discrete riemann surfaces and the ising model we define a new theory of discrete riemann surfaces and present its basic results the key idea is to consider not only a cellular decomposition of a surface but the union with its dual discrete holomorphy is defined by a straightforward discretisation of the cauchy riemann equation a lot of classical results in riemann theory have a discrete counterpart hodge star harmonicity hodge theorem weyl s lemma cauchy integral formula existence of holomorphic forms with prescribed holonomies giving a geometrical meaning to the construction on a riemann surface we define a notion of criticality on which we prove a continuous limit theorem we investigate its connection with criticality in the ising model we set up a dirac equation on a discrete universal spin structure and we prove that the existence of a dirac spinor is equivalent to criticality

how many cages midscribe an egg the midscribability theorem which was first proved by o schramm states that given a smooth strictly convex body k r 3 and a convex polyhedron p there exists a convex polyhedron q r 3 combi natorially equivalent to p which midscribes k here the word midscribe means that all its edges are tangent to the boundary surface of k b y using the intersection number technique together with the teichmuller theory of pack ings this paper provides an alternative approach to this theorem furthermore by combining schramm s method with the above ones we obtain a rigidity result as well that is such a polyhedron is unique under the normalization condition

a characterization of ideal polyhedra in hyperbolic 3 space the goals of this paper are to provide a characterization of dihedral angles of convex ideal those with all vertices on the sphere at infinity polyhedra in h3 and also of those convex polyhedra with some vertices on the sphere at infinity and some in the finite part of h3 these characterizations are given in respectively theorems 0 1 and 10 5 the first theorem is proved in detail while the proof of the second which is similar is only outlined the results of this paper grow out of the general framework of the author s doctoral dissertation 13 as published in 20 a lot of the language and some of the auxilary results come from there as well so familiarity with the latter reference is very helpful the necessity of the conditions postulated in theorem 0 1 has been shown in 14 hence only the sufficiency direction will be shown in the current paper in 1832 jakob steiner asked for a combinatorial characterization of convex polyhedra inscribed in the sphere this was considered intractable it took almost a hundred years to find a single example of an uninscribable combinatorial type by steinitz 24 however theorem 0 1 gives such a characterization section 11 is dedicated to a very brief historical survey and some purely graph theoretic and computational geometric consequences of theorem 0 1 in order to state this theorem suppose that a convex ideal polyhedron p in h3 is given let p denote the poincar dual of p and assign to each edge e of p a weight w e equal to the exterior dihedral angle at the corresponding edge e of p then the following result holds

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

on some nonunitary representations of the poincare group and their use for the construction of free quantum fields in this paper a class of nonunitary infinite dimensional hilbert space representations of a semidirect product is investigated the equivalence of this category with the category of finite dimensional representations of the stability subgroups is shown this theory is applied to the poincare group and to the construction of free quantum fields in an appendix a method is introduced for building an infinite family of finite dimensional indecomposable representations of the noncompact euclidean group in two dimensions such representations are used for carrying out the analysis of the massless fields

rigidity of inversive distance circle packings revisited abstract inversive distance circle packing metric was introduced by p bowers and k stephenson 7 as a generalization of thurston s circle packing metric 34 they conjectured that the inversive distance circle packings are rigid for nonnegative inversive distance guo 22 proved the infinitesimal rigidity and then luo 27 proved the global rigidity in this paper based on an observation of zhou 37 we prove this conjecture for inversive distance in 1 by variational principles we also study the global rigidity of a combinatorial curvature introduced in 14 16 19 with respect to the inversive distance circle packing metrics where the inversive distance is in 1

on the deformation of inversive distance circle packings ii abstract we show that the results in 8 are still true in hyperbolic background geometry setting that is the solution to chow luo s combinatorial ricci flow can always be extended to a solution that exists for all time furthermore the extended solution converges exponentially fast if and only if there exists a metric with zero curvature we also give some results about the range of discrete gaussian curvatures which generalize andreev thurston s theorem to some extent

