Some problems of linear differential equations on abstract spaces and unbounded perturbations of linear operator semigroup

This paper is a survey for development of linear distributed parameter system. At first we point out some questions existing in current study of control theory for the L^p linear system with an unbounded control operator and an unbounded observation operator, such as stabilization problem and observer theory that are closely relevant to state feedback operator. After then we survey briefly some results on relevant problems that are related to solvability of linear differential equations in general Banach space and semigroup perturbations. As a principle, we propose a concept of admissible state feedback operator for system (A, B). Finally we give an existence result of admissible state feedback operators, including semigroup generation and the equivalent conditions of admissibility of state feedback operators, for an L^p well-posed system.     

Limit cycles of a class of Liénard systems derived from state-dependent impulses

Discontinuous phenomena, in which objects may behave continuously and sometimes discretely are not only found in nature and under laboratory conditions but also in simple, familiar contexts. For example, this phenomenon is skillfully incorporated into the internal structure of mechanical wristwatches. Unless an extremely small amount of state-dependent impulse is applied intermittently, the reciprocating rotational movement of the balance and hairspring, which is the heart of the mechanical wristwatch, cannot be maintained. The small amount of state-dependent impulse, which is often overlooked, can make a significant difference; however, very few studies have examined this subject. This study assumes the underlying cause of discontinuous behaviors as impulses generated when an object reaches a particular state, assuming that the continuous behavior follows the Liénard system, which is widely studied in the field of electrical circuits. The main theorem provides the conditions under which the effect of the impulses causes a stable limit cycle in the Liénard system, even if no limit cycle exists when there are no impulses. The Poincaré–Bendixson theorem for discontinuous dynamical systems and phase plane analysis are used to prove the main theorem. Several examples and their simulations are provided to illustrate the main theorem.     

A genetic algorithm for the automated generation of molecules within constraints

Summary A genetic algorithm has been designed which generates molecular structures within constraints. The constraints may be any useful function, for example an enzyme active site, a pharmacophore or molecular properties from pattern recognition or rule-induction analyses. The starting point may be random or may utilise known molecules. These are modified to grow into families of structures which, using the evolutionary operators of selection, crossover and mutation evolve to better fit the constraints. The basis of the algorithm is described together with some applications in lead generation, 3D database construction and drug design. Genetic algorithms of this type may have wider applications in chemistry, for example in the design and optimisation of new polymers, materials (e.g. superconducting materials) or synthetic enzymes.     

Preparation and characterization of theβ-cyclodextrin inclusion complex with sulfafurazole

The 1 : 1 inclusion complex involving sulfafurazole (SF) and-cyclodextrin (-CD) is prepared by the freeze-drying method and characterized on the basis of its chemical analysis, thermal behavior, infrared spectrum, X-ray powder pattern and¹³C NMR spectrum in DMSO-d₆ solution. The stability constant of the inclusion complex was determined by the solubility method. The effect of cyclodextrin on the UV absorption spectrum of sulfafurazole was also observed.     

Mercury(II) complex with terpyridine: structure of the 2∶1 solvate of [Hg(terpy)2](CF3SO3)2 with acetone

[Hg(terpy)₂](CF₃SO₃)₂·0.5(CH₃)₂CO crystallizes in the triclinic space group witha=14.631(6),b=15.258(4),c=18.785(7) Å, =69.66(2), =70.72(1), =88.55(1)°. The crystal structure consists of two independent [Hg(terpy)₂]²⁺ cations, four trifluoromethanesulfonate anions and an acetone molecule in the asymmetric unit. Each mercury atom is coordinated by two tridentate terpyridine ligands forming an irregular six-coordination polyhedron. The Hg–N bond lengths range from 2.27(2) to 2.53(2) Å.     

Boundedness of solutions of nonlinear systems of differential equations

We first give an example to illustrate that the results in [12] concerning the boundedness of solutions of nonlinear oscillatory equations are not true. And then we obtain sufficient or necessary conditions for the boundedness of solutions of the nonlinear system of differential equations

Metastable decay rates,asymptotic expansions,and analytic continuation of thermodynamic functions

The grand potentialP(z)/kT of the cluster model at fugacityz, neglecting interactions between clusters, is defined by a power series _n Q _n z ⁿ , whereQ _n , which depends on the temperatureT, is the partition function of a cluster of sizen. At low temperatures this series has a finite radius of convergencez _s . Some theorems are proved showing that ifQ _n , considered as a function ofn, is the Laplace transform of a function with suitable properties, thenP(z) can be analytically continued into the complexz plane cut along the real axis fromz _s to + and that (a) the imaginary part ofP(z) on the cut is (apart from a relatively unimportant prefactor) equal to the rate of nucleation of the corresponding metastable state, as given by Becker-Döring theory, and (b) the real part ofP(z) on the cut is approximately equal to the metastable grand potential as calculated by truncating the divergent power series at its smallest term.     

The influence of cyclodextrin complexation on proton transfer in piperidinomethyl-2-naphthol

Summary Piperidinomethyl-2-naphthol forms inclusion complexes with -cyclodextrin with an equilibrium constant of 265M ^–1 in aqueous solution. The proton transfer equilibrium between the neutral and the zwitterionic form is strongly influenced by the association.

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Der Einfluß der Komplexierung mit -Cyclodextrin auf das Protontransfer-Gleichgewicht in Piperidino-2-naphthol (Kurze Mitt.)

Zusammenfassung Piperidinomethyl-2-naphthol bildet Einschlußkomplexe mit -Cyclodextrin mit einer Gleichgewichtskonstante von 265M ^–1. Das Protontransfer-Gleichgewicht zwischen neutraler Form und zwitterionischer Struktur wird durch diese Assoziation stark beeinflußt.

     

Titantetrafluorid – Eine überraschend einfache Kolumnarstruktur

Titanium Tetrafluoride – a Surprising Simple Column Structure For the first time single crystals of TiF₄ have been obtained by solvothermal decomposition of (O₂)₂Ti₇F₃₀ in anhydrous HF. The colourless, transparent needles crystallize orthorhombic in spacegroup Pnma–D (No. 62) with a = 2 281.1, b = 384.8, c = 956.8 pm, Z = 12. The new type of structure is dominated by isolated columns of corner-linked TiF₆-octahedra.     

NMR study of the stoichiometry,stability and exchange kinetics of complexation reaction between Pb2+ ion and 18-crown-6 in binary acetonitrile-water mixtures

Proton NMR was used to study the complexation reaction between lead ion and 18-crown-6 in a number of binary acetonitrile-water mixtures. Formation constant for the resulting 11 complexes in different solvent mixtures was determined by computer fitting of the chemical shift-mole ratio data. There is an inverse relationship between the complex stability and amount of water in the mixed solvent. The dissociative kinetics of the complex was studied by proton line-shape analysis. The Arrhenius plots showed a distinct isokinetic temperature at about 25°C at which the decomplexation rate is more or less independent of the solvent composition. the complexation rate and the activation parameters E _a , H and S, for the exchange have been determined and found to be strongly solvent dependent. There is actually a linear relationship between the mole fraction of acetonitrile in the mixed solvent and logarithm of the stability constant as well as activation parameters.