Universally Typical Sets for Ergodic Sources of Multidimensional Data

We lift important results about universally typical sets, typically sampled sets, and empirical entropy estimation in the theory of samplings of discrete ergodic information sources from the usual one-dimensional discrete-time setting to a multidimensional lattice setting. We use techniques of packings and coverings with multidimensional windows to construct sequences of multidimensional array sets which in the limit build the generated samples of any ergodic source of entropy rate below an $h_0$ with probability one and whose cardinality grows at most at exponential rate $h_0$.Comment: 15 pages, 1 figure. To appear in Kybernetika. This replacement corrects typos and slightly strengthens the main theore

Quantum Hall effect and the relative index for projections

We define the relative index, Index(P,Q), for a pair of infinite-dimensional projections on a Hilbert space to be the integer that is the natural generalization of dim(P)-dim(Q) in finite-dimensional vector spaces. We show that the Hall conductance for independent electrons in the plane is the relative index where P and Q project on the states below the Fermi energy for Hamiltonians that differ by a quantum flux and the Fermi energy is appropriately placed. This approach is closely related to, and sheds light on, Bellissard’s interpretation of the Hall conductance as an index

Bounds for the adiabatic approximation with applications to quantum computation

We present straightforward proofs of estimates used in the adiabatic approximation. The gap dependence is analyzed explicitly. We apply the result to interpolating Hamiltonians of interest in quantum computing.Comment: 15 pages, one figure. Two comments added in Secs. 2 and

A quantum version of Sanov's theorem

We present a quantum extension of a version of Sanov's theorem focussing on a hypothesis testing aspect of the theorem: There exists a sequence of typical subspaces for a given set $\Psi$ of stationary quantum product states asymptotically separating them from another fixed stationary product state. Analogously to the classical case, the exponential separating rate is equal to the infimum of the quantum relative entropy with respect to the quantum reference state over the set $\Psi$. However, while in the classical case the separating subsets can be chosen universal, in the sense that they depend only on the chosen set of i.i.d. processes, in the quantum case the choice of the separating subspaces depends additionally on the reference state.Comment: 15 page

Tree-ring stable isotopes and radiocarbon reveal pre- and post-eruption effects of volcanic processes on trees on Mt. Etna (Sicily, Italy)

Early detection of volcanic eruptions is of major importance for protecting human life. Ground deformation and changes in seismicity, geochemistry, petrology, and gravimetry are used to assess volcanic activity before eruptions. Studies on Mt. Etna (Italy) have demonstrated that vegetation can be affected by pre-eruptive activity before the onset of eruptions. During two consecutive years before Mt. Etna's 2002/2003 flank eruption, enhanced vegetation index (NDVI) values were detected along a distinct line which later developed into an eruptive fissure. However, the mechanisms by which volcanic activity can lead to changes in pre-eruption tree growth processes are still not well understood. We analysed ${\delta}^{13}$C, ${\delta}^{18}$O and $^{14}$C in the rings of the survived trees growing near to the line where the pre-eruptive increase in NDVI was observed in order to evaluate whether the uptake of water vapour or fossil volcanic CO2 could have contributed to the enhanced NDVI. We found a dramatic decrease in ${\delta}^{18}$O in tree rings formed before 2002/2003 in trees close to the eruption fissure, suggesting uptake of volcanic water by trees during pre-eruptive magma degassing. Moist conditions caused by outgassing of ascending magma may also have led to an observed reduction in tree-ring ${\delta}^{13}$C following the eruption. Furthermore, only ambiguous evidence for tree uptake of degassed CO2 was found. Our results suggest that additional soil water condensed from degassed water vapour may have promoted photosynthesis, explaining local increases in NDVI before the 2002/2003 Mt. Etna flank eruption. Tree-ring oxygen stable isotopes might be used as indicators of past volcanic eruptions

oHMint: Höhere Mathematik für MINT-Studierende - Onlinekurs und Lernplattform

Im oHMint-Projekt wird derzeit eine Online-Lernplattform zum Selbststudium der höheren Mathematik für Studierende der MINT-Studiengänge an deutschen Hochschulen entwickelt. Die aktuelle Pilotphase wird durch die Hamburg Open Online University (HOOU) finanziert, über die Universität Hamburg koordiniert und auf technischer Ebene durch die integral-learning GmbH umgesetzt. Das oHMint-Projekt wurde vom OMB+-Konsortium initiiert, welches bereits den Online Mathematik Brückenkurs Plu(OMB+) erstellt hat und damit wichtige Erfahrung in das Projekt bringt. Zunächst wird ein Kapitel, die Basiseinheit Differentialrechnung, des insgesamt vier Semester umfassenden Kurses als Prototyp erstellt. Dabei werden neue didaktische Herangehensweisen und innovative Aufgabentypen erprobt, deren Design sich möglichst stark an den Bedürfnissen der Zielgruppe orientiert. Es ist das Ziel, dass der komplette oHMint-Kurs einen zeitgemäßen Übergang von Schul- zu Hochschulmathematik bis zum Bachelorniveau ermöglicht. Die permanente Rücksprache mit dem OMB+-Konsortium im Laufe der Entwicklung liefert eine vielversprechende Ausgangsbasis für eine breite Akzeptanz und Verwendung des Endprodukts, das für die Öffentlichkeit frei zugänglich zur Verfügung gestellt werden soll. (DIPF/Orig.

Charge Deficiency, Charge Transport and Comparison of Dimensions

We study the relative index of two orthogonal infinite dimensional projections which, in the finite dimensional case, is the difference in their dimensions. We relate the relative index to the Fredholm index of appropriate operators, discuss its basic properties, and obtain various formulas for it. We apply the relative index to counting the change in the number of electrons below the Fermi energy of certain quantum systems and interpret it as the charge deficiency. We study the relation of the charge deficiency with the notion of adiabatic charge transport that arises from the consideration of the adiabatic curvature. It is shown that, under a certain covariance, (homogeneity), condition the two are related. The relative index is related to Bellissard's theory of the Integer Hall effect. For Landau Hamiltonians the relative index is computed explicitly for all Landau levels.Comment: 23 pages, no figure

Typical support and Sanov large deviations of correlated states

Discrete stationary classical processes as well as quantum lattice states are asymptotically confined to their respective typical support, the exponential growth rate of which is given by the (maximal ergodic) entropy. In the iid case the distinguishability of typical supports can be asymptotically specified by means of the relative entropy, according to Sanov's theorem. We give an extension to the correlated case, referring to the newly introduced class of HP-states.Comment: 29 pages, no figures, references adde