sequence folding lattice tiling and multidimensional coding folding a sequence b into a multidimensional box is a well known method which is used as a multidimensional coding technique the operation of folding is generalized in a way that the sequence b can be folded into various shapes and not just a box the novel definition of folding is based on a lattice tiling for the given shape s and a direction in the d dimensional integer grid necessary and sufficient conditions that a lattice tiling for s combined with a direction define a folding of a sequence into s are derived the immediate and most impressive applications are some new lower bounds on the number of dots in two dimensional synchronization patterns asymptotically optimal such patterns were known only for rectangular shapes we show asymptotically optimal such patterns for a large family of hexagons this is also generalized for multidimensional synchronization patterns the best known patterns in terms of dots for circles and other polygons are also given the technique and its application for two dimensional synchronization patterns raises some interesting problems in discrete geometry we will also discuss these problems it is also shown how folding can be used to construct multidimensional error correcting codes finally by using the new definition of folding new types of multidimensional pseudo random arrays with various shapes are generated

general affine surface areas abstract two families of general affine surface areas are introduced basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for l affine surface areas are established

dense mirroring surface recovery from 1d homographies and sparse correspondences in this work we recover the 3d shape of mirrors sunglasses and stainless steel implements a computer monitor displays several images of parallel stripes each image at a different angle reflections of these stripes in a mirroring surface are captured by the camera for every image point the direction of the displayed stripes and their reflections in the image are related by a 1d homography matrix computed with a robust version of the statistically accurate heteroscedastic approach by focusing on a sparse set of image points for which monitor image correspondence is computed the depth and the local shape may be estimated from these homographies the depth estimation relies on statistically correct minimization and provides accurate reliable results even for the image points where the depth estimation process is inherently unstable we are able to characterize this instability and develop an algorithm to detect and correct it after correcting the instability dense surface recovery of mirroring objects is performed using constrained interpolation which does not simply interpolate the surface depth values but also uses the locally computed 1d homographies to solve for the depth the correspondence and the local surface shape the method was implemented and the shape of several objects was densely recovered at submillimeter accuracy

smallest singular value of random matrices and geometry of random polytopes abstract we study the behaviour of the smallest singular value of a rectangular random matrix i e matrix whose entries are independent random variables satisfying some additional conditions we prove a deviation inequality and show that such a matrix is a good isomorphism on its image then we obtain asymptotically sharp estimates for volumes and other geometric parameters of random polytopes absolutely convex hulls of rows of random matrices all our results hold with high probability that is with probability exponentially in dimension close to 1

random polytopes and the efron stein jackknife inequality let k be a smooth convex body the convex hull of independent random points in k is a random polytope estimates for the variance of the volume and the variance of the number of vertices of a random polytope are obtained the essential step is the use of the efron stein jackknife inequality for the variance of symmetric statistics consequences are strong laws of large numbers for the volume and the number of vertices of the random polytope a conjecture of barany concerning random and best approximation of convex bodies is confirmed analogous results for random polytopes with vertices on the boundary of the convex body are given

recovering surface shape and orientation from texture texture provides an important source of information about the three dimensional structure of visible surfaces particularly for stationary monocular views to recover 3d structure the distorting effects of projection must be distinguished from properties of the texture on which the distortion acts this requires that assumptions must be made about the texture yet the unpredictability of natural textures precludes the use of highly restrictive assumptions the recovery method reported in this paper exploits the minimal assumption that textures do not mimic projective effects this assumption determines the strategy of attributing as much as possible of the variation observed in the image to projection equivalently the interpretation is chosen for which the texture prior to projection is made as uniform as possible this strategy was implemented using statistical methods first for the restricted case of planar surfaces and then by extension for curved surfaces the technique was applied successfully to natural images

a classification of sl n invariant valuations a classification of upper semicontinuous and sl n invariant valuations on the space of n dimensional convex bodies is established as a consequence complete characterizations of cen t ro affine and l p affine surface areas are obtained the proofs make use of a new sl n shaping process for convex bodies

