Existence of multiple solutions for second-order discrete boundary value problems

We give conditions on f involving pairs of discrete lower and discrete upper solutions which lead to the existence of at least three solutions of the discrete two-point boundary value problem yk+1 - 2yk + yk-1 + f (k, yk, vk) = 0, for k = 1,..., n - 1, y0 = 0 = yn,, where f is continuous and vk = yk - yk-1, for k = 1,..., n. In the special case f (k, t, p) = f (t) greater than or equal to 0, we give growth conditions on f and apply our general result to show the existence of three positive solutions. We give an example showing this latter result is sharp. Our results extend those of Avery and Peterson and are in the spirit of our results for the continuous analogue. (C) 2002 Elsevier Science Ltd. All rights reserved.

Difference equations associated with fully nonlinear boundary value problems for second order ordinary differential equations

We study the continuous problem y"=f(x,y,y'), xc[0,1], 0=G((y(0),y(1)),(y'(0), y'(1))), and its discrete approximation (y(k+1)-2y(k)+y(k-1))/h(2) =f(t(k), y(k), v(k)), k = 1,..., n-1, 0 = G((y(0), y(n)), (v(1), v(n))), where f and G = (g(0), g(1)) are continuous and fully nonlinear, h = 1/n, v(k) = (y(k) - y(k-1))/h, for k =1,..., n, and t(k) = kh, for k = 0,...,n. We assume there exist strict lower and strict upper solutions and impose additional conditions on f and G which are known to yield a priori bounds on, and to guarantee the existence of solutions of the continuous problem. We show that the discrete approximation also has solutions which approximate solutions of the continuous problem and converge to the solution of the continuous problem when it is unique, as the grid size goes to 0. Homotopy methods can be used to compute the solution of the discrete approximation. Our results were motivated by those of Gaines.

The association between auditory memory span and speech rate in children from kindergarten to sixth grade

This study reexamined the association between speech rate and memory span in children from kindergarten to sixth grade (N = 152) in order to potentially account for the inconsistencies within the published literature on this topic. Some of the inconsistencies in past research may reflect the different methods adopted in assessing speech rate. In particular, repeating word triples may itself involve memory demands, contaminating the correlation between speech rate and memory span in younger children. Analyses using composite speech rate and memory span measures showed that speech rate for word triples shared variance with memory span that was independent of speech rate for single words. Moreover, speech rate for word triples was largely redundant with age in explaining additional variation in memory span once the effects of speech rate for single words were controlled. (C) 2002 Elsevier Science.

Clinical evaluation of the WOMAC 3.0 OA Index in numeric rating scale format using a computerized touch screen version

Background: The Western Ontario and McMaster Universities (WOMAC) Osteoarthritis Index is a previously described self-administered questionnaire covering three domains: pain, stiffness and function. It has been validated in patients with osteoarthritis (OA) of the hip or knee in a paper-based format. Aim: To validate the WOMAC 3.0 using a numerical rating scale in a computerized touch screen format allowing immediate evaluation of the questionnaire. In the computed version cartoons, written and audio instruments were included in order facilitate application. Methods: Fifty patients, demographically balanced, with radiographically proven primary hip or knee OA completed the classical paper and the new computerized WOMAC version. Subjects were randomized either to paper format or computerized format first to balance possible order effects, Results: The intra-class correlation coefficients for pain, stiffness and function values were 0.915, 0.745 and 0.940, respectively. The Spearman correlation coefficients for pain, stiffness and function were 0.88, 0.77 and 0.87, respectively. Conclusion: These data indicate that the computerized WOMAC OA index 3.0 is comparable to the paper WOMAC in all three dimensions. The computerized version would allow physicians to get an immediate result and if present a direct comparison with a previous exam. (C) 2002 OsteoArthritis Research Society International. Published by Elsevier Science Ltd. All rights reserved.

How reliably do rheumatologists measure shoulder movement?

Objective: To assess the intrarater and interrater reliability among rheumatologists of a standardised protocol for measurement of shoulder movements using a gravity inclinometer. Methods: After instruction, six rheurnatologists independently assessed eight movements of the shoulder, including total and glenohumeral flexion, total and glenohumeral abduction, external rotation in neutral and in abduction, internal rotation in abduction and hand behind back, in random order in six patients with shoulder pain and stiffness according to a 6x6 Latin square design using a standardised protocol. These assessments were then repeated. Analysis of variance was used to partition total variability into components of variance in order to calculate intraclass correlation coefficients (ICCs). Results: The intrarater and interrater reliability of different shoulder movements varied widely. The movement of hand behind back and total shoulder flexion yielded the highest ICC scores for both intrarater reliability (0.91 and 0.83, respectively) and interrater reliability (0.80 and 0.72, respectively). Low ICC scores were found for the movements of glenohumeral abduction, external rotation in abduction, and internal rotation in abduction (intrarater ICCs 0.35, 0.43, and 0.32, respectively), and external rotation in neutral, external rotation in abduction, and internal rotation in abduction (interrater ICCs 0.29, 0.11, and 0.06, respectively). Conclusions: The measurement of shoulder movements using a standardised protocol by rheumatologists produced variable intrarater and interrater reliability. Reasonable reliability was obtained only for the movement of hand behind back and total shoulder flexion.

