On multiple solutions of the exterior neumann problem involving critical sobolev exponent

In this paper we consider the exterior Neumann problem involving a critical Sobolev exponent. We establish the existence of two solutions having a prescribed limit at infinity.

Dissolution of porcine incisor pulps in sodium hypochlorite solutions of varying compositions and concentrations

Background: The solubility of dental pulp tissue in sodium hypochlorite has been extensively investigated but results have been inconsistent; due most likely to variations in experimental design, the volume and/or rate of replenishment of the solutions used and the nature of the tissues assessed. Traditionally, the sodium hypochlorite solutions used for endodontic irrigation in Australia have been either Milton or commercial bleach, with Milton being the most common. Recently, a range of Therapeutic Goods Administration (TGA) approved proprietary sodium hypochlorite solutions, which contain surfactant, has become available. Some domestic chlorine bleaches now also contain surfactants. The purpose of this study was to perform new solubility assessments, comparing Milton with new TGA approved products, Hypochlor 1% and Hypochlor 4% forte, and with a domestic bleach containing surfactant (White King). Methods: Ten randomly assigned pulp samples of porcine dental pulp of approximately equal dimensions were immersed in the above solutions, as well as representative concentrations of sodium hydroxide. Time to complete dissolution was measured and assessed statistically. Results: White King 4% showed the shortest dissolution time, closely followed by Hypochlor 4% forte. White King 1% and Hypochlor 1% each took around three times as long to completely dissolve the samples of pulp as their respective 4% concentrations, while Milton took nearly 10 times as long. The sodium hydroxide solutions showed no noticeable dissolution of the pulp samples. Conclusions: The composition and content of sodium hypochlorite solutions had a profound effect on the ability of these solutions to dissolve pulp tissue in vitro. Greater concentrations provided more rapid dissolution of tissue. One per cent solutions with added surfactant and which contained higher concentrations of sodium hydroxide were significantly more effective in dissolution of pulp tissue than Milton.

Typical kernel size and number of sparse random matrices over Galois fields: a statistical physics approach

Using methods of statistical physics, we study the average number and kernel size of general sparse random matrices over GF(q), with a given connectivity profile, in the thermodynamical limit of large matrices. We introduce a mapping of GF(q) matrices onto spin systems using the representation of the cyclic group of order q as the q-th complex roots of unity. This representation facilitates the derivation of the average kernel size of random matrices using the replica approach, under the replica symmetric ansatz, resulting in saddle point equations for general connectivity distributions. Numerical solutions are then obtained for particular cases by population dynamics. Similar techniques also allow us to obtain an expression for the exact and average number of random matrices for any general connectivity profile. We present numerical results for particular distributions.

Comparative performance of analytical techniques for polyhexamethylene biguanide (PHMB) in ophthalmic solutions

Purpose: The use of PHMB as a disinfectant in contact lens multipurpose solutions has been at the centre of much debate in recent times, particularly in relation to the issue of solution induced corneal staining. Clinical studies have been carried out which suggest different effects with individual contact lens materials used in combination with specific PHMB containing care regimes. There does not appear to be, however, a reliable analytical technique that would detect and quantify with any degree of accuracy the specific levels of PHMB that are taken up and released from individual solutions by the various contact lens materials. Methods: PHMB is a mixture of positively charged polymer units of varying molecular weight that has maximum absorbance wavelength of 236 nm. On the basis of these properties a range of assays including capillary electrophoresis, HPLC, a nickelnioxime colorimetric technique, mass spectrophotometry, UV spectroscopy and ion chromatography were assessed paying particular attention to each of their constraints and detection levels. Particular interest was focused on the relative advantage of contactless conductivity compared to UV and mass spectrometry detection in capillary electrophoresis (CE). This study provides an overview of the comparative performance of these techniques. Results: The UV absorbance of PHMB solutions, ranging from 0.0625 to 50 ppm was measured at 236 nm. Within this range the calibration curve appears to be linear however, absorption values below 1 ppm (0.0001%) were extremely difficult to reproduce. The concentration of PHMB in solutions is in the range of 0.0002–0.00005% and our investigations suggest that levels of PHMB below 0.0001% (levels encountered in uptake and release studies) can not be accurately estimated, in particular when analysing complex lens care solutions which can contain competitively absorbing, and thus interfering, species in the solution. The use of separative methodologies, such as CE using UV detection alone is similarly limited. Alternative techniques including contactless conductivity detection offer greater discrimination in complex solutions together with the opportunity for dual channel detection. Preliminary results achieved by TraceDec1 contactless conductivity detection, (Gain 150%, Offset 150) in conjunction with the Agilent capillary electrophoresis system using a bare fused silica capillary (extended light path, 50 mid, total length 64.5 cm, effective length 56 cm) and a cationic buffer at pH 3.2, exhibit great potential with reproducible PHMB split peaks. Conclusions: PHMB-based solutions are commonly associated with the potential to invoke corneal staining in combination with certain contact lens materials. However this terminology ‘PHMBbased solution’ is used primarily because PHMB itself has yet to be adequately implicated as the causative agent of the staining and compromised corneal cell integrity. The lack of well characterised adequately sensitive assays, coupled with the range of additional components that characterise individual care solutions pose a major barrier to the investigation of PHMB interactions in the lenswearing eye.

