F-Thresholds versus a-Invariants for Standard Graded Toric Rings

In this paper, we establish inequalities between F-pure thresholds or F-thresholds and classical a-invariants. Moreover, we characterize Gorenstein toric rings in terms of F-pure thresholds and a-invariants.     

Anisotropic isoparametric hypersurfaces in Euclidean spaces

In this paper, by Nomizu’s method and some technical treatment of the asymmetry of the F-Weingarten operator, we obtain a classification of complete anisotropic isoparametric hypersurfaces, i.e., hypersurfaces with constant anisotropic principal curvatures, in Euclidean spaces, which is a generalization of the classical case for isoparametric hypersurfaces in Euclidean spaces. On the other hand, by an example of local anisotropic isoparametric surface constructed by B. Palmer, we find that in general anisotropic isoparametric hypersurfaces have both local and global aspects as in the theory of proper Dupin hypersurfaces, which differs from classical isoparametric hypersurfaces.     

Binomial arithmetical rank of lattice ideals

 In this paper, we will show that a lattice ideal is a complete intersection if and only if its binomial arithmetical rank equals its height, if the characteristic of the base field k is zero. And we will give the condition that a binomial ideal equals a lattice ideal up to radical in the case of char k=0. Further, we will study the upper bound of the binomial arithmetical rank of lattice ideals and give a sharp bound for the lattice ideals of codimension two. Received: 12 June 2001 / Revised version: 22 July 2002     

Integral curvature estimates for F-stable hypersurfaces

We consider immersed hypersurfaces in euclidean which are stable with respect to an elliptic parametric functional with integrand F = F(N) depending on normal directions only. We prove an integral curvature estimate provided that F is sufficiently close to the area integrand, extending the classical estimate of Schoen, Simon and Yau [19] for stable minimal hypersurfaces in , as well as the pointwise estimate of Simon [22] for F-minimizing hypersurfaces. As a crucial point of our analysis we derive a generalized Simons inequality for the laplacian of the length of a weighted second fundamental form with respect to an abstract metric associated with F. As an application, we obtain a new Bernstein result for complete F-stable hypersurfaces of dimension .Received: 14 July 2003, Accepted: 13 September 2004, Published online: 10 December 2004Mathematics Subject Classification: 53C42, 49Q10, 35J60     

Affine manifolds with diagonal holonomy

As first defined by Smillie, an affine manifold with diagonal holonomy is a manifold equipped with an atlas such that the changes of charts are restrictions of elements of the subgroup of Aff ( \mathbbRⁿ{\mathbb{R}^n}) formed by diagonal matrices. Refining Smillie’s theorem, Benoist proved that if a compact manifold M is split into manifolds with corners corresponding to complete simplicial fans of a fixed frame by its hypersurfaces with normal crossing, then the product of M and a torus of suitable dimension is a finite cover of an affine manifold with diagonal holonomy, and conversely. Motivated by the result of Benoist, we introduce a “Benoist manifold” and a natural definition of complexity for them. In particular, we study some properties of “Benoist manifolds”.     

Algebraic zero mean curvature hypersurfaces in de Sitter and anti de Sitter spaces

In this paper we provide a family of algebraic space-like surfaces in the three dimensional anti de Sitter space that shows that this Lorentzian manifold admits algebraic maximal examples of any order. Then, we classify all the space-like order two algebraic maximal hypersurfaces in the anti de Sitter N-dimensional space. Finally, we provide two families of examples of Lorentzian order two algebraic zero mean curvature hypersurfaces in the de Sitter space.     

Möbius Type Semigroups in SL(2,F)

In this work, we show that for an ordered field F with an order P, the semigroup S of elements in SL(2,F) having entries in P can be factored into upper triangular, diagonal and lower triangular matrices. Moreover, we show that the semigroup ±S is maximal in SL(2,F).     

Rings of quotients of C(X) induced by points

We study two rings of quotients of C(X), the ring of continuous functions on the Tychonoff space X. The first, F(X), is the ring of quotients induced by the filter of ideals consisting of dense finite intersections of fixed maximal ideals. The second, C[F], is the ring of quotients induced by the filter of dense cofinite subspaces of X. After some preliminary information we explicitly describe in §2 and §3 the constructions of these rings of quotients. In the third section, we use F(X)and C[F] to define and study the class of h-points and h-spaces. In particular, we show that C-spaces and P-spaces are h-spaces. In the last section we construct an ideal of C(X)which will be used to give an ideal theoretic characterization of h-points. This revised version was published online in June 2006 with corrections to the Cover Date.     

