A Note on the First Eigenvalue of Hypersurfaces

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＭａｉｎＲｅｓｕｌｔＬｅｔＭｂｅａｎｏｒｉｅｎｔｅｄｃｌｏｓｅｄｈｙｐｅｒｓｕｒｆａｃｅｏｆＭ,ａｎ（ｎ＋１）－ｄｉｍｅｎｓｉｏｎａｌｏｒｉｅｎｔｅｄｃｌｏｓｅｄＲｉｅｍａｎｎｍａｉｎｆｏｌｄ（ｃｌｏｓｅｄｍｅａｎｓｃｏｍｐａｃｔａｎｄｗｉｔｈｏｕｔｂｏｕｎｄａｒｙ）．ＤｅｎｏｔｅｔｈｅｆｉｒｓｔｅｉｇｅｎｖａｌｕｅｏｆｔｈｅＬａｐｌａｃｉａｎｆｏｒＭｂｙλ１．ＳｕｐｐｏｓｅｔｈａｔｔｈｅＲｉｃｃｉｃｕｒｖａｔｕｒｅｏｆＭ,Ｒｉｃ（Ｍ）,ｉｓｂｏｕｎｄ…     

关于代数连通度的一个注记

图G=(V,E)的次小的拉普拉斯特征值称为G的代数连通度,记为α(G).设δ(G)为G的最小度.Fiedler早在1973年便证明了α(G)≤δ(G),但他未能给出等号成立的极图刻划.后来,我们在[6]中确定了当δ(G)≤1/2|V(G)|时α(G)=δ(G)的充要条件.本文中,我们将确定任意情况下α(G)=δ(G)成立的所有极图.     

Algorithms of Intrinsic Complexity for Point Searching in Compact Real Singular Hypersurfaces

For a real square-free multivariate polynomial F, we treat the general problem of finding real solutions of the equation F=0, provided that the real solution set {F=0}_ℝ is compact. We allow that the equation F=0 may have singular real solutions. We are going to decide whether this equation has a non-singular real solution and, if this is the case, we exhibit one for each generically smooth connected component of {F=0}_ℝ. We design a family of elimination algorithms of intrinsic complexity which solves this problem. In the worst case, the complexity of our algorithms does not exceed the already known extrinsic complexity bound of (nd) ^O(n) for the elimination problem under consideration, where n is the number of indeterminates of F and d its (positive) degree. In the case that the real variety defined by F is smooth, there already exist algorithms of intrinsic complexity that solve our problem. However, these algorithms cannot be used in case when F=0 admits F-singular real solutions.     

关于代数逼近测度的注记

本文给出了一类代数数序列的极限的代数逼近测度.     

A Note on Octic Hypersurfaces with Many Nodes

In the present note we use rank-2-bundles over to construct octic hypersurfaces with many nodes. We give an example with 128 nodes.The author gratefully acknowledges the support of the DFG priority program ‘Global methods in complex geometry’.     

Interpolation by Convex Algebraic Hypersurfaces

LetΣbe the set of vertices of a convex non-degenerate polyhedron inRⁿ,n2. We suggest an algorithm to construct smooth convex algebraic hypersurfaces of degree as small as possible, going throughΣ.     

Focal Loci of Algebraic Hypersurfaces: A General Theory

The focal locus is traditionally defined for a differentiable submanifold of R ⁿ. However, since it depends essentially only on the notion of orthogonality, a focal locus can be also associated to an algebraic subvariety of the space , once we have chosen an orthogonal structure on this space. In this paper, we establish somebasic results in the theory of focal loci of algebraichypersurfaces in . Our main results concern the irreducibility of the ramification divisor of the end-point map and the dimension of the singular locus of this divisor, the birationality of the focal map and the degree of the focal locus of an algebraic hypersurface.     

A Note on Algebraic Solutions to Identification

Algebraic methods to establish the identification of structural equation models remain a viable option. However, sometimes it is unclear whether the algebraic solution establishes identification. One example is when there is more than one way to solve for the parameter, but one way leads to a single value and a second way leads to a function with more than one value. This article proves that one explicit and unique solution is sufficient for model identification even when other explicit solutions permit more than one solution. The results are illustrated with an example. The results are useful to attempts to use algebraic means to address model identification.     

