Virtual copies of semisimple Lie algebras in enveloping algebras of semidirect products and Casimir operators

Given a semidirect product $\frak{g}=\frak{s}\uplus\frak{r}$ of semisimple Lie algebras $\frak{s}$ and solvable algebras $\frak{r}$, we construct polynomial operators in the enveloping algebra $\mathcal{U}(\frak{g})$ of $\frak{g}$ that commute with $\frak{r}$ and transform like the generators of $\frak{s}$, up to a functional factor that turns out to be a Casimir operator of $\frak{r}$. Such operators are said to generate a virtual copy of $\frak{s}$ in $\mathcal{U}(\frak{g})$, and allow to compute the Casimir operators of $\frak{g}$ in closed form, using the classical formulae for the invariants of $\frak{s}$. The behavior of virtual copies with respect to contractions of Lie algebras is analyzed. Applications to the class of Hamilton algebras and their inhomogeneous extensions are given.Comment: 20 pages, 2 Appendice

Decomposing changes in income risk using consumption data

This paper concerns the decomposition of income risk into permanent and transitory components using repeated cross-section data on income and consumption. Our focus is on the detection of changes in the magnitudes of variances of permanent and transitory risks. A new approximation to the optimal consumption growth rule is developed. Evidence from a dynamic stochastic simulation is used to show that this approximation can provide a robust method for decomposing income risk in a nonstationary environment. We examine robustness to unobserved heterogeneity in consumption growth and to unobserved heterogeneity in income growth. We use this approach to investigate the growth in income inequality in the UK in the 1980s

Reciprocal relativity of noninertial frames: quantum mechanics

Noninertial transformations on time-position-momentum-energy space {t,q,p,e} with invariant Born-Green metric ds^2=-dt^2+dq^2/c^2+(1/b^2)(dp^2-de^2/c^2) and the symplectic metric -de/\dt+dp/\dq are studied. This U(1,3) group of transformations contains the Lorentz group as the inertial special case. In the limit of small forces and velocities, it reduces to the expected Hamilton transformations leaving invariant the symplectic metric and the nonrelativistic line element ds^2=dt^2. The U(1,3) transformations bound relative velocities by c and relative forces by b. Spacetime is no longer an invariant subspace but is relative to noninertial observer frames. Born was lead to the metric by a concept of reciprocity between position and momentum degrees of freedom and for this reason we call this reciprocal relativity. For large b, such effects will almost certainly only manifest in a quantum regime. Wigner showed that special relativistic quantum mechanics follows from the projective representations of the inhomogeneous Lorentz group. Projective representations of a Lie group are equivalent to the unitary reprentations of its central extension. The same method of projective representations of the inhomogeneous U(1,3) group is used to define the quantum theory in the noninertial case. The central extension of the inhomogeneous U(1,3) group is the cover of the quaplectic group Q(1,3)=U(1,3)*s H(4). H(4) is the Weyl-Heisenberg group. A set of second order wave equations results from the representations of the Casimir operators

Electrostatic propulsion system with a direct nuclear electrogenerator Patent

Nuclear electric generator for accelerating charged propellant particles in electrostatic propulsion syste

Neutrino masses in quartification schemes

The idea of quark-lepton universality at high energies has recently been explored in unified theories based upon the quartification gauge group SU(3)^4. These schemes encompass a quark-lepton exchange symmetry that results upon the introduction of leptonic colour. It has been demonstrated that in models in which the quartification gauge symmetry is broken down to the standard model gauge group, gauge coupling constant unification can be achieved, and there is no unique scenario. The same is also true when the leptonic colour gauge group is only partially broken, leaving a remnant SU(2)_\ell symmetry at the standard model level. Here we perform an analysis of the neutrino mass spectrum of such models. We show that these models do not naturally generate small Majorana neutrino masses, thus correcting an error in our earlier quartification paper, but with the addition of one singlet neutral fermion per family there is a realisation of see-saw suppressed masses for the neutrinos. We also show that these schemes are consistent with proton decay.Comment: 12 pages, minor changes. To appear in Phys. Rev.

