In the paper A non-implication between fragments of Martin’s Axiom related to a property which comes from Aronszajn trees [1], Proposition 2.7 is not true. To avoid this error and correct Proposition 2.7, the definition of the property R1,א1 is changed. In [1], all proofs of lemmas and theorems but Lemma 6.9 are valid about this definition without changing the proofs. We give a new statement an...

Let (∞, π∞) be an incident point-plane pair of PG(3, q). A tetrad with respect to (∞, π∞) is a set {X,Y, Z,W} of points of PG(3, q) \ π∞ such that {∞, X, Y, Z,W} is a cap of PG(3, q), ∞ ∈ 〈X,Y, Z〉 and W 6∈ 〈X,Y, Z〉. A set Θ of ovoids of PG(3, q) is tetradic with respect to (∞, π∞) if each ovoid contains ∞, has tangent plane π∞ and is such that each tetrad with respect to (∞, π∞) is contained in...

We consider the notion of property testing as applied to computational geometry. We aim at developing efficient algorithms which determine whether a given (geometrical) object has a predetermined property Q or is “far” from any object having the property. We show that many basic geometric properties have very efficient testing algorithms, whose running time is significantly smaller than the obj...

We consider the notion of property testing as applied to computational geometry. We aim at developing efficient algorithms which determine whether a given (geometrical) object has a predetermined property Q or is “far” from any object having the property. We show that many basic geometric properties have very efficient testing algorithms, whose running time is significantly smaller than the obj...

We study conjugacy closed loops (CC-loops) and power-associative CC-loops (PACC-loops). If Q is a PACC-loop with nucleus N , then Q/N is an abelian group of exponent 12; if in addition Q is finite, then |Q| is divisible by 16 or by 27. There are eight nonassociative PACC-loops of order 16, three of which are not extra loops. There are eight nonassociative PACC-loops of order 27, four of which h...