Approximate Formula Research Articles

Simulation Studies on the Dissipative Modified Kawahara Solitons in a Complex Plasma

In this work, a damped modified Kawahara equation (mKE) with cubic nonlinearity and two dispersion terms including the third- and fifth-order derivatives is analyzed. We employ an effective semi-analytical method to achieve the goal set for this study. For this purpose, the ansatz method is implemented to find some approximate solutions to the damped mKE. Based on the proposed method, two different formulas for the analytical symmetric approximations are formally obtained. The derived formulas could be utilized for studying all traveling waves described by the damped mKE, such as symmetric solitary waves (SWs), shock waves, cnoidal waves, etc. Moreover, the energy of the damped dressed solitons is derived. Furthermore, the obtained approximations are used for studying the dynamics of the dissipative dressed (modified Kawahara (mK)) dust-ion acoustic (DIA) solitons in an unmagnetized collisional superthermal plasma consisting of inertia-less superthermal electrons and inertial cold ions as well as immobile negative dust grains. Numerically, the impact of the collisional parameter that arises as a result of taking the ion-neutral collisions into account and the electron spectral index on the profile of the dissipative structures are examined. Finally, the analytical and numerical approximations using the finite difference method (FDM) are compared in order to confirm the high accuracy of the obtained approximations. The achieved results contribute to explaining the mystery of several nonlinear phenomena that arise in different plasma physics, nonlinear optics, shallow water waves, oceans, and seas, and so on.

DoS attack detection in identification of FIR systems with binary‐valued observations

AbstractTimely and effective detection of malicious attacks is a primary problem in cyber‐physical system security. This paper studies the detection of DoS (denial of service) attacks in identification of FIR (finite impulse response) systems with binary‐valued observations. Based on the prior information of the system, an attack detection algorithm is designed, and its online form is given. The detectability is discussed for the ability of detection algorithm, and the necessary and sufficient condition for the attack strategy to be detectable is obtained. The concepts of the missed detection rate and false alarm rate are introduced, and their calculation methods and approximate formulas under large samples are provided. Furthermore, it is analyzed the influence of prior information of system parameters and data length on the performance of detection algorithm. The rationality of theoretical results is verified by numerical simulation.

Una nueva manera de interpretar el método de tres puntos para el modelado del terreno en dos capas horizontales

Faced with the design of the grounding of an electrical installation, we must have an adequate model of the terrain. Modeling in two horizontal layers has proven to be the most useful, practical and accurate. In a previous presentation [1] a new approximate formula was proposed to be considered in IEEE 80-2000 and IEEE 81-2012 standards for terrain modeling when four-point measurement methods (Wenner or Schlumberger) were used. In this presentation, a similar procedure is used for the three-point method. It leads to the presentation of a new formula that relates the parameters of the two-layer model and simplifies their calculation from the series of field measurements. The form of application of the new procedure and the differences and complements with the other methods are shown. This method, by using rods during the measurements, is consistent with a common construction practice. This has notable advantages over the four-point methods in some types of terrain, especially those with high surface resistivity; while it can be used as a complement in other measurements except in those to be carried out on rocky terrain where it is difficult to drive rods.

Optimization of a combined tanker/platform type vessel based on simulation modeling

The choice of the height of the side of a cargo vessel, as a rule, is carried out on the basis of solving the equation of cargo capacity, while the remaining requirements for its size act as restrictions. However, if the ratios of the main dimensions and the requirements for the minimum freeboard height are somewhat conditional, then ensuring seaworthiness, which, in addition to capacity, when choosing the side height, should include unsinkability, requires high-quality accounting. The article presents a mathematical model for assigning the freeboard height, taking into account the requirements of the Rules of the Russian Classification Society for checking the landing and stability during flooding of one or several adjacent regulated compartments; an analysis of the effect on the height of the side of flooding of various compartments of an oil tanker of mixed (river-sea) navigation is carried out and an approximate formula for calculating it at the initial stages of design is proposed.

