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.     

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.     

Statistical and probabilistic approaches to confidence intervals of linear calibration in liquid chromatography.

The aim of this paper is to compare the reliability of two approaches to estimate the 95% confidence intervals of linear calibration in real situations. One is the statistical approach, which is well known in statistics, and the other is the probabilistic approach, which is based on a theory to predict the precision of instrumental analyses mainly from signal and noise, called FUMI (Function of Mutual Information) theory. The high-performance liquid chromatographic determination of quisalofop and maltose is taken as an example. Calibration lines obtained under the same experimental conditions are superimposed on the 95% confidence intervals to investigate whether the resulting confidence intervals can include all the calibration lines reasonably. A pair of 95% confidence intervals (upper and lower limits) can be calculated from each calibration line, but varies from calibration line to calibration line, although obtained under the same experimental conditions. The variability and reliability of the 95% intervals are also examined.     

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.     

A probabilistic evolution approach trilogy, part 2: spectral issues for block triangular evolution matrix, singularities, space extension

This is the second part of the trilogy on the probabilistic evolution approach and related to the quantum dynamical systems as the first part is. In this sense this work extends the content of the first part to the perhaps secondary but very important details. The spectral investigation of the evolution matrix reveals important issues first and brings the importance of the zero eigenvalues to the surface. The asymptotic convergence possibility and difficulties arising from there can be softened by redefining the state vector. Beside the redefinition, the dimensional extension by adding new elements to the state vector may facilitate the utilization of evolution matrix by bringing conicality or at least multinomiality. The space extension may also help us to deal with singular Hamiltonian systems. All these issues are focused on rather phenomenologically. Illustrative or not, no comprehensive implementation is given since the main purpose is just conceptuality.     

Hydrodynamics from statistical mechanics: combined dynamical-NEMD and conditional sampling to relax an interface between two immiscible liquids

We present a method to study hydrodynamic phenomena from atomistic simulations. In statistical mechanics, these fields are computed as the ensemble average over the time dependent probability density function corresponding to the time evolution of an initial conditional probability density function consistent with some initial conditions. These initial conditions typically consist in constraints on some macroscopic fields, e.g. the density field. We show how these processes can be studied by combining the dynamical approach to non-equilibrium molecular dynamics with the restrained simulation approach. As an illustration of our method, we study the relaxation to the equilibrium of an interface between two immiscible liquids. We show that, at a variance with the local time average method, the standard atomistic approach used in this field, our method is able to produce (macroscopic) fields satisfying the symmetry conditions of the problem.     

Usage de l'algebre de lie su(n) dans l'etude des systems à n etats. VI. Equations linéaires d'evolution positive

The n-level quantum system density-operators evolution is not usually given by the Liouville equation, but by a more general one. This equation must keep density-operators trace, hermiticity, and positiveness. These three conditions restrict the available kinds of evolution equations. In this paper we investigate linear equations for systems without memory effects. By using the first two conditions and the formalism we introduced in earlier papers, the evolution equation takes the form of a first order differential equation concerning the n² – 1 dimension “coherence-vector.” The third one is the essential object of this paper. Moreover, we obtain a cononical splitting of this equation into four parts that may be separately studied.     

Resolution and mass accuracy in matrix-assisted laser desorption ionization-time-of-flight

A mathematical model of time-of-flight mass analyzers employing uniform electric fields is presented that allows “exact” calculations of flight times as functions of mass-to-charge ratio, initial velocity and position, applied voltages, and instrument geometry. An “approximate” equation based on a series expansion of the “exact” result is derived which allows focusing conditions and limits on resolution to be determined for different instrument geometries and operating conditions. The fundamental theory is applied to predicting resolution and mass accuracy in matrix-assisted laser desorption ionization-time of flight. In this case higher order velocity focusing can provide excellent correction for the initial velocity distribution of a selected mass-to-charge ratio, but the focusing is mass-to-charge ratio dependent. There is generally a trade-off between ultimate resolution at a particular mass-to-charge ratio and resolution and mass accuracy over a broad mass range. In most practical applications the latter is more important. Calculations are compared with experimental results for a particular analyzer geometry, both at theoretical optimum velocity focus and at operating conditions where ultimate resolution is sacrificed for a broader range of relatively high resolution and better mass accuracy.     