the jackknife estimate of variance tukey s jackknife estimate of variance for a statistic s x 1 x 2 cdots x n which is a symmetric function of i i d random variables x i is investigated using an anova like decomposition of s it is shown that the jackknife variance estimate tends always to be biased upwards a theorem to this effect being proved for the natural jackknife estimate of operatorname var s x 1 x 2 cdots x n 1 based on x 1 x 2 cdots x n

random points and lattice points in convex bodies assume k subset of r d is a convex body and x is a large finite subset of k how many convex polytopes are there whose vertices belong to x is there a typical shape of such polytopes how well does the maximal such polytope which is actually the convex hull of x approximate k we are interested in these questions mainly in two cases the first is when x is a random sample of n uniform independent points from k in this case motivation comes from sylvester s famous four point problem and from the theory of random polytopes the second case is when x k boolean and z d where z d is the lattice of integer points in r d and the questions come from integer programming and geometry of numbers surprisingly or not so surprisingly the answers in the two cases are rather similar

constructions for perfect maps and pseudorandom arrays a construction of perfect maps i e periodic r v binary arrays in which each n m binary matrix appears exactly once is given a similar construction leads to arrays in which only the zero n m matrix does not appear and to a construction in which only a few n m binary matrices do not appear a generalization to the nonbinary case is given the constructions involve an interesting problem in shift register theory the solution is given for almost all the case of this problem

intrinsic volumes and f vectors of random polytopes on appelle polytope aleatoire k n l enveloppe convexe de n points choisis de facon aleatoire dans un corps convexe k on s interesse a l approximation stochastique de k par k n plus particulierement on obtient des resultats d approximation pour les esperances de v s k v s k n ou v s est le s ieme volume intrinseque du nombre de faces s dimensionnelles de k n et de la distance de hausdorff entre k et k n

structured mapping problems for linearly structured matrices abstract given an appropriate class of structured matrices s we characterize matrices x and b for which there exists a matrix a s such that a x b and determine all matrices in s mapping x to b we also determine all matrices in s mapping x to b and having the smallest norm we use these results to investigate structured backward errors of approximate eigenpairs and approximate invariant subspaces and structured pseudospectra of structured matrices

central limit theorems for random polytopes let k be a smooth convex set the convex hull of independent random points in k is a random polytope central limit theorems for the volume and the number of i dimensional faces of random polytopes are proved as the number of random points tends to infinity one essential step is to determine the precise asymptotic order of the occurring variances

the combinatorial structure of random polytopes abstract choose n random points in r d let p n be their convex hull and denote by f i p n the number of i dimensional faces of p n a general method for computing the expectation of f i p n i 0 d 1 is presented this generalizes classical results of efron in the case i 0 and renyi and sulanke in the case i d 1 to arbitrary i for random points chosen in a smooth convex body a limit law for f i p n is proved as n for random points chosen in a polytope the expectation of f i p n is determined as n this implies an extremal property for random points chosen in a simplex

variance asymptotics for random polytopes in smooth convex bodies let k subset mathbb r d be a smooth convex set and let mathcal p lambda be a poisson point process on mathbb r d of intensity lambda the convex hull of mathcal p lambda cap k is a random convex polytope k lambda as lambda rightarrow infty we show that the variance of the number of k dimensional faces of k lambda when properly scaled converges to a scalar multiple of the affine surface area of k similar asymptotics hold for the variance of the number of k dimensional faces for the convex hull of a binomial process in k

representation and shape matching of 3 d objects a three dimensional scene analysis system for the shape matching of real world 3 d objects is presented various issues related to representation and modeling of 3 d objects are addressed a new method for the approximation of 3 d objects by a set of planar faces is discussed the major advantage of this method is that it is applicable to a complete object and not restricted to single range view which was the limitation of the previous work in 3 d scene analysis the method is a sequential region growing algorithm it is not applied to range images but rather to a set of 3 d points the 3 d model of an object is obtained by combining the object points from a sequence of range data images corresponding to various views of the object applying the necessary transformations and then approximating the surface by polygons a stochastic labeling technique is used to do the shape matching of 3 d objects the technique matches the faces of an unknown view against the faces of the model it explicitly maximizes a criterion function based on the ambiguity and inconsistency of classification it is hierarchical and uses results obtained at low levels to speed up and improve the accuracy of results at higher levels the objective here is to match the individual views of the object taken from any vantage point details of the algorithm are presented and the results are shown on several unknown views of a complicated automobile casting

