We determine an orthogonal decomposition of the vector space of some curvature tensors on a co-Hermitian real vector space, in irreducible components with respect to the natural induced representation of c U(n)xl.One of the components is used to introduce a Bochner curvature tensor on a class of almost co-Hermitian manifolds (or almost contact metric manifolds), called C(ά)-manifolds, containing e.g. co-Kahlerian, Sasakian and Kenmotsu manifolds. Other applications of the decomposition are given.

Previous research (e.g., De Bock, 2002) has shown that-due to the extensive attention paid to proportional reasoning in elementary and secondary mathematics education-many students tend to overrely on proportional methods in diverse domains of mathematics (e.g., geometry, probability). We investigated the development of misapplication of proportional reasoning with the age and the educational

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The widespread interest in refactoring-transforming the source-code of an object-oriented program without changing its external behaviour-has increased the need for a precise definition of refactoring transformations and their properties. In this paper we explore the use of graph rewriting for specifying refactorings and their effect on programs. We introduce a graph representation for programs and show how two representative refactorings can be expressed by graph productions. Then we demonstrate that it is possible to prove that refactorings preserve certain program properties, and that graph rewriting is a suitable formalism for such proofs. FORMALIZING REFACTORINGS WITH GRAPH TRANSFORMATIONS 249(iii) Elegance. The notation should be convenient to read, so that tool builders will understand what is specified. Therefore, it is acceptable to define ad hoc notations for frequently used manipulations, as long as a sound formal basis is available. (iv) Expressiveness. The formalism should be able to specify the core concepts present in any classbased object-oriented language-namely classes, methods, and variables-and allow us to verify whether the core relationships-inheritance, message sends, variable access-between them are preserved after applying a series of rewriting rules.This paper presents a feasibility study to illustrate whether graph rewriting can be used to formalize refactorings as well as the properties to be preserved by these refactorings. The paper starts with a motivating example of a local area network (LAN) simulation (Section 2), and then proceeds with the different steps in the formal specification, namely: (a) a formal graph representation of an abstract syntax tree augmented with variable access and method invocation relations (Section 3); (b) rewrite rules for two representative refactorings (namely, EncapsulateVariable and PullUpmethod; Section 4); (c) preconditions for the two refactorings, which must be verified before the actual rewrite rules are fired (Section 5); (d) proofs, showing that a rewrite rule applied on a graph satisfying the necessary precondition actually preserves the necessary properties (Section 6); (e) validation, reporting on the experiment we performed converting Java programs into our graph representation and specifying the rewrite rules using two state-of-the-art graph transformation tools (Section 7). Subsequently we use the criteria (i)-(iv) to reflect on where the graph rewriting formalism worked well, but also where we encountered problems (Section 8). Finally, we conclude the paper with a summary of our main findings (Section 9). MOTIVATING EXAMPLEAs a motivating example, this paper uses a simulation of a LAN [16]. The example has been used successfully in several introductory programming courses to illustrate and teach good objectoriented design and also in advanced software engineering courses on refactoring [17]. The example is sufficiently simple for illustrative purposes, yet covers most of the interesting constructs of the objectoriented programming parad...