Fix a prime p. For nonnegative integers a, b, and d, we seek a formula for the number of subgroups of order pd in Z/paZ× Z/pbZ. Set Na,b,d = #{H ⊂ Z/pZ× Z/pZ : #H = p}. This is symmetric in a and b (Na,b,d = Nb,a,d), so when it is convenient we can limit attention to the case a ≤ b. Trivially Na,b,d = 0 if d > a + b, so we may assume 0 ≤ d ≤ a + b. For 1 ≤ a ≤ b, and a + b ≥ d, we will see that...

We apply the Dirac bracket quantization to open strings attached to branes in the presence of background antisymmetric field and recover an inherent noncommutativity in the internal coordinates of the brane. 1 E-mail: ardalan, arfaei, [email protected]

In this paper we develop a framework for semi-automated generation of test cases for object oriented programs from formal speci-cations. We use Object-Z as the formal speciication language. We have designed and prototyped a test system called OZTEST. It understands the characteristics of object-oriented speciications including inheritance. OZTEST maps the speciication to a corresponding impleme...

Recent research suggests that phonotactic probability (the likelihood of occurrence of a sound sequence) and neighborhood density (the number of words phonologically similar to a given word) influence spoken language processing and acquisition across the lifespan in both normal and clinical populations. The majority of research in this area has tended to focus on controlled laboratory studies r...

This article provides a formalization of the W3C Draft Core SHACL Semantics specification using Z notation. This formalization exercise has identified a number of quality issues in the draft. It has also established that the recursive definitions in the draft are well-founded. Further formal validation of the draft will require the use of an executable specification technology.

A key issue when reasoning with default rules is how to order them so as to derive plausible conclusions according to the more specific rules applicable to the situation under concern, to make sure that default rules are not systematically inhibited by more general rules, and to cope with the problem of irrelevance of facts with respect to exceptions. Pearl's system Z enables us to rank-order d...

Web Appendix A: Extended Models In the main paper we considered models with a separate survival probability for animals in their first year of life. We may wish to consider the survival probability separately for longer than a year, for example if an animal is juvenile for longer than a year. To include such cases we extend the models examined to consider survival separately for the first J yea...

RÉSUMÉ. La construction d'un analyseur syntaxique pour un langage de spécification formelle tel que l'Object Z n'est pas une tâche facile. En effet, elle nécessite une double compétence aussi bien dans le domaine de la compilation que dans le domaine de la spécification formelle. Dans cet article, nous présentons tout d'abord quelques outils de vérifications des spécifications écrites en Z et O...