The necessary optimality conditions with Lagrange multipliers  are studied and derived for a new class that includes the system of Caputo–Katugampola fractional derivatives to the optimal control problems with considering the end time free. The formula for the integral by parts has been proven for the left Caputo–Katugampola fractional derivative that contributes to the finding and deriving the necessary optimality conditions. Also, three special cases are obtained, including the study of the necessary optimality conditions when both the final time  and the final state  are fixed. According to convexity assumptions prove that necessary optimality conditions are sufficient optimality conditions.

It is widely accepted that early diagnosis of Alzheimer's disease (AD) makes it possible for patients to gain access to appropriate health care services and would facilitate the development of new therapies. AD starts many years before its clinical manifestations and a biomarker that provides a measure of changes in the brain in this period would be useful for early diagnosis of AD. Given the rapid increase in the number of older people suffering from AD, there is a need for an accurate, low-cost and easy to use biomarkers that could be used to detect AD in its early stages. Potentially, the electroencephalogram (EEG) can play a vital role in this but at present, no reliable EEG biomarker exists for early diagnosis of AD. The gradual s

Male reproductive health is intricately regulated by molecular and physiological processes, with the aryl hydrocarbon receptor (AhR) playing a crucial role. AhR is activated by various ligands and influences the onset and progression of diseases. The aim of this study was to evaluate the role of AhR on spermatogenesis in adult male rats were affected by resveratrol (RES) and CH223191, an AhR antagonist. The study include forty rats were randomly divided into four equal groups: Control group, DMSO group, RES group and AhR‾ group, the rats received respective treatments intraperitoneally twice weekly for 60 days, and various parameters related to male reproductive health were evaluated. The AhR that activation by the RES treatment w

Incorporating the LiDAR sensor in the most recent Apple devices represents a substantial development in 3D mapping technology. Meanwhile, Apple's Lidar is still a new sensor. Therefore, this article reviews the potential uses of the Apple Lidar sensor in various fields, including engineering and construction, focusing on indoor and outdoor as-built 3D mapping and cultural heritage conservation. The affordable cost and shorter observation times compared to traditional surveying and other remote sensing techniques make the Apple Lidar an attractive choice among scholars and professionals. This article highlights the need for continued research on the Apple LiDAR sensor technology while discussing its specifications and limitations. A