scholarly journals Existence of Dirichlet infinite harmonic measures on the unit disc

1995 ◽  
Vol 138 ◽  
pp. 141-167
Author(s):  
Mitsuru Nakai
Keyword(s):  
Euclidean Space ◽  
Harmonic Measure ◽  
Unit Disc ◽  
Affirmative Answer ◽  
Dimensional Euclidean Space ◽  
Dirichlet Integral ◽  
Harmonic Measures

The primary purpose of this paper is to give an affirmative answer to a problem posed by Ohtsuka [13] whether there exists a p-harmonic measure on the unit disc in the 2-dimensional Euclidean space R2 with an infinite p-Dirichlet integral for the exponent 1 < p < 2.

Interpolation by Linear Sums of Harmonic Measures

1975 ◽  
Vol 18 (2) ◽  
pp. 249-253
Author(s):  
Marvin Ortel
Keyword(s):  
Unit Circle ◽  
Harmonic Measure ◽  
Unit Disc ◽  
Boundary Values ◽  
Harmonic Measures ◽  
Image Position ◽  
Interior Points

Let α be an open arc on the unit circleand for z=reiθ, 0 ≤ r < 1, let(1)The function ω(z; α) is called the harmonic measure of the arc α with respect to the unit disc, (Nevanlinna 2); it is harmonic and bounded in the unit disc and possesses (Fatou) boundary values 1 and 0 at interior points of α and the complementary arc β respectively.

On A Generalization of the Dirichlet Integral

2002 ◽  
Vol 9 (3) ◽  
pp. 449-459
Author(s):  
T. Chantladze ◽  
N. Kandelaki ◽  
A. Lomtatidze ◽  
D. Ugulava
Keyword(s):  
Euclidean Space ◽  
Bounded Domain ◽  
Fixed Number ◽  
Dimensional Euclidean Space ◽  
Spline Functions ◽  
Boundary Values ◽  
Dirichlet Integral ◽  
Generalized Dirichlet

Abstract Using the theory of spline functions, we investigate the problem of minimization of a generalized Dirichlet integral where Ω is a bounded domain of an 𝑛-dimensional Euclidean space 𝑅𝑛, λ ≥ 0 is a fixed number, and 𝑢𝑥𝒊 is a generalized according to Sobolev with respect to 𝑥𝒊 derivative of the function 𝑢 defined on Ω. Minimization is realized with respect to the functions 𝑢 whose boundary values on Γ form a preassigned function, and for them 𝐹λ(𝑢) is finite.

On Vitali's Theorem for Groups of Motions of Euclidean Space

1999 ◽  
Vol 6 (4) ◽  
pp. 323-334
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Finite Dimensional

Abstract We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

scholarly journals A note on n-harmonic majorants

1986 ◽  
Vol 34 (3) ◽  
pp. 461-472
Author(s):  
Hong Oh Kim ◽  
Chang Ock Lee
Keyword(s):  
Harmonic Function ◽  
Complex Plane ◽  
Unit Disc ◽  
Dirichlet Integral ◽  
Harmonic Majorant ◽  
Harmonic Majorants

Suppose D (υ) is the Dirichlet integral of a function υ defined on the unit disc U in the complex plane. It is well known that if υ is a harmonic function in U with D (υ) < ∞, then for each p, 0 < p < ∞, |υ|p has a harmonic majorant in U.We define the “iterated” Dirichlet integral Dn (υ) for a function υ on the polydisc Un of Cn and prove the polydisc version of the well known fact above:If υ is an n-harmonic function in Un with Dn (υ) < ∞, then for each p, 0 < p < ∞, |υ|p has an n-harmonic majorant in Un.

Some properties of Wigner coefficients and hyperspherical harmonics

1956 ◽  
Vol 52 (3) ◽  
pp. 424-430 ◽  
Author(s):  
A. P. Stone
Keyword(s):  
Angular Momentum ◽  
Euclidean Space ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Hyperspherical Harmonics ◽  
Shift Operators ◽  
Analytical Relations

ABSTRACTGeneral shift operators for angular momentum are obtained and applied to find closed expressions for some Wigner coefficients occurring in a transformation between two equivalent representations of the four-dimensional rotation group. The transformation gives rise to analytical relations between hyperspherical harmonics in a four-dimensional Euclidean space.

On inequalities of the Tchebychev type

1963 ◽  
Vol 59 (1) ◽  
pp. 135-146 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Probability Theory ◽  
Euclidean Space ◽  
Random Variable ◽  
Dimensional Euclidean Space ◽  
Real Random Variable ◽  
Finite Dimensional

1. A type of problem which frequently occurs in probability theory and statistics can be formulated in the following way. We are given real-valued functions f(x), gi(x) (i = 1, 2, …, k) on a space (typically finite-dimensional Euclidean space). Then the problem is to set bounds for Ef(X), where X is a random variable taking values in , about which all we know is the values of Egi(X). For example, we might wish to set bounds for P(X > a), where X is a real random variable with some of its moments given.

Curve following in n—dimensional Euclidean space

SIMULATION ◽  
1973 ◽  
Vol 21 (5) ◽  
pp. 145-149 ◽  
Author(s):  
John Rees Jones
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space

scholarly journals Homology of moduli spaces of linkages in high-dimensional Euclidean space

2013 ◽  
Vol 13 (2) ◽  
pp. 1183-1224 ◽  
Author(s):  
Dirk Schütz
Keyword(s):  
Euclidean Space ◽  
Moduli Spaces ◽  
High Dimensional ◽  
Dimensional Euclidean Space

scholarly journals On the Volume of the Zero Cell of a Class of Isotropic Poisson Hyperplane Tessellations

2014 ◽  
Vol 46 (3) ◽  
pp. 622-642 ◽  
Author(s):  
Julia Hörrmann ◽  
Daniel Hug
Keyword(s):  
Euclidean Space ◽  
Asymptotic Behaviour ◽  
Explicit Formula ◽  
Dimensional Euclidean Space ◽  
Arbitrary Dimensions ◽  
Zero Cell ◽  
Parametric Class

We study a parametric class of isotropic but not necessarily stationary Poisson hyperplane tessellations in n-dimensional Euclidean space. Our focus is on the volume of the zero cell, i.e. the cell containing the origin. As a main result, we obtain an explicit formula for the variance of the volume of the zero cell in arbitrary dimensions. From this formula we deduce the asymptotic behaviour of the volume of the zero cell as the dimension goes to ∞.

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