This article explores two distinct function spaces: Hilbert spaces and mixed-Orlicz–Zygmund spaces with variable exponents. We first examine the relational properties of Hilbert spaces in a tensorial framework, utilizing self-adjoint operators to derive key results. Additionally, we extend a Maclaurin-type inequality to the tensorial setting using generalized convex mappings and establish various upper bounds. A non-trivial example involving exponential functions is also presented. Next, we introduce a new function space, the mixed-Orlicz–Zygmund space $\left({\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)\right)$, which unifies Orlicz–Zygmund spaces of integrability and sequence spaces. We investigate its fundamental properties including separability, compactness, and completeness, demonstrating its significance. This space generalizes the existing structures, reducing to mixed-norm Lebesgue spaces when $\beta =0$ and to classical Lebesgue spaces when $\mathtt{q}=\infty ,\beta =0$. Given the limited research on such spaces, our findings contribute valuable insights to the functional analysis.

This paper studies the effects of resource limitations, immunity decay, and delays on an Ebola epidemic model and an optimal control strategy. The model includes two types of delays: a delay in the incubation period of infected individuals and a delay in treatment. Conditions for a Hopf bifurcation at the endemic equilibrium are verified, with its direction and stability analyzed via normal form theory and the center manifold theorem. We also studied the optimal control problem for the SIRD delay model using educational campaigns and Ebola survivors’ immunity as control variables. Furthermore, we formulate an optimization problem based on Pontryagin’s maximum principle. This problem uses a modified Runge-Kutta approach with delays to discover the best control strategy to reduce infections and intervention costs. Finally, simulation results confirm analytical conclusions and show the practical implications of the optimum Ebola control plan using the dde23 MATLAB R2024a built-in solver and DDE-Biftool.

Plasma arc machining (PAM) has emerged as a cost-effective alternative to laser cutting forsheet-metal fabrication. However, thermo-fluid interactions during plasma cutting of ultra-thinbrass sheets remain insufficiently understood because of the material’s high thermal sensitivityand susceptibility to heat-induced damage. In particular, the combined effects of arc current andtraverse speed on kerf formation and surface integrity have not been systematically clarified forthin brass sections. Therefore, this study investigates the thermo-metallurgical behavior of 0.8mm brass sheets during plasma arc cutting using a Taguchi L16(4²) experimental design. Theinteraction between arc current and traverse speed was evaluated through kerf geometry, surfaceroughness, heat-affected zone (HAZ) evolution, microstructural changes, and hardnessdistribution to determine stable low-damage cutting conditions.The results showed that cutting performance is governed by the interaction between heat inputand melt-ejection stability. Higher arc current increased localized melting, resulting in greaterdross adhesion, kerf widening, and grain coarsening within the HAZ. Increasing traverse speedreduced thermal exposure and improved kerf quality; however, excessively high speeds slightlydegraded surface finish because of unstable molten-material evacuation. Microstructural analysisindicated that thermal accumulation controlled grain growth behavior, whereas hardnessremained nearly unchanged due to shallow thermal penetration in thin-sheet machining. Optimalcutting conditions were achieved at 30 A and 1500 cm/min, producing minimal kerf deteriorationand limited microstructural damage. These findings provide insight into heat-input control andmelt-flow stability during plasma cutting of thermally sensitive copper-based alloys.