## Basics

in** From Christoffel Words to Markoff Numbers**

November 2018; p ublished online January 2019 .

Chapter. Subjects: Pure Mathematics. 1225 words.

Definitions and basic results about words: alphabet, length, free monoid, concatenation, prefix, suffix, factor, conjugation, reversal, palindrome, commutative image, periodicity, ultimate...

## discrete structure

Overview page. Subjects: Computing.

A set of discrete elements on which certain operations are defined. Discrete implies noncontinuous and therefore discrete sets include finite and countable sets but not uncountable sets...

## An Application of Model Theory to Semimodules

in** Logic Journal of the IGPL**

February 2008; p ublished online February 2008 .

Journal Article. Subjects: Logic. 1102 words.

In this note, we prove that the theory *T* of cancellative semimodules over a semiring *R* has the amalgamation property. If *R* is an entire cancellative zerosumfree semiring, then *T* has no...

## semiring

Overview page. Subjects: Computing.

A set *S* (containing a 0 and a 1) on which there are defined two dyadic operations that are denoted by + and • and that obey certain properties: the set *S*, regarded as a set with a zero on...

## Stacky Lie Groups

in** International Mathematics Research Notices**

January 2008; p ublished online January 2008 .

Journal Article. Subjects: Mathematics; Pure Mathematics. 0 words.

Presentations of smooth symmetry groups of differentiable stacks are studied within the framework of the weak 2-category of Lie groupoids, smooth principal bibundles, and smooth...

## Generalized Virtual Braid Groups, Quasi-Shuffle Product and Quantum Groups

in** International Mathematics Research Notices**

January 2015; p ublished online January 2014 .

Journal Article. Subjects: Mathematics. 3689 words.

We introduce in this notice the generalized virtual braid group on *n* strands *GVB*
_{
n
}, generalizing simultaneously the braid groups and their virtual...

## Strongly involutive uninorm algebras

in** Journal of Logic and Computation**

June 2013; p ublished online June 2012 .

Journal Article. Subjects: Computing; Logic. 0 words.

We investigate uninorm algebras satisfying a strong version of involutiveness. More precisely, we require that negation is an order reversing monoid isomorphism between the positive cone...

## Scalars, Monads, and Categories

in** Quantum Physics and Linguistics**

February 2013; p ublished online May 2013 .

Chapter. Subjects: Applied Mathematics. 13626 words.

This chapter describes interrelations between: algebraic structure on sets of scalars, properties of monads associated with such sets of scalars, and structure in categories (especially...

## CATEGORIES OF DIAGRAMS

in** Category Theory**

May 2006; p ublished online September 2007 .

Chapter. Subjects: Algebra. 6900 words.

This chapter presents the proof for the Yoneda Lemma, which is probably the single most used result in category theory. It is interesting how often it comes up, especially in view of the...

## Comments on Solutions for Nonsingular Currents in Open String Field Theories

in** Progress of Theoretical Physics**

August 2007; p ublished online August 2007 .

Journal Article. 0 words.

We investigate analytic solutions to Witten's bosonic string field theory and Berkovits’ WZW-type superstring field theory. We construct solutions with parameters out of simpler ones, using...

## Kripke Resource Models of a Dependently‐typed, Bunched λ‐calculus

in** Journal of Logic and Computation**

December 2002; p ublished online December 2002 .

Journal Article. Subjects: Computing; Logic. 0 words.

The λΛ‐calculus is a dependent type theory with both linear and intuitionistic dependent function spaces. It can be seen to arise in two ways. Firstly, in logical frameworks, where it is the...

## Coxeter Groups

in** Descent in Buildings (AM-190)**

September 2015; p ublished online October 2017 .

Chapter. Subjects: Combinatorics and Graph Theory. 4887 words.

This chapter develops a theory of descent for buildings by assembling various results about Coxeter groups. It begins with the notation stating that *W* is an arbitrary group with a...

## Markoff Numbers

in** From Christoffel Words to Markoff Numbers**

November 2018; p ublished online January 2019 .

Chapter. Subjects: Pure Mathematics. 4429 words.

The Markoff equation is the diophantine equation x^{2} +y^{2} +z^{2} = 3xyz. A solution is called a Markoff triple. The main result in this chapter is a bijection between lower Christoffel words and...

## Higher-Order Signature Cocycles for Subgroups of Mapping Class Groups and Homology Cylinders

in** International Mathematics Research Notices**

January 2012; p ublished online January 2012 .

Journal Article. Subjects: Mathematics. 13724 words.

We define families of invariants for elements of the mapping class group of *Σ*, a compact-orientable surface. For any characteristic subgroup , let *J*(*H*) denote the subgroup of mapping...

## Polynomial Functors from Algebras Over a Set-Operad and Nonlinear Mackey Functors

in** International Mathematics Research Notices**

January 2015; p ublished online December 2013 .

Journal Article. Subjects: Mathematics. 24964 words.

In this paper, we give a description of polynomial functors from (finitely generated free) groups to abelian groups in terms of nonlinear Mackey functors, generalizing those given in a...

## Contraction of Hamiltonian [math]-Spaces

in** International Mathematics Research Notices**

October 2017; p ublished online September 2016 .

Journal Article. Subjects: Mathematics; Pure Mathematics. 18455 words.

**Abstract**

In the spirit of recent work of Harada–Kaveh and Nishinou–Nohara–Ueda, we study the symplectic geometry of Popov’s horospherical degenerations of...

## The syntactic concept lattice: Another algebraic theory of the context-free languages?

in** Journal of Logic and Computation**

October 2015; p ublished online July 2013 .

Journal Article. Subjects: Computing; Logic. 0 words.

The syntactic concept lattice is a residuated lattice associated with a given formal language; it arises naturally as a generalization of the syntactic monoid in the analysis of the...

## Models and Separation Logics for Resource Trees

in** Journal of Logic and Computation**

August 2007; p ublished online June 2007 .

Journal Article. Subjects: Computing; Logic. 15515 words.

In this article, we propose a new data structure, called resource tree, that is a node-labelled tree in which nodes contain resources which belong to a partial monoid. We define the...

## ‘GROUP-COMPLETION’, LOCAL COEFFICIENT SYSTEMS AND PERFECTION

in** The Quarterly Journal of Mathematics**

September 2013; p ublished online June 2013 .

Journal Article. Subjects: Pure Mathematics. 0 words.

It has been known since the foundation of higher algebraic *K*-theory that certain maps which are shown to induce isomorphisms on homology by ‘group-completion’ arguments are in fact acyclic...

## ring

Overview page. Subjects: Computing.

*R* on which there are defined two dyadic operations, normally denoted by + (addition) and · or juxtaposition (multiplication). With respect to addition, *null...*