A convenient synthesis of 3-aryl-5-methylidene-2-thiohydantoins

3-Aryl-5-methylidene-2-thiohydantoins were constructed in one-pot reaction of aryl isothiocyanates and 3-morpholino- alanine in alkaline medium with the subsequent treatment with boiling hydrochloric acid.     

Nonexistence Results of Minimal Immersions

Let M be a Riemannian m-dimensional manifold with m ≥ 3, endowed with non zero parallel p-form. We prove that there is no minimal isometric immersions of M in a Riemannian manifold N with constant strictly negative sectional curvature. Next we show that, under the conform flatness of the manifold N and some assumptions on the Ricci curvature of N, there is no α-pluriharmonic isometric immersion.     

On modular inequalities in variable L p spaces

We show that the Hardy-Littlewood maximal operator and a class of Calderón-Zygmund singular integrals satisfy the strong type modular inequality in variable L^p spaces if and only if the variable exponent p(x) ∼ const. Received: 15 September 2004     

Controlling rough paths

We formulate indefinite integration with respect to an irregular function as an algebraic problem which has a unique solution under some analytic constraints. This allows us to define a good notion of integral with respect to irregular paths with Hölder exponent greater than 1/3 (e.g. samples of Brownian motion) and study the problem of the existence, uniqueness and continuity of solution of differential equations driven by such paths. We recover Young's theory of integration and the main results of Lyons’ theory of rough paths in Hölder topology.     

Multilinear interpolation between adjoint operators

Multilinear interpolation is a powerful tool used in obtaining strong-type boundedness for a variety of operators assuming only a finite set of restricted weak-type estimates. A typical situation occurs when one knows that a multilinear operator satisfies a weak L^q estimate for a single index q (which may be less than one) and that all the adjoints of the multilinear operator are of similar nature, and thus they also satisfy the same weak L^q estimate. Under this assumption, in this note we give a general multilinear interpolation theorem which allows one to obtain strong-type boundedness for the operator (and all of its adjoints) for a large set of exponents. The key point in the applications we discuss is that the interpolation theorem can handle the case q?1. When q>1, weak L^q has a predual, and such strong-type boundedness can be easily obtained by duality and multilinear interpolation (cf. Interpolation Spaces, An Introduction, Springer, New York, 1976; Math. Ann. 319 (2001) 151; in: Function Spaces and Applications (Lund, 1986), Lecture Notes in Mathematics, Vol. 1302, Springer, Berlin, New York, 1988; J. Amer. Math. Soc. 15 (2002) 469; Proc. Amer. Math. Soc. 21 (1969) 441).     

Acetalation and Ketalation Catalyzed by SO42-/TiO2-MoO3

The catalytic activities of SO42-/TiO2-MoO3 in synthesizing cyclohexanone ethylene ketal,cyclohexanone 1,2-propanediol ketal, 2-propyl-1,3-dioxolane, 4-methyl-2-isopropyl-1,3-dioxolane,2-isopropyl-1,3-dioxolane, 4-methyl-2-isopropyl-1,3-dioxolane, butanone ethy-lene ketal and butanone 1,2-propanediol ketal were reported. It has been demonstrated that SO42-/TiO2-MoO3 is an excellent catalyst. Various factors concerned in this reaction have been investigated. The optimum conditions have been found, that is, the molar ratio of aldehyde/ketone to alcohol was 1:1.5 or 1:1.3,the mass ratio of the catalyst used to the reactants was 0.25～1.5%, and the reaction time was 45～60 min. Under this conditions, the yield of cyclohexanone ethylene ketal is 82.7%, cyclohexanone 1,2-propanediol ketal is 83.4%, the yield of 2-propyl-1,3-dioxolane is 68.1%,4-methyl-2-isopropyl-1,3-dioxolane is 87.5%, the yield of 2-isopropyl-1,3-dioxolane is 70.7%,4-methyl-2-isopropyl-1,3-dioxolane is 82.5%, the yield of butanone ethylene ketal is 74.1%, and butanone 1,2-propanediol ketal is 94.9%.Some equation and experiment results concerned of the synthetic acetals or ketals were listed as follows.     

Metric and Periodic Lines in Real Inner Product Space Geometries

The lines of euclidean and hyperbolic geometries are characterized as metric lines in the sense of Blumenthal–Menger, and the lines of spherical and elliptic geometry as periodic ones. In euclidean geometries there do not exist periodic lines.     

Contact hypersurfaces of Kaehler manifolds

Contact hypersurfaces of a Kaehler manifold have been characterized and classified, assuming the second fundamental form to be Codazzi (in particular, parallel). We have also discussed the special cases when the ambient space is a (i) Calabi-Yau manifold and (ii) a complex space-form.