Limit Theorems for the Compositions of Distributions in the Lobachevsky Plane and Space

Karpelevicii, F. I.; Tutubalin, V. N.

doi:10.1137/1104039

Cited by 52 publications

(25 citation statements)

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“…This procedure can be generalized to the space H + n . As λ, c → ∞ with c 2 /λ → 1, the vector process in (7.1) converges to X(t) = sinh B 1 (t), Y (t) = sinh B 2 (t), Z(t) = e B 3 (t) , The value of the normalizing constant, C = 1/2 √ π , in (7.4) is evaluated below (and slightly differs from that appearing in Karpelevich et al (1959) (η, t) = 1 2 √ π g(η, t), which coincides with (7.4).…”

Section: X(t) = Sinh U (T) Y (T) = Sinh V (T) Z(t) = E T (T)mentioning

confidence: 99%

Random motions at finite velocity in a non-Euclidean space

Orsingher

Gregorio

2007

Adv. Appl. Probab.

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In this paper telegraph processes on geodesic lines of the Poincaré half-space and Poincaré disk are introduced and the behavior of their hyperbolic distances examined. Explicit distributions of the processes are obtained and the related governing equations derived. By means of the processes on geodesic lines, planar random motions (with independent components) in the Poincaré half-space and disk are defined and their hyperbolic random distances studied. The limiting case of one-dimensional and planar motions together with their hyperbolic distances is discussed with the aim of establishing connections with the well-known stochastic representations of hyperbolic Brownian motion. Extensions of motions with finite velocity to the three-dimensional space are also hinted at, in the final section.

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Section: X(t) = Sinh U (T) Y (T) = Sinh V (T) Z(t) = E T (T)mentioning

confidence: 99%

Random motions at finite velocity in a non-Euclidean space

Orsingher

Gregorio

2007

Adv. Appl. Probab.

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“…(3.38). Formula (3.38) was obtained by Karpelevich et al [337], as well as the corresponding result for the three-dimensional analogue of H to be discussed in Volume II ( [667], where the fundamental solution of the heat equation on hyperbolic 3-space is considered).…”

Section: The Heat Equation On H Using Helgason's Transformmentioning

confidence: 81%

“…Then we shall discuss the central limit theorem on H, which was first proved by Karpelevich et al [337]. This can be used to study the power reflected by random inhomogeneities in a long transmission line, as was first noted by Gertsenshtein and Vasil'ev [208].…”

Section: Introductionmentioning

confidence: 79%

Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

Terras

2013

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Whenever there is a large earthquake the Earth vibrates for days afterwards. The vibrations consist of the superposition of the elastic-gravitational normal modes of the Earth that are excited by the earthquake.-From F. Gilbert [212, p. 107].A (surface or Laplace) spherical harmonic is an eigenfunction of the Laplacian on the sphere. These are the analogues of exponentials for Fourier analysis on the sphere. Laplace and Legendre introduced these functions in order to study gravitational theory in the 1780s. Spherical harmonics are necessary for the analysis of any phenomena with spherical symmetry; e.g., earthquakes, the hydrogen atom, and the solar corona. Some of these topics will be discussed later in this section.Since our treatment of harmonic analysis on the sphere is rather condensed, the reader may want to consult some of the following references for more information: Before discussing spherical harmonics, we need to understand something about the geometry of the sphere. This symmetric space is closely related to the orthogonal group O(n) of real n × n matrices U such that t UU = I, where t U denotes the transpose of U and I denotes the n × n identity matrix. The special orthogonal SO(n), is the subgroup of matrices U in O(n) such that the determinant of U is one. You can regard U in SO(n − 1) as an element of SO(n) by formingThe group O(n) is a compact group and the sphere is a compact symmetric space, making some of the analysis on it somewhat easier.Exercise 2.1.1 (The Sphere as a Quotient or Homogeneous Space). Consider the sphere S n−1 = {x ∈ R n | x = 1}. Show that S n−1 can be identified with the quotient space SO(n)/SO(n − 1), using the preceding identification of SO(n − 1) as a subgroup of SO(n). Hint.Map the coset gSO(n − 1) to the vector ge n for g in SO(n − 1) and e n = t (0,... ,0, 1).You may also want to read the discussion of the topology of spheres and orthogonal groups in Chevalley [85,. In particular, the fundamental (or Poincaré) group of SO(2) is isomorphic to Z, while that of SO(n), n ≥ 3, has order 2.From now on we shall consider only the sphere S 2 . See the references for the general case. Now the sphere S 2 is a differentiable manifold. This means that locally it looks like two-dimensional Euclidean space. To make this precise, we use the usual angular coordinates (θ , ϕ), 0 < ϕ < 2π, 0 < θ < π, pictured in Fig. 2.1, to parameterize S 2 except for the semi-circle C through the poles and (1, 0, 0). A similar coordinate patch can be constructed to cover the rest of the sphere. The equations for the rectangular coordinates (x, y, z) of a point on S 2 in terms of the angular coordinates are x = sin θ cos ϕ y = sin θ sin ϕ z = cos θ ⎫ ⎬ ⎭ (2.1)

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“…16 The notation usually used for Past(ξ) is J − (ξ). 17 Remember the set {q > 0} has two components.…”

Section: A Natural Boundary To Spacetimementioning

confidence: 99%