The mathematics of playing golf, or: a new class of difficult non-linear mixed integer programs

We consider a class of non-linear mixed integer programs with n integer variables and k continuous variables. Solving instances from this class to optimality is an NP-hard problem. We show that for the cases with k=1 and k=2, every optimal solution is integral. In contrast to this, for every k≥3 there exist instances where every optimal solution takes non-integral values. Received: August 2001 / Accepted: January 2002?Published online March 27, 2002     

Existence and Nonlinear Stability of Stationary States of the Schrödinger–Poisson System

We consider the Schrödinger–Poisson system in the repulsive (plasma physics) Coulomb case. Given a stationary state from a certain class we prove its nonlinear stability, using an appropriately defined energy-Casimir functional as Lyapunov function. To obtain such states we start with a given Casimir functional and construct a new functional which is in some sense dual to the corresponding energy-Casimir functional. This dual functional has a unique maximizer which is a stationary state of the Schrödinger–Poisson system and lies in the stability class. The stationary states are parameterized by the equation of state, giving the occupation probabilities of the quantum states as a strictly decreasing function of their energy levels.     

Conformal field algebras with quantum symmetry from the theory of superselection sectors

According to the theory of superselection sectors of Doplicher, Haag, and Roberts, field operators which make transitions between different superselection sectors—i.e. different irreducible representations of the observable algebra—are to be constructed by adjoining localized endomorphisms to the algebra of local observables. We find the relevant endomorphisms of the chiral algebra of observables in the minimal conformal model with central chargec=1/2 (Ising model). We show by explicit and elementary construction how they determine a representation of the braid groupB which is associated with a Temperley-Lieb-Jones algebra. We recover fusion rules, and compute the quantum dimensions of the superselection sectors. We exhibit a field algebra which is quantum group covariant and acts in the Hilbert space of physical states. It obeys local braid relations in an appropriate weak sense.     

Nachweise verwandter Elemente nebeneinander in einem Tropfen. II. Mitteilung: Der mikrochemische Nachweis von Aluminium und Beryllium mittels Ammoniummolybdats

Zusammenfassung Entgegen den Auffassungen von van Zijp über den von ihm veröffentlichten Nachweis des Aluminiums mit Ammoniummolybdat wurde gezeigt, daß ein derartiger Nachweis Empfindlichkeit und Störungsfreiheit in hohem Grade vereint.Der Nachweis durch Ammoniummolybdat konnte auf Beryllium ausgedehnt werden. Eine Reihe von Versuchen zeigte, daß Aluminium- und Berylliummolybdat in zwei verschiedenen Krystallformen ohne gegenseitige Beeinträchtigung selbst nebeneinander auszukrystallisieren vermögen.Es ist mir eine angenehme Pflicht, Herrn Prof. Dr. Wilhelm Böttger für sein stets färderndes Interesse und für die mikrochemische Arbeitsmäglichkeit überhaupt zu danken. Zu Dank verpflichtet bin ich auch der Fa. Otto Perutz, München für die freundliche überlassung von Pervola-Platten zu den Mikrophotographien.     

Untersuchungen an reagenzien zur photometrischen zinnbestimmung

The formation of complexes between tin(IV) and o-diphenols, and o-hydroxyquinones, has been studied spectrophotometrically. Simple o-hydroxyquinones do not react with tin(IV), chloranilic acid gives an SnR₂ complex, and o-diphenols give SnR₃ complexes. The reactions of tin(IV) with hydroxyxanthene dyes to give coloured products were considered for their possible analytical utility: the pH range, absorption spectrum, solubility and composition of the complexes were investigated, and the stability constants determined. Analytical procedures have been developed for tin and checked for precision and sensitivity: gallein and hydroxyhydroquinonephthalein (dihydroxyfluorescein) are to be recommended.     

Connectedness properties of polynomial maps between affine spaces

Let f:A^mAⁿ be a polynomial map between affine spaces. We give some sufficient conditions for the connectedness of the difference kernel of f and relate this to the Jacobian Conjecture.     

Preparation of 1-ferrocenyl-2-methyl-1-propylamine,a highly effective chiral template in asymmetrically induced synthesis

The 1-ferrocenyl-2-methyl-1-propylamine (2a)is the most effective chiral template in asymmetrically induced peptide synthesis by stereoselective four component condensation (4CC). Two routes for the synthesis of this amine via its N,N-dimethyl derivative (12a) an described. One route involves the conversion of 12a into the corresponding azide 14a by treatment with methyl iodide/sodium azide in diglyme/water and subsequent reduction of the azide. The preferred other route consists of treating 12a with thioglycolic acid/formic acid to yield the carboxymethylmercapto derivative 9a and transformation of the latter into 2a with aqueous ammonia/ammonium chloride/mercuric chloride. Some related reactions are also discussed.     

Fragmentation of 1-aryl-5-(2-dialkylaminovinyl)-1H-tetrazoles upon electron impact

The mass spectrometric behaviour of 1-aryl-5-(2-dialkylaminovinyI)-lH-tetrazoles was studied, especially 1-phenyl-5-(2-dimethylaminovinyl)-l H-tetrazole as a typical example of D- and ¹⁵N-labelled derivatives revealed a rearrangement via a carbodiimide-like intermediate ion.     

Heteroallen-komplexe des molybdänocens

Reactions of the extremely labile molybdenocene olefin complexe Mo(η⁵-C₅H₅)₂[(Z)-C₆H₅CH=CHC₆H₅] with heteroallenes X=C=Y (X=C=Y = CS₂, (p-tolyl)NCN(p-tolyl), (C₆H₅)₂CCO) gives the corresponding heteroallene complexes of molybdenocene Mo(η⁵-C₅H₅)₂(X=C=Y) in high yields. Spectroscopic data clearly indicate a dihapto-coordination of the heteroallenes via the C=X bond (X = O, S, N).     

Unexpected open-chain molecular structure of a titanocenyl-substituted dimethyl ether

Reaction of (η-C₅H₄CH₃)₂TiCl₂ with CH₃OCH₂MgCl yielded (η-C₅H₄-CH₃)₂Ti(Cl)CH₂OCH₃ (2a). Complex 2a crystallizes in space group P2₁/n with cell constants a 8.162(1), b 11.004(2), c 15.297(2) Å, β 96.40(1)°. In contrast to the three-membered metallacyclic zirconium compound of analogous composition (η-C₅H₅)₂Zr(Cl)CH₂OCH₃ (1), the titanocenyl-substituted ether 2a has an open-chain (η¹-O“-outside”) type structure.