Book Review: Differential Equations from the Algebraic Standpoint

Algebraic Differential Equations from Covering Maps

Let Y be a complex algebraic variety, G y Y an action of an algebraic group on Y , U ⊆ Y (C) a complex submanifold, Γ < G(C) a discrete, Zariski dense subgroup of G(C) which preserves U , and π : U → X(C) an analytic covering map of the complex algebraic variety X expressing X(C) as Γ\U . We note that the theory of elimination of imaginaries in differentially closed fields produces a generalize...

Nonsmooth Differential-algebraic Equations

A nonsmooth modeling paradigm for dynamic simulation and optimization of process operations is advocated. Nonsmooth differential-algebraic equations (DAEs) naturally model a wide range of physical systems encountered in chemical engineering conventionally viewed as exhibiting hybrid continuous/discrete behavior. Due to recent advancements in nonsmooth analysis, nonsmooth DAEs now have a suitabl...

Switched differential algebraic equations

In this chapter an electrical circuit with switches is modeled as a switched differential algebraic equation (switched DAE), i.e. each mode is described by a DAE of the form Ex′ = Ax+Bu where E is, in general, a singular matrix and u is the input. The resulting time-variance follows from the action of the switches present in the circuit, but can also be induced by faults occurring in the circui...

Index reduction of differential algebraic equations by differential algebraic elimination

High index differential algebraic equations (DAEs) are ordinary differential equations (ODEs) with constraints and arise frequently from many mathematical models of physical phenomenons and engineering fields. In this paper, we generalize the idea of differential elimination with Dixon resultant to polynomially nonlinear DAEs. We propose a new algorithm for index reduction of DAEs and establish...

Self-adjoint differential-algebraic equations