Two Stage SEREP Method for Condensation of Space Frame Structure for Dynamic Analysis

Analysis of frame structure can be done by the computer analysis, but there is requirement of simple and easy model which can give accurate result or with tolerable accuracy. The process of reducing the size of matrix is called condensation. It needs to be converted into system or model having less number of degrees of freedom as huge computational effort is required to analyze big models. Also, the models which requires analysis multiple times, for example, optimization of structural system, condensation is very useful. The single stiffness value can represent the stiffness of the whole story (elastic, rectangular frame with evenly distributed load), which give the responses accurately or tolerable accuracy. With this we can create a single lumped mass system of the frame structure which come very handy in further analysis of structure. For finding the single stiffness value, there are various approach like by using Approximate Story Stiffness Formulas, Static condensation, System Equivalent Reduction Expansion Process (SEREP) etc. Approximate Story Stiffness formulae don’t give very accurate result. To overcome its shortcoming Static Condensation and SEREP methods are used. Static Condensation gives better result but there is significant difference in the responses. Especially for space frame structures it is difficult to directly condense in to single degrees of freedom at each floor level. Hence, the present study proposes two stage SEREP method for condensation of spaceframe structure. Firstly, space frame is reduced to single degrees of freedom at each node, then it is further reduced to single degree of freedom at each floor. The proposed method gives very less (acceptable) error in comparison of natural frequencies.

Ultrasonic interference and critical attenuation in metal-plastic bilayer laminates

The resonance of ultrasonic waves in a metal-plastic bilayer laminate placed in an infinite medium is theoretically investigated, and critical attenuation is shown to occur under specific conditions. The bilayer laminate is subjected to normal wave incidence from the side of the plastic layer. Based on a linear viscoelastic model, the amplitude of the reflection spectrum for the laminate is formulated to obtain the exact results for wave resonance and critical attenuation by numerical calculations. Furthermore, the approximate formulae for the conditions of the resonance and critical attenuation are derived explicitly by Taylor expansions with respect to the loss factor of the plastic. As a result, the exact solution shows that the resonance frequencies of the plastic layer slightly increase with increasing loss factor. Their variation with the loss factor agrees well with the second-order approximation result but is insignificant in the cases of common plastics with relatively low loss factors. The exact results also show that the amplitude at each order resonance depends on the loss factor and has a minimum at a critical loss factor. The obtained critical loss factor decreases with increasing resonance order and is well reproduced by the zeroth-order approximation. Experiments are performed on two types of metal-plastic bonded laminates, demonstrating that the measured reflection spectra are in good agreement with the theoretical predictions. In particular, the amplitude of the measured reflection spectrum decreases significantly at a resonance frequency, which is theoretically predicted to satisfy the critical attenuation condition.

Measurement and mitigation of the “wild goose chase” phenomenon in taxi services

This paper examines the so-called wild goose chase phenomenon in taxi systems and proposes a new mitigation strategy based on dynamic vehicle swaps. An analytic queuing network model, including a system of differential equations, is derived to yield approximate formulas for the expected system performance in the steady-state equilibria under the vehicle swap strategy. This model is solved numerically to yield equilibrium points and the expected system performance metrics. To corroborate the theoretical predictions, a new data-processing method based on dynamic programming is proposed to identify and measure multiple distinct equilibrium states from observed data. Agent-based simulations are conducted to generate examples of such data, and to verify the formulas. Numerical results show that (i) the proposed measurement method works well in identifying the distinct equilibria, and (ii) the vehicle swap strategy works quite effectively in mitigating the wild goose chase phenomenon.

Two Approximation Formulas for Bateman’s G-Function with Bounded Monotonic Errors

Two new approximation formulas for Bateman’s G-function are presented with strictly monotonic error functions and we deduced their sharp bounds. We also studied the completely monotonic (CM) degrees of two functions involving G(r), deducing two of its inequalities and improving some of the recently published results.

