The phenomenon of dissipative self-organization is studied on the example of time-crystal networks. Particular attention was given to transient processes, attractor topologies, phase transitions, and asymptotic stability. New concepts were introduced, including topological phases, spinorial states, and bond flavors. Concepts such as ground states, chemical potentials, elastic forces, temperature, and statistical distributions were endowed with a new meaning associated with asymptotic stability. Phenomena usually attributed exclusively to quantum physics have been shown to occur in this essentially classical environment. They coexist with, and in some cases, such as charge quantization, are related to the phenomenon of time dilation. The approach was applied to model vacuum self-organization. We have shown that under the ac-tion of competing forces, such as gravity and antigravity, a cascade of phase transitions can transform an unorganized vacuum into phases in which interactions, fields and waves resemble electromagnetic, weak, and strong, and their elements can be used as building blocks for prototype particles. In addition, some interrelation field parameters and probabilities of particle transmutations were calculated, which are not predicted by the standard model. The results are consistent with the experiment. The presented material and methodology may be of interest for studying self-organization in different environments.

While traditional AC mechanical circuit breakers have been competent for protecting AC circuits, high penetration of DC power distribution technologies like DC microgrids (MGs) obligate better disruption performance features such as quick and reliable switch-ing speeds. However, a DC circuit breaker (DCCB) novel design is challenging due to the need to quickly break high currents within milliseconds, caused by the high fault current rise in DC grids compared to AC grids. In DC grids, the circuit breaker must provide zero current crossing and be designed for absorbing surges since the arc is not naturally extinguished by the system. Additionally, the DC breaker must mitigate the magnetic energy stored in the system inductance and withstand residual over-voltages after current interruption. These challenges require a fundamentally different topology for DCCBs, which are typically made using solid-state semiconductor technology, metal oxide varistors (MOV), and ultra-fast switches. This study aims to provide a comprehensive review on development, design, and performance description of DCCBs in parallel with a specific concentration on analysis of internal topology, energy absorption path and sub-circuits in Solid-State (SS)-based DCCBs. The research explores various novel designs that introduce different structures for energy dissipation solution. The classification of these designs is based on the fundamental principles of surge mitigation and a detailed analysis of the techniques employed in DCCBs. In addition, our framework offers an advantageous reference point for the future evolution of SS-circuit breakers in numerous developing power delivery systems.

We propose a non-invasive approach to control the quality of spheroids and their fusion into a complex bioconstruct. The proposed method is based on the union of the nutrient concentration change calculated using Fick's law and the "reaction-diffusion" equations, taking into account absorption coefficient with specifying the exact fusion geometry using implicit Function Repre-sentation (FRep) functions. The proposed approach allows us to analyze the viability of cells within the spheroid, predict the fusion of spheroids, and accurately model complex heteroge-neous biostructures as a future task for our research. These results will significantly accelerate the development of such a promising field of additive biotechnologies as 3D bioprinting.

Based on measured astronomical position data of heavenly objects in the Solar System and other planetary systems, all bodies in space seem to move in some kind of elliptical motion with respect to each other, whereas objects follow parabolic escape orbits while moving away from Earth and bodies asserting a gravitational pull, and some comets move in near-hyperbolic orbits when they approach the Sun. In this article, it is first mathematically proven that the “distance between points on any two different circles in three-dimensional space” is equivalent to the “distance of points on a vector ellipse from another fixed or moving point, as in two-dimensional space.” Then, it is further mathematically demonstrated that “distance between points on any two different circles in any number of multiple dimensions” is equivalent to “distance of points on a vector ellipse from another fixed or moving point”. Finally, two special cases when the “distance between points on two different circles in multi-dimensional space” become mathematically equivalent to distances in “parabolic” or “near-hyperbolic” trajectories are investigated. Concepts of “vector ellipse”, “vector hyperbola”, and “vector parabola” are also mathematically defined. The mathematical basis derived in this Article is utilized in the book “Everyhing Is A Circle: A New Model For Orbits Of Bodies In The Universe” in asserting a new Circular Orbital Model for moving bodies in the Universe, leading to further insights in Astrophysics.

This paper establishes methods that quantify the structure of statistical interactions within a given data set using the characterization of information theory in cohomology by finite methods, and provides their expression in terms of statistical physic and machine learning. Following [1–3], we show directly that k multivariate mutual-informations (I_k) are k-coboundaries. The k-cocycles are given by I_k = 0, which generalize statistical independence to arbitrary dimension k. The topological approach allows to investigate Shannon’s information in the multivariate case without the assumptions of independent identically distributed variables. We develop the computationally tractable subcase of simplicial information cohomology represented by entropy H_k and information I_k landscapes. The I₁ component defines a self-internal energy functional U_k, and (−1)^k I_k,k≥2 components define the contribution to a free energy functional G_k of the k-body interactions. The set of information paths in simplicial structures is in bijection with the symmetric group and random processes, provides a topological expression of the 2nd law and points toward a discrete Noether theorem (1st law). The local minima of free-energy, related to conditional information negativity and the non-Shannonian cone of Yeung [4], characterize a minimum free energy complex. This complex formalizes the minimum free-energy principle in topology, provides a definition of a complex system, and characterizes a multiplicity of local minima that quantifies the diversity observed in biology. Finite data size effects and estimation bias severely constrain the effective computation of the information topology on data, and we provide simple statistical tests for the undersampling bias and for the k-dependences following [5]. We give an example of application of these methods to genetic expression and cell-type classification. The maximal positive I_k identifies the variables that co-vary the most in the population, whereas the minimal negative I_k identifies clusters and the variables that differentiate-segregate the most. The methods unravel biologically relevant I₁₀ with a sample size of 41. It establishes generic methods to quantify the epigenetic information storage and a unified epigenetic unsupervised learning formalism.

