Derivation of the solution of certain singular integral equations

It is shown that the a;P?lication of the Poincare-Bertrand fcm~ulaw hen made in a suitable manner produces the s~lutiano f certain singular integral equations very quickly, thc method of arriving at which, otherwise, is too complicaled. Two singular integral equations are considered. One of these quaiions is with a Cauchy-tyge kcrnel arid the other is an equalion which appears in the a a w guide theory and the theory of dishcations. Adifferent approach i? alw made here to solve the singular integralquation> of the waveguide theor? ind this i ~ v o l v eth~e use of the inversion formula of the Cauchy-type singular integral equahn and dudion to a system of TIilberl problems for two unknowns which can be dwupled wry easily to obi& tbe closed form solutim of the irilegral equatlou at band. The methods of the prescnt paper avoid all the complicaled approaches of solving the singular integral equaticn of the waveguide theory knowr todate.

Generalized Burgers equations and Euler-Painlev transcendents. I

Initial-value problems for the generalized Burgers equation (GBE) ut+u betaux+lambdaualpha =(delta/2)uxx are discussed for the single hump type of initial data both continuous and discontinuous. The numerical solution is carried to the self-similar ``intermediate asymptotic'' regime when the solution is given analytically by the self-similar form. The nonlinear (transformed) ordinary differential equations (ODE's) describing the self-similar form are generalizations of a class discussed by Euler and Painlevé and quoted by Kamke. These ODE's are new, and it is postulated that they characterize GBE's in the same manner as the Painlev equations categorize the Kortweg-de Vries (KdV) type. A connection problem for some related ODE's satisfying proper asymptotic conditions at x=±[infinity], is solved. The range of amplitude parameter is found for which the solution of the connection problem exists. The other solutions of the above GBE, which display several interesting features such as peaking, breaking, and a long shelf on the left for negative values of the damping coefficient lambda, are also discussed. The results are compared with those holding for the modified KdV equation with damping. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.

Soil carbon stock in the tropical rangelands of Australia: Effects of soil type and grazing pressure, and determination of sampling requirement.

On-going, high-profile public debate about climate change has focussed attention on how to monitor the soil organic carbon stock (C(s)) of rangelands (savannas). Unfortunately, optimal sampling of the rangelands for baseline C(s) - the critical first step towards efficient monitoring - has received relatively little attention to date. Moreover, in the rangelands of tropical Australia relatively little is known about how C(s) is influenced by the practice of cattle grazing. To address these issues we used linear mixed models to: (i) unravel how grazing pressure (over a 12-year period) and soil type have affected C(s) and the stable carbon isotope ratio of soil organic carbon (delta(13)C) (a measure of the relative contributions of C(3) and C(4) vegetation to C(s)); (ii) examine the spatial covariation of C(s) and delta(13)C; and, (iii) explore the amount of soil sampling required to adequately determine baseline C(s). Modelling was done in the context of the material coordinate system for the soil profile, therefore the depths reported, while conventional, are only nominal. Linear mixed models revealed that soil type and grazing pressure interacted to influence C(s) to a depth of 0.3 m in the profile. At a depth of 0.5 m there was no effect of grazing on C(s), but the soil type effect on C(s) was significant. Soil type influenced delta(13)C to a soil depth of 0.5 m but there was no effect of grazing at any depth examined. The linear mixed model also revealed the strong negative correlation of C(s) with delta(13)C, particularly to a depth of 0.1 m in the soil profile. This suggested that increased C(s) at the study site was associated with increased input of C from C(3) trees and shrubs relative to the C(4) perennial grasses; as the latter form the bulk of the cattle diet, we contend that C sequestration may be negatively correlated with forage production. Our baseline C(s) sampling recommendation for cattle-grazing properties of the tropical rangelands of Australia is to: (i) divide the property into units of apparently uniform soil type and grazing management; (ii) use stratified simple random sampling to spread at least 25 soil sampling locations about each unit, with at least two samples collected per stratum. This will be adequate to accurately estimate baseline mean C(s) to within 20% of the true mean, to a nominal depth of 0.3 m in the profile.

Risk factors associated with corneal nerve alteration in type 1 diabetes in the absence of neuropathy: A longitudinal in vivo corneal confocal microscopy study

