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.

ATP-dependent K+ channels in renal ischemia reperfusion injury

ATP-dependent K+ channels (K-ATP) account for most of the recycling of K+ which enters the proximal tubules cell via Na, K-ATPase. In the mitochondrial membrane, opening of these channels preserves mitochondrial viability and matrix volume during ischemia. We examined KATP channel modulation in renal ischemia-reperfusion injury (IRI), using an isolated perfused rat kidney (IPRK) model, in control, IRI, IRI + 200 muM diazoxide (a K-ATP opener), IRI + 10 muM glibenclamide (a K-ATP blocker) and IRI + 200 muM diazoxide + 10 muM glibenclamide groups. IRI was induced by 2 periods of warm ischemia, followed by 45 min of reperfusion. IRI significantly decreased glomerular filtration rate (GFR) and increased fractional excretion of sodium (FENa) (p < 0.01). Neither diazoxide nor glibenclamide had an effect on control kidney function other than an increase in renal vascular resistance produced by glibenclamide. Pretreatment with 200 muM diazoxide reduced the postischemic increase in FENa (p < 0.05). Adding 10 muM glibenclamide inhibited the diazoxide effect on postischemic FENa (p < 0.01). Histology showed that kidneys pretreated with glibenclamide demonstrated an increase in injure in the thick ascending limb of outer medulla (p < 0.05). Glibenclamide significantly decreased post ischemic renal vascular resistance (p < 0.05). but had no significant effect on other renal function parameters. Our results suggest that sodium reabsorption is improved by K-ATP activation and blockade of K-ATP channels during IRI has an injury enhancing effect on renal epithelial function and histology. This may be mediated through K-ATP modulation in cell and or mitochondrial inner membrane.

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.

An augmented Lanczos algorithm for the efficient computation of a dot-product of a function of a large sparse symmetric matrix

The Lanczos algorithm is appreciated in many situations due to its speed. and economy of storage. However, the advantage that the Lanczos basis vectors need not be kept is lost when the algorithm is used to compute the action of a matrix function on a vector. Either the basis vectors need to be kept, or the Lanczos process needs to be applied twice. In this study we describe an augmented Lanczos algorithm to compute a dot product relative to a function of a large sparse symmetric matrix, without keeping the basis vectors.

Abnormal extracellular matrix protein transport associated with increased apoptosis of vascular smooth muscle cells in Marfan syndrome and bicuspid aortic valve thoracic aortic aneurysm

Background - Marfan syndrome (MS) is a genetic disorder caused by a mutation in the fibrillin gene FBN1. Bicuspid aortic valve (BAV) is a congenital heart malformation of unknown cause. Both conditions are associated with ascending aortic aneurysm and premature death. This study examined the relationship among the secretion of extracellular matrix proteins fibrillin, fibronectin, tenascin, and vascular smooth muscle cell (VSMC) apoptosis. The role of matrix metalloproteinase (MMP)- 2 in VSMC apoptosis was studied in MS aneurysm. Methods and Results - Aneurysm tissue was obtained from patients undergoing surgery ( MS: 4 M, 1 F, age 27 - 45 years; BAV: 3 M, 2 F, age 28 - 65 years). Normal aorta from subjects with nonaneurysm disease was also collected ( 4 M, 1 F, age 23 - 93 years). MS and BAV aneurysm histology showed areas of cystic medial necrosis (CMN) without inflammatory infiltrate. Immunohistochemical study of cultured MS and BAV VSMC showed intracellular accumulation and reduction of extracellular distribution of fibrillin, fibronectin, and tenascin. Western blot showed no increase in expression of fibrillin, fibronectin, or tenascin in MS or BAV VSMC and increased expression of MMP-2 in MS VSMCs. There was 4-fold increase in loss of cultured VSMC incubated in serum-free medium for 24 hours in both MS ( 27 +/- 8%) and BAV ( 32 +/- 14%) compared with control ( 7 +/- 5%). Conclusions - In MS and BAV there is alteration in both the amount and quality of secreted proteins and an increased degree of VSMC apoptosis. Up-regulation of MMP-2 might play a role in VSMC apoptosis in MS VSMC. The findings suggest the presence of a fundamental cellular abnormality in BAV thoracic aorta, possibly of genetic origin.

