Coupling of the relaxation and resonant elements in the autonomous chaotic relaxation oscillator (ACRO)

If a harmonic oscillator is embedded in a relaxation oscillator, the resulting system may behave like an autonomous chaotic relaxation oscillator (ACRO). The discharge transient of the relaxation oscillator excites sinusoidal oscillations in the harmonic oscillator and these sinusoids affect when the next discharge occurs. This can lead to chaotic intervals in the oscillator periods. A simple electronic model of the ACRO is studied over a wide range of parameters using numerical, analytic, and experimental techniques. The dynamics of the ACRO is found to be determined by three parameters: (1) tuning, (2) coupling, and (3) damping. Complex, intermittent outputs can always be inhibited by increasing the damping of the harmonic oscillator. For weak damping, strong coupling yields chaotic periods. With weak damping and weak coupling, complex behavior only occurs if the relaxation oscillator is tuned near a resonance of the harmonic oscillator. A new path to chaos, called a disruption bifurcation, is the source for intermittency in the ACRO. This bifurcation occurs when the amplitude of internal resonances is excited to the degree that existing limit cycles are disrupted.     

Spatially localized free vibrations for a non-linear string model

We prove the existence of infinitely many non-zero time-periodic solutions (breathers) to the dispersive wave equation of the form which are localized in the spatial variable, that is The main tool employed is the concentration compactness principle of P. L. Lions.     

Absolute solvation free energy of Li+ and Na+ ions in dimethyl sulfoxide solution: a theoretical ab initio and cluster-continuum model study

The solvation of the lithium and sodium ions in dimethyl sulfoxide solution was theoretically investigated using ab initio calculations coupled with the hybrid cluster-continuum model, a quasichemical theory of solvation. We have investigated clusters of ions with up to five dimethyl sulfoxide (DMSO) molecules, and the bulk solvent was described by a dielectric continuum model. Our results show that the lithium and sodium ions have four and five DMSO molecules into the first coordination shell, and the calculated solvation free energies are -135.5 and -108.6 kcal mol(-1), respectively. These data suggest a solvation free energy value of -273.2 kcal mol(-1) for the proton in dimethyl sulfoxide solution, a value that is more negative than the present uncertain experimental value. This and previous studies on the solvation of ions in water solution indicate that the tetraphenylarsonium tetraphenylborate assumption is flawed and the absolute value of the free energy of transfer of ions from water to DMSO solution is higher than the present experimental values.     

Rearrangement reactions of the transient lewis acids (CF(3))(3)B and (CF(3))(3)BCF(2): an experimental and theoretical study

Short-lived (CF(3))(3)B and (CF(3))(3)BCF(2) are generated as intermediates by thermal dissociation of (CF(3))(3)BCO and F(-) abstraction from the weak coordinating anion [B(CF(3))(4)](-), respectively. Both Lewis acids cannot be detected because of their instability with respect to rearrangement reactions at the B-C-F moiety. A cascade of 1,2-fluorine shifts to boron followed by perfluoroalkyl group migrations and also difluorocarbene transfer reactions occur. In the gas phase, (CF(3))(3)B rearranges to a mixture of linear perfluoroalkyldifluoroboranes C(n)()F(2)(n)()(+1)BF(2) (n = 2-7), while the respective reactions of (CF(3))(3)BCF(2) result in a mixture of linear (n = 2-4) and branched monoperfluoroalkyldifluoroboranes, e.g., (C(2)F(5))(CF(3))FCBF(2). For comparison, the reactions of [CF(3)BF(3)](-) and [C(2)F(5)BF(3)](-) with AsF(5) are studied, and the products in the case of [CF(3)BF(3)](-) are BF(3) and C(2)F(5)BF(2) whereas in the case of [C(2)F(5)BF(3)](-), C(2)F(5)BF(2) is the sole product. In contrast to reports in the literature, it is found that CF(3)BF(2) is too unstable at room temperature to be detected. The decomposition of (CF(3))(3)BCO in anhydrous HF leads to a mixture of the new conjugate Br?nsted-Lewis acids [H(2)F][(CF(3))(3)BF] and [H(2)F][C(2)F(5)BF(3)]. All reactions are modeled by density functional calculations. The energy barriers of the transition states are low in agreement with the experimental results that (CF(3))(3)B and (CF(3))(3)BCF(2) are short-lived intermediates. Since CF(2) complexes are key intermediates in the rearrangement reactions of (CF(3))(3)B and (CF(3))(3)BCF(2), CF(2) affinities of some perfluoroalkylfluoroboranes are presented. CF(2) affinities are compared to CO and F(-) affinities of selected boranes showing a trend in Lewis acidity, and its influence on the stability of the complexes is discussed. Fluoride ion affinities are calculated for a variety of different fluoroboranes, including perfluorocarboranes, and compared to those of the title compounds.     

