the c convergence of sg circle patterns to the riemann mapping abstract thurston conjectured that the riemann mapping function from a simply connected region onto the unit disk can be approximated by regular hexagonal packings schramm introduced circle patterns with combinatorics of the square grid sg and showed that sg circle patterns converge to meromorphic functions he and schramm proved that hexagonal disk packings converge in c to the riemann mapping in this paper we show a similar c convergence for sg circle patterns

fixed points koebe uniformization and circle packings a domain in the riemann sphere hat mathbb c is called a circle domain if every connected component of its boundary is either a circle or a point in 1908 p koebe ko1 posed the following conjecture known as koebe s kreisnormierungsproblem any plane domain is conformally homeomorphic to a circle domain in hat mathbb c when the domain is simply connected this is the content of the riemann mapping theorem

hexagonal circle patterns and integrable systems patterns with constant angles hexagonal circle patterns with constant intersection angles are introduced and studied it is shown that they are described by discrete integrable systems of toda type conformally symmetric patterns are classified circle pattern analogs of holomorphic mappings z sp c and log z are constructed as special isomonodromic solutions circle patterns studied in the paper include schramm s circle patterns with the combinatorics of the square grid as a special case

rigidity of infinite hexagonal triangulation of the plane in the paper we consider the rigidity problem of the infinite hexagonal triangulation of the plane under the piecewise linear conformal changes introduced by luo in 5 our result shows that if a geometric hexagonal triangulation of the plane is pl conformal to the regular hexagonal triangulation and all inner angles are in 2 for any constant 0 then it is the regular hexagonal triangulation this partially solves a conjecture of luo 4 the proof uses the concept of quasi harmonic functions to unfold the properties of the mesh mathematical sciences center tsinghua university beijing 100084 china email mike890505 gmail com department of computer science stony brook university new york 11794 usa email gu cs stonybrook edu mathematical sciences center tsinghua university beijing 100084 china email jsun math tsinghua edu cn 1 ar x iv 1 30 6 36 30 v1 m at h g t 1 6 ju n 20 13

robust stabilisation control for discrete time networked control systems we consider the analysis and synthesis of discrete time networked control systems ncss where the plant has additive uncertainty and the controller is updated with the sensor information at stochastic time intervals it is shown that the problem is linked to h control of linear discrete time stochastic systems and a new small gain theorem is established based on this result sufficient conditions are given for ensuring mean square stability of the ncs and the genetic algorithm is utilised to design the controller of the ncs based on a linear matrix inequality technique an illustrative example is given to demonstrate the effectiveness of our proposed method

moving target search a real time search for changing goals considers the case of heuristic search where the goal may change during the course of the search for example the goal may be a target that actively avoids the problem solver the authors present a moving target search algorithm mts to solve this problem the authors prove that if the average speed of the target is slower than that of the problem solver then the problem solver is guaranteed to eventually reach the target in a connected problem space the original mts algorithm was constructed with the minimum operations necessary to guarantee its completeness and hence is not very efficient to improve its efficiency the authors introduce ideas from the area of resource bounded planning into mts including 1 commitment to goals and 2 deliberation for selecting plans experimental results demonstrate that the improved mts is 10 to 20 times more efficient than the original mts in uncertain situations

a small gain theorem for linear stochastic systems abstract a well known result in linear control theory is the so called small gain theorem stating that if given two plants with transfer matrix functions t 1 and t 2 in h such that t 1 and t 2 when coupling t 2 to t 1 such that u 2 y 1 and u 1 y 2 one obtains an internally stable closed system the aim of the present paper is to describe a corresponding result for stochastic systems with state dependent white noise

state feedback control of systems with multiplicative noise via linear matrix inequalities we consider lti systems perturbed by parametric uncertainties modeled as white noise disturbances we show how to maximize via state feedback control the smallest norm of the noise intensity vector producing instability in the mean square sense using convex optimization over linear matrix inequalities we also show how to maximize performance robustness where performance is measured by expected output energy with either bounded initial conditions and zero inputs classical lqg cost or zero initial conditions and deterministic inputs of bounded energy a generalization of the h norm

on the teichmuller theory of circle patterns given a circle pattern on the riemann sphere in this paper we prove that its quasiconformal deformation space can be naturally identified with the product of the teichmuller spaces of its interstices

