# Showing papers in "Kodai Mathematical Journal in 2010"

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TL;DR: In this article, the authors introduce the notion of Newton non-degeneracy for mixed functions and develop a basic tool for the study of mixed hypersurface singularities, and show the existence of a canonical resolution of the singularity, and the presence of the Milnor fibration under the strong nonsmooth condition.

Abstract: Mixed functions are analytic functions in variables z1, ..., zn and their conjugates $\bar z_1$, ..., $\bar z_n$. We introduce the notion of Newton non-degeneracy for mixed functions and develop a basic tool for the study of mixed hypersurface singularities. We show the existence of a canonical resolution of the singularity, and the existence of the Milnor fibration under the strong non-degeneracy condition.

79 citations

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TL;DR: In this paper, the authors give a method to find every solution to the eikonal PDE on a Riemannian manifold locally, and give a local construction of arbitrary Euclidean helix submanifolds of any dimension and codimension.

Abstract: A submanifold of Rn whose tangent space makes constant angle with a fixed direction d is called a helix. Helix submanifolds are related with the eikonal PDE equation. We give a method to find every solution to the eikonal PDE on a Riemannian manifold locally. As a consequence we give a local construction of arbitrary Euclidean helix submanifolds of any dimension and codimension. Also we characterize the ruled helix submanifolds and in particular we describe those which are minimal.

30 citations

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TL;DR: In this paper, the moduli spaces of (semi)stable coherent sheaves on smooth projective surfaces were constructed as the modulus spaces of semi-stable quiver representations over a finite dimensional C-algebra.

Abstract: Let X be a smooth projective surface, Coh(X) the abelian category of coherent sheaves on X, and D(X) the bounded derived category of coherent sheaves on X. We study the Bridgeland stability conditions on D(X) and see that for some stability conditions on D(X) the moduli spaces of (semi)stable objects in D(X) coincide with the moduli spaces of (semi)stable coherent sheaves, while for some other stability conditions the moduli spaces of (semi)stable objects inD(X) coincide with the moduli spaces of (semi)stable modules over a finite dimensional C-algebra in the case of X with a full strong exceptional collection. In particular, we construct the moduli spaces of (semi)stable coherent sheaves on P2 as the moduli spaces of (semi)stable quiver representations. This gives another proof of Le Potier’s result [P1] and some variants.

23 citations

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TL;DR: In this paper, it was shown that any contact metric manifold whose Reeb vector field belongs to the (κ, μ)-nullity distribution canonically carries an almost bi-paracontact structure.

Abstract: We study almost bi-paracontact structures on contact manifolds. We prove that if an almost bi-paracontact structure is defined on a contact manifold (M, η), then under some natural assumptions of integrability M carries two transverse bi-Legendrian structures. Conversely, if two transverse bi-Legendrian structures are defined on a contact manifold, then M admits an almost bi-paracontact structure. We define a canonical connection on an almost bi-paracontact manifold and we study its curvature properties, which resemble those of the Obata connection of a para-hypercomplex (or complex-product) manifold. Further, we prove that any contact metric manifold whose Reeb vector field belongs to the (κ, μ)-nullity distribution canonically carries an almost bi-paracontact structure and we apply the previous results to the theory of contact metric (κ, μ)-spaces.

19 citations

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TL;DR: In this article, the Fekete-Szego problem for the class of k-uniformly convex functions and k-parabolic star-like functions is completely settled.

Abstract: By making use of an integral operator due to Noor, a new subclass of analytic functions, denoted by k–$\mathscr{UCV}_n$ (n $\in$ N0 := {0,1,2,};0 ≤ k < ∞); is introduced For this class the Fekete-Szego problem is completely settled The results obtained here also give the Fekete-Szego inequalities for the classes of k-uniformly convex functions and k-parabolic starlike functions

16 citations

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15 citations

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TL;DR: In this article, the authors studied gradient Ricci expanding solitons (X,g) satisfying Rc = cg + D2f, where Rc is the Ricci curvature, c 0 on X unless X,g is Ricci flat.

