For a Dickson pair $(q,n)$ we show that  $ \big \lbrace  \frac{q^k-1}{q-1}, 1 \leq k \leq  n \big  \rbrace $ forms a finite complete set of different residues modulo $n$. We also study the construction of a finite Dickson nearfield that arises from the Dickson pair $(q,n)$.

Background and objectives: Acquired morphological traits of seedlings in nurseries are as influential as their originality in their performance and vigor. The evaluation of these traits, including the Dickson quality index (DQI) is considered an efficient method for considering the vigor of various kinds of seedlings, including fruit and forest seedlings. The magnitude of this index for seedlings is a function of the amount and the distribution of dry mass, height, and diameter. Therefore, it is a destructive and time-consuming method for predicting seedling quality in nurseries. Consequently, in order to rank the morphological traits that determine the quality of apricot seedlings, this project was conducted to evaluate the correlation relations between such traits of apricot seedlings of the most important commercial varieties produced in different regions of Iran, based on path analysis and multiple linear regression analysis.Materials and methods: To evaluate the relationship between the morphological indicators in apricot seedlings, bare-root seedlings of 12 commercial cultivars with three replicates from 8 nurseries in four provinces were lifted during the transplanting seasons of 2020-2021. Sapling height, diameter above grafting line (DAGL), root length, number of roots and branches, shoot and root fresh and dry weights, height to diameter ratio, and DQI were evaluated, and simple correlations between them were estimated. The correlation coefficients were broken down into direct and indirect effects through path analysis, with DQI as the dependent variable.Results: The indices representing apricot seedling vigor, including DQI and some of its components containing DAGL, total dry mass, height to diameter ratio and shoot to root dry mass ratio were a function of the accumulated growth degree-days (AGDD) in different regions. So that by increasing AGDD from 2700 to about 4500 degree, the average DQI, DAGL and total dry mass were increased and height to diameter ratio was decreased. The shoot to root dry weight ratio of seedlings was also decreased in nurseries located in regions with more than 3400 growth degree days. Among the traits which can be evaluated at the nursery site through non-destructive methods, DAGL had a higher correlation with vigor and the other morphological traits, leading to being considered the most important index in the qualitative evaluation of apricot seedlings, followed by shoot and root dry weights. Although later can be evaluated by destructive methods, they had the highest correlations with seedling vigor. The shoot number, height, sturdiness quotient had the weakest correlations with vigor and other morphological traits. Conclusion: DAGL is considered the most efficient index in the quality assessment of the apricot seedlings due to having the highest correlation with the DQI and other morphological traits.

Safe water supply needs of living creatures; especially human life is the need to control it using a quality control laboratory tests and indexes is different. Standards for water quality parameters in water resources (including rivers, lakes and...) from different experts are used primarily to control the quality index calculated from results obtained for several tests that show water quality is. Water quality index to compare the quality of water resources with other resources and also to determine the health of a watershed in the different parts are used. Meanwhile, water quality control for a specific time period to understand the changes in water ecosystems are also used.

In this paper, a numerical method is presented using a Dickson-based collocations method to solve a fractional model of computer virus propagation. The model presented in this paper is a system of di erential equations of fraction. By using the Dickson-based collocation method and using Chebyshev's spatial points, we transform the system of de cit di erential equations into a system of algebraic equations. In this way, an approximate solution can be found for the proposed model. By introducing the error functions for the expressed fractional model, the accuracy and convergence of the obtained solutions are investigated. Some of the approximate results obtained using this method is displayed in the numerical results section.

Aims of the Study: The physical, chemical and biological characteristics of Karaj River (near the city of Tehran, Iran), has been studied to evaluate the suitability of its water for domestic uses.Materials & Methods: Samples of water were collected from various localities before, in and after Karaj Dam Lake and were analyzed based on needs for Water quality index (WQI) calculation. In this study, WQI were calculated for the best and the poor condition for quality zoning determination of Karaj water quality.Results: Comparison of water quality index in different sampling stations of Karaj River in different years showed that the water quality of this river was better in 2008 compared with years 2007 and 2009; however ANOVA analysis showed no significant difference between the data in these years (p<0.05). Conclusions: The range of water quality was different from medium to good (WQI=50-100), so this water can be used for domestic uses after suitable treatment.