asymptotic properties of input output operators norm associated with singularly perturbed systems with multiplicative white noise in this paper we study the problem of the asymptotic property of the norm of input output operators related to a class of singularly perturbed stochastic linear systems the system is under perturbation of multiplicative white noise by using reduction order and boundary layer techniques it is shown that the norm of the operator of the perturbed system is less than a given number gamma when the small perturbation varepsilon tends to zero if both the related norms of the reduced subsystem and the boundary layer subsystem are less than gamma furthermore a stabilizing robust controller is designed which is independent of perturbation varepsilon

constructing conformal maps of triangulated surfaces abstract we describe a means of computing the uniformizing conformal map from a triangulated surface whose triangles are realized as euclidean triangles in r 3 onto a fundamental domain in the unit disc d plane c or sphere s 2 mapping such triangulated surfaces arises in a number of applications such as conformal brain flattening we use the circle packing technique of bowers hurdal stephenson et al to first create a quasiconformal approximation to the conformal map then we apply a discrete form of conformal welding to reduce the distortion and converge to the conformal map in the limit

the c convergence of circle packings of bounded degree to the riemann mapping abstract thurston conjectured that hexagonal circle packings can be used to approximate the riemann mapping the corresponding convergence was proven by rodin and sullivan he and schramm showed that for hexagonal circle packings the convergence is c here the c convergence is generalized to the case of non hexagonal circle packings with bounded degree furthermore the estimation of the convergence rate is obtained for arbitrary order derivatives

hexagonal circle patterns with constant intersection angles and discrete painleve and riccati equations hexagonal circle patterns with constant intersection angles mimicking holomorphic maps zc and log z are studied it is shown that the corresponding circle patterns are immersed and described by special separatrix solutions of discrete painleve and riccati equations the general solution of the riccati equation is expressed in terms of the hypergeometric function global properties of these solutions as well as of the discrete zc and log z are established

minimal surfaces from infinitesimal deformations of circle packings abstract we study circle packings with the combinatorics of a triangulated disk in the plane and parametrize deformations of circle packings in terms of vertex rotation and cross ratios we show that there is a weierstrass representation formula relating infinitesimal deformations of circle packings to discrete minimal surfaces of koebe type furthermore every minimal surface of koebe type can be extended naturally to a discrete minimal surface of general type in this way discrete minimal surfaces via steiner s formula are unified

mean square stability analysis of stochastic continuous time linear networked systems in this technical note we study the mean square stability based analysis of stochastic continuous time linear networked systems the stochastic uncertainty is assumed to enter multiplicatively in system dynamics through input and output channels of the plant necessary and sufficient conditions for mean square exponential stability are expressed in terms of the input output property of deterministic or nominal system dynamics captured by the mean square system norm and variance of channel uncertainty the stability results can also be interpreted as a small gain theorem for continuous time stochastic systems linear matrix inequalities based optimization formulation is provided for the computation of mean square system norm for the stability analysis for a special case of single input channel uncertainty we also prove a fundamental limitation result that arises in the mean square exponential stabilization of the continuous time linear system overall the contributions in this technical note generalize the existing results on stability analysis from discrete time linear systems to continuous time linear systems with multiplicative uncertainty simulation results are presented for western system coordinating council 9 bus power system to demonstrate the application of the developed framework

minimal surfaces from circle patterns geometry from combinatorics the theory of polyhedral surfaces and more generally the field of discrete differential geometry are presently emerging on the border of differential and discrete geometry whereas classical differential geometry investigates smooth geometric shapes such as surfaces and discrete geometry studies geometric shapes with a finite number of elements polyhedra the theory of polyhedral surfaces aims at a development of discrete equivalents of the geometric notions and methods of surface theory the latter appears then as a limit of the refinement of the discretization current progress in this field is to a large extent stimulated by its relevance for computer graphics and visualization one of the central problems of discrete differential geometry is to find proper discrete analogues of special classes of surfaces such as minimal con stant mean curvature isothermic surfaces etc usually one can suggest vari ous discretizations with the same continuous limit which have quite different geometric properties the goal of discrete differential geometry is to find a dis cretization which inherits as many essential properties of the smooth geometry as possible our discretizations are based on quadrilateral meshes i e we discretize parametrized surfaces for the discretization of a special class of surfaces it is natural to choose an adapted parametrization in this paper we investigate conformai discretizations of surfaces i e discretizations in terms of circles and spheres and introduce a new discrete model for minimal surfaces see figures 1 and 2 in comparison with direct methods see in particular 23 leading

