A Linear Programming Approach to Weak Reversibility and Linear Conjugacy of Chemical Reaction Networks

15 páginas, 2 figuras.-- The final publication is available at www.springerlink.comA numerically effective procedure for determining weakly reversible chemical reaction networks that are linearly conjugate to a known reaction network is proposed in this paper. The method is based on translating the structural and algebraic characteristics of weak reversibility to logical statements and solving the obtained set of linear (in)equalities in the framework of mixed integer linear programming. The unknowns in the problem are the reaction rate coefficients and the parameters of the linear conjugacy transformation. The efficacy of the approach is shown through numerical examples.Matthew D. Johnston and David Siegel acknowledge the support of D. Siegel’s Natural Sciences and Engineering Research Council of Canada Discovery Grant. Gàbor Szederkényi acknowledges the support of the Hungarian National Research Fund through grant no. OTKA K-83440 as well as the support of project CAFE (Computer Aided Process for Food Engineering) FP7-KBBE-2007-1 (Grant no: 212754).Peer reviewe

Uniform semi-Latin squares and their pairwise-variance aberrations

For integers and , an semi-Latin square is an array of -subsets (called blocks) of an -set (of treatments), such that each treatment occurs once in each row and once in each column of the array. A semi-Latin square is uniform if every pair of blocks, not in the same row or column, intersect in the same positive number of treatments. It is known that a uniform semi-Latin square is Schur optimal in the class of all semi-Latin squares, and here we show that when a uniform semi-Latin square exists, the Schur optimal semi-Latin squares are precisely the uniform ones. We then compare uniform semi-Latin squares using the criterion of pairwise-variance (PV) aberration, introduced by J. P. Morgan for affine resolvable designs, and determine the uniform semi-Latin squares with minimum PV aberration when there exist mutually orthogonal Latin squares of order . These do not exist when , and the smallest uniform semi-Latin squares in this case have size . We present a complete classification of the uniform semi-Latin squares, and display the one with least PV aberration. We give a construction producing a uniform semi-Latin square when there exist mutually orthogonal Latin squares of order , and determine the PV aberration of such a uniform semi-Latin square. Finally, we describe how certain affine resolvable designs and balanced incomplete-block designs can be constructed from uniform semi-Latin squares. From the uniform semi-Latin squares we classified, we obtain (up to block design isomorphism) exactly 16875 affine resolvable designs for 72 treatments in 36 blocks of size 12 and 8615 balanced incomplete-block designs for 36 treatments in 84 blocks of size 6. In particular, this shows that there are at least 16875 pairwise non-isomorphic orthogonal arrays