Effective stability of the Trojan asteroids

We study the spatial circular restricted problem of three bodies in the light of Nekhoroshev theory of stability over large time intervals. We consider in particular the Sun-Jupiter model and the Trojan asteroids in the neighborhood of the Lagrangian point $L_4$. We find a region of effective stability around the point $L_4$ such that if the initial point of an orbit is inside this region the orbit is confined in a slightly larger neighborhood of the equilibrium (in phase space) for a very long time interval. By combining analytical methods and numerical approximations we are able to prove that stability over the age of the universe is guaranteed on a realistic region, big enough to include one real asteroid. By comparing this result with the one obtained for the planar problem we see that the regions of stability in the two cases are of the same magnitude.Comment: 9 pages, 2 figures, Astronomy & Astrophysics in pres

Determining the suitability of mass spectrometry for understanding the dissolution processes involved with pharmaceutical tablets

RATIONALE: A current challenge for analytical chemists is the development of measurement systems and approaches required to understand dynamic processes such as tablet dissolution. The design and development of oral tablets could be improved by the availability of detailed information about the rates of release of the individual tablet components. Small footprint mass spectrometry (MS) systems are gaining use for on-line reaction monitoring because of their ability to rapidly determine multiple reactant, intermediate, and product species. We have therefore assessed the utility of such MS systems to the study of dissolution processes. METHODS: Aqueous dissolution media containing phosphate and other non volatile buffer salts were pumped from a standard USPII dissolution vessel through an active splitter and back. The splitter sampled the dissolution stream and diluted it into a make-up flow which was pumped to a small single quadrupole mass spectrometer. Single ion monitoring was used to quantify the ions of interest. Three different bio-relevant dissolution media were studied to gauge the impact of the sample matrix. RESULTS: Individual dissolution profiles were obtained from a tablet containing three drugs and lactose as the soluble filler. This was successfully demonstrated with three different bio-relevant media designed to reflect the pH of the different sections of the human gastro-intestinal tract. Component concentrations as low as 0.06 μg/mL (representing 1% dissolution) were detected. The MS dissolution profiles correlated with the visual observation of tablet dissolution. MS gave linear responses with concentration for the individual components, although analysis of the tablet solution indicated that ion suppression is an area for further investigation CONCLUSIONS: An on-line MS system was used to determine the individual dissolution profiles of three drugs and lactose as they are released from the same tablet. The level of each of these components in solution was determined every 10 seconds, and each had a similar release profile. The dissolution profiles were determined using inorganic buffer solutions at three different bio-relevant pHs

Fractional compartmental models and multi-term Mittag–Leffler response functions

Systems of fractional differential equations (SFDE) have been increasingly used to represent physical and control system, and have been recently proposed for use in pharmacokinetics (PK) by (J Pharmacokinet Pharmacodyn 36:165–178, 2009) and (J Phamacokinet Pharmacodyn, 2010). We contribute to the development of a theory for the use of SFDE in PK by, first, further clarifying the nature of systems of FDE, and in particular point out the distinction and properties of commensurate versus non-commensurate ones. The second purpose is to show that for both types of systems, relatively simple response functions can be derived which satisfy the requirements to represent single-input/single-output PK experiments. The response functions are composed of sums of single- (for commensurate) or two-parameters (for non-commensurate) Mittag–Leffler functions, and establish a direct correspondence with the familiar sums of exponentials used in PK

Asymptotology of Chemical Reaction Networks

The concept of the limiting step is extended to the asymptotology of multiscale reaction networks. Complete theory for linear networks with well separated reaction rate constants is developed. We present algorithms for explicit approximations of eigenvalues and eigenvectors of kinetic matrix. Accuracy of estimates is proven. Performance of the algorithms is demonstrated on simple examples. Application of algorithms to nonlinear systems is discussed.Comment: 23 pages, 8 figures, 84 refs, Corrected Journal Versio

Approximate Solutions to Fractional Subdiffusion Equations: The heat-balance integral method

The work presents integral solutions of the fractional subdiffusion equation by an integral method, as an alternative approach to the solutions employing hypergeometric functions. The integral solution suggests a preliminary defined profile with unknown coefficients and the concept of penetration (boundary layer). The prescribed profile satisfies the boundary conditions imposed by the boundary layer that allows its coefficients to be expressed through its depth as unique parameter. The integral approach to the fractional subdiffusion equation suggests a replacement of the real distribution function by the approximate profile. The solution was performed with Riemann -Liouville time-fractional derivative since the integral approach avoids the definition of the initial value of the time-derivative required by the Laplace transformed equations and leading to a transition to Caputo derivatives. The method is demonstrated by solutions to two simple fractional subdiffusion equations (Dirichlet problems): 1) Time-Fractional Diffusion Equation, and 2) Time-Fractional Drift Equation, both of them having fundamental solutions expressed through the M-Write function. The solutions demonstrate some basic issues of the suggested integral approach, among them: a) Choice of the profile, b) Integration problem emerging when the distribution (profile) is replaced by a prescribed one with unknown coefficients; c) Optimization of the profile in view to minimize the average error of approximations; d) Numerical results allowing comparisons to the known solutions expressed to the M-Write function and error estimations.Comment: 15 pages, 7 figures, 3 table

Fractional dynamics pharmacokinetics–pharmacodynamic models

While an increasing number of fractional order integrals and differential equations applications have been reported in the physics, signal processing, engineering and bioengineering literatures, little attention has been paid to this class of models in the pharmacokinetics–pharmacodynamic (PKPD) literature. One of the reasons is computational: while the analytical solution of fractional differential equations is available in special cases, it this turns out that even the simplest PKPD models that can be constructed using fractional calculus do not allow an analytical solution. In this paper, we first introduce new families of PKPD models incorporating fractional order integrals and differential equations, and, second, exemplify and investigate their qualitative behavior. The families represent extensions of frequently used PK link and PD direct and indirect action models, using the tools of fractional calculus. In addition the PD models can be a function of a variable, the active drug, which can smoothly transition from concentration to exposure, to hyper-exposure, according to a fractional integral transformation. To investigate the behavior of the models we propose, we implement numerical algorithms for fractional integration and for the numerical solution of a system of fractional differential equations. For simplicity, in our investigation we concentrate on the pharmacodynamic side of the models, assuming standard (integer order) pharmacokinetics