Interconversion of Prony series for relaxation and creep

Various algorithms have been proposed to solve the interconversion equation of linear viscoelasticity when Prony series are used for the relaxation and creep moduli, G(t) and J(t). With respect to a Prony series for G(t), the key step in recovering the corresponding Prony series for J(t) is the determination of the coefficients {jk} of terms in J(t). Here, the need to solve a poorly conditioned matrix equation for the {jk} is circumvented by deriving elementary and easily evaluated analytic formulae for the {jk} in terms of the derivative dG(s)/ds of the Laplace transform G(s) of G(t)

On the scaling of molecular weight distribution functionals

When formulating a constitutive equation model or a mixing rule for some synthetic or biological polymer, one is essentially solving an inverse problem. However, the data will not only include the results obtained from simple step strain, oscillatory shear, elongational, and other experiments, but also information about the molecular weight scaling of key rheological parameters (i.e., molecular weight distribution functionals) such as zero-shear viscosity, steady-state compliance, and the normal stress differences. In terms of incorporating such scaling information into the formulation of models, there is a need to understand the relationship between various models and their molecular weight scaling, since such information identifies the ways in which molecular weight scaling constrains the choice of possible models. In Anderssen and Mead (1998) it was established formally that the members of a quite general class of reptation mixing rules all had the same molecular weight scaling. The purpose of this paper is to first introduce the concept of a generalized reptation mixing rule, which greatly extends the class examined by Anderssen and Mead, and then show that all such rules have the same molecular weight scaling. The proof is similar to that given by Anderssen and Mead, but uses the implicit function theorem to establish the uniqueness of the mean values which arise when invoking various integral mean-value representations for the molecular weight distribution functionals considered. The rheological significance of the new generalized two-parameter mixing rule, proposed in this paper, is examined in some detail in the conclusions. In particular, it is used to established how one must construct a mixing rule for a general polydispersed polymer where the molecular dynamics involves some single, some double and some higher levels of multiple reptation. The work of Maier et al. (1998) and Thimm et al. (2000) is then utilized to illustrate and validate this proposal

Evolution: Complexity, uncertainty and innovation

Complexity science provides a general mathematical basis for evolutionary thinking. It makes us face the inherent, irreducible nature of uncertainty and the limits to knowledge and prediction. Complex, evolutionary systems work on the basis of on-going, continuous internal processes of exploration, experimentation and innovation at their underlying levels. This is acted upon by the level above, leading to a selection process on the lower levels and a probing of the stability of the level above. This could either be an organizational level above, or the potential market place. Models aimed at predicting system behaviour therefore consist of assumptions of constraints on the micro-level – and because of inertia or conformity may be approximately true for some unspecified time. However, systems without strong mechanisms of repression and conformity will evolve, innovate and change, creating new emergent structures, capabilities and characteristics. Systems with no individual freedom at their lower levels will have predictable behaviour in the short term – but will not survive in the long term. Creative, innovative, evolving systems, on the other hand, will more probably survive over longer times, but will not have predictable characteristics or behaviour. These minimal mechanisms are all that are required to explain (though not predict) the co-evolutionary processes occurring in markets, organizations, and indeed in emergent, evolutionary communities of practice. Some examples will be presented briefly

Derivative spectroscopy and the continuous relaxation spectrum

Derivative spectroscopy is conventionally understood to be a collection of techniques for extracting fine structure from spectroscopic data by means of numerical differentiation. In this paper we extend the conventional interpretation of derivative spectroscopy with a view to recovering the continuous relaxation spectrum of a viscoelastic material from oscillatory shear data. To achieve this, the term “spectroscopic data” is allowed to include spectral data which have been severely broadened by the action of a strong low-pass filter. Consequently, a higher order of differentiation than is usually encountered in conventional derivative spectroscopy is required. However, by establishing a link between derivative spectroscopy and wavelet decomposition, high-order differentiation of oscillatory shear data can be achieved using specially constructed wavelet smoothing. This method of recovery is justified when the reciprocal of the Fourier transform of the filter function (convolution kernel) is an entire function, and is particularly powerful when the associated Maclaurin series converges rapidly. All derivatives are expressed algebraically in terms of scaling functions and wavelets of different scales, and the recovered relaxation spectrum is expressible in analytic form. An important feature of the method is that it facilitates local recovery of the spectrum, and is therefore appropriate for real scenarios where the oscillatory shear data is only available for a finite range of frequencies. We validate the method using synthetic data, but also demonstrate its use on real experimental data

