Analytical Derivation of Moment Equations in Stochastic Chemical Kinetics

The master probability equation captures the dynamic behavior of a variety of stochastic phenomena that can be modeled as Markov processes. Analytical solutions to the master equation are hard to come by though because they require the enumeration of all possible states and the determination of the transition probabilities between any two states. These two tasks quickly become intractable for all but the simplest of systems. Instead of determining how the probability distribution changes in time, we can express the master probability distribution as a function of its moments, and, we can then write transient equations for the probability distribution moments. In 1949, Moyal defined the derivative, or jump, moments of the master probability distribution. These are measures of the rate of change in the probability distribution moment values, i.e. what the impact is of any given transition between states on the moment values. In this paper we present a general scheme for deriving analytical moment equations for any N-dimensional Markov process as a function of the jump moments. Importantly, we propose a scheme to derive analytical expressions for the jump moments for any N-dimensional Markov process. To better illustrate the concepts, we focus on stochastic chemical kinetics models for which we derive analytical relations for jump moments of arbitrary order. Chemical kinetics models are widely used to capture the dynamic behavior of biological systems. The elements in the jump moment expressions are a function of the stoichiometric matrix and the reaction propensities, i.e. the probabilistic reaction rates. We use two toy examples, a linear and a non-linear set of reactions, to demonstrate the applicability and limitations of the scheme. Finally, we provide an estimate on the minimum number of moments necessary to obtain statistical significant data that would uniquely determine the dynamics of the underlying stochastic chemical kinetic system. The first two moments only provide limited information, especially when complex, non-linear dynamics are involved.     

Solution of the population balance equation using the sectional quadrature method of moments (SQMOM)

A discrete framework is introduced for simulating the particulate physical systems governed by population balance equations (PBE) with particle splitting (breakage) and aggregation based on accurately conserving (from theoretical point of view) an unlimited number of moments associated with the particle size distribution. The basic idea is based on the concept of primary and secondary particles, where the former is responsible for distribution reconstruction while the latter is responsible for different particle interactions such as splitting and aggregation. The method is found to track accurately any set of low-order moments with the ability to reconstruct the shape of the distribution. The method is given the name: the sectional quadrature method of moments (SQMOM) and has the advantage of being not tied to the inversion of large sized moment problems as required by the classical quadrature method of moments (QMOM). These methods become ill conditioned when a large number of moments are needed to increase their accuracy. On the contrary, the accuracy of the SQMOM increases by increasing the number of primary particles while using fixed number of secondary particles. Since the positions and local distributions for two secondary particles are found to have an analytical solution, no large moment inversion problems are anymore encountered. The generality of the SQMOM is proved by showing that all the related sectional and quadrature methods appearing in the literature for solving the PBE are merely special cases. The method has already been extended to bivariate PBEs.     

基于切比雪夫矩的抗几何攻击图像水印

旋转、缩放和平移(RST)等几何攻击能够破坏水印检测的同步性,而使水印检测失败。本文提出了基于切比雪夫矩的抗几何攻击图像水印算法.首先计算原始图像的切比雪夫矩,并将其作为原始矩,然后将水印图像在空域直接嵌入到原始图像中,并提取嵌入区域的切比雪夫矩。当图像未受攻击时,利用嵌入区域的矩与原始矩之差便可以提取出水印图像。当图像受到攻击后,利用上述方法同样可以提取出水印图像,但是我们需要将原始图像做相应的变换。与使用Zernike矩相比,切比雪夫矩具有更好的图像描述能力。当水印图像的目标物体较小时,切比雪夫矩可以较好的提取出水印图像,而Zernike矩则无法做到,同时该算法对于旋转、缩放、噪声攻击也具有较好的鲁棒性。     

Mathematical Properties of the Moments of Truncated Dynamic Magnetic Resonance Spectra and Their Potential Application

