This paper mainly concerns the area growth and bottom spectrum of complete stable minimal surfaces in a three-dimensional manifold with scalar curvature bounded from below. When ambient is Euclidean space, by an elementary argument, it shown directly stability inequality that such grows exactly as plane. Consequently, must be flat, well-known result due to Fisher-Colbrie Schoen well do Carmo Pe...

Let $$M^{n+1}$$ be a closed manifold of dimension $$3\le n+1\le 7$$ . We show that for $$C^\infty $$ -generic metric g on M, to any connected, closed, embedded, 2-sided, stable, minimal hypersurface $$S\subset (M,g)$$ corresponds sequence hypersurfaces $$\{\Sigma _k\}$$ scarring along S, in the sense area and Morse index $$\Sigma _k$$ both diverge infinity and, when properly renormalized, conve...

We propose a solution to a long-standing problem in the foundations of well-founded semantics (WFS) for logic programs. The problem addressed is this: which (non-modal) logic can be considered adequate for well-founded semantics in the sense that its minimal models (appropriately defined) coincide with the partial stable models of a logic program? We approach this problem by identifying the HT ...

-The set of all solutions of the minimal design problem (MDP) is presented in parametric form. This result is obtained by first deriving a parametric representation of all minimal bases of a particular vector space.From this the set of all solutions of the MDP are obtained. Then conditipns are placed on the. parameters, which conditions define set of stable solutions of the MDP. Solutions to th...

Quantitative stability of optimal values and solution sets to stochastic programming problems is studied when the underlying probability distribution varies in some metric space of probability measures. We give conditions that imply that a stochastic program behaves stable with respect to a minimal information (m.i.) probability metric that is naturally associated with the data of the program. ...