Abstract: In this paper, we study gradient Ricci expanding solitons (X,g) satisfying Rc = cg + D2f, where Rc is the Ricci curvature, c 0 on X unless (X,g) is Ricci flat. We also show that there is exponentially decay for scalar curvature on a complete non-compact expanding soliton with its Ricci curvature being e-pinched.

13 citations

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TL;DR: The main purpose of as discussed by the authors is to study almost Norden structures on 4-dimensional Walker manifolds and discuss the integrability and Kahler (holomorphic) conditions for these structures.

Abstract: The main purpose of the present paper is to study almost Norden structures on 4-dimensional Walker manifolds. We discuss the integrability and Kahler (holomorphic) conditions for these structures. The curvature properties for Norden-Walker metrics is also investigated. Then examples Norden-Walker metrics are constructed from an arbitrary harmonic function of two variables.

13 citations

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TL;DR: In this paper, the Izeki-Nayatani invariants of spaces in such a family are uniformly bounded from above by a constant strictly less than 1, and the invariants are shown to be invariant to the tangent cones of the Gromov-Hausdorff precompact family of spaces.

Abstract: In this paper, we will consider a family $\mathscr{Y}$ of complete CAT(0) spaces such that the tangent cone TCp Y at each point p $\in$ Y of each Y $\in$ $\mathscr{Y}$ is isometric to a (finite or infinite) product of the Euclidean cones Cone(Xα) over elements Xα of some Gromov-Hausdorff precompact family {Xα} of CAT(1) spaces. Each element of such $\mathscr{Y}$ is a space presented by Gromov [4] as an example of a "CAT(0) space with "bounded" singularities". We will show that the Izeki-Nayatani invariants of spaces in such a family are uniformly bounded from above by a constant strictly less than 1.

8 citations

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TL;DR: In this paper, the quotient curves of smooth plane curves with automorphisms are shown to be extremal curves in the sense of Castelnuovo's bound, and they can be divided into two types (type I and type II).

Abstract: We obtain several results of quotient curves of smooth plane curves with automorphisms. Such automorphisms can be divided into two types (type I and type II). The quotient curves of smooth plane curves with automorphisms of type I are extremal curves in the sense of Castelnuovo's bound. We also show some partial result on automorphisms of type II and give examples.

8 citations

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TL;DR: In this article, it was shown that M2n-1 is a real hypersurface whose geodesics orthogonal to the characteristic vector are mapped to circles of the same curvature 1 in an n-dimensional nonflat complex space form.

Abstract: We show that M2n-1 is a real hypersurface all of whose geodesics orthogonal to the characteristic vector ξ are mapped to circles of the same curvature 1 in an n-dimensional nonflat complex space form $\widetilde{M}_n$(c) (= CPn(c) or CHn(c)) if and only if M is a Sasakian manifold with respect to the almost contact metric structure from the ambient space $\widetilde{M}_n$(c). Moreover, this Sasakian manifold M is a Sasakian space form of constant φ-sectional curvature c + 1 for each c (≠0).

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TL;DR: In this paper, the authors improved Tanno's result that a conformal vector field on a contact metric manifold M is a strictly infinitesimal contact transformation, and thus it is an automorphism of M by waiving the "strictness" in the hypothesis.

Abstract: First we improve a result of Tanno that says "If a conformal vector field on a contact metric manifold M is a strictly infinitesimal contact transformation, then it is an infinitesimal automorphism of M" by waiving the "strictness" in the hypothesis. Next, we prove that a (k, μ)-contact manifold admitting a non-Killing conformal vector field is either Sasakian or has k = –n – 1, μ = 1 in dimension > 3; and Sasakian or flat in dimension 3. In particular, we show that (i) among all compact simply connected (k, μ)-contact manifolds of dimension > 3, only the unit sphere S2n+1 admits a non-Killing conformal vector field, and (ii) a conformal vector field on the unit tangent bundle of a space-form of dimension > 2 is necessarily Killing.

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TL;DR: In this paper, it was shown that the set consisting of homomorphisms Q(K) → Zq[t±1]/J is isomorphic to the set of Z[t−1]-homomorphisms H1(C∞(K)) → ZQ[t+1]-modules.