Discretization of variational regularization in Banach spaces

Consider a nonlinear ill-posed operator equation $F(u)=y$ where $F$ is defined on a Banach space $X$. In general, for solving this equation numerically, a finite dimensional approximation of $X$ and an approximation of $F$ are required. Moreover, in general the given data \yd of $y$ are noisy. In this paper we analyze finite dimensional variational regularization, which takes into account operator approximations and noisy data: We show (semi-)convergence of the regularized solution of the finite dimensional problems and establish convergence rates in terms of Bregman distances under appropriate sourcewise representation of a solution of the equation. The more involved case of regularization in nonseparable Banach spaces is discussed in detail. In particular we consider the space of finite total variation functions, the space of functions of finite bounded deformation, and the $L^\infty$--space

LHC Communication Infrastructure: Recommendations from the working group

The LHC Working Group for Communication Infrastructure (CIWG) was established in May 1999 with members from the accelerator sector, the LHC physics experiments, the general communication services, the technical services and other LHC working groups. It has spent a year collecting user requirements and at the same time explored and evaluated possible solutions appropriate to the LHC. A number of technical recommendations were agreed, and areas where more work is required were identified. The working group also put forward proposals for organizational changes needed to allow the design project to continue and to prepare for the installation and commissioning phase of the LHC communication infrastructure. This paper reports on the work done and explains the motivation behind the recommendations

Exact and explicit probability densities for one-sided Levy stable distributions

We study functions g_{\alpha}(x) which are one-sided, heavy-tailed Levy stable probability distributions of index \alpha, 0< \alpha <1, of fundamental importance in random systems, for anomalous diffusion and fractional kinetics. We furnish exact and explicit expression for g_{\alpha}(x), 0 \leq x < \infty, satisfying \int_{0}^{\infty} exp(-p x) g_{\alpha}(x) dx = exp(-p^{\alpha}), p>0, for all \alpha = l/k < 1, with k and l positive integers. We reproduce all the known results given by k\leq 4 and present many new exact solutions for k > 4, all expressed in terms of known functions. This will allow a 'fine-tuning' of \alpha in order to adapt g_{\alpha}(x) to a given experimental situation.Comment: 4 pages, 3 figures and 1 tabl

Circulating markers of arterial thrombosis and late-stage age-related macular degeneration: a case-control study.

PURPOSE: The aim of this study was to examine the relation of late-stage age-related macular degeneration (AMD) with markers of systemic atherothrombosis. METHODS: A hospital-based case-control study of AMD was undertaken in London, UK. Cases of AMD (n=81) and controls (n=77) were group matched for age and sex. Standard protocols were used for colour fundus photography and to classify AMD; physical examination included height, weight, history of or treatment for vascular-related diseases and smoking status. Blood samples were taken for measurement of fibrinogen, factor VIIc (FVIIc), factor VIIIc, prothrombin fragment F1.2 (F1.2), tissue plasminogen activator, and von Willebrand factor. Odds ratios from logistic regression analyses of each atherothrombotic marker with AMD were adjusted for age, sex, and established cardiovascular disease risk factors, including smoking, blood pressure, body mass index, and total cholesterol. RESULTS: After adjustment FVIIc and possibly F1.2 were inversely associated with the risk of AMD; per 1 standard deviation increase in these markers the odds ratio were, respectively, 0.62 (95% confidence interval 0.40, 0.95) and 0.71 (0.46, 1.09). None of the other atherothrombotic risk factors appeared to be related to AMD status. There was weak evidence that aspirin is associated with a lower risk of AMD. CONCLUSIONS: This study does not provide strong evidence of associations between AMD and systematic markers of arterial thrombosis, but the potential effects of FVIIc, and F1.2 are worthy of further investigation