Search Results (19)

Baroclinic instability is a fundamental mechanism of midlatitude atmospheric variability, and the Eady model remains one of its most useful idealized representations. In this work, we revisit the Eady configuration from the viewpoint of solution-space geometry rather than the classical normal-mode/growth-rate analysis. Starting from the reduced Eady vertical-structure equation, we show that the ratio of two independent solutions satisfies a Schwarzian-type relation that is invariant under homographic transformations, which naturally leads to an SL(2R) projective symmetry of the solution family. On this basis, we introduce a complex amplitude representation and reformulate coherence in terms of phase–amplitude synchronization constrained by projective invariants. Using Riccati-type constructions along geodesic parametrizations, the reduced dynamics are connected to a Stoler-type transform. Numerical exploration of the reduced model shows a systematic dependence on the control parameter $\omega$: small $\omega$ is associated with simple oscillatory or burst-like behavior, intermediate $\omega$ with period-doubling-like behavior, and large $\omega$ with strongly modulated dynamics and more intricate reconstructed attractors. These results should be interpreted as properties of the reduced symmetry-based model, and they suggest that projective invariants may provide a useful framework for classifying organization regimes in Eady-type disturbances, complementary to classical growth-rate analyses. Full article

Higher education is increasingly integrating artificial intelligence (AI) and sustainability-oriented learning, aligning with the United Nations Sustainable Development Goals (SDGs). Nonetheless, there is limited empirical evidence explaining how cognitive processes, AI-supported learning (AI-SL), and SDG awareness (SDGA) jointly relate to learner engagement (ENG) and learning outcomes (LO) across academic disciplines. This study examines the associations among working memory (WM), metacognition (MET), reasoning ability (REA), AI-SL, SDGA, ENG, and LO in higher education. Survey data were collected from undergraduate students across nine campuses of a public university system in Thailand. Multi-group structural equation modeling (MG-SEM) was employed to compare students enrolled in STEM and social science programs. Measurement model evaluation indicated satisfactory reliability and convergent and discriminant validity, with partial scalar invariance supported across groups, enabling meaningful structural comparisons. The results showed that ENG was strongly associated with LO in both groups. MET emerged as the strongest cognitive correlate of ENG overall, while REA was more strongly associated with ENG among STEM students, and SDGA showed a stronger association among social science students. AI-SL was positively associated with ENG and LO, with discipline-specific differences in the strength of the effect. The model explained substantial variance in ENG (R² = 0.48) and LO (R² = 0.52). These findings emphasize the value of integrating cognitive processes (WM, MET, and REA), AI-SL, and sustainability-oriented content within discipline-sensitive instructional designs in higher education. Full article

Assimilating complex systems to multifractal-type objects reveals continuous and non-differentiable curve dynamics, aligning with the Multifractal Theory of Motion. Two scenarios, a Schrödinger-type and a Madelung-type multifractal scenario, are possible in this setting. If the Madelung scenario employs maximized information entropy for a distribution density, then Newtonian and oscillator-type forces can be determined. In the presence of these forces and a matter background, we analyze the two-body problem. The obtained results are as follows: a generalized Hubble-type law, a dependence of Newton’s constant on the epoch and background density, a generalization of Lorentz transform (involving the Hubble constant, Newton’s constant, the speed of light, and cosmic matter density), etc. Moreover, in the same scenario, the functionality of a diffusion-type equation implies instabilities, such as period doubling, through an SL(2R) invariance. Thus, multiple infragalactic and extragalactic instabilities are exemplified. Full article

By assimilating any complex system into a multifractal, a new approach for describing the dynamics of such systems is proposed by means of the Multifractal Theory of Motion. In such context, the description of these dynamics is accomplished through continuous and non-differentiable curves (multifractal curves), giving rise to two scenarios. The first scenario is a Schrödinger-type multifractal scenario, a situation in which the motion laws can be related to the SL(2R) algebra invariant functions. The second scenario is a Madelung-type multifractal scenario, a situation in which if the differentiable and non-differentiable components of the velocity field satisfy a particular restriction, an SL(2R) symmetry can also be highlighted. Moreover, correlative dynamics in either of the two scenarios, based on the same SL(2R) symmetry, can be obtained by Riccati-type gauges, which imply Stoler coherent states. Several cases induced by the SL(2R) symmetry are also analyzed. Full article

Let $\Gamma$ be a non-uniform lattice in $SL(2,\mathbb{R})$. In this paper, we study various $L^2$-norms of automorphic representations of $SL(2,\mathbb{R})$. We bound these norms with intrinsic norms defined on the representation. Comparison of these norms can help us understand the growth of L-functions in a systematic way. Full article

