In this paper we prove that, given a compact four dimensional smooth Riemannian manifold (M, g) with smooth boundary there exists a metric conformal to g with constant T -curvature, zero Q-curvature and zero mean curvature under generic and conformally invariant assumptions. The problem amounts to solving a fourth order nonlinear elliptic boundary value problem (BVP) with boundary conditions gi...

The theory of spinors is developed for locally anisotropic (la) spaces, in brief la-spaces, which in general are modeled as vector bundles provided with nonlinear and distinguished connections and metric structures (such la-spaces contain as particular cases the Lagrange, Finsler and, for trivial nonlinear connections, Kaluza-Klein spaces). The la-spinor differential geometry is constructed. Th...

In this work, we construct the general solution to the Heat Equation (HE) and to many tensor structures associated to the Heat Equation, such as Symmetries, Lagrangians, Poisson Brackets (PB) and Lagrange Brackets (LB), using newly devised techniques that may be applied to any linear equation (e.g., Schrödinger Equation in field theory, or the small-oscillations problem in mechanics). In partic...

Necessary conditions of extremum (from the times of Fermat and Lagrange till our times) for extremal problems where smoothness is interlaced with convexity, and some type of regularity takes place, correspond to a unique general principle, which is due to Lagrange. This report is devoted to the Lagrange principle in the theory of optimization.

The study of fractional variational problems with derivatives in the sense of Caputo is a recent subject, the main results being Agrawal’s necessary optimality conditions of Euler-Lagrange and respective transversality conditions. Using Agrawal’s Euler-Lagrange equation and the Lagrange multiplier technique, we obtain here a Noether-like theorem for fractional optimal control problems in the se...

The idea of probabilistic metric space was introduced by Menger and he showed that probabilistic metric spaces are generalizations of metric spaces. Thus, in this paper, we prove some of the important features and theorems and conclusions that are found in metric spaces. At the beginning of this paper, the distance distribution functions are proposed. These functions are essential in defining p...

Affine quantum gravity involves (i) affine commutation relations to ensure metric positivity, (ii) a regularized projection operator procedure to accomodate firstand second-class quantum constraints, and (iii) a hard-core interpretation of nonlinear interactions to understand and potentially overcome nonrenormalizability. In this program, some of the less traditional mathematical methods employ...

The tradeoff between energy efficiency (EE) and spectral efficiency (SE) under the smart grid with joint energy harvesting and grid power supply is considered. The price factor in economics is used to propose the economic EE and SE. Then a new EE-SE tradeoff metric called the EE-SE adaptive tradeoff (ESAT) is built under the fluctuating harvesting energy. Furthermore, the offline optimization p...

We prove the existence of solutions to the 1-harmonic flow –i.e., the formal gradient flow of the total variation of a vector field with respect to the L-distance– from a domain of R into a hyper-octant of the N -dimensional unit sphere, S + , under homogeneous Neumann boundary conditions. In particular, we characterize the lower-order term appearing in the Euler-Lagrange formulation in terms o...

In the framework of Galilei classical and mechanics (i.e. , generally relativistic classical mechanics on a spacetime with absolute time) developed by Jadczyk and Modugno, we analyse systematically the relations between symmetries of the geometric objects. We show that the (holonomic) infinitesimal symmetries of the cosymplectic structure and of its potentials are also symmetries of spacelike m...