Random circles in the d-dimensional unit ball

1989 ◽  
Vol 26 (2) ◽  
pp. 408-412 ◽  
Author(s):  
Fernando Affentranger
Keyword(s):  
Euclidean Space ◽  
Unit Ball ◽  
Dimensional Unit

This note gives the solution of the following problem concerning geometric probabilities. What is the probability p(Bd; 2) that the circumference determined by three points P, P1 and P2 chosen independently and uniformly at random in the interior of a d-dimensional unit ball Bd in Euclidean space Ed (d ≧ 2) is entirely contained in Bd? From our result we conclude that p(Bd; 2) →π /(3√3) as d →∞.

Random circles in the d-dimensional unit ball

1989 ◽  
Vol 26 (02) ◽  
pp. 408-412 ◽  
Author(s):  
Fernando Affentranger
Keyword(s):  
Euclidean Space ◽  
Unit Ball ◽  
Dimensional Unit

This note gives the solution of the following problem concerning geometric probabilities. What is the probabilityp(Bd; 2) that the circumference determined by three pointsP, P1andP2chosen independently and uniformly at random in the interior of ad-dimensional unit ballBdin Euclidean spaceEd(d≧ 2) is entirely contained inBd? From our result we conclude thatp(Bd; 2) →π /(3√3) asd→∞.

scholarly journals The group of isometries on Hardy spaces of the n-ball and the polydisc

1980 ◽  
Vol 21 (2) ◽  
pp. 199-204 ◽  
Author(s):  
Earl Berkson ◽  
Horacio Porta
Keyword(s):  
Euclidean Space ◽  
Unit Ball ◽  
Complex Plane ◽  
Hardy Spaces ◽  
Inner Product ◽  
Euclidean Norm ◽  
Dimensional Euclidean Space ◽  
Open Unit ◽  
Open Unit Ball ◽  
Group Of Isometries

Let C be the complex plane, and U the disc |Z| < 1 in C. Cn denotes complex n-dimensional Euclidean space, <, > the inner product, and | · | the Euclidean norm in Cn;. Bn will be the open unit ball {z ∈ Cn:|z| < 1}, and Un will be the unit polydisc in Cn. For l ≤ p < ∞, p ≠ 2, Gp(Bn) (resp., Gp(Un)) will denote the group of all isometries of Hp(Bn) (resp., Hp(Un)) onto itself, where Hp(Bn) and HP(Un) are the usual Hardy spaces.

Holomorphic Mappings of the Hyperbolic Space into the Complex Euclidean Space and the Bloch Theorem

1975 ◽  
Vol 27 (2) ◽  
pp. 446-458 ◽  
Author(s):  
Kyong T. Hahn
Keyword(s):  
Euclidean Space ◽  
Unit Ball ◽  
Hyperbolic Space ◽  
Holomorphic Mapping ◽  
Continuous Functions ◽  
Holomorphic Mappings ◽  
Euclidean Metric ◽  
The Family ◽  
Complex Euclidean Space ◽  
Bounded Holomorphic

This paper is to study various properties of holomorphic mappings defined on the unit ball B in the complex euclidean space Cn with ranges in the space Cm. Furnishing B with the standard invariant Kähler metric and Cm with the ordinary euclidean metric, we define, for each holomorphic mapping f : B → Cm, a pair of non-negative continuous functions qf and Qf on B ; see § 2 for the definition.Let (Ω), Ω > 0, be the family of holomorphic mappings f : B → Cn such that Qf(z) ≦ Ω for all z ∈ B. (Ω) contains the family (M) of bounded holomorphic mappings as a proper subfamily for a suitable M > 0.

scholarly journals Algebras of Toeplitz Operators on the n-Dimensional Unit Ball

2018 ◽  
Vol 13 (2) ◽  
pp. 493-524 ◽  
Author(s):  
Wolfram Bauer ◽  
Raffael Hagger ◽  
Nikolai Vasilevski
Keyword(s):  
Unit Ball ◽  
Toeplitz Operators ◽  
Algebras Of Toeplitz Operators ◽  
Dimensional Unit