Skin-friction measurements in high-enthalpy hypersonic boundary layers

Skin-friction measurements are reported for high-enthalpy and high-Mach-number laminar, transitional and turbulent boundary layers. The measurements were performed in a free-piston shock tunnel with air-flow Mach number, stagnation enthalpy and Reynolds numbers in the ranges of 4.4-6.7, 3-13 MJ kg(-1) and 0.16 x 10(6)-21 x 10(6), respectively. Wall temperatures were near 300 K and this resulted in ratios of wall enthalpy to flow-stagnation enthalpy in the range of 0.1-0.02. The experiments were performed using rectangular ducts. The measurements were accomplished using a new skin-friction gauge that was developed for impulse facility testing. The gauge was an acceleration compensated piezoelectric transducer and had a lowest natural frequency near 40 kHz. Turbulent skin-friction levels were measured to within a typical uncertainty of +/-7%. The systematic uncertainty in measured skin-friction coefficient was high for the tested laminar conditions; however, to within experimental uncertainty, the skin-friction and heat-transfer measurements were in agreement with the laminar theory of van Driest (1952). For predicting turbulent skin-friction coefficient, it was established that, for the range of Mach numbers and Reynolds numbers of the experiments, with cold walls and boundary layers approaching the turbulent equilibrium state, the Spalding & Chi (1964) method was the most suitable of the theories tested. It was also established that if the heat transfer rate to the wall is to be predicted, then the Spalding & Chi (1964) method should be used in conjunction with a Reynolds analogy factor near unity. If more accurate results are required, then an experimentally observed relationship between the Reynolds analogy factor and the skin-friction coefficient may be applied.

A comparative study of iterative Chebyshev and Lanczos implementations of the boundary inhomogeneity method for quantum scattering

We have recently developed a scaleable Artificial Boundary Inhomogeneity (ABI) method [Chem. Phys. Lett.366, 390–397 (2002)] based on the utilization of the Lanczos algorithm, and in this work explore an alternative iterative implementation based on the Chebyshev algorithm. Detailed comparisons between the two iterative methods have been made in terms of efficiency as well as convergence behavior. The Lanczos subspace ABI method was also further improved by the use of a simpler three-term backward recursion algorithm to solve the subspace linear system. The two different iterative methods are tested on the model collinear H+H2 reactive state-to-state scattering.

The Lake Tekapo experiment (LTEX): An investigation of atmospheric boundary layer processes in complex terrain

A research program on atmospheric boundary layer processes and local wind regimes in complex terrain was conducted in the vicinity of Lake Tekapo in the southern Alps of New Zealand, during two 1-month field campaigns in 1997 and 1999. The effects of the interaction of thermal and dynamic forcing were of specific interest, with a particular focus on the interaction of thermal forcing of differing scales. The rationale and objectives of the field and modeling program are described, along with the methodology used to achieve them. Specific research aims include improved knowledge of the role of surface forcing associated with varying energy balances across heterogeneous terrain, thermal influences on boundary layer and local wind development, and dynamic influences of the terrain through channeling effects. Data were collected using a network of surface meteorological and energy balance stations, radiosonde and pilot balloon soundings, tethered balloon and kite-based systems, sodar, and an instrumented light aircraft. These data are being used to investigate the energetics of surface heat fluxes, the effects of localized heating/cooling and advective processes on atmospheric boundary layer development, and dynamic channeling. A complementary program of numerical modeling includes application of the Regional Atmospheric Modeling System (RAMS) to case studies characterizing typical boundary layer structures and airflow patterns observed around Lake Tekapo. Some initial results derived from the special observation periods are used to illustrate progress made to date. In spite of the difficulties involved in obtaining good data and undertaking modeling experiments in such complex terrain, initial results show that surface thermal heterogeneity has a significant influence on local atmospheric structure and wind fields in the vicinity of the lake. This influence occurs particularly in the morning. However, dynamic channeling effects and the larger-scale thermal effect of the mountain region frequently override these more local features later in the day.

The K-Th derivatives of the immanant and the X-symmetric power of an operator

In recent papers, formulas are obtained for directional derivatives, of all orders, of the determinant, the permanent, the m-th compound map and the m-th induced power map. This paper generalizes these results for immanants and for other symmetric powers of a matrix.

The norm of the K- Th derivative of the X-symmetric power of an operator

In this paper, the exact value for the norm of directional derivatives, of all orders, for symmetric tensor powers of operators on finite dimensional vector spaces is presented. Using this result, an upper bound for the norm of all directional derivatives of immanants is obtained.

