Search Results (107)

In the general linear Lie algebra of continuous linear transformations in n dimensions, we show that unequal Abelian scaling transformations on the components of a vector can stabilize the system information in the presence of Markov component transformations on the vector, which, alone, would lead to increasing entropy. The more interesting results follow from seeking Diophantine (integer) solutions, with the result that the system can be stabilized with constant information for each of a set of entropy rates ($k=\mathrm{1,2},3,\dots$ ). The first of these—the simplest—where $k=1$, results in the Fibonacci sequence, with information determined by the olden mean, and Fibonacci interpolating functions. Other interesting results include the fact that a new set of higher order generalized Fibonacci sequences, functions, golden means, and geometric patterns emerges for $k=2, 3, \dots$ Specifically, we define the $k$^th order golden mean as ${\Phi }_k=k/2+\sqrt{{(k/2)}^2+1}$ for $k =1, 2, 3, \dots .$. One can easily observe that one can form a right triangle with sides of 1 and $k/2$ and that this will give a hypotenuse of $\sqrt{{(k/2)}^2+1}$. Thus, the sum of the $k/2$ side plus the hypotenuse of these triangles so proportioned will give geometrically the exact value of the golden means for any value of k relative to the third side with a value of unity. The sequential powers of the matrix $(k^2+1,k,k,1)$ for any integer value of $k$ provide a generalized Fibonacci sequence. Also, using the general equation expressed as ${\Phi }_k={\displaystyle \frac{k}{2}}+\sqrt{{(k/2)}^2+1}$ for $k =\mathrm{1,2},3, \dots ,$ one can easily prove that ${\Phi }_k=k+1/{\Phi }_k$ which is a generalization of the familiar equation expressed as $\Phi =1+1/\Phi$. We suggest that one could look for these new ratios and patterns in nature, with the possibility that all of these systems are connected with the retention of information in the presence of increasing entropy. Thus, we show that two components of the general linear Lie algebra ($GL(n,\mathbb{R})$), acting simultaneously with certain parameters, can stabilize the information content of a vector over time. Full article

We give a variational characterization of the Born rule. For each measurement context, we project a quantum state ρ onto the corresponding abelian algebra by minimizing Umegaki relative entropy; Petz’s Pythagorean identity makes the dephased state the unique local minimizer, so the Born weights $p_C\left(i\right)=\mathrm{Tr}\left(\rho P_i\right)$ arise as a consequence, not an assumption. Globally, we measure contextuality by the minimum classical Kullback–Leibler distance from the bundle $\left\{p_C\left(\rho \right)\right\}$ to the noncontextual polytope, yielding a convex objective $\Phi \left(\rho \right)$. Thus, $\Phi \left(\rho \right)=0$ exactly when a sheaf-theoretic global section exists (noncontextuality), and $\Phi \left(\rho \right)>0$ otherwise; the closest noncontextual model is the classical I-projection of the Born bundle. Assuming finite dimension, full-rank states, and rank-1 projective contexts, the construction is unique and non-circular; it extends to degenerate PVMs and POVMs (via Naimark dilation) without change to the statements. Conceptually, the work unifies information-geometric projection, the presheaf view of contextuality, and categorical classical structure into a single optimization principle. Compared with Gleason-type, decision-theoretic, or envariance approaches, our scope is narrower but more explicit about contextuality and the relational, context-dependent status of quantum probabilities. Full article

The present work applies and extends the previously developed Quantitative Geometrical Thermodynamics (QGT) formalism to the derivation of a Hamiltonian for the damped harmonic oscillator (DHO) across all damping regimes. By introducing complex time, with the real part encoding entropy production and the imaginary part governing reversible dynamics, QGT provides a unified geometric framework for irreversible thermodynamics, showing that the DHO Hamiltonian can be obtained directly from the (complex) entropy production in a simple exponential form that is generalized across all damping regimes. The derived Hamiltonian preserves a modified Poisson bracket structure and embeds thermodynamic irreversibility into the system’s evolution. Moreover, the resulting expression coincides in form with the well-known Caldirola–Kanai Hamiltonian, despite arising from fundamentally different principles, reinforcing the validity of the QGT approach. The results are also compared with the GENERIC framework, showing that QGT offers an elegant alternative to existing approaches that maintains consistency with symplectic geometry. Furthermore, the imaginary time component is interpreted as isomorphic to the antisymmetric Poisson matrix through the lens of geometric algebra. The formalism opens promising avenues for extending Hamiltonian mechanics to dissipative systems, with potential applications in nonlinear dynamics, quantum thermodynamics, and spacetime algebra. Full article

