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...

We prove that there are computable enumerable incomplete btt-degres with the anticupping property. In fact no simple set is btt cuppable in the (not necessarily computable enumerable) btt-degrees. This solves a question of Odifreddi and settles the situation for the one remaining reducibility where the answer was unknown.

Grone and Merris [5] conjectured that the Laplacian spectrum of a graph is majorized by its conjugate vertex degree sequence. In this paper, we prove that this conjecture holds for a class of graphs including trees. We also show that this conjecture and its generalization to graphs with Dirichlet boundary conditions are equivalent.

We discuss the status of the problem of characterizing the finite (weak) lattices which can be embedded into the computably enumerable degrees. In particular, we summarize the current status of knowledge about the problem, provide an overview of how to prove these results, discuss directions which have been pursued to try to solve the problem, and present some related

Let d be a Turing degree containing differences of recursively enumerable sets (d.r.e.sets ) and R[d] be the class of less than d r.e.degrees in which d is relatively enumerable (r.e.). A.H.Lachlan proved that for any non-recursive d.r.e. d R[d] is not empty. We show that the r.e.degree defined by Lachlan for a d.r.e.set D ∈d is just the minimum degree in which D is r.e. Then we study for a giv...

We study ideals in the computably enumerable Turing degrees, and their upper bounds. Every proper Σ0 4 ideal in the c.e. Turing degrees has an incomplete upper bound. It follows that there is no Σ0 4 prime ideal in the c.e. Turing degrees. This answers a question of Calhoun (1993) [2]. Every properΣ0 3 ideal in the c.e. Turing degrees has a low2 upper bound. Furthermore, the partial order of Σ0...

Let $G$ be a finite group and $pi_{e}(G)$ be the set of orders of all elements in $G$. The set $pi_{e}(G)$ determines the prime graph (or Grunberg-Kegel graph) $Gamma(G)$ whose vertex set is $pi(G)$, the set of primes dividing the order of $G$, and two vertices $p$ and $q$ are adjacent if and only if $pqinpi_{e}(G)$. The degree $deg(p)$ of a vertex $pin pi(G)$, is the number of edges incident...

This paper introduces a new approach to classification which combines pairwise decomposition techniques with ideas and tools from fuzzy preference modeling. More specifically, our approach first decomposes a polychotomous classification problem involving m classes into an ensemble of binary problems, one for each ordered pair of classes. The corresponding classifiers are trained on the relevant...