fine hierarchy of regular ω languages by applying descriptive set theory we get several facts on the fine structure of regular languages considered by k wagner we present quite different shorter proofs for main his results and get new results our description of the fine structure is new very clear and automata free we prove also a closure property of the fine structure under boolean operations our results demonstrate deep interconnections between descriptive set theory and theory of languages

fick s law and fractality of nonequilibrium stationary states in a reversible multibaker map nonequilibrium stationary states are studied for a multibaker map a simple reversible chaotic dynamical system the probabilistic description is extended by representing a dynamical state in terms of a measure instead of a density function the equation of motion for the cumulative function of this measure is derived and stationary solutions are constructed with the aid of derham type functional equations to select the physical states the time evolution of the distribution under a fixed boundary condition is investigated for an open multibaker chain of scattering type this system corresponds to a diffusive flow experiment through a slab of material for long times any initial distribution approaches the stationary one obeying fick s law at stationarity the intracell distribution is singular in the stable direction and expressed by the takagi function which is continuous but has no finite derivatives the result suggests that singular measures play an important role in the dynamical description of non equilibrium states

borel classes and closed games wadge type and hurewicz type results for each countable ordinal and pair ao al of disjoint analytic subsets of 2 we define a closed game j aos a1 and a complete iit subset ht of 2 such that i a winning strategy for player i constructs a 24 set separating ao from a1 and ii a winning strategy for player ii constructs a continuous map p 2 a u a1 with p l ao h applications of this construction include a proof in second order arithmetics of the statement every iit non ss set is ii complete an extension to all levels of a theorem of hurewicz about 22 sets a new proof of results of kunugui novikov bourgain and the authors on borel sets with sections of given class extensions of results of stern and kechris our results are valid in arbitrary polish spaces and for the classes in lavrentieff s and wadge s hierarchies introduction let e 2 be the usual cantor space and following the standard modern terminology of addison let e and fit denote respectively the borel subsets of e of additive and multiplicative borel class starting with e open and ti closed the aim of this paper is to characterize given a set a c 2 and one of the borel classes say r whether a is in r or not by a characterization we have in mind the finding of some mathematical object associated with a and which positively witnesses that a e r or that a 0 r let r be the class e t if the set a is in e a natural candidate for such a witness is simply one particular construction of a as a e set a list of sequences of open sets together with instructions on the operations to be performed to get a in t steps the problem of how to find such constructions which is deeply connected to the structural properties of sets in the plane with e sections has been extensively investigated in papers by novikov n and kunugui kun for 1 see also dellacherie de saint raymond sr 1 for t 2 bourgain b 1 b 2 and louveau lo 1 for t 3 louveau lo 2 for t 3 in case a is known to be borel in case a ranges over larger classes the problem is studied in papers by stern st kechris ke and louveau lo 3 received by the editors december 26 1985 and in revised form november 19 1986 these results have been presented at the annual meeting of the american mathematical society new orleans january 1986 in a special session on determinacy and large cardinals 1980 mclthemcltics subject clclssificcltion 1985 revision primary o2k30 04a15 28a05 secondary 54h05 26a21

similarity invariant signatures for partially occluded planar shapes a methodology is described for associating local invariant signature functions to smooth planar curves in order to enable their translation rotation and scale invariant recognition from arbitrarily clipped portions the suggested framework incorporates previous approaches based on locating inflections curvature extrema breakpoints and other singular points on planar object boundaries and provides a systematic way of deriving novel invariant signature functions based on curvature or cumulative turn angle of curves these new signatures allow the specification of arbitrarily dense feature points on smooth curves whose locations are invariant under similarity transformations the results are useful for detecting and recognizing partially occluded planar objects a key task in low level robot vision

