Order Coefficients Research Articles

THERMODYNAMIC INTERACTION COEFFICIENTS IN LOW-CONCENTRATED LIQUID BINARY ALLOYS

The article considers basic expansion of thermodynamics and thermodynamic interaction coefficients of the first, second and third orders of low-concentrated binary alloys. The values of interaction coefficients of the first and second orders in 37 such systems were estimated according to experimental thermodynamic data on the concentration dependence of excess chemical potential of an impurity in liquid alloys of binary systems. Estimates were obtained by the numerical differentiation method. This method is based on Newton first interpolation formula. Calculation formulas for the corresponding estimates are given. A simple theory is proposed that relates the thermodynamic interaction coefficient of the second order with the first-order one in the liquid alloy of certain system. The theory is based on the lattice model of a solution and the principles of statistical mechanics. The FCC lattice is adopted as a model lattice. The model of pair interaction between metal atoms in the alloy was used. The radius of this interaction corresponds to radius of the nearest atomic shell. Using the proposed theory, thermodynamic interaction coefficients of the second-order for all 37 systems considered in this work, as well as the values of the third order interaction coefficients for 23 systems out of 37 mentioned above, were calculated. For these 23 systems, theoretical estimates of the second-order interaction coefficients are in agreement with experimental ones both by sign and by order of magnitude. This circumstance can be considered as evidence of applicability of the numerical differentiation method for estimation of thermodynamic interaction coefficients of the first and second orders in liquid binary alloys. The accuracy of estimating the values of the third derivative by numerical differentiation is insufficient. That makes it impossible to compare the calculated values of the interaction coefficients of the third order with the experimental ones, obtained by this method. It can be assumed that the theoretical calculations just give an idea of the magnitudes’ order of these coefficients.

The Study of Morphological Characteristics for Best Management Practices Over the Rampur Watershed of Mahanadi River Basin Using Prioritization

A morphometric analysis has been carried out for Rampur watershed of Mahanadi river basin, and its nine sub-watersheds have been prioritized according to the linear, aerial and relief characteristics. Based on digital elevation model, delineation of sub-watersheds (SWs) has been done by using ArcMap 10.2.2. The drainage analysis and their respective parameters such as stream order, stream length, stream frequency, drainage density, texture ratio, form factor, circulatory ratio, elongation ratio, bifurcation ratio and compactness coefficient for the sub-watersheds have been evaluated by applying remote sensing and geospatial techniques. Based on the behaviour of morphometric parameters of each sub-watershed (SW), the prioritized score has been allocated, and thereafter, the most sensitive parameters have been distinguished by the evaluated combined scores. The results achieved from the analysis show that the stream order varies from 1 to 6 and the first order covers 30.5% of the total catchment area. The watershed has 2087 stream segment numbers of all order. Based on morphometric analysis, prioritizing each SW has been the primary objective of this study. The final score of each sub-watershed has been allocated as per erosion threat. The highest priority has been allocated to the SW which shows the least compound parameter value. Eventually, the grouping of SWs has been done according to the high (4.0–4.7), medium (4.8–5.3) and low (> 5.4) priority classes depending on their prioritized score, maximum (6.0) and minimum (4.0).

Designing of IMC-PID Controller for Higher-order Process Based on Model Reduction Method and Fractional Coefficient Filter with Real-time Verification

AbstractAn improved model reduction scheme is proposed here for higher-order processes and subsequently an enhanced IMC-PID controller is designed based on the obtained reduced model. In the proposed scheme, higher-order processes are estimated as first-order-plus-dead-time (FOPDT) model. Designed IMC controller includes a filter having two separate time constants with fractional order coefficients. Efficacy of the proposed model reduction scheme is verified in terms of closed loop performance evaluation for higher-order minimum and non-minimum phase process models in comparison with improved SIMC (iSIMC) controller (Grimholt. Optimal PI and PID control of first-order plus delay processes and evaluation of the original and improved SIMC rules. J Process Control. 2018;70:36–46). Overall performance enhancement for the proposed method is demonstrated through simulation study as well as real-time experimentation on a level control loop.

