Limit theorems under the assumption of restricted convergence on a set with a finite limit point

We consider double sequences {{X_nk}} of independent and asymptotically constant random variables. For certain constants A_n we put The central problem is the following one: Assume that {F_n} for x ? I ? R₁ converges weakly to a nonconstant limit function ψ where I is a set with a finite limit point (restricted convergence Then additional conditions ensuring the complete convergence F_n ? F to a certain limit distribution function F are given. These additional conditions are weaker than the corresponding sufficient conditions known from the classical theory. Further, these results yield two new versions of the central limit theorem, see § 2. In the case of identically distributed summands with common distribution function V the assumption a sufficient to prove F_n ? F_α, provided that (F_α stands for a stable distribution with characteristic exponent α).     

Estimates from below for Lebesgue constants of Fourier series on compact Lie groups

In this paper the Lebesque constants (L _R ^K (G))_R>0 of Fourier series on compact Lie groups G corresponding to general one-dimensional groupings on the dual object G^{^} are estimated from below by the associated (abelian) Lebesgue constants (L _R ^K (T))_R>0 on a maximal torus T in G. For spherical groupings this leads to the estimate L _R (G)const.R^(l-1)/2, l=dimT2.     

Organometallic complex compounds,part 6(1). Property-specific ligand control of stability and reactivity of (1,3-dimethyl-η3-allyl)methylnickel-ligand complexes. Separation of electronic and steric effects

Summary The property-specific ligand control of 28 ligands on the decomposition temperatures in solution, measured by d.t.a. of a four-coordinate nickel(II)-complex is reported. A quantitative separation of electronic and steric effects by a multilinear regression analysis (75% electronic and 25% steric influence for the chosen ligands) is presented. The controlling effect of the selectivity on the decomposition (fraction of the C-C-linked product) (25 P-ligands) leads to an electronic: steric ratio of the property-specific ligand control of 5545 for the chosen ligands. An increase in the relative acceptor character of the P-ligands relatively destabilizes the complexes and thereby favours formation of a C-C-bond. An increase in steric hindrance also favours C-C-bond formation. A method for revising the steric parameter of P-ligands is presented and is used to correct the -value of (PhCH₂)₃P is corrected to 135°. SCCC-MO-calculations for testing the chemical reasoning of the separated electronic and steric ligand property control are shown.     

An extended ab initio QM/MM MD approach to structure and dynamics of Zn(II) in aqueous solution

Structural and dynamical properties of Zn(II) in aqueous solution were investigated, based on an ab initio quantum mechanical/molecular mechanical (QM/MM) molecular dynamics simulation at double-zeta Hartree-Fock quantum mechanical level including the first and second hydration shells into the QM region. The inclusion of the second shell in the QM region resulted in significant changes in the properties of the hydrate. The first shell coordination number was found to be 6, the second shell consists of approximately 14 water molecules. The structural properties were determined in terms of RDF, ADF, tilt and theta angle distributions, while dynamics were characterized by mean ligand residence times, ion-ligand stretching frequencies and the vibrational and librational motions of water ligands.     

Quantitative determination of the group of flavonoids and saponins from the extracts of the seeds of Glinus lotoides and tablet formulation thereof by high-performance liquid chromatography

The total flavonoids and saponins of the seeds of Glinus lotoides in the crude extracts and tablet formulation thereof were quantified by reversed-phase high-performance liquid chromatographic (RP-HPLC) methods with UV detection. The saponins were analyzed after acid hydrolysis in 3 M HCI at 100 degrees C for 1 h. Vicenin-2 and mollugogenol B were isolated and used as reference substances for the quantification of total flavonoids and saponins, respectively. The identity and purity (> 97%) of the standards were confirmed by spectroscopic (UV, MS, and NMR) and chromatographic (HPLC) methods. The flavonoids and saponins of the crude extract of the seeds and tablet formulation were separated by RP-HPLC (Nucleosil RP-18 column, 250 mm x 4.6 mm) using linear gradient elution systems of acetonitrile-water-0.1 M H3PO4 for flavonoids and methanol-water for saponins. Satisfactory separation of the compounds was obtained in less than 30 and 25 min, for the flavonoids and saponins, respectively. The methods were validated for linearity, repeatability, limits of detection (LOD) and limits of quantification (LOQ). Repeatability (inter- and intra-day, n = 6 and 9, respectively) showed less than 2% relative standard deviation (RSD). The LOD and LOQ were found to be 0.075 and 0.225 mg/mL, respectively, for vicenin-2 and 0.027 and 0.082 mg/100 mL, respectively, for mollugogenol B. The content of flavonoids and saponins of six single tablets was between 95 and 103% for flavonoids and 94-98% for saponins. The validated HPLC methods were employed to standardize a fingerprint of a laboratory produced purified extract, which could be used as a secondary standard for the routine quality control. Accordingly, the purified extract was found to contain 21.3% flavonoids (vicenin-2, 10%) and 25.4% saponins (glinuside G, 14.2%).     

