We describe Abello’s acyclic sets of linear orders [1] as the permutations visited by commuting equivalence classes of maximal reduced decompositions. This allows us to strengthen Abello’s structural result: we show that acyclic sets arising from this construction are distributive sublattices of the weak Bruhat order. This, in turn, shows that Abello’s acyclic sets are, in fact, the same as Cha...

Quantum orders in symmetric spin liquids. Quantum orders and.These lectures are an introduction to world of quantum fluids and to phenomena. To motivate the study of quantum turbulence one has to appreciate the.In order to understand how nucleation proceeds in quantum liquids such as. Cavitation in liquid helium, but I also consider water, liquid hydrogen, wet.As you can imagine, the number of ...

Data Envelopment Analysis (DEA) technique now widely use for efficiency evaluation of a set of Decision Making Units (DMUs). As regards of the necessity for ranking efficient units different DEA models presented each of which has advantages and rank efficient units from special aspects. Note that all the existing ranking models have disadvantages, as well and there is not a model in which all t...

This note provides an alternative proof of a known result on the existence of collections of orderings generating intersection quasi-orderings. Instead of proving the result from first principles, it is illustrated how it can be obtained by making use of an analogous relationship between partial orders and linear orders. Journal of Economic Literature Classification Nos.: C60, D11, D71. ∗ I tha...

In this paper, it is proved that all simple $K_4$-groups of type $L_2(q)$ can be characterized by their maximum element orders together with their orders. Furthermore, the automorphism groups of simple $K_4$-groups of type $L_2(q)$ are also considered.