A Langevin equation that governs the irregular stick-slip nano-scale friction

Friction force at the nanoscale, as measured from the lateral deflection of the tip of an atomic force microscope, usually shows a regular stick-slip behavior superimposed by a stochastic part (fluctuations). Previous studies showed the overall fluctuations to be correlated and multi-fractal, and thus not describable simply by e.g. a white noise. In the present study, we investigate whether one can extract an equation to describe nano-friction fluctuations directly from experimental data. Analysing the raw data acquired by a silicon tip scanning the NaCl(001) surface (of lattice constant 5.6 Å) at room temperature and in ultra-high vacuum, we found that the fluctuations possess a Markovian behavior for length scales greater than 0.7 Å. Above this characteristic length, the Kramers-Moyal approach applies. However, the fourth-order KM coefficient turns out to be negligible compared to the second order coefficients, such that the KM expansion reduces to the Langevin equation. The drift and diffusion terms of the Langevin equation show linear and quadratic trends with respect to the fluctuations, respectively. The slope 0.61 ± 0.02 of the drift term, being identical to the Hurst exponent, expresses a degree of correlation among the fluctuations. Moreover, the quadratic trend in the diffusion term causes the scaling exponents to become nonlinear, which indicates multifractality in the fluctuations. These findings propose the practical way to correct the prior models that consider the fluctuations as a white noise.

Synthesis of Wire Antennas Using the Multipole Expansion Method

A rigorous mathematical solution to the problem of synthesizing wire antennas using the multipole expansion method (MEM) is presented. The input current and required size of the antenna are estimated using the knowledge of the required field pattern. This solution is based on three steps. First, obtaining the expansion’s weighting coefficients that carry both the signature of the current and size of the antenna. In essence, the analysis of the direct modeling needs to be addressed first in order to define these coefficients. Second, extracting the coefficients from the required field pattern using the natural orthogonality property of the spherical harmonic functions. Finally, performing a numerical search for the argument of the spherical Bessel functions to satisfy the ratio between two successive moments. Once the size of the antenna is found, a recursive expression is used to obtain the input current. For the current distributions assumed herein, the odd order coefficients of the MEM are shown to be the only necessary coefficients to construct the radiation far-field. Also, it is found that the estimated model parameters are in excellent agreement with the analytical ones.

Predictions of diffusion rates of large organic molecules in secondary organic aerosols using the Stokes–Einstein and fractional Stokes–Einstein relations

Abstract. Information on the rate of diffusion of organic molecules within secondary organic aerosol (SOA) is needed to accurately predict the effects of SOA on climate and air quality. Diffusion can be important for predicting the growth, evaporation, and reaction rates of SOA under certain atmospheric conditions. Often, researchers have predicted diffusion rates of organic molecules within SOA using measurements of viscosity and the Stokes–Einstein relation (D∝1/η, where D is the diffusion coefficient and η is viscosity). However, the accuracy of this relation for predicting diffusion in SOA remains uncertain. Using rectangular area fluorescence recovery after photobleaching (rFRAP), we determined diffusion coefficients of fluorescent organic molecules over 8 orders in magnitude in proxies of SOA including citric acid, sorbitol, and a sucrose–citric acid mixture. These results were combined with literature data to evaluate the Stokes–Einstein relation for predicting the diffusion of organic molecules in SOA. Although almost all the data agree with the Stokes–Einstein relation within a factor of 10, a fractional Stokes–Einstein relation (D∝1/ηξ) with ξ=0.93 is a better model for predicting the diffusion of organic molecules in the SOA proxies studied. In addition, based on the output from a chemical transport model, the Stokes–Einstein relation can overpredict mixing times of organic molecules within SOA by as much as 1 order of magnitude at an altitude of ∼3 km compared to the fractional Stokes–Einstein relation with ξ=0.93. These results also have implications for other areas such as in food sciences and the preservation of biomolecules.