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

a nonlinear conjugate gradient method with a strong global convergence property conjugate gradient methods are widely used for unconstrained optimization especially large scale problems the strong wolfe conditions are usually used in the analyses and implementations of conjugate gradient methods this paper presents a new version of the conjugate gradient method which converges globally provided the line search satisfies the standard wolfe conditions the conditions on the objective function are also weak being similar to those required by the zoutendijk condition

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 characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere we describe a characterization of convex polyhedra in h 3 in terms of their dihedral angles developed by rivin we also describe some geometric and combinatorial consequences of that theory one of these consequences is a combinatorial characterization of convex polyhedra in e 3 all of whose vertices lie on the unit sphere that resolves a problem posed by jakob steiner in 1832

a unique representation of polyhedral types centering via mobius transformations for n 3 distinct points in the d dimensional unit sphere there exists a mobius transformation such that the barycenter of the transformed points is the origin this mobius transformation is unique up to post composition by a rotation we prove this lemma and apply it to prove the uniqueness part of a representation theorem for 3 dimensional polytopes as claimed by ziegler 1995 for each polyhedral type there is a unique representative up to isometry with edges tangent to the unit sphere such that the origin is the barycenter of the points where the edges touch the sphere

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

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

a variational principle for weighted delaunay triangulations and hyperideal polyhedra we use a variational principle to prove an existence and uniqueness theorem for planar weighted delaunay triangulations with non intersecting site circles with prescribed combinatorial type and circle intersection angles such weighted delaunay triangulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining vertices beyond the infinite boundary of hyperbolic space thus the main theorem states necessary and sufficient conditions for the existence and uniqueness of such polyhedra with prescribed combinatorial type and dihedral angles more generally we consider weighted delaunay triangulations in piecewise flat surfaces allowing cone singularities with prescribed cone angles in the vertices the material presented here extends work by rivin on delaunay triangulations and ideal polyhedra

a new conjugate gradient method with guaranteed descent and an efficient line search a new nonlinear conjugate gradient method and an associated implementation based on an inexact line search are proposed and analyzed with exact line search our method reduces to a nonlinear version of the hestenes stiefel conjugate gradient scheme for any inexact line search our scheme satisfies the descent condition bf g k sf t bf d k le frac 7 8 bf g k 2 moreover a global convergence result is established when the line search fulfills the wolfe conditions a new line search scheme is developed that is efficient and highly accurate efficiency is achieved by exploiting properties of linear interpolants in a neighborhood of a local minimizer high accuracy is achieved by using a convergence criterion which we call the approximate wolfe conditions obtained by replacing the sufficient decrease criterion in the wolfe conditions with an approximation that can be evaluated with greater precision in a neighborhood of a local minimum than the usual sufficient decrease criterion numerical comparisons are given with both l bfgs and conjugate gradient methods using the unconstrained optimization problems in the cute library

asymptotics of convex sets in euclidean and hyperbolic spaces we study convex sets c of finite but non zero volume in and we show that the intersection c8 of any such set with the ideal boundary of has minkowski and thus hausdorff dimension of at most n 1 2 and this bound is sharp at least in some dimensions n we also show a sharp bound when c8 is a smooth submanifold of in the hyperbolic case we show that for any k n 1 2 there is a bounded section s of c through any prescribed point p and we show an upper bound on the radius of the ball centered at p containing such a section we show similar bounds for sections through the origin of a convex body in and give asymptotic estimates as 1kn

a note about wyl s conjugate gradient method and its applications this paper reviews the development of different versions of nonlinear conjugate gradient methods shortly the special attentions were given to the wyl method which was proposed by zengxin wei et al the convergence properties of some conjugate gradient methods applied mathematics and computation 183 2006 1341 1350 and its applications

virtual coordinates in hyperbolic space based on ricci flow for wlans geographic routing is appropriate for sensor networks because of their special characteristics such as wireless local area networks however the existing holes in the practical networks may probably impact the performance of geographic routing in this paper we employ hyperbolic ricci flow to compute a new embedding of sensor nodes which could achieve the conformal map of the topology of the network to virtual coordinates in hyperbolic space even if its topology contains multiple holes and is easy to establish the corresponding routing in actual experiments and simulations our method has been proved valuable and feasible

