Exact Ones Research Articles

Reccurent-Iterative Scheme to approximate as Desired the Complete Elliptic Integrals

Two sets of closed analytic formulas are proposed for the approximate calculus of the complete elliptic integrals K(k) and E(k) in the normal form due to Legendre, their expressions having a remarkable simplicity and accuracy. The special usefulness of the newly proposed formulas consists in they allow performing the analytic study of variation of the functions in which they appear, using derivatives, being expressed in terms of elementary (especially algebraic) functions only, without any special function (this would mean replacing one difficulty by another of the same kind). Comparative tables of so found approximate values with the exact ones, reproduced from special functions tables, are given (wrt the elliptic integrals’ modulus k). The first set of formulas was suggested by Peano’s law on ellipse’s perimeter. The new functions and their derivatives coincide with the exact ones at k = 0 only. As for simplicity, the formulas in k / k don’t need mathematical tables nor advanced calculators, being purely algebraic. As for accuracy, the second set, something more intricate, gives more accurate values and extends more closely to k = 1. An original fast converging recurrent-iterative scheme to get sets of formulas with the desired accuracy is given in appendix 1. Using the results obtained by applying the newly proposed approximate formulas a method to approximate the complete elliptic integral Π(n, k) is given in appendix 2.

ProS: data series progressive k-NN similarity search and classification with probabilistic quality guarantees

Existing systems dealing with the increasing volume of data series cannot guarantee interactive response times, even for fundamental tasks such as similarity search. Therefore, it is necessary to develop analytic approaches that support exploration and decision making by providing progressive results, before the final and exact ones have been computed. Prior works lack both efficiency and accuracy when applied to large-scale data series collections. We present and experimentally evaluate ProS, a new probabilistic learning-based method that provides quality guarantees for progressive nearest neighbor (NN) query answering. We develop our method for k-NN queries and demonstrate how it can be applied with the two most popular distance measures, namely Euclidean and dynamic time warping. We provide both initial and progressive estimates of the final answer that are getting better during the similarity search, as well suitable stopping criteria for the progressive queries. Moreover, we describe how this method can be used in order to develop a progressive algorithm for data series classification (based on a k-NN classifier), and we additionally propose a method designed specifically for the classification task. Experiments with several and diverse synthetic and real datasets demonstrate that our prediction methods constitute the first practical solutions to the problem, significantly outperforming competing approaches.

Asymmetric response of inelastic circular plates to blast loading

The dynamic behaviour of clamped circular plates subjected to the concentrated blast loading is studied. The load is applied at a non-central point of the plate, non-axisymmetric deflections are taken into account. An approximate theoretical procedure developed earlier is applied for the evaluation of residual maximal deflections. The solution technique is based on the idea of equality of the power of the internal and external work, respectively. As it was shown earlier this concept leads to results which are close to exact ones in the case of axisymmetric loading of circular plates, also in the case of circular cylindrical shells.

Fetal Development in Anatomical Preparations of Ruysch and the Meckels in Comparison.

Anatomical collections have been used for centuries for research and teaching purposes. By the example of selected preparations of fetal development from the Ruysch collection (17th-18th centuries) and the Meckel collections (18th-19th century), this article aims to trace the changing purposes, specifics and methods of creating specimens as well as the development of anatomy during that period. The selected specimens are compared and analyzed implementing the historical-critical method. Regarding the appearance of the preparations, we see a transition from the visually aesthetic specimens (Ruysch) to exact ones (Meckel collections). If Ruysch's preparations were compared in his time to works of art, specimens created by three anatomists of the Meckel dynasty were primarily created for teaching and research purposes. However, Ruysch's preparations tracing fetal circulation were scientific discoveries of the time. As for preparations of fetal development from the Meckel collections, we see both specimens of physiological processes already known at that time and experimental ones. Regarding teratology, Ruysch and Meckel the Younger tried to explain malformations, but their anatomical preparations could hardly give answers to the cause of deviations from the norm. The differences between the collections can be explained by the different stages of development of anatomy of the time and by the research interests of the anatomists themselves.

