Research activity information

• A note on zeros of the Lagrange interpolation polynomial of the function 1/(z-c)

  Yasuo Kageyama

  In general, zeros of a Lagrange interpolation polynomial can be calculated only numerically, but if the interpolated function is given as the form 1/(z-c) and the sampling points are equally distributed on an ellipse, then the zeros can be represented explicitly and they are also equally distributed on an ellipse of common foci. We will show this fact and prove a theorem that presents a sufficient condition for this method to be generalized.

  The Japan Society for Industrial and Applied Mathematics, Sep. 2003, Transactions of the Japan Society for Industrial and Applied Mathematics, 13(3) (3), 391 - 402, Japanese

  [Refereed]

  Scientific journal


• Generalization of the Bernstein operator

  KAGEYAMA YASUO

  Dec. 2002, Analysis, Combinatorics and Computing, 265 - 274, English

  [Refereed]

  International conference proceedings


• Regularity of the coefficient matrix appearing in an interpolation based on the charge simulation method

  KAGEYAMA YASUO

  In a former paper, Inoue introduced an interpolation based on a scheme in the charge simulation method, but he did not mention the problem whether the coefficient matrix derived there is regular or not. We will first point out unnaturalness of his scheme and introduce another interpolation based on so-called Murota's invariant scheme, which is a slightly modified version of Inoue's scheme. Then we will guarantee that the coefficient matrix for our scheme is always regular.

  The Japan Society for Industrial and Applied Mathematics, Dec. 2001, 日本応用数理学会論文誌, 11(4) (4), 171 - 177, Japanese

  [Refereed]

  Scientific journal


• A new class of modified Bernstein operators

  Y Kageyama

  Nov. 1999, JOURNAL OF APPROXIMATION THEORY, 101(1) (1), 121 - 147, English

  [Refereed]

  Scientific journal

  Link to Kobe Univ. Repository (Kernel)


• Generalization of the left Bernstein quasi-interpolants

  Y Kageyama

  Aug. 1998, JOURNAL OF APPROXIMATION THEORY, 94(2) (2), 306 - 329, English

  [Refereed]

  Scientific journal

  Link to Kobe Univ. Repository (Kernel)

• Approximation Method by a Generalization of the Bernstein Polynomial

  Kageyama Yasuo

  Kobe University, Jul. 2001, Review of Kobe University of Mercantile Marine. Part II, Maritime studies, and science and engineering, 49, 33 - 40, Japanese

• Contemporary topics in mathematics and informatics

  KAGEYAMA YASUO

  Others, Kwansei Gakuin University Press, Jun. 2009, Japanese, 「等間隔標本点を用いた多項式近似は実用にならない」という従来の常識に対し，「収束精度・数値的安定性・計算量の観点からみて実用的な近似法が存在する」という新たな知見を提示した．

  Scholarly book

• Japan Society of Industrial and Applied Mathematics

• Analytical study for mean curvature flow by using an numerical algorithm

  ISHI Katsuyuki, MARUO Kenji, KAGEYAMA Yasuo, KUWAMURA Masataka, NAITO Yuki, ADACHI Tadayoshi

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Kobe University, 2005 - 2007

  Ishii studied the numerical algorithm of the mean curvature flow propsed by Benced , Merriman and Other in 1992. He prove the convergence of this algorithm such that the mean curvature flow equation is somehow directly derived form the algorithm. He also derived the optimal rate of convergence in the case of the smooth and compact motion by mean curvature. Ishii also studied the approximation of the noun curvature flow via the Allen -Cahn equation. He obtinaed the optimal rate of convergence to the smooth and compact motion by mean curvature. Maruo considered the structure of unbouned and radial viscosity solutions far semilinear elliptic equation He completely classified the structure of solutions in term of the asymptotic behavior at infinity of solutions Naito investigated the sel-similar solutions for semiliear heat equations with power nonlinearity and showed that the solutions behaves asymptotically like self-similar solutions. In the case of Soholev critical nonlinearity, he gave a sufficient condition for the existence of solutions accruing Type II blow up. He considered some semilinear elliptic equations. He obtained multiple existence of solutions. Kuwamura studied some reaction-diffusion equations with gradient/skew-gradient structure. Noting that such equations has Hamiltonian structure, he derived a necessary condition for the existence of nontrivial pattern wioth space periodicity in terms of Turing stability. By using some ODE with time-delay, he proposed a mathematical model fo an environmen problem. Adachi refined the resolvent estimates for N-body Stark Hamiltonian, which is known as the limit absorbing principle, by using some localization technique.

