a two variable approach to solve the polynomial lyapunov equation abstract a two variable polynomial approach to solve the one variable polynomial lyapunov equation is proposed lifting the problem from the one variable to the two variable context allows to use faddeev type recursions in order to solve the polynomial lyapunov equation in an iterative fashion the method is especially suitable for applications requiring exact or symbolic computation

symbolic computation of fisher information matrices for parametrized state space systems the asymptotic fisher information matrix fim has several applications in linear systems theory and statistical parameter estimation it occurs in relation to the cramer rao lower bound for the covariance of unbiased estimators it is explicitly used in the method of scoring and it determines the asymptotic convergence properties of various system identification methods it defines the fisher metric on manifolds of systems and it can be used to analyze questions on identifiability of parametrized model classes for many of these applications exact symbolic computation of the fim can be of great use in this paper two different methods are described for the symbolic computation of the asymptotic fim the first method applies to parametrized mimo state space systems driven by stationary gaussian white noise and proceeds via the solution of discrete time lyapunov and sylvester equations for which a method based on faddeev sequences is used this approach also leads to new short proofs of certain well known results on the structure of the fim for siso arma systems the second method applies to parametrized siso state space systems and uses an extended faddeev algorithm for both algorithms the concept of a faddeev reachability matrix and the solution of discrete time sylvester equations in controller companion form are central issues the methods are illustrated by two worked examples the first concerns three different parametrizations of the class of stable ar n systems the second concerns a model for an industrial mixing process in which the value of exact computation to answer identifiability questions becomes particularly clear

the relative neighborhood graph with an application to minimum spanning trees le graphe de voisinage relatif rng d un ensemble v de points dans un espace euclidien est un graphe v e tel que p q appartient a e si et seulement si il n existe pas de point z de v tel que d p z d p q et d q z d p q on montre que 1 le rng de n points du plan peut etre trouve en temps o n log n lequel est optimal a une constante multiplicative pres 2 le rng ainsi que l arbre minimum des sommets d un polygone a n sommets convexe peut etre trouve en un temps o n 3 sous l hypothese que trois points d entree ne forment pas un triangle isocele le rng de n points dans un espace dimension r peut etre trouve en un temps o n 2 pour r 3

neumann series and lattice sums we consider sums over the square lattice which depend only on radial distance and provide formulas which enable sums of functions with neumann series to be reexpressed as combinations of hypergeometric series we illustrate the procedure using trigonometric sums previously studied by borwein and borwein sums combining logarithms bessel functions j and powers of distance and sums of neumann functions we also exhibit sums which may be evaluated analytically and recurrence formulas linking sums

an iterative algorithm for finding a nearest pair of points in two convex subsets of rn abstract we present an algorithm for finding a nearest pair of points in two convex sets of r n and therefore their distance the algorithm is based on the fixed point theory of nonexpansive operators on a hilbert space its practical implementation requires a fast projection algorithm we introduce such a procedure for convex polyhedra this algorithm effects a local search in the faces using visibility as a guide for finding the global minimum after studying the convergence of both algorithms we detail computer experiments on polyhedra projection and distance in the case of distances these experiments show a sublinear time complexity relative to the total number of vertices

on projection algorithms for solving convex feasibility problems due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences most notably computerized tomography algorithms for solving convex feasibility problems continue to receive great attention to unify generalize and review some of these algorithms a very broad and flexible framework is investigated several crucial new concepts which allow a systematic discussion of questions on behaviour in general hilbert spaces and on the quality of convergence are brought out numerous examples are given

an optimal algorithm for constructing the weighted voronoi diagram in the plane let s denote a set of n points in the plane such that each point p has assigned a positive weight w p which expresses its capability to influence its neighbourhood in this sense the weighted distance of an arbitrary point x from p is given by de x p w p where de denotes the euclidean distance function the weighted voronoi diagram for s is a subdivision of the plane such that each point p in s is associated with a region consisting of all points x in the plane for which p is a weighted nearest point of s an algorithm which constructs the weighted voronoi diagram for s in o n2 time is outlined in this paper the method is optimal as the diagram can consist of n2 faces edges and vertices

