Variational formalism for generic shells in general relativity

We investigate the variational principle for the gravitational field in the presence of thin shells of completely unconstrained signature (generic shells). Such variational formulations have been given before for shells of timelike and null signatures separately, but so far no unified treatment exists. We identify the shell equation as the natural boundary condition associated with a broken extremal problem along a hypersurface where the metric tensor is allowed to be nondifferentiable. Since the second order nature of the Einstein-Hilbert action makes the boundary value problem associated with the variational formulation ill-defined, regularization schemes need to be introduced. We investigate several such regularization schemes and prove their equivalence. We show that the unified shell equation derived from this variational procedure reproduce past results obtained via distribution theory by Barrabes and Israel for hypersurfaces of fixed causal type and by Mars and Senovilla for generic shells. These results are expected to provide a useful guide to formulating thin shell equations and junction conditions along generic hypersurfaces in modified theories of gravity.

Meeting for Transformation: A Locality for Ritual Activities During the Early Neolithic Funnel Beaker Culture in Central Sweden

The former island of Södertörn, just south of Stock- holm, was intensively settled during the Early Neo- lithic. For more than twenty years a field at Stensborg, Grödinge parish, was surveyed for surface finds. Most numerous among the various artefact categories were axes, with stone axes of different shapes showing that they had been intentionally fragmented. Fragments of pointed-butted and thin-butted flint axes were also found, all of them changed by intensive heat. Just as the flint axes indicate contact with southern Scandi- navia, slate objects demonstrate the existence of net- works extending to northern Sweden. During the Early Neolithic the site was a slope situ- ated in the innermost part of a bay, delimited by the shoreline on one side and a ridge on the opposite side. Two small but pronounced ravines of streams also form part of the natural boundary of the site. During excavation of the field several small pits were found that were filled with fragments of axes, pottery and other objects, along with a considerable amount of carbonized seed. Most of the finds have in- dications of destruction, either directly or by the use of fire. The field seems to have been used as a place for assemblies, where rituals were an important part of the activities. The Stensborg site seems to represent yet another kind of natural enclosure involving ritual activities during the Early Neolithic. This presenta- tion is part of a project in progress.

A Complex-Lamellar Description Of Boundary Layer Transition

Flow transition is important, in both practical and phenomenological terms. However, there is currently no method for identifying the spatial locations associated with transition, such as the start and end of intermittency. The concept of flow stability and experimental correlations have been used, however, flow stability only identifies the location where disturbances begin to grow in the laminar flow and experimental correlations can only give approximations as measuring the start and end of intermittency is diffcult. Therefore, the focus of this work is to construct a method to identify the start and end of intermittency, for a natural boundary layer transition and a separated flow transition. We obtain these locations by deriving a complex-lamellar description of the velocity field that exists between a fully laminar and fully turbulent boundary condition. Mathematically, this complex-lamellar decomposition, which is constructed from the classical Darwin-Lighthill-Hawthorne drift function and the transport of enstrophy, describes the flow that exists between the fully laminar Pohlhausen equations and Prandtl's fully turbulent one seventh power law. We approximate the difference in enstrophy density between the boundary conditions using a power series. The slope of the power series is scaled by using the shape of the universal intermittency distribution within the intermittency region. We solve the complex-lamellar decomposition of the velocity field along with the slope of the difference in enstrophy density function to determine the location of the laminar and turbulent boundary conditions. Then from the difference in enstrophy density function we calculate the start and end of intermittency. We perform this calculation on a natural boundary layer transition over a flat plate for zero pressure gradient flow and for separated shear flow over a separation bubble. We compare these results to existing experimental results and verify the accuracy of our transition model.

A Complex-Lamellar Description Of Boundary Layer Transition

Flow transition is important, in both practical and phenomenological terms. However, there is currently no method for identifying the spatial locations associated with transition, such as the start and end of intermittency. The concept of flow stability and experimental correlations have been used, however, flow stability only identifies the location where disturbances begin to grow in the laminar flow and experimental correlations can only give approximations as measuring the start and end of intermittency is diffcult. Therefore, the focus of this work is to construct a method to identify the start and end of intermittency, for a natural boundary layer transition and a separated flow transition. We obtain these locations by deriving a complex-lamellar description of the velocity field that exists between a fully laminar and fully turbulent boundary condition. Mathematically, this complex-lamellar decomposition, which is constructed from the classical Darwin-Lighthill-Hawthorne drift function and the transport of enstrophy, describes the flow that exists between the fully laminar Pohlhausen equations and Prandtl's fully turbulent one seventh power law. We approximate the difference in enstrophy density between the boundary conditions using a power series. The slope of the power series is scaled by using the shape of the universal intermittency distribution within the intermittency region. We solve the complex-lamellar decomposition of the velocity field along with the slope of the difference in enstrophy density function to determine the location of the laminar and turbulent boundary conditions. Then from the difference in enstrophy density function we calculate the start and end of intermittency. We perform this calculation on a natural boundary layer transition over a flat plate for zero pressure gradient flow and for separated shear flow over a separation bubble. We compare these results to existing experimental results and verify the accuracy of our transition model.

The geographical distribution of soil elements determines the boundaries of the Daodi-zone of medicinal plant (Ligusticum chuanxiong Hort.)

Understanding the boundary of plants' best production zone is of great significance to the rational introduction cultivation. We set up 9 sample points in the center-zone and 28 sample points extending eastward along the longitude. Then, we were collecting the bulk soil of L.chuanxiong, testing soil properties and elements. Later, a multi-point cultivation test was carried out on L.chuanxiong, the content of elements in the plant measured by ICP-MS, and HPLC detected the active ingredients. Based on hierarchical clustering and PCA analysis, the study zone's soil element was divided into three groups. In the L.chuanxiong cultivation experiments, we found Ligustilide in rhizomes was positively related to Se and Sr and negatively related to Mn and the chemical elements. This study successfully verified that the medicinal plant's Daodi-zone has a natural boundary, and L.chuanxiong's Daodi-zone may be from east of Dujiangyan to the west edge of Longquan, rich in Na, Mg, K and Low Mn is the soil characteristics of the area. Moreover, we put forward a method to identify L.chuanxiong's Daodi-zone based on characteristic soil elements, which can also provide a practical basis for dividing other economic plants' growth environment.