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Research Article
On the Conditions of Ensuring Uniform Almost Periodic Functions of the Class of Entire Functions
Talbakzoda Farhodjon Mahmadsho*
Issue:
Volume 14, Issue 5, October 2025
Pages:
106-113
Received:
28 May 2025
Accepted:
13 June 2025
Published:
9 September 2025
Abstract: The study of almost periodic functions occupies an important place in functional analysis and the theory of differential equations, beginning with the classical works of H. Bohr, A. S. Besikovich and B. M. Levitan. Almost periodic functions, being a generalization of periodic functions, are characterized by the fact that they retain their structure under shifts, without being strictly periodic. On the other hand, entire functions are functions of a complex variable that are analytic in the entire complex plane. Their behavior, especially their growth and the location of their zeros, is studied in detail in the theory of functions of a complex variable. Of particular interest is the study of entire functions whose values on the real axis are almost periodic in the sense of Bohr. The question of under what conditions an entire function takes on values on the real axis that form a uniformly almost periodic function is a non-trivial problem at the intersection of function theory and spectral analysis. Such conditions can be formulated through the properties of the spectrum of the function, through the conditions on the coefficients of the Fourier series, and also through the growth properties of the function itself. These functions find application in spectral theory, quantum mechanics, oscillation theory, and other areas of mathematics and physics. In this section, we study the problems of approximation of functions f(x)∈B by entire functions of finite degree with arbitrary Fourier exponents. We establish necessary and sufficient conditions for functions f(x)∈B to belong to the class of entire functions of bounded degree.
Abstract: The study of almost periodic functions occupies an important place in functional analysis and the theory of differential equations, beginning with the classical works of H. Bohr, A. S. Besikovich and B. M. Levitan. Almost periodic functions, being a generalization of periodic functions, are characterized by the fact that they retain their structure...
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Research Article
Machine Learning (ML) and Artificial Intelligence (AI) Approaches to Unstructured Data
Farha Khan*,
Pratima Ojha,
Ghizal Firdous Ansari
Issue:
Volume 14, Issue 5, October 2025
Pages:
114-119
Received:
20 July 2025
Accepted:
12 August 2025
Published:
25 September 2025
Abstract: This study explores the application of machine learning (ML) and artificial intelligence (AI) techniques to analyze unstructured textual data, focusing on topic modeling, sentiment detection, and behavioral prediction. We employ multinomial document models and unsupervised learning strategies to extract latent topics and evaluate the emotional and conversational drivers behind social media posts. A major contribution is the implementation of Behavior Dirichlet Probability Model (BDPM) which analyzes user moods and behaviors through unstructured textual data. The results validate the hypothesis of the model's ability to identify and guess behavior patterns with high accuracy, providing actionable insights for digital marketing strategies, techniques to enhance user interaction and mental wellness evaluation.
Abstract: This study explores the application of machine learning (ML) and artificial intelligence (AI) techniques to analyze unstructured textual data, focusing on topic modeling, sentiment detection, and behavioral prediction. We employ multinomial document models and unsupervised learning strategies to extract latent topics and evaluate the emotional and ...
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Research Article
Computational Models for (M, K)-Quasi-*-Parahyponormal Operators
Issue:
Volume 14, Issue 5, October 2025
Pages:
120-129
Received:
7 August 2025
Accepted:
25 August 2025
Published:
25 September 2025
DOI:
10.11648/j.pamj.20251405.13
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Abstract: We study the two-parameter class of (M, k)-Quasi-∗-Parahyponormal operators on separable Hilbert spaces, which strictly enlarges the traditional parahyponormal and paranormal hierarchies. Analytically we prove three fundamental results: Every operator in the class has finite ascent and enjoys the single-valued extension property (SVEP); The Browder–Weyl partition holds, so Weyl’s theorem is valid; A non-trivial closed invariant subspace exists whenever the commutant contains a non-zero compact element. Complementing these theorems, we introduce a proposed computational framework that realises the abstract operators as large weighted-shift matrices, verifies the defining quadratic inequality, and computes eigenvalues as well aaccelerated pseudospectra. Together, the analytic results and the computational framework deepen the spectral theory of (M, k)-Quasi-∗-Parahyponormal Operators and supply the first large-scale numerical evidence for their structural properties.
Abstract: We study the two-parameter class of (M, k)-Quasi-∗-Parahyponormal operators on separable Hilbert spaces, which strictly enlarges the traditional parahyponormal and paranormal hierarchies. Analytically we prove three fundamental results: Every operator in the class has finite ascent and enjoys the single-valued extension property (SVEP); The Browder...
