Discrete Analytic Functions Research Articles

On discrete analytic functions: Products, rational functions and reproducing kernels

We introduce a family of discrete analytic functions, called expandable discrete analytic functions, which includes discrete analytic polynomials, and define two products in this family. The first one is defined in a way similar to the Cauchy-Kovalevskaya product of hyperholomorphic functions, and allows us to define rational discrete analytic functions. To define the second product we need a new space of entire functions which is contractively included in the Fock space. We study in this space some counterparts of Schur analysis.

A note on the uniqueness of Koebe-Andreev-Thurston theorem

In this paper, we present a new proof of the uniqueness of Koebe-Andreev-Thurston theorem. Our method is based on the argument principle in complex analysis and reviews the connection between the circle packing theorem and complex analysis.

On Beurling's boundary value problem in circle packing

We investigate a discrete counterpart of Beurling's boundary value problem for analytic functions in the framework of circle packing. In the case of discrete analytic functions modelled on arbitrary combinatorically closed disks existence of solutions is shown under rather general assumptions. As in the nondiscrete case finitely many branch circles can be prescribed, so the solutions include locally univalent as well as branched packings. The proof of existence rests on an application of Brouwer's fixed point theorem and a global parameterization of the differentiable manifold of circle packings. We also present some first results on the uniqueness of solutions. In the last two sections we propose an algorithm for the numerical solution of the problem, which is based on an embedded Newton method, and report on some test calculations.

Circle packings as differentiable manifolds

Circle packings are configurations of circles satisfying specified patterns of tangency and have emerged as the foundation for a fairly comprehensive theory of discrete analytic functions. Though many classical results found their counterpart in circle packing, other concepts have not yet been transferred, particularly those which require a linear structure. This paper puts circle packings in a framework of smooth manifolds, providing access to linear structures in their tangent spaces. Since we are especially interested in applications to boundary value problems (of Beurling and Riemann–Hilbert type), we do not only investigate the manifolds of circle packings and packing labels, but also manifolds formed by the centers of the boundary circles. The approach is elementary and rests on a detailed analysis of the contact equations which govern the tangency relation between neighboring circles.

Generalization of analytic functions

Abstract The concept of analyticity for complex functions on time scale complex plane was introduced by Bohner and Guseinov in 2005. They developed completely delta differentiability, delta analytic functions on products of two time scales, and Cauchy–Riemann equations for delta case. In this research paper we study on continuous, discrete and semi-discrete analytic functions and developed completely nabla differentiability, nabla analytic functions on products of two time scales, and Cauchy–Riemann equations for nabla case.

On the convergence of circle packings to the quasiconformal map

Rodin and Sullivan (1987) proved Thurston's conjecture that a scheme based on the Circle Packing Theorem converges to the Riemann mapping, thereby proved a refreshing geometric view of the Riemann Mapping Theorem. Naturally, we consider to use the ellipses to pack the bounded simply connected domain and obtain similarly a sequence simplicial homeomorphism between the ellipse packing and the circle packing. In this paper, we prove that these simplicial homeomorphism approximate a quasiconformal mapping from the bounded simply connected domain onto the unit disk with the modulus of their complex dilatations tending to 1 almost everywhere in the domain when the ratio of the longer axis and shorter axis of the ellipse tending to ∞.

Book Review: Introduction to circle packing: The theory of discrete analytic functions

Book Review: Introduction to circle packing: The theory of discrete analytic functions

The discrete Szëgo kernel

Of concern on this paper are complex-valued functions defined on the integer lattice (i.e. the set ) which are discrete analytic according to the definition given by Ferrand. In particular, we will study a Hilbert space consisting of the boundary values of discrete analytic functions defined on a finite simply connected union of unit squares of the integer lattice (a simple region), which is a discrete version of the Szëgo space. We will prove that this space admits a reproducing kernel, the discrete Szëgo kernel and will develop a general method to construct it. To sum up, the main merit of this paper is to present by means of an orthogonal projection operator a way to select among boundary values, those that can be extended to an analytic continuation.

