Probabilistic evolution theory for the solution of explicit autonomous ordinary differential equations: squarified telescope matrices

Probabilistic evolution theory (PREVTH) is used for the solution of initial value problems of first order explicit autonomous ordinary differential equation sets with second degree multinomial right hand side functions. It is an approximation method based on Kronecker power series: a rewriting of multivariate Taylor series using matrices having certain flexible parameters. Kronecker power series have matrices which are called telescope matrices: \(n \times n^{j+1}\) matrices where j is the index of summation. The additive terms of each telescope matrix is formed through Kronecker product from both sides by Kronecker powers of identity matrices. Recently, squarification is proposed in order to avoid the growing of the matrices in size at each additive term of the series. This paper explains the squarification procedure: the procedure used in order to avoid Kronecker multiplications within PREVTH so that the sizes of the matrices do not grow and so that the amount of necessary computation is reduced. The recursion between squarified matrices is also given. As a numerical application, the solution of a Hénon–Heiles system is provided.     

Probabilistic evolution theory for explicit autonomous ordinary differential equations: recursion of squarified telescope matrices and optimal space extension

Probabilistic evolution theory facilitates the solution of initial value problem of explicit autonomous ordinary differential equations with second degree multinomial right hand side functions. Its formulation has components we call telescope matrices. The matrices grow in size very rapidly and has many zeroes and repeating structures. In order to avoid the computational complexity coming from telescope matrices, squarified telescope matrices are utilized. Their calculation is through a recursion. This recursion has been used in several works by the authors and their colleagues but its proof was not given. This work gives the proof of the recursion and all the surrounding details. A second purpose of this work is to provide a method for most facilitative (optimal) space extension. Space extension is needed for using probabilistic evolution theory when degree of multinomiality of the right hand side functions is more than two. For this purpose, an approach using method of exhaustion (brute-force) is proposed.     

A high dimensional model representation based numerical method for solving ordinary differential equations

A new numerical method for solving ordinary differential equations by using High Dimensional Model Representation (HDMR) has been developed in this work. Higher order ordinary differential equations can be reduced to a set of first order ODEs. Although HDMR is generally used for multivariate functions, univariate functions are taken into account throughout the work because of the ODEs’ natures. Not the numerical solution but its image under an appropriately chosen linear ordinary differential operator is expressed as a linear combination of the positive deviation powers of independent variable from its initial value. The linear combination of these image functions are expected to form a basis set under consideration. The unknown constants in the linear combination are found by maximizing the constancy measurer formed in terms of the HDMR components after they are evaluated. Results are compared with well-known step size based numerical methods. A semi qualitative error analysis of the proposed method is also established.     

A probabilistic evolution approach trilogy, part 1: quantum expectation value evolutions, block triangularity and conicality, truncation approximants and their convergence

This is the first one of three companion papers focusing on the “probabilistic evolution approach (PEA)” which has been developed for the solution of the explicit ODE involving problems under certain consistent impositions. The main purpose here is the determination of the expectation value of a given operator in quantum mechanics by solving only ODEs, not directly using the wave function. To this end we first define a basis operator set over the Kronecker powers of an appropriately defined “system operator vector”. We assume that the target operator’s commutator with the system’s Hamiltonian can be expressed in terms of the above-mentioned basis operators. This assumption leads us to an infinite set of linear homogeneous ODEs over the expectation values of the basis operators. Its coefficient matrix is in block Hessenberg form when the target operator has no singularity, and beyond that, it may become block triangular when certain conditions over the system’s potential function are satisfied. The initial conditions are the basic determining agents giving the probabilistic nature to the solutions of the obtained infinite set of ODEs. They may or may not have fluctuations depending on the nature of the probability density. All these issues are investigated in a phenomenological and constructive theoretical manner in this paper. The remaining two papers are devoted to further details of PEA in quantum mechanics, and, the application of PEA to systems defined by Liouville equation.     

