Die Strommer-Zykliden des Flaggenraumes

According to H. BRAUNER a flag space is a three-dimensional affine space with an absolute {,f,F}, wheref is a line in the plane of infinity andF a point onf. A cyclide is a surface of order 4 that intersects only atf. A STROMMER-cyclide is a cyclide generated by translation of a circle in an isotropic plane. In this paper we proof that all STROMMER-cyclides can be generated by translation of an isotropic circle along a divergent Newton-parabola or a conic in a full isotropic plane. We give normal-forms and CAD-pictures of these surfaces. Finaly some geometrical results about STROMMER-cyclides are developed.

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Herrn o.Prof. Dr. Gy. Strommer zum 70. Geburtstag gewidmet.     

Discrepancies in the distribution of prime numbers

Chebyshev has noticed a certain predominance of primes of the form 4n + 3 over those of the form 4n + 1. He asserted that $\text{lim}{}_{\mathrm{x\to \infty }}\text{∑}\text{p > 2}\text{(?1)}{}^{\mathrm{<mtext>(p\; ?\; 1)</mtext><mtext>2</mtext>}}\text{e}{}^{\mathrm{<mtext>?p</mtext><mtext>x</mtext>}}\text{= ?∞}$. This was unproven until today. G. H. Hardy, J. E. Littlewood and E. Landau have shown its equivalence with an analogue to the famous Riemann hypothesis, namely, L(s, χ₁mod 4) ≠ 0, $\text{Re}\text{(s) >}\text{1}\text{2}$. S. Knapowski and P. Turán have given some similar (unproven) relations, e.g., , which are also equivalent to the above. Using Explixit Formulas the author shows that

$\text{lim}{}_{\mathrm{x\to \infty }}\text{∑}\text{p > 2}\text{(?1)}{}^{\mathrm{<mtext>(p\; ?\; 1)</mtext><mtext>2</mtext>}}\text{log}\text{pp}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{e}{}^{\mathrm{<mtext>?</mtext><mtext>(</mtext><mtext>log</mtext><mtext>2p)</mtext><mtext>x</mtext>}}\text{= ?∞ (1)}$

holds without any conjecture. (In addition, the order of magnitude of divergence is calculated.) It turns out that (1) is only a special case (in several respects). At first, it may be enlarged into

$\text{lim}{}_{\mathrm{x\to \infty }}\text{∑}\text{p > 2}\text{(?1)}{}^{\mathrm{<mtext>(p\; ?\; 1)</mtext><mtext>2</mtext>}}\text{log}\text{pp}{}^{\mathrm{?\alpha }}\text{e}{}^{\mathrm{<mtext>?</mtext><mtext>(</mtext><mtext>log</mtext><mtext>2p)</mtext><mtext>x</mtext><mtext>)</mtext>}}\text{= ?∞, 0?α?}\text{1}\text{2}\text{.}$

Then, it can be generalised to a wider class of progressions. For example, the same is true if one sums over the primes in the classes 3n + 2 and 3n + 1, with a “?” and a “+” sign, respectively. All results of this type depend on the location of the first nontrivial zero of the corresponding L-series. D. Shanks has given some arguments for the predominance of primes in residue classes of nonquadratic type. He conjectured “If m₁ mod k is a quadratic residue and m₂ mod k a non-residue, then there are “more” primes congruent m₂ than congruent m₁ mod k.” This indeed turns out to be true in the sense of (1), not only for k = 3, 4, but for some higher moduli as well. Finally, numerical calculations were made to investigate the behaviour of Δ₃(X) ? π(X, 2 mod 3) ? π(X, 1 mod 3) in the interval 2 ≤ X ≤ 18, 633, 261. No zero was found in this range. In the analogue case of Δ₄(X) ? π(X, 3 mod 4) ? π(X, 1 mod 4) the first sign change occurs at X = 26, 861.     

Zur Kennzeichnung von Lorentztransformationen in Endlichen Ebenen

Given a commutative field K we define d(A,B):= (a₁–b₁)²–(a₂–b₂)² for A=(a₁,a₂), B=(b₁,b₂) K². Given moreover a fixed k K^O, W. Benz has asked for all mappings : K²K² such that d(A,B)=k implies. This paper gives an answer if K=GF(p), p=5,7,11: must be a bijective collineation in case p = 7,11; there are non-injective mappings in case p=5. To obtain some of these results we have mads use of a computer.

