fuzzy transform and least squares approximation analogies differences and generalizations investigating the relations between the least squares approximation techniques and the fuzzy transform in this paper we show that the discrete fuzzy transform is invariant with respect to the interpolating and least squares approximation additionally the fuzzy transform is evaluated at any point by simply resampling the continuous approximation underlying the input data using numerical linear algebra we also derive new properties e g stability to noise additivity with respect to the input data and characterizations e g radial and dual membership maps of the discrete fuzzy transform finally we define the geometry and confidence driven discrete fuzzy transforms which take into account the intrinsic geometry and the confidence weights associated to the data

painleve equations nonlinear special functions the six painleve equations pi pvi were first discovered about a hundred years ago by painleve and his colleagues in an investigation of nonlinear second order ordinary differential equations recently there has been considerable interest in the painleve equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering consequently the painleve equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations despite being nonlinear equations although first discovered from strictly mathematical considerations the painleve equations have arisen in a variety of important physical applications including statistical mechanics plasma physics nonlinear waves quantum gravity quantum field theory general relativity nonlinear optics and fibre optics the painleve equations may be thought of a nonlinear analogues of the classical special functions they possess hierarchies of rational solutions and one parameter families of solutions expressible in terms of the classical special functions for special values of the parameters further the painleve equations admit symmetries under affine weyl groups which are related to the associated backlund transformations in this paper i discuss some of the remarkable properties which the painleve equations possess including connection formulae backlund transformations associated discrete equations and hierarchies of exact solutions in particular the second painleve equation pii is used to illustrate these properties and some of the applications of pii are also discussed

tubular neighborhoods of nodal sets and diophantine approximation we give upper and lower bounds on the volume of a tubular neighborhood of the nodal set of an eigenfunction of the laplacian on a real analytic closed riemannian manifold m as an application we consider the question of approximating points on m by nodal sets and explore analogy with approximation by rational numbers

the isomonodromy approach to matric models in 2d quantum gravity we consider the double scaling limit in the hermitian matrix model for 2d quantum gravity associated with the measure exp sum limits j 1 n t j z 2j n geqq 3 we show that after the appropriate modification of the contour of integration the cross migdal douglas shenker limit to the painleve i equation in the generic case of the pure gravity is valid and calculate the nonperturbative parameters of the corresponding painleve function our approach is based on the wkb analysis of the l a pair corresponding to the discrete string equation in the framework of the inverse monodromy method here we extend our results which were obtained before for the particular casesn 2 3 our analysis complements the isomonodromy approach proposed by g moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

the morphology of nodal lines random waves versus percolation in this paper we investigate the properties of nodal structures in random wave fields and in particular we scrutinize their recently proposed connection with short range percolation models we propose a measure which shows the difference between monochromatic random waves which are characterized by long range correlations and gaussian fields with short range correlations which are naturally assumed to be better modelled by percolation theory we also study the relevance of the quantities which we compute to the probability that nodal lines are in the vicinity of a given reference line

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

a neural network approach to the fuzzy transform the paper deals with the f transform technique which was introduced as a method for an approximate representation of continuous functions the same task is solved by many methods from different areas neural networks also belong to techniques which have been proved to be powerful for approximation objectives they provide us with many advantages especially incremental way of learning parameters the paper inherits neural approaches to the f transform method and presents experiments justifying the proposed approach incremental way of determination of certain parameters of the f transform method which were up to now given by batch formula enriches possible areas of application of the method by fast on line processes moreover the neural approach is demonstrated to be an appropriate one for finding such fuzzy partition of a domain which respects better a given set of measured samples which are to be approximated by a continuous function with no predetermined shape

an optimal algorithm for approximate nearest neighbor searching fixed dimensions consider a set of s of n data points in real d dimensional space r d where distances are measured using any minkowski metric in nearest neighbor searching we preprocess s into a data structure so that given any query point q r d is the closest point of s to q can be reported quickly given any positive real e data point p is a 1 e approximate nearest neighbor of q if its distance from q is within a factor of 1 e of the distance to the true nearest neighbor we show that it is possible to preprocess a set of n points in r d in o dn log n time and o dn space so that given a query point q r d and e 0 a 1 e approximate nearest neighbor of q can be computed in o c d e log n time where c d e d 1 6d e d is a factor depending only on dimension and e in general we show that given an integer k 1 1 e approximations to the k nearest neighbors of q can be computed in additional o kd log n time

