spacefilling curves and the planar travelling salesman problem to construct a short tour through points in the plane the points are sequenced as they appear along a spacefilling curve this heuristic consists essentially of sorting so it is easily coded and requires only o n memory and o n log n operations its performance is competitive with that of other fast methods

constructing competitive tours from local information abstract we consider the problem of a searcher exploring an initially unknown weighted planar graph g when the searcher visits a vertex v it learns of each edge incident to v the searcher s goal is to visit each vertex of g incurring as little cost as possible we present a constant competitive algorithm for this problem

submodularity and the traveling salesman problem in this paper we investigate the relationship between traveling salesman tour lengths and submodular functions this work is motivated by the one warehouse multi retailer inventory distribution problem with traveling salesman tour vehicle routing costs our goal is to find a submodular function whose values are close to those of optimal tour lengths through a central warehouse and a group of retailers our work shows that a submodular approximation to traveling salesman tour lengths whose error is bounded by a constant does not exist however we present heuristics that have errors which grow slowly with the number of retailers for the traveling salesman problem in the euclidean plane furthermore we perform computational tests that show that our submodular approximations of traveling salesman tour lengths have smaller errors than our theoretical worst case analysis would lead us to believe

generalized fixed point algebras and square integrable group actions we analyze rieffel s construction of generalized fixed point algebras in the setting of group actions on hilbert modules let g be a locally compact group acting on a c algebra b we construct a hilbert module f over the reduced crossed product of g and b using a pair e r where e is an equivariant hilbert module over b and r is a dense subspace of e with certain properties the generalized fixed point algebra is the c algebra of compact operators on f any hilbert module over the reduced crossed product arises by this construction for a pair e r that is unique up to isomorphism a necessary condition for the existence of r is that e be square integrable the consideration of square integrable representations of abelian groups on hilbert space shows that this condition is not sufficient and that different choices for r may yield different generalized fixed point algebras if b is proper in kasparov s sense there is a unique r with the required properties thus the generalized fixed point algebra only depends on e

morita rieffel equivalence and spectral theory for integrable automorphism groups of c algebras abstract given a c dynamical system a g we discuss conditions under which subalgebras of the multiplier algebra m a consisting of fixed points for are morita rieffel equivalent to ideals in the crossed product of a by g in case g is abelian we also develop a spectral theory giving a necessary and sufficient condition for to be equivalent to the dual action on the cross sectional c algebra of a fell bundle in our main application we show that a proper action of an abelian group on a locally compact space is equivalent to a dual action

pattern recognition with ordered labels abstract we consider a multi class pattern recognition problem with linearly ordered labels and a loss function which is a sum of squared deviations of decisions from true classes it is proved that the optimal in the bayesian sense decision rule is the a posteriori mean which is then rounded to the nearest integer then the plug in type estimator of the optimal decision rule is proposed we also investigate its asymptotic properties finally we discuss briefly the structure of a neural net which approximates the optimal decision rule

image analysis using space filling curves and 1d wavelet bases images are transformed into signals using a peano hilbert space filling curve this continuous mapping captures local informations when hilbert s curve is wandering inside the image then the obtained signal is analysed using 1 d one dimensional wavelet orthogonal bases the spaces of details of smaller and smaller scales are considered they are built from a scaling function an image can be computed by selecting wavelet coefficients corresponding to one or several scales and using the inverse peano hilbert procedure due to the good localization of wavelet analysis local and global patterns can be detected and global ones can be reinforced so the edge detection of global edges is prepared and easier with mathematical morphology

a note on the solvability of singular boundary value problems theorem suppose that 4 at u 1 e cw 11 x r pw e cp 11 and q t e w 11 withd0 q t 0 on 0 11 p o 0 and p t q t bounded on 0 11 b s l at df 00 s q t df ml m2 i iiwmo

on exponential bounds on the bayes risk of the kernel classification rule the exponential distribution free bounds for the kernel classification rule are derived the equivalence of all modes of the global convergence of the rule is established under optimal assumptions on the smoothing sequence also derived is the optimal global rate of convergence of the kernel regression estimate within the class of lipschitz distributions the rate is optimal for the nonparametric regression but not for classifications it is shown using the martingale device that weak strong and complete l sub 1 bayes risk consistencies are equivalent consequently the conditions on the smoothing sequence h sub n to 0 and nh sub n to infinity are necessary and sufficient for bayes risk consistency of the kernel classification rule the rate of convergence of the kernel classification rule is also given