We classify Weingarten conoids in the real special linear group $\mathrm{SL}(2,\mathbb{R})$. In particular, there is no linear Weingarten nontrivial conoids in $\mathrm{SL}(2,\mathbb{R})$. We also prove that the only conoids in $\mathrm{SL}(2,\mathbb{R})$ with constant Gaussian curvature are the flat ones. Finally, we show that any surface that is invariant under left translations of the subgroup $\mathcal{N}$ is a Weingarten surface. Full article

Some possible correspondences between the Scale Relativity Theory and the Space–Time Theory can be established. Since both the multifractal Schrödinger equation from the Scale Relativity Theory and the General Relativity equations for a gravitational field with axial symmetry accept the same SL(2R)-type invariance, an Ernst-type potential (from General Relativity) and also a multi-fractal tensor (from Scale Relativity) are highlighted in the description of complex systems dynamics. In this way, a non-differentiable description of complex systems dynamics can become functional, even in the case of standard theories (General Relativity and Quantum Mechanics). Full article

Dynamics in atmospheric structures are analyzed using the Scale Relativity Theory in Schrödinger-type and Madelung-type scenarios. In the Schrödinger-type scenario, the group invariances of the special linear group SL(2R)-type under Riccati-type gauges implies morphological atmospheric manifestations through frequency modulation, particularly through period doubling. In the Madelung-type scenario, the same group invariances type, manifested through harmonic mappings, implies the functionality of atmospheric mass conductions through mass superconducting-type by scale transition from nondifferentiable atmospheric dynamics to differentiable atmospheric dynamics. The compatibility of these two scenarios under the correlations of atmospheric morphologies-functionalities implies Stoler-type coherences of the atmospheric dynamics through the removal of atmospheric turbulence by means of laminar channels. Finally, these theories are successfully employed to analyze the vertical atmospheric dynamics of cases of insect swarms. Full article

The self-similar gravitational collapse solutions to the Einstein-axion–dilaton system have already been discovered. Those solutions become invariants after combining the spacetime dilation with the transformations of internal SL(2, R). We apply nonlinear statistical models to estimate the functions that appear in the physics of Black Holes of the axion–dilaton system in four dimensions. These statistical models include parametric polynomial regression, nonparametric kernel regression and semi-parametric local polynomial regression models. Through various numerical studies, we reached accurate numerical and closed-form continuously differentiable estimates for the functions appearing in the metric and equations of motion. Full article

Possible implications and consequences of using SL(2R) as invariance groups in the description at any scale resolution of the dynamics of any complex system are analyzed. From this perspective and based on Jaynes’ remark (any circumstance left unspecified in the description of any complex system dynamics has the concrete expression in the existence of an invariance group), in the present paper one specifies such unspecified circumstances that result directly from the consideration of the canonical formalism induced by the SL(2R) as invariance group. It follows that both the Hamiltonian function and the Guassian distribution acquire the status of invariant group functions, the parameters that define the Hamiltonian acquire statistical significances based on a principle of maximizing informational energy, the class of statistical hypotheses specific to Gaussians of the same average acts as transitivity manifolds of the group (transitivity manifolds which can be correlated with the multifractal-non-multifractal scale transitions), joint invariant functions induced through SL(2R) groups isomorphism (the SL(2R) variables group, and the SL(2R) parameters group, etc.). For an ensemble of oscillators of the same frequency, the unspecified circumstances return to the ignorance of the amplitude and phase of each of the oscillators, which forces the recourse to a statistical ensemble traversed by the transformations of the Barbilian-type group. Finally, the model is validated based on numerical simulations and experimental results that refer to transient phenomena in ablation plasmas. The novelty of our model resides in the fact that fractalization through stochasticization is imposed through group invariance, situation in which the group’s transitivity manifolds can be correlated with the scale resolution. Full article

Assimilating a complex fluid with a fractal object, non-differentiable behaviors in its dynamics are analyzed. Complex fluid dynamics in the form of hydrodynamic-type fractal regimes imply “holographic implementations” through velocity fields at non-differentiable scale resolution, via fractal solitons, fractal solitons–fractal kinks, and fractal minimal vortices. Complex fluid dynamics in the form of Schrödinger type fractal regimes imply “holographic implementations”, through the formalism of Airy functions of fractal type. Then, the in-phase coherence of the dynamics of the complex fluid structural units induces various operational procedures in the description of such dynamics: special cubics with SL(2R)-type group invariance, special differential geometry of Riemann type associated to such cubics, special apolar transport of cubics, special harmonic mapping principle, etc. In such a manner, a possible scenario toward chaos (a period-doubling scenario), without concluding in chaos (nonmanifest chaos), can be mimed. Full article

