Analysis of Melanoma Onset: Assessing Familial Aggregation by Using Estimating Equations and Fitting Variance Components via Bayesian Random Effects Models

We investigate whether relative contributions of genetic and shared environmental factors are associated with an increased risk in melanoma. Data from the Queensland Familial Melanoma Project comprising 15,907 subjects arising from 1912 families were analyzed to estimate the additive genetic, common and unique environmental contributions to variation in the age at onset of melanoma. Two complementary approaches for analyzing correlated time-to-onset family data were considered: the generalized estimating equations (GEE) method in which one can estimate relationship-specific dependence simultaneously with regression coefficients that describe the average population response to changing covariates; and a subject-specific Bayesian mixed model in which heterogeneity in regression parameters is explicitly modeled and the different components of variation may be estimated directly. The proportional hazards and Weibull models were utilized, as both produce natural frameworks for estimating relative risks while adjusting for simultaneous effects of other covariates. A simple Markov Chain Monte Carlo method for covariate imputation of missing data was used and the actual implementation of the Bayesian model was based on Gibbs sampling using the free ware package BUGS. In addition, we also used a Bayesian model to investigate the relative contribution of genetic and environmental effects on the expression of naevi and freckles, which are known risk factors for melanoma.

Predictive regression equations and clinical uses of peripheral pulse timing characteristics in children

Studies have shown that increased arterial stiffening can be an indication of cardiovascular diseases like hypertension. In clinical practice, this can be detected by measuring the blood pressure (BP) using a sphygmomanometer but it cannot be used for prolonged monitoring. It has been established that pulse wave velocity (PWV) is a direct measure of arterial stiffening but its usefulness is hampered by the absence of non-invasive techniques to estimate it. Pulse transit time (PTT) is a simple and non-invasive method derived from PWV. However, limited knowledge of PTT in children is found in the present literature. The aims of this study are to identify independent variables that confound PTT measure and describe PTT regression equations for healthy children. Therefore, PTT reference values are formulated for future pathological studies. Fifty-five Caucasian children (39 male) aged 8.4 +/- 2.3 yr (range 5-12 yr) were recruited. Predictive equations for PTT were obtained by multiple regressions with age, vascular path length, BP indexes and heart rate. These derived equations were compared in their PWV equivalent against two previously reported equations and significant agreement was obtained (p < 0.05). Findings herein also suggested that PTT can be useful as a continuous surrogate BP monitor in children.

Multiplicity results for second-order two-point boundary value problems with nonlinearities across several eigenvalues

We consider the boundary value problems for nonlinear second-order differential equations of the form u '' + a(t)f (u) = 0, 0 < t < 1, u(0) = u (1) = 0. We give conditions on the ratio f (s)/s at infinity and zero that guarantee the existence of solutions with prescribed nodal properties. Then we establish existence and multiplicity results for nodal solutions to the problem. The proofs of our main results are based upon bifurcation techniques. (c) 2004 Elsevier Ltd. All rights reserved.

Optical Bloch equations and enhanced decay of Rabi oscillations in strong driving fields

Optical Bloch equations are widely used for describing dynamics in a system consisting molecules, electromagnetic waves, and a thermal bath. We analyze applicability of these equations to a single molecule imbedded in a solid matrix. Classical Bloch equations and the limits of their applicability are derived from more general master equations. Simple and intuitively appealing picture based on stochastic Bloch equations shows that at low temperatures, contrary to common believes, a strong driving field can not only suppress but can also increase decay rates of Rabi oscillations. A physical system where predicted effects can be observed experimentally is suggested. (c) 2005 Elsevier B.V. All rights reserved.