the proof of the sufficient descent condition of the wei yao liu conjugate gradient method under the strong wolfe powell line search abstract in z wei et al the convergence properties of some conjugate gradient methods appl math comput 183 2006 1341 1350 wei et al proposed a new conjugate gradient method called wyl method which has good numerical experiments under the sufficient descent condition and wwp line search the given method s convergent results have been established by wei et al so the sufficient descent condition is important for the wyl method in this paper we prove that by restricting the parameter 1 4 under the swp line search condition the wyl method possess the sufficient descent condition which deduces the global convergent results of the wyl method under swp line search

duality structures and discrete conformal variations of piecewise constant curvature surfaces abstract a piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating the simplex has an isometric embedding into a constant curvature background geometry euclidean hyperbolic or spherical with the specified edge lengths additional geometric structure leads to a notion of discrete conformal structure generalizing circle packings and their generalizations as studied by thurston and others we analyze discrete conformal variations of piecewise constant curvature 2 manifolds giving particular attention to the variation of angles formulas are derived for the derivatives of angles in each background geometry which yield formulas for the derivatives of curvatures and to curvature functionals finally we provide a complete classification of possible definitions of discrete conformal structures in each of the background geometries

a new solution of the star triangle relation we obtain a new solution to the star triangle relation for an ising type model with two kinds of spin variables at each lattice site taking continuous real values and arbitrary integer values respectively the boltzmann weights are manifestly real and positive they are expressed through the euler gamma function and depend on sums and differences of spins at the ends of an edge of the lattice

hyperbolic manifolds with convex boundary let m m be a 3 manifold which carries a hyperbolic metric with convex boundary we consider the hyperbolic metrics on m such that the boundary is smooth and strictly convex we show that the induced metrics on the boundary are exactly the metrics with curvature k 1 and that the third fundamental forms of m are exactly the metrics with curvature k 2 each is obtained exactly once

gauss images of hyperbolic cusps with convex polyhedral boundary we prove that a 3 dimensional hyperbolic cusp with convex poly hedral boundary is uniquely determined by its gauss image furthermore any spherical metric on the torus with cone singularities of negative curvature and all closed contractible geodesics of length greater than 2 is the metric of the gauss image of some convex polyhedral cusp this result is an analog of the rivin hodgson theorem characterizing compact convex hyperbolic polyhedra in terms of their gauss images the proof uses a variational method namely a cusp with a given gauss image is identified with a critical point of a functional on the space of cusps with cone type singularities along a family of half lines the functional is shown to be concave and to attain maximum at an interior point of its domain as a byproduct we prove rigidity statements with respect to the gauss image for cusps with or without cone type singularities in a special case our theorem is equivalent to existence of a circle pattern on the torus with prescribed combinatorics and intersection angles this is the genus one case of a theorem by thurston in fact our theorem extends thurston s theorem in the same way as rivin hodgson s theorem extends an dreev s theorem on compact convex polyhedra with non obtuse dihedral angles the functional used in the proof is the sum of a volume term and curvature term we show that in the situation of thurston s theorem it is the potential for the combinatorial ricci flow considered by chow and luo our theorem represents the last special case of a general statement about isometric immersions of compact surfaces

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

global convergence properties of conjugate gradient methods for optimization this paper explores the convergence of nonlinear conjugate gradient methods without restarts and with practical line searches the analysis covers two classes of methods that are globally convergent on smooth nonconvex functions some properties of the fletcher reeves method play an important role in the first family whereas the second family shares an important property with the polak ribiere method numerical experiments are presented

the convergence properties of some new conjugate gradient methods abstract in this paper a new conjugate gradient formula k is given to compute the search directions for unconstrained optimization problems general convergence results for the proposed formula with some line searches such as the exact line search the wolfe powell line search and the grippo lucidi line search are discussed under the above line searches and some assumptions the global convergence properties of the given methods are discussed the given formula k 0 and has the similar form with k prp preliminary numerical results show that the proposed methods are efficient

a characterization of compact convex polyhedra in hyperbolic 3 space in this paper we study the extrinsic geometry of convex polyhedral surfaces in three dimensional hyperbolic spaceh 3 we obtain a number of new uniqueness results and also obtain a characterization of the shapes of convex polyhedra inh 3 in terms of a generalized gauss map this characterization greatly generalizes andre ev s theorem

