Bayesian Machine Learning (BML) and strong lensing time delay (SLTD) techniques are used in order to tackle the $$H_{0}$$H 0 tension in f(T) gravity. The power of BML relies on employing a model-based generative process which already plays an important role in different domains of cosmology and astrophysics, being the present work a further proof of this. Three viable f(T) models are considered: a power law, an exponential, and a squared exponential model. The learned constraints and respective results indicate that the exponential model, $$f(T)=\alpha T_{0}\left( 1-e^{-p T / T_{0}}\right) $$f ( T ) = α T 0 1 - e - p T / T 0 , has the capability to solve the $$H_{0}$$H 0 tension quite efficiently. The forecasting power and robustness of the method are shown by considering different redshift ranges and parameters for the lenses and sources involved. The lesson learned is that these values can strongly affect our understanding of the $$H_{0}$$H 0 tension, as it does happen in the case of the model considered. The resulting constraints of the learning method are eventually validated by using the observational Hubble data (OHD).

Seasonal snow cover can serve as an informative single-season indicator of atmospheric deposition in industrial urban areas because it accumulates airborne contaminants during the winter period. A total of 55 snow samples were collected across the urban area, and the liquid phase was analyzed for major and trace elements using instrumental elemental analysis with defined detection limits and measurement uncertainty. Descriptive statistics, background comparisons, and integrated pollution indicators were used to characterize the spatial variability and intensity of contamination. The results showed that the median concentrations of most analyzed elements did not exceed the reference limits; however, aluminum and iron exhibited elevated levels, with aluminum reaching 1.1–27 times and iron 1.0–3 times the reference values. Median concentrations included 270 μg L⁻¹ for Al, 118 μg L⁻¹ for Fe, 30 μg L⁻¹ for Zn, 11.5 μg L⁻¹ for Ni, and 7.3 μg L⁻¹ for Pb. The obtained data indicate a heterogeneous pollution pattern across Pavlodar and suggest the combined influence of mineral dust, urban-industrial emissions, road-dust resuspension, and natural inputs on snow chemistry. Because the study is based on one winter sampling campaign, the results should be interpreted as a single-season assessment of snow-cover contamination rather than as evidence of long-term temporal stability.

This paper describes the soliton surfaces approach to the Witten-Dijkgraaf-E.Verlinde-H. Verlinde (WDVV) equation. We constructed the surface associated with the WDVV equations using Sym-Tafel formula, which gives a connection between the classical geometry of manifolds immersed in R ^m and the theory of solitons. The so-called Sym-Tafel formula simplifies the explicit reconstruction of the surface from the knowledge of its fundamental forms, unifies various integrable nonlinearities and enables one to apply powerful methods of the soliton theory to geometrical problems. The soliton surfaces approach is very useful in construction of the so-called integrable geometries. Indeed, any class of soliton surfaces is integrable. Geometrical objects associated with soliton surfaces (tangent vectors, normal vectors, foliations by curves etc.) usually can be identified with solutions to some nonlinear models (spins, chiral models, strings, vortices etc.) [1], [2]. We consider the geometry of surfaces immersed in Euclidean spaces. Such soliton surfaces for the WDVV equation for n = 3 case with an antidiagonal metric η₁₁ = 0 are considered, and first and second fundamental forms of soliton surfaces are found for this case. Also, we study an area of surfaces for the WDVV equation for n = 3 case with an antidiagonal metric η₁₁ = 0.

Understanding the evolution of dark energy poses a significant challenge in modern cosmology, as it is responsible for the universe's accelerated expansion. In this study, we focus on a specific $f(T)$ cosmological model and analyze its behavior using observational data, including 31 data points from the CC dataset, 1048 points from the Pantheon SNe Ia samples, and 6 points from the BAO dataset. By considering a linear $f(T)$ model with an additional constant term, we derive the expression for the Hubble parameter as a function of cosmic redshift for non-relativistic pressureless matter. We obtain the best-fit values for the Hubble constant, $H_0$, and the model parameters $α$ and $β$, indicating a stable model capable of explaining late-time cosmic acceleration without invoking a dark energy component. This is achieved through modifying field equations to account for the observed accelerated expansion of the universe.

Motivated by anomalies in cosmic microwave background observations, we investigate the implications of $f(Q, T)$ gravity in Bianchi type-I spacetime, aiming to characterize the universe's spatially homogeneous and anisotropic properties. By using a linear combination of non-metricity $Q$ and the energy-momentum tensor trace $T$, we parametrize the deceleration parameter and derive the Hubble solution, which we then impose in the Friedmann equations of $f(Q, T)$ gravity. Bayesian analysis is employed to find the best-fit values of model parameters, with $1-σ$ and $2-σ$ contour plots illustrating the constraints from observational data, including $H(z)$ data and the Pantheon+ sample. Our analysis reveals a transition from a decelerated to an accelerated expansion phase, with the present deceleration parameter indicating an accelerating universe. The energy density gradually decreases over time, approaching zero for the present and future, indicating continuous expansion. The anisotropic pressure, initially notably negative, transitions to slightly negative values, suggesting the presence of dark energy. The evolving equation of state parameter $ω$ exhibits behavior akin to phantom energy, influenced by spacetime anisotropy. Violations of the null energy condition and the strong energy condition imply phantom-like behavior and accelerated expansion.