Synthesen und NMR‐Untersuchungen von Salzen mit den neuen Anionen [PtXn(CF3)6‐n]2— (n = 0 ‐ 5, X = F,OH, Cl,CN) und die Kristallstrukturanalyse von K2[(CF3)2F2Pt(μ‐OH)2PtF2(CF3)2]·2H2O

Syntheses and NMR Spectroscopic Ivestigations of Salts containing the Novel Anions [PtX_n(CF₃)_6‐n]^2— (n = 0 ‐ 5, X = F, OH, Cl, CN) and Crystal Structure of K₂[(CF₃)₂F₂Pt(μ‐OH)₂PtF₂(CF₃)₂]·2H₂O The first syntheses of trifluoromethyl‐complexes of platinum through fluorination of cyanoplatinates are reported. The fluorination of tetracyanoplatinates(II), K₂[Pt(CN)₄], and hexacyanoplatinates(IV), K₂[Pt(CN)₆], with ClF in anhydrous HF leads after working up of the products to K₂[(CF₃)₂F₂Pt(μ‐OH)₂PtF₂(CF₃)₂]·2H₂O. The structure of the salt is determined by a X‐ray structure analysis, P2₁/c (Nr. 14), a = 11.391(2), b = 11.565(2), c = 13.391(3)Å, β = 90.32(3)°, Z = 4, R₁ = 0.0326 (I > 2σ(I)). The reaction of [Bu₄N]₂[Pt(CN)₄] with ClF in CH₂Cl₂ generates mainly cis‐[Bu₄N]₂[PtCl₂(CF₃)₄] and fac‐[Bu₄N]₂[PtCl₃(CF₃)₃], but in contrast that of [Bu₄N]₂[Pt(CN)₆] with ClF in CH₂Cl₂ results cis‐[Bu₄N]₂[PtX₂(CF₃)₄], [Bu₄N]₂[PtX(CF₃)₅] (X = F, Cl) and [Bu₄N]₂[Pt(CF₃)₆]. In the products [Bu₄N]₂[PtX_n(CF₃)_6‐n] (X = F, Cl, n = 0—3) it is possibel to exchange the fluoro‐ligands into chloro‐ and cyano‐ligands by treatment with (CH₃)₃SiCl und (CH₃)₃SiCN at 50 °C. With continuing warming the trifluoromethyl‐ligands are exchanged by chloro‐ and cyano‐ligands, while as intermediates CF₂Cl and CF₂CN ligands are formed. The identity of the new trifluoromethyl‐platinates is proved by ¹⁹⁵Pt‐ and ¹⁹F‐NMR‐spectroscopy.     

Ferromagnetism and chirality in two-dimensional cyanide-bridged bimetallic compounds

The combination of hexacyanoferrate(III) anions, [Fe(CN)(6)](3)(-), with nickel(II) complexes derived from the chiral ligand trans-cyclohexane-1,2-diamine (trans-chxn) affords the enantiopure layered compounds [Ni(trans-(1S,2S)-chxn)(2)](3)[Fe(CN)(6)](2).2H(2)O (1) and [Ni(trans-(1R,2R)-chxn)(2)](3)[Fe(CN)(6)](2).2H(2)O (2). These chiral systems behave as ferromagnets (T(c) = 13.8 K) with a relatively high coercive field (H(c) = 0.17 T) at 2 K. They also exhibit an unusual magnetic behavior at low temperatures that has been attributed to the dynamics of the magnetic domains in the ordered phase.     

Preparation,purification, and analysis of alkylated cyclodextrins

Methods of alkylation of α-, β- and γ-cyclodextrins have been optimized with regard to the parameters of reaction, degree of alkylation and yields. The analysis of the reaction mixtures and of the isolated single species has been performed by high temperature GC and HPLC. The phase systems of the preferably applied HPLC have been carefully adjusted by variation of both the stationary and mobile phases to the very different hydrophobicities of the various alkylated CD species which have been synthesized. Several partially or fully alkylated CD species were isolated from preparative scale HPLC separations in high purity.     

Commutative and noncommutative representations of matrix-valued herglotz-nevanlinna functions

Operator realizations of matrix-valued Herglotz-Nevanlinna functions play an important and essential role in system theory, in the spectral theory of bounded nonselfadjoint operators, and in interpolation problems. Here, a generalization for realization results of the Brodskiǐ-Livsic type is given for Herglotz-Nevanlinna functions whose spectral measures are compactly supported.     

[2 + 2] Photodimerization of the acenaphthylene ruthenium complex [(C5Me4CH2OMe)Ru(η6-C12H8)]+

The [2 + 2] photodimerization of the complex [(C₅Me₄CH₂OMe)Ru(η⁶-C₁₂H₈)]⁺ under visible-light irradiation leads to a mixture of the head-to-head heptacyclene products [(μ-η⁶: η⁶-C₂₄H₁₆)Ru₂(C₅Me₄CH₂OMe)₂]²⁺ (syn- and anti-) with the predominant formation of the syn-isomer; the structures of both isomers were established by X-ray diffraction analysis.