Research Article
An Innovative Result on the Radius of Convexity and Uniform Convexity of Normalised Bessel Function
Natarajan Saraswathi*
,
Murugesan Kasthuri
Issue:
Volume 14, Issue 3, June 2026
Pages:
115-119
Received:
4 April 2026
Accepted:
20 April 2026
Published:
30 April 2026
Abstract: This paper investigates the geometric properties of normalised Bessel functions of the first kind, focusing on the radii of convexity and uniform convexity. Using analytic techniques such as logarithmic differentiation and properties of zeros of the Bessel function. Additionally, we provide new proofs related to the order convexity and compare our results with existing results to reveal interesting relationships, and we provide a graphical interpretation that supports the analytical findings.
Abstract: This paper investigates the geometric properties of normalised Bessel functions of the first kind, focusing on the radii of convexity and uniform convexity. Using analytic techniques such as logarithmic differentiation and properties of zeros of the Bessel function. Additionally, we provide new proofs related to the order convexity and compare our ...
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Research Article
Zia Theory of Temporal Reality (ZTTR): A Unified Deterministic–Probabilistic Model of Time and Reality
Md. Ziaur Rahman*
Issue:
Volume 14, Issue 3, June 2026
Pages:
120-128
Received:
18 April 2026
Accepted:
3 May 2026
Published:
18 May 2026
DOI:
10.11648/j.ajam.20261403.12
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Abstract: Time and reality are central concepts in both philosophy and physics. While classical mechanics describes the universe as deterministic, where the present emerges from past conditions, quantum theory introduces probabilistic interpretations suggesting inherent uncertainty at microscopic scales. These two perspectives have led to an ongoing debate regarding whether reality is fundamentally deterministic, probabilistic, or a combination of both. The Zia Theory of Temporal Reality (ZTTR) integrates these frameworks by proposing that the present state of reality is determined by accumulated past causes, while the future evolves probabilistically under uncertainty. To formalize this concept, the Zia Unified Continuity Equation (ZUCE) is expressed as a stochastic model where the state of reality represents a combination of initial conditions, integrated causal influences, and stochastic fluctuations represented by a Wiener process. To enhance generality and address limitations of fixed functional assumptions, the uncertainty term is further extended as a generalized function, allowing different forms such as diffusion-like, logarithmic, or nonlinear growth depending on system dynamics. In this framework, deterministic causal accumulation represents the structured continuity of past influences, while the stochastic component reflects uncertainty arising from complexity, interaction, and incomplete knowledge of system variables. This interpretation aligns with developments in statistical mechanics, information theory, and complex systems research. The applicability of the model is illustrated through conceptual and numerical examples drawn from physical processes, biological systems, and learning dynamics. These examples demonstrate that deterministic historical structures and stochastic variations can coexist within a single temporal framework. The Zia Theory of Temporal Reality thus provides a generalized and mathematically consistent perspective on temporal evolution and contributes to interdisciplinary research in physics, complexity science, and artificial intelligence.
Abstract: Time and reality are central concepts in both philosophy and physics. While classical mechanics describes the universe as deterministic, where the present emerges from past conditions, quantum theory introduces probabilistic interpretations suggesting inherent uncertainty at microscopic scales. These two perspectives have led to an ongoing debate r...
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Research Article
The First Notions of Descriptive Statistics Applications with EXCEL, EVIEWS and R STUDIO
Alain Mbayo Yamwembo*
Issue:
Volume 14, Issue 3, June 2026
Pages:
129-138
Received:
23 February 2026
Accepted:
21 April 2026
Published:
21 May 2026
DOI:
10.11648/j.ajam.20261403.13
Downloads:
Views:
Abstract: This paper defines and explains, in simple and clear terms, key concepts of descriptive statistics and connects statistical theory with applied statistical practice through real cases. It begins by clarifying the meaning of statistics and the basic notions used daily in many fields, while distinguishing statistics as a science and an art from statistics understood as data and measures. The study adopts the following definition: “Statistics is simply an art and a science that allows for the collection, analysis, presentation, and interpretation of data for the purpose of making decisions about any phenomenon under study.” To illustrate these concepts, practical applications were carried out using real data from two institutions: AL BANK and a non-governmental organization. For the calculations and applications, EXCEL, EVIEWS, and R STUDIO were used. In the AL BANK case, relative frequencies were calculated step by step with the three software programs and then verified manually using the mathematical frequency formula. In Excel, the process involved entering the data, computing the total number of observations, and calculating the relative frequencies. In EVIEWS, the observations were entered, descriptive statistical measures including the total were obtained, and relative frequencies were then calculated. In R STUDIO, the data were entered as a vector and relative frequencies were computed. The three methods produced identical results, showing that 66% of AL BANK employees are single, 31% are married, 2% are widowed, and 1% is divorced. In the non-governmental organization case, the variable “business opening” was used to calculate cumulative frequencies. The procedure consisted of constructing the absolute frequency table, the relative frequency table, and then the cumulative absolute and cumulative relative frequency tables. The analysis showed that, in terms of cumulative absolute frequency, ten individuals reported that they sometimes or often open their businesses. Regarding relative frequencies, six out of sixteen individuals reported often opening their businesses (37.5%), while 25% sometimes open their businesses, 25% rarely open them, and 12.5% never open them. These findings highlight the importance of frequency as the one of the fundamental concepts of applied statistics.
Abstract: This paper defines and explains, in simple and clear terms, key concepts of descriptive statistics and connects statistical theory with applied statistical practice through real cases. It begins by clarifying the meaning of statistics and the basic notions used daily in many fields, while distinguishing statistics as a science and an art from stati...
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