326 research outputs found
On functional module detection in metabolic networks
Functional modules of metabolic networks are essential for understanding the metabolism of an organism as a whole. With the vast amount of experimental data and the construction of complex and large-scale, often genome-wide, models, the computer-aided identification of functional modules becomes more and more important. Since steady states play a key role in biology, many methods have been developed in that context, for example, elementary flux modes, extreme pathways, transition invariants and place invariants. Metabolic networks can be studied also from the point of view of graph theory, and algorithms for graph decomposition have been applied for the identification of functional modules. A prominent and currently intensively discussed field of methods in graph theory addresses the Q-modularity. In this paper, we recall known concepts of module detection based on the steady-state assumption, focusing on transition-invariants (elementary modes) and their computation as minimal solutions of systems of Diophantine equations. We present the Fourier-Motzkin algorithm in detail. Afterwards, we introduce the Q-modularity as an example for a useful non-steady-state method and its application to metabolic networks. To illustrate and discuss the concepts of invariants and Q-modularity, we apply a part of the central carbon metabolism in potato tubers (Solanum tuberosum) as running example. The intention of the paper is to give a compact presentation of known steady-state concepts from a graph-theoretical viewpoint in the context of network decomposition and reduction and to introduce the application of Q-modularity to metabolic Petri net models
Unification in the Datastructure Multisets
In a forthcoming paper A. Herold and J. Siekmann generalize "pure" AC-unification ([ST 75], [LS 76]) to terms containing additional function symbols (see also [ST 81], [FA 84]). Generalized AC-unification thus attains practical relevance for a broad range of applications. Pure AC-unification is used as a basic mechanism and it is this key role that has motivated our research.
We have improved upon earlier approaches by basing (pure) AC-unification on a firm theoretical basis and presenting algorithms which fully exploit the properties of the underlying mathematical structure.
In particular, the high degree of parallelism for AC-unification will become apparent. Our algorithms have been designed for parallel hardware but still yield significant improvements over earlier algorithms when used in sequential mode
A Euclid style algorithm for MacMahon's partition analysis
Solutions to a linear Diophantine system, or lattice points in a rational
convex polytope, are important concepts in algebraic combinatorics and
computational geometry. The enumeration problem is fundamental and has been
well studied, because it has many applications in various fields of
mathematics. In algebraic combinatorics, MacMahon's partition analysis has
become a general approach for linear Diophantine system related problems. Many
algorithms have been developed, but "bottlenecks" always arise when dealing
with complex problems. While in computational geometry, Barvinok's important
result asserts the existence of a polynomial time algorithm when the dimension
is fixed. However, the implementation by the LattE package of De Loera et. al.
does not perform well in many situations. By combining excellent ideas in the
two fields, we generalize Barvinok's result by giving a polynomial time
algorithm for MacMahon's partition analysis in a suitable condition. We also
present an elementary Euclid style algorithm, which might not be polynomial but
is easy to implement and performs well. As applications, we contribute the
generating series for magic squares of order 6.Comment: 28 pages, some modification, add the link to the Maple package
CTEucli
Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving
We derive efficient algorithms for coarse approximation of algebraic
hypersurfaces, useful for estimating the distance between an input polynomial
zero set and a given query point. Our methods work best on sparse polynomials
of high degree (in any number of variables) but are nevertheless completely
general. The underlying ideas, which we take the time to describe in an
elementary way, come from tropical geometry. We thus reduce a hard algebraic
problem to high-precision linear optimization, proving new upper and lower
complexity estimates along the way.Comment: 15 pages, 9 figures. Submitted to a conference proceeding
Algorithms in algebraic number theory
In this paper we discuss the basic problems of algorithmic algebraic number
theory. The emphasis is on aspects that are of interest from a purely
mathematical point of view, and practical issues are largely disregarded. We
describe what has been done and, more importantly, what remains to be done in
the area. We hope to show that the study of algorithms not only increases our
understanding of algebraic number fields but also stimulates our curiosity
about them. The discussion is concentrated of three topics: the determination
of Galois groups, the determination of the ring of integers of an algebraic
number field, and the computation of the group of units and the class group of
that ring of integers.Comment: 34 page
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