Showing papers in "Tokyo Journal of Mathematics in 2002"

Real Hypersurfaces of Complex Space Forms in Terms of Ricci $*$-Tensor

Abstract: It is known that there are no Einstein real hypersurfaces in complex space forms equipped with the K\"ahler metric. In the present paper we classified the $*$-Einstein real hypersurfaces $M$ in complex space forms $M_{n}(c)$ and such that the structure vector is a principal curvature vector.

On the Geometry of the Tangent Bundle with the Cheeger-Gromoll Metric

Abstract: . Let (M,g) be aRiemannian manifold of constant sectional curvature κ and (TM,g ˜ ) be the tangentbundle of M equipped with the Cheeger-Gromoll metric induced by g . We give necessary and sufficient conditionsfor TM having positive scalar curvature. This gives counterexamples to a stated theorem of Sekizawa. 1. Introduction. ARiemannianmetric g onasmoothmanifold M givesrisetoseveralnaturalRiemannianmetrics on the tangent bundle TM of M . The best known example is the Sasaki metric g ˆintroduced in [6], see also [2]. In the present paper we study tangent bundles equipped withthe so called Cheeger-Gromoll metric. Its construction was suggested in [1] but the firstexplicit description was given by Musso and Tricerri in [5].In [7] Sekizawa calculates the Levi-Civita connection ∇˜, the curvature tensor R ˜,thesectional curvatures K ˜ and the scalar curvature S ˜ of the Cheeger-Gromoll metric. He thenstates in his Theorem 6.3 that if (M,g) is an m -dimensional manifold of constant sectionalcurvature

Multiplicity and Hilbert-Kunz Multiplicity of Monoid Rings

Abstract: In this paper, we will give a method to compute the multiplicity and the Hilbert-Kunz multiplicity of monoid rings. The multiplicity and the Hilbert-Kunz multiplicity are fundamental invariants of rings. For example, the multiplicity (resp. the Hilbert-Kunz multiplicity) of a regular local ring equals to one. Monoid rings are defined by lattice ideals, which are binomial ideals I in a polynomial ring R over a field such that any monomial is a non zero divisor on R/I. Affine semigrouprings are monoidrings. Hencewe want to extendthe thoery of affine semigroup rings to that of monoid rings. 1. Main Result. Let N> 0b e an integer andZ the ring of integers. For α ∈ Z N , we denote the i-th entry of α by αi .W e say α> 0i fα � 0a ndαi ≥ 0 for each i .A nd α>α � if α − α � > 0. Let R = k[X1, ··· ,X N ] be a polynomial ring over a field k .F or α> 0, we simply write X α in place of N=1 X αi i . For a positive submodule V of Z N of rank r, we define an ideal I( V )of R ,w hich is generatedby all binomials X α −X β with α−β ∈ V (we say that V is positiveif it is contained in the kernel of a map Z N → Z which is defined by positive integers). Put d = N − r .T hen R/I (V ) is naturally a Z d -graded ring, which is called a monoid ring. Further, there is a

Classification of Sextics of Torus Type

Abstract: In [7], the second author classified configurations of the singularities on tame sextics of torus type. In this paper, we give a complete classification of the singularities on irreducible sextic of torus type, without assuming the tameness of the sextics. We show that there exist 121 configurations and there are 5 pairs and a triple of configurations for which the corresponding moduli spaces coincide, ignoring the respective torus decomposition.

Realization of Vassiliev Invariants by Unknotting Number One Knots

Abstract: We show that for any natural number $n$ and any knot $K$, there are infinitely many unknotting number one knots, all of whose Vassiliev invariants of order less than or equal to $n$ coincide with those of $K$.

Spacelike Maximal Surfaces in 4-dimensional\\Space Forms of Index 2

Abstract: We give necessary and sufficient conditions for the existence of spacelike maximal surfaces in 4-dimensional space forms of index 2. We also discuss spacelike maximal surfaces with constant Gaussian curvature or constant normal curvature, and a rigidity type problem.

A Remark on the Existence of Steady Navier-Stokes Flows in a Certain Two-Dimensional Infinite Channel

Abstract: We consider the steady Navier-Stokes equations $$ \left\{ \begin{array}{@{}l@{\hspace{2pt}}ll} (\mathbf{u}\cdot abla)\mathbf{u} & = u \Delta\mathbf{u} - abla p & \text{in}~\Omega\,,\\ \text{div}\,\mathbf{u} & =0 & \text{in} ~\Omega\,, \end{array} \right. $$ in a 2-dimensional unbounded multiply-connected domain $\Omega$ contained in an infinite straight channel $\mathbf{R}\times(-1,1)$, under general outflow condition. We look for a solution which tends to a Poiseuille flow at infinity. In this note, we shall show the existence of solution to this problem under the assumption of symmetry with respect to the axis for the domain and the boundary value, and for small Poiseuille flow. We do not assume that the boundary value is small. The regularity and the asymptotic behavior of the solution are also discussed.