a new approach to vector median filtering based on space filling curves the availability of a wide set of multidimensional information sources in different application fields e g color cameras multispectral remote sensing imagery devices etc is the basis for the interest of image processing research on extensions of scalar nonlinear filtering approaches to multidimensional data filtering a new approach to multidimensional median filtering is presented the method is structured into two steps absolute sorting of the vectorial space based on peano space filling curves is proposed as a preliminary step in order to map vectorial data onto an appropriate one dimensional 1 d space then a scalar median filtering operation is applied the main advantage of the proposed approach is the computational efficiency of the absolute sorting step which makes the method globally faster than existing median filtering techniques this is particularly important when dealing with a large amount of data e g image sequences presented results also show that the filtering performances of the proposed approach are comparable with those of vector median filters presented in the literature

monotone method for singular bvp in the presence of upper and lower solutions in this paper we examine existence of monotone approximations of solutions of singular boundary value problem p x y x q x f x y py for 0 0 p x y x 0 a 1y b email protected 1p b y b email protected 1 under quite general conditions on f x y py we show that solution of the singular two point boundary value problem is unique here p 0 0 q x is allowed to have integrable singularity at x 0 and we do not assume 0 bdsp s

asymptotic efficiency of classifying procedures using the hermite series estimate of multivariate probability densities corresp pattern recognition procedures derived from a nonparametric estimate of multivariate probability density functions using the orthogonal hermite system are examined for sufficiently regular densities the convergence rate of the mean integrated square error mise is o n l epsilon epsilon 0 where n is the number of observations and is independent of the dimension as a consequence the rate at which the probability of misclassification converges to the bayes probability of error as the length n of the learning sequence tends to infinity is also independent of the dimension of the class densities and equals o n 1 2 delta delta o

a distribution free tsp tour length estimation model for random graphs abstract traveling salesman problem tsp tour length estimations can be used when it is not necessary to know an exact tour e g when using certain heuristics to solve location routing problems the best estimation models in the tsp literature focus on random instances where the node dispersion is known those that do not require knowledge of node dispersion are either less accurate or slower in this paper we develop a new regression based tour length estimation model that is distribution free accurate and fast with a small standard deviation of the estimation errors when the distribution of the node coordinates is known it provides a close estimate of the well known asymptotic tour length estimation formula of beardwood et al 1959 more importantly when the distribution is unknown or non integrable so beardwood et al s estimation cannot be used our model still provides good fast tour length estimates

a new methodology of simulated annealing for the optimisation problems abstract complex optimisation problems with many degrees of freedom are often characterised by the enormously large configuration space typically o en or o n the idea of simulated annealing sa proposed by kirkpatrick has been applied to the complex optimisation problems which can be treated as annealing a statistical mechanical system from high temperature to low temperature however the sa is terribly slow for large problem sizes in typically o n3lnn time we discover the hybrid algorithm ha which is based on a hybrid mechanism which combines conventional heuristics with low temperature simulated annealing ltsa which could be parallelised easily the ha is a new approach of resolving optimisation problems with o n complexity where information propagation can be inhibited by restraining the range of searches in the configuration space we use the ha to resolve several famous combinatorial optimisation problems including the travelling salesman problem tsp of large sizes up to 1 000 000 cities within 3 to 5 percent of the optimal value in linear time and other nonuniformly distributed tsps as well we shall also discuss the applicability of the ha to the optimisation problems in general

continuous spectral decompositions of abelian group actions on c algebras let g be a locally compact abelian group following ruy exel we view fell bundles over the pontrjagin dual group of g as continuous spectral decompositions of g actions on c algebras we classify such spectral decompositions using certain dense subspaces related to marc rieffel s theory of square integrability there is a unique continuous spectral decomposition if the group acts properly on the primitive ideal space of the c algebra but there are also examples of group actions without or with several inequivalent spectral decompositions

solvability of some nonlinear boundary value problems on etablit l existence des solutions positives de problemes a deux points limites singuliers pour des equations differentielles de la forme y f t y y o o t 1

index information algorithm with local tuning for solving multidimensional global optimization problems with multiextremal constraints multidimensional optimization problems where the objective function and the constraints are multiextremal non differentiable lipschitz functions with unknown lipschitz constants and the feasible region is a finite collection of robust nonconvex subregions are considered both the objective function and the constraints may be partially defined to solve such problems an algorithm is proposed that uses peano space filling curves and the index scheme to reduce the original problem to a holder one dimensional one local tuning on the behaviour of the objective function and constraints is used during the work of the global optimization procedure in order to accelerate the search the method neither uses penalty coefficients nor additional variables convergence conditions are established numerical experiments confirm the good performance of the technique

routing heuristics for automated pick and place machines we examine the problem of sequencing the placements of multiple part types for a computer controlled placement machine the problem is modeled as a collection of interdependent traveling salesman problems a heuristic based on a space filling curve is shown to be easy to compute and quite effective numerical experiments show that on problems of realistic size our heuristics show very little divergence from optimality the probabilistic analysis of the proposed heuristic indicates that the proposed heuristics are asymptotically optimal finally we use our heuristic to perform the sequencing operation on real placement machines the results of this experiment are in accord with our numerical simulations the main ideas of this study have become part of the control system currently in use in a large electronic card assembly facility that produces approximately 12 000 boards per day improvements in throughput of between 4 and 9 have been reported

