Spectral Properties of Grain Boundaries at Small Angles of Rotation

We study some spectral properties of a simple two-dimensional model for small angle defects in crystals and alloys. Starting from a periodic potential $V \colon \R^2 \to \R$, we let $V_\theta(x,y) = V(x,y)$ in the right half-plane $\{x \ge 0\}$ and $V_\theta = V \circ M_{-\theta}$ in the left half-plane $\{x < 0\}$, where $M_\theta \in \R^{2 \times 2}$ is the usual matrix describing rotation of the coordinates in $\R^2$ by an angle $\theta$. As a main result, it is shown that spectral gaps of the periodic Schr\"odinger operator $H_0 = -\Delta + V$ fill with spectrum of $R_\theta = -\Delta + V_\theta$ as $0 \ne \theta \to 0$. Moreover, we obtain upper and lower bounds for a quantity pertaining to an integrated density of states measure for the surface states.Comment: 22 pages, 3 figure

Resolution of simple singularities yielding particle symmetries in a space-time

A finite subgroup of the conformal group SL(2,C) can be related to invariant polynomials on a hypersurface in C^3. The latter then carries a simple singularity, which resolves by a finite iteration of basic cycles of deprojections. The homological intersection graph of this cycles is the Dynkin graph of an ADE Lie group. The deformation of the simple singularity corresponds to ADE symmetry breaking. A 3+1-dimensional topological model of observation is constructed, transforming consistently under SL(2,C), as an evolving 3-dimensional system of world tubes, which connect ``possible points of observation". The existence of an initial singularity for the 4-dimensional space-time is related to its global topological structure. Associating the geometry of ADE singularities to the vertex structure of the topological model puts forward the conjecture on a likewise relation of inner symmetries of elementary particles to local space-time structure.Comment: 16 pages, LaTe

The rich demystified: A reply to Bach, Corneo, and Steiner (2008)

The contribution Bach, Corneo and Steiner (2008) has argued that "the rich" do not pay taxes adequately in relation to their income, finding, for instance, an effective tax rate of only 38.1% for the 0.001% fractile of German income taxpayers in 2001. This result contrasts sharply with the legislated top marginal income tax rate of 48.5%. We subject the results contained in Bach, Corneo and Steiner (2008) to a rigorous analysis: We find major flaws and inconsistencies with regard to methodology, i.e. the omission of corporate taxes and inter-temporal aspects of taxation. Restating basic rules for the measurement of effective tax rates, we provide values for what we term the "comprehensive nominal tax rate" (CNTR) and show that the headline result in Bach, Corneo and Steiner (2008) of 38.1% is underestimated by over 12 percentage points. As an important distributional result, the CNTR increases with increasing taxable income. --Top Incomes,Income Taxation,Taxing the Rich,Comprehensive Tax Burden

The Rich Demystified - A Reply to Bach, Corneo, and Steiner (2008)

The contribution Bach, Corneo, and Steiner (2008) has argued that “the rich” do not pay taxes adequately in relation to their income, finding, for instance, an effective tax rate of only 38.1% for the 0.001% fractile of German income taxpayers in 2001. This result contrasts sharply with the legislated top marginal income tax rate of 48.5%. We subject the results contained in Bach, Corneo, and Steiner (2008) to a rigorous analysis: We find major flaws and inconsistencies with regard to methodology, i.e. the omission of corporate taxes and inter-temporal aspects of taxation. Restating basic rules for the measurement of effective tax rates, we provide values for what we term the “comprehensive nominal tax rate” (CNTR) and show that the headline result in Bach, Corneo, and Steiner (2008) of 38.1% is underestimated by over 12 percentage points. As an important distributional result, the CNTR increases with increasing taxable income.top incomes, income taxation, taxing the rich, comprehensive nominal tax rate