a single shot structured light means by encoding both color and geometrical features color encoded structured light is an important means for single shot 3d reconstruction in this article we present a pseudorandom array based structured light method the proposed pattern is composed with four colors and one geometric shape is inlaid to the color elements to extend the coding element from four to eight as a result a relatively small coding window with the size of 2 2 can be obtained grid point between each two adjacent pattern elements is defined as the pattern feature and an improved feature detector is proposed for the robust and accurate grid point localization the grid points are divided into two types and represented by a topological structure the decoding procedure is performed in two steps the shape decoding and color decoding finally the epipolar constraint and neighboring constraint are introduced to improve the decoding reliability accuracy and robustness of the proposed method are evaluated in the experiments some real objects including human face are reconstructed to demonstrate the 3d reconstruction quality of the proposed structured light method compared with conventional color sl pattern a novel color geometrical pattern was proposed the pattern consists of four colors and one embedded geometrical feature with a coding capacity of 65 63 a relative small unique window size of 2 2 was obtained the decoding procedure was implemented as color decoding and geometrical decoding a robust and accurate pattern feature detection algorithm was also presented

stochastical approximation of convex bodies enveloppe convexe de points aleatoires sur la sphere enveloppe convexe de points aleatoires sur la frontiere d un corps convexe lisse

beta polytopes and poisson polyhedra f vectors and angles abstract we study random polytopes of the form x 1 x n defined as convex hulls of independent and identically distributed random points x 1 x n in r d with one of the following densities f d x c d 1 x 2 x 1 beta distribution 1 or f d x c d 1 x 2 x r d beta distribution d 2 here c d and c d are normalizing constants this setting also includes the uniform distribution on the unit sphere and the standard normal distribution as limiting cases we derive exact and asymptotic formulae for the expected number of k faces of x 1 x n for arbitrary k 0 1 d 1 we prove that for any such k this expected number is strictly monotonically increasing with n also we compute the expected internal and external angles of these polytopes at faces of every dimension and more generally the expected conic intrinsic volumes of their tangent cones by passing to the large n limit in the beta case we compute the expected f vector of the convex hull of poisson point processes with power law intensity function using convex duality we derive exact formulae for the expected number of k faces of the zero cell for a class of isotropic poisson hyperplane tessellations in r d this family includes the zero cell of a classical stationary and isotropic poisson hyperplane tessellation and the typical cell of a stationary poisson voronoi tessellation as special cases in addition we prove precise limit theorems for this f vector in the high dimensional regime as d finally we relate the d dimensional beta and beta distributions to the generalized pareto distributions known in extreme value theory

finitary simple lie algebras abstract an algebra is called finitary if it consists of finite rank transformations of a vector space we classify finitary simple lie algebras over a field of characteristic 0 we also describe finitary irreducible lie algebras

on 0 1 polytopes with many facets there exist n dimensional 0 1 polytopes with as many as cn logn n 4 facets this is our main result it answers a question of komei fukuda and gunter m ziegler c 2001 academic press

lyapunov exponents for some isotropic random matrix ensembles a random matrix with rows distributed as a function of their length is said to be isotropic when these distributions are gaussian beta type i or beta type ii previous work has from the viewpoint of integral geometry obtained the explicit form of the distribution of the determinant we use these result to evaluate the sum of the lyapunov spectrum of the corresponding random matrix product and we further give explicit expressions for the largest lyapunov exponent generalisations to the case of complex or quaternion entries are also given for standard gaussian matrices x the full lyapunov spectrum for products of random matrices i n 1 over c x is computed in terms of a generalised hypergeometric function in general and in terms of a single single integral involving a modified bessel function for the largest lyapunov exponent