群代数作用遍历性的一点注记

证明了由自由群整数环上一类元素确定的代数作用的遍历性,计算了Heisenberg群因子中特定元素的Fuglede-Kadison行列式值.     

On the Algebraic Cohomology of Real Algebraic M-Varieties

For a real algebraic M-variety X, the canonical homomorphism of the algebraic cohomology group of the set of real points into the algebraic cohomology group of the set of complex points $$\varrho _k :H_{alg}^k (X(\mathbb{R}),\mathbb{Z}/2) \to H_{alg}^{2k} (X(\mathbb{C}),\mathbb{Z}/2)$$ is considered. This homomorphism agrees with the cycle mappings. Estimates for the dimension of the kernel of this homomorphism are given.     

An Improved Upper Complexity Bound for the Topology Computation of a Real Algebraic Plane Curve

The computation of the topological shape of a real algebraic plane curve is usually driven by the study of the behavior of the curve around its critical points (which includes also the singular points). In this paper we present a new algorithm computing the topological shape of a real algebraic plane curve whose complexity is better than the best algorithms known. This is due to the avoiding, through a sufficiently good change of coordinates, of real root computations on polynomials with coefficients in a simple real algebraic extension of to deal with the critical points of the considered curve. In fact, one of the main features of this algorithm is that its complexity is dominated by the characterization of the real roots of the discriminant of the polynomial defining the considered curve.     

Real Hypersurfaces Having Many Pseudo-hyperplanes

Abstract

Let nand dbe natural integers satisfying n ≥ 3 and d ≥ 10. Let Xbe an irreducible real hypersurface Xin ? ⁿ of degree dhaving many pseudo-hyperplanes. Suppose that Xis not a projective cone. We show that the arrangement ? of all d ? 2 pseudo-hyperplanes of Xis trivial, i.e., there is a real projective linear subspace Lof ? ⁿ (?) of dimension n ? 2 such that L ? Hfor all H ∈ ?. As a consequence, the normalization of Xis fibered over ?¹in quadrics. Both statements are in sharp contrast with the case n = 2; the first statement also shows that there is no Brusotti-type result for hypersurfaces in ? ⁿ , for n ≥ 3.     

A Note on the Canonical Divisor of Quasi-homogeneous Affine Algebraic Varieties

We give a necessary and sufficient condition for the canonical divisor to vanish on a quasi-homogeneous affine algebraic variety.     

Algebraicity of Global Real Analytic Hypersurfaces

Let X be an algebraic manifold without compact component and let V be a compact coherent analytic hypersurface in X, with finite singular set. We prove that V is diffeotopic (in X) to an algebraic hypersurface in X if and only if the homology class represented by V is algebraic and singularities are locally analytically equivalent to Nash singularities. This allows us to construct algebraic hypersurfaces in X with prescribed Nash singularities.     

Filling Real Hypersurfaces by Pseudoholomorphic Discs

We study pseudoholomorphic discs with boundaries attached to a real hypersurface E in an almost complex manifold. We give sufficient conditions for filling a one-sided neighborhood of E by the discs.      

Computational Complexity in Algebraic Systems

Computability and computational complexity were first considered over the fields of real and complex numbers and generalized to arbitrary algebraic systems. This article approaches the theory of computational complexity over an arbitrary algebraic system by taking computability over the list extension for a computational model of it. We study the resultant polynomial complexity classes and mention some NP-complete problems.     

Real Hypersurfaces in Complex Two-Plane Grassmannians

 The complex two-plane Grassmannian G ₂(C ^m+2 in equipped with both a K?hler and a quaternionic K?hler structure. By applying these two structures to the normal bundle of a real hypersurface M in G ₂(C ^m+2 one gets a one- and a three-dimensional distribution on M. We classify all real hypersurfaces M in G ₂ C ^m+2 , m≥3, for which these two distributions are invariant under the shape operator of M. Received 13 November 1996; in revised form 3 March 1997     

A Note on the k-very Ampleness of Semi-stable Bundles on an Algebraic Curve

We give a criterion for k-very ampleness of semi-stable bundles on an algebraic curve.