Π1 classes are important to the logical analysis of many parts of mathematics. The Π1 classes form a lattice. As with the lattice of computably enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality, or more precisely, hyperhypersimplicity, namely the notion of a thin class. We prove a number of results relatin...

A computably enumerable (c.e.) degree is a maximal contiguous degree if it is contiguous and no c.e. degree strictly above it is contiguous. We show that there are infinitely many maximal contiguous degrees. Since the contiguous degrees are definable, the class of maximal contiguous degrees provides the first example of a definable infinite anti-chain in the c.e. degrees. In addition, we show t...

Abstract In 1980 Carleson posed a question on the minimal regularity of an initial data function in Sobolev space $H^s({\mathbb {R}}^n)$ that implies pointwise convergence for solution linear Schrödinger equation. After progress by many authors, this was recently resolved (up to endpoint) Bourgain, whose counterexample construction maximal operator proved necessary condition regularity, and Du ...

Cdf(x) = sup deg(p)=d ∣∣∣∣p.v. ∫ f(x− y)e(p(y)) dy y ∣∣∣∣ in which d is an integer, p is a polynomial of degree d, e(u) := e, f is a Schwarz function and the integral is understood in the principal value sense. This definition is motivated principally by the case d = 1. C1f controls the maximal partial Fourier integrals of f and it extends to a bounded map from L into itself for 1 < p < ∞. The ...

Let us write f(n, ∆; C2k+1) for the maximal number of edges in a graph of order n and maximum degree ∆ that contains no cycles of length 2k + 1. For n 2 ≤ ∆ ≤ n − k − 1 and n sufficiently large we show that f(n, ∆; C2k+1) = ∆(n −∆), with the unique extremal graph a complete

Semi-hyperhypersimple c.e. sets, also known as diagonals, were introduced by Kummer. He showed that by considering an analogue of hyperhypersimplicity, one could characterize the sets which are the Halting problem relative to arbitrary computable numberings. One could also consider half of splittings of maximal or hyperhypersimple sets and get another variant of maximality and hyperhypersimplic...

In this paper we present fixed-parameter algorithms for the problem Dual—given two hypergraphs, decide if one is the transversal hypergraph of the other—and related problems. In the first part, we give algorithms for the parameters number of edges of the hypergraphs, the maximum degree of a vertex, and vertex complementary degrees. In the second part, we use an Apriori approach to obtain FPT re...

A subset of vertices in a graph is called a dissociation set if it induces a subgraph with vertex degree at most 1. A dissociation set D is maximal if no other dissociation set contains D. The complexity of finding a dissociation set of maximum size in line graphs and finding a maximal dissociation set of minimum size in general graphs is considered.

There are several approaches addressing the problem of understanding the mechanism of color confinement in non-Abelian gauge theories. The most popular share the idea that only a subset of the degrees of freedom are relevant for confinement. In the Abelian projection approach, one takes into account the maximal Abelian subgroup U(1) of the gauge group SU(N) and monopoles are the effective degre...