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0225U004437,
(0125U002652)
,
Analysis of the spectra of countably generated algebras of symmetric polynomials and possible applications in quantum mechanics and computer science
Субсеметричні і слабко-симетричні аналітичні функції та їх зв'язок з теорією нормальних операторів
Zahorodniuk Andrii V., д.ф.-м.н.
05-12-2025
Vasyl Stefanyk Precarpathian National University
The goal is to obtain new theoretical results in nonlinear analysis and operator theory, which can become a significant contribution to world science. New approaches to investigations of algebras of analytic invariants of symmetries on Banach spaces will be developed, and the description of algebraic and topological structures on spectra will be obtained. The results will provide new tools for applications to problems in quantum mechanics, coding, and the optimization of neural networks.
Keywords: symmetric polynomials on Banach spaces, algebras of analytic functions, spectral analysis of operators, spectrum of topological algebras, invariants of semigroups of symmetry of a Banach space, quantum ideal gas models, coding and encryption algorithms. Object of research: countably generated algebras of symmetric polynomials in Banach space, spectra. The goal is to obtain new theoretical results in nonlinear analysis and operator theory, which can become a significant contribution to world science. New approaches to investigations of algebras of analytic invariants of symmetries on Banach spaces will be developed, and the description of algebraic and topological structures on spectra will be obtained. The results will provide new tools for applications to problems in quantum mechanics, coding, and the optimization of neural networks. Research methodology. Methods of algebra, functional analysis, and theory of functions and methods of dynamical systems theory. Results and their novelty. The expected results provide answers to problems that have remained open in modern functional analysis in recent years, despite numerous attempts to solve them by the world's leading experts in this field. In particular, this concerns the problem of describing the bases of the algebra of subsymmetric polynomials both on sequence spaces and on function spaces and describing the spectra of countably generated algebras of symmetric analytic functions. The expected results of the Project are aimed at the development of new research methods in the spectral theory of operators and in applications to statistical quantum mechanics. The value of the expected results lies in the possibility of further application to applied problems of mathematics, mathematical physics, computer cryptography regarding the development of new coding and encryption algorithms, which in the future can increase the level of security of data processing, storage and transmission.
Roman I. Dmytryshyn
Andriy V. Zagorodnyuk
Taras V. Vasylyshyn
Cherneha Iryna V.
Kravtsiv Viktoriia V.
Serhii V. Sharyn
Novosad Zoriana H.
Viktoriia V. Artysh
Mariia M. Paliichuk
2025-12-05
Updated: 2025-12-06
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