Purpose The aim of this study was to determine alterations to the corneal subbasal nerve plexus (SNP) over four years using in vivo corneal confocal microscopy (IVCM) in participants with type 1 diabetes and to identify significant risk factors associated with these alterations. Methods A cohort of 108 individuals with type 1 diabetes and no evidence of peripheral neuropathy at enrollment underwent laser-scanning IVCM, ocular screening, and health and metabolic assessment at baseline and the examinations continued for four subsequent annual visits. At each annual visit, eight central corneal images of the SNP were selected and analyzed to quantify corneal nerve fiber density (CNFD), branch density (CNBD) and fiber length (CNFL). Linear mixed model approaches were fitted to examine the relationship between risk factors and corneal nerve parameters. Results A total of 96 participants completed the final visit and 91 participants completed all visits. No significant relationships were found between corneal nerve parameters and time, sex, duration of diabetes, smoking, alcohol consumption, blood pressure or BMI. However, CNFD was negatively associated with HbA1c (β=-0.76, P<0.01) and age (β=-0.13, P<0.01) and positively related to high density lipids (HDL) (β=2.01, P=0.03). Higher HbA1c (β=-1.58, P=0.04) and age (β=-0.23, P<0.01) also negatively impacted CNBD. CNFL was only affected by higher age (β=-0.06, P<0.01). Conclusions Glycemic control, HDL and age have significant effects on SNP structure. These findings highlight the importance of diabetic management to prevent corneal nerve damage as well as the capability of IVCM for monitoring subclinical alterations in the corneal SNP in diabetes.

A new type of high-order elements based on the mesh-free interpolations

We propose a new type of high-order elements that incorporates the mesh-free Galerkin formulations into the framework of finite element method. Traditional polynomial interpolation is replaced by mesh-free interpolations in the present high-order elements, and the strain smoothing technique is used for integration of the governing equations based on smoothing cells. The properties of high-order elements, which are influenced by the basis function of mesh-free interpolations and boundary nodes, are discussed through numerical examples. It can be found that the basis function has significant influence on the computational accuracy and upper-lower bounds of energy norm, when the strain smoothing technique retains the softening phenomenon. This new type of high-order elements shows good performance when quadratic basis functions are used in the mesh-free interpolations and present elements prove advantageous in adaptive mesh and nodes refinement schemes. Furthermore, it shows less sensitive to the quality of element because it uses the mesh-free interpolations and obeys the Weakened Weak (W2) formulation as introduced in [3, 5].

Steady mixed convection stagnation-point flow of upper convected Maxwell fluids with magnetic field

The steady MHD mixed convection flow of a viscoelastic fluid in the vicinity of two-dimensional stagnation point with magnetic field has been investigated under the assumption that the fluid obeys the upper-convected Maxwell (UCM) model. Boundary layer theory is used to simplify the equations of motion. induced magnetic field and energy which results in three coupled non-linear ordinary differential equations which are well-posed. These equations have been solved by using finite difference method. The results indicate the reduction in the surface velocity gradient, surface heat transfer and displacement thickness with the increase in the elasticity number. These trends are opposite to those reported in the literature for a second-grade fluid. The surface velocity gradient and heat transfer are enhanced by the magnetic and buoyancy parameters. The surface heat transfer increases with the Prandtl number, but the surface velocity gradient decreases.

Co(II) and Cr(III) complexes of formate-formamide mixed ligands: synthesis, structures, single crystal-to-single crystal transformation and magnetic behaviour

Solvothermal treatment of an equimolar mixture of Co(NO3)(2)center dot 6H(2)O, HCONH2 and NaN3 in MeOH at 100 degrees C yielded a three-dimensional NaCl type network Co(HCOO)(2)(HCONH2)(2) center dot HCONH2 (1a) containing formamides in the pores of the structure. Solvated pink 1a undergoes single crystal-to-single crystal (SCSC) transformation at 215 degrees C to form the desolvated dark brown product Co(HCOO)(2)-( HCONH2)(2) (1b) with the retention of the original framework. Reversible single crystal-to-single crystal transformation of 1b (brown) to 1a (pink) in the presence of excess formamide was also established at room temperature. The coordination environment around Co(II) in both 1a and 1b is octahedral with a CoN2O4 coordination composition. A similar reaction replacing Co(II) by Cr(III) produced a heterometallic 3D extended network Na[Cr(HCOO)(4)(HCONH2)(2)]center dot 2H(2)O (2a) at 100 degrees C. An increase in reaction temperature to 150 degrees C produced a simple mononuclear complex Cr(HCOO)(3)(HCONH2)(3) center dot 3H(2)O (2b). Variable temperature magnetic studies revealed the presence of a canting phenomena in both 1a and 1b, and hysteresis loop in the field dependent magnetisation plot at 2 K whereas complex 2a is simply paramagnetic in nature.

Conservation law with the flux function discontinuous in the space variable- II - Convex-concave type fluxes and generalized entropy solutions

We deal with a single conservation law with discontinuous convex-concave type fluxes which arise while considering sign changing flux coefficients. The main difficulty is that a weak solution may not exist as the Rankine-Hugoniot condition at the interface may not be satisfied for certain choice of the initial data. We develop the concept of generalized entropy solutions for such equations by replacing the Rankine-Hugoniot condition by a generalized Rankine-Hugoniot condition. The uniqueness of solutions is shown by proving that the generalized entropy solutions form a contractive semi-group in L-1. Existence follows by showing that a Godunov type finite difference scheme converges to the generalized entropy solution. The scheme is based on solutions of the associated Riemann problem and is neither consistent nor conservative. The analysis developed here enables to treat the cases of fluxes having at most one extrema in the domain of definition completely. Numerical results reporting the performance of the scheme are presented. (C) 2006 Elsevier B.V. All rights reserved.