Longitudinal study of veterinarians from entry to the veterinary course to 10 years after graduation: attitudes to work, career and profession

Objective To describe the attitudes of veterinarians to their work, career and profession during the 10 years after graduation. Design Longitudinal study of students who started their course at The University of Queensland in 1985 and 1986, and who completed questionnaires in their first and fifth year as students, and after one, five and 10 years as veterinarians. Methods Data from 129 (96%) questionnaires completed after 10 years as a veterinarian were coded numerically then analysed, together with data from previous questionnaires, with SAS System 7 for Windows 95. Results After 10 years, almost all respondents were either very glad they had done the veterinary course (57%) or generally glad, though with some misgivings (37%). Despite this, only 55% would definitely become a veterinarian if they 'had to do it over again'. The responses for about one-third were different from those given five years earlier. The views of many were related to the level of support and encouragement received in their first job after graduation. There were 42% who were working less than half-time as veterinarians, and their main reasons were, in order, raising children, long hours of work, attitudes of bosses and clients, and poor pay. A majority was concerned about the ethics and competence of some colleagues, and almost all believed that consideration of costs must influence the type of treatment animals receive. Conclusions Most veterinarians were glad to have done the veterinary course, but for about one-quarter their career had not lived up to expectations and almost half would not do it again in another incarnation. Stress, hours of work, difficulties in balancing personal life with career and low income were important concerns for many. Low income may contribute to the low number of males entering the veterinary profession.

The development and psychometric properties of a measure of social and adaptive functioning for children and adolescents

Developed, piloted, and examined the psychometric properties of the Child and Adolescent Social and Adaptive Functioning Scale (CASAFS), a self-report measure designed to examine the social functioning of young people in the areas of school performance, peer relationships, family relationships, and home duties/self-care. The findings of confirmatory and exploratory factor analysis support a 4-factor solution consistent with the hypothesized domains. Fit indexes suggested that the 4-correlated factor model represented a satisfactory solution for the data, with the covariation between factors being satisfactorily explained by a single, higher order factor reflecting social and adaptive functioning in general. The internal consistency and 12-month test-retest reliability of the total scale was acceptable. A significant, negative correlation was found between the CASAFS and a measure of depressive symptoms, showing that high levels of social functioning are associated with low levels of depression. Significant differences in CASAFS total and subscale scores were found between clinically depressed adolescents and a matched sample of nonclinical controls. Adolescents who reported elevated but subclinical levels of depression also reported lower levels of social functioning in comparison to nonclinical controls.

Robust model-order reduction of complex biological processes

This paper addresses robust model-order reduction of a high dimensional nonlinear partial differential equation (PDE) model of a complex biological process. Based on a nonlinear, distributed parameter model of the same process which was validated against experimental data of an existing, pilot-scale BNR activated sludge plant, we developed a state-space model with 154 state variables in this work. A general algorithm for robustly reducing the nonlinear PDE model is presented and based on an investigation of five state-of-the-art model-order reduction techniques, we are able to reduce the original model to a model with only 30 states without incurring pronounced modelling errors. The Singular perturbation approximation balanced truncating technique is found to give the lowest modelling errors in low frequency ranges and hence is deemed most suitable for controller design and other real-time applications. (C) 2002 Elsevier Science Ltd. All rights reserved.

Stress correlation function evolution in lattice solid elasto-dynamic models of shear and fracture zones and earthquake prediction

It has been argued that power-law time-to-failure fits for cumulative Benioff strain and an evolution in size-frequency statistics in the lead-up to large earthquakes are evidence that the crust behaves as a Critical Point (CP) system. If so, intermediate-term earthquake prediction is possible. However, this hypothesis has not been proven. If the crust does behave as a CP system, stress correlation lengths should grow in the lead-up to large events through the action of small to moderate ruptures and drop sharply once a large event occurs. However this evolution in stress correlation lengths cannot be observed directly. Here we show, using the lattice solid model to describe discontinuous elasto-dynamic systems subjected to shear and compression, that it is for possible correlation lengths to exhibit CP-type evolution. In the case of a granular system subjected to shear, this evolution occurs in the lead-up to the largest event and is accompanied by an increasing rate of moderate-sized events and power-law acceleration of Benioff strain release. In the case of an intact sample system subjected to compression, the evolution occurs only after a mature fracture system has developed. The results support the existence of a physical mechanism for intermediate-term earthquake forecasting and suggest this mechanism is fault-system dependent. This offers an explanation of why accelerating Benioff strain release is not observed prior to all large earthquakes. The results prove the existence of an underlying evolution in discontinuous elasto-dynamic, systems which is capable of providing a basis for forecasting catastrophic failure and earthquakes.

