This paper has no novel learning or statistics: it is concerned with making a wide class of preexisting statistics and learning algorithms computationally tractable when faced with data sets with massive numbers of records or attributes. It briefly reviews the static AD-tree structure of Moore and Lee (1998), and offers a new structure with more attractive properties: (1) the new structure scal...

Classically, null designs were defined on the poset of subsets of a given finite set (boolean algebra). A null design is defined as a collection of weighted k-subsets such that the sum of the weights of ksubsets containing a t-subset is 0 for every t-subset, where 0 ≤ t < k ≤ n. Null designs are useful to understand designs or to construct new designs from a known one. They also deserve researc...

In this paper we deal with the null controllability problem for the heat equation with a memory term by means of boundary controls. For each positive final time T and when the control is acting on the whole boundary, we prove that there exists a set of initial conditions such that the null controllability property fails.

The additivity spectrum ADD(I) of an ideal I ⊂ P(I) is the set of all regular cardinals κ such that there is an increasing chain {Aα : α < κ} ⊂ I with ∪α<κAα / ∈ I. We investigate which set A of regular cardinals can be the additivity spectrum of certain ideals. Assume that I = B or I = N , where B denotes the σ-ideal generated by the compact subsets of the Baire space ωω , and N is the ideal o...

A Group Labeled Graph is a pair (G,Λ) where G is an oriented graph and Λ is a mapping from the arcs of G to elements of a group. A (not necessarily directed) cycle C is called non-null if for any cyclic ordering of the arcs in C, the group element obtained by ‘adding’ the labels on forward arcs and ‘subtracting’ the labels on reverse arcs is not the identity element of the group. Non-null cycle...

With each of the classical tree–like forcings adjoining a new real, one can associate a σ– ideal on the reals in a natural way. For example, the ideal of Marczewski null sets s corresponds to Sacks forcing S, while the ideal of nowhere Ramsey sets r corresponds to Mathias forcing R. We show (in ZFC) that none of these ideals is included in any of the others. We also discuss Mycielski’s ideal P2...

We work in set-theory without choice ZF. Given a commutative field K, we consider the statement D(K): “On every non null K-vector space there exists a non-null linear form.” We investigate various statements which are equivalent to D(K) in ZF. Denoting by Z2 the two-element field, we deduce that D(Z2) implies the axiom of choice for pairs. We also deduce that D(Q) implies the axiom of choice fo...