Topological Field Theory of the Initial Singularity of Space-Time

Here we suggest a possible resolution of the initial space-time singularity. In this novel approach, the initial singularity of space-time corresponds to a 0 size singular gravitational instanton, characterised by a Riemannian metric configuration (++++) in dimension D = 4. Associated with the 0 scale of space-time, the initial singularity is thus not considered in terms of divergences of physical fields but can be resolved in terms of topological field symmetries and associated invariants (in particular the first Donaldson invariant ). In this perspective, we here introduce a new topological invariant, associated with 0 scale, of the form Z = Tr (-1)s which we call "singularity invariant"

Vibrational entropy of L12 Cu3Au measured by inelastic neutron scattering

The phonon density of states of elemental Au, Cu, and Cu3Au with L12 chemical order were measured by inelastic neutron scattering and used to calculate the vibrational entropy of formation of the ordered compound from the elemental metals. A vibrational entropy of formation of (0.06±0.03) kB/atom at 300 K was obtained, with the vibrational entropy of the ordered alloy being larger than that of the elemental metals. The phonon DOS of the disordered Cu3Au was simulated by adding the phonon DOS curves of fcc Cu, L12 Cu3Au, and fcc Au to match the numbers of first-nearest-neighbor pairs in a disordered alloy. The vibrational entropy obtained with this simulated DOS disagrees with calorimetric data and theoretical estimates, indicating that the phonon DOS of disordered Cu3Au depends on chemical order at spatial lengths larger than is set by first-nearest-neighbor pairs

Investigation of the utility of mean square approximation systems and in system response predictions

A method is presented for estimating the variability of a system's natural frequencies arising from the variability of the system's parameters. The only information required to obtain the estimates is the member variability, in the form of second order properties, and the natural frequencies and mode shapes of the mean system. Several examples are worked out in detail to illustrate how the method is applied

Topological Origin of Inertia

The purpose of the present article is to propose a topological interpretation of the inertial interaction. In this context, the instantaneous propagation and the infinite range of the inertial interaction can be explained as a topological amplitude associated to a 0 size singular gravitational instanton.We suggest here that the topological charge of the singular gravitational instanton of 0 size represents the source of the inertial interaction. As the topological charge, the inertial interaction is equally invariant and propagates itself instantaneously from one point to another in spacetime. Such a property is not explicable within the frame of field theory but may find a solution in the topological theory as proposed by E.Witten in 1988. In our model, the initial singularity is the source of a topological amplitude corresponding to the charge of the 0 size singular gravitational instanton detectable at the boundary S3 of the singular gravitational instanton having the topology of the 4-dimensional Euclidean ball B4. The pseudo-observables of Riemannien spacetime at the origin are here interpreted as co-cycles on instanton moduli space and are associated to Ksi-cycles of the B4 4-manifold (Donaldson application)

The KMS State of Space-Time at the Planck Scale

Considering the expected thermal equilibrium characterizing the physics at the Planck scale, it is here stated, for the first time, that, as a system, the space-time at the Planck scale must be considered as subject to the Kubo-Martin-Schwinger (KMS) condition. Consequently, in the interior of the KMS strip, i.e. from the scale ß = 0 to the scale ß = Planck, the fourth coordinate g44 must be considered as complex, the two real poles being ß = 0 and ß = Planck. This means that within the limits of the KMS strip, the Lorentzian and the Euclidean metric are in a "quantum superposition state" (or coupled), this entailing a "unification" (or coupling) between the topological (Euclidean) and the physical (Lorentzian) states of space-time

Construction of Cocycle Bicrossproducts by Twisting

We consider here a generalisation of the Majid's mirror product when one of the components of the product is replaced by a twist. This leads to a new "twisted mirror product" construction for cocycle bicrossproduct Hopf algebra and as an application we obtain a canonical cocycle bicrossproduct for any Hopf algebra H

Phonon entropy of alloying and ordering of Cu-Au

Inelastic neutron scattering spectra were measured with a time-of-flight spectrometer on six disordered Cu-Au alloys at 300 K. The neutron-weighted phonon density of states was obtained from a conventional analysis of these spectra. Several methods were developed to account for this neutron weighting and obtain the phonon entropy of the disordered alloys. The phonon entropies of formation of disordered fcc Cu-Au alloys obtained in this way were generally mutually consistent, and were also consistent with predictions from a cluster approximation obtained from ab-initio calculations by Ozolin[underaccent cedilla [below] s-breve, Wolverton, and Zunger. We estimate a phonon entropy of disordering of 0.15±0.05kB/atom in Cu3Au at 300 K. A resonance mode associated with the motions of the heavy Au atoms in the Cu-rich alloys was observed at 9 meV. An analysis of the resonance mode provided a check on the partial phonon entropy of Au atoms

Temperature dependence of the phonon entropy of vanadium

The phonon density-of-states (DOS) of elemental vanadium was measured at elevated temperatures by inelastic neutron scattering. The phonon softening predicted by thermal expansion against the bulk modulus is much larger than the measured shifts in phonon energies. We conclude that the phonon anharmonicities associated with thermal expansion are largely canceled by effects from phonon-phonon scattering. Prior measurements of the heat capacity and calculations of the electronic entropy of vanadium are assessed, and consistency requires an explicit temperature dependence of the phonon DOS. Using data from the literature, similar results are found for chromium, niobium, titanium, and zirconium