scholarly journals Some remarks on the spaceR2(E)

1983 ◽  
Vol 6 (3) ◽  
pp. 459-466
Author(s):  
Claes Fernström
Keyword(s):  
Compact Subset ◽  
Complex Plane ◽  
Rational Functions ◽  
Compact Set ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Point Evaluation ◽  
Bounded Point Evaluation ◽  
Point Evaluations ◽  
Necessary And Sufficient

LetEbe a compact subset of the complex plane. We denote byR(E)the algebra consisting of the rational functions with poles offE. The closure ofR(E)inLp(E),1≤p<∞, is denoted byRp(E). In this paper we consider the casep=2. In section 2 we introduce the notion of weak bounded point evaluation of orderβand identify the existence of a weak bounded point evaluation of orderβ,β>1, as a necessary and sufficient condition forR2(E)≠L2(E). We also construct a compact setEsuch thatR2(E)has an isolated bounded point evaluation. In section 3 we examine the smoothness properties of functions inR2(E)at those points which admit bounded point evaluations.

scholarly journals Smoothness properties of functions inR2(x)at certain boundary points

1979 ◽  
Vol 2 (3) ◽  
pp. 415-426
Author(s):  
Edwin Wolf
Keyword(s):  
Lebesgue Measure ◽  
Compact Subset ◽  
Complex Plane ◽  
Rational Functions ◽  
Point Evaluation ◽  
Boundary Points ◽  
Bounded Point Evaluation ◽  
Dimensional Lebesgue Measure

LetXbe a compact subset of the complex planeℂ. We denote byR0(X)the algebra consisting of the (restrictions toXof) rational functions with poles offX. Letmdenote2-dimensional Lebesgue measure. Forp≥1, letRp(X)be the closure ofR0(X)inLp(X,dm).In this paper, we consider the casep=2. Letxϵ∂Xbe both a bounded point evaluation forR2(X)and the vertex of a sector contained inIntX. LetLbe a line which passes throughxand bisects the sector. For thoseyϵL∩Xthat are sufficiently nearxwe prove statements about|f(y)−f(x)|for allfϵR2(X).

scholarly journals Functions in the spaceR2(E)at boundary points of the interior

1983 ◽  
Vol 6 (2) ◽  
pp. 363-370
Author(s):  
Edwin Wolf
Keyword(s):  
Lebesgue Measure ◽  
Compact Subset ◽  
Complex Plane ◽  
Limit Point ◽  
Rational Functions ◽  
Point Evaluation ◽  
Boundary Points ◽  
Bounded Point Evaluation ◽  
Dimensional Lebesgue Measure

LetEbe a compact subset of the complex planeℂ. We denote byR(E)the algebra consisting of (the restrictions toEof) rational functions with poles offE. Letmdenote2-dimensional Lebesgue measure. Forp≥1, letRp(E)be the closure ofR(E)inLp(E,dm).In this paper we consider the casep=2. Letx ϵ ∂Ebe a bounded point evaluation forR2(E). Suppose there is aC>0such thatxis a limit point of the sets={y|y ϵ Int E,Dist(y,∂E)≥C|y−x|}. For thosey ϵ Ssufficiently nearxwe prove statements about|f(y)−f(x)|for allf ϵ R(E).

scholarly journals Algebraic independence properties of the Fredholm series

1978 ◽  
Vol 26 (1) ◽  
pp. 31-45 ◽  
Author(s):  
J. H. Loxton ◽  
A. J. van der Poorten
Keyword(s):  
Algebraic Independence ◽  
Rational Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Algebraic Numbers ◽  
Independence Properties ◽  
Image Position ◽  
Necessary And Sufficient

AbstractWe consider algebraic independence properties of series such as We show that the functions fr(z) are algebraically independent over the rational functions Further, if αrs (r = 2, 3, 4, hellip; s = 1, 2, 3, hellip) are algebraic numbers with 0 < |αrs|, we obtain an explicit necessary and sufficient condition for the algebraic independence of the numbers fr(αrs) over the rationals.

scholarly journals Proximinal subspaces ofA(K)of finite codimension

2003 ◽  
Vol 2003 (39) ◽  
pp. 2501-2505
Author(s):  
T. S. S. R. K. Rao
Keyword(s):  
Compact Subset ◽  
Convex Set ◽  
Compact Convex ◽  
Continuous Functions ◽  
Finite Codimension ◽  
Sufficient Condition ◽  
Choquet Simplex ◽  
Necessary And Sufficient Condition ◽  
Compact Convex Set ◽  
Necessary And Sufficient

