On the approximation by three consecutive continued fraction convergents

Denote by _pn/_qn,n=1,2,3,…$p_n/q_n,n=1,2,3,\dots$, the sequence of continued fraction convergents of the real irrational number x$x$. Define the sequence of approximation coefficients by _θn:=_qn|_qnx−_pn|,n=1,2,3,…${\theta }_n:=q_n|q_{n}x-p_n|,n=1,2,3,\dots$. A laborious way of determining the mean value of the sequence |_θn+1−_θn−1|,n=2,3,…$|{\theta }_{n+1}-{\theta }_{n-1}|,n=2,3,\dots$, is simplified. The method involved also serves for showing that for almost all x$x$ the pattern _θn−1<_θn<_θn+1${\theta }_{n-1}<{\theta }_n<{\theta }_{n+1}$ occurs with the same asymptotic frequency as the pattern _θn+1<_θn<_θn−1${\theta }_{n+1}<{\theta }_n<{\theta }_{n-1}$, namely 0.12109?$0.12109?$. All the four other patterns have the same asymptotic frequency 0.18945?$0.18945?$. The constants are explicitly given.     

A novel and easy route to 1,3,4-thiadiazine derivatives via the three-component reaction of phenylhydrazine, α-bromo aryl ketones and aryl isothiocyanates

The three-component condensation of phenylhydrazine, α-bromo aryl ketones and aryl isothiocyanates leading to 1,3,4-thiadiazine derivatives is described.     

Precise Control of Intramolecular Charge‐Transport: The Interplay of Distance and Conformational Effects

A new series of donor–bridge–acceptor (D–B–A) compounds consisting of π‐conjugated oligofluorene (oFL) bridges between a ferrocene (Fc) electron‐donor and a fullerene (C₆₀) electron‐acceptor have been synthesized. In addition to varying the length of the bridge (i.e., mono‐ and bi‐fluorene derivatives), four different ways of linking ferrocene to the bridge have been examined. The Fc moiety is linked to oFL: 1) directly without any spacer, 2) by an ethynyl linkage, 3) by a vinylene linkage, and 4) by a p‐phenylene unit. The electronic interactions between the electroactive species have been characterized by cyclic voltammetry, absorption, fluorescence, and transient absorption spectroscopy in combination with quantum chemical calculations. The calculations reveal exceptionally close energy‐matching between the Fc and the oFL units, which results in strong electronic‐coupling. Hence, intramolecular charge‐transfer may easily occur upon exciting either the oFLs or Fcs. Photoexcitation of Fc–oFL–C₆₀ conjugates results in transient radical‐ion‐pair states. The mode of linkage of the Fc and FL bridge has a profound effect on the photophysical properties. Whereas intramolecular charge‐separation is found to occur rather independently of the distance, the linker between Fc and oFL acts (at least in oFL) as a bottleneck and significantly impacts the intramolecular charge‐separation rates, resulting in beta values between β_CS 0.08 and 0.19 Å^?1. In contrast, charge recombination depends strongly on the electron‐donor–acceptor distance, but not at all on the linker. A value of β_CR (0.35±0.01 Å^?1) was found for all the systems studied. Oligofluorenes prove, therefore, to be excellent bridges for probing how small structural variations affect charge transport in D–B–A systems.     

Catalytic Reactions in Heterophosphole Complex Chemistry

Catalytic reaction of 2 H -azaphosphirene complex 1 with ortho -, meta -, and para -benzodinitrile ( 2a-c ) led in all cases to the 2 H -1,4,2-diazaphosphole complexes 3a-c if ferrocenium hexafluorophosphate was used as catalyst. In the case of the meta - and para -benzodinitriles 2b-c , the bis-2 H -1,4,2-diazaphosphole complexes 4b-c were additionally obtained. Under the same reaction conditions, acetone, diethylketone and cyclohexanone ( 5a-c ) reacted with complex 1 to yield j 3 -1,3,2-oxazaphospholene complexes 6a-c in good yields.     

