We follow the work of [7] and study the online bin packing problem, where every item has one of two possible sizes which are known in advance. We focus on the parametric case, where both item sizes are bounded from above by 1 k for some natural number k ≥ 1. We show that for every possible pair of item sizes, there is an algorithm with competitive ratio of at most (k+1) 2 k2+k+1 . We prove that...

We consider the two-dimensional bin packing problem (BPP): given a set of rectangular items, find the minimal number of rectangular bins needed to pack all items. Rotation of the items is not permitted. We show for any integer k ≥ 3 that at most k − 1 bins are needed to pack all items if every item fits into a bin and if the total area of items does not exceed k/4-times the bin area. Moreover, ...

Balls of two diierent radii r 1 < r 2 can trivially be packed in E 3 with density better than the classical packing density 3 L = 0:74048. .. , if r 1 =r 2 0:41::. We show that this even holds for r 1 =r 2 = 0:623. .. Further we show, under strong restrictions, that r 1 =r 2 close to 1 does not yield a density > 3 L. These results correspond to similar results on circle packings in the plane

G. Wegner [12] gave a geometric characterization of all so–called Groemer packing of n ≥ 2 unit discs in E2 that are densest packings of n unit discs with respect to the convex hull of the discs. In this paper we provide a number theoretic characterization of all n satisfying that such a “Wegner packing” of n unit discs exists, and show that the proportion of these n is 23 24 among all natural ...

This paper summarizes recent results about the Hausdorff measure of the image, graph and level sets of Gaussian random fields, the packing dimension and packing measure of the image of fractional Brownian motion, the local times and multiple points of Gaussian random fields. Some open problems are also pointed out.

We show that the dimension adim introduced by R. Kaufman (1987) coincides with the packing dimension Dim, but the dimension aDim introduced by Hu and Taylor (1994) is different from the Hausdorff dimension. These results answer questions raised by Hu and Taylor. AMS Classification numbers: Primary 28A78, 28A80.

KATAPAK-S is a type of structured catalytic packing, which is used in reactive distillation processes. The dry pressure drop characteristic (the pressure drop in the absence of liquid flow) is of significant importance for the investigation of process hydrodynamics. In this paper, the dry pressure drop within the catalyst packed channels of KATAPAK-S has been investigated using Computational Fl...

In this paper we consider a two dimensional strip packing problem. The problem consists of packing a set of rectangular items in one strip of width W and infinite height. They must be packed without overlapping, parallel to the edge of the strip and we assume that the items are oriented, i.e. they cannot be rotated. To solve this problem, we use three exact methods: a branch and bound method, a...

A chain packing H in a graph is a subgraph satisfying given degree constraints at the vertices. Its size is the number of odd degree vertices in the subgraph. An odd subtree packing is a chain packing which is a forest in which all non-isolated vertices have odd degree in the forest. We show that for a given graph and degree constraints, the size of a maximum chain packing and a maximum odd sub...

To the best of our knowledge, this is the first algorithm unifying arbitrary rectilinear block packing and soft block packing. Furthermore, this algorithm handles arbitrary convex or concave rectilinear block packing in the most efficient way compared to other sequence pair-based approaches. At the same time, the algorithm can handle rectangle soft block effectively. The concept of non-redundan...