Ordering of Energy Levels in Heisenberg Models and Applications

In a recent paper we conjectured that for ferromagnetic Heisenberg models the smallest eigenvalues in the invariant subspaces of fixed total spin are monotone decreasing as a function of the total spin and called this property ferromagnetic ordering of energy levels (FOEL). We have proved this conjecture for the Heisenberg model with arbitrary spins and coupling constants on a chain. In this paper we give a pedagogical introduction to this result and also discuss some extensions and implications. The latter include the property that the relaxation time of symmetric simple exclusion processes on a graph for which FOEL can be proved, equals the relaxation time of a random walk on the same graph. This equality of relaxation times is known as Aldous' Conjecture.Comment: 20 pages, contribution for the proceedings of QMATH9, Giens, September 200

Hierarchical model for the scale-dependent velocity of seismic waves

Elastic waves of short wavelength propagating through the upper layer of the Earth appear to move faster at large separations of source and receiver than at short separations. This scale dependent velocity is a manifestation of Fermat's principle of least time in a medium with random velocity fluctuations. Existing perturbation theories predict a linear increase of the velocity shift with increasing separation, and cannot describe the saturation of the velocity shift at large separations that is seen in computer simulations. Here we show that this long-standing problem in seismology can be solved using a model developed originally in the context of polymer physics. We find that the saturation velocity scales with the four-third power of the root-mean-square amplitude of the velocity fluctuations, in good agreement with the computer simulations.Comment: 7 pages including 3 figure

Active Faulting in Lake Constance (Austria, Germany, Switzerland) Unraveled by Multi-Vintage Reflection Seismic Data

Probabilistic seismic hazard assessments are primarily based on instrumentally recorded and historically documented earthquakes. For the northern part of the European Alpine Arc, slow crustal deformation results in low earthquake recurrence rates and brings up the necessity to extend our perspective beyond the existing earthquake catalog. The overdeepened basin of Lake Constance (Austria, Germany, and Switzerland), located within the North-Alpine Molasse Basin, is investigated as an ideal (neo-) tectonic archive. The lake is surrounded by major tectonic structures and constrained via the North Alpine Front in the South, the Jura fold-and-thrust belt in the West, and the Hegau-Lake Constance Graben System in the North. Several fault zones reach Lake Constance such as the St. Gallen Fault Zone, a reactivated basement-rooted normal fault, active during several phases from the Permo-Carboniferous to the Mesozoic. To extend the catalog of potentially active fault zones, we compiled an extensive 445 km of multi-channel reflection seismic data in 2017, complementing a moderate-size GI-airgun survey from 2016. The two datasets reveal the complete overdeepened Quaternary trough and its sedimentary infill and the upper part of the Miocene Molasse bedrock. They additionally complement existing seismic vintages that investigated the mass-transport deposit chronology and Mesozoic fault structures. The compilation of 2D seismic data allowed investigating the seismic stratigraphy of the Quaternary infill and its underlying bedrock of Lake Constance, shaped by multiple glaciations. The 2D seismic sections revealed 154 fault indications in the Obersee Basin and 39 fault indications in the Untersee Basin. Their interpretative linkage results in 23 and five major fault planes, respectively. One of the major fault planes, traceable to Cenozoic bedrock, is associated with a prominent offset of the lake bottom on the multibeam bathymetric map. Across this area, high-resolution single channel data was acquired and a transect of five short cores was retrieved displaying significant sediment thickness changes across the seismically mapped fault trace with a surface-rupture related turbidite, all indicating repeated activity of a likely seismogenic strike-slip fault with a normal faulting component. We interpret this fault as northward continuation of the St. Gallen Fault Zone, previously described onshore on 3D seismic data