The moments of truncated dynamic magnetic resonance spectra, M_n(L), are expanded in terms of power series of the integration range −L to +L. The expansion consists of three contributions: 1) an L-independent term that, for the first three moments (n=1 to 3), is independent of the motion and equals the corresponding moment, M_n, of the rigid powder spectrum; 2) a limited number of (positive) terms, diverging as L^k (k<n, odd), reflecting the broadening effect due to motion; these terms vanish for the first three moments and become nonzero, with motion-dependent coefficients, only from the fourth moment on; and 3) an infinite series of converging (negative) terms, in powers of 1/L^k (k is odd), reflecting the reduction of the moments due to the truncation of the spectra; these terms are motion dependent for all moments. The convergence properties of this series are discussed and expressions for the lower (truncated) moments in the slow and fast motion limits of a secular Hamiltonian are derived. For the slow motion limit, it is shown how the L-dependence of the moments can be used to estimate the magnetic and dynamic parameters. The procedure is demonstrated using computer-simulated spectra. In the fast motion regime, closed expressions are obtained in a similar form to those of the relaxation equations. The effect of natural line width and strong-collision dynamics on the various moments as well as that of nonsecular terms in the Hamiltonian are also briefly discussed.     

ON AVERAGED DIFFUSION EQUATIONS

Methods of obtaining averaged diffusion equations are considered in case of non-uniform profile of the velocity in a channel. With the flow of Couette as an example, the comparison of exact and approximate solutions (obtained by means of perturbation method) has been carried out. The peculiarities of function of residence time distribution of liquid in the flows with non-uniform velocity field are noted. It is shown that the distribution function moments values including the zero and first moments values would depend on the degree of the velocity profile irregularity, on efficiency of radial mixing in a system as well as on the averaging method. The averaged diffusion equations which have been found by means of perturbation method are the most general of proposed ones at present in the appropriate literature. The Taylor's model and Goldstein's hyperbolic equations are included in said averaged equations as particular cases. The table of the numerical values of first three moments of RTD-function necessary for determining of the model parameters is given. The problems of application of the obtained averaged equation for calculating real chemical apparatuses, e.g. reactors, are discussed.     

Heat and mass transfer in packed beds—IV: Use of weighted moments to determine axial dispersion coefficients

The method of weighted moments is analysed on the basis of measured axial dispersion in a methanehydrogen mixture flowing through a bed of glass beads at various velocities. The optimum value of the Laplace parameter S is chosen as the one giving the minimum deviation between experiment and model in the integrand of the second weighted moment. The corresponding axial dispersion coefficients are lower than those determined with ordinary moments, and agree very closely with the best-fit values in the time domain. An equation proposed by Anderssen and White based on other considerations is shown to give a relatively good approximation of the optimum value of S. With the aid of this near-optimum S, calculating time for weighted moments can be shorter than for curve fitting in the time domain.     

Planar and Nonplanar Unsaturation. Preparation,Properties and Molecular-Orbital Characterization of Some Fluoro-Derivatives of Anthracene and Anthraquinone

Several fluoroanthracenes (I-III) and fluoroanthraquinones (IV-VII) were prepared, and their PMR and UV spectra, as well as dipole moments, recorded. These are discussed in terms of molecular-orbital data, both of the pi-electronic type (including nonbonding 2p-orbitals) and of the all-valence-electron type. PMR signals can be assigned on the basis of such calculations. The interpretation of UV spectra is straightforward in the case of fluoroanthracenes, but other techniques are required for fluoroanthraquinones. Measured dipole moments are close to the theoretical, except in the case of 1, 4, 5, 8-tetrafluoroanthraquinone. A distorted structure is proposed for this molecule.     

The reduction of the order of state variable models using the method of moments

High order state variable models of chemical processes can be reduced to lower order models by matching the moments of the impules responses for those states which it is desired to retain. The moments of the full model are calculated directly from the state equations and the parameters of the reduced model are calculated from these moments. The method places no restriction on the choice of outputs and all the inputs are retained.     

Integral formulation of the population balance equation: Application to particulate systems with particle growth

Numerical solution of the population balance equation (PBE) is widely used in many scientific and engineering applications. Available numerical methods, which are based on tracking population moments instead of the distribution, depend on quadrature methods that destroy the distribution itself. The reconstruction of the distribution from these moments is a well-known ill-posed problem and still unresolved question. The present integral formulation of the PBE comes to resolve this problem. As a closure rule, a Cumulative QMOM (CQMOM) is derived in terms of the monotone increasing cumulative moments of the number density function, which allows a complete distribution reconstruction. Numerical analysis of the method show two unique properties: first, the method can be considered as a mesh-free method. Second, the accuracy of the targeted low-order cumulative moments depends only on order of the CQMOM, but not on the discrete grid points used to sample the cumulative moments.     

