Non-closed Form Research Articles

Physics-Inspired Evolutionary Machine Learning Method: From the Schrödinger Equation to an Orbital-Free-DFT Kinetic Energy Functional.

We introduce a machine learning (ML)-supervised model function (which is in fact a functional rather than a regular function) that is inspired by the variational principle of physics. This ML hypothesis evolutionary method, termed ML-Ω, allows us to go from data to differential equation(s) underlying the physical (chemical, engineering, etc.) phenomena from which the data are derived from. The fundamental equations of physics can be derived from this ML-Ω evolutionary method when the proper training data is used. By training the ML-Ω model function with only three hydrogen-like atom energies, the method can find Schrödinger's exact functional and, from it, Schrödinger's fundamental equation. Then, in the field of density functional theory (DFT), when the model function is trained with the energies from the known Thomas-Fermi (TF) formula , it correctly finds the exact TF functional. Finally, the method is applied to find a local orbital-free (OF) functional expression of the independent electron kinetic energy functional Ts based on the γTFλvW model. By considering the theoretical energies of only five atoms (He, Be, Ne, Mg, and Ar) as the training set, the evolutionary ML-Ω method finds an ML-Ω-OF-DFT local Ts functional (γTFλvW(0.964,1/4)) that outperforms all the OF-DFT functionals of a representative group. Moreover, our ML-Ω-OF functional overcomes the difficulty of LDA's and some local generalized gradient approximation (GGA)-DFT's functionals to describe the stretched bond region at the correct spin configuration of diatomic molecules. Nonsmooth and nonclosed form functionals can be considered in the ML-Ω model function and still be effectively trained. Although our evolutionary ML-Ω model function can work without an explicit prior-form functional, by using the techniques of symbolic regression, in this work, we exploit prior-form functional expressions to make the training process simpler and faster. The ML-Ω method can be considered at the intersection of ML and the natural sciences.

Deriving the new cotangent Fréchet distribution with real data analysis

This study introduces the family of the novel ”cot-generator” distributions based on the cotangent function. The construction of the probability density function based on the cot-generator and its corresponding moments, reliability indicators, and order statistics are derived. Regarding inferential procedures, the maximum likelihood estimation method is implemented to estimate the model parameters in a non-closed form. The novel Cot-Fréchet distribution (NCFD) is then focused on a specific class member, constructed using the Fréchet distribution as the baseline. The estimators’ bias, variance, and mean square error (MSE) efficiency are examined and evaluated through Monte Carlo simulation experiments. The novel model is also applied to some real data sets to assess its efficacy and is compared with other competing distributions.

Dynamic Amplification of Railway Bridges under Varying Wagon Pass Frequencies

Train configurations give rise to a primary wagon pass forcing frequency and their multiples. When any one of these frequencies coincides with the natural frequency of vibration of the bridge, a resonant response can occur. This condition can amplify the dynamic response of the bridge, leading to increased levels of displacement, stresses and acceleration. Increased stress levels on critical bridge structural elements increases the rate at which fatigue damage accumulates. Increased bridge acceleration levels can affect passenger comfort, noise levels, and can also compromise train safety. For older bridges the effects of fatigue, and being able to predict the remaining life, has become a primary concern for bridge engineers. Better understanding of the sensitivity of fatigue damage to the characteristics of the passing train will lead to more accurate remaining life predictions and can also help to identify optimal train speeds for a given train–bridge configuration. In this paper, a mathematical model which enables the dynamic response of railway bridges to be assessed for different train configurations is presented. The model is based on the well established closed from solution of the Euler–Bernoulli Beam (EBB) model, for a series of moving loads, using the inverse Laplace–Carson transform. In this work the methodology is adapted to allow different train configurations to be easily implemented into the formulation in a generalised form. A generalised equation, which captures the primary wagon pass frequency for any train configuration, is developed and verified by presenting the results of the bridge response in the frequency domain. The model, and the accuracy of the equation for predicting the primary wagon pass frequency, is verified using independently obtained measured field train–bridge response data. The main emphasis of this work is to enable the practicing engineer, railway operators and bridge asset owners, to easily and efficiently make an initial assessment of dynamic amplification, and the optimal train speeds, for a given bridge and train configuration. This is visually presented in this work using a Campbell diagram, which shows dynamic amplification and compares this with those calculated based on the design code, across a range of train speeds. The diagram is able to identify train speeds at which a resonance response can occur, and the wagon pass frequency, or its multiples, which are causing the increased dynamic amplification. The model is implemented in Matlab and demonstrated by analysing a range of short- to medium-single span simply supported plate girder railway bridges, typically found on the UK railway network, using the standard BS-5400 train configurations. The model does not consider the effects of the train mass and suspension system as this would require a non-closed form numerical solution of the problem which is not practical for the purposes of an initial assessment of the train–bridge interaction problem.

