Discrete Analytic Functions Research Articles

On the convergence of circle packings to the Riemann map

Rodin and Sullivan (1987) proved Thurston’s conjecture that a scheme based on the Circle Packing Theorem converges to the Riemann mapping, thereby providing a refreshing geometric view of Riemann’s Mapping Theorem. We now present a new proof of the Rodin–Sullivan theorem. This proof is based on the argument principle, and has the following virtues. 1. It applies to more general packings. The Rodin–Sullivan paper deals with packings based on the hexagonal combinatorics. Later, quantitative estimates were found, which also worked for bounded valence packings. Here, the bounded valence assumption is unnecessary and irrelevant. 2. Our method is rather elementary, and accessible to non-experts. In particular, quasiconformal maps are not needed. Consequently, this gives an independent proof of Riemann’s Conformal Mapping Theorem. (The Rodin–Sullivan proof uses results that rely on Riemann’s Mapping Theorem.) 3. Our approach gives the convergence of the first and second derivatives, without significant additional difficulties. While previous work has established the convergence of the first two derivatives for bounded valence packings, now the bounded valence assumption is unnecessary.

Circle Packing: Experiments In Discrete Analytic Function Theory

Circle packings are configurations of circles with specified patterns of tangency, and lend themselves naturally to computer experimentation and visualization. Maps between them display, with surprising faithfulness, many of the geometric properties associated with classical analytic functions. This paper introduces the fundamentals of an emerging “discrete analytic function theory” and investigates connections with the classical theory. It then describes several experiments, ranging from investigation of a conjectured discrete Koebe ¼ theorem to a multigrid method for computing discrete approximations of classical analytic functions. These experiments were performed using CirciePack, a software package described in the paper and available free of charge.

Average kissing numbers for non-congruent sphere packings

The Koebe circle packing theorem states that every finite planar graph can be realized as the nerve of a packing of (non-congruent) circles in R^3. We investigate the average kissing number of finite packings of non-congruent spheres in R^3 as a first restriction on the possible nerves of such packings. We show that the supremum k of the average kissing number for all packings satisfies 12.566 ~ 666/53 <= k < 8 + 4*sqrt(3) ~ 14.928 We obtain the upper bound by a resource exhaustion argument and the upper bound by a construction involving packings of spherical caps in S^3. Our result contradicts two naive conjectures about the average kissing number: That it is unbounded, or that it is supremized by an infinite packing of congruent spheres.

A polynomial time circle packing algorithm

The Andreev-Koebe-Thurston circle packing theorem is generalized and improved in two ways. Simultaneous circle packing representations of the map and its dual map are obtained such that any two edges dual to each other cross at the right angle. The necessary and sufficient condition for a map to have such a primal-dual circle packing representation is that its universal cover graph is 3-connected. A polynomial time algorithm is obtained that given such a map M and a rational number ε>0 finds an ε-approximation for the primal-dual circle packing representation of M. In particular, there is a polynomial time algorithm that produces simultaneous geodesic line convex drawings of a given map and its dual in a surface with constant curvature, so that only edges dual to each other cross.

Algorithms for orthogonal drawings (abstract)

The Andreev-Thurston circle packing theorem is generalized and improved in three ways. First, to arbitrary maps on closed surfaces. Second, we get the simultaneous circle packing 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 cross at the right angle. The necessary and sufficient condition for a map to have such a primal-dual circle packing representation is that its universal cover is 3-connected (the map has no "planar" 2-separations). Finally, a polynomial time algorithm is obtained that given a map M and a rational number &amp;#949; &amp;gt; 0 finds an &amp;#949;-approximation for the primal-dual circle packing representation of M. In particular, we get a polynomial time algorithm for geodesic convex representation of reduced maps on arbitrary surfaces.

Existence and uniqueness of packings with specified combinatorics

Generalizations of the Andreev-Thurston circle packing theorem are proved. One such result is the following.

Discrete analysis on a radial lattice

A discrete model for analytic functions is constructed using lattice points of the complex plane arranged in radial form. The discrete analytic functions are defined as solutions of a finite-difference approximation to the polar Cauchy-Riemann equations. The resulting discrete power z ( n) (an analogue of z n ) has a simple algebraic form (a direct analogue of ϱ n exp{ inθ}) and has some surprising properties. For example every discrete polynomial ∑ 0 m a n z ( n) has a factorization in terms of the zeros of its classical counterpart ∑ 0 m a n z n every discrete entire function has a power series representation ∑ a n z ( n) .

