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.

the minimal prime decomposition for semiprime ideals is defined and studied on z-ideals of c(x). the necessary and sufficient condition for existence of the minimal prime decomposition of a z-ideal / is given, when / satisfies one of the following conditions: (i) / is an intersection of maximal ideals. (ii) i is an intersection of o , s, when x is basically disconnected. (iii) i=o , when x x ha...

Abstract. A topological group H is called ω -narrow if for every neighbourhood V of it’s identity element there exists a countable set A such that V A = H = AV. A semigroup G is called a generalized group if for any x ∈ G there exists a unique element e(x) ∈ G such that xe(x) = e(x)x = x and for every x ∈ G there exists x − 1 ∈ G such that x − 1x = xx − 1 = e(x). Also le...

The program Reverse Mathematics (RM for short) seeks to identify the axioms necessary prove theorems of ordinary mathematics, usually working in language second-order arithmetic L 2 . A major theme RM is therefore study structures that are countable or can be approximated by sets. Now, sets must represented sequences here, because higher-order definition ‘countable set’ involving injections/bij...

We characterize the ω-stable theories all of whose countable models admit decidable presentations. In particular, we show that for a countable ω-stable T , every countable model of T admits a decidable presentation if and only if all n-types in T are recursive and T has only countably many countable models. We further characterize the decidable models of ω-stable theories with countably many co...

1 Probability Theory over Countable Spaces Definition 1 (Countable Probability Space). A Countable Probability Space is composed of the tuple (Ω,Pr). • Countable Sample Space Ω The sample space is a set of outcomes for a particular experiment. “Countable” means that either the size of Ω is finite or there is a one-to-one and onto correspondence between the the elements of Ω and the set of Natur...

In recent years, much work in descriptive set theory has been focused on the Borel complexity of naturally occurring classification problems, in particular, the study of countable Borel equivalence relations and their structure under the quasi-order of Borel reducibility. Following the approach of Louveau and Rosendal in [8] for the study of analytic equivalence relations, we study countable Bo...