Further results on monoids acting on trees

This paper further develops the theory of arbitrary semigroups acting on trees via elliptic mappings. A key tool is the Lyndon-Chiswell length function L for the semigroup S which allows one to construct a tree T and an action of S on T via elliptic maps. Improving on previous results, the length function of the action will also be L

On the lattice of subgroups of a free group: complements and rank

A $\vee$-complement of a subgroup $H \leqslant \mathbb{F}_n$ is a subgroup $K \leqslant \mathbb{F}_n$ such that $H \vee K = \mathbb{F}_n$. If we also ask $K$ to have trivial intersection with $H$, then we say that $K$ is a $\oplus$-complement of $H$. The minimum possible rank of a $\vee$-complement (resp. $\oplus$-complement) of $H$ is called the $\vee$-corank (resp. $\oplus$-corank) of $H$. We use Stallings automata to study these notions and the relations between them. In particular, we characterize when complements exist, compute the $\vee$-corank, and provide language-theoretical descriptions of the sets of cyclic complements. Finally, we prove that the two notions of corank coincide on subgroups that admit cyclic complements of both kinds.Comment: 27 pages, 5 figure

The Phillips - Barger model for the elastic cross section and the Odderon

Inspired by the recent TOTEM data for the elastic proton -- proton ($pp$) scattering at $\sqrt{s} =$ 8 and 13 TeV, we update previous studies of the differential cross sections using the Phillips -- Barger (PB) model, which parametrizes the amplitude in terms of a small number of free parameters. We demonstrate that this model is able to describe the recent $pp$ data on a statistically acceptable way. Additionally, we perform separate fits of the $pp$ data for each center - of - mass energy and propose a parametrization for the energy dependence of the parameters present in the PB model. As a consequence, we are able to present the PB predictions for the elastic proton - proton cross section at $\sqrt{s} = 546$ GeV and $1.8$ TeV, which are compared with the existing antiproton -- proton ($\bar{p}p$) data. We show that the PB predictions, constrained by the $pp$ data, are not able to describe the $\bar{p}p$ data. In particular, the PB model predicts a dip in the differential cross section that is not present in the $\bar{p}p$ data. Such result suggests the contribution of the Odderon exchange at high energies.Comment: 6 pages, 4 tables, 2 figures, results updated, matches published versio

On periodic points of free inverse monoid endomorphisms

It is proved that the periodic point submonoid of a free inverse monoid endomorphism is always finitely generated. Using Chomsky's hierarchy of languages, we prove that the fixed point submonoid of an endomorphism of a free inverse monoid can be represented by a context-sensitive language but, in general, it cannot be represented by a context-free language.Comment: 18 page

Analytic solutions in the dyon black hole with a cosmic string: scalar fields, Hawking radiation and energy flux

Charged massive scalar fields are considered in the gravitational and electromagnetic field produced by a dyonic black hole with a cosmic string along its axis of symmetry. Exact solutions of both angular and radial parts of the covariant Klein-Gordon equation in this background are obtained, and are given in terms of the confluent Heun functions. The role of the presence of the cosmic string in these solutions is showed up. From the radial solution, we obtain the exact wave solutions near the exterior horizon of the black hole, and discuss the Hawking radiation spectrum and the energy flux.Comment: 21 pages. arXiv admin note: substantial text overlap with arXiv:1405.7846, arXiv:1401.5397, arXiv:1406.688

Effective Lower Bounding Techniques for Pseudo-Boolean Optimization

Linear Pseudo-Boolean Optimization (PBO) is a widely used modeling framework in Electronic Design Automation (EDA). Due to significant advances in Boolean Satisfiability (SAT), new algorithms for PBO have emerged, which are effective on highly constrained instances. However, these algorithms fail to handle effectively the information provided by the cost function of PBO. This paper addresses the integration of lower bound estimation methods with SAT-related techniques in PBO solvers. Moreover, the paper shows that the utilization of lower bound estimates can dramatically improve the overall performance of PBO solvers for most existing benchmarks from EDA. 1

Satisfiability-Based Algorithms for Boolean Optimization

This paper proposes new algorithms for the Binate Covering Problem (BCP), a well-known restriction of Boolean Optimization. Binate Covering finds application in many areas of Computer Science and Engineering. In Artificial Intelligence, BCP can be used for computing minimum-size prime implicants of Boolean functions, of interest in Automated Reasoning and Non-Monotonic Reasoning. Moreover, Binate Covering is an essential modeling tool in Electronic Design Automation. The objectives of the paper are to briefly review branch-and-bound algorithms for BCP, to describe how to apply backtrack search pruning techniques from the Boolean Satisfiability (SAT) domain to BCP, and to illustrate how to strengthen those pruning techniques by exploiting the actual formulation of BCP. Experimental results, obtained on representative instances indicate that the proposed techniques provide significant performance gains for a large number of problem instances