This work takes an interesting diversion, revealing the extraordinary capacity to determine the precise number of primes in a space tripled over another. Exploring the domain of K-almost prime numbers, this paper provides a clear explanation of the complex idea. In addition to outlining the conditions under which odd K-almost prime numbers must exist, it presents a novel method for figuring out how often odd numbers are as 2-almost prime, 3-almost prime, 4-almost prime, and so on, up to a specified limit n. The work goes one step further and offers useful advice on how to use these approaches to precisely calculate the prime counting function, π(n). Essentially, it offers a comprehensive exploration of the mathematical fabric, where primes reveal their mysteries in both large and small spaces.

— In this paper, we investigate the existence, the boundedness, the asymptotic behavior, and the oscillatory behavior of the positive solutions of the fuzzy di ff erence equation

In this paper, we investigate the existence, the boundedness, the asymptotic behavior, and the oscillatory behavior of the positive solutions of the fuzzy difference equation z_n+1=A+B/z_n-m , where n∈ℕ_0=ℕ∪{0} , (z_n) is a sequence of positive fuzzy numbers, A , B , and the initial conditions z_-j , j=1, 2,…,m , are positive fuzzy numbers, and m is a positive integer.

In this investigation the front door design is considered for the LADA VESTA car. The investigation is aimed at solving the issue of the weight decreasing and at the same time increasing its crashworthiness, and applied to the practice situation in racing that whether this car door will be light and strong enough to withstand the different crashing forces which it can bear during the practice conditions. Although there isn’t any practical testing of a car but simulation through FEA by ABAQUS is performed to find out its strength and most importantly strength to weight ratio as compared to a conventional steel door which faces the same force with large weight.

In this paper, a hybrid technique called the homotopy analysis Sumudu transform method has been implemented solve fractional-order partial differential equations. This technique is the amalgamation of Sumudu transform method and the homotopy analysis method. Three examples are considered to validate and demonstrate the efficacy and accuracy of the present technique. It is also demonstrated that the results obtained from the suggested technique are in excellent agreement with the exact solution which shows that the proposed method is efficient, reliable and easy to implement for various related problems of science and engineering.

In present scenario, light weighting becomes a main issue for energy efficiency in automotive industry. The emission of gases and fuel efficiency of vehicles are two important issues. The best way to improve the fuel efficiency is to decrease the weight of vehicle parts. Research and development played an important role in lightweight materials for decreasing cost, increasing ability to be recycled, enabling their integration into vehicles, and maximizing their fuel economy efficacy. There arises a need for developing a novel generation of materials that will combine both weight reduction and safety issues. The application of carbon fibre reinforced plastic material offers the best lightweight potential to realize lightweight concepts. Carbon fibre reinforced plastic has outstanding specific stiffness, specific strength, and fatigue properties compared to commonly used metals. In automotive industry, the advantages of carbon fibre reinforced plastic are reduction in weight, part integration and reduction, crashworthiness, durability, toughness, and aesthetic appealing. Carbon fibre reinforced plastic is a composite material that has been used extensively in various applications such as aerospace industry, sports equipment, oil and gas industry, and automotive industry. Keeping in view the aforementioned advantages of carbon fibre reinforced plastic, the authors have presented a brief review on carbon fibre for automotive industrial applications.