Heuristic Formula Research Articles

Edge-to-center plasma density ratios in two-dimensional plasma discharges

Edge-to-center plasma density ratios—so-called h factors—are important parameters for global models of plasma discharges as they are used to calculate the plasma losses at the reactor walls. There are well-established theories for h factors in the one-dimensional (1D) case. The purpose of this paper is to establish h factors in two-dimensional (2D) systems, with guidance from a 2D particle-in-cell (PIC) simulation. We derive analytical solutions of a 2D fluid theory that includes the effect of ion inertia, but assumes a constant (independent of space) ion collision frequency (using an average ion velocity) across the discharge. Predicted h factors from this 2D fluid theory have the same order of magnitude and the same trends as the PIC simulations when the average ion velocity used in the collision frequency is set equal to the ion thermal velocity. The best agreement is obtained when the average ion velocity varies with pressure (but remains independent of space), going from half the Bohm velocity at low pressure, to the thermal velocity at high pressure. The analysis also shows that a simple correction of the widely-used 1D heuristic formula may be proposed to accurately incorporate 2D effects.

Study of a Controllable Queueing System with Unreliable Heterogeneous Servers

We consider a two-channel Markov queueing system with unreliable heterogeneous servers and a common queue. The claims are distributed among the servers with a threshold control policy. According to this policy, a server with the smaller average usage cost must be busy if the system itself is not empty, and the other server is used if the number of customers in the queue exceeds a certain threshold. We analyze the system in stationary mode. We present a method for computing the probabilities of system states and expressions for average performance and reliability characteristics. For the problem of minimizing average losses per unit of time, we obtain a heuristic formula that approximately computes the optimal threshold policy and proposes a method for computing the stationary distribution of the claim waiting time in the system.

Memory-induced acceleration and slowdown of barrier crossing.

We study the mean first-passage time τMFP for the barrier crossing of a single massive particle with non-Markovian memory by Langevin simulations in one dimension. In the Markovian limit of short memory time τΓ, the expected Kramers turnover between the overdamped (high-friction) and the inertial (low-friction) limits is recovered. Compared to the Markovian case, we find barrier crossing to be accelerated for intermediate memory time, while for long memory time, barrier crossing is slowed down and τMFP increases with τΓ as a power law τMFP∼τΓ2. Both effects are derived from an asymptotic propagator analysis: while barrier crossing acceleration at intermediate memory can be understood as an effective particle mass reduction, slowing down for long memory is caused by the slow kinetics of energy diffusion. A simple and globally accurate heuristic formula for τMFP in terms of all relevant time scales of the system is presented and used to establish a scaling diagram featuring the Markovian overdamped and the Markovian inertial regimes, as well as the non-Markovian intermediate memory time regime where barrier crossing is accelerated and the non-Markovian long memory time regime where barrier crossing is slowed down.

Explaining the Transition from Diffusion Limited to Reaction Limited Surface Assembly of Molecular Species through Spatial Variations.

Surface assembly is often decomposed into two classes: diffusion and reaction limited processes. The transition between the two cases is complex because the dynamics are so different. In this article, we simulate, explain, and experimentally discuss the evolution of the spatial distribution for surface assemblies with diffusion limited and reaction limited processes. Explicitly, we demonstrate that diffusion limited and reaction limited processes show some temporal differences, but more importantly, we show that the spatial arrangements are different enough to discriminate between the two cases. Using fundamental properties, such as the diffusion constant, we calculate the evolution of the spatial profile and derive from physical, heuristic models the assembly rate for reaction and diffusion limited processes based on the individual particle’s interactions with the surface. Finally, we confirm the spatial profile differences between diffusion and reaction limited cases by experimentally measuring the surface assembly between two molecules of similar size, but having different assembly routes. Unique to our description is that we have derived and simulated everything through the particle picture in place of ensemble descriptions such as the diffusion equation, and we show the equivalence between our heuristic formulas and those derived from the diffusion equation.

A statistical model of the wave field in a bounded domain

Numerical simulations of plasma heating with radiofrequency waves often require repetitive calculations of wave fields as the plasma evolves. To enable effective simulations, bench marked formulas of the power deposition have been developed. Here, a statistical model applicable to waves with short wavelengths is presented, which gives the expected amplitude of the wave field as a superposition of four wave fields with weight coefficients depending on the single pass damping, as. The weight coefficient for the wave field coherent with that calculated in the absence of reflection agrees with the coefficient for strong single pass damping of an earlier developed heuristic model, for which the weight coefficients were obtained empirically using a full wave code to calculate the wave field and power deposition. Antennas launching electromagnetic waves into bounded domains are often designed to produce localised wave fields and power depositions in the limit of strong single pass damping. The reflection of the waves changes the coupling that partly destroys the localisation of the wave field, which explains the apparent paradox arising from the earlier developed heuristic formula that only a fraction as2(2−as) and not as of the power is absorbed with a profile corresponding to the power deposition for the first pass of the rays. A method to account for the change in the coupling spectrum caused by reflection for modelling the wave field with ray tracing in bounded media is proposed, which should be applicable to wave propagation in non-uniform media in more general geometries.

