Real hypersurfaces in complex two-plane Grassmannians with commuting restricted Jacobi operators

In this paper, we have considered a new commuting condition, that is, $(R_\xi\phi) S = S (R_\xi\phi)$ \big(resp. (\Bar{R}_N\phi) S = S (\Bar{R}_N\phi)\big) between the restricted Jacobi operator~$R_\xi\phi$ (resp. \Bar{R}_N\phi), and the Ricci tensor $S$ for real hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$. In terms of this condition we give a complete classification for Hopf hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$

Outlines of presentations

Illinois Custom Spray Operators' School Outline of Presentations. January 12-14, 1949. This was the very first Spray Operator's School conference

Final report: Future management of the aquatic charter industry in Western Australia.

The Tour Operator\u27s Fishing Working Group (TOFWG) was established by the Hon Monty House MLA, Minister for Fisheries, in recognition of the growing importance of this element of the tourism industy and in recognition of the need to ensure that the development of this industry occurs in an ecologically sustainable framework across the Sate. The recommendations outlined in this report establish a management framework within which the aquatic charter industry may continue to develop, and which will ensure the long-term sustainability of the natural resources on which the industry is based

Discussion paper: future management of the aquatic charter industry in Western Australia

The Tour Operators Fishing Working Group (TOFWG) was established by the Hon Monty House MLA, Minister for Fisheries, in recognition of the growing importance of this element of the tourism industry and in recognition of the need to ensure that the development of this industry occurs in an ecologically sustainable framework across the State. This discussion paper is the result of extensive consultation with the aquatic charter operators\u27 industry by the TOFWG. Proposals outlined here for community discussion are intended to establish a management framework within which the aquatic charter operators’ industry may continue to develop, and which will ensure the long-term sustainability of the natural resources on which the industry is based

Logarithmic intertwining operators and vertex operators

This is the first in a series of papers where we study logarithmic intertwining operators for various vertex subalgebras of Heisenberg vertex operator algebras. In this paper we examine logarithmic intertwining operators associated with rank one Heisenberg vertex operator algebra $M(1)_a$, of central charge $1-12a^2$. We classify these operators in terms of {\em depth} and provide explicit constructions in all cases. Furthermore, for $a=0$ we focus on the vertex operator subalgebra L(1,0) of $M(1)_0$ and obtain logarithmic intertwining operators among indecomposable Virasoro algebra modules. In particular, we construct explicitly a family of {\em hidden} logarithmic intertwining operators, i.e., those that operate among two ordinary and one genuine logarithmic L(1,0)-module.Comment: 32 pages. To appear in CM