optimal partitioning for multi vehicle systems using quadratic performance criteria we consider the problem of characterizing a generalized voronoi diagram that is relevant to a special class of area assignment problems for multi vehicle systems it is assumed that the motion of each vehicle is described by a second order mechanical system with time varying linear or affine dynamics the proposed generalized voronoi diagram encodes information regarding the proximity relations between the vehicles and arbitrary target points in the plane these proximity relations are induced by an anisotropic generalized distance function that incorporates the vehicle dynamics in particular the generalized distance is taken to be the minimum control effort required for the transition of a vehicle to an arbitrary target point with a small terminal speed at a fixed final time the space we wish to partition corresponds to the union of all the terminal positions that can be attained by each vehicle using finite control effort consequently the partition space has lower dimension than the state space of each vehicle we show that in the general case the solution to the proposed partitioning problem can be associated with a power voronoi diagram generated by a set of spheres in a five dimensional euclidean space for the computation of which efficient techniques exist in the relevant literature

coverage control in constant flow environments based on a mixed energy time metric in this paper we study a multi vehicle coverage control problem in constant flow environments while taking into account both energy consumption and traveling time more specifically the metric called the mixed energy time metric is a weighted sum of the energy consumption and the traveling time for a vehicle to travel from one point to another in constant flows when using the minimum energy control and the objective is to find vehicle locations that can minimize the expected mixed energy time required for the set of vehicles to cover a region we propose a gradient based control law which is calculated based on refined approximated voronoi cells induced by the mixed energy time metric and of which the convergence is proved via hybrid systems theory simulations show that the refined gradient based control can achieve similar performance as the exact gradient based control

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

decentralized spatial partitioning algorithms for multi vehicle systems based on the minimum control effort metric we consider the problem of characterizing a spatial partition of the position space of a team of vehicles with linear time varying kinematics the generalized metric that determines the proximity relations between the vehicles and an arbitrary target point in the partition space is the minimum control effort required for each vehicle to reach the latter point with zero miss distance and exactly zero velocity at a prescribed final time we show that the solution to the generalized voronoi partitioning problem can be associated with a special class of spatial partitions known as affine diagrams because the combinatorial complexity of the affine diagrams is comparable to the one of the standard voronoi diagrams their computation does not pose a significant challenge in applications of multi vehicle systems subsequently we propose an algorithm for the computation of the spatial partition which is decentralized in the sense that each vehicle can compute an approximation of its own cell independently from the other vehicles from the same team without utilizing a common spatial mesh numerical simulations that illustrate the theoretical developments are also presented

stable marker particle method for the voronoi diagram in a flow field the voronoi diagram in a flow field is a tessellation of water surface into regions according to the nearest island in the sense of a boat sail distance which is a mathematical model of the shortest time for a boat to move from one point to another against the flow of water the computation of the diagram is not easy because the equi distance curves have singularities to overcome the difficulty this paper derives a new system of equations that describes the motion of a particle along the shortest path starting at a given point on the boundary of an island and thus gives a new variant of the marker particle method in the proposed method each particle can be traced independently and hence the computation can be done stably even though the equi distance curves have singular points

conditioned response time distribution for a large closed processor sharing system with multiple classes in very heavy usage a large closed processor sharing system with multiple job classes is considered the system consists of a bank of terminals grouped according to job classes where the class structure allows for distinctions in the user s behavior in the terminal and in the service requirements from the central processing unit our earlier results for the single class system in very heavy usage are extended to the multiple class systems for large systems asymptotic approximations to the distribution of the equilibrium response time conditioned on the required service time tau are derived the first order correction term to the leading normal approximation which leads to a skewness of the distribution is shown to have the same structure as the single class system equations are obtained for determining the asymptotic approximations to the mean variance and skewness of the response time as functions of the required service time tau

on the hyers ulam stability of functional equations connected with additive and quadratic mappings abstract we investigate some inequalities connected with the hyers ulam stability of three functional equations which have a solution of the form a q where a is an additive mapping and q is a quadratic one

on the hyers ulam stability of the functional equations that have the quadratic property abstract the hyers ulam stability of the quadratic functional equation 1 on a restricted domain shall be investigated and the result shall be applied to the study of an asymptotic behavior of that equation furthermore the hyers ulam stability problems of another quadratic equation 4 on a restricted domain shall also be treated under the approximately even or odd condition and some asymptotic behaviors of the quadratic mappings and the additive mappings shall be investigated

hyers ulam rassias stability of a jensen type functional equation abstract in this paper we solve the jensen type functional equation 1 1 likewise we investigate the hyers ulam rassias stability of this equation