Probabilistic evolution approach for the solution of explicit autonomous ordinary differential equations. Part 1: Arbitrariness and equipartition theorem in Kronecker power series

Probabilistic evolution approach is a newly developed theory which may be utilized for the solution of ordinary differential equations. The approach may directly be applied for initial value problems of explicit first order autonomous ordinary differential equation sets with analytic right hand side functions. Analyticity plays an important role since it facilitates the expansion into direct power series which is the key element of the approach. Direct power series appear not only in all applications of probabilistic evolution but also show themselves as a promising tool for novel approximation methods. In this work, similarities and differences between Taylor series and direct power series are rigorously studied. Arbitrariness in transposed vector coefficients of direct power series is detailed. Equipartition theorem of direct power series is conjectured and proven in order to obtain unique transposed vector coefficients.     

Solutions to linear matrix ordinary differential equations via minimal,regular, and excessive space extension based universalization

This work focuses on the solution of the linear matrix ordinary differential equations where the first derivative of the unknown matrix is equal to the same unknown matrix premultiplied by a given matrix polynomially varying with the independent variable. Work aims to get a universal form for this equation by using the space extension concept where new unknowns are defined to get more amenable form for the equation. The convergence of the series solution to this equation obtained via minimal, regular, and excessive space extension is also investigated with the aid of an appropriate norm analysis which also enables us to get error estimates for the truncated series solutions. A few illustrative examples are presented for practical convergence issues like approximation quality.     

A probabilistic foundation for dynamical systems: phenomenological reasoning and principal characteristics of probabilistic evolution

This paper is the second in a series of two. The first paper has been devoted to the detailed explanation of the mathematical formulation of the underlying theoretical framework. Specifically, the first paper shows that it is possible to construct an infinite linear ODE set, which describes a probabilistic evolution. The evolution is probabilistic because the unknowns are expectations, with appropriate initial conditions. These equations, which we name, Probabilistic Evolution Equations (PEE) are linear at the level of ODEs and initial conditions. In this paper, we first focus on the phenomenological reasoning that lead us to the derivation of PEE. Second, the aspects of the PEE construction is revisited with a focus on the spectral nature of the probabilistic evolution. Finally, we postulate fruitful avenues of research in the fields of dynamical causal modeling in human neuroimaging and effective connectivity analysis. We believe that this final section is a prime example of how the rigorous methods developed in the context of mathematical chemistry can be influential in other fields and disciplines.     

Vibronic coupling simulations for linear and nonlinear optical processes: theory

A comprehensive vibronic coupling model based on the time-dependent wavepacket approach is derived to simulate linear optical processes, such as one-photon absorbance and resonance Raman scattering, and nonlinear optical processes, such as two-photon absorbance and resonance hyper-Raman scattering. This approach is particularly well suited for combination with first-principles calculations. Expressions for the Franck-Condon terms, and non-Condon effects via the Herzberg-Teller coupling approach in the independent-mode displaced harmonic oscillator model are presented. The significance of each contribution to the different spectral types is discussed briefly.     

Liquid-liquid equilibria for ternary systems of water-phenol and solvents: data and representation with models

Alvarez Gonzalez, J.R., Macedo, E.A., Soares, M.E. and Medina, A.G., 1986. Liquid-liquid equilibria for ternary systems of water-phenol and solvents: data and representation with models. Fluid Phase Equilibria, 26: 289–302.Liquid-liquid equilibrium data are presented for the systems water-phenol-benzene, water-phenol-ethylbenzene, water-phenol-nonanoic acid, water-phenol-ethyl acetate, water-phenol-isopropyl acetate, water-phenol-n-butyl acetate, water-phenol-isoamyl acetate and water-phenol-cyclohexyl acetate at 25°C and water-phenol-n-hexyl acetate at 25, 35 and 45°C. These systems are of practical interest for solvent extraction studies, as phenol constitutes a valuable raw material in the production of various chemicals and is recoverable from residual waters.The experimental data presented in this paper are correlated in terms of the NRTL and UNIQUAC models and the relevant parameters are presented.     

