L. On determinant notation

L-determinant class and approximation of L-Betti numbers

A standing conjecture in L-cohomology is that every finite CW complex X is of L-determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class G of groups containing e.g. all extensions of residually finite groups with amenable quotients, all residually amenable groups and free products of these. If, in addition, X is L-acyclic, we also prove that the L-...

Sentence Object Notation: Multilingual sentence notation based on Wordnet

The representation of sentences is a very important task. It can be used as a way to exchange data interapplications. One main characteristic, that a notation must have, is a minimal size and a representative form. This can reduce the transfer time, and hopefully the processing time as well. Usually, sentence representation is associated to the processed language. The grammar of this language a...

Relationships Among PL, #L, and the Determinant

Recent results by Toda, Vinay, Damm, and Valiant have shown that the complexity of the determinant is characterized by the complexity of counting the number of accepting computations of a nondeterministic logspace-bounded machine. (This class of functions is known as #L.) By using that characterization and by establishing a few elementary closure properties, we give a very simple proof of a the...

Relationships Among PL, #L, and the Determinant

The Leibniz notation on manifolds

which is extremely strange the first time one encounters it. After all, why not just write V (x, y) = ye1 − xe2 where e1 = (1, 0), e2 = (0, 1)? What do partial derivatives have to do with anything? (That was the question I asked.) However, if viewed in the right way, it is not so strange. For if a vector field V is given, then one natural thing that can be done with it is to differentiate a sca...