A contemporary linear representation theory for ordinary differential equations: probabilistic evolutions and related approximants for unidimensional autonomous systems

In this paper, we build on our previous research on probabilistic foundations of dynamical systems and introduce a theory of linear representation for ordinary differential equations. The theory is developed for explicit ODEs and can be further extended to cover implicit cases. In this report, we investigate the case of a canonical single unknown autonomous system. First we construct a linear representation to get an infinite linear ODE set with a constant coefficient matrix which can be transformed into an upper triangular form. Then we find its approximate truncated solutions. We describe a number of properties of the theory using this framework. The companion of this paper expands this canonical approach to cover multidimensional cases using the theory of folded arrays which is another line of research established by our research group.     

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.     

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.     

A probabilistic evolution approach trilogy, part 3: temporal variation of state variable expectation values from Liouville equation perspective

This is the third and therefore the final part of a trilogy on probabilistic evolution approach. The work presented here focuses on the probabilistic evolution determination for the state variables of a many particle system from classical mechanical point of view. Probabilistic evolution involves the expected value evolutions for all natural number Kronecker powers of the state variables, positions and momenta. We use the phase space distribution of the Liouville equation perspective to construct the expected values of the state variables’ Kronecker powers to define unknown temporal functions. The infinite number homogeneous linear ODEs with an infinite constant coefficient matrix are constructed by following the same steps as in the previous two works on quantum mechanics. The only difference is in the definitions of the expected values here. We also focus on a system of many harmonic oscillators to illustrate the block triangularity.     

Probabilistic evolution approach to the expectation value dynamics of quantum mechanical operators,part I: integral representation of Kronecker power series and multivariate Hausdorff moment problems

This is the first one of two companion papers focusing on the establishment of a new path for the expectation value dynamics of the quantum mechanical operators. The main goal of these studies is to do quantum mechanics without explicitly solving Schrödinger wave equation, in other words, without using wave functions except their initially given forms. This goal is achieved by using Ehrenfest theorem and utilizing probabilistic evolution approach (PEA). PEA, first introduced by Metin Demiralp, is a method providing solutions to the nonlinear ordinary differential equations by transforming them to a set of linear ODEs at the cost of denumerably infinite dimensionality. It is recently shown that this method produces analytic solutions, if the initial conditions are given appropriately at some special cases. However, generalization of these conditions to the quantum mechanical applications is not straightforward due to the dispersion of the quantum mechanical systems. For this purpose, multivariate moment problems for the integral representation of the Kronecker power series are introduced and then solved yielding to more specific and precise convergence analysis for the quantum mechanical applications.     

Transition state barriers in multidimensional Marcus theory

Multidimensional Marcus theory is the extension of traditional Marcus theory to systems in which multiple particles are transferred. Rather than the intersecting parabolas of Marcus theory, multidimensional Marcus theory involves the intersection of paraboloids. In this paper, we examine the conditions under which a full multidimensional treatment of these paraboloids is necessary and when it is possible to use a simpler one-dimensional formalism. In particular, we examine transition state barrier energies, which are essential parameters in many reaction rate equations, and which depend on the formalism used. We find, based on both analytic calculations and numerical simulation, that the reduced one-dimensional treatment yields excellent agreement with the exact, multidimensional results over a wide variety of conditions for one particular choice of the single collective reaction coordinate. We also outline a procedure for calculating accurate multidimensional transition state barrier energies and apply it to a two-dimensional model of proton-coupled electron transfer.     

Quantum dynamics calculations using symmetrized, orthogonal Weyl-Heisenberg wavelets with a phase space truncation scheme. III. Representations and calculations

In a previous paper [J. Theo. Comput. Chem. 2, 65 (2003)], one of the authors (B.P.) presented a method for solving the multidimensional Schrodinger equation, using modified Wilson-Daubechies wavelets, and a simple phase space truncation scheme. Unprecedented numerical efficiency was achieved, enabling a ten-dimensional calculation of nearly 600 eigenvalues to be performed using direct matrix diagonalization techniques. In a second paper [J. Chem. Phys. 121, 1690 (2004)], and in this paper, we extend and elaborate upon the previous work in several important ways. The second paper focuses on construction and optimization of the wavelength functions, from theoretical and numerical viewpoints, and also examines their localization. This paper deals with their use in representations and eigenproblem calculations, which are extended to 15-dimensional systems. Even higher dimensionalities are possible using more sophisticated linear algebra techniques. This approach is ideally suited to rovibrational spectroscopy applications, but can be used in any context where differential equations are involved.     

