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The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic hierarchy σ = (σψ, σ∧), where the second component takes value in operators on the infinite model cone of the local wedges. In general understanding of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying the elliptcity of the principal edge symbol σ∧ which includes the (in general not explicitly known) number of additional conditions on the edge of trace and potential type. We focus here on these queations and give explicit answers for a wide class of elliptic operators that are connected with the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation and ellipticity for Dirichlet-Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues of the Agranovich-Dynin formula for edge boundary value problems, and we establish relations of elliptic operators for different weights, via the spectral flow of the underlying conormal symbols.
Central Asia is located at the confluence of large-scale atmospheric circulation systems. It is thus likely to be highly susceptible to changes in the dynamics of those systems; however, little is still known about the regional paleoclimate history. Here we present carbon and hydrogen isotopic compositions of n-alkanoic acids from a late Holocene sediment core from Lake Karakuli (eastern Pamir, Xinjiang Province, China). Instrumental evidence and isotopeenabled climate model experiments with the Laboratoire de Meteorologie Dynamique Zoom model version 4 (LMDZ4) demonstrate that delta D values of precipitation in the region are influenced by both temperature and precipitation amount. We find that these parameters are inversely correlated on an annual scale, i.e., the climate has varied between relatively cool and wet and more warm and dry over the last 50 years. Since the isotopic signals of these changes are in the same direction and therefore additive, isotopes in precipitation are sensitive recorders of climatic changes in the region. Additionally, we infer that plants use year-round precipitation (including snowmelt), and thus leaf wax delta D values must also respond to shifts in the proportion of moisture derived from westerly storms during late winter and early spring. Downcore results give evidence for a gradual shift to cooler and wetter climates between 3.5 and 2.5 cal kyr BP, interrupted by a warm and dry episode between 3.0 and 2.7 kyr BP. Further cool and wet episodes occur between 1.9 and 1.5 and between 0.6 and 0.1 kyr BP, the latter coeval with the Little Ice Age. Warm and dry episodes from 2.5 to 1.9 and 1.5 to 0.6 kyr BP coincide with the Roman Warm Period and Medieval Climate Anomaly, respectively. Finally, we find a drying tend in recent decades. Regional comparisons lead us to infer that the strength and position of the westerlies, and wider northern hemispheric climate dynamics, control climatic shifts in arid Central Asia, leading to complex local responses. Our new archive from Lake Karakuli provides a detailed record of the local signatures of these climate transitions in the eastern Pamir.