A new Heuristic Method RPW Method and Genetic Algorithm Techniques (Hybrid Method) for Evaluating Multi Product Assembly Line Balancing – A Case Study

Assembly line production is one of the widely used production systems. The problem of Assembly Line Balancing deals with the distribution of activities among the workstations which lead to the maximum utilization of Human Resources and facilities without disturbing the work sequence. The problem is motivated by a vehicle-sequencing problem at an Automobile company, Gurugram. A new Heuristic Method i.e. RPW Method and Genetic algorithm techniques(hybrid method) for the Type-I of Multi Product of Assembly Line Balancing Problem(MALBP) have been used. The hybrid method has been developed which is based on the heuristic rules and formulae. The programme is coded in C#(C Sharp). The hybrid method was gave good solutions for straight line balancing problem. The present work of case study provided new knowledge to develop a better tool of assembly line balancing to solve real world problems more efficiently at the shortest possible time and it leads to increase line efficiency and production rate by reducing the number of workstations and also reducing balance delay and smoothness index.

Stochastic Periodic-Review Models with Duration- and Quantity-Dependent Inventory Costs: Properties and Approximations

In this paper, we formulate a stochastic periodic-review inventory model with lead-time that considers both duration- and quantity-dependent inventory costs, and show that the optimality equation has the same format as the standard newsvendor problem with a modified demand distribution. We suggest (i) an approximation of the optimal order-up-to level, based on discretization of the inventory cost accounting, and (ii) two heuristic formulas, based on the two-moment normal approximation of the modified demand distribution, and on an approximated deterministic model. We evaluate the performances of these formulas using a Brownian motion demand process under different scenarios, and discuss their advantages and disadvantages.

Quantifying the origins of life on a planetary scale

A simple, heuristic formula with parallels to the Drake Equation is introduced to help focus discussion on open questions for the origins of life in a planetary context. This approach indicates a number of areas where quantitative progress can be made on parameter estimation for determining origins of life probabilities, based on constraints from Bayesian approaches. We discuss a variety of "microscale" factors and their role in determining "macroscale" abiogenesis probabilities on suitable planets. We also propose that impact ejecta exchange between planets with parallel chemistries and chemical evolution could in principle amplify the development of molecular complexity and abiogenesis probabilities. This amplification could be very significant, and both bias our conclusions about abiogenesis probabilities based on the Earth and provide a major source of variance in the probability of life arising in planetary systems. We use our heuristic formula to suggest a number of observational routes for improving constraints on origins of life probabilities.

Collisional damping rates for plasma waves

The distinction between the plasma dynamics dominated by collisional transport versus collective processes has never been rigorously addressed until recently. A recent paper [P. H. Yoon et al., Phys. Rev. E 93, 033203 (2016)] formulates for the first time, a unified kinetic theory in which collective processes and collisional dynamics are systematically incorporated from first principles. One of the outcomes of such a formalism is the rigorous derivation of collisional damping rates for Langmuir and ion-acoustic waves, which can be contrasted to the heuristic customary approach. However, the results are given only in formal mathematical expressions. The present brief communication numerically evaluates the rigorous collisional damping rates by considering the case of plasma particles with Maxwellian velocity distribution function so as to assess the consequence of the rigorous formalism in a quantitative manner. Comparison with the heuristic (“Spitzer”) formula shows that the accurate damping rates are much lower in magnitude than the conventional expression, which implies that the traditional approach over-estimates the importance of attenuation of plasma waves by collisional relaxation process. Such a finding may have a wide applicability ranging from laboratory to space and astrophysical plasmas.

Combining inertia and constriction technique in the PSO applied to fault identification in a hydraulic system

A novel methodology for simultaneous parametric fault isolation and mode switching identification in the spool motion of a Directional Control Valve (DCV), under multi fault assumption, has been reported in this paper. The shape of the profile traversed by the DCV spool was assumed to be trapezoidal in both healthy and faulty condition, but the slope of trapezoidal may change due to fault. Under this assumption, the real valued fault parameter and binary mode switching were identified by real valued Particle Swarm Optimization (PSO) alone instead of a combined real and binary valued PSO (Hybrid PSO). A novel PSO algorithm by combining the concepts of varying inertia weight (both increasing and decreasing trend) and constriction factor has been proposed in the article to achieve more accurate identification. Its validity was examined using an existing heuristic formula and by conducting several tests on a benchmark function used for fault identification. Superior improvement was observed in the identification with increasing inertia weight than that of widely used decreasing inertia weight, when combined with the constriction coefficient. A high pressure hydraulic circuit used in dumper and several other lifting machines was used as a simulation example.

