Mappings and meshes

Introduction. From the point of view of conformal mapping, it is unsatisfactory to consider the individual points of the boundary of a simply connected region as the primitive constituents of this boundarv. When such a region is mapped conformally onto the unit disk, in accordance with the Riemann mapping theorem, the points of the unit circumference correspond, indeed, to the prime ends of the region. These prime ends are here considered as equivalence classes of sequences of points of the region under a relation P*. The principal tool we employ to introduce this relation is extremal length, a conformally invariant quantity associated with a family of curves. Its definition was first given by Ahlfors and Beurling ([2], [3]). The term prinle end originated with Caratheodory [4], who initiated the systematic study of the structure of the boundary of a simply connected region. His approach was topological, and it dealt with conceptssubregions, crosscuts, etc.-which are defined with reference to the given region. It is easily seen that with Caratheodory's definition the prime ends of the unit disk correspond to points on the circumference in the sense that an equivalence class is associated with each boundary point and that two sequences belong to the same class if and only if they converge to the same point of the circumference. The problem that arises under his approach, however, and this is solved by one of Caratheodory's fundamental theorems, is that one must show that prime ends are preserved under conformal mapping. Lindelif [6] circumvented this difficulty by defining prime ends by reference to the conformal map of the disk onto the region; namely in terms of the set of indetermination or cluster set. However, his method does not obviate an explicit analysis of the topological situation in the region itself. Lindelof also obtained the following result, which is related to the classification of prime ends. Caratheodory had distinguished two classes of points, principal and subsidiary (see Section 2. 7, below), on the point set of a prime end, i. e. on the

Our goal is to provide a novel method of representing 2D shapes, where each shape will be assigned a unique fingerprint - a computable approximation to a conformal map of the given shape to a canonical shape in 2D or 3D space (see page 22 for a few examples). In this paper, we make the first significant step in this program where we address the case of simply, and doubly-connected planar domains. We prove uniform convergence of our approximation scheme to the appropriate conformal mapping. Along the way, we affirm a conjecture raised by Ken Stephenson in the 90's which predicts that the Riemann mapping can be approximated by a sequence of electrical networks. In fact, we first treat a more general case. Consider a planar annulus, i.e., a bounded, two-connected, Jordan domain, endowed with a sequence of triangulations exhausting it. We construct a corresponding sequence of maps which converge uniformly on compact subsets of the domain, to a conformal homeomorphism onto the interior of a Euclidean annulus bounded by two concentric circles. The resolution of Stephenson's Conjecture then follows by a limiting argument. Our methods involve harmonic mappings and boundary value problems: discrete and analytic in the plane. The scheme we constructed is programmable.

In this note, we give an explicit construction of iterative approximation of Riemann mappings. Our method is motivated by Koebe's proof of the Riemann mapping theorem.

The Riemann mapping theorem, that an arbitrary simply connected region of the plane can be mapped one-to-one and conformally onto a circle, first appeared in the Inaugural dissertation of Riemann (1826-1866) in 1851. The theorem is important, for by it a result proved for the circle can often be transformed from the circle to a more general region. The proof is difficult, as involving both behavior of a function in the small (conformal mapping) and behavior in the large (one-toone mapping). Riemann's proof was open to criticism and in the following decades numerous mathematicians sought for a proof, e.g., H. A. Schwarz (1843-1921), A. Harnack (1851-1888), H. Poincare (1854-1912), etc., until the first rigorous proof was given in 1900 by W. F. Osgood. The proof of Osgood represented, in my opinion, the coming of age of mathematics in America. Until then, numerous American mathematicians had gone to Europe for their doctorates, or for other advanced study, as indeed did Osgood. But the mathematical productivity in this country in quality lagged behind that of Europe, and no American before 1900 had reached the heights that Osgood then reached. William Fogg Osgood (1864-1943) was born in Boston in 1864, graduated from Harvard College in 1886, stayed in Cambridge for a year of graduate work, and then went to Gottingen with a Harvard fellowship for further study, especially with Felix Klein (1849-1925). According to gossip, Osgood became so enamored of a Gottingen lady that his work suffered and Klein sent him to Erlangen for his doctorate. In any case, he was accorded the degree from Erlangen in 1890 for a thesis on Abelian integrals, and one or two days later he married the girl in G6ttingen, and one or two days still later they sailed for the United States of America. His

Before proving the Riemann Mapping Theorem, we examine the relation between conformal mapping and the theory of fluid flow. Our main goal is to motivate some of the results of the next section and the treatment here will be less formal than that of the remainder of the book.

