Abstract: This paper elucidates the Lorentz group, a fundamental subgroup of the Poincaré group. The orbits and little groups associated with the Lorentz group are described in detail, along with their corresponding properties. The Poincaré group is presented. Another of the fundamental aspects of the Poincaré group is Wigner's little groups obtained from this group. An in-depth discussion both for the cases of massive and massless relativistic particles within the context of little groups is given. Our examination extends to the properties of various special groups associated with the Poincaré group. Applications of these groups are elaborated by physical examples taken from high-energy physics and optics from both classical and quantum domains. Specifically, covariant harmonic oscillators including entangled states, proton form factors, and the parton picture as proposed by Feynman are discussed. In this context, laser cavities and shear states are also addressed. We lay out the underlying mathematics that connects these apparently disparate realms of physics.

Abstract: This work presents a formal axiomatization of emergent physical laws derived from the local decision-making of discrete agents. By postulating a set of foundational axioms—including Locality, Internal Persistence, Minimal Consensus, Strategic Indeterminacy, Temporal Compatibility, Equitable Exchange, and Historical Optimization—the framework rigorously establishes how microscopic stochastic interactions and variational principles give rise to macroscopic phenomena such as conservation laws and wave dynamics. The approach integrates concepts from classical mechanics, variational calculus, and game theory to bridge local agent behavior with global emergent order, providing a unified description applicable to both classical and quantum regimes.

Abstract: One of the most challenging tasks in studying precipitation is quantifying how the complexities of individual components contribute to the overall system complexity. To address this, we employed information measures based on Kolmogorov complexity (KC), specifically the Kolmogorov complexity spectrum (KC spectrum) and the Kolmogorov complexity plane (KC plane). We applied these measures to monthly time series data, both measured and simulated by the EBU POM regional climate model, spanning the period from 1982 to 2005 for Sombor (45.78°N, 19.12°E) in Serbia. The variables analyzed included precipitation—a complex physical system—and its individual components: mean temperature, minimum and maximum temperatures, humidity, wind speed, and global radiation. By applying the listed measures to the all-time series, we calculated normalized KC spectra for each position in the KC plane, displaying interactive master amplitudes against individual amplitudes. We proposed a simplified four-step method to compute the relative change in complexities within the overlapping area beneath the KC spectra. Our results facilitated a discussion on the relationship between the complexity of precipitation and that of its individual components.

Abstract: We construct the conformable actuated pendulum model in the conformable Lagrangian formalism. We solve the equations of motion in the absence of force and in the case of a specific force resulting from torques which generalizes a well known mechanical model. Our study shows that the conformable model captures essential information about the physical system encoded in the parameters which depend on the conformability factor α. This dependence can describe internal variations such as viscous friction, transmission or environmental effects. We solve the equations of motion analytically for α=1/2 and by Frobenius’ method for α≠1/2.

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Assembly theory bridges the gap between evolutionary biology and physics by providing a framework to quantify the generation and selection of novelty in biological systems. We formalize the assembly space as an acyclic digraph of strings with 2-in-regular assembly steps vertices and provide a novel definition of the assembly index. In particular, we show that the upper bound of the assembly index depends quantitatively on the number b of unit-length strings, and the longest length N of a string that has the assembly index of N − k is given by N_(N−1) = b² + b + 1 and by N_(N−k) = b² + b + 2k for 2 ≤ k ≤ 9. We also provide particular forms of such maximum assembly index strings. For k = 1, such odd-length strings are nearly balanced. We also show that each k copies of an n-plet contained in a string decrease its assembly index at least by k(n − 1) − a, where a is the assembly index of this n-plet. We show that the minimum assembly depth satisfies d _min^(N) = ⌈log₂(N)⌉, for all b, and is the assembly depth of a maximum assembly index string. We also provide the general formula for the lengths of the minimum assembly index strings having only one independent assembly step in their assembly spaces. Since these results are also valid for b = 1, assembly theory subsumes information theory.

Abstract: This survey provides a review of the theoretical research on the classic system of matrix equations AX = C and XB = D, which has wide-ranging applications across fields such as control theory, optimization, image processing, and robotics. The paper discusses various solution methods for the system, focusing on specialized approaches, including generalized inverse methods, matrix decomposition techniques, and solutions in the forms of Hermitian, extreme rank, reflexive, and conjugate solutions. Additionally, specialized solving methods for specific algebraic structures, such as Hilbert spaces, Hilbert C∗-modules, and quaternions, are presented. The paper explores the existence conditions and explicit expressions for these solutions, along with examples of their application in electronic network and color image.

