A note on the generalized Rayleigh equation: limit cycles and stability

The aim of the present paper is to give an analytical proof on the existence and stability of the limit cycles in the generalized Rayleigh equation, which models diabetic chemical processes through a constant area duct where the effect of heat addition or rejection is considered, ${\frac{d^{2}x}{dt^{2}}+x = \varepsilon(1-(\frac{dx}{dt}) ^{2n})\,\frac{dx}{dt}}$ where n is a positive integer and ε a small real parameter. The main tool used for it is the averaging theory.     

On boundary conditions to the diffusion equation

The boundary conditions to the diffusion equation are obtained in the case of a diffusing particle which, in a single collision, may either recoil from or stick to the boundary wall with a priori known probabilities q and p. It is shown that the value p may be determined from diffusion penetration measurements in a circular tube.     

Bistability in the chemical master equation for dual phosphorylation cycles

Dual phospho/dephosphorylation cycles, as well as covalent enzymatic-catalyzed modifications of substrates are widely diffused within cellular systems and are crucial for the control of complex responses such as learning, memory, and cellular fate determination. Despite the large body of deterministic studies and the increasing work aimed at elucidating the effect of noise in such systems, some aspects remain unclear. Here we study the stationary distribution provided by the two-dimensional chemical master equation for a well-known model of a two step phospho/dephosphorylation cycle using the quasi-steady state approximation of enzymatic kinetics. Our aim is to analyze the role of fluctuations and the molecules distribution properties in the transition to a bistable regime. When detailed balance conditions are satisfied it is possible to compute equilibrium distributions in a closed and explicit form. When detailed balance is not satisfied, the stationary non-equilibrium state is strongly influenced by the chemical fluxes. In the last case, we show how the external field derived from the generation and recombination transition rates, can be decomposed by the Helmholtz theorem, into a conservative and a rotational (irreversible) part. Moreover, this decomposition allows to compute the stationary distribution via a perturbative approach. For a finite number of molecules there exists diffusion dynamics in a macroscopic region of the state space where a relevant transition rate between the two critical points is observed. Further, the stationary distribution function can be approximated by the solution of a Fokker-Planck equation. We illustrate the theoretical results using several numerical simulations.     

Decoupling conditions for the vibronic equation in dimers

The analytical conditions are obtained and discussed under which the vibronic equations for dimers can be decoupled.     

Study of adsorption processes of model drugs at supercritical conditions using partial least squares regression

The characterisation of adsorption or impregnation processes using conventional or supercritical fluid technologies becomes an increasing part of the research on drug formulations. The complexity of the relationships between these adsorption processes and the experimental variables potentially influencing them, however, makes these studies more problematic. In this paper, a chemometric approach based on nonlinear partial least squares (NL-PLS) modelling is applied to characterise the effect of the experimental variables on the supercritical impregnation process. Various adsorbent materials such as silica gel, zeolite and amberlite were investigated using the following model compounds as adsorbates: benzoic, salicylic and acetylsalicylic acids.     

Linear calibrations in chromatography: The incorrect use of ordinary least squares for determinations at low levels,and the need to redefine the limit of quantification with this regression model

Ordinary least squares is widely applied as the standard regression method for analytical calibrations, and it is usually accepted that this regression method can be used for quantification starting at the limit of quantification. However, it requires calibration being homoscedastic and this is not common. Different calibrations have been evaluated to assess whether ordinary least squares is adequate to quantify estimates at low levels. All calibrations evaluated were linear and heteroscedastic. Despite acceptable values for precision at limit of quantification levels were obtained, ordinary least squares fitting resulted in significant and unacceptable bias at low levels. When weighted least squares regression was applied, bias at low levels was solved and accurate estimates were obtained. With heteroscedastic calibrations, limit values determined by conventional methods are only appropriate if weighted least squares are used. A “practical limit of quantification” can be determined with ordinary least squares in heteroscedastic calibrations, which should be fixed at a minimum of 20 times the value calculated with conventional methods. Biases obtained above this “practical limit” were acceptable applying ordinary least squares and no significant differences were obtained between the estimates measured using weighted and ordinary least squares when analyzing real‐world samples.     