Unsteady mixed convection on the stagnation-point flow adjacent to a vertical plate with a magnetic field

An analysis is performed to study the unsteady combined forced and free convection flow (mixed convection flow) of a viscous incompressible electrically conducting fluid in the vicinity of an axisymmetric stagnation point adjacent to a heated vertical surface. The unsteadiness in the flow and temperature fields is due to the free stream velocity, which varies arbitrarily with time. Both constant wall temperature and constant heat flux conditions are considered in this analysis. By using suitable transformations, the Navier-Stokes and energy equations with four independent variables (x, y, z, t) are reduced to a system of partial differential equations with two independent variables (eta, tau). These transformations also uncouple the momentum and energy equations resulting in a primary axisymmetric flow, in an energy equation dependent on the primary flow and in a buoyancy-induced secondary flow dependent on both primary flow and energy. The resulting system of partial differential equations has been solved numerically by using both implicit finite-difference scheme and differential-difference method. An interesting result is that for a decelerating free stream velocity, flow reversal occurs in the primary flow after certain instant of time and the magnetic field delays or prevents the flow reversal. The surface heat transfer and the surface shear stress in the primary flow increase with the magnetic field, but the surface shear stress in the buoyancy-induced secondary flow decreases. Further the heat transfer increases with the Prandtl number, but the surface shear stress in the secondary flow decreases.

Family of Mixed-Valence Oxovanadium(IV/V) Dinuclear Entities Incorporating N4O3-Coordinating Heptadentate Ligands: Synthesis, Structure, and EPR Spectra

Dinuclear ((VVV)-V-IV) oxophenoxovanadates of general formula [V2O3L] have been synthesized in excellent yields by reacting bis(acetylacetonato)oxovanadium(IV) with H3L in a 2:1 ratio in acetone under an N-2 atmosphere. Here L3- is the deprotonated form of 2,6-bis[{{(2-hydroxybenzyl)(N',N'-(dimethylamino)ethyl)}amino}methyl]-4-methylphenol (H3L1), 2,6-bis[{{(5-methyl-2-hydroxybenzyl)(N',N'-(dimethylamino)ethyl)}amino}methyl]-4-methylphenol (H3L2) 2,6-bis[ {{(5-tert-butyl-2-hydroxybenzyl)(N',N'-(dimethylamino)ethyl)}amino}methyl]-4-methylphenoI (H3L3), 2,6-bis[{{(5-chloro-2-hydroxybenzyl)(N',N'-(dimethylamino)ethyl)}amino}methyl]-4-methylphenol (H3L4) , 2,6-bis[{{(5-bromo-2-hydroxybenzyl)(N',N'-(dimethylamino)ethyl)}amino}methyl]-4-methylphenol (H3L5), or 2,6-bis[{{(5-methoxy-2-hydroxybenzyl)(N',N'-(dimethylamino)ethyl)}amino}methyl]-4-methylphenol (H3L6). In [V2O3L1], both the metal atoms have distorted octahedral geometry. The relative disposition of two terminal V=O groups in the complex is essentially cis. The O=V...V=O torsion angle is 24.6(2)degrees. The V-O-oxo-V and V-O-phenoxo-V angles are 117.5(4) and 93.4(3)degrees, respectively. The V...V bond distance is 3.173(5) Angstrom. X-ray crystallography, IR, UV-vis, and H-1 and V-51 NMR measurements show that the mixed-valence complexes contain two indistinguishable vanadium atoms (type 111). The thermal ellipsoids of O2, O4, C10, C14, and C15 also suggests a type III complex in the solid state. EPR spectra of solid complexes at 77 K display a single line indicating the localization of the odd electron (3d(xy)(1)). Valence localization at 77 K is also consistent with the V-51 hyperfine structure of the axial EPR spectra (3d(xy)(1) ground state) of the complexes in frozen (77 K) dichloromethane solution: S = 1/2, g(parallel to) similar to 1.94, g(perpendicular to) similar to 1.98, A(parallel to) similar to 166 x 10(-4) cm(-1), and A(perpendicular to) similar to 68 x 10(-4) cm(-1). In contrast isotropic room-temperature solution spectra of the family have 15 hyperfine lines (g(iso) similar to 1.974 and A(iso) similar to 50 x 10(-4) cm(-1)) revealing that the unpaired electron is delocalized between the metal centers. Crystal data for the [V2O3L1].CH2Cl2 complex are as follows: chemical formula, C32H43O6N4C12V2; crystal system, monoclinic; space group, C2/c; a = 18.461(4), b = 17.230(3), c = 13.700(3) Angstrom; beta = 117.88(3)degrees; Z = 8.