Genetic studies of Cu-Pb-Zn mineralisation in Triassic red beds of Western Europe

Continental red bed sequences are host, on a worldwide scale, to a characteristic style of mineralisation which is dominated by copper, lead, zinc, uranium and vanadium. This study examines the features of sediment-hosted ore deposits in the Permo-Triassic basins of Western Europe, with particular reference to the Cu-Pb-Zn-Ba mineralisation in the Cheshire Basin, northwest England, the Pb-Ba-F deposits of the Inner Moray Firth Basin, northeast Scotland, and the Pb-rich deposits of the Eifel and Oberpfalz regions, West Germany. The deposits occur primarily but not exclusively in fluvial and aeolian sandstones on the margins of deep, avolcanic sedimentary basins containing red beds, evaporites and occasionally hydrocarbons. The host sediments range in age from Permian to Rhaetian and often contain (or can be inferred to have originally contained) organic matter. Textural studies have shown that early diagenetic quartz overgrowths precede the main episode of sulphide deposition. Fluid inclusion and sulphur isotope data have significantly constrained the genetic hypotheses for the mineralisation and a model involving the expulsion of diagenetic fluids and basinal brines up the faulted margins of sedimentary basins is favoured. Consideration of the development of these sedimentary basins suggests that ore emplacement occurred during the tectonic stage of basin evolution or during basin inversion in the Tertiary. ð34S values for barite in the Cheshire Basin range from 13.8% to 19.3% and support the theory that the Upper Triassic evaporites were the principal sulphur source for the mineralisation and provided the means by which mineralising fluids became saline. In contrast, δ34S values for barite in the Inner Moray Firth Basin (mean δ34S = + 29%) are not consistent with simple derivation of sulphur from the evaporite horizons in the basin and it is likely that sulphur-rich Jurassic shales supplied the sulphur for the mineralisation at Elgin. Possible sources of sulphur for the mineralisation in West Germany include hydrothermal vein sulphides in the underlying Devonian sediments and evaporites in the overlying Muschelkalk. Textural studies of the deeply buried sandstones in the Cheshire Basin reveal widespread dissolution and replacement of detrital phases and support the theory that red bed diagenetic processes are responsible for the release of metals into pore fluids. The ore solutions are envisaged as being warm (60-150%C), saline (9-22 wt % equiv NaCl) fluids in which metals were transported as chloride complexes. The distribution of δ34S values for sulphides in the Cheshire Basin (-1.8% to + 16%), the Moray Firth Basin (-4.8% to + 27%) and the German Permo-Triassic Basins (-22.2% to -12.2%) preclude a magmatic source for the sulphides and support the contention that sulphide precipitation is thought to result principally from sulphate reduction processes, although a decrease in temperature of the ore fluid or reaction with carbonates may also be important. Methane is invoked as the principal reducing agent in the Cheshire Basin, whilst terrestrial organic debris and bacterial reduction processes are thought to have played a major part in the genesis of the German ore deposits.

Integral Manifolds and Bounded Solutions of Singularly Perturbed Systems of Impulsive Differential Equations

Sufficient conditions for the existence of bounded solutions of singularly perturbed impulsive differential equations are obtained. For this purpose integral manifolds are used.

Continuous Dependence of Solutions of Quasidifferential Equations with Non-Fixed Time of Impulses

In this article on quasidifferential equation with non-fixed time of impulses we consider the continuous dependence of the solutions on the initial conditions as well as the mappings defined by these equations. We prove general theorems for quasidifferential equations from which follows corresponding results for differential equations, differential inclusion and equations with Hukuhara derivative.

Extremal Solutions for a Class of Functional Differential Equations

We study, in Carathéodory assumptions, existence, continuation and continuous dependence of extremal solutions for an abstract and rather general class of hereditary differential equations. By some examples we prove that, unlike the nonfunctional case, solved Cauchy problems for hereditary differential equations may not have local extremal solutions.

