The outer-distance of nodes in random trees

The outer-distance of a nodeu in a rooted treeT _n is the height of the subtree determined byu and all nodesv such thatu is on the path joiningv and the root ofT. We show that the expected outer-distance of nodes of treesT _n in certain families is asymptotic toB logn where the constantB depends on .     

A result on hypothesis testing for a multivariate normal distribution when some observations are missing

Let U_i = (X_i, Y_i), i = 1, 2,…, n, be a random sample from a bivariate normal distribution with mean μ = (μ_x, μ_y) and covariance matrix

. Let X_i, i = n + 1,…, N represent additional independent observations on the X population. Consider the hypothesis testing problem H₀ : μ = 0 vs. H₁ : μ ≠ 0. We prove that Hotelling's T² test, which uses (X_i, Y_i), i = 1, 2,…, n (and discards X_i, i = n + 1,…, N) is an admissible test. In addition, and from a practical point of view, the proof will enable us to identify the region of the parameter space where the T²-test cannot be beaten. A similar result is also proved for the problem of testing μ_x ? μ_y = 0. A Bayes test and other competitors which are similar tests are discussed.     

Sur la forme canonique des formes extérieures

Sans résumé

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The article is the text of a talk given at the Symposium on Differential Geometry in Debrecen, Hungary, on August 28–September 3, 1975.     

取代基物化参数及其在药物定量构效关系中的应用

把取代基电性、立体及疏水性物化参数组合建立一种新取代基描述方法, 对环尿素类和N,N-二甲基-2-溴苯乙胺类衍生物进行结构表征. 对训练样本集通过逐步回归筛选变量, 所建多元线性回归方程R2分别为0.853和0.960, 留一法交互检验Rcv2分别为0.723和0.901；用预测集样本作外部预测, 所得Qext2分别为0.7617和0.7653. 结果显示：环尿素类化合物结构中苯环邻位立体、间位疏水、对位疏水及立体因素对该类药物抗HIV活性产生阻抑作用； N,N-二甲基-2-溴苯乙胺类苯环上取代基立体因素及对位给电子效应有利于提高肾上腺素能阻断活性.     

Reaktive E = C(p-p)π-Systeme. XLII [1]. Neue Koordinationsverbindungen des 2-(Diisopropylamino)-1-phosphaethins: [{η4-(iPr2NCP)2}Ni{η2-(iPr2NCP)}], [(Ph3P)2Pt{η2-(iPr2NCP)}] und [Co2(CO)6{η2-η2-(iPr2NCP)}]

Reactive E = C(pp)π-Systems. XLII [1]. Novel Coordination Compounds of 2-(Diisopropylamino)-1-phosphaethyne: [{η⁴-(iPr₂NCP)₂}Ni{η²-(iPr₂NCP)}], [(Ph₃P)₂Pt{η²-(iPr₂NCP)}], and [Co₂(CO)₆{η²-μ²-(iPr₂NCP)}] 2-(Diisopropylamino)-phosphaethyne iPr₂N? C?P ( 2 ) reacts with the Ni(0)-complexes [Ni(1,5-cyclooctadiene)₂] and [Ni(CO)₃(1-azabicyclo[2.2.2]octane)], respectively, to give the novel complex [{η⁴-(iPr₂NCP)₂}Ni{η²-(iPr₂NCP)}] ( 5 ), with the 1,3-diphosphacyclobutadiene derivative and 2 (side-on) as π-ligands. The molecular structure of 5 determined by X-ray diffraction on single crystals proves the spin systems and rotational barriers deduced from NMR-data (¹H, ¹³C-, ³¹P). The PC distances of the four-membered ring of 1.817(2) and 1.818(2) Å – as expected – are considerably longer than the PC bond of the η²-coordinated phosphaalkyne 2 [1.671(2) Å]. – In the reactions of 2 with [(Ph₃P)₂Pt(C₂H₄)] or [Co₂(CO)₈] the ligand properties of 2 resemble those of alkynes affording the complexes [(Ph₃P)₂Pt{η²-(iPr₂NCP)}] ( 7 ) with side-on coordinated 2 and [Co₂(CO)₆{η²-μ²-(iPr₂NCP)}] with 2 acting as a 4e donor bridge in quantitative yield. In attempts to prepare copper(I) complexes of the aminophosphaalkyne 2 by reaction with CuCl or CuI the only isolable product formed in reasonable amounts under the influence of air and moisture is the 1 λ³, 3 λ⁵-diphosphetene (iPr₂N) ( 10 ) (isolated yield: ca. 20%). The crystal structure analysis of 10 indicates a strong structural relationship to the diamino-2-phosphaallyl cation [Me(iPr₂N)]⁺ ( 12 ), the 1,3-diphosphacyclobutadiene ligand (iPr₂NCP)₂ in the binuclear complex [{η¹, μ²-(iPr₂NCP)₂}Ni₂(CO)₆] ( 3a ) as well as to the heterocycles (dme)₂LiOE₂′ (E′ = S, 11a ; E′ = Se, 11b ) prepared by Becker et al. [11b, 35].     

