Stability is a cornerstone in the analysis and design of complex dynamical systems, typically evaluated through the linearization of non-stationary system dynamics around a reference or attractive trajectory. While this process yields a linear time-varying (LTV) system—a representation central to modern frameworks such as contraction theory and adaptive control—general stability analysis for LTV systems remains a significant theoretical and practical challenge. Existing methodologies are often restricted to narrow cases, such as periodic or slowly varying systems, or rely on finding solutions to the time-dependent Lyapunov equation, which are frequently computationally demanding or analytically intractable for high-dimensional systems. This work introduces an alternative dynamical eigenstructure approach designed to derive explicit analytical solutions and define robust stability criteria for LTV systems. By generalizing classical time-invariant eigenvalue analysis, we define a dynamical eigenstructure consisting of time-varying eigenvalues and eigenfunctions that satisfy a fundamental differential dynamical equation. This unified perspective allows for the direct assessment of stability properties from the system’s time-evolving spectral characteristics without requiring the construction of a Lyapunov function. We provide a rigorous derivation of the stability conditions based on these dynamical trajectories. Finally, the effectiveness and versatility of this framework are demonstrated through several illustrative examples, showcasing its ability to provide deeper analytical insights into the transient behavior, and facilitate the stability analysis, of non-autonomous systems.

In this presentation are shown the mathematical fundamentals to obtain robust control Lyapunov functions for fractional order uncertain Lagrangian and Hamiltonian systems. As it is known, Lagrangian and Hamiltonian mathematical representations are important to model different types of physical systems such as mechanical, electrical, chemical, biological. It is important to consider the types of uncertainties in this kind of systems, but in this case this presentation is focused in multiplicative and additive uncertainties. It is important that a complete topological analysis is performed for both kinds of uncertainties. These uncertainties are studied in the case of Lagrangian and Hamiltonian systems. Then the implementation of the design methods to obtain robust control Lyapunov functions are evinced in this presentation. The robust control Lyapunov functions are designed considering the fractional order representation of the Lagrangian and Hamiltonian systems. These study finish with two case of study, one the modeling of a fractional order electrical machine and a fractional order mechanical system, in order to obtain appropriate fractional order robust control Lyapunov functions for robust control law design. In this presentation is evinced how the modeling of different types of physical systems are modeled, so the advantages of implementing the fractional order approach has advantages over the integer order one.

The development of autonomous unmanned aerial vehicle (UAV) spraying systems requires a mathematically rigorous control framework to ensure stability, efficiency, and robustness under uncertain field conditions. This study presents a hybrid mathematical–AI approach that integrates nonlinear optimization and control theory with adaptive machine learning for precision agriculture. The UAV path and spray scheduling problems are formulated as a constrained multi-objective optimization model that minimizes energy expenditure and chemical use while satisfying coverage and drift constraints. Using graph-theoretic modeling and nonlinear programming, optimal trajectories are derived that guarantee convergence toward feasible solutions under bounded uncertainty. Reinforcement learning is then employed to provide adaptive control, while maintaining mathematically verifiable stability through Lyapunov-based performance analysis. The proposed hybrid controller couples the learning dynamics of the AI model with the deterministic properties of the mathematical control law, ensuring predictable behavior even under varying canopy and wind conditions. Simulation experiments demonstrate significant improvements in deposition uniformity and chemical savings compared with conventional fixed-rate strategies. By grounding UAV spraying in the principles of optimization theory, stability analysis, and control mathematics, this work moves beyond heuristic AI control to a mathematically interpretable and generalizable framework. The study illustrates how integrating rigorous mathematical formulation with intelligent control design can advance sustainable and precision-driven agricultural automation.

This study explores a system of Yang–Baxter-type matrix equations, XAX=BXB and XBX=AXA, which generalize the classical matrix Yang–Baxter equation. This work focuses on analyzing the existence of intertwining and commuting solutions using geometric and topological methods. To support this analysis, the notions of anti-diagonally dominant matrices (ADMs) and strictly anti-diagonally dominant matrices (SADMs) are introduced. It is shown that strictly anti-diagonally dominant matrices are nonsingular, ensuring stability and uniqueness in the associated linear systems. Furthermore, if the coefficient matrices of the system satisfy the SADM condition, then an intertwining solution X exists that fulfills both Yang–Baxter-type relations. When the matrices A and B are invertible, the corresponding solution X is proved to be a commuting one. These findings extend the algebraic framework of Yang–Baxter systems and provide new insights into the dominance properties that govern the solvability of matrix equations.

The equation was formally introduced by C. N. Yang in two landmark papers published in late 1967 on a one-dimensional quantum many-body problem. Yang established the form A(u)B(u+v)A(v) = B(u)A(u+v)B(v), where A(u) and B(v) are rational functions of the spectral parameters u and v. Later, in 1972, R. J. Baxter employed the same relation while solving the eight-vertex model in two-dimensional statistical mechanics.

