open 3 manifolds whose fundamental groups have infinite center and a torus theorem for 3 orbifolds our main result is a characterization of open seifert fibered 3 manifolds in terms of the fundamental group and large scale geometric properties of a triangulation as an application we extend the seifert fiber space theorem and the torus theorem to a class of 3 orbifolds

centroid neural network with a divergence measure for gpdf data clustering an unsupervised competitive neural network for efficient clustering of gaussian probability density function gpdf data of continuous density hidden markov models cdhmms is proposed in this paper the proposed unsupervised competitive neural network called the divergence based centroid neural network dcnn employs the divergence measure as its distance measure and utilizes the statistical characteristics of observation densities in the hmm for speech recognition problems while the conventional clustering algorithms used for the vector quantization vq codebook design utilize only the mean values of the observation densities in the hmm the proposed dcnn utilizes both the mean and the covariance values when compared with other conventional unsupervised neural networks the dcnn successfully allocates more code vectors to the regions where gpdf data are densely distributed while it allocates fewer code vectors to the regions where gpdf data are sparsely distributed when applied to korean monophone recognition problems as a tool to reduce the size of the codebook the dcnn reduced the number of gpdfs used for code vectors by 65 3 while preserving recognition accuracy experimental results with a divergence based k means algorithm and a divergence based self organizing map algorithm are also presented in this paper for a performance comparison

cusp size bounds from singular surfaces in hyperbolic 3 manifolds singular maps of surfaces into a hyperbolic 3 manifold are utilized to find upper bounds on meridian length l curve length and maximal cusp volume for the manifold this allows a proof of the fact that there exist hyperbolic knots with arbitrarily small cusp density and that every closed orientable 3 manifold contains a knot whose complement is hyperbolic with maximal cusp volume less than or equal to 9 we also find particular upper bounds on meridian length l curve length and maximal cusp volume for hyperbolic knots in s 3 depending on crossing number particular improved bounds are obtained for alternating knots

fixed smoothing asymptotics for time series 1 in this paper we derive higher order edgeworth expansions for the finite sample distributions of the subsampling based t statistic and the wald statistic in the gaussian location model under the so called fixed smoothing paradigm in particular we show that the error of asymptotic approximation is at the order of the reciprocal of the sample size and obtain explicit forms for the leading error terms in the expansions the results are used to justify the second order correctness of a new bootstrap method the gaussian dependent bootstrap in the context of gaussian location model 1 introduction many economic and financial applications involve time series data with autocorrelation and heteroskedasticity properties often the unknown dependence structure is not the chief object of interest but the inference on the parameter of interest involves the estimation of unknown dependence in stationary time series models estimated by generalized method of moments gmm robust inference is typically accomplished by consistently estimating the asymptotic covariance matrix which is proportional to the long run variance lrv matrix of the estimating equations or moment conditions defining the estimator using a kernel smoothing method in the econometrics and statistics literature the bandwidth parameter truncation lag involved in the kernel smoothing method is assumed to grow slowly with sample size in order to achieve consistency the inference is conducted by plugging in a covariance matrix estimator that is consistent under heteroskedasticity and autocorrelation this approach dates back to newey and west 25 and andrews 1 recently kiefer and vogelsang 13 kv hereafter developed an alternative first order asymptotic theory for the hac heteroskedasticity and autocorrelation consistent based robust inference where the proportion of the bandwidth involved in the hac estimator to the sample size t denoted as b is held fixed in the asymptotics under the fixed b asymptotics the hac estimator converges to a nondegenerate yet nonstandard limiting distribution the tests based on the fi xed b asymptotic approximation were shown to enjoy better finite sample properties than the tests based on the small b asymptotic theory

weighted centroid neural network for edge preserving image compression an edge preserving image compression algorithm based on an unsupervised competitive neural network is proposed the proposed neural network the called weighted centroid neural network wcnn utilizes the characteristics of image blocks from edge areas the mean residual vector quantization m rvq scheme is utilized in this proposed approach as the framework of the proposed algorithm the edge strength of image block data is utilized as a tool to allocate the proper code vectors in the proposed wcnn the wcnn successfully allocates more code vectors to the image block data from edge area while it allocates less code vectors to the image black data from shade or non edge area when compared to conventional neural networks based on vq algorithm as a result a simple application of wcnn to an image compression problem gives improved edge characteristics in reconstructed images over conventional neural network based on vq algorithms such as self organizing map som and adaptive som