In 1999, Molodtsov initiated the theory of soft sets as a new mathematical tool for dealing with uncertainties in many fields of applied sciences. In 2011, Shabir and Naz introduced and studied the notion of soft topological spaces, also defining and investigating many new soft properties as generalization of the classical ones. In this paper, we introduce the notions of soft separation between soft points and soft closed sets in order to obtain a generalization of the well-known Embedding Lemma to the class of soft topological spaces.

In this study, we present the concept of interval-valued fuzzy soft point and then introduce the notions of neighborhood and quasi-neighbourhood of it in interval-valued fuzzy soft topological spaces. Separation axioms in interval-valued fuzzy soft topology, so-called $q$-$T_{i}$ for $ i=0,1,2,3,4 ,$ is introduced and some of its basic properties are also studied.

Vibration suppression in a field of frequencies and the creation of band gaps which do not allow the propagation of the waves is studied in this paper by means of microstructures designed through topology optimization. Topology optimization is formulated in frequency domain. Band gap design is based on Floquet-Bloch theory and genetic optimization. The result is appealing in view of modern 3D printing techniques.

In this article, we first defined a stronger form of (r,s)-generalized fuzzy semi-closed sets (briefly, (r,s)-gfsc sets) called (r,s)-g*fsc sets and investigated some of its features. Moreover, we showed that (r,s)-fsc set → (r,s)-g*fsc set → (r,s)-gfsc set, but the converse may not be true. In addition, we explored novel types of fuzzy generalized mappings between double fuzzy topological spaces (U, τ, τ*) and (V, η, η*), and the relationships between these classes of mappings were examined with the help of some illustrative examples. Thereafter, we introduced novel types of higher separation axioms called (r,s)-GFS-regular and (r,s)-GFS-normal spaces with the help of (r,s)-gfsc sets and discussed some topological properties of them. Finally, some novel types of compactness via (r,s)-gfso sets were defined and the relationships between them were introduced.

The Hyper-Torus Universe Model (HTUM) proposes a novel framework to unify quantum mechanics, cosmology, and consciousness. This paper introduces HTUM's core concept: a higher-dimensional hyper-torus encompassing all possible states of existence. HTUM suggests that the universe functions as a quantum system where all outcomes are inherently connected, with consciousness playing a significant role in actualizing reality. The model introduces several key innovations, including the universe's 4-dimensional toroidal structure (4DTS), providing a new geometric interpretation of spacetime. HTUM presents dark matter and dark energy as nonlinear probabilistic phenomena within this toroidal framework, offering a unique perspective on their nature and interactions. The model proposes a mechanism for wave function collapse through universal self-observation, bridging quantum mechanics and gravity. It introduces a Topological Vacuum Energy Modulator (TVEM) function, offering a potential resolution to the cosmological constant problem. Furthermore, HTUM integrates consciousness as a fundamental aspect of the universe, influencing quantum state actualization and potentially resolving the hard problem of consciousness. The model explores the implications of a timeless singularity and the emergence of time as a product of causal relationships within the toroidal structure. This framework addresses critical challenges in modern physics, including the nature of quantum entanglement, the origin and fate of the universe, and the relationship between mind and matter. The paper discusses the mathematical formulation of HTUM, its implications for quantum mechanics and cosmology, and its potential to bridge the gap between science and philosophy. We present detailed predictions for the cosmic microwave background (CMB) power spectrum, gravitational wave signatures, and large-scale structure formation, demonstrating how HTUM differs from the standard ΛCDM model. Our analysis reveals distinctive oscillatory patterns in the relative difference between HTUM and ΛCDM predictions, with deviations reaching up to ±20% at certain scales. The paper also explores the unification of fundamental forces within the HTUM framework, proposing novel approaches to particle physics and quantum field theory. It introduces the concept of unified mathematical operations, suggesting a more holistic approach to mathematical thinking that aligns with the model's interconnected view of the universe. Additional testable predictions and proposed experimental approaches are discussed, including advanced space-based tests and laboratory experiments probing quantum gravity effects. These specific, quantifiable predictions offer multiple avenues for empirical validation, highlighting HTUM's potential to be distinguished from ΛCDM and other alternative cosmological models. HTUM represents a significant new perspective in our understanding of the universe, inviting further research and exploration into the nature of reality, consciousness, and the fundamental structure of the cosmos. By providing a comprehensive framework that unifies various aspects of physics, cosmology, and consciousness studies, HTUM offers a promising path toward a more complete understanding of our universe.