Radial Limits Research Articles

On the two questions of Lohwater and Piranian

The problem we are dealing with consists in the following: find the necessary and sufficient conditions for the zero measure subset of the circumference at which points the bounded analytic function has no radial limits.

Boundary limits of pluri-green potentials in the unit ball of

In this paper we investigate radial and tangential limits of pluri-Green potentials Vμ on the unit ball B in , which for suitable μ are defined by where for z ∊ B,φ z denotes the holomorphic automorphism of B satisfying φz(0)=z, φz(z) = 0, and φz(φz(w))=w. The main result of this paper is as follows: Let λ be the M-invariant measure on B and let f be a non-negative measurable function on B satisfying , and for some p<1, then the pluri-Green potential V f has tangential limit at almost all points of the boundary S of B. It is shown that the result is best possible.

Admissible limits of bloch functions on bounded strongly pseudoconvex domains

Let be a bounded strongly pseudoconvex domain with C2 boundary . In this paper we prove that for a Bloch function in the existance of radial limits at almost all implies the existence of admissible limits almost everywhere on .

Radial correlation limits of helium and heliumlike atoms

Radial correlation limits of two-electron atoms with atomic numbers Z = 1–10 are calculated by using modified Kinoshita wave functions in which all the parameters are optimized. The optimal Kinoshita functions show a rapid energy convergence with the increasing number N of constituent terms, and the radial energies convergent to 10 significant figures are obtained. The results show that both the calculated and estimated values of the radial correlation limits in the literature are insufficiently accurate. In the case of He, for example, the present calculation gives −2.879 028 764 hartrees with N = 40, while the best literature value is −2.879 028 6 hartrees.

Substorm onsets observed by CRRES: Determination of energetic particle source regions

We examine data from the Electron/Proton Angle Spectrometer instrument, also known as Medium Energy B, flown on the CRRES satellite (July 1990 to August 1991). Because of its strongly eccentric orbit, CRRES can detect substorm onsets in the region L = 4 to L = 8, covering the time sectors 1600–0800 MLT. Substorm onsets which are virtually dispersionless in energy have commonly been associated with acceleration processes which occur in the immediate vicinity of the satellite. Data from CRRES show that such virtually dispersionless onsets can occur far in the inner magnetosphere down to L = 4.3 and are distributed up to ±5 hours around local magnetic midnight. Results from a statistical study are presented. We develop a method of using single‐satellite measurements of near‐dispersionless substorm onsets, together with the observed “drift echoes” of these injections, to establish a “source region” for the energetic particles observed at onset. This method is here applied to electrons only. A database of model drift paths has been set up and is used to calculate the possible interceptions between drift paths of various energies from a source region with the satellite trajectory. Our results indicate that the source region can have a sharp boundary in MLT and can extend over several RE. For some events it is possible to deduce an outer radial limit for the injection region, favoring a more local substorm onset mechanism. Our modeling further shows that the injection and drift echo signature observed at the satellite is quite insensitive to the actual shape of the injection boundary.

Interpolating Blaschke products

where ||/||oo is the L°° = L°°(dD,dθ) norm of the non-tangential limit /. Thus we can view H°° as a uniformly closed subalgebra of L°°. An inner function is a function / G H°° with \I(e) = 1, a.e. dθ. For example S(z) = e ^ is an inner function, though it has radial limit 0 at ζ = 1. Inner functions are of central importance in the study of analytic functions on D. For example, Beurling used inner functions (and coined the name) to characterize the translation invariant subspaces of the Hardy space H. Rudin and Carleson used inner functions to characterize the ideals of the disk algebra, and Newman used them to characterize the Shilov boundary of if. If {zn} C D with Σ n ( l — \zn\) < oo then a Blaschke product with zero set {zn} is an inner function of the form

On functions in the little Bloch space and inner functions

We prove that analytic functions in the little Bloch space assume every value as a radial limit on a set of Hausdorff dimension one, unless they have radial limits on a set of positive measure. The analogue for inner functions in the little Bloch space is also proven, and characterizations of various classes of Bloch functions in terms of their level sets are given.

ON BOUNDARY BEHAVIOR OF TYPICAL MAPPINGS

In this article the structure of radial limit sets is investigated for typical mappings defined in the unit ball in -dimensional Euclidean space for and belonging to various function spaces. The structure of the sets of all such mappings in these function spaces is also studied. As an auxiliary result we obtain a description of the structure of typical compact sets in for .

Application of genetic algorithms to service restoration in distribution systems

AbstractService restoration in distribution systems can be formulated as a combinatorial optimization problem to determine power sources for each load considering radial network constraints and power source limits. Until now, the problem has been considered using conventional methods, e.g., the branch‐and‐bounds method, expert system, neural networks, and fuzzy reasoning.Recently, Genetic Algorithms (GA) have been recognized as one of the efficient methods for solving large combinatorial optimization problems. The method can perform parallel search, and can more easily search optimal solutions.This paper presents an application of GA to service restoration in distribution systems. The feasibility of the proposed method is demonstrated on a typical distribution system model. The result shows that the method can solve the problem efficiently, and this tendency becomes dominant by increasing problem dimensions.

Toeplitz algebras and Rieffel deformations

We establish a representation theorem for Toeplitz operators on the Segal-Bargmann (Fock) space ofC n whose “symbols” have uniform radial limits. As an application of this result, we show that Toeplitz algebras on the open ball inC n are “strict deformation quantizations”, in the sense of M. Rieffel, of the continuous functions on the corresponding closed ball.

