Kinematical analysis of overconstrained and underconstrained mechanisms by means of computational algebraic geometry

We investigate and explain several exceptional phenomena appearing in mechanism kinematics. The starting point for the kinematical analysis of a mechanism is the formation of the relevant constraint map defining the constraint equations for the coordinates of the particular system. The constraint equations define the configuration space of the mechanism, which reveals the essential kineamtic characteristics. But in some cases the properties of the map, and not the configuration space itself, are important. This is true for example for so called under- and overconstrained mechanisms for which the standard formulation of constraints gives usually not enough or too many constraints when considering the dimension of their configuration space. These concepts also naturally lead to the concept of kinematotropic mechanisms which posses motion modes of different dimension. In this context the concept of a kinematotropy as a motion between such modes is introduced in this paper. We present a general approach to the kinematic analysis of mechanisms using the theory of algebraic geometry and tools of computational algebraic geometry. The configuration space is considered as a real algebraic variety defined by the constraints. The phenomena and needed theory are explained and several illustrative examples are given. In particular the underconstrained phenomenon is explained by considering the real and complex dimension of the configuration space variety.     

Rapid Screening of Indole-3-Acetic Acid and Other Indole Derivatives in Bacterial Culture Broths by Planar Chromatography

JPC – Journal of Planar Chromatography – Modern TLC - The aim of this study was to find suitable separation conditions for the rapid screening of indole derivatives in bacterial culture...     

Dynamo Effect in the Kraichnan Magnetohydrodynamic Turbulence

The existence of a dynamo effect in a simplified magnetohydrodynamic model of turbulence is considered when the magnetic Prandtl number approaches zero or infinity. The magnetic field is interacting with an incompressible Kraichnan-Kazantsev model velocity field which incorporates also a viscous cutoff scale. An approximate system of equations in the different scaling ranges can be formulated and solved, so that the solution tends to the exact one when the viscous and magnetic-diffusive cutoffs approach zero. In this approximation we are able to determine analytically the conditions for the existence of a dynamo effect and give an estimate of the dynamo growth rate. Among other things we show that in the large magnetic Prandtl number case the dynamo effect is always present. Our analytical estimates are in good agreement with previous numerical studies of the Kraichnan-Kazantsev dynamo by Vincenzi (J. Stat. Phys. 106:1073–1091, 2002).     

Tensor invariants in numerical geometric integration of ODEs

We consider geometric numerical integration (GI) of ordinary differential equations (ODEs). We propose that in GI one needs concepts which are both geometric and algebraic. In this paper we start from an algebraic point of view: we introduce tensor invariants attached to an ODE as well as to an integrator. The notion of “sharing a tensor invariant” generalizes the well known notion of conserving a symplectic structure by an integrator. Several examples are given.     

A holomorphic representation approach to the regularization of model field theories in coupled cluster form

Summary We explain in detail the so-called Bargmann or holomorphic representation, and apply it to the general class of single-mode bosonic field theories. Since these model field theories have no attribute of separability and are, in some sense, maximally nonlocal, they are an especially severe test of the capability of coupled cluster methods to parametrize them satisfactorily. They include the cases of anharmonic oscillators of order 2K (K=2, 3,...), for which ordinary perturbation theory is known to diverge, and we therefore make a special study of such systems. We demonstrate for the first time for any quantum-mechanical problem with infinite Hilbert space that both the normal and extended coupled cluster methods (NCCM and ECCM) have phase spaces which rigorously exist. We analyze completely the asymptotic properties of the complete sets of the NCCM and ECCM amplitudes, either of which fully characterizes the system. It is thereby shown how the holomorphic representation can be used to regularize completely all otherwise formally divergent series that appear. We demonstrate in detail how the entire NCCM and ECCM programmes can be carried through for these systems, including the diagonalization of the classically mapped Hamilitonians in the respective classical NCCM and ECCM phase spaces.     

Rotation Planar Extraction and Medium-Pressure Solid—Liquid Extraction of Onion (Allium cepa)

JPC – Journal of Planar Chromatography – Modern TLC - Extraction of onion (Allium cepa L.) with 80:20 (v/v) methanol—water in water by rotation planar extraction (RPE) and medium-...     

On the numerical solution of involutive ordinary differential systems

We consider a Galerkin finite element method that uses piecewisebilinears on a modified Shishkin mesh for a model singularlyperturbed convection–diffusion problem on the unit square.The method is shown to be convergent, uniformly in the perturbationparameter , of order N^–1in a global energy norm, providedonly that N^–1, where O(N²)mesh points are used. Thuson the new mesh the method yields more accurate results thanon Shishkin’s original piecewise uniform mesh, whereit is convergent of order N^–1lnN. Numerical experiments support our theoretical results. Received 14 September, 1998. Revised 24 September, 1999.