Some Co-Birkhoff Type Theorems

Poincaré-birkhoff-witt Theorems

We sample some Poincaré-Birkhoff-Witt theorems appearing in mathematics. Along the way, we compare modern techniques used to establish such results, for example, the Composition-Diamond Lemma, Gröbner basis theory, and the homological approaches of Braverman and Gaitsgory and of Polishchuk and Positselski. We discuss several contexts for PBW theorems and their applications, such as Drinfeld-Jim...

On Hopf algebra structures over free operads

The operad Lie can be constructed as the operad of primitives PrimAs from the operad As of associative algebras. This is reflected by the theorems of Friedrichs, Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore. We replace the operad As by families of free operads P, which include the operad Mag freely generated by a noncommutative non-associative binary operation and the operad of Stasheff poly...

On New Forms of the Ergodic Theorem

We present generalizations of the classical Birkhoff and von Neumann ergodic theorems, where the time average is replaced by a more general average, including some density.

Some Fixed Point Theorems for Weakly Compatible Multivalued Mappings Satisfying Some General Contractive Conditions of Integral Type

Birkhoff-kellogg Theorems on Invariant Directions for Multimaps

This paper presents Birkhoff-Kellogg type theorems on invariant directions for a large class of maps. A number of results which will enable to deduce results for upper semicontinuous maps which are either (a) Kakutani, (b) acyclic, (c) O’Neill, or (d) admissible (strongly) in the sense of Gorniewicz are given. The results in this paper, when the map is compact, complement and extend the previou...