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Variational geometries describing corrugated graphene sheets are proposed. The isothermal thermomechanical properties of these sheets are described by a two-dimensional Weyl space. The equation that couples the Weyl geometry with isothermal distributions of the temperature of graphene sheets, is formulated. This material space is observed in a three-dimensional orthogonal configurational point space as regular surfaces which are endowed with a thermal state vector field fulfilling the isothermal thermal state equation. It enables to introduce a non-topological dimensionless thermal shape parameter of non-developable graphene sheets. The properties of the congruence of lines generated by the thermal state vector field are discussed.
For each simple euclidean Jordan algebra $V\!,$ we introduce the analogue of hamiltonian, angular momentum and Laplace-Runge-Lenz vector in the Kepler problem. Being referred to as the universal hamiltonian, universal angular momentum and universal Laplace-Runge-Lenz vector respectively, they are elements in (essentially) the TKK (Tits-Kantor-Koecher) algebra of $V$ and satisfy commutation relations similar to the ones for the hamiltonian, angular momentum and Laplace-Runge-Lenz vector in the Kepler problem. We also give some examples of Poisson realization of the TKK algebra, along with the resulting classical generalized Kepler problems. For the simplest simple euclidean Jordan algebra (i.e., $\mathbb R$), we give examples of operator realization for the TKK algebra, along with the resulting quantum generalized Kepler problems.
This paper explores some applications of Lusternik-Schnirelmann theory and its recent offshoots. In particular, we show how the LS category of real projective space leads to the Borsuk-Ulam theorem and the Brouwer fixed point theorem. After the development of some LS categorical tools, we also show the importance of LS category in understanding the Arnold conjecture on fixed points of Hamiltonian diffeomorphisms. We then examine ways in which LS category fits into the framework of differential geometry. In particular, we give a refinement of Bochner's theorem on the first Betti number of a non-negatively Ricci-curved space and a Bochner-like corollary to a recent theorem of Kapovitch-Petrunin-Tuschmann. Finally, we survey the new LS categorical notion of topological complexity and its relation to the motion planning problem in robotics.
The governing equation of the Helfrich spontaneous-curvature model is the Helfrich equation. It is a coordinate free equation that describes the equilibrium shapes of biological (fluid) membranes. We make use of the conformal metric representation of the Helfrich equation and by applying the symmetry group reduction method we obtain a translationally invariant solution. Based on that solution, we derive analytic expressions for the position vector of special cylindrical equilibrium shapes. Plots of the graphs of some closed directrices of these shapes are presented.