Stochastic modeling of a billiard in a gravitational field: Power law behavior of Lyapunov exponents

We consider the motion of a point particle (billiard) in a uniform gravitational field constrained to move in a symmetric wedge-shaped region. The billiard is reflected at the wedge boundary. The phase space of the system naturally divides itself into two regions in which the tangent maps are respectively parabolic and hyperbolic. It is known that the system is integrable for two values of the wedge half-angle ₁ and ₂ and chaotic for ₁<< ₂. We study the system at three levels of approximation: first, where the deterministic dynamics is replaced by a random evolution; second, where, in addition, the tangent map in each region is, replaced by its average; and third, where the tangent map is replaced by a single global average. We show that at all three levels the Lyapunov exponent exhibits power law behavior near ₁ and ₂ with exponents 1/2 and 1, respectively. We indicate the origin of the exponent 1, which has not been observed in unaccelerated billiards.     

Comparison of procedures for correction of matrix interferences in the multi-element analysis of soils by ICP-AES with a CCD detection system

Three procedures, matrix matching, plasma optimisation and single-point standard-addition, have been evaluated to ascertain the best procedure for simultaneous multi-element analysis of industrial soils by ICP-AES with CCD detection. A standard reference material, CRM143 from the Bureau Communautaire de Réference (BCR), has been analysed for Cd, Cr, Cu, Mn, Ni and Pb using the three different matrix interference correction procedures. All three procedures give comparable results which are in good agreement with the BCR values, except for Cr. The single-point standard addition procedure was chosen, on the basis of economy and ease of implementation, to correct for matrix interferences in the determination of Cd, Co, Cr, Cu, Mn, Mo, Ni, Pb, Sn and V in soil samples collected from an industrial site in England. Concentrations of some of the elements were found to vary greatly with sampling depth. For example, the concentration of Mn, determined using the atomic line at 279.920 nm, increased from 426 ± 3 g/g at a depth of 18–28 cm to 5996 ± 144 g/g at 60–85 cm.     

Polychronous kinetics with nonstationary rate constants. Effect of a medium

Characteristic features of the kinetics of solid-state cage reactions with distributed parameters of the relaxing matrix were considered. Depending on the ratio of the constants of the reaction rate and relaxation of environment, the kinetics of chemical conversions can be either exponential or nonexponential. Plausible reasons for the unsteady-state character of the kinetics of the processes of two types,viz., the reactions of alkyl radicals in amorphous alcohol matrices and conversions in biological systems, were discussed. The main reason for the unsteady-state character of the reactions of the first type is a dispersion of the equilibrium distances between the reagents. Kinetics of the reactions of the second type, such as rebinding of the ligands in the heme-containing proteins (e.g., in myoglobin), is determined by the distances in the pairs of reagents and the relaxation transitions. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 3, pp. 469–476, March, 1997.     

Determination of Organophosphorus Pesticides in Soil by MMSPD-GC-NPD and Confirmation by GC-MS

A novel method, modified matrix solid-phase dispersion (MMSPD), has been developed for quantitative analysis of organophosphorus pesticide residues in soil. It was based on matrix solid-phase dispersion (MSPD) and continuous liquid-solid extraction (continuous LSE), using Florisil as sorbent and dichloromethane as the recycling solvent. Two soils with different texture and physicochemical properties were studied to validate the method. The effect of residence time of pesticides in soil was also studied. MMSPD was compared with MSPD and continuous LSE respectively. Determination was carried out by gas chromatography with nitrogen-phosphorus detection (GC-NPD). The method gave recoveries ranging from 72–105% with relative standard deviations (RSDs) lower than 15% for the pesticides studied. The limits of detection (LODs) ranged from 0.1 to 0.6 ng g⁻¹. Two pesticide residues have been detected in real soil samples from Fujian, China, using MMSPD. The pesticides were confirmed by gas chromatography-mass spectrometry (GC-MS) in a selected-ion monitoring (SIM) mode. Revised: 4 and 9 April 2006     

Exact results for a generalized classicalO(n) matrix spin model

A generalizedO(n) matrix version of the classical Heisenberg model, introduced by Fuller and Lenard as a classical limit of a quantum model, is solved exactly in one dimension. The free energy is analytic and the pair correlation functions decay exponentially for all finite temperatures. It is shown, however, that even for a finite number of spins the model has a phase transition in then limit. The transition features a specific heat jump, zero long-range order at all temperatures, and zero correlation length at the critical point. The Curie-Weiss version of the model is also solved exactly and shown to have standard mean-field type behavior for all finiten and to differ from the one-dimensional results in then limit.     

