Reproducing kernel Hilbert spaces and the law of the iterated logarithm for Gaussian processes

In this paper, we find the limit set of a sequence (2 log n)^?1/2 X _n (t), n≧3) of Gaussian processes in C [0,1], where the processes X _n (t) are defined on the same probability space and have the same distribution. Our result generalizes the theorems of Oodaira and Strassen, and we also apply it to obtain limit theorems for stationary Gaussian processes, moving averages of the type \(\int\limits_0^t {f\left( {t - s} \right)dW\left( s \right)} \) , where W(s) is the standard Wiener process, and other Gaussian processes. Using certain properties of the unit ball of the reproducing kernel Hubert space of X _n (t), we derive the usual law of the iterated logarithm for Gaussian processes. The case of multidimensional time is also considered.     

Quaternion analyticity and conformally Kählerian structure in Euclidean gravity

Starting from the fact that the d=4 Euclidean flat spacetime is conformally related to the Kähler manifold H ²×S ², we show the Euclidean Schwarzschild metric to be conformally related to another Kähler manifold M ²×S ² with M ² being conformal to H ² in two dimensions. Both metrics which are conformally Kählerian, are form-invariant under the infinite parameter Fueter group, the Euclidean counterpart of Milne's group of clock regraduation. The associated Einstein's equations translate into Fueter's quaternionic analyticity. The latter leads to an infinite number of local continuity equations.     

A minor new monoterpene glycoside from paeonia veitchii

A new flavone, 6, 2?-dihydroxy-5, 7, 8, 6?-tetramethoxyflavone, was isolated from the roots of Scutellaria baicalensis. Its structure was established on the basis of spectral evidences.     

Hydrogen‐Assisted Unique Incorporation of 1,4‐Divinylbenzene in Copolymerization with Propylene Using an Isospecific Zirconocene Catalyst

Summary: Propylene was copolymerized with 1,4‐divinylbenzene (1,4‐DVB) using rac‐Me₂Si(2‐Me‐4‐Ph‐Ind)₂ZrCl₂ and MAO in the presence of hydrogen. ¹H NMR spectra of the obtained copolymer confirmed the successful incorporation of 1,4‐DVB. ¹³C NMR analysis revealed a unique copolymer structure with the incorporated 1,4‐DVB units predominantly forming 1,4‐substituted phenyl repeating units in the polymer main chain. A plausible mechanism involving spontaneous insertion of terminal styryl into Zr‐H species was proposed to rationalize the unique incorporation of 1,4‐DVB.

Copolymerization of propylene with 1,4‐divinylbenzene in the presence of hydrogen.     

"Ladder" structure in tonal noise generated by laminar flow around an airfoil

The presence of a "ladder" structure in the airfoil tonal noise was discovered in the 1970s, but its mechanism hitherto remains a subject of continual investigation in the research community. Based on the measured noise results and some numerical analysis presented in this letter, the variations of four types of airfoil tonal noise frequencies with the flow velocity were analyzed individually. The ladder structure is proposed to be caused by the acoustic/hydrodynamic frequency lag between the scattering of the boundary layer instability noise and the discrete noise produced by an aeroacoustic feedback loop.     

Benzotriazole‐Containing D–π–A Conjugated Organic Dyes for Dye‐Sensitized Solar Cells

A series of metal‐free benzotriazole‐based dipolar dyes have been developed as sensitizers for dye‐sensitized solar cells (DSSCs). Different heteroaromatic rings such as furan, thiophene, and selenophene, were used in combination with benzotriazole as the conjugated spacer group. Light harvesting, charge recombination, and electron injection of the cells fabricated are affected by the heteroaromatic ring used in the spacer. The DSSC with the thiophene‐containing dye has the highest conversion efficiency of 6.20 %, which reaches 85 % of the standard cell based on N719.     

Algébre de lie de type witt

Let ρ be an abelian group and k a commutative field. Inspired by the example of the Witt algebra, we introduce a family of ρ-graded Lie algebras (called Witt type Lie algebras) V = ⊕α?r Vα. where the dimension of each homogeneous component is 1. We characterise those which are simple and we give a complete classification of these Lie algebras. When P is free and the characteristic of k is zero, we show that the universal central extension of simple Witt type Lie algebras is 1-dimensional, and the corresponding cocycle is similar to the cocycle of the Virasoro algebra.     

Rigid representations of a double quiver of type A, and Richardson elements in seaweed Lie algebras

In this paper, we show that there is always an open adjointorbit in the nilpotent radical of a seaweed Lie algebra in gl_n(k),thus answering positively in this gl_n(k) case to a questionraised independently by Michel Duflo and Dmitri Panyushev. Theproof gives an explicit construction, using -filtered modulesof quasi-hereditary algebras arising from quotients of the doubleof quivers of type A. An example of a seaweed Lie algebra ina simple Lie algebra of type E₈ not admitting an open orbitin its nilpotent radical is given.