The topological structure of the vortices in the O(n) symmetric TDGL model

In the light of $\phi$--mapping method and topological current theory, the topological structure of the vortex state in TDGL model and the topological quantization of the vortex topological charges are investigated. It is pointed out that the topological charges of the vortices in TDGL model are described by the Winding numbers of $\phi$--mapping which are determined in terms of the Hopf indices and the Brouwer degrees of $\phi$--mapping.Comment: 9 pages, LaTe

Quantum four-body system in D dimensions

By the method of generalized spherical harmonic polynomials, the Schr\"{o}dinger equation for a four-body system in $D$-dimensional space is reduced to the generalized radial equations where only six internal variables are involved. The problem on separating the rotational degrees of freedom from the internal ones for a quantum $N$-body system in $D$ dimensions is generally discussed.Comment: 19 pages, no figure, RevTex, Submitted to J. Math. Phy

Computing the elliptic genus of higher rank E-strings from genus 0 GW invariants

We show that the elliptic genus of the higher rank E-strings can be computed based solely on the genus 0 Gromov-Witten invariants of the corresponding elliptic geometry. To set up our computation, we study the structure of the topological string free energy on elliptically fibered Calabi-Yau manifolds both in the unrefined and the refined case, determining the maximal amount of the modular structure of the partition function that can be salvaged. In the case of fibrations exhibiting only isolated fibral curves, we show that the principal parts of the topological string partition function at given base-wrapping can be computed from the knowledge of the genus 0 Gromov-Witten invariants at this base-wrapping, and the partition function at lower base-wrappings. For the class of geometries leading to the higher rank E-strings, this leads to the result stated in the opening sentence.Comment: 40 page

Effects of Finite Deformed Length in Carbon Nanotubes

The effect of finite deformed length is demonstrated by squashing an armchair (10,10) single-walled carbon nanotube with two finite tips. Only when the deformed length is long enough, an effectual metal-semiconductor-metal heterojunction can be formed in the metallic tube. The effect of finite deformed length is explained by the quantum tunnelling effect. Furthermore, some conceptual designs of nanoscale devices are proposed from the metal-semiconductor-metal heterojunction.Comment: 4 pages, 4 figure