A convenient synthetic route to 2,4,6-tris(chlorosulfonyl)- and 2,4,6-tris(fluorosulfonyl)phenol, aniline and chlorobenzene

Convenient and preparative synthetic procedures of 2,4,6-tris(chlorosulfonyl)- and 2,4,6-tris(fluorosulfonyl)phenol, -chlorobenzene and -aniline have been elaborated. Chlorine exchange for fluorine by KF interaction on 2,4,6-tris(chlorosulfonyl)aniline and especially 2,4,6-tris(chlorosulfonyl)phenol proceeds easily and selectively under anhydrous conditions in dioxane. Unlike, 2,4,6-tris(chlorosulfonyl)chlorobenzene transformation requires the presence of water. On the basis of 2,4,6-tris(fluorosulfonyl)phenol and some of its salts, XRD measurements demonstrated the structural similarity to picric acid and its derivatives in the solid state.     

Contact passivation in silicon solar cells using atomic‐layer‐deposited aluminum oxide layers

Atomic‐layer‐deposited aluminum oxide (AlO_x) layers are implemented between the phosphorous‐diffused n⁺‐emitter and the Al contact of passivated emitter and rear silicon solar cells. The increase in open‐circuit voltage V_oc of 12 mV for solar cells with the Al/AlO_x/n⁺‐Si tunnel contact compared to contacts without AlO_x layer indicates contact passivation by the implemented AlO_x. For the optimal AlO_x layer thickness of 0.24 nm we achieve an independently confirmed energy conversion efficiency of 21.7% and a V_oc of 673 mV. For AlO_x thicknesses larger than 0.24 nm the tunnel probability decreases, resulting in a larger series resistance. (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Insight into the Metabolic Profiles of Pb(II) Removing Microorganisms

The objective of the study was to gather insight into the metabolism of lead-removing microorganisms, coupled with Pb(II) removal, biomass viability and nitrate concentrations for Pb(II) bioremoval using an industrially obtained microbial consortium. The consortium used for study has proven to be highly effective at removing aqueous Pb(II) from solution. Anaerobic batch experiments were conducted with Luria-Bertani broth as rich growth medium over a period of 33 h, comparing a lower concentration of Pb(II) with a higher concentration at two different nutrient concentrations. Metabolite profiling and quantification were conducted with the aid of both liquid chromatography coupled with tandem mass spectroscopy (UPLC-HDMS) in a “non-targeted” fashion and high-performance liquid chromatography (HPLC) in a “targeted” fashion. Four main compounds were identified, and a metabolic study was conducted on each to establish their possible significance for Pb(II) bioremoval. The study investigates the first metabolic profile to date for Pb(II) bioremoval, which in turn can result in a clarified understanding for development on an industrial and microbial level.     

Can a polynomial interpolation improve on the Kaplan-Yorke dimension?

The Kaplan-Yorke dimension can be derived using a linear interpolation between an h-dimensional Lyapunov exponent λ(h)>0 and an h+1-dimensional Lyapunov exponent λ(h+1)<0. In this Letter, we use a polynomial interpolation to obtain generalized Lyapunov dimensions and study the relationships among them for higher-dimensional systems.     

Coupled map lattices as spatio-temporal fitness functions: Landscape measures and evolutionary optimization

Coupled Map Lattices (CML) can be interpreted as spatio-temporal fitness landscapes which may pose a dynamic optimization problem. In this paper, we analyze such dynamic fitness landscapes in terms of the landscape measures modality, ruggedness, information content and epistasis. These measures account for different aspects of problem hardness. We use an evolutionary algorithm to solve the dynamic optimization problem and study the relationship between performance criteria of the algorithm and the landscape measures. In this way we relate problem hardness to expectable performance.     

On the algebra of symmetries of Laplace and Dirac operators

We consider a generalization of the classical Laplace operator, which includes the Laplace–Dunkl operator defined in terms of the differential-difference operators associated with finite reflection groups called Dunkl operators. For this Laplace-like operator, we determine a set of symmetries commuting with it, in the form of generalized angular momentum operators, and we present the algebraic relations for the symmetry algebra. In this context, the generalized Dirac operator is then defined as a square root of our Laplace-like operator. We explicitly determine a family of graded operators which commute or anticommute with our Dirac-like operator depending on their degree. The algebra generated by these symmetry operators is shown to be a generalization of the standard angular momentum algebra and the recently defined higher-rank Bannai–Ito algebra.