a linear time combinatorial algorithm to find the orthogonal hull of an object on the digital plane a combinatorial algorithm to compute the orthogonal hull of a digital object imposed on a background grid is presented in this paper the resolution and complexity of the orthogonal hull can be controlled by varying the grid size which may be used for a multiresolution analysis of a given object existing algorithms on finding the convex hull are based on divide and conquer strategy sweepline approach etc whereas the proposed algorithm is combinatorial in nature whose time complexity is linear on the object perimeter instead of the object area for a larger grid size the perimeter of an object decreases in length in terms of grid units and hence the runtime of the algorithm reduces significantly the algorithm uses only comparison and addition in the integer domain thereby making it amenable to usage in real world applications where speed is a prime factor experimental results including the cpu time demonstrate the elegance and efficacy of the proposed algorithm

on matrix equations x axf c and x axf c with the help of the concept of kronecker map an explicit solution for the matrix equation x axf c is established this solution is neatly expressed by a symmetric operator matrix a controllability matrix and an observability matrix in addition the matrix equation email protected f c is also studied an explicit solution for this matrix equation is also proposed by means of the real representation of a complex matrix this solution is neatly expressed by a symmetric operator matrix two controllability matrices and two observability matrices

closed form solutions to the nonhomogeneous yakubovich conjugate matrix equation some closed form solutions are provided for the nonhomogeneous yakubovich conjugate matrix equation email protected f by r with x and y being unknown matrices the presented solutions can offer all the degrees of freedom which is represented by an arbitrarily chosen parameter matrix the primary feature of the solutions is that the matrices f and r are not restricted to be in any canonical form or may be even unknown a priori one of the solutions is neatly expressed in terms of controllability matrices and observability matrices

representation complexity of adaptive 3d distance fields in this paper we study the representation complexity of a kind of data structure that stores the information necessary to compute the distance from a point to a geometric body these data structures called adaptive splitting based on cubature distance fields asbcdf are binary search trees generated by the adaptive splitting based on cubature asbc algorithm that adaptively subdivides the space surrounding the body into tetrahedra their representation complexity is measured by the number of nodes in the tree two times the number of tetrahedra in the resulting tessellation in the case of convex polyhedra we prove that this quantity remains bounded as the number of vertices of the polyhedra increases to infinity experimental results show that the number of tetrahedra in the tessellations is almost independent of the combinatorial complexity of the polyhedra this means that the average compute time of the distance to arbitrary convex polyhedra is almost constant therefore asbcdfs are especially suitable for real time applications involving rapidly changing environments modelized with complex polyhedra

an o kn log n algorithm for decomposing a set of polygons into d separable components abstract given a set of polygons p p 1 p 2 p n in the plane a d separable component with respect to a given set of directions d d 1 d 2 d k is a subset of polygons q p which can be moved freely as a group in any of the directions d without colliding with any polygon not in q we present an o kn log n algorithm for computing the d separable components of a set of n polygons the algorithm is based on the concept of d closure defined here and can be viewed as a certain orderly application of the well known plane sweep method to specially formed subgroups of polygons an improper choice of the order of the plane sweep operations or of the subgroups of polygons can increase the complexity to o n 2 in the worst case the notion of separable components is useful for partitioning the free space in vlsi design and for motion planning in robotics

on the definition and computation of rectilinear convex hulls abstract recently the computation of the rectilinear convex hull of a collection of rectilinear polygons has been studied by a number of authors from these studies three distinct definitions of rectilinear convex hulls have emerged we examine these three definitions for point sets in general pointing out some of their consequences and we give optimal algorithms to compute the corresponding rectilinear convex hulls of a finite set of points in the plane

