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An Explicit Formula for the Discrete Power Function Associated with Circle Patterns of Schramm Type

Hisashi Ando, Mike Hay, Kenji Kajiwara, Tetsu Masuda

2014 Volume 57 Issue 1 Pages 1-41
Published: 2014
Released on J-STAGE: April 15, 2014

DOIhttps://doi.org/10.1619/fesi.57.1

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We present an explicit formula for the discrete power function introduced by Bobenko, which is expressed in terms of the hypergeometric τ functions for the sixth Painlevé equation. The original definition of the discrete power function imposes strict conditions on the domain and the value of the exponent. However, we show that one can extend the value of the exponent to arbitrary complex numbers except even integers and the domain to a discrete analogue of the Riemann surface. Moreover, we show that the discrete power function is an immersion when the real part of the exponent is equal to one.

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On the Stability of a Linear Retarded Differential-Difference Equation

Hiroyuki Nakajima

2014 Volume 57 Issue 1 Pages 43-56
Published: 2014
Released on J-STAGE: April 15, 2014

DOIhttps://doi.org/10.1619/fesi.57.43

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In the present paper, we give a necessary and sufficient condition for the zero solution of a linear retarded system dx(t)/dt = Ax(t) + Bx(t − τ) to be asymptotically stable. Here A is a real-valued n × n matrix and B = bI, where b is a scalar parameter and I is the n × n unit matrix.
The stability analysis is reduced to deriving a necessary and sufficient condition for all the roots of a characteristic equation z − α − βe^−z = 0 to have negative real parts. Here α is a complex number defined by α = τλ with an eigenvalue λ of A, and β = τb. Our stability criterion is a natural extension of that for the widely-known case where α is a real number.

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Higher Order Asymptotic Expansion for the Heat Equation with a Nonlinear Boundary Condition

Tatsuki Kawakami

2014 Volume 57 Issue 1 Pages 57-89
Published: 2014
Released on J-STAGE: April 15, 2014

DOIhttps://doi.org/10.1619/fesi.57.57

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We consider the heat equation with a nonlinear boundary condition: (P) ∂_tu = Δu in R₊^N × (0,∞), ∂_νu = κ|u|^p−1u on ∂ R₊^N × (0,∞), u(x,0) = φ (x) in R₊^N, where R₊^N = {x= (x′,x_N) ∈ R^N: x_N > 0}, N ≥ 2, ∂_t = ∂/∂_t, ∂_ν = −∂/∂x_N, κ ∈ R, and p > 1 + 1/N. Let u be a solution of (P) satisfying sup_t>0(1 + t)^{(N/2)(1−1/q)}[||u(t)||_L^q(R₊^N) + t^1/(2q)||u(t)||_{L^q(∂R₊^N)}] < ∞, q ∈ [1,∞]. In this paper, under suitable assumptions of the initial function φ, we establish the method of obtaining higher order asymptotic expansions of the solution u as t → ∞.

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On Delay-Dependent Stability Conditions for a Three-Term Linear Difference Equation

Jan Čermák, Petr Tomášek

2014 Volume 57 Issue 1 Pages 91-106
Published: 2014
Released on J-STAGE: April 15, 2014

DOIhttps://doi.org/10.1619/fesi.57.91

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We present a new type of necessary and sufficient conditions for asymptotic stability of a three-term linear delay difference equation. The derived conditions are quite explicit and involve, among others, a critical value of the delay, when the studied equation loses its asymptotic stability property.

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A Discrete Analogue of Periodic Delta Bose Gas and Affine Hecke Algebra

Yoshihiro Takeyama

2014 Volume 57 Issue 1 Pages 107-118
Published: 2014
Released on J-STAGE: April 15, 2014

DOIhttps://doi.org/10.1619/fesi.57.107

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We consider an eigenvalue problem for a discrete analogue of the Hamiltonian of the non-ideal Bose gas with delta-potentials on a circle. It is a two-parameter deformation of the discrete Hamiltonian for joint moments of the partition function of the O'Connell-Yor semi-discrete polymer. We construct the propagation operator by using integral-reflection operators, which give a representation of the affine Hecke algebra. We also construct eigenfunctions by means of the Bethe ansatz method. In the case where one parameter of our Hamiltonian is equal to zero, the eigenfunctions are given by specializations of the Hall-Littlewood polynomials.

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Propagation of Regular Singularities in a Complex Analytic Characteristic Initial Value Problem

Toru Tsutsui

2014 Volume 57 Issue 1 Pages 119-161
Published: 2014
Released on J-STAGE: April 15, 2014

DOIhttps://doi.org/10.1619/fesi.57.119

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We consider a characteristic initial value problem of a class of second order linear partial differential equation with regular singular initial data in the complex domain. We express the solution by means of series of hypergeometric functions, and clarify the behavior of the solution near the singularities. Our construction owes much to the contiguity relations of hypergeometric functions.

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Liouville Theorem for Lichnerowicz Equation on Complete Noncompact Manifolds

Liang Zhao

2014 Volume 57 Issue 1 Pages 163-172
Published: 2014
Released on J-STAGE: April 15, 2014

DOIhttps://doi.org/10.1619/fesi.57.163

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In this paper, we study the gradient estimates for positive solutions to the Lichnerowicz equation Δ_fu + hu = Au^p + Bu^−q, where h, p, q, A, B are real constants and p > 1, q > 1. Moreover, by these gradient estimates, we can get some very interesting Liouville theorems.

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