Approximation of conformal mappings by circle patterns

A circle pattern is a configuration of circles in the plane whose combinatorics is given by a planar graph G such that to each vertex of G corresponds a circle. If two vertices are connected by an edge in G, the corresponding circles intersect with an intersection angle in (0, π). Two sequences of circle patterns are employed to approximate a given conformal map g and its first derivative. For the domain of g we use embedded circle patterns where all circles have the same radius decreasing to 0 and with uniformly bounded intersection angles. The image circle pattern has the same combinatorics and intersection angles and is determined from boundary conditions (radii or angles) according to the values of g′ (|g′| or arg g′). For quasicrystallic circle patterns the convergence result is strengthened to C ^∞-convergence on compact subsets.      

Circle Patterns on Singular Surfaces

We consider “hyperideal” circle patterns, i.e., patterns of disks appearing in the definition of the weighted Delaunay decomposition associated with a set of disjoint disks, possibly with cone singularities at the centers of those disks. Hyperideal circle patterns are associated with hyperideal hyperbolic polyhedra. We describe the possible intersection angles and singular curvatures of those circle patterns on Euclidean or hyperbolic surfaces with cone singularities. This is related to results on the dihedral angles of ideal or hyperideal hyperbolic polyhedra. The results presented here extend those in Schlenker (Math. Res. Lett. 12(1), 82–112, [2005]), however, the proof is completely different (and more intricate) since Schlenker (Math. Res. Lett. 12(1), 82–112, [2005]) used a shortcut which is not available here. The author would like to thank the RIP program at Oberwolfach, where part of the research presented here was conducted. Partially supported by the “ACI Jeunes Chercheurs” Métriques privilégiés sur les variétés à bord, 2003-06, and the ANR program Representations of surface groups, 2007-09.     

Hyperbolic polygons of minimal perimeter with given angles

We prove that, among all convex hyperbolic polygons with given angles, the perimeter is minimized by the unique polygon with an inscribed circle. The proof relies on work of Schlenker (Trans Am Math Soc 359(5): 2155–2189, 2007).     

Rigidity of Quasicrystallic and <Emphasis Type="Italic">Z</Emphasis><Superscript><Emphasis Type="Italic">γ</Emphasis></Superscript>-Circle Patterns

The uniqueness of the orthogonal Z ^γ -circle patterns as studied by Bobenko and Agafonov is shown, given the combinatorics and some boundary conditions. Furthermore we study (infinite) rhombic embeddings in the plane which are quasicrystallic, that is, they have only finitely many different edge directions. Bicoloring the vertices of the rhombi and adding circles with centers at vertices of one of the colors and radius equal to the edge length leads to isoradial quasicrystallic circle patterns. We prove for a large class of such circle patterns which cover the whole plane that they are uniquely determined up to affine transformations by the combinatorics and the intersection angles. Combining these two results, we obtain the rigidity of large classes of quasicrystallic Z ^γ -circle patterns.     

Multivalued Quasimöbius Mappings from Circle to Circle

Siberian Mathematical Journal - We prove that if a&nbsp;multivalued mapping&nbsp; $ F $ of circle to circle has the $ \eta $ -BAD property (bounded distortion of generalized angles with...     

The Circle Semigroup of a Ring

Let R be a ring and define x ○ y = x + y - xy, which yields a monoid (R, ○), called the circle semigroup of R. This paper investigates the relationship between the ring and its circle semigroup. Of particular interest are the cases where the semigroup is simple, 0-simple, cancellative, 0-cancellative, regular, inverse, or the union of groups, or where the ring is simple, regular, or a domain. The idempotents in R coincide with the idempotents in (R, ○) and play an important role in the theory developed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Complete Upper Angle Around a Point on the Minkowski Plane

The dependence of the complete upper angle in the sense of A. D. Aleksandrov about a point on the Minkowski plane on the form of the “unit circle” (the centrally symmetric convex curve Φ determining the Minkowski metric ρ_Φ) is studied.The complete upper angle is computed in three cases: if Φ is a square, a “cut circle,” or a “rounded rhombus.” Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 42–53.     

Imbedded Circle Patterns with the Combinatorics of the Square Grid and Discrete Painlevé Equations

   Abstract. A discrete analogue of the holomorphic maps z ^γ and log(z) is studied. These maps are given by Schramm's circle pattern with the combinatorics of the square grid. It is shown that the corresponding circle patterns are imbedded and described by special separatrix solutions of discrete Painlevé equations. Global properties of these solutions, as well as of the discrete z ^γ and log(z) , are established.     

Measures on circle coarse-grained systems of sets

We show that a (non-negative) measure on a circle coarse-grained system of sets can be extended, as a (non-negative) measure, over the collection of all subsets of the circle. This result contributes to quantum logic probability (de Lucia in Colloq Math 80(1):147–154, 1999; Gudder in Quantum Probability, Academic Press, San Diego, 1988; Gudder in SIAM Rev 26(1):71–89, 1984; Harding in Int J Theor Phys 43(10):2149–2168, 2004; Navara and Pták in J Pure Appl Algebra 60:105–111, 1989; Pták in Proc Am Math Soc 126(7):2039–2046, 1998, etc.) and completes the analysis of coarse-grained measures carried on in De Simone and Pták (Bull Pol Acad Sci Math 54(1):1–11, 2006; Czechoslov Math J 57(132) n.2:737–746, 2007), Gudder and Marchand (Bull Pol Acad Sci Math 28(11–12):557–564, 1980) and Ovchinnikov (Construct Theory Funct Funct Anal 8:95–98, 1992).     