approximating generalized distance functions on weighted triangulated surfaces with applications given p a simple connected possibly non convex polyhedral surface composed of positively weighted triangular faces we consider paths from generalized sources points segments polygonal chains or polygonal regions to points on p that stay on p and avoid obstacles segments polygonal chains or polygonal regions the distance function defined by a generalized source is a function that assigns to each point of p the cost of the shortest path from the source to the point in this paper we present an algorithm for computing approximate generalized distance functions we also provide an algorithm that computes a discrete representation of the approximate distance function and as applications algorithms for computing discrete order k voronoi diagrams and for approximately solving facility location problems finally we present experimental results obtained with our implementation of the provided algorithms

on the stability of a functional equation deriving from an inequality of popoviciu for convex functions abstract in this paper we solve the jensen type functional equation n n 2 k 2 f x 1 x n n n 2 k 1 i 1 n f x i k 1 i 1 k n f x i 1 x i k k deriving from an inequality of popoviciu for convex functions likewise we investigate the stability of this functional equation in the spirit of hyers ulam rassias and g vru

conditioned response time distribution for a large closed processor sharing system in very heavy usage the service of an interactive computer system is characterized by the random response waiting time perceived by users the system consists of a bank of user terminals each of which asynchronous

brief paper the zermelo voronoi diagram a dynamic partition problem we consider a voronoi like partition problem in the plane for a given finite set of generators each element in this partition is uniquely associated with a particular generator in the following sense an agent that resides within a set of the partition at a given time will arrive at the generator associated with this set faster than any other agent that resides anywhere outside this set at the same instant of time the agent s motion is affected by the presence of a temporally varying drift which is induced by local winds currents as a result the minimum time to a destination is not equivalent to the minimum distance traveled this simple fact has important ramifications over the partitioning problem it is shown that this problem can be interpreted as a dynamic voronoi diagram problem where the generators are not fixed but rather they are moving targets to be reached in minimum time the problem is solved by first reducing it to a standard voronoi diagram by means of a time varying coordinate transformation we then extend the approach to solve the dual problem where the generators are the initial locations of a given set of agents distributed over the plane such that each element in the partition consists of the terminal positions that can be reached by the corresponding agent faster than any other agent

decentralized spatial partitioning for multi vehicle systems in spatiotemporal flow field we consider the problem of characterizing a generalized voronoi diagram partition of a convex polygon in a two dimensional euclidean space that encodes information about the proximity relations between a team of aerial marine vehicles and arbitrary points in the partition space these proximity relations are determined by the time required for each vehicle to reach an arbitrary point time to go in the partition space when driven by a locally optimal feedback control law in the presence of a spatiotemporal drift field the main contribution of this work is the presentation of a partitioning algorithm which is decentralized in the sense that each vehicle can independently compute its corresponding cell from the generalized voronoi partition without computing or receiving information about the cells of the other vehicles finally we present numerical simulations using data from real drift fields to illustrate the key features of the decentralized solution to the proposed class of spatial partitioning problems

on the stability of a quadratic jensen type functional equation abstract in this paper we obtain the general solution of the quadratic jensen type functional equation 9f x y z 3 f x f y f z 4 f x y 2 f y z 2 f z x 2 and prove the stability of this equation in the spirit of hyers ulam rassias and g vruta

on the performance of state dependent single server queues new asymptotic methods for the analysis of queueing systems are introduced and applied to state dependent m g 1 queues the methods are used to compute approximations to the stationary density of the queue length the mean length of a busy period the mean number of customers served during a busy period as well as other quantities of interest we obtain results that are superior to those obtained from diffusion approximations in that they are uniformly valid for all values of the traffic intensity while diffusion approximations are adequate only when this quantity is close to one when specialized to state independent queues our approximations are shown to agree with the asymptotic expansions of known exact results finally we show that the behavior of the state dependent systems is markedly different from that of the corresponding state independent systems

a fixed point approach to stability of functional equations abstract we prove a simple fixed point theorem for some not necessarily linear operators and derive from it several quite general results on the stability of a very wide class of functional equations in single variable

on a direct method for proving the hyers ulam stability of functional equations abstract we study the stability of an equation in a single variable of the form f x a f h x b f h x with some conditions imposed on constants a b and function h the results are later applied for proving the stability by use of a direct method of a couple of functional equations in several variables they let us not only shorten various approaches obtained during the last years but also improve the approximating constants

