A Generic Framework for Engineering Graph Canonization Algorithms

The state-of-the-art tools for practical graph canonization are all based on the individualization-refinement paradigm, and their difference is primarily in the choice of heuristics they include and in the actual tool implementation. It is thus not possible to make a direct comparison of how individual algorithmic ideas affect the performance on different graph classes. We present an algorithmic software framework that facilitates implementation of heuristics as independent extensions to a common core algorithm. It therefore becomes easy to perform a detailed comparison of the performance and behaviour of different algorithmic ideas. Implementations are provided of a range of algorithms for tree traversal, target cell selection, and node invariant, including choices from the literature and new variations. The framework readily supports extraction and visualization of detailed data from separate algorithm executions for subsequent analysis and development of new heuristics. Using collections of different graph classes we investigate the effect of varying the selections of heuristics, often revealing exactly which individual algorithmic choice is responsible for particularly good or bad performance. On several benchmark collections, including a newly proposed class of difficult instances, we additionally find that our implementation performs better than the current state-of-the-art tools

N=5 three-algebras and 5-graded Lie superalgebras

We discuss a generalization of N=6 three-algebras to N=5 three-algebras in connection to anti-Lie triple systems and basic Lie superalgebras of type II. We then show that the structure constants defined in anti-Lie triple systems agree with those of N=5 superconformal theories in three dimensions.Comment: 15 pages, v3: typos corrected, references added, and published versio

A realization of the Lie algebra associated to a Kantor triple system

We present a nonlinear realization of the 5-graded Lie algebra associated to a Kantor triple system. Any simple Lie algebra can be realized in this way, starting from an arbitrary 5-grading. In particular, we get a unified realization of the exceptional Lie algebras f_4, e_6, e_7, e_8, in which they are respectively related to the division algebras R, C, H, O.Comment: 11 page

Inferring Chemical Reaction Patterns Using Rule Composition in Graph Grammars

Modeling molecules as undirected graphs and chemical reactions as graph rewriting operations is a natural and convenient approach tom odeling chemistry. Graph grammar rules are most naturally employed to model elementary reactions like merging, splitting, and isomerisation of molecules. It is often convenient, in particular in the analysis of larger systems, to summarize several subsequent reactions into a single composite chemical reaction. We use a generic approach for composing graph grammar rules to define a chemically useful rule compositions. We iteratively apply these rule compositions to elementary transformations in order to automatically infer complex transformation patterns. This is useful for instance to understand the net effect of complex catalytic cycles such as the Formose reaction. The automatically inferred graph grammar rule is a generic representative that also covers the overall reaction pattern of the Formose cycle, namely two carbonyl groups that can react with a bound glycolaldehyde to a second glycolaldehyde. Rule composition also can be used to study polymerization reactions as well as more complicated iterative reaction schemes. Terpenes and the polyketides, for instance, form two naturally occurring classes of compounds of utmost pharmaceutical interest that can be understood as "generalized polymers" consisting of five-carbon (isoprene) and two-carbon units, respectively

Study of compensation process of ion beams

For investigation of space charge compensation process due to residual gas ionization and the experimentally study of the rise of compensation, a Low Energy Beam Transport (LEBT) system consisting of an ion source, two solenoids, a decompensation electrode to generate a pulsed decompensated ion beam and a diagnostic section was set up. The potentials at the beam axis and the beam edge were ascertained from time resolved measurements by a residual gas ion energy analyzer. A numerical simulation of self-consistent equilibrium states of the beam plasma has been developed to determine plasma parameters which are difficult to measure directly. The temporal development of the kinetic and potential energy of the compensation electrons has been analyzed by using the numerically gained results of the simulation. To investigate the compensation process the distribution and the losses of the compensation electrons were studied as a function of time. The acquired data show that the theoretical estimated rise time of space charge compensation neglecting electron losses is shorter than the build up time determined experimentally. To describe the process of space charge compensation an interpretation of the achieved results is given

Generic Strategies for Chemical Space Exploration

Computational approaches to exploring "chemical universes", i.e., very large sets, potentially infinite sets of compounds that can be constructed by a prescribed collection of reaction mechanisms, in practice suffer from a combinatorial explosion. It quickly becomes impossible to test, for all pairs of compounds in a rapidly growing network, whether they can react with each other. More sophisticated and efficient strategies are therefore required to construct very large chemical reaction networks. Undirected labeled graphs and graph rewriting are natural models of chemical compounds and chemical reactions. Borrowing the idea of partial evaluation from functional programming, we introduce partial applications of rewrite rules. Binding substrate to rules increases the number of rules but drastically prunes the substrate sets to which it might match, resulting in dramatically reduced resource requirements. At the same time, exploration strategies can be guided, e.g. based on restrictions on the product molecules to avoid the explicit enumeration of very unlikely compounds. To this end we introduce here a generic framework for the specification of exploration strategies in graph-rewriting systems. Using key examples of complex chemical networks from sugar chemistry and the realm of metabolic networks we demonstrate the feasibility of a high-level strategy framework. The ideas presented here can not only be used for a strategy-based chemical space exploration that has close correspondence of experimental results, but are much more general. In particular, the framework can be used to emulate higher-level transformation models such as illustrated in a small puzzle game