Consistency of a rank test against general alternatives of change points (surfaces) and continuous trend

Considered are modifications of a rank test of randomness for the one- and multi-dimensional regular design cases as well as for the one- and multi-dimensional random design cases. The null hypothesis is that all observations are independent and identically distributed. The main result is the proof of consistency of the test in each of the above cases against two general alternatives.Alternative 1: there exists a pairwise disjoint partion U _i =1 ^m D _i =D, where D d1, is a bounded domain inside which one makes observations, such that (1) if an observation point falls insideD _i , then the corresponding observed value is the realization of a random variable _i i = l,...,m; (2) there exists an ordering % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamXvP5wqonvsaeHbfv3ySLgzaGqbaiab-Tha7jabe67a4Hqbdiab% +LgaPnaaBaaaleaacaWGRbaabeaakiab-1ha9naaDaaaleaacaWGRb% Gaeyypa0JaaGymaaqaaiaad2gaaaaaaa!4C2D!\[\{ \xi i_k \} _{k = 1}^m \], where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabe67a4nXvP5wqonvsaeHbfv3ySLgzaGqbdiab-LgaPnaaBaaa% leaacaWGRbaabeaaaaa!454D!\[\xi i_k \] is stochastically smaller than % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabe67a4nXvP5wqonvsaeHbfv3ySLgzaGqbdiab-LgaPnaaBaaa% leaacaWGRbaabeaakmaaBaaaleaacqGHRaWkcaaIXaaabeaakiaacY% cacaWGRbGaeyypa0JaaGymaiaacYcacaGGUaGaaiOlaiaac6cacaGG% SaGaamyBaiabgkHiTiaaigdaaaa!509B!\[\xi i_k _{ + 1} ,k = 1,...,m - 1\], (3) the partition is independent of the number of observation points. Note thatm, this ordering, and the sets D _i are not known a priori: one tests only for the existence of such a partition. Note also that in the one-dimensional case the initial sequence need not be stochastically monotone under the alternative.Alternative 2: there exists an arbitrary asymptotically continuous trend in location. Asymptotically continuous means that the trend converges to some continuous, not identically constant function as the number of data points goes to infinity. This function need not be monotone.A numerical example illustrating the use of the obtained results for image analysis (edge detection) is presented.     

Sol-gel polycondensation of methyltrimethoxysilane in ethanol studied by <Superscript>29</Superscript>Si NMR spectroscopy using a two-step acid/base procedure

Sol-gel polymerization of methyltrimethoxysilane (MTMS) in ethanol using a two-step acid/base catalyzed procedure (B2) is followed by ²⁹Si NMR spectroscopy. Analysis of the structural evolution of the B2 system shows that esterification of monomeric and end silicon species is rate-limited while that of linear and cyclic species is able to reach pseudoequilibrium in the second basic step. Condensation reactivity is reduced with increasing network connectivity, however, to a much less degree under B2 conditions than MTMS polymerization under acidic conditions. Steric effects as well as many other factors are attributed to this trend. The concentration of cyclic and polycyclic species of the B2 system is nearly 3 times lower compared to the acid-catalyzed system. The empirical degree of condensation at the gel point is determined to be 0.88. The effects of cyclization and phase separation on MTMS gelation are discussed for both B2 and acid-catalyzed systems. Based on these results it is believed that MTMS-based gels form for B2 and not acid-catalyzed conditions due to reduced cyclization, rapid hydrolysis and condensation, effective use of functional groups, and effective contribution of branched and polycyclic species as crosslinking points to connect polymeric chains in the B2 system.     

Computing the Fukui function from ab initio quantum chemistry: approaches based on the extended Koopmans’ theorem

The extended Koopmans’ theorem is related to Fukui function, which measures the change in electron density that accompanies electron attachment and removal. Two approaches are used, one based on the extended Koopmans’ theorem differential equation and the other based directly on the expression of the ionized wave function from the extended Koopmans’ theorem. It is observed that the Fukui function for electron removal can be modeled as the square of the first Dyson orbital, plus corrections. The possibility of useful generalizations to the extended Koopmans’ theorem is considered; some of these extensions give approximations, or even exact expressions, for the Fukui function for electron attachment.     

Solvent and substitution effects upon the phosphorescent triplet state of carbonyl pyridines

The effects of perturbing nitrogen upon localized triplet excitations are observed for several carbonyl pyridines. Conventional and pulse optically detected magnetic resonance (ODMR) techniques are used to characterize the spin-relaxation processes in these aromatic carbonyl molecules.     

Iron(II) and zinc(II) monohelical binaphthyl salen complexes

A new chiral binaphthyl salen ligand with rigid polyaromatic sidearms gives monohelical complexes (Fe(II) and Zn(II)) of predetermined handedness.     

