Multiplicity of closed characteristics on symmetric convex hypersurfaces in $\mathbb{R}^{2n}$

Let be a compact hypersurface in bounding a convex set with non-empty interior. In this paper it is proved that there always exist at least n geometrically distinct closed characteristics on if is symmetric with respect to the origin. Received: 18 May 1999 / Published online: 4 April 2002     

Some remarks on factoriality of certain hypersurfaces in \mathbb{P}^4

Let V be a reduced and irreducible hypersurface of degree k 3. In this paper we prove that if the singular locus of V consists of ₂ ordinary double points, ₃ ordinary triple points and if ₂ + 4₃ < (k – 1)², then any smooth surface contained in V is a complete intersection on V.Received: 7 January 2004     

Factoriality condition of some nodal threefolds in {\mathbb{P}^4}

We prove that for n = 8, 9, 10, 11, a nodal hypersurface of degree n in is factorial if it has at most (n − 1)² − 1 nodes. The author is grateful to Ivan Cheltsov for valuable comments and suggestions.     

Optimal $\mathcal{L}^2$ error bounds for MITC3 type element

Summary. In this paper, we derive the optimal error bounds for the stabilized MITC3 element [3], the MIN3 type element [7] and the T3BL element [8]. In this way we have solved the problem proposed recently in [5] in a positive manner. Moreover, we estimate the difference between stabilized MITC3 and MIN3 and show it is of order uniform in the plate thickness. Received May 31, 2000 / Revised version received April 2, 2001 / Published online September 19, 2001     

Smooth threefolds in ${\Bbb P}^5$ without apparent triple or quadruple points and a quadruple-point formula

For a projective variety X of codimension 2 in defined over the complex number field , it is traditionally said that X has no apparent -ple points if the -secant lines of X do not fill up the ambient projective space , equivalently, the locus of -ple points of a generic projection of X to ${\Bbb P}^{n+1}$ is empty. We show that a smooth threefold in has no apparent triple points if and only if it is contained in a quadric hypersurface. We also obtain an enumerative formula counting the quadrisecant lines of X passing through a general point of and give necessary cohomological conditions for smooth threefolds in without apparent quadruple points. This work is intended to generalize the work of F. Severi [fSe] and A. Aure [Au], where it was shown that a smooth surface in has no triple points if and only if it is either a quintic elliptic scroll or contained in a hyperquadric. Furthermore we give open questions along these lines. Received: 24 January 2000 / Published online: 18 June 2001     

Asymptotics for the zeros and poles of normalized Pad\'{e} approximants to{\rm e}^{z}

Summary. With denoting the -th partial sum of ${\rm e}^{z}$, the exact rate of convergence of the zeros of the normalized partial sums, , to the Szeg\"o curve was recently studied by Carpenter et al. (1991), where is defined by Here, the above results are generalized to the convergence of the zeros and poles of certain sequences of normalized Pad\'{e} approximants to , where is the associated Pad\'{e} rational approximation to . Received February 2, 1994     

Laguerre geometry of hypersurfaces in $$\mathbb{R}^{n}$$

Laguerre geometry of surfaces in is given in the book of Blaschke [Vorlesungen über Differentialgeometrie, Springer, Berlin Heidelberg New York (1929)], and has been studied by Musso and Nicolodi [Trans. Am. Math. soc. 348, 4321–4337 (1996); Abh. Math. Sem. Univ. Hamburg 69, 123–138 (1999); Int. J. Math. 11(7), 911–924 (2000)], Palmer [Remarks on a variation problem in Laguerre geometry. Rendiconti di Mathematica, Serie VII, Roma, vol. 19, pp. 281–293 (1999)] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in . For any umbilical free hypersurface with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator : TM → TM, and show that is a complete Laguerre invariant system for hypersurfaces in with n≥ 4. We calculate the Euler–Lagrange equation for the Laguerre volume functional of Laguerre metric by using Laguerre invariants. Using the Euclidean space , the semi-Euclidean space and the degenerate space we define three Laguerre space forms , and and define the Laguerre embeddings and , analogously to what happens in the Moebius geometry where we have Moebius space forms S ⁿ , and (spaces of constant curvature) and conformal embeddings and [cf. Liu et al. in Tohoku Math. J. 53, 553–569 (2001) and Wang in Manuscr. Math. 96, 517–534 (1998)]. Using these Laguerre embeddings we can unify the Laguerre geometry of hypersurfaces in , and . As an example we show that minimal surfaces in or are Laguerre minimal in .C. Wang Partially supported by RFDP and Chuang-Xin-Qun-Ti of NSFC.     

Constructing infinitely many smooth structures on $3\mathbb{CP}^2 \# n\overline{\mathbb{CP}}^2$

Using Seiberg-Witten theory and rational blow-down procedures of R. Fintushel and R.J. Stern, we construct infinitely many irreducible smooth structures, both symplectic and non-symplectic, on the four-manifold for each integer n lying in the interval . Received: 17 January 2000 / Published online: 18 January 2002     

Ordinal arithmetic and \Sigma_{1}-elementarity

We will introduce a partial ordering on the class of ordinals which will serve as a foundation for an approach to ordinal notations for formal systems of set theory and second-order arithmetic. In this paper we use to provide a new characterization of the ubiquitous ordinal . Received: 18 August 1997     

Syzygies of Multi-Variable Higher Spin Dirac Operators on $$\mathbb{R}^{3}$$

This paper is a short report on the generalization of some results of our previous paper [12] to the case of spin j/2 Dirac operators in real dimension three for arbitrary odd integer j. We use an explicit formula for the local expression of such operators to study their algebraic properties, construct the compatibility conditions of the overdetermined system associated to the operator in several spatial variables, and we prove that its associated algebraic complex, dual do the BGG sequence coming from representation theory, has substantially the same pattern as the Cauchy-Fueter complex. The author is a member of the Eduard Čech Center and his research is supported by the relative grants.     

On non-projective normal surfaces

In this note we construct examples of non-projective normal proper algebraic surfaces and discuss the somewhat pathological behaviour of their Neron–Severi group. Our surfaces are birational to the product of a projective line and a curve of higher genus. Received: 29 January 1999     

Some observations on Babu\vs}ka and Brezzi theories

Summary. Some observations are made on abstract error estimates for Galerkin approximations based on Babuška-Brezzi conditions. A basic error estimate due to Babuška is sharpened by means of an identity that for any nontrivial idempotent operator P. Some remarks are also made on the Brezzi's theory for mixed variational problems and their Galerkin approximations. Received March 1, 2000 / Revised version received September 28, 2000 / Published online June 17, 2002 RID="*" ID="*" This work was partially supported by NSF DMS-9706949, NSF ACI-9800244 and NASA NAG2-1236 Correspondence to: J. Xu     

Four positive solutions for the semilinear elliptic equation: -Delta u+u=a(x)u^p+f(x) in {mathbb R}^N

We consider the existence of positive solutions of the following semilinear elliptic problem in : where , , , and . Under the conditions: 1° for all , 2° as , 3° there exist and such that 4°, we show that (*) has at least four positive solutions for sufficiently small but . Received December 11, 1998 / Accepted July 16, 1999 / Published online April 6, 2000     

The Albanese map of a 3-fold of general type whose canonical map is composed with a pencil

Received: 15 October 1999; in final form: 13 June 2000 / Published online: 29 April 2002