computation of surface orientation and structure of objects using grid coding in this correspondence algorithms are introduced to infer surface orientation and structure of visible object surfaces using grid coding we adopt the active lighting technique to spatially encode the scene for analysis the observed objects which can have surfaces of arbitrary shape are assumed to rest on a plane base plane in a scene which is encoded with light cast through a grid plane two orthogonal grid patterns are used where each pattern is obtained with a set of equally spaced stripes marked on a glass pane the scene is observed through a camera and the object surface orientation is determined using the projected patterns on the object surface if the surfaces under consideration obey certain smoothness constraints a dense orientation map can be obtained through proper interpolation the surface structure can then be recovered given this dense orientation map both planar and curved surfaces can be handled in a uniform manner the algorithms we propose yield reasonably accurate results and are relatively tolerant to noise especially when compared to shape from shading techniques in contrast to other grid coding techniques reported which match the grid junctions for depth reconstruction under the stereopsis principle our techniques use the direction of the projected stripes to infer local surface orientation and do not require any correspondence relationship between either the grid lines or the grid junctions to be specified the algorithm has the ability to register images and can therefore be embedded in a system which integrates knowledge from multiple views

dual mixed volumes and the slicing problem we develop a technique using dual mixed volumes to study the isotropic constants of some classes of spaces in particular we recover strengthen and generalize results of ball and junge concerning the isotropic constants of subspaces and quotients of lp and related spaces an extension of these results to negative values of p is also obtained using generalized intersection bodies in particular we show that the isotropic constant of a convex body which is contained in an intersection body is bounded up to a constant by the ratio between the latter s mean radius and the former s volume radius we also show how type or cotype 2 may be used to easily prove inequalities on any isotropic measure

the spherical convex floating body abstract for a convex body on the euclidean unit sphere the spherical convex floating body is introduced the asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated this gives rise to a new spherical area measure the floating area remarkably this floating area turns out to be a spherical analogue to the classical affine surface area from affine differential geometry several properties of the floating area are established

psychophysical and computational studies towards a theory of human stereopsis psychophysical studies are described which pose a strong challenge to models of human stereopsis based on the processing of disparity information within independent spatial frequency tuned binocular channels these studies support instead the proposal that the processes of human binocular combination integrally relate the extraction of disparity information with the construction of raw primal sketch assertions this proposal implies global binocular combination rules using principles of figural continuity and cross channel correspondence to disambiguate local matches found independently within spatial frequency channels exploratory small scale computational experiments with stereo algorithms based on these rules are described and found to be successful in dealing with a variety of stereo inputs the constraints presented by objects which are exploited by these algorithms are discussed

a state of the art in structured light patterns for surface profilometry shape reconstruction using coded structured light is considered one of the most reliable techniques to recover object surfaces having a calibrated projector camera pair a light pattern is projected onto the scene and imaged by the camera correspondences between projected and recovered patterns are found and used to extract 3d surface information this paper presents an up to date review and a new classification of the existing techniques some of these techniques have been implemented and compared obtaining both qualitative and quantitative results the advantages and drawbacks of the different patterns and their potentials are discussed

block tridiagonal reduction of perturbed normal and rank structured matrices abstract it is well known that if a matrix a c n n solves the matrix equation f a a h 0 where f x y is a linear bivariate polynomial then a is normal a and a h can be simultaneously reduced in a finite number of operations to tridiagonal form by a unitary congruence and moreover the spectrum of a is located on a straight line in the complex plane in this paper we present some generalizations of these properties for almost normal matrices which satisfy certain quadratic matrix equations arising in the study of structured eigenvalue problems for perturbed hermitian and unitary matrices

variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points we show that the random point measures induced by vertices in the convex hull of a poisson sample on the unit ball when properly scaled and centered converge to those of a mean zero gaussian field we establish limiting variance and covariance asymptotics in terms of the density of the poisson sample similar results hold for the point measures induced by the maximal points in a poisson sample the approach involves introducing a generalized spatial birth growth process allowing for cell overlap

a second wedderburn type theorem for some classes of linearly structured matrices abstract we consider some classes s of structured matrices endowed each one with a structure of lie or jordan algebra we show that any s decomposes as the direct sum s s s s of well described minimal ideals being each one a class of structured matrices of the same type as s