Choice of Regularization Methods in Experiment Processing: Solving Inverse Problems of Thermal Conductivity

This work is aimed at numerical studies of inverse problems of experiment processing (identification of unknown parameters of mathematical models from experimental data) based on the balanced identification technology. Such problems are inverse in their nature and often turn out to be ill-posed. To solve them, various regularization methods are used, which differ in regularizing additions and methods for choosing the values of the regularization parameters. Balanced identification technology uses the cross-validation root-mean-square error to select the values of the regularization parameters. Its minimization leads to an optimally balanced solution, and the obtained value is used as a quantitative criterion for the correspondence of the model and the regularization method to the data. The approach is illustrated by the problem of identifying the heat-conduction coefficient on temperature. A mixed one-dimensional nonlinear heat conduction problem was chosen as a model. The one-dimensional problem was chosen based on the convenience of the graphical presentation of the results. The experimental data are synthetic data obtained on the basis of a known exact solution with added random errors. In total, nine problems (some original) were considered, differing in data sets and criteria for choosing solutions. This is the first time such a comprehensive study with error analysis has been carried out. Various estimates of the modeling errors are given and show a good agreement with the characteristics of the synthetic data errors. The effectiveness of the technology is confirmed by comparing numerical solutions with exact ones.

Neural-network-based Riemann solver for real fluids and high explosives; application to computational fluid dynamics

The Riemann problem is fundamental to most computational fluid dynamics (CFD) codes for simulating compressible flows. The time to obtain the exact solution to this problem for real fluids is high because of the complexity of the fluid model, which includes the equation of state; as a result, approximate Riemann solvers are used in lieu of the exact ones, even for ideal gases. We used fully connected feedforward neural networks to find the solution to the Riemann problem for calorically imperfect gases, supercritical fluids, and high explosives and then embedded these network into a one-dimensional finite volume CFD code. We showed that for real fluids, the neural networks can be more than five orders of magnitude faster than the exact solver, with prediction errors below 0.8%. The same neural networks embedded in a CFD code yields very good agreement with the overall exact solution, with a speed-up of three orders of magnitude with respect to the same CFD code that use the exact Riemann solver to resolve the flux at the interfaces. Compared to the Rusanov flux reconstruction method, the neural network is half as fast but yields a higher accuracy and is able to converge to the exact solution with a coarser grid.

Inexact gradient projection method with relative error tolerance

A gradient projection method with feasible inexact projections is proposed in the present paper. The inexact projection is performed using a general relative error tolerance. Asymptotic convergence analysis under quasiconvexity assumption and iteration-complexity bounds under convexity assumption of the method employing constant and Armijo step sizes are presented. Numerical results are reported illustrating the potential advantages of considering inexact projections instead of exact ones in some medium scale instances of a least squares problem over the spectrohedron.

Application of Differential Transform Method with Adomian Polynomial for solving RLC Circuits Problems and Higher Order differential equations

. The differential transform method (DTM) for solving linear and nonlinear higher order differential equations especially which arising in the field of electric circuit problems is used. In almost cases, DTM gives the series of solutions which can be easily converted to exact ones. In nonlinear problems it is useful to use the Adomian polynomial to overheads the nonlinear terms. Also, this method is useful in both initial and boundary value problems.This method introduces a semi-analytic solution for differential equations, in polynomial form, it differs from the regular high order Taylor series technique, which necessitates the symbolic calculation of the necessary derivatives for the data functions. Comparison is made between numerical solutions obtained using DTM and other numerical methods, which trusses up the efficiency of the DTM method. Some illustrative examples are presented with some RLC circuit problems to show the simplicity and efficiency of the used method. All calculations are made with the help of computer algebra software Mathematica 11.2.

Event-Triggered Resilient L∞ Control for Markov Jump Systems Subject to Denial-of-Service Jamming Attacks.