Existing studies on ontology evolution lack automated mechanisms to balance semantic coherence and adaptability under real-time uncertainties, particularly in resolving spatiotemporal asymmetry and multidimensional coupling imbalances in dam-break scenarios. Traditional methods such as WordNet’s tree symmetry and FrameNet’s frame symmetry fail to formalize dynamic adjustments through quantitative metrics, leading to path dependency and delayed responses. This study addresses this gap by introducing a novel symmetry-entropy-constrained matrix fusion algorithm, which integrates algebraic direct sum operations and Hadamard product with entropy-driven adaptive weighting. The original contribution lies in the symmetry entropy metric, which quantifies structural deviations during fusion to systematically balance semantic stability and adaptability. This work formalizes ontology evolution as a symmetry-driven optimization process. Experimental results demonstrate that shared concepts between ontologies (s = 3) reduce structural asymmetry by 25% compared to ontologies (s = 1), while case studies validate the algorithm’s ability to reconcile discrepancies between theoretical models and practical challenges in evacuation efficiency and crowd dynamics. This advancement promotes the evolution of traditional emergency management systems towards an adaptive intelligent form. Full article

The explanation of thermodynamics through geometric models was initiated by seminal figures such as Carnot, Gibbs, Duhem, Reeb, and Carathéodory. Only recently, however, has the symplectic foliation model, introduced within the domain of geometric statistical mechanics, provided a geometric definition of entropy as an invariant Casimir function on symplectic leaves—specifically, the coadjoint orbits of the Lie group acting on the system, where these orbits are interpreted as level sets of entropy. We present a symplectic foliation interpretation of thermodynamics, based on Jean-Marie Souriau’s Lie group thermodynamics. This model offers a Lie algebra cohomological characterization of entropy, viewed as an invariant Casimir function in the coadjoint representation. The dual space of the Lie algebra is foliated into coadjoint orbits, which are identified with the level sets of entropy. Within the framework of thermodynamics, dynamics on symplectic leaves—described by the Poisson bracket—are associated with non-dissipative phenomena. Conversely, on the transversal Riemannian foliation (defined by the level sets of energy), the dynamics, characterized by the metric flow bracket, induce entropy production as transitions occur from one symplectic leaf to another. Full article

We re-examine the mathematical properties of the kink and antikink soliton solutions to the Logarithmic Schrödinger Equation (LogSE), a nonlinear logarithmic version of the Schrödinger Equation incorporating Everett–Hirschman entropy. We devise successive approximations with increasing accuracy. From the most successful forms, we formulate an analytical solution that provides a very accurate solution to the LogSE. Finally, we consider combinations of such solutions to mathematically model kink and antikink bound states, which can serve as a possible candidate for modeling dilatonic quantum gravity states. Full article

Many information-theoretic quantities have corresponding representations in terms of sets. Many of these information quantities do not have a fixed sign—for example, the co-information can be both positive and negative. In previous work, we presented a signed measure space for entropy where the smallest sets (called atoms) all have fixed signs. In the present work, we demonstrate that these atoms have natural algebraic behaviour which can be expressed in terms of ideals (characterised here as upper sets), and we show that this behaviour allows us to make bounding arguments and describe many fixed-sign information quantity expressions. As an application, we give an algebraic proof that the only completely synergistic system of three finite variables X, Y and $Z=f(X,Y)$ is the XOR gate. Full article

This note aims to offer a non-technical and self-contained introduction to gravitational algebras and their applications in the nonequilibrium physics of gravitational systems. We begin by presenting foundational concepts from operator algebra theory and exploring their relevance to perturbative quantum gravity. Additionally, we provide a brief overview of the theory of nonequilibrium dynamical systems in finite dimensions and discuss its generalization to gravitational algebras. Specifically, we focus on entropy production in black hole backgrounds and fluctuation theorems in de Sitter spacetime. Full article

The main aim of this study is to achieve the numerical solution for the Navier–Stokes equations for incompressible, non-turbulent, and subsonic fluid flows with some Gaussian physical uncertainties. The higher-order stochastic finite volume method (SFVM), implemented according to the iterative generalized stochastic perturbation technique and the Monte Carlo scheme, are engaged for this purpose. It is implemented with the aid of the polynomial bases for the pressure–velocity–temperature (PVT) solutions, for which the weighted least squares method (WLSM) algorithm is applicable. The deterministic problem is solved using the freeware OpenFVM, the computer algebra software MAPLE 2019 is employed for the LSM local fittings, and the resulting probabilistic quantities are computed. The first two probabilistic moments, as well as the Shannon entropy spatial distributions, are determined with this apparatus and visualized in the FEPlot software. This approach is validated using the 2D heat conduction benchmark test and then applied for the probabilistic version of the 3D coupled lid-driven cavity flow analysis. Such an implementation of the SFVM is applied to model the 2D lid-driven cavity flow problem for statistically homogeneous fluid with limited uncertainty in its viscosity and heat conductivity. Further numerical extension of this technique is seen in an application of the artificial neural networks, where polynomial approximation may be replaced automatically by some optimal, and not necessarily polynomial, bases. Full article