Evolution of one-dimensional wind-driven sea spectra

We analyze modern operational models of wind wave prediction on the subject for compliance dissipation. Our numerical simulations from the "first principle" demonstrate that heuristic formulas for damping rate of free wind sea due to "white capping" (or wave breaking) dramatically exaggerates the role of this effect in these models.

Relative index of inequality and slope index of inequality: a structured regression framework for estimation.

The relative index of inequality and the slope index of inequality are the two major indices used in epidemiologic studies for the measurement of socioeconomic inequalities in health. Yet the current definitions of these indices are not adapted to their main purpose, which is to provide summary measures of the linear association between socioeconomic status and health in a way that enables valid between-population comparisons. The lack of appropriate definitions has dissuaded the application of suitable regression methods for estimating the slope index of inequality. We suggest formally defining the relative and slope indices of inequality as so-called least false parameters, or more precisely, as the parameters that provide the best approximation of the relation between socioeconomic status and the health outcome by log-linear and linear models, respectively. From this standpoint, we establish a structured regression framework for inference on these indices. Guidelines for implementation of the methods, including R and SAS codes, are provided. The new definitions yield appropriate summary measures of the linear association across the entire socioeconomic scale, suitable for comparative studies in epidemiology. Our regression-based approach for estimation of the slope index of inequality contributes to an advancement of the current methodology, which mainly consists of a heuristic formula relying on restrictive assumptions. A study of the educational inequalities in all-cause and cause-specific mortality in France is used for illustration. The proposed definitions and methods should guide the use and estimation of these indices in future studies.

Edge-to-center density ratios in low-temperature plasmas

The ion flux leaving a plasma at a boundary can be given by: Γi = hLn0uB, where n0 is the maximum central plasma density, uB is the Bohm velocity, and hL is the sheath edge-to-center plasma density ratio. Such hL factors have become synonymous with global modeling of plasma discharges, where they play a vital role in the prediction of plasma losses to bounding surfaces. By performing one-dimensional (1D) particle-in-cell simulations of inductively and capacitively coupled plasmas (ICPs and CCPs) over a wide pressure range, we explicitly test the validity of standard heuristic formulae commonly used to estimate hL. The ICP simulation results are found to be in very good agreement, while a large discrepancy is present for the CCP results at high pressures. The onset of this discrepancy is found to be correlated with the bulk-to-sheath edge ionization transition that occurs in CCPs at high pressures. Consequently, global models will strongly underestimate plasma losses in this regime.

Heuristics on pairing-friendly abelian varieties

We discuss heuristic asymptotic formulae for the number of isogeny classes of pairing-friendly abelian varieties of fixed dimension $g\geqslant 2$ over prime finite fields. In each formula, the embedding degree $k\geqslant 2$ is fixed and the rho-value is bounded above by a fixed real ${\it\rho}_{0}>1$. The first formula involves families of ordinary abelian varieties whose endomorphism ring contains an order in a fixed CM-field $K$ of degree $g$ and generalizes previous work of the first author when $g=1$. It suggests that, when ${\it\rho}_{0}<g$, there are only finitely many such isogeny classes. On the other hand, there should be infinitely many such isogeny classes when ${\it\rho}_{0}>g$. The second formula involves families whose endomorphism ring contains an order in a fixed totally real field $K_{0}^{+}$ of degree $g$. It suggests that, when ${\it\rho}_{0}>2g/(g+2)$ (and in particular when ${\it\rho}_{0}>1$ if $g=2$), there are infinitely many isogeny classes of $g$-dimensional abelian varieties over prime fields whose endomorphism ring contains an order of $K_{0}^{+}$. We also discuss the impact that polynomial families of pairing-friendly abelian varieties has on our heuristics, and review the known cases where they are expected to provide more isogeny classes than predicted by our heuristic formulae.

Evaluation of normalisation methods for uniaxial bias extension tests on engineering fabrics

An investigation has been conducted to evaluate the performance of normalisation methods for the uniaxial bias extension test. The predictions of two published theories for rate-independent behaviour are examined and a third heuristic method is proposed. Using hypothetical test data, the predictions of the two rate-independent theories are shown to be equivalent for specimen dimensions of high aspect ratio; in this case the predictions can be well-represented using the simple heuristic formula. The predictions diverge for specimens of low aspect ratio, when specimens are sheared to very high shear angles. In order to examine the significance of this divergence on real data, results from tests on several different engineering fabrics are normalised. Differences in the predictions of the normalisation methods are observed and the question of the significance of these differences is discussed. The paper also examines the applicability of rate-independent theory for normalising rate-dependent materials.