APPROXIMATE POLYNOMIAL EXPANSION METHOD FOR INVERTING LAPLACE TRANSFORMS OF IMPULSE RESPONSES

We evaluate a method for inversion of Laplace transforms based on analytical expressions of temporal moments substituted into generalized Laguerre polynomial expansions. The moment expressions are derived from the Laplace transform of an impulse response function, a computation that can be performed by a symbolic manipulation computer program. Only the lower-order moments that contribute to the Poisson function are necessary for quite accurate approximations of heat and mass transfer, as well as chemical mixing and reaction problems, in the range of models between a stirred tank and a long column.     

Quadrature-based moment methods for the population balance equation: An algorithm review

The dispersed phase in multiphase flows can be modeled by the population balance model (PBM). A typical population balance equation (PBE) contains terms for spatial transport, loss/growth and breakage/coalescence source terms. The equation is therefore quite complex and difficult to solve analytically or numerically. The quadrature-based moment methods (QBMMs) are a class of methods that solve the PBE by converting the transport equation of the number density function (NDF) into moment transport equations. The unknown source terms are closed by numerical quadrature. Over the years, many QBMMs have been developed for different problems, such as the quadrature method of moments (QMOM), direct quadrature method of moments (DQMOM), extended quadrature method of moments (EQMOM), conditional quadrature method of moments (CQMOM), extended conditional quadrature method of moments (ECQMOM) and hyperbolic quadrature method of moments (HyQMOM). In this paper, we present a comprehensive algorithm review of these QBMMs. The mathematical equations for spatially homogeneous systems with first-order point processes and second-order point processes are derived in detail. The algorithms are further extended to the inhomogeneous system for multiphase flows, in which the computational fluid dynamics (CFD) can be coupled with the PBE. The physical limitations and the challenging numerical problems of these QBMMs are discussed. Possible solutions are also summarized.     

Solution of population balance equations using the direct quadrature method of moments

The implementation of a population balance equation (PBE) in computational fluid dynamics (CFD) represents a crucial element in the simulation of multiphase flows. Some of the available methods, such as classes methods (CM) and Monte Carlo (MC) methods, are computationally expensive and simulation of real cases of practical interest requires intractable CPU times. On the other hand, other methods such as the method of moments (MOM) are computationally affordable but have proven to be inaccurate for a number of cases. In recent work a new closure, the quadrature method of moments (QMOM), has been introduced, applied and validated. In our earlier work, QMOM was shown to be an efficient and accurate method for tracking the moments of the particle size distribution (PSD) in a CFD simulation. However, QMOM presents two main disadvantages: (i) if applied to multi-variate distributions it loses simplicity and efficiency, and (ii) by tracking only the moments of the PSD, it does not represent realistically polydisperse systems with strong coupling between the internal coordinates and phase velocities. In order to address these issues, in this work the direct quadrature method of moments (DQMOM) is formulated, validated, and tested. DQMOM is based on the idea of tracking directly the variables appearing in the quadrature approximation, rather than tracking the moments of the PSD. Nevertheless, for monovariate cases we show that QMOM and DQMOM yield identical results. In addition, we show how it is possible to extend the DQMOM to multivariate cases and some of relevant theoretical and numerical issues are discussed. These issues are discussed in the present work for homogeneous and one-dimensional flows. References to recent CFD applications of DQMOM to multiphase flows are provided as further proof of the utility of the method.     

Studying the Squeeze Forces and Tilting Moments in Misaligned Radial Face Seals Coning

The Reynolds equation for misaligned radial face seals coning is solved numerically, and the squeeze effects on the separating forces and tilting moments are studied. Also, the variations in the seal characteristics with eccentricity, thin film thickness and radius ratio are presented. The numerical results for the forces and moments are given in nondimensional form.     

Dipolmomente und Konformation von 3- und 4-Brom- und Chlorheteracyclohexanen

Dipole Moments and Conformation of 3- and 4-Bromo and Chloro Heteracyclohexanes The dipole moments of some 3- and 4-bromo and chloro heteracyclohexanes are measured by the method of Debye and calculated both by the quantum-mechanical method CNDO/2 and vector addition method. The determination of conformational free energy is made on the basis of measured dipole moments μ of the conformation mixture and the calculated dipole moments μ_e and μ_a of the pure conformers. The results are discussed in relation to n.m.r.-results.     

