Sequence spaces are mathematical structures that play a pivotal role in studying functional analysis, topology, and sequence theory. These spaces consist of sequences of elements from a given set, typically the set of real or complex numbers, equipped withspecific topological or algebraic properties. It explores sequence spaces defined by the statistical convergence of fuzzy numbers, focusing on the development and analysis of new sequence spaces that extend classical sequence spaces in the context of fuzzyset theory. Employing a difference operator, furthermore, provides a sequence space of fuzzy numbers, F (c) I (S) and F(c) I (S)0, determined via I-statistical convergence. Research investigates the basic algebraic and topological features of these spaces, offering athorough examination of their structural features. Additionally, it explores crucial links related to these spaces, including symmetry, solidity, and convergence-free features, and it establishes several significant inclusion outcomes. The research advances knowledgeof I-statistical convergence in fuzzy number sequence space by expanding on traditional ideas and providing guidance on using them in fuzzy set theory and uncertainty-related fields.

The study of BCK/BCI algebras was initiated by Imai Iseki in 1966. These concepts of two classes of abstract algebras studied by eminent authors. Here we study properties of a non-singular BCI algebra which has been introduced by Prasad and Abid under the semi commutative BCI-algebra.

Classical sequence spaces such as l∞ , c, and c0 have been extensively studied and recognized as fundamental in the advancement of functional analysis and related areas of mathematics. In this article, we investigate the algebraic properties together with para- norm structures of the sequence spaces W0 (∆, f )(C2), W (∆, f )(C2), and W∞ (∆, f )(C2) in a bi-complex setting, which are induced by a non-negative real-valued function ϕ.

The important effort of Pawlak established Rough Set Theory to robust paradigm for handling uncertainty by defining approximation of concepts within an equivalence-based approximation space . Subsequent research has explored the connections between rough sets, algebraic structures, and topology. Though, a unified framework that seamlessly integrates these three domains for complex algebraic structures like rings and fields remains largely undeveloped. This paper introduce a novel hybrid structure, the Topological-Rough Algebraic Framework, which generalizes the concept of a topological approximation space to the realm of rings and fields. We construct an approximation space ) where the equivalence relation is induce for Ideal of a Ring , partitioning to Ring into cossets. This partition naturally serves as a aim for a Topol gy to , leading to the integrated triple ), Within the proposed framework we investigate lower and upper approximation of subsets, characterizing the conditions under which a set is exact or rough. We further analyze key topological properties of the space ) establishing their relationships with the underlying algebraic properties of the ring and its ideals. The framework not only provides a theoretical generalization of existing models but also demonstrates practical applicability in coding theory, cryptographic analysis, and data mining. This work bridges a significant gap between algebraic structures, rough set theory, and general topology, offering a powerful new lens for mathematical analysis.

In this paper, the concept of exponentially $m$-isometry \cite{Hedayatian} on a Hilbert space is generalized when an additional semi-inner product is considered. We present a comprehensive study of the algebraic properties and characterizations of operators within this extended class. Furthermore, we explore the dynamical behavior of these operators and conclude with an analysis of their spectral properties.

Units of measure with prefixes and conversion rules are given a formal semantic model in terms of categorial group theory. Basic structures and both natural and contingent semantic operations are defined. Conversion rules are represented as a class of ternary relations with both group-like and category-like properties. A hierarchy of subclasses is explored, each satisfying stronger useful algebraic properties than the preceding, culminating in a direct efficient conversion-by-rewriting algorithm.

This paper presents new aggregation operators for p,q,r-fractional fuzzy sets based on the Frank t-norm and t-conorm. We introduce the p,q,r-fractional fuzzy Frank weighted average and p,q,r-fractional fuzzy Frank weighted geometric operators and discuss their algebraic properties, including closure, boundedness, idempotency, and monotonicity. Based on new operations, we develop a multi-criteria group decision-making framework that integrates the evaluations of multiple experts via the proposed Frank operators and ranks the alternatives under p,q,r-fractional fuzzy information. The model is applied to a cryptocurrency stability assessment problem, where four coins are evaluated with respect to six criteria. The results show that both aggregation operators yield consistent rankings with good discriminatory power among the alternatives. A sensitivity analysis is conducted to check the stability of the model under parameter variations. A comparative study further demonstrates the compatibility and advantages of the proposed method over several existing decision-making approaches. The proposed framework is well suited to decision-making scenarios in which multiple experts’ opinions must be integrated within a complex fuzzy information environment.