Global time‐dependent chorus maps from low‐Earth‐orbit electron precipitation and Van Allen Probes data

AbstractSubstorm injected electrons (several–100 s keV) produce whistler‐mode chorus waves that are thought to have a major impact on the radiation belts by causing both energization and loss of relativistic electrons in the outer belt. High‐altitude measurements, such as those from the Van Allen Probes, provide detailed wave measurements at a few points in the magnetosphere. But physics‐based models of radiation‐belt dynamics require knowledge of the global distribution of chorus waves. We demonstrate that time‐dependent, global distributions of near‐equatorial chorus wave intensities can be inferred from low‐Earth‐orbit (LEO) measurements of precipitating low‐energy electrons. We compare in situ observations of near‐equatorial chorus waves with LEO observations of precipitating electrons and derive a heuristic formula that relates, quantitatively, electron precipitation fluxes to chorus wave intensities. Finally, we demonstrate how that formula can be applied to LEO precipitation measurements and in situ Van Allen Probes wave measurements to provide global, data‐driven inputs for radiation belt models.

A comparison of perturbation formulas for a square electromagnetic resonator

Several perturbation formulas are available for the resonant frequency of an electromagnetic cavity subjected to shape and material perturbations. Each is based on a mathematical identity from vector calculus, a choice of trial fields, and the manner in which the boundary conditions are met. The limits of usefulness of a given perturbation formula depend on these factors. Slater proposed a perturbation formula for the resonant frequency of cavities that maintained currency for a long time. However, what remained obscure was that Dombrowski also proposed a perturbation formula, making a correction to that of Slater. A critical performance comparison of various perturbation formulas for a fixed geometry was not undertaken during the time they were proposed, for want of an exact solution to validate their accuracy. They were assessed only in some landmark or limiting cases. Today, several efficient numerical packages are available, and the need for perturbation formulas is rather grim or even completely obviated. If there is any interest in revisiting these formulas, it is all the more a matter of curiosity until there comes a revival in this area. The aim of this tutorial is then to apply about five different perturbation formulas to the same geometry (in the present case, a square cavity perturbed in shape); to obtain a closed-form expression for each; and to compare the limits of their accuracy. The performance of these formulas is rated with the help of commercially available HFSS software, using the eigenmode solver, both visually by plotting, and also using a power series. Along with these formulas, a new heuristic perturbation formula is proposed in this paper, which ranks number one among all of these in terms of accuracy as far as the present geometry is concerned.

Distribution of elliptic twin primes in isogeny and isomorphism classes

Motivated by a recent application to hash functions suggested by O. Chevassut, P.-A. Fouque, P. Gaudry and D. Pointcheval, we study the frequency with which groups of points on an elliptic curve over a finite field and its quadratic twist are both of prime order. We obtain a heuristic asymptotic formula for the number of isogeny and isomorphism classes of such curves and present some supporting computational results.

Analytical model-based energy and exergy analysis of a gas microturbine at part-load operation

In this paper a universal analytical model for part-load operation of gas microturbines has been elaborated which is subsequently used in the energy and exergy analysis of a sample device. The model, based on the Brayton cycle and heuristic part-load performance formulas, takes into account: the temperature variation of working fluid specific heat at constant pressure in calculations of adiabatic processes, enthalpy, and exergy, the non-linear dependence of pressure drop on flow rate, and the cooling of generator by intake air. The model is validated using the manufacturer data for a commercially available microturbine of 30 kWe and results of measurements. The agreement is very good as for such a general simple analytical model. Exergy calculations based on the elaborated model show that the greatest potential for improving the efficiency of the microturbine lies in the combustion chamber and recuperator, as these components are characterized by the largest exergy destruction and loss.

Applying an accurate spherical model to gamma-ray burst afterglow observations

We present results of model fits to afterglow data sets of GRB970508, GRB980703 and GRB070125, characterized by long and broadband coverage. The model assumes synchrotron radiation (including self-absorption) from a spherical adiabatic blast wave and consists of analytic flux prescriptions based on numerical results. For the first time it combines the accuracy of hydrodynamic simulations through different stages of the outflow dynamics with the flexibility of simple heuristic formulas. The prescriptions are especially geared towards accurate description of the dynamical transition of the outflow from relativistic to Newtonian velocities in an arbitrary power-law density environment. We show that the spherical model can accurately describe the data only in the case of GRB970508, for which we find a circumburst medium density consistent with a stellar wind. We investigate in detail the implied spectra and physical parameters of that burst. For the microphysics we show evidence for equipartition between the fraction of energy density carried by relativistic electrons and magnetic field. We also find that for the blast wave to be adiabatic, the fraction of electrons accelerated at the shock has to be smaller than 1. We present best-fit parameters for the afterglows of all three bursts, including uncertainties in the parameters of GRB970508, and compare the inferred values to those obtained by different authors.