evaluation of stability of k means cluster ensembles with respect to random initialization many clustering algorithms including cluster ensembles rely on a random component stability of the results across different runs is considered to be an asset of the algorithm the cluster ensembles considered here are based on k means clusterers each clusterer is assigned a random target number of clusters k and is started from a random initialization here we use 10 artificial and 10 real data sets to study ensemble stability with respect to random k and random initialization the data sets were chosen to have a small number of clusters two to seven and a moderate number of data points up to a few hundred pairwise stability is defined as the adjusted rand index between pairs of clusterers in the ensemble averaged across all pairs nonpairwise stability is defined as the entropy of the consensus matrix of the ensemble an experimental comparison with the stability of the standard k means algorithm was carried out for k from 2 to 20 the results revealed that ensembles are generally more stable markedly so for larger k to establish whether stability can serve as a cluster validity index we first looked at the relationship between stability and accuracy with respect to the number of clusters k we found that such a relationship strongly depends on the data set varying from almost perfect positive correlation 0 97 for the glass data to almost perfect negative correlation 0 93 for the crabs data we propose a new combined stability index to be the sum of the pairwise individual and ensemble stabilities this index was found to correlate better with the ensemble accuracy following the hypothesis that a point of stability of a clustering algorithm corresponds to a structure found in the data we used the stability measures to pick the number of clusters the combined stability index gave best results

pl equivariant surgery and invariant decompositions of 3 manifolds using normal surface theory h j2 we introduce the notion of least weight normal surfaces the weight of a normal surface is a nonnegative integer invariant of the normal isotopy class of the surface if we focus on a particular class of normal surfaces and choose representatives which minimize the weight over the class then we have least weight normal surfaces it is remarkable how these least weight normal surfaces exhibit many of the same useful properties as least area minimal surfaces they provide a piecewise linear pl environment to obtain the recent topological results coming from the analysis and geometry of least area surfaces in an impressive series of papers meeks and yau m y i m y z m y 3 m y m y meeks simon and yau m s y scott s meeks and scott m s and freedman hass and scott f h s introduced the analysis and geometry of least area surfaces into the study of topological questions about 3 manifolds the consequences have led to the resolution of many outstanding problems in the topology of 3 manifolds since most of the problems being solved with these new techniques are topological in nature it has been felt that there should be a more topological approach to the proofs of these results besides the bulk of the work in the theory of

asymptotic expansions for sums of block variables under weak dependence let x i be a sequence of random vectors and y in f in x i l be zero mean block variables where x i l x i x i l 1 i 1 are overlapping blocks of length l and where f in are borel measurable functions this paper establishes valid joint asymptotic expansions of general orders for the joint distribution of the sums n i 1 x i and n i 1 y in under weak dependence conditions on the sequence x i i when the block length t grows to infinity in contrast to the classical edgeworth expansion results where the terms in the expansions are given by powers of n 1 2 the expansions derived here are mixtures of two series one in powers of n 1 2 and the other in powers of n l 1 2 applications of the main results to i expansions for studentized statistics of time series data and ii second order correctness of the blocks of blocks bootstrap method are given

nonzero degree maps between closed orientable three manifolds this paper adresses the following problem given a closed orientable three manifold m are there at most finitely many closed orientable three manifolds 1 dominated by m we solve this question for the class of closed orientable graph manifolds more precisely the main result of this paper asserts that any closed orientable graph manifold 1 dominates at most finitely many orientable closed three manifolds satisfying the poincare thurston geometrization conjecture to prove this result we state a more general theorem for haken manifolds which says that any closed orientable three manifold m1 dominates at most finitely many haken manifolds whose gromov simplicial volume is sufficiently close to that of m

asymptotic expansions for sums of weakly dependent random vectors it is shown that formal edgeworth expansions are valid for sums of weakly dependent random vectors the error of approximation has ordero n s 2 2 if the strong mixing coefficients in iii are decreasing at an exponential rate the above conditions can easily be checked and are often satisfied when the sequence of random vectors is a gaussian or a markov or an autoregressive process explicit formulas are given for the distribution of finite fourier transforms of a strictly stationary time series

