quantum field theory grassmannians and algebraic curves this paper is devoted in part to clarifying some aspects of the relation between quantum field theory and infinite grassmannians and in part to pointing out the existence of a close analogy between conformal field theory on riemann surfaces and the modern theory of automorphic representations along the way we develop a multiplicative analog of the usual additive ward identities of current algebra we also reformulate the additive ward identities in a way which may be useful in terms of the residues of operator valued differential forms a concluding section is devoted to some remarks on string field theory in an appendix we attempt to clarify the recent construction by beilinson manin and schechtman of what might be called global virasoro algebras

interval completion is fixed parameter tractable we present an algorithm with runtime o k 2k n 3m for the following np complete problem m garey and d johnson computers and intractability a guide to the theory of np completeness w h freeman and co san francisco 1979 problem gt35 given an arbitrary graph g on n vertices and m edges can we obtain an interval graph by adding at most k new edges to g this resolves the long standing open question h kaplan r shamir and r e tarjan siam j comput 28 1999 pp 1906 1922 r g downey and m r fellows parameterized complexity springer verlag new york 1999 m serna and d thilikos bull eur assoc theory comput sci eatcs 86 2005 pp 41 65 g gutin s szeider and a yeo in proceedings iwpec 2006 lecture notes in comput sci 4169 springer verlag berlin 2006 pp 60 71 first posed by kaplan shamir and tarjan of whether this problem was fixed parameter tractable the problem has applications in profile minimization for sparse matrix computations j a george and j w h liu computer solution of large sparse positive definite systems prentice hall englewood cliffs nj 1981 r e tarjan in sparse matrix computations j r bunch and d j rose eds academic press 1976 pp 3 22 and our results show tractability for the case of a small number k of zero elements in the envelope our algorithm performs bounded search among possible ways of adding edges to a graph to obtain an interval graph and combines this with a greedy algorithm when graphs of a certain structure are reached by the search

discrete dynamical systems in l topological spaces in this paper we investigate iteration of zadeh image operators and their discrete dynamical systems in l topological spaces some basic concepts such as limit set invariant set periodic point nonwandering point and recurrent point are discussed in weakly induced topological spaces we also study topological mixing topological transitivity and minimality for such dynamical systems while comparing with those corresponding ones in crisp spaces

from integrable nets to integrable lattices inspired by the results of jonas einsenhart demoulin and bianchi on the permutability property of classical geometrical transformations of conjugate nets and its reductions of pseudo orthogonal pseudo symmetric and pseudo egorov types dressing transformations of the n component kp hierarchy described within the grassmannian are used to generate quadrilateral lattices and its corresponding reductions as a byproduct we get the corresponding discrete dressing transformations in particular we characterize the vectorial fundamental discrete transformations preserving the symmetric lattice

on the topological structure of spaces of fuzzy compacta we prove that the space of fuzzy compacta in a peano continuum is homeomorphic to the hilbert space 2 as a corollary we find that the spaces of fuzzy compacta in r n and 2 are also homeomorphic to hilbert space

dressing methods for geometric nets ii orthogonal and egorov nets the grassmannian formalism of kp hierarchies is used to study geometric nets of orthogonal type and their subclass of egorov nets efficient dressing methods for cauchy propagators are provided which lead to wide families of explicit nets frobenius manifolds and solutions to the witten dijkgraff verlinde verlinde associativity equations are also constructed

the topological structure of fuzzy sets with endograph metric for a non degenerate convex subset y of the n dimensional euclidean space r n let k y be the family of all fuzzy sets of r n which are upper semicontinuous fuzzy convex and normal with compact supports contained in y we show that the space k y with the topology of endograph metric is homeomorphic to the hilbert cube q 1 1 w iff y is compact and the space k y is homeomorphic to x n q sup x n 1 iff y is non compact and locally compact

certifying algorithms for recognizing interval graphs and permutation graphs a certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces the certificate is a piece of evidence that proves that the answer has not been compromised by a bug in the implementation we give linear time certifying algorithms for recognition of interval graphs and permutation graphs and for a few other related problems previous algorithms fail to provide supporting evidence when they claim that the input graph is not a member of the class we show that our certificates of nonmembership can be authenticated in o v time

dressing methods for geometric nets i conjugate nets the formalism of multicomponent kp hierarchies is applied to deriving efficient dressing methods for conjugate nets the notion of the cauchy propagator is used for characterizing these nets in terms of spectral data explicit examples in dimensions n 2 and 3 are given in particular periodic nets and cartesian nets with a gaussian localized deformation are exhibited

charged free fermions vertex operators and the classical theory of conjugate nets we show that the quantum field theoretical formulation of the function theory has a geometrical interpretation within the classical transformation theory of conjugate nets in particular we prove that i the partial charge transformations preserving the neutral sector are laplace transformations ii the basic vertex operators are levy and adjoint levy transformations and iii the diagonal soliton vertex operators generate fundamental transformations we also show that the bilinear identity for the multicomponent kadomtsev petviashvili hierarchy becomes through a generalized miwa map a bilinear identity for the multidimensional quadrilateral lattice equations