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

on the isotropy constant of projections of polytopes the isotropy constant of any d dimensional polytope with n vertices is bounded by cn d where c 0 is a numerical constant

structured light using pseudorandom codes we solve the correspondence problem in active stereo vision using a novel pseudorandom coded structured light sl this coding scheme performs well in the presence of occlusion in settings where color coding is feasible 3d information can be obtained using a single image

variance asymptotics and scaling limits for random polytopes abstract let k be a convex set in r d and let k be the convex hull of a homogeneous poisson point process p of intensity on k when k is a simple polytope we establish scaling limits as for the boundary of k in a vicinity of a vertex of k and we give variance asymptotics for the volume and k face functional of k k 0 1 d 1 resolving an open question posed in 17 the scaling limit of the boundary of k and the variance asymptotics are described in terms of a germ grain model consisting of cone like grains pinned to the extreme points of a poisson point process on r d 1 r having intensity d e d h d h d v

numerical shape from shading and occluding boundaries an iterative method for computing shape from shading using occluding boundary information is proposed some applications of this method are shown we employ the stereographic plane to express the orientations of surface patches rather than the more commonly used gradient space use of the stereographic plane makes it possible to incorporate occluding boundary information but forces us to employ a smoothness constraint different from the one previously proposed the new constraint follows directly from a particular definition of surface smoothness we solve the set of equations arising from the smoothness constraints and the image irradiance equation iteratively using occluding boundary information to supply boundary conditions good initial values are found at certain points to help reduce the number of iterations required to reach a reasonable solution numerical experiments show that the method is effective and robust finally we analyze scanning electron microscope sem pictures using this method other applications are also proposed

a class of simple matrix algebras abstract fix an infinite set i and consider the associative matrix algebra m i f where f is a base field with char f 2 for any couple of bijective maps i i such that and 2 2 we introduce a linear subspace of m i f we endow it with a structure of non associative algebra for a certain bilinear product and obtain a wide class of non associative algebras containing in particular the lie algebras lie m i f t we show that each algebra is simple

the visual interpretation of surface contours this article examines the computational problems underlying the 3 d interpretation of surface contours a surface contour is the image of a curve across a physical surface such as the edge of a shadow cast across a surface a gloss contour wrinkle seam or pigmentation marking surface contours by and large are not as restricted as occluding contours and therefore pose a more difficult interpretation problem nonetheless we are adept at perceiving a definite 3 d surface from even simple line drawings e g graphical depictions of continuous functions of two variables the solution of a specific surface shape comes by assuming that the physical curves are particularly restricted in their geometric relationship to the underlying surface these geometric restrictions are examined

an analytic solution to the busemann petty problem on sections of convex bodies we derive a formula connecting the derivatives of parallel section functions of an origin symmetric star body in r n with the fourier transform of powers of the radial function of the body a parallel section function or ni 1 dimensional x ray gives the ni 1 dimensional volumes of all hyperplane sections of the body orthogonal to a given direction this formula provides a new characterization of intersection bodies in r n and leads to a unifled analytic solution to the busemann petty problem suppose that k and l are two origin symmetric convex bodies inr n such that the ni 1 dimensional volume of each central hyperplane section of k is smaller than the volume of the corresponding section of l is the n dimensional volume of k smaller than the volume of l in conjunction with earlier established connections between the busemann petty problem intersection bodies and positive deflnite distributions our formula shows that the answer to the problem depends on the behavior of the ni 2 nd derivative of the parallel section functions the a rmative answer to the busemann petty problem for n 4 and the negative answer for n 5 now follow from the fact that convexity controls the second derivatives but does not control the derivatives of higher orders

convex hulls of random walks expected number of faces and face probabilities abstract consider a sequence of partial sums s i 1 i 1 i n starting at s 0 0 whose increments 1 n are random vectors in r d d n we are interested in the properties of the convex hull c n conv s 0 s 1 s n assuming that the tuple 1 n is exchangeable and a certain general position condition holds we prove that the expected number of k dimensional faces of c n is given by the formula e f k c n 2 k n l 0 n 1 d 2 l d 2 l k 1 for all 0 k d 1 where n m and n m are stirling numbers of the first and second kind respectively further we compute explicitly the probability that for given indices 0 i 1 i k 1 n the points s i 1 s i k 1 form a k dimensional face of conv s 0 s 1 s n this is done in two different settings for random walks with symmetrically exchangeable increments and for random bridges with exchangeable increments these results generalize the classical one dimensional discrete arcsine law for the position of the maximum due to e sparre andersen all our formulae are distribution free that is do not depend on the distribution of the increments k s the main ingredient in the proof is the computation of the probability that the origin is absorbed by a joint convex hull of several random walks and bridges whose increments are invariant with respect to the action of direct product of finitely many reflection groups of types a n 1 and b n this probability in turn is related to the number of weyl chambers of a product type reflection group that are intersected by a linear subspace in general position

