Sur les points autour desquels le mouvement brownien plan tourne beaucoup

Résumé Nous étudions le comportement asymptotique de l'aireA _n de l'ensemble des points autour desquels le mouvement brownien plan a tourné environn fois sur un intervalle de temps fixé [0,t]. Nous montrons en particulier que lorsquen tend vers l'infini,A _n est équivalent dansL ² àt/(2n ²) asn

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Summary We study the asymptotic behaviour of the areaA _n of the set of points around which the planar Brownian motion winds about n times on a given timeinterval [0,t]. We prove thatA _n is equivalent (in theL ²-sense) tot/(2n ²) asn tends to infinity.

     

Über die Reaktionen von monosubstituierten Guanidinen mit 1-Phenyl-1,3-butandion

Methyl-, benzyl- and phenylguanidine (2 b–d) react with 1-phenyl-1,3-butanedione to yield exclusively N²-substituted 4-methyl-6-phenyl-2-pyrimidinamines10 b–d. The formation of isomeric N¹-substituted 2(1H)-pyrimidinimines11 or12 cannot be observed. The structural formulae of10 b and c were proved by spectroscopical methods. The structure of the phenylguanidine-phenylbutanedione-condensate was determined by comparison and establishment of the identity of its picrate with an authentical sample of10 d-picrate, which had been synthetized from pyrimidinthione13 (via methylthiopyrimidine16 · HI). Boiling13 with aniline in butanol yields thiodipyrimidine15 (and not10 d).     

Synthesis and Structure Elucidation of Pyrimidobenzimidazoles and Fused Derivatives II [1]. Dodecahydrobenzimidazo-[2,1-b]quinazolines and Decahydrobenzimidazo-[2,1-b]benzo[h]quinazolines [2]

Summary.  The cyclization reactions of trans-3a-hexahydro-2-benzimidazolamine with 2-alkylidene- and 2-benzylidenecyclohexanones and α-tetralones, respectively, yield mixtures of two isomeric condensates each. Thorough high resolution NMR analyses (¹H and ¹³C NMR, HH-COSY, gs-HSQC, gs-HMBC, 1D TOCSY, and 1D NOE difference spectra) revealed that the corresponding isomers are in all cases linearly fused diastereomeric 12α- and 12β-substituted trans-6aα-dodecahydrobenzimidazo[2,1-b]quinazolines and 7α-substituted trans-8aβ- and trans-8aα-decahydrobenzimidazo[2,1-b]benzo[h]quinazolines, respectively. The formation of corresponding angularly fused regioisomers was not observed so far. The stereochemistry and the tautomerism of some bases and their hydrochlorides as well as the regiospecific course of the cyclization reactions are discussed. Biological tests showed that the novel compounds don’t exert remarkable antibacterial and antimycotic effects. Received November 3, 1999. Accepted November 30, 1999     

Synthese und Strukturaufklärung von Pyrimidobenzimidazolen und kondensierten Derivaten, 3. Mitt. [1,2]. Decahydropyrimido[1,2-<Emphasis Type="Italic">a</Emphasis>]benzimidazol-2-ole und Octahydropyrimido[1,2-<Emphasis Type="Italic">a</Emphasis>]benzimidazole

Zusammenfassung.  Die Reaktion von racemischem trans-Hexahydrobenzimidazol-2-amin mit drei ausgew?hlten vinylogen Ketonen unter milden Bedingungen wurde untersucht. Nur im Fall der Umsetzung mit 4-Phenyl-3-buten-2-on konnten Cycloadditionsprodukte isoliert werden. Diese sind laut NMR-Analysen Gemische von zwei der acht m?glichen Diastereomeren, n?mlich von rac-2β-Methyl-4β-phenyl-trans-5aα- und trans-5aβ-decahydropyrimido[1,2-a]benzimidazol-2α-ol. Bei Umsetzungen des Amins mit Phenylbutenon bei erhoühten Temperatur sowie mit, 2-Methyl-1-phenyl-1-penten-3-on und 4^′-Chlorchalkon bildeten sich Gemische entsprechend substituierter rac-4α-Phenyl-trans-5aβ- und trans-5aα-octahydropyrimido[1,2-a]benzimidazole. Struktur und Stereochemie der Titelverbindungen und ihrer Hydrochloride wurden anhand NMR-spektroskopischer Untersuchungen aufgekl?rt. Die Ringschlu?reaktionen verlaufen in allen F?llen gleichgerichtet regio-, jedoch nicht diastereoselektiv. Versuche zur Beeinflussung der Regio- und/oder Diastereoselektivit?t der Cyclisierungsreaktion durch Variation der Reaktionsbedingungen hatten keinen Erfolg.

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Synthesis and Structure Elucidation of Pyrimidobenzimidazoles and Fused Derivatives III [1,2]. Decahydropyrimido[1,2- a ]benzimidazol-2-oles and Octahydropyrimido[1,2- a ]benzimidazoles

Summary.  The reaction of racemic trans-hexahydrobenzimidazol-2-amine with three vinylogous ketones under mild conditions was studied. Only in the case of 4-phenyl-3-buten-2-one cycloaddition products could be isolated. According to NMR spectroscopy they consist of mixtures of two of eight possible diastereomers: rac-2β-methyl-4β-phenyl-trans-5aα- and trans-5aβ-decahydropyrimido[1,2-a]benzimidazol-2α-ole. Reaction of the amine with the butenone at higher temperature and with, 2-methyl-1-phenyl-1-penten-3-one, and 4^′-chlorochalcone afforded mixtures of two diastereomers each, which turned out as rac-4α-phenyl-trans-5aβ- and trans-5aα-octahydropyrimido[1,2-a]benzimidazoles. Complete structural and stereochemical assignments of the title compounds and their hydrochlorides were established by NMR spectroscopic investigations. The results showed that all investigated cyclization reactions proceeded regioselectively with equal orientation of the components, but not diastereoselectively. Variation of the reaction conditions did influence neither regio- nor diastereoselectivity.