Lissajous Curves as A'Campo Divides, Torus Knots and Their Fiber Surfaces

Abstract: We present a new form of the fiber surface $F$ for links arising from isolated complex plane curve singularities, in particular, the torus knot $T(p, q)$, where $F$ is a smoothing of a long thin band which has as many clasp-singularities as the unknotting number of $T(p, q)$. As an application, we give a visual proof that the Lissajous curve $(\cos{p\theta}, \cos{q\theta})$ regarded as a divide corresponds to the torus link $T(p, q)$.

Chaos and Entropy for Circle Maps

Abstract: Our aim is to check that the notions of positive entropy, chaos in the sense of Devaney and $\omega$-chaos are equivalent for the circle maps.

Projectively Invariant Cocycles of Holomorphic Vector Fields on an Open Riemann Surface

Abstract: Let $\Sigma$ be an open Riemann surface and $Hol (\Sigma)$ be the Lie algebra of holomorphic vector fields on $\Sigma.$ We fix a projective structure (i.e. a local $SL_2(C)-$structure) on $\Sigma.$ We calculate the first group of cohomology of $Hol(\Sigma)$ with coefficients in the space of linear holomorphic operators acting on tensor densities, vanishing on the Lie algebra $SL_2 (C).$ The result is independant on the choice of the projective structure. We give explicit formulae of 1-cocycles generating this cohomology group.

The Bound of the Difference between Parameter Ideals and their Tight Closures

Abstract: Let R be a Noetherian local ring with the maximal ideal m and assume that R possesses positive characteristic p > 0. For each m-primary ideal I in R let eI (R) and I∗ denote the multiplicity of R with respect to I and the tight closure of I , respectively. Here let us briefly recall the definition of tight closures. For an ideal a in R let a[q] = (a | a ∈ a)R where q = p with e ≥ 0. Let R0 = R \ ⋃ ∈MinR p. Let a∗ denote the set of elements x ∈ R for which there exists c ∈ R0 such that cx ∈ a[q] for all q 0. Then the set a∗ forms an ideal in R containing a, which we call the tight closure of a. The ideal a is said to be tightly closed if a∗ = a. A local ring R is called F -rational if every parameter ideal in R is tightly closed. In this paper we investigate the behavior of sup R(q ∗/q) and sup {e (R)− R(R/q∗)}, where q moves all parameter ideals in R. These two values are very closely related. In fact, they agree if R is Cohen-Macaulay. By definition of F -rationality, it immediately follows that sup R(q ∗/q) = 0 if and only if R is F -rational. The second author studied sup R(q∗/q) in [N2] and he gave some necessary conditins for sup R(q ∗/q) to be finite. On the other hand, Watanabe and Yoshida posed in [WY] the conjecture that the difference e (R) − R(R/q∗) is not negative for every prameter ideal q in R if R is unmixed. Moreover, they conjectured that e (R) − R(R/q∗) = 0 for some parameter ideal q, then R is Cohen-Macaulay and F -rational. This problem is affirmatively solved in [GN] under the condition that R is a homomorphic image of a Cohen-Macaulay ring of chracteristic p > 0 and AssR = AsshR. We further investigate the finiteness of the supremum of the difference e (R) − R(R/q∗) in this paper. The first result gives a necessary and sufficient condition for sup R(q ∗/q) to be finite. Namely,

Deforming Ricci Positive Metrics

Abstract: We find conditions under which a Ricci positive metric can be deformed in a tubular neighbourhood of some submanifold to agree with another given Ricci positive metric. We insist that this final metric has everywhere positive Ricci curvature.

A Note on the Construction of Metacyclic Extensions

Abstract: Let $p$ be an odd prime and $r$ a divisor of $p-1$. We present a characterization of metacyclic extensions of degree $pr$ containing a given cyclic extension of degree $r$ over a field of characteristic other than $p$. Furthermore, we give a method of constructing polynomials with Galois groups which are Frobenius groups of degree $p$.

$q$-linear Functions and Algebraic Independence

Abstract: We define $q$-linear arithmetical functions and $-q$-linear ones and show the algebraic independence over $\mathbb{C}(z)$ of their generating functions.

Mean-Variance Hedging for Discontinuous Semimartingales

Abstract: Mean-variance hedging is well-known as one of hedging methods for incomplete markets. Our end is leading to mean-variance hedging strategy for incomplete market models whose asset price process is given by a discontinuous semimartingale and whose mean-variance trade-off process is not deterministic. In this paper, on account, we focus on this problem under the following assumptions: (1) the local martingale part of the stock price process is a process with independent increments; (2) a certain condition restricting the number and the size of jumps of the asset price process is satisfied; (3) the mean-variance trade-off process is uniformly bounded; (4) the minimal martingale measure coincides with the variance-optimal martingale measure.

A Note on Hurwitzian Numbers

Abstract: In this note Hurwitzian numbers are defined for the nearest integer, and backward continued fraction expansions, and Nakada's $\alpha$-expansions. It is shown that the set of Hurwitzian numbers for these continued fractions coincides with the classical set of such numbers.

On the Diophantine Equations ${n\choose 2}=cx^l$ and ${n\choose 3}=cx^l$

Abstract: Upper bounds for the solution $l$ of the equations of the title are derived by using results concerning the equation $ax^l-by^l=c$ with $a,b,c$ non-zero integers. These solutions are also determined in some special cases.