Words in Linear Groups, Random Walks, Automata and P-Recursiveness

Fix a finite set $S \subset {GL}(k,\mathbb{Z})$. Denote by $a_n$ the number of products of matrices in $S$ of length $n$ that are equal to 1. We show that the sequence $\{a_n\}$ is not always P-recursive. This answers a question of Kontsevich.Comment: 10 pages, 1 figur

Automatic enumeration of regular objects

We describe a framework for systematic enumeration of families combinatorial structures which possess a certain regularity. More precisely, we describe how to obtain the differential equations satisfied by their generating series. These differential equations are then used to determine the initial counting sequence and for asymptotic analysis. The key tool is the scalar product for symmetric functions and that this operation preserves D-finiteness.Comment: Corrected for readability; To appear in the Journal of Integer Sequence

Fuzzy stability analysis of regenerative chatter in milling

During machining, unstable self-excited vibrations known as regenerative chatter can occur, causing excessive tool wear or failure, and a poor surface finish on the machined workpiece. Consequently it is desirable to predict, and hence avoid the onset of this instability. Regenerative chatter is a function of empirical cutting coefficients, and the structural dynamics of the machine-tool system. There can be significant uncertainties in the underlying parameters, so the predicted stability limits do not necessarily agree with those found in practice. In the present study, fuzzy arithmetic techniques are applied to the chatter stability problem. It is first shown that techniques based upon interval arithmetic are not suitable for this problem due to the issue of recursiveness. An implementation of fuzzy arithmetic is then developed based upon the work of Hanss and Klimke. The arithmetic is then applied to two techniques for predicting milling chatter stability: the classical approach of Altintas, and the time-finite element method of Mann. It is shown that for some cases careful programming can reduce the computational effort to acceptable levels. The problem of milling chatter uncertainty is then considered within the framework of Ben-Haim's information-gap theory. It is shown that the presented approach can be used to solve process design problems with robustness to the uncertain parameters. The fuzzy stability bounds are then compared to previously published data, to investigate how uncertainty propagation techniques can offer more insight into the accuracy of chatter predictions

Modelling the discrete and infrequent official interest rate change in the UK

This paper is an empirical analysis of the manner in which official interest rates are determined by the Bank of England. We use a nonlinear framework that allow for the separate study of factors affecting the magnitude of positive and negative interest rate changes as well as their probabilities. Using this approach, new kinds of monetary shocks are defined and used to evaluate their impact on the UK economy. Among them, unanticipated negative interest rate changes are especially important. The model generalizes previous approaches in the literature and provides a rich methodology to understand central banks' decisions and their consequences

Kronecker product identities from D-finite symmetric functions

Using an algorithm for computing the symmetric function Kronecker product of D-finite symmetric functions we find some new Kronecker product identities. The identities give closed form formulas for trace-like values of the Kronecker product.Comment: 6 page

MODELLING THE DISCRETE AND INFREQUENT OFFICIAL INTEREST RATE CHANGE IN THE UK

This paper is an empirical analysis of the manner in which official interest rates are determined by the Bank of England. We use a nonlinear framework that allow for the separate study of factors affecting the magnitude of positive and negative interest rate changes as well as their probabilities. Using this approach, new kinds of monetary shocks are defined and used to evaluate their impact on the UK economy. Among them, unanticipated negative interest rate changes are especially important. The model generalizes previous approaches in the literature and provides a rich methodology to understand central banks’ decisions and their consequences.