Classic formulae for entropy and cross-entropy contain operations ${\displaystyle \frac{x}{0}}$ and ${log}_{2}x$ that are not defined on all inputs. This can lead to calculations with problematic subexpressions such as $0{log}_{2}0$ and uncertainties in large scale calculations; partiality also introduces complications in logical analysis. Instead of adding conventions or splitting formulae into cases, we create a new algebra of real numbers with two symbols $\pm \infty$ for signed infinite values and a symbol named ⊥ for the undefined. In this resulting arithmetic, entropy, cross-entropy, Kullback–Leibler divergence, and Shannon divergence can be expressed without concerning any further conventions. The algebra may form a basis for probability theory more generally. Full article

An algebraic graph is defined in terms of graph theory as a graph with related algebraic structures or characteristics. If the vertex set of a graph $\mathcal{G}$ is a group, a ring, or a field, then $\mathcal{G}$ is called an algebraic structure graph. This work uses an algebraic structure graph based on the modular ring ${\mathbb{Z}}_n$, known as a hyper-chordal ring network. The lower and upper bounds of the local fractional metric dimension are computed for certain families of hyper-chordal ring networks. Utilizing the cardinalities of local fractional resolving sets, local fractional resolving (LFR)M-polynomials are computed for hyper-chordal ring networks. Further, new topological indices based on (LFR)M-polynomials are established for the proposed networks. The local fraction entropies are developed by modifying the first three kinds of Zagreb entropies, which are calculated for the chosen hyper-chordal ring networks. Furthermore, numerical and graphical comparisons are discussed to observe the order between newly computed topological indices. Full article

In this paper, a problem of random disturbance attenuation capabilities for linear time-invariant continuous systems, affected by random disturbances with bounded $\sigma$-entropy, is studied. The $\sigma$-entropy norm defines a performance index of the system on the set of the aforementioned input signals. Two problems are considered. The first is a state-space $\sigma$-entropy analysis of linear systems, and the second is an optimal control design using the $\sigma$-entropy norm as an optimization objective. The state-space solution to the $\sigma$-entropy analysis problem is derived from the representation of the $\sigma$-entropy norm in the frequency domain. The formulae of the $\sigma$-entropy norm computation in the state space are presented in the form of coupled matrix equations: one algebraic Riccati equation, one nonlinear equation over log determinant function, and two Lyapunov equations. Optimal control law is obtained using game theory and a saddle-point condition of optimality. The optimal state-feedback control, which minimizes the $\sigma$-entropy norm of the closed-loop system, is found from the solution of two algebraic Riccati equations, one Lyapunov equation, and the log determinant equation. Full article

We present a hyperchaotic oscillator with two linear terms and seven nonlinear terms that displays special algebraic properties. Notably, the introduced oscillator features distinct equilibrium types: single-point, line, and spherical equilibria. The introduced oscillator exhibits attractive dynamics like hyperchaos with two wing attractors. To gain a better understanding, we provide the bifurcation and Lyapunov exponents. The Kolmogorov–Sinai entropy is applied to show the complexity of the oscillator. A microcontroller realization confirms the reliability of the oscillator. The proposed oscillator is successfully applied for RNG. Full article

The paper is devoted to the modeling of nonlinear viscoelastic materials. The constitutive equations are considered in differential form via relations between strain, stress, and their derivatives in the Lagrangian description. The thermodynamic consistency is established by using the Clausius–Duhem inequality through a procedure that involves two uncommon features. Firstly, the entropy production is regarded as a positive-valued constitutive function per se. This view implies that the inequality is in fact an equation. Secondly, this statement of the second law is investigated by using an algebraic representation formula, thus arriving at quite general results for rate terms that are usually overlooked in thermodynamic analyses. Starting from strain-rate or stress-rate equations, the corresponding finite equations are derived. It then emerges that a greater generality of the constitutive equations of the classical models, such as those of Boltzmann and Maxwell, are obtained as special cases. Full article