correlation functions on the border lines of transient chaos chaotic properties of a repeller strongly influence the transient properties of a system close to it in particular the correlations in the transient regime in this paper results are presented for repellers of one dimensional maps having a fixed point whose lyapunov exponent agrees with the escape rate from the repeller it is proven that the corresponding natural measure of the repeller is a ensuremath delta function at the origin eigenfunctions of the frobenius perron operator are computed the correlation function is calculated near the situation of permanent chaos and anomalous decay of the correlations is found scaling properties are given on the route from a weak repeller to a nonrepeller the analytic results are supported by numerical calculations textcopyright 1996 the american physical society

on automatic threshold selection for polygonal approximations of digital curves polygonal approximation is a very common representation of digital curves a polygonal approximation depends on a parameter e which is the error value in this paper we present a method for an automatic selection of the error value e let e be a polygonal approximation of the original curve with an error value e we define a set of function ns e s s s such that for a given value of s ns e is the number of edges that contain at least s vertices in e the time complexity for computing the set of functions ns e s ss s is almost linear in n the number of vertices in in this paper we analyse the ns graph and show that for adequate values of s a wide plateau is expected to appear at the top of the graph this plateau corresponds to a stable state in the multi scale representation of e we show that the functions ns s s are a statistical representation of some kind of scale space image

sets of filter convergence of sequences of continuous functions abstract we consider sets of filter convergence and divergence to infinity of sequences of continuous functions for borel filters we characterize the sets of filter convergence of sequences of continuous functions on the real line for borel filters we also give sufficient and necessary conditions for results involving sets of filter divergence to infinity in particular we give the full description of such sets for the statistical convergence for metric spaces

shape representation by multiscale contour approximation an approach is presented for deriving qualitative descriptions of contours containing structures at different unknown scales the descriptions are in terms of straight arcs curved arcs with sign of curvature corners and points delimiting the arcs inflexion points and transitions from straight to curved furthermore the tangents at these points are derived the approach is based on the construction of a hierarchic family of polygons having the scale space property of causality structure can only disappear as scale goes from fine to coarse using the principle that structures that are stable over scale represent significant properties the features of the descriptive representations are then derived

fast decomposition of digital curves into polygons using the haar transform the perimeters of features in digital images are decomposed into sets of straight lines using the haar transform the haar basis functions preserve sharp changes in direction along the perimeter while still allowing smoothing the coefficients of the haar transform are simply related to the error between the polygonal decomposition and the true perimeter the haar transform can be calculated efficiently with a computational overhead linear in the number of data points

properties of border states of transient chaos transient chaos with absolutely continuous conditionally invariant measure is studied and a general solution of the frobenius perron equation is presented for complete one dimensional maps in particular properties of borderline situations exhibiting phase transitions within the framework of thermodynamical formalism are investigated

scale based detection of corners of planar curves a technique for detecting and localizing corners of planar curves is proposed the technique is based on gaussian scale space which consists of the maxima of absolute curvature of the boundary function presented at all scales the scale space of isolated simple and double corners is first analyzed to investigate the behavior of scale space due to smoothing and interactions between two adjacent corners the analysis shows that the resulting scale space contains line patterns that either persist terminate or merge with a neighboring line next the scale space is transformed into a tree that provides simple but concise representation of corners at multiple scales finally a multiple scale corner detection scheme is developed using a coarse to fine tree parsing technique the parsing scheme is based on a stability criterion that states that the presence of a corner must concur with a curvature maximum observable at a majority of scales experiments were performed to show that the scale space corner detector is reliable for objects with multiple size features and noisy boundaries and compares favorably with other corner detectors tested

the conjugacy problem in ergodic theory all common probability preserving transformations can be represented as elements of mpt the group of measure preserving transformations of the unit interval with lebesgue measure this group has a natural polish topology and the induced topology on the set of ergodic transformations is also polish our main result is that the set of ergodic elements t in mpt that are isomorphic to their inverse is a complete analytic set this has as a consequence the fact that the isomorphism relation is also a complete analytic set and in particular is not borel this is in stark contrast to the situation of unitary operators where the spectral theorem can be used to show that conjugacy relation in the unitary group is borel this result explains perhaps why the problem of determining whether ergodic transformations are isomorphic or not has proven to be so intractable the construction that we use is general enough to show that the set of ergodic t s with nontrivial centralizer is also complete analytic on the positive side we show that the isomorphism relation is borel when restricted to the rank one transformations which form a generic subset of mpt it remains an open problem to nd a good explicit method of checking when two rank one transformations are isomorphic in memoriam prior to the

