By Abe T.

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The e-book constitutes the joint refereed lawsuits of the ninth foreign convention on Relational tools in desktop technology, RelMiCS 2006, and the 4th foreign Workshop on functions of Kleene Algebras, AKA 2006, held in Manchester, united kingdom in August/September 2006. The 25 revised complete papers offered including invited papers and the summary of an invited speak have been conscientiously reviewed and chosen from forty four submissions.

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On the 2d ∞ 1 24 , η(τ )2 η(2τ )η( τ2 ) = n=1 with q = e2π iτ , where η(τ ) = q 24 other hand, we have 2d η(2τ ) η(τ ) = n = η(τ )2d , (1 − q ) n=1 tr SF(θ ) θ q d L0 + 12 = q 2d ∞ 1 − 48 (1 − q n− 12 ) = n=1 η( τ2 ) η(τ ) 2d . 2) (see [25, Chap. 4]). 3) φ1 (τ )2d ± φ2 (τ )2d . 5) = φ2 (τ ) which follow from the well known modular transformation lows πi η(τ + 1) = e 12 η(τ ), η − 1 τ = (−iτ )1/2 η(τ ). By using the formula we have the following proposition. 6) 790 T. 5 The modular transformations of SSF ± (τ ) and SSF(θ )± (τ ) with respect to the transformations τ → τ + 1 and τ → − τ1 are given by SSF ± (τ + 1) = e SSF ± 1 − τ = 1 2d+1 dπi 6 SSF(θ )+ (τ ) − SSF(θ )− (τ ) ± SSF(θ )± (τ + 1) = ±e− SSF(θ )± − 1 τ = SSF ± (τ ), dπi 12 (−iτ )d (SSF + (τ ) − SSF − (τ )), 2 SSF(θ )± (τ ), 1 SSF(θ )+ (τ ) + SSF(θ )− (τ ) ± 2d−1 SSF + (τ ) + SSF − (τ ) .

4]). 3) φ1 (τ )2d ± φ2 (τ )2d . 5) = φ2 (τ ) which follow from the well known modular transformation lows πi η(τ + 1) = e 12 η(τ ), η − 1 τ = (−iτ )1/2 η(τ ). By using the formula we have the following proposition. 6) 790 T. 5 The modular transformations of SSF ± (τ ) and SSF(θ )± (τ ) with respect to the transformations τ → τ + 1 and τ → − τ1 are given by SSF ± (τ + 1) = e SSF ± 1 − τ = 1 2d+1 dπi 6 SSF(θ )+ (τ ) − SSF(θ )− (τ ) ± SSF(θ )± (τ + 1) = ±e− SSF(θ )± − 1 τ = SSF ± (τ ), dπi 12 (−iτ )d (SSF + (τ ) − SSF − (τ )), 2 SSF(θ )± (τ ), 1 SSF(θ )+ (τ ) + SSF(θ )− (τ ) ± 2d−1 SSF + (τ ) + SSF − (τ ) .

Math. Phys. 202, 169–195 (1999) 8. : Classification of irreducible modules for the vertex operator algebra M(1)+ II. Higher Rank, J. Algebra 240, 389–325 (2001) 9. : Induced modules for vertex operator algebras. Commun. Math. Phys. 179 (1), 157–183 (1996) 10. : On quantum Galois theory. Duke Math. J. 86 (2), 305–321 (1997) 11. : On axiomatic approaches to vertex operator algebras and modules. Mem. Am. Math. Soc. 104, (1993) 12. : Vertex operator algebras and the monster. Pure and Applied Mathematics, 134, Academic Boston (1988) 13.

### A Z 2-orbifold model of the symplectic fermionic vertex operator superalgebra by Abe T.

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