unitarizable massless fields of higher order and higher spin explicit fock spaces for massless multispinors and tensors satisfying n 0 are constructed it is found that they carry indecomposable representations which allow gupta bleuler type quantization for symmetric tensors and odd n there is a unitary space of physical states

combinatorial ricci flows on surfaces we show that the analogue of hamilton s ricci flow in the combinatorial setting produces solutions which converge exponentially fast to thurston s circle packing on surfaces as a consequence a new proof of thurston s existence of circle packing theorem is obtained as another consequence ricci flow suggests a new algorithm to find circle packings

some remarks on the gupta bleuler triplet indecomposable representations of the poincare group associated to infrared singular field theory models are discussed in the framework of the general theory of the gupta bleuler triplet formulated by araki it is shown that the definition of maximal hilbert space structures related to the infrared properties of the states of the models can be exploited to construct representation spaces for the gupta bleuler triplet the examples of the two dimensional massless scalar field and of the electromagnetic field in the landau gauge are discussed in particular in the first example the relation between the gupta bleuler triplet and the algebraic treatment of the massless scalar field is investigated in the case of electromagnetism the structure of the representation of the poincare group in the landau gauge is clarified the explicit form of the corresponding gupta bleuler triplet for the one particle space of the electromagnetic field is exhibited

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

global convergence of the polak ribiere polyak conjugate gradient method with an armijo type inexact line search for nonconvex unconstrained optimization problems we propose two algorithms for nonconvex unconstrained optimization problems that employ polak ribiere polyak conjugate gradient formula and new inexact line search techniques we show that the new algorithms converge globally if the function to be minimized has lipschitz continuous gradients preliminary numerical results show that the proposed methods for particularly chosen line search conditions are very promising

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

rigidity of polyhedral surfaces ii we study the rigidity of polyhedral surfaces using variational principles the action functionals are derived from the cosine laws the main focus of this paper is on the cosine law for a nontriangular region bounded by three possibly disjoint geodesics several of these cosine laws were first discovered and used by fenchel and nielsen by studying the derivative of the cosine laws we discover a uniform approach to several variational principles on polyhedral surfaces with or without boundary as a consequence the work of penner bobenko and springborn and thurston on rigidity of polyhedral surfaces and circle patterns are extended to a very general context

volume optimization normal surfaces and thurston s equation on triangulated 3 manifolds we propose a finite dimensional variational principle on triangulated 3 manifolds so that its critical points are related to solutions to thurston s gluing equation and haken s normal surface equation the action functional is the volume

the asymptotic behavior of the discrete holomorphic map z a via the riemann hilbert method we study the asymptotic behavior of the discrete analogue of the holomorphic map za the analysis is based on the use of the riemann hilbert approach specifically using the deift zhou nonlinear steepest descent method we prove the asymptotic formulas which were conjectured in 2000

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

variational principles for circle patterns and koebe s theorem we prove existence and uniqueness results for patterns of circles with prescribed intersection angles on constant curvature surfaces our method is based on two new functionals one for the euclidean and one for the hyperbolic case we show how colin de verdiere s bragger s and rivin s functionals can be derived from ours

on the volume of hyperbolic polyhedra where v1k is the volume vol 2 sj of the apex sjk sj c s to wjk schl ifli proved this formula for spherical simplices in 1936 h kneser gave a second very skilful proof of l see i l l and 4 sect 5 1 for both the spherical and hyperbolic cases up to a change of sign in the latter case but even for a three dimensional non euclidean simplex the integration of this schl li differential is practically impossible in fact the most basic objects in polyhedral geometry are orthoschemes or orthogonal simplices first introduced by schl fli an n orthoscheme r is an n simplex with vertices po pn such that

