introduction. Parametrized p-adic integrals are being better understood recently by the work of Denef in an algebraic context and the work of the author in an analytic context (and, by the theory of motivic integrals, but we will not treat uniformity questions here). In this paper we give a contribution to the theory of p-adic integrals, based on the Cell Decomposition Theorem of [3]. In [3], a framework of constructible functions which is stable under integration is introduced. Here we extend the notion of constructible functions such that parametrized (possibly twisted) local zeta functions, which are p-adic integrals of quite a general form, are included, and we prove that it is stable under p-adic integration. We control the possible order of poles of the zeta functions, uniform in the parameters.

This book has developed over a 20-year period during which the calendrical algorithms and our presentation of them have continually evolved. Our initial motivation wasaneffort by one of us (E.M.R.) to create Emacs-Lisp code that would provide calendar and diary features for GNU Emacs [8]; this version of the code included the Gregorian, Islamic, and Hebrew calendars (the Hebrew implemented by N.D.). A deluge of inquiries from around the globe soon made it clear to us that there was keen interest in an explanation that would go beyond the code itself, leading to our article [2] and encouraging us to rewrite the code completely, this time in Common Lisp [9]. The subsequent addition—by popular demand—of the Mayan and French Revolutionary calendars to GNU Emacs prompted a second article [6]. We received many hundreds of reprint requests for these articles. This response far exceeded our expectations and provided the impetus to write a book in which we could more fully address the multifaceted subject of calendars and their implementation.

The subject of calendars has always fascinated us with its cultural, historical, and mathematical wealth, and we have occasionally employed calendars as accessible examples in introductory programming courses.

Introduction. A variety of sensorimotor coordination mechanisms are being successfully modelled on the basis of continuous dynamical system approaches (Beer [1997]; Steinhage and Bergener [2000]; Turvey and Carello [1995]). This work is invoked as empirical support for a sweeping “dynamicist” thesis: intelligent and adaptive biological behaviours are to be ultimately accounted for in terms of continuous (dynamical) systems; properly computational investigations make approximate simulation tools available for dynamical theories but play no essential theoretical role (Port and Gelder [1995]). Similar claims about mathematical theorizing in cognitive ethology and biology at large can be found in (Beer [1995], Steels [1995]) and (Longo [2003]), respectively. These claims are critically examined here, in the light of theoretical models of simple sensorimotor adaptive behaviours. These case studies are particularly suited to our purposes, for continuous dynamical approaches are supposedly at their best in the modelling of sensorimotor coordination mechanisms. Computational approaches, we submit, are being fruitfully pursued there too.

The Pawukon calendar of Bali is a complex example of a calendar based on concurrent cycles (see Section 1.11). The whole calendar repeats every 210 days, but these 210-day “years” are unnumbered. The calendar comprises 10 subcycles of lengths 1 through 10, all running simultaneously. The subcycles that determine the calendar are those of length 5, 6, 7; the others are altered to fit, by repetitions or other complications.

Like many other cultures in the region, the Balinese also have a lunisolar calendar of the old Hindu style (see Chapter 9), but leap months have been added erratically; we do not describe its details. This lunisolar calendar is used to determine only one holiday: Nyepi, a “New Year's Day” marking the start of the tenth lunar month, near the onset of spring.

Structure and Implementation

The main subcycles of the Pawukon calendar are those of length 5, 6, and 7, and the whole calendar repeats every 210 days, the least common multiple of 5, 6, and 7.

The Mayans, developers of an ancient Amerindian civilization in Central America, employed three separate, overlapping calendrical systems called by scholars the long count, the haab, and the tzolkin. Their civilization reached its zenith during the period 250–900 C.E., and the Mayans survive to this day in Guatemala and in the Yucatan peninsula of Mexico and Belize; some groups have preserved parts of the calendar systems.

As developed in, stability theory is based on the notion of an invariant type, more specifically a definable type, and the closely related theory of independence of substructures. We will review the definitions in Chapter 2 below; suffice it to recall here that an (absolutely) invariant type gives a recipe yielding, for any substructure A of any model of T, a type p│A, in a way that respects elementary maps between substructures; in general one relativizes to a set C of parameters, and considers only A containing C. Stability arose in response to questions in pure model theory, but has also provided effective tools for the analysis of algebraic and geometric structures. The theories of algebraically and differentially closed fields are stable, and the stability-theoretic analysis of types in these theories provides considerable information about algebraic and differential-algebraic varieties. The model companion of the theory of fields with an automorphism is not quite stable, but satisfies the related hypothesis of simplicity; in an adapted form, the theory of independence remains valid and has served well in applications to difference fields and definable sets over them. On the other hand, such tools have played a rather limited role, so far, in o-minimality and its applications to real geometry.

Where do valued fields lie? Classically, local fields are viewed as closely analogous to the real numbers. We take a “geometric” point of view however, in the sense of Weil, and adopt the model completion as the setting for our study.

Structure and History

The Hindus have both solar and lunisolar calendars. In the Hindu lunisolar system, as in other lunisolar calendars, months follow the lunar cycle and are synchronized with the solar year by introducing occasional leap months. Unlike the Hebrew lunisolar calendar (described in Chapter 7), Hindu intercalated months do not follow a short cyclical pattern. Moreover, unlike other calendars, a day can be omitted any time in a lunar month.

Modern Hindu calendars are based on close approximations to the true times of the sun's entrance into the signs of the zodiac and of lunar conjunctions (new moons). Before about 1100 C.E., however, Hindu calendars used mean times. Though the basic structure of the calendar is similar for both systems, the mean (madhyama) and true (spaṣṭa) calendars can differ by a few days or can be shifted by a month. In this chapter we implement the mean system, as described in [4, pp. 360–446], which is arithmetical; Chapter 18 is devoted to the more recent astronomical version. For an ancient description of Hindu astronomy, calendars, and holidays, see the book on India by al-Bīrūnā[1]; a more modern reference is [3].

In this chapter, we use formulas from Section 1.12 to cast a number of the calendars presented in Part I into a unified framework. Years must be determined by the occurrence of some “critical” mean annual event, like a mean equinox or mean solstice. Months must also follow a uniform pattern. In single-cycle calendars, new years begin on the day the critical annual event happens before (or possibly at) some critical time of day. In double-cycle calendars, months begin on the day of a critical mensual event, and years begin with the month associated with the critical annual event.

Single Cycle Calendars

There are four “single-cycle” paradigms for calendars, as we independently allow

1. the determining critical annual event to occur strictly before, or to occur at or before, some critical time of day, and

2. the pattern of months to follow a fixed yearly pattern, according to equations (1.78)–(1.81) or to follow a mean monthly pattern, in tune with the yearly pattern.

Introduction. We start by briefly describing the strong colouring problem for permutation groups that was investigated in [13] and [12]; this problem has its origins in the classification of maximal partial clones over a finite non-empty set [14, 15, 16]. We shall immediately reformulate this decision problem as a Constraint Satisfaction Problem (CSP) in order to exploit various universal algebraic tools to study its algorithmic complexity.

The calendars in the second part of this book are based on accurate astronomical calculations. This chapter defines the essential astronomical terms and describes the necessary astronomical functions. Fuller treatment can be found in the references—an especially readable discussion is given in [9].

We begin with an explanation of how positions of locations on Earth and of heavenly bodies are specified, followed by an examination of the notion of time itself.