STOCHASTIC SOLUTIONS FOR FRACTIONAL WAVE EQUATIONS.

A fractional wave equation replaces the second time derivative by a Caputo derivative of order between one and two. In this paper, we show that the fractional wave equation governs a stochastic model for wave propagation, with deterministic time replaced by the inverse of a stable subordinator whose index is one half the order of the fractional time derivative.

Age and gender-dependent bone density changes of the human skull disclosed by high-resolution flat-panel computed tomography.

INTRODUCTION: The objective of this article was to estimate the age at death in forensic or anthropologic applications based on human skull investigation. Sex-dependent differences were analyzed. METHODS: Digital, high-resolution, flat-panel-based volumetric computed tomography (eXplore Locus Ultra scanner) images (165,920) of 244 European human skulls-163 males, 81 females-were analyzed according to their radiological bone density, based on Hounsfield units (H) that are directly related to the x-ray attenuation of the scanned material. Data were collected by the Department of Forensic Medicine at the University Hospital Giessen and Marburg during 2007 and 2008. Correlation analysis was used for data description. RESULTS: Human skull density estimates are widely scattered as a function of age for both sexes. Male skull bone density remains constant during lifetime, whereas female skull bone density decays slowly from approximately 20 years onwards. CONCLUSIONS: Bone density decay only theoretically provides a new method to determine age at death for adult females. Due to the scattering of the data, an accuracy of approximately ±18 years is found at a confidence interval of 75%, which is, unfortunately, of limited practical interest. We found new sex differences of bone density decay in the skull that are potentially of relevance for the general understanding of bone degradation processes.

High-resolution flat-panel volumetric CT images show no correlation between human age and sagittal suture obliteration--independent of sex.

This study investigated whether digital, high-resolution CT-images of the internal human sagittal suture structure include information that enables a novel method of age at death (aad) determination. To accomplish this, coronal, flat-panel-based volumetric computed tomography (eXplore Locus Ultra scanner) images were automatically analyzed by a software implementation of an algorithm that determines user independent whether a suture is open or closed. 29,205 images of the local vicinity of the sagittal suture of 164 males and 85 females of European descent were investigated separately for both sexes. We used conditional probabilities and a chi(2)-test to investigate whether there is a correlation between aad and suture obliteration or not. The computer-aided analysis enables us to handle huge volumes of data that could not otherwise be analyzed within a reasonable time frame. The implemented algorithm ensured a strongly reproducible, reliable, accurate, and fast differentiation between closed and open sutures. The evaluation of various statistical parameters suggests that there is no reason to assume a correlation between age and suture closure--with equal findings for both sexes. Therefore, we conclude that determination of aad based on the evaluation of sagittal suture obliteration is not possible. This agrees well - thus unsatisfactorily - with the recent literature.

Stochastic calculus for uncoupled continuous-time random walks.

The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications not only in physics but also in insurance, finance, and economics. A definition is given for a class of stochastic integrals driven by a CTRW, which includes the Ito and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is a martingale. It is proved that, as a consequence of the martingale transform theorem, if the CTRW is a martingale, the Ito integral is a martingale too. It is shown how the definition of the stochastic integrals can be used to easily compute them by Monte Carlo simulation. The relations between a CTRW, its quadratic variation, its Stratonovich integral, and its Ito integral are highlighted by numerical calculations when the jumps in space of the CTRW have a symmetric Lévy alpha -stable distribution and its waiting times have a one-parameter Mittag-Leffler distribution. Remarkably, these distributions have fat tails and an unbounded quadratic variation. In the diffusive limit of vanishing scale parameters, the probability density of this kind of CTRW satisfies the space-time fractional diffusion equation (FDE) or more in general the fractional Fokker-Planck equation, which generalizes the standard diffusion equation, solved by the probability density of the Wiener process, and thus provides a phenomenologic model of anomalous diffusion. We also provide an analytic expression for the quadratic variation of the stochastic process described by the FDE and check it by Monte Carlo.