the existence of solutions to nonlinear second order periodic boundary value problems abstract we consider the existence of periodic boundary value problems of nonlinear second order ordinary differential equations under certain half lipschitzian type conditions several existence results are obtained as applications positive periodic solutions of some laplacian type equations and duffing type equations are investigated

complete convergence of short paths and karp s algorithm for the tsp let xi 1 i 0 sum infty n 1 p left t n sqrt n beta gt epsilon right lt infty this result is essential in justifying karp s algorithm for the traveling salesman problem under the independent model and it settles a question posed by b w weide

pattern recognition algorithms based on space filling curves and orthogonal expansions the classical problem of constructing a multidimensional pattern classifier in the bayesian framework is considered preprocessing of the learning sequence by a quasi inverse of a space filling curve is proposed and properties of space filling curves which are necessary to obtain bayes risk consistency are indicated the learning sequence transformed into the unit interval is used to estimate the coefficients in an orthogonal expansion of the bayes decision rule to transform a new observation into the unit interval requires o d elementary operations where d is the dimension of the observation space strong bayes risk consistency of the classifiers is proved when distributions of the random pair of the observation vector and its class are absolutely continuous with respect to the lebesgue measure attainable convergence rate of the bayes risk is discussed details of the classification algorithm based on the haar series and its properties are presented distribution free consistency of the classifier is established the performance of such a classifier is tested both on simulated data and on the standard benchmarks providing misclassification errors comparable to or even better than the k nearest neighbors k nn method

scanning and sequential decision making for multidimensional data part i the noiseless case we investigate the problem of scanning and prediction ldquoscandiction rdquo for short of multidimensional data arrays this problem arises in several aspects of image and video processing such as predictive coding for example where an image is compressed by coding the error sequence resulting from scandicting it thus it is natural to ask what is the optimal method to scan and predict a given image what is the resulting minimum prediction loss and whether there exist specific scandiction schemes which are universal in some sense specifically we investigate the following problems first modeling the data array as a random field we wish to examine whether there exists a scandiction scheme which is independent of the field s distribution yet asymptotically achieves the same performance as if this distribution were known this question is answered in the affirmative for the set of all spatially stationary random fields and under mild conditions on the loss function we then discuss the scenario where a nonoptimal scanning order is used yet accompanied by an optimal predictor and derive bounds on the excess loss compared to optimal scanning and prediction this paper is the first part of a two part paper on sequential decision making for multidimensional data it deals with clean noiseless data arrays the second part deals with noisy data arrays namely with the case where the decision maker observes only a noisy version of the data yet it is judged with respect to the original clean data

b spline collocation methods and their convergence for a class of nonlinear derivative dependent singular boundary value problems abstract this paper is concerned with the construction and convergence analysis of two b spline collocation methods for a class of nonlinear derivative dependent singular boundary value problems ddsbvp the first method is based on uniform mesh while the second method is based on non uniform mesh for the second method we use a grading function to construct the non uniform grid we prove that the method based on uniform mesh is of second order accuracy and the method based on non uniform mesh is of fourth order accuracy three nonlinear examples with derivative dependent source functions are considered to verify the performance and theoretical rate of convergence of present methods moreover we consider some special cases of the problem under consideration in order to compare our methods with other existing methods it is shown that our second method based on cubic b spline basis functions has the same order of convergence as quartic b spline collocation method 1 moreover our methods yield more accurate results and are computationally attractive than the methods developed in 1 2 3 4 5 6 7 8 the proposed methods are applied on three real life problems the first problem describes the distribution of radial stress on a rotationally shallow membrane cap the second problem arises in the study of thermal explosion in cylindrical vessel and the third problem arises in astronomy

manipulation and presentation of multidimensional image data using the peano scan a fractal curve first discussed by giuseppe peano has useful statistical properties in image processing the curve can act as a transform between a line and n dimensional space retaining some of the spatially associative properties of the space applications of the curve include multispectral image display compression in the color and spa tial domain classification and color display the technique enables the display of high quality color images on a frame store capable of displaying only 256 colors simultaneously a logically ordered and complete set of colors for false color work can be selected an im provement in compression by about 3x for color images can be ob tained by using this method in conjunction with an existing spatial compression technique a simple algorithm for constructing the curves is shown

heuristics based on spacefilling curves for combinatorial problems in euclidean space we describe a family of heuristics to solve combinatorial problems such as routing and partitioning these heuristics exploit geometry but ignore specific distance measures consequently they are simple and fast but nonetheless fairly accurate and so seem well suited to operational problems where time or computing resources are limited we survey promising new application areas and show how procedures may be customized to reflect the structure of particular applications