on the minimum ropelength of knots and links the ropelength of a knot is the quotient of its length by its thickness the radius of the largest embedded normal tube around the knot we prove existence and regularity for ropelength minimizers in any knot or link type these are c 1 1 curves but need not be smoother we improve the lower bound for the ropelength of a nontrivial knot and establish new ropelength bounds for small knots and links including some which are sharp

automatic image segmentation by dynamic region growth and multiresolution merging image segmentation is a fundamental task in many computer vision applications in this paper we propose a new unsupervised color image segmentation algorithm which exploits the information obtained from detecting edges in color images in the cie l a b color space to this effect by using a color gradient detection technique pixels without edges are clustered and labeled individually to identify some initial portion of the input image content elements that contain higher gradient densities are included by the dynamic generation of clusters as the algorithm progresses texture modeling is performed by color quantization and local entropy computation of the quantized image the obtained texture and color information along with a region growth map consisting of all fully grown regions are used to perform a unique multiresolution merging procedure to blend regions with similar characteristics experimental results obtained in comparison to published segmentation techniques demonstrate the performance advantages of the proposed method

principles of multivariate analysis a user s perspective part i looking at multivariate data part ii samples populations and models part iii analysing ungrouped data part iv analysing grouped data part v analysing association among variables appendix some basic matrix theory a1 definitions a2 elementary arithmetic operations a3 determinants and inverses a4 quadratic forms a5 latent roots and vectors a6 matrix square root a7 partitioned matrices a8 vector differentiation references index

a cellular nerve for higher categories abstract we realise joyal cell category as a dense subcategory of the category of categories the associated cellular nerve of an category extends the well known simplicial nerve of a small category cellular sets like simplicial sets carry a closed model structure in quillen s sense with weak equivalences induced by a geometric realisation functor more generally there exists a dense subcategory a of the category of a algebras for each operad a in batanin s sense whenever a is contractible the resulting homotopy category of a algebras i e weak categories is equivalent to the homotopy category of compactly generated spaces

weak complicial sets i basic homotopy theory this paper develops the foundations of a simplicial theory of weak categories which builds upon the insights originally expounded by ross street in his 1987 paper on oriented simplices the resulting theory of weak complicial sets provides a common generalisation of the theories of strict categories kan complexes and joyal s quasi categories we generalise a number of results due to the current author with regard to complicial sets and strict categories to provide an armoury of well behaved technical devices such as joins and gray tensor products which will be used to study the weak category theory of these structures in a series of companion papers in particular we establish their basic homotopy theory by constructing a quillen model structure on the category of stratified simplicial sets whose fibrant objects are the weak complicial sets as a simple corollary of this work we provide an independent construction of joyal s model structure on simplicial sets for which the fibrant objects are the quasi categories

inner horns for 2 quasi categories abstract dimitri ara s 2 quasi categories which are certain presheaves over andre joyal s 2 cell category 2 are an example of a concrete model that realises the abstract notion of 2 category in this paper we prove that the 2 quasi categories and the fibrations into them can be characterised using the inner horn inclusions and the equivalence extensions introduced by david oury these maps are more tractable than the maps that ara originally used and therefore our result can serve as a combinatorial foundation for the study of 2 quasi categories

iterated wreath product of the simplex category and iterated loop spaces generalising segal s approach to 1 fold loop spaces the homotopy theory of n fold loop spaces is shown to be equivalent to the homotopy theory of reduced tn spaces where tn is an iterated wreath product of the simplex category a sequence of functors from tn to g allows for an alternative description of the segal spectrum associated to a g space this yields a canonical reduced tn set model for each eilenberg maclane space

non parametric detection of the number of signals hypothesis testing and random matrix theory detection of the number of signals embedded in noise is a fundamental problem in signal and array processing this paper focuses on the non parametric setting where no knowledge of the array manifold is assumed first we present a detailed statistical analysis of this problem including an analysis of the signal strength required for detection with high probability and the form of the optimal detection test under certain conditions where such a test exists second combining this analysis with recent results from random matrix theory we present a new algorithm for detection of the number of sources via a sequence of hypothesis tests we theoretically analyze the consistency and detection performance of the proposed algorithm showing its superiority compared to the standard minimum description length mdl based estimator a series of simulations confirm our theoretical analysis