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Prof. R. Artzy zum 70. Geburtstag gewidmet.     

Confirmation of the Structures of Lutein and Zeaxanthin

A series of new esters of lutein ( 1a ) have been prepared with the aim of confirming the structure of lutein via an X‐ray crystal‐structure analysis. Although well crystallized, only one of the derivatives, the (?)‐(1R)‐menthyl carbonate ( 1i ) proved to be useful for a complete structure analysis. The same derivative of zeaxanthin ( 2a ) also allowed its crystal structure to be determined. Both analyses represent the first successful X‐ray crystal structure analyses of the most important xanthophylls. At the same time, they confirm both the constitution and absolute configuration of 1a and 2a that had been deduced earlier by classical methods.     

Interaction of nonionic detergents and swelling clays

Summary The 25° sorption isotherms of a polyoxyethylated n-dodecanol with 14 ethylene oxide units (C₁₂EO₁₄) on sodium and calcium montmorillonite were determined. Sorption complexes of the two clays were also prepared with C₁₂EO₃₀, n-dodecanol, polyethylene glycol and polypropylene glycol. X-ray powder diagrams show that the sorbed organic molecules are intercalated with their chains parallel to the montmorillonite lamellae, in layers one or two molecules thick. Sodium montmorillonite sorbs C₁₂ EO₁₄ and C₁₂EO₃₀ in excess of close-packed double layers: the excess detergent is occluded in interstices and absorbs moisture at 85% relative humidity to the same extent as the bulk detergents. The double layer complexes are not swelled by water; the single layer complexes sorb a single layer of water molecules at 85% relative humidity.Apparent densities of the sorbed organic molecules decrease with increasing EO content while bulk densities increase. This is explained in terms of chain flexibility and polarity.

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Zusammenfassung Die 25°-Absorptions-Isothermen von polyoxyäthyliertem n-Dodekanol mit 14Äthylenoxyd-Einheiten(C₁₂EO₁₄) an Na- und Ca-Montmorillonit wurden bestimmt. Sorptions-Komplexe der 2 Tone wurden ebenfalls mit nDodekanol, Polyäthylenglykol und Polypropylenglykol präpariert. Pulverdiagramme zeigen, daß die sorbierten organischen Moleküle mit ihren Ketten parallel zu den Montmorillonitlamellen eingelagert sind, in Schichten 1 oder 2 Moleküle dick. Na-Montmorillonit sorbiert C₁₂EO₁₄ und C₁₂EO₃₀ im Überschuß zur dichtgepackten Doppelschicht. Der Überschuß ist in Hohlräumen eingelagert und absorbiert Wasser bei 85° relativer Feuchtigkeit im selben Ausmaß wie die reinen Detergentien. Die Doppelschichtkomplexe werden nicht durch Wasser gequollen. Die Einschichtkomplexe sorbieren eine Monoschicht von Wasser-Molekülen bei 85° relativer Luftfeuchtigkeit.Scheinbare Dichten der sorbierten organischen Moleküle nehmen mit steigendem Äthylenoxydgehalt ab, während die Dichten der reinen Substanz ansteigen. Dies läßt sich auf Grund der Kettenbeweglichkeit und Polarität erklären.

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Presented at the 145^th Meeting of the American Chemical Society, New York, N. Y., September 1963.     

Zur Kenntnis der anomalen Reaktionen bei der Aminostickstoff-Bestimmung. IX. Die Reaktionen der Pyrrolin- und Oxazolinverbindungen mit salpetriger Säure

