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1

Nonlinear Choquard equations on hyperbolic space

He Haiyang

Opuscula Mathematica

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2022

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Vol. 42, no. 5

691--708

EN

In this paper, our purpose is to prove the existence results for the following nonlinear Choquard equation [formula] on the hyperbolic space BN, where ΔBN denotes the Laplace-Beltrami operator on BN, [formula] λ is a real parameter, 0 < μ < N, 1 < p ≤ 2∗μ,N ≥ 3 and [formula] is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.

2

On Potential Theory of Hyperbolic Brownian Motion with Drift

Serafin Grzegorz

Probability and Mathematical Statistics

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2020

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Vol. 40, Fasc. 1

1--22

EN

Consider the λ-Green function and the λ-Poisson kernel of a Lipschitz domain U ⊂ Hn = {x ∈ Rn: xn > 0} for hyperbolic Brownian motion with drift. We provide several relationships that facilitate studying those objects and explain somewhat their nature. As an application, we yield uniform estimates for sets of the form Sa,b = {x ∈ Hn : xn > a, x1 ∈ (0, b)}, a, b > 0, which covers and extends existing results of that kind.

3

Ergodic properties of random infinite products of nonexpansive mappings

Reich S., Zaslavski A. J.

Journal of Mathematics and Applications

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2017

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Vol. 40

149--159

EN

In this paper we are concerned with the asymptotic behavior of random (unrestricted) infinite products of nonexpansive selfmappings of closed and convex subsets of a complete hyperbolic space. In contrast with our previous work in this direction, we no longer assume that these subsets are bounded. We first establish two theorems regarding the stability of the random weak ergodic property and then prove a related generic result. These results also extend our recent investigations regarding nonrandom infinite products.

4

De la Vallée Poussin Summability, the Combinatorial Sum [wzór] and the de la Vallée Poussin Means Expansion

Ali Z. S.

Journal of Mathematics and Applications

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2017

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Vol. 40

5--20

EN

In this paper we apply the de la Vallee Poussin sum to a combinatorial Chebyshev sum by Ziad S. Ali in [1]. One outcome of this consideration is the main lemma proving the following combinatorial identity: with Re(z) standing for the real part of z we have (wzór). Our main lemma will indicate in its proof that the hypergeometric factors 2F1(1, 1/2 + n; 1 + n; 4); and 2F1(1, 1/2 + 2n; 1 + 2n; 4) are complex, each having a real and imaginary part. As we apply the de la Vallee Poussin sum to the combinatorial Chebyshev sum generated in the Key lemma by Ziad S. Ali in [1], we see in the proof of the main lemma the extreme importance of the use of the main properties of the gamma function. This represents a second important consideration. A third new outcome are two interesting identities of the hypergeometric type with their new Meijer G function analogues. A fourth outcome is that by the use of the Cauchy integral formula for the derivatives we are able to give a dierent meaning to the sum: (wzór). A fifth outcome is that by the use of the Gauss-Kummer formula we are able to make better sense of the expressions (wzór) by making use of the series denition of the hypergeometric function. As we continue we notice a new close relation of the Key lemma, and the de la Vallee Poussin means. With this close relation we were able to talk about P the de la Vallee Poussin summability of the two innite series (wzór). Furthermore the application of the de la Vallee Poussin sum to the Key lemma has created two new expansions representing the following functions: (wzór).

5

Intersection of Generic Rotations in Some Classical Spaces

Nowak K., Tomkowicz G.

Bulletin of the Polish Academy of Sciences. Mathematics

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2016

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Vol. 64, no. 2/3

105--107

EN

Consider an o-minimal structure on the real field R and two definable subsets A, B of the Euclidean space Rn, of the unit sphere Sn or of the hyperbolic space Hn, n ≥ 2, which are of dimensions k, l ≤ n−1, respectively. We prove that the dimension of the intersection σ(A) ∩ B is less than min{k, l} for a generic rotation σ of the ambient space; here we set dim ∅ = −1.

6

Hitting hyperbolic half-space

Małecki J., Serafin G.

Demonstratio Mathematica

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2012

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Vol. 45, nr 2

337-360

EN

Let (…) be the n-dimensional hyperbolic Brownian motion with drift, that is a diffusion on the real hyperbolic space (…) having the Laplace-Beltrami operator with drift as its generator. We prove the reflection principle for (…) which enables us to study the process (…) killed when exiting the hyperbolic half space, that is the set (…). We provide formulae, uniform estimates and describe asymptotic behavior of the Green function and the Poisson kernel of D for the process (…). Finally, we derive formula for the (…) Poisson kernel of the set D.

7

Weingarten surfaces of revolution in 3-dimensional hyperbolic space

Yoon D.

Demonstratio Mathematica

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2005

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Vol. 38, nr 4

917--922

EN

The purpose of this paper is to construct a family of Weingarten surfaces of revolution satisfying the Weingarten relation K1= f(K2) in 3-dimensional hyperbolic space H3, where K1,K2 are principal curvatures and f is a some function.