Projective Hjelmslev planes (PH-planes) are a generalization of projective planes in which each point-pair is joined by at least one line and, dually, each line-pair has a nontrivial intersection. Multiply joined points (and multiply intersecting lines) are called neighbor points (and neighbor lines). By hypothesis, the neighbor relations of a PH-plane A are equivalence relations which induce a...

Abstract. Szemerédi’s Theorem states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman generalized it, showing that sets of integers with positive upper density contain arbitrarily long polynomial configurations; Szemerédi’s Theorem corresponds to the linear case of this polynomial theorem. We focus on the case farthest f...

We make explicit the exponential bound on degrees of polynomials appearing in Effective Quillen-Suslin Theorem, and apply it jointly with Hilbert-Burch Theorem to show that syzygy module a sequence m n variables defining complete intersection ideal grade two is free, basis can be computed bounded degrees. In known cases, these bounds improve previous results.

We introduce intersection schemes (a generalization of uniform oriented matroids of rank 3) to describe the combinatorial properties of arrangements of pseudocircles in the plane and on closed orientable surfaces. Similar to the FolkmanLawrence topological representation theorem for oriented matroids we show that there is a one-to-one correspondence between intersection schemes and equivalence ...

Oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses. By self-intersection number one means the minimum number of transversal self-intersection points of representatives of the class. We prove that if a class is chosen at random from among all classes of m le...

The purpose of this paper is to define the notion of “real” intersection between paths drawn on the 3d digital boundary of a connected object. We consider two kinds of paths for different adjacencies, and define the algebraic number of oriented intersections between these two paths. We show that this intersection number is invariant under any homotopic transformation we apply on the two paths. ...

We show that the infinite matroid intersection conjecture of NashWilliams implies the infinite Menger theorem proved recently by Aharoni and Berger. We prove that this conjecture is true whenever one matroid is nearly finitary and the second is the dual of a nearly finitary matroid, where the nearly finitary matroids form a superclass of the finitary matroids. In particular, this proves the inf...

Although intersection homology lacks a ring structure, certain expressions (called uniform) in the intersection homology of an irreducible projective variety X always give the same value, when computed via the decomposition theorem on any resolution Xr → X . This paper uses uniform (and non-uniform) expressions to define what is believed to be the usual intersection homology (and its local-glob...

(i= 1 . 2, . . .) such that the intersection n A, contains a perfect subset i=1 (and is therefore of power 2No) . They asked for what Hausdorff measure functions (k(i) is it possible to choose the subsequence to make the intersection set (1 A„,, of positive -measure . In the present note We show that the strongest possible result in this direction is true . This is given by the following; theor...

We prove that for a topological spaceX with the property that H∗(U) = 0 for ∗ ≥ d and every open subset U of X, a finite family of open sets in X has nonempty intersection if for any subfamily of size j, 1 ≤ j ≤ d+1, the (d−j)-dimensional homology group of its intersection is zero. We use this theorem to prove new results concerning transversal affi ne planes to families of convex sets