A smoothing spline based test of model adequacy in polynomial regression

For the regression model % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-qqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xHapdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa% aajeaWbaGaamyAaaWcbeaakiabeccaGiabg2da9iabeccaGiaabAga% caGGOaGaamiDamaaBaaajeaWbaGaamyAaaWcbeaakiaacMcacaqGGa% Gaey4kaSIaaeiiamaavababeqcbaCaaiaadMgaaSqab0qaaiabew7a% LbaakiaabccacaGGOaGaeqyTduMaai4jaiaadohacaqGGaGaamyAai% aadMgacaWGKbGaaeiiaiaad6eacaGGOaGaam4taiaacYcacaqGGaGa% eq4Wdm3aaWbaaSqabKqaGgaacaaIYaaaaOGaaiykaiaacMcaaaa!57B9!\[y_i = {\rm{f}}(t_i ){\rm{ }} + {\rm{ }}\mathop \varepsilon \nolimits_i {\rm{ }}(\varepsilon 's{\rm{ }}iid{\rm{ }}N(O,{\rm{ }}\sigma ^2 ))\], it is proposed to test the null hypothesis that f is a polynomial of degree less than some given value m. The alternative is that f is such a polynomial plus a scale factor b ^1/2 times an (m–1)-fold integrated Wiener process. For this problem, it is shown that no uniformly (in b) most powerful test exists, but a locally (at b=0) most powerful test does exist. Derivation and calculation of the test statistic is based on smoothing spline theory. Some approximations of the null distribution of the test statistic for the locally most powerful test are described. An example using real data is shown along with a computing algorithm.This author's research was supported by the National Science Foundation under grants numbered DMS-8202560 and DMS-8603083.     

Solid phase synthesis of benzamidine and butylamine-derived hydantoin libraries

We have constructed a number of benzamidine- and butylamine-based hydantoin compounds by means of an efficient route using solid phase synthesis in which neat diisopropylamine was employed for a novel cyclization/traceless cleavage step. All library compounds were obtained in excellent yield and high purity. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the grading numbers of direct products of chains

For a finite lattice L, denote by $\text{l}{}^1\text{(L)}$ and $\text{l}{}_1\text{(L)}$ respectively the upper length and lower length of L. The grading number g(L) of L is defined as $\text{g(L) = l}{}^1\text{(}\text{Sub}\text{(L))-l}{}_1\text{(}\text{Sub}\text{(L))}$ where Sub(L) is the sublattice-lattice of L. We show that if K is a proper homomorphic image of a distributive lattice L, then $\text{l}{}_1\text{(}\text{Sub}\text{(K)) < l}{}_1\text{(}\text{Sub}\text{(L))}$; and derive from this result, formulae for $\text{l}{}_1\text{(}\text{Sub}\text{(L))}$ and g(L) where L is a product of chains.     

Two theorems on graceful trees

Let T be a tree on n vertices which are labelled by the integers in N = {1,2,…,n} such that each vertex of T is associated with a distinct number in N. The weight of an edge is defined to be the absolute value of the difference between the two numbers labelled at its end vertices. If the weights of all edges of T are distinct, we call T a graceful tree. In this note, two methods for constructing bigger graceful trees from a given one and a given pair of graceful trees are provided.     

Methionine synthase reductase polymorphisms are associated with serum osteocalcin levels in postmenopausal women

Homocysteine (Hcy) is thought to play an important role in the development of osteoporosis and fracture. Methionine synthase reductase (MTRR) is an enzyme involved in the conversion of Hcy to methionine. We hypothesized that certain genetic polymorphisms of MTRR leading to reduced enzyme activity may cause hyperhomocysteinemia and affect bone metabolism. We therefore examined the associations of the A66G and C524T polymorphisms of the MTRR gene with bone mineral density (BMD) and serum osteocalcin levels in postmenopausal women. Although we did not detect any significant associations between MTRR polymorphisms and BMD or serum osteocalcin levels, we found that the 66G/524C haplotype, which has reduced enzyme activity, was significantly associated with serum osteocalcin levels in a gene-dose dependent manner (P = 0.002). That is, the highest osteocalcin levels (34.5 +/- 16.8 ng/ml) were observed in subjects bearing two copies, intermediate osteocalcin levels (32.6 +/- 14.4 ng/ml) were observed in subjects bearing one copy, and the lowest levels of osteocalcin (28.8 +/- 10.9 ng/ml) were observed in subjects bearing no copies. These results suggest that the 66G/524C haplotype of the MTRR gene affect bone turn over rate.     

Eta1 and eta2 complexes of lambda3-1,2,4,6-thiatriazinyls with CpCr(CO)x

The title heterocyclic radicals coordinate to either 17e CpCr(CO)3 or 15e CpCr(CO)2 moieties as one-electron or as three-electron donors, respectively; in the former the bonding is via the perpendicular p orbital of the sulfur atom, while in the latter bonding is via p(pi) orbitals on both sulfur and nitrogen.