ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS TO BOUNDARY VALUE PROBLEMS ON TIME SCALES

This work formulates existence, uniqueness, and uniqueness-implies-existence theorems for solutions to two-point vector boundary value problems on time scales. The methods used include maximum principles, a priori bounds on solutions, and the nonlinear alternative of Leray-Schauder

Wiman’s formula for a second order dynamic equation

We derive Wiman’s asymptotic formula for the number of generalized zeros of (nontrivial) solutions of a second order dynamic equation on a time scale. The proof is based on the asymptotic representation of solutions via exponential functions on a time scale. By using the Jeffreys et al. approximation we prove Wiman’s formula for a dynamic equation on a time scale. Further we show that using the Hartman-Wintner approximation one can derive another version of Wiman’s formula. We also prove some new oscillation theorems and discuss the results by means of several examples

HILLE-KNESER-TYPE CRITERIA FOR SECOND-ORDER DYNAMIC EQUATIONS ON TIME SCALES

We consider the pair of second-order dynamic equations, (r(t)(xΔ)γ)Δ + p(t)xγ(t) = 0 and (r(t)(xΔ)γ)Δ + p(t)xγσ (t) = 0, on a time scale T, where γ \u3e 0 is a quotient of odd positive integers. We establish some necessary and sufficient conditions for nonoscillation of Hille-Kneser type. Our results in the special case when T = R involve the well known Hille-Kneser-type criteria of second-order linear differential equations established by Hille. For the case of the second-order half-linear differential equation, our results extend and improve some earlier results of Li and Yeh and are related to some work of Dosly and Rehak and some results of Rehak for half-linear equations on time scales. Several examples are considered to illustrate the main results

Nabla discrete fractional Grüss type inequality

Properties of the discrete fractional calculus in the sense of a backward difference are introduced and developed. Here, we prove a more general version of the Grüss type inequality for the nabla fractional case. An example of our main result is given

Survey of the qualitative properties of fractional difference operators: monotonicity, convexity, and asymptotic behavior of solutions

In this article we discuss some of the qualitative properties of fractional difference operators. We especially focus on the connections between the fractional difference operator and the monotonicity and convexity of functions. In the integer-order setting, these connections are elementary and well known. However, in the fractional-order setting the connections are very complicated and muddled. We survey some of the known results and suggest avenues for future research. In addition, we discuss the asymptotic behavior of solutions to fractional difference equations and how the nonlocal structure of the fractional difference can be used to deduce these asymptotic properties

Quantitative Reasoning in Environmental Science: Rasch Measurement to Support QR Assessment

Original work is hosted at USF Libraries Scholar Commons publisher of Numeracy, the electronic journal of the National Numeracy Network (NNN). Abstract : The ability of middle and high school students to reason quantitatively within the context of environmental science was investigated. A quantitative reasoning (QR) learning progression, with associated QR assessments in the content areas of biodiversity, water, and carbon, was developed based on three QR progress variables: quantification act, quantitative interpretation, and quantitative modeling. Diagnostic instruments were developed specifically for the progress variable quantitative interpretation (QI), each consisting of 96 Likert-scale items. Each content version of the instrument focused on three scale levels (macro scale, micro scale, and landscape scale) and four elements of QI identified in prior research (trend, translation, prediction, and revision). The QI assessments were completed by 362, 6th to 12th grade students in three U.S. states. Rasch (1960/1980) measurement was used to determine item and person measures for the QI instruments, both to examine validity and reliability characteristics of the instrument administration and inform the evolution of the learning progression. Rasch methods allowed identification of several QI instrument revisions, including modification of specific items, reducing number of items to avoid cognitive fatigue, reconsidering proposed item difficulty levels, and reducing Likert scale to 4 levels. Rasch diagnostics also indicated favorable levels of instrument reliability and appropriate targeting of item abilities to student abilities for the majority of participants. A revised QI instrument is available for STEM researchers and educators

