0 1 classes are important to the logical analysis of many parts of mathematics. The 0 1 classes form a lattice. As with the lattice of computable enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality namely the notion of a thin class. We prove a number of results relating automorphisms, invariance and thin cla...

Deformations of maximal supergravity theories induced by gauging non-abelian subgroups of the duality group reveal the presence of charged M-theory degrees of freedom that are not necessarily contained in supergravity. The relation with Mtheory degrees of freedom is confirmed by the representation assignments under the duality group of the gauge charges and the ensuing vector and tensor gauge f...

Yang-Mills theory is studied in a variant of 't Hooft's maximal Abelian gauge. In this gauge magnetic monopoles arise in the Abelian magnetic field. We show, however, that the full (non-Abelian) magnetic field does not possess any monopoles, but rather strings of magnetic fluxes. We argue that these strings are the relevant infrared degrees of freedom. The properties of the magnetic strings whi...

We study generalizations of shortest programs as they pertain to Schaefer’s MIN∗ problem. We identify sets of m-minimal and T-minimal indices and characterize their truth-table and Turing degrees. In particular, we show MIN ⊕ ∅′′ ≡T ∅′′′, MIN (n) ⊕ ∅(n+2) ≡T ∅(n+4), and that there exists a Kolmogorov numbering ψ satisfying both MINψ ≡tt ∅′′′ and MIN (n) ψ ≡T ∅(n+4). This Kolmogorov numbering al...

Alon, Kleitman, Lipton, Meshulam, Rabin and Spencer (Graphs. Combin. 7 (1991), no. 2, 97-99) proved, that for any hypergraph F = fF1; F2; : : : ; Fd(q 1)+1g, where q is a primepower, and d denotes the maximal degree of the hypergraph, there exists an F0 F , such that jSF2F0 F j 0 (mod q). We give a direct, alternative proof for this theorem, and we also give an explicit construction of a hyperg...

Let A be a finitely generated domain of GK dimension less than 3 over a field K and let Q(A) denote the quotient division algebra of A. Using the ideas of Smoktunowicz, we show that if D is a finitely generated division subalgebras of Q(A) of GK dimension at least 2, then Q(A) is finite dimensional as a left D-vector space. We use this to show that if A is a finitely generated domain of GK dime...

Given an affine domain of Gelfand–Kirillov dimension 2 over an algebraically closed field, it is shown that the centralizer of any non-scalar element of this domain is a commutative domain of Gelfand–Kirillov dimension 1 whenever the domain is not polynomial identity. It is shown that the maximal subfields of the quotient division ring of a finitely graded Goldie algebra of Gelfand– Kirillov di...