This paper is an exposition of some applications of Stochastic Processes to boundary behavior problems for harmonic functions. As an illustration, we give a proof of Fatou's theorem in simply connected plane domains which is probabilistic and does not use the Riemann mapping theorem. The paper closes with some remarks on further related work and open questions.

The regular topological structure endows quadrilateral meshes with advantages such as high computational accuracy, reduced element requirements, and enhanced computational efficiency, granting them irreplaceable application value in high‐end engineering simulation domains. However, significant bottlenecks persist in the fully automatic generation technology of quadrilateral meshes for complex geometric models. Existing methods often encounter key issues such as high mesh distortion rates and insufficient algorithm stability when handling high‐curvature boundaries and multiscale feature regions, limiting their widespread application in engineering simulations. This article proposes a quad‐dominant mesh generation method based on gradient fields. Firstly, the three‐dimensional triangular mesh is transformed into a two‐dimensional parametric domain through conformal mapping. Subsequently, based on the compression ratio measured from three‐dimensional to two‐dimensional, the direction and size of the advancing front are calculated to draw contour lines in the parametric domain. A threshold is set to automatically generate high‐quality grids. Based on the extracted contour lines, high‐quality quadrilateral grids are generated pairwise along the gradient field directions. For regions that do not meet the threshold requirements, triangular mesh filling is employed to generate a two‐dimensional quad‐dominant grid. Due to the bijective nature of the mapping, the quad‐dominant surface mesh is obtained through the inverse transformation from the 2D parametric domain back to the 3D space.

We consider Riemann mappings from bounded Lipschitz domains in the plane to a triangle. We show that in this case the Riemann mapping has a linear variational principle: It is the minimizer of the Dirichlet energy over an appropriate affine space. By discretizing the variational principle in a natural way we obtain discrete conformal maps which can be computed by solving a sparse linear system. We show that these discrete conformal maps converge to the Riemann mapping in [Formula: see text], even for non-Delaunay triangulations. Additionally, for Delaunay triangulations the discrete conformal maps converge uniformly and are known to be bijective. As a consequence we show that the Riemann mapping between two bounded Lipschitz domains can be uniformly approximated by composing the discrete Riemann mappings between each Lipschitz domain and the triangle.

This thesis falls naturally into two distinct parts. Both come under the general heading of the theory of conformal mapping though the later part incorporates work in potential theory. We first study the growth of means of the logarithmic derivative of a univalent function in the disc. Here results have been obtained by Hayman and by Baernstein and Brown. Hayman has shown that an elementary upper bound for the growth of these means is best possible in general. Later on, Baernstein and Brown showed that the means of certain classes of monotone slit mappings, including support points of the class S of normalised univalent functions, grow no faster than those of the Koebe function up to a multiplicative constant. The question remained open for unrestricted monotone slit mappings. We settle this question by constructing a monotone slit mapping the mean of whose logarithmic derivative grows faster than that of the Koebe function. Following this, we discuss some recent work by Burdzy on the boundary behaviour of positive harmonic functions in Lipschitz domains and applications of this work to the angular derivative problem. Burdzy obtains his results on the angular derivative by probabilistic methods. Rodin and Warschawski later gave a classical proof of part of Burdzy’s main result and related his criteria for the existence of an angular derivative to criteria which they had used previously. They were, however, unable to obtain a non-probabilistic proof of the full theorem. Using a new non-probabilistic method, we prove a theorem on the growth of positive harmonic functions vanishing near a boundary point of a Lipschitz domain. The plane case of this result and some special cases in space were proved by Burdzy in a series of articles. He went on to prove the full result in space in a later paper with R. J. Williams. Our result enables us to give an elementary proof of the remainder of Burdzy’s theorem on the angular derivative and so complements Rodin and Warschawski’s work. We complete our study of this problem by proving two further related results on the boundary behaviour of positive harmonic functions in Lipschitz domains. It is likely that the methods used will be helpful in problems of a similar nature.