computing quasiconformal maps using an auxiliary metric and discrete curvature flow surface mapping plays an important role in geometric processing which induces both area and angular distortions if the angular distortion is bounded the mapping is called a quasiconformal mapping qc mapping many surface mappings in our physical world are quasiconformal the angular distortion of a qc mapping can be represented by the beltrami differentials according to qc teichmuller theory there is a one to one correspondence between the set of beltrami differentials and the set of qc surface mappings under normalization conditions therefore every qc surface mapping can be fully determined by the beltrami differential and reconstructed by solving the so called beltrami equation in this work we propose an effective method to solve the beltrami equation on general riemann surfaces the solution is a qc mapping associated with the prescribed beltrami differential the main strategy is to define an auxiliary metric am on the domain surface such that the original qc mapping becomes conformal under the auxiliary metric the desired qc mapping can then be obtained by using the conventional conformal mapping method in this paper we first formulate a discrete analogue of qc mappings on triangular meshes then we propose an algorithm to compute discrete qc mappings using the discrete yamabe flow method to the best of our knowledge it is the first work to compute the discrete qc mappings for general riemann surfaces especially with different topologies numerically the discrete qc mapping converges to the continuous solution as the mesh grid size approaches to 0 we tested our algorithm on surfaces scanned from real life with different topologies experimental results demonstrate the generality and accuracy of our auxiliary metric method

a modified wei yao liu conjugate gradient method for unconstrained optimization in this paper we give a modified wei yao liu conjugate gradient method wei et al 2006 18 which will reduce to the original wei yao liu method and possess the sufficient descent property without any line search furthermore we prove that the presented method is globally convergent for nonconvex functions with the weak wolfe powell line search in a similar way we also extend these results to the modified liu storey method preliminary numerical results show that the proposed methods are effective for the given test problems

angled decompositions of arborescent link complements this paper describes a way to subdivide a 3 manifold into angled blocks namely polyhedral pieces that need not be simply connected when the individual blocks carry dihedral angles that fit together in a consistent fashion we prove that a manifold constructed from these blocks must be hyperbolic the main application is a new proof of a classical unpublished theorem of bonahon and siebenmann that all arborescent links except for three simple families of exceptions have hyperbolic complements in the 1990s andrew casson introduced a powerful technique for constructing and studying cusped hyperbolic 3 manifolds his idea was to subdivide a manifold m into angled ideal tetrahedra that is tetrahedra whose vertices are removed and whose edges carry prescribed dihedral angles when the dihedral angles of the tetrahedra add up to 2 around each edge of m the triangulation is called an angled triangulation casson proved that every orientable cusped 3 manifold that admits an angled triangulation must also admit a hyperbolic metric and outlined a possible way to find the hyperbolic metric by studying the volumes of angled tetrahedra an idea also developed by rivin 17 the power of casson s approach lies in the fact that the defining equations of an angled triangulation are both linear and local making angled triangulations relatively easy to find and deform much easier than to study an actual hyperbolic triangulation as in 14 20 or in some aspects of thurston s seminal approach 22 the aim of this paper is to extend this approach to larger and more complicated building blocks these blocks can be ideal polyhedra instead of tetrahedra but they may also have nontrivial topology in general an angled block will be a 3 manifold whose boundary is subdivided into faces looking locally like the faces of an ideal polyhedron in a sense to be defined the edges between adjacent faces carry prescribed dihedral angles in section 2 we will describe the precise combinatorial conditions that the dihedral angles must satisfy these conditions will imply the following generalization of a result by lackenby 11 corollary 4 6

new nonlinear conjugate gradient formulas for large scale unconstrained optimization problems abstract we propose new conjugate gradient formulas for computing the search directions for unconstrained optimization problems the new formulas turn out to be the conjugate descent formula if exact line searches are made some formulas possess the sufficient descent property without any line searches general convergence results for the proposed formulas with the weak wolfe powell conditions are studied we prove that some of the formulas with the steplength technique which ensures the zoutendijk condition to be held are globally convergent in addition the global convergence results for some other formulas with the standard armijo line search are also given preliminary numerical results show that the proposed methods are very promising

