Algorithmic counting of nonequivalent compact Huffman codes

It is known that the following five counting problems lead to the same integer sequence~$f_t(n)$: the number of nonequivalent compact Huffman codes of length~$n$ over an alphabet of $t$ letters, the number of `nonequivalent' canonical rooted $t$-ary trees (level-greedy trees) with $n$~leaves, the number of `proper' words, the number of bounded degree sequences, and the number of ways of writing $1= \frac{1}{t^{x_1}}+ \dots + \frac{1}{t^{x_n}}$ with integers $0 \leq x_1 \leq x_2 \leq \dots \leq x_n$. In this work, we show that one can compute this sequence for \textbf{all} $n<N$ with essentially one power series division. In total we need at most $N^{1+\varepsilon}$ additions and multiplications of integers of $cN$ bits, $c<1$, or $N^{2+\varepsilon}$ bit operations, respectively. This improves an earlier bound by Even and Lempel who needed $O(N^3)$ operations in the integer ring or $O(N^4)$ bit operations, respectively

GPU Accelerated Explicit Time Integration Methods for Electro-Quasistatic Fields

Electro-quasistatic field problems involving nonlinear materials are commonly discretized in space using finite elements. In this paper, it is proposed to solve the resulting system of ordinary differential equations by an explicit Runge-Kutta-Chebyshev time-integration scheme. This mitigates the need for Newton-Raphson iterations, as they are necessary within fully implicit time integration schemes. However, the electro-quasistatic system of ordinary differential equations has a Laplace-type mass matrix such that parts of the explicit time-integration scheme remain implicit. An iterative solver with constant preconditioner is shown to efficiently solve the resulting multiple right-hand side problem. This approach allows an efficient parallel implementation on a system featuring multiple graphic processing units.Comment: 4 pages, 5 figure

Coloring decompositions of complete geometric graphs

A decomposition of a non-empty simple graph $G$ is a pair $[G,P]$, such that $P$ is a set of non-empty induced subgraphs of $G$, and every edge of $G$ belongs to exactly one subgraph in $P$. The chromatic index $\chi'([G,P])$ of a decomposition $[G,P]$ is the smallest number $k$ for which there exists a $k$-coloring of the elements of $P$ in such a way that: for every element of $P$ all of its edges have the same color, and if two members of $P$ share at least one vertex, then they have different colors. A long standing conjecture of Erd\H{o}s-Faber-Lov\'asz states that every decomposition $[K_n,P]$ of the complete graph $K_n$ satisfies $\chi'([K_n,P])\leq n$. In this paper we work with geometric graphs, and inspired by this formulation of the conjecture, we introduce the concept of chromatic index of a decomposition of the complete geometric graph. We present bounds for the chromatic index of several types of decompositions when the vertices of the graph are in general position. We also consider the particular case in which the vertices are in convex position and present bounds for the chromatic index of a few types of decompositions.Comment: 18 pages, 5 figure

Moving towards a "COAL-PEC"?

Coal has for many years been considered as a resource of the past and as a result its importance has been underestimated. Yet coal still is the main pillar for generating electricity in most countries: A quarter of the worldwide primary energy consumption is provided by coal. While the world's largest coal producers, China, the USA and India, are at the same time the largest consumers of coal. Smaller producers and consumers of coal engage extensively in international trade. In particular the seaborne coal trade has increased significantly since the 1990's. In the past two years prices of import coal also have increased considerably. In September 2008, importers in Europe had to pay prices of more than 200 US dollars per ton, a price level many times higher than the historical average. In this context, fears have increasingly been voiced that the international coal market - analogous to the oil market which continues to be dominated by the OPEC-might witness the emergence of a supplier cartel, a "COAL-PEC". A strong tendency towards the concentration of companies has in fact been observed in the international coal market in the past years. Increased prices could have resulted from the use of market power. Drivers for the price increase were the strong rise in demand, in particular from China and India, capacity bottlenecks in production and shipment as well as a lack of investments. In the future a tight market and high coal prices have to be expected.Coal, Energy, Market structure, Simulation model

COALMOD-World: A Model to Assess International Coal Markets until 2030

Coal continues to be an important fuel in many countries' energy mix and, despite the climate change concerns, it is likely to maintain this position for the next decades. In this paper a numerical model is developed to investigate the evolution of the international market for steam coal, the coal type used for electricity generation. The main focus is on future trade ows and investments in production and transport infrastructure until 2030. "COALMOD-World" is an equilibrium model, formulated in the complementarity format. It includes all major steam coal exporting and importing countries and represents the international trade as one globalized market. Some suppliers of coal are at the same time major consumers, such as the USA and China. Therefore, domestic markets are also included in the model to analyze their interaction with the international market. Because of the different qualities of steam coal, we include different heating values depending on the origin of the coal. At the same time we observe the mass-specific constraints on production, transport and export capacity. The time horizon of our analysis is until 2030, in 5-year steps. Production costs change endogenously over time. Moreover, endogenous investments are included based on a net present value optimization approach and and the shadow prices of capacities constraints. Investments can be carried out in production, inland freight capacities (rail in most countries), and export terminals. The paper finishes with an application of the model to a base case scenario and suggestions for alternative scenarios.coal, energy, numerical modeling, investments, international trade