Introduction to circle packing: the theory of discrete analytic functions

Introduction to circle packing: the theory of discrete analytic functions

A local search approach to a circle cutting problem arising in the motor cycle industry

This paper is concerned with the development of a customized circle packing algorithm for a manufacturer of sprockets for the motor cycle industry. Practical constraints mean that the problem differs somewhat from those tackled elsewhere in the literature. In particular, the layouts need to conform to a given structure. This is achieved by using a local search algorithm with an appropriate starting solution and a series of neighbourhoods designed to preserve the layout structure. Empirical evidence based on real data shows that the quality of the resulting solutions closely matches that of cutting patterns currently produced by human experts. Computation times average around 20–30 s per order as compared to several hours for an equivalent manual solution.

Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green’s function

Two discretizations, linear and nonlinear, of basic notions of the complex analysis are considered. The underlying lattice is an arbitrary quasicrystallic rhombic tiling of a plane. The linear theory is based on the discrete Cauchy-Riemann equations, the nonlinear one is based on the notion of circle patterns. We clarify the role of the rhombic condition in both theories: under this condition the corresponding equations are integrable (in the sense of 3D consistency, which yields also the existense of zero curvature representations, Backlund transformations etc.). We demonstrate that in some precise sense the linear theory is a linearization of the nonlinear one: the tangent space to a set of integrable circle patterns at an isoradial point consists of discrete holomorphic functions which take real (imaginary) values on two sublattices. We extend solutions of the basic equations of both theories to Z d , where d is the number of different edge slopes of the quasicrystallic tiling. In the linear theory, we give an integral representation of an arbitrary discrete holomorphic function, thus proving the density of discrete exponential functions. We introduce the d-dimensional discrete logarithmic function which is a generalization of Kenyon's discrete Green's function, and uncover several new properties of this function. We prove that it is an isomonodromic solution of the discrete Cauchy-Riemann equations, and that it is a tangent vector to the space of integrable circle patterns along the family of isomonodromic discrete power functions.

Dimers, tilings and trees

Generalizing results of Temperley (London Mathematical Society Lecture Notes Series 13 (1974) 202), Brooks et al. (Duke Math. J. 7 (1940) 312) and others (Electron. J. Combin. 7 (2000); Israel J. Math. 105 (1998) 61) we describe a natural equivalence between three planar objects: weighted bipartite planar graphs; planar Markov chains; and tilings with convex polygons. This equivalence provides a measure-preserving bijection between dimer coverings of a weighted bipartite planar graph and spanning trees of the corresponding Markov chain. The tilings correspond to harmonic functions on the Markov chain and to “discrete analytic functions” on the bipartite graph. The equivalence is extended to infinite periodic graphs, and we classify the resulting “almost periodic” tilings and harmonic functions.

Optimization Strategies for Optimization Procedures and Algorithms

AbstractNew theory of circle patterns (or packing) is now well developed. Uniformization result, the circle packing theorem [1], is based on analytic functions, while Doyle spirals are analoques of entire functions.Doyle constructed entire immersed hexagonal circle patterns analoques to the exponential map and conjectured that these immersed packings which came to be known as Doyle spirals, are the only entire immersed hexagonal circle packings [2, 3].Schramm had found that a framework of circle patterns based on the square grid which is called SG patterns, is more tractable than the traditional hexagonal patterns [4].We present a new approach concerned with a theory of discrete multistage optimization [5, 6]. Optimization strategies for optimization procedures are good tools for modelling. Consequently calculations are not far from empirical observations and experimental results. This clearly demonstrates the broad interest in our field and the role which mathematical modelling and analysis is expected to play with regard to the solution of the challenging problems that we are facing at the beginning of this millennium.

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.

