Discrete Analytic Functions Research Articles

On circle patterns and spherical conical metrics

The Koebe-Andreev-Thurston circle packing theorem, as well as its generalization to circle patterns due to Bobenko and Springborn, holds for Euclidean and hyperbolic metrics possibly with conical singularities, but fails for spherical metrics because of the nonuniqueness coming from Möbius transformations. In this paper, we show that a unique existence result for circle pattern with spherical conical metric holds if one prescribes the total geodesic curvature of each circle instead of the cone angles.

Schur analysis and discrete analytic functions: Rational functions and co-isometric realizations

We define and study rational discrete analytic functions and prove the existence of a coisometric realization for discrete analytic Schur multipliers.

Stress-field driven conformal lattice design using circle packing algorithm

Reliable extreme lightweight is the pursuit in many high-end manufacturing areas. Aided by additive manufacturing (AM), lattice material has become a promising candidate for lightweight optimization. Configuration of lattice units at the material level and the distribution of lattice units at the structure level are the two main research directions recently. This paper proposes a generative strategy for lattice infilling optimization using organic strut-based lattices. A sphere packing algorithm driven by von Mises stress fields determines the lattice distribution density. Two typical configurations, Voronoi polygons and Delaunay triangles, are adopted to constitute the frames, respectively. Based on finite element analysis, a simplified truss model is utilized to evaluate the lattice distribution in terms of mechanical properties. Optimization parameters, including node number, mapping gradient, and the range of varying circle size, are investigated through the genetic algorithm (GA). Multiple feasible solutions are obtained for further solidification modelling. To avoid the stress concentration, the organic strut-based lattice units are created by the iso-surface modelling method. The effectiveness of the proposed generative approach is illustrated through a classical 3-point bending beam. The stiffness of the optimized structure, verified through experimental testing, has increased 80% over the one using the traditional uniform body center cubic (BCC) lattice distribution.

On dynamical gaskets generated by rational maps, Kleinian groups, and Schwarz reflections

According to the Circle Packing Theorem, any triangulation of the Riemann sphere can be realized as a nerve of a circle packing. Reflections in the dual circles generate a Kleinian group H H whose limit set is a generalized Apollonian gasket Λ H \Lambda _H . We design a surgery that relates H H to a rational map g g whose Julia set J g \mathcal {J}_g is (non-quasiconformally) homeomorphic to Λ H \Lambda _H . We show for a large class of triangulations, however, the groups of quasisymmetries of Λ H \Lambda _H and J g \mathcal {J}_g are isomorphic and coincide with the corresponding groups of self-homeomorphisms. Moreover, in the case of H H , this group is equal to the group of Möbius symmetries of Λ H \Lambda _H , which is the semi-direct product of H H itself and the group of Möbius symmetries of the underlying circle packing. In the case of the tetrahedral triangulation (when Λ H \Lambda _H is the classical Apollonian gasket), we give a quasiregular model for the above actions which is quasiconformally equivalent to g g and produces H H by a David surgery. We also construct a mating between the group and the map coexisting in the same dynamical plane and show that it can be generated by Schwarz reflections in the deltoid and the inscribed circle.

Energy-Efficient Initial Deployment and ML-Based Postdeployment Strategy for UAV Network With Guaranteed QoS

Nowadays, the extensive use of unmanned aerial vehicle (UAV)-enabled networks in different applications demands intelligent deployment planning to utilize several benefits of UAVs. This article proposes a complete solution for deploying a UAV network over an unprecedented public meet-up area that offers a guaranteed quality-of-service demand with no interference and capacity limit violation. We call that the proposed solution is complete as it includes both initial and postdeployment planning. Under initial deployment, we offer three different placement algorithms, known as anticlockwise spiral algorithm, clockwise spiral algorithm, and hexagonal circle packing algorithm, to determine the energy-efficient 3-D positions of capacity-limited UAVs with no inter-UAV interference. After deployment, the random walk by users demands postdeployment planning for UAVs. We propose a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$Q$</tex-math></inline-formula> -learning-based algorithm to realign the existing UAVs to maintain the outage. In order to do a more realistic performance assessment of the proposed algorithms, we model the user distribution for a hotspot region by the Thomas cluster point process and the Matern cluster point process. The obtained results exhibit that all three initial deployment algorithms show better performance than a random deployment. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$Q$</tex-math></inline-formula> -learning algorithm under postdeployment offers network lifetime enhancement in addition to outage improvement.