We study an analogue of Garkavi's result on proximinal subspaces ofC(X)of finite codimension in the context of the spaceA(K)of affine continuous functions on a compact convex setK. We give an example to show that a simple-minded analogue of Garkavi's result fails for these spaces. WhenKis a metrizable Choquet simplex, we give a necessary and sufficient condition for a boundary measure to attain its norm onA(K). We also exhibit proximinal subspaces of finite codimension ofA(K)when the measures are supported on a compact subset of the extreme boundary.

On the residual Julia sets of rational functions

1997 ◽  
Vol 17 (1) ◽  
pp. 205-210 ◽  
Author(s):  
SHUNSUKE MOROSAWA
Keyword(s):  
Kleinian Group ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Functions ◽  
Sufficient Condition ◽  
Limit Set ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

We consider the subset of the Julia set called the residual Julia set, which comes from an analogy of the residual limit set of a Kleinian group. We give a necessary and sufficient condition in order that the residual Julia set is empty in the case of hyperbolic rational functions.

\(\mathcal{I}\)-convergence and \(\tau^{*}\)-closedness of \(\mathcal{I}\)-compact sets

2017 ◽  
Vol 8 (1) ◽  
pp. 78
Author(s):  
Navpreet Singh Noorie ◽  
Nitakshi Goyal
Keyword(s):  
Compact Set ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Sets ◽  
Accumulation Points ◽  
Necessary And Sufficient ◽  
The Relationship

We introduce the convergence and accumulation points of a filter with respect to an ideal and also give the relationship between them and with the usual convergence and accumulation points of a filter. We use these results to obtain necessary and sufficient condition for an \(\mathcal{I}\)-compact set to be \(\tau^{*}\)-closed in \(S_2\) and normal spaces. Finally the sufficient condition for an \(\mathcal{I}\)-compact set to be \(\tau^{*}\)-closed in \(S_2\) mod $\mathcal{I}$ spaces are obtained.

A Proposed Framework Emphasizing Auditor Reliability over Auditor Independence

2003 ◽  
Vol 17 (3) ◽  
pp. 257-266 ◽  
Author(s):  
Mark H. Taylor ◽  
F. Todd DeZoort ◽  
Edward Munn ◽  
Martha Wetterhall Thomas
Keyword(s):  
Public Interest ◽  
Auditor Independence ◽  
Public Confidence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accounting Profession ◽  
The Public ◽  
Necessary And Sufficient

This paper introduces an auditor reliability framework that repositions the role of auditor independence in the accounting profession. The framework is motivated in part by widespread confusion about independence and the auditing profession's continuing problems with managing independence and inspiring public confidence. We use philosophical, theoretical, and professional arguments to argue that the public interest will be best served by reprioritizing professional and ethical objectives to establish reliability in fact and appearance as the cornerstone of the profession, rather than relationship-based independence in fact and appearance. This revised framework requires three foundation elements to control subjectivity in auditors' judgments and decisions: independence, integrity, and expertise. Each element is a necessary but not sufficient condition for maximizing objectivity. Objectivity, in turn, is a necessary and sufficient condition for achieving and maintaining reliability in fact and appearance.

The Power of Public Positions

2018 ◽  
Author(s):  
Thomas Sinclair
Keyword(s):  
Detailed Analysis ◽  
Political Authority ◽  
The State ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
State Institutions ◽  
State Of Nature ◽  
Necessary And Sufficient

The Kantian account of political authority holds that the state is a necessary and sufficient condition of our freedom. We cannot be free outside the state, Kantians argue, because any attempt to have the “acquired rights” necessary for our freedom implicates us in objectionable relations of dependence on private judgment. Only in the state can this problem be overcome. But it is not clear how mere institutions could make the necessary difference, and contemporary Kantians have not offered compelling explanations. A detailed analysis is presented of the problems Kantians identify with the state of nature and the objections they face in claiming that the state overcomes them. A response is sketched on behalf of Kantians. The key idea is that under state institutions, a person can make claims of acquired right without presupposing that she is by nature exceptional in her capacity to bind others.

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