Cu(II) in liquid ammonia: an approach by hybrid quantum-mechanical/molecular-mechanical molecular dynamics simulation.

To investigate the solvation structure of the Cu(II) ion in liquid ammonia, ab initio quantum-mechanical/molecular-mechanical (QM/MM) molecular dynamics (MD) simulations were carried out at Hartree Fock (HF) and hybrid density functional theory (B3 LYP) levels. A sixfold-coordinated species was found to be predominant in the HF case whereas five- and sixfold-coordinated complexes were obtained in a ratio 2:1 from the B3 LYP simulation. In contrast to hydrated Cu(II), which exhibits a typical Jahn-Teller distortion, the geometrical arrangement of ligand molecules in the case of ammonia can be described as a [2 + 4] ([2 + 3]) configuration with 4 (3) elongated copper-nitrogen bonds. First shell solvent exchange reactions at picosecond rate took place in both HF and B3 LYP simulations, again in contrast to the more stable sixfold-coordinated hydrate. NH3 ligands apparently lead to strongly accelerated dynamics of the Cu(II) solvate due to the "inverse" [2 + 4] structure with its larger number of elongated copper-ligand bonds. Several dynamical properties, such as mean ligand residence times or ion-ligand stretching frequencies, prove the high lability of the solvated complex.     

C[triple‐bond]C—H systems as hydrogen‐bond donors and acceptors: trans‐1,2‐diethynylcyclo­hexane‐1,2‐diol and trans‐1,4‐diprop‐2‐ynylcyclo­hexane‐1,4‐diol monohydrate

The title 1,2‐diol derivative, C₁₀H₁₂O₂, crystallizes with two independent but closely similar mol­ecules in the asymmetric unit. Only two of the four OH groups are involved in classical hydrogen bonding; the mol­ecules thereby associate to form chains parallel to the short c axis. The other two OH groups are involved in O—H⋯(C[triple‐bond]C) systems. Additionally, three of the four C[triple‐bond]C—H groups act as donors in C—H⋯O inter­actions. The 1,4‐diol derivative crystallizes with two independent half‐mol­ecules of the diol (each associated with an inversion centre) and one water mol­ecule in the asymmetric unit, C₁₂H₁₆O₂·H₂O. Both OH groups and one water H atom act as classical hydrogen‐bond donors, leading to layers parallel to the ac plane. The second water H atom is involved in a three‐centre contact to two C[triple‐bond]C bonds. One acetyl­enic H atom makes a very short `weak' hydrogen bond to a hydr­oxy O atom, and the other is part of a three‐centre system in which the acceptors are a hydroxy O atom and a C[triple‐bond]C bond.     

Photocatalyzed [2 + 2 + 2]-cycloaddition of nitriles with acetylene: an effective method for the synthesis of 2-pyridines under mild conditions

The photocatalyzed [2 + 2 + 2]-cycloaddition of nitriles with 2 equiv of acetylene to 2-pyridines can be carried out under mild conditions and represents a valuable extension to common synthetical methods. For the ideal wavelength range (350-500 nm), lamps as well as sunlight can be used. Working at room temperature and in organic solvents such as toluene or hexane as well as in water gives satisfying results in many cases. However, it is also possible to vary the solvent and the reaction temperature of the photocatalyzed synthesis and to choose, with respect to the specific substrate, specific requirements for this particular reaction and general requirements of the method. This simple and selective method derives its potential mainly from the large variety of applicable nitriles. Suitable substrates include (functionalized) aliphatic and aromatic nitriles as well as cyanamides derived from secondary amines.