Dynamics of a family of piecewise-linear area-preserving plane maps II. Invariant circles

This paper studies the behavior under iteration of the maps T_{ab}(x,y)=(F_{ab}(x)-y,x) of the plane R^2, in which F_{ab}(x)=ax if x>=0 and bx if x<0. The orbits under iteration correspond to solutions of the nonlinear difference equation x_{n+2}= 1/2(a-b)|x_{n+1}| + 1/2(a+b)x_{n+1} - x_n. This family of maps has the parameter space (a,b)\in R^2. These maps are area-preserving homeomorphisms of R^s that map rays from the origin into rays from the origin. This paper shows the existence of special parameter values where T_{ab} has every nonzero orbit an invariant circle with irrational rotation number, and these invariant circles are piecewise unions of arcs of conic sections. Numerical experiments suggest the possible existence of many other parameter values having invariant circles.Comment: v2 corresponds to second half of old part I; 27 pages latex, 9 ps figure files. Current part I is math.DS/0301294, part III is math.DS/0505103; v3 reflects prior work of Beardon, Bullett and Rippo

Adaptation kinetics in bacterial chemotaxis

Cells of Escherichia coli, tethered to glass by a single flagellum, were subjected to constant flow of a medium containing the attractant alpha-methyl-DL-aspartate. The concentration of this chemical was varied with a programmable mixing apparatus over a range spanning the dissociation constant of the chemoreceptor at rates comparable to those experienced by cells swimming in spatial gradients. When an exponentially increasing ramp was turned on (a ramp that increases the chemoreceptor occupancy linearly), the rotational bias of the cells (the fraction of time spent spinning counterclockwise) changed rapidly to a higher stable level, which persisted for the duration of the ramp. The change in bias increased with ramp rate, i.e., with the time rate of change of chemoreceptor occupancy. This behavior can be accounted for by a model for adaptation involving proportional control, in which the flagellar motors respond to an error signal proportional to the difference between the current occupancy and the occupancy averaged over the recent past. Distributions of clockwise and counterclockwise rotation intervals were found to be exponential. This result cannot be explained by a response regular model in which transitions between rotational states are generated by threshold crossings of a regular subject to statistical fluctuation; this mechanism generates distributions with far too many long events. However, the data can be fit by a model in which transitions between rotational states are governed by first-order rate constants. The error signal acts as a bias regulator, controlling the values of these constants

Majorana Fermions and Non-Abelian Statistics in Three Dimensions

We show that three dimensional superconductors, described within a Bogoliubov de Gennes framework can have zero energy bound states associated with pointlike topological defects. The Majorana fermions associated with these modes have non-Abelian exchange statistics, despite the fact that the braid group is trivial in three dimensions. This can occur because the defects are associated with an orientation that can undergo topologically nontrivial rotations. A new feature of three dimensional systems is that there are "braidless" operations in which it is possible to manipulate the groundstate associated with a set of defects without moving or measuring them. To illustrate these effects we analyze specific architectures involving topological insulators and superconductors.Comment: 4 pages, 2 figures, published versio

Distribution and sighting frequency of reef fishes in the Florida Keys National Marine Sanctuary

This study analyzed species richness, distribution, and sighting frequency of selected reef fishes to describe species assemblage composition, abundance, and spatial distribution patterns among sites and regions (Upper Keys, Middle Keys, Lower Keys, and Dry Tortugas) within the Florida Keys National Marine Sanctuary (FKNMS) barrier reef ecosystem. Data were obtained from the Reef Environmental Education Foundation (REEF) Fish Survey Project, a volunteer fish-monitoring program. A total of 4,324 visual fish surveys conducted at 112 sites throughout the FKNMS were used in these analyses. The data set contained sighting information on 341 fish species comprising 68 families. Species richness was generally highest in the Upper Keys sites (maximum was 220 species at Molasses Reef) and lowest in the Dry Tortugas sites. Encounter rates differed among regions, with the Dry Tortugas having the highest rate, potentially a result of differences in the evenness in fishes and the lower diversity of habitat types in the Dry Tortugas region. Geographic coverage maps were developed for 29 frequently observed species. Fourteen of these species showed significant regional variation in mean sighting frequency (%SF). Six species had significantly lower mean %SF and eight species had significantly higher mean %SF in the Dry Tortugas compared with other regions. Hierarchical clustering based on species composition (presence-absence) and species % SF revealed interesting patterns of similarities among sites that varied across spatial scales. Results presented here indicate that phenomena affecting reef fish composition in the FKNMS operate at multiple spatial scales, including a biogeographic scale that defines the character of the region as a whole, a reef scale (~50-100 km) that include meso-scale physical oceanographic processes and regional variation in reef structure and associated reef habitats, and a local scale that includes level of protection, cross-shelf location and a suite of physical characteristics of a given reef. It is likely that at both regional and local scales, species habitat requirements strongly influence the patterns revealed in this study, and are particularly limiting for species that are less frequently observed in the Dry Tortugas. The results of this report serve as a benchmark for the current status of the reef fishes in the FKNMS. In addition, these data provide the basis for analyses on reserve effects and the biogeographic coupling of benthic habitats and fish assemblages that are currently underway. (PDF contains 61 pages.