Simulating vibrational dynamics is essential for understanding molecular structure, unlocking useful applications such as vibrational spectroscopy for high-fidelity chemical detection. Quantum algorithms for vibrational dynamics are emerging as a promising alternative to resource-demanding classical approaches, but this domain is largely underdeveloped compared with quantum simulations of electronic structure. In this work, we describe in detail three distinct forms of the vibrational Hamiltonian: canonical bosonic quantization, real space representation, and the Christiansen second-quantized form. Leveraging the Lie algebraic properties of each, we develop efficient fragmentation schemes to enable the use of Trotter product formulas for simulating time evolution. We introduce circuits required to implement time evolution in each form and highlight factors that contribute to the simulation cost, including the choice of vibrational coordinates. Using a perturbative approach for the Trotter error, we obtain tight estimates of the T gate cost for the simulation of time evolution in each form, enabling their quantitative comparison. Combining tight Trotter error estimates and efficient fragmentation schemes, we find that for the CH4 molecule with 9 vibrational modes, time evolution for approximately 1.8 ps may be simulated using as few as 36 qubits and approximately 3 × 108 T gates─an order-of-magnitude speedup over state-of-the-art algorithms. Finally, we present calculations of vibrational spectra using each form to demonstrate the fidelity of our algorithms. This work presents a unified and highly optimized framework that makes simulating vibrational dynamics an attractive use case for quantum computers.

Algebraic properties of geometric-type operator means and related centrality characterizations

By using the geometric and algebraic properties of Bernstein polynomials, a composite method for solving the initial value problems (IVP's) with first-order, singular and nonlinear ordinary differential equations (ODE's) has been developed. The Newton’s method is incorporated into the method to solve the resulting system of nonlinear equations. The algorithm of the problem solving is reduced to the calculation of the unknown Bernstein coefficients of the approximate solution. The effectiveness of the proposed method is verified by comparing the present numerical results by other existing ones. The proposed method reduces the computation cost and gives a better approximation to the exact solution even for small degrees of approximation. Another advantage of the present method is the ability to calculate the approximate solution at each point of the solution interval in addition to the grid points.

In this article, we have introduced the notion of generalized difference sequence spaces within the framework of bi-complex numbers. Further we have established their algebraic, topological and geometric properties including some inclusion relations.

ABSTRACT To achieve an efficient time‐frequency representation of higher‐dimensional signals, we introduce the notion of the Clifford‐valued bendlet transform, utilizing the geometric and algebraic properties intrinsic to bendlets and Clifford algebras. The bendlet system, a second‐order shearlet with bent elements, enables the precise characterization of the curvature of discontinuities. Clifford algebras, meanwhile, convert geometric objects into fundamental computational elements and define universal operators applicable to all types of geometric elements. In this work, we rigorously investigate the essential properties of the proposed transform using operator theory and Clifford‐valued Fourier transforms. We present a detailed analysis of the convergence of inversion formulae and the boundedness of the associated localization operators. Furthermore, we thoroughly explore certain classes of uncertainty principles for the Clifford‐valued bendlet transform. Our findings establish a robust theoretical foundation for the practical application of this transform in multidimensional signal processing.