matrix orthogonal laurent polynomials on the unit circle and toda type integrable systems matrix orthogonal laurent polynomials in the unit circle and the theory of toda like integrable systems are connected using the gauss borel factorization of two left and a right cantero morales velazquez block moment matrices which are constructed using a quasi definite matrix measure a block gauss borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal laurent polynomials and matrix szeg polynomials which can be expressed in terms of schur complements of bordered truncations of the block moment matrix the corresponding block extension of the christoffel darboux theory is derived deformations of the quasidefinite matrix measure leading to integrable systems of toda type are studied the integrable theory is given in this matrix scenario wave and adjoint wave functions lax and zakharov shabat equations bilinear equations and discrete flows connected with darboux transformations we generalize the integrable flows of the cafasso s matrix extension of the toeplitz lattice for the verblunsky coefficients of szeg polynomials an analysis of the miwa shifts allows for the finding of interesting connections between christoffel darboux kernels and miwa shifts of the matrix orthogonal laurent polynomials

fuzzy differential equations without fuzzy convexity classical fuzzy differential equations defined in terms of the hukuhara derivative depend critically on the convexity of the level sets and result in expanding level sets here hullermeier s suggestion of defining fuzzy differential equations at each level set via differential inclusions is combined with ideas of aubin on morphological equations which allow nonlocal set evolution to remove the assumption of fuzzy convexity and thus to allow fuzzy differential equations to be defined for non convex level sets this approach uses reachable sets as a more general form of set integration and in contrast to the aumann set integral does not necessarily give rise to convex sets the results presented in this paper are even more general since they concern fuzzy sets that need to be only closed without additional assumptions of convexity compactness or even normality in particular an existence and uniqueness theorem is established under the assumption that the right hand sides satisfy a one sided lipschitz condition rather than a much stronger lipschitz condition fuzzy delay differential equations are also considered from this new perspective

analytic bilinear approach to integrable hierarchies ii multicomponent kp and 2d toda lattice hierarchies an analytic bilinear approach for the construction and study of integrable hierarchies is discussed generalized multicomponent kp and 2d toda lattice hierarchies are considered this approach allows us to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations a generalized hierarchy incorporates basic hierarchy modified hierarchy singularity manifold equation hierarchy and corresponding linear problems different levels of generalized hierarchy are connected via invariants of combescure symmetry transformation the resolution of functional equations also leads to the function and addition formulas to it

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

on graphs associated to sets of rankings in this paper we analyze families of rankings by studying structural properties of graphs given a finite number of elements and a set of rankings of those elements two elements compete when they exchange their relative positions in at least two rankings and we can associate an undirected graph to a set of rankings by connecting elements that compete we call this graph a competitivity graph competitivity graphs have already appeared in the literature as co comparability graphs f graphs or intersection graphs associated to a concatenation of permutation diagrams we introduce certain important sets of nodes in a competitivity graph for example nodes that compete among them form a competitivity set and nodes connected by chains of competitors form a set of eventual competitors these sets are analyzed and a method to obtain sets of eventual competitors directly from a set of rankings is shown

the topological structure of the space of fuzzy compacta abstract a fuzzy compactum in a space x is an upper semi continuous map f a 0 1 from a nonempty compact subspace a x to the unit closed interval 0 1 such that x a f x 0 is dense in a in this paper we investigate the topological structure of the space k e x of fuzzy compacta of a separable metric space x with the hausdorff metric in particular we prove that k e x l 2 if x is an infinite locally compact locally connected connected and separable metric space

on a topological fuzzy fixed point theorem and its application to non ejective fuzzy fractals abstract a topological fuzzy fixed point theorem is given when generalizing and improving the main result by diamond kloeden and pokrovskii 1997 1 apart from the sole existence a weak local stability property called non ejectivity in the sense of browder of fuzzy fixed points is established this theorem is then applied for obtaining non ejective fuzzy fractals an alternative approach via the knaster tarski theorem is also presented

an o nm time certifying algorithm for recognizing hhd free graphs in this paper we consider the recognition problem on a class of perfectly orderable graphs namely the hhd free graphs such graphs do not contain any induced subgraph isomorphic to a house a hole or a domino we prove properties of the hhd free graphs which enable us to present an o nm time and o n m space algorithm for determining whether a graph on n vertices and m edges is hhd free currently this is the fastest algorithm for this problem we also describe how the algorithm can be augmented to provide a certificate an induced house hole or domino whenever it decides that the input graph is not hhd free thus answering an open question posed by hoang and sritharan theoretical computer science 259 2001 233 244 the certificate computation requires o n m additional time and o n space

