Let {X(t)}t≥0 denote a Lévy process in R with exponent Ψ. Taylor (1986) proved that the packing dimension of the range X([0 , 1]) is given by the index (0.1) γ′ = sup { α ≥ 0 : lim inf r→0+ ∫ 1 0 P {|X(t)| ≤ r} rα dt = 0 } . We provide an alternative formulation of γ′ in terms of the Lévy exponent Ψ. Our formulation, as well as methods, are Fourier-analytic, and rely on the properties of the Ca...

In the context of Kolmogorov’s algorithmic approach to the foundations of probability, Martin-Löf defined the concept of an individual random sequence using the concept of a constructive measure 1 set. Alternate characterizations use constructive martingales and measures of impossibility. We prove a direct conversion of a constructive martingale into a measure of impossibility and vice versa, s...

We shall use the \iterated Venetian blind" construction to show that the packing dimensions of plane sections of subsets of R n can depend essentially on the directions of the planes. We shall also establish the instability of the packing dimension of sections under smooth dieomorphisms.

The two-dimensional discrete bin-packing problem (2BP ) consists in minimizing the number of identical rectangles used to pack a set of smaller rectangles. This problem is NP-hard. It occurs in industry if pieces of steel, wood, or paper have to be cut from larger rectangles. It belongs to the family of cutting and packing problems, more precisely to the class of TwoDimensional Single Bin Size ...

Classically it is known that any set with packing dimension less than 1 is meager in the sense of Baire category. We establish a resource-bounded extension: if a class X has ∆-strong dimension less than 1, then X is ∆-meager. This has the applications of explaining some of Lutz’s simultaneous ∆-meager, ∆-measure 0 results and providing a new proof of a Gu’s strong dimension result on infinitely...

In this paper, we present improved bounds for the First Fit algorithm for the bin-packing problem.We prove C FF (L) ≤ 17 10C (L)+ 7 10 for all lists L, and the absolute performance ratio of FF is at most 12 7 . © 2010 Elsevier B.V. All rights reserved.

In this article we describe a practical planning problem, the potted plant packing problem, and our first approaches to solve this problem. The transportation of plants is comparably expensive as they require careful treatment due to there sensitivity. For standardised transport, potted plants are loaded on transport trollies. The cost of transportation depends on the number of these trollies. ...

In the bin packing problem, we get a sequence of packets of size r 1 ; r 2 ; r 3 ; : : :, each r i 1, and we must place the packets into bins of size 1 using as few bins as possible. The size of a bin is the accumulated size of all packets in the bin, and the free capacity of a bin is 1 minus its size.

An induced packing of odd cycles in a graph is a packing such that there is no edge in the graph between any two odd cycles in the packing. We prove that the problem is solvable in time O(n ) when the input graph is planar. We also show that deciding if a graph has an induced packing of two odd cycles is NP-complete.