iosr-Jm   Volume-21 ~ Issue-3 ~ May – Jun. 2025

Abstract : The theory has been generally accepted and established in recent years but a thorough introduction is required to make the concept accessible to a wider audience. Probabilities provide a detailed introduction to incorrect likelihoods, including hypotheses and implementations that illustrate the current state of the art. Important reading for academic researchers, institutes of science and other organisations and professionals working in fields such as risk.......

Keywords: Applications, Imprecise Probabilities, Operations Research, Game-theoretic probability, Non-parametric statistics.

[1] Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent Measures Of Risk. Mathematical Finance, 9, 203–228.
[2] Ben-Tal, A., El Ghaoui, L., & Nemirovski, A. (2009). Robust Optimization. Princeton: Princeton University Press.
[3] Bertsimas, D., & Thiele, A. (2006). Robust And Data-Driven Optimization: Modern Decision Making Under Uncertainty. Tutorials On Operations Research, INFORMS, Chap. 4, 195–122.
[4] Embrechts, P., Kl¨Uppelberg, C., & Mikosch T. (1997). Modelling Extremal Events For Insurance And Finance. New York: Springer..

Abstract : Solidification and fusion are important processes applied in several fields of science and technology. Recently, far beyond the realms of materials science and metallurgy, many applications have risen in latent heat thermal energy storage and melting and growth of ice plates. Due to the relative difficulty in obtaining numerical solutions for moving boundary problems for a wide range of space and time scales. No studies in the literature consider a comprehensive first and second-order treatment of Biot number for phase change. This work proposes four closed-form solutions for the.......

Keywords: Partial differential equation; Convective boundary condition; Unsteady analytical solution; Moving boundary problem; Semi-infinite slab.

[1] Rappaz, M. Modeling And Characterization Of Grain Structures Of Defects In Solidification. Current Opinion In Solid State And Materials Science, 2016; 20: 37-45.
[2] Iten M, Liu S, Shukla A. A Review On The Air-PCM-TES Application For Free Cooling And Heating In The Buildings, Renewable Sustainable Energy Rev 2016; 61: 175-186.
[3] Flemings, MC. Solidification Processing. New York: Mcgraw-Hill, 1974.
[4] Wang D, Zhou C, Xu G, Huaiyuan A. Heat Transfer Behavior Of Top Side-Pouring Twin-Roll Casting. Journal Of Materials Processing Technology, 2014; 214: 1275-84.
[5] Ozisik MN, Finite Difference Methods In Heat Transfer, 1st Edn. CRC Press, Boca Raton, 1994.

Abstract : Cryptography, the science of encoding and decoding information, has a rich history that spans thousands of years, evolving from simple substitution ciphers to complex modern encryption algorithms. This paper explores the chronological development of cryptographic techniques, beginning with early systems used by ancient civilizations, such as the Egyptians and Greeks, and advancing through key milestones like the Caesar cipher and the Vigenère cipher. .......

Keywords: Cryptography, ancient ciphers, Caesar cipher, Vigenère cipher, World War.

[1]. Adams, C., & Lloyd, S. (2003). Understanding Public-Key Infrastructure: Concepts, Standards, and Deployment Considerations (2nd ed.). Addison-Wesley.
[2]. Al-Kindi. (9th century). Risalah fi Istikhraj al-Mu'amma [Manuscript for the Deciphering Cryptographic Messages].
[3]. Antonopoulos, A. M., & Wood, G. (2018). Mastering Ethereum: Building Smart Contracts and DApps. O’Reilly Media.
[4]. Babbage, D., & McKenna, R. (2019). Quantum computing and its implications for modern cryptography. Journal of Cryptographic Engineering, 9(3), 45–62. https://doi.org/10.1007/s13389-019-00231-0
[5]. Boneh, D., Franklin, M. K., & Venkatesan, R. (2001). Identity-based encryption from the Weil pairing. SIAM Journal on Computing, 32(3), 586–615. https://doi.org/10.1137/S0097539700384746.