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

polynomial alias higher degree fuzzy transform of complex valued functions abstract in this article we propose a general approach to the computation of components of the direct higher degree fuzzy transform apart from the orthogonal bases of the subspaces of polynomials of weighted hilbert spaces with respect to a generalized uniform fuzzy partition which are used in all papers on fuzzy transform of higher degree we admit also the non orthogonal bases an advantage of using non orthogonal bases consists in the possibility of replacing orthogonal polynomials derivation of which by the gram schmidt orthogonalization process can be questionable difficult or imprecise by suitable non orthogonal polynomials of much simpler form we present a simple matrix calculus and show how it can be used to introduce the components of the direct higher degree fuzzy transform with the help of the monomial basis we prove a convergence theorem and an approximation theorem for the higher degree fuzzy transform the results are illustrated by examples including a comparison with standard methods

eigenfunctions with prescribed nodal sets in this paper we consider the problem of prescribing the nodal set of low energy eigenfunctions of the laplacian our main result is that given any separating closed hypersurface in a compact n manifold m there is a riemannian metric on m such that the nodal set of its first nontrivial eigenfunction is we present a number of variations on this result which enable us to show in particular that the first nontrivial eigenfunction can have as many non degenerate critical points as one wishes

a generalization of determinant formulae for the solutions of painleve ii and xxxiv equations a generalization of determinant formulae for the classical solutions of painleve xxxiv and painleve ii equations are constructed using the technique of darboux transformation and hirota s bilinear formalism it is shown that the solutions admit determinant formulae even for the transcendental case

amalgamations of the painleve equations new hierarchies of nonlinear ordinary differential equations that are generalizations of the painleve equations are presented these hierarchies are shown to contain the painleve equations as special cases emphasis is on the sixth order ordinary differential equations special solutions for one of them are expressed via the general solutions of the p1 p2 equations and special cases of the p3 and p5 equations special solutions of another sixth order ode are determined by the general solutions of the p2 p4 p3 and p5 equations five of six painleve equations can be considered as special cases of these sixth order odes isomonodromic linear problems to solve the cauchy problems of hierarchies by the inverse monodromy transform are given

fuzzy data mining for interesting generalized association rules due to the increasing use of very large databases and data warehouses mining useful information and helpful knowledge from transactions is evolving into an important research area most conventional data mining algorithms identify the relationships among transactions using binary values and find rules at a single concept level transactions with quantitative values and items with hierarchy relation are however commonly seen in real world applications in this paper we thus introduce the problem of mining fuzzy generalized association rules from quantitative data a fuzzy mining algorithm based on srikant and agrawal s method is proposed for extracting implicit generalized knowledge from transactions stored as quantitative values it integrates fuzzy set concepts and generalized data mining technologies to achieve this purpose items in rules may be from any level of the given taxonomy the effect of numbers of fuzzy regions on the performance of the proposed algorithm is also discussed

non additive interval valued f transform this article proposes a new interval valued fuzzy transform its construction is based on a possibilistic interpretation of the partition on which the fuzzy transform is built the main advantage of this approach is that it provides specific interval valued functions whose interpretation is straightforward this interpretation relates to a traditional sampling reconstruction framework where little is known about the sampling and or reconstructing kernels numerous properties of the proposed approach are proved that could be useful for function analysis and comparison in the experimental section we illustrate some properties of the proposed transform while highlighting interesting features of the obtained framework

multi dimensional fuzzy transform and projection fuzzy and inverse fuzzy transforms introduced by i perfilieva is an important tool for signal and image fuzzy representation it led to several variants as the natural least squares minimization named fuzzy projection we deal with this transform proposing first to show its simplicity and its easy numerical implementation in any dimension of space secondly we point out that a key parameter permits to control its numerical robustness thirdly we show that this parameter also governs the stability of the representation by fuzzy projection when choosing different partitions of the space we conclude in discussing linear representation especially triangular representation for signals and pyramidal representation for images

differentiation by the f transform and application to edge detection in this contribution we consider integrable functions of many in particular two variables and extend for them the technique of f transform to f1 transform we prove that linear coefficients in the representation of the f1 transform component approximate the first partial derivatives we give estimates for this approximation finally we discuss the application to the edge detection problem we propose a new algorithm where the technique of f1 transform is combined with the canny detector a theoretical justification and advantages of the proposed approach are emphasized and illustrative examples are presented

first degree birational transformations of the painleve equations and their contiguity relations we present a consistent truncation allowing us to obtain the first degree birational transformation found by okamoto for the sixth painleve equation the discrete equation arising from its contiguity relation is then just the sum of six simple poles an algebraic solution is presented which is equivalent to but simpler than the umemura solution finally the well known confluence provides a unified picture of all first degree birational transformations for the lower painleve equations ranging them in two distinct sequences

a new reconstruction from the f transform components a new notion of adjoint fuzzy partition is introduced and the reconstruction of a function from its f transform components is analyzed an analogy with the nyquistshannonkotelnikov sampling theorem is discussed special attention is paid to the case where sample values of a function come with noise we show that the double reconstruction through the f transform components significantly reduces the influence of noise

development of autofocusing algorithm based on fuzzy transforms in this paper we propose a new passive auto focusing algorithm to acquire images with enhanced quality through a digital camera the focus measure operations play a key role in a passive auto focusing method the proposed focus measure operation is based on a localized version of image variance method the localization of the image variance focus measure operation can be implemented with the use of fuzzy partition and fuzzy transform in order to validate our approach experimental comparative studies are conducted on test images with the use of different focus measure operators the selected operators have been chosen from an extensive review of the state of the art approaches a comparative analysis demonstrates that the proposed algorithm is superior to the conventional methods