Zusammenfassung Es wurde die Reaktion von Pyrrolin- und Oxazolinverbindungen mit salpetriger Säure näher studiert, zumalTh. Wagner- Jauregg die Beobachtung machte, daß diese Verbindungen mit salpetriger Säure 1 Mol Stickstoff entwickeln, ohne daß eine primäre Aminogruppe vorhanden ist. Es zeigte sich, daß die Stickstoffentwicklung wie bei einem leicht desaminierbaren Amin schon bei 20° C schnell und vollständig vonstatten geht (Reaktion 1). Bei den Pyrrohnverbindungen ist ab 50° C noch eine weitere gasbildende Reaktion feststellbar, die bei 80 °C 1,40 Mole (N₂ + N₂O) liefert (Reaktion 2). Für die Reaktion 1 wird wahrscheinlich gemacht, daß sich zunächst eine in wäßriger Lösung unbeständige N-Nitrosoverbindung bildet. Bei deren Hydrolyse entsteht ein Diazoniumsalz, das unter Stickstoffentwicklung zerfällt. Dieses Reaktionsschema wird durch die Tatsache gestützt, daß bei Nitrosierung inwasserfreier Phase keine Stickstoffentwicklung erfolgt.Es wird weiters gezeigt, daß die Reaktion 2 (Gasentwicklung bei erhöhter Temperatur) durch die Folgeprodukte der Reaktion 1 verursacht wird. Bei diesen Folgeprodukten handelt es sich, wieWagner-Jauregg fand, um Acetonylverbindungen. Diese werden, wie wir bereits früher gezeigt haben, nitrosiert, wobei sich Isonitrosoverbindungen bilden. Die Isonitrosogruppe wird von überschüssiger salpetriger Säure unter Bildung von N₂ und N₂O zersetzt.Für die Aminostickstoffbestimmung ergibt sich daraus, daß Pyrrolinund Oxazolinverbindungen nur mit Nitrosylbromid in Eisessig (vgl.Kainz, Huber undKasler ¹⁵) richtig analysiert werden künnen, während die Bestimmung mit wäßriger Nitritlösung(van Slyke) ein Fehlresultat verursacht.

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Summary A study was made of the reaction of pyrrolin- and oxazoline compounds with nitrous acid, especially sinceTh. Wagner- Jauregg observed that these compounds generate 1 mol of nitrogen with nitrous acid, though no primary amino group is present. It was observed that the evolution of nitrogen, as in the case of a readily deaminable amine, occurs rapidly and completely even at 20° C (reaction 1). With the pyrroline compounds, it was possible to demonstrate another gas-producing reaction from 50° C on, which yields 1.40 mols (N₂ + N₂O) at 80° C (reaction 2). It is probable that in reaction (1) there is formation of a N-nitroso compound instable in water solution, and that the hydrolysis of this primary product yields a diazonium salt, which decomposes with evolution of nitrogen. This reaction scheme is supported by the fact that no formation of nitrogen is observed on nitrosation in solutions containing no water.It was also shown that the reaction (2) (evolution of gas at raised temperature) is caused by the secondary product of reaction (1). In these latter products there are involved, as shown byWagner- Jauregg, acetonyl compounds, which as we have proved previously, are nitrosated with formation ofiso-nitroso compounds. Theiso-nitroso group is decomposed by excess nitrous acid with production of N₂ and N₂O.With respect to the determination of amino nitrogen, it was found that pyrroline- and oxazoline compounds can be correctly analyzed only with nitrosyl bromide in glacial acetic acid (compareKainz, Huber, andKasler ¹⁵), whereas the determination with aqueous nitrite solution(van Slyke) brings about an incorrect result.

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Résumé Les auteurs ont poussé l'étude de la réaction des dérivés de la pyrroline et de l'oxazoline avec l'acide nitreux à partir de l'observation deTh. Wagner-Jauregg suivant laquelle ces combinaisons réagissent sur l'acide nitreux en dégageant une molécule d'azote en l'absence de tout groupe aminé primaire. Il est apparu que le dégagement d'azote se produit rapidement et complètement à 20° C comme pour une amine réagissant facilement (Réaction 1). Pour les combinaisons de la pyrroline une autre réaction commençant à partir de 50° C et donnant lieu à la formation de gaz a pu être établie; à 80° C, elle donne naissance à 1,40 molécule du mélange (N₂ + N₂O) (Réaction 2). Pour la réaction 1 il est vraisemblable qu'il se produit tout d'abord en solution aqueuse une combinaison N-nitrosée instable. Son hydrolyse produit un sel de diazonium qui se décompose avec dégagement d'azote. Ce schéma réactionnel est confirmé par le fait que par nitrosation en phase non aqueuse il ne se produit aucun dégagement d'azote.Il a été en outre montré que la réaction 2 (dégagement de gaz par élévation de température) est dûe aux produits formés dans la réaction 1. Parmi ces produits, se trouvent, comme l'ont montréWagner-Jauregg, des combinaisons acétonylées. Ces dernières, comme nous l'avons déjà montré antérieurement, se nitrosent en donnant lieu à la formation de combinaisons isonitrosées. Le groupement isonitrosé est décomposé par l'acide nitreux en excès avec formation de N₂ et de N₂O. Pour la détermination de l'azote aminé, il en résulte que les dérivés de la pyrroline et de l'oxazoline ne peuvent être correctement analysés qu'en employant du bromure de nitrosyle en solution dans l'acide acétique (cf.Kainz, Huber etKasler ¹⁵) tandis que la détermination par action d'une solution aqueuse de nitrite (van Slyke) fournit un résultat erroné.

