Axioms

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 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 $3.6\times$ 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.

In this paper, we introduce a generalized $(\epsilon$-$\delta )$-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.