A KNOTTED MINIMAL TREE

1999 ◽  
Vol 01 (01) ◽  
pp. 71-86
Author(s):  
KRYSTYNA KUPERBERG
Keyword(s):  
Unit Ball ◽  
Three Dimensional ◽  
Minimal Tree ◽  
Dimensional Unit ◽  
Finite Set ◽  
Set Of Points

There is a finite set of points on the boundary of the three-dimensional unit ball whose minimal tree is knotted. This example answers a problem posed by Michael Freedman.

scholarly journals Clifford algebra-valued orthogonal polynomials in the open unit ball of Euclidean space

2004 ◽  
Vol 2004 (52) ◽  
pp. 2761-2772 ◽  
Author(s):  
Fred Brackx ◽  
Nele De Schepper ◽  
Frank Sommen
Keyword(s):  
Euclidean Space ◽  
Orthogonal Polynomials ◽  
Clifford Algebra ◽  
Unit Ball ◽  
Real Axis ◽  
Jacobi Polynomials ◽  
Orthogonality Relation ◽  
Weight Functions ◽  
Open Unit ◽  
Open Unit Ball

A new method for constructing Clifford algebra-valued orthogonal polynomials in the open unit ball of Euclidean space is presented. In earlier research, we only dealt with scalar-valued weight functions. Now the class of weight functions involved is enlarged to encompass Clifford algebra-valued functions. The method consists in transforming the orthogonality relation on the open unit ball into an orthogonality relation on the real axis by means of the so-called Clifford-Heaviside functions. Consequently, appropriate orthogonal polynomials on the real axis give rise to Clifford algebra-valued orthogonal polynomials in the unit ball. Three specific examples of such orthogonal polynomials in the unit ball are discussed, namely, the generalized Clifford-Jacobi polynomials, the generalized Clifford-Gegenbauer polynomials, and the shifted Clifford-Jacobi polynomials.

scholarly journals The Closest Packing of Spherical Caps in n Dimensions

1955 ◽  
Vol 2 (3) ◽  
pp. 139-144 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Euclidean Space ◽  
Unit Sphere ◽  
Closest Packing ◽  
Dimensional Unit ◽  
Spherical Caps ◽  
Image Position

Let Sn denote the “surface” of an n-dimensional unit sphere in Euclidean space of n dimensions. We may suppose that the sphere is centred at the origin of coordinates O, so that the points P(x1, x2, …, xn) of Sn satisfyWe suppose that n≥2.

Conformal Metrics on the Unit Ball in Euclidean Space

1998 ◽  
Vol 77 (3) ◽  
pp. 635-664 ◽  
Author(s):  
M Bonk ◽  
P Koskela ◽  
S Rohde
Keyword(s):  
Euclidean Space ◽  
Unit Ball ◽  
Conformal Metrics

scholarly journals The group of isometries on Hardy spaces of the n-ball and the polydisc

1980 ◽  
Vol 21 (1) ◽  
pp. 199-204
Author(s):  
Earl Berkson ◽  
Horacio Porta
Keyword(s):  
Euclidean Space ◽  
Unit Ball ◽  
Complex Plane ◽  
Hardy Spaces ◽  
Inner Product ◽  
Euclidean Norm ◽  
Dimensional Euclidean Space ◽  
Open Unit ◽  
Open Unit Ball ◽  
Group Of Isometries

Let C be the complex plane, and U the disc |z| < 1 in C. Cn denotes complex n-dimensional Euclidean space, <, > the inner product, and | · | the Euclidean norm in Cn. Bn will be the open unit ball {z ∈ Cn: |z| < 1}, and Un will be the unit polydisc in Cn. For 1 ≤p<∞, p≠2, Gp(Bn) (resp., Gp (Un)) will denote the group of all isometries of Hp (Bn) (resp., Hp (Un)) onto itself, where Hp (Bn) and Hp (Un) are the usual Hardy spaces.

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