An efficient zero-order evolutionary method for solving the orbital-free density functional theory problem by direct minimization.

A differential evolution (DE) global optimization method for all-electron orbital-free density functional theory (OF-DFT) is presented. This optimization method does not need information about function derivatives to find extreme solutions. Results for a series of known orbital-free energy functionals are presented. Ground state energies of atoms (H to Ar) are obtained by direct minimization of the energy functional without using either Lagrange multipliers or damping procedures for reaching convergence. Our results are in agreement with previous OF-DFT calculations obtained using the standard Newton-Raphson and trust region methods. Being a zero-order method, the DE method can be applied to optimization problems dealing with non-differentiable functionals or functionals with non-closed forms.

Studying Multi-Stage Diffusion Dynamics using Epidemic Modeling Framework

Buying process has always carried a two-fold perspective with itself. On one hand, it is important for individuals and on other hand it is equally important for the firms to deliver the perfect need and want to the customer. Amongst this entire process, awareness along with positive motivation towards the product; plays an equally significant role in strategizing the plans for any company. Plenty of models have been proposed and many would be in the pipeline that have talked about the connectivity of these processes and their impact on the final adoption. In the current work, these processes have been studied through the analogy taken from epidemic modelling framework. Furthermore, an approximation method; Range Kutta of 4th order has been utilized to come to a near approximate solution to the otherwise available non-closed form solution. The proposed modelling framework is validated on real-life data sets and the results depict the existence and presence of various stages under consideration.

On the geometry of nearly trans-Sasakian manifolds

The geometry of nearly trans-Sasakian manifolds is researched in this paper. The complete group of structural equations and the components of the Lee vector on the space of the associated $G$-structure are obtained for such manifolds. Conditions are found under which a nearly trans-Sasakian structure is a trans-Sasakian, a cosymplectic, a closely cosymplectic, a Sasakian structure or a Kenmotsu structure. The conditions are obtained when the nearly trans-Sasakian structure is a special generalized Kenmotsu structure of the second kind. A complete classification of nearly trans-Sasakian manifolds is obtained, i.e. it is proved that a nearly trans-Sasakian manifold is either a trans-Sasakian manifold or has a closed contact form. It is proved that the nearly trans-Sasakian structure with a nonclosed contact form is homothetic to the Sasakian structure. The criterion of ownership of a nearly trans-Sasakian structure is obtained. It is proved that the class of nearly trans-Sasakian manifolds with a closed contact form and a closed Lee form coincides with the class of almost contact metric manifolds with a closed contact form, which are locally conformal to the closely cosymplectic manifolds. Examples of such manifolds are given. The necessary and sufficient conditions for the complete integrability of the first fundamental distribution of a nearly trans-Sasakian manifold are obtained. It is proved that a nearly K\"ahler structure on the leaves of the first fundamental distribution of a nearly trans-Sasakian manifold is induced.

Static analysis of box-girder bridge under the influence of Indian railway vehicle loading using ANSYS finite element model

The box-girder bridge has recently gained a lot of popularity because of its serviceability, stability, and structural efficiency. The box-girder bridge also has a lower structural weight than any other type of bridge. However, the analysis of such a bridge is too complex and challenging for the designers. This paper offers a modelling process for the study of a box- girder bridge with a ballastless sub-track system using the finite element method and evaluates the different static response characteristics of the bridge when it is loaded according to Indian Railway standards. The modelling and the evaluation of the 3D model of the bridge have been done using non-closed form finite element method (FEM) based ANSYS software and loadings have been applied symmetrically and un- symmetrically. Static analysis is carried out. The model has been simulated, and the resultants of deflection and stresses have been determined, taking into account the effect of different combinations of loading from the Indian Railways. The present modelling process is applied to analyze the box-girder bridge for 5 spans of 32 m each. For analysis of any box-girder bridge, though, researchers can use the modelling process described above.

