We prove that all holomorphic vertex operator algebras of central charge $24$ with non-trivial weight one subspaces are unitary. The main method is to use the orbifold construction a VOA $V$ directly from Niemeier lattice $V_N$. show it possible extend unitary form for $V_N$ by using and some information automorphism group $\mathrm{Aut}(V)$.

It is proved that the vertex operator algebra V is isomorphic to the moonshine VOA V ♮ of Frenkel-Lepowsky-Meurman if it satisfies conditions (a,b,c,d) or (a,b,c,d). These conditions are: (a) V is the only irreducible module for itself and V is C2-cofinite; (a) dimVn ≤ dimV ♮ n for n ≥ 3; (b) the central charge is 24; (c) V1 = 0; (d) V2 (under the first product on V ) is isomorphic to the Gries...

Let V be a vertex operator algebra. It is shown that the categories of weak, admissible and ordinary g-twisted modules for the tensor product VOA V ⊗k are isomorphic to the categories of weak, admissible and ordinary V -modules respectively where g is a k cycle automorphism of V ⊗k. For arbitrary permutation automorphism g of V ⊗k the category of admissible g-twisted modules for V ⊗k is semi si...

Aluminium-coated micromirrors driven by electrothermal and electromagnetic actuations have been demonstrated for 3-D variable optical attenuation applications. Three types of attenuation schemes based on electrothermal, electromagnetic and hybrid, i.e. combination of electrothermal and electromagnetic, actuations have been developed. In addition, two different designs have been fabricated and c...

We discuss the notion of a Batalin-Vilkovisky (BV) algebra and give several classical examples from differential geometry and Lie theory. We introduce the notion of a quantum operator algebra (QOA) as a generalization of a classical operator algebra. In some examples, we view a QOA as a deformation of a commutative algebra. We then review the notion of a vertex operator algebra (VOA) and show t...

ABSTRACT In designing a tool to aid Volkswagen of America in its vehicle distribution system reengineering effort, it was necessary to compute performance measures in terms of both transportation cost and customer service that are functions of probabilistic and dynamic elements. These elements include inventory control policies, demand seasonality, customer demand, customer choice, and transpor...