Sequential self-assembly in metal-organic frameworks

This perspective discusses the use of sequential self-assembly in the construction of metal-organic frameworks through the systematic insertion, replacement, and removal of organic structural building units. We review previous works that can be classified as such sequential self-assembly in multidimensional MOFs.     

Fast and reliable numerical methods to simulate complex chemical kinetic mechanisms

Models that simulate atmospheric photochemistry require the use of a stiff ordinary differential equations (ODEs) solver. Since the simulation of the chemical transformations taking place in the system takes up to 80 percent of the CPU time, the numerical solver must be computationally fast. Also, the residual error from the solver must be small. Because most accurate solvers are relatively slow, modelers continue to search for timely, yet accurate integration methods. Over the past years an extensive number of articles have been dedicated to this subject. One of the highly debated questions is whether one should construct specialized algorithms or instead use general methods for stiff ODEs. In the present article we use the second alternative. We apply three linearly (semi-)implicit methods from the classical stiff ODE literature which we modified to implement the sparse routines to solve the system of equations describing a complex kinetic mechanism. © 1998 John Wiley & Sons, Inc. Int J Chem Kinet 30: 349–358, 1998     

A physical model for the longitudinal polarizabilities of polymer chains

The aim of this work is to provide a physical model to relate the polarizability per unit cell of oligomers to that of their corresponding infinite polymer chains. For this we propose an extrapolation method for the polarizability per unit cell of oligomers by fitting them to a physical model describing the dielectric properties of polymer chains. This physical model is based on the concept of a dielectric needle in which we assume a polymer chain to be well described by a cylindrically shaped nonconducting rod with a radius much smaller than its length. With this model we study in which way the polarizability per unit cell approaches the limit of the infinite chain. We show that within this model the macroscopic contribution of the induced electric field to the macroscopic electric field vanishes in the limit of an infinite polymer chain, i.e., there is no macroscopic screening. The macroscopic electric field becomes equal to the external electric field in this limit. We show that this identification leads to a relation between the polarizability per unit cell and the electric susceptibility of the infinite polymer chain. We test our dielectric needle model on the polarizability per unit cell of oligomers of the hydrogen chain and polyacetylene obtained earlier using time-dependent current-density-functional theory in the adiabatic local-density approximation and with the Vignale-Kohn functional. We also perform calculations using the same theory on truly infinite polymer chains by employing periodic boundary conditions. We show that by extrapolating the oligomer results according to our dielectric needle model we get good agreement with our results from calculations on the corresponding infinite polymer chains.     

Association-dissociation equilibrium of loop structures in single-chain folding into a toroidal condensate

Recently, it has been revealed that a semiflexible polyelectrolyte chain can form a partially folded conformation stably as a result of an electrostatic interaction. Interestingly, there are cases where the appearance of this structure requires a high-salt condition of a solution. In order to solve this problem, we consider the double equilibrium of the formation of loops and their aggregation on a single-chain polymer. First, an aggregate with a typical surface energy is examined as a test case. The basic nature of the folding transition is discussed with regard to the chemical potential of loop structures. Next, we consider a charged aggregate for which the interior is completely neutralized by counter ions. In this model, a partially folded chain appears with a high-salt condition. Based on this model, screened interactions between surface charges and a toroidal shape of a folded structure are considered essential factors bihind this phenomenon.     