asymptotic distribution of statistics in time series veritable conditions are given for the validity of formal edgeworth expansions for the distribution of sums x 1 x n where x i f z i z i p 1 and z 1 z 2 is a strict sense stationary sequence that can be written as z j g e j k k 0 with an iid sequence e i of innovations these models include nonlinear functions of arma processes z i as well as certain nonlinear ar processes the results apply to many statistics in nonlinear time series models

on mapping class groups of contractible open 3 manifolds let w be an irreducible eventually end irreducible contractible open 3 manifold other than r 3 and let v be a good exhaustion of w let h w v be the subgroup of the mapping class group h w which is eventually carried by v this paper shows how to compute h w v in terms of the mapping class groups of certain compact 3 manifolds associated to v the computation is carried out for a genus two example and for the classical genus one example of whitehead for these examples h w h w v

second order correctness of the blockwise bootstrap for stationary observations we show that the blockwise bootstrap approximation for the distribution of a studentized statistic computed from dependent data is second order correct provided we choose an appropriate variance estimator we also show how to adapt the bc a confidence interval of efron to the dependent case for the proofs we extend the results of gotze and hipp on the validity of the formal edgeworth expansion for a sum to the studentized mean

refinements in asymptotic expansions for sums of weakly dependent random vectors let s0 denote the nth normalized partial sum of a sequence of mean zero weakly dependent random vectors this paper gives asymptotic expansions for ef sn under weaker moment conditions than those of g6tze and hipp 1983 it is also shown that an expansion for ef sn with an error term o n s 2 2 is valid without any cramer type condition if f has partial derivatives of order s 1 only this settles a conjecture of g6tze and hipp in their 1983 paper

hyperbolic knots spanning accidental seifert surfaces of arbitrarily high genus a method for constructing hyperbolic knots each of which bounds accidental incompressible seifert surfaces of arbitrarily high genus is given

edgeworth expansions for studentized statistics under weak dependence in this paper we derive valid edgeworth expansions for studentized versions of a large class of statistics when the data are generated by a strongly mixing process under dependence the asymptotic variance of such a statistic is given by an infinite series oflag covariances and therefore studentizing factors i e estimators of the asymptotic standard error typically involve an increasing number say l of lag covariance estimators which are themselves quadratic functions of the observations the unboundedness of the dimension l of these quadratic functions makes the derivation and the form of the expansions nonstandard it is shown that in contrast to the case of the studentized means under independence the derived edgeworth expansion is a superposition of three distinct series respectively given by one in powers of n 1 2 one in powers of n l 1 2 resulting from the standard error of the studentizing factor and one in powers of the bias of the studentizing factor where n denotes the sample size

finite group actions on 3 manifolds if g is a finite group acting smoothly on a closed surface f it is well known that g leaves invariant some riemannian metric of constant curvature on f thus any action of g on the 2 sphere s 2 is conjugate in diff s 2 to an orthogonal action if g acts on the torus sx s 1 there is a g invariant flat metric on s a s 1 and if g acts on a surface f with negative euler number then f admits a g invariant hyperbolic metric recently thurston th 1 th 2 th 3 has described the eight 3 dimensional geometries which provide geometric structures for closed 3 manifolds in the same way that the 2 sphere s 2 the euclidean plane e 2 and the hyperbolic plane h 2 provide geometric structures for surfaces see also the survey article by scott sc4 thurston also conjectured that if m is a closed 3 manifold with a geometric structure modelled on one of these eight geometries say x then any smooth action of a finite group g on m should leave invariant some metric on m inducing the geometry x we will say that g preserves the geometric structure on m in this case it should be noted that the restriction to smooth actions of g on m is essential for bing bi showed that there are involutions of s 3 whose fixed set is a wild 2 sphere however in dimension two it was proved by eilenberg ei that any action of a finite group on a surface is conjugate to a smooth action in this paper our main result asserts that thurston s conjecture holds for five of the eight geometries the result is the following

a census of cusped hyperbolic 3 manifolds the census provides a basic collection of noncompact hyperbolic 3 manifolds of finite volume it contains descriptions of all hyperbolic 3 manifolds obtained by gluing the faces of at most seven ideal tetrahedra additionally various geometric and topological invariants are calculated for these manifolds the findings are summarized and a listing of all manifolds appears in the microfiche supplement

nielsen realization for 3 manifolds containing two sided projective planes for compact irreducible sufficiently large 3 manifolds containing 2 sided projective planes we consider the following realization problem given a finite subgroup of the outer automorphism group of the fundamental group is there a finite group of homeomorphisms which induces this subgroup