An investigation of the plastic fracture of high strength steels

Three generally recognized stages of plastic fracture in high strength steels are considered in detail. These stages consist of void initiation, void growth, and void coalescence. A brief review of the existing literature on plastic fracture is included along with an outline of the experimental approach used in the investigation

The effect of microstructure and strength on the fracture toughness of an 18 Ni, 300 grade maraging steel

Fractography and metallographic sectioning were used to investigate the influence of microstructure and strength on the fracture toughness (KIc) and fracture mechanism of an 18 Ni, 300 grade maraging steel. Increased yield strength from 1442 to 2070 MN/m squared through precipitation hardening results in a KIc loss from 143 to 55 MN/m superscript 3/2. Ti (C,N) Ti2S, and TiC inclusions in sizes from 1 to 8, 1 to 15, and 0.1 to 2 microns respectively serve as sites for void nucleation and lead to fracture by the dimpled rupture process in all strength levels considered. TiC nucleated dimples occupy more than half the fracture in all conditions. Void nucleation rate and resultant number of dimples per unit area of fracture increase with increasing yield strength. Average dimple size decreases with increasing strength and/or overaging which follows from the decreasing amount of stable void growth measured by sectioning tensile specimens. Void growth is assisted by crack branching along a path of TiC inclusions. Coalescence occurs in the highest strength materials by a combination of TiC void nucleation and premature separation at strengthening precipitates

Projective Representations of the Inhomogeneous Hamilton Group: Noninertial Symmetry in Quantum Mechanics

Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these representations are equivalent to the unitary representations of the central extension of the group. The formulation of the inertial states of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group, and its nonrelativistic limit in terms of the Galilei group, are fundamental examples. Interestingly, neither of these symmetries includes the Weyl-Heisenberg group; the hermitian representations of its algebra are the Heisenberg commutation relations that are a foundation of quantum mechanics. The Weyl-Heisenberg group is a one dimensional central extension of the abelian group and its unitary representations are therefore a particular projective representation of the abelian group of translations on phase space. A theorem involving the automorphism group shows that the maximal symmetry that leaves invariant the Heisenberg commutation relations are essentially projective representations of the inhomogeneous symplectic group. In the nonrelativistic domain, we must also have invariance of Newtonian time. This reduces the symmetry group to the inhomogeneous Hamilton group that is a local noninertial symmetry of Hamilton's equations. The projective representations of these groups are calculated using the Mackey theorems for the general case of a nonabelian normal subgroup

Processing peracetic acid treated bloodmeal into bioplastic

Renewable and biodegradable bioplastics can be produced from biopolymers such as proteins. Animal blood is a by-product from meat processing and is rich in protein. It is dried into low value bloodmeal and is used as animal feed or fertiliser. Previous work has shown that bloodmeal can be converted into a thermoplastic using water, urea, sodium dodecyl sulphate (SDS), sodium sulphite and triethylene glycol (TEG). To increase its range of applications and acceptance from consumers, the colour and odour was removed from bloodmeal using peracetic acid (PAA). The aim of this study was to investigate the bioplastic processing of 3-5% (w/w) PAA treated bloodmeal. 3-5% PAA treated bloodmeal powder was compression moulded using different combinations of water, TEG, glycerol, SDS, sodium sulphite, urea, borax, salt and sodium silicate at concentrations up to 60 parts per hundred bloodmeal (pphBM). Partially consolidated extrudates and fully consolidated compression moulded sheets were obtained using a combination of water, TEG and SDS. 4% PAA treated bloodmeal produced the best compression moulded sheets and extrudates and was chosen for investigating the effects of water, TEG and SDS concentration on consolidation, specific mechanical energy input (SME) and product colour during extrusion. Analysis of variance (ANOVA) showed SDS was the most important factor influencing its ability to be extruded because it detangled protein chains and allowed them to form new stabilising interactions required for consolidation. The best extruded sample, which was 98% consolidated and 49% white, contained 40 pphBM water, 10 pphBM TEG and 6 pphBM SDS

The effect of microstructure and strength on the fracture toughness of an 18 Ni, 300 grade maraging steel

A 300 grade maraging steel was chosen as a vehicle by which to understand the inverse relationship between strength and toughness in high strength alloys such as the 18 Ni maraging steels. The 18 Ni, 300 grade maraging material was a commercial grade consumable-electrode, vacuum arc remelted heat obtained in the form of forged and annealed plate. The matrix contained a population of second-phase impurity inclusions which was a product of the casting and hot working processes. These inclusions did not change with subsequent precipitation hardening. Changes in microstructure resulting in strength increases were brought about by variations in aging temperature and time. Maximum strength was attained in the 300 grade maraging steel by aging at 427 C (800 F) for 100 hours. Tensile, fatigue precracked Charpy impact, and plane-strain fracture toughness tests were performed at room temperature, 20 C (68 F). With increasing strength the fracture toughness decreases as smaller and smaller inclusions act as sites for void initiation