a large class of integrable reductions of the equations characterizing the multidimensional quadrilateral lattice an n dimensional lattice in m n m whose elementary quadrilaterals are planar and whose continuous limit describes submanifolds parametrized by conjugate lines 11 we also show that generically in the limit of the small lattice parameter half of these reductions lead to the darboux equations for symmetric fields and the second half lead to the generalized lame equations describing n dimensional submanifolds of m parametrized by conjugate orthogonal systems of coordinates we finally show that a distinguished example of the second class of reductions corresponds to the multidimensional circular lattice an n dimensional lattice in m n m whose elementary quadrilaterals are inscribed in circles 7

bounded situation calculus action theories in this paper 1 we investigate bounded action theories in the situation calculus a bounded action theory is one which entails that in every situation the number of object tuples in the extension of fluents is bounded by a given constant although such extensions are in general different across the infinitely many situations we argue that such theories are common in applications either because facts do not persist indefinitely or because the agent eventually forgets some facts as new ones are learned we discuss various classes of bounded action theories then we show that verification of a powerful first order variant of the calculus is decidable for such theories notably this variant supports a controlled form of quantification across situations we also show that through verification we can actually check whether an arbitrary action theory maintains boundedness

propositional knowledge base revision and minimal change abstract the semantics of revising knowledge bases represented by sets of propositional sentences is analyzed from a model theoretic point of view a characterization of all revision schemes that satisfy the gardenfors rationality postulates is given in terms of minimal change with respect to an ordering among interpretations revision methods proposed by various authors are surveyed and analyzed in this framework the correspondences between gardenfors like rationality postulates and minimal changes with respect to other orderings are also investigated

transformations of quadrilateral lattices motivated by the classical studies on transformations of conjugate nets we develop the general geometric theory of transformations of their discrete analogs the multidimensional quadrilateral lattices i e lattices x zn rm n m whose elementary quadrilaterals are planar our investigation is based on the discrete analog of the theory of the rectilinear congruences which we also present in detail we study in particular the discrete analogs of the laplace combescure levy radial and fundamental transformations and their interrelations the composition of these transformations and their permutability is also investigated from a geometric point of view the deep connections between transformations and discretizations is also investigated for quadrilateral lattices we finally interpret these results within the formalism

nonlinear differential difference equations as backlund transformations shows that the best known nonlinear differential difference equations associated with the discrete schrodinger spectral problem and also with the discrete zakharov shabat spectral problem can be interpreted as backlund transformations for some continuous nonlinear evolution equations

darboux transformations for linear operators on two dimensional regular lattices darboux transformations for linear operators on regular two dimensional lattices are reviewed the six point scheme is considered as the master linear problem whose various specifications reductions and sublattice combinations lead to other linear operators together with the corresponding darboux transformations the second part of the review deals with multidimensional aspects of basic reductions of the four point scheme as well as the three point scheme

iterated belief change in the situation calculus john mccarthy s situation calculus has left an enduring mark on artificial intelligence research this simple yet elegant formalism for modelling and reasoning about dynamic systems is still in common use more than forty years since it was first proposed the ability to reason about action and change has long been considered a necessary component for any intelligent system the situation calculus and its numerous extensions as well as the many competing proposals that it has inspired deal with this problem to some extent in this paper we offer a new approach to belief change associated with performing actions that addresses some of the shortcomings of these approaches in particular our approach is based on a well developed theory of action in the situation calculus extended to deal with belief moreover by augmenting this approach with a notion of plausibility over situations our account handles nested belief belief introspection mistaken belief and handles belief revision and belief update together with iterated belief change

on the logic of iterated belief revision abstract we show in this paper that the agm postulates are too weak to ensure the rational preservation of conditional beliefs during belief revision thus permitting improper responses to sequences of observations we remedy this weakness by proposing four additional postulates which are sound relative to a qualitative version of probabilistic conditioning contrary to the agm framework the proposed postulates characterize belief revision as a process which may depend on elements of an epistemic state that are not necessarily captured by a belief set we also show that a simple modification to the agm framework can allow belief revision to be a function of epistemic states we establish a model based representation theorem which characterizes the proposed postulates and constrains in turn the way in which entrenchment orderings may be transformed under iterated belief revision

discrete line complexes and integrable evolution of minors based on the classical plcker correspondence we present algebraic and geometric properties of discrete integrable line complexes in cp3 algebraically these are encoded in a discrete integrable s

