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The effect of a heat source and temperature gradient on Brinkman–Bènard Triple-Diffusive magneto-Marangoni (BBTDMM) convection in a two-layer system is investigated. The two-layer system is horizontally infinite and is surrounded on all sides by adiabatic boundaries. It is exposed to basic uniform and non-uniform temperature profiles and heat sources. The appropriate eigenvalues and thermal Marangoni numbers (TMNs), which depend on temperature and concentration, are obtained for the temperature profiles (TPs) for lower rigid and higher free boundaries with surface tension. The transformed system of ordinary differential equations is solved by using an exact technique. For all three TPs, the impact of significant relevant parameters on these eigenvalues, and hence on BBTDMM convection, are investigated versus the thermal ratio. It is observed that, by increasing the values of the modified internal Rayleigh number for the fluid layer and the solute Marangoni numbers, the Darcy number, and the viscosity ratio for the set of physical parameters chosen in the study, one can postpone BBTDMM convection. Higher values of the modified internal Rayleigh numbers for the porous layer augment BBTDMM convection.

In this paper, we concentrate on a control system with a non-local condition that is governed by a Hilfer fractional neutral stochastic evolution hemivariational inequality (HFNSEHVI). By using concepts of the generalized Clarke sub-differential and a fixed point theorem for multivalued maps, we first demonstrate adequate requirements for the existence of mild solutions to the concerned control system. Then, using limited Lagrange optimal systems, we demonstrate the existence of optimal state-control pairs that are regulated by an HFNSEHVI with a non-local condition. In order to demonstrate the existence of fixed points, the symmetric structure of the spaces and operators that we create is essential. Without considering the uniqueness of the control system’s solutions, the best control results are established. Lastly, an illustration is used to demonstrate the major result.

The resolution of the acceleration and jerk vectors of a particle moving on a space curve in the Euclidean 3-space is considered. By applying this resolution and Siacci’s theorem, alternative resolutions of acceleration and jerk vectors are derived based on the quasi-frame. In the osculating plane, the acceleration vector is resolved as the sum of its tangential and radial components. In addition, in the osculating and rectifying planes, the jerk vector is resolved along the tangential direction and two special radial directions. The maximum permissible speed on a space curve at all trajectory points is established via the jerk vector formula. Finally, some examples are presented to illustrate how the results work.