Numerical procedure for second order non-linear ordinary differential equations and application to heat transfer problem

In this paper, we have first given a numerical procedure for the solution of second order non-linear ordinary differential equations of the type y″ = f (x;y, y′) with given initial conditions. The method is based on geometrical interpretation of the equation, which suggests a simple geometrical construction of the integral curve. We then translate this geometrical method to the numerical procedure adaptable to desk calculators and digital computers. We have studied the efficacy of this method with the help of an illustrative example with known exact solution. We have also compared it with Runge-Kutta method. We have then applied this method to a physical problem, namely, the study of the temperature distribution in a semi-infinite solid homogeneous medium for temperature-dependent conductivity coefficient.

Unsteady mixed convection flow of a thermomicropolar fluid on a long thin vertical cylinder

The unsteady laminar mixed convection boundary layer flow of a thermomicropolar fluid over a long thin vertical cylinder has been studied when the free stream velocity varies with time. The coupled nonlinear partial differential equations with three independent variables governing the flow have been solved numerically using an implicit finite difference scheme in combination with the quasilinearization technique. The results show that the buoyancy, curvature and suction parameters, in general, enhance the skin friction, heat transfer and gradient of microrotation, but the effect of injection is just opposite. The skin friction and heat transfer for the micropolar fluid are considerably less than those for the Newtonian fluids. The effect of microrotation parameter is appreciable only on the microrotation gradient. The effect of the Prandtl number is appreciable on the skin friction, heat transfer and gradient of microtation.

Studies on optical absorption and photoluminescenceproperties of Pr3+ and Nd3+ doped mixed alkali chloroborate glasses

This article presents the optical absorption and emission properties of Pr3+ and Nd3+ doped two different mixed alkali chloroborate glass matrices of the type 70B(2)O(3)center dot xLiCl center dot(30 - x)NaCl and 70B(2)O(3)center dot xLiCl center dot(30 - x)KCl (x = 5, 10, 15.20 and 25). The variation of Judd-Ofelt parameters (Omega(2), Omega(4) and Omega(6)), total radiative transition probabilities (A(T)), radiative lifetimes (tau(R)) and emission cross-sections (sigma(p)) with the variation of alkali contents in the glass matrix have been discussed in detail. The changes in the peak wavelengths of the hypersensitive transition and intensity parameters with x are correlated to the structural changes in the host matrix. The estimated radiative lifetimes of certain excited states of Pr3+ and Nd3+ in these two glass matrices are reported. Peak stimulated emission cross-sections (sigma(p)) are reported for the observed emission transitions of Pr3+ and Nd3+ ions. Branching ratios (beta) of the observed emission transitions obtained from the Judd-Ofelt theory are compared with the values obtained from the emission spectra. (C) 2010 Elsevier B.V. All rights reserved.

Nonclassical rotational inertia in a two-dimensional bosonic solid containing grain boundaries

We study the occurrence of nonclassical rotational inertia (NCRI) arising from superfluidity along grain boundaries in a two-dimensionalbosonic system. We make use of a standard mapping between the zero-temperature properties of this system and the statistical mechanics of interacting vortex lines in the mixed phase of a type-II superconductor. In the mapping, the liquid phase of the vortex system corresponds to the superfluid bosonic phase. We consider numerically obtained polycrystalline configurations of the vortex lines in which the microcrystals are separated by liquidlike grain-boundary regions which widen as the vortex system temperature increases. The NCRI of the corresponding zero-temperature bosonic systems can then be numerically evaluated by solving the equations of superfluid hydrodynamics in the channels near the grain boundaries. We find that the NCRI increases very abruptly as the liquid regions in the vortex system (equivalently, superfluid regions in the bosonic system) form a connected, system-spanning structure with one or more closed loops. The implications of these results for experimentally observed supersolid phenomena are discussed.

Unsteady mixed convection with double diffusion over a horizontal cylinder and a sphere within a porous medium

The combined effects of the permeability of the medium, magnetic field, buoyancy forces and dissipation on the unsteady mixed convection flow over a horizontal cylinder and a sphere embedded in a porous medium have been studied. The nonlinear coupled partial differential equations with three independent variables have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. The skin friction, heat transfer and mass transfer increase with the permeability of the medium, magnetic field and buoyancy parameter. The heat and mass transfer continuously decrease with the stream-wise distance, whereas the skin friction increases from zero, attains a maximum and then decreases to zero. The skin friction, heat transfer and mass transfer are significantly affected by the free stream velocity distribution. The effect of dissipation parameter is found to be more pronounced on the heat transfer than on the skin friction and mass transfer