Correlation of current quenching and occurrence of metal vapor in apseudospark discharge

The quenching phenomenon, i.e., a sudden interrupt of the discharge current, was investigated in a pseudospark discharge with charging voltage of 2.5 kV, maximum current of 2 kA and discharge duration of 3 μs. The working gas was hydrogen at a pressure of 40 Pa. Concerning electrode material and geometric parameters, molybdenun electrodes were chosen with hole diameters of 5 mm; the electrode distance was 3 mm. In this parameter range, a temporal correlation of current quenching and the occurrence of metal vapor could be detected by means of time-resolved optical spectroscopy. With each current interruption a sudden increase of emission from neutral molybdenum atoms as well as an increase of cathode spot emission, which is spatially localized on the cathode, occurs. Also oxygen ions were observed which show a similar time-dependence, however with a significant delay of the order of 200 ns. The results are discussed in the scope of the mechanism proposed for quenching, i.e., ion depletion in the plasma boundary layer, and the mechanisms occurring in the high current phase of a pseudospark discharge     

Convex Drawings of Planar Graphs and the Order Dimension of 3-Polytopes

We define an analogue of Schnyder's tree decompositions for 3-connected planar graphs. Based on this structure we obtain: Let G be a 3-connected planar graph with f faces, then G has a convex drawing with its vertices embedded on the (f–1)×(f–1) grid. Let G be a 3-connected planar graph. The dimension of the incidence order of vertices, edges and bounded faces of G is at most 3.The second result is originally due to Brightwell and Trotter. Here we give a substantially simpler proof.     

A Theorem on Higher Bruhat Orders

We show that inclusion order and single-step inclusion coincide for higher Bruhat orders B(n,2) , i.e., . Received April 7, 1998.     

Coding and Counting Arrangements of Pseudolines

Arrangements of lines and pseudolines are important and appealing objects for research in discrete and computational geometry. We show that there are at most 2^{0.657> n2}2^{0.657\> n^{2}} simple arrangements of n pseudolines in the plane. This improves on previous work by Knuth who proved an upper bound of 3^\binomn2 @ 2^{0.792> n2}3^{\binom{n}{2}} \cong 2^{0.792\> n^{2}} in 1992 and the first author, who obtained 2^{0.697> n2}2^{0.697\> n^{2}} in 1997. The argument uses surprisingly little geometry. The main ingredient is a lemma that was already central to the argument given by Knuth.     

On the Number of Arrangements of Pseudolines

Given a simple arrangement of n pseudolines in the Euclidean plane, associate with line i the list σ _i of the lines crossing i in the order of the crossings on line i. is a permutation of . The vector (σ ₁ ,σ ₂ , ...,σ_n) is an encoding for the arrangement. Define if and , otherwise. Let , we show that the vector (τ ₁ , τ ₂ , ... , τ_n) is already an encoding. We use this encoding to improve the upper bound on the number of arrangements of n pseudolines to . Moreover, we have enumerated arrangements with 10 pseudolines. As a byproduct we determine their exact number and we can show that the maximal number of halving lines of 10 point in the plane is 13. Received December 20, 1995, and in revised form March 8, 1996.     

Posets and planar graphs

A k-decomposition (G₁,…,G_k) of a graph G is a partition of its edge set to form k spanning subgraphs G₁,…,G_k. The classical theorem of Nordhaus and Gaddum bounds χ(G₁) + χ(G₂) and χ(G₁)χ(G₂) over all 2-decompositions of K_n. For a graph parameter p, let p(k;G) denote the maximum of over all k-decompositions of the graph G. The clique number ω, chromatic number χ, list chromatic number χℓ, and Szekeres–Wilf number σ satisfy ω(2;K_n) = χ(2;K_n) = χℓ(2;K_n) = σ(2;K_n) = n + 1. We obtain lower and upper bounds for ω(k;K_n), χ(k;K_n), χℓ(k;K_n), and σ(k;K_n). The last three behave differently for large k. We also obtain lower and upper bounds for the maximum of χ(k;G) over all graphs embedded on a given surface. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Tolerance graphs,and orders

We show that, if a tolerance graph is the complement of a comparability graph, it is a trapezoid graph, i.e., the complement of an order of interval dimension at most 2. As consequences we are able to give obstructions for the class of bounded tolerance graphs and to give an example of a graph that is alternatingly orientable but not a tolerance graph. We also characterize the tolerance graphs among complements of trees. © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 129–140, 1998     

Investigation of the different discharge mechanisms in pseudosparkdischarges

The intention of this paper is to give an overview of recent experiments explaining the development and transition of the discharge phases in a pseudospark. The reported experiments include single gap pseudospark discharges in ultra-high-vacuum systems with hydrogen as working gas, as well as multigap pseudospark discharges in argon. Temporally and spatially resolved framing photography, spectrometry, raster electron microscopy and time resolved electrical measurements are presented. The experiments comprise a current range of some hundreds of amps to 60 kA. The results are used to specify the four characteristic phases of the pseudospark: Townsend-, hollow cathode-, high current- and metal vapor arc phase     

On the Complexity of Partial Order Properties

The recognition complexity of ordered set properties is considered in terms of how many questions must be put to an adversary to decide if an unknown partial order has the prescribed property. We prove a lower bound of order n ² for properties that are characterized by forbidden substructures of fixed size. For the properties being connected, and having exactly k comparable pairs, k n ² / 4 we show that the recognition complexity is (n\choose 2). The complexity of interval orders is exactly (n\choose 2) - 1. We further establish bounds for being a lattice, being of height k and having width k.