random points on the boundary of smooth convex bodies the convex hull of n independent random points chosen on the boundary of a convex body k c r d according to a given density function is a random polytope the expectation of its i th intrinsic volume for i 1 d is investigated in the case that the boundary of k is sufficiently smooth asymptotic expansions for these expected intrinsic volumes as n oo are derived

central limit theorems for random polytopes in a smooth convex set let k be a smooth convex set with volume one in choose n random points in k independently according to the uniform distribution the convex hull of these points denoted by kn is called a random polytope we prove that several key functionals of kn satisfy the central limit theorem as n tends to infinity

cones generated by random points on half spheres and convex hulls of poisson point processes let u 1 u 2 ldots be random points sampled uniformly and independently from the d dimensional upper half sphere we show that as n rightarrow infty the f vector of the d 1 dimensional convex cone c n generated by u 1 ldots u n weakly converges to a certain limiting random vector without any normalization we also show convergence of all moments of the f vector of c n and identify the limiting constants for the expectations we prove that the expected grassmann angles of c n can be expressed through the expected f vector this yields convergence of expected grassmann angles and conic intrinsic volumes and answers thereby a question of barany et al random struct algorithms 50 1 3 22 2017 https doi org 10 1002 rsa 20644 our approach is based on the observation that the random cone c n weakly converges after a suitable rescaling to a random cone whose intersection with the tangent hyperplane of the half sphere at its north pole is the convex hull of the poisson point process with power law intensity function proportional to vert x vert d gamma where gamma 1 we compute the expected number of facets the expected intrinsic volumes and the expected t functional of this random convex hull for arbitrary gamma 0

on the variance of random polytopes abstract a random polytope is the convex hull of uniformly distributed random points in a convex body k a general lower bound on the variance of the volume and f vector of random polytopes is proved also an upper bound in the case when k is a polytope is given for polytopes as for smooth convex bodies the upper and lower bounds are of the same order of magnitude the results imply a law of large numbers for the volume and f vector of random polytopes when k is a polytope

intersection bodies and dual mixed volumes the setting for this article is euclidean n space ky n 2 let x denote the set of convex bodies compact convex subsets with nonempty interiors in r and let x denote the subset of x that contains the centered centrally symmetric with respect to the origin bodies for ue s let e denote the hyperplane through the origin that is orthogonal to u and let ki e denote the image of the orthogonal projection of the body ke x onto e we shall use v to denote i dimensional lebesgue measure for v and v we shall usually write v and u shephard 38 poses the question if k l e xe and

random convex hulls and extreme value statistics in this paper we study the statistical properties of convex hulls of n random points in a plane chosen according to a given distribution the points may be chosen independently or they may be correlated after a non exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem we present a unifying approach based on the notion of support function of a closed curve and the associated cauchy s formulae that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics as an example of correlated points we study here in detail the case when the points represent the vertices of n independent random walks in the continuum time limit this reduces to n independent planar brownian trajectories for which we compute exactly for all n the mean perimeter and the mean area of their global convex hull our results have relevant applications in ecology in estimating the home range of a herd of animals some of these results were announced recently in a short communication phys rev lett 103 140602 2009

perfect maps given positive integers r s u and spl upsi an r s u spl upsi perfect map pm is defined to be a periodic r spl times s binary array in which every u spl times spl upsi binary array appears exactly once as a periodic subarray perfect maps are the natural extension of the de bruijn sequences to two dimensions in the paper the existence question for perfect maps is settled by giving constructions for perfect maps for all parameter sets subject to certain simple necessary conditions extensive use is made of previously known constructions by finding new conditions which guarantee their repeated application these conditions are expressed as bounds on the linear complexities of the periodic sequences formed from the rows and columns of perfect maps

