Dynamics with Infinitely Many Derivatives: The Initial Value Problem

Differential equations of infinite order are an increasingly important class of equations in theoretical physics. Such equations are ubiquitous in string field theory and have recently attracted considerable interest also from cosmologists. Though these equations have been studied in the classical mathematical literature, it appears that the physics community is largely unaware of the relevant formalism. Of particular importance is the fate of the initial value problem. Under what circumstances do infinite order differential equations possess a well-defined initial value problem and how many initial data are required? In this paper we study the initial value problem for infinite order differential equations in the mathematical framework of the formal operator calculus, with analytic initial data. This formalism allows us to handle simultaneously a wide array of different nonlocal equations within a single framework and also admits a transparent physical interpretation. We show that differential equations of infinite order do not generically admit infinitely many initial data. Rather, each pole of the propagator contributes two initial data to the final solution. Though it is possible to find differential equations of infinite order which admit well-defined initial value problem with only two initial data, neither the dynamical equations of p-adic string theory nor string field theory seem to belong to this class. However, both theories can be rendered ghost-free by suitable definition of the action of the formal pseudo-differential operator. This prescription restricts the theory to frequencies within some contour in the complex plane and hence may be thought of as a sort of ultra-violet cut-off.Comment: 40 pages, no figures. Added comments concerning fractional operators and the implications of restricting the contour of integration. Typos correcte

Computer simulation of recrystallization in non-uniformly deformed metals

The classical Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation [F = 1 - exp(- kt)n] for nucleation and growth transformations works very well for most solid state transformations but fails regularly when applied to recrystallization of plastically deformed metals. Under conditions of near constant growth rate, a high exponent (n [ges] 3) is predicted but low exponents (n [les] 2) are typically measured. Another common observation is that the slope of a JMAK plot, from which the exponent is inferred, decreases as recrystallization proceeds. Analysis of the published data suggested the hypothesis that the failure of the JMAK theory as applied to recrystallization is because of the lack of uniformity of the stored energy of plastic deformation on the grain size scale. This hypothesis was tested by use of Monte Carlo simulations of the type previously used successfully to model grain growth and recrystallization. The earlier simulations of recrystallization used uniform stored energies whereas the simulations presented here varied the stored energy from grain to grain. The kinetics were plotted on JMAK plots which exhibited low and varying exponents closely resembling experimental data. Specific simulations were performed to test the basic JMAK assumption that makes a correction for the effect of impingement under conditions of random nucleation, namely dF/dFe = (1 - F), where F is the actual volume fraction and Fe is the extended volume fraction--that which would obtain in the absence of impingement and overlap between new grains. It was found the assumption is accurate under conditions of uniform stored energy. With non-uniform stored energy, however, the correction underestimated the effect of impingement by a factor that rapidly increased (to over two orders of magnitude) during recrystallization.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/28082/1/0000528.pd

Cosmological perturbations in SFT inspired non-local scalar field models

We study cosmological perturbations in models with a single non-local scalar field originating from the string field theory description of the rolling tachyon dynamics. We construct the equation for the energy density perturbations of the non-local scalar field and explicitly prove that for the free field it is identical to a system of local cosmological perturbation equations in a particular model with multiple (maybe infinitely many) local free scalar fields.Comment: 21 pages, no figures, v3: presentation improved, results unchanged, references adde

Comprehensive analysis of epigenetic clocks reveals associations between disproportionate biological ageing and hippocampal volume

The concept of age acceleration, the difference between biological age and chronological age, is of growing interest, particularly with respect to age-related disorders, such as Alzheimer’s Disease (AD). Whilst studies have reported associations with AD risk and related phenotypes, there remains a lack of consensus on these associations. Here we aimed to comprehensively investigate the relationship between five recognised measures of age acceleration, based on DNA methylation patterns (DNAm age), and cross-sectional and longitudinal cognition and AD-related neuroimaging phenotypes (volumetric MRI and Amyloid-β PET) in the Australian Imaging, Biomarkers and Lifestyle (AIBL) and the Alzheimer’s Disease Neuroimaging Initiative (ADNI). Significant associations were observed between age acceleration using the Hannum epigenetic clock and cross-sectional hippocampal volume in AIBL and replicated in ADNI. In AIBL, several other findings were observed cross-sectionally, including a significant association between hippocampal volume and the Hannum and Phenoage epigenetic clocks. Further, significant associations were also observed between hippocampal volume and the Zhang and Phenoage epigenetic clocks within Amyloid-β positive individuals. However, these were not validated within the ADNI cohort. No associations between age acceleration and other Alzheimer’s disease-related phenotypes, including measures of cognition or brain Amyloid-β burden, were observed, and there was no association with longitudinal change in any phenotype. This study presents a link between age acceleration, as determined using DNA methylation, and hippocampal volume that was statistically significant across two highly characterised cohorts. The results presented in this study contribute to a growing literature that supports the role of epigenetic modifications in ageing and AD-related phenotypes

Ex-prime ministers and Parliament A riposte to mythology

SIGLEAvailable from British Library Document Supply Centre-DSC:7751.534(4/96) / BLDSC - British Library Document Supply CentreGBUnited Kingdo