We define the Alpha Group as a four-dimensional real associative algebra A = spanR{1, i, μ, iμ}, with relations i² = −1, μ² = μ (idempotent invariant), and iμ = −μi (noncommutative). Under the regular representation, elements of A act as 4×4 real matrices. Division is defined projectively via right multiplication by invertible elements of A.

The angular deformation operator M(θ) is the original 4×4 matrix introduced in the Alpha Group framework, whose entries depend analytically on θ and encode the coupling between the imaginary and μ-components of the algebra. This operator governs the deformation of basis directions and induces a θ-dependent metric structure on the associated orbit space.

To study the induced topology, we construct ε-graphs over point clouds generated by iterated application of M(θ). Vertices are connected whenever the induced metric satisfies d(x,y) ≤ ε. The resulting filtration defines a Vietoris–Rips simplicial complex.

Persistent homology groups Hk for k = 0, 1, 2, 3 are computed along the filtration. For 0.4 ≤ ε ≤ 0.8, the second homology group H² exhibits sustained growth, indicating stable 2-cycles generated by the θ-dependent deformation. The third homology group H³ remains constant across the filtration, acting as a structural invariant.

A structural transition occurs near θ ≈ π/2, where the antisymmetric coupling encoded in M(θ) maximizes the generation of higher-order cycles. These results demonstrate that the algebraic structure of the Alpha Group induces a dynamically coupled topology with persistent higher-dimensional invariants, structurally distinct from classical Riemannian models.

This work presents a rigorous analysis of central modular phases in finite-dimensional Tomita–Takesaki theory and develops an explicit construction of the non-standard case in which the modular conjugation fails to be involutive. Focusing on the four-dimensional Dirac spinor algebra, we show through direct computation that the corresponding charge-conjugation operator squares to minus the identity rather than the conventional plus identity assumed in standard modular theory. Interpreting this operator as a modular conjugation demonstrates the existence of a modular structure whose square equals negative one, providing a concrete counterexample to the usual assumption that modular conjugations must be involutions.

We prove that this sign is a genuine algebraic invariant: modular conjugations with positive and negative square values cannot be related by any unitary similarity transformation, and therefore represent two fundamentally distinct modular phases. We further show that these two possibilities exhaust all central modular phases in finite-dimensional fermionic systems, yielding a complete binary classification. The construction is fully explicit, representation-independent, and formulated within the standard algebraic setting.

Finally, we extend the analysis to pseudo-Hermitian quantum systems by defining a twisted Tomita operator that respects closability and admits a well-defined polar decomposition. This establishes that non-standard modular phases persist in modified inner-product settings and remain compatible with the general modular framework. The results provide a mathematically controlled foundation for modular structures beyond the Hermitian paradigm and contribute new algebraic invariants relevant to modern mathematical physics.

*-envelopping algebras of the functionals of stochastic automata from topological subshifts are newly written as the restriction of all the functions of isometric isomorphism on the weak open set. The Margulis Theory is here applied to theweak open set: the sigma-algebra generated from the weak topology coincides when transforming a strong topology: the original functional is smooth (therefore, a homeomorphism in the weak topology is a diffeomorphism in strong topology). The Frechet operator is descended to ordinary differentials in finite-dimensional spaces, after which the measure from the functional is written based on the Radon-Nikodym derivative dμ/dv: the transition probabilities are the derivatives of the functionals, which are density functions with respect to the measure v. The Frechet operator is a map that is used to define isomorphisms between Frechet spaces; after having constructed the isomorphism using the weak *-topology in a dual space, this map is used to transfer the isomorphism to a weak-topology-norm space. The isometric isomorphism is a specification of the Rokhlin aperiodic endomorphism. The non-empty compact space, the distance to it, and the functional constitute a triplet, which forms a Smale space. The newly found dynamical system is an integrable dynamical system of newly 'weakened' hyperbolicity; this dynamical system is compared with that of the Ruelle diffeomorphism [D. Ruelle, Non-commutative algebras for hyperbolic diffeomorphisms, Invent. Math. 93, 1-13(1988).] when the Sinai–Ruelle–Bowen measure is substituted with the new opportune uniform-entropy measure. The new corresponding Miatello–Wallach family of meromorphic functions is written: the variances are indicated to be calculated after application of generic potential theory. The star-algebras are proven to be reduced to envelopping algebras, to which the Tomita–Takesaki paradigm is proven to apply.

Introduction: We develop a spectral–topological framework linking the spectral radius of data matrices to the emergence of higher homotopy groups in associated simplicial complexes. Using a Hurewicz-type principle, we show that spectral growth governs both the birth and collapse of higher-order topological structure. The approach provides a computationally tractable alternative to persistent homology, with rigorous results for low dimensions and principled conjectures in general.