solution of projection problems over polytopes in this paper we present variants of a convergent projection and contraction algorithm 25 for solving projection problems over polytope by using the special struture of the projection problems an iterative algorithm with constant step size is given which is globally linearly convergent these algorithms are simple to implement and each step of the method requires only a few matrix vector multiplications especially for minimums norm problems over transportation or general network polytopes onlyo n additions ando n multiplications are needed at each iteration numerical results for randomly generated test problems over network polytopes up to 10000 variables indicate that the presented algorithms are simple and efficient even for large problems

computational methods for linear matrix equations given the square matrices a b d e and the matrix c of conforming dimensions we consider the linear matrix equation a mathbf x e d mathbf x b c in the unknown matrix mathbf x our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing

guarding a line segment a line segment on the plane equipped with the maximum norm is guarded by the defender d the invader i wants to reach the line segment but he has to keep the distance from the defender no less than a given constant 0 the defender and the invader can move on the whole plane with maximal speed of rd and ri respectively where ri rd no further kinematic or dynamic constrains are imposed on the defender and the invader motions the maximal length of the line segment which can be guarded by the defender is established in this paper

finding the projection on a polytope an iterative method abstract an iterative algorithm is developed for the problem of finding the projection of a point on a polytope described as intersection of half spaces a proof of the convergence is given and its efficiency is investigated

the discrete and mixed minimax 2 center problems abstract letting p be a set of n points in the plane the discrete minimax 2 center problem d m m 2 c p is to find two disks centered at p 1 p 2 p that minimize the maximum of two terms namely the euclidean distance between two centers and the distance of any other point to the closer center the mixed minimax 2 center problem m m m 2 c p is when one of the two centers is not in p we present algorithms solving the d m m 2 c p and m m m 2 c p the time complexities of solving the d m m 2 c p and m m m 2 c p are o n 2 log n and o n 2 log 2 n respectively furthermore we consider two steiner minimum sum dipolar spanning tree problems in which one of the two dipoles is a steiner point and the dipoles are both steiner points these two problems are shown to be solvable in o n log n and o n time respectively

balanced realizations canonical form parametrization model reduction abstract balanced realizations are used to construct a canonical form for the set of minimal and asymptotically stable mimo systems of a given order a parametrization resutt is derived that allows us to show a truncation property of this canonical form

a state space calculus for rational probability density functions and applications to non gaussian filtering we propose what we believe to be a novel approach to performing calculations for rational density functions using state space representations of the densities by standard results from realization theory a rational probability density function is considered to be the transfer function of a linear system with generally complex entries the stable part of this system is positive real which we call the density summand the existence of moments is investigated using the markov parameters of the density summand moreover explicit formulae are given for the existing moments in terms of these markov parameters some of the main contributions of the paper are explicit state space descriptions for products and convolutions of rational densities as an application which is of interest in its own right the filtering problem is investigated for a linear time varying system whose noise inputs have rational probability density functions in particular state space formulations are derived for the calculation of the prediction and update equations the case of cauchy noise is treated as an illustrative example

streaming with minimum space an algorithm for covering by two congruent balls in this paper we design a simple streaming algorithm for maintaining two smallest balls of equal radius in d dimension to cover a set of points in an on line fashion different from most of the traditional streaming models at any step we use the minimum amount of space by only storing the locations and the common radius of the balls previously such a geometric algorithm is only investigated for covering with one ball one center by zarrabi zadeh and chan 2006 16 we give an analysis of our algorithm which is significantly different from the one center algorithm due to the obvious possibility of grouping points wrongly under this streaming model we show that our algorithm has an approximation ratio 2 for d 1 and at most 5 611 for any fixed d 1 we also present lower bounds of 1 5 and 1 604 for the problem in the d 1 and d 1 cases respectively

linear time algorithms for linear programming in r 3 and related problems linear time algorithms for linear programming in r 2 and r 3 are presented the methods used are applicable for other graphic and geometric problems as well as quadratic programming for exam

