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.

Abstract : This research paper is related to the concept of approximation of a function belonging to the Lipschitz class by the Euler-matrix triple product summability method of the Fourier series. The Euler-matrix product summability method has been used while working in this direction. So, many known results may become particular cases of our result. Our result may be useful for future researchers......

Keywords: Euler sum, Matrix mean, product summability method, Lipschitz class, Fourier series, Lebesgue integral.

[1] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, Cambridge, 3rd Rev. Ed., 2002.
[2] A.A.Das, S.K, Paikray, P.Parida, U.K.Misra, Degree Of Approximation In The Generalized Class (E,Q)A-Product Summability Means Of Fourier Series, Twms J.App. And Eng. Math. V.10, Special Issue,2020, Pp. 53-62
[3] B. E. Rhoades, On The Degree Of Approximation Of Functions Belonging To Lipschitz Class By Hausdorff Means Of Its Fourier Series, Tamkang Journal Of Mathematics, 34 (2003), No. 3, 245-247. Mr2001920(2004g:41023). Zbl 1039.42001.
[4] B. P. Padhy, Banitamani Mallik, U.K. Misra And Mahendra Mishra, On Degree Of Approximation Of Fourier Series By Product Means (E,Q)A, International Journal Of Mathematics And Computation, Issn 097- 570x(Online), Issn 0974-5718 (Print), Vol19; Issue No.2; Year 2013 Copyright C 2013 By Ceser Publications.
[5] E.C. Titechmalch: The Theory Of Functions, Oxford University Press,P.P. 402-403(1939).

Abstract : In this paper, we investigate the concept of closure within the framework of single-valued neutrosophic primal topological spaces. We introduce and define the single-valued neutrosophic primal closure operator and examine its fundamental properties, such as idempotency, extensivity, and preservation under primal continuous mappings. Furthermore, we explore the relationships between the primal closure and other classical closure operators in neutrosophic and fuzzy topologies. Several characterizations are provided to distinguish primal closed sets from general closed sets. Our results contribute to a deeper understanding of the structural behavior of neutrosophic primal topologies and offer a foundation for further applications in uncertainty modeling and decision-making systems.

Keywords: Single-valued neutrosophic primal; closure and interior operators (including per-primal, α-primal, β-primal forms); single-valued neutrosophic primal continuity; α-continuity; compactness.

[1]. Smarandache, F. A Unifying Field In Logics, Neutrosophy: Neutrosophic Probability, Set And Logic; American Research Press: Rehoboth, NM, USA.
[2]. Salama, A.A.; Alblowi, S.A. Neutrosophic Set And Neutrosophic Topological Spaces. IOSR J. Math. 2012, 3, 31–35.
[3]. Salama, A.A.; Smarandache, F.; Kroumov, V. Neutrosophic Crisp Sets And Neutrosophic Crisp Topological Spaces. Neutrosophic Sets Syst. 2014, 2, 25–30.
[4]. Chang, C.L. Fuzzy Topological Spaces. J. Math. Anal. Appl. 1968, 24, 182–190.
[5]. Hur, K.; Lim, P.K.; Lee, J.G.; Kim, J. The Category Of Neutrosophic Sets. Neutrosophic Sets Syst. 2016, 14, 12–20.

Abstract : Ramanujan's modular equations and continued fractions have been a cornerstone in number theory, particularly in relation to theta functions. This paper explores the connection between Ramanujan's cubic continued fraction and Jacobi's theta functions, emphasizing their transformation properties and modular equations. A key focus is the derivation of modular equations........

[1] Andrews, G. E. (1979). The Theory Of Partitions. Cambridge University Press.
[2] Apostol, T. M. (1990). Modular Functions And Dirichlet Series In Number Theory (2nd Ed.). Springer.
[3] Berndt, B. C. (1991). Ramanujan's Notebooks, Part III. Springer.
[4] Berndt, B. C. (1994). Ramanujan's Notebooks, Part IV. Springer-Verlag.
[5] Berndt, B. C., & Bhargava, S. (1999). Ramanujan's Theta Functions And Their Applications. Memoirs Of The American Mathematical Society, 99(477), 1–106

Abstract : This paper explores the various factors that can cause service interruptions in queueing models and their consequences on system performance. We consider a single server queueing model with interruptions. These interruptions arise from a finite number of environmental factors. For a subset of these factors, the interrup- tion is mild, allowing service to continue at a reduced rate. There is a possibility for self-correction of these mild interruptions, after which the server resumes service at the normal rate. The duration of uninterrupted service with a mild interrup- tion is measured by an interruption clock. When this clock triggers........

[1] B. Deepa, Kalidass K., Vijayalakshmi V.(2021): An M/M/1/N Queue With Working Breakdowns And Vacations-An Approach To Cost Anal- Ysis. Journal Of Physics: Conference Series, Vol.1850(1). Doi:10.1088/1742- 6596/1850/1/012069.
[2] Chen J., Yen T., Wang K.(2016): Cost Optimization Of A Single-Server Queue With Working Breakdowns Under The N Policy. Journal Of Testing And Evaluation, 44(5),2059-2067. Doi:10.1520/Jte20150080.
[3] C. D. Liou (2015): Markovian Queue Optimisation Analysis With An Unreliable Server Subject To Working Breakdowns And Impatient Customers.International Journal Of Systems Science46(12), 2165-2182. Doi:10.1080/00207721.2013.859326.
[4] C.S.Kim, Klimenok V., A. Kuznetsov (2006): A Multi-Server Queue With Negative Customers And Partial Protection Of Service. Proceedings Of The 13th International Conference On Analytical And Stochastic Modelling Techniques And Applications (Asmta06), 28-31 May 2006, Bonn, Germany, Eds. K. Al-Begain, Pp 143-148.
[5] Gaver Jr. D.P.(1962): A Waiting Line With Interrupted Service, Including Priorities. Journal Of The Royal Statistical Society,24(1),73-90. Doi:10.1111/J.2517- 6161.1962. Tb00438.X

Abstract : In this paper, we introduced the concepts of.......

[1] F. R. Jarullah , "An Orthogonal Left Centralizer And Reverse Left Centralizer Semiprime Rings", Turkish Journal Of Computer And Mathematics Education ,13(2),P.P.625-630,2022.
[2] F.R.Jarullah And S.M. Salih, "A Jordan Higher Reverse Left(Resp. Right) Centralizer On Prime Rings", Academic Journal For Engineering And Science,3(4),P.P.12-18, 2021.
[3] B. Zalar, "On Centralizers Of Semiprime Ring", Comment. Math. Univ. Carol.,32(4), P.P.609-614, 1991.