Thin films composed of gold nanoparticles dispersed in a dielectric matrix: the influence of the host matrix on the optical and mechanical responses

Gold nanoparticles were dispersed in two different dielectric matrices, TiO2 and Al2O3, using magnetron sputtering and a post-deposition annealing treatment. The main goal of the present work was to study how the two different host dielectric matrices, and the resulting microstructure evolution (including both the nanoparticles and the host matrix itself) promoted by thermal annealing, influenced the physical properties of the films. In particular, the structure and morphology of the nanocomposites were correlated with the optical response of the thin films, namely their localized surface plasmon resonance (LSPR) characteristics. Furthermore, and in order to scan the future application of the two thin film system in different types of sensors (namely biological ones), their functional behaviour (hardness and Young's modulus change) was also evaluated. Despite the similar Au concentrations in both matrices (~ 11 at.%), very different microstructural features were observed, which were found to depend strongly on the annealing temperature. The main structural differences included: (i) the early crystallization of the TiO2 host matrix, while the Al2O3 one remained amorphous up to 800 °C; (ii) different grain size evolution behaviours with the annealing temperature, namely an almost linear increase for the Au:TiO2 system (from 3 to 11 nm), and the approximately constant values observed in the Au:Al2O3 system (4–5 nm). The results from the nanoparticle size distributions were also found to be quite sensitive to the surrounding matrix, suggesting different mechanisms for the nanoparticle growth (particle migration and coalescence dominating in TiO2 and Ostwald ripening in Al2O3). These different clustering behaviours induced different transmittance-LSPR responses and a good mechanical stability, which opens the possibility for future use of these nanocomposite thin film systems in some envisaged applications (e.g. LSPR-biosensors).

Effect of acid or alkaline catalyst and of different capping agents on the optical properties of CdS nanoparticles incorporated within a diureasil hybrid matrix

CdS nanoparticles (NPs) were synthesized using colloidal methods and incorporated within a diureasil hybrid matrix. The surface capping of the CdS NPs by 3-mercaptopropyltrimethoxysilane (MPTMS) and 3-aminopropyltrimethoxysilane (APTMS) organic ligands during the incorporation of the NPs within the hybrid matrix has been investigated. The matrix is based on poly(ethylene oxide)/poly(propylene oxide) chains grafted to a siliceous skeleton through urea bonds and was produced by sol–gel process. Both alkaline and acidic catalysis of the sol–gel reaction were used to evaluate the effect of each organic ligand on the optical properties of the CdS NPs. The hybrid materials were characterized by absorption, steady-state and time-resolved photoluminescence spectroscopy and High Resolution Transmission Electron Microscopy (HR-TEM). The preservation of the optical properties of the CdS NPs within the diureasil hybrids was dependent on the experimental conditions used. Both organic ligands (APTMS and MPTMS) demonstrated to be crucial in avoiding the increase of size distribution and clustering of the NPs within the hybrid matrix. The use of organic ligands was also shown to influence the level of interaction between the hybrid host and the CdS NPs. The CdS NPs showed large Stokes shifts and long average lifetimes, both in colloidal solution and in the xerogels, due to the origin of the PL emission in surface states. The CdS NPs capped with MPTMS have lower PL lifetimes compared to the other xerogel samples but still larger than the CdS NPs in the original colloidal solution. An increase in PL lifetimes of the NPs after their incorporation within the hybrid matrix is related to interaction between the NPs and the hybrid host matrix.

Carbon isotope excursions and microfacies changes in marine Permian-Triassic boundary sections in Hungary

Several Permian-Triassic boundary sections occur in various structural units within Hungary. These sections represent different facies zones of the western Palaeotethys margin. The Gardony core in the NE part of the Transdanubian Range typically represents the inner ramp, while the Balvany section in the Bukk Mountains of northern Hungary represents an outer ramp setting. The two sections have different patterns for their delta(13)C values. The Balvany section shows a continuous change towards more negative delta(13)C values starting at the first biotic decline, followed by a sharp, quasi-symmetric negative peak at the second decline. The appearance of the delta(13)C peak has no relationship to the lithology and occurs within a shale with low overall carbonate content, indicating that the peak is not related to diagenesis or other secondary influences. Instead, the shift and the peak reflect primary processes related to changes in environmental conditions. The continuous shift in delta(13)C values is most probably related to a decrease in bioproductivity, whereas the sharp peak can be attributed to an addition of C strongly depleted in (13)C to the ocean-atmosphere system. The most plausible model is a massive release of methane-hydrate. The quasi-symmetric pattern suggests a rapid warming-cooling cycle or physical unroofing of sediments through slope-failure and releasing methane-hydrate. The Gidony-1 core shows a continuous negative delta(13)C shift starting below the P-T boundary. However, the detailed analyses revealed a sharp delta(13)C peak in the boundary interval, just below the major biotic decline, although its magnitude doesn't reach that observed in the Balvany section. Based on careful textural examination and high-resolution stable isotope microanalyses we suggest that the suppression of the delta(13)C peak that is common in the oolitic boundary sections is due to combined effects of condensed sedimentation, sediment reworking and erosion, as well as perhaps diagenesis. (c) 2005 Elsevier B.V All rights reserved.