In the present paper we initiate the study of a certain kind of partition inequality, by showing, for example, that if M 5 is an integer and the integers a and b are relatively prime to M and satisfy 1  a < b < M/2, and the c(m,n) are defined by 1 (sqa, sqM a; qM )1 1 (sqb, sqM b; qM )1 := X

In incremental development strategies, modelers frequently refine Statecharts models to satisfy requirements and changes. Although several solutions exist to the problem of Statecharts refinement, they provide such levels of freedom that a statechart cannot make assumptions or guarantees about its future structure. In this paper, we propose a set of bounding rules to limit the allowable Statech...

Ontologies expressed in Description Logics or extensions of Datalog are gradually used for describing the domain of many research and industrial strength applications. They provide a formal semantically rich and data-independent layer over which user queries can be posed. A prominent technique for query answering in ontology-based applications is query rewriting, where the given user query Q an...

The drawbacks of using refinement alone in the construction of specifications from simple abstract models is used as the spur for the introduction of retrenchment, a method based on the main ideas of refinement, but one which is more liberal in character. The basics of the retrenchment mechanism are reviewed in preparation for exploring its integration with refinement. The particular aspect of ...

Let A be a set and let B be a non empty set. Observe that B is non empty. In this article we present several logical schemes. The scheme FraenkelInvolution deals with a non empty set A, subsets B, C of A, and a unary functor F yielding an element of A, and states that: B = {F(a); a ranges over elements of A : a ∈ C} provided the parameters have the following properties: • C = {F(a); a ranges ov...

In this paper, we give several new refinements of generalized Aczél’s inequality. Moreover, as applications, some new refinements of integral form of generalized Aczél-type inequality are given. Mathematics subject classification (2010): Primary 26D15; Secondary 26D10.

Type refinements allow invariants about algebraic datatypes to be expressed through the type system. We present a small functional language and type system that elegantly combines datasort refinements (commonly called refinement types) and dependent index refinements, so that one can specify invariants using whatever refinement is most suitable. Our type system has intersections (novel in the p...

There are a growing number of popular web sites where users submit and review instructions for completing tasks as varied as building a table and baking a pie. In addition to providing their subjective evaluation, reviewers often provide actionable refinements. These refinements clarify, correct, improve, or provide alternatives to the original instructions. However, identifying and reading all...