on the two block h problem for a class of unstable distributed systems this paper deals with the two block h control problem for distributed plants with finitely many unstable modes we assume that weighting filters in the h mixed sensitivity problem are finite dimensional then the corresponding optimal two block problem can be solved by finding the schmidt pairs of a hankel operator whose symbol is of the form m 2 m 1u u where u r h u h and m2 rh and m1 h are inner and the suboptimal two block problem can be solved by finding the solutions of certain functional equations very similar to the ones satisfied by the schmit pairs of the above mentioned hankel operator in this paper a unified approach is proposed for solving both the optimal and suboptimal two block problems we obtain two systems of linear equations expressed in terms of state space realizations of u and m2 whose solutions give the schmidt pairs of the associated hankel operator and the functions needed for the parametrization of all the suboptimal solutions respectively

krein space approach to h sup spl infin mixed sensitivity minimization for a class of infinite dimensional systems motivated by the computation of the h sup spl infin optimal performance and optimal mixed sensitivity for a class of infinite dimensional systems the author considers the explicit computation of the norm and minimal symbols of a class of hankel plus toeplitz type operators with irrational symbols using the krein space approach the author shows how to explicitly compute the norm and parameterize all the minimal symbols the results are used to obtain a complete solution to the h sup spl infin optimal mixed sensitivity synthesis problem for a class of infinite dimensional systems an explicit formula for computing the h sup spl infin optimal performance and explicit parameterization of all optimal and suboptimal mixed sensitivities an illustrative example is given

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

generic mean curvature flow i generic singularities it has long been conjectured that starting at a generic smooth closed embedded surface in r 3 the mean curvature ow remains smooth until it arrives at a singularity in a neighborhood of which the ow looks like concentric spheres or cylinders that is the only singularities of a generic ow are spherical or cylindrical we will address this conjecture here and in a sequel the higher dimensional case will be addressed elsewhere the key to showing this conjecture is to show that shrinking spheres cylinders and planes are the only stable self shrinkers under the mean curvature ow we prove this here in all dimensions an easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away

computational design and additive manufacturing of periodic conformal metasurfaces by synthesizing topology optimization with conformal mapping abstract in this paper we present a computational framework for computational design and additive manufacturing of free form periodic metasurfaces the proposed scheme rests on the level set based topology approach and the conformal mapping theory a metamaterial with pre specified performance is created using a level set based topology optimization method the achieved unit cell is further mapped to the 3d quad meshes on a free form surface by applying the conformal mapping method which can preserve the local shape and angle during the mapping with embedded geometric information the proposed level set based optimization methods not only can act as a motivator for design synthesis but also can be seamlessly hooked with additive manufacturing without the need of cad reconstructions the current computational framework provides a solution to increasing applications involving innovative metamaterial designs on free form surfaces for different fields of interests the performance of the proposed scheme is illustrated through two benchmark examples where a negative poisson s ratio unit cell pattern and a stiff and light inner structure are mapped to 3d free form surfaces and fabricated through additive manufacturing

a discrete uniformization theorem for polyhedral surfaces a notion of discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper this discrete conformality is shown to be computable it is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying gauss bonnet formula furthermore the hyperbolic polyhedral metric with given curvature can be obtained using a discrete yamabe flow with surgery in particular each hyperbolic polyhedral metric on a closed surface with negative euler characteristic is discrete conformal to a unique hyperbolic metric

a note on essential spectra and norms of mixed hankel toeplitz operators in this brief hankel toeplitz operators which occur in feedback theory e g in the minimization of mixed h sensitivity and complementary sensitivity will be considered a method of computing their spectra eigenvectors and norms will be presented for infinite dimensional systems subject to continuous weightings