from continuous to discrete painleve equations abstract discrete analogues of the painleve equations have recently been derived using either a method based on orthogonal polynomials or the singularity confinement method here we analyze additional methods for deriving integrable non autonomous discrete equations these methods are based on the discrete akns approach on the use of backlund and schlesinger transforms of the continuous painleve equations and on discrete analogues of miura transformations

new contiguity relation of the sixth painleve equation from a truncation for the master painleve equation p6 u we define a consistent method adapted from the weiss truncation for partial differential equations which allows us to obtain the first degree birational transformation of okamoto two new features are implemented to achieve this result the first one is the homography between the derivative of the solution u and a riccati pseudopotential the second one is an improvement of a conjecture by fokas and ablowitz on the structure of this birational transformation we then build the contiguity relation of p6 which yields one new second order non autonomous discrete equation

multivariate interpolation of arbitrarily spaced data by moving least squares methods abstract given a function f in scattered data points x1 xn r s we solve the least squares problem i 1 n j o q a j x b j x i f x i 2 v i x min in order to generate interpolants aj x bj x here the bj denote basis functions e g polynomials and the vi x are inverse distance weight functions we express the unknowns aj x the interpolant and the interpolation error in terms of moving convex combinations of functions corresponding to those of classical interpolation at q 1 points with respect to these basis functions further we discuss properties of moving least squares methods and in the case of polynomial basis functions we test various methods and give perspective plots

nodal sets of eigenfunctions on riemannian manifolds soit m n une variete connexe compacte a metrique riemannienne c soit le laplacien de m soit f une fonction propre reelle de de valeur propre on demontre que la fonction propre f s annule au plus a l ordre c 1 2 pour tout point dans m

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

fuzzy transforms theory and applications the technique of direct and inverse fuzzy f transforms of three different types is introduced and approximating properties of the inverse f transforms are described all three types of the direct f transform are transformations from a function space to a finite dimensional vector space the first ordinary f transform is constructed on the basis of the ordinary algebra of reals while the other two types of the f transform are constructed on the basis of residuated lattice the core idea of the technique of f transforms is a fuzzy partition of a universe into fuzzy subsets factors clusters granules etc we claim that for a sufficient representation of a function defined on this universe we may consider its average values over fuzzy subsets from the partition thus a function can be associated with a mapping from a set of fuzzy subsets to the set of its thus obtained average values a number of theorems establishing best approximation properties of the inverse f transforms are proved in fact three types of the inverse f transform are the best approximations in average from below and from above respectively as one of many possible applications we present a method of image compression and reconstruction on the basis of the f transform

random unitary matrices permutations and painleve this paper is concerned with certain connections between the ensemble of n n unitary matrices specifically the characteristic function of the random variable tr u and combinatorics specifically ulam s problem concerning the distribution of the length of the longest increasing subsequence in permutation groups and the appearance of painleve functions in the answers to apparently unrelated questions among the results is a representation in terms of a painleve v function for the characteristic function of tr u and using recent results of baik deift and johansson an expression in terms of a painleve ii function for the limiting distribution of the length of the longest increasing subsequence in the hyperoctahedral groups

rational surfaces associated with affine root systems and geometry of the painleve equations we present a geometric approach to the theory of painleve equations based on rational surfaces our starting point is a compact smooth rational surface x which has a unique anti canonical divisor d of canonical type we classify all such surfaces x to each x there corresponds a root subsystem of e 1 8 inside the picard lattice of x we realize the action of the corresponding affine weyl group as the cremona action on a family of these surfaces we show that the translation part of the affine weyl group gives rise to discrete painleve equations and that the above action constitutes their group of symmetries by backlund transformations the six painleve differential equations appear as degenerate cases of this construction in the latter context x is okamoto s space of initial conditions and d is the pole divisor of the symplectic form defining the hamiltonian structure

an interval valued inversion of the non additive interval valued f transform as shown in a recent paper the fuzzy transform can be reinterpreted as a sampling process with an ill known sampling kernel or point spread function this reinterpretation leads to an interval valued direct fuzzy transform the non additive fuzzy transform it provides the convex set of all sampled values that should have been obtained by sampling a continuous signal with a convex set of sampling kernels its dual transform the non additive inverse fuzzy transform allows computation of an interval valued reconstruction of a sampled signal which is the set of all values that would have been obtained by interpolating or approximating this sampled signal when using a convex set of reconstruction kernels in this paper we consider using a new interval valued inversion process within this new framework to upsample a signal i e reconstruct a high resolution signal with a low resolution signal in a semi blind context as illustrated in the experimental part of the paper the main advantage of using this method compared to previous techniques is its inherent robustness with respect to modeling the sampling process