map learning with uninterpreted sensors and effectors this dissertation presents a set of methods by which a learning agent called a critter can learn a sequence of increasingly abstract and powerful interfaces to control a robot whose sensorimotor apparatus and environment are initially unknown the result of the learning is a rich hierarchical model of the robot s world its sensorimotor apparatus and environment the learning methods rely on generic properties of the robot s world such as almost everywhere smooth effects of actions on sensory features at the lowest level of the hierarchy the critter analyzes the effects of its actions in order to define control signals one for each of the robot s degrees of freedom it uses a generate and test approach to define sensory features that capture important aspects of the environment it uses linear regression to learn action models that characterize context dependent effects of the control signals on the learned features it uses these models to define high level control laws for finding and following paths defined using constraints on the learned features the critter abstracts these control laws which interact with the continuous environment to a finite set of actions that implement discrete state transitions at this point the critter has abstracted the robot s world to a finite state machine and can use existing methods to learn its structure

embeddedness of minimal surfaces with total boundary curvature at most 4 pi this paper proves that classical minimal surfaces of arbitrary topological type with total boundary curvature at most 47r must be smoothly embedded related results are proved for varifolds and for soap film surfaces

a homotopy coherent cellular nerve for bicategories abstract the subject of this paper is a nerve construction for bicategories introduced by leinster which defines a fully faithful functor from the category of bicategories and normal pseudofunctors to the category of presheaves over joyal s category 2 we prove that the nerve of a bicategory is a 2 quasi category a model for 2 categories due to ara and moreover that the nerve functor restricts to the right part of a quillen equivalence between lack s model structure for bicategories and a bousfield localisation of ara s model structure for 2 quasi categories we deduce that lack s model structure for bicategories is quillen equivalent to rezk s model structure for 2 2 spaces on the category of simplicial presheaves over 2 to this end we construct the homotopy bicategory of a 2 quasi category and prove that a morphism of 2 quasi categories is an equivalence if and only if it is essentially surjective on objects and fully faithful we also prove a quillen equivalence between ara s model structure for 2 quasi categories and the hirschowitz simpson pellissier model structure for quasi category enriched segal categories from which we deduce a few more results about 2 quasi categories including a conjecture of ara concerning weak equivalences of 2 categories

reedy categories and the vartheta construction we use the notion of multi reedy category to prove that if mathcal c is a reedy category then vartheta mathcal c is also a reedy category this result gives a new proof that the categories vartheta n are reedy categories we then define elegant reedy categories for which we prove that the reedy and injective model structures coincide

on k hulls and related problems for any set x of points in any dimension and any k 1 2 cdots we introduce the concept of the k hull of x the k hull is the set of points p such that for any hyperplane containing p there

rao s distance between normal populations that have common principal components summary a theoretical derivation is provided of rao s distance between two normal populations that have different means and whose dispersion matrices satisfy the common principal component model according to this model the dispersion matrices share the same eigenvectors but are permitted to have arbitrary eigenvalues this structure is useful for modeling a variety of real data sets so the theoretical result has important practical implications it is illustrated on a biometric data set

new perspectives on self linking abstract we initiate the study of classical knots through the homotopy class of the nth evaluation map of the knot which is the induced map on the compactified n point configuration space sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth mapping space model for knots we compute the homotopy types of the first three mapping space models showing that the third model gives rise to an integer valued invariant we realize this invariant in two ways in terms of collinearities of three or four points on the knot and give some explicit computations we show this invariant coincides with the second coefficient of the conway polynomial thus giving a new geometric definition of the simplest finite type invariant finally using this geometric definition we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot

