Abstract We establish the equivalence of Gromov ellipticity and subellipticity in algebraic category.

A variant of compactness designated as statistical has been introduced. Statistically compact subsets n-dimensional Euclidean space Rn have characterized by using Cantor-Bendixson derivative and it is further extended to the product RN.

In this paper, we extend the concept of scaled Gromov hyperbolic graph, originally developed for the Thin Triangle Condition (TTC), to the computationally simplified, but less intuitive, Four-Point Condition (FPC). The original motivation was that, for a large but finite network graph to enjoy some of the typical properties to be expected in negatively curved Riemannian manifolds, the delta mea...

In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes what has until now been an ad hoc collection of results by many authors. In addition to providing much greater generality than any prior work, our results also give new insight into the nature of the connection between Dio...

Hausdorff and Gromov distances are introduced and treated in the context of categories enriched over a commutative unital quantale V . The Hausdorff functor which, for every V-category X, provides the powerset of X with a suitable V-category structure, is part of a monad on V-Cat whose Eilenberg-Moore algebras are order-complete. The Gromov construction may be pursued for any endofunctor K of V...

The Palais-Smale Condition C holds for the Yang-Mills functional on principal bundles over compact manifolds of dimension ≤ 3. This was established by S. Sedlacek [17] and C. Taubes [18] Proposition 4.5 using the compactness theorem of K. Uhlenbeck [20]; see also [23]. It is well known that Condition C fails for Yang-Mills over compact manifolds of dimension ≥ 4. The example of SO(3)-invariant ...

We establish an existence result for semilinear elliptic problems with the associated functional not satisfying the Palais-Smale condition. The nonlinearity of our problem does not satisfy the Ambrosetti-Rabinowitz condition.