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    Finite nilpotent semigroups of small coclass

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    The parameter coclass has been used successfully in the study of nilpotent algebraic objects of different kinds. In this paper a definition of coclass for nilpotent semigroups is introduced and semigroups of coclass 0, 1, and 2 are classified. Presentations for all such semigroups and formulae for their numbers are obtained. The classification is provided up to isomorphism as well as up to isomorphism or anti-isomorphism. Commutative and self-dual semigroups are identified within the classification.Comment: 11 page

    Spectral Properties of Grain Boundaries at Small Angles of Rotation

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    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 ⁣:R2RV \colon \R^2 \to \R, we let Vθ(x,y)=V(x,y)V_\theta(x,y) = V(x,y) in the right half-plane {x0}\{x \ge 0\} and Vθ=VMθV_\theta = V \circ M_{-\theta} in the left half-plane {x<0}\{x < 0\}, where MθR2×2M_\theta \in \R^{2 \times 2} is the usual matrix describing rotation of the coordinates in R2\R^2 by an angle θ\theta. As a main result, it is shown that spectral gaps of the periodic Schr\"odinger operator H0=Δ+VH_0 = -\Delta + V fill with spectrum of Rθ=Δ+VθR_\theta = -\Delta + V_\theta as 0θ00 \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
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