Let M be a connected oriented surface and let G'2 be the Grassmannian of oriented 2-planes in Euclidean (2 + c)-space. E2 + l. Smooth maps t: M -» (7f are studied to determine whether or not they are Gauss maps. Both local and global results are obtained. If í is a Gauss map of an immersion X: M -» E2 + 1, we study the extent to which / uniquely determines X under certain circumstances. Let X: ...

where we view Gm(S) as the subgroup of invertible scalars in GLn(S). Define PGLn = s ◦ P̃GLn to be its sheafification (taken in the sense of Problem 8 on PS 2). Our goal will be to show that PGLn is an affine group scheme that represents automorphisms of projective space in the sense that PGLn(S) ' AutS(S × Pn−1). Before we begin, we require some preliminaries about Zariski sheaves. Let F : Sch→...

A new method is presented for identifying and comparing closed, bounded, free-form curves that are defined by even implicit polynomial (IP) equations in the Cartesian coordinates x and y. The method provides a new expression for an IP involving a product of conic factors with unique conic factor centers. The critical points for an IP curve also are defined. The conic factor centers and the crit...

This paper is devoted to studying boundedness criteria for extended-real-valued functions. The study is done in the framework of dual vector spaces, using new objects such as self-equilibrated sets and functions. We establish two boundedness new major criteria. The first one says that an extended-real-valued function is bounded below provided it is minorized by an affine map on one of its self-...

A preconditioning procedure is developed for the L2 and more general optimal transport problems. The procedure is based on a family of affine map pairs which transforms the original measures into two new measures that are closer to each other while preserving the optimality of solutions. It is proved that the preconditioning procedure minimizes the remaining transportation cost among all admiss...

We extend the knowledge compilation map introduced by Darwiche and Marquis with three influential propositional fragments, the Krom CNF one (also known as the bijunctive fragment), the Horn CNF fragment and the affine fragment (also known as the biconditional fragment) as well as seven additional languages based on them, and composed respectively of Krom or Horn CNF formulas, renamable Horn CNF...

Let D be an affine difference set of order n in an abelian group G relative to a subgroup N . Set H̃ = H \ {1, ω}, where H = G/N and ω = ∏ σ∈H σ. Using D we define a two-to-one map g from H̃ to N . The map g satisfies g(σ) = g(σ) and g(σ) = g(σ−1) for any multiplier m of D and any element σ ∈ H̃. As applications, we present some results which give a restriction on the possible order n and the grou...

Let G be an affine algebraic group over an algebraically closed field k of characteristic zero. In this paper, we consider finite G-equivariant morphisms F : X → Y of irreducible affine varieties. First we determine under which conditions on Y the induced map FG : X//G → Y//G of quotient varieties is also finite. This result is reformulated in terms of kernels of derivations on k-algebras A ⊂ B...

In this paper we focus on the identification of discrete-time hybrid systems in the Mixed-Logic Dynamical (MLD)/Piece-Wise Affine (PWA) form. This problem can be formulated as the reconstruction of a possibly discontinuous PWA map with a multi-dimensional domain. In order to achieve our goal, we propose an algorithm that exploits the combined use of clustering, linear identification, and classi...

A preconditioning procedure is developed for the L2 and more general optimal 3 transport problems. The procedure is based on a family of affine map pairs which transforms the 4 original measures into two new measures that are closer to each other, while preserving the optimality 5 of solutions. It is proved that the preconditioning procedure minimizes the remaining transportation 6 cost among a...