Georg Regensburger

Solving and factoring boundary problems for linear ordinary differential equations in differential algebras

We present a new approach for expressing and solving boundary problems for linear ordinary differential equations in the language of differential algebras. Starting from an algebra with a derivation and integration operator, we construct an algebra of linear integro-differential operators that is expressive enough for specifying regular boundary problems with arbitrary Stieltjes boundary conditions as well as their solution operators. On the basis of these structures, we define a new multiplication on regular boundary problems in such a way that the resulting Greens operator is the reverse composition of the constituent Greens operators. We provide also a method for lifting any factorization of the underlying differential operator to the level of boundary problems. Since this method only needs the computation of initial value problems, it can be used as an effective alternative for computing Greens operators in the case where one knows how to factor the given differential operators.

Sign Conditions for Injectivity of Generalized Polynomial Maps with Applications to Chemical Reaction Networks and Real Algebraic Geometry

We give necessary and sufficient conditions in terms of sign vectors for the injectivity of families of polynomial maps with arbitrary real exponents defined on the positive orthant. Our work relates and extends existing injectivity conditions expressed in terms of Jacobian matrices and determinants. In the context of chemical reaction networks with power-law kinetics, our results can be used to preclude as well as to guarantee multiple positive steady states. In the context of real algebraic geometry, our work recognizes a prior result of Craciun, Garcia-Puente, and Sottile, together with work of two of the authors, as the first partial multivariate generalization of the classical Descartes’ rule, which bounds the number of positive real roots of a univariate real polynomial in terms of the number of sign variations of its coefficients.

Generalized Mass Action Systems: Complex Balancing Equilibria and Sign Vectors of the Stoichiometric and Kinetic-Order Subspaces

Mass action systems capture chemical reaction networks in homogeneous and dilute solutions. We suggest a notion of generalized mass action systems that admits arbitrary power-law rate functions and serves as a more realistic model for reaction networks in intracellular environments. In addition to the complexes of a network and the related stoichiometric subspace, we introduce corresponding kinetic complexes, which represent the exponents in the rate functions and determine the kinetic-order subspace. We show that several results of chemical reaction network theory carry over to the case of generalized mass action kinetics. Our main result essentially states that if the sign vectors of the stoichiometric and kinetic-order subspace coincide, there exists a unique complex balancing equilibrium in every stoichiometric compatibility class. However, in contrast to classical mass action systems, multiple complex balancing equilibria in one stoichiometric compatibility class are possible in general.

Enzyme allocation problems in kinetic metabolic networks: optimal solutions are elementary flux modes.

The survival and proliferation of cells and organisms require a highly coordinated allocation of cellular resources to ensure the efficient synthesis of cellular components. In particular, the total enzymatic capacity for cellular metabolism is limited by finite resources that are shared between all enzymes, such as cytosolic space, energy expenditure for amino-acid synthesis, or micro-nutrients. While extensive work has been done to study constrained optimization problems based only on stoichiometric information, mathematical results that characterize the optimal flux in kinetic metabolic networks are still scarce. Here, we study constrained enzyme allocation problems with general kinetics, using the theory of oriented matroids. We give a rigorous proof for the fact that optimal solutions of the non-linear optimization problem are elementary flux modes. This finding has significant consequences for our understanding of optimality in metabolic networks as well as for the identification of metabolic switches and the computation of optimal flux distributions in kinetic metabolic networks.

On integro-differential algebras

Abstract The concept of integro-differential algebra has been introduced recently in the study of boundary problems of differential equations. We generalize this concept to that of integro-differential algebra with a weight, in analogy to the differential Rota–Baxter algebra. We construct free commutative integro-differential algebras with weight generated by a differential algebra. This gives in particular an explicit construction of the integro-differential algebra on one generator. Properties of the free objects are studied.

Integro-differential polynomials and operators

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

A skew polynomial approach to integro-differential operators

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

A Symbolic Framework for Operations on Linear Boundary Problems

We describe a symbolic framework for treating linear boundary problems with a generic implementation in the Theorema system. For ordinary differential equations, the operations implemented include computing Greens operators, composing boundary problems and integro-differential operators, and factoring boundary problems. Based on our factorization approach, we also present some first steps for symbolically computing Greens operators of simple boundary problems for partial differential equations with constant coefficients. After summarizing the theoretical background on abstract boundary problems, we outline an algebraic structure for partial integro-differential operators. Finally, we describe the implementation in Theorema, which relies on functors for building up the computational domains, and we illustrate it with some sample computations including the unbounded wave equation.

Parametrizing compactly supported orthonormal wavelets by discrete moments

We discuss parametrizations of filter coefficients of scaling functions and compactly supported orthonormal wavelets with several vanishing moments. We introduce the first discrete moments of the filter coefficients as parameters. The discrete moments can be expressed in terms of the continuous moments of the related scaling function. To solve the resulting polynomial equations we use symbolic computation and in particular Gröbner bases. The cases of four to ten filter coefficients are discussed and explicit parametrizations are given.

Regular and singular boundary problems in maple

We describe a new Maple package for treating boundary problems for linear ordinary differential equations, allowing two-/multipoint as well as Stieltjes boundary conditions. For expressing differential operators, boundary conditions, and Greens operators, we employ the algebra of integro-differential operators. The operations implemented for regular boundary problems include computing Greens operators as well as composing and factoring boundary problems. Our symbolic approach to singular boundary problems is new; it provides algorithms for computing compatibility conditions and generalized Greens operators.