A theoretical study of five nitrates: Electronic structure and bond dissociation energies

Three density-functional methods (B3P86, B3PW91, and B3LYP) are employed to investigate the O–NO₂ bond lengths, frontier orbital energies, and O–NO₂ bond dissociation energies (BDEs) of n-propyl nitrate (NPN), isopropyl nitrate (IPN), 2-ethylhexyl nitrate (EHN), triethylene glycol dinitrate (Tri-EGDN), and tetraethylene glycol dinitrate (Tetra-EGDN). It is found that the O–NO₂ bond lengthens (destabilizes) in the order of IPN, NPN, EHN, Tetra-EGDN, and Tri-EGDN. From the data of frontier orbital energies (E_HOMO, E_LUMO), and energy gaps (ΔE), we estimate the relative thermal stability ordering of five nitrates and their corresponding radicals. The predicted BDEs of O–NO₂ bond in NPN, IPN, EHN, Tri-EGDN, and Tetra-EGDN, are 176.6, 174.5, 168.1, 156.1, and 159.3 kJ mol⁻¹, respectively. Based on the finding that the present results of BDEs are well coincident with the experimental results of apparent activation energies from the literature, we can draw a conclusion that the experimental thermolysis of five nitrates is only unimolecular homolytical cleavage of the O–NO₂ bonds.     

Spektroskopische Eigenschaften von Di(phthalocyaninato)metallaten(III) der Seltenen Erden. Teil 2: Die Multikern(1H-, 13C-, 15N-, 17O- und 31P)-NMR-Spektren und Nachweis komplexgebundenen Wassers

Spectroscopical Properties of Di(phthalocyaninato)metalates(III) of the Rare Earth Elements. Part 2: The Multinuclear (¹H, ¹³C, ¹⁵N, ¹⁷O, and ³¹P) NMR Spectra and Determination of Complex Bound Water The ¹H, ¹³C, ¹⁵N, ¹⁷O and ³¹P NMR spectra of (PNP)[Ln(Pc^2?)₂] and (TDOA)[Ln(Pc^2?)₂] (Ln = La …? (—Pm) …? Lu); PNP: di(triphenylphosphine)-iminium; TDOA: Tri(n-dodecyl)n-octylammonium dissolved in CD₂Cl₂ are reported. With the exception of the nitrogen atoms of the cations all light atoms of both homologues complex salt rows have been detected and assigned. Proof of the presence of additional water contained in the (solid) complex salts has been given unambigously. The otherwise strong Fermi contact interactions diminuish rapidly with growing distance from the paramagnetic centre favouring dipolar (pseudo-contact) interactions. As a consequence the mostly element independent paramagnetic shifts of the atoms situated on the periphery of the anion as well as those of the cations and water are determined by the distance factor. Therefore a first glance of the structure of the ion-pair present in solution emerges, in which the water molecule seems to play an important role.     

Gould-Jacobs reaction of 6-amino-2,3-diphenylquinoxaline

Summary Catalytic hydrogenation of 2,3-diphenyl-6-nitroquinoxaline led to the corresponding amine1 which in turn afforded products3a-i on reaction with alkoxymethylene derivatives2a-i. Thermal cyclization of3b and3f yielded substituted pyrazinoquinolones5b and5f, respectively. Optimal conditions for the successful hydrolysis of ester5b were found. The structures of all products were deduced from their IR, UV,¹H, and¹³C NMR spectroscopic data.Dedicated to Prof.F. Sauter on the occasion of his 65th birthday