Nilpotency of skew braces and multipermutation solutions of the Yang-Baxter equation

We study relations between different notions of nilpotency in the context of skew braces and applications to the structure of solutions to the Yang-Baxter equation. In particular, we consider annihilator nilpotent skew braces, an important class that turns out to be a brace-theoretic analog to the class of nilpotent groups. In this vein, several well-known theorems in group theory are proved in the more general setting of skew braces.Comment: 18 pages. Postprint versio

Factorizations of skew braces

We introduce strong left ideals of skew braces and prove that they produce non-trivial decomposition of set-theoretic solutions of the Yang–Baxter equation. We study factorization of skew left braces through strong left ideals and we prove analogs of Itô’s theorem in the context of skew left braces. As a corollary, we obtain applications to the retractability problem of involutive non-degenerate solutions of the Yang–Baxter equation. Finally, we classify skew braces that contain no non-trivial proper characteristic ideals.Fil: Jespers, E.. Vrije Unviversiteit Brussel; BélgicaFil: Kubat, L.. Vrije Unviversiteit Brussel; BélgicaFil: Van Antwerpen, A.. Vrije Unviversiteit Brussel; BélgicaFil: Vendramin, Claudio Leandro. Institute of Mathematical Sciences at NYU Shanghai; China. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

On various types of nilpotency of the structure monoid and group of a set-theoretic solution of the Yang--Baxter equation

Given a finite bijective non-degenerate set-theoretic solution $(X,r)$ of the Yang--Baxter equation we characterize when its structure monoid $M(X,r)$ is Malcev nilpotent. Applying this characterization to solutions coming from racks, we rediscover some results obtained recently by Lebed and Mortier, and by Lebed and Vendramin on the description of finite abelian racks and quandles. We also investigate bijective non-degenerate multipermutation (not necessarily finite) solutions $(X,r)$ and show, for example, that this property is equivalent to the solution associated to the structure monoid $M(X,r)$ (respectively structure group $G(X,r)$) being a multipermuation solution and that $G=G(X,r)$ is solvable of derived length not exceeding the multipermutation level of $(X,r)$ enlarged by one, generalizing results of Gateva-Ivanova and Cameron obtained in the involutive case. Moreover, we also prove that if $X$ is finite and $G=G(X,r)$ is nilpotent, then the torsion part of the group $G$ is finite, it coincides with the commutator subgroup $[G,G]_+$ of the additive structure of the skew left brace $G$ and $G/[G,G]_+$ is a trivial left brace.Comment: 35 page

Preserving the gauge invariance of meson production currents in the presence of explicit final-state interactions

A comprehensive formalism is developed to preserve the gauge invariance of currents describing the photo- or electroproduction of mesons off the nucleon when the final-state interactions of mesons and nucleons is taken into account explicitly. Replacing exchange currents by auxiliary currents, it is found that all contributions due to explicit final-state interactions are purely transverse and do not contain a Kroll-Ruderman-type contact current. The relation of the present formulation to tree-level-type prescriptions is shown.Comment: 6 pages, 2 figures; published versio

Gauge-invariant theory of pion photoproduction with dressed hadrons

Based on an effective field theory of hadrons in which quantum chromodynamics is assumed to provide the necessary bare cutoff functions, a gauge-invariant theory of pion photoproduction with fully dressed nucleons is developed. The formalism provides consistent dynamical descriptions of pi-N --> pi-N scattering and Gamma-N --> pi-N production mechanisms in terms of nonlinear integral equations for fully dressed hadrons. Defining electromagnetic currents via the gauging of hadronic n-point Green's functions, dynamically detailed currents for dressed nucleons are introduced. The dressed hadron currents and the pion photoproduction current are explicitly shown to satisfy gauge invariance in a self-consistent manner. Approximations are discussed that make the nonlinear formalism manageable in practice and yet preserve gauge invariance. This is achieved by recasting the gauge conditions for all contributing interaction currents as continuity equations with ``surface'' terms for the individual particle legs coming into or going out of the hadronic interaction region. General procedures are given that approximate any type of (global) interaction current in a gauge-invariance preserving manner as a sum of single-particle ``surface'' currents. It is argued that these prescriptions carry over to other reactions, irrespective of the number or type of contributing hadrons or hadronic systems.Comment: 33 pages, RevTeX; includes 8 postscript figures (requires psfig.sty). This version corrects some minor errors, etc.; contains updated references. Accepted for publication in Phys. Rev. C56 (Oct. 97

Ontwikkeling prototype residuverwijdering met VAM-Residuce : eindrapportage 1 Januari 2010- 15 Mei 2011

Voor het succesvol implementeren van een residuverwijderingsprototype in de huidige sorteerlijnen is het belangrijk dat tijdens het sorteerproces niet te veel schuim ontstaat in het sorteerwater en dat de concentratie van het gebruikte reducerend middel (VAM-Residuce) in het sorteerproces op peil gehouden kan worden. Een tweetal titratiemethodes en een schuim-onderdrukkende techniek zijn ontwikkeld om respectievelijk de concentratie VAM-Residuce te controleren en overmatige schuimvorming tijdens het sorteerproces tegen te gaan

Solution of the Bethe-Salpeter equation for pion-nucleon scattering

A relativistic description of pion-nucleon scattering based on the four-dimensional Bethe-Salpeter equation is presented. The kernel of the equation consists of s- and u-channel nucleon and delta pole diagrams, as well as rho and sigma exchange in the t-channel. The Bethe-Salpeter equation is solved by means of a Wick rotation, and good fits are obtained to the s- and p-wave phase shifts up to 360 MeV pion laboratory energy. The coupling constants determined by the fits are consistent with the commonly accepted values in the literature.Comment: 34 pages, RevTeX; 7 figures. Several references added, a few typos corrected. Accepted for publication in Physical Review

A Gauge Invariant Unitary Theory for Pion Photoproduction

A covariant, unitary and gauge invariant theory for pion photoproduction on a single nucleon is presented. To achieve gauge invariance at the operator level one needs to include both the $\pi N$ and $\gamma\pi N$ thresholds. The final amplitude can be written in terms of a distorted wave in the final $\pi N$ channel provided one includes additional diagrams to the standard Born term in which the photon is coupled to the final state pion and nucleon. These additional diagrams are required in order to satisfy gauge invariance.Comment: 4 pages, LaTeX, 1 figure as a separate uuencoded compressed tar fil