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The error function occurs widely in multiple areas of mathematics, mathematical physics and natural sciences. There has been no work in this area for the past four decades. In this article, we estimate the coefficient bounds with q-difference operator for certain classes of the spirallike starlike and convex error function associated with convolution product using subordination as well as quasi-subordination. Though this concept is an untrodden path in the field of complex function theory, it will prove to be an encouraging future study for researchers on error function.

The error function takes place in a wide range in the fields of mathematics, mathematical physics and natural sciences. The aim of the current paper is to investigate certain properties such as univalence and close-to-convexity of normalized imaginary error function, which its region is symmetric with respect to the real axis. Some other outcomes are also obtained.Mathematics Subject Classification (2010): 30C45, 30C50, 33B20.Received 27 August 2021; Accepted 27 August 2021.

Sorption of synthetic dyes on low-cost solid sorbents is a simple technique for their removal from wastewater. Recent initiatives in the sorption process have sought the use of activated carbon derived from agricultural wastes as it provides an attractive and cheaper alternative to commercial activated carbon, which is usually expensive. This research investigates the sorption kinetics and equilibrium of two synthetic cationic dyes, crystal violet and methylene blue from aqueous media using activated carbon prepared from an agro-waste, Millettia thonningii seed pods. Sorption experiments were carried out using the batch process. The kinetic data were analyzed using the pseudo-first-order, pseudo-second-order, and intraparticle diffusion models while the equilibrium data were analyzed using the Langmuir, Freundlich, and Redlich–Peterson isotherm models. Nonlinear regression method was used to fit the data to the isotherm models in order to determine model parameters and the best-fit isotherms. Thus, three error functions; coefficient of determination, Chi-square statistic test, and the sum of error squares were applied to evaluate the sorption data. The pseudo-second-order model best described the sorption kinetics of both dyes while the Redlich–Peterson model described the equilibrium data the most, followed closely by the Freundlich isotherm model indicating a heterogeneous sorbent surface. The experimental results indicate that the agro-waste derived activated carbon is a viable adsorbent for the remediation of dye-contaminated water.

In recent years, special functions have played a significant role in the investigation of different subclasses within the class of bi-univalent functions. In this work, we present and investigate two new subclasses of bi-univalent functions defined in U=ς∈C:|ς|<1, characterized by Bernoulli polynomials associated with imaginary error functions. For functions belonging to these subclasses, we establish bounds for their initial coefficients. For these classes, we also tackle the Fekete–Szegö problem. Several new results are also obtained as special cases by specifying certain parameter values in the general findings.

We present algorithms to approximate the scaled complementary error function, exp x 2 e r f c ( x ) , and the Dawson integral, e - x 2 ∫ x 0 e t 2 d t , to the best accuracy in the standard single, double, and quadruple precision arithmetic. The algorithms are based on expansion in Chebyshev subinterval polynomial approximations together with expansion in terms of Taylor series and/or Laplace continued fraction. The present algorithms, implemented as Fortran elemental modules, have been benchmarked versus competitive algorithms available in the literature and versus functions built-in in modern Fortran compilers, in addition to comprehensive tables generated with variable precision computations using the Matlab™ symbolic toolbox . The present algorithm for calculating the scaled complementary error function showed an overall significant efficiency improvement (factors between 1.3 and 20 depending on the compiler and tested dataset) compared to the built-in function “ Erfc_Scaled ” in modern Fortran compilers, whereas the algorithm for calculating the Dawson integral is exceptional in calculating the function to 32 significant digits (compared to 19 significant digits reported in the literature) while being more efficient than competitive algorithms as well.

In this work, we develop a method for rational approximation of the Fourier transform (FT) based on the real and imaginary parts of the complex error function w(z)=e−z2(1−erf(−iz))=K(x,y)+iL(x,y), z=x+iy, where K(x,y) and L(x,y) are known as the Voigt and imaginary Voigt functions, respectively. In contrast to our previous rational approximation of the FT, the expansion coefficients in this method are not dependent on the values of a sampled function. As the values of the Voigt functions remain the same, this approach can be used for rapid computation with help of look-up tables. Mathematica codes with some examples are presented.

The suitability of beechwood chip biochar(BC1) and garden green waste biochar (BC2) for Cr(VI) removal was explored in this study. Green waste biochar (BC2) was found to be the most effective for Cr(VI) removal (84.6%) at room temperature at an adsorbent dose of 10 g [L.sup.-1] at an optimum pH of 5. The Freundlich isotherm model exhibited the best fit, followed closely by the Langmuir model, suggesting a heterogeneous adsorption process with a potential contribution from monolayer adsorption, as supported by nine error functions. Adsorption kinetics was best explained by pseudo nth-order model (PNO). Gibbs free energy ([DELTA]G) was found to vary between -24.19 to -29.48 (BC1) and -24.18 to -28.39 483 kJ [mol.sup.-1](BC2), indicating a spontaneous reaction. Enthalpy ([DELTA]H) was 74.483 kJ [mol.sup.-1]for BC1 and 59.51 483 kJ [mol.sup.-1] for BC2, which indicates chemisorption in the current study. Keywords: beechwood, garden green waste, chromium, error function analysis, PNO model.

In this present investigation, we will concern with the family of normalized analytic error function which is defined by By making the use of the trigonometric polynomials , , e ) as well as the rule of subordination, we introduce several new classes that consist of 𝔮-starlike and 𝔮-convex error functions. Afterwards, we derive some coefficient inequalities for functions in these classes.

In Constraint Programming, constraints are usually represented as predicates allowing or forbidding combinations of values. However, some algorithms can exploit a finer representation: error functions. By associating a function to each constraint type to evaluate the quality of an assignment, it extends the expressiveness of regular Constraint Satisfaction Problem/Constrained Optimization Problem formalisms. Their usage comes with a price though: it makes problem modeling significantly harder, since users must provide a set of error functions that are not always easy to define. Here, we propose a method to automatically learn an error function corresponding to a constraint, given its predicate version only. This is, to the best of our knowledge, the first attempt to automatically learn error functions for hard constraints. In this paper, we also give for the first time a formal definition of combinatorial problems with hard constraints represented by error functions. Our method aims to learn error functions in a supervised fashion, trying to reproduce either the Hamming or the Manhattan distance, by using a graph model we named Interpretable Compositional Networks. This model allows us to get interpretable results. We run experiments on 7 different constraints to show its versatility. Experiments show that our system can learn functions that scale to high dimensions, and can learn fairly good functions over incomplete spaces. We also show that learned error functions can be used efficiently to represent constraints in different classic problems.

In adsorption study, to describe sorption process and evaluation of best-fitting isotherm model is a key analysis to investigate the theoretical hypothesis. Hence, numerous statistically analysis have been extensively used to estimate validity of the experimental equilibrium adsorption values with the predicted equilibrium values. Several statistical error analysis were carried out. In the present study, the following statistical analysis were carried out to evaluate the adsorption isotherm model fitness, like the Pearson correlation, the coefficient of determination and the Chi-square test, have been used. The ANOVA test was carried out for evaluating significance of various error functions and also coefficient of dispersion were evaluated for linearised and non-linearised models. The adsorption of phenol onto natural soil (Local name Kalathur soil) was carried out, in batch mode at 30 ± 20 C. For estimating the isotherm parameters, to get a holistic view of the analysis the models were compared between linear and non-linear isotherm models. The result reveled that, among above mentioned error functions and statistical functions were designed to determine the best fitting isotherm.