The discrete Paley-Wiener theorem

then the function is called discrete entire (d.e.). If f is written in terms of its real and imaginary parts, Eq. (1) s h ows the initial analogy between regular analytic functions and d.e. functions, since it is equivalent to a pair of partial difference equations analogous to the Cauchy-Riemann equations. Discrete analytic functions were first developed by Ferrand [3] and then Duffin [2], and many similarities between d.e. functions and entire functions have been established recently by Zeilberger and others [5-71. In particular, in [Sj, Zeilberger established one form of a discrete Paley-Wiener theorem for d.e. functions of exponential growth, where a function is said to be of exponential growth (R, 5’) if there exists a constant C such that if(m, n)l < CRI”z’Slni for every pair of integers (m, n). In connecting d.e. functions of exponential growth with the Paley-Wiener theorem, two different approaches could be taken. The approach used by Zeilberger is to concentrate on the discrete nature of the function and make a connection with Fourier series, A different approach, which is explored in this paper, is to make a closer analogy with entire functions of exponential type themselves, and use a discrete analog of a Fourier transform along the real line. One distinct advantage of this approach is that the exponential growth of the function can be connected with the domain of integration in a manner that is strikingly analogous to the way that the exponential type of an entire function is connected with the domain of integration of its transform. In this paper, the initial emphasis will be on developing properties of the discrete exponential functions so as to approach the theory of transforms using the discrete exponentials. Then a very intriguing bilinear transformation will be used to connect this theory with Zeilbergcr’s result, achieving a direct analogy with the Paley-Wiener theorem. Finally, an application of the theorem 172

Discrete geometric function theory II

The foundations are laid here for a discrete analytic function theory which can be applied to geometric difference (or q-difference) functions. Using the concept of a q-analytic function. a discrete contour integral is defined and analogues are found for Cauchy integral theorems. Appropriate discrete analytic continuation and multiplication operators are outlined and their properties examined. The development of the theory of q-analytic functions is seen to be closely allied to that monodiffric functions, but significant advantages and distinctive differences are noted

Methods for constructing linear difference operators which simulate analyticity†

Two methods of construction are presented for obtaining families of linear difference operators simulating the operator . These operators are then applied to suggest the construction of corresponding discrete analogues of Laplace's harmonic operator . Other applications include the generation of additional solutions of the linear difference equations under consideration via indefinite discrete line integrals. The results may also be viewed as extensions of monodiffric and discrete analytic function theories

Discrete geometric function theory !

The foundations are laid here for a discrete analytic function theory which can be applied to geometric difference (or q-difference) functions. Using the concept of a q-analytic function. a discrete contour integral is defined and analogues are found for Cauchy integral theorems. Appropriate discrete analytic continuation and multiplication operators are outlined and their properties examined. The development of the theory of q-analytic functions is seen to be closely allied to that monodiffric functions, but significant advantages and distinctive differences are noted

A discrete analogue of the Paley-Wiener theorem for bounded analytic functions in a half plane

In this note we prove a discrete analogue to the following Paley–Weiner theorem: Let f be the restriction to the line of a bounded analytic function in the upper half plane; then the spectrum of f is contained in ([0, ∈). The discrete analogue of complex analysis is the theory of discrete analytic functions invented by Lelong-Ferrand (1944) and developed by Duffin (1956) and others.

Further properties of discrete analytic functions

Analogs of the classical theorems of Liouville, Phragmén-Lindelöf, and Paley-Wiener are proved in the class of discrete analytic functions.

Discrete Analytic Functions of Exponential Growth

Discrete Analytic Functions of Exponential Growth

A new basis for discrete analytic polynomials

AbstractA new basis for discrete analytic polynomials is presented for which the series converges absolutely to a discrete analytic function in the upper right quarter lattice whenever

A new approach to the theory of discrete analytic functions

A new approach to the theory of discrete analytic functions

Discrete analytic functions of exponential growth

Analogues of classical representation formulas for entire functions of exponential type are proved in the class of discrete analytic functions.

A new definition of discrete analytic functions

The concept of a tetradiffric function is introduced. This new scheme for defining discrete analytic functions is shown to retain the algebraic simplicity of monodiffric functions, while introducing to the theory a symmetry similar to the Schwarz Reflection Principle.

A note on a discrete analytic function

An unsolved problem in discrete analytic function theory has been to find a suitable analogue of the function . An analogue z(α), of the function zα, is found here for discrete analytic functions of the first kind (or monodiffric functions). This function resolves a conjecture of Isaacs in the negative, and at the same time it introduces multi-valued functions into the discrete analytic theory.

The discrete Green's function and the discrete kernel function

In this paper discrete analogs of the ∂/ ∂z and ∂/∂ z ̄ operators and a new line integral are defined and used to establish the relationship between the discrete Green's function and the Bergman kernel for the Hilbert space of discrete analytic functions on a discrete region which is the union of net squares.