Usually in the study of limit problems, will encounter more complex problems, in this paper, we discuss how to use the concept of equivalent infinitesimal better limit operation. At the same time, in the process of research, we re-explore the proof of Taylor's formula, and find that some functions have a similar expansion form to Taylor's formula, that is, 'fractional expansion'. It is also found that after the linear combination of Taylor expansion and fractional expansion, the obtained expansion is more accurate, which helps us to have a better understanding of the approximation of function expansion.

Leaf litter decomposition and carbon release patterns in five homegarden tree species of Kumaun Himalaya viz. Ficus palmata, Ficus auriculata, Ficus hispida, Grewia optiva and Celtis austalaris were investigated. The study was carried out for 210 days by using litter bag technique. In the current investigation, the duration needed for desertion of the original biomass of diverse leaf litter varied from 150 to 210 days and specifies a varying pattern of decomposition and carbon release among the species. Grewia optiva took the longest time to decompose (210 days) while Ficus hispida decomposed more quickly than rest of the species (150 days). The relative decomposition rate (RDR) was reported highest in Ficus hispida (0.009-0.02 g-1d-1) and lowest in Grewia optiva (0.008-0.004 g-1d-1). Carbon (%) in remaining litter was in the order: Ficus auriculata (24.4 %) >Ficus hispida (24.3%) > Celtis austaralis (19.8%) > Ficus palmata (19.7%) > Grewia optiva (19%). The relationship between percentage weight loss and time elapsed showed the significant negative correlation with carbon release pattern in all the species. Releasing nutrients into the soil through the decomposition of homegarden tree residuals is a crucial ecological function that also regulates the nutrient recycling in homegarden agroforestry practices.

This study examines the spatial distribution of socioeconomic conditions in Colombia, using Moran's Index as a tool for spatial autocorrelation analysis. Key indicators related to education, health, infrastructure, access to basic services, employment, and housing conditions are addressed, allowing the identification of inequalities and structural barriers. The research reveals patterns of positive autocorrelation in several socioeconomic dimensions, suggesting a concentration of poverty and underdevelopment in certain geographic areas of the country. The results show that municipalities with more unfavorable conditions tend to cluster spatially, particularly in the northern, northwestern, western, eastern, and southern regions of the country, while the central areas exhibit better conditions. Permutation analyses are employed to validate the statistical significance of the findings, and LISA cluster maps highlight the regions with the highest concentration of poverty and social vulnerability. This work contributes to the literature on inequality and regional development in emerging economies, demonstrating that public policies should prioritize intervention in territories that exhibit significant spatial clustering of poverty. The methodology and findings provide a foundation for future studies on spatial correlation and economic planning in both local and international contexts.

In this paper, the characteristic behavior of the disc consisting of thermoplastic composite CF/PA6 material was considered. Analysis was made by taking into account the usage areas of the materials and referring to certain temperatures between 30 ℃ and 150 ℃. Composite materials are lightweight; they show high strength. For these reasons, they are preferred in technology, especially in the aircraft and aerospace industry. With this study, the radial and tangential stresses determined within a certain temperature The temperatures were determined and compared with previous studies in the literature. According to the results obtained, it is believed that the thermoplastic composite CF/PA6 disc design can be used in engineering.

Identify and diagnosis of homogenous units and separating them and eventually planning separately for each unit are considered the most principled way to manage units of forests and creating these trustable maps of forest’s types, plays important role in making optimum decisions for managing forest ecosystems in wide areas. Field method of circulation forest and Parcel explore to determine type of forest require to spend cost and much time. In recent years, providing these maps by using digital classification of remote sensing’s data has been noticed. The important tip to create these units is scale of map. To manage more accurate, it needs larger scale and more accurate maps. Purpose of this research is comparing observed classification of methods to recognize and determine type of forest by using data of Land Cover of Modis satellite with 1 kilometer resolution and on images of OLI sensor of LANDSAT satellite with 30 kilometers resolution by using vegetation indicators and also timely PCA and to create larger scale, better and more accurate resolution maps of homogenous units of forest. Eventually by using of verification, the best method was obtained to classify forest in Golestan province’s forest located on north-east of country.

Every year, hundreds of fires occur in the forests and rangelands across the world and damage thousands hectare of trees, shrubs, and plants which cause environmental and economic damages. This study aims to establish a real time forest fire alert system for better forest management and monitoring in Golestan Province. In this study, in order to prepare fire hazard maps, the required layers were produced based on fire data in Golestan forests and MODIS sensor data. At first, the natural fire data was divided into two categories of training and test samples randomly. Then, the vegetation moisture stresses and greenness were considered using six indexes of NDVI, MSI, WDVI, OSAVI, GVMI and NDWI in natural fire area of training category on the day before fire occurrence and a long period of 15 years, and the risk threshold of the parameters was considered in addition to selecting the best spectral index of vegetation. Finally, the model output was validated for fire occurrences of the test category. The results showed the possibility of prediction of fire site before occurrence of fire with more than 80 percent accuracy.