Mass action systems capture chemical reaction networks in homogeneous and dilute solutions. We suggest a notion of generalized mass that admits arbitrary power-law rate functions serves as more realistic model for intracellular environments. In addition to the complexes network related stoichiometric subspace, we introduce corresponding kinetic complexes, which represent exponents determine kinetic-order subspace. show several results theory carry over case kinetics. Our main result...

Background: The optimization of metabolic rates (as linear objective functions) represents the methodical core flux-balance analysis techniques which have become a standard tool for study genome-scale models. Besides (growth and synthesis) rates, yields are key parameters characterization biochemical transformation processes, especially in context biotechnological applications. However, ratios hence nonlinear under arbitrary constraints is not possible with current techniques. Despite...

Designing mechanical devices, called linkages, that draw a given plane curve has been topic interested engineers and mathematicians for hundreds of years, recently also computer scientists. Already in 1876, Kempe proposed procedure solving the problem full generality, but his constructions tend to be extremely complicated. We provide novel algorithm produces much simpler works only parametric curves. Our approach is transform into factorization task over some noncommutative algebra. show how...

A fundamental result in metabolic pathway analysis states that every flux mode can be decomposed into a sum of elementary modes. However, only decomposition without cancelations is biochemically meaningful, since reversible reaction cannot have different directions the contributing This essential requirement has been largely overlooked by community. Indeed, modes cancelations. The an immediate consequence theorem Rockafellar which element linear subspace conformal (a cancelations) vectors...

In this paper we develop a symbolic technique to obtain asymptotic expressions for ruin probabilities and discounted penalty functions in renewal insurance risk models when the premium income depends on present surplus of portfolio. The analysis is based boundary problems linear ordinary differential equations with variable coefficients. algebraic structure Green's operators allows us an intuitive way tackling behavior solutions, leading exponential-type expansions Cram\'er-type asymptotics....

Chemical reaction networks with generalized mass-action kinetics lead to power-law dynamical systems. As a simple example, we consider the Lotka reactions and resulting planar ODE. We characterize parameters (positive coefficients real exponents) for which unique positive equilibrium is center.

We propose two algebraic structures for treating integral operators in conjunction with derivations: The algebra of integro-differential polynomials describes nonlinear and differential together initial values. can be used to solve boundary problems linear ordinary equations. In both cases, we describe canonical/normal forms algorithmic simplifiers.

We construct the algebra of integro-differential operators over an ordinary directly in terms normal forms. In case polynomial coefficients, we use skew polynomials for defining Weyl as a natural extension classical one variable. Its forms, algebraic properties and its relation to localization differential are studied. Fixing integration constant, regain with coefficients.

We start from a parametrized system of $d$ generalized polynomial equations (with real exponents) for positive variables, involving $n$ monomials with parameters. Existence and uniqueness solution all parameters right-hand sides is equivalent to the bijectivity (every element of) family polynomial/exponential maps. characterize exponential maps in terms two linear subspaces arising coefficient exponent matrices, respectively. In particular, we obtain conditions sign vectors nondegeneracy...

It is well known that, for mass-action systems, complex-balanced equilibria are asymptotically stable. For generalized even if there exists a unique equilibrium (in every stoichiometric class and all rate constants), it need not be We first discuss several notions of matrix stability (on linear subspace) such as D-stability diagonal stability, then we apply our abstract results to systems. In particular, show that the subspace constants) implies uniqueness. cyclic networks, characterize...

Elementary flux mode (EFM) analysis allows an unbiased description of metabolic networks in terms minimal pathways (involving a set reactions). To date, the enumeration EFMs is impracticable genome-scale models. In complementary approach, we introduce concept tope (FT), involving maximal reactions (with fixed directions), which one to study coordination reaction directions and opens new way for EFM enumeration.