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.

This article is concerned with the question of whether an energy bound implies a genus for pseudo-holomorphic curves in almost complex manifolds. After reviewing what known dimensions other than six, we establish new result this direction dimension six; particular, symplectic Calabi–Yau 3–folds. The proof relies on compactness and regularity theorems currents.

We examine the gravitational wave frequencies from neutron stars during thermal evolution, adopting relativistic Cowling approximation. particularly focus on star models, in which direct Urca (rapid cooling process) does not work, without superfluidity and superconductivity. For such curve hardly depends equation of state (EOS) as well mass star, while we show that strongly depend both properti...

We introduce split principles and show that their negations provide simple combinatorial characterizations of large cardinal properties. As examples, we show how inaccessiblity, weak compactness, subtlety, almost ineffability and ineffability can be characterized. Two-cardinal versions of these principles allow us to characterize when the two-cardinal versions of these properties hold: when κ i...