Abstract: Let K be an n-dimensional knot (n ≥ 1), Q(K) the knot quandle of K, Zq[t±1]/J an Alexander quandle, and C∞(K) the infinite cyclic covering space of Sn+2$\backslash$K. We show that the set consisting of homomorphisms Q(K) → Zq[t±1]/J is isomorphic to Zq[t±1]/J ⊕ HomZ[t±1] (H1(C∞(K)), Zq[t±1]/J) as Z[t±1]-modules. Here, HomZ[t±1](H1(C∞(K)), Zq[t±1]/J) denotes the set consisting of Z[t±1]-homomorphisms H1(C∞(K)) → Zq[t±1]/J.

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TL;DR: In this article, it was shown that a Hopf hypersurface M in a non-flat complex space form M-n(c) with constant mean curvature and with eta-recurrent Ricci tensor is locally congruent to one of real hypersurfaces of type A and B.

Abstract: Baikoussis, Lyu and Suh [1] showed that a Hopf hypersurface M in a non-flat complex space form M-n(c) with constant mean curvature and with eta-recurrent Ricci tensor is locally congruent to one of real hypersurfaces of type A and B. They also conjectured that the same result can be obtained even without the constancy assumption on the mean curvature (cf. [I, Remark 5.1.]). The purpose of this paper is to answer this question in the affirmative.

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TL;DR: In this paper, a global version of the Le-Ramanujam μ-constant theorem for polynomials is proposed, where the Euler characteristic of a generic fiber of a polynomial is constant and the global monodromy fibrations of the fiber are all isomorphic.

Abstract: We are interested in a global version of Le-Ramanujam μ-constant theorem for polynomials. We consider an analytic family {fs}, s $\in$ [0, 1], of complex polynomials in two variables, that are Newton non-degenerate. We suppose that the Euler characteristic of a generic fiber of fs is constant, then we show that the global monodromy fibrations of fs are all isomorphic, and that the degree of fs is constant (up to an algebraic automorphism of C2).

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TL;DR: In this article, the authors considered the Julia sets of rational functions with irrationally indifferent fixed points and showed that the points can be biaccessible in the Julia set of quadratic polynomials.

Abstract: We are interested in biaccessible points in the Julia sets of rational functions. D. Schleicher and S. Zakeri studied which points can be biaccessible in the Julia sets of quadratic polynomials with irrationally indifferent fixed points [SZ, Za]. In this paper, we consider the two polynomial families fc(z) = zd + c, gθ(z) = e2πiθz + zd and the cubic rational family hθ,a(z) = e2πiθz2$\frac{z-a}{1-\bar{a}z}$.

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TL;DR: In this paper, the authors investigated the exponent of convergence of the zero-sequence of solutions of the second order linear differential equation, and obtained some results which improved and generalize the previous results.

Abstract: In this paper, we investigate the exponent of convergence of the zero-sequence of solutions of the second order linear differential equation $$f''+\left(\sum_{j=1}^{l}Q_j(z)e^{P_j(z)}\right)f=0,$$ where Pj (z) (j= 1,2, ..., l ≥ 3) are polynomials of degree n ≥ 1, Qj (z) are entire functions of order less than n, and obtain some results which improve and generalize the previous results in [8, 9, 13].

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TL;DR: In this paper, the authors extended the work of Ruscheweyh and Ahlfors on sufficient conditions for a normalized analytic function on the unit disk to be univalent there.

Abstract: Ruscheweyh extended the work of Becker and Ahlfors on sufficient conditions for a normalized analytic function on the unit disk to be univalent there. In this paper we refine the result to a quasiconformal extension criterion with the help of Becker's method. As an application, a positive answer is given to an open problem proposed by Ruscheweyh.

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TL;DR: In this paper, it was shown that if the second fundamental form A of M satisfies ∫M |A|3 < ∞, then M is an affine n-dimensional plane.

Abstract: Let Mn (n ≤ 7) be an n-dimensional complete immersed super stable minimal submanifold in an (n + p)-dimensional Euclidean space Rn+p with flat normal bundle. We prove that if the second fundamental form A of M satisfies ∫M |A|3 < ∞, then M is an affine n-dimensional plane.

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TL;DR: In this article, the dual spaces of restrictions of a dual pair of reproducing kernel Hilbert spaces in a discrete set were characterized and a canonical dense subset to the restriction spaces was given.