In this article, the event-triggered resilient L∞ control problem is concerned for the Markov jump systems in the presence of denial-of-service (DoS) jamming attacks. First, a fixed lower bound-based event-triggering scheme (ETS) is presented in order to avoid the Zeno problem caused by exogenous disturbance. Second, when DoS jamming attacks are involved, the transmitted data are blocked and the old control input is kept by using the zero-order holder (ZOH). On the basis of this process, the effect of DoS attacks on ETS is further discussed. Next, by utilizing the state-feedback controller and multiple Lyapunov functions method, some criteria incorporating the restriction of DoS jamming attacks are proposed to guarantee the L∞ control performance of the event-triggered Markov closed-loop jump system. In particular, the bounded transition rates rather than the exact ones are taken into account. That is appropriate for the practical environment in which transition rates of the Markov process are difficult to measure accurately. Correspondingly, some criteria are proposed to obtain state-feedback gains and event-triggering parameters simultaneously. Finally, we provide two examples to show the effectiveness of the proposed method.

A highly efficient method for structural model reduction

AbstractModel reduction is a commonly used method for quickly computing the lower‐order eigenvalues and eigenvectors of a structure. This article proposed a direct model reduction method without any iterations. Compared with the existing methods, the proposed technique has two main advantages. The first one is that the explicit expression of the transformation matrix can be directly obtained using a recurrence formula. The second one is that the calculation efficiency of the proposed method is very high because there is no need for iterative calculation. Three types of structure are used as examples to verify the proposed method. It is found that the lower‐frequency eigenpairs calculated by the reduced model are very close to the exact ones obtained by the full model. Moreover, the calculation time of the proposed method is much less than that of the existing methods. It has been shown that the proposed method is reliable and effective.

Assessing the quality of relaxation-time approximations with fully automated computations of phonon-limited mobilities

The mobility of carriers, as limited by their scattering with phonons, can now routinely be obtained from first-principles electron-phonon coupling calculations. However, so far, most computations have relied on some form of simplification of the linearized Boltzmann transport equation based on either the self-energy, the momentum- or constant relaxation time approximations. Here, we develop a high-throughput infrastructure and an automatic workflow and we compute 69 phonon-limited mobilities in semiconductors. We compare the results resorting to the approximations with the exact iterative solution. We conclude that the approximate values may deviate significantly from the exact ones and are thus not reliable. Given the minimal computational overhead, our work encourages to rely on this exact iterative solution and warns on the possible inaccuracy of earlier results reported using relaxation time approximations.

Interpolation method for solving Volterra integral equations with weakly singular kernel using an advanced barycentric Lagrange formula

We presented an interpolation method for solving weakly singular Volterra integral equations of the second kind (SVK2). The method based on the barycentric Lagrange interpolation.. For the chosen nodes of the two singular kernel variables, we created two rules that confirm that the denominator of the kernel will never become zero or have an imaginary value. The product of four matrices, one of which is the square coefficients matrix of the barycentric functions, expresses both the unknown and data functions. The singular kernel is interpolated twice with to its two variables. Furthermore, the kernel is represented by five matrices: two of which are the monomial basis of the kernel's variables, and one matrix expresses the kernel's values at the two sets of nodes for the kernel's variables. Without using the collocation approach, we get the matrix of the unknown coefficients by substituting the interpolated solution into the integral equation to obtain an equivalent algebraic system. By solving the algebraic system, we get the interpolated solution. We solved five examples, two for non-singular equations and three for weakly singular equations. By adopting lower interpolation degrees, the interpolated solutions for non-singular equations are shown to be equal to the exact ones, whereas, for singular equations, they are found to strongly converge to the exact ones and are superior to those obtained by other cited methods. This ensures the proposed method's uniqueness and high accuracy.

Free Vibration and Buckling Characteristics of Uniform Beam: A Modified Segmented Rod Method

This paper proposes a modified segmented rod method (MSRM) in order to facilitate the introduction of boundary conditions in the analysis of free vibrations and buckling of Euler–Bernoulli beams. First, the basic idea of proposed modifications is presented in the case of elementary length beam, and then the idea is generalized to the full-length beam. In addition to facilitating the introduction of boundary conditions, by comparing the results of MSRM with segmented rod method (SRM) and Hencky bar-chain method (HBM), a faster convergence of MSRM results to the exact ones was observed. Although the case of uniform beam is considered in this paper, proposed MSRM may contribute to greater use of segmented beam approach in solving of various static, free vibration or buckling problems of non-uniform beams. Also, it was shown that MSRM can be easily adapted for the analysis of beams that resting on partial elastic foundation.