3 manifolds which admit finite group actions we prove several results which support the following conjectures 1 any smooth action of a finite group on a geometric 3 manifold can be conjugated to preserve the geometric structure 2 every irreducible closed 3 manifold m with infinite 1 m is finitely covered by a haken 3 manifold

general m estimates for contaminated p th order autoregressive processes consistency and asymptotic normality this work is concerned with simultaneous estimation of coefficients and a scale parameter of a p th order autoregressive process x t the observations are y t v t z t 1 v t x t where z t is a contaminating process and v t represents the proportion of contamination if x t or z t have heavy tails both least squares estimates and ordinary m estimates are seriously affected under general conditions we prove consistency and asymptotic normality of a general class of m estimates which contains some m estimates studied by denby and martin 6

the existence of maps of nonzero degree between aspherical 3 manifolds for any two closed orientable surfaces there is a map of nonzero degree from one to another this result is based on the fact that the orientable closed surface are classified completely there is no such a classification in dimension 3 in this paper all closed aspherical 3 manifolds are divided into eight classes a complete answer of the existences of nonzero degree maps between 3 manifolds in the different classes is obtained in a reasonable sense the existence of nonzero degree maps between 3 manifolds in the same class are also discussed there are several results of independent interest it is noted that the existence of a map of nonzero degree between two 3 manifolds is a very restrictive condition posed on those two 3 manifolds which is much different from the case of dimension 2 the shape of our work is certainly influenced by thurston however both the statements and the proofs of our theorems do not involve the thurston s geometries we understand that if we state the results in the category of haken manifolds and thurston s geometrical manifolds and invoke the g r o m o v norm of hyperbol ic 3 manifold see chap 6 of r t l and chap 4 of gro the proof of some result in the paper can be simplified we present our work in the way below for two reasons a we prefer to prove all results in a uniform elementary way in the sense of using only the fundamental groups and the incompressible surfaces of 3 manifolds b since the thurston s most famous conjecture that every atoroidal closed irreducible 3 manifold has a hyperbolic structure is still open we get the results in a theoretically wider category there are 5 sections after the introduction all manifolds and maps under consideration are piece wise linear there is nothing lost under this assumption see chap 1 of hi in the first four sections of these five all 3 manifolds are orientable and irreducible section 1 states the theorems proved in the paper section 2 provides results which are used to prove the theorems section 3 and sect 4 are the proofs of the theorems various examples are given in sect 4

minimal index torsion free subgroups of kleinian groups a kleinian group is a discrete subgroup of psl 2 c the full group of orientation preserving isometries of 3 dimensional hyperbolic space in the language of t1 q h3 is a hyperbolic 3 orbifold that is a metric 3 orbifold in which all sectional curvatures are 1 and for which is the orbifold fundamental group see t1 for further details a fuchsian group is a discrete subgroup of psl 2 r and as such acts discontinuously on the hyperbolic plane we define to be of finite co volume resp co area if the volume resp area of the quotient orbifold q is finite by selberg s lemma if is a kleinian resp fuchsian group of finite covolume resp co area it contains a torsion free subgroup of finite index by definition any torsion free subgroup cannot contain any finite subgroups of so the index must be a multiple of the lowest common multiple of all orders of finite subgroups of see cfjr for a proof being of finite co volume implies there are only finitely many conjugacy classes of finite subgroups thus we make the following

the π orbifold group of a link on associe a un enlacement l dans s 3 un groupe o l appele le groupe de variete d orbite de l on etudie certaines proprietes du groupe o l reliees a la geometrie de l enlacement l

