A frequency-dependent basis function applied to microstrip

Spheroidal wave functions (SWF) and the spectral-domain method (SDM) are used to compute the effective dielectric constant for microstrip. The authors explore the ability of the SWF to change shape as a function of a parameter, e.g. frequency, while maintaining orthogonality, completeness, edge condition, and aperture limit. The authors introduce the SDM equations, provide a brief overview of the SWF, and study the effective dielectric constant as a function of frequency for several commonly used basis functions. A single-term expansion for the vector current density provides excellent results over a broad spectrum (1-100 GHz). Numerical results compare favorably with other commonly used techniques     

MetateM: An introduction

In this paper a methodology for the use of temporal logic as an executable imperative language is introduced. The approach, which provides a concrete framework, calledMetateM, for executing temporal formulae, is motivated and illustrated through examples. In addition, this introduction provides references to further, more detailed, work relating to theMetateM approach to executable logics.     

A New Approach to Abstract Syntax with Variable Binding

The permutation model of set theory with atoms (FM-sets), devised by Fraenkel and Mostowski in the 1930s, supports notions of ‘name-abstraction’ and ‘fresh name’ that provide a new way to represent, compute with, and reason about the syntax of formal systems involving variable-binding operations. Inductively defined FM-sets involving the name-abstraction set former (together with Cartesian product and disjoint union) can correctly encode syntax modulo renaming of bound variables. In this way, the standard theory of algebraic data types can be extended to encompass signatures involving binding operators. In particular, there is an associated notion of structural recursion for defining syntax-manipulating functions (such as capture avoiding substitution, set of free variables, etc.) and a notion of proof by structural induction, both of which remain pleasingly close to informal practice in computer science. Received October 2000 / Accepted in revised form April 2001     

Contrary to time conditionals in Talmudic logic

We consider conditionals of the form A ? B where A depends on the future and B on the present and past. We examine models for such conditional arising in Talmudic legal cases. We call such conditionals contrary to time conditionals. Three main aspects will be investigated:

1. Inverse causality from future to past, where a future condition can influence a legal event in the past (this is a man made causality).
2. Comparison with similar features in modern law.
3. New types of temporal logics arising from modelling the Talmudic examples.

We shall see that we need a new temporal logic,which we call Talmudic temporal logic with linear open advancing future and parallel changing past, based on two parameters for time.     

A study of substitution,using nominal techniques and Fraenkel–Mostowksi sets

Fraenkel–Mostowski (FM) set theory delivers a model of names and alpha-equivalence. This model, now generally called the ‘nominal’ model, delivers inductive datatypes of syntax with alpha-equivalence — rather than inductive datatypes of syntax, quotiented by alpha-equivalence.The treatment of names and alpha-equivalence extends to the entire sets universe. This has proven useful for developing ‘nominal’ theories of reasoning and programming on syntax with alpha-equivalence, because a sets universe includes elements representing functions, predicates, and behaviour.Often, we want names and alpha-equivalence to model capture-avoiding substitution. In this paper we show that FM set theory models capture-avoiding substitution for names in much the same way as it models alpha-equivalence; as an operation valid for the entire sets universe which coincides with the usual (inductively defined) operation on inductive datatypes.In fact, more than one substitution action is possible (they all agree on sets representing syntax). We present two distinct substitution actions, making no judgement as to which one is ‘right’ — we suspect this question has the same status as asking whether classical or intuitionistic logic is ‘right’. We describe the actions in detail, and describe the overall design issues involved in creating any substitution action on a sets universe.Along the way, we think in new ways about the structure of elements of FM set theory. This leads us to some interesting mathematical concepts, including the notions of planes and crucial elements, which we also describe in detail.