This paper proposes an adaptation, to the two-dimensional irregular bin packing problem of the Djang and Finch heuristic (DJD), originally designed for the one-dimensional bin packing problem. In the two-dimensional case, not only is it the case that the piece’s size is important but its shape also has a significant influence. Therefore, DJD as a selection heuristic has to be paired with a plac...

We consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with exponent h equal to the Hausdorff dimension of the Julia set. Both h-dimensional Hausdorff and packing measures are finite and positive on the Julia set and ...

Several of the best known problems and conjectures in graph theory arise in studying the behavior of a graphical invariant on a graph product. Examples of this are Vizing’s conjecture, Hedetniemi’s conjecture and the calculation of the Shannon capacity of graphs, where the invariants are the domination number, the chromatic number and the independence number on the Cartesian, categorical and st...

Figure 1: An Apollonian Circle Packing Apollonius’s Theorem states that given three mutually tangent circles, there are exactly two circles which are tangent to all three. Apollonian circle packings are produced by repeating the construction of mutually tangent circles to fill all remaining spaces. A remarkable consequence of Descartes’ Theorem is if the initial four tangent circles have integr...

We consider online bin packing where in addition to the opening cost of each bin, the arriving items collect delay costs until their assigned bins are released (closed), and the open bins themselves collect holding costs. Besides being of practical interest, this problem generalises several previously unrelated online optimisation problems. We provide a general online algorithm for this problem...

conclusions the mssrhs could be considered as a more suitable packing material for long-term biofilter operation to remove h2s pollution. results in this study, after 53 days of operating time and 54 g/m3h mass loading, the ph values in the ss and mssrhs packed filters were reduced from 6.8 to 1.8 and from 6.9 to 3.4, respectively. the mssrhs packed filter showed more buffering capacity to resi...

Laparoscopy offers great exposure and surgical detail, reduces blood loss and the need for excessive abdominal packing-- and bowel manipulation making it an excellent modality to perform pelvic floor surgery. Laparoscopic repair of level I or apical vaginal prolapse may be challenging, due to the need for extensive dissection and advanced suturing skills. However, it offers the efficacy of open...

while w(d) = 4=15 and we conclude that, in any event, w(B 3) 1. Case 3. Suppose jB 3 j = 4. B 3 cannot contain an X 1 item, since 1=2 + 3(1=5) > 1. Neither can it contain two X 2 items, since 2(1=3) + 2(1=5) > 1. Similarly, it cannot contain four X 3 items, since each has size greater than 1/4. However, if it contains three items of type X 3 and one of type X 4 , then w(B 3) 3(4=15) + 1=5 = 1. ...