Asymptotic behaviour of pth means of analytic and subharmonic functions in the unit disc and angular distribution of zeros

2020 ◽  
Vol 236 (2) ◽  
pp. 931-957
Author(s):  
Igor E. Chyzhykov
Keyword(s):  
Angular Distribution ◽  
Asymptotic Behaviour ◽  
Unit Disc ◽  
Distribution Of Zeros ◽  
Subharmonic Functions

scholarly journals Riesz-Martin representation for positive super-polyharmonic functions in A Riemannian manifold

2006 ◽  
Vol 2006 ◽  
pp. 1-9
Author(s):  
V. Anandam ◽  
S. I. Othman
Keyword(s):  
Riemannian Manifold ◽  
Complex Plane ◽  
Unit Disc ◽  
Biharmonic Function ◽  
Subharmonic Functions ◽  
Polyharmonic Functions ◽  
Superharmonic Functions ◽  
Martin Representation

Letube a super-biharmonic function, that is,Δ2u≥0, on the unit discDin the complex plane, satisfying certain conditions. Then it has been shown thatuhas a representation analogous to the Poisson-Jensen representation for subharmonic functions onD. In the same vein, it is shown here that a functionuon any Green domainΩin a Riemannian manifold satisfying the conditions(−Δ)iu≥0for0≤i≤mhas a representation analogous to the Riesz-Martin representation for positive superharmonic functions onΩ.

scholarly journals The exact spectral asymptotic of the logarithmic potential on harmonic function space

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1701-1714
Author(s):  
Djordjije Vujadinovic
Keyword(s):  
Harmonic Function ◽  
Asymptotic Behaviour ◽  
Function Space ◽  
Unit Disc ◽  
Logarithmic Potential ◽  
Bergman Projection ◽  
Potential Type

In this paper we consider the product of the harmonic Bergman projection Ph: L2(D) ? L2h (D) and the operator of logarithmic potential type defined by L f(z)=- 1/2? ?D ln |z-?|f(?)dA(?), where D is the unit disc in C. We describe the asymptotic behaviour of the eigenvalues of the operator (PhL)+(PhL). More precisely, we prove that limn?+? n2sn(PhL) = ?4?2/3-1.

Cercles De Remplissage and Asymptotic Behaviour along Circuitous Paths

1970 ◽  
Vol 22 (2) ◽  
pp. 389-393 ◽  
Author(s):  
P. M. Gauthier
Keyword(s):  
Meromorphic Function ◽  
Asymptotic Behaviour ◽  
Metric Space ◽  
Unit Disc ◽  
Riemann Sphere ◽  
Holomorphic Functions ◽  
Weak Form ◽  
Value Distribution ◽  
Hyperbolic Metric ◽  
Chordal Metric

In this paper we consider the value distribution of a meromorphic function whose behaviour is prescribed along a spiral. The existence of extremely wild holomorphic functions is established. Indeed a very weak form of one of our results would be that there are holomorphic functions (in the unit disc or the plane) for which every curve “tending to the boundary” is a Julia curve.The theorems in this paper generalize results of Gavrilov [7], Lange [9], and Seidel [11].I wish to express my thanks to Professor W. Seidel for his guidance and encouragement.2. Preliminaries. For the most part we will be dealing with the metric space (D, ρ) where D is the unit disc, |z| < 1, and ρ is the non-Euclidean hyperbolic metric on D. The chordal metric on the Riemann sphere will be denoted by x.

scholarly journals Complex asymptotics of Poincaré functions and properties of Julia sets

2008 ◽  
Vol 145 (3) ◽  
pp. 699-718 ◽  
Author(s):  
GREGORY DERFEL ◽  
PETER J. GRABNER ◽  
FRITZ VOGL
Keyword(s):  
Functional Equation ◽  
Asymptotic Expansion ◽  
Periodic Function ◽  
Asymptotic Behaviour ◽  
Harmonic Measure ◽  
Julia Set ◽  
Julia Sets ◽  
Real Polynomial ◽  
Distribution Of Zeros ◽  
Geometric Properties

AbstractThe asymptotic behaviour of the solutions of Poincaré's functional equation f(λz) = p(f(z)) (λ > 1) for p a real polynomial of degree ≥ 2 is studied in angular regions W of the complex plain. It is known [9, 10] that f(z) ~ exp(zρF(logλz)), if f(z) → ∞ for z ∞ and z ∈ W, where F denotes a periodic function of period 1 and ρ = logλ deg(p). In this paper we refine this result and derive a full asymptotic expansion. The constancy of the periodic function F is characterised in terms of geometric properties of the Julia set of p. For real Julia sets we give inequalities for multipliers of Pommerenke-Levin-Yoccoz type. The distribution of zeros of f is related to the harmonic measure on the Julia set of p.

Electron Scattering in Solids

1990 ◽  
Vol 48 (2) ◽  
pp. 4-5
Author(s):  
Ryuichi Shimizu ◽  
Ze-Jun Ding
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Angular Distribution ◽  
Elastic Scattering ◽  
Electron Scattering ◽  
Cross Sections ◽  
Expansion Method ◽  
Electron Excitation ◽  
Partial Wave Expansion ◽  
Backscattered Electrons

Monte Carlo simulation has been becoming most powerful tool to describe the electron scattering in solids, leading to more comprehensive understanding of the complicated mechanism of generation of various types of signals for microbeam analysis.The present paper proposes a practical model for the Monte Carlo simulation of scattering processes of a penetrating electron and the generation of the slow secondaries in solids. The model is based on the combined use of Gryzinski’s inner-shell electron excitation function and the dielectric function for taking into account the valence electron contribution in inelastic scattering processes, while the cross-sections derived by partial wave expansion method are used for describing elastic scattering processes. An improvement of the use of this elastic scattering cross-section can be seen in the success to describe the anisotropy of angular distribution of elastically backscattered electrons from Au in low energy region, shown in Fig.l. Fig.l(a) shows the elastic cross-sections of 600 eV electron for single Au-atom, clearly indicating that the angular distribution is no more smooth as expected from Rutherford scattering formula, but has the socalled lobes appearing at the large scattering angle.

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