Approximate Bayesian techniques for inference in stochastic dynamical systems

This thesis is concerned with approximate inference in dynamical systems, from a variational Bayesian perspective. When modelling real world dynamical systems, stochastic differential equations appear as a natural choice, mainly because of their ability to model the noise of the system by adding a variant of some stochastic process to the deterministic dynamics. Hence, inference in such processes has drawn much attention. Here two new extended frameworks are derived and presented that are based on basis function expansions and local polynomial approximations of a recently proposed variational Bayesian algorithm. It is shown that the new extensions converge to the original variational algorithm and can be used for state estimation (smoothing). However, the main focus is on estimating the (hyper-) parameters of these systems (i.e. drift parameters and diffusion coefficients). The new methods are numerically validated on a range of different systems which vary in dimensionality and non-linearity. These are the Ornstein-Uhlenbeck process, for which the exact likelihood can be computed analytically, the univariate and highly non-linear, stochastic double well and the multivariate chaotic stochastic Lorenz '63 (3-dimensional model). The algorithms are also applied to the 40 dimensional stochastic Lorenz '96 system. In this investigation these new approaches are compared with a variety of other well known methods such as the ensemble Kalman filter / smoother, a hybrid Monte Carlo sampler, the dual unscented Kalman filter (for jointly estimating the systems states and model parameters) and full weak-constraint 4D-Var. Empirical analysis of their asymptotic behaviour as a function of observation density or length of time window increases is provided.

Simultaneous biochemical reaction and separation in a continuous rotating annular chromatograph

The aim of this work has been to investigate the behaviour of a continuous rotating annular chromatograph (CRAC) under a combined biochemical reaction and separation duty. Two biochemical reactions have been employed, namely the inversion of sucrose to glucose and fructose in the presence of the enzyme invertase and the saccharification of liquefied starch to maltose and dextrin using the enzyme maltogenase. Simultaneous biochemical reaction and separation has been successfully carried out for the first time in a CRAC by inverting sucrose to fructose and glucose using the enzyme invertase and collecting continuously pure fractions of glucose and fructose from the base of the column. The CRAC was made of two concentric cylinders which form an annulus 140 cm long by 1.2 cm wide, giving an annular space of 14.5 dm3. The ion exchange resin used was an industrial grade calcium form Dowex 50W-X4 with a mean diameter of 150 microns. The mobile phase used was deionised and dearated water and contained the appropriate enzyme. The annular column was slowly rotated at speeds of up to 240°h-1 while the sucrose substrate was fed continuously through a stationary feed pipe to the top of the resin bed. A systematic investigation of the factors affecting the performance of the CRAC under simultaneous biochemical reaction and separation conditions was carried out by employing a factorial experimental procedure. The main factors affecting the performance of the system were found to be the feed rate, feed concentrations and eluent rate. Results from the experiments indicated that complete conversion could be achieved for feed concentrations of up to 50% w/v sucrose and at feed throughputs of up to 17.2 kg sucrose per m3 resin/h. The second enzymic reaction, namely the saccharification of liquefied starch to maltose employing the enzyme maltogenase has also been successfully carried out on a CRAC. Results from the experiments using soluble potato starch showed that conversions of up to 79% were obtained for a feed concentration of 15.5% w/v at a feed flowrate of 400 cm3/h. The product maltose obtained was over 95% pure. Mathematical modelling and computer simulation of the sucrose inversion system has been carried out. A finite difference method was used to solve the partial differential equations and the simulation results showed good agreement with the experimental results obtained.