matching of partially occluded planar curves abstract an algorithm for curves partial matching up to a similarity transformation is presented the matching is based on the total curvature graph which is the direction of the first derivative of the curve no higher derivatives are required the algorithm uses dynamic programming it is not dependent on reference or feature points the base step consists of a comparison of the total curvatures of neighborhoods the neighborhoods are dynamically chosen according to local characteristics of the curve in the recursive step ratios of lengths of matched curve segments are compared in order to extend the matches the matching is not sensitive to local deviations of the curves the algorithm is fast and simple its worst case time complexity is o mn where m and n are the number of points in the two digital curves experimental results are presented

simple maps with fractal diffusion coefficients we consider chains of one dimensional piecewise linear chaotic maps with uniform slope we study the diffusive behavior of an initially nonuniform distribution of points as a function of the slope of the map by solving the frobenius perron equation for markov partition values of the slope we relate the diffusion coefficient to eigenvalues of the topological transition matrix the diffusion coefficient obtained shows a fractal structure as a function of the slope of the map this result may be typical for a wide class of maps such as two dimensional sawtooth maps

on the detection of dominant points on digital curves a parallel algorithm is presented for detecting dominant points on a digital closed curve the procedure requires no input parameter and remains reliable even when features of multiple sizes are present on the digital curve the procedure first determines the region of support for each point based on its local properties then computes measures of relative significance e g curvature of each point and finally detects dominant points by a process of nonmaximum suppression this procedure leads to the observation that the performance of dominant points detection depends not only on the accuracy of the measure of significance but also on the precise determination of the region of support this solves the fundamental problem of scale factor selection encountered in various dominant point detection algorithms the inherent nature of scale space filtering in the procedure is addressed and the performance of the procedure is compared to those of several other dominant point detection algorithms using a number of examples

uniformisations partielles et criteres a la hurewicz dans le plan we give characterizations of the borel sets potentially in some wadge class among the borel sets with countable vertical sections of a product of two polish spaces to do this we use some partial uniformization results

optimum uniform piecewise linear approximation of planar curves two dimensional digital curves are often uniformly approximated by polygons or piecewise linear curves several algorithms have been proposed in the literature to find such curves we present an algorithm that finds a piecewise linear curve with the minimal number of segments required to approximate a curve within a uniform error with fixed initial and final points we compare our optimal algorithm to several suboptimal algorithms with respect to the number of linear segments required in the approximation and the execution time of the algorithm

diffusion effusion and chaotic scattering an exactly solvable liouvillian dynamics we study diffusion and chaotic scattering in a chain of baker maps coupled together which forms an area preserving mapping of an infinitely extended strip onto itself this exactly solvable mapping sustains chaotic behaviors and diffusion processes the relationship between the diffusion coefficient the lyapunov exponent and the entropy per unit time is derived the long lived classical resonances of the liouville evolution operator are proved to converge toward the eigenvalues of the phenomenological diffusion equation in this sense there is a quasi isomorphism between the resonance spectrum of the liouville evolution and the eigenvalue spectrum of the phenomenological diffusion equation furthermore we show that a fractal repeller is associated to each non equilibrium state in the isolated and finite multibaker chain the nonequilibrium states are all unstable with respect to the equilibrium validating a weak form of the second principle of thermodynamics for the present dynamical system consequences of nonequilibrium fractals on classical measurements are discussed we then describe the open multibaker chain as a scattering system fractal properties of chaotic scattering are here shown to be related to diffusion in the chain

