Products in Residue Classes

We consider a problem of P. Erdos, A. M. Odlyzko and A. Sarkozy about the representation of residue classes modulo m by products of two not too large primes. While it seems that even the Extended Riemann Hypothesis is not powerful enough to achieve the expected results, here we obtain some unconditional results ``on average'' over moduli m and residue classes modulo m and somewhat stronger results when the average is restricted to prime moduli m = p. We also consider the analogous question wherein the primes are replaced by easier sequences so, quite naturally, we obtain much stronger results.Comment: 18 page

Fish Habitat Utilization Patterns and Evaluation of the Efficacy of Marine Protected Areas in Hawaii: Integration of NOAA Digital Benthic Habitat Mapping and Coral Reef Ecological Studies

Over the past four decades, the state of Hawaii has developed a system of eleven Marine Life Conservation Districts (MLCDs) to conserve and replenish marine resources around the state. Initially established to provide opportunities for public interaction with the marine environment, these MLCDs vary in size, habitat quality, and management regimes, providing an excellent opportunity to test hypotheses concerning marine protected area (MPA) design and function using multiple discreet sampling units. NOAA/NOS/NCCOS/Center for Coastal Monitoring and Assessment’s Biogeography Team developed digital benthic habitat maps for all MLCD and adjacent habitats. These maps were used to evaluate the efficacy of existing MLCDs for biodiversity conservation and fisheries replenishment, using a spatially explicit stratified random sampling design. Coupling the distribution of habitats and species habitat affinities using GIS technology elucidates species habitat utilization patterns at scales that are commensurate with ecosystem processes and is useful in defining essential fish habitat and biologically relevant boundaries for MPAs. Analysis of benthic cover validated the a priori classification of habitat types and provided justification for using these habitat strata to conduct stratified random sampling and analyses of fish habitat utilization patterns. Results showed that the abundance and distribution of species and assemblages exhibited strong correlations with habitat types. Fish assemblages in the colonized and uncolonized hardbottom habitats were found to be most similar among all of the habitat types. Much of the macroalgae habitat sampled was macroalgae growing on hard substrate, and as a result showed similarities with the other hardbottom assemblages. The fish assemblages in the sand habitats were highly variable but distinct from the other habitat types. Management regime also played an important role in the abundance and distribution of fish assemblages. MLCDs had higher values for most fish assemblage characteristics (e.g. biomass, size, diversity) compared with adjacent fished areas and Fisheries Management Areas (FMAs) across all habitat types. In addition, apex predators and other targeted resources species were more abundant and larger in the MLCDs, illustrating the effectiveness of these closures in conserving fish populations. Habitat complexity, quality, size and level of protection from fishing were important determinates of MLCD effectiveness with respect to their associated fish assemblages. (PDF contains 217 pages

Analysis of nuclear waste disposal in space, phase 3. Volume 2: Technical report

The options, reference definitions and/or requirements currently envisioned for the total nuclear waste disposal in space mission are summarized. The waste form evaluation and selection process is documented along with the physical characteristics of the iron nickel-base cermet matrix chosen for disposal of commercial and defense wastes. Safety aspects of radioisotope thermal generators, the general purpose heat source, and the Lewis Research Center concept for space disposal are assessed as well as the on-pad catastrophic accident environments for the uprated space shuttle and the heavy lift launch vehicle. The radionuclides that contribute most to long-term risk of terrestrial disposal were determined and the effects of resuspension of fallout particles from an accidental release of waste material were studied. Health effects are considered. Payload breakup and rescue technology are discussed as well as expected requirements for licensing, supporting research and technology, and safety testing

Analysis of nuclear waste disposal in space, phase 3. Volume 1: Executive summary of technical report

The objectives, approach, assumptions, and limitations of a study of nuclear waste disposal in space are discussed with emphasis on the following: (1) payload characterization; (2) safety assessment; (3) health effects assessment; (4) long-term risk assessment; and (5) program planning support to NASA and DOE. Conclusions are presented for each task

An inviscid dyadic model of turbulence: the fixed point and Onsager's conjecture

Properties of an infinite system of nonlinearly coupled ordinary differential equations are discussed. This system models some properties present in the equations of motion for an inviscid fluid such as the skew symmetry and the 3-dimensional scaling of the quadratic nonlinearity. It is proved that the system with forcing has a unique equilibrium and that every solution blows up in finite time in $H^{5/6}$-norm. Onsager's conjecture is confirmed for the model system

Representations of reductive normal algebraic monoids

The rational representation theory of a reductive normal algebraic monoid (with one-dimensional center) forms a highest weight category, in the sense of Cline, Parshall, and Scott. This is a fundamental fact about the representation theory of reductive normal algebraic monoids. We survey how this result was obtained, and treat some natural examples coming from classical groups.Comment: 10 pages. To appear in a volume of the Fields Communications Series: "Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics," edited by Mahir Can, Zhenheng Li, Benjamin Steinberg, and Qiang Wan

Hybrid Deterministic-Stochastic Methods for Data Fitting

Many structured data-fitting applications require the solution of an optimization problem involving a sum over a potentially large number of measurements. Incremental gradient algorithms offer inexpensive iterations by sampling a subset of the terms in the sum. These methods can make great progress initially, but often slow as they approach a solution. In contrast, full-gradient methods achieve steady convergence at the expense of evaluating the full objective and gradient on each iteration. We explore hybrid methods that exhibit the benefits of both approaches. Rate-of-convergence analysis shows that by controlling the sample size in an incremental gradient algorithm, it is possible to maintain the steady convergence rates of full-gradient methods. We detail a practical quasi-Newton implementation based on this approach. Numerical experiments illustrate its potential benefits.Comment: 26 pages. Revised proofs of Theorems 2.6 and 3.1, results unchange

Loop space homology associated with the mod 2 Dickson invariants

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