Received October 19, 2000. Accepted October 30, 2000     

Special Reactions of α,β-Unsaturated Ketones III [1]. Formation and Structure Elucidation of Dimers of 2-Aryl-methyleneketones: (5E)-5-Benzylidene-9β-phenyl-trans-4a-1,2,3,4,4a,5,6,7,8,9a-decahydroxanthen-4aα-ol and 3β,5β-Bis-(4-methoxyphenyl)-2α,4β,6α-trimethyl-4α-propionyl-cyclohexanone [2]

Summary.  The reaction of 2-benzylidenecyclohexanone and 1-(4-methoxyphenyl)-2-methyl-1-penten-3-one with guanidine did not yield the expected 4-phenylhexahydro-2-quinazolinamine and 4-(p-methoxyphenyl)-dihydro-2-pyrimidinamine, respectively, but nitrogen-free products which turned out as [3+3]- and [4+2]-cycloadducts of two molecules of the applied vinylogous ketone each. According to elemental analyses, mass spectra, and, in particular, NMR analyses (¹H and ¹³CNMR, HH-COSY, gs-HSQC, gs-HMBC, 1D TOCSY, NOESY, and 1D NOE difference spectra), the prepared dimers were identified as racemic (5E)-5-benzylidene-9β-phenyl-trans-4a-1,2,3,4,4a,5,6,7,8,9a-decahydroxanthen-4aα-ol and 3β,5β-bis-(4-methoxyphenyl)-2α,4β,6α-trimethyl-4α- propionylcyclohexanone, respectively. Structure and stereochemistry of the dimers are elucidated, and mechanisms for their formation are proposed. Received April 11, 2001. Accepted (revised) May 18, 2001     

On Some Annihilating and Coalescing Systems

In the present paper we continue the investigation of the so-called coalescing ideal gas in one dimension, initiated by Ermakov. The model consists of point-like particles moving with velocities ±1 which coalesce and choose a fresh velocity with the same distribution when colliding. In the previous paper various identities in law were derived for the infinitely extended system. In the present paper we consider the scaling limit of the model in its various guises. The main result is the derivation of the scaling limit (invariance principle) for the joint law of an arbitrary finite number of individual particle trajectories in this system. We also obtain the scaling limit of the density profile of the system, which strongly resembles earlier results of Belitsky and Ferrari.     

Verallgemeinerung eines Interpolationsproblems

Ohne Zusammenfassung     

Über 1.4-Dialkyl-3-acetoxy-7-acetamido-2-oxabicyclo[2.2.1]heptane

Zusammenfassung 2.5-Dialkyl-3.4-dihydro-2H-pyran-2-aldehyde (1 c–1 f) setzen sich mit Acetamid und Essigsäureanhydrid zu 2,5-Dialky-3.4-dihydro-2H-pyran-2-methandioldiacetaten (2 c–2 f), N.N-[(2.5-Dialkyl-3.4-dihydro-2H-pyran-2-yl)-methylen]bisacetamiden (3 c–3 f) und zu 1.4-Dialkyl-3-acetoxy-7-acetamido-2-oxabicyclo[2.2.1]heptanen (6 c–6 f) um. Ausdem 3.4-Dihydro-2H-pyran-2-aldehyd (1 a) entsteht unter gleichen Bedingungen kein Oxabicycloheptan. Mit Methylharnstoff reagieren die genannten Pyranaldehyde zu Pyranylmethylenbismethylharnstoffen9 c–9 f.

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1.4-dialkyl-3-acetoxy-7-acetamido-2-oxabicyclo[2.2.1]heptanes

2.5-Dialkyl-3.4-dihydro-2H-pyran-2-carboxaldehydes (1 c to1 f) are transformed by acetamide and acetic anhydride to 2.5-dialkyl-3.4-dihydro-2H-pyran-2-methanediol diacetates (2 c to2 f), N.N-[(2.5-dialkyl-3.4-dihydro-2H-pyran-2-yl)-methylen] bisacetamides (3 c–3 f) and to 1.4-dialkyl-3-acetoxy-7-acetamido-2-oxabicyclo[2.2.1]heptanes (6 c–6 f). 3.4-dihydro-2H-pyran-2-carboxaldehyde (1 a) is not converted to an oxabicycloheptane under the same conditions. With methylurea the pyranaldehydes mentioned react to pyranylmethylenebismethylureas9 c–9 f.

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Herrn Prof. Dr.M. Pailer zum 60. Geburtstag gewidmet.     

Über den Crotonaldehyd

Zusammenfassung Crotonaldehyd reagiert beim Erhitzen unter Druck nicht zu 3,4-Dimethyl-3,4-dihydro-2H-pyran-2-aldehyd, sondern zu Methylcyclohexadien- und Tolylaldehyden.

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Dimerization of crotonic aldehyde

Crotonaldehyde is not converted by heating under pressure into 3,4-dimethyl-3,4-dihydro-2H-pyran-2-aldehyde, but gives methylcyclohexadiene- and tolyl aldehydes.