one fixed point actions and homology 3 spheres one of the best known three dimensional manifolds is the poincare homology three sphere 1 2 3 5 this closed 3 manifold has the same homology groups as the standard sphere s3 but is not simply connected see pnc p 106 or br2 section 1 8 numerous investigations during the past century have shown that this manifold has many remarkable properties several equivalent descriptions of 1 2 3 5 are summarized in kis from the viewpoint of transformation groups one noteworthy property is that 1 2 3 5 is the only nonsimply connected homology sphere admitting a transitive action of a compact lie group brl the manifold 1 2 3 5 also figures importantly in regularity questions for group actions on spherelike manifolds linear or orthogonal actions of finite groups on spheres are generally viewed as the simplest and most regular examples of such actions and one of the themes of transformation groups is to determine the extent to which arbitrary actions on spherelike manifolds resemble linear actions one distinguishing feature of linear actions is that their fixed point sets if nonempty are also spheres the well known results of p a smith cf br2 show that continuous p group actions on homology spheres all resemble linear actions in this respect i e the fixed point sets are always zp homology cohomology spheres on the other hand outside the family of p groups there are many exotic group actions perhaps the most elementary example of this type involves 1 2 3 5 if a is the subgroup of the rotation group so3 given by the symmetries of a regular icosahedron so that a is isomorphic to the alternating group a5 then the quotient space s03 a is the poincare homology 3 sphere and the induced action of a5 a has exactly one fixed point this example suggests that general actions of finite groups on homology spheres can be quite different from linear actions and much of the work on exotic finite group actions during the past three decades was at least partially motivated by the existence of this group action

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

sutured manifold hierarchies essential laminations and dehn surgery x0 introduction a compact orientable surface f with nonnegative euler characteristic is either a sphere a disk a torus or an annulus if a 3 manifold m contains such an essential surface then it is said to be reducible reducible toroidal or annular respectively any such surface can be used to decompose the manifold further into simpler manifolds we say that m is a simple manifold if it has no such surfaces a simple manifold is expected to have a nice geometric structure if m has nonempty boundary then the geometrization theorem of thurston for haken manifolds says that m with boundary tori removed admits a nite volume hyperbolic structure with totally geodesic boundary when m has no boundaries thurston s geometrization conjecture asserts that m is either hyperbolic or is a seifert b er space with orbifold a sphere with at most 3 cone points suppose t is a torus boundary component of m we use m to denote the manifold obtained by dehn lling on t so that the slope on t bounds a disk in the dehn lling solid torus when m e k is the exterior of a knot k in s 3 denote m by k and call it the manifold obtained by surgery on the knot k it is well known that if m is simple then there are only nitely many dehn llings on t which produce non simple manifolds if m 1 and m 2 are non simple manifolds then the geometric intersection number between 1 and 2 denoted by 1 2 is proved to be at most 8 by gordon gor1 there are 10 dieren t cases according to the types of nonnegative euler characteristic surfaces in m i in many cases the upper bounds for 1 2 have been established see table 0 1

least area incompressible surfaces in 3 manifolds let m be a riemannian manifold and let f be a closed surface a map f f m is called least area if the area of f is less than the area of any homotopic map from f to m note that least area maps are always minimal surfaces but that in general minimal surfaces are not least area as they represent only local stationary points for the area function the existence of least area immersions in a homotopy class of maps has been established when the homotopy class satisfies certain injectivity conditions on the fundamental group 18 17 in this paper we shall consider the possible singularities of such immersions our results show that the general philosophy is that least area surfaces intersect least meaning that the intersections and self intersections of least area immersions are as small as their homotopy classes allow when measured correctly one should note that evidence supporting this view had been found by meeks yau in their embedding theorems for minimal disks and 2 spheres 13 143 our main result asserts that if a least area immersion is homotopic to an embedding then it has no self intersections which clearly exemplifies the above philosophy the precise result is the following

tapered block bootstrap summary we introduce and study tapered block bootstrap methodology that yields an improvement over the well known block bootstrap for time series of kiinsch 1989 the asymptotic validity and the favourable bias properties of the tapered block bootstrap are shown the important practical issues of optimally choosing the window shape and the block size are addressed in detail while some finite sample simulations are presented validating the good performance of the tapered block bootstrap