scholarly journals Linear measure on plane continua of finite linear measure

1989 ◽  
Vol 27 (1-2) ◽  
pp. 169-177
Author(s):  
H. Alexander
Keyword(s):  
Linear Measure ◽  
Finite Linear

scholarly journals A question of Gross and the uniqueness of entire functions

1995 ◽  
Vol 138 ◽  
pp. 169-177 ◽  
Author(s):  
Hong-Xun yi
Keyword(s):  
Entire Function ◽  
Nevanlinna Theory ◽  
Entire Functions ◽  
Linear Measure ◽  
Finite Linear

For any set S and any entire function f letwhere each zero of f — a with multiplicity m is repeated m times in Ef(S) (cf. [1]). It is assumed that the reader is familiar with the notations of the Nevanlinna Theory (see, for example, [2]). It will be convenient to let E denote any set of finite linear measure on 0 < r < ∞, not necessarily the same at each occurrence. We denote by S(r, f) any quantity satisfying .

scholarly journals NONDECREASING FUNCTIONS, EXCEPTIONAL SETS AND GENERALIZED BOREL LEMMAS

2010 ◽  
Vol 88 (3) ◽  
pp. 353-361
Author(s):  
R. G. HALBURD ◽  
R. J. KORHONEN
Keyword(s):  
Maximum Modulus ◽  
Point Of View ◽  
Modulus Function ◽  
Linear Measure ◽  
Real Analysis ◽  
Exceptional Set ◽  
Exceptional Sets ◽  
Finite Linear ◽  
Increasing Function ◽  
Nondecreasing Functions

AbstractAccording to the classical Borel lemma, any positive nondecreasing continuous function T satisfiesT(r+1/T(r))≤2T(r) outside a possible exceptional set of finite linear measure. This lemma plays an important role in the theory of entire and meromorphic functions, where the increasing function T is either the logarithm of the maximum modulus function, or the Nevanlinna characteristic. As a result, exceptional sets appear throughout Nevanlinna theory, in particular in Nevanlinna’s second main theorem. In this paper, we consider generalizations of Borel’s lemma. Conversely, we consider ways in which certain inequalities can be modified so as to remove exceptional sets. All results discussed are presented from the point of view of real analysis.

On continuous functions with graphs of σ-finite linear measure

1974 ◽  
Vol 76 (1) ◽  
pp. 33-43
Author(s):  
James Foran
Keyword(s):  
Bounded Variation ◽  
Finite Length ◽  
Continuous Functions ◽  
Linear Measure ◽  
Image Position ◽  
Approximate Differentiability ◽  
Finite Linear

In this paper comparisons are made between the class of continuous functions of generalized bounded variation and the class of continuous functions with graphs having σ-finite length i.e. linear measure. An investigationof the differentiability and approximate differentiability of such functions discloses the fact that the latter class is considerably more extensivethan the former one. The following definitions will be needed:(1) A function f is said to be of bounded variation (VB) on a set E ifwhere the supremum is taken over all sequences {[ai, bi]} of non-overlapping intervals with endpoints in E.

On the definition of tangents to sets of infinite linear measure

1956 ◽  
Vol 52 (1) ◽  
pp. 20-29 ◽  
Author(s):  
A. S. Besicovitch
Keyword(s):  
Linear Measure ◽  
Upper Density ◽  
Definition Of ◽  
Almost All ◽  
Finite Linear ◽  
Strong Claim ◽  
Infinite Linear ◽  
Angle Vertex

Given a plane set E of positive and finite linear measure, the tangent at a point x at which the upper density of E is positive (which is so at almost all points of E and is not so at almost all points outside E) is defined in the following way. The line l through x is said to be the tangent to the set at the point x if in any angle vertex x that leaves the line outside the density of E is zero. This definition when applied to sets of infinite linear measure leads often to the conclusion that no tangent exists in cases when the structure of the set singles out some lines that have strong claim to be tangents to the set.

Continua of Finite Linear Measure I

1943 ◽  
Vol 65 (1) ◽  
pp. 137 ◽  
Author(s):  
Samuel Eilenberg ◽  
O. G. Harrold
Keyword(s):  
Linear Measure ◽  
Finite Linear

Analysis of tangential properties of curves of infinite length

1957 ◽  
Vol 53 (1) ◽  
pp. 69-72 ◽  
Author(s):  
A. S. Besicovitch
Keyword(s):  
Measure Zero ◽  
Finite Measure ◽  
Infinite Length ◽  
Linear Measure ◽  
Regular Component ◽  
Infinite Measure ◽  
Almost All ◽  
Finite Linear ◽  
Tangent Set ◽  
Complementary Study

In a recent paper (1) I have studied tangential properties of sets of infinite measure. This note represents a complementary study of sets and, in particular of arcs, of σ-finite linear measure. Suppose we have a linearly measurable set E of infinite measure that can be represented as the sum of sets of finite measure. Write En = En, 1 + En, 2, where En, 1 is the set of regular points of En and En, 2 that of irregular ones. The set F = ΣEn, 1 is a regular component of E and the set G = ΣEn, 2 an irregular one. A tangent toEn exists at almost all points of En, 1 and at almost no points of En, 2. Denote by Tn the set of all tangents to En and by T the sum-set ΣTn. T will be called a tangent-set to E. Lines of T are defined corresponding to almost all points of F and to the points of a subset of G of measure zero.

Spaces of σ-finite linear measure

2013 ◽  
Vol 133 (2) ◽  
pp. 245-252
Author(s):  
Ihor Stasyuk ◽  
Edward D. Tymchatyn
Keyword(s):  
Linear Measure ◽  
Finite Linear
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