Secret sharing is the process of distributing a secret among n shareholders, in such a way that only a subset of them can recover the secret, while unauthorized subsets, referred to as dishonest shareholders, cannot access the secret. During the secret reconstruction phase, when shareholders present their shares, a dishonest shareholder or shareholders can always obtain the secret exclusively by presenting fake shares, thus leaving the honest shareholders with nothing but a fake secret. Detecting cheating is crucial for achieving a fair secret reconstruction. In this paper, it has been proposed a cheating-detecting secret sharing scheme that utilizes polynomial coefficients for cheat detection in secret reconstruction. It is leveraged the invertibility property of polynomial coefficients in the z_q field to detect cheat and employ relationships that follow a linear equation.

A Sturm-Liouville problem with n-potential functions in the second order differential equation and which contains spectral parameter depending on linearly in one boundary condition is considered. The asymptotic formulas for the eigenvalues, nodal parameters (nodal points and nodal lengths) of this problem are calculated by the Prüfer's substitutions. Also, using these asymptotic formulas, an explicit formula for the potential functions are given. Finally, a numerical example is given.

In this paper, the two-observational  percentile, percentile and maximum likelihood estimation of the probability density function of  Inverse Weibull random variable are studied. Finally, these estimates are compared using simulation studies and a real data.

In this paper, we introduce the definition of a new general integral transform which call it "A new general polynomial transform ". Also, we introduce properties, theorems, proofs and transforms of the Logarithmic functions, polynomials functions and other functions. As well as, we discuss how we can apply this integral transform and its inverse to solve the ordinary differential equations with variable coefficients.

Background: Scoliosis is a health problem that causes a side-to-side curvature in the spine. The curvature may have an “ S” or “ C” shape. To evaluate scoliosis, the Cobb angle has been commonly used. However, digital image processing allows the Cobb angle to be obtained easily and quickly, several researchers have determined that Cobb angle contains high variations (errors) in the measurements. Therefore, a more reproducible computer aided-method to evaluate scoliosis is presented. Material and Methods: In this analytical study, several polynomial curves were fitted to the spine curvature (4th to 8th order) of thirty plain films of scoliosis patients to obtain the Curvature-Length of the spine. Each plain film was evaluated by 3 physician observers. Curvature was measured twice using the Cobb method and the proposed Curvature-Length Technique (CLT). Data were analyzed by a pairedsample Student t-test and Pearson correlation method using SPSS Statistics 25. Results: The curve of 7th order polynomial had the best fit on the spine curvature and was also used for our proposed method (CLT) obtaining a significant positive correlation when compared to Cobb measurements (r=0. 863, P<0. 001). The Intraclass Correlation (ICC) was between 0. 863 and 0. 948 for Cobb method and0. 974 to 0. 984 for CLT method. In addition, mean measurement of the inter-observer COV (Coefficient of Variation) for Cobb method was of 0. 185, that was significantly greater than the obtained with CLT method of 0. 155, this means that CLT method is 16. 2% more repeatable than Cobb Method. Conclusion: Based on results, it was concluded that CLT method is more reproducible than the Cobb method for measuring spinal curvature.

In this paper, estimation of second order polynomial systems’ domain of attraction (DA) via rational Lyapunov function is investigated. One of the methods for estimating DA is to find the greatest level set. In this study, in order to obtain the greatest level set, cost function based on increasing the region enclosed to the level set has been offered instead of using shape factor.Estimating DA has been converted into solving bilinear matrix inequality optimization problem.Capacity of this method compared to other methods in recent studies has been shown through some examples.

To determine the best function for describing lactation of Najdi cow in Khuzestan, the day-based milk production test records, collected by Shushtar support station from 1990 to 2014, were used. Records included 2369 data related to 444 first parity cows. To determine the best mathematical function, six functions includind incomplete gamma, wilmink, inverse polynomial, complex logarithmic, Cobby and Le Du, and polynomial regression were evaluated. All functions were fitted using PROC NLIN procedure of SAS software and Gauss-Newton iteration method on day-based milk production test records; and compared based on coefficient of determination R2, corrected coefficient of determination, the mean square error and Akaike index. The results showed that the inverse polynomial function offers a better description for lactation curve in Najdi cattle.  The lactation curve parameters (a, b, c) were estimated 4.4841, 0.2514, 0.00273, and their heritability were 0.354±0.120, 0.280±0.11 and 0.276±0.155, respectively. Moderate heritability suggests the opportunity of genetic response via selection program. Heritability of these traits can be improved by correcting the known environmental impacts.

Probability density functions of the involved random variables are essential for the reliability-based design of offshore structures. The objective of present research was the derivation of probability density function (PDF) for the local joint flexibility (LJF) factor, fLJF, in two-planar tubular DK-joints commonly found in jacket-type offshore structures. A total of 162 finite element (FE) analyses were carried out on 81 FE models of DK-joints subjected to two types of axial loading. Generated FE models were validated using available experimental data, FE results, and design formulas. Based on the results of parametric FE study, a sample database was prepared for the fLJF values and density histograms were generated for respective samples based on the Freedman-Diaconis rule. Nine theoretical PDFs were fitted to the developed histograms and the maximum likelihood (ML) method was applied to evaluate the parameters of fitted PDFs. In each case, the Kolmogorov-Smirnov and chi-squared tests were used to evaluate the goodness of fit. Finally, the Inverse Gaussian model was proposed as the governing probability distribution function for the fLJF. After substituting the values of estimated parameters, two fully defined PDFs were presented for the fLJF in tubular DK-joints subjected to two types of axial loading.