ABELIAN 3-FOLDS IN PRODUCTS OF PROJECTIVE SPACES

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An embedding theorem of Sobolev type for an operator with singularity

We discuss spaces of Sobolev type which are defined by the operator with singularity: , where and . This operator appears in a one-dimensional harmonic oscillator governed by Wigner's commutation relations. We study smoothness of and continuity of () where is in each space of Sobolev type, and obtain a generalization of the Sobolev embedding theorem. On the basis of a generalization of the Fourier transform, the proof is carried out. We apply the result to the Cauchy problems for partial differential equations with singular coefficients.

     

Electronic structure and conductance of large molecules and DNA

Using a new technique based on embedding in a local orbital formalism, the electronic structure and electron transmission properties of long biological molecules are calculated, in particular DNA. The electronic structure is found by adding one structural unit at a time to the molecule, and calculating an embedding potential for adding the next structural unit. At present, an extended Hückel scheme is used for the Hamiltonian. The transmission is also calculated within the embedding scheme, taking the molecule–metal contacts into account. The results for transmission depend greatly on the orbitals to which contact is made, and also on energy. The implications of these calculations for conductance are discussed.     

On the behavior of the homogeneous self-dual model for conic convex optimization

There is a natural norm associated with a starting point of the homogeneous self-dual (HSD) embedding model for conic convex optimization. In this norm two measures of the HSD model's behavior are precisely controlled independent of the problem instance: (i) the sizes of ɛ-optimal solutions, and (ii) the maximum distance of ɛ-optimal solutions to the boundary of the cone of the HSD variables. This norm is also useful in developing a stopping-rule theory for HSD-based interior-point solvers such as SeDuMi. Under mild assumptions, we show that a standard stopping rule implicitly involves the sum of the sizes of the ɛ-optimal primal and dual solutions, as well as the size of the initial primal and dual infeasibility residuals. This theory suggests possible criteria for developing starting points for the homogeneous self-dual model that might improve the resulting solution time in practice. This research has been partially supported through the MIT-Singapore Alliance.     

类圈图的亏格分布

一个图 G 的亏格分布是指序列{g_k}, g_k表示 G 嵌入亏格为 k 的闭的可定向曲面的数目. 该文给出了标准类圈图的亏格分布的递推公式, 并得到类圈图的嵌入多项式的计算公式.