a general soft method for learning svm classifiers with l 1 norm penalty based on the geometric interpretation of support vector machines svms this paper presents a general technique that allows almost all the existing l 2 norm penalty based geometric algorithms including gilbert s algorithm schlesinger kozinec s sk algorithm and mitchell dem yanov malozemov s mdm algorithm to be softened to achieve the corresponding learning l 1 svm classifiers intrinsically the resulting soft algorithms are to find e optimal nearest points between two soft convex hulls theoretical analysis has indicated that our proposed soft algorithms are essentially generalizations of the corresponding existing hard algorithms and consequently they have the same properties of convergence and almost the identical cost of computation as a specific example the problem of solving n svms by the proposed soft mdm algorithm is investigated and the corresponding solution procedure is specified and analyzed to validate the general soft technique several real classification experiments are conducted with the proposed l 1 norm based mdm algorithms and numerical results have demonstrated that their performance is competitive to that of the corresponding l 2 norm based algorithms such as sk and mdm algorithms

closed form solutions to sylvester conjugate matrix equations some explicit closed form solutions of homogeneous and nonhomogeneous sylvester conjugate matrix equations are provided in this paper one of the solutions is expressed in terms of controllability matrices and observability matrices the proposed approach does not require all the coefficient matrices to be in any canonical forms and the solutions provide a significant degree of freedom which is represented by an arbitrarily chosen parameter matrix by specifying the solutions of the homogeneous sylvester conjugate equation some new expressions of the solutions of the normal sylvester normal sylvester conjugate and sylvester equations are given this fact reveals that the sylveter conjugate matrix equations are a more general class of some previously investigated matrix equations

computing the shortest network under a fixed topology we show that in any given uniform orientation metric plane the shortest network interconnecting a given set of points under a fixed topology can be computed by solving a linear programming problem whose size is bounded by a polynomial in the number of terminals and the number of legal orientations when the given topology is restricted to a steiner topology our result implies that the steiner minimum tree under a given steiner topology can be computed in polynomial time in any given uniform orientation metric with spl lambda legal orientations for any fixed integer spl lambda spl ges 2 this settles an open problem posed by brazil thomas and weng 2000

analytical results for a class of sums involving bessel functions and square arrays we consider a class of sums involving bessel functions summed over the square array the sums involve the length of an arbitrary vector lying within the central unit cell we establish conditions under which the sums reduce to polynomial forms in possibly with a single logarithmic term in addition and show how these polynomials may be conveniently evaluated

semi online maintenance of geometric optima and measures we give the first nontrivial worst case results for dynamic versions of various basic geometric optimization and measure problems under the semi online model where during the insertion of an object we are told when the object is to be deleted problems that we can solve with sublinear update time include the hausdorff distance of two point sets discrete 1 center largest empty circle convex hull volume in three dimensions volume of the union of axis parallel cubes and minimum enclosing rectangle the decision versions of the hausdorff distance and discrete 1 center problems can be solved fully dynamically some applications are mentioned

on determining optimal strategies in pursuit games in the plane pursuit games have application to robotics the pursuer models a moving obstacle and the evader models a robot that tries to reach a goal region without colliding with the moving obstacle at each moment the robot does not know the future trajectory of the obstacle the motion of the pursuer and the evader is controlled by their sets of permissible velocities called indicatrices we allow indicatrices that are more general than the simple motion i e velocities are bounded by an l2 norm circle we provide sufficient condition for a pursuit game to have value in this case we give optimal strategies for the pursuer and the evader we prove that the pursuit game in which the pursuer and the evader are convex objects moving with simple motion has value

bottleneck steiner trees in the plane a steiner tree with maximum weight edge minimized is called a bottleneck steiner tree bst the authors propose a theta mod rho mod log mod rho mod time algorithm for constructing a bst on a point set rho with points labeled as steiner or demand a lower bound in the linear decision tree model is also established it is shown that if it is desired to minimize further the number of used steiner points then the problem becomes np complete it is shown that when locations of steiner points are not fixed the problem remains np complete however if the topology of the final tree is given then the problem can be solved in theta mod rho mod log mod rho mod time the bst problem can be used for example in vlsi layout communication network design and facility location problems

finding the minimum vertex distance between two disjoint convex polygons in linear time abstract let v v 1 v 2 v m and w w 1 w 2 w n be two linearly separable convex polygons whose vertices are specified by their cartesian coordinates in order an algorithm with o m n worst case time complexity is described for finding the minimum euclidean distance between a vertex v 1 in v and a vertex w j in w it is also shown that the algorithm is optimal