partitioning algorithms for multi agent systems based on finite time proximity metrics we address a generalized voronoi partitioning problem for a team of mobile agents with nonlinear dynamics with respect to a state dependent proximity metric in particular the proximity pseudo metric corresponds to the reduction of a generalized energy metric that occurs during the transfer of an agent to an arbitrary destination with zero terminal velocity in finite time the realization of every finite time state transition takes place by means of a class of continuous feedback control laws that render the closed loop dynamics of each mobile agent non lipschitzian the arrival time also turns out to be a state dependent quantity whose functional description is not prescribed a priori we show that the partitioning problem studied in this work can admit a decentralized solution that is each agent can compute its own cell independently from its teammates provided that is aware of the positions and velocities of its neighboring agents numerical simulations that illustrate the theoretical developments are also presented

fast marching methods fast marching methods are numerical schemes for computing solutions to the nonlinear eikonal equation and related static hamilton jacobi equations based on entropy satisfying upwind schemes and fast sorting techniques they yield consistent accurate and highly efficient algorithms they are optimal in the sense that the computational complexity of the algorithms is o n log n where n is the total number of points in the domain the schemes are of use in a variety of applications including problems in shape offsetting computing distances from complex curves and surfaces shape from shading photolithographic development computing first arrivals in seismic travel times construction of shortest geodesics on surfaces optimal path planning around obstacles and visibility and reflection calculations in this paper we review the development of these techniques including the theoretical and numerical underpinnings provide details of the computational schemes including higher order versions and demonstrate the techniques in a collection of different areas

ulam s stability of a generalization of the frechet functional equation abstract we prove some general stability and hyperstability results for a generalization of the well known frechet equation stemming from the characterization of the inner product spaces due to jordan and von neumann the main result yields several stability outcomes for various other well known functional equations thus we obtain in particular some new inequalities characterizing the inner product spaces as the main tool we use a fixed point theorem for some function spaces

optimal partitioning for spatiotemporal coverage in a drift field abstract we consider the problem of partitioning an area in the plane populated by a team of aerial marine vehicles into a finite collection of non overlapping sets the sets of this partition are in an one to one correspondence with the vehicles under the following rule each point in the given set of the partition can be reached by the corresponding vehicle in this set faster than any other vehicle in the presence of a spatiotemporal drift field consequently a voronoi like partition results which encodes the proximity relations between the vehicles and arbitrary points in the plane with respect to the minimum time to go the construction of this voronoi like partition is based on its interpretation as the intersection of a forest of the cost minimum time to go surfaces emanating from each generator with their common lower envelope the characterization of each cost surface is achieved by means of an efficient expansion scheme of the level sets of the minimum time to go function which utilizes in turn the structure of the optimal synthesis of the minimum time problem without resorting to exhaustive numerical techniques e g fast marching methods we examine the topological characteristics of the partition by using control system theoretic concepts and tools the theoretical developments are illustrated with a number of numerical examples

high resolution flux based level set method a new high resolution flux based finite volume method for general advection equations in nondivergent form including a level set equation for moving interfaces is introduced the method is applicable to the case of nondivergence free velocity and to general unstructured grids in higher dimensions we show that the method is consistent and that the numerical solution fulfills the discrete minimum maximum principle numerical experiments show its second order accuracy for smooth solutions as well as for solutions with discontinuous derivatives and on general unstructured meshes numerical examples for passive transport and shrinking of dynamic interfaces including examples with topological changes are presented using locally adapted two dimensional and three dimensional grids

coverage optimization and spatial load balancing by robotic sensor networks this technical note studies robotic sensor networks performing static coverage optimization with area constraints given a density function describing the probability of events happening and a performance function measuring the cost to service a location the objective is to position sensors in the environment so as to minimize the expected servicing cost moreover because of load balancing considerations the area of the region assigned to each robot is constrained to be a pre specified amount we characterize the optimal configurations as center generalized voronoi configurations the generalized voronoi partition depends on a set of weights one per robot assigned to the network we design a jacobi iterative algorithm to find the weight assignment whose corresponding generalized voronoi partition satisfies the area constraints this algorithm is distributed over the generalized delaunay graph we also design the move to center and compute weight strategy to steer the robotic network towards the set of center generalized voronoi configurations while monotonically optimizing coverage

asymptotic expansions for a closed multiple access system a closed multiple access system composed of m sources and a single service facility is analyzed the closed network is modeled as a finite source m g 1 queueing system we consider systems with a large number of sources i e m gg 1 and we assume that the mean time required to process an individual request is short o 1 m we then construct asymptotic approximations to the stationary distribution of the number of requests in the service facility by using the method of matched asymptotic expansions we give formulas for the first and second moments of the number of requests for all traffic intensities

