The Essential Spectral Radius and Asymptotic Properties of Transfer Operators

Abs t r ac t The statistical behavior of deterministic and stochastic dynamical sys-tems may be described using transfer operators, which generalize the no-tion of Frobenius-Perron and Koopman operators. Since numerical tech-niques to analyse dynamical systems based on eigenvalues problems for the corresponding transfer operator have emerged, bounds on its essential spectral radius became of interest. This article shows that they are also of great theoretical interest. We give an analytical representation of the essential spectral radius in L\fj,), which then is exploited to analyse the asymptotical properties of transfer operators by combining results from functional analysis, Markov operators and Markov chain theory. In par ticular, it is shown that an essential spectral radius less than 1, uniform constrictiveness and some "weak form " of the so-called Doeblin condition are equivalent. Finally, we apply the results to study three main prob-lem classes: deterministic systems stochastically perturbed deterministic systems and stochastic systems K e y w o r d s, uniformly constrictive, asymptotically stable, exact, asymptotically pe

Investigation of the Interactions of Cationic Polyelectrolytes with Anionic Surfactants: Effects of Polymer, Surfactant and Solution Properties

The intent of this research is to explore and understand the effects that a range of polymer, surfactant and solution parameters have on the interaction of oppositely-charged polymers and surfactants. Cationic polysaccharides were chosen for this research because they are known to interact with anionic surfactants, and they offer a wide range of adjustable polymer properties, including molecular weight, charge substitution, and backbone structure. Cationic poly(vinylpyridinium hydrochloride) polymers were chosen for these studies because they provide the opportunity to explore the effects of charge position on the interaction of cationic polymers with anionic surfactants and how this influences the mechanism of interaction. The overall goal of this research is to define the effects of polymer and surfactant structural properties, and solution properties, on the interaction between cationic polymers and anionic surfactants, and the subsequent formation of coacervate in these systems. The interaction of cationic polymers with varying properties with anionic surfactant was studied using conventional microscopic and macroscopic methodologies to probe the mechanism of interaction in these systems. Polyquaternium-10 systems interacted with anionic surfactant in accordance with the cooperative mechanism of interaction and coacervate formation as described by Goddard. The mechanism of interaction between poly(vinylpyridinium hydrochloride) polymers and anionic surfactant was found to be dependent on the position of the cationic charge relative to the hydrophobic polymer backbone. Polymer-surfactant interaction with poly(4- vinylpyridinium hydrochloride) and anionic surfactant occurred via the site-specific cooperative mechanism of interaction. However, the interaction of poly(2- vinylpyridinium hydrochloride) with anionic surfactant exhibited characteristics of the site-specific cooperative interaction mechanism as well as the macroion-macroion interaction mechanism. A high-throughput screening method was developed to facilitate systematic studies of the effects of polymer, surfactant and solution properties on the macroscopic property of coacervate formation. This method allowed rapid and reproducible preparation and analysis of multi-component systems and representation of the amount of coacervate and compositional range of coacervate formation in these systems in easily understood contour phase diagrams. In the cationic polysaccharide systems, the amount of coacervate and the compositional range of coacervate formation displayed a dependence on both the polymer charge density and molecular weight. Also, the polymer critical overlap concentration was observed to affect coacervate amount with higher coacervate formation observed above c*. Coacervate formation with the poly(vinylpyridinium hydrochloride) polymers was found to be dependent not only on the position of the cationic charge on the polymer, but also on the structure of the surfactant tail group. Coacervate formed initially with P4VP and P2 VP and sodium capryl sulfonate and sodium xylene sulfonate was not stable over 24 hours, however coacervate formed between these polymers and sodium dodecylbenzene sulfonate was stable over 24 hours. This indicates that a hydrophobic chain with sufficient length and/or an aromatic group is necessary to form thermodynamically stable coacervate. The effect of salt in solution on polymer-surfactant interaction was studied with both classes of polymer. A dependence of coacervate amount and compositional range of coacervate formation on salt concentration was observed. The effect of salt was dependent on the degree of polymer charge substitution. The order of addition of polymer, surfactant, and salt also affected coacervate formation. This was consistent for both low and high molecular weight polymers, as well as low and high charge substituted polymers. Although an effect of addition order was observed in all systems, the specific effects differed depending on the polymer properties

Self-monitoring : an efficient and effective intervention for academic and behavioral targets in the school