A self-consistent variation-perturbation theory for open shell atomic systems: Calculation of polarizabilities and shielding factors

In this paper a fully coupled variation-perturbation theory has been introduced for handling open-shell atomic systems in the presence of static perturbations. The static dipolar and quadrupolar polarizabilities and shielding factors for Li, Be⁺, B²⁺, B, C⁺, Na and Al have been calculated and compared with available results. Important sources of discrepancies in previous calculations of shielding factors have been noticed and critically examined. The dipolar shielding factors calculated in this paper show satisfactory agreement with the theoretical N/Z ratio. In the absence of any theoretical test for the quadrupolar shielding factor, the accuracy of the calculations of γ_∞, should be judged in the light of the accuracy of the corresponding β∞ values.     

A dynamic mean field theory for dissipative interacting many-electron systems: Markovian formalism and its implementation

To demonstrate its applicability for realistic open systems, we apply the dynamic mean field quantum dissipative theory to simulate the photo-induced excitation and nonradiative decay of an embedded butadiene molecule. The Markovian approximation is adopted to further reduce the computational time, and the resulting Markovian formulation assumes a variation of Lindblad's semigroup form, which is shown to be numerically stable. In the calculation, all 22 valence electrons in the butadiene molecule are taken as the system and treated explicitly while the nuclei of the molecules are taken as the immediate bath of the system. It is observed that (1) various excitations decay differently, which leads to different peak widths in the absorption spectra; and (2) the temperature dependences of nonradiative decay rates are distinct for various excitations, which can be explained by the different electron-phonon couplings.     

Curing of linear and crosslinked epoxy systems: A fluorescence study with dansyl derivatives

The curing of diglycidyl ether of bisphenol A (DGEBA) with N,N′-dimethylethylenediamine (N,N′-DMEDA) or ethylenediamine (EDA) was monitored by fluorescence spectroscopy and Fourier transform infrared (in the near-infrared region). 5-Dimethylamino-naphthalene-1-sulfonamide (DNS) derivatives were used as probes (fluorophores added to the reaction mixture) and labels (fluorophores attached by covalent bonds to diglycidyl reactants). The term containing the ratio of the reaction rate constants for the addition of the secondary and primary amine hydrogens to the epoxide was included in the reduced reaction rate term for the autocatalyzed and catalyzed epoxide curing reactions. The changes in the integrated fluorescence intensities of the labels during the epoxy group conversion indicated, in some cases, the most important changes in the chemical transformations of the reaction mixture: the epoxy group conversion, during which a rapid increase in the tertiary amino group concentration was first observed; the gel point (for EDA); and the entry of the system into the glassy state (for N,N′-DMEDA and EDA). The fluorescence probes monitored neither the gel point nor the threshold of the glassy state. For the DGEBA–N,N′-DMEDA system, a wavy dependence of the integrated fluorescence intensities of the DNS labels on the epoxy group conversion might reflect the molar concentrations of polymer homologues (referred to the initial number of moles in the system) in the reaction mixtures of the diepoxide and secondary diamine. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 64–78, 2004     

Viscosity,refractive index,and surface tension of multicomponent systems: Mathematical representation and estimation from data for binary systems

An equation previously developed for estimation of the excess thermodynamic properties of multicomponent systems from binary mixing data has been applied to other physical properties through extrathermodynamic properties such as the excess Gibbs free energy of activation for viscous flow, molar refractivity, and exess surface free energy. This equation provides reasonably accurate predictions for viscosity, refractive index and surface tension of ternary and quaternary systems, given the properties of the various binary combinations of the components. The equation also serves quite well as a point-of-departure for mathematical representation of experimental data, in that all of the data considered could be represented within experimental uncertainty with the aid of no more than one adjustable parameter for each multicomponent system.     

364A - Multicomponent redox systems: analytical representation A(Uh)9 method of analysis and theoretical resolution limits

In order to analyse spectrometric and potentiometric data for a multicomponent system we have derived a relationship A(U_h) relating directly absorbance and redox potential changes. The advantages of such a representation A(U_h) are discussed with regard to the usual one [log(ox/red), U_h] and compared to other methods.The theoretical limits of the resolution of the midpoint potentials of two components are determined in the case of one and two-electron processes. These limits, which are function of the physical parameters and of the experimental precisions, can be less than 5 mV. The limit for the detection of a minor absorbing species is also discussed.The analysis technique was tested experimentally with mixtures of dyes. In the following paper of this issue the potentiometric-resolution of the c type cytochromes in yeast mitochondria is presented.