The division algorithm for multivariate polynomials over fields has been introduced not very long ago, in connection with algorithmic and computational problems in the rings since the work of Buchberger. The well-known fact is a term order must be a well-ordering and the division procedure in the ring of multivariate polynomials over a field terminates even if the division term is not leading term, but is freely chosen. In this paper, we will show the original geometric approach to the classification of all possible orders on the ring of polynomials in two variables is given. The connection between this classification and the well-known classification of Robiano is exposed in details.

Heavy metals can be released from cooking stewpots into food, causing food contamination and posing a risk to human health. This research study investigated the release of heavy metals into food from stewpot cookers used in Gharyan, Libya. As well as, investigating the effect of scratches, acidic food, and storage on the concentrations of those metals. Eight new and used cooking stewpots (with scratches) made of: (Tefal, stainless steel, copper and aluminum) were collected, and then fresh tomato juice was cooked in these stewpots and left in the refrigerator for six days, the samples were digested and analyzed by atomic absorption spectrophotometer (AAS) to measure the concentration of the heavy metals: (cadmium, lead, iron, copper and zinc). Results showed that cadmium and lead concentrations were less than the detection limit of the AAS instrument, except for one sample (old Tefal stewpot with scratches), in which the lead concentration was (0.046ppm). The concentration of iron, copper and zinc in the samples was less than the maximum limit allowed by WHO/FAO; except for one sample in which the iron concentration exceeded that value (old copper pot with scratches) and its value was (2.892ppm). The results of the descriptive statistical analysis showed that the average concentration of iron (0.493 ppm) was higher than that of zinc and copper (0.054 ppm), (0.016 ppm), respectively. In addition, the rate of leakage of those elements from old (scratched) cookware was higher than that from the new ones made from the same material, and the concentration of these elements increases significantly when these stewpots contain scratches, since that increases the rate of leakage of those metals, especially in acidic food such as tomatoes.

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An optimized image watermarking technique based on LU factorization and entropy analysis in combination with lifting wavelet transform and discrete cosine transform is presented in this paper. At first, the original image is decomposed by a 2-level lifting wavelet transform for obtaining the coefficients of a high-frequency subband followed by discrete cosine transform. Afterward, non-overlapping blocks are obtained by dividing the coefficients of discrete cosine transform whereas LU factorization is applied to each nonoverlapping blocks based on pseudo-random sequences. Then, the watermark is embedded into the first row, the first column element of the upper triangular matrix of LU factorization. The normalized cross-correlation (NC), and the peak signal-to-noise ratio (PSNR) are used to evaluate the invisibility and robustness of the presented technique. The experimental results have indicated that the presented technique fulfills all watermarking requirements in terms of invisibility, robustness, security, and capacity. The comparison with the existing scheme has shown that the proposed watermarking technique has a superior performance in terms of invisibility than the existing scheme.

The theory of operators is an important subject in analytic function theory, geometric function theory and univalent function theory. It also is an important subject in applied sciences. It is still an active field of research and various types of problems, which can be solved by generalising operators. Recently, many researchers have shown great interests in the study of differential operators in the theory of univalent functions and various subclasses of analytic functions defined in the open unit disc. In this paper a generalised derivative operator will be used to derive some results concerning the subordination and superordination of analytic function in the open unit disc.