Liquid-vapor density profiles from equilibrium limit of diffusion equation for interacting particles

Liquid-vapor density profiles are derived from the equilibrium limit of diffusion equation for interacting particles. These profiles are in good agreement with classical hyperbolic tangent relation. For simple Lennard-Jones fluids, predicted density distributions agree with computer simulation data, but have a slightly sharper transition zone. For alkali metals with Lennard-Jones-like potentials, the new equations predict a very good average distribution with quite satisfactory agreement with Monte Carlo simulation results. For liquid metals and water surfaces, accurate interfacial profile predictions also can be achieved by using effective two-body potential data instead of Lennard-Jones parameters.     

Relative positions of limit cycles in the continuous culture vessel with variable yield

To estimate the relative position of limit cycles for a continuous culture vessel is always useful in the qualitative study of the system. In this paper, we construct an annular region containing all the limit cycles for the chemostat with variable yield model that was studied by Huang (J. Math. Chem. 5, 151–166. 1990), and by Pilyugin and Waltman (Math. Biosci. 182, 151–166. 2003).AMS subject classification 34A34, 92D25     

Hopf bifurcation and multiple limit cycles in bio-chemical reaction of the morphogenesis process

Morphogenetic process is an interesting but very hard bio-chemical problem. In this paper, we consider a bio-chemical model in temporal morphogenesis which is a generalization of the model studied by Gierer–Meinhardt. By using the theory of ordinary differential equations, it is shown that the model undergoes a Hopf bifurcation if the parameters in the model satisfy the following relationship: λ = 2/(ρ ₂ + x*)−1. It is also proved that the close orbit created by the Hopf bifurcation is stable. The conditions that guarantee the system has three closely nested limit cycles are also obtained in the paper.     

Comparison of methods of determining Arrhenius equation parameters by the least squares method

The parameters of the Arrhenius equation determined by the linear, weighted linear and non-linear least squares methods and by the simplex method are compared. Since the non-linear least squares method permits the consideration of statistical weights of both the dependent (k) and independent (T) variables and does not involve logarithmic transformation, it is advisable to calculate the parameters of the Arrhenius equation by means of the non-linear least squares method.

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Zusammenfassung Arrhenius-Parameter bestimmt nach der linearen, mit Gewichten operierenden linearen und nicht-linearen Methode der kleinsten Fehlerquadrate oder nach der Simplex-Methode werden miteinander verglichen. Da bei der nicht-linearen Methode der kleinsten Fehlerquadrate sowohl die abhÄngige (K) als auch die unabhÄngige (T) Variable mit statistischen Gewichten versehen werden kann und keine logarithmische Transformation vorgenommen wird, ist es zweckmÄ\ig, die Berechnung der Parameter der Arrhenius-Gleichung nach der nicht-linearen Methode der kleinsten Fehlerquadrate auszuführen.

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, , . , k, T, , .

     

Finite volume solution of the modified Poisson-Boltzmann equation for two colloidal particles

The double layer forces between spherical colloidal particles, according to the Poisson-Boltzmann (PB) equation, have been accurately calculated in the literature. The classical PB equation takes into account only the electrostatic interactions, which play a significant role in colloid science. However, there are at, and above, biological salt concentrations other non-electrostatic ion specific forces acting that are ignored in such modelling. In this paper, the electrostatic potential profile and the concentration profile of co-ions and counterions near charged surfaces are calculated. These results are obtained by solving the classical PB equation and a modified PB equation in bispherical coordinates, taking into account the van der Waals dispersion interactions between the ions and both surfaces. Once the electrostatic potential is known we calculate the double layer force between two charged spheres. This is the first paper that solves the modified PB equation in bispherical coordinates. It is also the first time that the finite volume method is used to solve the PB equation in bispherical coordinates. This method divides the calculation domain into a certain number of sub-domains, where the physical law of conservation is valid, and can be readily implemented. The finite volume method is implemented for several geometries and when it is applied to solve PB equations presents low computational cost. The proposed method was validated by comparing the numerical results for the classical PB calculations with previous results reported in the literature. New numerical results using the modified PB equation successfully predicted the ion specificity commonly observed experimentally.     

An equation equivalent to the Meyer equation

Conclusions The derivation of an equation to the Meyer equation for tautomers possessing two forms in equilibrium one of which absorbs in the IR or UV spectra, was given.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 11, pp. 2611–2612, November, 1969.     

