The determination of all syzygies for the dependent polynomial invariants of the Riemann tensor. I. Pure Ricci and pure Weyl invariants

In this paper, we shall consider all pure Ricci and pure Weyl scalar invariants of any degree, in a four-dimensional Lorentzian space. We present a general graph-theoretic based reduction algorithm which decomposes, using syzygies, any pure invariant in terms of the independent base invariants {r₁,r₂,r₃} or {w₁,w₂}

The determination of all syzygies for the dependent polynomial invariants of the Riemann Tensor. II. mixed invariants of Eeven degree in the Ricci Spinor

We continue our analysis of the polynomial invariants of the Riemann tensor in a four-dimensional Lorentzian space. We concentrate on the mixed invariants of even degree in the Ricci spinor Φ_ABȦḂ and show how, using constructive graph-theoretic methods, arbitrary scalar contractions between copies of the Weyl spinor ψ_ABCD, its conjugate ψ_ȦḂĊḊ and an even number of Ricci spinors can be expressed in terms of paired contractions between these spinors. This leads to an algorithm for the explicit expression of dependent invariants as polynomials of members of the complete set. Finally, we rigorously prove that the complete set as given by Sneddon [J. Math. Phys. 39, 1659-1679 (1998)] for this case is both complete and minimal.

Three-dimensional atom probe analysis of solute distribution in thermomechanically processed TRIP steels

Complex multiphase microstructures were obtained in transformation induced plasticity C–Mn–Si–(Nb–Al–Mo) steels by simulated controlled thermomechanical processing. These microstructures were characterized using transmission electron microscopy, X-ray diffraction and three-dimensional atom probe tomography (APT), which was used to determine the partitioning of elements between different phases and microconstituents. The measured carbon concentration (not, vert, similar0.25 at%) in the ferrite of carbide-free bainite was higher than expected from para-equilibrium between the austenite and ferrite, while the concentrations of substitutional elements were the same as in the parent austenite suggesting that incomplete bainite transformation occurred. In contrast, the distribution of substitutional elements between the ferrite lath and austenite in carbide-containing bainite indicated a complete bainite reaction. The average carbon content in the retained austenite (3.2 ± 1.6 at%) was somewhat higher than the T0 limit. On the basis of the APT measured composition, the calculated Ms temperatures for retained austenite were above room temperature, indicating its low chemical stability.

Precipitate characterisation of an advanced high-strength low-alloy (HSLA) steel using atom probe tomography

Increased fuel economy, combined with the need for the improved safety has generated the development of new hot-rolled high-strength low-alloy (HSLA) and multiphase steels such as dual-phase or transformation-induced plasticity steels with improved ductility without sacrificing strength and crash resistance. However, the modern multiphase steels with good strength-ductility balance showed deteriorated stretch-flangeability due to the stress concentration region between the soft ferrite and hard martensite phases [1]. Ferritic, hot-rolled steels can provide good local elongation and, in turn, good stretch-flangeability [2]. However, conventional HSLA ferritic steels only have a tensile strength of not, vert, similar600 MPa, while steels for the automotive industry are now required to have a high tensile strength of not, vert, similar780 MPa, with excellent elongation and stretch-flangeability [1]. This level of strength and stretch-flangeability can only be achieved by precipitation hardening of the ferrite matrix with very fine precipitates and by ferrite grain refinement. It has been suggested that Mo [3] and Ti [4] should be added to form carbides and decrease the coiling temperature to 650 °C since only a low precipitation temperature can provide the precipitation refinement [4]. These particles appeared to be (Ti, Mo)C, with a cubic lattice and a parameter of 0.433 nm, and they were aligned in rows [4]. It was reported [4] that the formation of these very fine carbides led to an increase in strength of not, vert, similar300 MPa. However, the detailed analysis of these particles has not been performed to date due to their nanoscale size. The aim of this work was to carry out a detailed investigation using atom probe tomography (APT) of precipitates formed in hot-rolled low-carbon steel containing additions Ti and Mo.

The investigated low-carbon steel, containing Fe–0.1C–1.24Mn–0.03Si–0.11Cr–0.11Mo–0.09Ti–0.091Al at.%, was produced by hot rolling. The processing route has been described in detail elsewhere [5] European Patent Application, 1616970 A1, 18.01.2006.[5]. The microstructure was characterised by transmission electron microscopy (TEM) on a Philips CM 20, operated at 200 kV using thin foil and carbon replica techniques. Qualitative energy dispersive X-ray spectroscopy (EDXS) was used to analyse the chemical composition of particles. The atomic level of particle characterisation was performed at the University of Sydney using a local electrode atom probe [6]. APT was carried out using a pulse repetition rate of 200 kHz and a 20% pulse fraction on the sample with temperature of 80 K. The extent of solute-enriched regions (radius of gyration) and the local solute concentrations in these regions were estimated using the maximum separation envelope method with a grid spacing of 0.1 nm [7]. A maximum separation distance between the atoms of interest of dmax = 1 nm was used.

The microstructure of the steel consisted of two types of fine ferrite grains: (i) small recrystallised grains with an average grain size of 1.4 ± 0.2 μm; and (ii) grains with a high dislocation density (5.8 ± 1.4 × 1014 m−2) and an average grain size of 1.9 ± 0.1 μm in thickness and 2.7 ± 0.1 μm in length (Fig. 1a). Some grains with high dislocation density displayed an elongated shape with Widmanstätten side plates and also the formation of cells and subgrains (Fig. 1a). The volume fraction of recrystallised grains was 34 ± 8%.

