A graph is called circulant if it is a Cayley graph on a cyclic group, i.e. its adjacency matrix is circulant. Let D be a set of positive, proper divisors of the integer n > 1. The integral circulant graph ICGn (D) has the vertex set Z n and the edge set E (ICGn (D)) = {{a, b}; gcd (a – b, n) Î D}. Let n = p1p2….pkm, where p1, p2, …., pk are distinct prime numbers and gcd (p1p2…pk, m) = 1. The open problem posed in paper [A. Ilic, The energy of unitary Cayley graphs, Linear Algebra Appl., 431 (2009) 1881-1889] about calculating the energy of an arbitrary integral circulant ICGn (D) is completely solved in this paper, where D = {p1, p2, …., pk}.

IntroductionSoil moisture or water content, and the fraction of that water available to plants, are among the most critical aspects of soil water management. The concept of plant-available water was first introduced nearly a century ago by Veihmeyer and Hendrickson (1927), derived from the difference between Field Capacity (FC) and Permanent Wilting Point (PWP). The concepts of the Unrestricted Water Content Range and the Minimum Restricted Water Content Range were proposed by DaSilva et al. (1994). In this framework, in addition to the two moisture limits (FC and PWP), soil aeration and the effect of soil penetration resistance on water availability to the plant are considered using simple relationships. A limitation or defect of the LLWR concept is that it treats the boundary values for aeration porosity, penetration resistance, and soil water potential as abrupt or discontinuous in restricting water availability. In an effort to overcome the shortcomings of these preceding concepts, Minasny and McBratney (2003) proposed the Soil Water Integral Energy (IE) as a criterion for estimating plant-available water in soil, replacing the focus on soil moisture content. Soil water integral energy is a measure of the energy required to extract water from the soil over a specified range of soil water content. Under this concept: firstly, the plant-available water is not solely confined to the PWP and FC range,secondly, the effect of rapid drainage at high moisture contents, which reduces the opportunity for soil water supply to the plant, is taken into account,and thirdly, the limitation imposed by soil hydraulic conductivity at low moisture contents on water flow towards the root and subsequent absorption is incorporated. They utilized various weighting functions across a wide range of soil water potentials, encompassing the potential effect of all limiting physical characteristics on soil water availability. The most significant limitation in employing this index is the time-consuming and costly process of obtaining the soil moisture characteristic curve, the soil penetration resistance curve, and the accuracy or reliability of the coefficients used in defining the proposed weighting functions. Furthermore, in addition to time and expense, errors present in soil sampling and measurement can impose constraints on the application of IE (Integral Energy).  Materials and MethodsThe study area includes a part of the Tabriz plain. For estimation of IE using the deep learning method, Artificial Neural Network (ANN), and Multiple Linear Regression (MLR), Mathematica Wolfram software version 14. 1. 0 was utilized. The input features for all three models (MLR, ANN, and Deep ANN) were identical. The data were randomly divided into two groups: training (67 data points) and testing (30 data points). The input features for the models included: 1-Percentage of water-stable aggregates 2-Soil bulk density 3-Porosity 4-Saturated hydraulic conductivity of the soil 5-Percentage of soil texture particles 6-Equivalent calcium carbonate 7-Penetration resistance at saturated moisture 8-Saturated moisture.  Results and DiscussionThe created models were evaluated using the evaluation statistics of the coefficient of determination R2, the adjusted coefficient of determination R2adj, the root mean square error RMSE, the relative error RMSEr, the model efficiency coefficient NSE, and the average percentage of relative error RME. The results showed that the deep learning method with the highest adjusted coefficient of determination (training: 0. 998, test: 0. 661) and the lowest root mean square error (training: 15. 943, test: 118. 593), the artificial neural network method (training: 0. 945, test: 0. 514) and root mean square error (training: 45. 347, test: 139. 267), and the linear multivariate regression method (training: 0. 544, test: 0. 317) and root mean square error (training: 126. 955, test: 239. 264), respectively, provide the best estimate of the IE index.  ConclusionThis study underscores the importance of soil water management and the precise assessment of Plant Available Water (PAW). Given the limitations of traditional concepts like PAW and LLWR, particularly their reliance on discontinuous boundaries, the newer the soil water Integral Energy (IE) criterion is adopted as a more accurate measure for estimating plant water availability in soil. The results demonstrated that IE can be effectively and accurately estimated in the studied area (the Tabriz plain) using deep learning techniques (Artificial Neural Networks), relying on a comprehensive set of key soil properties, including the percentage of water-stable aggregates, bulk density, porosity, and saturated hydraulic conductivity. These findings pave the way for applying advanced, data-driven modeling approaches to optimize soil water resource management in arid and semi-arid regions.

