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...
Show More
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 ...
Show More
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
Downloads:
Views:
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...
Show More