Bioreaction and separation in batch chromatographic columns

The objective of this work has been to study the behaviour and performance of a batch chromatographic column under simultaneous bioreaction and separation conditions for several carbohydrate feedstocks. Four bioreactions were chosen, namely the hydrolysis of sucrose to glucose and fructose using the enzyme invertase, the hydrolysis of inulin to fructose and glucose using inulinase, the hydrolysis of lactose to glucose and galactose using lactase and the isomerization of glucose to fructose using glucose isomerase. The chromatographic columns employed were jacketed glass columns ranging from 1 m to 2 m long and the internal diameter ranging from 0.97 cm to 1.97 cm. The stationary phase used was a cation exchange resin (PUROLITE PCR-833) in the Ca2+ form for the hydrolysis and the Mg2+ form for the isomerization reactions. The mobile phase used was a diluted enzyme solution which was continuously pumped through the chromatographic bed. The substrate was injected at the top of the bed as a pulse. The effect of the parameters pulse size, the amount of substrate solution introduced into the system corresponding to a percentage of the total empty column volume (% TECV), pulse concentration, eluent flowrate and the enzyme activity of the eluent were investigated. For the system sucrose-invertase complete conversions of substrate were achieved for pulse sizes and pulse concentrations of up to 20% TECV and 60% w/v, respectively. Products with purity above 90% were obtained. The enzyme consumption was 45% of the amount theoretically required to produce the same amount of product as in a conventional batch reactor. A value of 27 kg sucrose/m3 resin/h for the throughput of the system was achieved. The systematic investigation of the factors affecting the performance of the batch chromatographic bioreactor-separator was carried out by employing a factorial experimental procedure. The main factors affecting the performance of the system were the flowrate and enzyme activity. For the system inulin-inulinase total conversions were also obtained for pulses sizes of up to 20 % TECV and a pulse concentration of 10 % w/v. Fructose rich fractions with 100 % purity and representing up to 99.4 % of the total fructose generated were obtained with an enzyme consumption of 32 % of the amount theoretically required to produce the same amount of product in a conventional batch reactor. The hydrolysis of lactose by lactase was studied in the glass columns and also in an SCCR-S unit adapted for batch operation, in co-operation with Dr. Shieh, a fellow researcher in the Chemical Engineering and Applied Chemistry Department at Aston University. By operating at up to 30 % w/v lactose feed concentrations complete conversions were obtained and the purities of the products generated were above 90%. An enzyme consumption of 48 % of the amount theoretically required to produce the same amount of product in a conventional batch reactor was achieved. On working with the system glucose-glucose isomerase, which is a reversible reaction, the separation obtained with the stationary phase conditioned in the magnesium form was very poor although the conversion obtained was compatible with those for conventional batch reactors. By working with a mixed pulse of enzyme and substrate, up to 82.5 % of the fructose generated with a purity of 100 % was obtained. The mathematical modelling and computer simulation of the batch chromatographic bioreaction-separation has been performed on a personal computer. A finite difference method was used to solve the partial differential equations and the simulation results showed good agreement with the experimental results.

Physics and mathematics of dispersion-managed optical solitons

We review the main physical and mathematical properties of dispersion-managed (DM) optical solitons. Theory of DM solitons can be presented at two levels of accuracy: first, simple, but nevertheless, quantitative models based on ordinary differential equations governing evolution of the soliton width and phase parameter (the so-called chirp); and second, a comprehensive path-average theory that is capable of describing in detail both the fine structure of DM soliton form and its evolution along the fiber line. An analogy between DM soliton and a macroscopic nonlinear quantum oscillator model is also discussed. © 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. All rights reserved.

A Differential Game Described by a Hyperbolic System

An antagonistic differential game of hyperbolic type with a separable linear vector pay-off function is considered. The main result is the description of all ε-Slater saddle points consisting of program strategies, program ε-Slater maximins and minimaxes for each ε ∈ R^N > for this game. To this purpose, the considered differential game is reduced to find the optimal program strategies of two multicriterial problems of hyperbolic type. The application of approximation enables us to relate these problems to a problem of optimal program control, described by a system of ordinary differential equations, with a scalar pay-off function. It is found that the result of this problem is not changed, if the players use positional or program strategies. For the considered differential game, it is interesting that the ε-Slater saddle points are not equivalent and there exist two ε-Slater saddle points for which the values of all components of the vector pay-off function at one of them are greater than the respective components of the other ε-saddle point.

On a Differential Equation with Left and Right Fractional Derivatives

Mathematics Subject Classification: 26A33; 70H03, 70H25, 70S05; 49S05

Properties of Lp (K)-Solutions of Linear Nonhomogeneous Impulsive Differential Equations with Unbounded Linear Operator

Sufficient conditions for the existence of Lp(k)-solutions of linear nonhomogeneous impulsive differential equations with unbounded linear operator are found. An example of the theory of the linear nonhomogeneous partial impulsive differential equations of parabolic type is given.

Solutions of Tikhonov functional equations and applications to multiplication operators on Szegö spaces

We consider a natural representation of solutions for Tikhonov functional equations. This will be done by applying the theory of reproducing kernels to the approximate solutions of general bounded linear operator equations (when defined from reproducing kernel Hilbert spaces into general Hilbert spaces), by using the Hilbert-Schmidt property and tensor product of Hilbert spaces. As a concrete case, we shall consider generalized fractional functions formed by the quotient of Bergman functions by Szegö functions considered from the multiplication operators on the Szegö spaces.

Análise estequiométrica da dinâmica não-linear de redes de reações químicas

Dissertação (mestrado)—Universidade de Brasília, Instituto de Química, Programa de Pós-Graduação em Química, 2016.