In this article we completely determine the possible dimensions of integral points and holomorphic curves on the complement of a union of hyperplanes in projective space. Our main theorems generalize a result of Evertse and Györy, who determined when all sets of integral points (over all number fields) on the complement of a union of hyperplanes are finite, and a result of Ru, who determined wh...

Conventional binary classiication trees such as CART either split the data using axis-aligned hyperplanes or they perform a compu-tationally expensive search in the continuous space of hyperplanes with unrestricted orientations. We show that the limitations of the former can be overcome without resorting to the latter. For every pair of training data-points, there is one hyperplane that is orth...

Let V be a vector space of dimension d over a field K and let A be a central arrangement of hyperplanes in V . To answer a question posed by K. Aomoto, P. Orlik and H. Terao construct a commutative K-algebra U(A) in terms of the equations for the hyperplanes ofA. In the course of their work the following question naturally occurred: ◦ Is U(A) determined by the intersection lattice L(A) of the h...

One says that Veldkamp lines exist for a point-line geometry Γ if, for any three distinct (geometric) hyperplanes A, B and C (i) A is not properly contained in B and (ii) A ∩ B ⊆ C implies A ⊂ C or A ∩ B = A ∩ C. Under this condition, the set V of all hyperplanes of Γ acquires the structure of a linear space – the Veldkamp space – with intersections of distinct hyperplanes playing the role of l...

Abstract In hyperbolic space, the angle of intersection and distance classify pairs totally geodesic hyperplanes. A similar algebraic invariant classifies hyperplanes in Einstein universe. dimension 3, symplectic splittings a 4-dimensional real vector space model is determinant. The classification contributes to complete disjointness criterion for crooked surfaces 3-dimensional