subset selection in multiple linear regression models a hybrid of genetic and simulated annealing algorithms the question of variable selection in a multiple linear regression model is a major open research topic in statistics the subset selection problem in multiple linear regression deals with the selection of a minimal subset of input variables without loss of explanatory power in this paper we adapt the genetic and simulated annealing algorithms for variable selection in multiple linear regression the performance of this hybrid heuristic method is compared to those obtained by forward selection backward elimination and classical genetic algorithm search a comparative analysis on the literature data sets and simulation data shows that our hybrid heuristic method may suggest efficient alternative to traditional subset selection methods for the variable selection problem in multiple linear regression models

biological shape analysis by digital curvature univ sao paulo ifsc inst phys sao carlos cybernet vis res grp br 13560970 sao carlos sp brazil

solving the protein sequence metric problem biological sequences are composed of long strings of alphabetic letters rather than arrays of numerical values lack of a natural underlying metric for comparing such alphabetic data significantly inhibits sophisticated statistical analyses of sequences modeling structural and functional aspects of proteins and related problems herein we use multivariate statistical analyses on almost 500 amino acid attributes to produce a small set of highly interpretable numeric patterns of amino acid variability these high dimensional attribute data are summarized by five multidimensional patterns of attribute covariation that reflect polarity secondary structure molecular volume codon diversity and electrostatic charge numerical scores for each amino acid then transform amino acid sequences for statistical analyses relationships between transformed data and amino acid substitution matrices show significant associations for polarity and codon diversity scores transformed alphabetic data are used in analysis of variance and discriminant analysis to study dna binding in the basic helix loop helix proteins the transformed scores offer a general solution for analyzing a wide variety of sequence analysis problems

the second hull of a knotted curve the convex hull of a set k in space consists of points which are in a certain sense su rrounded by k when k is a closed curve we define its higher hulls consisting of po ints which are multiply surrounded by the curve our main theorem shows that if a curve is knotted then it has a nonempty second hull this provides a new proof of the fary milnor theorem that every knotted curve has total curvature at least 4 a space curve must loop around at least twice to become knotted this intuitive idea was captured in the celebrated fary milnor theorem which says the total curvature of a knot ted curve k is at least 4 we prove a stronger result that there is a certain region of space doubly enclosed in a preci se sense by k we call this region the second hull of k the convex hull of a connected set k is characterized by the fact that every plane through a point in the hull must in teresect k if k is a closed curve then a generic plane inter sects k an even number of times so the convex hull is the set of points through which every plane cuts k twice extending this we introduce a definition which measures the degree to which points are enclosed by k a point p is said to be in the n th hull of k if every plane through p intersects k at least 2n times a space curve is convex and planar if and only if it has the two piece property every plane cuts it into at most two pieces this has been generalized by kuiper among others see cc97 in the study of tight and taut submanifolds it was milnor mil50 who first observed that for a knotted curve there are planes in every direction which cut it four times our main theorem strengthens this to find points through which every plane cuts the knot four times more precisely it says that any curve k with a knotted inscribed polygon in particular any knotted curved of finite total curvature has a nonempty second hull as shown for a trefoil in figure 1 fur thermore in a certain sense this hull must extend across th e knot using quadrisecants we get an alternate approach to second hulls and in particular we show that the second hull of a generic prime knot k is not contained in any embedded normal tube aroundk we use some ideas from integral geometry to show that the total curvature of a space curve with nonempty n th hull is at least 2 n thus our theorem provides an alternate proof of the fary milnor theorem we do not yet know any topological knot type which is guaranteed to have a nonempty third hull though we suspect that this is true for the 3 4 torus knot see figure 5 it w ould

a distance between elliptical distributions based in an embedding into the siegel group this paper describes two different embeddings of the manifolds corresponding to many elliptical probability distributions with the informative geometry into the manifold of positive definite matrices with the siegel metric generalizing a result published previously elsewhere these new general embeddings are applicable to a wide class of elliptical probability distributions in which the normal t student and cauchy are specific examples a lower bound for the rao distance is obtained which is itself a distance and through these embeddings a number of statistical tests of hypothesis are derived