The concept of entropy is not uniquely relevant to the statistical mechanics but, among others, it can play pivotal role in the analysis of a time series, particularly the stock market data. In this area, sudden events are especially interesting as they describe abrupt data changes with potentially long-lasting effects. Here, we investigate the impact of such events on the entropy of financial time series. As a case study, we assume data of the Polish stock market, in the context of its main cumulative index, and discuss it for the finite time periods before and after outbreak of the 2022 Russian invasion of Ukraine. This analysis allows us to validate the entropy-based methodology in assessing changes in the market volatility, as driven by the extreme external factors. We show that some qualitative features of such market variations can be well captured in terms of the entropy. In particular, the discussed measure appears to highlight differences between data of the two considered timeframes in agreement with the character of their empirical distributions, which is not always the case in terms of the conventional standard deviation. Moreover, the entropy of cumulative index averages, qualitatively, the entropies of composing assets, suggesting capability for describing interdependencies between them. The entropy is also found to exhibit signatures of the upcoming extreme events. To this end, the role of recent war in shaping the current economic situation is briefly discussed.

In the framework of the mimetic approach, we study the \(f(R,R_{\mu \nu }R^{\mu \nu })\) gravity with the Lagrange multiplier constraint and the scalar potential. We introduce field equations for the discussed theory and overview their properties. By using the general reconstruction scheme we obtain the power law cosmology model for the \(f(R,R_{\mu \nu }R^{\mu \nu })=R+d(R_{\mu \nu }R^{\mu \nu })^p\) case as well as the model that describes symmetric bounce. Moreover, we reconstruct model, unifying both matter dominated and accelerated phases, where ordinary matter is neglected. Using inverted reconstruction scheme we recover specific \(f(R,R_{\mu \nu }R^{\mu \nu })\) function which give rise to the de-Sitter evolution. Finally, by employing the perfect fluid approach, we demonstrate that this model can realize inflation consistent with the bounds coming from the BICEP2/Keck array and the Planck data. We also discuss the holographic dark energy density in terms of the presented \(f(R,R_{\mu \nu }R^{\mu \nu })\) theory. Thus, it is suggested that the introduced extension of the mimetic regime may describe any given cosmological model.

Motivated by the growing interest in the nonmetricity-matter couplings, we develop the scalar–tensor formulation of recently introduced f(Q, T) gravity, where Q is the nonmetricity and T is the trace of the energy–momentum tensor. The main properties of the scalar–tensor formalism for the Friedmann–Lemaître–Robertson–Walker (FLRW) Universe are discussed, and we introduce an appropriate set of dynamical variables to analyze the cosmic evolution of the scalar–tensor f(Q, T) cosmology as a dynamical system. By considering two distinct cosmic fluids, namely matter and radiation, we have demonstrated that the cosmological phase space exhibits the typical curvature-dominated, radiation-dominated, matter-dominated, and exponentially accelerated fixed points. Furthermore, under an appropriate set of initial conditions compatible with the current observations from the Planck satellite, our analysis shows that the scalar–tensor f(Q, T) successfully yields models indistinguishable from the $$\Lambda $$ Λ CDM cosmology and compatible with the weak-field solar system dynamics, without the inclusion of a cosmological constant $$\Lambda $$ Λ . Thus, the theory introduced herein may be regarded as a suitable candidate to describe the cosmological dynamics of the Universe.

Recently introduced $$f(\mathcal {G},T)$$f ( G , T ) theory is generalized by adding dependence on the arbitrary scalar field $$\phi $$ϕ and its kinetic term $$(\nabla \phi )^2$$( ∇ ϕ ) 2 , to explore non-minimal interactions between geometry, scalar and matter fields in context of the Gauss–Bonnet theories. The field equations for the resulting $$f\left( \mathcal {G},\phi ,(\nabla \phi )^2,T\right) $$f G , ϕ , ( ∇ ϕ ) 2 , T theory are obtained and show that particles follow non-geodesic trajectories in a perfect fluid surrounding. The energy conditions in the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime are discussed for the generic function $$f\left( \mathcal {G},\phi ,(\nabla \phi )^2,T\right) $$f G , ϕ , ( ∇ ϕ ) 2 , T . As an application of the introduced extensions, using the reconstruction techniques we obtain functions that satisfy common cosmological models, along with the equations describing energy conditions for the reconstructed $$f\left( \mathcal {G},\phi ,(\nabla \phi )^2,T\right) $$f G , ϕ , ( ∇ ϕ ) 2 , T gravity. The detailed discussion of the energy conditions for the de Sitter and power-law spacetimes is provided in terms of the fixed kinetic term i.e. in the $$f\left( \mathcal {G},\phi ,T\right) $$f G , ϕ , T case. Moreover, in order to check viability of the reconstructed models, we discuss the energy conditions in the specific cases, namely the $$f(R,\phi ,(\nabla \phi )^2)$$f ( R , ϕ , ( ∇ ϕ ) 2 ) and $$f=\gamma (\phi ,X)\mathcal {G}+\mu T^{1/2}$$f = γ ( ϕ , X ) G + μ T 1 / 2 approaches. We show, that for the appropriate choice of parameters and constants, the energy conditions can be satisfied for the discussed scenarios.