While recent years have witnessed a fast growth in mathematical artificial intelligence (AI). One of the most successful mathematical AI approaches is topological data analysis via persistent homology (PH) that provides explainable AI by extracting multiscale structural features from complex datasets. Interpretability is crucial for world models, the new frontier in AI that can understand and simulate reality. This article investigates the interpretability and representability of three foundational mathematical AI methods, PH, persistent Laplacians (PL) derived from topological spectral theory, and persistent commutative algebra (PCA) rooted in Stanley–Reisner theory. We apply these methods to a set of data, including geometric shapes, synthetic complexes, fullerene structures, and biomolecular systems to examine their geometric, topological, and algebraic properties. PH captures topological invariants such as connected components, loops, and voids through persistence barcodes. PL extends PH by incorporating spectral information, quantifying topological invariants, geometric stiffness, and connectivity via harmonic and nonharmonic spectra. PCA introduces algebraic invariants such as graded Betti numbers, facet persistence, and ‐vectors, offering combinatorial, topological, geometric, and algebraic perspectives on data over scales. Comparative analysis reveals that while PH offers computational efficiency and intuitive visualization, PL provides enhanced geometric sensitivity, and PCA delivers rich algebraic interpretability. Together, these methods form a hierarchy of mathematical representations, enabling explainable and generalizable AI for real‐world data.

Let 𝜏 be a faithful normal semifinite trace on a von Neumann algebra ℳ. The block projection operator 𝒫̃︀𝑛 (𝑛 ≥ 2) on the *-algebra 𝑆(ℳ, 𝜏 ) of all 𝜏 -measurable operators is investigated. It is shown that 𝑓(𝒫̃︀𝑛(𝐴)) ≥ 𝒫̃︀𝑛(𝑓(𝐴)) for any operator monotone function 𝑓 on R + and 𝐴 ∈ 𝑆(ℳ, 𝜏 ) + . For an operator convex function 𝑓 on R + , we have 𝑓(𝒫̃︀𝑛(𝐴)) ≤ 𝒫̃︀𝑛(𝑓(𝐴)) for 𝐴 ∈ 𝑆(ℳ, 𝜏 ) + . Conditions are established under which 𝒫̃︀𝑛(𝐴) belongs to the class 𝑆0(ℳ, 𝜏 ) of 𝜏 -compact operators, to the class 𝐹(ℳ, 𝜏 ) of elementary operators, to the classes 𝐿𝑝(ℳ, 𝜏 ) of operators 𝜏 -integrable with 𝑝-th power, or to the ℳ algebra itself. If 𝐴, 𝐵 ∈ 𝑆(ℳ, 𝜏 ) and 𝒫̃︀𝑛(𝐵) is a left (right) inverse for the operator 𝐴, then 𝒫̃︀𝑛(𝐵) is also a left (respectively, right) inverse for the operator 𝒫̃︀𝑛(𝐴).

Algebraic Properties of the Primitive Idempotent in Clifford Analysis

We study forcing properties of the Boolean algebras P(ω)/I, where I is a Borel ideal on ω. We show (Theorem 2.12) that (under a large cardinal hypothesis) P(ω)/I does not add reals if and only if it has a dense σ-closed subset. For analytic P-ideals I we show (Theorem 3.3) that either P(ω)/I is ωω-bounding or it is not proper. We also investigate the existence of completely separable I-MAD families.

This article introduces and investigates a new integer sequence, termed the higher-order Mersenne sequence, defined in analogy with higher-order Fibonacci numbers and closely related to classical Mersenne numbers. We establish a range of fundamental algebraic properties of this sequence, including its Binet-type formula, Catalan’s identity, d’Ocagne’s identity, generating function, and several finite and binomial summation identities. Further, we explore its connections with both Mersenne and Jacobsthal numbers. The study also examines the sequence obtained via the binomial transform of higher-order Mersenne numbers, deriving its recurrence relation and algebraic characteristics. In addition, matrix generators and a tridiagonal matrix representation are developed to enrich the structural understanding of these numbers.

We investigate certain weak conditions that yield Fritz John necessary optimality conditions in smooth vector optimization. By leveraging specific properties of the positive dual of a closed and convex cone, we improve upon several results found in the existing literature. These methods are then extended to set optimization through the introduction of a special type of differential tailored for set-valued maps whose values exhibit a particular structure. This approach relies on an embedding procedure that allows sets with specific topological and algebraic cone properties to be represented as points in a normed vector space. Additionally, this study provides the opportunity to offer an important clarification regarding the concept of so-called cone-compact set.