The asymptotic distribution of circles in the orbits of Kleinian groups

Let P\mathcal{P} be a locally finite circle packing in the plane ℂ invariant under a non-elementary Kleinian group Γ and with finitely many Γ-orbits. When Γ is geometrically finite, we construct an explicit Borel measure on ℂ which describes the asymptotic distribution of small circles in P\mathcal{P}, assuming that either the critical exponent of Γ is strictly bigger than 1 or P\mathcal{P} does not contain an infinite bouquet of tangent circles glued at a parabolic fixed point of Γ. Our construction also works for P\mathcal{P} invariant under a geometrically infinite group Γ, provided Γ admits a finite Bowen-Margulis-Sullivan measure and the Γ-skinning size of P\mathcal{P} is finite. Some concrete circle packings to which our result applies include Apollonian circle packings, Sierpinski curves, Schottky dances, etc.     

Quadrature formula and zeros of para-orthogonal polynomials on the unit circle

Given a probability measure μ on the unit circle T, we study para-orthogonal polynomials B_n(^.,w) (with fixed w ∈ T) and their zeros which are known to lie on the unit circle. We focus on the properties of zeros akin to the well known properties of zeros of orthogonal polynomials on the real line, such as alternation, separation and asymptotic distribution. We also estimate the distance between the consecutive zeros and examine the property of the support of μ to attract zeros of para-orthogonal polynomials. This revised version was published online in June 2006 with corrections to the Cover Date.     

Ginzburg-Landau vortex analogues

We consider a static one-dimensional Ginzburg-Landau equation (on a line segment or a circle) involving a large parameter λ. We show that as λ→∞, there exist solutions whose asymptotic behavior resembles the behavior of the two-dimensional vortex solutions. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 1, pp. 18–35, July, 2000.     

Area minimizing sets subject to a volume constraint in a convex set

For a given convex subset Ω of Euclidean n-space, we consider the problem of minimizing the perimeter of subsets of Ω subject to a volume constraint. The problem is to determine whether in general a minimizer is also convex. Although this problem is unresolved, we show that if Ω satisfies a “great circle” condition, then any minimizer is convex. We say that Ω satisfies a great circle condition if the largest closed ball B contained in Ω has a great circle that is contained in the boundary of Ω. A great circle of B is defined as the intersection of the boundary of B with a hyperplane passing through the center of B.     

Derivatives of Circle Rotations and the Similarity of Orbits

It is proved that the derivative of a circle rotation on an arbitrary interval is either a circle rotation or a noncyclic exchange of three intervals. In the former case, all possible values of the new angle of rotation are computed. It is shown that the restriction of the orbit of a circle rotation to an eigeninterval of differentiation is similar to the orbit of another circle rotation. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 314, 2004, pp. 272–284.     

Cross-ratio distortion and Douady-Earle extension: II. Quasiconformality and asymptotic conformality are local

Let f be an orientation-preserving circle homeomorphism and Φ the Douady-Earle extension of f. In this paper, we show that the quasiconformality and asymptotic conformality of Φ are local properties; i.e., if f is quasisymmetric or symmetric on an arc of the unit circle, then Φ is quasiconformal or asymptotically conformal nearby. Furthermore, our methods enable us to conclude the global quasiconformality and asymptotic conformality from local properties. In the quasiconformal case, our methods also enable us to provide an upper bound for the maximal dilatation of Φ on a neighborhood of the arc in the open unit disk in terms of the cross-ratio distortion norm of f on the arc.     

Differentiable structure of ω-subsets of symplectic groups

In this paper, we study the differentiable structure of the ω-subset of Sp(2n), which is formed by all matrices in Sp(2n) possessing ω as an eigenvalue, for ω on the unit circle in the complex plane. Based on this result the ω -index theory parametrized by all ω on the unit circle for arbitrary symplectic paths is defined.     

Advancing front circle packing to approximate conformal strips

A circle packing is a set of tangent and disjoint discs. Maps between circle packings with the same tangency are discrete analogues of conformal mappings, which have application for example in mechanical, fluid, and thermal engineering. We describe an advancing front algorithm to compute the circle packing of a strip around a closed planar curve. Conformal mappings preserve local angles and shapes; our algorithm uses these properties to obtain via the fast Fourier transform the centers and radii for the circle packing of successive trigonometric Lagrange curves in a strip. To check the algorithm, different results are compared with well-known conformal mappings. Real time deformations of circle packings are possible by changing the shape of the initial closed curve.     

Version of the Grothendieck theorem for subspaces of analytic functions in lattices

A version of Grothendieck’s inequality says that any bounded linear operator acting from a Banach lattice X to a Banach lattice Y acts from X(ℓ²) to Y (ℓ²) as well. A similar statement is proved for Hardy-type subspaces in lattices of measurable functions. Namely, let X be a Banach lattice of measurable functions on the circle, and let an operator T act from the corresponding subspace of analytic functions X_A to a Banach lattice Y or, if Y is also a lattice of measurable functions on the circle, to the quotient space Y/Y_A. Under certain mild conditions on the lattices involved, it is proved that T induces an operator acting from X_A(ℓ²) to Y (ℓ²) or to Y/Y_A(ℓ2), respectively. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 5–16.     

Strengthening theC t -closing lemma for dynamical systems and foliations on the torus

We prove a strengthenedC ^r -closing lemma (r≥1) for wandering chain recurrent trajectories of flows without equilibrium states on the two-dimensional torus and for wandering chain recurrent orbits of a diffeomorphism of the circle. The strengthenedC ^r -closing lemma (r≥1) is proved for a special class of infinitely smooth actions of the integer lattice ℤ ^k on the circle. The result is applied to foliations of codimension one with trivial holonomy group on the three-dimensional torus. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 323–331, March, 1997. Translated by S. K. Lando