on a combinatorial curvature for surfaces with inversive distance circle packing metrics abstract in this paper we introduce a new combinatorial curvature on triangulated surfaces with inversive distance circle packing metrics then we prove that this combinatorial curvature has global rigidity to study the yamabe problem of the new curvature we introduce a combinatorial ricci flow along which the curvature evolves almost in the same way as that of scalar curvature along the surface ricci flow obtained by hamilton 20 then we study the long time behavior of the combinatorial ricci flow and obtain that the existence of a constant curvature metric is equivalent to the convergence of the flow on triangulated surfaces with nonpositive euler number we further generalize the combinatorial curvature to curvature and prove that it is also globally rigid which is in fact a generalized bowers stephenson conjecture 6 we also use the combinatorial ricci flow to study the corresponding yamabe problem

well centered triangulation meshes composed of well centered simplices have nice orthogonal dual meshes the dual voronoi diagram this is useful for certain numerical algorithms that prefer such primal dual mesh pairs we prove that well centered meshes also have optimality properties and relationships to delaunay and minmax angle triangulations we present an iterative algorithm that seeks to transform a given triangulation in two or three dimensions into a well centered one by minimizing a cost function and moving the interior vertices while keeping the mesh connectivity and boundary vertices fixed the cost function is a direct result of a new characterization of well centeredness in arbitrary dimensions that we present ours is the first optimization based heuristic for well centeredness and the first one that applies in both two and three dimensions we show the results of applying our algorithm to small and large two dimensional meshes some with a complex boundary and obtain a well centered tetrahedralization of the cube we also show numerical evidence that our algorithm preserves gradation and that it improves the maximum and minimum angles of acute triangulations created by the best known previous method

an improved wei yao liu nonlinear conjugate gradient method for optimization computation in this paper we take a little modification to the wei yao liu nonlinear conjugate gradient method proposed by wei et al z wei s yao l liu the convergence properties of some new conjugate gradient methods appl math comput 183 2006 1341 1350 such that the modified method possesses better convergence properties in fact we prove that the modified method satisfies sufficient descent condition with greater parameter email protected 0 12 in the strong wolfe line search and converges globally for nonconvex minimization we also extend these results to the hestenes stiefel method and prove that the modified hs method is globally convergent for nonconvex functions with the standard wolfe conditions numerical results are reported by using some test problems in the cute library

ricci flow for 3d shape analysis ricci flow is a powerful curvature flow method which is invariant to rigid motion scaling isometric and conformal deformations we present the first application of surface ricci flow in computer vision previous methods based on conformal geometry which only handle 3d shapes with simple topology are subsumed by the ricci flow based method which handles surfaces with arbitrary topology we present a general framework for the computation of ricci flow which can design any riemannian metric by user defined curvature the solution to ricci flow is unique and robust to noise we provide implementation details for ricci flow on discrete surfaces of either euclidean or hyperbolic background geometry our ricci flow based method can convert all 3d problems into 2d domains and offers a general framework for 3d shape analysis we demonstrate the applicability of this intrinsic shape representation through standard shape analysis problems such as 3d shape matching and registration and shape indexing surfaces with large nonrigid anisotropic deformations can be registered using ricci flow with constraints of feature points and curves we show how conformal equivalence can be used to index shapes in a 3d surface shape space with the use of teichmuller space coordinates experimental results are shown on 3d face data sets with large expression deformations and on dynamic heart data

a fock krein realization of the landau gauge an explicit fock realization of the landau gauge of the free electromagnetic field with a hilbert space structure which is simply the same as in the gupta bleuler gauges is given

a descent nonlinear conjugate gradient method for large scale unconstrained optimization in this paper a new nonlinear conjugate gradient method was proposed for large scale unconstrained optimization which possesses the following three properties i the sufficient descent property holds without any line searches ii employing some steplength technique which ensures the zoutendijk condition to be held this method is globally convergent iii this method inherits an important property of the polak ribiere polyak prp method the tendency to turn towards the steepest descent direction if a small step is generated away from the solution preventing a sequence of tiny steps from happening preliminary numerical results show that this method is very promising

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

how to cage an egg this paper proves that given a convex polyhedron p subset mathbb r 3 and a smooth strictly convex body k subset mathbb r 3 there is some convex polyhedron q combinatorically equivalent to p which midscribes k that is all the edges of q are tangent to k furthermore with some stronger smoothness conditions on k the space of all such q is a six dimensional differentiable manifold