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 some distance problems in fixed orientations in vlsi design technology requirements often dictate the use of only two orthogonal orientations determining both the shape of objects and the distance function the l 1 metric to be used for wiring objects more recent vlsi fabrication technology is capable of creating edges and wires in both the orthogonal and diagonal orientations we generalize the distance concept to the case where any fixed set of orientations is allowed and introduce a family of naturally induced metrics and the subsequent generalization of geometrical concepts a shortest connection between two points is in this case a path composed of line segments with only the given orientations we derive optimal solutions for various basic planar distance problems in this setting such as the computation of a voronoi diagram a minimum spanning tree and the minimum and maximum distance between two convex polygons many other theoretically interesting and practically relevant problems remain to be solved in particular the new famil

minimum diameter spanning trees and related problems the problem of finding a minimum diameter spanning tree mdst of a set of n points in the euclidean space is considered the diameter of a spanning tree is the maximum distance between any two poi

stable and efficient evaluation of periodized green s functions for the helmholtz equation at high frequencies a difficulty that arises in the context of infinite d periodic rough surface scattering relates to the effective numerical evaluation of the corresponding quasi periodic green function g q p due to its relevance in a variety of applications this problem has generated significant interest over the last 40 years and a variety of numerical methods have been devised for this purpose none of these methods to evaluate g q p however were designed for high frequency calculations as a result in this regime these methods become prohibitively expensive and or unstable here we present a novel scheme that can be shown to outperform every alternative numerical evaluation procedure and is especially effective for high frequency calculations our new algorithm is based on the use of some exact integrals that arise on judicious manipulation of the integral representation of g q p and which reduce the overall problem to that of evaluation of a sequence of simpler integrals that can be effectively handled by standard quadrature formulas we include a variety of numerical results that confirm that indeed our algorithm compares favorably with alternative methods

iterative solutions to the kalman yakubovich conjugate matrix equation abstract two operations are introduced for complex matrices in terms of these two operations an infinite series expression is obtained for the unique solution of the kalman yakubovich conjugate matrix equation based on the obtained explicit solution some iterative algorithms are given for solving this class of matrix equations convergence properties of the proposed algorithms are also analyzed by using some properties of the proposed operations for complex matrices

the fisher information matrix for linear systems estimation of parameters of linear systems is a problem often encountered in applications the cramer rao lower bound gives a lower bound on the variance of any unbiased parameter estimation method and therefore provides an important tool in the assessment of a parameter estimation method and for experimental design here we study the calculation of the fisher information matrix the inverse of the cramer rao lower bound from a system theoretic point of view a number of results appear in the literature that deal with the case where the stationary data is given as the output of a linear system driven by gaussian noise the non stationary situation where the data is the output of a linear system with gaussian measurement noise is rarely considered despite its importance in applications a general description will be given for fisher information for such data in terms of a derivative system for a uniformly sampled data set of impulse response type a closed form expression can be given for the fisher information using the solution of a lyapunov equation

on solutions of the matrix equations xf ax c and xf ax c with the help of the concept of kronecker map an explicit solution for the matrix equation xf ax c is established this solution is neatly expressed by a symmetric operator matrix a controllability matrix and an observability matrix in addition the matrix equation xf ax c is also studied an explicit solution for this equation is also proposed by means of the real representation of a complex matrix this solution is neatly expressed by a symmetric operator matrix two controllability matrices and two observability matrices

