Kodai Mathematical Journal

Estimates of eigenvalues of a clamped problem

Tao Zheng

2015 Volume 38 Issue 2 Pages 249-269
Published: 2015
Released on J-STAGE: July 11, 2015

DOIhttps://doi.org/10.2996/kmj/1436403889

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In this paper, for the eigenvalue problem of a clamped plate problem on complex projective space with holomorphic sectional curvature c(> 0) and n(≥ 3)-dimensional noncompact simply connected complete Riemannian manifold with sectional curvature Sec satisfying −a² ≤ Sec ≤ −b², where a ≥ b ≥ 0 are constants, we obtain universal eigenvalue inequalities. Moreover, we deduce the estimates of the upper bounds of eigenvalues.

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Weierstrass semigroups on double covers of plane curves of degree 5

Seon Jeong Kim, Jiryo Komeda

2015 Volume 38 Issue 2 Pages 270-288
Published: 2015
Released on J-STAGE: July 11, 2015

DOIhttps://doi.org/10.2996/kmj/1436403890

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We investigate Weierstrass semigroups of ramification points on double covers of plane curves of degree d. Using the results we determine all the Weierstrass semigroups in the case d = 5 when the genus of the covering curve is greater than 17 and the ramification point is on a non-ordinary flex.

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Entire functions and their first derivatives sharing simple β-points for a small function β

Amer H. H. Al-Khaladi

2015 Volume 38 Issue 2 Pages 289-301
Published: 2015
Released on J-STAGE: July 11, 2015

DOIhttps://doi.org/10.2996/kmj/1436403891

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The following theorem has been proved by A. Schweizer [7]. If a nonconstant entire function f and its derivative f′ share their simple zeros and if every simple a-point of f is a (not necessarily simple) a-point of f′ for some nonzero constant a, then f ≡ f′. In this paper we shall prove that the above result is also true when the nonzero constant a is replaced by a meromorphic small function β($\not\equiv$ 0, ∞).

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A lower diameter bound for compact domain manifolds of shrinking Ricci-harmonic solitons

Homare Tadano

2015 Volume 38 Issue 2 Pages 302-309
Published: 2015
Released on J-STAGE: July 11, 2015

DOIhttps://doi.org/10.2996/kmj/1436403892

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In this paper, we shall give a lower diameter bound for compact domain manifolds of shrinking Ricci-harmonic solitons. Our result may be regarded as a generalization to Ricci-harmonic geometry of the recent works by Fernández-López and García-Río (Q. J. Math. 61, 319-327, 2010), Futaki and Sano (Asian J. Math. 17, 17-32, 2013), and Futaki et al. (Ann. Global Anal. Geom. 44, 105-114, 2013).

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Pseudo asymptotically periodic solutions of two-term time fractional differential equations with delay

Zhinan Xia

2015 Volume 38 Issue 2 Pages 310-332
Published: 2015
Released on J-STAGE: July 11, 2015

DOIhttps://doi.org/10.2996/kmj/1436403893

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In this paper, the existence and uniqueness of pseudo $\mathcal{S}$-asymptotically ω-periodic mild solutions of class r for the nonlinear two-term time fractional differential equations with delay are investigated. The working tools are based on the generalization of the semigroup theory and fixed point theory. Finally, we present an application to a fractional partial differential equation with delay.

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On ideals of rings of fractions and rings of polynomials

Yuan Ting Nai, Dongsheng Zhao

2015 Volume 38 Issue 2 Pages 333-342
Published: 2015
Released on J-STAGE: July 11, 2015

DOIhttps://doi.org/10.2996/kmj/1436403894

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We investigate the links between the lattice Idl(R) of ideals of a commutative ring R and the lattices Idl(R′) of ideals of various new rings R′ constructed from R, in particular, the ring S⁻¹R of fractions and the ring R[X] of polynomials. For any partially ordered set P, we construct another poset N(P) and show that P satisfies the ascending chain condition if and only if N(P) satisfies the ascending chain condition. As an application of this result, we give an order version proof for Hilbert's Basis Theorem.

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On a conjecture of Beltrametti-Sommese for polarized 4-folds

Yoshiaki Fukuma

2015 Volume 38 Issue 2 Pages 343-351
Published: 2015
Released on J-STAGE: July 11, 2015

DOIhttps://doi.org/10.2996/kmj/1436403895

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Let (X,L) be a polarized manifold of dimension 4. In this paper, we prove that h⁰(K_X + 3L) > 0 if K_X + 3L is nef, which is a conjecture of Beltrametti-Sommese for polarized 4-folds.

