Discrete Approximations of Metric Measure Spaces of Controlled Geometry

We find a necessary and sufficient condition for a doubling metric space to carry a (1,p)-Poincare inequality. The condition involves discretizations of the metric space and Poincare inequalities on graphs.Comment: 23 Page

Boron determination in steels by Inductively-Coupled Plasma spectometry (ICP)

The sample is treated with 5N H2SO4 followed by concentrated HNO3 and the diluted mixture is filtered. Soluble B is determined in the filtrate by Inductively-Coupled Plasma (ICP) spectrometry after addition HCl and extraction of Fe with ethyl-ether. The residue is fused with Na2CO3 and, after treatment with HCl, the insoluble B is determined by ICP spectrometry as before. The method permits determination of ppm amounts of B in steel

Study of the double non linear quantum resonances in diatomic molecules

We study the quantum dynamics of diatomic molecule driven by a circularly polarized resonant electric field. We look for a quantum effect due to classical chaos appearing due to the overlapping of nonlinear resonances associated to the vibrational and rotational motion. We solve the Schr\"odinger equation associated with the wave function expanded in term of proper stationary states, $|n>\otimes|lm>$ (vibrational$\otimes$angular momentum states). Looking for quantum-classic correspondence, we consider the Liouville dynamics in the two dimensional phase space defined by the coherent -like state of vibrational states, and it is found some similarities when the quantum dynamics is pictured in terms of number and phase operators.Comment: 15 pages, 4 figure

Order Reduction of the Chemical Master Equation via Balanced Realisation

We consider a Markov process in continuous time with a finite number of discrete states. The time-dependent probabilities of being in any state of the Markov chain are governed by a set of ordinary differential equations, whose dimension might be large even for trivial systems. Here, we derive a reduced ODE set that accurately approximates the probabilities of subspaces of interest with a known error bound. Our methodology is based on model reduction by balanced truncation and can be considerably more computationally efficient than the Finite State Projection Algorithm (FSP) when used for obtaining transient responses. We show the applicability of our method by analysing stochastic chemical reactions. First, we obtain a reduced order model for the infinitesimal generator of a Markov chain that models a reversible, monomolecular reaction. In such an example, we obtain an approximation of the output of a model with 301 states by a reduced model with 10 states. Later, we obtain a reduced order model for a catalytic conversion of substrate to a product; and compare its dynamics with a stochastic Michaelis-Menten representation. For this example, we highlight the savings on the computational load obtained by means of the reduced-order model. Finally, we revisit the substrate catalytic conversion by obtaining a lower-order model that approximates the probability of having predefined ranges of product molecules.Comment: 12 pages, 6 figure