We show that there are only finitely many Diophantine quadruples, that is, sets of four positive integers {a1, a2, a3, a4} such that aiaj +1 is a square for all 1 ≤ i < j ≤ 4, consisting of Fibonacci numbers.

Modern variational analysis provides a sophisticated unification of convex and smooth optimization theory, achieving striking generality but at the expense of possible pathology. The general theory must handle highly irregular and oscillatory functions and sets, and yet, on the other hand, a rich family of concrete instances involve no such pathology. In particular, from a variety of variationa...

In this paper, we study the rate of convergence of the cyclic projection algorithm applied to finitely many semi-algebraic convex sets. We establish an explicit convergence rate estimate which relies on the maximum degree of the polynomials that generate the semi-algebraic convex sets and the dimension of the underlying space. We achieve our results by exploiting the algebraic structure of the ...

A division ring of positive characteristic with countably many pure types is a field. Wedderburn showed in 1905 that finite fields are commutative. As for infinite fields, we know that superstable [1, Cherlin, Shelah] and supersimple [4, Pillay, Scanlon, Wagner] ones are commutative. In their proof, Cherlin and Shelah use the fact that a superstable field is algebraically closed. Wagner showed ...

Endocannabinoids act as retrograde messengers that, by inhibiting neurotransmitter release via presynaptic CB(1) cannabinoid receptors, regulate the functionality of many synapses. In addition, the endocannabinoid system participates in the control of neuron survival. Thus, CB(1) receptor activation has been shown to protect neurons from acute brain injury as well as in neuroinflammatory condit...

A theorem of Sperner [2] states that a collection of subsets of (l,..., n), no two ordered by inclusion, contains at most ( &,) sets. How many twoelement chains A cB of subsets of {I,..., n} can be found such that sets in different chains are not related? More generally, we seek to determinef,(n), defined to be the maximum m such that there exist subsets A(i,j) 5 {l,..., n}, 1 < i < m, 0 <j < k...

Disjoint union is a partial binary operation returning the union of two sets if they are disjoint and undefined otherwise. A disjoint-union partial algebra of sets is a collection of sets closed under disjoint unions, whenever they are defined. We provide a recursive first-order axiomatisation of the class of partial algebras isomorphic to a disjointunion partial algebra of sets but prove that ...

This paper is one in a series that investigates topological measures on locally compact spaces. A measure defined open and closed sets; it finitely additive the collection of sets, inner regular outer sets. We examine semisolid sets give way constructing from solid-set functions compact, Hausdorff, connected, connected For spaces we present two simpler methods than currently available one. exam...

The purpose of this paper is to prove that the symbolic Rees rings ideals defining certain finite sets points in projective plane over an algebraically closed field are finitely generated using a ring theoretical criterion which known as Huneke's criterion.

The asymmetric unit of the title polymeric compound, [Cu(C8H4NO2)2(C10H14N2O)(H2O)] n , contains one Cu(II) atom, one coordinating water mol-ecule, two 4-cyano-benzoate (CB) ligands and one coordinating N,N-di-ethyl-nicotinamide (DENA) mol-ecule. The DENA ligand acts as a bis-monodentate ligand, while the CB anions are monodentate. Two O atoms of the CB ligands, one O atom of the water mol-ecul...