On a Question of Seidel Concerning Holomorphic Functions Bounded on a Spiral

1969 ◽  
Vol 21 ◽  
pp. 1255-1262 ◽  
Author(s):  
K. F. Barth ◽  
W. J. Schneider
Keyword(s):  
Oral Communication ◽  
Holomorphic Functions ◽  
Asymptotic Value ◽  
Asymptotic Values

Let S be a spiral contained in D = {|z| < 1} such that S tends to C = {|z| = 1}. For the sake of brevity, by “f is bounded on S” we shall mean that f is holomorphic in D, unbounded, and bounded on S. The existence of such functions was first discussed by Valiron (9; 10); see also (1; 3; 8). Valiron also proved that any function that is “bounded on a spiral” must have the asymptotic value ∞ (10, p. 432). Functions that are bounded on a spiral may also have finite asymptotic values (1, p. 1254). In view of the above, Seidel has raised the question (oral communication): “Does there exist a function bounded on a spiral that has only the asymptotic value ∞?”. The following theorem answers this question affirmatively.

scholarly journals Time Evolution of Initial Errors in Lorenz’s 05 Chaotic Model

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Hynek Bednář ◽  
Aleš Raidl ◽  
Jiří Mikšovský
Keyword(s):  
Weather Prediction ◽  
Prediction Method ◽  
Atmospheric Model ◽  
Ensemble Prediction ◽  
Asymptotic Value ◽  
Time Limits ◽  
Initial Errors ◽  
Asymptotic Values ◽  
Chaotic Model ◽  
Almost All

Initial errors in weather prediction grow in time and, as they become larger, their growth slows down and then stops at an asymptotic value. Time of reaching this saturation point represents the limit of predictability. This paper studies the asymptotic values and time limits in a chaotic atmospheric model for five initial errors, using ensemble prediction method (model’s data) as well as error approximation by quadratic and logarithmic hypothesis and their modifications. We show that modified hypotheses approximate the model’s time limits better, but not without serious disadvantages. We demonstrate how hypotheses can be further improved to achieve better match of time limits with the model. We also show that quadratic hypothesis approximates the model’s asymptotic value best and that, after improvement, it also approximates the model’s time limits better for almost all initial errors and time lengths.

Boundary Behavior and Quasi-Normality of Finitely Valent Holomorphic Functions

1973 ◽  
Vol 25 (4) ◽  
pp. 812-819
Author(s):  
David C. Haddad
Keyword(s):  
Complex Number ◽  
Unit Disc ◽  
Boundary Behavior ◽  
Dense Subset ◽  
Holomorphic Functions ◽  
Asymptotic Values

A function denned in a domain D is n-valent in D if f(z) — w0 has at most n zeros in D for each complex number w0. Let denote the class of nonconstant, holomorphic functions f in the unit disc that are n-valent in each component of the set . MacLane's class is the class of nonconstant, holomorphic functions in the unit disc that have asymptotic values at a dense subset of |z| = 1.

Lignin and holocellulose relations during long-term decomposition of some forest litters. Long-term decomposition in a Scots pine forest. IV

1984 ◽  
Vol 62 (12) ◽  
pp. 2540-2550 ◽  
Author(s):  
Bjorn Berg ◽  
Gunnar Ekbohm ◽  
Charles McClaugherty
Keyword(s):  
United States ◽  
Scots Pine ◽  
Field Experiments ◽  
The United States ◽  
Pine Forest ◽  
Chemical Components ◽  
Asymptotic Value ◽  
Asymptotic Values ◽  
Scots Pine Forest

We investigated the relative changes in celluloses and lignin during decomposition of leaf and needle litters and wood in field experiments. The litter came from two different forest systems: one in the United States and one in Sweden. We found that the fraction of holocellulose in the total lignocellulose (Q) during decomposition approached an asymptotic value at which the disappearance of both the chemical components proceeded at the same rate. Different litter types approached different asymptotic values of Q. Possible implications of the finding are discussed.

