On equations over sets of integers

Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition S+T=\makeset{m+n}{m \in S, \: n \in T} and with ultimately periodic constants is exactly the class of hyper-arithmetical sets. Equations using addition only can represent every hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can also be represented by equations over sets of natural numbers equipped with union, addition and subtraction S \dotminus T=\makeset{m-n}{m \in S, \: n \in T, \: m \geqslant n}. Testing whether a given system has a solution is $\Sigma^1_1$-complete for each model. These results, in particular, settle the expressive power of the most general types of language equations, as well as equations over subsets of free groups.Comment: 12 apges, 0 figure

Correlation functions of integrable models: a description of the ABACUS algorithm

Recent developments in the theory of integrable models have provided the means of calculating dynamical correlation functions of some important observables in systems such as Heisenberg spin chains and one-dimensional atomic gases. This article explicitly describes how such calculations are generally implemented in the ABACUS C++ library, emphasizing the universality in treatment of different cases coming as a consequence of unifying features within the Bethe Ansatz.Comment: 30 pages, 8 figures, Proceedings of the CRM (Montreal) workshop on Integrable Quantum Systems and Solvable Statistical Mechanics Model

Copying equations to assess mathematical competence: An evaluation of pause measures using graphical protocol analysis

Can mathematical competence be measured by analyzing the patterns of pauses between written elements in the freehand copying of mathematical equations? Twenty participants of varying levels of mathematical competence copied sets of equations and sequences of numbers on a graphics tablet. The third quartile of pauses is an effective measure, because it re- flects the greater number of chunks and the longer time spent per chunk by novices as they processed the equations. To compensate for individual differences in speeds of elementary operations and skill in writing basic mathematical symbols, variants on the measure were devised and tested