Intermediate planar algebra revisited

Solomon's Descent Algebra Revisited

Starting from a non-standard definition, the descent algebra of the symmetric group is investigated. Homomorphisms into the tensor product of smaller descent algebras are defined. They are used to construct the irreducible representations and to obtain the nilpotency index of the radical.

Planar Subgraph Isomorphism Revisited

The problem of Subgraph Isomorphism is defined as follows: Given a pattern H and a host graph G on n vertices, does G contain a subgraph that is isomorphic to H? Eppstein [SODA 95, J’GAA 99] gives the first linear time algorithm for subgraph isomorphism for a fixed-size pattern, say of order k, and arbitrary planar host graph, improving upon the O(n √ )-time algorithm when using the “Color-codi...

Hu’s Primal Algebra Theorem revisited

It is shown how Lawvere’s one-to-one translation between Birkhoff’s description of varieties and the categorical one (see [6]) turns Hu’s theorem on varieties generated by a primal algebra (see [4], [5]) into a simple reformulation of the classical representation theorem of finite Boolean algebras as powerset algebras.

Elementary algebra revisited: Randomized algorithms

We look at some simple algorithms for elementary problems in algebra that yield dramatic efficiency improvements over standard methods through randomization. The randomized algorithms are, in a sense, “obvious”. Their formal statement was delayed partly by the need for rigorous analysis, but more so by the need to re-think traditional approaches to elementary algorithms. We illustrate this phil...

Random cubic planar graphs revisited