Integrability and geometry of the Wynn recurrence

We show that the Wynn recurrence (the missing identity of Frobenius of the Padé approximation theory) can be incorporated into the theory of integrable systems as a reduction of the discrete Schwarzian Kadomtsev–Petviashvili equation. This allows, in particular, to present the geometric meaning of the recurrence as a construction of the appropriately constrained quadrangular set of points. The interpretation is valid for a projective line over arbitrary skew field what motivates to consider non-commutative Padé theory. We transfer the corresponding elements, including the Frobenius identities, to the non-commutative level using the quasideterminants. Using an example of the characteristic series of the Fibonacci language we present an application of the theory to the regular languages. We introduce the non-commutative version of the discrete-time Toda lattice equations together with their integrability structure. Finally, we discuss application of the Wynn recurrence in a different context of the geometric theory of discrete analytic functions.

Discrete analytic functions, structured matrices and a new family of moment problems

Using Zeilberger generating function formula for the values of a discrete analytic function in a quadrant we make connections with the theory of structured reproducing kernel spaces, structured matrices and a generalized moment problem. An important role is played by a Krein space realization result of Dijksma, Langer and de Snoo for functions analytic in a neighborhood of the origin. A key observation is that, using a simple Moebius transform, one can reduce the study of discrete analytic functions in the upper right quadrant to problems of function theory in the open unit disk. As an example, we associate to each finite positive measure on [0,2π] a discrete analytic function on the right-upper quarter plane with values on the lattice defining a positive definite function. Emphasis is put on the rational case, both when an underlying Carathéodory function is rational and when, in the positive case, the spectral function is rational. The rational case and the general case are linked via the existence of a unitary dilation, possibly in a Krein space.

A Positive Answer for 3IM+1CM Problem with a General Difference Polynomial

In this paper, the 3IM+1CM theorem with a general difference polynomial L z , f will be established by using new methods and technologies. Note that the obtained result is valid when the sum of the coefficient of L z , f is equal to zero or not. Thus, the theorem with the condition that the sum of the coefficient of L z , f is equal to zero is also a good extension for recent results. However, it is new for the case that the sum of the coefficient of L z , f is not equal to zero. In fact, the main difficulty of proof is also from this case, which causes the traditional theorem invalid. On the other hand, it is more interesting that the nonconstant finite-order meromorphic function f can be exactly expressed for the case f ≡ − L z , f . Furthermore, the sharpness of our conditions and the existence of the main result are illustrated by examples. In particular, the main result is also valid for the discrete analytic functions.

Ball packings for links

The ball number of a link L, denoted by ball(L), is the minimum number of solid balls (not necessarily of the same size) needed to realize a necklace representing L. In this paper, we show that ball(L)≤5cr(L) where cr(L) denotes the crossing number of a nontrivial nonsplittable link L. To this end, we use the connection of the Lorentz geometry with the ball packings. The well-known Koebe–Andreev–Thurston circle packing Theorem is also an important brick for the proof. Our approach yields an algorithm to construct explicitly the desired necklace representation of L in R3.