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

variational principles for branched circle patterns abstract we are concerned with variational principles for branched circle patterns we use critical point theory to establish existence and uniqueness furthermore we apply the ricci flow method to give an approximation scheme which converges exponentially to the branched circle pattern metric

feedback minimax sensitivity and optimal robustness in this paper we look for feedbacks that minimize the sensitivity function of a linear single variable feedback system represented by its frequency responses sensitivity to disturbances and robustness under plant perturbations are measured in a weighted h infty norm in an earlier paper zames proposed an approach to feedback design involving the measurement of sensitivity by multiplicative seminorms which have certain advantages over the widely used quadratic norm in problems where there is plant uncertainty or where signal power spectra are not fixed but belong to sets the problem was studied in a general setting and some h infty examples were solved here a detailed study of the single variable case is undertaken the results are extended to unstable plants and explicit formulas for the general situation of a finite number of right half plane rhp plant zeros or poles are provided the q or approximate inverse parametrization of feedbacks that maintain closed loop stability is extended to the ease of unstable plants the h infty and wiener hopf approaches are compared

discrete riemann surfaces based on quadrilateral cellular decompositions abstract our aim in this paper is to provide a theory of discrete riemann surfaces based on quadrilateral cellular decompositions of riemann surfaces together with their complex structure encoded by complex weights previous work in particular of mercat mainly focused on real weights corresponding to quadrilateral cells having orthogonal diagonals we discuss discrete coverings discrete exterior calculus and discrete abelian differentials our presentation includes several new notions and results such as branched coverings of discrete riemann surfaces the discrete riemann hurwitz formula double poles of discrete one forms and double values of discrete meromorphic functions that enter the discrete riemann roch theorem and a discrete abel jacobi map

the volume of hyperbolic alternating link complements if a hyperbolic link has a prime alternating diagram d then we show that the link complement s volume can be estimated directly from d we define a very elementary invariant of the diagram d its twist number t d and show that the volume lies between v 3 t d 2 2 and v 3 10t d 10 where v 3 is the volume of a regular hyperbolic ideal 3 simplex as a consequence the set of all hyperbolic alternating and augmented alternating link complements is a closed subset of the space of all complete finite volume hyperbolic 3 manifolds in the geometric topology

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

closed formulae for a parametric mixed sensitivity problem abstract mixed sensitivity optimization is a two block problem which normally requires an iterative solution the purpose of this paper is to give an explicit closed formula for the optimal cost in the case of a parametric mixed sensitivity problem we also show that all the controllers which satisfy an arbitrary h norm bound may be parameterized in terms of the solutions to the standards h 2 riccati equations we hope that these results will help progress towards closed formulae for the optimal cost in a wider class of problems

recurrence of planar graph limits we prove that any distributional limit of nite planar graphs in which the degree of the root has an exponential tail is almost surely recurrent as a corollary we obtain that the uniform innite planar triangulation and quadrangulation uipt and uipq are almost surely recurrent resolving a conjecture of angel benjamini and schramm we also settle another related problem of benjamini and schramm we show that in any bounded degree nite planar graph the probability that the simple random walk started at a uniform random vertex avoids its initial location for t steps is at most c logt

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

a branched andreev thurston theorem for circle packings of the sphere andreev thurston theorem let r be a triangulation of s which is not simplicially equivalent to a tetrahedron and let r 0 n be given where t denotes the set of edges of x suppose that a if eu e2 e3 e r 1 form a closed loop on s and if 2 i e n then eif e2 e3 are the three edges of a two simplex of x b if ex e2 e3 ea e r 1 form a closed loop on s and if 2 i e 2k then ex e2 e2 e4 form the boundary of the union of two adjacent twosimplexes of x

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

a note on rigidity theorem of λ hypersurfaces self shrinkers are an important class of solutions to the mean curvature flow and their generalization is hypersurfaces in this paper we study hypersurfaces and give a rigidity result about complete hypersurfaces

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

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

the general distance problem in h optimal control theory abstract the general distance problem that arises in h optimal control theory is considered transformations that reduce the general h problem to the general distance problem are reviewed and an iterative scheme called iteration is considered in which the general distance problem is reduced to a standard best approximation problem the iteration is viewed as a problem of finding the zero crossing of a function this function is shown to be continuous monotonically decreasing convex and bounded by some very simple functions these properties make it possible to obtain very rapid convergence of the iterative process the issues of model reduction and approximation in h synthesis will also be addressed

characterizations of circle patterns and finite convex polyhedra in hyperbolic 3 space the aim of this paper is to study finite convex polyhedra in three dimensional hyperbolic space mathbb h 3 we characterize the quasiconformal deformation space of each finite convex polyhedron as a corollary we obtain some results on finite circle patterns in the riemann sphere with dihedral angle 0 le theta pi that is for any circle pattern on hat mathbb c its quasiconformal deformation space can be naturally identified with the product of the teichmuller spaces of its interstices