Bayesian Method for Hurdle Regression

Hurdle model is an alternative model to overcome overdispersion caused by excess zero. The model consists of two stages: a binary process that determines whether the response variable has zero values or positive values, and the second stage to model only the positive counts. The first stage is modelled using binary logistic regression, and the next stage is modeled with the zero-truncated model using Poisson regression. Bayesian method was employed to estimate the models’ parameters. Non-informative priors were specified for the parameters, and combined with the likelihood from the data, the non-closed form of posterior distributions were obtained, thus leading to the use of Markov Chain Monte Carlo (MCMC) with Gibbs Sampling to obtain samples from the posterior distributions. This method was applied to model the frequency of motoric complication in people with Parkinson’s disease. The result showed that subtotal scores from the three parts of Movement Disorder Society-Unified Parkinson’s Disease Rating Scale (MDS-UPDRS) could explain the frequency of motoric complication well, implied by the significance of the regression coefficients.

Tan-G class of trigonometric distributions and its applications

In this paper, we introduce a new general class of trigonometric distributions based on the tangent function, called the Tan-G class. A mathematical procedure of the Tan-G class is carried out, including expansions for the probability density function, moments, central moments and Renyi entropy. The estimates are acquired in a non-closed form by the maximum likelihood estimation method. Then, an emphasis is put on a particular member of this class defined with the Burr XII distribution as baseline, called the Tan-BXII distribution. The inferential properties of the Tan-BXII model are investigated. Finally, the Tan-BXII model is applied to a practical data set, illustrating the interest of the Tan-G class for the practitioner.

Recursive Modified Pattern Search on High-Dimensional Simplex : A Blackbox Optimization Technique

In this paper, a novel derivative-free pattern search based algorithm for Black-box optimization is proposed over a simplex constrained parameter space. At each iteration, starting from the current solution, new possible set of solutions are found by adding a set of derived step-size vectors to the initial starting point. While deriving these step-size vectors, precautions and adjustments are considered so that the set of new possible solution points still remain within the simplex constrained space. Thus, no extra time is spent in evaluating the (possibly expensive) objective function at infeasible points (points outside the unit-simplex space); which being the primary motivation of designing a customized optimization algorithm specifically when the parameters belong to a unit-simplex. While minimizing any objective function of m parameters, within each iteration, the objective function is evaluated at 2m new possible solution points. So, upto 2m parallel threads can be incorporated which makes the computation even faster while optimizing expensive objective functions over high-dimensional parameter space. Once a local minimum is discovered, in order to find a better solution, a novel ‘re-start’ strategy is considered to increase the likelihood of finding a better solution. Unlike existing pattern search based methods, a sparsity control parameter is introduced which can be used to induce sparsity in the solution in case the solution is expected to be sparse in prior. A comparative study of the performances of the proposed algorithm and other existing algorithms are shown for a few low, moderate and high-dimensional optimization problems. Upto 338 folds improvement in computation time is achieved using the proposed algorithm over Genetic algorithm along with better solution. The proposed algorithm is used to estimate the simultaneous quantiles of North Atlantic Hurricane velocities during 1981–2006 by maximizing a non-closed form likelihood function with (possibly) multiple maximums.

Stochastic dominance relations for generalised parametric distributions obtained through composition

Investigating stochastic dominance within flexible multi-parametric families of distributions is often complicated, owing to the high number of parameters or non-closed functional forms. To simplify the problem, we use the T–X method, making it possible to obtain generalised models through the composition of cumulative distributions and quantile functions. We derive conditions for the second-order stochastic dominance and for the increasing convex order within multi-parametric families in two steps, namely: (i) breaking them down via the T–X approach and (ii) checking dominance conditions of the (more) manageable distributions composing the model. We apply our method to some special distributions and focus on the beta-generated family, which enables the comparisons of order statistics of i.i.d. samples from (possibly) different random variables.

Capacity of Cooperative Spatial Modulation (CSM) System with Optimum Relay Selection

In this work, the capacity of the cooperative spatial modulation (CSM) system is analysed. Relay-aided transmission is considered to improve the coverage and the reliability using decode and forward scheme. Due to the nonclosed form of the spatial modulation capacity function, its upper and lower bounds are derived. The lower bound of the capacity is exploited to formulate the optimal relay selection problem to maximize the channel utilization, where the optimal relay selection algorithm (ORSA) and modified maximum harmonic mean (MMHM) are proposed to select single or multiple relays. A comparison study in terms of the performance and the computational complexity is presented for the evaluation purpose. Simulation results demonstrate that the proposed algorithm significantly improves the capacity of CSM compared to other most popular relay selection strategies.