Application of probabilistic neural network in the clinical diagnosis of cancers based on clinical chemistry data

As a recently developed and powerful classification tool, probabilistic neural network was used to distinguish cancer patients from healthy persons according to the levels of nucleosides in human urine. Two datasets (containing 32 and 50 patterns, respectively) were investigated and the total consistency rate obtained was 100% for dataset 1 and 94% for dataset 2. To evaluate the performance of probabilistic neural network, linear discriminant analysis and learning vector quantization network were also applied to the classification problem. The results showed that the predictive ability of the probabilistic neural network is stronger than the others in this study. Moreover, the recognition rate for dataset 2 can achieve to 100% if combining these three methods together, which indicated the promising potential of clinical diagnosis by combining different methods.     

Using probabilistic relational learning to support bronchial carcinoma diagnosis based on ion mobility spectrometry

Ion Mobility Spectrometry (IMS) provides a means for analyzing the substances a person exhales. In this paper, we report on an approach to support early diagnosis of bronchial carcinoma based on such IMS measurements. Given the peaks in a set of ion mobility spectra, we first cluster these peaks with a modified k-means algorithm. We then apply probabilistic relational modelling and learning methods to a logical representation of the data obtained from the ion mobility spectra and the peak clusters. Markov Logic Networks and the MLN system Alchemy are employed for various modelling and learning scenarios. These scenarios are evaluated with respect to ease of use, classification accuracy, and knowledge representation aspects.     

A semi-supervised learning approach for RNA secondary structure prediction

RNA secondary structure prediction is a key technology in RNA bioinformatics. Most algorithms for RNA secondary structure prediction use probabilistic models, in which the model parameters are trained with reliable RNA secondary structures. Because of the difficulty of determining RNA secondary structures by experimental procedures, such as NMR or X-ray crystal structural analyses, there are still many RNA sequences that could be useful for training whose secondary structures have not been experimentally determined. In this paper, we introduce a novel semi-supervised learning approach for training parameters in a probabilistic model of RNA secondary structures in which we employ not only RNA sequences with annotated secondary structures but also ones with unknown secondary structures. Our model is based on a hybrid of generative (stochastic context-free grammars) and discriminative models (conditional random fields) that has been successfully applied to natural language processing. Computational experiments indicate that the accuracy of secondary structure prediction is improved by incorporating RNA sequences with unknown secondary structures into training. To our knowledge, this is the first study of a semi-supervised learning approach for RNA secondary structure prediction. This technique will be useful when the number of reliable structures is limited.     

Correlation and response functions with non-Markovian dissipation: a reduced Liouville-space theory

Based on a recently developed quantum dissipation formulation [R. X. Xu and Y. J. Yan, J. Chem. Phys. 116, 9196 (2002)], we present a reduced Liouville-space approach to evaluate the response and correlation functions of dissipative systems. The weak system-bath interaction is treated properly for its effects on the initial state, the evolution, and the correlation between coherent driving and non-Markovian dissipation. Numerical demonstration shows this correlated effect cannot be neglected even in the calculation of linear response quantities that do not explicitly depend on external fields. Highlighted in this paper is also the proper choice of theory among various formulations in the weak system-bath interaction regime.     

Classifying molecules using a sparse probabilistic kernel binary classifier

The central idea of supervised classification in chemoinformatics is to design a classifying algorithm that accurately assigns a new molecule to one of a set of predefined classes. Tipping has devised a classifying scheme, the Relevance Vector Machine (RVM), which is in terms of sparsity equivalent to the Support Vector Machine (SVM). However, unlike SVM classifiers, the RVM classifiers are probabilistic in nature, which is crucial in the field of decision making and risk taking. In this work, we investigate the performance of RVM binary classifiers on classifying a subset of the MDDR data set, a standard molecular benchmark data set, into active and inactive compounds. Additionally, we present results that compare the performance of SVM and RVM binary classifiers.     

