Two-point Padé approximants for formal Stieltjes series

We prove that certain two-point Padé approximants occupying the diagonal of the Padé table form monotone sequences of lower and upper bounds uniformly converging to a Stieltjes function. The results can be applied to the theory of inhomogeneous media for the calculation of the bounds on the effective transport coefficients of heterogeneous materials.     

Fuzzy quantum logics and infinite-valued Łukasiewicz logic

A new, physically more plausible definition of a fuzzy quantum logic is proposed. It is shown that this definition coincides with the previously studied definition of a fuzzy quantum logic; therefore it defines objects which are traditional quantum logics with ordering sets of states. The new definition is expressed exclusively in terms of fuzzy set operations which are generated by connectives of multiple-valued logic studied by ukasiewicz at the beginning of the 20th century. Therefore, the logic of quantum mechanics is recognized as a version of infinite-valued ukasiewicz logic.     

Separately harmonic and subharmonic functions

Let u(x, y) be defined in B ₁×B ₂ where B ₁ ^m and B ₂ ⁿ , and assume that u(x, ·) harmonic for every fixed x and u(·, y) is subharmonic for every fixed y. We show that if u(·, y) is, in addition, C ² for each y then u is subharmonic in B ₁×B ₂ in both variables jointly.     

A new heuristic algorithm solving the linear ordering problem

The linear ordering problem is an NP-hard combinatorial problem with a large number of applications. Contrary to another very popular problem from the same category, the traveling salesman problem, relatively little space in the literature has been devoted to the linear ordering problem so far. This is particularly true for the question of developing good heuristic algorithms solving this problem.In the paper we propose a new heuristic algorithm solving the linear ordering problem. In this algorithm we made use of the sorting through insertion pattern as well as of the operation of permutation reversal. The surprisingly positive effect of the reversal operation, justified in part theoretically and confirmed in computational examples, seems to be the result of a unique property of the problem, called in the paper the symmetry of the linear ordering problem. This property consists in the fact that if a given permutation is an optimal solution of the problem with the criterion function being maximized, then the reversed permutation is a solution of the problem with the same criterion function being minimized.     

Jauch-Piron system of imprimitivities for phonons. I. Localizability in discrete space

This paper is devoted to a discussion of the notion of localizability for phonons, i.e., quasiparticles arising from the harmonic vibrations of a system ofn atoms bound to one another by elastic forces. The natural tools for the analysis of localizability are the projection operatorsÊ() acting on the Hilbert space of one-phonon states, where is an arbitrary subset of the set that consists ofn vectors specifying the equilibrium positions ofn atoms. The expectation value ofÊ() is the probability that the phonon belongs to the atoms whose equilibrium positions are characterized by the elements of . For a strongly localizable phonon all of the projection operatorsÊ() commute with one another, whereas in the case of a weakly localizable phonon the operatorsÊ(₁) andÊ(₂) do not commute when ₁ and ₂ overlap. With the aid of the Jauch-Piron quantum theory of localization in space, the present paper describes the method of obtainingÊ() and also shows that if in the system ofn atoms there exist normal modes of zero frequency, then the phonon is only weakly localizable. Given the explicit expression forÊ(), one can define the number-of-phonons operator as well as the quasiparticle analogue (given in a companion paper) of the Wigner distribution function.     

Domination inequality for martingale transforms of a Rademacher sequence

Letf _n = Σ _k=1 ⁿ v _k r _k ,n=1,…, be a martingale transform of a Rademacher sequence (r _n)and let (r _n ^′ ) be an independent copy of (r _n).The main result of this paper states that there exists an absolute constantK such that for allp, 1≤p<∞, the following inequality is true: In order to prove this result, we obtain some inequalities which may be of independent interest. In particular, we show that for every sequence of scalars (a _n)one has where is theK-interpolation norm between ℓ₁ and ℓ₂. We also derive a new exponential inequality for martingale transforms of a Rademacher sequence. This research was supported in part by an NSF grant and an FRPD grant at NCSU.     