multivariate orthogonal polynomials and integrable systems multivariate orthogonal polynomials in d real dimensions are considered from the perspective of the cholesky factorization of a moment matrix the approach allows for the construction of corresponding multivariate orthogonal polynomials associated second kind functions jacobi type matrices and associated three term relations and also christoffel darboux formulae the multivariate orthogonal polynomials their second kind functions and the corresponding christoffel darboux kernels are shown to be quasi determinants as well as schur complements of bordered truncations of the moment matrix quasi tau functions are introduced it is proven that the second kind functions are multivariate cauchy transforms of the multivariate orthogonal polynomials discrete and continuous deformations of the measure lead to toda type integrable hierarchy being the corresponding flows described through lax and zakharov shabat equations bilinear equations are found varying size matrix nonlinear partial difference and differential equations of the 2d toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials the discrete flows which are shown to be connected with a gauss borel factorization of the jacobi type matrices and its quasi determinants lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasi tau matrices which generalize to the multidimensional realm those that relate the baker and adjoint baker functions to ratios of miwa shifted tau functions in the 1d scenario in this context the multivariate extension of the elementary darboux transformation is given in terms of quasi determinants of matrices built up by the evaluation at a poised set of nodes lying in an appropriate hyperplane in r d of the multivariate orthogonal polynomials the multivariate christoffel formula for the iteration of m elementary darboux transformations is given as a quasi determinant it is shown using congruences in the space of semi infinite matrices that the discrete and continuous flows are intimately connected and determine nonlinear partial difference differential equations that involve only one site in the integrable lattice behaving as a kadomstev petviashvili type system finally a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given in particular it is shown that the toda times that preserve the invariance condition lay in a secant variety of the veronese variety of the fixed point set of the linear isometry

on the problem and integrable equations using the problem and dual problem we derive bilinear relations which allow us to construct integrable hierarchies in different parametrizations their darboux backlund transformations and to analyse constraints for them in a very simple way scalar kp bkp and ckp hierarchies are considered as examples

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

fully dynamic representations of interval graphs abstract we present a fully dynamic algorithm that maintains three different representations of an interval graph a minimal interval model of the graph the pq tree of its maximal cliques and its modular decomposition after each vertex or edge modification insertion or deletion the algorithm determines whether the new graph is an interval graph in o n time and in the positive updates the three representations within the same complexity furthermore we state that the modular decomposition tree of an interval graph and the pq tree of its maximal cliques are algorithmically equivalent and we completely explicit the relationship between the two structures

an efficient certifying algorithm for the hamiltonian cycle problem on circular arc graphs abstract a certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces the certificate is an evidence that can be used to authenticate the correctness of the answer a hamiltonian cycle in a graph is a simple cycle in which each vertex of the graph appears exactly once the hamiltonian cycle problem is to determine whether or not a graph contains a hamiltonian cycle the best result for the hamiltonian cycle problem on circular arc graphs is an o n 2 log n time algorithm where n is the number of vertices of the input graph in fact the o n 2 log n time algorithm can be modified as a certifying algorithm although it was published before the term certifying algorithms appeared in the literature however whether there exists an algorithm whose time complexity is better than o n 2 log n for solving the hamiltonian cycle problem on circular arc graphs has been opened for two decades in this paper we present an o n time certifying algorithm to solve this problem where represents the maximum degree of the input graph the certificates provided by our algorithm can be authenticated in o n time

fuzzy dynamical systems a fuzzy dynamical system on an underlying complete locally compact metric state space x is defined axiomatically in terms of a fuzzy attainability set mapping on x this definition includes as special cases crisp single and multivalued dynamical systems on x it is shown that the support of such a fuzzy dynamical system on x is a crisp multivalued dynamical system on x and that such a fuzzy dynamical system can be considered as a crisp dynamical system on a state space of nonempty compact fuzzy subsets of x in addition fuzzy trajectories are defined their existence established and various properties investigated

bihamiltonian hierarchies in 2d topological field theory at one loop approximation we compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2d topological field theory the bihamiltonian structure of the hierarchy is given by a classical w algebra we compute the central charge of this algebra we also express the generating function of elliptic gromov witten invariants via tau function of the isomonodromy deformation problem arising in the theory of wdvv equations of associativity

a possibilistic argument for irreversibility abstract our aim is to explicate the idea of irreversibility by a mathematical representation of irreversible processes occurring in possibilistic dynamical systems with this in mind solutions of the cauchy problem in the space of fuzzy points are examined to meet the correspondence argument solutions of the cauchy problem in the space of concentrated points are examined too

evolution of systems under fuzzy dynamic laws abstract the global behavior of a system with information on fuzzy described short term evolution laws is proposed and investigated in the paper for deterministic or stochastic models such problems are solved by use of differential equations describing the short term laws but if the short term laws of evolution are known only in a fuzzy form and the information on the initial state of the system is also fuzzy or if we do not have a reasonable statistical model describing the systems environment the fuzzy approach is more appropriate we will show that the introduction of the fuzzy state fuzzy velocity and use of the fuzzy logic rules for obtaining the equation of evolution of the system leads to a consistent description of the fuzzy dynamics our approach is based on the fuzzy master equation an analog of the equation describing the markov process in stochastic dynamics for continuous evolution we obtain differential equations describing the flows of the fuzzy state two cases of weak and strong fuzziness are considered a comparison with other approaches to fuzzy dynamics is performed