Quasigroups and tactical systems

A quasigroupQ is a set together with a binary operation which satisfies the condition that any two elements of the equationxy =z uniquely determines the third. A quasigroup is in indempotent when any elementx satisfies the indentityxx =x. Several types of Tactical Systems are defined as arrangement of points into “blocks” in such a way as to balance the incidence of (ordered or unordered) pairs of points, and shown to be coexistent with idempotent quasigroups satisfying certain identifies. In particular the correspondences given are: 1. totally symmetric idempotent quasigroups and Steiner triple systems, 2. semi-symmetric idempotent quasigroups and directed triple systems, 3. idempotent quasigroups satisfying Schröder's Second Law, namely (xy)(yx)=x, and triple tourna-ments, and 4. idempotent quasigroups satisfying Stein's Third Law, namely (xy)(yx)=y, and directed tournaments. These correspondences are used to obtain corollaries on the existence of such quasig-roups from constructions of the Tactical Systems. In particular this provides a counterexample to an ”almost conjecture“ of Norton and Stein (1956) concerning the existence of those quasigroups in 3 and 4 above. Indeed no idempotent qnasigroups satisfying Stein's Third Law and with order divisible by four were known to N. S. Mendelsohn when he wrote a paper on such quasigroups for the Third Waterloo Conference on Combinatorics (May, 1968). Finally, a construction for triple tournaments is interpreted as a Generalized Semi-Direct Product of idempotent quasigroups.     

Heat transfer modeling for turbulent shear flows on curved surfaces

This paper examines two models for two-dimensional curved flow heat transfer. Both models are based on the assumption of local-equilibrium turbulence. Yet, one model predicts an increase for the turbulent Prandtl number with convex curvature, while the other gives the opposite behavior. Through examination of the near wall temperature profiles, it is possible to identify the correct behavior for turbulent Prandtl number. It is found that the turbulent Prandtl number for curved flows increases with convex curvature. The discrepancy between the two models is traced to the modeling terms proposed for the pressure-temperature-gradient correlation and for wall corrections. While the introduction of the rapid component and the wall corrections in the modeling of the pressure-strain correlation do not affect the behavior of the resultant shear stress, these modeling terms in the pressure-temperature-gradient correlation cause the turbulent Prandtl number to decrease with convex curvature.

---

Zusammenfassung In dieser Arbeit werden zwei Modelle für zweidimensionale Wärmeübertragung in gekrümmten Strömungen untersucht. Beide Modelle basieren auf der Annahme von lokalem Gleichgewicht innerhalb der turbulenten Strömung. Trotzdem resultiert das eine Modell in einer Erhöhung der turbulenten Prandtl-Zahl für konvexe Krümmungen, während das zweite Modell genau das Gegenteil vorhersagt. Es ist möglich, durch Untersuchung des Temperaturprofiles in Wandnähe das korrekte Verhalten der turbulenten Prandtl-Zahl zu identifizieren. Es stellt sich dabei heraus, daß die turbulente Prandtl-Zahl für gekrümmte Strömungen in konvexen Kurven zunimmt. Der Widerspruch zwischen den beiden Modellen läßt sich auf die Abbildungsausdrücke zurückführen, die für die Wechselbeziehungen zwischen den Druck- und Temperaturgradienten und die für die Wandkorrekturen angesetzt worden sind. Während die Einführung der schnellen Komponente und die Wandkorrekturen in der Darstellung der Wechselbeziehung zwischen Druck und Beanspruchung keinen Einfluß auf das Verhalten des resultierenden Schubdruckes haben, verursachen diese Darstellungsausdrücke für die Wechselbeziehung zwischen Druck- und Temperaturgradienten ein Abnehmen der turbulenten Prandtl-Zahl für konvexe Kurven.

     

Upper bound on total cross sections

It is shown, under certain assumptions, that if the Nth derivative of the absorptive part of an elastic scattering amplitude exists in the interval t ₁t0, then, the total cross section has the asymptotic upper bound      

Chemisorption of carbon monoxide on the platinum(100) surface: A definition of electronic states by photoelectron spectroscopy

UPS studies are reported for carbon monoxide chemisorbed, at varying coverages, on a clean Pt(100) surface. A new assignment of surface states is obvious from the broadening and resolution of the emission peak, positioned at ca. 8 eV below the Fermi level, following pre-adsorption of propene and pyridine. The valence band density of states, determined by He(I) and He(II) emission spectroscopy, is compared with that reported earlier by X-ray electron emission spectroscopy.     

