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- Dirac operators (1)
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- Mathematisch-Naturwissenschaftliche Fakultät (3) (remove)

We show that the residue density of the logarithm of a generalized Laplacian on a closed manifold definesan invariant polynomial-valued differential form. We express it in terms of a finite sum of residues ofclassical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulas providea pedestrian proof of the Atiyah–Singer formula for a pure Dirac operator in four dimensions and for atwisted Dirac operator on a flat space of any dimension. These correspond to special cases of a moregeneral formula by Scott and Zagier. In our approach, which is of perturbative nature, we use either aCampbell–Hausdorff formula derived by Okikiolu or a noncommutative Taylor-type formula.

It has been suggested that coronal mass ejections (CMEs) remove the magnetic he-licity of their coronal source region from the Sun. Such removal is often regarded to be necessary due to the hemispheric sign preference of the helicity, which inhibits a simple annihilation by reconnection between volumes of opposite chirality. Here we monitor the relative magnetic he-licity contained in the coronal volume of a simulated flux rope CME, as well as the upward flux of relative helicity through horizontal planes in the simulation box. The unstable and erupting flux rope carries away only a minor part of the initial relative helicity; the major part remains in the volume. This is a consequence of the requirement that the current through an expanding loop must decrease if the magnetic energy of the configuration is to decrease as the loop rises, to provide the kinetic energy of the CME.

The Takab complex is composed of a variety of metamorphic rocks including amphibolites, metapelites, mafic granulites, migmatites and meta-ultramafics, which are intruded by the granitoid. The granitoid magmatic activity occurred in relation to the subduction of the Neo-Tethys oceanic crust beneath the Iranian crust during Tertiary times. The granitoids are mainly granodiorite, quartz monzodiorite, monzonite and quartz diorite. Chemically, the magmatic rocks are characterized by ASI < 1.04, AI < 0.87 and high contents of CaO (up to ∼ 14.5 wt %), which are consistent with the I-type magmatic series. Low FeO t /(FeO t +MgO) values (< 0.75) as well as low Nb, Y and K 2 O contents of the investigated rocks resemble the calc-alkaline series. Low SiO 2 , K 2 O/Na 2 O and Al 2 O 3 accompanied by high CaO and FeO contents indicate melting of metabasites as an appropriate source for the intrusions. Negative Ti and Nb anomalies verify a metaluminous crustal origin for the protoliths of the investigated igneous rocks. These are comparable with compositions of the associated mafic migmatites, in the Takab metamorphic complex, which originated from the partial melting of amphibolites. Therefore, crustal melting and a collision-related origin for the Takab calc-alkaline intrusions are proposed here on the basis of mineralogy and geochemical characteristics. The P–T evolution during magmatic crystallization and subsolidus cooling stages is determined by the study of mineral chemistry of the granodiorite and the quartz diorite. Magmatic crystallization pressure and temperature for the quartz-diorite and the granodiorite are estimated to be P ∼ 7.8 ± 2.5 kbar, T ∼ 760 ± 75 ◦C and P ∼ 5 ± 1 kbar, T ∼ 700 ◦C, respectively. Subsolidus conditions are consistent with temperatures of ∼ 620 ◦C and ∼ 600 ◦C, and pressures of ∼ 5 kbar and ∼ 3.5 kbar for the quartz-diorite and the granodiorite, respectively.