We consider rectangular graph superlattices of sides l 1, l 2 with the wave-function coupling at the junctions either of the type, when they are continuous and the sum of their derivatives is proportional to the common value at the junction with a coupling constant , or the s type with the roles of functions and derivatives reversed; the latter corresponds to the situations where the junctions are realized by complicated geometric scatterers. We show that the band spectra have a hidden fractal structure with respect to the ratio := l 1/l 2. If the latter is an irrational badly approximable by rationals, lattices have no gaps in the weak-coupling case. We show that there is a quantization for the asymptotic critical values of at which new gap series open, and explain it in terms of number-theoretic properties of . We also show how the irregularity is manifested in terms of Fermi-surface dependence on energy, and possible

Let <i>A</i> and <i>B</i> be non-negative self-adjoint operators in a separable Hilbert space such that their form sum <i>C</i> is densely defined. It is shown that the Trotter product formula holds for imaginary parameter values in the <i>L</i> ²-norm, that is, one has <div class="gsh_dspfr"> \lim_{no+\infty} \int\limits^T_{-T} \left\|\left(e^{-itA/n}e^{-itB/n} ight)^nh - e^{-itC}hight\|^2dt = 0

Simple-homotopy for cell complexes is a special type of topological homotopy constructed by elementary collapses and elementary expansions. In this paper, we introduce graph homotopy for graphs and Graham homotopy for hypergraphs and study the relation between the two homotopies and the simple-homotopy for cell complexes. The graph homotopy is useful to describe topological properties of discretized geometric figures, while the Graham homotopy is essential to characterize acyclic hypergraphs and acyclic relational database schemes.

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The chapter discusses the equivariant loop theorem for three-dimensional manifolds that is needed in settling the Smith conjecture and reviews the existence theorems for minimal surfaces. The equivariant version of the loop theorem says that there are a finite number of properly embedded disks in <i>M</i> that satisfy the required properties. The loop theorem respects the action of the group <i>G</i> in a suitable manner. The chapter puts a metric on <i>M</i> so that the group <i>G</i> acts isometrically and so that <i>M</i> is convex with respect to the outward normal. Then with respect to this metric, the existence of an immersed disk <i>D</i>₁ is demonstrated, in <i>M</i> whose boundary <i> D₁</i>, represents a nontrivial element in ₁(<i>S</i>) and whose area is minimal among all such disks. The chapter describes Morrey's solution for the plateau problem in a general Riemannian manifold. The existence theorem for manifolds with boundary