Extremal Distances Research Articles

CNED sets: countably negligible for extremal distances

CNED sets: countably negligible for extremal distances

Rigidity and continuous extension for conformal maps of circle domains

We present sufficient conditions so that a conformal map between planar domains whose boundary components are Jordan curves or points has a continuous or homeomorphic extension to the closures of the domains. Our conditions involve the notions of cofat domains and C N E D CNED sets, i.e., countably negligible for extremal distances, recently introduced by the author. We use this result towards establishing conformal rigidity of a class of circle domains. A circle domain is conformally rigid if every conformal map onto another circle domain is the restriction of a Möbius transformation. We show that circle domains whose point boundary components are C N E D CNED are conformally rigid. This result is the strongest among all earlier works and provides substantial evidence towards the rigidity conjecture of He–Schramm [Invent. Math. 115 (1994), no. 2, 297–310], relating the problems of conformal rigidity and removability.

Extremal distance and conformal radius of a CLE4 loop

Consider CLE$_4$ in the unit disk and let $\ell$ be the loop of the CLE$_4$ surrounding the origin. Schramm, Sheffield and Wilson determined the law of the conformal radius seen from the origin of the domain surrounded by $\ell$. We complement their result by determining the law of the extremal distance between $\ell$ and the boundary of the unit disk. More surprisingly, we also compute the joint law of these conformal radius and extremal distance. This law involves first and last hitting times of a one-dimensional Brownian motion. Similar techniques also allow us to determine joint laws of some extremal distances in a critical Brownian loop-soup cluster.

Criteria of Removable Sets for Harmonic Functions in the Sobolev Spaces L1p,w

Ahlfors and Beurling [16] proved that set 𝐸 is removable for class 𝐴𝐷2 of analytic functions with the finite Dirichlet integral if and only if 𝐸 does not change extremal distances. Their proof uses the conformal invariance of class 𝐴𝐷2, so it does not immediately generalize to 𝑝 ̸= 2 and to the relevant classes of harmonic functions in the space. In 1974 Hedberg [19] proposed new approaches to the problem of describing removable singularities in the function theory. In particular he gave the exact functional capacitive conditions for a set to be removable for class 𝐻𝐷𝑝(𝐺). Here 𝐻𝐷𝑝(𝐺) is the class of real-valued harmonic functions 𝑢 in a bounded open set 𝐺 ⊂ 𝑅𝑛, 𝑛 ≥ 2, and such that ∫︁ 𝐺 |∇𝑢|𝑝 𝑑𝑥 < ∞, 𝑝 > 1. In this paper we extend Hedberg’s results on class 𝐻𝐷𝑝,𝑤(𝐺) of harmonic functions 𝑢 in 𝐺 and such that ∫︁ 𝐺 |∇𝑢|𝑝 𝑤𝑑𝑥 < ∞. Here a locally integrable function 𝑤 : 𝑅𝑛 → (0,+∞) satisfies the Muckenhoupt condition [20] sup 1 |𝑄| ∫︁ 𝑄 𝑤𝑑𝑥 ⎛ ⎝ 1 |𝑄| ∫︁ 𝑄 𝑤1−𝑞𝑑𝑥 ⎞ ⎠ 𝑝−1 < ∞, where the supremum is taking over all coordinate cubes 𝑄 ⊂ 𝑅𝑛, 𝑞 ∈ (1,+∞) and 1 𝑝 + 1 𝑞 = 1; by ℒ𝑛(𝑄) = |𝑄| we denote the 𝑛-dimensional Lebesgue measure of 𝑄. We denote by 𝐿1 𝑞 , ˜ 𝑤(𝐺) the Sobolev space of locally integrable functions 𝐹 on 𝐺, whose generalized gradient in 𝐺 are such that ‖𝑓‖𝐿1 𝑞 , ˜ 𝑤(𝐺) = ⎛ ⎝ ∫︁ 𝐺 |∇𝑓|𝑞 ˜ 𝑤𝑑𝑥 ⎞ ⎠ 1 𝑞 < ∞, where ˜ 𝑤 = 𝑤1−𝑞. The closure of 𝐶∞ 0 (𝐺) in ‖ · ‖𝐿1 𝑞 , ˜ 𝑤(𝐺) is denoted by ∘L 1 𝑞, ˜ 𝑤(𝐺). For compact set 𝐾 ⊂ 𝐺 (𝑞, ˜ 𝑤)-capacity regarding 𝐺 is defined by 𝐶𝑞, ˜ 𝑤(𝐾) = inf 𝑣 ∫︁ 𝐺 |∇𝑣|𝑞 ˜ 𝑤𝑑𝑥, where the infimum is taken over all 𝑣 ∈ 𝐶∞ 0 (𝐺) such that 𝑣 = 1 in some neighbourhood of 𝐾. Note that 𝐶𝑞, ˜ 𝑤(𝐾) = 0 is independent from the choice of bounded set 𝐺 ⊂ 𝑅𝑛. We set 𝐶𝑞, ˜ 𝑤(𝐹) = 0 for arbitrary 𝐹 ⊂ 𝑅𝑛 if for every compact 𝐾 ⊂ 𝐹 there exists a bounded open set 𝐺 such that 𝐶𝑞, ˜ 𝑤(𝐾) = 0 regarding 𝐺. To conclude, we formulate the main results. Theorem 1. Compact 𝐸 ⊂ 𝐺 is removable for 𝐻𝐷𝑝,𝑤(𝐺) if and only if 𝐶∞ 0 (𝐺 ∖ 𝐸) is dense in ∘L 1 𝑞, ˜ 𝑤(𝐺). Theorem 2. Compact 𝐸 ⊂ 𝐺 is removable for 𝐻𝐷𝑝,𝑤(𝐺) if and only if 𝐶𝑞, ˜ 𝑤(𝐸) = 0. Corollary. The property of being removable for 𝐻𝐷𝑝,𝑤(𝐺) is local, i.e. compact 𝐸 ⊂ 𝐺 is removable if and only if every 𝑥 ∈ 𝐸 has a compact neighbourhood, whose intersection with 𝐺 is removable. Theorem 3. If 𝐺 is an open set in 𝑅𝑛 and 𝐶𝑞, ˜ 𝑤(𝑅𝑛 ∖𝐺) = 0. Then 𝐶∞ 0 (𝐺) is dense in ∘L 1 𝑞, ˜ 𝑤(𝑅𝑛).