Abstract: High-dimensional data often contain noise and redundancy, which can significantly undermine the performance of machine learning. To address this challenge, we propose an advanced robust principal component analysis (RPCA) model that integrates bidirectional graph Laplacian constraints alongwith the anchor point technique. This approach constructs two graphs from both the sample and feature perspectives for a more comprehensive capture of the underlying data structure. Moreover, the anchor point technique serves to substantially reduce computational complexity, making the model more efficient and scalable. Experiments conducted on the GTdatabase demonstrate that our model maintains high accuracy and improves efficiency, particularly under challenging conditions like varying illumination and pose. The method enhances dimensionality reduction and robustness in face recognition, making it suitable for large-scale applications.

Abstract: Inspired by the author 's Riemann hypothesis, this paper attempts to solve the contradiction between general relativity and quantum mechanics in physics. Under the guidance of Euler identity, two important ideas of collision and vibration are introduced. It is concluded that quantum mechanics cannot describe gravity because gravity cannot constitute this dimension of matter. The document deeply discusses the relationship between material dimension and energy, including the stability and change of dimension, the relationship between energy and material, and the relationship between time and dimension. Through detailed assumptions and explanations, this paper provides a new perspective for us to understand the complexity of the material world. It mainly introduces how different dimensions of matter interact, the generation and transformation of energy, and the influence of dimensional changes on matter. The following is a summary of the core content of the paper : the influence of material dimension and energy, the change of dimension, the stability and change of dimension, the relationship between gravitation and material, time and dimension, and the realization of dimension change.

Abstract: We present a geometric field theory in which the action and field equation are constructed from a vector field and its covariant derivative and have full general covariance in a higher-dimensional spacetime. The field equation is the simplest possible generalisation of the Poisson equation for gravity consistent with general covariance and the equivalence principle. It contains the Ricci tensor and metric acting as operators on the vector field. If the symmetrised covariant derivative is diagonalisable across a neighbourhood under real changes of coordinate basis, spacetime coincides with a product manifold. The dimensionalities of the factor spaces are determined by its eigenvalues and hence by its algebraic invariants. Tensors decompose into multiplets which have both Lorentz and internal symmetry indices. The vector field decomposes into conformal Killing vector fields for each of the factor spaces.The field equation has a `classical vacuum' solution which is a Cartesian product of factor spaces. The factor spaces are all Einstein manifolds or two-dimensional Riemannian manifolds. All have a Ricci curvature of roughly the same order of magnitude, or are Ricci-flat. A worked example is provided in six dimensions.Away from this classical vacuum, connection components in appropriate coordinates include $SO(N)$ gauge fields. The Riemann tensor includes their field strength. Unitary gauge symmetries act indirectly on tensor fields and some or all of the unitary gauge fields are found amongst the $SO(N)$ gauge fields. Symmetry restoration occurs at the zero-curvature `decompactification limit', in which all dimensions appear on the same footing.

Abstract: In this paper an underlying perturbed Ricci flow construction is made, within the metric operator space, originated from the Heisenberg dynamical equations, to formulate a method which appears to provide a new geometric approach for the geometric formulation of the quantum mechanical dynamics. A quantum mechanical notion of stability and local instability is introduced within the quantum mechanical theory, based on the quantum mechanical dynamical equations governing the evolution of the tensor metric operator. The stability analysis is carried in the topology of little H¨older spaces of metrics which the tensor metric operator acts on. Finally, a conjecture is introduced in attempt to characterize the stability properties of the quantum mechanical system such that it brings in the quantum mechanical dynamics into the analysis.

Abstract: In this paper, based on the novel generalized Hamilton-real (GHR) calculus, we propose for the first time a quaternion Nesterov’s accelerated projected gradient algorithm for computing the dominant eigenvalue and eigenvector of quaternion Hermitian matrices. By introducing momentum terms and look-ahead updates, the algorithm achieves a faster convergence rate. We theoretically prove the convergence of the quaternion Nesterov’s accelerated projected gradient algorithm. Numerical experiments show that the proposed method outperforms the quaternion projected gradient ascent method and the traditional algebraic methods in terms of computational accuracy and runtime efficiency.

