Locally nilpotent derivations of affine domains
Any locally nilpotent derivation of an affine domain is equivalent to a restriction of a Jacobian type derivation of a polynomial ring.
On weighted projective planes and their affine rulings
(Here, and throughout this paper, all algebraic varieties are over an algebraically closed field k of characteristic zero.) As we will see in Section 1, the weighted projective planes P(a; b; ) satisfy (‡). Paper [2] investigates the problem of finding all affine rulings of a given surface X satisfying (†). In particular, it shows that if X satisfies (†) then the problem reduces to that of describing a certain set T0(X) of triples (m; T1; T2), where m is a positive integer and each Ti is a 2 hi matrix with entries in N (0 hi 2). The aim of the present paper is to give an explicit description of the set T0(X) in the case where X is a weighted projective plane; this is achieved by Corollary 7.1 and Propositions 7.3, 7.4 and 7.7. Thus [2] and this paper solve the above mentioned problem for weighted projective planes. Let us also point out the following characterization of weighted projective planes, which we prove in the form of Corollary 6.12, below (see 1.19 for the notion of resolution graph of a normal surface):
Inhomogeneous Boundary Value Problems for Compressible Navier-Stokes Equations: Well-Posedness and Sensitivity Analysis
In this paper compressible, stationary Navier–Stokes equations are considered. A framework for analysis of such equations is established. In particular, the well-posedness for inhomogeneous boundary value problems of elliptic-hyperbolic type is shown. Analysis is performed for small perturbations of the so-called approximate solutions that take form (1.12). The approximate solutions are determined from Stokes problem (1.11). The small perturbations are given by solutions to (1.13). The uniqueness of solutions for problem (1.13) is proved, and, in addition, the differentiability of solutions with respect to the coefficients of differential operators is shown. The results on the well-posedness of nonlinear problems are interesting on their own and are used to obtain the shape differentiability of the drag functional for incompressible Navier–Stokes equations. The shape gradient of the drag functional is derived in the classical and useful for computations form; an appropriate adjoint state is introduced to ...