Extremal Distances for Subtree Transfer Operations in Binary Trees

Three standard subtree transfer operations for binary trees, used in particular for phylogenetic trees, are: tree bisection and reconnection (TBR), subtree prune and regraft (SPR), and rooted subtree prune and regraft (rSPR). We show that for a pair of leaf-labelled binary trees with n leaves, the maximum number of such moves required to transform one into the other is n-Theta (sqrt{n}), extending a result of Ding, Grünewald, and Humphries, and this holds also if one of the trees is fixed arbitrarily. If the pair is chosen uniformly at random, then the expected number of moves required is n-Theta (n^{2/3}). These results may be phrased in terms of agreement forests: we also give extensions for more than two binary trees.

Limit distributions of extremal distances to the nearest neighbor

Abstract Theorems on the limit distributions of the minimal and maximal distances to the nearest neighbor in a sample of random independent points having a uniform distribution on a metric space are proved. As examples of such spaces a multidimensional torus and a binary cube are considered.

Lazy orbits: An optimization problem on the sphere

Non-transitive subgroups of the orthogonal group play an important role in the non-Euclidean geometry. If $G$ is a closed subgroup in the orthogonal group such that the orbit of a single Euclidean unit vector does not cover the (Euclidean) unit sphere centered at the origin then there always exists a non-Euclidean Minkowski functional such that the elements of $G$ preserve the Minkowskian length of vectors. In other words the Minkowski geometry is an alternative of the Euclidean geometry for the subgroup $G$. It is rich of isometries if $G$ is "close enough" to the orthogonal group or at least to one of its transitive subgroups. The measure of non-transitivity is related to the Hausdorff distances of the orbits under the elements of $G$ to the Euclidean sphere. Its maximum/minimum belongs to the so-called lazy/busy orbits, i.e. they are the solutions of an optimization problem on the Euclidean sphere. The extremal distances allow us to characterize the reducible/irreducible subgroups. We also formulate an upper and a lower bound for the ratio of the extremal distances. As another application of the analytic tools we introduce the rank of a closed non-transitive group $G$. We shall see that if $G$ is of maximal rank then it is finite or reducible. Since the reducible and the finite subgroups form two natural prototypes of non-transitive subgroups, the rank seems to be a fundamental notion in their characterization. Closed, non-transitive groups of rank $n-1$ will be also characterized. Using the general results we classify all their possible types in lower dimensional cases $n=2, 3$ and $4$. Finally we present some applications of the results to the holonomy group of a metric linear connection on a connected Riemannian manifold.

Bifurcations at the dawn of Modern Science

In this article we review two classical bifurcation problems: the instability of an axisymmetric floating body studied by Archimedes, 2300 years ago and the multiplicity of images observed in curved mirrors, a problem which has been solved by Alhazen in the 11th century. We will first introduce these problems in trying to keep some of the flavor of the original analysis and then, we will show how they can be reduced to a question of extremal distances studied by Apollonius.