The object of the present paper is to give a new set of proofs of some fundamental existence theorems of conformal mapping and potential theory by using the notion of a kernel function introduced by Bergman [2, 3, 4]. The proofs which we present here are a more or less natural outcome of the recent researches done in this theory [5, 6, 7, 11, 12]. Our new attack upon the existence problem has on the one hand the advantage of generality, since it is applicable in the case of partial differential equations of elliptic type and since for functions of several complex variables the theory of the kernel function is well developed, while on the other hand the approach is simple and elementary in comparison with, say, the method using the Dirichlet principle. For potential theory and conformal mapping in multiply-connected plane domains, our proof is probably as simple as any known, and even for simply-connected domains the method has the advantage that it is based on a general theory rather than upon special artifices. A further point which we gain is that the kernel functions associated with the various norms of function theory and elliptic partial differential equations are well adapted to numerical computation, in view of their relation to orthogonal functions, so that we obtain at one and the same time an existence proof and a computational algorithm. The most serious drawback in our method is, perhaps, that we must make assumptions upon the smoothness of the boundary of the domains we consider, so that the general case is reached only after a topological approximation argument is given. From the broader point of view, our treatment is important in that it yields a unified attack upon various existence problems. An extensive class of existence theorems of function theory of one or more variables and of partial differential equations of elliptic type can be developed using one basic procedure. This procedure can be outlined as follows. We set up the reproducing kernel function in a given function space by solving a suitable extremal problem, or alternatively by means of orthogonal functions, and we investigate the scalar product of the kernel function with what may be described as the fundamental singularity associated with this space. A local argument is used to study the behavior of this scalar product as the infinity of the fundamental singularity crosses the boundary of the domain of definition of the functions in our class, and the desired existence theorems follow from the geometric properties thus obtained. The local investigations depend in a general way upon a knowledge of our problem in simple domains, such as, for example, the unit circle.

Optimal Mesh Algorithms for the Voronoi Diagram of Line Segments and Motion Planning in the Plane

formula omitted]MST: A robust unified algorithm for quadrilateral mesh adaptation in 2D and 3D

Conformal Mapping of Nonsmooth Domains via the Kerzman–Stein Integral Equation

The work of Banach [1],2 Mazur [2], and others has emphasized the fundamental role played by convex functions and sets in the study of Banach spaces and (continuous) linear transformations between them. It is natural to suspect that similar considerations underlie the theory of general linear transformations, in which closure rather than the restricted notion of continuity is predominant. In this paper, therefore, we study linear transformations T from a Banach space B1 , to a Banach space B2, where T denotes a single-valued, additive and homogeneous operation with linear domain DT in B1, and linear range RT in B2. Our definition of linearity is purely algebraic and will necessarily imply continuity and total definition (DT = B1) cnly in the case of linear functionals. With these exceptions our terminology conforms to that of Banach. When T is linear the expression K(x) = 11 Tx 11, xe DT is a convex function, and E[K(x) _ a], a > 0, is a convex set, where E[K(x) ? a] denotes the set of points x e DK = DT such that K(x) ? a. An examination of the connections between the closure of T, the closure of E[K(x) < a], and the structure of convex K's yields a number of theorems with interesting applications, including as special cases theorems of Gelfand [3] and Bosanquet and Kestelmann [4] of the Banach-Steinhaus type (cf. Zygmund [5], pp. 97-100). Upon specializing B1 and B2 to Hilbert space we obtain a characterization of the domain of a selfadjoint transformation. A by-product of the analysis is a surprisingly simple proof of Wintner's theorem [6, ?108]: every definite symmetric transformation possesses a self-adjoint extension.

A classical theorem of Spencer shows that any set system with $n$ sets and $n$ elements admits a coloring of discrepancy $O(\sqrt{n})$. Recent exciting work of Bansal, Lovett, and Meka shows that such colorings can be found in polynomial time. In fact, the Lovett--Meka algorithm finds a half integral point in any “large enough” polytope. However, their algorithm crucially relies on the facet structure and does not apply to general convex sets. We show that for any symmetric convex set $K$ with Gaussian measure at least $e^{-n/500}$, the following algorithm finds a point $y \in K \cap [-1,1]^n$ with $\Omega(n)$ coordinates in $\pm 1$: (1) take a random Gaussian vector $x$; (2) compute the point $y$ in $K \cap [-1,1]^n$ that is closest to $x$; (3) return $y$. This provides another truly constructive proof of Spencer's theorem and the first constructive proof of a theorem of Gluskin and Giannopoulos.