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Research Article
Pointwis Biflatness as an Extension of Banach Algebras
Majid Ghorbani*,
Davood Ebrahimi Bagha
Issue:
Volume 14, Issue 5, October 2025
Pages:
130-134
Received:
1 September 2025
Accepted:
12 September 2025
Published:
22 October 2025
DOI:
10.11648/j.pamj.20251405.14
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Abstract: In this paper, we introduce and systematically study the concept of pointwise biflatness in Banach algebras, which generalizes classical biflatness by localizing the homological structure to individual elements. Unlike global biflatness, this localized approach captures finer algebraic and module-theoretic behaviors that remain invisible under classical definitions. We prove that a pointwise biflatBanach algebra is super-amenable if and only if it possesses an identity element, providing a precise criterion linking local biflatness with classical amenability. Additionally, we explore the interrelations between pointwise biflatness, pointwise flatness, and pointwise amenability, clarifying their mutual influence and delineating boundaries between local and global homological properties. Applications to group algebras and classical Banach algebras are presented, illustrating scenarios where global biflatness fails but pointwise biflatness holds. These results provide concrete examples, highlight the practical relevance of the localized approach, and establish a foundation for further study in operator algebras, Segal algebras, and harmonic analysis. This study opens new directions for theoretical research and offers refined tools for understanding module structures, cohomologicalbehaviors, and approximation properties in Banach algebras.
Abstract: In this paper, we introduce and systematically study the concept of pointwise biflatness in Banach algebras, which generalizes classical biflatness by localizing the homological structure to individual elements. Unlike global biflatness, this localized approach captures finer algebraic and module-theoretic behaviors that remain invisible under clas...
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Research Article
Same Values Analysis Attack over Binary Elliptic Curves
Issue:
Volume 14, Issue 5, October 2025
Pages:
135-156
Received:
6 August 2025
Accepted:
21 August 2025
Published:
22 October 2025
DOI:
10.11648/j.pamj.20251405.15
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Abstract: In this work, we are interested in studying a particular class of Side Channel Attacks on elliptic curves defined over binary fields. Side Channel Attacks exploit physical leakages such as power consumption or electromagnetic emanations during cryptographic computations in order to recover secret information. Among these attacks, the one we focus on is known as Same Values Analysis (SVA). This method does not rely directly on distinguishing the sequence of operations, but rather on detecting situations where different inputs lead to identical intermediate values inside the formulas used for point addition and point doubling. Since these formulas are usually well known and publicly available, an adversary can exploit such collisions in order to reveal sensitive information, in particular the secret scalar used during scalar multiplication. The objective of our study is therefore to identify the points on elliptic curves that produce identical intermediate variables during addition or doubling steps, and to determine the algebraic conditions under which such coincidences occur. By analyzing these conditions, one can highlight vulnerabilities in scalar multiplication algorithms and evaluate their potential exploitation. We will focus our investigation on three important families of elliptic curves over binary fields: Weierstrass curves, Edwards curves, and Hessian curves. For completeness, we will also verify the effectiveness of the SVA attack on standardized curves recommended by NIST and SECG. The results of our analysis clearly show that all these curves are vulnerable to Same Values Analysis, which implies that their use in cryptographic protocols requires a careful reassessment of security guarantees.
Abstract: In this work, we are interested in studying a particular class of Side Channel Attacks on elliptic curves defined over binary fields. Side Channel Attacks exploit physical leakages such as power consumption or electromagnetic emanations during cryptographic computations in order to recover secret information. Among these attacks, the one we focus o...
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Research Article
On a Family of Congruent Numbers Defined Modulo 72: A Conjectural Study
Issue:
Volume 14, Issue 5, October 2025
Pages:
157-160
Received:
14 September 2025
Accepted:
18 October 2025
Published:
22 October 2025
DOI:
10.11648/j.pamj.20251405.16
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Abstract: We investigate the interplay between the Chinese Remainder Theorem and the theory of congruent numbers through a modular approach to expressing integers as sums of three cubes. By analyzing congruence systems arising from specific residue classes modulo 8 and modulo 9, we classify the possible integers likely to be representable as sums of cubes based on their modular residues. We also explore computational methods applying this modular framework to identify explicit cube decompositions within these classes.
Abstract: We investigate the interplay between the Chinese Remainder Theorem and the theory of congruent numbers through a modular approach to expressing integers as sums of three cubes. By analyzing congruence systems arising from specific residue classes modulo 8 and modulo 9, we classify the possible integers likely to be representable as sums of cubes ba...
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