Simulation of factors impeding water quality trading market performance

Over the past several decades, market-based approaches to natural resource management have received increased attention as a means to cost-effectively achieve environmental quality goals. Following on what has been hailed a success for reducing air pollution, water quality trading (WQT) has more recently been seen as the next great opportunity for reducing water pollution, especially for nutrient loading. Numerous trading programs have been pilot tested and/or adopted in states throughout the nation, with more than 70 programs now in operation (Breetz et al., 2004). WQT would allow multiple contributors to surface water degradation to determine how best to meet an overarching collective goal related to pollution reduction. WQT takes advantage of differences in pollution abatement costs. In the case of point/nonpoint source trading, such as between wastewater treatment plants (WWTPs) and agricultural producers, it is often the agricultural producers who can achieve a given level of nutrient reduction at less cost through their adoption of various best management practices that reduce sedimentation and nutrient loading to surface waters. Trading would allow WWTPs to purchase “credits” generated by producers who reduced their pollution loading to achieve an equivalent level of reduction as might be required by a regulatory discharge permit at a lower overall cost. While there is substantial evidence that nonpoint sources have lower nutrient reduction costs than point sources, experience with WQT reveals a common theme: little or no trading activity. The success of WQT seems, in part, to depend on the structure of the market created to bring buyers and sellers together to transact exchanges. These outcomes suggest the presence of obstacles to trading that were not recognized in the design of existing programs. To examine the ways that various market imperfections may impact the performance of a WQT market, an agent-based model was constructed which simulated a hypothetical point-nonpoint market. In particular, the market was modeled using a variant of the sequential, bilateral trading algorithm proposed by Atkinson and Tietenberg (1991). Our proposed paper first presents an overview of the simulation modeling technique and then analyzes the effects of two prominent market impediments identified in the WQT literature: information levels and trading ratios. Information levels refer to buyers’ and sellers’ knowledge of each others’ bid prices. A frictionless WQT market would be one where all of the potential buyers (i.e., point sources) would know all of the sellers’ (i.e., nonpoint sources) offer prices and vice versa. In this full information environment, we can expect that trades would be consummated in the order of their gains. That is, first buyers and sellers to be paired together for trading would be the buyers with the highest offer prices and the sellers with the lowest bid prices. Successive trades will have successively smaller gains until the gap between bid and offer prices reaches zero. This is the textbook Walrasian market and would closely approximate a double auction institution, where all buyers and sellers submit their offers and bids, which are then sorted and matched by a centralized market manager. While the full information scenario serves as a useful benchmark, most existing WQT markets are decentralized in nature, so that limited information causes traders to be matched in a less efficient sequence. A variety of information levels are possible. One side of the market may have more information than the other (limited information) or neither side having any knowledge of the other side’s bid or offer prices (low information). Each of these scenarios leads to a different sequencing of trades. This paper analyzes the effect of different information levels on market performance. Market performance is measured in terms of cost savings, the number of credits traded, and the average reduction costs under different information scenarios. Trading ratios are a common component of many existing WQT programs. A typical trading ratio of 2:1 requires a nonpoint source to reduce two pounds of expected nutrient loading in order to receive one pound of trading credit. These ratios serve as a “safety factor” and are incorporated to account for the uncertainty in the measurement and monitoring of nonpoint source loading. Because nonpoint traders must reduce loading by 2 pounds for every 1 pound emitted by point source traders, there will be a net reduction of 1 pound of expected loading for each trade. So, while inhibiting some trades from ever occurring, trading ratios also have the potential to improve water quality beyond trading with a 1:1 trading ratio. This paper examines these tradeoffs in terms of effects on market performance and then describes procedures that can be used to characterize an optimal trading ratio if one exists. Because WQT programs, by nature, involve complex interactions between economics and the biophysical world, accurately simulating a real-world WQT market requires at minimum a basic understanding of the types of data that watershed models can provide. This paper concludes by briefly discussing data requirements, points of consideration, and integrative techniques used in the simulation of WQT in real-world watersheds.water quality trading, market based, trading ratio, information levels, point source, nonpoint source, simulation, Environmental Economics and Policy, Resource /Energy Economics and Policy,

Palladium-Catalyzed Asymmetric Conjugate Addition of Arylboronic Acids to Five-, Six-, and Seven-Membered β-Substituted Cyclic Enones: Enantioselective Construction of All-Carbon Quaternary Stereocenters

The first enantioselective Pd-catalyzed construction of all-carbon quaternary stereocenters via 1,4-addition of arylboronic acids to β-substituted cyclic enones is reported. Reaction of a wide range of arylboronic acids and cyclic enones using a catalyst prepared from Pd(OCOCF_3)_2 and a chiral pyridinooxazoline ligand yields enantioenriched products bearing benzylic stereocenters. Notably, this transformation is tolerant to air and moisture, providing a practical and operationally simple method of synthesizing enantioenriched all-carbon quaternary stereocenters

Frequency-Domain Stochastic Modeling of Stationary Bivariate or Complex-Valued Signals

There are three equivalent ways of representing two jointly observed real-valued signals: as a bivariate vector signal, as a single complex-valued signal, or as two analytic signals known as the rotary components. Each representation has unique advantages depending on the system of interest and the application goals. In this paper we provide a joint framework for all three representations in the context of frequency-domain stochastic modeling. This framework allows us to extend many established statistical procedures for bivariate vector time series to complex-valued and rotary representations. These include procedures for parametrically modeling signal coherence, estimating model parameters using the Whittle likelihood, performing semi-parametric modeling, and choosing between classes of nested models using model choice. We also provide a new method of testing for impropriety in complex-valued signals, which tests for noncircular or anisotropic second-order statistical structure when the signal is represented in the complex plane. Finally, we demonstrate the usefulness of our methodology in capturing the anisotropic structure of signals observed from fluid dynamic simulations of turbulence.Comment: To appear in IEEE Transactions on Signal Processin