Controlling diffusion in sol-gel derived monoliths

Redox probes were trapped within a silica monolith prepared in part with organoalkoxysilanes containing a quaternary ammonium functional group. The diffusion coefficients of the entrapped molecules were measured as the gels were slowly dried using chronoamperometry and cyclic voltammetry with ultramicroelectrodes. Gel-entrapped cobalt(II) tris(bipyridine) (Co(bpy)(3)(2+)) diffuses at rates similar to that measured in the sols by incorporating a small amount of the positively charged functional group in the matrix. In comparison, the diffusion coefficient of gel-entrapped ferricyanide (Fe(CN)(6)(3-)) drops an order of magnitude relative to its value in the sol soon after gelation. These results demonstrate the ease at which diffusion in hydrated gels can be easily controlled by simply changing the charge on the walls of the silica host.     

A Bloembergen–Purcell–Pound <Superscript>13</Superscript>C NMR relaxation study of the ionic liquid 1-butyl-3-methylimidazolium hexafluorophosphate

The molecular structure and rotational motion of the ionic liquid 1-butyl-3-methylimidazolium hexafluorophosphate ([BMIM][PF6]) were studied over a wide temperature range using the Bloembergen–Purcell–Pound ¹³C NMR spin–lattice relaxation method and NOE factors. Examination of the spin–lattice relaxation times (T ₁) and the rates (R ₁=1/T ₁) of the 1-butyl-3-methylimidazolium cation reveals the relative motions of each carbon in the imidazolium cation. The rotational characteristics of the [BMIM] cation are supported by ab-initio molecular structures of [BMIM][PF6] using density functional theory (DFT) and Hartree–Fock (HF) methods. The ab-initio gas phase structures of [BMIM][PF6] indicate that the 1-butyl-3-methylimidazolium C2 hydrogen, the ring methyl group, and the butyl side-chain hydrogen atoms form hydrogen bonds with the hexafluorophosphate anion.     

Through-shell alkyllithium additions and borane reductions

The through-shell borane reduction and methyllithium addition to benzaldehyde (1), benzocyclobutenone (2), and benzocyclobutenedione (3) incarcerated inside a hemicarcerand (4) with four tetramethylenedioxy bridges are reported. All guests could be reduced and methylated. Selective monoreduction and monomethylation were observed for 3. In the methyllithium addition to 4[symbol: see text]3, the initially formed lithium alcoholate underwent a Moore rearrangement. The reactivity of the incarcerated guests toward methyllithium increased in the order 1 < 2 < 3 and toward borane in the order 1 < 2 approximately equal 3. Guest reactivity was correlated with the inner-phase location of the reacting carbonyl group in the preferred guest inner-phase orientation. The latter was determined from the X-ray structures of 4[symbol: see text]1, 4[symbol: see text]2, and 4[symbol: see text]3, from molecular mechanical calculations, and from the hemicarcerand-induced upfield shift of the guest proton resonances. In the methyllithium and n-butyllithium addition to 4[symbol: see text]1 and 4[symbol: see text]3 at elevated temperatures, selective cleavage of a host's spanner or tetramethylenedioxy bridge, respectively, was observed. The cleavage of one spanner also took place in the methyllithium addition to the 1-methyl-2-pyrrolidinone hemicarceplex. These scission reactions are initiated by the initially formed lithium alcoholates, which show enhanced basicity and nucleophilicity in the inner phase as compared to the bulk phase. Mechanisms for the host scission reactions are discussed.     

Linear combination of concomitants of order statistics with application to testing and estimation

Summary Let (X ₁,Y ₁), (X ₂,Y ₂),…, (X _n,Y _n) be i.i.d. as (X, Y). TheY-variate paired with therth orderedX-variateX _rn is denoted byY _rn and terms the concomitant of therth order statistic. Statistics of the form are considered. The asymptotic normality ofT _n is established. The asymptotic results are used to test univariate and bivariate normality, to test independence and linearity ofX andY, and to estimate regression coefficient based on complete and censored samples.     

Probing the donor-acceptor proximity on the physicochemical properties of porphyrin-fullerene dyads: "tail-on" and "tail-off" binding approach.

A new approach of probing proximity effects in porphyrin-fullerene dyads by using an axial ligand coordination controlled "tail-on" and "tail-off" binding mechanism is reported. In the newly synthesized porphyrin-fullerene dyads for this purpose, the donor-acceptor proximity is controlled either by temperature variation or by an axial ligand replacement method. In o-dichlorobenzene, 0.1 M (TBA)ClO(4), the synthesized zincporphyrin-fullerene dyads exhibit seven one-electron reversible redox reactions within the accessible potential window of the solvent and the measured electrochemical redox potentials and UV-visible absorption spectra reveal little or no ground-state interactions between the C(60) spheroid and porphyrin pi-system. The proximity effects on the photoinduced charge separation and charge recombination are probed by both steady-state and time-resolved fluorescence techniques. It is observed that in the "tail-off" form the charge-separation efficiency changes to some extent in comparison with the results obtained for the "tail-on" form, suggesting the presence of some through-space interactions between the singlet excited zinc porphyrin and the C(60) moiety in the "tail-off" form. The charge separation rates and efficiencies are evaluated from the fluorescence lifetime studies. The charge separation via the singlet excited states of zinc porphyrin in the studied dyads is also confirmed by the quick rise-decay of the anion radical of the C(60) moiety within 20 ns. Furthermore, a long-lived ion pair with lifetime of about 1000 ns is also observed in the investigated zinc porphyrin-C(60) dyads.