PMCMC for Term Structure of Interest Rates under Markov Regime Switching and Jumps

AbstractA parameter estimation method, called PMCMC in this paper, is proposed to estimate a continuous-time model of the term structure of interests under Markov regime switching and jumps. There is a closed form solution to term structure of interest rates under Markov regime. However, the model is extended to be a CKLS model with non-closed form solutions which is a typical nonlinear and non-Gaussian state-space model(SSM) in the case of adding jumps. Although the difficulty of parameter estimation greatly prevents from researching models, we prove that the nonlinear and non-Gaussian state-space model has better performances in studying volatility. The method proposed in this paper will be implemented in simulation and empirical study for SHIBOR. Empirical results illustrate that the PMCMC algorithm has powerful advantages in tackling the models.

A Geometric Algebra Based Higher Dimensional Approximation Method for Statics and Kinematics of Robotic Manipulators

Evaluation of a robotic (serial or parallel) manipulator’s static and kinematics performance requires the solution of closed and complicated non-closed form systems of equations for both the forward and inverse problem types. In this work, a robotic system or manipulator is represented as a network whose motion generating kinematic pairs are represented as the network’s inter-connected nodes. Within this network theoretic context, we develop a formulation employing higher dimensional multivectors defined in Clifford Algebra that approximates the computational outcomes of such complicated systems of equations, for both inverse and forward problem types. The statics and kinematics performance of these mechanical networks (serial and parallel robots) are rapidly evaluated without the need to solve their respective traditional systems of equations using what we call ‘screw hypervolumes’, which capture the algebro-geometric structures generated by a robot’s physical design parameters and end-effector position in terms of kinematics and statics. These dimensionless hypervolume functions enable the specialist to produce a very computationally efficient function/index that approximates the kinematic and static output spaces of robots. We find that this formulation inadvertently presents a curious mathematical analogue to these systems of equations. A 6R serial robot, the Delta robot and the 3-RRR manipulator are analysed using the hypervolume methods for the case studies. This approach can be adapted and applied to other multi-agent physical systems displaying non-commutative interactions as well.

Heat conduction in harmonic chains with Lévy-type disorder.

We consider heat transport in a one-dimensional harmonic chain attached at its ends to Langevin heat baths. The harmonic chain has mass impurities where the separation d between any two successive impurities is randomly distributed according to a power-law distribution P(d)∼1/d^{α+1}, being α>0. In the regime where the first moment of the distribution is well defined (1<α<2) the thermal conductivity κ scales with the system size N as κ∼N^{(α-3)/α} for fixed boundary conditions, whereas for free boundary conditions κ∼N^{(α-1)/α} if N≫1. When α=2, the inverse localization length λ scales with the frequency ω as λ∼ω^{2}lnω in the low-frequency regime, due to the logarithmic correction, the size scaling law of the thermal conductivity acquires a nonclosed form. When α>2, the thermal conductivity scales as in the uncorrelated disorder case. The situation α<1 is only analyzed numerically, where λ(ω)∼ω^{2-α}, which leads to the following asymptotic thermal conductivity: κ∼N^{-(α+1)/(2-α)} for fixed boundary conditions and κ∼N^{(1-α)/(2-α)} for free boundary conditions.

Characterization of Glucokinase Catalysis from a Pseudo-Dimeric View.

Glucose phosphorylation by glucokinase exhibits a sigmoidal dependency on substrate concentration regardless of its simple structure. Dimorph mechanism suggested the existence of two enzymatic states with different catalytic properties, which has been shown to be plausible by structural analysis. However, the dimorph mechanism gives rise to a complicated or non-explicit non-closed mathematical form. It is neither feasible to apply the dimorph mechanism in effector characterizations. To improve the area of glucokinase study with stronger theoretical support and less complication in computation, we proposed the investigation of the enzyme from a pseudo-dimeric angle. The proposed mechanism started from the idealization of two monomeric glucokinase as a dimeric complex, which significantly simplified the glucose phosphorylation kinetics, while the differences in enzyme reconfiguration caused by variable substrates and effectors have been successfully characterized. The study presented a simpler and more reliable way in studying the properties of glucokinase and its effectors, providing guidelines of effector developments for hyperglycemia and hypoglycemia treatment.