the c convergence of hexagonal disk packings to the riemann map let omega mathop subset limits ne c be a simply connected domain the rodin sullivan theorem states that a sequence of disk packings in the unit disk u converges in a well defined sense to a conformal map from to u moreover it is known that the first and second derivatives converge as well here it is proven that for hexagonal disk packings the convergence is c this is done by studying mobius invariants of the disk packings that are discrete analogs of the schwarzian derivative

rigidity of infinite disk patterns let p be a locally flnite disk pattern on the complex plane c whose combinatorics are described by the one skeleton g of a triangulation of the open topological disk and whose dihedral angles are equal to a function e 0 2 on the set of edges let p be a combinatorially equivalent disk pattern on the plane with the same dihedral angle function we show that p and p difier only by a euclidean similarity in particular when the dihedral angle function is identically zero this yields the rigidity theorems of b rodin and d sullivan and of o schramm whose arguments rely essentially on the pairwise disjointness of the interiors of the disks the approach here is analytical and uses the maximum principle the concept of vertex extremal length and the recurrency of a family of electrical networks obtained by placing resistors on the edges in the contact graph of the pattern a similar rigidity property holds for locally flnite disk patterns in the hyperbolic plane where the proof follows by a simple use of the maximum principle also we have a uniformization result for disk patterns in a future paper the techniques of this paper will be extended to the case when 0 in particular we will show a rigidity property for a class of inflnite convex polyhedra in the 3 dimensional hyperbolic space

stochastic h sub spl infin tracking with preview for state multiplicative systems the problem of finite horizon h sub spl infin tracking for linear time varying systems with stochastic parameter uncertainties is investigated we consider three tracking patterns depending on the nature of the reference signal i e whether it is perfectly known in advance measured on line or previewed in a fixed time interval ahead the stochastic uncertainties appear in both the dynamic and measurement matrices of the system for each of the above three cases a game theory approach is applied for the state feedback case where given a specific reference signal the controller plays against nature which chooses the initial condition and the energy bounded disturbance the problems are solved using an expected value of the standard performance index over the stochastic parameters where necessary and sufficient conditions are found for the existence of a saddle point equilibrium the infinite horizon time invariant tracking problem is also solved the theory developed is demonstrated by a simple tracking example

on the convergence of circle packings to the riemann map rodin and sullivan 1987 proved thurston s conjecture that a scheme based on the circle packing theorem converges to the riemann mapping thereby providing a refreshing geometric view of riemann s mapping theorem we now present a new proof of the rodin sullivan theorem this proof is based on the argument principle and has the following virtues 1 it applies to more general packings the rodin sullivan paper deals with packings based on the hexagonal combinatorics later quantitative estimates were found which also worked for bounded valence packings here the bounded valence assumption is unnecessary and irrelevant 2 our method is rather elementary and accessible to non experts in particular quasiconformal maps are not needed consequently this gives an independent proof of riemann s conformal mapping theorem the rodin sullivan proof uses results that rely on riemann s mapping theorem 3 our approach gives the convergence of the first and second derivatives without significant additional difficulties while previous work has established the convergence of the first two derivatives for bounded valence packings now the bounded valence assumption is unnecessary

planar relative schottky sets and quasisymmetric maps a relative schottky set in a planar domain is a subset of obtained by removing from open geometric discs whose closures are in and are pairwise disjoint in this paper we study quasisymmetric and related maps between relative schottky sets of measure zero we prove in particular that quasisymmetric maps between such sets in jordan domains are conformal locally bi lipschitz and that their first derivatives are locally lipschitz we also provide a locally bi lipschitz uniformization result for relative schottky sets in jordan domains and establish rigidity with respect to local quasisymmetric maps for relative schottky sets in the unit disc

on unified concepts of detectability and observability for continuous time stochastic systems this paper is concerned with detectability and observability of continuous time stochastic linear systems by adopting the idea used in defining these two concepts for time varying systems and markovian jump linear systems that have been studied in the literature corresponding definitions for continuous time stochastic linear systems are proposed these two definitions are not only able to unify some recent definitions on these two concept reported in the literature but also allow us to propose an efficient rank criterion to test observability of continuous time stochastic linear systems it seems that this rank criterion is quite analogous to the rank criterion for deterministic linear systems with the help of these two concepts and the new criteria the stochastic lyapunov equation is revisited and some recent published work on this equation are generalized numerical examples are given to illustrate the effectiveness of the proposed approach