Laminar flow in epicycloidal corrugated pipes

AbstractDoes longitudinal surface corrugation or roughness affect viscous fluid flow? To answer this question, we study viscous flows through the entire family of pipes with epicycloidal walls, which have n cusps, . We find that the flow becomes insensitive to the longitudinal wall corrugations due to the epicycloidal cusps as : surface roughness of the walls does not affect the flow if it is longitudinal. Exact flows in these corrugated pipes are compared with three ‘rules of thumb’ which give general but approximate formulae for the rate of flow through a pipe of given cross‐sectional shape. Upper and lower bounds on flow, based on circumscribed and inscribed circular boundaries, give even better estimates than the three rules of thumb. All the flow formulae, exact and approximate, apply also to the twisting of solid beams.

An Approximation Formula for Nielsen’s Beta Function Involving the Trigamma Function

We prove that the function σ(s) defined by β(s)=6s2+12s+53s2(2s+3)−ψ′(s)2−σ(s)2s5,s>0, is strictly increasing with the sharp bounds 0<σ(s)<49120, where β(s) is Nielsen’s beta function and ψ′(s) is the trigamma function. Furthermore, we prove that the two functions s↦(−1)1+μβ(s)−6s2+12s+53s2(2s+3)+ψ′(s)2+49μ240s5, μ=0,1 are completely monotonic for s>0. As an application, double inequality for β(s) involving ψ′(s) is obtained, which improve some recent results.

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.

Automated evaluation of ECG parameters during the COVID-19 pandemic

Algorithms for prompt automated evaluation of electrocardiogram parameters in the absence of specialized equipment and specialized specialists are considered. The patient’s electrocardiogram is recorded on a paper tape, then it is photographed on the primary care doctor’s mobile phone and processed by a specialized application. The application digitizes the photographed image of the electrocardiogram, evaluates its main parameters as well as calculates criteria for the differential diagnosis of certain diseases using approximate formulas. In addition, the digitized electrocardiogram image is transmitted to the server and processed using a machine learning system. Algorithms for digitizing and analyzing an electrocardiogram have been developed that make it possible to evaluate its elements that are important for diagnosis, and the average error in determining the position of the most complex (smoothed) peaks — P and T waves — was no more than 0.1 mm. An algorithm for the criteria analysis of an electrocardiogram is proposed to support the differential diagnosis of acute myocardial infarction with ST segment elevation and early ventricular repolarization syndrome which provides accuracy values of 0.85 and F-scores of 0.74. An alternative algorithm based on a deep neural network is proposed which provides the best values — 0.96 and 0.88, respectively, but requires large computing resources and is executed on the server. The algorithms are implemented as a set of library functions. They can be used both independently and as part of a full-scale clinical decision support system for automated evaluation of electrocardiogram parameters based on a client-server architecture. In addition, all calculation results, together with a photograph of the original electrocardiogram, can be promptly transferred to a qualified cardiologist in order to receive an advisory opinion remotely.

The VIX Future in Bergomi Models: Fast Approximation Formulas and Joint Calibration with S&P 500 Skew

Using the exponential generating function of Hermite polynomials, we expand the prices of VIX power payoffs (including VIX futures) in various Bergomi models at any order in the volatility-of-volatility. We introduce the notion of volatility of the VIX squared implied by a VIX power payoff, which we call “implied VIX$^2$ volatility” and also expand at any order. We cover the one-factor and (skewed) two-factor Bergomi models and allow for maturity-dependent and/or time-dependent parameters. When the initial term-structure of variance swaps is flat, we provide the expansions up to order 8 in closed form; otherwise, they simply involve one-dimensional integrals, which are extremely fast to compute. Extensive numerical experiments show that the implied volatility expansion converges much faster than the price expansion and is extremely accurate for a wide range of model parameters, including typical market calibrating parameters with very large volatilities-of-volatility. It leads to new, simple approximation formulas for the price of a VIX power payoff that shed light on how those prices depend on model parameters. We combine the new expansion and the Bergomi--Guyon expansion of the vanilla smile [L. Bergomi and J. Guyon, Risk, May (2012), pp. 60--66] to calibrate the two-factor Bergomi model jointly to the term-structures of S&P 500 at-the-money skew and VIX futures. Very interestingly, the joint fit selects (1) much larger values of volatility-of-volatility and mean reversion than those previously reported in [L. Bergomi, Risk, October (2005), pp. 67--73] and [L. Bergomi, Stochastic Volatility Modeling, CRC Press, 2016], and (2) fully correlated Brownian motions, thus producing a (Markovian) pure path-dependent volatility model with rough-like paths.

