For a general subcritical second-order elliptic operator P in a domain Ω⊂Rn (or noncompact manifold), we construct Hardy-weight W which is optimal in the following sense.
If ( M, g ) is an open manifold having a limiting Carleman weight, then some conformal multiple of the metric g admits a parallel unit vector field.
LLE preserves the local properties of the manifold by means of a 'reconstruction weights operation', which locally linearizes by solving a constrained least-squares problem the manifold in the neighborhood of each sample (Roweis and Saul, 2000).
In the automotive industry, 3XX.X castings have replaced cast iron in transmission cases, intake manifolds, engine blocks, and cylinder heads because the reduced weight improves fuel economy.
The consequences are manifold and comprise phenomena as different as body size and weight, glucose metabolism and homeostasis or female fertility.
end{cases} (1.2) In [1], Wu proved that equation (1.2) involving a sign-changing weight function has at least two solutions by using the Nehari manifold.
These manifold requirements result in a complex design process with optimisation scenarios often involving contrary objectives in terms of weight, stiffness and sound radiation.
Instead, Seligmann [7] used the molecular weight as a proxy for biosynthetic cost, reasoning that this may take into account all the manifold effects of producing larger, more complex amino acids.
The selected Gaussian weight is optimal in a certain sense, and it has a deep connection to the heat kernel on a manifold that gives the general solution to the heat equation.
In this paper those Riemannian manifolds which admit limiting Carleman weights, were characterized.