Steiner t-designs for large t

One of the most central and long-standing open questions in combinatorial design theory concerns the existence of Steiner t-designs for large values of t. Although in his classical 1987 paper, L. Teirlinck has shown that non-trivial t-designs exist for all values of t, no non-trivial Steiner t-design with t > 5 has been constructed until now. Understandingly, the case t = 6 has received considerable attention. There has been recent progress concerning the existence of highly symmetric Steiner 6-designs: It is shown in [M. Huber, J. Algebr. Comb. 26 (2007), pp. 453-476] that no non-trivial flag-transitive Steiner 6-design can exist. In this paper, we announce that essentially also no block-transitive Steiner 6-design can exist.Comment: 9 pages; to appear in: Mathematical Methods in Computer Science 2008, ed. by J.Calmet, W.Geiselmann, J.Mueller-Quade, Springer Lecture Notes in Computer Scienc

Is There a Walrasian Equilibrium in Exchange Markets with Endowment Effect

We provide an axiomatic framework for exchange markets with a willingness- to-pay/willingness-to-accept discrepancy. First, we obtain a two parameter family of market invariants under price-scaling representing the excess demand. One of the parameters can be identified as endowment. The other is a new feature, called demand-supply gap, that leads to classical general equilibrium if zero. Second, we provide representations of price and demand as unbounded operators on an infinite dimensional Hilbert space. We prove that neither can this space be finite dimensional nor can these operators be bounded. Third, if the demand-supply gap is not zero we obtain that price and demand are not simultaneously sharply measurable and consequently a Walrasian equilibrium does not exist

Classical and Quantum Discrete Dynamical Systems

We study deterministic and quantum dynamics from a constructive "finite" point of view, since the introduction of a continuum, or other actual infinities in physics poses serious conceptual and technical difficulties, without any need for these concepts to physics as an empirical science. For a consistent description of the symmetries of dynamical systems at different times and the symmetries of the various parts of such systems, we introduce discrete analogs of the gauge connections. Gauge structures are particularly important to describe the quantum behavior. We show that quantum behavior is the result of a fundamental inability to trace the identity of indistinguishable objects in the process of evolution. Information is available only on invariant statements and values, relating to such objects. Using mathematical arguments of a general nature we can show that any quantum dynamics can be reduced to a sequence of permutations. Quantum interferences occur in the invariant subspaces of permutation representations of symmetry groups of dynamical systems. The observable values can be expressed in terms of permutation invariants. We also show that for the description of quantum phenomena, instead of a nonconstructive number system --- the field of complex numbers, it is enough to use cyclotomic fields --- the minimal extentions of natural numbers suitable for quantum mechanics. Finite groups of symmetries play a central role in this article. The interest in such groups has an additional motivation in physics. Numerous experiments and observations in particle physics point to an important role of finite groups of relatively low orders in a number of fundamental processes.Comment: author's English translation added; 74 pages in English, 80 pages in Russian; English translation in Phys. Part. Nucl. ISSN 1063-7796 (2013) 44, No. 1, pp 47-91 was done without author's control; V3: Section 4 revised; V4: some correction

Near-field seismic displacement and tilt associated with the explosive activity of Stromboli

Broadband seismic recordings in the near-field of Strombolian explosions, at 500 m distance, show pronounced effects of tilt. The tilt signal is predominant in the horizontal components beyond about 50 s period while it is negligible in the vertical component. The waveform of the tilt signal at the seismometer output is a double time integral of the waveform due to ground displacement. Since the waveform of the displacement is known from the vertical component, the waveform of the tilt signal in the horizontal seismogram can be reconstructed and both contributions can be separated from each other with a linear regression. We have analyzed data recorded in the summit region of Stromboli in 1995 and 1996. The regional tilt can be determined from the differential vertical displacement between instruments a few tens of meters apart. Local tilts determined with individual instruments scatter around the regional value, most probably due to local strain-tilt-coupling. Mogi's (1958) formulae for a pressure source in a homogeneous halfspace are used to interpret the results. The source displaces a volume of several tens of cubic meters of the surrounding rock before the explosive discharge; typical volumes were 25 m3 in July 1995 and 60 m3 in September 1996