This research paper will review current research concerning the use of self-monitoring as an intervention technique for academic and behavioral targets in the school. The history and theory will be described, as well as the elements and implementation of self-monitoring as an intervention technique. Classroom applications with academic and behavioral targets are reviewed, as well as caveats on the use of self-monitoring as a classroom intervention technique. Recommendations concerning the need for additional research on the applicability of self-monitoring as an intervention technique are also described

Bayesian data assimilation to support informed decision-making in individualized chemotherapy

An essential component of therapeutic drug/biomarker monitoring (TDM) is to combine patient data with prior knowledge for model-based predictions of therapy outcomes. Current Bayesian forecasting tools typically rely only on the most probable model parameters (maximum a-posteriori (MAP) estimate). This MAP-based approach, however, does neither necessarily predict the most probable outcome nor does it quantify the risks of treatment inefficacy or toxicity. Bayesian data assimilation (DA) methods overcome these limitations by providing a comprehensive uncertainty quantification. We compare DA methods with MAP-based approaches and show how probabilistic statements about key markers related to chemotherapy-induced neutropenia can be leveraged for more informative decision support in individualized chemotherapy. Sequential Bayesian DA proved to be most computational efficient for handling interoccasion variability and integrating TDM data. For new digital monitoring devices enabling more frequent data collection, these features will be of critical importance to improve patient care decisions in various therapeutic areas

Solving the Chemical Master Equation for Monomolecular Reaction Systems Analytically

The stochastic dynamics of a well-stirred mixture of molecular species interacting through different biochemical reactions can be accurately modelled by the chemical master equation (CME). Research in the biology and scientific computing community has concentrated mostly on the development of numerical techniques to approximate the solution of the CME via many realizations of the associated Markov jump process. The domain of exact and/or efficient methods for directly solving the CME is still widely open, which is due to its large dimension that grows exponentially with the number of molecular species involved. In this article, we present an exact solution formula of the CME for arbitrary initial conditions in the case where the underlying system is governed by monomolecular reactions. The solution can be expressed in terms of the convolution of multinomial and product Poisson distributions with time-dependent parameters evolving according to the traditional reaction-rate equations. This very structured representation allows to deduce easily many properties of the solution. The model class includes many interesting examples. For more complex reaction systems, our results can be seen as a first step towards the construction of new numerical integrators, because solutions to the monomolecular case provide promising ansatz functions for Galerkin-type methods

Lumping of Physiologically-Based Pharmacokinetic Models and a Mechanistic Derivation of Classical Compartmental Models

In drug discovery and development, classical compartment models and physiologically based pharmacokinetic (PBPK) models are successfully used to analyze and predict the pharmacokinetics of drugs. So far, however, both approaches are used exclusively or in parallel, with little to no cross-fertilization. An approach that directly links classical compartment and PBPK models is highly desirable. We derived a new mechanistic lumping approach for reducing the complexity of PBPK models and establishing a direct link to classical compartment models. The proposed method has several advantages over existing methods: Perfusion and permeability rate limited models can be lumped; the lumped model allows for predicting the original organ concentrations; and the volume of distribution at steady state is preserved by the lumping method. To inform classical compartmental model development, we introduced the concept of a minimal lumped model that allows for prediction of the venous plasma concentration with as few compartments as possible. The minimal lumped parameter values may serve as initial values for any subsequent parameter estimation process. Applying our lumping method to 25 diverse drugs, we identified characteristic features of lumped models for moderate-to-strong bases, weak bases and acids. We observed that for acids with high protein binding, the lumped model comprised only a single compartment. The proposed lumping approach established for the first time a direct derivation of simple compartment models from PBPK models and enables a mechanistic interpretation of classical compartment models

Physiologically Based Pharmacokinetic Modelling: A Sub-Compartmentalized Model of Tissue Distribution

We present a sub-compartmentalized model of drug distribution in tissue that extends existing approaches based on the well-stirred tissue model. It is specified in terms of di®erential equations that explicitly account for the drug concentration in erythrocytes, plasma, interstitial and cellular space. Assuming, in addition, steady state drug distribution and by lumping the different sub-compartments, established models to predict tissue-plasma partition coe±cients can be derived in an intriguingly simple way. This direct link is exploited to explicitly construct and parameterize the sub-compartmentalized model for moderate to strong bases, acids, neutrals and zwitterions. The derivation highlights the contributions of the different tissue constituents and provides a simple and transparent framework for the construction of novel tissue distribution models