Let Ω be a disk of radius R in the plane. A set F of closed unit disks contained in Ω forms a maximal packing if the unit disks are pairwise disjoint and the set is maximal: i.e., it is not possible to add another disk to F while maintaining the packing property. A point p is hidden within the “forest” defined by F if any ray with apex p intersects some disk of F : that is, a person standing at...

An open (resp., closed) subset A of a topological space (X, T ) is called F-open F-closed) set if cl(A)\A A\int(A)) finite set. In this work, we study the main properties these definitions and examine relationships between F-closed sets with other kinds such as regularly open, closed, sets. Then, establish some operators F-interior, F-closure, F-derived...etc., using At end introduce F-continuo...

We call a foliation (M,F) on manifold M chaotic if it is topologically transitive and the union of closed leaves dense in M. A foliated not assumed to be compact. The foliations can considered as multidimensional generalization dynamical systems sense Devaney. For covered by fibrations we prove that only its global holonomy group chaotic. introduce concept integrable Ehresmann connection for na...

A pseudo-disk is a subset of the plane that is homeomorphic to a closed disk. Consider a family F of pseudo-disks whose interiors are pairwise disjoint, and such that any two pseudo-disks intersect in at most one point. If any point of the plane is contained in at most k elements of F (with k sufficiently large), then we show that the elements of F can be colored with at most k + 1 colors so th...

Scott conjectured in [6] that the class of graphs with no induced subdivision of a given graph is χ-bounded. We verify his conjecture for maximal triangle-free graphs. Let F be a graph. We denote by Forb∗(F ) the class of graphs with no induced subdivision of F . A class G of graphs is χ-bounded if there exists a function f such that every graph G of G satisfies χ(G) ≤ f(ω(G)), where χ and ω re...

In this paper, new probability estimates are derived for ideal lattice codes from totally real number fields using ideal class Dedekind zeta functions. In contrast to previous work on the subject, it is not assumed that the ideal in question is principal. In particular, it is shown that the corresponding inverse norm sum depends not only on the regulator and discriminant of the number field, bu...

If F ⊆ N N is an analytic family of pairwise eventually different functions then the following strong maximality condition fails: For any countable H ⊆ N N, no member of which is covered by finitely many functions from F , there is f ∈ F such that for all h ∈ H there are infinitely many integers k such that f (k) = h(k). However if V = L then there exists a coanalytic family of pairwise eventua...

The paper addresses the question of when the least-xed-point operator, in a cartesian closed category of domains, is characterised as the unique dinatural transformation from the exponentiation bifunctor to the identity functor. We give a suucient condition on a cartesian closed full subcategory of the category of algebraic cpos for the characterisation to hold. The condition is quite mild, and...

An irreducible open 3-manifold W is R 2-irreducible if it contains no non-trivial planes, i.e. given any proper embedded plane in W some component of W ? must have closure an embedded halfspace R 2 0; 1). In this paper it is shown that if M is a connected, P 2-irreducible, open 3-manifold such that 1 (M) is nitely generated and the universal covering space f M of M is R 2-irreducible, then eith...