P versus NP and geometry

I describe three geometric approaches to resolving variants of P v. NP, present several results that illustrate the role of group actions in complexity theory, and make a first step towards completely geometric definitions of complexity classes.Comment: 20 pages, to appear in special issue of J. Symbolic. Comp. dedicated to MEGA 200

Invariance: a Theoretical Approach for Coding Sets of Words Modulo Literal (Anti)Morphisms

Let $A$ be a finite or countable alphabet and let $\theta$ be literal (anti)morphism onto $A^*$ (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under $\theta$ ($\theta$-invariant for short).We establish an extension of the famous defect theorem. Moreover, we prove that for the so-called thin $\theta$-invariant codes, maximality and completeness are two equivalent notions. We prove that a similar property holds in the framework of some special families of $\theta$-invariant codes such as prefix (bifix) codes, codes with a finite deciphering delay, uniformly synchronized codes and circular codes. For a special class of involutive antimorphisms, we prove that any regular $\theta$-invariant code may be embedded into a complete one.Comment: To appear in Acts of WORDS 201

Curriculum deregulation in England and Scotland - Different directions of travel?

This chapter explores the balance in curricular policy between input regulation (for example prescription of content) and output regulation (for example accountability mechanisms). The chapter draws upon two case studies, England and Scotland, which have adopted diverging approaches to curriculum regulation, identifying the current balance in each country between input and output regulation. Drawing upon an ecological understanding of teacher agency, we conclude the chapter with an analysis of the extent to which England and Scotland are centralised or decentralised systems, and the relative freedom of teachers in each case to engage in school-based curriculum development

Constructing Carmichael numbers through improved subset-product algorithms

We have constructed a Carmichael number with 10,333,229,505 prime factors, and have also constructed Carmichael numbers with k prime factors for every k between 3 and 19,565,220. These computations are the product of implementations of two new algorithms for the subset product problem that exploit the non-uniform distribution of primes p with the property that p-1 divides a highly composite \Lambda.Comment: Table 1 fixed; previously the last 30 digits and number of digits were calculated incorrectl

Quivers of monoids with basic algebras

We compute the quiver of any monoid that has a basic algebra over an algebraically closed field of characteristic zero. More generally, we reduce the computation of the quiver over a splitting field of a class of monoids that we term rectangular monoids (in the semigroup theory literature the class is known as $\mathbf{DO}$) to representation theoretic computations for group algebras of maximal subgroups. Hence in good characteristic for the maximal subgroups, this gives an essentially complete computation. Since groups are examples of rectangular monoids, we cannot hope to do better than this. For the subclass of $\mathscr R$-trivial monoids, we also provide a semigroup theoretic description of the projective indecomposables and compute the Cartan matrix.Comment: Minor corrections and improvements to exposition were made. Some theorem statements were simplified. Also we made a language change. Several of our results are more naturally expressed using the language of Karoubi envelopes and irreducible morphisms. There are no substantial changes in actual result