Erdos-Turan with a moving target, equidistribution of roots of reducible quadratics, and Diophantine quadruples

A Diophantine $m$-tuple is a set $A$ of $m$ positive integers such that $ab+1$ is a perfect square for every pair $a,b$ of distinct elements of $A$. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are bounded by $x$. In doing so, we extend two existing tools in ways that might be of independent interest. The Erd\H os-Tur\'an inequality bounds the discrepancy between the number of elements of a sequence that lie in a particular interval modulo 1 and the expected number; we establish a version of this inequality where the interval is allowed to vary. We also adapt an argument of Hooley on the equidistribution of solutions of polynomial congruences to handle reducible quadratic polynomials.Comment: 24+epsilon page

Evolutionary plant breeding for low input systems

Heritable variation is at the heart of the process of evolution. However, variation is restricted in breeding for uniform crop populations using the pedigree line approach. Pedigree lines are successful in agriculture because synthetic inputs are used to raise fertility and control weeds, pests and diseases. An alternative method promoted for exploring the value of variation and evolutionary fitness in crops is to create composite cross populations. Composite cross populations are formed by assembling seed stocks with diverse evolutionary origins, recombination of these stocks by hybridization, the bulking of F1 progeny, and subsequent natural election for mass sorting of the progeny in successive natural cropping environments. Composite cross populations can provide dynamic gene pools, which in turn provide a means of conserving germplasm resources: they can also allow selection of heterogeneous crop varieties. The value of composite cross populations in achieving these aims is dependent on the outcome of mass trials by artificial and natural selection acting upon the heterogeneous mixture. There is evidence to suggest that composite cross populations may be an efficient way of providing heterogeneous crops and of selecting superior pure lines for low input systems characterized by unpredictable stress conditions

Origami building blocks: generic and special 4-vertices

Four rigid panels connected by hinges that meet at a point form a 4-vertex, the fundamental building block of origami metamaterials. Here we show how the geometry of 4-vertices, given by the sector angles of each plate, affects their folding behavior. For generic vertices, we distinguish three vertex types and two subtypes. We establish relationships based on the relative sizes of the sector angles to determine which folds can fully close and the possible mountain-valley assignments. Next, we consider what occurs when sector angles or sums thereof are set equal, which results in 16 special vertex types. One of these, flat-foldable vertices, has been studied extensively, but we show that a wide variety of qualitatively different folding motions exist for the other 15 special and 3 generic types. Our work establishes a straightforward set of rules for understanding the folding motion of both generic and special 4-vertices and serves as a roadmap for designing origami metamaterials.Comment: 8 pages, 9 figure

Rail Track Maintenance Planning: An Assessment Model

In Australia, railway track maintenance costs comprise between 25-35 percent of total freight train operating costs. Track maintenance planning models have been shown to reduce maintenance costs by 5 to 10 percent though improved planning. This paper describes a model which has been developed to deal with the track maintenance planning function at the medium to long-term levels. This model simulates the impacts of degrading railway track conditions and related maintenance work, in contrast to tradition models that mainly use expert systems. The model simulates the degrading track condition using an existing track degradation model. Track condition data from that model is used to determine if safety related speed restrictions are needed and what immediate maintenance work may be required for safe train operations. The model outputs the net present value of the benefits of undertaking a given maintenance strategy, when compared with a base-case scenario. The model approach has advantages over current models in investigating what if scenarios. The track engineer can assess the possible benefits in reduced operating costs from upgrading track infrastructure or from the use of improved maintenance equipment. After describing the model inputs and the assumptions used, the paper deals with the simulation of track maintenance and of train operating costs over time. The results of applying the model to a test track section using a number of different maintenance strategies are also given

A Characteristic Scale on the Cosmic Microwave Sky

The current suite of results from Cosmic Microwave Background anisotropy experiments is fulfilling the promise of providing extraordinary levels of discrimination between cosmological models. We calculate a binned anisotropy power spectrum, which we tabulate, along with error bars and bin-to-bin correlations, so that it can be easily used for constraining models. The resulting power spectrum is flat at large angles, with a gradual rise to a prominent peak at around 0.5 degrees and a decrease thereafter. This is precisely the shape predicted by inflationary-inspired adiabatic models. Within that class of cosmologies, this characteristic scale imprinted on the CMB sky can be used to infer that the geometry of the Universe is very close to flat. The next wave of CMB results should add fuel to the debate about whether or not the Universe once inflated, as well as beginning in earnest the task of measuring cosmological parameters.Comment: 6 pages, 1 figure. A less technical article based on the same work has appeared in Science Perspectives under the title "How Flat is the Universe?" (Science, Mar 24, 2000, 2171-2172