Statistical Study of Hard X-Ray Spectral Breaks in Solar Flares

The Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) provides hard X-ray spectral observations with ${\approx}\,1~\mbox{keV}$ resolution to study flare-accelerated ( ${>}\,10~\mbox{keV}$ ) electrons through their bremsstrahlung emission. Here we report on a statistical study of RHESSI flares with emission above 150 keV, focusing on the spectral shape at the hard X-ray peak. Spectral parameters are derived by fitting the photon spectrum with a broken power law and by the standard thick-target fit. Consistent with previous studies, the most common spectral shape of the photon spectrum (52 out of 65 events) is a double power law with a downward break (“knee”), with ten events showing a single power law and three events having an upward break (“ankle”). The spectral breaks occur typically around 55 keV and the difference of the spectral index above and below the break, $\gamma _{2}$ and $\gamma _{1}$ , is typically between 0.3 and 1. We show correlations between the downward break parameters. The most prominent correlation, with a rank order coefficient of $\rho =0.92$ , is between the power-law indices above and below the break: $\gamma _{1} = (0.74\pm 0.04)\gamma _{2} + (0.34 \pm 0.14)$ . Applying a thick target fit to the photon spectrum, a similar correlation is also found for the flare-accelerated electron spectra with $\delta _{1} =(0.85\pm 0.08)\delta _{2} - (0.3\pm 0.3)$ ( $\rho =0.67$ ). Spectral breaks could be a property of the acceleration mechanism itself or they could be a secondary effect produced by particle transport or wave-particle interactions. Any theoretical models should be consistent with these correlations. In addition, we find that one upward and 23 (49%) downward breaks are consistent with nonuniform ionization within the thick target.

Boundary value problems in Lipschitz domains for equations with lower order coefficients

We use the method of layer potentials to study the R 2 R_2 regularity problem and the D 2 D_2 Dirichlet problem for second order elliptic equations with lower order coefficients in bounded Lipschitz domains. For R 2 R_2 we establish existence and uniqueness by assuming that L \mathcal {L} is of the form L u = − div ( A ∇ u + b u ) + c ∇ u + d u \mathcal {L}u=-\text {div}(A\nabla u+bu)+c\nabla u+du , where the matrix A A is uniformly elliptic and Hölder continuous, b b is Hölder continuous, and c , d c,d belong to Lebesgue classes and satisfy either the condition d ≥ div b d\geq \ \text {div}\,b or d ≥ div c d\geq \ \text {div}\,c in the sense of distributions. In particular, A A is not assumed to be symmetric, and there is no smallness assumption on the norms of the lower order coefficients. We also show existence and uniqueness for D 2 D_2 for the adjoint equations L t u = 0 \mathcal {L}^tu=0 .

Moser’s estimates for degenerate Kolmogorov equations with non-negative divergence lower order coefficients

We prove Lloc∞ estimates for positive solutions to the following degenerate second order partial differential equation of Kolmogorov type with measurable coefficients of the form ∑i,j=1m0∂xiaij(x,t)∂xju(x,t)+∑i,j=1Nbijxj∂xiu(x,t)−∂tu(x,t)++∑i=1m0bi(x,t)∂iu(x,t)−∑i=1m0∂xiai(x,t)u(x,t)+c(x,t)u(x,t)=0 where (x,t)=(x1,…,xN,t)=z is a point of RN+1, and 1≤m0≤N. (aij) is a uniformly positive symmetric matrix with bounded measurable coefficients, (bij) is a constant matrix. We apply the Moser’s iteration method to prove the local boundedness of the solution u under minimal integrability assumption on the coefficients.