Master equation approach to vibrational energy transfer between two diatomic molecules

In this paper a semiclassical non-markovian master equation is derived. We begin by using the well-known tetradic form of the Liouville equation for a reduced density operator. By projecting the diagonal matrix elements of the operator, we obtain an infinite-order master equation. This equation is then applied in the lowest-order approximation to collinear collisions between the diatomic molecules: H₂H₂, N₂N₂ and Cl₂Cl₂. With an assumed form of the interaction potential for such a problem we have also derived an analytical expression for the V—V transition probabilities. They are then calculated over a wide range of velocities of the colliding molecules and compared with exact semiclassical ones. An excellent agreement of the results is found for small velocities (i.e. υ ≈ 10⁴ cm/s). For larger values of υ (≈ 10⁵ cm/s) the results obtained from the master equation approach agree with the exact ones only in the low-velocity range for light molecules and low oscillatory states.     

The geometric isomers of methyl-2,4,6-decatrienoate, including pheromones of at least two species of stink bugs

All eight geometric isomers of methyl 2,4,6-decatrienoate were synthesized from readily accessible starting materials by fully exploiting Wittig-type olefinations, and taking advantage of an easy separation of 2E and 2Z unsaturated esters. The aggregation pheromone of the brown-winged green bug, Plautia stali, methyl (E,E,Z)-2,4,6-decatrienoate (also a cross-attractant for the brown marmorated stink bug, Halyomorpha halys), was expediently produced in two easy steps from (E)-4,4-dimethoxy-2-butenal in 55% yield. The sex pheromone of the red-shouldered stink bug, Thyanta pallidovirens, methyl (E,Z,Z)-2,4,6-decatrienoate, was conveniently synthesized from 2,4-octadiyn-1-ol in 32% yield using in situ manganese dioxide oxidation-Wittig condensation in a key step.     

Bulk melting of ice at the limit of superheating

The ice-water phase transition after an ultrafast temperature jump is studied in HDO:D2O (15 M) ice with use of 2-color IR spectroscopy. The OH-stretching vibration is applied for rapid heating of the sample and for fast and sensitive probing of local temperature and structure. For energy depositions beyond the limit of superheating (330 +/- 10 K) partial melting in two steps is observed and assigned to (i) catastrophic melting within the thermalization time of the excited ice lattice of 5 +/- 2 ps and (ii) secondary melting with a time constant of 33 +/- 5 ps that is assigned to interfacial melting at the generated phase boundaries. The latter process is found to consume energy amounts in agreement with the latent heat of melting and is accompanied by an accelerated temperature and pressure decrease of the residual ice component.     

SO2 removal performances of Al- and Mg-modified carbide slags from CO2 capture cycles at calcium looping conditions

Journal of Thermal Analysis and Calorimetry - In this work, the SO2 removal performances of the aluminum- and the magnesium-modified carbide slags fabricated by the combustion synthesis from CO2...     

共沸点存在的充分条件

以纯液体蒸气压和亨利系数为出发点,讨论了二组分气液相图存在共沸点的充分条件。结合纯液体蒸气压和亨利系数的物理意义,对共沸点存在的条件进行了说明。     

Polymer fibers drawn to the limit

The orientational drawing of polymers is known to be terminated because of sample rupture. The limiting draw ratio λ_lim reached may be different (either large or small) depending on the polymer and the actual drawing conditions. The purpose of the present work is to identify the change of supermolecular structure of polymer fibers which results in the termination of orientational drawing. Small-and wide-angle x-ray diffraction were used to study the variation of geometrical parameters of this structure with increasing draw ratio λ. The geometrical parameters discussed are the dispersions (fluctuation) of long periods and of longitudinal sizes of crystalline as well as amorphous regions. In this study we used fibers of poly(vinyl alcohol), poly(ε-caprolactam), polyoxymethylene, and poly(4,4′-diphenyloxide) pyromellitimide. It is found that the long period dispersion of these polymers, drawn under different conditions, increases to approximately the same value for different samples drawn to the limit, this relative standard deviation δ_L of long periods being 0.30-0.40. It is also found that the crystallite size dispersion does not increase with increasing λ; the increase of λ_L is due to increasing dispersion of the amorphous region lengths. For poly(vinyl alcohol) fibers drawn to the limit under different conditions and which have different λ_lim, the relative standard deviation of the sizes of amorphous regions δ_A turned out to be about the same (ca. 0.60). The latter evidence gives grounds to suggest that the rupture of polymers under drawing is associated with reaching a high degree of amorphous region size dispersion. In those regions which are considerably below the average size there probably will appear local overstress and molecular ruptures because the relative deformation of these regions is much larger than that of the adjacent regions in the cross section of the sample.