using simple decomposition for smoothing and feature point detection of noisy digital curves this correspondence presents an algorithm for smoothed polygonal approximation of noisy digital planar curves and feature point detection the resulting smoothed polygonal representation preserves the signs of the curvature function of the curve the algorithm is based on a simple decomposition of noisy digital curves into a minimal number of convex and concave sections the location of each separation point is optimized yielding the minimal possible distance between the smoothed approximation and the original curve curve points within a convex concave section are discarded if their angle signs do not agree with the section sign and if the resulted deviations from the curve are less than a threshold spl epsi which is derived automatically inflection points are curve points between pairs of convex concave sections and cusps are curve points between pairs of convex convex or concave concave sections corners and points of local minimal curvature are detected by applying the algorithm to respective total curvature graphs the detection of the feature points is based on properties of pairs of sections that are determined in an adaptive manner rather than on properties of single points that are based on a fixed size neighborhood the detection is therefore reliable and robust complexity analysis and experimental results are presented

transient chaos and critical states in generalized baker maps generalized multibaker maps are introduced to model dissipative systems which are spatially extended only in certain directions and escape of particles is allowed in other ones effects of nonlinearity are investigated by varying a control parameter emphasis is put on the appearance of the critical state representing the borderline of transient chaos where anomalous behavior sets in the investigations extend to the conditionally invariant and the related natural measures and to transient diffusion in normal and critical states as well permanent chaos is also considered as a special case

the curvature primal sketch in this paper we introduce a novel representation of the significant changes in curvature along the bounding contour of planar shape we call the representation the curvature primal sketch because of the close analogy to the primal sketch representation advocated by marr for describing significant intensity changes we define a set of primitive parameterized curvature discontinuities and derive expressions for their convolutions with the first and second derivatives of a gaussian we describe an implemented algorithm that computes the curvature primal sketch by matching the multiscale convolutions of a shape and illustrate its performance on a set of tool shapes several applications of the representation are sketched

generalized markov coarse graining and spectral decompositions of chaotic piecewise linear maps spectral decompositions of the evolution operator for probability densities are obtained for the most general one dimensional piecewise linear markov maps and a large class of repellers the eigenvalues obtained with respect to the space of functions piecewise analytic over the minimal markov partition equal the reciprocals of the zeros of the ruelle zeta functions the logarithms of the zeros correspond to the decay rates of time correlation functions of analytic observables when the system is mixing the space can also be extended to include piecewise analytic observables permitted to have discontinuities at the elements of any given periodic orbit s so that local behavior of observables can be considered the new spectra associated with the extension are surprisingly simple and are related to the relative stability factors of the given orbit s finally arbitrarily slowly decaying periodic and aperiodic nonanalytic eigenmodes are constructed

evaluation of interest point detectors many different low level feature detectors exist and it is widely agreed that the evaluation of detectors is important in this paper we introduce two evaluation criteria for interest points repeatability rate and information content repeatability rate evaluates the geometric stability under different transformations information content measures the distinctiveness of features different interest point detectors are compared using these two criteria we determine which detector gives the best results and show that it satisfies the criteria well

repellers semi attractors and long lived chaotic transients abstract we study the chaotic transients observed in many deterministic systems in general they are related to strange repellers or semi attractors if they are repelling in some and attracting in other directions we propose formulas relating the average life time of the transient to dimensions of the repeller and to lyapunov exponents of the flow on it the formulas are tested numerically in a number of cases

motion estimation of two dimensional objects based on the straight line hough transform a new approach abstract in this paper we propose a novel technique for estimating motion of two dimensional objects the technique involves representing objects uniquely by straight line approximations of the boundary p and estimating motion from shifts in the p space the straight line approximation of the boundary is obtained by detecting dominant points during contour tracing and then approximating the pixels between adjacent dominant points along the contour by a straight line using the hough transform we have developed a computationally efficient way of detecting dominant points in a single pass during contour tracing the proposed object representation enables us to determine the translation and rotation with significantly fewer computations compared with existing methods our method can also detect the motion parameters correctly for partially overlapping objects