Derived Brackets

We survey the many instances of derived bracket construction in differential geometry, Lie algebroid and Courant algebroid theories, and their properties. We recall and compare the constructions of Buttin and of Vinogradov, and we prove that the Vinogradov bracket is the skew-symmetrization of a derived bracket. Odd (resp., even) Poisson brackets on supermanifolds are derived brackets of canonical even (resp., odd) Poisson brackets on their cotangent bundle (resp., parity-reversed cotangent bundle). Lie algebras have analogous properties, and the theory of Lie algebroids unifies the results valid for manifolds on the one hand, and for Lie algebras on the other. We outline the role of derived brackets in the theory of Poisson structures with background'.     

Pure quantitative characterization of finite simple groups

Let G be a finite group, and let π _e (G) be the set of all element orders of G. In this short paper we prove that π _e (B _n (q)) ≠ π _e (C _n (q)) for all odd q.      

Dupin hypersurfaces with four principal Curvatures,II

If M is an isoparametric hypersurface in a sphere S ⁿ with four distinct principal curvatures, then the principal curvatures κ₁, . . . , κ₄ can be ordered so that their multiplicities satisfy m ₁ = m ₂ and m ₃ = m ₄, and the cross-ratio r of the principal curvatures (the Lie curvature) equals −1. In this paper, we prove that if M is an irreducible connected proper Dupin hypersurface in R ⁿ (or S ⁿ ) with four distinct principal curvatures with multiplicities m ₁ = m ₂ ≥ 1 and m ₃ = m ₄ = 1, and constant Lie curvature r = −1, then M is equivalent by Lie sphere transformation to an isoparametric hypersurface in a sphere. This result remains true if the assumption of irreducibility is replaced by compactness and r is merely assumed to be constant.      

Differential equations uniquely determined by algebras of point symmetries

We continue to investigate strongly and weakly Lie remarkable equations, which we defined in a recent paper. We consider some relevant algebras of vector fields on ℝ^k (such as the isometric, affine, projective, or conformal algebras) and characterize strongly Lie remarkable equations admitted by the considered Lie algebras. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 486–494, June, 2007.     

Geometric and Combinatorial Realizations of Crystal Graphs

For irreducible integrable highest weight modules of the finite and affine Lie algebras of type A and D, we define an isomorphism between the geometric realization of the crystal graphs in terms of irreducible components of Nakajima quiver varieties and the combinatorial realizations in terms of Young tableaux and Young walls. For type A_n⁽¹⁾, we extend the Young wall construction to arbitrary level, describing a combinatorial realization of the crystals in terms of new objects which we call Young pyramids. Presented by P. Littleman Mathematics Subject Classifications (2000) Primary 16G10, 17B37. Alistair Savage: This research was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada and was partially conducted by the author for the Clay Mathematics Institute.     

Asymmetry in the Multiprotein Systems of Molecular Biology

Signaling in living systems needs to achieve high specificity, to be reversible, and to achieve high signal to noise. Signaling mediated by multiprotein systems has evolved that avoids the requirement for high-affinity binary complexes that would be difficult to reverse and which, in the overcrowded cell, would lead to excessive noise in the system. Symmetrical structures are only occasionally formed. When they are, it is principally to colocate components, for example, the tyrosyl kinases of growth factors, where dimers form. Symmetry is, however, often broken, presumably to create more sensitivity and specificity in the signaling system by assembling other components, into higher-order multiprotein systems. The binding of a single heparin to two 1:1 FGF:FGFR complexes is an example, as is the binding of a single ligase to the Xrcc4 dimer, perhaps so creating a further DNA-binding site.     

负折射指数物质中金属线对电磁波的影响

构造两种结构负折射指数物质(NRM)，利用数值模拟计算两种结构的S参数.由于这种结构尺度比波长小，为精确利用有限积分技术(FIT)进行模拟.通过对这两种结构中金属线位置变化对金属谐振频率影响进行模拟，最后得出由于金属线与谐振环耦合，金属线对称破缺影响谐振环谐振频率和通带，并对等效负折射指数为负的频带产生影响. 关键词： 负折射率 左手物质 通带 金属线对称破缺     

关于由衍射片的背面反射光形成的衍射图样的讨论

在光的衍射实验中,我们看到的一般是由通过衍射片的直射光形成的衍射图样,但在衍射片的另一面,同样存在着另一个衍射图样,它是由衍射片的背面反射光形成的,两个衍射图样关于衍射片对称.本文给出了这种实验现象,并用巴比涅原理解释了这种现象.     

Lie Symmetry and Conserved Quantities for Nonholonomic Vacco Dynamical Systems

In this paper the Lie symmetry and conserved quantities for nonholonomic Vacco dynamical systems are studied. The determining equation of the Lie symmetry for the system is given. The general Hojman conserved quantity and the Lutzky conserved quantity deduced from the symmetry are obtained.     

场源基本微元的对称性与场的性质

直接由标量基本微元（点源）和矢量基本微元（电流元等）的对称性和两个基本假设讨论基本微元产生的场矢量的性质。