The transitive closure of a reflexive, symmetric, analytic relation is an analytic equivalence relation. Does some smaller class contain the transitive closure of every reflexive, symmetric, closed relation? An essentially negative answer is provided here. Every analytic equivalence relation on an arbitrary Polish space is Borel embeddable in the transitive closure of the union of two smooth Bo...

The concepts of closure systems and closure operations in lattice theory are basic and applied to many fields in mathematics and theoretical computer science. In this paper authors find out a suitable definition of closure systems in Dedekind categories, and thereby give an equivalence proof for closure systems and closure operations in Dedekind categories.

Characterizations of Egalitarian Characterizations of Egalitarian Characterizations of Egalitarian Characterizations of Egalitarian Binary Relations as Transitive Closures Binary Relations as Transitive Closures Binary Relations as Transitive Closures Binary Relations as Transitive Closures-with a special reference to Lo with a special reference to Lo with a special reference to Lo with a speci...

Recent studies show that the underlying amnioserosa works actively to help bring together the two epithelial sheets and close the embryonic hole during dorsal closure in fruitfly development.

We prove that the notions of finite strict singularity, strict singularity and weak compactness coincide for operators defined on various spaces: the disc algebra, subspaces of C(K) with reflexive annihilator and subspaces of the Morse-TransueOrlicz space Mq (Ω, μ) with q > 2.

BACKGROUND The role of percutaneous closure of patent foramen ovale (PFO) in patients with cryptogenic stroke or transient ischemic attack remains controversial. Registry data have suggested considerable benefit of closure over medical therapy, but the prospective, randomized CLOSURE I trial found no benefit for device closure. METHODS We compared patients enrolled into CLOSURE I to off-label...