Let (Ln)n≥1 be the sequence of Lucas numbers, defined recursively by L1≔1, L2≔3, and Ln+2≔Ln+1+Ln, for every integer n≥1. We determine asymptotic behavior loglcm(L1+s1,L2+s2,…,Ln+sn) as n→+∞, (sn)n≥1 a periodic in {−1,+1}. also carry out same analysis independent uniformly distributed random variables These results are numbers-analogs previous obtained author Fibonacci numbers.

This contribution investigates Ln(III) complexes formed with a small ditopic ligand, L1, and their structural, thermodynamic and photophysical properties. The spectrophotometric and NMR titrations evidence the triangular assemblies [Ln(3)(L1-H)(3)](6+) at stoichiometric conditions and their properties are discussed in relation to L2-containing analogues. In addition, the dinuclear species, [Ln(...

In cervical cancer, high frequencies of regulatory T cells (Tregs) and immunosuppressive PD-L1+CD14+ antigen-presenting cells dominate the microenvironment of tumor-positive lymph nodes (LN+). It is unknown whether this is restricted to LN+ or precedes metastasis, emanating from the primary tumor and spreading through tumor-draining lymph nodes (TDLNs). To investigate immunosuppression in the l...

In this paper we study topology of the variety of closed planar n-gons with given side lengths l1, . . . , ln. The moduli space M` where ` = (l1, . . . , ln), encodes the shapes of all such n-gons. We describe the Betti numbers of the moduli spaces M` as functions of the length vector ` = (l1, . . . , ln). We also find sharp upper bounds on the sum of Betti numbers of M` depending only on the n...

We give a set of trace functions which give a global parametrization of the Teichmüller space T (g, n)(L1, . . . , Ln) of hyperbolic surfaces of genus g with n geodesic boundary components of lengths L1,. . . , Ln such that the action of the mapping class group on the Teichmüller space can be represented by rational transformations in the parameters.

BACKGROUND The immunotherapeutic role of programmed death-ligand 1 (PD-L1) in life expectancy in many cancers has been highlighted. However, data regarding PD-L1 expression in papillary thyroid carcinoma (PTC) are limited. In this study, we describe the PD-L1 and programmed cell death protein 1 (PD-1) expressions in PTC and analyze their correlation with lymph node (LN) metastasis. METHODS Cl...

The density of primes dividing at least one term of the Lucas sequence defined by L0(P) = 2, L1(P)= P and Ln(P) = PLn-1(P) + Ln-2(P) for n ~ 2, with P an arbitrary integer, is determined.

In this paper we study topology of the variety of closed pla-nar n-gons with given side lengths l1,. .. , ln. The moduli space M ℓ where ℓ = (l1,. .. , ln), encodes the shapes of all such n-gons. We describe the Betti numbers of the moduli spaces M ℓ as functions of the length vector ℓ = (l1,. .. , ln). We also find sharp upper bounds on the sum of Betti numbers of M ℓ depending only on the num...

A "basophilic method" for phenolic hydroxyl bridged lanthanide coordination polymers (CPs) was developed. With this method, eleven CPs with the general formula of [Ln(HL1)L1·H(2)O](n) (Ln = Tb (1), Nd (2), Eu (3), Gd (4), La (5), Er (6), Y (7), H(2)L1 = 4-methyl salicylic acid) and [Ln(HL2)L2·2MeOH](n) (Ln = Eu (8), Tb (9), Gd (10), La (11), H(2)L2 = 3-hydroxy-2-naphthoic acid) were synthesized...

We present a Zariski pair in affine complex plane consisting of two line arrangements, each of which has six lines. In the seminal paper [3], Zariski started the study of the fundamental groups of the complements of plane algebraic curves. Among other things, he constructed a pair of plane curves with the same degree and the same local singularities, but non-isomorphic fundamental groups, which...