Foundations

In this paper, we investigate a new class of nonlinear fractional boundary value problems (BVPs) involving -Caputo fractional derivative operators subject to multipoint closed boundary conditions. Such a formulation of boundary data generalizes classical closure constraints in terms of nonlocal dependence of the unknown function at several interior points, giving rise to a flexible mechanism for describing physical and engineering phenomena governed by nonlocal and memory effects. The proposed problem is first transformed into an equivalent fixed-point formulation, enabling the application of standard analytical tools. Results concerning the existence and uniqueness of solutions to the problem are obtained through the application of fixed-point principles, specifically those of Banach, Krasnosel’skiĭ, and the Leray–Schauder nonlinear alternative. The obtained results extend and generalize several known findings. Illustrative examples are presented to demonstrate the applicability of the theoretical findings. Moreover, the introduction incorporates a succinct review of boundary value problems associated with fractional differential equations and inclusions subject to closed boundary conditions.

Perceptual Control Theory (PCT) and the Free Energy Principle (FEP) are two foundational, principle-based frameworks originally developed to explain brain function. However, since their initial proposals, both frameworks have been generalized to account for the behavior of living systems more broadly. Despite their conceptual overlap and practical successes, a mathematical comparison of the two frameworks has yet to be undertaken. In this article, we briefly introduce and compare the philosophical foundations underlying PCT and FEP. We then introduce and compare their experimental and mathematical foundations concretely in the context of bacterial chemotaxis. With these foundations in place, we can use tools from category theory to argue that PCT can be formally understood as a subset of the FEP framework; however, it is worth noting that the mathematical machinery unique to FEP is not required to successfully model bacterial chemotaxis. Finally, we conclude with a proposal for a mathematical synthesis where each framework plays an orthogonal yet complementary role.

A probabilistic version of geometry is introduced. The fifth postulate of Euclid (Playfair’s axiom) is adopted in the following probabilistic form: consider a line and a point not on the line—there is exactly one line through the point with probability P, where  . Playfair’s axiom is logically independent of the rest of the Hilbert system of axioms of the Euclidian geometry. Thus, the probabilistic version of the Playfair axiom may be combined with other Hilbert axioms. $P=1$ corresponds to the standard Euclidean geometry; $P=0$ corresponds to the elliptic- and hyperbolic-like geometries.   corresponds to the introduced probabilistic geometry. Parallel constructions in this case are Bernoulli trials. Theorems of the probabilistic geometry are discussed. Given a triangle and a line drawn from a vertex parallel to the opposite side, the event that this line is actually parallel occurs with probability P. Otherwise, the line may intersect the side or diverge. Parallelism is not transitive in the probabilistic geometry. Probabilistic geometry occurs on the surface with a stochastically variable Gaussian curvature. Alternative geometries adopting various versions of the probabilistic Playfair axiom are introduced. Probabilistic non-Archimedean geometry is addressed. Applications of the probabilistic geometry are discussed.