On Linearizing Systems of Diffusion Equations

We consider systems of diffusion equations that have considerable interest in Soil Science and Mathematical Biology and focus upon the problem of finding those forms of this class that can be linearized. In particular we use the equivalence transformations of the second generation potential system to derive forms of this system that can be linearized. In turn, these transformations lead to nonlocal mappings that linearize the original system.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Isotropy, shear, symmetry and exact solutions for relativistic fluid spheres

The symmetry method is used to derive solutions of Einstein's equations for fluid spheres using an isotropic metric and a velocity four vector that is non-comoving. Initially the Lie, classical approach is used to review and provide a connecting framework for many comoving and so shear free solutions. This provides the basis for the derivation of the classical point symmetries for the more general and mathematicaly less tractable description of Einstein's equations in the non-comoving frame. Although the range of symmetries is restrictive, existing and new symmetry solutions with non-zero shear are derived. The range is then extended using the non-classical direct symmetry approach of Clarkson and Kruskal and so additional new solutions with non-zero shear are also presented. The kinematics and pressure, energy density, mass function of these solutions are determined.Comment: To appear in Classical and Quantum Gravit

Carbon residence time dominates uncertainty in terrestrial vegetation responses to future climate and atmospheric CO2.

Future climate change and increasing atmospheric CO2 are expected to cause major changes in vegetation structure and function over large fractions of the global land surface. Seven global vegetation models are used to analyze possible responses to future climate simulated by a range of general circulation models run under all four representative concentration pathway scenarios of changing concentrations of greenhouse gases. All 110 simulations predict an increase in global vegetation carbon to 2100, but with substantial variation between vegetation models. For example, at 4 °C of global land surface warming (510-758 ppm of CO2), vegetation carbon increases by 52-477 Pg C (224 Pg C mean), mainly due to CO2 fertilization of photosynthesis. Simulations agree on large regional increases across much of the boreal forest, western Amazonia, central Africa, western China, and southeast Asia, with reductions across southwestern North America, central South America, southern Mediterranean areas, southwestern Africa, and southwestern Australia. Four vegetation models display discontinuities across 4 °C of warming, indicating global thresholds in the balance of positive and negative influences on productivity and biomass. In contrast to previous global vegetation model studies, we emphasize the importance of uncertainties in projected changes in carbon residence times. We find, when all seven models are considered for one representative concentration pathway × general circulation model combination, such uncertainties explain 30% more variation in modeled vegetation carbon change than responses of net primary productivity alone, increasing to 151% for non-HYBRID4 models. A change in research priorities away from production and toward structural dynamics and demographic processes is recommended.The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7 2007-2013) under Grant 238366. R.B., R.K., R.D., A.W., and P.D.F. were supported by the Joint Department of Energy and Climate Change/Department for Environment, Food and Rural Affairs Met Office Hadley Centre Climate Programme (GA01101). A.I. and K.N. were supported by the Environment Research and Technology Development Fund (S-10) of the Ministry of the Environment, Japan. We acknowledge the World Climate Research Programme’s Working Group on Coupled Modelling, which is responsible for the Coupled Model Intercomparison Project (CMIP), and we thank the climate modeling groups responsible for the GFDL-ESM2M, HadGEM2-ES, IPSL-CM5A-LR, MIROC-ESM-CHEM, and NorESM1-M models for producing and making available their model output. For CMIP, the US Department of Energy’s Program for Climate Model Diagnosis and Intercomparison provides coordinating support and led development of software infrastructure in partnership with the Global Organization for Earth System Science Portals. This work has been conducted under the framework of the Inter-Sectoral Impact Model Intercomparison Project (ISI-MIP). The ISI-MIP Fast Track project was funded by the German Federal Ministry of Education and Research (BMBF) with project funding Reference 01LS1201A.This is the author accepted manuscript. The final version is available from PNAS via http://dx.doi.org/10.1073/pnas.122247711

Effective radiative forcing and adjustments in CMIP6 models

The effective radiative forcing, which includes the instantaneous forcing plus adjustments from the atmosphere and surface, has emerged as the key metric of evaluating human and natural influence on the climate. We evaluate effective radiative forcing and adjustments in 17 contemporary climate models that are participating in the Coupled Model Intercomparison Project (CMIP6) and have contributed to the Radiative Forcing Model Intercomparison Project (RFMIP). Present-day (2014) global-mean anthropogenic forcing relative to pre-industrial (1850) levels from climate models stands at 2.00 (&plusmn;0.23) W&thinsp;m&minus;2, comprised of 1.81 (&plusmn;0.09) W&thinsp;m&minus;2 from CO2, 1.08 (&plusmn; 0.21) W&thinsp;m&minus;2 from other well-mixed greenhouse gases, &minus;1.01 (&plusmn; 0.23) W&thinsp;m&minus;2 from aerosols and &minus;0.09 (&plusmn;0.13) W&thinsp;m&minus;2 from land use change. Quoted uncertainties are 1&nbsp;standard deviation across model best estimates, and 90&thinsp;% confidence in the reported forcings, due to internal variability, is typically within 0.1 W&thinsp;m&minus;2. The majority of the remaining 0.21&thinsp;W&thinsp;m&minus;2 is likely to be from ozone. In most cases, the largest contributors to the spread in effective radiative forcing (ERF) is from the instantaneous radiative forcing (IRF) and from cloud responses, particularly aerosol&ndash;cloud interactions to aerosol forcing. As determined in previous studies, cancellation of tropospheric and surface adjustments means that the stratospherically adjusted radiative forcing is approximately equal to ERF for greenhouse gas forcing but not for aerosols, and consequentially, not for the anthropogenic total. The spread of aerosol forcing ranges from &minus;0.63 to &minus;1.37&thinsp;W&thinsp;m&minus;2, exhibiting a less negative mean and narrower range compared to 10 CMIP5 models. The spread in 4&times;CO2 forcing has also narrowed in CMIP6 compared to 13 CMIP5 models. Aerosol forcing is uncorrelated with climate sensitivity. Therefore, there is no evidence to suggest that the increasing spread in climate sensitivity in CMIP6 models, particularly related to high-sensitivity models, is a consequence of a stronger negative present-day aerosol forcing and little evidence that modelling groups are systematically tuning climate sensitivity or aerosol forcing to recreate observed historical warming. </div

New upper solution bounds for perturbed continuous algebraic Riccati equations applied to automatic control

In dynamical systems studies, the so-called Riccati and Lyapunov equations play an important role in stability analysis, optimal control and filtering design. In this paper, upper matrix bounds for the perturbation of the stabilizing solution of the continuous algebraic Riccati equation (CARE) are derived for the case when one, or all the coefficient matrices are subject to small perturbations. Comparing with existing works on this topic, the proposed bounds are less restrictive. In addition to these bounds, iterative algorithms are also derived to obtain more precise estimates