Equivariant Birch-Swinnerton-Dyer Conjecture for the Base Change of Elliptic Curves: An Example

Let E be an elliptic curve defined over Q and let K/Q be a finite Galois extension with Galois group G. The equivariant Birch-Swinnerton-Dyer conjecture for h(1)(E x(Q) K)(1) viewed as amotive over Q with coefficients in Q[G] relates the twisted L-values associated with E with the arithmetic invariants of the same. In this paper I prescribe an approach to verify this conjecture for a given data. Using this approach, we verify the conjecture for an elliptic curve of conductor 11 and an S-3-extension of Q.

Studies in metal-ammonia reduction-5: Reduction and reductive methylation of some naphthoic acids

Birch reductio and reductive methylations of some substituted naphtholic acids have been examined. The factors influencing the mechanism of reduction process have been discussed. Some of the reduced naphthoic acids are useful synthons for synthesis.

Studies in metal-ammonia reduction. Part 4. Reduction and reductive methylation of 7-methoxy-3,4-dihydrophenanthren-1(2H)-one and 7-methoxy-1,2-dihydrophenanthren-4(3H)-one

Birch reduction and reductive methylations of the title compounds have been investigated. 7-Methoxy-3,4-dihydrophenanthren-1(2H)-one (2) yields the cis-3,4,9,10,11,12-hexahydro-derivative (15) while the 7-methoxy-1,2-dihydrophenanthren-4(3H)-one (5) is reduced to the corresponding 1,2,9,10-tetrahydro-derivative (7). The factors influencing the mechanism of the reduction process have been discussed. The reductive methylation products of the ketone (2) are useful substrates in the synthesis of 9-methyl steroids.

Strategies of synthesis of aromatic polyketides using cyclohexa-1,4-and-1,3-dienes in Alder Rickett reactions

Cyclohexa-1, 4-dienes with appropriate substituents, obtained by birch reduction of the substituted benzene, react directly with derivatives of propiolic ester or aldchyde to yield aromatic polyketides. The following compounds have been synthesized; mycophenolic acid, nidulol methyl other, the root growth hormone 3, 5-dihydroxy-2-formyl-4-mythyl-benzoic acid, antibiotic DB 2073, the macrocyclic lactones lasiodiplodin and dihydrozearalenone and the biphenyl derivatives alternario and altenusin. Polyketide anthraquinones can be made from naphthoquinone precursors.

Direct conversion of 13-beta-alkylgonatetraenes into 13-beta-alkylgon-4-en-3-ones

Birch reduction of 8,9-didehydroestradiol-17 beta 3-methyl ether 1 or 9(11)-didehydroestradiol-17 beta 3-methyl ether 2 followed by acid hydrolysis results in a mixture of 19-nortestosterone 8 and 19-nor-9 beta, 10 alpha-testosterone 9 in varying amounts. However, reduction of their acetates with sodium or lithium, tert-butyl alcohol in liquid ammonia and in the presence of aniline affords exclusively 19-nortestosterone. Similarly, 18a-homo-19-nortestosterone 12 is prepared from the acetate of 18a-homoestradiol-17 beta 3-methyl ether, 10.

ChemInform Abstract: Synthesis Based on Cyclohexadienes. Part 16. Synthesis of Methyl 2,7-Dimethyltricyclo (5.2.2.01,5)undec-5-ene-6-carboxylates

Synthesis of methyl 2, 7-dimethyltricyclo{5.2.2.0(1,5}undec-5-en-6-carboxylates, the tricyclic skeleton present in (+)-allo-cedrol (1) is described using the Diels-Alder strategy. Thus, Birch reduction of the aromatic acid 8 gives 5, the methyl ester of which is isomerised with DBU to a 1:1 mixture of the dienes 6 and 4. Cycloaddition of this mixture with 2-chloroacrylonitrile followed by hydrolysis yields the ketone 60 having the tricyclo{5.2.2.0(1.5)}undec-5-ene framework. Similar reaction with methyl vinyl ketone affords the regioisomeric adducts 61 and 62.

Synthesis based on cyclohexadienes. Part-27. Synthesis of functionalized tricyclo[5.2.2.0(1,5)]undecanes

A new methodology for the construction of tricyclo[5.2.2.0(1,5)]undecanes is described from indane-4-carboxylic acids. Birch reduction of indane-4-carboxylic acids followed by conjugation and cycloaddition with alpha-chloroacrylonitrile and hydrolysis lead to the tricyclic compounds 36 and 48 which are intermediates in the eremolactone synthesis.

Stereoselective total synthesis of (+/-)-tochuinyl acetate and (+/-)-dihydrotochuinyl acetates

Details of the first total synthesis of the marine natural product dihydrotochuinyl acetate is described. Cyclopentenone annulation of p-methylacetophenone via a Claisen rearrangement-Wacker oxidation based sequence generated the cyclopentenone 3, a known precursor for the sesquiterpenes cuparene, laurene, alpha-cuparenone and beta-cuparenones. Conversion of the ketone moiety into a carboxylate followed by stereoselective alkylation and reduction transformed the cyclopentenone 3 into the primary alcohol 19. Birch reduction of the alcohol 19 followed by acetylation furnished (+/-)-dihydrotochuinyl acetate, whereas direct acetylation of 19 furnished (+/-)-tochuinyl acetate. (C) 1998 Elsevier Science Ltd. All rights reserved.

Clustering Based Large Margin Classification: A Scalable Approach using SOCP Formulation

This paper presents a novel Second Order Cone Programming (SOCP) formulation for large scale binary classification tasks. Assuming that the class conditional densities are mixture distributions, where each component of the mixture has a spherical covariance, the second order statistics of the components can be estimated efficiently using clustering algorithms like BIRCH. For each cluster, the second order moments are used to derive a second order cone constraint via a Chebyshev-Cantelli inequality. This constraint ensures that any data point in the cluster is classified correctly with a high probability. This leads to a large margin SOCP formulation whose size depends on the number of clusters rather than the number of training data points. Hence, the proposed formulation scales well for large datasets when compared to the state-of-the-art classifiers, Support Vector Machines (SVMs). Experiments on real world and synthetic datasets show that the proposed algorithm outperforms SVM solvers in terms of training time and achieves similar accuracies.

High-pressure synchrotron x-ray diffraction study of RMnO3 (R=Eu, Gd, Tb and Dy) upto 50 GPa

We have carried out synchrotron based high-pressure x-ray diffraction study of orthorhombic EuMnO3, GdMnO3, TbMnO3 and DyMnO3 up to 54.4, 41.6, 47.0 and 50.2 GPa, respectively. The diffraction peaks of all the four manganites shift monotonically to higher diffraction angles and the crystals retain the orthorhombic structure till the highest pressure. We have fitted the observed volume versus pressure data with the Birch-Murnaghan equation of state and determined the bulk modulus to be 185 +/- 6 GPa, 190 +/- 16 GPa, 188 +/- 9 GPa and 192 +/- 8 GPa for EuMnO3, GdMnO3, TbMnO3 and DyMnO3, respectively. The bulk modulus of EuMnO3 is comparable to other manganites, in contrast to theoretical predictions.