Stable and Convergent Difference Schemes for Weakly Singular Convolution Integrals

We obtain new numerical schemes for weakly singular integrals of convolution type called Caputo fractional order integrals using Taylor and fractional Taylor series expansions and grouping terms in a novel manner. A fractional Taylor series expansion argument is utilized to provide fractional-order approximations for functions with minimal regularity. The resulting schemes allow for the approximation of functions in Cγ [0, T], where 0 \u3c γ \u3c= 5. A mild invertibility criterion is provided for the implicit schemes. Consistency and stability are proven separately for the whole-number-order approximations and the fractional-order approximations. The rate of convergence in the time variable is shown to be O(γ), 0 \u3c γ ≤ 5 for u ∈ Cγ [0, T], where is the size of the partition of the time mesh. Crucially, the assumption of the integral kernel K being decreasing is not required for the scheme to converge in second-order and below approximations. Optimal convergence results are then proven for both sets of approximations, where fractional-order approximations can obtain up to whole-number rate of convergence in certain scenarios. Finally, numerical examples are provided that illustrate our findings

Uniform l1 Behavior of a Time Discretization Method for a Volterra Integrodifferential Equation With Convex Kernel; Stability

We study stability of a numerical method in which the backward Euler method is combined with order one convolution quadrature for approximating the integral term of the linear Volterra integrodifferential equation u\u27(t) + ∫0 β (t - s)Au(s) ds = 0, t ≥ 0, u(0) = u0, which arises in the theory of linear viscoelasticity. Here A is a positive self-adjoint densely defined linear operator in a real Hilbert space, and β (t) is locally integrable, nonnegative, nonincreasing, convex, and -β\u27(t) is convex. We establish stability of the method under these hypotheses on β(t). Thus, the method is stable for a wider class of kernel functions β(t) than was previously known. We also extend the class of operators A for which the method is stable. © 2011 Society for Industrial and Applied Mathematics

Numerical Solutions for Weakly Singular Hammerstein Equations and Their Superconvergence

In the recent paper [7], it was shown that the solutions of weakly singular Hammerstein equations satisfy certain regularity properties. Using this result, the optimal convergence rate of a standard piecewise polynomial collocation method and that of the recently proposed collocationtype method of Kumar and Sloan [10] are obtained. Superconvergence of both of these methods are also presented. In the final section, we discuss briefly a standard productintegration method for weakly singular Hammerstein equations and indicate its superconvergence property. © 1992 Rocky Mountain Mathematics Consortium

Superconvergence of the Iterated Collocation Methods for Hammerstein Equations

In this paper, we analyse the iterated collocation method for Hammerstein equations with smooth and weakly singular kernels. The paper expands the study which began in [16] concerning the superconvergence of the iterated Galerkin method for Hammerstein equations. We obtain in this paper a similar superconvergence result for the iterated collocation method for Hammerstein equations. We also discuss the discrete collocation method for weakly singular Hammerstein equations. Some discrete collocation methods for Hammerstein equations with smooth kernels were given previously in [3, 18]

Late Holocene climate reorganisation and the North American Monsoon

The North America Monsoon (NAM) provides the majority of rainfall for central and northern Mexico as well as parts of the south west USA. The controls over the strength of the NAM in a given year are complex, and include both Pacific and Atlantic systems. We present here an annually resolved proxy reconstruction of NAM rainfall variability over the last ~6ka, from an inwash record from the Laguna de Juanacatlán, Mexico. This high resolution, exceptionally well dated record allows changes in the NAM through the latter half of the Holocene to be investigated in both time and space domains, improving our understanding of the controls on the system. Our analysis shows a shift in conditions between c. 4 and 3 ka BP, after which clear ENSO/PDO type forcing patterns are evident

Anthropogenic Threats to Wild Cetacean Welfare and a Tool to Inform Policy in This Area