Further studies on the chemistry and structure of n-oxides of sparteine and its derivatives—V: Synthesis and structure of 2-phenylsparteine-n16-oxide,a new case of “sponge” proton

The synthesis of 2-phenylsparteine-N₁₆-oxide (7) and its perchlorate salt (7-H⁺) was carried out. On the basis of spectral data, and by comparison with appropriate sparteine-N-oxides, the mechanism of formation and the structures of the two new compounds were proposed. It was found, the basicity of the new N-oxide is unexpectedly high and comparable to the basicity of quaternary ammonium hydroxides. The structure and the strength of intramolecular H-bond in 7-H⁺ makes 7 an excellent “catcher” proton or specific ”sponge” proton.     

Development of a capillary electrophoretic method for the analysis of amino acids containing tablets

Ketosteril is an enteral medicinal product indicated for prevention and therapy in chronic renal insufficiency in connection with a low protein diet. Tablets of Ketosteril contain five essential amino acids like: Lys, His, Thr, Trp, Tyr and another five amino acids in the form of their hydroxy and keto analogues as calcium salts, that are: alpha-ketoleucine, alpha-ketoisoleucine, alpha-ketovaline, alpha-ketophenylalanine and alpha-hydroxymethionine. The composition of Ketosteril tablets is routinely tested with three LC methods. Capillary electrophoretic method seems to be a good alternative for amino acids and their analogues determination in multicomponent pharmaceuticals because of short analysis time and the possibility to assay all components during a single run without any pretreatment. Electrophoresis was performed in 50 microm I.D. fused-silica capillaries with 65 cm distance to the detector. Capillaries were installed in Waters Quanta 4000 electrophoretic equipment with a positive power supply and on-line UV detection at 214 nm. Separations were done in a buffer containing 40 mM Tris and 160 mM boric acid titrated with NaOH to pH 10. The method developed allows the separation of all investigated analytes with an efficiency of n = 230,000 and 20 min analysis time. The method was applied for determination of all components of Ketosteril in commercial tablets.     

Electronic structure of octacyanides of molybdenum IV and V according to the SCCC MO method

The SCCC MO method has been used to calculate the electronic structure and spectrum of Mo(CN) ₈ ^4– and Mo(CN) ₈ ^3– ions of a D _4d symmetry. All metal-ligand and ligand-ligand interactions and all overlap integrals (over products of HF atomic orbitals) have been included within the SCCC MO scheme. Basing on these calculations a new assignment of some absorption bands is proposed. The overall agreement between the calculated and observed electronic spectrum is satisfactory.

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Zusammenfassung Elektronenstruktur und Spektrum der [Mo(CN)₈]^–4- und [Mo(CN)₈]^–3-Ionen in D _4d -Symmetrie wurden mit der SCCC-MO-Methode berechnet. Alle Metall-Ligand- und Ligand-Ligand-Wechsel-wirkungen und Überlappungsintegrale wurden berücksichtigt. Die Bandenzuordnung wurde neu vorgenommen. Die Übereinstimmung der berechneten und gemessenen Spektren ist befriedigend.

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Résumé Application à Mo(CN) ₈ ^4– et Mo(CN) ₈ ^3– de la méthode SCCC MO basée sur des orbitales atomiques HF approchées, en tenant compte du recouvrement et de toutes les interactions ligand-ligand. Dans la partie I on discute le cas de la symétrie D _4d . L'accord global avec l'expérience est satisfaisant en ce qui concerne les spectres. On montre aussi que toutes les attributions antérieures basées sur la théorie du champ cristallin semblent inadéquates.

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The authors are grateful to Prof. W. Jakób, Dr. Z. Stasicka and Dr. A. Samotus for many interesting discussions, to Prof. J. Chojnacki for providing us with new X-ray results before publication and to Dr. A. J. Sadlej for a help in dealing with the computer.