In this paper, we address a new version of Darboux frame using a common tangent vector
field to a surface along a curve and call this frame the rotation minimizing Darboux frame (RMDF).
Then, we give the parametric equation due to the RMDF frame for a sweeping surface and show that the
parametric curves on this surface are curvature lines. Consequently, necessary and sucient conditions
for sweeping surfaces to be developable ruled surfaces are derived. Also, we analyze the conditions
when the resulting developable surface is a cylinder, cone or tangential surface. We also provide some
examples to illustrate the main results.

In this paper using the Blaschke approach we generalized the Bertrand curves to spacelike
ruled and developable surfaces. It is proved that, every spacelike ruled surface have a Bertrand oset
if and only if an equation should be fulfilled among their dual integral invariants. Consequently, some
new relationships and theorems for the developability of the Bertrand osets of spacelike ruled surfaces
are outlined.

In this work, we introduce a time-like ruled surface in one-parameter hyperbolic dual spherical motions. This provides the ability
to derive some formulae of surface theory into line spaces. Then, a time-like Plücker conoid associated with the motion has been
obtained, and its kinematic geometry is researched in detail. Consequently, a characterization for a time-like ruled surface to be a
constant Disteli-axis is derived and investigated in detail. At last, we have discussed some special cases which lead to some special
time-like ruled surfaces such as the time-like helicoids, Lorentzian sphere, and time-like cone.

In this paper, the relationships among the instantaneous invariants of a one-parameter Lorentzian spatial movement and the
local invariants of the axodes are studied. New proofs for Euler–Savary, and Disteli formulae are given which demonstrate
the elegance and efficiency of the E. Study map in Lorentzian spatial kinematics. Consequently, two spacelike line congruences
are introduced and their spatial equivalent are examined in detail.

Two-dimensional nanomaterials such as graphene nanosheets and molybdenum disulfide nanoflakes were easily hybridized with spinel Mn₃O₄ in a single deposition process at low temperature by a facile kinetic spray approach under low vacuum conditions using a nano-particle deposition system. The heterostructured Mn₃O₄–2D hybrid nanocomposites (NCs) were utilized to enhance the reaction kinetics of oxygen evolution. Hybrid NCs of Mn₃O₄ with MoS₂ and graphene showed nanoflakes and nanosheet morphologies, respectively. The synergy improvement in nanocomposites of Mn₃O₄-graphene nanosheets and Mn₃O₄₋MoS₂ nanoflakes was examined by x-ray photoemission spectroscopy and analysis of Raman spectra that results in a decrease in OER overpotential to 289 mV at 10 mA⋅cm⁻² for Mn₃O₄-graphene nanosheets and 295 mV for Mn₃O₄₋MoS₂ nanoflakes. The estimated Tafel slope strongly decreased from 88 mV⋅dec⁻¹of pure Mn₃O₄ nanorods to around 44 mV⋅dec⁻¹ for hybrid NCs. This behavior indicates improved charge transfer kinetics due to the decrease in charge transfer resistance. Furthermore, all fabricated nanostructure electrocatalysts exhibited very good stability for prolonged galvanostatic polarization up to 50 h.