Stability Analysis of Third-order Magnetic Gradient Tensor

So far, magnetic gradient tensor has been studied in the field of exploration. The third-order magnetic gradient tensor(MGT) can obtain more target information. But its stability is poor. It will lead to a large error under the condition of noise. Aiming at the above shortcomings, this paper puts forward an improved method. The approximate calculation formula of the third-order MGT is analyzed and improved. It can be got from the simulation that the proposed method is more excellent.

Projectile Motion in Midair Using Simple Analytical Approximations

The study of the motion of a projectile, thrown at an angle to the horizon, is a wonderful classical problem. This issue has been the subject of great interest to investigators for centuries. Currently, the study of parabolic motion, in the absence of any drag force, is a common example in introductory physics courses. The theory of parabolic motion allows you to analytically determine the trajectory and all important characteristics of the movement of the projectile. Introduction of air resistance forces into the study of the motion, however, complicates the problem and makes it difficult to obtain analytical solutions. This especially applies to the movement of the projectile, subjected to quadratic air drag force. Numerical studies of projectile motion with quadratic dependence on projectile speed have been performed in many works. From an educational point of view, many students are not confident with numerical methods and gain little insight from using them. So the description of the projectile motion by means of simple approximate analytical formulas under air resistance has great methodological and educational importance. The main goal of this work is to give analytical approximations for the projectile trajectory and movement characteristics as simple as possible from a technical point of view, in order to be grasped even by first-year undergraduates.

An Approximate Formula for the Vapor Pressure Deficit Calculation to Estimate the Evaporation in Runoff Formation Models of Russian Rivers

An Approximate Formula for the Vapor Pressure Deficit Calculation to Estimate the Evaporation in Runoff Formation Models of Russian Rivers

Implementation of Radiation and Soil Losses in VFTO Simulation by Frequency-Domain TL Model

Transmission line (TL) models are widely used within the simulation of the very fast transient over-voltages (VFTO) for gas-insulated substations (GISs); however, due to quasi-TEM approximation and model complexity, energy losses related to bushing radiation and frequency-dependent soil parameters are either not available or simply neglected, limiting the simulation accuracy of TL models. In this study, a new frequency-domain TL model is proposed, enabling the model to consider the influences of both radiation and soil losses in a VFTO simulation. Derived from the radiation power of a line antenna model, the radiation resistance of a “lossy” TL is put forward as an equivalent of the radiation loss for GIS bushing. For the soil loss, the Sunde approximation formula and the L-S soil dispersion expression are involved, and the parameters of the overhead busbar are also analyzed. The model gets validation by the measured VFTO generated at a full-scale 1100 kV GIS testing circuit. By considering radiation loss to suppress high-frequency response and soil loss to improve low-frequency response accuracy, the calculated and measured waveforms have the best consistency, which may greatly help assess the risk of insulation breakdown and the level of very fast transient disturbance in GISs.

Estimating the Number of Molecules in Molecular Junctions Merely Based on the Low Bias Tunneling Conductance at Variable Temperature

Temperature (T) dependent conductance G=G(T) data measured in molecular junctions are routinely taken as evidence for a two-step hopping mechanism. The present paper emphasizes that this is not necessarily the case. A curve of lnG versus 1/T decreasing almost linearly (Arrhenius-like regime) and eventually switching to a nearly horizontal plateau (Sommerfeld regime), or possessing a slope gradually decreasing with increasing 1/T is fully compatible with a single-step tunneling mechanism. The results for the dependence of G on T presented include both analytical exact and accurate approximate formulas and numerical simulations. These theoretical results are general, also in the sense that they are not limited, e.g., to the (single molecule electromigrated (SET) or large area EGaIn) fabrication platforms, which are chosen for exemplification merely in view of the available experimental data needed for analysis. To be specific, we examine in detail transport measurements for molecular junctions based on ferrocene (Fc). As a particularly important finding, we show how the present analytic formulas for G=G(T) can be utilized to compute the ratio f=Aeff/An between the effective and nominal areas of large area Fc-based junctions with an EGaIn top electrode. Our estimate of f≈0.6×10-4 is comparable with previously reported values based on completely different methods for related large area molecular junctions.