Long-range automaton models of earthquakes: Power-law accelerations, correlation evolution, and mode-switching

We introduce a conceptual model for the in-plane physics of an earthquake fault. The model employs cellular automaton techniques to simulate tectonic loading, earthquake rupture, and strain redistribution. The impact of a hypothetical crustal elastodynamic Green's function is approximated by a long-range strain redistribution law with a r(-p) dependance. We investigate the influence of the effective elastodynamic interaction range upon the dynamical behaviour of the model by conducting experiments with different values of the exponent (p). The results indicate that this model has two distinct, stable modes of behaviour. The first mode produces a characteristic earthquake distribution with moderate to large events preceeded by an interval of time in which the rate of energy release accelerates. A correlation function analysis reveals that accelerating sequences are associated with a systematic, global evolution of strain energy correlations within the system. The second stable mode produces Gutenberg-Richter statistics, with near-linear energy release and no significant global correlation evolution. A model with effectively short-range interactions preferentially displays Gutenberg-Richter behaviour. However, models with long-range interactions appear to switch between the characteristic and GR modes. As the range of elastodynamic interactions is increased, characteristic behaviour begins to dominate GR behaviour. These models demonstrate that evolution of strain energy correlations may occur within systems with a fixed elastodynamic interaction range. Supposing that similar mode-switching dynamical behaviour occurs within earthquake faults then intermediate-term forecasting of large earthquakes may be feasible for some earthquakes but not for others, in alignment with certain empirical seismological observations. Further numerical investigation of dynamical models of this type may lead to advances in earthquake forecasting research and theoretical seismology.

Secondary structure of the third extracellular loop responsible for ligand selectivity of a mammalian gonadotropin-releasing hormone receptor

The extracellular loop 3 (ECL3) of the mammalian gonadotropin-releasing hormone receptor (GnRH-R) contains an acidic amino acid (Glu(301) in the mouse GnRH-R,) that confers agonist selectivity for Are in mammalian GnRH. It is proposed that a specific conformation of ECL3 is necessary to orientate the carboxyl side chain of the acidic residue for interaction with Arg(8) of GnRH, which is supported by decreased affinity for Arg(8) GnRH but not Gln(8) GnRH when an adjacent Pro is mutated to Ala. To probe the structural contribution of the loop domain to the proposed presentation of the carboxyl side chain, we synthesized a model peptide (CGPEMLNRVSEPGC) representing residues 293-302 of mouse ECL3, where Cys and Gly residues are added symmetrically at the N and C termini, respectively, allowing the introduction of a disulfide bridge to simulate the distances at which the ECL3 is tethered to the transmembrane domains 6 and 7 of the receptor. The ability of the ECL3 peptide to bind GnRH with low affinity was demonstrated by its inhibition of GnRH stimulation of inositol phosphate production in cells expressing the GnRH-R. The CD bands of the ECL3 peptides exhibited a superposition of predominantly unordered structure and partial contributions from beta-sheet structure. Likewise, the analysis of the amide I and amide III bands from micro-Raman and FT Raman experiments revealed mainly unordered conformations of the cyclic and of the linear peptide. NMR data demonstrated the presence of a beta-hairpin among an ensemble of largely disordered structures in the cyclic peptide. The location of the turn linking the two strands of the hairpin was assigned to the three central residues L-296, N-297, and R-298. A small population of structured species among an ensemble of predominantly random coil conformation suggests that the unliganded receptor represents a variety of structural conformers, some of which have the potential to make contacts with the ligand. We propose a mechanism of receptor activation whereby binding of the agonist to the inactive receptor state induces and stabilizes a particular structural state of the loop domain, leading to further conformational rearrangements across the transmembrane domain and signal propagating interaction with G proteins. Interaction of the Glu(301) of the receptor with Arg(8) of GnRH induces a folded configuration of the ligand. Our proposal thus suggests that conformational changes of both ligand and receptor result from this interaction.

Existence of multiple solutions for finite difference approximations to second-order boundary value problems

Error condition detected We consider discrete two-point boundary value problems of the form D-2 y(k+1) = f (kh, y(k), D y(k)), for k = 1,...,n - 1, (0,0) = G((y(0),y(n));(Dy-1,Dy-n)), where Dy-k = (y(k) - Yk-I)/h and h = 1/n. This arises as a finite difference approximation to y" = f(x,y,y'), x is an element of [0,1], (0,0) = G((y(0),y(1));(y'(0),y'(1))). We assume that f and G = (g(0), g(1)) are continuous and fully nonlinear, that there exist pairs of strict lower and strict upper solutions for the continuous problem, and that f and G satisfy additional assumptions that are known to yield a priori bounds on, and to guarantee the existence of solutions of the continuous problem. Under these assumptions we show that there are at least three distinct solutions of the discrete approximation which approximate solutions to the continuous problem as the grid size, h, goes to 0. (C) 2003 Elsevier Science Ltd. All rights reserved.