vehicle routing with a sparse feasibility graph abstract in this paper we introduce the concept of a feasibility graph for vehicle routing problems a graph where two customers are linked if and only if it is possible for them to be successive adjacent customers on some feasible vehicle route we consider the problem of designing vehicle routes when the underlying feasibility graph is sparse i e when any customer has only a few other customers to which they can be adjacent on a vehicle route this problem arose during a consultancy study that involved the design of fixed vehicle routes delivering to contiguous adjacent postal districts a heuristic algorithm for the problem is presented and computational results given for a number of test problems involving up to 856 customers

anisotropic diagrams labelle shewchuk approach revisited f labelle and j shewchuk have proposed a discrete definition of anisotropic voronoi diagrams these diagrams are parametrized by a metric field under mild hypotheses on the metric field such voronoi diagrams can be refined so that their dual is a triangulation with elements shaped according to the specified anisotropic metric field we propose an alternative view of the construction of these diagrams and a variant of labelle and shewchuk s meshing algorithm this variant computes the voronoi vertices using a higher dimensional power diagram and refines the diagram as long as dual triangles overlap we see this variant as a first step toward a 3 dimensional anisotropic meshing algorithm

distribution of processor sharing customers for a large closed system with multiple classes a closed processor sharing ps system with multiple customer classes is considered the system consists of one infinite server is station and one ps station for a system with a large number of customers a saturated ps station and an arbitrary number of customer classes asymptotic approximations to the stationary distribution of the total number of customers at the ps station are derived the asymptotics for the probability mass function is described by a quasi potential function which defines the exponential decay for the distribution and a state dependent preexponential factor both functions have an explicit expression in terms of the solution at each point x of a polynomial equation whose order equals the number of classes and whose coefficients are explicit functions of x the quasi potential function at its minimum point provides the logarithmic asymptotics for the normalization constant and the asymptotic approximation for the variance is inversely proportional to the second derivative of

sequential pursuit of multiple targets under external disturbances via zermelo voronoi diagrams abstract we address the problem of pursuit between a collection of targets and a team of pursuers distributed in the plane subject to an environmental disturbance e g wind sea current the objective of the pursuers is to intercept the moving targets which however are not affected by the presence of the flow field we first solve the multiple pursuers single target problem by assigning only one pursuer to chase the target at every instant of time based on a voronoi like partition of the plane during the pursuit the pursuer assignment changes dynamically based on this partition we then present an algorithm to efficiently update this voronoi like partition on line the results are then extended to the multiple pursuers multiple targets case simulations are included to illustrate the theoretical results

power diagrams properties algorithms and applications the power pow x s of a point x with respect to a sphere s in euclidean d space e d is given by d 2 x z r 2 where d denotes the euclidean distance function and z and r are the center and the radius of s the power diagram of a finite set s of spheres in e d is a cell complex that associates each s in s with the convex domain x in e d operatorname pow x s operatorname pow x t text for all t in s s the close relationship to convex hulls and arrangements of hyperplanes is investigated and exploited efficient algorithms that compute the power diagram and its order k modifications are obtained among the applications of these results are algorithms for detecting k sets for union and intersection problems for cones and paraboloids and for constructing weighted voronoi diagrams and voronoi diagrams for spheres upper space bounds for these geometric problems are derived

voronoi diagram in the laguerre geometry and its applications we extend the concept of voronoi diagram in the ordinary euclidean geometry for n points to the one in the laguerre geometry for n circles in the plane where the distance between a circle and a point is defined by the length of the tangent line and show that there is an o n log n algorithm for this extended case the voronoi diagram in the laguerre geometry may be applied to solving effectively a number of geometrical problems such as those of determining whether or not a point belongs to the union of n circles of finding the connected components of n circles and of finding the contour of the union of n circles as in the case with ordinary voronoi diagrams the algorithms proposed here for those problems are optimal to within a constant factor some extensions of the problem and the algorithm from different viewpoints are also suggested

a hybrid particle level set method for improved interface capturing in this paper we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field our method uses lagrangian marker particles to rebuild the level set in regions which are underresolved this is often the case for flows undergoing stretching and tearing the overall method maintains a smooth geometrical description of the interface and the implementation simplicity characteristic of the level set method our method compares favorably with volume of fluid methods in the conservation of mass and purely lagrangian schemes for interface resolution the method is presented in three spatial dimensions

variations on the drygas equation and its stability abstract we study the stability of the drygas functional equation g x y g x y 1 2 g x g y g y 1 mixing the direct method of the proof with the method of the invariant means dropping the assumption about the domain to be an abelian group we assume that the function we are dealing with is central i e g x y g y x is approximatively central i e g x y g y x satisfies the kannappan condition i e g x y z g x z y or that the group is amenable