on the singularity of multivariate hermite interpolation in this paper we study the singularity of multivariate hermite interpolation of type total degree we present two methods to judge the singularity of the interpolation schemes considered and by methods to be developed we show that all hermite interpolation of type total degree on m d k points in r d is singular if d 2k and then we solve the hermite interpolation problem on email protected d 3 nodes completely precisely all hermite interpolations of type total degree on email protected d 1 points with d 2 are singular only three cases for m d 2 and one case for m d 3 can produce regular hermite interpolation schemes respectively besides we also present a method to compute the interpolation space for hermite interpolation of type total degree

on integrability of weingarten surfaces a forgotten class rediscovered by a systematic search a forgotten class of integrable surfaces is shown to disprove the finkel wu conjecture the associated integrable nonlinear partial differential equation possesses a zero curvature representation a third order symmetry and a nonlocal transformation to the sine gordon equation sin we leave open the problem of finding a backlund autotransformation and a recursion operator that would produce a local hierarchy

the structure of spectral problems and geometry hyperbolic surfaces in e3 working in the framework of sym s soliton surfaces approach we point out that some simple assumptions about the structure of linear spectral problems of the theory of solitons lead uniquely to the geometry of some special immersions in this paper we consider general su 2 spectral problems under some very weak assumptions they turn out to be associated with hyperbolic surfaces surfaces of negative gaussian curvature immersed in three dimensional euclidean space and especially with the so called bianchi surfaces

some contributions to the metatheory of the situation calculus we focus on a rich axiomatization for actions in the situation calculus that includes among other features a solution to the frame problem for deterministic actions our work is foundational in nature directed at simplifying the entailment problem for these axioms specifically we make four contributions to the metatheory of situation calculus axiomatizations of dynamical systems 1 we prove that the above mentioned axiomatization for actions has a relative satisfiability property the full axiomatization is satisfiable iff the axioms for the initial state are 2 we define the concept of regression relative to these axioms and prove a soundness and completeness theorem for a regression based approach to the entailment problem for a wide class of queries 3 our formalization of the situation calculus requires certain foundational axioms specifying the domain of situations these include an induction axiom whose presence complicates human and automated reasoning in the situation calculus we characterize various classes of sentences whose proofs do not require induction and in some cases some of the other foundational axioms 4 we prove that the logic programming language golog never requires any of the foundational axioms for the evaluation of programs

lattice and q difference darboux zakharov manakov systems via delta dressing method a general scheme is proposed for the introduction of lattice and q difference variables to integrable hierarchies in the frame of the delta dressing method using this scheme lattice and q difference darboux zakharov manakov systems of equations are derived darboux backlund and combescure transformations and exact solutions for these systems are studied

on the unification of classical and novel integrable surfaces ii difference geometry a novel class of discrete integrable surfaces is recorded this class of discrete o surfaces is shown to include discrete analogues of classical surfaces such as isothermic linear weingarten guichard and petot surfaces moreover natural discrete analogues of the gaussian and mean curvatures for surfaces parametrized in terms of curvature coordinates are used to define surfaces of constant discrete gaussian and mean curvatures and discrete minimal surfaces remarkably these turn out to be prototypical examples of discrete o surfaces it is demonstrated that the construction of a backlund transformation for discrete o surfaces leads in a natural manner to an associated parameter dependent linear representation canonical discretizations of the classical pseudosphere and breather pseudospherical surfaces are generated connections with pioneering work by bobenko and pinkall are established

belief revision and projection in the epistemic situation calculus abstract this article considers defeasible beliefs in dynamic settings in particular we examine the belief projection problem what is believed after performing an action and or receiving new information the approach is based on an epistemic variant of reiter s situation calculus where actions not only have physical effects but may also provide new information to the agent the preferential belief structure is initially determined using conditional statements new information is then incorporated using two popular belief revision schemes namely natural and lexicographic revision the projection problem is solved twofold in this formalism by goal regression and by knowledge base progression

bivariate hermite interpolation and applications to algebraic geometry we try to solve the bivariate interpolation problem 1 3 for polynomials 1 1 wheres is a lower set of lattice points and for theq th interpolation knot a q is the set of orders of derivatives that appear in 1 3 the number of coefficients s is equal to the number of equations a q if this is possible for all knots in general position the problem is almost always solvable a a s we seek to determine whether 1 3 is a a s an algorithm is given which often gives a positive answer to this it can be applied to the solution of a problem of hirschowitz in algebraic geometry we prove that for hermite conditions 1 3 when alla q are lower triangles of orderp andp is of total degreen 1 3 is a a s for allp 1 2 3 and alln except for the two casesp 1 n 2 andp 1 n 4