Performances Based on Ability Estimation of the Methods of Detecting Differential Item Functioning: A Simulation Study

The aim of the study is to examine differential item functioning (DIF) detection methods—the simultaneous item bias test (SIBTEST), Item Response Theory likelihood ratio (IRT-LR), Lord chi square (χ2), and Raju area measures—based on ability estimates when purifying items with DIF from the test, considering conditions of ratio of the items with DIF, effect size of DIF, and type of DIF. This study is a simulation study and 50 replications were conducted for each condition. In order to compare DIF detection methods, error (RMSD) and coefficient of concordance (Pearson’s correlation coefficient) were calculated according to estimated and initial abilities for the reference group. As a result of the study, the lowest error and the highest concordance were seen in the case of 10% uniform DIF in the test and the method of IRT-LR, considering all other conditions. Moreover, for the method of SIBTEST and IRT-LR in all conditions, it was found that the error obtained by purifying items with C level DIF is lower than the error obtained by purifying items with both B and C level DIF. Similarly, for the method of SIBTEST and IRT-LR in all conditions, it was seen that the concordance coefficient found by purifying C level DIF is higher than the coefficient by purifying items with both B and C level DIF.

Computing the roots of sparse high–degree polynomials that arise from the study of random simplicial complexes

The problem of computing the roots of a particular sequence of sparse polynomials pn(t) is considered. Each instance pn(t) incorporates only the n + 1 monomial terms $t,t^{2},t^{4},\ldots ,t^{2^{n}}$ associated with the binomial coefficients of order n and alternating signs. It is shown that pn(t) has (in addition to the obvious roots t = 0 and 1) precisely n − 1 simple roots on the interval (0,1) with no roots greater than 1, and a recursion relating pn(t) and pn+ 1(t) is used to show that they possess interlaced roots. Closed–form expressions for the Bernstein coefficients of pn(t) on [0,1] are derived and employed to compute the roots in double–precision arithmetic. Despite the severe variation of the graph of pn(t), and tight clustering of roots near t = 1, it is shown that for n ≤ 10, the roots on (0,1) are remarkably well–conditioned and can be computed to high accuracy using both the power and Bernstein forms.

Support Staff in the Caribbean:How Job Satisfaction is Related to Organizational Commitment?

The purpose of this quantitative, correlational research was to examine to what degree a relationship existed between job satisfaction and organizational commitment among support staff (cafeteria employees) in a higher educational institution in the Caribbean. The study included a sample of 27 full-time cafeteria employees who worked in a private for profit higher educational institution in the Caribbean. Study participants completed two survey instruments, the Minnesota Satisfaction Questionnaire (MSQ) short form and the Klein Unidimensional Target-free (KUT) scale. The questionnaires were in the form of hard copies. The Spearman rank order coefficient was used to measure the strength and direction of the relationship between the variables of interestjob satisfaction and organizational commitment. Results found that there was a moderate significant positive relationship between overall job satisfaction and overall organizational commitment among cafeteria employees at the for profit higher educational institution.

Dominant Cubic Coefficients of the ‘1/3-Rule’ Reduce Contest Domains

Antagonistic exploitation in competition with a cooperative strategy defines a social dilemma, whereby eventually overall fitness of the population decreases. Frequency-dependent selection between two non-mutating strategies in a Moran model of random genetic drift yields an evolutionary rule of biological game theory. When a singleton fixation probability of co-operation exceeds the selectively neutral value being the reciprocal of population size, its relative frequency in the population equilibrates to less than 1/3. Maclaurin series of a singleton type fixation probability function calculated at third order enables the convergent domain of the payoff matrix to be identified. Asymptotically dominant third order coefficients of payoff matrix entries were derived. Quantitative analysis illustrates non-negligibility of the quadratic and cubic coefficients in Maclaurin series with selection being inversely proportional to population size. Novel corollaries identify the domain of payoff matrix entries that determines polarity of second order terms, with either non-harmful or harmful contests. Violation of this evolutionary rule observed with non-harmful contests depends on the normalized payoff matrix entries and selection differential. Significant violations of the evolutionary rule were not observed with harmful contests.

