21,439 research outputs found

    Structural characterization of decomposition in rate-insensitive stochastic Petri nets

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    This paper focuses on stochastic Petri nets that have an equilibrium distribution that is a product form over the number of tokens at the places. We formulate a decomposition result for the class of nets that have a product form solution irrespective of the values of the transition rates. These nets where algebraically characterized by Haddad et al.~as SΠ2S\Pi^2 nets. By providing an intuitive interpretation of this algebraical characterization, and associating state machines to sets of TT-invariants, we obtain a one-to-one correspondence between the marking of the original places and the places of the added state machines. This enables us to show that the subclass of stochastic Petri nets under study can be decomposed into subnets that are identified by sets of its TT-invariants

    On functional module detection in metabolic networks

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    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

    Invariant Generation for Multi-Path Loops with Polynomial Assignments

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    Program analysis requires the generation of program properties expressing conditions to hold at intermediate program locations. When it comes to programs with loops, these properties are typically expressed as loop invariants. In this paper we study a class of multi-path program loops with numeric variables, in particular nested loops with conditionals, where assignments to program variables are polynomial expressions over program variables. We call this class of loops extended P-solvable and introduce an algorithm for generating all polynomial invariants of such loops. By an iterative procedure employing Gr\"obner basis computation, our approach computes the polynomial ideal of the polynomial invariants of each program path and combines these ideals sequentially until a fixed point is reached. This fixed point represents the polynomial ideal of all polynomial invariants of the given extended P-solvable loop. We prove termination of our method and show that the maximal number of iterations for reaching the fixed point depends linearly on the number of program variables and the number of inner loops. In particular, for a loop with m program variables and r conditional branches we prove an upper bound of m*r iterations. We implemented our approach in the Aligator software package. Furthermore, we evaluated it on 18 programs with polynomial arithmetic and compared it to existing methods in invariant generation. The results show the efficiency of our approach

    Rational invariants of even ternary forms under the orthogonal group

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    In this article we determine a generating set of rational invariants of minimal cardinality for the action of the orthogonal group O3\mathrm{O}_3 on the space R[x,y,z]2d\mathbb{R}[x,y,z]_{2d} of ternary forms of even degree 2d2d. The construction relies on two key ingredients: On one hand, the Slice Lemma allows us to reduce the problem to dermining the invariants for the action on a subspace of the finite subgroup B3\mathrm{B}_3 of signed permutations. On the other hand, our construction relies in a fundamental way on specific bases of harmonic polynomials. These bases provide maps with prescribed B3\mathrm{B}_3-equivariance properties. Our explicit construction of these bases should be relevant well beyond the scope of this paper. The expression of the B3\mathrm{B}_3-invariants can then be given in a compact form as the composition of two equivariant maps. Instead of providing (cumbersome) explicit expressions for the O3\mathrm{O}_3-invariants, we provide efficient algorithms for their evaluation and rewriting. We also use the constructed B3\mathrm{B}_3-invariants to determine the O3\mathrm{O}_3-orbit locus and provide an algorithm for the inverse problem of finding an element in R[x,y,z]2d\mathbb{R}[x,y,z]_{2d} with prescribed values for its invariants. These are the computational issues relevant in brain imaging.Comment: v3 Changes: Reworked presentation of Neuroimaging application, refinement of Definition 3.1. To appear in "Foundations of Computational Mathematics

    Higher dimensional Automorphic Lie Algebras

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    The paper presents the complete classification of Automorphic Lie Algebras based on sln(C)\mathfrak{sl}_n (\mathbb{C}), where the symmetry group GG is finite and the orbit is any of the exceptional GG-orbits in C\overline{\mathbb{C}}. A key feature of the classification is the study of the algebras in the context of classical invariant theory. This provides on one hand a powerful tool from the computational point of view, on the other it opens new questions from an algebraic perspective, which suggest further applications of these algebras, beyond the context of integrable systems. In particular, the research shows that Automorphic Lie Algebras associated to the TOY\mathbb{T}\mathbb{O}\mathbb{Y} groups (tetrahedral, octahedral and icosahedral groups) depend on the group through the automorphic functions only, thus they are group independent as Lie algebras. This can be established by defining a Chevalley normal form for these algebras, generalising this classical notion to the case of Lie algebras over a polynomial ring.Comment: 43 pages, standard LaTeX2
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