Methods: Given a sequence of symmetric matrices A_n, we constructed clique complexes X_n via thresholding. Spectral radius ρ(A_n) was used as the governing control parameter. Algebraic-topological tools (Hurewicz theorem, homology–homotopy correspondence) and probabilistic asymptotics were combined to study limiting behavior as n → ∞.

Results: Theorem 1 Spectral Threshold for π₂: There exist deterministic thresholds 0 < ρ₂^– < ρ₂⁺ such that π₂(X_n) is nontrivial with high probability if and only if ρ(A_n) ∈ (ρ₂^–, ρ₂⁺). Full formal proof will be provided in the paper.

Theorem 2 Limiting Law for Homotopy Rank: For k = 2, n^-1rank(π₂(X_n)) converges in probability to a continuous function of ρ(A_n). Full formal proof will be given in the paper.

Conjecture 1: Spectral thresholds (ρ_k^–, ρ_k⁺) exist ∀k ≥ 2.

Conjecture 2: A deterministic limiting law holds for n^-1rank(π_k).

Conjecture 3: Centered homotopy ranks satisfy asymptotic normality.

Conclusion: The results establish spectral radius as a unifying scalar invariant governing higher-order topology. The framework connects spectral graph theory, Hurewicz-type arguments, and random topology, opening a path toward scalable inference of homotopy without full persistence. Related frontier work includes Kahle (random complexes), Linial–Meshulam models, and recent spectral-TDA (topological data analysis) connections in applied topology.

This article introduces the concept of (α, β)-intuitionistic fuzzy positive implicative ideals in BCK-algebras, a novel extension of fuzzy set theory applied to algebraic structures. BCK-algebras are a class of algebraic structures that have been widely studied in the context of fuzzy logic and fuzzy set theory, with applications in information sciences, decision-making processes, and artificial intelligence. We utilize the relationships that belong to (∈) and quasi-coincidence (q) between intuitionistic fuzzy points and intuitionistic fuzzy sets, where α and β can be any of {∈, q, ∈ ∨ q, ∈ ∧ q} except ∈ ∧ q. This approach allows for a more nuanced and flexible treatment of fuzzy ideals in BCK-algebras, encompassing various existing concepts as special cases. The proposed notion generalizes various existing concepts in fuzzy BCK-algebras, providing a unified framework for studying implicative ideals. Key properties of (α, β)-intuitionistic fuzzy positive implicative ideals are investigated, including characterization theorems and relationships with other types of fuzzy ideals in BCK-algebras. We establish several important results, including necessary and sufficient conditions for an intuitionistic fuzzy set to be an (α, β)-intuitionistic fuzzy positive implicative ideal, and examine the relationships between these ideals and other types of fuzzy ideals, such as fuzzy implicative ideals and fuzzy positive implicative ideals. Several examples are provided to illustrate the concepts and demonstrate their significance, highlighting the applicability and relevance of the proposed notion. This work aims to enrich the theory of fuzzy BCK-algebras and contribute to the broader field of fuzzy algebraic structures, with potential applications in information sciences and decision-making processes. The introduced concepts and results are expected to stimulate further research in this direction, providing new avenues for exploration and discovery in fuzzy mathematics and its applications.

The notion of reflexivity is one of the very well-known notions of ring theory, which was introduced by Meson [Comm. Algebra, 9, 1709-1724 (1981)]. He called a ring R reflexive if aRb=0 implies bRa=0 for any a, b ∈ R. During the period, many ring theorists studied and generalized the notion of reflexive rings. In 2015, Chakraborty [Asian-European J. Math., 8, 1550003:1-15, (2015)] introduced the notion of central reflexive rings as a generalization of reflexive rings. He called a ring R central reflexive if aRb = 0 implies bRa ⊆ C(R) for any a, b ∈ R, where C(R) denotes the set of all central elements of R. Recently, Calci et al. [Math. Bohemica, 149, 225-235 (2024)] introduced the notion of J-reflexive rings as a generalization of central reflexive rings. He called a ring R J-reflexive if aRb = 0 implies bRa ⊆ J(R) for any a, b ∈ R, where J(R) denotes the Jacobson radical of R.

Here, we introduce the notion of H-reflexive rings, which lies strictly between the classes of central reflexive rings and J-reflexive rings. In support, we give some examples and counterexamples. We find that the notions of reflexive, central reflexive, H-reflexive, and J-reflexive rings are equivalent to the class of J-semisimple rings; and the notions of H-reflexive and central reflexive rings are equivalent to the class of rings with no nil ideals. We prove that a ring R is H-reflexive if and only if, for every a, b ∈ R, < a >< b >= 0 implies < b >< a > ⊆ T(R), where T(R) denotes the hypercenter of R. We also give some characterizations of H-reflexive rings with the help of their extension rings. Finally, we discuss some of their basic properties and their relationship with the class of Armendariz rings.