We present an analytical solution for the isothermal gas sphere under the truncated M-fractional derivative (TMDISO), a two-parameter generalization of the classical Lane-Emden equation, introducing the derivative order α and truncation parameter β via the Mittag–Leffler function. Using a recursive power-series scheme accelerated by the Euler–Abel transformation and Padé approximants, we derive the M-truncated Emden function u( x;α ,β) and validate the method against the classical ( α = β = 1 ) solution to within | Δ u| < 10^ - 3for x ≤ 13 . Decreasing α or β systematically damps the growth of the Emden function and broadens the density distribution relative to the classical isothermal result. We apply the model to compute the visual extinction coefficient A_V as a function of projected impact parameter b for a Bonnor–Ebert-like geometry and fit the profiles to the near-infrared extinction map of Barnard 68. A systematic grid search over α∈[ 0.60,1.00] and β∈[ 0.60,1.00] (45 models, 9 × 5 grid) identifies a global minimum at α = 1.00, β = 0.70, yielding χ^2 = 7.394 (reduced χ^2 = 0.569 ), a 21 percent improvement over the classical model ( χ^2 = 9.363 ). The best-fit parameters ρ_c = 3.21 × 10^ - 19 gcm^ - 3 , n_c = 8.22 × 10^4 cm^ - 3 , and R = 18,410AU are fully consistent with independent dust-emission estimates for Barnard 68. The preference for α = 1.00 establishes that the improvement arises solely from the Mittag–Leffler truncation through Γ( β + 1)^2 in the series coefficients, encoding non-local physical processes such as sub-thermal turbulence or magnetic pressure support without invoking a fractional-order spatial derivative. A linear radial perturbation analysis yields a fractional Sturm–Liouville eigenvalue equation showing that all α < 1 configurations are dynamically unstable ( σ_0^2 < 0 ), while the best-fit case yields σ_0^2 = + 0.003889 and D = 14.045 , placing Barnard 68 precisely on the Bonnor–Ebert marginal stability threshold and independently corroborating the photometric best-fit parameters.
Drilling has made it possible to explore the Earth’s interior, but still only to very shallow depths on the scale of the planet. Since 1960s, scientists have worked to reach the crust–mantle boundary known as the Moho. This boundary representing the foundations of continents is usually beyond the reach of our present-day technology. However, in some places on Earth, it is possible to reach the crust–mantle frontier without drilling to its original depth. The Ivrea-Verbano Zone in the Italian Alps is the prime target to explore the Moho at 2-km depth for the first time in the history of science. This is because the collision between tectonic plates generating the Alps brought the crust–mantle boundary along with a cohesive and preserved crustal section to such a shallow depth. Thus, under conditions of low pressure and temperature, our present-day drilling technology can put down a borehole to this depth. Based on the collaborative effort involving > 100 scientists worldwide, the ICDP (International Continental Scientific Drilling Program) DIVE (Drilling the Ivrea-Verbano zonE) project tackles key questions on the chemistry and architecture of the lithosphere, the geophysical signatures, insights into the deep biosphere, and the nature of the Moho. Here our perspective contribution presents new speculations about the Ivrea-Verbano Zone Moho transition zone based on the lithological column explored and sampled in the two boreholes and the petrophysical properties (rocks’ compressional wave velocities and densities) of the pre-Permian and Lower Permian lower crust lithologies.
We explore and compare the dynamics of two transcomplex recombination reactions XeI– + Cs+ → CsI + Xe and XeCs+ + I– → CsI + Xe in the collision energy range from 0.1 to 2.5 eV. The reactions are simulated by the quasiclassical trajectory method on a semiempirical diabatic potential energy surface proposed in 2021. Recombination of an alkali cation M+ and a halide anion X– via the weakly bound negative ionic complex RX– with a neutral atom R has never been studied before. The results of trajectory simulation include the excitation and opacity functions, the distributions of the vibrational, rotational, and total internal energies of the cesium iodide molecule, and the distributions of the energy of the flying apart of the products CsI and Xe. All the principal dynamical characteristics of the two reactions mentioned turn out to be quite close. However, at almost all the collision energies, the cross section of ion recombination via XeI– is larger than the cross section of recombination via XeCs+. The main motivation for this work is the recently discovered peculiar deterministic chaos in the transcomplex recombination reactions RCs+ + Br– → CsBr + R with R = Kr, Xe, Hg.
Agricultural practices are increasingly challenged by biotic and abiotic stresses, which compromise crop yield and quality. Weeds, in particular, present a persistent problem, necessitating the exploration of sustainable alternatives to chemical herbicides. This study investigates the allelopathic effects of aqueous extracts from three wild plant species on common purslane (Portulaca oleracea L.). The research involved comprehensive phytochemical analyses and an evaluation of 19 germination and growth parameters. Notably, the Arisarum vulgare (O.Targ.Tozz). extract (AVE) exhibited strong inhibitory effects on purslane germination and growth, closely mirroring the action of glyphosate, a widely used synthetic herbicide. The inhibitory effects of AVE were attributed to its rich content of polyphenols, flavonoids, and condensed tannins. In contrast, extracts from Urtica urens (L.) (UUE) and Chamaemelum mixtum (L.) (CME) showed weaker and variable effects, with some parameters suggesting potential stimulatory responses under the tested conditions. The originality of this research lies in the novel selection of Arisarum vulgare (O.Targ.Tozz) and the thorough investigation of its herbicidal properties, offering new insights for sustainable weed management.
The (2+1)-dimensional Boussinesq equation describes complex wave propagation in shallow water with varying amplitudes, where dispersion and nonlinearity are important factors. The equation captures wave interactions in multiple dimensions. The Hirota bilinear method and the extended hyperbolic function approach are used to construct lump wave solutions, their interactions with periodic, strip and double-strip waves, and traveling wave solutions. Stability analysis is carried out to determine whether the solutions develop singularities or remain bounded. Bifurcation theory and sensitivity analysis for dynamical systems are applied to obtain phase diagrams of the governing model. Bifurcation theory of planar dynamical systems is also used to examine the model’s qualitative analysis. Three-dimensional graphical representations of the exact solutions show periodic features such dark V-shaped, singular bell-shaped, bright, unique periodic, and periodic soliton solutions. The findings provide analytical insight into the nonlinear dynamics and stability features of the model. Both approaches highlight the rich solution structure and its applications under various physical conditions.