ON SETS OF NONEXISTENCE OF RADIAL LIMITS OF BOUNDED ANALYTIC FUNCTIONS

Let be a function defined in the unit disc ; the unit circle ; the set of points of at which has no radial limits. In the paper a complete characterization is given of the sets for bounded analytic functions in . It is proved that for any set of linear measure zero there exists a function , bounded and analytic in , such that . Bibliography: 6 titles.

Radial Limits and growths of fractional Cauchy transforms

for |z| < 1. Fα is a Banach space with respect to the norm defined by ‖f‖Fα = inf ‖μ‖ where μ varies over all measures in M for which (1) or (2) holds and where ‖μ‖ is the total variation norm of μ. The spaces Fα have been studied in a number of recent papers and earlier work was extensively done corresponding to the case α = 1. This paper gives a survey of some recent results about Fα which focus on questions about radial limits. We will summarize work contained in [3], [4] and [5]. This material was presented in a talk by the author at the Complex Analysis Semester held at the Banach Center in Warsaw during the fall, 1992. The author wishes to thank the organizers for their invitation to participate in that conference and for their hospitality, especially to Professor Jan Krzyz.

Different types of voltage instability

In reported incidents, different types of voltage instability have occurred and each has presented in a quite distinct manner. To better acquaint power engineers with the different types, two incidents are described as each commenced with a type of system voltage instability and culminated with radial voltage instability. In neither of these two incidents were operators fully aware of the situation. The analysis of the incidents provided in this paper can give important operational insights. In particular it shows the value of timely load shedding and the need to allow a 20% margin from the radial voltage stability limit. >

Invariant Subspaces of the Dirichlet Shift and Pseudocontinuations

In this paper we study extremal functions for invariant subspaces $\mathcal {M}$ of the Dirichlet shift, i.e., solutions $\varphi$ of the extremal problem \[ \sup \{ |{f^{(n)}}(0)|/{\left \| f \right \|_D}:f \in \mathcal {M},f \ne 0\} \]. Here $n$ is the smallest nonnegative integer such that the sup is positive. It is known that such a function $\varphi$ generates $\mathcal {M}$. We show that the derivative $(z\varphi )\prime$ has a pseudocontinuation to the exterior disc. This pseudocontinuation is an analytic continuation exactly near those points of the unit circle where $\varphi$ is bounded away from zero. We also show that the radial limit of the absolute value of an extremal function exists at every point of the unit circle. Some of our results are valid for all functions that are orthogonal to a nonzero invariant subspace.

Radial limits of superharmonic functions in the plane

Radial limits of superharmonic functions in the plane

Invariant subspaces of the Dirichlet shift and pseudocontinuations

In this paper we study extremal functions for invariant subspaces M \mathcal {M} of the Dirichlet shift, i.e., solutions φ \varphi of the extremal problem \[ sup { | f ( n ) ( 0 ) | / ‖ f ‖ D : f ∈ M , f ≠ 0 } \sup \{ |{f^{(n)}}(0)|/{\left \| f \right \|_D}:f \in \mathcal {M},f \ne 0\} \] . Here n n is the smallest nonnegative integer such that the sup is positive. It is known that such a function φ \varphi generates M \mathcal {M} . We show that the derivative ( z φ ) ′ (z\varphi )\prime has a pseudocontinuation to the exterior disc. This pseudocontinuation is an analytic continuation exactly near those points of the unit circle where φ \varphi is bounded away from zero. We also show that the radial limit of the absolute value of an extremal function exists at every point of the unit circle. Some of our results are valid for all functions that are orthogonal to a nonzero invariant subspace.

On the Minimal Invariant Subspaces of the Hyperbolic Composition Operator

The composition operator induced by a hyperbolic Möbius transform $\phi$ on the classical Hardy space ${H^2}$ is considered. It is known that the invariant subspace problem for Hilbert space operators is equivalent to the fact that all the minimal invariant subspaces of this operator are one- dimensional. In connection with that we try to decide by the properties of a given function $u$ in ${H^2}$ if the corresponding cyclic subspace is minimal or not. The main result is the following. If the radial limit of $u$ is continuously extendable at one of the fixed points of $\phi$ and its value at the point is nonzero, then the cyclic subspace generated by $u$ is minimal if and only if $u$ is constant.

On the Analytic Radon-Nikodym Property for Bounded Subsets in Banach Spaces

We define the analytic Radon–Nikodym property for bounded subsets in complex Banach spaces and show the equivalence between the analytic Radon–Nikodym property, convergence of a class of analytic (or alternatively, Hardy) martingales related to the subset and the existence of radial limits for a class of analytic functions related to the subset.

Is Proxima really in orbit about α Cen A/B?

Despite the widespread acceptance that Proxima is in a bound orbit about α Cen A/B, widely quoted kinematic data for these stars do not support such a conclusion. We set radial velocity limits for Proxima consistent with the bound-orbit hypothesis, and show that a reasonably precise radial velocity obtained by the ESO Coravel programme lies close to these limits. However, given the importance of Proxima for a number of astrophysical questions, more accurate kinematic data are still merited.

Radial limits and radial growth of cauchy-stieltjes transforms

This paper studies the radial limit and me radial growth of functions defined by where μ is a complex-valued Borel measure. The results depend on local properties of the measure μ. Various nontangential limits associated with f are shown to hold off exceptional sets whose s -capacity is zero.