Instantons and unitarity in quantum cosmology with fixed four-volume

We find a number of complex solutions of the Einstein equations in the so-called unimodular version of general relativity, and we interpret them as saddle points yielding estimates of a gravitational path integral over a space of almost everywhere Lorentzian metrics on a spacetime manifold with topology of the "no-boundary" type. In this setting, the compatibility of the no-boundary initial condition with the definability of the quantum measure reduces reduces to the normalizability and unitary evolution of the no-boundary wave function \psi. We consider the spacetime topologies R^4 and RP^4 # R^4 within a Taub minisuperspace model with spatial topology S^3, and the spacetime topology R^2 x T^2 within a Bianchi type I minisuperspace model with spatial topology T^3. In each case there exists exactly one complex saddle point (or combination of saddle points) that yields a wave function compatible with normalizability and unitary evolution. The existence of such saddle points tends to bear out the suggestion that the unimodular theory is less divergent than traditional Einstein gravity. In the Bianchi type I case, the distinguished complex solution is approximately real and Lorentzian at late times, and appears to describe an explosive expansion from zero size at T=0. (In the Taub cases, in contrast, the only complex solution with nearly Lorentzian late-time behavior yields a wave function that is normalizable but evolves nonunitarily, with the total probability increasing exponentially in the unimodular "time" in a manner that suggests a continuous creation of new universes at zero volume.) The issue of the stability of these results upon the inclusion of more degrees of freedom is raised.Comment: 32 pages, REVTeX v3.1 with amsfonts. (v2: minor typos etc corrected.

The behavior of adaptive bone-remodeling simulation models

The process of adaptive bone remodeling can be described mathematically and simulated in a computer model, integrated with the finite element method. In the model discussed here, cortical and trabecular bone are described as continuous materials with variable density. The remodeling rule applied to simulate the remodeling process in each element individually is, in fact, an objective function for an optimization process, relative to the external load. Its purpose is to obtain a constant, preset value for the strain energy per unit bone mass, by adapting the density. If an element in the structure cannot achieve that, it either turns to its maximal density (cortical bone) or resorbs completely.\ud \ud It is found that the solution obtained in generally a discontinuous patchwork. For a two-dimensional proximal femur model this patchwork shows a good resemblance with the density distribution of a real proximal femur.\ud \ud It is shown that the discontinuous end configuration is dictated by the nature of the differential equations describing the remodeling process. This process can be considered as a nonlinear dynamical system with many degrees of freedom, which behaves divergent relative to the objective, leading to many possible solutions. The precise solution is dependent on the parameters in the remodeling rule, the load and the initial conditions. The feedback mechanism in the process is self-enhancing; denser bone attracts more strain energy, whereby the bone becomes even more dense. It is suggested that this positive feedback of the attractor state (the strain energy field) creates order in the end configuration. In addition, the process ensures that the discontinuous end configuration is a structure with a relatively low mass, perhaps a minimal-mass structure, although this is no explicit objective in the optimization process.\ud \ud It is hypothesized that trabecular bone is a chaotically ordered structure which can be considered as a fractal with characteristics of optimal mechanical resistance and minimal mass, of which the actual morphology depends on the local (internal) loading characteristics, the sensor-cell density and the degree of mineralization