On braid words and irreflexivity

Presented by J. Mycielski.     

A class of Garside groupoid structures on the pure braid group

We construct a class of Garside groupoid structures on the pure braid groups, one for each function (called labelling) from the punctures to the integers greater than 1. The object set of the groupoid is the set of ball decompositions of the punctured disk; the labels are the perimeters of the regions. Our construction generalises Garside's original Garside structure, but not the one by Birman-Ko-Lee. As a consequence, we generalise the Tamari lattice ordering on the set of vertices of the associahedron.

     

The adjoint braid arrangement as a combinatorial Lie algebra via the Steinmann relations

We study a certain discrete differentiation of piecewise-constant functions on the adjoint of the braid hyperplane arrangement, defined by taking finite-differences across hyperplanes. In terms of Aguiar-Mahajan's Lie theory of hyperplane arrangements, we show that this structure is equivalent to the action of Lie elements on faces. We use layered binary trees to encode flags of adjoint arrangement faces, allowing for the representation of certain Lie elements by antisymmetrized layered binary forests. This is dual to the well-known use of (delayered) binary trees to represent Lie elements of the braid arrangement. The discrete derivative then induces an action of layered binary forests on piecewise-constant functions, which we call the forest derivative. Our main result states that forest derivatives of functions factorize as external products of functions precisely if one restricts to functions which satisfy the Steinmann relations, which are certain four-term linear relations appearing in the foundations of axiomatic quantum field theory. We also show that the forest derivative satisfies the Lie properties of antisymmetry the Jacobi identity. It follows from these Lie properties, and also crucially factorization, that functions which satisfy the Steinmann relations form a left comodule of the Lie cooperad, with the coaction given by the forest derivative. Dually, this endows the adjoint braid arrangement modulo the Steinmann relations with the structure of a Lie algebra internal to the category of vector species. This work is a first step towards describing new connections between Hopf theory in species and quantum field theory.     

Simple and immune relations on countable structures

 Let 𝒜 be a computable structure and let R be a new relation on its domain. We establish a necessary and sufficient condition for the existence of a copy ℬ of 𝒜 in which the image of R (?R, resp.) is simple (immune, resp.) relative to ℬ. We also establish, under certain effectiveness conditions on 𝒜 and R, a necessary and sufficient condition for the existence of a computable copy ℬ of 𝒜 in which the image of R (?R, resp.) is simple (immune, resp.). Received: 4 February 2001 Published online: 5 November 2002 RID="*" ID="*" The first three authors gratefully acknowledge support of the NFS Binational Grant DMS-0075899. RID="*" ID="*" The first three authors gratefully acknowledge support of the NFS Binational Grant DMS-0075899. RID="*" ID="*" The first three authors gratefully acknowledge support of the NFS Binational Grant DMS-0075899.     

Definable relations in Turing degree structures

In this paper we investigate questions about the definability of classes of n-computably enumerable (c. e.) sets and degrees in the Ershov difference hierarchy. It is proved that the class of all c. e. sets is definable under the set inclusion ? in all finite levels of the difference hierarchy. It is also proved the definability of all m-c. e. degrees in each higher level of the hierarchy. Besides, for each level n, n ≥ 2, of the hierarchy a definable non-trivial subset of n-c. e. degrees is established.     

Symplectomorphisms and discrete braid invariants

Area and orientation preserving diffeomorphisms of the standard 2-disc, referred to as symplectomorphisms of \({\mathbb {D}^{2}}\), allow decompositions in terms of positive twist diffeomorphisms. Using the latter decomposition, we utilize the Conley index theory of discrete braid classes as introduced in Ghrist et al. (Invent. Math. 152:369–432, 2003) and Ghrist et al. (C. R. Acad. Sci. Paris Sér. I Math. 331:861–865, 2000) to obtain a Morse type forcing theory of periodic points: a priori information about periodic points determines a mapping class which may force additional periodic points.     

Binary relations, interval structures and join spaces

We study correspondences between interval structures and hyperstructures.     

Involution and symmetry reductions

After reviewing some notions of the formal theory of differential equations, we discuss the completion of a given system to an involutive one. As applications to symmetry theory, we study the effects of local solvability and of gauge symmetries, respectively. We consider nonclassical symmetry reductions and more general reductions using differential constraints.     

Waldspurger's Involution and Types

Waldspurger's involution for the genuine irreducible supercuspidalrepresentations of SL₂(F) is parametrized in terms of typesin the case F p-adic with p odd. In particular, it is shownthat the in-volution is given by conjugating by an element ofGL₂(F) and twisting one of the defining parameters of an associatedtype by a quadratic character, the relevant parameter beinga character on the norm one elements of a quadratic extension.     

Direct Ramsey theorem for structures involving relations and functions

We prove the direct structural Ramsey theorem for structures with relations as well as functions. The result extends the theorem of Abramson and Harrington and of Nešet?il and Rödl.     

Equivalence relations in semihypergroups and the corresponding quotient structures

In this paper, we introduce and study two equivalence relations in semihypergroups, for which the corresponding quotient structures are monoids and commutative monoids.     

The polynomial hierarchy for some structures over the binary words

For each k > 0 we construct an algebraic structure over which the polynomial hierarchy collapses at level k. We also find an algebraic structure over which the polynomial hierarchy does not collapse. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Ramsey theorem for structures with both relations and functions

We prove a generalization of Prömel's theorem to finite structures with both relations and functions.     

Nonlinear 0–1 programming: II. Dominance relations and algorithms

A nonlinear 0–1 program can be restated as a multilinear 0–1 program, which in turn is known to be equivalent to a linear 0–1 program with generalized covering (g.c.) inequalities. In a companion paper [6] we have defined a family of linear inequalities that contains more compact (smaller cardinality) linearizations of a multilinear 0–1 program than the one based on the g.c. inequalities. In this paper we analyze the dominance relations between inequalities of the above family. In particular, we give a criterion that can be checked in linear time, for deciding whether a g.c. inequality can be strengthened by extending the cover from which it was derived. We then describe a class of algorithms based on these results and discuss our computational experience. We conclude that the g.c. inequalities can be strengthened most of the time to an extent that increases with problem density. In particular, the algorithm using the strengthening procedure outperforms the one using only g.c. inequalities whenever the number of nonlinear terms per constraint exceeds about 12–15, and the difference in their performance grows with the number of such terms. Research supported by the National Science Foundation under grant ECS 7902506 and by the Office of Naval Research under contract N00014-75-C-0621 NR 047-048.     

On braid groups and right-angled Artin groups

In this article we prove a special case of a conjecture of A. Abrams and R. Ghrist about fundamental groups of certain aspherical spaces. Specifically, we show that the \(n\) -point braid group of a linear tree is a right-angled Artin group for each \(n\) .     

Zariski theorems and diagrams for braid groups

Empirical properties of generating systems for complex reflection groups and their braid groups have been observed by Orlik-Solomon and Broué-Malle-Rouquier, using Shephard-Todd classification. We give a general existence result for presentations of braid groups, which partially explains and generalizes the known empirical properties. Our approach is invariant-theoretic and does not use the classification. The two ingredients are Springer theory of regular elements and a Zariski-like theorem. Oblatum 7-XII-2000 & 22-III-2001?Published online: 20 July 2001