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On θ-congruent numbers on real quadratic number fields

Ali S. Janfada, Sajad Salami

2015 Volume 38 Issue 2 Pages 352-364
Published: 2015
Released on J-STAGE: July 11, 2015

DOIhttps://doi.org/10.2996/kmj/1436403896

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Let K = Q ($\sqrt{m}$) be a real quadratic number field, where m > 1 is a squarefree integer. Suppose that 0 < θ < π has rational cosine, say cos(θ) = s/r with 0 < |s| < r and gcd(r,s) = 1. A positive integer n is called a (K,θ)-congruent number if there is a triangle, called the (K,θ, n)-triangles, with sides in K having θ as an angle and nα_θ as area, where α_θ = $\sqrt{r^2-s^2}$. Consider the (K,θ)-congruent number elliptic curve E_n,θ: y² = x(x + (r + s)n) (x − (r − s)n) defined over K. Denote the squarefree part of positive integer t by sqf(t). In this work, it is proved that if m ≠ sqf(2r(r − s)) and mn ≠ 2, 3, 6, then n is a (K,θ)-congruent number if and only if the Mordell-Weil group E_n,θ(K) has positive rank, and all of the (K, θ, n)-triangles are classified in four types.

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Monomorphisms in categories of log schemes

Shinichi Mochizuki

2015 Volume 38 Issue 2 Pages 365-429
Published: 2015
Released on J-STAGE: July 11, 2015

DOIhttps://doi.org/10.2996/kmj/1436403897

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In the present paper, we study category-theoretic properties of monomorphisms in categories of log schemes. This study allows one to give a purely category-theoretic reconstruction of the log scheme that gave rise to the category under consideration. We also obtain analogous results for categories of schemes of locally finite type over the ring of rational integers that are equipped with "archimedean structures". Such reconstructions were discussed in two previous papers by the author, but these reconstructions contained some errors, which were pointed out to the author by C. Nakayama and Y. Hoshi. These errors revolve around certain elementary combinatorial aspects of fan decompositions of two-dimensional rational polyhedral cones—i.e., of the sort that occur in the classical theory of toric varieties—and may be repaired by applying the theory developed in the present paper.

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Minimaxness and finiteness properties of formal local cohomology modules

Shahram Rezaei

2015 Volume 38 Issue 2 Pages 430-436
Published: 2015
Released on J-STAGE: July 11, 2015

DOIhttps://doi.org/10.2996/kmj/1436403898

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Let $\mathfrak{a}$ be an ideal of local ring (R,$\mathfrak{m}$) and M a finitely generated R-module and n an integer. We prove some results concerning minimaxness and finiteness of formal local cohomology modules. We discuss the maximum and minimum integers such that $\mathfrak{F}_\mathfrak{a}^i$(M) is minimax and also we obtain the maximum and minimum integers such that $\mathfrak{F}_\mathfrak{a}^i$(M) is finitely gnerated.

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Universal inequalities for eigenvalues of a system of sub-elliptic equations on Heisenberg group

Feng Du, Chuanxi Wu, Guanghan Li, Changyu Xia

2015 Volume 38 Issue 2 Pages 437-450
Published: 2015
Released on J-STAGE: July 11, 2015

DOIhttps://doi.org/10.2996/kmj/1436403899

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In this paper, we study the eigenvalue problem of a system of sub-elliptic equations on abounded domain in the Heisenberg group and obtain some universal inequalities. Moreover, for the lower order eigenvalues of this eigenvalue problem, we also give some universal inequalities.

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Sovereign and ribbon weak Hopf algebras

Xiaohui Zhang, Xiaofan Zhao, Shuanhong Wang

2015 Volume 38 Issue 2 Pages 451-469
Published: 2015
Released on J-STAGE: July 11, 2015

DOIhttps://doi.org/10.2996/kmj/1436403900

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In this paper, we study the Deligne's sovereign structure theorem on the finite dimensional weak Hopf algebras, and give a necessary and sufficient condition for a finite dimensional (co)quasitriangular weak Hopf algebra H with bijective antipode to admit a (co)ribbon structure. As an application we discuss the ribbon structures over the Drinfeld doubles of some weak Hopf algebras to verify our theory.

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Solvability of the initial value problem to a model system for water waves

Yuuta Murakami, Tatsuo Iguchi

2015 Volume 38 Issue 2 Pages 470-491
Published: 2015
Released on J-STAGE: July 11, 2015

DOIhttps://doi.org/10.2996/kmj/1436403901

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We consider the initial value problem to a model system for water waves. The model system is the Euler-Lagrange equations for an approximate Lagrangian which is derived from Luke's Lagrangian for water waves by approximating the velocity potential in the Lagrangian. The model are nonlinear dispersive equations and the hypersurface t = 0 is characteristic for the model equations. Therefore, the initial data have to be restricted in an infinite dimensional manifold in order to the existence of the solution. Under this necessary condition and a sign condition, which corresponds to a generalized Rayleigh-Taylor sign condition for water waves, on the initial data, we show that the initial value problem is solvable locally in time in Sobolev spaces.

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