Estimating common parameters of growth curve models

Suppose that we have two independent random matrices X ₁ and X ₂ having multivariate normal distributions with common unknown matrix of parameters (q×m) and different unknown covariance matrices ₁ and ₂, given by N _{p1, N1} (B ₁ A ₁; ₁, I) and N _{p2, N2} (B ₂ A ₂; ₂, I) respectively. Let % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaigdaaaa!391C!\[\hat \xi 1\] (% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaikdaaaa!391D!\[\hat \xi 2\]) be the MLE of based on X ₁ (X ₂) only. When q=1, necessary and sufficient conditions that a combined estimator of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaigdaaaa!391C!\[\hat \xi 1\] and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaikdaaaa!391D!\[\hat \xi 2\] has uniformly smaller covariance matrix than those of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaigdaaaa!391C!\[\hat \xi 1\] and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaikdaaaa!391D!\[\hat \xi 2\] are given. The k-sample problem as well as one-sample problem is also discussed. These results are extensions of those of Graybill and Deal (1959, Biometrics, 15, 543–550), Bhattacharya (1980, Ann. Statist., 8, 205–211; 1984, Ann. Inst. Statist. Math., 36, 129–134) to multivariate case.Dedicated to Professor Yukihiro Kodama on his 60th birthday.Bowling Green State UniversityVisiting Professor on leave from the University of Tsukuba, Japan. Now at Department of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan.This research was partially supported by University of Tsukuba Project Research 1986.     

A characterization of cut locus for $$C^1$$ hypersurfaces

Let \(\Omega \) be an open set in \(\mathbb {R}^n\) with \(C^1\)-boundary and \(\Sigma \) be the skeleton of \(\Omega \), which consists of points where the distance function to \(\partial \Omega \) is not differentiable. This paper characterizes the cut locus (ridge) \(\overline{\Sigma }\), which is the closure of the skeleton, by introducing a generalized radius of curvature and its lower semicontinuous envelope. As an application we give a sufficient condition for vanishing of the Lebesgue measure of \(\overline{\Sigma }\).     

Ultra-fast One-Handed Helix Induction and Its Static Helicity Memory in a Poly(biphenylylacetylene) with a Catalytic Amount of Chiral Ammonium Salts

We report an ultra-fast helix induction and subsequent static helicity memory in poly(biphenylylacetylene) (PBPA- A ) assisted by a catalytic amount of nonracemic ammonium salts comprised of non-coordinating tetrakis[3,5-bis(trifluoromethyl)phenyl]borate (BArF⁻) as a counter anion. The remarkable acceleration of the helix-induction rate in PBPA- A accompanied by the significant amplification of the asymmetry relies on the two methoxymethoxy groups of the biphenyl pendants, which can gain access to enfold the chiral ammoniums in a crown-ether manner in specific aromatic solvents, leading to ultra-fast helicity induction, which is completed within 30 s. In aromatic solvents, helicity memory is lost rapidly, but is quite stable in long-chain hydrocarbons. The best use of specific solvents for helicity induction and static helicity memory, respectively, provides a highly sensitive chirality sensing system toward a small amount of chiral amines and amino acids when complexed with BArF⁻.     

Self-assembled cyclic boron-dipyrrin oligomers

Self-assembled cyclic boron-dipyrrin oligomers 3-5 obtained by the reaction of catecholyldipyrrin with boron trichloride were characterized by 1H NMR, MALDI-TOF MS, and X-ray measurements and the interaction between 3 and alkali metal ions was examined.     

Reaction mechanism of direct episulfidation of caryophyllene and humulene

Direct episulfidations of caryophyllene or humulene with elemental sulfur were examined by means of gas chromatography. Caryophyllene-6,7-episulfide was formed at an early stage in a reaction of the caryophyllene and elemental sulfur at 120 degrees C. Caryophyllene-3,6-sulfide and polymer compounds were formed after the episulfidation. Formations of the these compounds were related to the disappearance of the caryophyllene-6,7-episulfide. Isomerization from the caryophyllene to isocaryophyllene was also observed during the reaction. In the reaction of humulene with elemental sulfur, humulene-6,7-episulfide was initially produced and then converted to humulene-9,10-episulfide. It was assumed that the polymer compound in the reaction of humulene with sulfur was related to the disappearance of the both humulene episulfides.