This paper investigates the steady gas flow in a finite-length straight nozzle governed by two-dimensional non-isentropic compressible Euler equations with heat transfer effects, known in engineering as Rayleigh flow. We formulate a boundary value problem derived from transonic shocks and construct a class of non-trivial subsonic flow special solutions in a constant cross-section straight nozzle. By proving the structural stability of this special solution under two-dimensional perturbations of upstream flow and downstream back pressures satisfying certain symmetry conditions, the rationality of the proposed boundary value problem is demonstrated. For subsonic flows, the two-dimensional steady compressible Euler equations represent typical quasi-linear elliptic-hyperbolic complex systems. We present a decomposition lemma for the mathematical model, which effectively separates the elliptic and hyperbolic modes in the steady subsonic Euler equations. It is shown that heat transfer effects introduce significant interactions between elliptic and hyperbolic modes, leading to a class of second-order elliptic equations with multiple integral-type non-local terms. The solution to the linearized problem is obtained by analyzing the algebraic and analytical properties of infinitely weakly coupled boundary value problems for integro-differential equations via Fourier analysis methods.

Abstract The dissertation consists of an introductory chapter and eight published articles centered around the following topics: (1) calibration of the interpretability strength of weak first-order theories; (2) determination of the (un)decidability of fine-grained fragments of the intended models of the aforementioned theories; and (3) investigation of algebraic properties of the lattice of interpretability degrees of computably enumerable essentially undecidable theories. 1. J. Murwanashyaka, A weak theory of building blocks. Mathematical Logic Quarterly , vol. 70 (2024), pp. 233–254. doi: 10.1002/malq.202300015 2. J. Murwanashyaka, Hilbert’s tenth problem for term algebras with a substitution operator. Computability , vol. 13 (2024), nos. 3–4, pp. 433–457. doi: 10.3233/COM-230444 3. J. Murwanashyaka, Weak essentially undecidable theories of concatenation, part II. Archive for Mathematical Logic , vol. 63 (2024), pp. 353–390. doi: 10.1007/s00153-023-00898-y 4. J. Murwanashyaka, F. Pakhomov, and A. Visser, There are no minimal essentially undecidable theories. Journal of Logic and Computation , vol. 34 (2024), no. 6, pp. 1159–1171. doi: 10.1093/logcom/exad005 5. J. Murwanashyaka, Weak essentially undecidable theories of concatenation. Archive for Mathematical Logic , vol. 61 (2022), nos. 7–8, pp. 939–976. doi: 10.1007/s00153-022-00820-y . 6. J. Murwanashyaka, Weak sequential theories of finite full binary trees, Revolutions and Revelations in Computability (U. Berger, J. N. Y. Franklin, F. Manea, and A. Pauly, editors), CiE 2022, Lecture Notes in Computer Science, Vol. 13359, Springer International Publishing, 2022, pp. 208–219. doi: 10.1007/978-3-031-08740-0_18 . 7. L. Kristiansen and J. Murwanashyaka, First-order concatenation theory with bounded quantifiers. Archive for Mathematical Logic , vol. 60 (2021), nos. 1–2, pp. 77–104. doi: 10.1007/s00153-020-00735-6 . 8. L. Kristiansen and J. Murwanashyaka, On interpretability between some weak essentially undecidable theories, Beyond the Horizon of Computability (M. Anselmo, G. Della Vedova, F. Manea, and A. Pauly, editors), CiE 2020, Lecture Notes in Computer Science, Vol. 12098, Springer International Publishing, 2020, pp. 63–74. doi: 10.1007/978-3-030-51466-2_6 . Abstract prepared by Juvenal Murwanashyaka E-mail : murwanashyaka@math.cas.cz URL : http://hdl.handle.net/10852/104677