ON THE CONVERGENCE OF THE PARABOLIC APPROXIMATION OF A CONSERVATION LAW IN SEVERAL SPACE DIMENSIONS

1.IntrodnctionWegiveaproofofthestrongconvergenceinofthesolutionoftheparabolicapproximationtowardstheentropicsolutiontothescalarconservationlawwhereuo(RN),udenotessomeapproximationofuosuchthatandthefluxfsatisfiesTheconvergenceoftheapproximatesolutions...     

Further studies on the sesquiterpene ketone germazone

Germazane type sesquiterpenoids germazol (2), 7(11)-dihydrogermazol (3), 7(11)-dihydrogermazone (4), germazene-7(11) (17) and germazane (18) have been prepared from germazone (1). The presence and location of the cyclobutane ring in germazone (1) was confirmed by the acid-catalyzed cleavage of 2 to the cis-eudesmane derivatives 6,7 and 8.     

Transannular cyclisation of germacrone and isogermacrone via oxymercuration-demercuration

Germacrone 1 undergoes transannular cyclisation by OM-DM yielding the trans-decalin derivatives 3,4 and 5. Isogermacrone 2 similarly yields 3, 4 and 5 and, as a minor product, the ketone 8. The stereochemistry and the mechanism of the intramolecular olefinic cyclisation are discussed.     

ON LACUNARY STATISTICAL CONVERGENCE OF ORDER α

In this article, we introduce the concept of lacunary statistical convergence of order α of real number sequences and give some inclusion relations between the sets of lacunary statistical convergence of order α and strong Nαθ(p)-summability. Furthermore, some relations between the spaces Nαθ(p) and Sαθare examined.     

Spirodepressolide: an unusual bis-norsesquiterpene lactone from Achillea depressa Janka

A new bis-norsesquiterpene lactone, spirodepressolide (1), has been isolated from the aerial parts of Achillea depressa Janka and its structure determined by spectroscopic methods to have a new 4-methyl-7-methylethyl-spirobicyclo[4,4]nonane skeleton.     

Intramolecular cyclisation of α,β-epoxy-artemisia ketone

α,β-epoxy-artemisia ketone, $\text{2}$ undergoes intramolecular cyclisation by OM-DM yielding the tetrahydrofuran derivatives $\text{3}$ – $\text{6}$, while on treatment with BF₃,-etherate $\text{2}$ cyclises to the cyclopentanones $\text{8}$ and $\text{9}$. The structure of all products has been elucidated by spectral methods. The mechanism of the cyclisation reaction is discussed briefly.     

The configuration and conformation of isogermacrone. The significance of crossed conformations in olefinic cyclisations

The 9,10-double bond of isogermacrone was assigned the Z configuration based on¹HNMR nOe measurements and the ¹³CNMR shifts of the methyl groups. An X-ray crystal structure determination verified this conclusion and the structure of isogermacrone was shown to be (2Z, 7E)-3,7-dimethyl-10(1-methylethyl-idene)-2,7-cyclodecadien-1-one (1). The molecule crystallised in an anti conformation, which according to ¹HNMR is the predominant conformation in solution at room temperature.¹H NMR spectroscopy revealed the presence of syn and anti conformers in the ratio 1:2 at about -60°. The stereochemistry of the base-induced formation of isogermacrone 1 from germacrone 2 and that of the transannular cyclisation of isogermacrone are discussed.