Human activities and anthropogenic environmental changes are having a profound effect on biodiversity and the sustainability and health of many populations and species of wild mammals. There has been less attention devoted to the impact of human activities on the welfare of individual wild mammals, although ethical reasoning suggests that the welfare of an individual is important regardless of species abundance or population health. There is growing interest in developing methodologies and frameworks that could be used to obtain an overview of anthropogenic threats to animal welfare. This paper shows the steps taken to develop a functional welfare assessment tool for wild cetaceans (WATWC) via an iterative process involving input from a wide range of experts and stakeholders. Animal welfare is a multidimensional concept, and the WATWC presented made use of the Five Domains model of animal welfare to ensure that all areas of potential welfare impact were considered. A pilot version of the tool was tested and then refined to improve functionality. We demonstrated that the refined version of the WATWC was useful to assess real-world impacts of human activity on Southern Resident killer whales. There was close within-scenario agreement between assessors as well as between-scenario differentiation of overall welfare impact. The current article discusses the challenges raised by assessing welfare in scenarios where objective data on cetacean behavioral and physiological responses are sparse and proposes that the WATWC approach has value in identifying important information gaps and in contributing to policy decisions relating to human impacts on whales, dolphins, and porpoises

Meta-analyses of whale-watching impact studies : Comparisons of cetacean responses to disturbance

Acknowledgements. The International Whaling Commission funded this study through a grant assigned to D.L. D.L. was also funded by the Scottish Funding Council for funding through grant HR09011 to the Marine Alliance for Science and Technology for Scotland. While writing the manuscript, V.S. was sponsored by a Fulbright scholarship. We thank the many people that replied to the 2 MAR - MAM calls and Dr. Stankowich for his previous comments on the manuscript.Peer reviewedPublisher PD

A 6,000-year record of environmental change from the eastern Pacific margin of central Mexico

The transition from the mid- to late-Holocene in MesoAmerica saw increasing complexity in spatial patterns of change. Records from the western part of the region are sparse, with lacustrine sequences affected by long term anthropogenic disturbance or lacking chronological resolution. Here, we present a continuous palaeoecological and geochemical record from Laguna de Juanacatlán, a remote lake in the mountains of the western TMVB. Diatom assemblages, XRF scanning data and bulk organic geochemistry from a well-dated, 7.25-m laminated sequence were combined with summary pollen data from a 9-m partially laminated core to provide a continuous record of catchment and lake ecosystem changes over the last c. 6,000 years. Relatively humid conditions prevailed prior to c. 5.1 cal ka, which supported dense oak-pine forest cover around a deep, stratified lake. A trend towards drier conditions began c. 5.1 cal ka, intensifying after 4.0 cal ka, consistent with weakening of the North American Monsoon. Between 3.0 and 1.2 cal ka, lower lake levels and variable catchment run-off are consistent with increasing ENSO influence observed in the Late Holocene in the neotropics. From 1.2 to 0.9 cal ka, a marked change to catchment stability and more intense stratification reflected drier conditions and / or reduced rainfall variability and possibly warmer temperatures. After 0.9 cal ka, conditions were wetter, with an increase in catchment disturbance associated with the combined effects of climate and human activity. In recent decades, the lake ecosystem has changed markedly, possibly in response to recent climate change as well as local catchment dynamics

Evaluation of Phage Display Discovered Peptides as Ligands for Prostate-Specific Membrane Antigen (PSMA)

The aim of this study was to identify potential ligands of PSMA suitable for further development as novel PSMA-targeted peptides using phage display technology. The human PSMA protein was immobilized as a target followed by incubation with a 15-mer phage display random peptide library. After one round of prescreening and two rounds of screening, high-stringency screening at the third round of panning was performed to identify the highest affinity binders. Phages which had a specific binding activity to PSMA in human prostate cancer cells were isolated and the DNA corresponding to the 15-mers were sequenced to provide three consensus sequences: GDHSPFT, SHFSVGS and EVPRLSLLAVFL as well as other sequences that did not display consensus. Two of the peptide sequences deduced from DNA sequencing of binding phages, SHSFSVGSGDHSPFT and GRFLTGGTGRLLRIS were labeled with 5-carboxyfluorescein and shown to bind and co-internalize with PSMA on human prostate cancer cells by fluorescence microscopy. The high stringency requirements yielded peptides with affinities KD∼1 μM or greater which are suitable starting points for affinity maturation. While these values were less than anticipated, the high stringency did yield peptide sequences that apparently bound to different surfaces on PSMA. These peptide sequences could be the basis for further development of peptides for prostate cancer tumor imaging and therapy. © 2013 Shen et al