Joins and meets in the structure of Ceers

We study computably enumerable equivalence relations (abbreviated as ceers) under computable reducibility, and we investigate the resulting degree structure Ceers, which is a poset with a smallest and a greatest element. We point out a partition of the ceers into three classes: the finite ceers, the light ceers, and the dark ceers. These classes yield a partition of the degree structure as well, and in the language of posets the corresponding classes of degrees are first order definable within Ceers. There is no least, no maximal, no greatest dark degree, but there are infinitely many minimal dark degrees. We study joins and meets in Ceers, addressing the cases when two incomparable degrees of ceers X,Y have or do not have join or meet according to where X,Y are located in the classes of the aforementioned partition: in particular no pair of dark ceers has join, and no pair in which at least one ceer is dark has meet. We also exhibit examples of ceers X,Y having join which coincides with their uniform join, but also examples when their join is strictly less than the uniform join. We study join-irreducibility and meet-irreducibility. In particular we characterize the property of being meet-irreducible for a ceer E, by showing that it coincides with the property of E being self-full, i.e. every reducibility from E to itself is in fact surjective on its equivalence classes (this property properly extends darkness). We then study the quotient structure obtained by dividing the poset Ceers by the degrees of the finite ceers, and study joins and meets in this quotient structure. We look at automorphisms of Ceers, and show that there are continuum many automorphisms fixing the dark ceers, and continuum many automorphisms fixing the light ceers. Finally, we compute the complexity of the index sets of the classes of ceers studied in the paper

Bathymetric terrain model of the Atlantic margin for marine geological investigations.

Bathymetric terrain models of seafloor morphology are an important component of marine geological investigations. Advances in acquisition and processing technologies of bathymetric data have facilitated the creation of high-resolution bathymetric surfaces that approach the resolution of similar surfaces available for onshore investigations. These bathymetric terrain models provide a detailed representation of the Earth’s subaqueous surface and, when combined with other geophysical and geological datasets, allow for interpretation of modern and ancient geological processes. The purpose of the bathymetric terrain model presented in this report is to provide a high-quality bathymetric surface of the Atlantic margin of the United States that can be used to augment current and future marine geological investigations. The input data for this bathymetric terrain model, covering almost 305,000 square kilometers, were acquired by several sources, including the U.S. Geological Survey, the National Oceanic and Atmospheric Administration National Geophysical Data Center and the Ocean Exploration Program, the University of New Hampshire, and the Woods Hole Oceanographic Institution. These data have been edited using hydrographic data processing software to maximize the quality, usability, and cartographic presentation of the combined terrain model

A LOCAL CHARACTERIZATION OF VC-MINIMALITY

Abstract. We show VC-minimality is Π 0 4 -complete. In particular, we give a local characterization of VC-minimality. We also show dp-smallness is Π 1 1 -complete

Size distribution of submarine landslides and its implication to tsunami hazard in Puerto Rico

Author Posting. © American Geophysical Union, 2006. This article is posted here by permission of American Geophysical Union for personal use, not for redistribution. The definitive version was published in Geophysical Research Letters 33 (2006): L11307, doi:10.1029/2006GL026125.We have established for the first time a size frequency distribution for carbonate submarine slope failures. Using detailed bathymetry along the northern edge of the carbonate platform north of Puerto Rico, we show that the cumulative distribution of slope failure volumes follows a power-law distribution. The power-law exponent of this distribution is similar to those for rock falls on land, commensurate with their interpreted failure mode. The carbonate volume distribution and its associated volume-area relationship are significantly different from those for clay-rich debris lobes in the Storegga slide, Norway. Coupling this relationship with tsunami simulations allows an estimate of the maximum tsunami runup and the maximum number of potentially damaging tsunamis from landslides to the north shore of Puerto Rico

Trial and error mathematics: Dialectical systems and completions of theories

This paper is part of a project that is based on the notion of a dialectical system, introduced by Magari as a way of capturing trial and error mathematics. In Amidei et al. (2016, Rev. Symb. Logic, 9, 1–26) and Amidei et al. (2016, Rev. Symb. Logic, 9, 299–324), we investigated the expressive and computational power of dialectical systems, and we compared them to a new class of systems, that of quasi-dialectical systems, that enrich Magari’s systems with a natural mechanism of revision. In the present paper we consider a third class of systems, that of p-dialectical systems, that naturally combine features coming from the two other cases. We prove several results about p-dialectical systems and the sets that they represent. Then we focus on the completions of first order theories. In doing so, we consider systems with connectives, i.e. systems that encode the rules of classical logic. We show that any consistent system with connectives represents the completion of a given theory. We prove that dialectical and q-dialectical systems coincide with respect to the completions that they can represent. Yet, p-dialectical systems are more powerful; we exhibit a p-dialectical system representing a completion of Peano Arithmetic that is neither dialectical nor q-dialectical