Previous Issue
Volume 5, September
 
 

Foundations, Volume 5, Issue 4 (December 2025) – 3 articles

  • Issues are regarded as officially published after their release is announced to the table of contents alert mailing list.
  • You may sign up for e-mail alerts to receive table of contents of newly released issues.
  • PDF is the official format for papers published in both, html and pdf forms. To view the papers in pdf format, click on the "PDF Full-text" link, and use the free Adobe Reader to open them.
Order results
Result details
Section
Select all
Export citation of selected articles as:
14 pages, 376 KB  
Article
Probabilistic Geometry Based on the Fuzzy Playfair Axiom
by Edward Bormashenko
Foundations 2025, 5(4), 34; https://doi.org/10.3390/foundations5040034 - 1 Oct 2025
Abstract
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  [...] Read more.
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 0P1. 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. 0<P<1 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. Full article
(This article belongs to the Section Mathematical Sciences)
Show Figures

Figure 1

7 pages, 224 KB  
Article
On Relative Stability for Strongly Mixing Sequences
by Adam Jakubowski and Zbigniew Stanisław Szewczak
Foundations 2025, 5(4), 33; https://doi.org/10.3390/foundations5040033 - 25 Sep 2025
Abstract
We consider a class of strongly mixing sequences with infinite second moment. This class contains important GARCH processes that are applied in econometrics. We show the relative stability for such processes and construct a counterexample. We apply these results and obtain a new [...] Read more.
We consider a class of strongly mixing sequences with infinite second moment. This class contains important GARCH processes that are applied in econometrics. We show the relative stability for such processes and construct a counterexample. We apply these results and obtain a new CLT without the requirement of exponential decay of mixing coefficients, and provide a counterexample to this as well. Full article
(This article belongs to the Section Mathematical Sciences)
14 pages, 1990 KB  
Hypothesis
From Magnetic Field Seeds to Planetary and Galactic Magnetism
by Angelo De Santis, Roberto Dini and Gianfranco Cianchini
Foundations 2025, 5(4), 32; https://doi.org/10.3390/foundations5040032 - 23 Sep 2025
Viewed by 102
Abstract
This study investigates the origin and amplification of magnetic fields in planets and galaxies, emphasizing the foundational role of a seed magnetic field (SMF) in enabling dynamo processes. We propose a universal mechanism whereby an SMF arises naturally in systems where an orbiting [...] Read more.
This study investigates the origin and amplification of magnetic fields in planets and galaxies, emphasizing the foundational role of a seed magnetic field (SMF) in enabling dynamo processes. We propose a universal mechanism whereby an SMF arises naturally in systems where an orbiting body rotates non-synchronously with respect to its central mass. Based on this premise, we derive a general equation for the SMF applicable to both planetary and galactic scales. Incorporating parameters such as orbital distance, rotational velocity, and core radius, we then introduce a dimensionless factor to characterize the amplification of this seed field via dynamo processes. By comparing model predictions with magnetic field data from the solar system and the Milky Way, we find that the observed magnetic fields can be interpreted as the product of a universal gravitationally induced SMF and a body-specific amplification factor. Our results offer a novel perspective on the generation of magnetic fields in a wide range of astrophysical contexts and suggest new directions for theoretical investigation, including the environments surrounding black holes. Full article
(This article belongs to the Section Physical Sciences)
Show Figures

Figure 1

Previous Issue
Back to TopTop