The general Randi´c index of a graph G = (V,E) was defined asR_α=∑_(u,v∈V)(d_u d_v )^α , where du is the degree of vertex u and α isan arbitrary real number. In this paper we define the Randi´c indexof a uniform hypergraph and obtain lower and upper bounds for Rαdepending different values of α.

Understanding the structure of a forest is crucial for detecting forest dynamics, describing ecosystem stability, and designing effective management plans and protection and restoration measures. This study primarily aims to investigate the spatial pattern and structural characteristics of beech forests in the optimal developmental stage using variable area sample plots. In compartment 327 of the Grozban district of Kheyrood forest, eleven variable area sample plots with geometric shapes, exhibiting characteristics of the optimal developmental stage, were selected. The location, diameter, and height of all trees larger than 7.5 cm were measured in each sample plot. The results revealed that the largest contribution of trees among the investigated sample plots was related to mid-diameter and low-diameter classes. The Clark and Evans index and uniform angle results indicated a regular distribution of trees in all studied sample plots. The mingling index averaged 0.19 across all sample plots, suggesting low mixing and a minimal tendency of beech trees to blend with other species. The average diameter and height differentiation index were calculated as 0.31 and 0.30, respectively, indicating an average difference between neighboring trees. Overall, the use of indices based on the nearest neighbor with variable area sample plots and geometric shapes demonstrated that these indices effectively quantify the structural characteristics of forest stands in the optimal developmental stage. Consequently, it is recommended to conduct similar research in other developmental stages.

his study aimed to evaluate the structural characteristics of Quercus macranthera forests in the Arasbaran region, Hatam Mashe Si, Meshgin-Shahr. To achieve this, all trees and shrubs were full calipered in three one-hectare plots. Additionally, the structural characteristics were measured using the nearest neighbor’ s method. To measure intensity of trees, several indices like nearest neighbor distance index (density of trees), uniform angle index and Clark and Evans index (variety of location), Mingling index and Shannon-Wiener index (tree diversity in diameter classes), diameter and height differentiation indices (dimensional differences of trees), and complex structural diversity index (tree diversity) were used. The results showed that Q. macranthera represented a relatively homogeneous structure in its main site, and the mean of distance to the neighbors was 2. 59 m. The results of uniform angle and Clark and Evans indices showed that the spatial pattern of trees was regular. Mingling and Shannon-Wiener indices showed a low tendency of Q. macranthera to mix with other tree species so that the majority of trees were Q. macranthera. The height and diameter differentiation indices illustrated that the dimensions of the trees had a slight difference. According to complex structural diversity index, tree diversity was at the medium level. It can be concluded that the Q. macranthera communities have a homogeneous structure in a study region with a low mixture and single canopy layer. The information can provide the opportunity to monitor future changes in Q. macranthera communities in this region. To promote ecosystem stability, it is suggested to create heterogeneity of stand structure and support other tree species in this area.

Background and Objectives: Forest structure is a key factor in growth, yield and how changes occur in the forest stands, and evaluating the structure of plantations can play an important role in improving afforestation and timely implementation of forestry treatments in each stand. This study aimed to evaluate structural characteristics of plantation forests in the Saib Tabrizi forest park located on Tabriz, after 25 years of planting have passed, and abandon its management. Materials and Methods: For this purpose, after initial field surveys, three one-hectare plots were selected and based on quantitative indicators, the structure of each plot-sample was investigated. In each plot, structural characteristics of all trees (species type, diameter at breast height, tree height, canopy height, and canopy diameter) were measured and the distance method (based on nearest neighbor’ s) were used to quantify stand structure. In order to study of stand structural some index including nearest neighbors distance index (density of trees), uniform angle index and Clark and Evans index (variety of location), Mingling index (species mixture), diameter and height differentiation indices (dimensional differences of trees), and complex structural diversity index (tree diversity) were used. Results: Based on the results, the average density of trees was 448 per hectare. In the studied sample plots, the average DBH was equaled to be 12. 9 cm, the average height of the trees was 5. 1 m and the average basal area was calculated to be 0. 013 m2 per hectare. The highest abundance among the species belonged to the species of Pinus nigra with 50. 4% and Robinia pseudoacacia 32. 4%, respectively. The results of the uniform angle index showed that a cluster pattern in spatial distribution of trees, however Clark and Evans index showed a regular pattern. According to the Mingling index, the studied stands have relatively moderate mixing. The height and diameter differentiation indices showed that the dimensions of the trees had a slight difference, and have a homogeneous structure. According to complex structural diversity index, tree diversity was obtained at the low level. Conclusion: In conclusion, it can be said that after nearly three decades of planting trees in this park, tree communities have maintained the homogeneity of their structure, and trees have moderate mixture. Therefore, it can be stated that the development of forest structure in the research area is slow. Hence it is recommended to increase the biological sustainability, silvicultural treatments are needed to increase mixture and heterogeneity of structure. The results of this study can provide objective justifications for the importance of consciously creating forestry strategies in the country's forestry as a missing link in the forest management.