A probabilistic derivation of the partial least-squares algorithm

Traditionally the partial least-squares (PLS) algorithm, commonly used in chemistry for ill-conditioned multivariate linear regression, has been derived (motivated) and presented in terms of data matrices. In this work the PLS algorithm is derived probabilistically in terms of stochastic variables where sample estimates calculated using data matrices are employed at the end. The derivation, which offers a probabilistic motivation to each step of the PLS algorithm, is performed for the general multiresponse case and without reference to any latent variable model of the response variable and also without any so-called "inner relation". On the basis of the derivation, some theoretical issues of the PLS algorithm are briefly considered: the complexity of the original motivation of PLS regression which involves an "inner relation"; the original motivation behind the prediction stage of the PLS algorithm; the relationship between uncorrelated and orthogonal latent variables; the limited possibilities to make natural interpretations of the latent variables extracted.     

A probabilistic theory on the time to fracture of amorphous crosslinked polymers

The energy probability theory of fracture proposed by Valanis has been extended to apply to the global fracture of a large material region and has been generalized to take into account the probabilistic distribution among bonds of the free energy of the region. In contrast to the theory of fracture initiation of the first author, it is assumed, in the present paper, that global fracture is brought about by the fracture of a critical number N_cr of primary bonds. Whereas this condition alleviates some of the problems encountered in the application of the theory to fracture initiation and propagation—such as determination of the local energy field at the root of a developed crack—it introduces the difficulty that N_cr may, in general, depend on the geometry of the specimen and the directionality of the stress field, though it was found to be relatively insensitive to the rate of deformation, as corroborated in this paper. The theory is used in this work to predict the time of global fracture, under conditions of constant load, constant strain, and constant strain rate.     

Q-fit: a probabilistic method for docking molecular fragments by sampling low energy conformational space

A new method is presented that docks molecular fragments to a rigid protein receptor. It uses a probabilistic procedure based on statistical thermodynamic principles to place ligand atom triplets at the lowest energy sites. The probabilistic method ranks receptor binding modes so that the lowest energy ones are sampled first. This allows constraints to be introduced to limit the depth of the search leading to a computationally efficient method of sampling low energy conformational space. This is combined with energy minimization of the initial fragment placement to arrive at a low energy conformation for the molecular fragment. Two different search methods are tested involving (i) geometric hashing and (ii) pose clustering methods. Ten molecular fragments were docked that have commonly been used to test docking methods. The success rate was 8/10 and 10/10 for generating a close solution ranked first using the two different sampling procedures. In general, all five of the top ranked solutions reproduce the observed binding mode, which increases confidence in the predictions. A set of ten molecular fragments that have previously been identified as problematic were docked. Success was achieved in 3/10 and 4/10 using the two different methods. Again there is a high level of agreement between the two methods and again in the successful cases the top ranked solutions are correct whilst in the case of the failures none are. The geometric hashing and pose clustering methods are fast averaging 13 and 11 s per placement respectively using conservative parameters. The results are very encouraging and will facilitate the process of finding novel small molecule lead compounds by virtual screening of chemical databases.     

Entropically driven phase separation of highly branched/linear polyolefin blends

Small‐angle neutron scattering (SANS) was used to examine the melt phase behavior of a heavily branched comb PEE polymer blended separately with two linear PEE copolymers. In this case, PEE refers to poly(ethylene‐r‐ethylethylene) with 10% ethylene units; therefore, the molecular architecture was the only difference between the two components of the blends. The molecular weights of the two linear random copolymers were 60 and 220 kg/mol, respectively. The comb polymer contained an average of 54 long branches, with a molecular weight of 13.7 kg/mol, attached to a backbone with a molecular weight of 10 kg/mol. Three different volume compositions (25/75, 50/50, and 75/25) were investigated for both types of blends. SANS results indicate that all the blends containing the lower molecular weight linear polymer formed single‐phase mixtures, whereas all the blends containing the high molecular weight linear polymer phase‐separated. These results are discussed in the context of current theories for polymer blend miscibility. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 2965–2975, 2000