Centering Koebe polyhedra via Möbius transformations

A variant of the Circle Packing Theorem states that the combinatorial class of any convex polyhedron contains elements, called Koebe polyhedra, midscribed to the unit sphere centered at the origin, and that these representatives are unique up to Möbius transformations of the sphere. Motivated by this result, various papers investigate the problem of centering spherical configurations under Möbius transformations. In particular, Springborn proved that for any discrete point set on the sphere there is a Möbius transformation that maps it into a set whose barycenter is the origin, which implies that the combinatorial class of any convex polyhedron contains an element midsribed to a sphere with the additional property that the barycenter of the points of tangency is the center of the sphere. This result was strengthened by Baden, Krane and Kazhdan who showed that the same idea works for any reasonably nice measure defined on the sphere. The aim of the paper is to show that Springborn’s statement remains true if we replace the barycenter of the tangency points by many other polyhedron centers. The proof is based on the investigation of the topological properties of the integral curves of certain vector fields defined in hyperbolic space. We also show that most centers of Koebe polyhedra cannot be obtained as the center of a suitable measure defined on the sphere.

Packing Disks by Flipping and Flowing

We provide a new type of proof for the Koebe–Andreev–Thurston (KAT) planar circle packing theorem based on combinatorial edge-flips. In particular, we show that starting from a disk packing with a maximal planar contact graph G, one can remove any flippable edge $$e^-$$ of this graph and then continuously flow the disks in the plane, so that at the end of the flow, one obtains a new disk packing whose contact graph is the graph resulting from flipping the edge $$e^-$$ in G. This flow is parameterized by a single inversive distance.

The Dirichlet problem for orthodiagonal maps

We prove that the discrete harmonic function corresponding to smooth Dirichlet boundary conditions on orthodiagonal maps, that is, plane graphs having quadrilateral faces with orthogonal diagonals, converges to its continuous counterpart as the mesh size goes to 0. This provides a convergence statement for discrete holomorphic functions, similar to the one obtained by Chelkak and Smirnov [4] for isoradial graphs. We observe that by the double circle packing theorem [3], any finite, simple, 3-connected planar map admits an orthodiagonal representation.Our result improves the work of Skopenkov [41] and Werness [47] by dropping all regularity assumptions required in their work and providing effective bounds. In particular, no bound on the vertex degrees is required. Thus, the result can be applied to models of random planar maps that with high probability admit orthodiagonal representation with mesh size tending to 0. In a companion paper [21], we show that this can be done for the discrete mating-of-trees random map model of Duplantier, Gwynne, Miller and Sheffield [15,25].

A linearized circle packing algorithm

This paper presents a geometric algorithm for approximating radii and centers for a variety of univalent circle packings, including maximal circle packings on the unit disc and the sphere and certain polygonal circle packings in the plane. This method involves an iterative process which alternates between estimates of circle radii and locations of circle centers. The algorithm employs sparse linear systems and in practice achieves a consistent linear convergence rate that is far superior to traditional packing methods. It is deployed in a MATLAB® package which is freely available. This paper gives background on circle packing, a description of the linearized algorithm, illustrations of its use, sample performance data, and remaining challenges.

Circle packing with generalized branching

There is a fairly comprehensive theory of discrete analytic functions based on circle packing. In this theory, discrete analytic functions are represented as maps between circle packings that share combinatorial tangency patterns. Branching behavior, however, has until now been restricted by the need to place branch points at circle centers. In this paper, the authors introduce mechanisms for generalized branching which remove this restriction. Their use is illustrated by overcoming combinatorial obstructions to branching in the construction of a discrete Ahlfors function.

On a new discretization of complex analysis

This paper develops an approach to discretization of complex analysis proposed by S. P. Novikov and the author in 2003. Under this approach discrete analytic functions are real-valued. It is shown that a large class of such functions on a lattice admits a canonical multiplication by the imaginary unit. Arbitrary lattices are considered for a triangular discretization and rhombic lattices for a quadrangular discretization. Bibliography: 24 titles.