Soil pore size distribution (SPSD) is one of the most important soil physical quality indices which reflect soil inherent characteristics and its management system. The soil water integral energy (EI) is an index that represents the amount of energy needed to uptake a mass unit of soil water by plants. In this research, we studied the effects of SPSD curves location and shape parameters on EI index of different soil water ranges in medium to coarse-textured soils of Torogh Agricultural and Natural Resources Research and Education Station in Khorasan-Razavi province. Thirty points with different soil textures and organic carbon contents were selected. After conducting required laboratory and field measurements using standard methods, the soil moisture release curve (SMRC) parameters, the SPSD curve parameters, plant available water (PAW) and least limiting water range (LLWR) which were measured in matric heads of 100 and 330 cm for the field capacity, integral water capacity (IWC) and EI of the mentioned soil water ranges were calculated and the relationships between the SPSD curve location and shape parameters and EI values (for PAW100, PAW330, LLWR100, LLWR330 and IWC) were statistically analyzed. The results showed that, in medium to coarse-textured soils, increasing the equivalent pore diameter and reducing the diversity of soil pore sizes along with the tendency of the SPSD curves to be more peaked in the center and more tailed at the two ends compared with lognormal distribution could lead to lower EI and easier uptake of water by plants in different soil water ranges.

Objective: The objective of this research is to optimize the crude distillation unit (CDU) in oil refineries by reducing energy consumption and improving operational efficiency through the application of a Proportional-Integral-Plus (PIP) control system within a Non-Minimal State Space (NMSS) framework. Material and Method: Simulations of the CDU were carried out using Aspen Plus for modeling the distillation process and MATLAB for implementing the PIP control structure. The controller was tuned by an economic cost function, optimizing key parameters such as furnace duty, side-draw rates, and condenser heat removal. The PIP control system was compared to traditional control methods, with performance evaluated under various disturbances, including feed rate, temperature, and composition changes. Results: The PIP control strategy significantly improved the CDU’s performance, reducing operating costs by up to 100% compared to traditional control methods optimized by the Integral of Time-weighted Absolute Error (ITAE). The PIP system demonstrated superior disturbance handling and energy efficiency while maintaining product quality. Conclusion: The findings indicate that the PIP control system is a highly effective tool for optimizing energy consumption and process stability in modern refineries, especially under fluctuating operational conditions. Its application could lead to substantial cost savings and improved efficiency in CDU operations.

This research focuses on the effect of passive methods in the field of energy storage, produce energy by using renewable energies, and also to achieve the construction patterns of residential buildings in the climate conditions of Shiraz, with the aim reducing energy consumption. Statistics show that in Iran, fuel consumption in residential and commercial buildings with an average annual growth of 7. It is about 2.37% of the country's total fuel consumption. One way to minimize energy consumption in this sector is to increase the role of zero-energy buildings. Zero-energy buildings provide technical ways to consume less energy in buildings. These buildings are designed and built so that all or a large part of these provide the energy they need through the combination of energy saving and the use of renewable energy technology. In this research, after studying the climate condition of the city of Shiraz, a building in the Energy Plus software, the simulation and climatic data of the Shiraz station was introduced to the software. To achieve the most optimal conditions of energy consumption in the climatic conditions of Shiraz, each of the building elements Residential was optimized. The simulation shows that it is possible to reduce the energy consumption of the building by using passive building materials and designs, including double-glazed glass, Wall insulation, etc. On the other hand, by using solar energy, using pvsyst software, you can supply the energy needed by the building.

To evaluate water, soil and plant relationships, soil water availability for plants is very important for irrigation management and scheduling. Integral energy shows the readily available water for plant in the range of soil available water. The purpose of this study was to investigate the concept of integral energy in 40 samples of paddy soils in Guilan province and to study the relationship between this index and various soil properties. The results showed that all soil samples had moderate to heavy texture. The mean Dexter S index in two soil textures including clay and clay loam was less than 0. 35 (27% of the soil samples). The mean values of integral energy in the all studied soil samples, assuming the upper limit of soil water available at saturation, field capacity (at soil suctions of 100 and 330 cm) were 226. 65, 270. 21 and 336. 91 J kg-1, respectively. The results showed that the soil texture classes with the highest value of integral energy were associated with the lowest values of S index. Moreover, the results confirm the better correlation between the calculated integral energy, assuming the upper limit at field capacity in soil suction of 330 cm, with the other soil properties. In addition, the integral energy values can be accurately estimated using soil saturation moisture and organic carbon to clay content ratio for silty clay loam (R2= 0. 70) and clay loam (R2=0. 95) textured soils, respectively.