Sample Moments and Weak Convergence to Multivariate Stochastic Power Integrals

This work considers sample moments arising from least squares, least absolute deviation, and extremum estimators of linear and nonlinear multivariate systems with I(1) regressors. The sample moments are shown to converge weakly to multivariate stochastic power integrals, and these results can be considered as a multivariate generalization of the univariate results reported earlier.     

Hyperfine Interactions of cis and trans Octahedral Fe2+ Sites in the Layered Silicate Annite

Spin-polarized first-principles calculations based on density functional theory were performed for 23- and 101-atom embedded clusters representing annite. Hyperfine interactions and magnetic moments were obtained and compared with experimental Mössbauer spectra and SQUID magnetometry measurements. It was found that the electric field gradients and the magnetic fields are profoundly affected by the position of the hydroxyls. The calculated magnetic moments are close to 4 μ_B in both octahedral sites.     

Description of Aerosol Dynamics by the Quadrature Method of Moments

ABSTRACT

The method of moments (MOM) may be used to determine the evolution of the lower-order moments of an unknown aerosol distribution. Previous applications of the method have been limited by the requirement that the equations governing the evolution of the lower-order moments be in closed form. Here a new approach, the quadrature method of moments (QMOM), is described. The dynamical equations for moment evolution are replaced by a quadrature-based approximate set that satisfies closure under a much broader range of conditions without requiring that the size distribution or growth law maintain any special mathematical form. The conventional MOM is recovered as a special case of the QMOM under those conditions, e.g., free-molecular growth, for which conventional closure is satisfied. The QMOM is illustrated for the growth of sulfuric acid-water aerosols and simulations of diffusion-controlled cloud droplet growth are presented.     

THREE-DIMENSIONAL FINITE ELEMENT ANALYSIS OF STRESS RESPONSE IN ADHESIVE BUTT JOINTS SUBJECTED TO IMPACT BENDING MOMENTS

The stress wave propagation and the stress distribution in adhesive butt joints of T-shaped similar adherends subjected to impact bending moments are calculated using a three-dimensional finite-element method (FEM). An impact bending moment is applied to a joint by dropping a weight. The FEM code employed is DYNA3D. The effects of the Young's modulus of adherends, the adhesive thickness, and the web length of T-shaped adherends on the stress wave propagation at the interfaces are examined. It is found that the highest stress occurs at the interfaces. In the case of T-shaped adherends, it is seen that the maximum principal stress at the interfaces increases as Young's modulus of the adherends increases. In the special case where the web length of T-shaped adherends equals the flange length, the maximum principal stress at the interfaces increases as Young's modulus of the adherends decreases. The maximum principal stress at the interfaces increases as the adherend thickness decreases. The characteristics of the T-shaped adhesive joints subjected to static bending moments are also examined by FEM and compared with those under impact bending moments. Furthermore, strain response of adhesive butt joints was measured using strain gauges. A fairly good agreement is observed between the numerical and the experimental results.     

Conformational Studies on Xanthene,Thioxanthene and Acridan

A study of the dipole moments of xanthone, thioxanthone and their halogeno-derivatives leads to the conclusion that these molecules are not planar. In the xanthene and thioxanthene series, the dipole moments as well as the NMR spectra can be explained by a planar or a non-planar but rapidly inverting model. In the acridan series, the NMR spectra of N-acylacridans show that the molecule is non-planar and inverts. The barrier is 11.7, 13.1 and 13.1 Kcal/mole for the N-acetyl-, N-chloroacetyl- and N-iodoacetyl derivatives, respectively. Equally, it has been observed that at low temperature the rotation around the XCH₂ CO bond in the halogenoacetylacridans is hindered.     

A second‐order moment method applied to gas–solid risers

Second‐order moment method of particles is proposed on the basis of the kinetic theory of granular flow. Closure equations for the third‐order velocity moments are presented to account for the increase of the probability of collisions of particles on the basis of the elementary kinetic theory and order of magnitude analysis. The boundary conditions for the set of equations describing flow of particles are proposed with the consideration of the momentum exchange by collisions between the wall and the particles. The distributions of velocity, concentration and moments of particles are predicted. Simulated results are compared with experimental data measured by Tartan and Gidaspow and Bhusarapu et al. in risers, and Tsuji et al. in a vertical pipe. The effects of the closure equations for the third‐order velocity moments and the fluid‐particle velocity correlation tensor on flow behavior of particles are analyzed. © 2012 American Institute of Chemical Engineers AIChE J, 2012