Abstract: We characterize the dual spaces of restrictions of a dual pair of reproducing kernel Hilbert spaces in a discrete set Consequently, we give a canonical dense subset to the restriction spaces As applications, we reprove a variational principle in a dual pair of reproducing kernel Hilbert spaces Also we give a geometric representation for the existence and ergodicity condition of equilibrium Glauber and Kawasaki dynamics for some determinantal point processes

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TL;DR: In this paper, it was shown that the ∂-transforms of the Riemann sphere into the complex projective space are energy bounded and that their subsequences converge to harmonic bubble tree maps.

Abstract: When harmonic maps from the Riemann sphere into the complex projective space are energy bounded, it contains a subsequence converging to a bubble tree map fI: TI → CPn. We show that their ∂-transforms and $\overline{\partial}$-transforms are also energy bounded. Hence their subsequences converge to harmonic bubble tree maps $f_1^{I_1}:T^{I_1}$ → CPn and $f_{-1}^{I_{-1}}:T^{I_{-1}}$ → CPn respectively. In this paper, we show relations between fI, $f_1^{I_1}$ and $f_{-1}^{I_{-1}}$.

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TL;DR: In this paper, the authors used the theory of normal families and differential equations to obtain a uniqueness theorem of entire function which was an improvement of Chang and Fang [1] and improved some results of Pang and Zalcman.

Abstract: In the paper, we have two purposes. Firstly, we prove two theorems and two corollaries of normal families which improve and generalize some results of Pang and Zalcman [9], Zhang, Sun and Pang [13], Chang and Fang [2]. Secondly, we use the theory of normal families and differential equations to obtain a uniqueness theorem of entire function which is an improvement of Chang and Fang [1].

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TL;DR: In this paper, a complete description of the global monodromy of a Lefschetz fibration arising from the Fermat surface of degree 4 is given and a positive relation among right hand Dehn twists in the mapping class group of a closed orientable surface of genus 3 is obtained.

Abstract: A complete description of the global monodromy of a Lefschetz fibration arising from the Fermat surface of degree 4 is given. As a by-product we get a positive relation among right hand Dehn twists in the mapping class group of a closed orientable surface of genus 3.

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TL;DR: In this paper, the authors extended Isaev's characterization of the unit polydisc Δn in Cn to any bounded symmetric domain in C n. In particular, they showed that Δn can be characterized for any symmetric manifold M with the property that every isotropy subgroup of the holomorphic automorhism group of M is compact.

Abstract: In 2008, we obtained an intrinsic characterization of the unit polydisc Δn in Cn from the viewpoint of the holomorphic automorphism group. In connection with this, A. V. Isaev investigated the structure of a complex manifold M with the property that every isotropy subgroup of the holomorphic automorhism group of M is compact, and obtained the same characterization of Δn as ours among the class of all such manifolds. In this paper, we establish some extensions of these results. In particular, Isaev's characterization of the unit polydisc Δn is extended to that of any bounded symmetric domain in Cn.

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TL;DR: In this article, it was shown that afforested surfaces do not belong to the family of hyperbolic Riemann surfaces on which there are no nonzero singular harmonic functions.

Abstract: An afforested surface W := , N being the set of positive integers, is an open Riemann surface consisting of three ingredients: a hyperbolic Riemann surface P called a plantation, a sequence (Tn)n$\in$N of hyperbolic Riemann surfaces Tn each of which is called a tree, and a sequence (σn)n$\in$N of slits σn called the roots of Tn contained commonly in P and Tn which are mutually disjoint and not accumulating in P. Then the surface W is formed by foresting trees Tn on the plantation P at the roots for all n $\in$ N, or more precisely, by pasting surfaces Tn to P crosswise along slits σn for all n $\in$ N. Let ${\mathscr O}_s$ be the family of hyperbolic Riemann surfaces on which there are no nonzero singular harmonic functions. One might feel that any afforested surface W := belongs to the family ${\mathscr O}_s$ as far as its plantation P and all its trees Tn belong to ${\mathscr O}_s$. The aim of this paper is, contrary to this feeling, to maintain that this is not the case.