The aim of the study was to create a mathematical model useful for monitoring the release of bioactive aldehydes covalently bonded to the chitosan by reversible imine linkage, considered as a polymer–drug system. For this purpose, two hydrogels were prepared by the acid condensation reaction of chitosan with the antifungal 2-formyl-phenyl-boronic acid and their particularities; influencing the release of the antifungal aldehyde by shifting the imination equilibrium to the reagents was considered, i.e., the supramolecular nature of the hydrogels was highlighted by polarized light microscopy, while scanning electron microscopy showed their microporous morphology. Furthermore, the in vitro fungicidal activity was investigated on two fungal strains and the in vitro release curves of the antifungal aldehyde triggered by the pH stimulus were drawn. The theoretical model was developed starting from the hypothesis that the imine-chitosan system, both structurally and functionally, can be assimilated, from a mathematical point of view, with a multifractal object, and its dynamics were analyzed in the framework of the Scale Relativity Theory. Thus, through Riccati-type gauges, two synchronous dynamics, one in the scale space, associated with the fungicidal activity, and the other in the usual space, associated with the antifungal aldehyde release, become operational. Their synchronicity, reducible to the isomorphism of two SL(2R)-type groups, implies, by means of its joint invariant functions, bioactive aldehyde compound release dynamics in the form of “kink–antikink pairs” dynamics of a multifractal type. Finally, the theoretical model was validated through the experimental data. Full article

The construction of first integrals for $SL(2,\mathbb{R})$-invariant nth-order ordinary differential equations is a non-trivial problem due to the nonsolvability of the underlying symmetry algebra $\mathfrak{sl}(2,\mathbb{R}).$ Firstly, we provide for $n=2$ an explicit expression for two non-constant first integrals through algebraic operations involving the symmetry generators of $\mathfrak{sl}(2,\mathbb{R}),$ and without any kind of integration. Moreover, although there are cases when the two first integrals are functionally independent, it is proved that a second functionally independent first integral arises by a single quadrature. This result is extended for $n>2,$ provided that a solvable structure for an integrable distribution generated by the differential operator associated to the equation and one of the prolonged symmetry generators of $\mathfrak{sl}(2,\mathbb{R})$ is known. Several examples illustrate the procedures. Full article

Traffic microsimulation models use the movement of individual driver-vehicle-units (DVUs) and their interactions, which allows a detailed estimation of the traffic noise using Common Noise Assessment Methods (CNOSSOS). The Dynamic Traffic Noise Assessment (DTNA) methodology is applied to real traffic situations, then compared to on-field noise levels from measurement campaigns. This makes it possible to determine the influence of certain local traffic factors on the evaluation of noise. The pattern of distribution of vehicles along the avenue is related to the logic of traffic light control. The analysis of the inter-cycles noise variability during the simulation and measurement time shows no influence from local factors on the prediction of the dynamic traffic noise assessment tool based on CNOSSOS. A multifractal approach of acoustic waves propagation and the source behaviors in the traffic area are implemented. The novelty of the approach also comes from the multifractal model’s freedom which allows the simulation, through the fractality degree, of various behaviors of the acoustic waves. The mathematical backbone of the model is developed on Cayley–Klein-type absolute geometries, implying harmonic mappings between the usual space and the Lobacevsky plane in a Poincaré metric. The isomorphism of two groups of SL(2R) type showcases joint invariant functions that allow associations of pulsations–velocities manifolds type. Full article

The dynamics of a transient plasma generated by laser ablation on a Mg target was investigated by means of the Langmuir probe method and fractal analysis. The empirical data showcased the presence of an oscillatory behavior at short expansion times (<1 μs) characterized by two oscillation frequencies and a classical behavior for longer evolution times. Space- and time-resolved analysis was implemented in order to determine main plasma parameters like the electron temperature, plasma potential, or charged particle density. In the motion fractal paradigm, a theoretical model was built for the description of laser-produced plasma dynamics expressed through fractal-type equations. The calibration of such dynamics was performed through a fractal-type tunneling effect for physical systems with spontaneous symmetry breaking. This allows both the self-structuring of laser-produced plasma in two structures based on its separation on different oscillation modes and the determination of some characteristics involved in the self-structuring process. The mutual conditionings between the two structures are given as joint invariant functions on the action of two isomorph groups of SL(2R) type through the Stoler-type transformation, explicitly given through amplitude self-modulation. Full article