schlomilch series and grating sums we consider sums over the set of positive integers relevant to construction of periodic green s functions for diffraction gratings and similar problems and provide a general formula for a combination of bessel functions of complex order and complex powers of distance from the origin this general formula is investigated in a number of particular cases and in particular we provide expressions which enable sums of functions with neumann series to be re expressed as combinations of hypergeometric series we also investigate sums of neumann functions of integer order using analytic continuation techniques to provide formulae for their evaluation which we demonstrate are accurate and efficient in both the high and low frequency regions we also exhibit sums which may be evaluated analytically and recurrence formulae linking sums

optimal algorithms for the intersection and the minimum distance problems between planar polygons two planar geometric problems relating to a convex n gon p and a simple nonconvex m gon q are considered

kronecker maps and sylvester polynomial matrix equations the concepts of kronecker maps and sylvester polynomial matrix equations are proposed some properties of the kronecker maps are deduced it is shown that many kinds of matrix equations can be formulated into the so called sylvester polynomial matrix equations with the help of the kronecker maps general explicit solutions of the sylvester polynomial matrix equations are established when the polynomial matrices possess f primeness for the nonsquare case the proposed solutions can offer all the degrees of freedom represented by a free parameter matrix

a recursive algorithm for finding the minimum norm point in a polytope and a pair of closest points in two polytopes for a given pair of finite point setsp andq in some euclidean space we consider two problems problem 1 of finding the minimum euclidean norm point in the convex hull ofp and problem 2 of finding a minimum euclidean distance pair of points in the convex hulls ofp andq we propose a finite recursive algorithm for these problems the algorithm is not based on the simplicial decomposition of convex sets and does not require to solve systems of linear equations

a faddeev sequence method for solving lyapunov and sylvester equations lyapunov and sylvester equations play an important role in linear systems theory this paper deals with a method of solving such equations of the form ap pb k and p apb k with a rm m b rn n and p k rm n by exploiting the matrix algebra structure of the problem no use is made of kronecker products and the largest matrices occurring in the algorithms are of sizes m m n n and m n the faddeev method for matrix inversion lies at the very heart of the algorithms presented it occurs on several levels of the problem for the matrices a and b and for the lyapunov and sylvester operators the resulting algorithms are capable of solving the equations in a finite number of recursion steps they are very much apt for symbolic calculation it is shown how a solution can be quickly obtained for an equation with an arbitrary right hand side k provided a solution is known for a right hand side xyt of rank 1 where a x and bt y are reachable pairs the concept of a faddeev reachability matrix introduced here turns out to be very useful it establishes a close connection between the controller canonical companion form of a reachable pair a b and the faddeev sequence of a if a is already on controller form then its faddeev sequence takes on an especially simple form also in the symmetric case where a bt many important simplifications arise for this case alternative algorithms that require less iterations are developed the paper concludes with some examples concerning the symbolic solution of the lyapunov equation ap pat bbt with a b on controller form showing the potential of the algorithms

solution to generalized sylvester matrix equations an explicit solution to the generalized sylvester matrix equation ax by exf is established this solution is expressed in terms of the r controllability matrix of e a b a generalized symmetric operator matrix and an observability matrix moreover based on this solution solutions to some other matrix equations are also derived the results may provide great convenience for the analysis and synthesis problems related to these equations

fundamentals of restricted orientation convexity abstract a restricted orientation convex set also called an o convex set is a set of points whose intersection with lines from some fixed set is empty or connected the notion of o convexity generalizes standard convexity and orthogonal convexity we explore some of the basic properties of o convex sets in two and higher dimensions we also study o connected sets which are restricted o convex sets with several special properties we introduce and investigate restricted orientation analogs of lines flats and hyperplanes and characterize o convex and o connected sets in terms of their intersections with hyperplanes we then explore properties of o connected curves in particular we show when replacing a segment of an o connected curve with a new curvilinear segment yields an o connected curve and when the catenation of several curvilinear segments forms an o connected segment we use these results to characterize an o connected set in terms of o connected segments joining pairs of its points that are wholly contained in the set we also identify some of the major properties of standard convex sets that hold for o convexity in particular we establish the following results the intersection of a collection of o convex sets is an o convex set every o connected curvilinear segment is a segment of some o connected curve for every two points of an o convex set there is an o convex segment joining them that is wholly contained in the set