brief paper on detectability of stochastic systems we discuss notions of detectability for stochastic linear control systems of ito type a natural concept of detectability requires a non zero output if the state process is unstable we show that this property can equivalently be characterized by a generalized version of the hautus test the proof is based on spectral theory for positive operators

schwarz s lemma for circle packings burt rodin department of mathematics university of california la jolla ca 92093 usa 1 introduction conformal mappings can be approximated by circle packing isomorphisms this is proved in 10 roughly speaking see sect 5 for a more detailed description a bounded region f2 is almost filled by e circles 7 from the regular hexagonal e circle packing of the plane denote this approximate circle pack ing of q by q by results of andreev and thurston there is a combinatorially isomorphic circle packing d of the unit disk d denote this isomorphism suitably normalized by 7 7 f2 d let f be the piecewise linear quasicon formal mapping determined by the associated triangulations as e 0 f con verges uniformly on compacta to the riemann mapping function f f2 d let

a new computational search technique for ai based on the cantor set abstract this paper discusses the development of a new search algorithm for artificial intelligence systems the search technique is based on the mathematical theory of the cantor set the search algorithm will be efficient for specialized search domains where the distribution of the data elements to be searched is approximately normal the approach uses the iterative procedure of deleted middle thirds this facilitates quick pruning of a search space the new search technique has potential applications in computational systems with large input output data handling extensive experiments comparing the cantor set search css to binary search indicate that the new search technique holds good promise

static h2 and h output feedback of discrete time lti systems with state multiplicative noise abstract a parameter dependent approach for designing static output feedback controller for linear time invariant systems with state multiplicative noise is introduced which achieves a minimum bound on either the stochastic h 2 or the h performance levels a solution is obtained also for the case where in addition to the stochastic parameters the system matrices reside in a given polytope in this case a parameter dependent lyapunov function is described which enables the derivation of the required constant feedback gain via a solution of a set of linear matrix inequalities that correspond to the vertices of the uncertainty polytope the stochastic parameters appear in both the dynamics and the input matrices of the state space model of the system the problems are solved using the expected value of the standard performance indices over the stochastic parameters the theory developed is demonstrated by a simple example

solving beltrami equations by circle packing we use andreev thurston s theorem on the existence of circle packings to construct approximating solutions to the beltrami equations on riemann surfaces the convergence of the approximating solutions on compact subsets will be shown this gives a constructive proof of the existence theorem for beltrami equations

rigidity of infinite circle packings the nerve of a packing is a graph that encodes its combinatorics the vertices of the nerve correspond to the packed sets and an edge occurs between two vertices in the nerve precisely when the corresponding sets of the packing intersect

rigidity of circle domains whose boundary hasσ finite linear measure let be a circle domain in the riemann sphere whose boundary has finite linear measure we show that is rigid in the sense that any conformal homeomorphism of onto any other circle domain is equal to the restriction of a mobius transformation previously kaufman and bishop have independently found examples of non rigid circle domains whose boundary is a cantor set of hausdorff dimension one

robust h sub spl infin filtering of stationary continuous time linear systems with stochastic uncertainties the problem of applying h sub spl infin filters on stationary continuous time linear systems with stochastic uncertainties in the state space signal model is addressed these uncertainties are modeled via white noise processes the relevant cost function is the expected value of the standard h sub spl infin performance index with respect to the uncertain parameters the solution is obtained via a stochastic bounded real lemma that results in a modified riccati inequality this inequality is expressed in the form of a linear matrix inequality whose solution provides the filter parameters the method proposed is also applied to the case where in addition to the stochastic uncertainty other deterministic parameters of the system are not perfectly known and are assumed to lie in a given polytope the problem of mixed h sub 2 h sub spl infin filtering for the above system is also treated the theory developed is demonstrated by a practical example

surveillance search for a moving target the surveillance search problem is a generalization of the detection search problem studied by brown and stone among others in the surveillance version of the problem the search terminates only if the time and location of a detection satisfy certain problem specific conditions the objective is to maximize the expected value of a payoff received during and at the end of the search for instance we may wish to maximize the probability of finding the target in a specified subset of the state space or at a specified time consequently we might detect the target several times before the search terminates a search strategy i e an allocation of the available search effort must account for this possibility in this paper we formulate the surveillance search problem for applications in which the target moves according to a not necessarily time homogeneous markov process we then derive a set of necessary conditions for the optimality of a search strategy and develop algorithms based on the algorithm of brown for the detection search problem to find strategies that satisfy these necessary conditions for problems in which the search effort is infinitely divisible