Assessment of reliability-based design of stable slopes

The viability of applying reliability-based design in the limit equilibrium evaluation of slope stability is assessed. Two model slopes are designed according to the limit states specification of European standard EN 1997:2004 and reliability assessed using full Monte Carlo calculation with 106trials. Comparison of these results with reliability estimates obtained via a spectrum of different analysis techniques, broadly grouped into first-order reliability methods (FORM) and performance function moment estimation methods, indicates that FORM-based techniques are superior, with a first-order response surface method found to provide the best combination of accuracy, stability, and speed of convergence. In this way, accurate reliability indices for nonclosed form problems of the nature considered here can be obtained within 20–30 function calls, depending on the number of free parameters and required precision.

Parameter estimation for load‐sharing system subject to Wiener degradation process using the expectation‐maximization algorithm

AbstractIn practice, many systems exhibit load‐sharing behavior, where the surviving components share the total load imposed on the system. Different from general systems, the components of load‐sharing systems are interdependent in nature, in such a way that when one component fails, the system load has to be shared by the remaining components, which increases the failure rate or degradation rate of the remaining components. Because of the load‐sharing mechanism among components, parameter estimation and reliability assessment are usually complicated for load‐sharing systems. Although load‐sharing systems with components subject to sudden failures have been intensely studied in literatures with detailed estimation and analysis approaches, those with components subject to degradation are rarely investigated. In this paper, we propose the parameter estimation method for load‐sharing systems subject to continuous degradation with a constant load. Likelihood function based on the degradation data of components is established as a first step. The maximum likelihood estimators for unknown parameters are deduced and obtained via expectation‐maximization (EM) algorithm considering the nonclosed form of the likelihood function. Numerical examples are used to illustrate the effectiveness of the proposed method.

F1-ATPase conformational cycle from simultaneous single-molecule FRET and rotation measurements

Despite extensive studies, the structural basis for the mechanochemical coupling in the rotary molecular motor F1-ATPase (F1) is still incomplete. We performed single-molecule FRET measurements to monitor conformational changes in the stator ring-α3β3, while simultaneously monitoring rotations of the central shaft-γ. In the ATP waiting dwell, two of three β-subunits simultaneously adopt low FRET nonclosed forms. By contrast, in the catalytic intermediate dwell, two β-subunits are simultaneously in a high FRET closed form. These differences allow us to assign crystal structures directly to both major dwell states, thus resolving a long-standing issue and establishing a firm connection between F1 structure and the rotation angle of the motor. Remarkably, a structure of F1 in an ε-inhibited state is consistent with the unique FRET signature of the ATP waiting dwell, while most crystal structures capture the structure in the catalytic dwell. Principal component analysis of the available crystal structures further clarifies the five-step conformational transitions of the αβ-dimer in the ATPase cycle, highlighting the two dominant modes: the opening/closing motions of β and the loosening/tightening motions at the αβ-interface. These results provide a new view of tripartite coupling among chemical reactions, stator conformations, and rotary angles in F1-ATPase.

Development of an Accurate Quintic Equation of State: The Necessity of the Irreducible Quadratic Polynomial in the Attractive Term

In conventional van der Waals type equation of state (EOS) modeling, the denominator of the attractive contribution is regarded as being a necessarily factorable quadratic polynomial for the closed-form expression of the corresponding Helmholtz free energy. This study evaluates the effect of the opposite case, an irreducible quadratic polynomial, on the description of the volumetric behavior of real fluid through a comparison with the critical isotherm data, coexistence curve, and PVT isotherm data. In a generalized cubic EOS, an irreducible quadratic polynomial was found to yield an improved description of the flattened region around the critical density while a nonclosed form of the Helmholtz free energy results. For an improved volumetric description using an irreducible quadratic polynomial and a derivation of the closed form of the Helmholtz free energy, we developed a quintic EOS containing seven parameters, two of which are determined by correlating the critical isotherm data and others, by regressing the sub- and supercritical properties. In the description of the critical isotherms of nine pure compounds, a comparison of the present model with the Benedict–Webb–Robinson–Soave (BWRS) EOS showed that the present model exhibits slightly larger deviations than those with BWRS EOS; however, a good agreement was obtained with the reference model of REFPROP 8.0 in the description of the saturated vapor pressure, saturated density, and PVT isotherm data over a wide range of temperatures.