Bilateral XVA pricing under stochastic default intensity: PDE modelling and computation

Adjusting derivative prices to take into account default risk has attracted the attention of several researchers and practitioners, especially after the 2007-2008 financial crisis. We derive a novel partial differential equation (PDE) model for derivative pricing including the adjustment for default risk, assuming that the default risk of one of the counterparties (the buyer) follows a Cox-Ingersoll-Ross (CIR) process, while the other party has constant default risk. The time-dependent PDE derived is of Black-Scholes type and involves two “space” variables, namely the asset price and the buyer default intensity, as well as a nonlinear source term. We formulate boundary conditions appropriate for the default intensity variable. The numerical solution of the PDE is based on standard finite differences, and a penalty-like iteration for handling the nonlinearity. We also develop and analyze a novel asymptotic approximation formula for the adjusted price of derivatives, resulting in a very efficient, accurate, and convenient for practitioners formula. We present numerical results that indicate stable second order convergence for the 2D PDE solution in terms of the discretization size. We compare the effectiveness of the 2D PDE and asymptotic approximations. We study the effect of various numerical and market parameters to the values of the adjusted prices and to the accuracy of the computed solutions.

Specificity of fault tree-based functional safety indicator definition in emergency shutdown systems

Aim. The paper aims to analyse the specifics of the use of commercial fault tree (FT)-based software suites as part of engineering practice for the purpose of dependability calculation of emergency shutdown systems (ESS). Standards of the IEC 61508 Functional safety series stress that, in such cases, there is a possibility of incorrect and non-conservative estimates of the mean probability of failure on demand of an ESS safety feature. Incorrect results are primarily caused by the use of approximate and simplified formulas for identifying the dependability indicators of ESS circuit components and calculating the ESS mean unavailability for safety function performance based on the mean unavailability values of its components. In order to correct the FT simulation results, correction factors can be used that take into account the ESS circuit structure along with exact formulas per IEC 61508-6 for calculating the mean probability of failure on demand of the ESS circuit components. Additionally, the type of common cause failure (ССF) model can be chosen. Methods. A comparative analysis was performed as regards the effects of components of hazardous failures that may be detected or not detected by internal diagnostics on the assessment of the mean probability of failure on demand of an ESS circuit components. It was shown that in less dependable components this dependence significantly affects the unavailability value. The efficiency of correction coefficients that take into account the ESS circuit architecture also depends on the dependability of components, and their introduction is justified for those components whose safety integrity level is between 1 and 2. Engineering estimation of the functional safety indicators can be done using a beta-factor model of common cause failures that is employed as part of design analysis of ESS functional safety. Results. An analysis of simplified and approximate formulas for calculating the mean unavailability of the non-redundant elements of an ESS circuit has shown that in the case of an over 90-percent diagnostic coverage the use of simplified formulas causes an underestimation of the unavailability indicator caused by the increased effect of detected hazardous failures on the probability of ESS misoperation. If the FT analysis is used for the purpose of deducing a conservative estimate of an ESS circuit unavailability indicator, correction factors should be used, whose values depend on the ESS channels redundancy parameters. Two models of accounting for CCF were examined that are used when calculating ESS functional safety. It was shown that under any ESS model the system’s dependability indicators decrease. This decrease is defined by the value of the beta factor and the dependability of the ESS system elements.Conclusion. The information presented in the paper indicates the limited applicability of the simplified formula for calculating the mean unavailability of non-redundant ESS elements as the input data for FT construction. When identifying the safety integrity level of an ESS circuit that includes elements with a low dependability, it should be taken into consideration that, if a FT is used, commercial software suites may overestimate the dependability, which is undesirable in respect to functional safety analysis.