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The study aims to explain the different effects between multiplication, addition, division and subtraction equations. In this sense, multiplication and addition equations develop a new element or fact, while division equations express information within the element or fact, subtraction is the selection of the element or fact. Furthermore, for a multiplication and addition equation to result in the new element or fact, it is necessary to depend on the intensity of a specific physical concept, without extreme change, as this would result in the alteration of the element or fact. It is worth highlighting the importance of fixed values ​​in the equation to describe the specific nature of the element or fact. The study makes it possible to understand open equations (without solutions) in mathematics and physics, in addition to understanding existing equations. The study helps in understanding the theory of obligatory necessity, the theory of differences between elements, the theory of limited numbers and the theory of greater and lesser progressions.

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The goal of this paper is to establish Dugundji’s extension theorem in p-normed spaces, and then as applications, fixed theorems in p-normed spaces are given for p ∈ (0, 1]. The results in this paper would provide a general fundamental basis for the development of fixed point theory, in supporting for the study of Schauder’s conjecture and related nonlinear analysis in p-vector spaces which are either finite or infinite-dimensional. In particularly, how important the Dugundgji type extension and fixed-point theorem in p-normed spaces can be found from Yuan’s recent work in [41]-[42] and related discussion wherein.

Abstract: In this paper, we attempt to interpret gravity by entropy. We first introduce generalized entropy, acceleration of its entropy and its partial entropy, and assume that generalized entropy can represent as a second-order polynomial by applying the idea of the logistic function to its entropy. Besides, we define the inverse of partial entropy as the gravitational potential. By applying these concepts, we attempt to explain that 1) gravity becomes constant values within small distance under certain conditions. It is possible that gravity has 5-states within small enough distance. There may exist anti-gravity which is the opposite of Newton's gravity among 5-states. Furthermore, within small distance, we show the possibility that the gravitational potential and the Coulomb potential can treat in the same way, that 2) the rotation speed of the galaxy does not depend on its radius if the radius is within the size level of the universe (The galaxy rotation curve problem), and that 3) the gravitational acceleration toward the center may change at long distance compared to Newton's gravity. We show that it becomes an expansion of Newton's gravity, and that the possibility of the existence of certain constants which control gravity and the speed of galaxies, and that gravity may relate to entropy. It also describes the relationship between the Yukawa type potential and generalized partial entropy with negative. Using equations proposed in this paper, it attempts to propose 11-types of forces (accelerations) including the gravitational acceleration g and compare the ratios of the fundamental 4-forces in nature (strong-force, electromagnetic force, weak-force, and gravity). Furthermore, it suggests that there may exist new forces, and that the gravitational constant G can fluctuate if entropy changes. Thermodynamics, quantum, gravity, electromagnetic and ecology may unify through entropy.

Abstract: This study examines the flow dynamics of synovial fluid within a lubricated knee joint during movement, incorporating a linear re-absorption rate of water and nutrients at the synovium. The fluid behavior is modeled using a couple-stress fluid framework which accounts for inertial forces and employing a slip boundary condition, which plays a crucial role in reducing drag and enhancing joint lubrication for the formation of a uniform lubrication layer over the cartilage surfaces. Mathematically, the nonlinear governing equations are transformed into a system of linear partial differential equations using a recursive approach and inverse method is applied to further reduce these equations to a system of ordinary differential equations, which are solved using software Mathematica. The results indicate that synovial fluid flow generates high pressure and shear stress at the synovium due to the combined effects of inertial forces, linear re-absorption, and micro-rotation within the couple-stress fluid. Axial flow intensifies at the center of the joint capsule during activity driven by linear re-absorption and molecular rotation, while transverse flow increases near the synovium due to its permeability. These findings provide critical insights for biomedical engineers to quantify re-absorption rates and stress distributions in synovial fluid under normal physiological conditions.

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The gravitational and electrostatic fields are both conservative fields, thus their forces exhibit similar forms. But there are also differences best seen in Gauss's law, where the sources of these fields are leveraged by the vacuum differently: the source of gravity is enhanced by a modest factor of $4\pi$, whereas the Coulomb source is strongly amplified by a factor of $\sim\!\! 10^{11}$. Discontented by such vexing disparities, we cast Newton's gravitational law and Coulomb's law in the same form that allows for categorical comparisons. The conformity of these force laws suggests that the effective universal gravitational constant is $4\pi\varepsilon_0 G$, where $\varepsilon_0$ is the vacuum permittivity and $G$ is the Newtonian gravitational constant. Furthermore, there is no need for adopting an equivalence principle. The numerical value of $4\pi\varepsilon_0 G$ appears also in the deep limit of MOND and in varying-$G$ gravity, where it specifies (apart from units) the magnitude of the mysterious constant $\mathcal{A}_0$, the only constant in such theories besides their gravitationally interacting masses. The same methodology also offers self-consistent definitions and insightful clarifications concerning dimensionless constants in general and some particular fundamental constants of cosmology and particle physics.

