scholarly journals SINGULAR INNER FUNCTION AND ANALYTIC BOUNDED POINT EVALUATION

Author(s):  
K. Al-Hami
1983 ◽  
Vol 6 (3) ◽  
pp. 459-466
Author(s):  
Claes Fernström

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.


1995 ◽  
Vol 347 (6) ◽  
pp. 2169
Author(s):  
R. M. Range ◽  
M. I. Stessin

1979 ◽  
Vol 2 (3) ◽  
pp. 415-426
Author(s):  
Edwin Wolf

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).


1983 ◽  
Vol 6 (2) ◽  
pp. 363-370
Author(s):  
Edwin Wolf

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).


Energies ◽  
2021 ◽  
Vol 14 (8) ◽  
pp. 2301
Author(s):  
Yun-Sung Cho ◽  
Yun-Hyuk Choi

This paper describes a methodology for implementing the state estimation and enhancing the accuracy in large-scale power systems that partially depend on variable renewable energy resources. To determine the actual states of electricity grids, including those of wind and solar power systems, the proposed state estimation method adopts a fast-decoupled weighted least square approach based on the architecture of application common database. Renewable energy modeling is considered on the basis of the point of data acquisition, the type of renewable energy, and the voltage level of the bus-connected renewable energy. Moreover, the proposed algorithm performs accurate bad data processing using inner and outer functions. The inner function is applied to the largest normalized residue method to process the bad data detection, identification and adjustment. While the outer function is analyzed whether the identified bad measurements exceed the condition of Kirchhoff’s current law. In addition, to decrease the topology and measurement errors associated with transformers, a connectivity model is proposed for transformers that use switching devices, and a transformer error processing technique is proposed using a simple heuristic method. To verify the performance of the proposed methodology, we performed comprehensive tests based on a modified IEEE 18-bus test system and a large-scale power system that utilizes renewable energy.


2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Dionysios Anninos ◽  
Beatrix Mühlmann

Abstract We explore the conjectured duality between a class of large N matrix integrals, known as multicritical matrix integrals (MMI), and the series (2m − 1, 2) of non-unitary minimal models on a fluctuating background. We match the critical exponents of the leading order planar expansion of MMI, to those of the continuum theory on an S2 topology. From the MMI perspective this is done both through a multi-vertex diagrammatic expansion, thereby revealing novel combinatorial expressions, as well as through a systematic saddle point evaluation of the matrix integral as a function of its parameters. From the continuum point of view the corresponding critical exponents are obtained upon computing the partition function in the presence of a given conformal primary. Further to this, we elaborate on a Hilbert space of the continuum theory, and the putative finiteness thereof, on both an S2 and a T2 topology using BRST cohomology considerations. Matrix integrals support this finiteness.


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