The Lonngren wave equation governs the propagation of electrical signals in telegraph lines containing tunnel diodes. In this paper, wave solutions of the equation are constructed and the dynamics of its reduced form are analysed. The generalized projective Riccati equation method provides hyperbolic and trigonometric solution families corresponding to localized and periodic waves. A Riccati-based sub-equation approach is then applied, in which the trial function is composed of Riccati solutions arranged in layers, with the spatial and temporal variables entering independently, and the unknown coefficients are determined by coefficient matching. This approach produces bright, singular, dark-singular, periodic and rational wave profiles. The influence of the model parameters on the amplitude, width and propagation of these waves is shown graphically for representative cases. The reduced equation is also studied as a planar Hamiltonian system, whose phase portraits change qualitatively at a critical relation between the wave parameters and the model coefficients, where the two equilibria merge into a single degenerate point. Under external periodic forcing, the system exhibits both regular and chaotic responses depending on the forcing parameters and initial conditions, as confirmed by Poincaré sections and Lyapunov exponents.
Hydnum repandum is an edible mushroom with reported antimicrobial and antiproliferative activities. In this study, selenium nanoparticles synthesized using H. repandum extract (Hr-SeNPs) were physicochemically characterized and evaluated for antimicrobial and anticancer activity. The optimized formulation was obtained at a 2:1 ascorbic acid: sodium selenite molar ratio. UV spectroscopy showed a maximum at 224 nm, ICP-MS detected 1.271 ± 0.061 mg/mL selenium in the post-centrifugation supernatant, and EDX confirmed elemental selenium. DSC showed thermal transitions at approximately 30, 50, 100, and 223 °C, while DLS revealed a mean particle diameter of 118.9 nm with a PDI of 0.254; SEM showed predominantly spherical particles. Hr-SeNPs exhibited broad antimicrobial activity, producing inhibition zones of 20.4–39.2 mm against bacteria and 23.6–38.4 mm against yeasts, markedly exceeding those of empty SeNPs (9.6–14.1 and 8.6–15.1 mm, respectively). The strongest responses were observed against Escherichia coli (39.2 mm) and Candida glabrata (38.4 mm). In HepG2 cells, Hr-SeNPs reduced viability in a dose- and time-dependent manner, with IC50 values of 227.2 µg/mL at 24 h and 136.2 µg/mL at 48 h, while HUVEC viability remained above 50
Sustainable fertilization strategies have the potential to improve the nutritional quality of vegetables while reducing dependence on synthetic fertilizers. This study evaluated the effects of five fertilizer treatments (control, chemical fertilizer, organomineral fertilizer, cattle manure, and vermicompost) on the nutraceutical composition of fresh bean (Phaseolus vulgaris L. cv. Sazova 1949) over two growing seasons using a randomized complete block design. The objective was to determine how different fertilizer sources influence leaf and seed mineral composition, total antioxidant capacity, total phenolic content, organic acid profiles, and phenolic compound composition. Fertilizer effects varied among traits and years. Vermicompost significantly increased leaf manganese and zinc concentrations, whereas organomineral fertilizer significantly increased seed protein content and total phenolic content in 2019. Organic acid and phenolic compound profiles showed compound-specific responses, with organomineral fertilizer generally producing higher concentrations of citric, malic, succinic, chlorogenic, and ellagic acids. In contrast, fertilizer effects on total antioxidant capacity and several mineral traits were not consistently significant across years, indicating a strong influence of environmental conditions. Overall, organomineral fertilizer and vermicompost generally produced more favorable nutraceutical profiles than the control and conventional chemical fertilizer. Because the experiment was conducted at a single location over two growing seasons, further multi-environment studies are needed to validate the consistency and wider applicability of these findings.
This study investigates a (3+1)-dimensional modified Korteweg-de Vries equation describing the interaction of cubic nonlinearity and third-order dispersion. Two complementary analytical approaches are employed to characterize its nonlinear wave structures. First, the Riccati mapping method is applied to the nonlinear ordinary differential equation obtained through a wave transformation. In addition, the Riccati mapping neural network method is implemented directly on the governing partial differential equation using a layered trial function with a Riccati-generated activation function, while symbolic computation determines the associated network parameters. The proposed framework yields various exact solutions, including dark waves, kink structures, and singular periodic waves, together with the parametric conditions for their existence. Three-dimensional surfaces, contour plots, and two-dimensional profiles illustrate representative solutions and their propagation characteristics. The reduced ordinary differential equation is then reformulated as an equivalent planar Hamiltonian system, and its equilibrium points are systematically classified. Bifurcation analysis shows that the sign of a composite parameter involving the dispersion relation and three directional coefficients determines whether the origin behaves as a saddle or a centre. These regimes correspond to homoclinic and heteroclinic trajectories and connect directly with solitary and kink-type solutions. Finally, an external periodic perturbation is introduced to examine complex dynamics. The perturbed system displays quasi-periodic and irregular responses, analyzed through phase portraits, time series, Poincaré sections, sensitivity analysis, and Lyapunov exponents. The study integrates exact solution techniques, bifurcation theory, and dynamical analysis into a unified framework for nonlinear wave characterization and clarifies the underlying dynamical mechanisms.
We study linear systems governed by the weighted Caputo fractional derivative of order α∈ (0,1) with weight w and increasing function g. The weight is an arbitrary positive function, and σ (t)=g(t)-g(a) . A known conjugation relation and a change of variable turn the system into a classical Caputo system for the weighted state w(t)x(t) in the intrinsic time σ (t) . Matignon’s theorem therefore transfers to the whole weighted class. The weighted state tends to zero for every initial value if and only if every eigenvalue λ of the system matrix satisfies λ 0 and |λ |>απ /2 . It stays bounded if and only if every eigenvalue satisfies λ =0 or |λ |≥απ /2 and every critical eigenvalue is semisimple. The new results lie beyond this reduction. We prove that σ (t)^αw(t)x(t) converges to an explicit nonzero limit, so the decay rate σ (t)^-α is attained by every nontrivial solution, and we give an explicit exponential growth bound when some eigenvalue λ 0 satisfies |λ |<απ /2 . Under the stability condition, x(t) itself tends to zero if and only if w(t)σ (t)^α→∞ : the weight can create or destroy asymptotic decay of the unweighted state, while the stability sector of the weighted state stays fixed. Numerical experiments confirm the theory, including the sharp constants. A two-layer model of anomalous thermal relaxation serves as a model-admissibility illustration, without experimental data: it shows which pairs (w, g) are compatible with a return to equilibrium.
