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Abstract  

In the present paper lattice packings of open unit discs are considered in the Euclidean plane. Usually, efficiency of a packing is measured by its density, which in case of lattice packings is the quotient of the area of the discs and the area of the fundamental domain of the packing. In this paper another measure, the expandability radius is introduced and its relation to the density is studied. The expandability radius is the radius of the largest disc which can be used to substitute a disc of the packing without overlapping the rest of the packing. Lower and upper bounds are given for the density of a lattice packing of given expandability radius for any feasible value. The bounds are sharp and the extremal configurations are also presented. This packing problem is related to a covering problem studied by Bezdek and Kuperberg [BK97].

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We consider finite packings of unit-balls in Euclidean 3-spaceE 3 where the centres of the balls are the lattice points of a lattice polyhedronP of a given latticeL 3⊃E3. In particular we show that the facets ofP induced by densest sublattices ofL 3 are not too close to the next parallel layers of centres of balls. We further show that the Dirichlet-Voronoi-cells are comparatively small in this direction. The paper was stimulated by the fact that real crystals in general grow slowly in the directions normal to these dense facets.

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Abstract  

K. Bezdek and T. Odor proved the following statement in [1]: If a covering ofE 3 is a lattice packing of the convex compact bodyK with packing lattice Λ (K is a Λ-parallelotopes) then there exists such a 2-dimensional sublattice Λ′ of Λ which is covered by the set ∪(K+z∣z ∈ Λ′). (KL(Λ′) is a Λ′-parallelotopes). We prove that the statement is not true in the case of the dimensionsn=6, 7, 8.

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] contacts are, of course, essential for lattice packing ( Fig. 4 , Table 3 ). In the region of νO–H stretching, the broad absorption bands at 3406 cm −1 suggest hydrogen bonding in the Co complex ( Table 4 ). The band at 889 cm −1 is due to the

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