Clustering of rapidly settling, low-inertia particle pairs in isotropic turbulence. Part 2. Comparison of theory and DNS

Part 1 (Rani et al. J. Fluid Mech., vol. 871, 2019, pp. 450–476) of this study presented a stochastic theory for the clustering of monodisperse, rapidly settling, low-Stokes-number particle pairs in homogeneous isotropic turbulence. The theory involved the development of closure approximations for the drift and diffusion fluxes in the probability density function (p.d.f.) equation for the pair relative positions $\boldsymbol{r}$. In this part 2 paper, the theory is quantitatively analysed by comparing its predictions of particle clustering with data from direct numerical simulations (DNS) of isotropic turbulence containing particles settling under gravity. The simulations were performed at a Taylor micro-scale Reynolds number $Re_{\unicode[STIX]{x1D706}}=77.76$ for three Froude numbers $Fr=\infty ,0.052,0.006$, where $Fr$ is the ratio of the Kolmogorov scale of acceleration and the magnitude of gravitational acceleration. Thus, $Fr=\infty$ corresponds to zero gravity, and $Fr=0.006$ to the highest magnitude of gravity among the three DNS cases. For each $Fr$, particles of Stokes numbers in the range $0.01\leqslant St_{\unicode[STIX]{x1D702}}\leqslant 0.2$ were tracked in the DNS, and particle clustering quantified both as a function of separation and the spherical polar angle. We compared the DNS and theory values for the exponent $\unicode[STIX]{x1D6FD}$ characterizing the power-law dependence of clustering on separation. The $\unicode[STIX]{x1D6FD}$ from the $Fr=0.006$ DNS case are in reasonable agreement with the theoretical predictions obtained using the second drift closure (referred to as DF2). To quantify the anisotropy in clustering, we calculated the leading–order coefficient in the spherical harmonics expansion of the p.d.f. of pair relative positions. The coefficients predicted by the theory (DF2) again show reasonable agreement with those calculated from the DNS clustering data for $Fr=0.006$. However, we note that in spite of the high magnitude of gravity, the clustering is only marginally anisotropic both in DNS and theory. The theory predicts that the spherical harmonic coefficient scales with $\unicode[STIX]{x1D6FD}(=\unicode[STIX]{x1D6FD}_{2}St_{\unicode[STIX]{x1D702}}^{2})$, where $\unicode[STIX]{x1D6FD}_{2}$ is the ratio of the drift and diffusion flux coefficients. Since the drift flux, and thereby $\unicode[STIX]{x1D6FD}_{2}$, is seen to decrease with gravity for $St_{\unicode[STIX]{x1D702}}<1$, the anisotropy is also correspondingly diminished.

A Novel Fractional-Order Grey Prediction Model and Its Modeling Error Analysis

Based on the grey prediction model GM(1,1), a novel fractional-order grey prediction model is proposed and its modeling error is systematically studied. In this paper, exponential data sequences are generated for numerical simulation. Via the numerical simulation method, the mean absolute percentage error (MAPE) of the fractional-order GM(1,1) with different values of order and development coefficient is compared to the GM(1,1) and the discrete GM(1,1). The error distribution of the sequences of exponential data is given. The GM(1,1) and the direct modeling GM(1,1) are both special cases of the fractional-order GM(1,1). The conclusion is helpful to further optimize the grey model using fractional-order operators and to expand the applicable bound of GM(1,1).

Inverse Problem of Finding the Coefficient of the Lowest Term in Two-Dimensional Heat Equation with Ionkin-Type Boundary Condition

We consider an inverse problem of determining the time-dependent lowest order coefficient of two-dimensional (2D) heat equation with Ionkin boundary and total energy integral overdetermination condition. The global well-posedness of the problem is obtained by generalized Fourier method combined with the unique solvability of the second kind Volterra integral equation. For obtaining a numerical solution of the inverse problem, we propose the discretization method from a new combination. On the one hand, it is known the traditional method of uniform finite difference combined with numerical integration on a uniform grid (trapezoidal and Simpson’s), on the other hand, we give the method of non-uniform finite difference is combined by a numerical integration on a non-uniform grid (with Gauss–Lobatto nodes). Numerical examples illustrate how to implement the method.