On the Correctness of Polynomial Difference Operators

Difference equations arise in various areas of mathematics. Together with the method of generating functions, they yield a powerful machinery for studying enumeration problems in combinatorial analysis (see, e.g. [1]). Another source of difference equations is the discretization of differential equations. For instance, the discretization of Cauchy-Riemann equations led to the appearance of the theory of discrete analytic functions (see, e.g. [2, 3]) which found applications in the theory of Reimannian surfaces and combinatorial analysis (see, e.g. [4, 5]). The methods of discretization of a differential problem comprise an important part of the theory of difference schemes, and they also lead to difference equations (see, e.g. [6]). The theory of difference schemes is exploring ways of constructing difference schemes, explores the challenges posed difference and the convergence of the solution of the difference problem to the solution of the original differential problem, engaged justification of algorithms for solving problems of difference. An important place among these are correct. We define the shift operators δj with respect to the variables xj : δjf (x) = f (x1, . . . , xj−1, xj + 1, xj+1, . . . , xn), and polynomial difference operator of the form

A Möbius-Invariant Power Diagram and Its Applications to Soap Bubbles and Planar Lombardi Drawing

We use three-dimensional hyperbolic geometry to define a form of power diagram for systems of circles in the plane that is invariant under Mobius transformations. By applying this construction to circle packings derived from the Koebe---Andreev---Thurston circle packing theorem, we show that every planar graph of maximum degree three has a planar Lombardi drawing (a drawing in which the edges are drawn as circular arcs, meeting at equal angles at each vertex). We use circle packing to construct planar Lombardi drawings of a special class of 4-regular planar graphs, the medial graphs of polyhedral graphs, and we show that not every 4-regular planar graph has a planar Lombardi drawing. We also use these power diagrams to characterize the graphs formed by two-dimensional soap bubble clusters (in equilibrium configurations) as being exactly the 3-regular bridgeless planar multigraphs, and we show that soap bubble clusters in stable equilibria must in addition be 3-connected.

Content-Aware Photo Collage Using Circle Packing

In this paper, we present a novel approach for automatically creating the photo collage that assembles the interest regions of a given group of images naturally. Previous methods on photo collage are generally built upon a well-defined optimization framework, which computes all the geometric parameters and layer indices for input photos on the given canvas by optimizing a unified objective function. The complex nonlinear form of optimization function limits their scalability and efficiency. From the geometric point of view, we recast the generation of collage as a region partition problem such that each image is displayed in its corresponding region partitioned from the canvas. The core of this is an efficient power-diagram-based circle packing algorithm that arranges a series of circles assigned to input photos compactly in the given canvas. To favor important photos, the circles are associated with image importances determined by an image ranking process. A heuristic search process is developed to ensure that salient information of each photo is displayed in the polygonal area resulting from circle packing. With our new formulation, each factor influencing the state of a photo is optimized in an independent stage, and computation of the optimal states for neighboring photos are completely decoupled. This improves the scalability of collage results and ensures their diversity. We also devise a saliency-based image fusion scheme to generate seamless compositive collage. Our approach can generate the collages on nonrectangular canvases and supports interactive collage that allows the user to refine collage results according to his/her personal preferences. We conduct extensive experiments and show the superiority of our algorithm by comparing against previous methods.

Reproducing kernel Hilbert spaces generated by the binomial coefficients

We study a reproducing kernel Hilbert space of functions defined on the positive integers and associated to the binomial coefficients. We introduce two transforms, which allow us to develop a related harmonic analysis in this Hilbert space. Finally, we mention connections with the theory of discrete analytic functions, statistics, and with the quantum case.

The boundary value problem for discrete analytic functions

This paper is on further development of discrete complex analysis introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We consider a graph lying in the complex plane and having quadrilateral faces. A function on the vertices is called discrete analytic, if for each face the difference quotients along the two diagonals are equal.We prove that the Dirichlet boundary value problem for the real part of a discrete analytic function has a unique solution. In the case when each face has orthogonal diagonals we prove that this solution uniformly converges to a harmonic function in the scaling limit. This solves a problem of S. Smirnov from 2010. This was proved earlier by R. Courant–K. Friedrichs–H. Lewy and L. Lusternik for square lattices, by D. Chelkak–S. Smirnov and implicitly by P.G. Ciarlet–P.-A. Raviart for rhombic lattices.In particular, our result implies uniform convergence of the finite element method on Delaunay triangulations. This solves a problem of A. Bobenko from 2011. The methodology is based on energy estimates inspired by alternating-current network theory.