Femtosecond time-resolved absorption spectroscopy of astaxanthin in solution and in alpha-crustacyanin

Steady-state absorption and femtosecond time-resolved spectroscopic studies have been carried out on astaxanthin dissolved in CS2, methanol, and acetonitrile, and in purified alpha-crustacyanin. The spectra of the S0 --> S2 and S1 --> S(n) transitions were found to be similarly dependent on solvent environment. The dynamics of the excited-state decay processes were analyzed with both single wavelength and global fitting procedures. In solution, the S1 lifetime of astaxanthin was found to be approximately 5 ps and independent of solvent. In alpha-crustacyanin, the lifetime was noticeably shorter at approximately 1.8 ps. Both fitting procedures led to the conclusion that the lifetime of the S2 state was either comparable to or shorter than the instrument response time. The data support the idea that dimerization of astaxanthin in alpha-crustacyanin is the primary molecular basis for the bathochromic shift of the S0 --> S2 and S1 --> S(n) transitions. Planarization of the astaxanthin molecule, which leads to a longer effective pi-electron conjugated chain and a lower S1 energy, accounts for the shorter tau1 in the protein.     

Jahn‐Teller‐Ordnung in pipzH2[Mn2F8], einem Fluoromanganat(III) mit neuer Schichtstruktur

Jahn‐Teller Ordering in pipzH₂[Mn₂F₈], a Fluoromanganate(III) with a New Layer Structure From a solution of (pipzH₂)[MnF₂(HPO₄)(H₂O)]·(H₂PO₄) in concentrated hydrofluoric acid the title compound could be crystallized and was characterized by X‐ray crystallography: monoclinic, space group P2₁/n, Z = 8, a = 13.275(1), 10.400(1), c = 14.928(1) Å, β = 112.337(8), R = 0.0265. The structure shows a new type of anionic layers [Mn₂F₈] built of dimers of edge‐sharing [MnF₆] octahedra linked via common vertices, alternating with layers of piperazinium(2+) cations. A dense network of hydrogen bonds N–H···F achieves a strong 3D interconnection. Strong elongation of the [MnF₆] octahedra is observed due to the Jahn‐Teller effect.The long axes show parallel (ferrodistortive) orientation within the dimers, but the ordering between the dimers is antiferrodistorive. A possible mechanism of formation is discussed on the basis of precursor structures.     

Catalytic osmylation and oxypalladation of khellin. Two useful methods for furan ring degradation. Replacement of the furan ring by an isoxazole ring.

Two oxidation reactions are described which result in the selective degradation of the furan ring in khellin. Conversion of these products to previously inaccessible analogues is described.     

Correlation of ring nitrogen substituents with carbon-13 nuclear magnetic resonance data in azoloazines

Carbon–13 nuclear magnetic resonance data have been acquired on 22 azoloazines. Chemical shifts have been correlated by a step–wise linear multiple regression with nitrogen substituents in both the 5- and 6-membered rings using pyrrolo[1,2-α]pyridine as the reference for chemical shift correlation. The data demonstrate that a highly correlated set of chemical shift parameters exist. Nitrogen substitution in the five–membered ring produces larger cross–ring effects than are oberved in the five–membered ring when substitution occurs in the six–membered ring. Within the six–membered ring a constant para- substituent parameter is noted. The meta- and para- parameters are more complex and fall into two groups for each parameter. Within the five–membered ring, a highly regular chemical shift pattern is observed which reflects an attenuated perturbation from nitrogen substitution in the six–membered ring.     

Fluoro-olefin chemistry Part 20 [1]. Reaction of hexafluoropropene with alcohols

Reaction of hexafluoropropene (HFP) with a series of alcohols under thermal, photochemical or peroxide-initiated conditions affords the 1:1 adducts CF₃CHFCF₂CR¹R²OH (R¹ = H, R² = H, Me, Prⁿ or CF₃; R¹ = Me, R² = Me or Et) in high yield via a radical chain mechanism. Adduct are not formed with the alcohols (CF₃)₂CHOH and CF₃CHFCF₂CH₂OH. Other 1:1 adducts of structure CHF₂CF(CF₃)CH₂OH and CH₃(C₂H₃CF₂CHFCF₃)CH₂OH are formed as minor products in the methanol and $\text{n}$-butanol reactions, respectively.