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VIII. Mitteilung: Anomalie einiger Aminosäuren¹.     

Potent and selective structure-based dibenzofuran inhibitors of transthyretin amyloidogenesis: kinetic stabilization of the native state

Transthyretin (TTR) amyloidogenesis requires rate-limiting tetramer dissociation and partial monomer denaturation to produce a misassembly competent species. This process has been followed by turbidity to identify transthyretin amyloidogenesis inhibitors including dibenzofuran-4,6-dicarboxylic acid (1). An X-ray cocrystal structure of TTR.1(2) reveals that it only utilizes the outer portion of the two thyroxine binding pockets to bind to and inhibit TTR amyloidogenesis. Herein, structure-based design was employed to append aryl substituents at C1 of the dibenzofuran ring to complement the unused inner portion of the thyroxine binding pockets. Twenty-eight amyloidogenesis inhibitors of increased potency and dramatically increased plasma TTR binding selectivity resulted. These function by imposing kinetic stabilization on the native tetrameric structure of TTR, creating a barrier that is insurmountable under physiological conditions. Since kinetic stabilization of the TTR native state by interallelic trans suppression is known to ameliorate disease, there is reason to be optimistic that the dibenzofuran-based inhibitors will do the same. Preventing the onset of amyloidogenesis is the most conservative strategy to intervene clinically, as it remains unclear which of the TTR misassembly intermediates results in toxicity. The exceptional binding selectivity enables these inhibitors to occupy the thyroxine binding site(s) in a complex biological fluid such as blood plasma, required for inhibition of amyloidogenesis in humans. It is now established that the dibenzofuran-based amyloidogenesis inhibitors have high selectivity, affinity, and efficacy and are thus excellent candidates for further pharmacologic evaluation.     

Synthese des Dysidins

Synthesis of Dysidin The synthesis of dysidin ((?)- 1 ), the enantiomer of a metabolite of the marine sponge Dysidea herbacea, is described. To effect the synthesis, (±)-5-isopropyl-4-methoxy-3-pyrrolin-2-one ( 7 ) is converted to its lithium salt and reacted with (?)-(5R,2E)-3-methoxy-5-trichloromethyl-2-hexenoyl chloride ((-)- 11 ) to give (?)- 1 and its diastereoisomer (+)-5-epidysidin ((+)- 12 ) epimeric at C(5) of the pyrrolinone ring. The (?)-acyl chloride (?)- 11 has been synthesized from (+)-(R)-3-(trichloromethyl)butanoic acid ((+)- 8 ) via the intermediates (+)- 9 and (?)- 10 , the pyrrolinone 7 from N-benzyl-oxycarbonyl-L-valine via the intermediate 5 . The enantiomers of acid 8 have been resolved by fractional crystallization of their diastereoisomeric N-(1-phenylethyl)amides. The (R)-chirality of (+)- 8 was determined by comparing the ¹H-NMR spectra of the diastereoisomeric N-(1-phenylethyl)amides 16 and 17 , made from (+)- 8 by substituting deuterium for chlorine, with the spectra of the N-(1-phenylethyl)amides 14 and 15 of known absolute configuration. This correlation shows that literature value (R) for (?)- 8 is in error. Therefore, the structural formulae of (?)-dysidenin and (+)-isodysidenin, two other metabolites of D.herbacea, have to be changed to their mirror images as shown in formulae (?)- 3 and (+)- 4 , respectively.