Role of opposite charges in protein electrospray ionization mass spectrometry

The conformation dependence of protein spectra recorded by electrospray ionization mass spectrometry (ESI-MS) is an interesting and useful phenomenon, whose origin is still the object of debate. Different mechanisms have been invoked in the attempt to explain the lower charge state of folded versus unfolded protein ions in ESI-MS, such as electrostatic repulsions, solvent accessibility, charge availability, and native-like interactions. In this work we try to subject to direct experimental test the hypothesis that conformation-dependent neutralization of charges with polarity opposite to the net charge of the protein ion could play a critical role in such an effect. We present results of time-of-flight nano-ESI-MS on the peptide angiotensin II, indicating that negative charges of carboxylate groups can contribute to spectra recorded in positive-ion mode when stabilized by favorable electrostatic interactions, which is the central assumption of our hypothesis. Comparison of horse and spermwhale myoglobin (Mb) shows that changing the total number of basic residues within a given three-dimensional structure shifts the charge-state distribution (CSD) of the folded protein in positive-ion mode. This result appears to be in contrast to models in which electrostatic repulsions or availability of charges in the ESI droplets represent the limiting factor for the ionization of folded protein ions in ESI-MS. At the same time, it suggests a role of acidic residues in conformational effects in positive-ion mode. Furthermore, an attempt is made to rationalize those cases in which, in contrast, the main charge state observed in ESI-MS under non-denaturing conditions deviates considerably from the net charge expected on the basis of the amino-acid composition. These cases usually correspond to proteins with quite balanced content in basic and acidic residues, suggesting that this might be a factor influencing their charging behavior in ESI-MS. Experiments on mutants of ribonuclease Sa (RNase Sa) reveal that progressively reducing the excess of acidic residues, replacing them by lysine, causes almost no shift in the spectrum of the folded protein in negative-ion mode. Analogously, variants with an excess of three or five basic residues give similar spectra in positive-ion mode. These results indicate a lower limit to the extent of ionization observable by ESI-MS (6- or 8+ in the case of RNase Sa in water). Below such limit of net charge, changes in the relative amount of ionizable side chains do not affect the qualitative features of the observed CSDs. A progressive loss of signal intensity caused by the mutations in negative-ion mode suggests that low charge states might also be counterselected, even within the m/z range theoretically accessible to the instrument.     

Theoretical advantages and drawbacks of on-line,multidimensional liquid chromatography using multiple columns operated in parallel

The theoretical advantages and drawbacks of using a multiple-, parallel column approach in on-line multidimensional liquid chromatography systems were investigated. Much time or peak capacity can be gained with the use of multiple parallel columns at the second-dimension while the aggregate time of separation increases only by the increment of the gradient time of the second-dimension. Multidimensional chromatographic systems are now used to perform many tasks ranging from routine, fast analyses to specialized, arduous separations. In this work, we focus on the advantages of a multiple, parallel columns approach to on-line multidimensional liquid chromatography systems. Calculations of the achievable peak capacities were made as functions of the number of columns operated in parallel. Increasing the number of second-dimension columns from one to two or three causes the largest increase in peak capacity with only a slight increase of aggregate time. We also present some practical aspects to consider when attempting multidimensional separations with multiple columns operated in parallel.     

On identifiability for chemical systems from measurable variables

The dynamics of the composition of chemical species in reacting systems can be characterized by a set of autonomous differential equations derived from mass conservation principles and some elementary hypothesis related to chemical reactivity. These sets of ordinary differential equations (ODEs) are basically non-linear, their complexity grows as much increases the number of substances present in the reacting media and can be characterized by a set of phenomenological constants (kinetic rate constants) which contains all the relevant information about the physical system. The determination of these kinetic constants is critical for the design or control of chemical systems from a technological point of view but the non-linear nature of the ODEs implies that there are hidden correlations between the parameters which maybe can be revealed with a identifiability analysis.     

超临界CO2中甲醇和乙醇无限稀释扩散系数的分子动力学模拟与实验测定

采用分子动力学模拟(MD)方法对甲醇和乙醇在超临界二氧化碳中的无限稀释扩散系数进行了模拟计算, 并应用泰勒分散理论, 采用超临界色谱仪对模拟结果进行了实验验证. 模拟计算值与实验值较吻合, 且变化规律基本一致, 表明采用这种新方法可以准确有效地预测超临界体系的扩散性质, 能够方便地应用于工程设计.     

Folding of a supramolecular framework based on a tetrametallic clip driven by π-π interactions

Supramolecular extended frameworks based on a rotational molecular clip and having the same "primary structure" can exhibit either an unfolded or a folded framework. This process is driven by intramolecular π-π interactions, resulting in the formation of an infinite π-stacked column within the folded supramolecular framework.