Prediction of ion-induced nanopattern formation using Monte Carlo simulations and comparison to experiments

Ion induced nanopattern formation has been experimentally investigated for many different ion-target combinations and different ion irradiation conditions. Several theories and models have been developed throughout the past few years to explain the observed boundary conditions for pattern formation as well as features of the patterns like wavelengths, growth rates, shapes, and amplitudes. To compare specific experiments with the predictions of analytical theories, it is necessary to calculate the linear and non-linear coefficients of the respective equation of motion of a surface profile. Monte Carlo simulations of ion–solid interactions based on the binary collision approximation provide a very fast, rather universal, and accurate way to calculate these coefficients. The universality expresses the broad range of ion species, ion energies, and target compositions accessible by the simulations. The coefficients are obtained from the moments of calculated crater functions, describing ion erosion, mass redistribution, and ion implantation. In this contribution, we describe how most linear, non-linear, and higher order coefficients can be determined from crater function moments. We use the obtained data to compare the results of selected experimental studies with the predictions of theoretical models. We find good quantitative agreement, e.g., for irradiation of Si with Ar and Kr ions, Al2O3 with Ar and Xe ions, and amorphous carbon with Ne ions.

A general trace formula for a differential operator on a segment with zero order coefficient perturbed by a finite signed measure

A general trace formula for a differential operator on a segment with zero order coefficient perturbed by a finite signed measure

Principal component analysis of up-the-ramp sampled infrared array data

We describe the results of principal component analysis (PCA) of up-the-ramp sampled IR array data from the HST WFC3 IR, JWST NIRSpec, and prototype WFIRST WFI detectors. These systems use respectively Teledyne H1R, H2RG, and H4RG-10 near-IR detector arrays with a variety of IR array controllers. The PCA shows that the Legendre polynomials approximate the principal components of these systems (i.e. they roughly diagonalize the covariance matrix). In contrast to the monomial basis that is widely used for polynomial fitting and linearization today, the Legendre polynomials are an orthonormal basis. They provide a quantifiable, compact, and (nearly) linearly uncorrelated representation of the information content of the data. By fitting a few Legendre polynomials, nearly all of the meaningful information in representative WFC3 astronomical datacubes can be condensed from 15 up-the-ramp samples down to 6 compressible Legendre coefficients per pixel. The higher order coefficients contain time domain information that is lost when one projects up-the-ramp sampled datacubes onto 2-dimensional images by fitting a straight line, even if the data are linearized before fitting the line. Going forward, we believe that this time domain information is potentially important for disentangling the various non-linearities that can affect IR array observations, i.e. inherent pixel non-linearity, persistence, burn in, brighter-fatter effect, (potentially) non-linear inter-pixel capacitance (IPC), and perhaps others.

Approximation of Ensemble Boundary Using Spectral Coefficients.

A spectral analysis of a Boolean function is proposed for approximating the decision boundary of an ensemble of classifiers, and an intuitive explanation of computing Walsh coefficients for the functional approximation is provided. It is shown that the difference between the first- and third-order coefficient approximations is a good indicator of optimal base classifier complexity. When combining neural networks, the experimental results on a variety of artificial and real two-class problems demonstrate under what circumstances ensemble performance can be improved. For tuned base classifiers, the first-order coefficients provide performance similar to the majority vote. However, for weak/fast base classifiers, higher order coefficient approximation may give better performance. It is also shown that higher order coefficient approximation is superior to the Adaboost logarithmic weighting rule when boosting weak decision tree base classifiers.