Generating Surface Geometry in Higher Dimensions using Local Cell Tilers

In two dimensions contour elements surround two dimensional objects, in three dimensions surfaces surround three dimensional objects and in four dimensions hypersurfaces surround hyperobjects. These surfaces can be represented by a collection of connected simplices, hence, continuous n dimensional surfaces can be represented by a lattice of connected n-1 dimensional simplices. The lattice of connected simplices can be calculated over a set of adjacent n-dimensional cubes, via for example the Marching Cubes Algorithm. These algorithms are often named local cell tilers. We propose that the local-cell tiling method can be usefully-applied to four dimensions and potentially to N-dimensions. We present an algorithm for the generation of major cases (cases that are topologically invariant under standard geometrical transformations) and introduce the notion of a sub-case which simplifies their representations. Each sub-case can be easily subdivided into simplices for rendering and we describe a backtracking tetrahedronization algorithm for the four dimensional case. An implementation for surfaces from the fourth dimension is presented and we describe and discuss ambiguities inherent within this and related algorithms

Revising Z: part I - logic and semantics

This is the first of two related papers. We introduce a simple specification logic ZC comprising a logic and a semantics (in ZF set theory) within which the logic is sound. We then provide an interpretation for (a rational reconstruction of) the specification language Z within ZC. As a result we obtain a sound logic for Z, including a basic schema calculus

Solvability and regularity for an elliptic system prescribing the curl, divergence, and partial trace of a vector field on Sobolev-class domains

We provide a self-contained proof of the solvability and regularity of a Hodge-type elliptic system, wherein the divergence and curl of a vector field are prescribed in an open, bounded, Sobolev-class domain, and either the normal component or the tangential components of the vector field are prescribed on the boundary. The proof is based on a regularity theory for vector elliptic equations set on Sobolev-class domains and with Sobolev-class coefficients.Comment: 49 Pages, improved exposition and corrected typo

'Behind Enemy Lines' Menzies, Evatt and Passports for Peking

This article focuses primarily on Australian government responses to the 1952 Peace Conference for Asia and the Pacific Regions. Because the conference was to be held in Peking, it was the subject of immense controversy: Chinese communists were fighting Australian soldiers in Korea and Australian peace activists, most communist or 'fellow travellers', sought to travel behind the 'bamboo curtain'. In this context, the Menzies government's policies on passports were sharply silhouetted. Although this conference has been overlooked in the literature, we can infer from the trajectory of relevant Cold War historiography that Prime Minister Menzies would adopt restrictive, even draconian, policies. This article argues otherwise. It suggests that it was that consistent champion of civil liberties, former deputy prime minister, attorney-general and secretary of the General Assembly of the United Nations and now, in 1952, Leader of the Opposition, Dr Evatt, who favoured more repressive action towards prospective delegates. In contrast, Menzies and his Cabinet were more lenient and shifted towards a harsher policy belatedly and reluctantly. This episode, therefore, challenges some comfortable assumptions about how the early Cold War was fought in Australia

Results on formal stepwise design in Z

Stepwise design involves the process of deriving a concrete model of a software system from a given abstract one. This process is sometimes known as refinement. There are numerous refinement theories proposed in the literature, each of which stipulates the nature of the relationship between an abstract specification and its concrete counterpart. This paper considers six refinement theories in Z that have been proposed by various people over the years. However, no systematic investigation of these theories, or results on the relationships between them, have been presented or published before. This paper shows that these theories fall into two important categories and proves that the theories in each category are equivalent

An analysis of total correctness refinement models for partial relation semantics I

This is the first of a series of papers devoted to the thorough investigation of (total correctness) refinement based on an underlying partial relational model. In this paper we restrict attention to operation refinement. We explore four theories of refinement based on an underlying partial relation model for specifications, and we show that they are all equivalent. This, in particular, sheds some light on the relational completion operator (lifted-totalisation) due to Wookcock which underlines data refinement in, for example, the specification language Z. It further leads to two simple alternative models which are also equivalent to the others