Extremal classical interpolation problems (matrix case)

AbstractIn this paper, we try to find amongst the solutions w(z) of the corresponding interpolation problem the solution which satisfies an additional extremal condition. We show that degenerate interpolation problems play an important role in the theory of extremal interpolation problems. At the end of the paper we accomplish a comparison of our approach with former known approaches

The lithospheric mantle and lower crust-mantle relationships under Scotland: a xenolithic perspective

In the British Isles the majority of volcanic rocks containing upper mantle and lower crustal xenoliths occur in Scotland. Most of the occurrences are of Carboniferous–Permian age. This paper presents new data on the mineral chemistry of spinel lherzolite xenoliths from the five principal Scottish tectonic terranes. Compositional variations among the minerals emphasize the broad lateral heterogeneity of the subcontinental lithospheric mantle across the region. The remarkable range of Al2O3 v. CaO exhibited by the clinopyroxenes compared with data from other ‘xenolith provinces' emphasizes the extremely complex tectonomagmatic history of the Scottish lithosphere. The generalized age increase from southern and central Scotland to the Northern Highland and Hebridean terranes of the north and NW, with concomitant complexity of geological history, is reflected also by trace element and isotopic studies. Reaction relationships in lherzolites from the Hebridean Terrane, owing to pervasive metasomatism, involve secondary growth of sodic feldspar. This, and light REE enrichment of clinopyroxenes, points to involvement of a natro-carbonatitic melt. Most pyroxenitic xenoliths are inferred to form a basal crustal layer with a generally sharp discontinuity above the underlying (dominantly lherzolitic) mantle. A second discontinuity is inferred to separate these ultramafic cumulates from overlying, broadly cognate metagabbroic cumulates

Semiseparable integral operators and explicit solution of an inverse problem for the skew-self-adjoint Dirac-type system

Inverse problem to recover the skew-self-adjoint Dirac-type system from the generalized Weyl matrix function is treated in the paper. Sufficient conditions under which the unique solution of the inverse problem exists, are formulated in terms of the Weyl function and a procedure to solve the inverse problem is given. The case of the generalized Weyl functions of the form $\phi(\lambda)\exp\{-2i\lambda D\}$, where $\phi$ is a strictly proper rational matrix function and $D=D^* \geq 0$ is a diagonal matrix, is treated in greater detail. Explicit formulas for the inversion of the corresponding semiseparable integral operators and recovery of the Dirac-type system are obtained for this case

Evidence of the proto-Iceland plume in northwestern Ireland at 42 Ma from helium isotopes

Combined K-Ar and Ar-Ar dating of a xenolith-bearing alkaline dyke, previously considered to be of late Palaeozoic age, suggest a mid-Eocene age of c. 43 Ma, indicating that it is part of extensive rift-related Tertiary intrusive activity. Hig

The petrogenesis of Carboniferous-Permian dyke and sill intrusions across northern Europe

The presence or absence of a thermally anomalous mantle plume during the formation of the widespread Carboniferous-Permian magmatism of northern Europe is examined. The geochemistry of representative samples from the extensive Carboniferous-Permian dyke and sill intrusions across northern Europe are reported in order to ascertain whether they have a common 'plume' source. Both tholeiitic and alkaline magmas have diverse trace element compositions. Alkaline samples with relatively low Ti and Nb/La 1 from the asthenosphere. The tholeiites have a close affinity to E-MORB but have mixed with variable amounts of lithosphere and upper crust. Tectonic reorganisation and decompression melting of a trace element-enriched mantle is considered to have controlled the Carboniferous-Permian magmatism, which contains no coherent geochemical evidence for a single plume-related thermo-chemical anomaly. © Springer-Verlag 2006

Felsic lavas or rheomorphic ignimbrites: is there a chemical distinction?

In the environment of felsic magma generation associated with continental flood basalt (CFB) provinces there is a close association between magma composition, phenocryst assemblage, temperature and eruption mechanism. In this paper we propose that by examining the chemical composition and, in particular the high field strength element (Zr and Nb) contents of the high silica rocks, we can identify those that contained halogen-rich volatiles and which degassed at different levels (deep versus shallow). The degassing depth has a direct influence on the type of eruption, with shallow degassing promoting explosive rather than effusive behaviour and consequentially the former is more likely to give rise to ignimbrites. Thus, we can infer likely eruption mechanisms, as high concentrations of F and Cl dramatically decrease magma viscosities and should favour deep degassing and hence promote lava effusion rather than explosive pyroclastic eruption. This hypothesis is tested by constraining the other possible variables which have an effect on the mode of eruption, and by examining the nature and composition of melt and fluid inclusions in quartz from CFB-associated felsic volcanic rocks.</p

Tectonic forcing of longitudinal valleys in the Himalaya: morphological analysis of the Ladakh Batholith, North India

Longitudinal valleys form first order topographic features in many mountain belts. They are commonly located along faults that separate tectonic zones with varying uplift histories. The Indus Valley of Ladakh, northern India, runs northwestwards following the boundary between the relatively undeformed Ladakh Batholith to the north–east and the folded and thrusted Zanskar mountains to the south–west. In this region the Shyok Valley, on the northern side of the batholith, approximately parallels the course of the Indus. This study investigates geomorphic variations in transverse catchments that drain the Ladakh Batholith, into the Indus and Shyok rivers. The batholith has been divided into three zones based on varying structural characteristics of its northeastern and southwestern boundaries. Morphometric analysis of 62 catchments that drain into the Indus and Shyok valleys was carried out using three digital datasets, and supported by field observations. Morphometric asymmetry is evident in the central zone where the Shyok valley is considered tectonically inactive, but the Indus Valley is bound by the northeastwardly thrusting Indus Molasse and the batholith. In this zone the catchments that drain into the Indus Valley are more numerous, shorter, thinner and have lower hypsometric integrals than those that drain into the Shyok. By linking these observations with the regional geology and thermochronological data it is proposed that high sediment discharge from the deformed Indus Molasse Indus Valley has progressively raised base levels in the Indus Valley and resulted in sediment blanketing of the opposing tectonically quiescent catchments that drain southwestwards off the batholith. The Indus Molasse thrust front has propagated at least 36 km towards the Ladakh Batholith over the last 20 Ma. Hence it is proposed that this long term asymmetric structural deformation and exhumation has forced the Indus longitudinal valley laterally into the Ladakh Batholith resulting in the morphometric asymmetry of its transverse catchments