Zur projektiven Differentialgeometrie der Flächen,die von einer einparametrigen Schar von Kegelschnitten erzeugt werden

Ohne Zusammenfassung     

The true self-repelling motion

We construct and study a continuous real-valued random process, which is of a new type: It is self-interacting (self-repelling) but only in a local sense: it only feels the self-repellance due to its occupation-time measure density in the `immediate neighbourhood' of the point it is just visiting. We focus on the most natural process with these properties that we call `true self-repelling motion'. This is the continuous counterpart to the integer-valued `true' self-avoiding walk, which had been studied among others by the first author. One of the striking properties of true self-repelling motion is that, although the couple (X _t , occupation-time measure of X at time t) is a continuous Markov process, X is not driven by a stochastic differential equation and is not a semi-martingale. It turns out, for instance, that it has a finite variation of order 3/2, which contrasts with the finite quadratic variation of semi-martingales. One of the key-tools in the construction of X is a continuous system of coalescing Brownian motions similar to those that have been constructed by Arratia [A1, A2]. We derive various properties of X (existence and properties of the occupation time densities L _t (x), local variation, etc.) and an identity that shows that the dynamics of X can be very loosely speaking described as follows: −dX _t is equal to the gradient (in space) of L _t (x), in a generalized sense, even though x↦L _t (x) is not differentiable. Received: 15 April 1997 / Revised version: 30 January 1998     

Die zweifachen Blutelschen Kegelschnittflächen

A surface in projective space generated by a one parameter family of conics is called a conic surface of Blutel if the tangent planes of taken along a generating conic, envelop a quadratic cone. If the conjugate curves (with respect to the generating conics) are conics, too, we call a two-fold Blutel's conic surface. In an earlier paper [4] it was shown that the planes of both conic families, the generating and the conjugate one, belong to a pencil, each. The present paper completes these investigations by integrating the derivative equations (3), (8), (9), (10). As a final result, a complete classification of all these surfaces is given. They are all algebraic of at most fourth order and furthermore—besides the quadrics and certain degenerate cases—they are complex projectively equivalent to the cyclides of Dupin.     

Über die Darstellung von substituierten 3,4-Dihydro-2(1H)-pyrimidin-iminen und 2-Aminopyrimidinen aus Guanidin und α,β-ungesättigten Ketonen

Guanidine reacts with the 2-alken-1-ones4 a-f and5 to give the unstable dihydropyrimidinimines (or-amines respectively)8 a-f (I, II or III respectively) and the hexahydroquinazolinimine (-amine)9 (I, II or III);8 a-f lose H₂, partly in the course of the reaction, partly during recrystallization to yield the 2-pyrimidinamines10 a-c, e, f, and the 4-methyl-5,6,7,8-tetrahydro-2-quinazolinamine11. With picric acid the unstable compounds8 d, f and9, resp. are converted into the stable 2-amino-3,4-dihydro-1H-2-pyrimidinylium picrates12 d and into the 2-amino-4-methyl-3,5,6,7,8,8 a-hexahydro-1H-2-quinazolinium picrate (13 resp.14, 15)¹⁸, whereas8 e reacts with HCl to give the chloride12 e. The structures of12 d-f follow from their NMR-spectra, and of10 a-c, e, f and11 by alternative syntheses by known methods^12–17 (e.g. from -diketones and guanidine). The reaction of8 d-f and9 to10 d-f and11 respectively is compared with previously described dehydrogenation and disproportionation reactions, especially of 3,4-dihydro-2(1H)-pyrimidinones (6 d ⁴,23 ²⁰) and-thiones (30 ¹) of similar structure and formulated as a base-catalysed elimination of H₂.

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HerrnA. Fuchsgruber danken wir für die Aufnahme und Hilfe bei der Interpretation der NMR-Spektren.     

Zur Kenntnis der Struktur der Chalkon-Guanidin-Kondensate Über Heterocyclen, 75. Mitt.

Guanidine reacts with chalkone1 a, 4-methylchalkone1 b and 4′-methylchalkone1 c resp. to yield mixtures of pyrimidinamines2 a,3 b and3 c (=3 b) resp. with (2:1)-condensatesA,B andC resp. The structures of the compoundsA-C (whicha priori could be dihydropyrimidopyrimidines4 a-c or5 a-c or6 a-c) are elucidated. NMR-investigations show that the saltsA-C · HCl must be symmetrically substituted pyrimidopyrimidinyliumchlorides4 a-c · HCl or5 a-c · HCl (and not6 a-c · HCl). Furthermore, it is proved by chemical methods that the condensatesB · HCl andC · HCl are pyrimidopyrimidinyliumchlorides4 b andc · HCl (and not5 b andc · HCl): The structure ofB · HCl (=4 b · HCl) was established by total synthesis of dimethylpyrimidopyrimidinyliumpicrate9 b-Pi from10 c (via13 c · HI-18 · HCl) and transformation ofB · HCl into an identical salt9 b-Pi via hexahydropyrimidopyrimidine8 b · HCl. The structure ofC · HCl (=4 c · HCl) was determined by comparison of its hydrogenation product (=8c · HCl) with8 b · HCl. The structure of condensateA · HCl (=4 a · HCl) results from conclusion by analogy. The spatial structure of4 a-c · HCl and8 a-c · HCl is discussed; it was established by NMR that the salts are racemic mixtures of stereoisomers4 a-c K · HCl and8 a-c K · HCl resp. and their antipodes (with C₂ symmetry).     