Determination of all syzygies for the dependent polynomial invariants of the Riemann Tensor. III. mixed invariants of arbitrary degree in the Ricci Spinor

In this paper, we rigorously prove that the complete set of Riemann tensor invariants given by Sneddon [J. Math. Phys. 40, 5905 (1999)] is both minimal and complete. Furthermore, we provide a two-stage algorithm for the explicit construction of polynomial syzygies relating any dependent Riemann tensor invariant to members of the complete set.

Conditions to ensure the 2-D characteristic polynomial of uncertain matrix is very strict Hurwitz

This article derives some new conditions for the bivariate characteristic
polynomial of an uncertain matrix to be very strict Hurwitz. The uncertainties are assumed of the structured and unstructured type. Using the two-dimensional inverse Laplace transform, we derive the bounds on the uncertainties, which will ensure that the bivariate characteristic polynomial is very strict Hurwitz. Two numerical examples are given to illustrate the results.

Robust strictly positive real synthesis of polynomial segments for discrete time systems

By using the result of robust strictly positive real synthesis of polynomial segments for continuous time systems, it is proved that, for any two n-th order polynomials a(z) and b(z), the Schur stability of their convex combination is necessary and sufficient for the existence of an n-th order polynomial c(z) such that c(z)/a(z) and c(z)/b(z) are both strictly positive real. We also provide the construction method of c(z). Illustrative examples are provided to show the effectiveness of this method.

Cycles of genocide, stories of denial : Atom Egoyan's Ararat

This article focuses on Atom Egoyan's Ararat and explores how, through a convoluted narrative structure, Egoyan grapples with denial of the Armenian Genocide and the consequences of those denials for present generations—both Turkish and Armenian—illuminated in the film as an extension of the genocide. Egoyan uses a film-within-a-film to move beyond a popular definition of genocide as mass killing alone and links the understanding of stories, truths, and perspectives in everyday life to the dehumanizing acts of genocide. Employing the philosopher Emmanuel Levinas’ ethical theory of the Other (the ethical) and philosophical understandings of ontology (dehumanization) to illuminate the genocide and its ongoing denial, this article contends that Egoyan's focus on the generations of genocide survivors points to the ethical responsibility to one another that underlies everyday lives and sits at the heart of what is absent in the acts of genocide.

Syzygies of the polynomial invariants of the Riemann tensor

This thesis introduces a novel way of writing polynomial invariants as network graphs, and applies this diagrammatic notation scheme, in conjunction with graph theory, to derive algorithms for constructing relationships (syzygies) between different invariants. These algorithms give rise to a constructive solution of a longstanding classical problem in invariant theory.

Atom probe tomography of solute distributions in mg-based alloys

Atom probe tomography (APT) has been carried out on three magnesium-based alloys: M1 (Mg-1 wt pct Mn), AZ31(Mg-3 pct Al-1 pct Zn), and ME10 (Mg-1 pct Mn- 0.4 pct misch metal). The aims of this experiment were to measure the composition of the matrix and to investigate solute clustering in the matrix of the three different alloys. For AZ31, the matrix composition was variable but close to the bulk composition. For ME10 and M1, the matrix was depleted in alloying additions, with the remainder residing in precipitates. Most alloying additions were found to exhibit clustering to some extent, with misch metal having the strongest partitioning behavior to clusters. Solute clusters did not appear to affect mechanical twinning. It has been proposed that the clustering behavior of misch metal contributes to its ability to modify the recrystallization texture.

Atom probe microscopy of self-assembled monolayers : preliminary results

We have achieved three-dimensional imaging of decanethiol self-assembled monolayers (SAMs) on metal surfaces by atom probe tomography (APT). The present Letter provides preliminary results on Ni [001] and Au [111], shows the analytical potential of APT analysis of SAMs, and details developments in specimen preparation and in data-treatment methodologies. Importantly, the investigation of the mass spectra from analysis of the SAMs revealed no combination of sulfur and hydrogen at the interface between the metal substrates and the organic materials, potentially providing insight about the bonding of the thiols on the substrate.

Laguerre Runge-Kutta-Fehlberg method for simulating laser pulse propagation in biological tissue

An efficient algorithm for solving the transient radiative transfer equation for laser pulse propagation in biological tissue is presented. A Laguerre expansion is used to represent the time dependency of the incident short pulse. The Runge–Kutta– Fehlberg method is used to solve the intensity. The discrete ordinates method is used to discretize with respect to azimuthal and zenith angles. This method offers the advantages of representing the intensity with a high accuracy using only a few Laguerre polynomials, and straightforward extension to inhomogeneous media. Also, this formulation can be easily extended for solving the 2-D and 3-D transient radiative transfer equations.

Understanding the behavior of advanced high-strength steels using atom probe tomography

The key evidence for understanding the mechanical behavior of advanced high strength steels was provided by atom probe tomography (APT). Chemical overstabilization of retained austenite (RA) leading to the limited transformation-induced plasticity (TRIP) effect was deemed to be the main factor responsible for the low ductility of nanostructured bainitic steel. Appearance of the yield point on the stress-strain curve of prestrained and bake-hardened transformationinduced plasticity steel is due to the unlocking from weak carbon atmospheres of newly formed during prestraining dislocations.