fuzzy distances for proximity characterization under uncertainty models of objects used in planning motions for robots are considered in this paper a fuzzy approach to handle imprecision in estimated object location is presented procedures for computing fuzzy distances between a pair of objects are developed the procedures are analyzed for their computational complexity the utility of the distance computations in reasoning about spatial relationships is illustrated

spatial planning a configuration space approach this paper presents algorithms for computing constraints on the position of an object due to the presence of ther objects this problem arises in applications that require choosing how to arrange or how to move objects without collisions the approach presented here is based on characterizing the position and orientation of an object as a single point in a configuration space in which each coordinate represents a degree of freedom in the position or orientation of the object the configurations forbidden to this object due to the presence of other objects can then be characterized as regions in the configuration space called configuration space obstacles the paper presents algorithms for computing these configuration space obstacles when the objects are polygons or polyhedra

efficient computation of schlomilch type series a class of schlomilch type series which are functions of two spatial parameters x and y are transformed into series which are more suitable for numerical computation when x is small the resulting formulae involve a free parameter a which does not affect the value of the original series but does affect the convergence characteristics of the new representation by choosing a appropriately we can improve the convergence of the series significantly and calculations are presented which demonstrate the efficiency of the new representations when evaluating the series numerically mellin transforms are used to derive the asymptotic form of the series in the limit as x 0 y constant and this also leads to efficient methods for computing certain image series

on the multimodality of distances in convex polygons abstract examples are given of n vertex convex polygons for which the distances between a fixed vertex and the remaining vertices visited in order form a multi modal function we show that this function may have as many as n 2 modes or local maxima further examples are given of n vertex convex polygons in which n 2 8 pairs of vertices are local maxima of their corresponding distance functions these results are used to construct an example that shows that a general algorithm of dobkin and snyder may not in fact be used to find the diameter of a convex polygon

on the complexity of some common geometric location problems given n demand points in the plane the p center problem is to find p supply points anywhere in the plane so as to minimize the maximum distance from a demand point to its respective nearest supply point the p median problem is to minimize the sum of distances from demand points to their respective nearest supply points we prove that the p center and the p median problems relative to both the euclidean and the rectilinear metrics are np hard in fact we prove that it is np hard even to approximate the p center problems sufficiently closely the reductions are from 3 satisfiability

optimal point location in a monotone subdivision point location often known in graphics as hit detection is one of the fundamental problems of computational geometry in a point location query we want to identify which of a given collection of geometric objects contains a particular point let mathcal s denote a subdivision of the euclidean plane into monotone regions by a straight line graph of m edges in this paper we exhibit a substantial refinement of the technique of lee and preparata siam j comput 6 1977 pp 594 606 for locating a point in mathcal s based on separating chains the new data structure called a layered dag can be built in o m time uses o m storage and makes possible point location in o log m time unlike previous structures that attain these optimal bounds the layered dag can be implemented in a simple and practical way and is extensible to subdivisions with edges more general than straight line segments

factored forms for solutions of ax xb c and x axb c in companion matrices abstract the main concern of this paper is linear matrix equations with block companion matrix coefficients it is shown that general matrix equations ax xb c and x axb c can be transformed to equations whose coefficients are block companion matrices c l x xc m diag i 0 0 and x c l xc m diag i 0 0 respectively where l and cm stand for the first and second block companion matrices of some monic r r matrix polynomials l si s 1j 0 jlj and m ti t7minus 1j 0 jmj the solution of the equat with block companion coefficients is reduced to solving vector equations sx where the matrix s is r2l r2l l max s t and enjoys some symmetry properties

comment on computing the shortest network under a fixed topology a linear programming formulation was given for the problem of computing a shortest network under a fixed topology under the lambda metric we point out a nontrivial error in this paper and give a correct and simpler linear programming formulation we also show that the result can be generalized to any distance function given by a minkowski unit circle that is a centrally symmetric polygon