Challenges in quantifying changes in the global water cycle

Human influences have likely already impacted the large-scale water cycle but natural variability and observational uncertainty are substantial. It is essential to maintain and improve observational capabilities to better characterize changes. Understanding observed changes to the global water cycle is key to predicting future climate changes and their impacts. While many datasets document crucial variables such as precipitation, ocean salinity, runoff, and humidity, most are uncertain for determining long-term changes. In situ networks provide long time-series over land but are sparse in many regions, particularly the tropics. Satellite and reanalysis datasets provide global coverage, but their long-term stability is lacking. However, comparisons of changes among related variables can give insights into the robustness of observed changes. For example, ocean salinity, interpreted with an understanding of ocean processes, can help cross-validate precipitation. Observational evidence for human influences on the water cycle is emerging, but uncertainties resulting from internal variability and observational errors are too large to determine whether the observed and simulated changes are consistent. Improvements to the in situ and satellite observing networks that monitor the changing water cycle are required, yet continued data coverage is threatened by funding reductions. Uncertainty both in the role of anthropogenic aerosols, and due to large climate variability presently limits confidence in attribution of observed changes

A Climate Index Optimized for Longshore Sediment Transport Reveals Interannual and Multidecadal Littoral Cell Rotations

A recent 35-year endpoint shoreline change analysis revealed significant counterclockwiserotations occurring in north-central Oregon, USA, littoral cells that extend 10s of kilometers in length.While the potential for severe El Niños to contribute to littoral cell rotations at seasonal to interannual scalewas previously recognized, the dynamics resulting in persistent (multidecadal) rotation were unknown,largely due to a lack of historical wave conditions extending back multiple decades and the difficulty ofseparating the timescales of shoreline variability in a high energy region. This study addresses this questionby (1) developing a statistical downscaling framework to characterize wave conditions relevant for longshoresediment transport during data-poor decades and (2) applying a one-line shoreline change model toquantitatively assess the potential for such large embayed beaches to rotate. A climateINdex was optimizedto capture variability in longshore wave power as a proxy for potentialLOngshore Sediment Transport(LOST_IN), and a procedure was developed to simulate many realizations of potential wave conditions fromthe index. Waves were transformed dynamically with Simulating Waves Nearshore to the nearshore asinputs to a one-line model that revealed shoreline rotations of embayed beaches at multiple time and spatialscales not previously discernible from infrequent observations. Model results indicate that littoral cellsrespond to both interannual and multidecadal oscillations, producing comparable shoreline excursions toextreme El Niño winters. The technique quantitatively relates morphodynamic forcing to specific climatepatterns and has the potential to better identify and quantify coastal variability on timescales relevant to achanging climate.This work would not have been possible without funding from the NSF Graduate Research Fellowship Program (GRFP) through NSF grant DGE-1314109, the Coastal and Ocean Climate Applications (COCA) program through NOAA grant NA15OAR4310243, NOAA’s Regional Integrated Sciences and Assessments Program (RISA), under NOAA grant NA15OAR4310145, and the Spanish Ministerio de Educación Cultura y Deporte FPU (Formación del Profesorado Universitario) studentship BOE-A-2013-12235. Beach survey data collection undertaken on the Oregon coast was made possible by the Northwest Association of Networked Ocean Observing Systems (NANOOS) through NOAA grant NA16NOS0120019

Rainfall variability at decadal and longer time scales: signal or noise?

Rainfall variability occurs over a wide range of temporal scales. Knowledge and understanding of such variability can lead to improved risk management practices in agricultural and other industries. Analyses of temporal patterns in 100 yr of observed monthly global sea surface temperature and sea level pressure data show that the single most important cause of explainable, terrestrial rainfall variability resides within the El Nino-Southern Oscillation (ENSO) frequency domain (2.5-8.0 yr), followed by a slightly weaker but highly significant decadal signal (9-13 yr), with some evidence of lesser but significant rainfall variability at interclecadal time scales (15-18 yr). Most of the rainfall variability significantly linked to frequencies tower than ENSO occurs in the Australasian region, with smaller effects in North and South America, central and southern Africa, and western Europe. While low-frequency (LF) signals at a decadal frequency are dominant, the variability evident was ENSO-like in all the frequency domains considered. The extent to which such LF variability is (i) predictable and (ii) either part of the overall ENSO variability or caused by independent processes remains an as yet unanswered question. Further progress can only be made through mechanistic studies using a variety of models