Normal Bundles to Laufer Rational Curves in Local Calabi–Yau Threefolds

We prove a conjecture by F. Ferrari. Let X be the total space of a nonlinear deformation of a rank two holomorphic vector bundle on a smooth rational curve, such that X has trivial canonical bundle and has sections. Then the normal bundle to such sections is computed in terms of the rank of the Hessian of a suitably defined superpotential at its critical points.     

The Sheffer group and the Riordan group

We define the Sheffer group of all Sheffer-type polynomials and prove the isomorphism between the Sheffer group and the Riordan group. An equivalence of the Riordan array pair and generalized Stirling number pair is also presented. Finally, we discuss a higher dimensional extension of Riordan array pairs.     

The Low Energy Limit of String Theory and its Compactifications with Background Fluxes

String theory is consistently defined in ten dimensions. In order to extract any information about four-dimensional physics, we need to understand the properties of the six-dimensional compact manifold orthogonal to our four-dimensional universe. A possibility that is being very much explored lately is to look at manifolds on which background fluxes are turned on. In this article, we present an introduction to string theory, focusing on its massless sector. We then review traditional compactifications to four–dimensions, and finally motivate and describe the so-called flux compactifications. We interpret the allowed six-dimensional manifolds from the point of view of generalized complex geometry.     

A Convenient Synthesis of Phosphorines and Phospholidins by <Emphasis Type="Italic">Lawesson</Emphasis>’s Reagent and Microwave Irradiation

Summary. A new and simple method for the synthesis of 1,3,2-diazaphosphorines and 1,3,2-oxazaphosphorine was developed based on the reactions of Lawesson83tfhjn2g9qqmjdf/xxlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">s reagent with 83tfhjn2g9qqmjdf/xxlarge946.gif" alt="beta" align="MIDDLE" BORDER="0">-aminopropionitriles and 83tfhjn2g9qqmjdf/xxlarge946.gif" alt="beta" align="MIDDLE" BORDER="0">-hydroxypropionitrile. In addition, a rapid and facile synthesis of 1,3,2-diazaphospholidin-4-ones by the reaction of P(NEt₂)₃ with hindered diamino substrates under microwave irradiation was also achieved. The prepared phosphorines show herbicidal activity to some extent in the preliminary bioassays.     

Cosmological principle and primordial spinor fields

An exact solution of Dirac's equation in an open Robertson-Walker universe is constructed such that the corresponding energy-momentum densityT _µv obeys the cosmological principle. However, in anopen universe, this principlemust always be violated by the other physical densities of the cosmological solution (e.g. pseudoscalar, (axial-) current and polarization). If the current densityj _µ = isrequired to obey the cosmological principle, then the universe must beflat.     

Four-dimensional BF theory as a topological quantum field theory

Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that four-dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal G-bundle. The case G=GL(4,83t15/xxlarge8477.gif" alt="Ropf" align="BASELINE" BORDER="0">) is especially interesting because every 4-manifold is then naturally equipped with a principal G-bundle, namely its frame bundle. In this case, the partition function of a compact oriented 4-manifold is the exponential of its signature, and the resulting TQFT is isomorphic to that constructed by Crane and Yetter using a state sum model, or by Broda using a surgery presentation of four-manifolds.     

Mirror supersymmetry in the space of (super) extended Poincaré algebras p(p,q)

We classify extended Poincaré Lie superalgebras and Lie algebras of any signature (p, q), i.e. Lie superalgebras and ₂-graded Lie algebras g = g₀ + g₁, where g₀ = s₀(V) + V is the (generalized) Poincaré Lie algebra of the pseudo Euclidean vector space V = ^{p, q} of signature (p, q) and g₁ is a spin 1/2 s₀(V)-module extended to a s₀-module with kernel V.As a result of the classification, we obtain, if g₁ = S is the spinor module, the numbers L ⁺(n, s) (resp. L ^–(n, s)) of independent such Lie super algebras (resp. Lie algebras), which are periodic functions of the dimension n=p+q (mod 8) and the signature s=p–q (mod 8) and satisfy: L ⁺(–n, s)=L ^– (n, s).Supported by Max-Planck-Institut für Mathematik (Bonn).Supported by the Alexander von Humboldt Foundation, MSRI (Berkeley) and SFB 256 (Bonn University).     

Transformed steroids

The approaches to the synthesis of various 1683/xxlarge945.gif" alt="agr" align="BASELINE" BORDER="0">,1783/xxlarge945.gif" alt="agr" align="BASELINE" BORDER="0">-disubstituted pregna-4,9-dien-3,20-diones from the corresponding 983/xxlarge945.gif" alt="agr" align="BASELINE" BORDER="0">-derivatives with labile epoxide, dioxolane, and oxathiolane ringsD83/xxlarge8242.gif" alt="prime" align="BASELINE" BORDER="0"> have been studied. The transformation has been found to proceed efficiently when the 983/xxlarge945.gif" alt="agr" align="BASELINE" BORDER="0">-sulfinic esters are used at the intermediate stage and then elimination of sulfinic acid by TsOH/SiO₂ is carried out.For part 193, see ref. 7.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 5, pp. 983–986, May, 1993.     

Synthesis of the four enantiomers of diethyl 1,2-di(N-Boc-amino)propylphosphonates

A simple and efficient synthetic strategy to all four enantiomerically pure diethyl 1,2-di(N-Boc-amino)propylphosphonates has been elaborated starting from the corresponding N-[(R)-(1-phenylethyl)]aziridine-(2S)- and N-[(S)-(1-phenylethyl)]aziridine-(2R)-carboxaldehydes, employing a one-pot three-components Kabachnik-Fields reaction followed by the hydrogenolytic removal of the chiral auxiliary and aziridine ring opening with simultaneous protection of the amino groups as the N-Boc derivatives.