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Multiscale Modeling and Homogenization of Reaction-Diffusion Systems Involving Biological Surfaces
Many complex chemical processes are responsible for the proper functioning of the human body. A prime example is the finely structured endoplasmic reticulum, which plays an important role in the metabolisms of human cells. To handle mathematical models that account for this fine structure, periodic homogenization methods are derived and applied. Previous results on homogenization of partial differential equations on finely structured manifolds are extended: Using the periodic unfolding method, diffusion terms on manifolds with different scalings with powers of the homogenization parameter, in particular in case of fast diffusion, are homogenized and are applied in three different biological systems: a linear model of carcinogenesis of cells, a nonlinear extension of the linear carcinogenesis model and a model considering T-cell signaling. Simulations and interpretations of the homogeneous T-cell signaling model give an insight into the related biological mechanisms.
Frontiers of Life
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Global Regularity and Uniqueness of Solutions in a Surface Growth Model Using Rigorous A-Posteriori Methods
The use of rigorous numerical methods to approach problems which can not be solved using standard methods (yet) has increased signifiantly in recent years. In this book, riogorous a-posteriori methods are used to study the time evolution of a surface growth model, given by a fourth order semi-linear parabolic partial differential equation, where standard methods fail to verify global uniqueness and smoothness of solutions. Based on an arbitrary numerical approximation, a-posteriori error-analysis is applied in order to prevent a blow up analytically. This is a method that in a similar way also applies to the three dimensional Navier-Stokes equations. The main idea consists of energy-estimate...
Adaptive Finite Elements in the Discretization of Parabolic Problems
Adaptivity is a crucial tool in state-of-the-art scientific computing. However, its theoretical foundations are only understood partially and are subject of current research. This self-contained work provides theoretical basics on partial differential equations and finite element discretizations before focusing on adaptive finite element methods for time dependent problems. In this context, aspects of temporal adaptivity and error control are considered in particular. Based on the gained insights, a specific adaptive algorithm is designed and analyzed thoroughly. Most importantly, it is proven that the presented adaptive method terminates within any demanded error tolerance. Moreover, the developed algorithm is analyzed from a numerical point of view and its performance is compared to well-known standard methods. Finally, it is applied to the real-life problem of concrete carbonation, where two different discretizations are compared.
Mine Wastes and Water, Ecological Engineering and Metals Extraction
The book reviews past and present mine waste management processes. It estimates global water consumption by major mining resources per annum. This consumption will lead land use resources (agriculture and water) to collide with mining interests expected in the near future. With the application of novel metal extraction processes and the adoption of ecological engineering as an approach to waste and water management, a reduction in water and land consumption can be achieved. Using these methodologies would make mining more sustainable. Together with ore and metal recycling, mining methods can be brought into the 21st century. The book describes natural weathering processes and the microbiolog...
Colloidal Transport in Porous Media
This book covers the basics of abiotic colloid characterization, of biocolloids and biofilms, the resulting transport phenomena and their engineering aspects. The contributors comprise an international group of leading specialists devoted to colloidal sciences. The contributions include theoretical considerations, results from model experiments, and field studies. The information provided here will benefit students and scientists interested in the analytical, chemical, microbiological, geological and hydrological aspects of material transport in aquatic systems and soils.
Commutability of Gamma-limits in problems with multiple scales
In the calculus of variations, the goal is to explore extrema of a given integral functional. From origins of the problem, it might be expected that the functional can be adequately simplified by neglecting some small quantities. A way to rigorously justify such an approximation is the Γ-convergence that ensures convergence of corresponding (global) extrema. The main motivation of this work is to investigate properties of doubly indexed integral functionals that Γ-converge for one index fixed. In other words, for two possible approximations we would like to determine whether we may perform them consecutively and if they commute. Our examples are taken from material science with homogenization being one of these two processes. In the first part we are considering a setting related to the elastic regime. However, our assumptions are fairly general and allow for applications in different areas. The second part is devoted to problems in the Hencky plasticity. They are considerably different due to special growth properties of the density.
Advances in Microbial Physiology
This volume contains advances in microbial physiology, particularly: factors affecting the production of l-phenylacetylcarbinol by yeast.
