A result of Balas and Yu (1989) states that the number of maximal independent sets of a graph G is at most δ+1, where δ is the number of pairs of vertices in G at distance 2, and p is the cardinality of a maximum induced matching inG. In this paper, we give an analogue of this result for hypergraphs and, more generally, for subsets of vectors B in the product of n lattices L = L1 × · · · × Ln, ...

The Fibonacci sequence {Fn} is defined by the recurrence relation Fn = Fn−1+ Fn−2, for n ≥ 2 with F0 = 0 and F1 = 1. The Lucas sequence {Ln} , considered as a companion to Fibonacci sequence, is defined recursively by Ln = Ln−1 + Ln−2, for n ≥ 2 with L0 = 2 and L1 = 1. It is well known that F−n = (−1)Fn and L−n = (−1)Ln, for every n ∈ Z. For more detailed information see [9],[10]. This paper pr...

Aims and Objectives: This article proposes the “scalpel model,” a new model of third and additional language (L3/Ln) acquisition. The model aims to identify and examine what happens beyond the initial state of acquisition and what factors may influence change from one state of knowledge to another. Methodology: The article briefly examines the currently proposed hypotheses and models and evalua...

A series of pentanuclear Ln(III) clusters, [Ln5(μ4-OH)(μ3-OH)4(L1)(acac)6] (H4L1 = p-tert-butylthiacalix[4]arene; acac = acetylacetonate; Ln = Dy, Ho, Er) and [Ln5(μ5-OH)(μ3-OH)4(L1)(L2)2(acac)2(CH3OH)2] (H3L2 = 5,11,17,23-tetrakis(1,1-dimethylethyl)-25,26,27-trihydroxy-28-methoxy thiacalix[4]arene; Ln = Dy, Ho, Er), have been synthesized based on the thiacalix[4]arene ligand. All of these comp...

In this paper, we …nd families of (0; 1; 1) tridiagonal matrices whose determinants and permanents equal to the negatively subscripted Fibonacci and Lucas numbers. Also we give complex factorizations of these numbers by the …rst and second kinds of Chebyshev polynomials. 1. Introduction The well-known Fibonacci sequence, fFng ; is de…ned by the recurrence relation, for n 2 Fn+1 = Fn + Fn 1 (1.1...

We have developed a new class of lanthanide nano-clusters that self-assemble using flexible Schiff base ligands. Cd-Ln and Ni-Ln clusters, [Ln8Cd24(L1)12(OAc)39Cl7(OH)2] (Ln = Nd, Eu), [Eu8Cd24(L1)12(OAc)44], [Ln8Cd24(L2)12(OAc)44] (Ln = Nd, Yb, Sm) and [Nd2Ni4(L3)2(acac)6(NO3)2(OH)2], were constructed using different types of flexible Schiff base ligands. These molecular nano-clusters exhibit ...

Using Smolyak’s construction [5], we derive a new algorithm for approximating multivariate functions over bounded or unbounded regions in R with the error measured in a weighted L1-norm. We provide upper bounds for the algorithm’s cost and error for a class of functions whose mixed first order partial derivatives are bounded in L1-norm. In particular, we prove that the error and the cost (measu...

The unsymmetrical tridentate benzimidazole-pyridine-carboxamide units in ligands L1-L4 react with trivalent lanthanides, Ln(III), to give the nine-co-ordinate triple-helical complexes [Ln(Li)3]3+ (i = 1-4) existing as mixtures of C3-symmetrical facial and C1-symmetrical meridional isomers. Although the beta13 formation constants are 3-4 orders of magnitude smaller for these complexes than those...

We consider a positive measure on [0,∞) and a sequence of nested spaces L0 ⊂ L1 ⊂ L2 · · · of rational functions with prescribed poles in [−∞, 0]. Let {φk}k=0, with φ0 ∈ L0 and φk ∈ Lk \ Lk−1, k = 1, 2, . . . be the associated sequence of orthogonal rational functions. The zeros of φn can be used as the nodes of a rational Gauss quadrature formula that is exact for all functions in Ln · Ln−1, a...