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 II - logical development

This is the second of two related papers. In "Revising Z: Part I - logic and semantics" (this journal) we introduced a simple specification logic ZC comprising a logic and a semantics (in ZF set theory). We then provided an interpretation for (a rational reconstruction of) the specification language Z within ZC. As a result we obtained a sound logic for Z, including the basic schema calculus. In this paper we extend the basic framework with more sophisticated features (including schema operations) and we mount a critique of a number of concepts used in Z. We further demonstrate that the complications and confusions which these concepts introduce can be avoided without compromising expressibility

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

Global existence of near-affine solutions to the compressible Euler equations

We establish global existence of solutions to the compressible Euler equations, in the case that a finite volume of ideal gas expands into vacuum. Vacuum states can occur with either smooth or singular sound speed, the latter corresponding to the so-called physical vacuum singularity when the enthalpy vanishes on the vacuum wave front like the distance function. In this instance, the Euler equations lose hyperbolicity and form a degenerate system of conservation laws, for which a local existence theory has only recently been developed. Sideris found a class of expanding finite degree-of-freedom global-in-time affine solutions, obtained by solving nonlinear ODEs. In three space dimensions, the stability of these affine solutions, and hence global existence of solutions, was established by Had\v{z}i\'{c} \& Jang with the pressure-density relation $p = \rho^\gamma$ with the constraint that $1< \gamma\le {\frac{5}{3}}$. They asked if a different approach could go beyond the $\gamma > {\frac{5}{3}}$ threshold. We provide an affirmative answer to their question, and prove stability of affine flows and global existence for all $\gamma >1$, thus also establishing global existence for the shallow water equations when $\gamma=2$.Comment: 51 pages, details added to Section 4.7, to appear in Arch. Rational Mech. Ana

Site-saturation studies of β-lactamase: Production and characterization of mutant β-lactamases with all possible amino acid substitutions at residue 71

A mutagenic technique that "saturates" a particular site in a protein with all possible amino acid substitutions was used to study the role of residue 71 in β-lactamase (EC 3.5.2.6). Threonine is conserved at residue 71 in all class A β-lactamases and is adjacent to the active site Ser-70. All 19 mutants of the enzyme were characterized by the penam and cephem antibiotic resistance they provided to Escherichia coli LS1 cells. Surprisingly, cells producing any of 14 of the mutant β-lactamases displayed appreciable resistance to ampicillin; only cells with mutants having Tyr, Trp, Asp, Lys, or Arg at residue 71 had no observable resistance to ampicillin. However, the mutants are less stable to cellular proteases than wild-type enzyme is. These results suggest that Thr-71 is not essential for binding or catalysis but is important for stability of the β-lactamase protein. An apparent change in specificity indicates that residue 71 influences the region of the protein that accommodates the side chain attached to the β-lactam ring of the substrate

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

Affine motion of 2d incompressible fluids surrounded by vacuum and flows in SL(2,R){\rm SL}(2,{\mathbb R})

The affine motion of two-dimensional (2d) incompressible fluids surrounded by vacuum can be reduced to a completely integrable and globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient in ${\rm SL}(2,{\mathbb R})$. In the case of perfect fluids, the motion is given by geodesic flow in ${\rm SL}(2,{\mathbb R})$ with the Euclidean metric, while for magnetically conducting fluids (MHD), the motion is governed by a harmonic oscillator in ${\rm SL}(2,{\mathbb R})$. A complete classification of the dynamics is given including rigid motions, rotating eddies with stable and unstable manifolds, and solutions with vanishing pressure. For perfect fluids, the displacement generically becomes unbounded, as $t\to\pm\infty$. For MHD, solutions are bounded and generically quasi-periodic and recurrent.Comment: 60 pages, 7 figure