Explicit Solutions of Nonlocal Boundary Value Problems, Containing Bitsadze-Samarskii Constraints

MSC 2010: 44A35, 35L20, 35J05, 35J25

Exact Solutions of Nonlocal Boundary Value Problems for One- and Two-Dimensional Heat Equation

Иван Хр. Димовски, Юлиан Ц. Цанков - Предложен е метод за намиране на явни решения на клас двумерни уравнения на топлопроводността с нелокални условия по пространствените променливи. Методът е основан на директно тримерно операционно смятане. Класическата дюамелова конволюция е комбинирана с две некласически конволюции за операторите ∂xx и ∂yy в една тримерна конволюция. Съответното операционно смятане използва мултипликаторни частни. Мултипликаторните частни позволяват да се продължи принципът на Дюамел за пространствените променливи и да се намерят явни решения на разглежданите гранични задачи. Общите разглеждания са приложени в случая на гранични условия от типа на Йонкин. Намерени са експлицитни решения в затворен вид.

Singular Solutions of Protter’s Problem for a Class of 3-D Hyperbolic Equations

Недю Иванов Попиванов, Алексей Йорданов Николов - През 1952 г. М. Протър формулира нови гранични задачи за вълновото уравнение, които са тримерни аналози на задачите на Дарбу в равнината. Задачите са разгледани в тримерна област, ограничена от две характеристични конуса и равнина. Сега, след като са минали повече от 50 години, е добре известно, че за безброй гладки функции в дясната страна на уравнението тези задачи нямат класически решения, а обобщеното решение има силна степенна особеност във върха на характеристичния конус, която е изолирана и не се разпространява по конуса. Тук ние разглеждаме трета гранична задача за вълновото уравнение с младши членове и дясна страна във формата на тригонометричен полином. Дадена е по-нова от досега известната априорна оценка за максимално възможната особеност на решенията на тази задача. Оказва се, че при по-общото уравнение с младши членове възможната сингулярност е от същия ред като при чисто вълновото уравнение.

Remarks on blow up time for solutions of a nonlinear diffusion system with time dependent coefficients

2000 Mathematics Subject Classification: 35K55, 35K60.

Random extremal solutions of differential inclusions

Given a Lipschitz continuous multifunction $F$ on ${\mathbb{R}}^{n}$, we construct a probability measure on the set of all solutions to the Cauchy problem $\dot x\in F(x)$ with $x(0)=0$. With probability one, the derivatives of these random solutions take values within the set $ext F(x)$ of extreme points for a.e.~time $t$. This provides an alternative approach in the analysis of solutions to differential inclusions with non-convex right hand side.

Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators: the Riemann-Liouville case

In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator $\Delta_+^{(\alpha,\beta,\gamma)}:= D_{x_0^+}^{1+\alpha} +D_{y_0^+}^{1+\beta} +D_{z_0^+}^{1+\gamma},$ where $(\alpha, \beta, \gamma) \in \,]0,1]^3$, and the fractional derivatives $D_{x_0^+}^{1+\alpha}$, $D_{y_0^+}^{1+\beta}$, $D_{z_0^+}^{1+\gamma}$ are in the Riemann-Liouville sense. Applying operational techniques via two-dimensional Laplace transform we describe a complete family of eigenfunctions and fundamental solutions of the operator $\Delta_+^{(\alpha,\beta,\gamma)}$ in classes of functions admitting a summable fractional derivative. Making use of the Mittag-Leffler function, a symbolic operational form of the solutions is presented. From the obtained family of fundamental solutions we deduce a family of fundamental solutions of the fractional Dirac operator, which factorizes the fractional Laplace operator. We apply also the method of separation of variables to obtain eigenfunctions and fundamental solutions.

Enhancing the numerical performance of fire field models (CMS Paper)

This paper describes two new techniques designed to enhance the performance of fire field modelling software. The two techniques are "group solvers" and automated dynamic control of the solution process, both of which are currently under development within the SMARTFIRE Computational Fluid Dynamics environment. The "group solver" is a derivation of common solver techniques used to obtain numerical solutions to the algebraic equations associated with fire field modelling. The purpose of "group solvers" is to reduce the computational overheads associated with traditional numerical solvers typically used in fire field modelling applications. In an example, discussed in this paper, the group solver is shown to provide a 37% saving in computational time compared with a traditional solver. The second technique is the automated dynamic control of the solution process, which is achieved through the use of artificial intelligence techniques. This is designed to improve the convergence capabilities of the software while further decreasing the computational overheads. The technique automatically controls solver relaxation using an integrated production rule engine with a blackboard to monitor and implement the required control changes during solution processing. Initial results for a two-dimensional fire simulation are presented that demonstrate the potential for considerable savings in simulation run-times when compared with control sets from various sources. Furthermore, the results demonstrate the potential for enhanced solution reliability due to obtaining acceptable convergence within each time step, unlike some of the comparison simulations.