Codimension-one minimal projections onto Haar subspaces

Let H_n be an n-dimensional Haar subspace of and let H_n−1 be a Haar subspace of H_n of dimension n−1. In this note we show (Theorem 6) that if the norm of a minimal projection from H_n onto H_n−1 is greater than 1, then this projection is an interpolating projection. This is a surprising result in comparison with Cheney and Morris (J. Reine Angew. Math. 270 (1974) 61 (see also (Lecture Notes in Mathematics, Vol. 1449, Springer, Berlin, Heilderberg, New York, 1990, Corollary III.2.12, p. 104) which shows that there is no interpolating minimal projection from C[a,b] onto the space of polynomials of degree n, (n2). Moreover, this minimal projection is unique (Theorem 9). In particular, Theorem 6 holds for polynomial spaces, generalizing a result of Prophet [(J. Approx. Theory 85 (1996) 27), Theorem 2.1].     

The number of linear extensions of bipartite graphs

The number of linear extensions among the orientations of a bipartite graph is maximum just if the orientation itself is bipartite, the natural one.     

Acid-base reactions of 1,3,4-thiadiazolines and thiocarbonyl ylides; 1,3,4-thiadiazoline-2-spiro-2′-adamantane

The surprising formation of C₂₂H₃₂N₂S₂ from the title compound 1 at 45°C involves the interaction of the basic adamantanethione S-methylide ($\text{2}$) with its acidic precursor $\text{1}$, in the course of which the anion $\text{6}$ undergoes electrocyclic ring opening; the acid and base functions offer the clue to a prolific chemistry of the thiadiazoline $\text{1}$ and the thiocarbonyl ylide $\text{2}$.     

ContinuousK-theory

We study arithmetical properties of homotopy groups of thel-adic completion of Quillen'sK-theory space of number field, with a view on the Dwyer-Friedlander comparison map into étaleK-theory. The relation of these groups toK-theory is a complete analogy to the relation of continuous étale cohomology to étale cohomology. We identify the torsion subgroup of the resulting term with the subgroup of divisible elements inK _2n (F). We prove that this term is sent isomorphically into étaleK-theory, giving some further evidence for the Lichtenbaum-Quillen conjectures.     

V. Thioanalogues of sparteine lactams. (+)-2-Thiono-17-oxosparteine and (+)-2,17-dithionosparteine

The X-ray and spectroscopic results clearly indicate that the (+)-2-thiono-17-oxosparteine (1) and (+)-2,17-ditihionosparteine (2) are conformationally rigid. In order to analyze deviations of lactam/thiolactam groups from planarity induced by ring constraints, the Dunitz-Winkler approach has been used. The lactam and thiolactam groups are close to planarity, only the lactam group in one of the two independent molecules of 1 is markedly non-planar. The bond angles in the thiolactam and lactam groups are highly diverse. Rings A and C adopt a distorted sofa conformation in both compounds. The distortions in the molecules of 2 as compared with those in the related monothiolactams correspond to the unusual chemical shifts of H5(eq), H5(ax) and H11, as well as to the extremely low J_5ax-6 and extremely large J_5eq-6 coupling constants. Also chemical shifts show a similar regularity being extremely high and low for C2 and C17, respectively.     

The First Reaction of Dimethoxycarbene with an Imine Moiety

The nucleophilic dimethoxycarbene (DMC; 2 ) generated by thermal decomposition of 2,5‐dihydro‐1,3,4‐oxadiazole derivative 1 in boiling toluene reacts smoothly with N‐(9H‐fluoren‐9‐ylidene)‐4‐methylbenzenesulfonamide ( 7b ) to yield carbonimidoate derivative 10 . A multi‐step reaction pathway, initiated by the attack of DMC onto the C?N bond and followed by the migration of the sulfonyl group (or via a sulfinate anion) is proposed to explain the formation of the final product. In contrast to the formal ketimine 7b , N‐benzylidene‐4‐methylbenzenesulfonamide ( 7a ), a formal aldimine, does not react with DMC under comparable conditions.     

Association equlibria in acetonitrile solution of tetracyanoquinodimethanides

Trimetric ion (TCNQ)^2?₃ has been discovered in mixed solutions of TCNQ and LiTCNQ in acetonitrile. The estimated association constant isK_T = 5.6 × 10¹⁰ øl² mol^?2.     

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.     

Proton transfer reactions from N-H acid [5,10,15,20-tetrakis(pentafluorophenyl)-21-H, 23-H-porphyrin] to strong bases in acetonitrile

Deprotonation of 5,10,15,20-tetrakis(pentafluorophenyl)-21-H, 23-H-porphyrin (PhF₅PorH₂) by various bases has been studied by ¹H NMR and kinetic methods. The kinetic parameters in acetonitrile were defined for proton transfer reactions yielding [NH]⁺ protonated bases and [NHN]⁻ anions with intramolecular hydrogen-bonded chains.