The Laplacian and Dirac operators on critical planar graphs

On a periodic planar graph whose edge weights satisfy a certain simple geometric condition, the discrete Laplacian and d-bar operators have the property that their determinants and inverses only depend on the local geometry of the graph. We obtain explicit expressions for the logarithms of the (normalized) determinants, as well as the inverses of these operators. We relate the logarithm of the determinants to the volume plus mean curvature of an associated hyperbolic ideal polyhedron. In the associated dimer and spanning tree models, for which the determinants of d-bar and the Laplacian respectively play the role of the partition function, this allows us to compute the entropy and correlations in terms of the local geometry. In addition, we define a continuous family of special discrete analytic functions, which, via convolutions gives a general process for constructing discrete analytic functions and discrete harmonic functions on critical planar graphs.

Recurrence of Distributional Limits of Finite Planar Graphs

Suppose that $G_j$ is a sequence of finite connected planar graphs, and in each $G_j$ a special vertex, called the root, is chosen randomly-uniformly. We introduce the notion of a distributional limit $G$ of such graphs. Assume that the vertex degrees of the vertices in $G_j$ are bounded, and the bound does not depend on $j$. Then after passing to a subsequence, the limit exists, and is a random rooted graph $G$. We prove that with probability one $G$ is recurrent. The proof involves the Circle Packing Theorem. The motivation for this work comes from the theory of random spherical triangulations.

Approximation of Quasisymmetries Using Circle Packings

The Universal Teichmüller Space has recently gained the attention of physicists as a setting for developing string theory. Two common models for this space are the set of quasicircles and the set of quasisymmetries. The link between these models lies in the fact that the Riemann maps from \(\mathbb{D}\) and \(\mathbb{D}^*\) to the complementary domains of quasicircles induce quasisymmetric automorphisms of \(\partial \mathbb{D}\). We develop a means of approximating these quasisymmetries given their associated quasicircles. This provides a concrete method for switching from the quasicircle to the quasisymmetry model of the Universal Teichmüller Space. Our approach uses discrete analytic functions induced by circle packings to approximate the Riemann maps.

Circle packings of maps —The Euclidean case

In an earlier work, the author extended the Andreev-Koebe-Thurston circle packing theorem. Additionally, a polynomial time algorithm for constructing primal-dual circle packings of arbitrary (essentially) 3-connected maps was found. In this note, additional details concerning surfaces of constant curvature 0 (with special emphasis on planar graphs where a slightly different treatment is necessary) are presented.

Circle Packings of Maps in Polynomial Time

The Andreev–Koebe–Thurston circle packing theorem is generalized and improved in two ways. First, we obtain simultaneous circle packings of the map and its dual map so that, in the corresponding straight-line representations of the map and the dual, any two edges dual to each other are perpendicular. Necessary and sufficient condition for a map to have such a primal–dual circle packing representation in a surface of constant curvature is that its universal cover is 3-connected (the map has no ‘planar’ 2-separations). Second, an algorithm is obtained that given a mapMand a rational number ϵ < 0 finds an ϵ-approximation for the radii and the co-ordinates of the centres for the primal–dual circle packing representation ofM. The algorithm is polynomial in |E(M)| and log(1/ϵ). In particular, for a map without planar 2-separations on an arbitrary surface, we have a polynomial time algorithm for simultaneous geodesic convex representations of the map and its dual, so that only edges dual to each other cross, and the angles at the crossings are arbitrarily close to π/2.

Harmonic functions on planar and almost planar graphs and manifolds, via circle packings

The circle packing theorem is used to show that on any bounded valence transient planar graph there exists a non constant, harmonic, bounded, Dirichlet function. If \(P\) is a bounded circle packing in \({\Bbb R}^2\) whose contacts graph is a bounded valence triangulation of a disk, then, with probability \(1\), the simple random walk on \(P\) converges to a limit point. Moreover, in this situation any continuous function on the limit set of \(P\) extends to a continuous harmonic function on the closure of the contacts graph of \(P\); that is, this Dirichlet problem is solvable. We define the notions of almost planar graphs and manifolds, and show that under the assumptions of transience and bounded local geometry these possess non constant, harmonic, bounded, Dirichlet functions. Let us stress that an almost planar graph is not necessarily roughly isometric to a planar graph.