Antipodal convex hypersurfaces

Motivated by a conjecture of Steinhaus, we consider the mapping F, associating to each point x of a convex hypersurface the set of all points at maximal intrinsic distance from x. We first provide two large classes of hypersurfaces with the mapping F single-valued and involutive. Afterwards we show that a convex body is smooth and has constant width if its double has the above properties of F, and we prove a partial converse to this result. Additional conditions are given, to characterize the (doubly covered) balls.     

Complete maximal spacelike hypersurfaces ofH 1 4 (c)*

In this paper, we prove that the hyperbolic cylinderH ¹(c ₁)×H ²(c ₂) is the only complete maximal spacelike hypersurfaces inH ₁ ⁴ (c) with nonzero constant Gauss-Kronecker curvature and give a characterization of complete maximal spacelike hypersurfaces ofH ₁ ⁴ (c) with constant scalar curvature. The project Supported by NNSFC, FECC and CPF     

Liouville theorem,conformally invariant cones and umbilical surfaces for Grushin-type metrics

We prove a classification theorem for conformal maps with respect to the control distance generated by a system of diagonal vector fields in ℝ ⁿ . It turns out that in many cases all such maps can be obtained as compositions of suitable dilations, inversions and isometries. Our methods involve a study of the singular Riemannian metric associated with the vector fields. In particular, we identify some conformally invariant cones related to the Weyl tensor. The knowledge of such cones enables us to classify all umbilical hypersurfaces.     

On threshold schemes from large sets

An anonymous (t, w)-threshold scheme is one of the schemes for secret sharing. Combinatorial designs and especially large sets of Steiner systems and of Steiner systems “with holes” have an important role in the design of perfect (t, w)-threshold schemes. In this article we investigate perfect (4, 4)-threshold schemes. We use large sets to form such systems with a large number of keys. In particular we construct the first known infinite families of large sets of H-designs with block size 4. © 1996 John Wiley & Sons, Inc.     

A normal form and schemes of quadratic forms

We present a solution of the problem of construction of a normal diagonal form for quadratic forms over a local principal ideal ring R = 2R with a QF-scheme of order 2. We give a combinatorial representation for the number of classes of projective congruence quadrics of the projective space over R with nilpotent maximal ideal. For the projective planes, the enumeration of quadrics up to projective equivalence is given; we also consider the projective planes over rings with nonprincipal maximal ideal. We consider the normal form of quadratic forms over the field of p-adic numbers. The corresponding QF-schemes have order 4 or 8. Some open problems for QF-schemes are mentioned. The distinguished finite QF-schemes of local and elementary types (of arbitrarily large order) are realized as the QF-schemes of a field. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 1, pp. 161–178, 2007.     

Regularity of averages over hypersurfaces

Averages over smooth measures on smooth compact hypersurfaces inR ⁿ are studied. With assumptions on the decay of the Fourier transform of the measure we obtain mixed norm estimates for these means, for exampleL ^p estimates of multiparameter maximal functions over compact hypersurfaces.     

On submanifolds of Lxf F satisfying Chen's basic equality

It is well known that the warped product Lx_f F of a line L and a Kaehler manifold F is an almost contact Riemannian manifold which is characterized by some tensor equations appeared in (1.7) and (1.8). In this paper we determine submanifolds of Lx_f F which are tangent to the structure vector field and satisfy Chen's basic equality. Also, we investigate tubular hypersurfaces of Lx_f CE ^m which satisfy Chen's basic equality where CE ^m is a complex Euclidean m-space. This revised version was published online in June 2006 with corrections to the Cover Date.     

Maximal functions and ergodic averages related to Waring’s problem

We study the arithmetic analogue of maximal functions on diagonal hypersurfaces. This paper is a natural step following the papers of [13], [14] and [16]. We combine more precise knowledge of oscillatory integrals and exponential sums to generalize the asymptotic formula in Waring’s problem to an approximation formula for the Fourier transform of the solution set of lattice points on hypersurfaces arising in Waring’s problem and apply this result to arithmetic maximal functions and ergodic averages. In sufficiently large dimensions, the approximation formula, ? ²-maximal theorems and ergodic theorems were previously known. Our contribution is in reducing the dimensional constraint in the approximation formula using recent bounds of Wooley, and improving the range of ? ^p spaces in the maximal and ergodic theorems. We also conjecture the expected range of spaces.     

Frobenius Endomorphisms in the Umbral Calculus

Frobenius operators F_n are introduced on sequences of binomial type. The Laguerre polynomials are essentially characterized by the property that F_n coincides with n-fold binomial convolution.     

Gorenstein binomial edge ideals associated with scrolls

Let I_G be the binomial edge ideal on the generic 2×n - Hankel matrix associated with a closed graph G on the vertex set [n]. We characterize the graphs G for which I_G has maximal regularity and is Gorenstein.