The goals of the present paper are to initiate a program to systematically study and rigorously establish what a physicist might refer to as the “operator content of orbifold models.” To explain what this might mean, and to clarify the title of the paper, we will assume that the reader is familiar with the algebraic formulation of 2-dimensional CFT in the guise of vertex operator algebras (VOA)...

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive o...

In this article we prove the following: A topos with a point is connected atomic if and only if it is the classifying topos of a localic group, and this group can be taken to be the locale of automorphisms of the point. We explain and give the necessary definitions to understand this statement. The hard direction in this equivalence was first proved in print in [4], Theorem 1, Section 3, Chapte...

Many different programs are the implementation of the same algorithm. This makes the collection of algorithms a quotient of the collection of programs. Similarly, there are many different algorithms that implement the same computable function. This makes the collection of computable functions into a quotient of the collection of algorithms. Algorithms are intermediate between programs and funct...

DISCLAIMER: This is just a draft. I am not sure of the final purpose of this document, so at the moment I have included lots of known results, in particular proofs of the usual Fundamental Theorem of Galois theory (in the finite and then the algebraic case). Seeing such proofs helps (me) to understand the analogies between the Fundamental Theorem presented here and the usual cases. Also, CAVEAT...

These notes give a concise exposition of the theory of fields, including the Galois theory of finite and infinite extensions and the theory of transcendental extensions. They are an expansion of those handed out during a course taught to first-year graduate students. v2.01 (August 21, 1996). First version on the web. v2.02 (May 27, 1998). Fixed about 40 minor errors; 57 pages. v3.00 (April 3, 2...

The aim of the talk is to explain how to make these statements precise and prove them, using differential Galois theory. The reference to all of the material here is [vdPS03]. By solving a differential equation we mean obtaining a solution via a finite number of operations of the following kind (starting with a rational function): • Adding a function algebraic over the functions we already have...

1 The theory of equations 3 1.1 Primitive question . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Quadratic equations . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 The remainder theorem . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Relation between coefficients and roots . . . . . . . . . . . . . 5 1.5 Complex roots of 1 . . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 Cub...