The terminology and notation used here have been introduced in the following articles: [9], [8], [7], [3], [4], [6], [5], [2], and [1]. For simplicity we adopt the following rules: W , X, Y , Z will denote sets, D will denote a non-empty set, f will denote a function, and x, y will be arbitrary. Let B be a set. We say that B is a Tarski-Class if and only if: for all X, Y such that X ∈ B and Y ⊆...

Finding matching customers for a given product based on individual user’s preference is critical for many applications, especially in e-commerce. Recently, the reverse top-k query is proposed to return a number of customers who regard a given product as one of the k most favorite products based on a linear model. Although a few “hot” products can be returned to some customers via reverse top-k ...

Let N be a k-net of prime order p. We find bounds on the p-rank of (the point-line incidence matrix of) N for k ∈ {3, 4}, and observe connections between the p-rank and certain structural properties of N . Implications for the study of finite projective planes are described. 1. Loops and 3-Nets of Prime Order Let (L, ∗) be a loop of prime order p. The 3-net N = N (L) coordinatized by L is the p...

Young flattenings, introduced by Landsberg and Ottaviani, give determinantal equations for secant varieties and provide lower bounds for border ranks of tensors. We find special monomial-optimal Young flattenings that provide the best possible lower bound for all monomials up to degree 6. For degree 7 and higher these flattenings no longer suffice for all monomials. To overcome this problem we ...

In problems involving approximation, completion, denoising, dimension reduction, estimation, interpolation, modeling, order reduction, regression, etc, we argue that the near-universal practice of assuming that a function, matrix, or tensor (which we will see are all the same object in this context) has low rank may be ill-justified. There are many natural instances where the object in question...