drawing graphs in the plane with high resolution this paper presents the problem of drawing a graph in the plane so that edges appear as straight lines and the minimum angle formed by any pair of incident edges is maximized the resolution of a layout is defined to be the size of the minimum angle formed by incident edges of the graph and the resolution of a graph to be the maximum resolution of any layout of the graph the resolution r of a graph is characterized in terms of the maximum node degree d of the graph by proving that omega frac 1 d 2 leqslant r leqslant frac 2 pi d for any graph moreover it is proved that r theta frac 1 d for many graphs including planar graphs complete graphs hypercubes multidimensional meshes and tori and other special networks it is also shown that the problem of deciding if r frac 2 pi d for a graph is np hard for d 4 and by using a counting argument that r o frac log d d 2 for many graphs

isothermic triangulated surfaces we found a class of triangulated surfaces in euclidean space which have similar properties as isothermic surfaces in differential geometry we call a surface isothermic if it admits an infinitesimal isometric deformation preserving the mean curvature integrand locally we show that this class is mobius invariant isothermic triangulated surfaces can be characterized either in terms of circle patterns or based on conformal equivalence of triangle meshes this definition generalizes isothermic quadrilateral meshes a consequence is a discrete analog of minimal surfaces here the weierstrass data needed to construct a discrete minimal surface consist of a triangulated plane domain and a discrete harmonic function

sensitivity minimization in an h norm parametrization of all suboptimal solutions a number of problems in the control of linear feedback systems can be reduced to the following we are given three stable rational matrix functions k of sizes p 1 x q 1 p 1 x q 2 and p 2 x q 1 respectively and seek a stable rational q 2 x p 2 matrix function s so as to minimize k s we assume that p 1 q 2 p 2 q 1 and that and have maximal rank q 2 and p 2 respectively on the j axis given a tolerance level sufficiently large we obtain a linear fractional map g f 0 11 g 0 12 0 13 0 21 g 0 22 0 23 1 such that f k s with s stable and f if and only if g is a stable q 2 x p 2 matrix function with g 1 the computation of 0 0 ij 1 i 2 1 j 3 reduces to solving a pair of symmetric wiener hopf factorization problems for the special case where i q2 0 t i p2 0 and k not necessarily stable to which the general case can be reduced we provide explicit state space formulae for 0 in terms of a state space realization of k and the

branched circle packings and discrete blaschke products in this paper we introduce the notion of discrete blaschke products via circle packing we first establish necessary and sufficient conditions for the existence of finite branched circle packings next discrete blaschke products are defined as circle packing maps from univalent circle packings that properly fill d z izi 1 to the corresponding branched circle packings that properly cover d it is verified that such maps have all geometric properties of their classical counterparts finally we show that any classical finite blaschke product can be approximated uniformly on compacta of d by discrete ones

on the optimal and suboptimal solutions of the mixed sensitivity problem abstract in this note using elementary operator theoretic methods we give a method for the explicit computation of the suboptimal and optimal solutions to the mixed sensitivity problem for stable single input single output distributed systems

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

rigidity of schottky sets we call the complement of a union of at least three disjoint round open balls in the unit sphere bbb s n a schottky set we prove that every quasisymmetric homeomorphism of a schottky set of spherical measure zero to another schottky set is the restriction of a m obius transformation on bbb s n in the other direction we show that every schottky set in bbb s 2 of positive measure admits nontrivial quasisymmetric maps to other schottky sets these results are applied to establish rigidity statements for convex subsets of hyperbolic space that have totally geodesic boundaries

an operator theoretic approach to the mixed sensitivity minimization problem in this paper we consider the mixed sensitivity minimization problem scalar case lt gives rise to the so called two block problem on the algebra he we analyze this problem from an operator point of view using krein space theory we obtain a necessary and sufficient condition for the uniqueness of the solution and a parameterization of all solutions in the non uniqueness case moreover an interpolation interpretation is given for the finite dimensional case

a gap theorem of self shrinkers in this paper we study complete self shrinkers in euclidean space and prove that an n dimensional complete self shrinker with polynomial volume growth in euclidean space r n 1 is isometric to either r n s n n or r n m s m m 1 m n 1 if the squared norm s of the second fundamental form is constant and satisfies s 10 7

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

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