Extremal distance, hyperbolic distance, and convex hulls over domains with smooth boundary

Given a simply connected planar domain › we develop estimates for boundary derivatives on @› and estimates for hyperbolic and extremal distances in › and the hyperbolic convex hull boundary S›. We focus on the case when the underlying domain has smooth boundary; this allows very explicit formulas in terms of a collection of invariants which clarify behavior even in the generic case. In particular, we are able to obtain very explicit estimates using the intimate connection between the convex hull boundary and the geometry of the medial axis. As applica- tions, we include here a refinement and alternate proof of the Thurston-Sullivan conjecture that the nearest-point retraction is 2-Lipschitz in the hyperbolic metrics and a variant of the Ahlfors distortion theorem which works as an integral along branches of the medial axis. In this paper, we study the special case of the hyperbolic convex hull boundary S› over a domain › ‰ C with smooth boundary; in particular we obtain a number of estimates and invariants which relate extremal lengths and hyperbolic distances upstairs in the convex hull boundary to the same quantities downstairs in the domain via the nearest-point retraction map r: › ! S›. The study of the smooth case was originally motivated, perhaps unexpectedly, by computer vision, where con- formal mapping techniques were introduced by Sharon and Mumford (see (25)) to construct metric spaces of smooth curves. In this application, smooth Jordan curves are represented by their associated welding maps, and a Riemannian metric on curves is induced by the Weil-Peterson metric on the dieomorphism group of the circle. The curves of interest are typically at least C 2 , and careful geometric estimates for boundary derivatives and extremal distances are required in order to understand geodesic distances between curves. We obtain these by working with the convex hull boundary S›. The surface S› admits an explicit construction for the Riemann map ¶: S› ! D (see, for example, (6) or (18)) and this mapcan act as a sort of poor-man's Riemann map from the disk to › itself. Among the advantages is that S› is a ruled surface, and this special geometry permits easier analysis; in addition, the Riemann mapis an entirely local construction, unlike the global nature of the Riemann map to ›. The resulting formulas are, for the most part, simple and explicit, and we feel that working in the smooth case has the potential to simplify many computations

Extremal Distances between Sections of Convex Bodies

Let K, D be centrally symmetric convex bodies in \(\mathbb{R}^n .\) Let k < n and let dk(K, D) be the smallest Banach–Mazur distance between k-dimensional sections of K and D. Define $$ \Delta (k,n) = \sup d_k (K,D), $$ where the supremum is taken over all n-dimensional convex symmetric bodies K, D. We prove that, for any k < n, $$ \Delta (k,n) \sim _{{\text{log}}\,n} \left\{ {\begin{array}{*{20}l} {\sqrt k } & {{\text{if}}\;k \leq n^{2/3} ,} \\ {\frac{{k^2 }} {n}} & {{\text{if}}\;k > n^{2/3} ,} \\ \end{array} } \right. $$ where \(A \sim _{\log n} B\) means that \(1/(C\log ^a n) \cdot A \leq B \leq (C\log ^a n) \cdot A\) for some absolute constants C, a > 0.

Computing extremal and approximate distances in graphs having unit cost edges

Using binary search and a Strassen-like matrix multiplication algorithm we obtain efficient algorithms for computing the diameter, the radius, and other distance-related quantities associated with undirected and directed graphs having unit cost (unweighted) edges. Similar methods are used to find approximate values for the distances between all pairs of vertices, and if the graph satisfies certain regularity conditions to find the exact distances.

On null sets for extremal distances of order 2 and harmonic functions

On null sets for extremal distances of order 2 and harmonic functions

Extremal length and harmonic functions on Riemann surfaces

Expressions for several conformally invariant pseudometrics on a Riemann surface R R are given in terms of three new forms of reduced extremal distance. The pseudometrics are defined by means of various subclasses of the set of all harmonic functions on R R having finite Dirichlet integral. The reduced extremal distance between two points is defined on R R , on the Alexandroff one-point compactification of R R and on the Kerékjártó-Stoïlow compactification of R R . These reduced extremal distances are computed in terms of harmonic functions having specified singularities and boundary behavior. The key to establishing this connection with harmonic functions is a general theorem dealing with extremal length on a compact bordered Riemann surface and its extensions to noncompact bordered surfaces. These results are used to obtain new tests for degeneracy in the classification theory of Riemann surfaces. Finally, some of the results are illustrated for a hyperbolic simply connected Riemann surface.

An iterative algorithm for optimum control problems with inequality constraints

This paper presents an iterative algorithm to approximate inequality constrained optimal control problems. The method uses Pontryagin's necessary conditions of optimality with a penalty method. The initial values of the adjoint vector <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Psi(t)</tex> , and the penalty coefficients are evaluated in such a way that the final conditions are satisfied and the extremal distances between the obtained trajectory and the constraints are imposed. The computing time is remarkably small. This method can treat linear problems with fixed or variable final time with mixed or simple constraints. A test problem is solved.

On the null-sets for extremal distances

On the null-sets for extremal distances