Here, the effect of rotating cavity filled by porous materials exposed to two variable magnetic fields on the heat transfer natural convection is investigated. The two hot semicylinder are cooled by the heat transfer through the porous medium. The two magnetic fields affect the nanofluid flow and the heat transfer of the natural convection inside the porous cavity. The characteristic equations related to the fluid flow including the continuum equation, momentum and two nano-fluid and the solid-state matrix energy equations of the porous medium have been solved to predict the problem behavior. The influence of the cavity rotation angle on the streamlines and temperature field is investigated. The results show that increasing the rotation angle has oscillatory effects on the magnitude of the streamlines. For the rotation angles = 20 and = 100 the heat transfer via both phases is intensified. The rotation of the cavity depends on how the magnetic field-induced Lorentz and Kelvin forces amplify or weaken the heat transfer, thereby altering the Nusselt number of both phases of porous medium.

Statement of Problem: In Orthodontic diagnosis and treatment planning one of the most important issue is an accurate anteroposterior measurement of jaw relationship. The most commonly used angular and linear measurements that have been proposed, such as ANB angle and Wit's appraisal can be inaccurate and misleading due to their dependence on various factors.Purpose: This study was aimed at establishing a new cephalometric measurement, named the m angle, to assess the sagittal jaw relationship with accuracy and reproducibility. The m angle uses 3 skeletal landmarks, point A, point B and a perpendicular line from point A to the mandibular plan.Materials and Method: One hundred and fifteen pretreatment cephalometric radiographs of orthodontic patients (mean age 16±2 years) were selected on the basic of inclusion criteria according to their skeletal and profile patterns. The patients compromised of 30 class I, 45 class II and 40 class III skeletal relationships and profile pattern, the mean and standard deviation for the new angel (b) and the previous reported angle (m) were evaluated. One way analysis of variance (ANOVA) and the Newman- Keuls test were used for data analysis as well as, dermination of the new angle's specifity and sensitivity comparing with b angle.Results: The results obtained showed that a patient with a m angle between 16.1° and 23.9° can be considered to have a class I skeletal pattern. A more acute m angle is an indication of a class II skeletal pattern and a more obtuse m angle indicates a class III skeletal pattern.Conclusion: The new angle (m angle) has an acceptable specificity and sensitivity in determining the anteroposterior relation of jaws and thus can be used for assessing the type and severity of jaw discrepancies, besides the other diagnostic parameters.

Spatial patterns of tree species can provide valuable insights into forest dynamics, including processes such as forest establishment, tree growth, species competition, and mortality. This study aimed to investigate the structure, spatial pattern, and interactions of beech trees during the volume accumulation stage of forest evolution in the climax stage of succession. Three one-hectare sample plots with characteristics representative of this stage were selected, and tree characteristics, including species type, DBH, and location, were measured and recorded for all living trees. The results of the uniform angle index indicated a regular pattern among neighboring trees. The average mingling index was 0.08, suggesting intense intra-species competition and a low tendency for beech trees to be located near other species. The mean diameter and height differentiation indices were 0.36 and 0.28, respectively, indicating an average difference between neighboring trees. According to the results of the univariate L11 function, in sample plots one, two, and three, trees exhibited a regular pattern up to distances of 7, 4, and 6 meters between trees, respectively. At greater distances, trees were located within the Monte Carlo range. Based on the results of the L12 bivariate function, in all three sample plots, small diameter trees had a negative interaction with medium and large diameter trees up to a distance of at least 5 meters. Any intervention based on silvicultural programs should take into account the preservation of the structural features of stands at this stage of beech evolution.