standard triples of structured matrix polynomials the notion of standard triples plays a central role in the theory of matrix polynomials we study such triples for matrix polynomials p lambda with structure mathcal s where mathcal s is the hermitian symmetric star even star odd star palindromic or star antipalindromic structure with star t we introduce the notion of mathcal s structured standard triple with the exception of t anti palindromic matrix polynomials of even degree with both 1 and 1 as eigenvalues we show that p lambda has structure mathcal s if and only if p lambda admits an mathcal s structured standard triple and moreover that every standard triple of a matrix polynomial with structure mathcal s is mathcal s structured we investigate the important special case of mathcal s structured jordan triples

intersections of random hypersurfaces and visibility for a wide class of stationary random hypersurfaces in d the notion of the projection body is introduced it turns out that this convex body a very special case of which is matheron s steiner compact associated with a poisson process of hyperplanes contains most of the information concerning certain intersection properties of the random hypersurface while its polar reciprocal set is closely connected with the behaviour of the random hypersurface in visibility problems this enables one to give a unified treatment of several intersection and visibility problems for random hypersurfaces a detailed investigation of the case where the random hypersurface is generated by a poisson process is given separately

monotonicity of facet numbers of random convex hulls abstract let x 1 x n be independent random points that are distributed according to a probability measure on r d and let p n be the random convex hull generated by x 1 x n n d 1 for natural classes of probability distributions and by means of blaschke petkantschin formulae from integral geometry it is shown that the mean facet number of p n is strictly monotonically increasing in n

poisson polyhedra in high dimensions abstract the zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated as the space dimension tends to infinity the model includes the zero cell of stationary and isotropic poisson hyperplane tessellations as well as the typical cell of a stationary poisson voronoi tessellation as special cases it is shown that asymptotically in the space dimension with overwhelming probability these cells satisfy the hyperplane conjecture if the distance exponent and the intensity parameter are suitably chosen dimension dependent functions also the high dimensional limits of the mean number of faces are explored and the asymptotic behaviour of an isoperimetric ratio is analysed in the background are new identities linking the f vector of the zero cell to certain dual intrinsic volumes

random polytopes in a convex polytope independence of shape and concentration of vertices write d for the set of all convex bodies convex compact sets with nonempty interior in d define o g l d as the set of those k e 5 b d with vol k 1 fix k e g i d and choose points x l x e k randomly independently and according to the uniform distribution on k then k c o n v x l xn is a random polytope in k write e k n for the expectation of the random variable v o l k k n e k n shows how well k approximates k in volume on the average groemer grl proved that among all convex bodies k e o g l d the ellipsoids are approximated worst i e

on the mean width of random polytopes on the boundary of a d dimensional convex body a probability distribution with a positive continuous density function g is given the convex hull of n points chosen independently according to g is a random polytope the asymptotic behaviour n of the expected value of the mean width of the random polytope is determined

an algebraic approach to shape from image problems abstract this paper presents a new method for recovering three dimensional shapes of polyhedral objects from their single view images the problem of recovery is formulated in a constrained optimization problem in which the constraints reflect the assumption that the scene is composed of polyhedral objects and the objective function to be minimized is a weighted sum of quadric errors of surface information such as shading and texture for practical purpose it is decomposed into the two more tractable problems a linear programming problem and an unconstrained optimization problem in the present method the global constraints placed by the polyhedron assumption are represented in terms of linear algebra whereas similar constraints have usually been represented in terms of a gradient space moreover superstrictness of the constraints can be circumvented by a new concept position free incidence structure for this reason the present method has several advantages it can recover the polyhedral shape even if image data are incorrect due to vertex position errors it can deal with perspective projection as well as orthographic projection the number of variables in the optimization problem is very small three or a little greater than three and any kinds of surface information can be incorporated in a unifying manner

limit theorems for convex hulls it is shown that the process of vertices of the convex hull of a uniform sample from the interior of a convex polygon converges locally after rescaling to a strongly mixing markov process as the sample size tends to infinity the structure of the limiting markov process is determined explicitly and from this a central limit theorem for the number of vertices of the convex hull is derived similar results are given for uniform samples from the unit disk