完备α度量的相对抛物型仿射球(英文)

摘要:设y:M→Rⁿ⁺¹是一个光滑连通流形到实仿射空间Rⁿ⁺¹的局部强凸浸入超曲面,而且是一个定义在区域Ω（?）Rⁿ上的严格凸函数x_n+1=f（x₁,x₂,…,x_n）的图.在α相对法化下,相对抛物型仿射球满足一个四阶非线性偏微分方程组.本文证明了这类抛物型仿射球的一个新的Bernstein性质.     

Affine Algebraic Varieties Relative to an Algebraic Theory

Affine algebraic varieties relative to an algebraic theory are introduced and described as irreducible components of affine algebraic sets. Their category is shown to be dually equivalent to the category of irreducible functional algebras.     

The Relative Brauer Group of an Affine Double Plane

We study algebra classes and divisor classes on a normal affine surface of the form z ² = f(x, y). The affine coordinate ring is T = k[x, y, z]/(z ² ? f), and if R = k[x, y][f ^?1] and S = R[z]/(z ² ? f), then S is a quadratic Galois extension of R. If the Galois group is G, we show that the natural map H¹(G, Cl(T)) → H¹(G, Pic(S)) factors through the relative Brauer group B(S/R) and that all of the maps are onto. Sufficient conditions are given for H¹(G, Cl(T)) to be isomorphic to B(S/R). The groups and maps are computed for several examples.     

Affine Surfaces which are Both Affine Harmonic and Affine Maximal

We study affine surfaces which are both affine maximal and affine harmonic. We prove that an indefinite surface satisfying both conditions is affine equivalent to an open part $(u,{1\over 2}u^2,P_1(u)+\upsilon,P_2(u)+{1\over 2}\upsilon^2)$ , where P₁ and P₂ are arbitrary functions of one variable.     

Affine Quasi-Heredity of Affine Schur Algebras

In this paper we prove that the affine Schur algebra \(\widehat {S}(n,r)\) is affine quasi-hereditary. This result is then used to show that the centralizer subquotient algebras of \(\widehat {S}(n,r)\) are Laurent polynomial algebras. Moreover, we give a parameter set of simple \(\widehat {S}(n,r)\)-modules and identify this parameter set with that given in [7]. In the Appendix, the affine quasi-heredity of affine quantum Schur algebras is studied.     

Representations of Affine Multifunctions by Affine Selections

The paper deals with affine selections of affine (both convex and concave) multifunctions acting between finite-dimensional real normed spaces. It is proved that each affine multifunction with compact values possesses an exhaustive family of affine selections and, consequently, can be represented by its affine selections. Moreover, a convex multifunction with compact values possesses an exhaustive family of affine selections if and only if it is affine. Thus the existence of an exhaustive family of affine selections is the characteristic feature of affine multifunctions which differs them from other convex multifunctions with compact values. Besides a necessary and sufficient condition for a concave multifunction to be affine on a given convex subset is also proved. Finally it is shown that each affine multifunction with compact values can be represented as the closed convex hull of its exposed affine selections and as the convex hull of its extreme affine selections. These statements extend the Straszewicz theorem and the Krein–Milman theorem to affine multifunctions. Dedicated to Boris Mordukhovich in honour of his 60th birthday.     

Vector Hulls of Affine Spaces and Affine Bundles

Every affine space A can be canonically immersed as a hyperplane into a vector space  , which is called the vector hull of A. This immersion satisfies a universal property for affine functions defined on A. In the same way, every affine map between affine spaces has a linear prolongation to their vector hulls. Though not much known, this construction is greatly clarifying, both for affine geometry and for its applications. The goal of this paper is to perform a thorough study of the vector hull functor and to describe its counterpart in the framework of affine bundles. With this respect, it is shown that the vector hull of some interesting affine bundles, and more specifically some jet bundles, can be identified with certain vector bundles.      

Euclidean Complete Affine Surfaces with Constant Affine Mean Curvature

The purpose of this paper is to prove that alocally strongly convex, Euclidean complete surface with constantaffine mean curvature is also affine complete. Consequently weobtain a classification of locally strongly convex, Euclideancomplete surfaces with constant affine mean curvature.     

Affine Frames of Multivariate Box Splines and their Affine Duals

We give a simple and explicit construction of primal and dual wavelet filters based on refinable multivariate splines (with respect to dilation matrices M) such that the corresponding wavelet functions generate dual affine frames of arbitrarily high regularity. Furthermore, the number of wavelets does not depend on the regularity. We apply the method also to generalized B-splines.     

Affine Klingenbergsche Strukturen

In this paper we define the affine Klingenberg structures (AK-structures) as a generalization of the affine Klingenberg planes (AK-planes) [2]. By means of moduls over the local rings (which need not be free as with the coordinate AK-planes is the case) there is constructed a class of the coordinate AK-structures. In II and III we define the Desarguesian AK-structures. Any coordinate AK-structure is a Desarguesian AK-structure. With the methods established by E.Artin and applied by W.Klingenberg in [4] we show that any Desarguesian AK-structure is isomorphic with the coordinate AK-structure. Some of the results obtained have been applied to the theory of the AK-planes. Thus, for example, it has been shown that it is possible to assume less with the definition of the Desarguesian AK-planes than with [4] or even with [5].     

Affine cartesian codes

We compute the basic parameters (dimension, length, minimum distance) of affine evaluation codes defined on a cartesian product of finite sets. Given a sequence of positive integers, we construct an evaluation code, over a degenerate torus, with prescribed parameters of a certain type. As an application of our results, we recover the formulas for the minimum distance of various families of evaluation codes.