Kodai Mathematical Journal

A finite generating set for the level 2 mapping class group of a nonorientable surface

Błażej Szepietowski

2013 Volume 36 Issue 1 Pages 1-14
Published: 2013
Released on J-STAGE: April 04, 2013

DOIhttps://doi.org/10.2996/kmj/1364562714

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We obtain a finite set of generators for the level 2 mapping class group of a closed nonorientable surface of genus g ≥ 3. This set consists of isotopy classes of Lickorish's Y-homeomorphisms also called crosscap slides.

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Banach spaces of bounded Dirichlet finite harmonic functions on Riemann surfaces

Mitsuru Nakai

2013 Volume 36 Issue 1 Pages 15-37
Published: 2013
Released on J-STAGE: April 04, 2013

DOIhttps://doi.org/10.2996/kmj/1364562715

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The Banach space of bounded Dirichlet finite harmonic functions on an open Riemann surface will be seen to be reflexive and also separable if and only if the underlying Riemann surface does not carry any unbounded Dirichlet finite harmonic function.

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Positive Toeplitz operators of finite rank on the parabolic Bergman spaces

Masaharu Nishio, Noriaki Suzuki, Masahiro Yamada

2013 Volume 36 Issue 1 Pages 38-49
Published: 2013
Released on J-STAGE: April 04, 2013

DOIhttps://doi.org/10.2996/kmj/1364562716

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We define the Toeplitz operators on the parabolic Bergman spaces by using a positive bilinear form. In this setting we characterize finite rank Toeplitz operators. A relation with the Carleson inclusion is also discussed.

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On Hermitian modular forms of small weight over imaginary quadratic fields

Hisashi Kojima, Yasuhide Miura, Hiroshi Sakata, Yasushi Tokuno

2013 Volume 36 Issue 1 Pages 50-55
Published: 2013
Released on J-STAGE: April 04, 2013

DOIhttps://doi.org/10.2996/kmj/1364562717

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In this paper, we prove that an Hermitian modular form with small weight over the quadratic field with class number one is a linear combination of theta series associated with Hermitian quadratic forms.

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On meromorphic functions sharing a one-point set and three two-point sets CM

Manabu Shirosaki

2013 Volume 36 Issue 1 Pages 56-68
Published: 2013
Released on J-STAGE: April 04, 2013

DOIhttps://doi.org/10.2996/kmj/1364562718

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We show that if two meromorphic functions sharing a one-point set and three two-point sets CM, then one of them is a Möbius transform of the other.

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Another improvement of Montel's criterion

Yan Xu

2013 Volume 36 Issue 1 Pages 69-76
Published: 2013
Released on J-STAGE: April 04, 2013

DOIhttps://doi.org/10.2996/kmj/1364562719

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Let ${\cal F}$ be a family of meromorphic functions defined in a domain D ⊂ C, let ψ₁, ψ₂ and ψ₃ be three meromorphic functions such that ψ_i(z) $\not\equiv$ ψ_j(z) (i ≠ j) in D, one of which may be ∞ identically, and let l₁, l₂ and l₃ be positive integers or ∞ with 1/l₁ + 1/l₂ + 1/l₃ < 1. Suppose that, for each f ∈ ${\cal F}$ and z ∈ D, (1) all zeros of f – ψ_i have multiplicity at least l_i for i = 1,2,3; (2) f(z₀) ≠ ψ_i(z₀) if there exist i, j ∈ {1,2,3} (i ≠ j) and z₀ ∈ D such that ψ_i(z₀) = ψ_j(z₀). Then ${\cal F}$ is normal in D. This improves and generalizes Montel's criterion.

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Bifurcation set, M-tameness, asymptotic critical values and Newton polyhedrons

Tat Thang Nguyen

2013 Volume 36 Issue 1 Pages 77-90
Published: 2013
Released on J-STAGE: April 04, 2013

DOIhttps://doi.org/10.2996/kmj/1364562720

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Let F = (F₁, F₂, ..., F_m): Cⁿ → C^m be a polynomial dominant mapping with n > m. In this paper we give the relations between the bifurcation set of F and the set of values where F is not M-tame as well as the set of generalized critical values of F. We also construct explicitly a proper subset of C^m in terms of the Newton polyhedrons of F₁, F₂, ..., F_m and show that it contains the bifurcation set of F. In the case m = n – 1 we show that F is a locally C^∞-trivial fibration if and only if it is a locally C⁰-trivial fibration.

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Leaf-wise intersections in coisotropic submanifolds

Satoshi Ueki

2013 Volume 36 Issue 1 Pages 91-98
Published: 2013
Released on J-STAGE: April 04, 2013

DOIhttps://doi.org/10.2996/kmj/1364562721

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The leaf-wise intersection on a coisotropic submanifold of a symplectic manifold is a generalization of the Lagrangian intersection investigated by Weinstein. In a similar way as Weinstein's argument, we replace the leaf-wise intersections by zero points of some closed 1-form, and show the same result as Moser's on the existence of leaf-wise intersections under different conditions.

