Multinuclear magnetic resonance study of some mesoionic oxatriazoles containing nitrogenous exocyclic groups

¹H,¹³C,¹⁴N and¹⁵N NMR measurements are reported for four mesoionic 1-oxa-2, 3, 4-triazoles containing exocyclic nitrogenous groups. The NMR signal assignments are discussed and compared with those previously published for some corresponding oxatriazoles. The results obtained support the proposed cyclic mesoionic structures for the compounds studied. The questions of possible charge delocalization and valence tautomerism are addressed. Compound with N^– H as a exocyclic group (Fig. 1) is found to be relatively unstable, this is attributed to proton migration in the corresponding non-cyclic form of this molecule.Published in Khimiya Geterotsikiicheskikh Soedinenii, No. 9, pp. 1260–1263, September, 1995.     

Solution of the problem of combinatorial characterization of the dimension of the kernel of a starshaped set

It is proved for a nonempty proper subset S of a real topological linear space L that kerS ={convA_z: z bdryS} and for a close connected nonconvex subset S of L that kerS ={convA_z: z slncS}, where bdryS and slncS denote the sets of boundary points and strong local nonconvexity points of S, respectively, and A_z = {s S: z is clearly visible from s via S}.This extends previous results and, combined with standard techniques, yields among others two Krasnosel'skii-type characterizations for the dimension of kerS in R^d in case of a nonempty proper set S with bdryS bounded and a closed connected nonconvex set S with lncS bounded.The assumption of boundedness of S turns out to be irrelevant in these criteria.Herrn Professor Dr. Wilhelm Stoll gewidmetResearch partially supported by the grant PB 2 1140 91 01     

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.     

Reactions of Polycyclic Ketones with Dimethoxycarbene; a Convenient Route for a ‘One‐Pot’ Preparation of Some α‐Hydroxycarboxylic Acid Esters

Polycyclic ‘cage’ ketones, such as pentacyclo[5.4.0.0^2,6.0^3,10.0^5,9]undecan‐8‐one ( 10 ), pentacyclo[5.4.0.0^2,6.0^3,10.0^5,9]undecane‐8,11‐dione ( 11 ), and adamantan‐2‐one ( 16 ) were treated with the nucleophilic dimethoxycarbene (DMC; 1 ), which was generated thermally from 2,5‐dihydro‐2,2‐dimethoxy‐5,5‐dimethyl‐1,3,4‐oxadiazole ( 4a ) in boiling toluene. In this ‘one‐pot’ procedure, the α‐hydroxycarboxylic acid ester 12 or a corresponding derivative 15 or 17 was obtained (Schemes 4–7). Additionally, ‘cage’ thione 21 was treated with DMC under the same conditions yielding dimethoxythiirane 22 (Scheme 8). Subsequent hydrolysis or desulfurization (followed by hydrolysis on silica gel) of 22 gave α‐mercaptocarboxylate 25 and the corresponding desulfurized ester 24 , respectively. In all cases, the addition of DMC occurred stereoselectively, and the addition from the exo‐face is postulated to explain the structures of the isolated products.