Abstract: The difference in velocity between the electrons and protons in an electrical conductor is said to produce a magnetic force due to Special Relativity effects. According to established theory, the magnetic force observed from a stationary reference frame is that of the electrostatic force viewed from a moving reference frame in consideration of Special Relativity effects. The traditional method of calculating the electrostatic force from a moving reference frame and considering Special Relativity effects is laborious. The approach includes calculation of the force between two parallel electrical conductors. This paper presents a more direct approach for calculating the electrostatic field that involves only a single electrical conductor. The new approach also provides a clearer understanding of the underlying physics.

Abstract: We investigate in this paper the static radial coordinate dependent spherically symmetric spacetime in teleparallel F(T) gravity for a scalar field source. We begin by setting the static field equations (FEs) to be solved and solve the conservation laws for scalar field potential solutions. We simplify the FEs and then find a general formula for computing the new teleparallel F(T) solutions applicable for any scalar field potential V(T) and coframe ansatz. We compute new non-trivial teleparallel F(T) solutions by using a power-law coframe ansatz for each scalar potential case arising from the conservation laws. We apply this formula to find new exact teleparallel F(T) solutions for several cases of coframe ansatz parameter. The new F(T) solution classes may be relevant for various astrophysical applications inside any dark energy (DE) source described by a fundamental scalar field such as quintessence, phantom energy or quintom system to name only those types.

Abstract: We present new insights into the van der Waals (vdW) model, focusing on two key aspects: 1) the concept of maximum possible density (MPD) n=1/b, and 2) the Fano factor’s role in analyzing supercritical fluids and also in determining limits on the applicability of the van der Waals approach. Our aim in these investigations is to bridge the complexity of statistical mechanics with practical applications, targeting researchers and engineers in thermodynamic modeling and supercritical fluid systems. Our MPD, derived from the vdW excluded volume parameter b, offers a theoretical framework to explore packing constraints imposed by molecular interactions. At n=1/b, we observe a vanishing Fano factor ω, signifying suppressed particle number fluctuations. This behavior highlights the reduced configurational complexity due to geometric constraints, with implications for dense granular systems and high-pressure materials design. We use an auxiliary temperature Tw that emerges as a critical diagnostic tool, reflecting the scaling of fluctuations and bridging microscopic interactions with macroscopic stability. In supercritical fluids, we show that the Fano factor provides a sensitive parameter to detect instability thresholds near the spinodal line, revealing the intricate transitions between liquid-like and gas-like states. Our findings underscore the importance of fluctuations in modeling the nuanced thermodynamics of confined fluids and industrial supercritical reservoirs. Our results advance the vdW framework by connecting fundamental thermodynamic principles to actionable insights for practical technologies, particularly in the optimization of supercritical extraction and drying processes. This dual emphasis enhances our understanding of complex systems, marking a meaningful contribution to both theoretical and applied domains. We will see that a putative solid phase is predicted by the van der Waals equation at high densities. Importantly enough, we find that the Fano factor is able to detect the limits of applicability for the van der Waals method.

Abstract: In the framework of the Robertson–Walker metric, we investigate the behavior of a subordinated cosmic scale factor a(t). The parent process a˜(τ) is modeled as a sharp phase transition between two Einstein universes, each with different total energies E1 and E2, at a specific moment τc of the operational time τ. To account for the effect of random clocks, which may arise from differences in the physical properties of dark matter and ordinary matter, this model incorporates a random operational time in the form of an inverse, strictly increasing Lévy-type subordinator. As an observable in physical time t, a(t) is defined as the mean of the parent process over an ensemble of realizations of the inverse subordinator. Specifically, we show that, under certain parameter regimes, the evolution of a(t) mimics primordial inflation, subsequent decelerated expansion, and cosmic acceleration. Employing energy conservation in a closed, expanding universe, we discuss the possibility of incorporating matter density evolution. Furthermore, we establish the emergence of dark energy. We believe that this work introduces a novel application of fractional calculus to cosmology.