Abstract. We establish the lower bound p2t(e, e) exp(−t1/3), for the large times asymptotic behaviours of the probabilities p2t(e, e) of return to the origin at even times 2t, for random walks associated with finite symmetric generating sets of solvable groups of finite Prüfer rank. (A group has finite Prüfer rank if there is an integer r , such that any of its finitely generated subgroup admit...

We consider the problem of the presence of short cycles in the graphs of nonzero elements of matrices which have sublinear rank and nonzero entries on the main diagonal, and analyze the connection of these properties to the rigidity of matrices. In particular, we exhibit a family of matrices which shows that sublinear rank does not imply the existence of triangles. This family can also be used ...

In this paper I propose B-Rank, an efficient ranking algorithm for recommender systems. B-Rank is based on a random walk model on hypergraphs. Depending on the setup, B-Rank outperforms other state of the art algorithms in terms of precision, recall ∼ (19%− 50%) and inter list diversity ∼ (20%− 60%). B-Rank captures well the difference between popular and niche objects. The proposed algorithm p...

A group of finite Morley rank is a group whose definable subsets have a notion of dimension satisfying several basic axioms [BN94, p. 57]. Such groups arise naturally in model theory; initially as א1-categorical groups, i.e. as groups determined up to isomorphism by their first order theory, and here the simple groups correspond exactly. More recently groups of finite Morley rank have appeared ...

We define the determinantal divisor rank of an integral matrix to be the number of invariant factors which equal 1. Some properties of the determinantal divisor rank are proved, which are analogous to known properties of the usual rank. These include the Frobenious inequality for the rank of a product and a relation between the rank of a submatrix of a matrix and that of its complementary subma...

We show that determining Kapranov rank of tropical matrices is not only NP-hard over any field but if Diophantine equations over the rational numbers is undecidable, determining Kapranov rank over the rational numbers is also undecidable. We prove that Kapranov rank of tropical matrices is not bounded in terms of tropical rank, answering a question of Develin, Santos, and Sturmfels [4].

The purpose of this research is to investigate appropriate collaborative forecasting models, both for acceptable accuracy and effectiveness for information sharing via the Internet to partners. Several forecasting models were investigated in this research. For comparing the models, five years historical data are collected from a lathe machine manufacturer in Taiwan, and one of its partners – a ...