A novel trilinear decomposition algorithm： Three-dimension non-negative matrix factorization

Non-negative matrix factorization(NMF)is a technique for dimensionality reduction by placing non-negativity constraints onthe matrix.Based on the PARAFAC model,NMF was extended for three-dimension data decomposition.The three-dimension non-negative matrix factorization(NMF3)algorithm,which was concise and easy to implement,was given in this paper.The NMF3algorithm implementation was based on elements but not on vectors.It could decompose a data array directly without unfolding,which was not similar to that the traditional algorithms do.It has been applied to the simulated data array decomposition andobtained reasonable results.It showed that NMF3 could be introduced for curve resolution in chemometrics.     

Exact electronic properties in the classically forbidden region of a metal surface

We derive exact properties of the inhomogeneous electron gas in the asymptotic classically forbidden region at a metal–vacuum interface within the framework of local effective potential energy theory. We derive a new expression for the asymptotic structure of the Kohn–Sham density functional theory (KS‐DFT) exchange‐correlation potential energy v_xc(r) in terms of the irreducible electron self‐energy. We also derive the exact asymptotic structure of the orbitals, density, the Dirac density matrix, the kinetic energy density, and KS exchange energy density. We further obtain the exact expression for the Fermi hole and demonstrate its structure in this asymptotic limit. The exchange‐correlation potential energy is derived to be v_xc(z → ∞) = ?α_KS,xc/z, and its exchange and correlation components to be v_x(z → ∞) = ?α_KS,x/z and v_c(z → ∞) = ?α_KS,c/z, respectively. The analytical expressions for the coefficients α_KS,xc and α_KS,x show them to be dependent on the bulk‐metal Wigner–Seitz radius and the barrier height at the surface. The coefficient α_KS,c = 1/4 is determined in the plasmon‐pole approximation and is independent of these metal parameters. Thus, the asymptotic structure of v_xc(z) in the vacuum region is image‐potential‐like but not the commonly accepted one of ?1/4z. Furthermore, this structure depends on the properties of the metal. Additionally, an analysis of these results via quantal density functional theory (Q‐DFT) shows that both the Pauli W_x(z → ∞) and lowest‐order correlation‐kinetic W(z → ∞) components of the exchange potential energy v_x(z → ∞), and the Coulomb W_c(z → ∞) and higher‐order correlation‐kinetic components of the correlation potential energy v_c(z → ∞), all contribute terms of O(1/z) to the structure. Hence correlations attributable to the Pauli exclusion principle, Coulomb repulsion, and correlation‐kinetic effects all contribute to the asymptotic structure of the effective potential energy at a metal surface. The relevance of the results derived to the theory of image states and to KS‐DFT is also discussed. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Separating Ag, B, Cd, Dy, Eu, and Sm in a Gd matrix using 2-ethylhexyl phosphonic acid mono-2-ethylhexyl ester extraction chromatography for ICP-AES analysis

The separation procedure for Ag, B, Cd, Dy, Eu and Sm as impurities in Gd matrix using ICP-AES technique with an extraction chromatographic column has been developed. The spectral interference of the Gd matrix on the elements was eliminated using a chromatography technique with 2-ethylhexyl phosphonic acid mono-2-ethylhexyl ester (PC-88A) as the mobile phase and XAD-16 resin as the stationary phase. Ag⁺, B₄O₇²⁻, and Cd²⁺ were eluted with 0.1 M HNO₃, while rare earth ions were not. The best eluent for separating Eu and Sm in the Gd matrix was 0.3 M HNO₃. The limit of quantitation for these elements was 0.6-3.0 ng mL⁻¹. The recovery of Ag, B, and Cd was 90-104% using 0.1 M HNO₃ as the eluent, while that of Eu, Gd, and Sm ranged from 100 to 102% with 0.3 M HNO₃. Dy was recovered quantitatively with 4 M HNO₃. The relative standard deviation of the methods for a set of three replicates was between 1.0 and 15.4% for the synthetic and standard Gd solutions. The proposed separation procedure was used to measure Ag, B, Cd, Dy, Eu, and Sm in a standard Gd solution.