Über 4,4,6-trisubstituierte 3,4-Dihydro-2(1H)-pyrimidinimine,Hexahydro-2(1H)-chinazolinimine bzw.-one bzw.-thione und Spiro[cycloalkan-1,4′(1′H)-cycloalkeno-pyrimidin]-2′ (3′H)-imine

The 2-alken-1-ones2 b, 2 c and3–8, which have very different structures, were reacted with guanidine to give cyclic compounds:2 b and2 c resp. are transformed by guanidine to the trisubstituted 3,4-dihydro-2(1H)-pyrimidinimines1 b and1 c resp., action of guanidine on the cyclo-hexylidenaceton (3) yields 4-methyl-1,3-diaza-4-spiro[5,5]-undecen-2-imine (11); isopropylidencyclohexanone4 and the isomeric (methylcyclohexenyl)ethanone5 condense with guanidine resp. urea resp. NH₄CNS to give 4,4-dimethyl-3,4,5,6,7,8-hexahydro-2(1H)-quinazolinimine (15), resp.-one (16), resp.-thione (17) and 4,8a-dimethyl-3,4,6,7,8,8a-hexahydro-2(1H)-quinazolinthione (19). With the cycloalkyliden-alkanones6–8 guanidine reacts to yield the spiro[cycloalkan-1,4(1H)-cycloalkenopyrimidin]-2(3H)-imines24–26. The structure of the compounds cited is derived from their NMR-and (partially) mass spectra; from most of the bases picrates were also synthesized.

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Herrn Prof. Dr.H. Grubitsch zum 70. Geburtstag gewidmet.     

Über die Bromierung von substituierten 3,4-Dihydro- und 6-Hydroxy- bzw. 6-Äthoxy-3,4,5,6-tetrahydro-2(1H)-pyrimidinonen

Bromination of 1-benzyl-4-methyl-3.4-dihydro-2(1H)-pyrimidinone (9 a) with 1 mole Br₂ in CHCl₃ yields 1-benzyl-5-bromo-6-hydroxy-4-methyltetrahydro-2(1H)-pyrimidinone,12 a, or the 6-ethoxypyrimidinone13 a, according to whether H₂O orEtOH is used in working up. With 2 moles Br₂,9 a analogously affords the 5.5-dibromopyrimidinnes14 a or15 a. Bromination of the 6-hydroxypyrimidinone10 a yields the same products,12 a and13 a, or14 a and15 a respectively, while the 4-phenyl-pyrimidinones9 b and11 b yield the corresponding 5-bromo-and 5.5-dibromopyrimidinones13 b and15 b. The structures of the compounds12 a-15 b are confirmed by their NMR data and chemical properties: the oxopyrimidinylmethylureas16 a and17 a are formed by the action of methylurea on12 a and13 a, or on14 a and15 a respectively; with hexamethylenetetramine,12 a reacts to give the 5.6-dihydroxypyrimidinone18 a, while13 b is transformed to the 4-phenylpyrimidinone19 b. 13 b was also synthesized from α-bromocinnamaldehyde. The mechanism of bromination is discussed.     

Über polycyclische 2-Alken-1-on—Guanidin-Kondensate

Guanidine reacts with 1,3-diphenyl-2-propen-1-one to yield not only the dihydropyrimidine2 a ² and the aromatic compound3 a ², but also a bicyclic base, 2,4,6,8-tetraphenyl-2,8-dihydro-1H-pyrimido[1,2–a]pyrimidine (4). Action of 4-phenyl-3-buten-2-one on guanidine generates a 22-condensate, 7-methyl-4,5-diphenyl-4,4a,5,6,7,8,10,10a-octahydro-7,10a-methano-pyrimido[4,5–d]diazocine-2,9(1H, 3H)-diimine12 R besides the methylphenylpyrimidinamine3 b ². The structural formulae4 and12 are proved by NMR-spectra. The mass spectra and the mechanisms of the formation of4 and12 are also discussed.

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Herrn Prof. Dr.O. Hromatka zum 70. Geburtstag gewidmet.     

Some mathematical aspects of the scaling limit of critical two-dimensional systems

It has been observed long ago that many systems from statistical physics behave randomly on macroscopic level at their critical temperature. In two dimensions, these phenomena have been classified by theoretical physicists thanks to conformal field theory, that led to the derivation of the exact value of various critical exponents that describe their behavior near the critical temperature. In the last couple of years, combining ideas of complex analysis and probability theory, mathematicians have constructed and studied a family of random fractals (called ‘Schramm-Loewner evolutions’ or SLE) that describe the only possible conformally invariant limits of the interfaces for these models. This gives a concrete construction of these random systems, puts various predictions on a rigorous footing, and leads to further understanding of their behavior. The goal of this paper is to survey some of these recent mathematical developments, and to describe a couple of basic underlying ideas. We will also briefly describe some very recent and ongoing developments relating SLE, Brownian loop soups and conformal field theory.     

Flame synthesis of calcium carbonate nanoparticles

Calcium carbonate nanoparticles of 20-50 nm size were obtained from a flame spray process where combustion of specific calcium-containing precursors results in amorphous or crystalline calcium carbonate particles depending on the spray flow conditions.