Stress intensity factor calculation of the cracks interacted by the oval-holes in anisotropic elastic solids under remote and non-uniform surface stresses

In underground rock engineering, natural defects (such as cracks, joints, cavities of arbitrary shape) and anisotropy of rock mass are main factors influencing the engineering strength. Calculating stress intensity factors (SIFs) of anisotropic elastic solids with multiple oval-holes and cracks is of great significance not only for designing the stop-holes to improve the stability of mining exploitation, but also for optimizing the hydraulic fracturing parameters to improve the productivity of geothermal/shale-gas utilization. Current studies are mainly focused on the loading case of remote stresses, regardless of the surface stresses applied onto the holes and cracks. In this paper, both remote uniform stresses and non-uniform surface stresses are taken into account to calculate the multi-crack SIFs interacted by the multiple oval-holes in anisotropic elastic solids. Two elementary solutions for traction components of single oval-hole and single crack subjected to the surface point-forces in the anisotropic solids are analytically deduced based on Cauchy integral approach in complex variable theory. A new integral equation approach is established to solve the interacting SIFs of multiple oval holes and cracks in the two-dimensional anisotropic elastic solids, without any limitation of the number, sizes, orientations and locations for oval-holes and cracks. Its accuracy and feasibility are validated by comparing our SIF solutions against the available exact solutions, approximated solutions and finite element solutions. The new method has high accuracy (with almost the same SIF solutions as the exact ones), simple formulation (without any singularity) and wide applicability (under both remote uniform stresses and non-uniform surface stresses). It can be further developed for three-dimensional anisotropic hole-crack problem under the same complex stress states.

Analytical model for viscous and elastic Rayleigh–Taylor instabilities in convergent geometries at static interfaces

Great attention has been attracted to study the viscous and elastic Rayleigh–Taylor instability in convergent geometries, especially for their low mode asymmetries that behave distinctively from the planar counterparts. However, most analyses have focused on the instability at static interfaces that excludes the studies of the Bell–Plesset effects and the elastic–plastic transition since they involve too complex mathematics. Herein, we perform detailed analyses on the dispersion relations by applying the viscous and elastic potential flow method to obtain their approximate growth rates compared with the exact ones to demonstrate: (i) The approximate growth rates based on potential flow method generally coincide with the exact ones. (ii) An alternative expression is proposed to overcome the discrepancy for the low mode asymmetries at fluid/fluid interface. (iii) Extra care must be taken in solids since the maximum discrepancies occur at the n = 1 mode and at the mode proximate to the cutoff. This analytical method of great simplicity is essential to describe the dynamic interface by including the overall motion of the interface based on the static construction, while the exact analysis involves too complex mathematics to be extended by including the Bell–Plesset effects and the elastic–plastic properties. To sum up, the approximate analytical dispersion relations derived in convergent geometries, have the potential for dealing with dynamic interfaces where Bell–Plesset effects are combined with elastic–plastic transition.

Approximated beats exact Interface Jacobians at model-based co-simulation - explanation and impact

This work deals with the effect of approximated and exact Interface Jacobians according the stability of the overall co-simulation, which is coupled via the Model-based Pre-Step Stabilization Method. At the co-simulation of a helicopter and its controller, the exact Interface Jacobian based co-simulation results in an instable behaviour, whereas with approximated ones the results are accurate and stable. This leads to the question why is there this paradoxical behaviour? For a detailed analysis, the dual mass oscillator, a co-simulation benchmark example, is additionally investigated and also a simulation study of more than 2500 runs has been performed. Leading to the conclusion that for industrial use cases the usage of approximated Interface Jacobians is favourable against exact ones due to the so-called mutual influence between the subsystems.