Some results of Lindelöf type involving the segmental behaviour of holomorphic functions

1993 ◽  
Vol 114 (1) ◽  
pp. 57-65 ◽  
Author(s):  
Francisca Bravo ◽  
Daniel Girela
Keyword(s):  
Analytic Function ◽  
Unit Disc ◽  
Slow Growth ◽  
Holomorphic Functions ◽  
Maximum Modulus ◽  
Classical Theorem ◽  
Asymptotic Value

AbstractA classical theorem of Lindelöf asserts that if ƒ is a function analytic and bounded in the unit disc δ which has the asymptotic value L at a point ξ ε ∂ δ then it has the non-tangential limit L at ξ. This result does not remain true for functions f analytic in δ whose maximum modulus grows to infinity arbitrarily slowly. However, the second author has recently obtained some results of Lindelöf type valid for these functions. In this paper we obtain new results of this kind. We prove that if f is an analytic function of slow growth in δ and ξ ε ∂ δ, then certain restrictions on the growth of ƒ′ along a segment which ends at ξ do imply that ƒ has a non-tangential limit at ξ.

Note on Carnap's relational asymptotic relative frequencies

1958 ◽  
Vol 23 (3) ◽  
pp. 257-260 ◽  
Author(s):  
Frank Harary
Keyword(s):  
Equivalence Relation ◽  
Relative Frequency ◽  
Isomorphism Class ◽  
Binary Relations ◽  
Asymptotic Results ◽  
Asymptotic Value ◽  
Unpublished Work ◽  
Asymptotic Values ◽  
Special Cases ◽  
Linear Graphs

A (binary) relation is a collection of ordered couples. Two relations are isomorphic if there is a 1–1 correspondence between their fields which preserves the ordered couples. Isomorphism between relations is itself an equivalence relation, and a structure is an isomorphism class of binary relations.Carnap ([1], p. 124) asks certain questions concerning (both the exact and) the asymptotic value of the relative frequency that a relation on p objects satisfies certain properties. Among the most interesting special cases are the asymptotic value of the relative frequency that a binary relation on p objects be (a) symmetric, (b) reflexive, (c) transitive irreflexive anti-symmetric, and (d) symmetric irreflexive. These results already appear either in, or almost in, the literature, in various disguises. The object of this expository note is to bring them to light, especially since some of the references may not be generally known to logicians.Carnap ([1], p. 124) points out explicitly the correspondence between binary relations and linear graphs. Davis [2] and Harary [4] obtains precise results for the number of structures with certain properties using the language of relations and graphs respectively; but these do not supply the asymptotic values directly. The asymptotic results which are most useful here are either contained in, or are straight-forward extensions of, the formulas presented in Ford and Uhlenbeck [3], which in turn are based on unpublished work of G. Pólya. In addition, we utilize a result of L. Moser on transitive relations, which appears in Wine and Freund [5].

Integrals and derivatives of functions in MacLane's class and of normal functions

1972 ◽  
Vol 71 (1) ◽  
pp. 111-121 ◽  
Author(s):  
K. F. Barth ◽  
W. J. Schneider
Keyword(s):  
Holomorphic Function ◽  
Dense Subset ◽  
Asymptotic Value ◽  
Normal Functions ◽  
Asymptotic Values ◽  
Derivatives Of

If f is meromorphic in D = {,|z| < 1}, we say that f has the asymptotic value a at ζ, ζ ∈ C(={|z| = 1}), if there exists an arc Γ such that Γ ends at ζ, (Γ − {ζ}) ⊂ D, and f has the limit a as |z| → 1 on Γ. The class (M) originally defined by G. R. MacLane ((8), p. 8), consists of those functions that are non-constant and holomorphic (meromorphic) in D and that have asymptotic values at a dense subset of C. In (8), p. 36, MacLane has shown the following three conditions are sufficient for a non-constant holomorphic function f to be in :

Sign in / Sign up

Export Citation Format

Share Document