The Yang–Baxter equation is one of the most celebrated equations in science. There exist some correspondences between Yang–Baxter operators and Hopf algebras. For example, the quasitriangular Hopf algebras are used to define ribbon Hopf algebras and knot invariants. Finding new solutions for the Yang–Baxter equation and relating it to other algebraic structures is of great interest for scientists, as the classification of its solutions is out of reach for the moment. In the current paper, the new solutions arise from Boolean algebras and from racks. Our new solutions of the Yang–Baxter equation are generalizations of well-known solutions and they are related to the Euler formula. There exist self-dual Yang–Baxter operators as well as self-dual geometry theorems. There exists a partial dualization in the definition of entwining structures and there exists a partial dualization in our new theorem. Finding geometric representations for the phenomena from the Yang–Baxter theory is very useful at conferences and congresses. In geometry, the Morley’s trisector theorem was proved by the Fields medalist Alain Connes. We propose a similar theorem, which is a many circles version of a famous theorem. Connecting celebrated topics, the current paper also deals with an Euler formula proof for the original Johnson–Tzitzeica theorem.
Motivated by sparse principal component analysis (PCA), this paper proposes ATE-ManPG, a safeguarded transport-extrapolated proximal gradient method for minimizing the sum of a smooth function and a convex, possibly nonsmooth function over the Stiefel manifold. The method augments the manifold proximal gradient method (ManPG) with two ingredients: momentum obtained by transporting the previous proximal search direction to the current tangent space, protected by a monotone restart safeguard, and a bounded adaptive proximal parameter. For the exact idealized iteration, global convergence to stationary points and O(1/ε2) iteration complexity are proved; under a proximal Kurdyka–Łojasiewicz condition with exponent 1/2 and a relative-error condition, both stated as explicit hypotheses, the convergence is R-linear. Experiments on a sparse-PCA model, validated by an external residual certificate at the returned point, show a 3.6× mean iteration-count reduction relative to fixed-parameter ManPG; a factorial ablation attributes most of this reduction to the adaptive parameter, while transported momentum adds a further 6.4% on average, concentrated in one paired trial.
This work addresses the issue of robust iterative learning tracking for heterogeneous multi-agent systems in the presence of random impulsive disturbances. Unlike existing literature focusing solely on bounded deterministic impulsive perturbations, this work explicitly models the stochastic nature of practically prevalent random impulses, which emerge randomly in the iteration domain and impair tracking performance. A key mathematical observation is that these persistent non-repetitive random impulses preclude asymptotic zero convergence of the tracking error. To tackle this fundamental limitation, a distributed iterative learning control strategy is proposed to realize robust ultimate bounded tracking. We establish sufficient conditions that guarantee the ultimate boundedness of the tracking error in expectation by applying the Gronwall inequality and mathematical analysis tools. The derived theoretical results generalize the classical convergence criteria developed under deterministic impulsive disturbance frameworks. Finally, numerical simulations are conducted to verify the effectiveness of the established theoretical results.
In this paper, we introduce a generalized (ε-δ)-type contractive condition and develop a unified framework that extends a wide spectrum of classical fixed-point results for self-mappings in complete metric spaces, with or without continuity assumptions. The framework accommodates both contractive and nonexpansive mappings and unifies a broad class of classical fixed-point theorems, including those of Banach, Kannan, Chatterjea, Bianchini, Matkowski, Boyd–Wong, Meir–Keeler, Reich, Ćirić, Ćirić–Jachymski–Matkowski, rational contractions, and Pant, within a multiple-fixed-point setting. One of the established theorems yields a characterization of metric completeness that is different from that of Subrahmanyam in his research in 1975, which covers mappings that are discontinuous at their fixed points as well as mappings admitting multiple fixed points. The theoretical developments are illustrated through several nontrivial examples, including a geometric example in the complex plane whose fixed points are the Nth roots of unity. Following the initial 2024 resolution by Pant and collaborators, this paper provides an alternative constructive solution to the strong version of Rhoades’ 1988 open problem. An example on the integer solutions of a nonlinear Diophantine equation illustrates the results in a discrete arithmetic setting.
We formulate an Information-Geometric Grammar Principle (IGGP) for adaptive systems in which grammatical admissibility, probabilistic realization, and statistical geometry are treated as distinct but connected levels of description. A grammar is represented by a state space together with a set of admissible productions and a probabilistic rule assigning weights to those productions. The resulting family of probability distributions defines a statistical manifold endowed with the Fisher information metric. We distinguish probabilistic adaptation, which changes probabilities while preserving the admissible grammar, from structural adaptation, which changes the set of admissible productions itself. For a fixed statistical manifold, an information-geometric action leads to geodesic equations and provides a coordinate-invariant criterion for locally economical statistical evolution. Interacting components generate a collective statistical manifold, where off-diagonal Fisher components quantify statistical coupling. We further identify the additional convexity and integrability conditions required for a Legendre representation and emphasize that a thermodynamic interpretation is conditional rather than automatic. The resulting framework provides a hierarchy from grammar to statistics, Fisher geometry, adaptive dynamics, collective organization, and, when additional conditions hold, information-geometric thermodynamics.