Background and objectives: Hyrcanian forests are very important in terms of preserving genetic reserves and the existence of some plant species that are unique to this vegetation area. Therefore, further investigation of the characteristics of these diverse forest communities for the purpose of ideal management and multi-purpose planning, with the aim of maintaining diversity in these forests, is always an inevitable necessity. Among these features is the dynamics of the structure, which affects the functioning of the ecosystem and is corrected by forest management and natural disturbances. Understanding its structure and dynamics through predicting the future structure of the forest and its evolution is very important for implementing management goals, reducing forestry costs and increasing the productivity of forest ecosystems. Based on this, this research was carried out with the main purpose of investigating the structural diversity of development stages in the managed eastern mixed beech forests.Methodology: a 10-hectare area in the managed stands of parcel No. 514, series 5 of the Safaroud forestry plan, which has 42 mosaics of initial, optimal and decay development stages, which are separated based on the indicators and criteria for determining the development stages in forest were identified and selected regardless of its size and shape. The trees inside each mosaic were numbered and measured with full callipering method. The location of all the trees with a diameter at breast height (dbh) greater than 7.5 cm was recorded using the distance and azimuth of the bases relative to each other for each tree. After numbering the trees, characteristics such as dbh and total height of the trees were measured. Then, based on the dbh, the trees were divided into four classes: small (less than 35 cm), medium (35-50 cm), large (50-75 cm) and extra large (more than 75 cm). To quantify the structure of each mosaic in the studied stands, a set of different structural indices was used based on the three characteristics of spatial pattern, mingling and dimensions of tree. After analyzing the data, the structural characteristics of the stands, including the indices of diameter and height differentiation, uniform angle, mean distance to the nearest neighbor, mingling and Clark and Evans index in the mosaics were calculated. Uniform angle and Clarke and Evans indices were used to check the diversity of tree spatial patterns. Also, to check the species admixture, the mingling index was used, and to check the diversity of tree dimensions in each development stage mosaic, the diameter and height differentiation indices were used. The density of trees was also measured using the distance index to the nearest neighbors. Analyzes related to the type of distribution of trees and the type of mixing in the studied stands were done using Crancod software (ver. 1.4).Results: A total of 3105 trees of beech and other tree species were identified and recorded, and their quantitative characteristics were presented according to the mosaics of development stages. The 10-hectare area had a total of 42 mosaics of different development stages with irregular shapes and variable areas that had been separated. The initial development stage with 18 mosaics had the most and the optimal development stage with 11 mosaics had the least frequency. The area of the mosaics varied from 292 to 5145 square meters. In total, in the investigated stands, the initial stage had the highest surface and the decay stage had the lowest surface in the studied area. The initial and decay development stages accounted for the highest and lowest number of stands dendity with an average of 427 and 212 stems per hectare, respectively. The average basal area was measured as 38.75, 42.26, and 37.63 square meters per hectare in the initial, optimal, and decay development stages, respectively. The mean distance to the nearest neighbor’s index in the mosaics of the initial, optimal and decay development stages, was obtained 4.39, 4.96 and 5.76 meters, respectively. The value of Clark and Evans index was less than one in all three development stages, which indicates that the trees in all mosaics have a cluster distribution pattern. The value of the uniform angle index was calculated as 0.5 and more than 0.5 in all stages, which indicates the cluster distribution pattern of trees. The mean mingling index in the mosaics of initial, optimal and decay development stages, was obtained as 0.31, 0.27 and 0.18, respectively. Also, the mean diameter differentiation index was calculated at 0.43, 0.37 and 0.43 in the initial, optimal and decay stages, respectively. The mean height differentiation index was 0.31, 0.28 and 0.33 in the initial, optimal and decay stages, respectively, which indicates the low to medium height difference of the neighboring trees.Conclusion: According to the management activities in the studied beech stands, the results of the quantification of the stand structure according to the mosaics of different development stages can be used as a guide and template for the management of other similar forest stands. Also, the characteristics and functions used in this research can be used by forest managers as indicators to measure the interventions made (like tending and marking) as well as to monitor the development stages in these stands over time. The structural diversity of uneven-aged managed forests is different among different development stages. Therefore, in order to understand how management activities, affect the short-term structural dynamics in beech forests, similar researches should be conducted in five to 10-year time series in the investigated stands. Also, to get better results, these studies should be done over time and by monitoring a fixed stand.

Stand structure and species diversity are two useful parameters for complex assessment of forest biodiversity, and provide important information for management and decision making for appropriate silvicultural system. For this purpose, five one ha plots were selected randomly in homogeneous ecological units of beech-hornbeam type in Nav-e Asalem, Gilan province. In order to determine complex structural diversity index (SI) in the studied forest type, uniform angle, mingling and DBH dimensions difference indicators were calculated. Also in order to determine density of trees, the nearest neighbors distance index was used. According to the results, the average value of nearest neighbors distance index was calculated to 5.58 meter. Mean uniform angle index was 0.52, which indicates clumped distribution of trees. Mean mingling index (0.45) indicates moderate mixture of the studied mixed stand. The amount of DBH dimensions difference index was 0.47, indicates moderate difference between trees diameter at breast height. The complex structural diversity index which presents three dimensions of structure, including diversity of spatial pattern, species diversity, and diversity of DBH dimensions) was calculated to 0.475. This value confirms a high level of tree diversity. The results of this study provide key information for management and maintaining of tree diversity in the studied mixed beech-hornbeam stand. Also with monitoring and management of tree diversity level we can reduce the negative impact of natural and human factors.