Chebyshev Action on Finite Fields

Given a polynomial f and a finite field F one can construct a directed graph where the vertices are the values in the finite field, and emanating from each vertex is an edge joining the vertex to its image under f. When f is a Chebyshev polynomial of prime degree, the graphs display an unusual degree of symmetry. In this paper we provide a complete description of these graphs, and also provide some examples of how these graphs can be used to determine the decomposition of primes in certain field extensions

Simplest random K-satisfiability problem

We study a simple and exactly solvable model for the generation of random satisfiability problems. These consist of $\gamma N$ random boolean constraints which are to be satisfied simultaneously by $N$ logical variables. In statistical-mechanics language, the considered model can be seen as a diluted p-spin model at zero temperature. While such problems become extraordinarily hard to solve by local search methods in a large region of the parameter space, still at least one solution may be superimposed by construction. The statistical properties of the model can be studied exactly by the replica method and each single instance can be analyzed in polynomial time by a simple global solution method. The geometrical/topological structures responsible for dynamic and static phase transitions as well as for the onset of computational complexity in local search method are thoroughly analyzed. Numerical analysis on very large samples allows for a precise characterization of the critical scaling behaviour.Comment: 14 pages, 5 figures, to appear in Phys. Rev. E (Feb 2001). v2: minor errors and references correcte

Maximum Skew-Symmetric Flows and Matchings

The maximum integer skew-symmetric flow problem (MSFP) generalizes both the maximum flow and maximum matching problems. It was introduced by Tutte in terms of self-conjugate flows in antisymmetrical digraphs. He showed that for these objects there are natural analogs of classical theoretical results on usual network flows, such as the flow decomposition, augmenting path, and max-flow min-cut theorems. We give unified and shorter proofs for those theoretical results. We then extend to MSFP the shortest augmenting path method of Edmonds and Karp and the blocking flow method of Dinits, obtaining algorithms with similar time bounds in general case. Moreover, in the cases of unit arc capacities and unit ``node capacities'' the blocking skew-symmetric flow algorithm has time bounds similar to those established in Even and Tarjan (1975) and Karzanov (1973) for Dinits' algorithm. In particular, this implies an algorithm for finding a maximum matching in a nonbipartite graph in $O(\sqrt{n}m)$ time, which matches the time bound for the algorithm of Micali and Vazirani. Finally, extending a clique compression technique of Feder and Motwani to particular skew-symmetric graphs, we speed up the implied maximum matching algorithm to run in $O(\sqrt{n}m\log(n^2/m)/\log{n})$ time, improving the best known bound for dense nonbipartite graphs. Also other theoretical and algorithmic results on skew-symmetric flows and their applications are presented.Comment: 35 pages, 3 figures, to appear in Mathematical Programming, minor stylistic corrections and shortenings to the original versio

Stein structures: existence and flexibility

This survey on the topology of Stein manifolds is an extract from our recent joint book. It is compiled from two short lecture series given by the first author in 2012 at the Institute for Advanced Study, Princeton, and the Alfred Renyi Institute of Mathematics, Budapest.Comment: 29 pages, 11 figure

Perturbation expansions at large order: Results for scalar field theories revisited

The question of the asymptotic form of the perturbation expansion in scalar field theories is reconsidered. Renewed interest in the computation of terms in the epsilon-expansion, used to calculate critical exponents, has been frustrated by the differing and incompatible results for the high-order behaviour of the perturbation expansion reported in the literature. We identify the sources of the errors made in earlier papers, correct them, and obtain a consistent set of results. We focus on phi^4 theory, since this has been the most studied and is the most widely used, but we also briefly discuss analogous results for phi^N theory, with N>4. This reexamination of the structure of perturbation expansions raises issues concerning the renormalisation of non-perturbative effects and the nature of the Feynman diagrams at large order, which we discuss.Comment: 14 page

Abelianisation of Logarithmic sl2\mathfrak{sl}_2-Connections

We prove a functorial correspondence between a category of logarithmic $\mathfrak{sl}_2$-connections on a curve $X$ with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover $\pi : \Sigma \to X$. The proof is by constructing a pair of inverse functors $\pi^{\text{ab}}, \pi_{\text{ab}}$, and the key is the construction of a certain canonical cocycle valued in the automorphisms of the direct image functor $\pi_\ast$.Comment: Comments are always welcome! Version edits: Added a running example (2.5, 2.10, 2.23, 2.25, 2.27, 2.35, 2.39, 2.43, 3.13), Lemma 2.9, and more figures (1, 2, 4, 7, 8, 11). Expanded the discussion after Definition 2.46. Journal reference to follo