Directions in hyperbolic lattices

Abstract

Translation of the Bible or any other text unavoidably involves a determination about its meaning. There have been different views of meaning from ancient times up to the present, and a particularly Enlightenment and Modernist view is that the meaning of a text amounts to whatever the original author of the text intended it to be. This article analyzes the authorial-intent view of meaning in comparison with other models of literary and legal interpretation. Texts are anchors to interpretation but are subject to individualized interpretations. It is texts that are translated, not intentions. The challenge to the translator is to negotiate the meaning of a text and try to choose the most salient and appropriate interpretation as a basis for bringing the text to a new audience through translation.

Entry and return times for semi-flows

“…The first hitting time of η is τ 1 (η) = ℓ 1 a.s., and so Eτ 1 (η) = ℓ 1 = p 1 I −1 ξ , which confirms (38).…”

“…variable ℓ takes the value ℓ i with probability p i . The first hittingtime of η is τ 1 (η) d = ℓ, and so Eτ 1 (η) = I −1ξ , which is precisely(38).Appendix B. An illustrative example for Propositions 6 and 7Consider the sequence of stationary, simple point processes (2n+1)(m+s)+a n k ,…”

Statistical Regularity of Apollonian Gaskets

“…In macroscopic physics, cosmologists use pair correlations to study the distribution of stars and galaxies. Within mathematics, there is also a rich literature on the spatial statistics of point processes arising from various settings, such as Riemann zeta zeros [24], fractional parts of { √ n, n ∈ Z + } [14], directions of lattice points [9], [8], [18], [27], [21], [13], Farey sequences and their generalizations [17], [7], [5], [29], [4], [2], and translation surfaces [1], [3], [33]. Our list of interesting works here is far from inclusive.…”

“…Figures 3-7 were produced in a research project of Illinois Geometry Lab (IGL) [12], where Weiru Chen, Calvin Kessler and Mo Jiao were the undergraduate investigators, Amita Malik was the graduate mentor, and the author of this paper was the faculty mentor. Although we didn't use the results from [21] directly, that paper together with the data produced from the IGL project gave us the main inspiration of this paper. The technique employed in this paper is mainly from [34], [26], [23].…”

Spatial Statistics of Apollonian Gaskets

“…Spatial statistics from some other point processes have otherwise been rigorously established: gap distribution of the fractional parts of p ? nq by Elkies and McMullen [8], distribution of directions of Euclidean or hyperbolic lattices [6], [5], [13], [21], [17], distribution of Farey sequences [12], [4], [3], and gap distribution of saddle connection directions in translation surfaces [1], [2]. Our list of interesting works here is far from inclusive.…”