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On vanishing Fermat quotients and a bound of the Ihara sum

Igor E. Shparlinski

2013 Volume 36 Issue 1 Pages 99-108
Published: 2013
Released on J-STAGE: April 04, 2013

DOIhttps://doi.org/10.2996/kmj/1364562722

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We improve an estimate of A. Granville (1987) on the number of vanishing Fermat quotients q_p (ℓ) modulo a prime p when ℓ runs through primes ℓ ≤ N. We use this bound to obtain an unconditional improvement of the conditional (under the Generalised Riemann Hypothesis) estimate of Y. Ihara (2006) on a certain sum, related to vanishing Fermat quotients. In turn this sum appears in the study of the index of certain subfields of cyclotomic fields Q(exp(2πi/p²)).

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Formal group laws for multiple sine functions and applications

Shin-ya Koyama, Nobushige Kurokawa

2013 Volume 36 Issue 1 Pages 109-118
Published: 2013
Released on J-STAGE: April 04, 2013

DOIhttps://doi.org/10.2996/kmj/1364562723

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We investigate addition relations for multiple sine functions from the view point of formal group laws. We find that the functions which appear in the coefficients are related to classical Eisenstein serires. As application we obtain a limit formula for automorphic forms.

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Recurrence relations for Super-Halley's method with Hölder continuous second derivative in Banach spaces

Maroju Prashanth, Dharmendra K. Gupta

2013 Volume 36 Issue 1 Pages 119-136
Published: 2013
Released on J-STAGE: April 04, 2013

DOIhttps://doi.org/10.2996/kmj/1364562724

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The aim of this paper is to study the semilocal convergence of the Super-Halley's method used for solving nonlinear equations in Banach spaces by using the recurrence relations. This convergence is established under the assumption that the second Frëchet derivative of the involved operator satisfies the Hölder continuity condition which is milder than the Lipschitz continuity condition. A new family of recurrence relations are defined based on two constants which depend on the operator. An existence-uniqueness theorem and a proori error estimates are provided for the solution x*. The R-order of the method equals to (2 + p) for p ∈ (0,1] is also established. Three numerical examples are worked out to demonstrate the efficacy of our approach. On comparison with the results obtained for the Super-Halley's method in [3] by using majorizing sequence, we observed improved existence and uniqueness regions for the solution x* by our approach.

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A notion of Δ-multigenus for certain rank two ample vector bundles

Enrique Arrondo, Antonio Lanteri, Carla Novelli

2013 Volume 36 Issue 1 Pages 137-153
Published: 2013
Released on J-STAGE: April 04, 2013

DOIhttps://doi.org/10.2996/kmj/1364562725

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A notion of "delta-genus" for ample vector bundles $\mathcal{E}$ of rank two on a smooth projective threefold X is defined as a couple of integers (δ₁, δ₂). This extends the classical definition holding for ample line bundles. Then pairs (X, $\mathcal{E}$) with low δ₁ and δ₂ are classified under suitable additional assumptions on $\mathcal{E}$.

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Contact metric structures on S³

Michael Markellos, Charalambos Tsichlias

2013 Volume 36 Issue 1 Pages 154-166
Published: 2013
Released on J-STAGE: April 04, 2013

DOIhttps://doi.org/10.2996/kmj/1364562726

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In this paper, we construct a new family of contact metric structures on the unit sphere S³. Especially, the above family has the property that ∇_ξτ = 2ατφ.

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Conformally natural extensions in view of dynamics

Yunping Jiang, Sudeb Mitra, Zhe Wang

2013 Volume 36 Issue 1 Pages 167-173
Published: 2013
Released on J-STAGE: April 04, 2013

DOIhttps://doi.org/10.2996/kmj/1364562727

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We give an easy description of the barycentric extension of a map of the unit circle to the closed unit disk using some ideas from dynamical systems. We then prove that every circle endomorphism of the unit circle of degree d ≥ 2 (with a topological expansion condition) has a conformally natural extension to the closed unit disk which is real analytic on the open unit disk. If the endomorphism is uniformly quasisymmetric, then the extension is quasiconformal.

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Chern classes and the Rost cohomological invariant

Nobuaki Yagita

2013 Volume 36 Issue 1 Pages 174-178
Published: 2013
Released on J-STAGE: April 04, 2013

DOIhttps://doi.org/10.2996/kmj/1364562728

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Extensions of the Euler-Satake characteristic determine point singularities of orientable 3-orbifolds

Ryan Carroll, Christopher Seaton

2013 Volume 36 Issue 1 Pages 179-188
Published: 2013
Released on J-STAGE: April 04, 2013

DOIhttps://doi.org/10.2996/kmj/1364562729

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We compute the extensions of the Euler-Satake characteristic of a closed, effective, orientable 3-orbifold corresponding to free and free abelian groups in terms of the number and type of point singularities of the orbifold. Using these computations, we show that the free Euler-Satake characteristics determine the number and type of point singularities, and that it takes an infinite collection of free Euler-Satake characteristics to do so. Additionally, we show that the stringy orbifold Euler characteristic determines all of the free abelian Euler-Satake characteristics for an orbifold in this class.

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