Random vs. Best-First: Impact of Sampling Strategies on Decision Making in Model-Based Diagnosis

Statistical samples, in order to be representative, have to be drawn from a population in a random and unbiased way. Nevertheless, it is common practice in the field of model-based diagnosis to make estimations from (biased) best-first samples. One example is the computation of a few most probable fault explanations for a defective system and the use of these to assess which aspect of the system, if measured, would bring the highest information gain. In this work, we scrutinize whether these statistically not well-founded conventions, that both diagnosis researchers and practitioners have adhered to for decades, are indeed reasonable. To this end, we empirically analyze various sampling methods that generate fault explanations. We study the representativeness of the produced samples in terms of their estimations about fault explanations and how well they guide diagnostic decisions, and we investigate the impact of sample size, the optimal trade-off between sampling efficiency and effectivity, and how approximate sampling techniques compare to exact ones.

Effect of Kerr nonlinearity on signal and pump intensities in EDFA comprising single-mode step index fiber: Estimation by a simple but accurate mathematical formalism

The paper presents simple but accurate analytical expressions for the estimation of variation of signal and pump intensity with normalised radial distance in case of erbium doped fiber amplifier. Choosing two typical single-mode step index fibers in our investigation, we hereby predict the said variations relating to both pump and signal in presence as well as absence of Kerr nonlinearity. The stated formulation uses power series expression for the fundamental modal field of graded index optical fiber as derived by Chebyshev formalism. Method of iteration is applied for the prediction of concerned parameters in presence of Kerr type nonlinearity. Our results have been shown to be agreeing excellently with the numerical exact ones in absence and also in presence of Kerr nonlinearity. Exact results in case of Kerr nonlinearity are determined using the cumbersome finite element method. It is noteworthy that execution of our proposed method requires very less computation and as such it will be remarkably user friendly for researchers and technologists associated with nonlinear optical engineering.

Nature of chemical bond and potential barrier in an invariant energy-orbital picture.

Physical nature of the chemical bond and potential barrier is studied in terms of energy natural orbitals (ENOs), which are extracted from highly correlated electronic wavefunctions. ENO provides an objective one-electron picture about energy distribution in a molecule, just as the natural orbitals (NOs) represent one electron view about electronic charge distribution. ENO is invariant in the same sense as NO is, that is, ENOs converge to the exact ones as a series of approximate wavefunctions approach the exact one, no matter how the methods of approximation are adopted. Energy distribution analysis based on ENO can give novel insights about the nature of chemical bonding and formation of potential barriers, besides information based on the charge distribution alone. With ENOs extracted from full configuration interaction wavefunctions in a finite yet large enough basis set, we analyze the nature of chemical bonding of three low-lying electronic states of a hydrogen molecule, all being in different classes of the so-called covalent bond. The mechanism of energy lowering in bond formation, which gives a binding energy, is important, yet not the only concern for this small molecule. Another key notion in chemical bonding is whether a potential basin is well generated stiff enough to support a vibrational state(s) on it. Based on the virial theorem in the adiabatic approximation and in terms of the energy and force analyses with ENOs, we study the roles of the electronic kinetic energy and its nuclear derivative(s) on how they determine the curvature (or the force constant) of the potential basins. The same idea is applied to the potential barrier and, thereby, the transition states. The rate constant within the transition-state theory is formally shown to be described in terms of the electronic kinetic energy and the nuclear derivatives only. Thus, the chemical bonding and rate process are interconnected behind the scenes. Besides this aspect, we pay attention to (1) the effects of electron correlation that manifests itself not only in the orbital energy but also in the population of ENOs and (2) the role of nonadiabaticity (diabatic state mixing), resulting in double basins and a barrier on a single potential curve in bond formation. These factors differentiate a covalent bond into subclasses.

High accuracy power series method for solving scalar, vector, and inhomogeneous nonlinear Schrödinger equations

We develop a high accuracy power series method for solving partial differential equations with emphasis on the nonlinear Schrödinger equations. The accuracy and computing speed can be systematically and arbitrarily increased to orders of magnitude larger than those of other methods. Machine precision accuracy can be easily reached and sustained for long evolution times within rather short computing time. In-depth analysis and characterisation for all sources of error are performed by comparing the numerical solutions with the exact analytical ones. Exact and approximate boundary conditions are considered and shown to minimise errors for solutions with finite background. The method is extended to cases with external potentials and coupled nonlinear Schrödinger equations.