This is a companion paper of “Finite Euclidean graphs and Ramanujan graphs” by the same authors. Finite analogues of the Poincaré upper half plane, i.e., finite upper half plane graphs, were studied by many researchers, including Terras, Evans etc. Finally, it was proved by combining works of A. Weil, P. Deligne, R. Evans, H. Stark, N. Katz, W. Li and many others, that the finite upper half pla...

In this paper, we define the concept of Jacobi forms of half-integral weight using Takase’s automorphic factor of weight 1/2 for a two-fold covering group of the symplectic group on the Siegel upper half plane and find covariant maps for the SchrödingerWeil representation. Using these covariant maps, we construct Jacobi forms of half integral weight with respect to an arithmetic subgroup of the...

Introduction Let Γ be a fuchsian subgroup of P SL(2, R). In this paper we consider the Γ-equivariant form of the Berezin's quantization of the upper half plane which will correspond to a deformation quantization of the (singular) space H/Γ. Our main result is that the von Neumann algebra associated to the Γ− equivariant form of the quantization is stable isomorphic with the von Neumann algebra ...

and Applied Analysis 3 Let β > 0. The weighted-type space or growth space on the upper half-planeA∞ β Π consists of all f ∈ H Π such that ∥ ∥f ∥ ∥ A∞ β Π sup z∈Π Iz β ∣ ∣f z ∣ ∣ < ∞. 1.7 It is easy to check thatA∞ β Π is a Banach space with the norm defined above. For weightedtype spaces on the unit disk, polydisk, or the unit ball see, for example, papers 10, 32, 33 and the references therein....

For the finite field IFq of q elements (q odd) and a quadratic nonresidue α ∈ IFq, we define the distance function δ ( u+ v √ α, x+ y √ α ) = (u− x)2 − α(v − y)2 vy on the upper half plane Hq = {x + y √ α | x ∈ IFq, y ∈ IFq} ⊆ IFq2 . For two sets E ,F ⊂ Hq with #E = E, #F = F and a non-trivial additive character ψ on IFq, we give the following estimate ∣∣∣∣∣ ∑ w∈E,z∈F ψ(δ(w, z)) ∣∣∣∣∣ ≤ min {√ ...

Abstract Schramm-Loewner Evolution describes the scaling limits of interfaces in certain statistical mechanical systems. These interfaces are geometric objects that are not equipped with a canonical parametrization. The standard parametrization of SLE is via half-plane capacity, which is a conformal measure of the size of a set in the reference upper half-plane. This has useful harmonic and com...

Mathematical morphology is a nonlinear image processing methodology based on the application of complete lattice theory to spatial structures. Let us consider an image model where at each pixel is given a univariate Gaussian distribution. This model is interesting to represent for each pixel the measured mean intensity as well as the variance (or uncertainty) for such measurement. The aim of th...

and Applied Analysis 3 In two main theorems in 20 , the authors proved the following results, which we now incorporate in the next theorem. Theorem A. Assume p ≥ 1 and φ is a holomorphic self-map of Π . Then the following statements true hold. a The operator Cφ : H Π → A∞ Π is bounded if and only if sup z∈Π Im z ( Imφ z )1/p < ∞. 1.8 b The operator Cφ : H Π → B∞ Π is bounded if and only if sup ...