  Competitive research funding


• Ｒでの非線形退化楕円型偏微分方程式の非有界な粘性解の構造の研究

  丸尾 健二

  科学研究費補助金／基盤研究(C), 2005

  Competitive research funding


• Studies on the applications of the theory of viscosity solutions to some singular perturbation problems

  ISHII Katsuyuki, MARUO Kenji, KAGEYAMA Yasuo

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), 2002 - 2004

  Katsuyuki Ishii studied a numerical algorithm for motion by mean curvature, which is proposed by Bence, Merriman and Osher and obtained the following results 1.I gave a proof of convergence showing how the mean curvature flow equation is derived from this algorithm. (This is a joint work with Yoko Goto and Takayoshi Ogawa.) 2.I obtained the rate of convergence of this algorithm in the case of smoth motion by mean curvature. I also showed the optimality in the case of a circle evolving by curvature. Kenji Maruo studied semilinear degenerate elliptic partial differential equations in the plane. He obtained the following 3.Assuming that the coefficients of the equation are radially symmetric, he proved that, under some growth conditions at infinity, the continuous viscosity solutions are radially symmetric Yasuo Kageyama obtained the rate of convergence and some properties of modified Bernstein polynomials


• 関数近似の新しい手法の開発

  影山 康夫

  日本学術振興会, 科学研究費助成事業, 若手研究(B), 神戸大学(海事科学部), 2002 - 2003

  1.代用電荷法は、一種の「radial basis function」による関数近似法と見なすことができる。そして、代用電荷法における電荷点配置問題が、関数1/(z-c)に対するLagrange補間多項式の零点分布と関連があるように思えるため、これについて詳しく研究した。まだ発展の余地はあるが、標本点が楕円上に等間隔に配置されている場合については顕著な結果を導くことができたので、昨年度、それを中心とした論文「関数1/(z-c)のLagrange補間多項式の零点に関する一考察」を日本応用数理学会論文誌に投稿し、今年度、掲載された。 2.Bernstein作用素は代数多項式で表現され、形状保存性を持つ線形近似作用素の中で(ある意味で)最良であることが知られている.そこで、この代数多項式に関する結果に対し、三角多項式で表現される正線形近似作用素に関しても、何らかの最適化問題を考察することができるだろうという着想を得た。そして実際、その最適化問題を定式化した上で「Fejer-Korovkin作用素」と呼ばれるものが最良であるという結果を導くことができた。この結果に対し、Fourier級数・Fourier積分・離散Fourier変換との関連や物理的意味等について、現在研究している最中であり、結果がまとまり次第、Journal of Approximation Theoryのような専門誌に投稿する予定である。 3.他にも、「最良の関数近似法は何か」という問題に対し、様々なアイディアや手法を発見した。まだ、それらを具体化する段階には至っていないが、今後順調に事が運べば非常に意義深い成果を出すことができそうだという手応えを得ている。


• Research in viscosity solutions using the method of Functional Analysis.

  MARUO Kenji, INOUE Tetuo, ISHII Katsuyuki, TOMITA Yoshihito, MIYAKODA Tuyako, KAGEYAMA Yasuo

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Kobe University of Mercantile Marine, 1998 - 1999

  We consider the Dirichelet problem for a semilinear degenerate elliptic equation (DP) : -g(|x|)Δu+f(|x|, u(x)) = 0, and Boundary Condition where N【greater than or equal】2 and g(|x|), f(|x|, u) are continuous. We discuss the problem (DP) under the following assumption : 1)g is nonnegative. 2)f is strictly monotone for u. We first define a standard viscosity solution by the viscosity solution such that if g(|x|) = 0 then f(|x|, u(x)) = 0. Then we can prove that the any continuous standard viscosity solution is the radial solution and it is unique. We add an assumption : 3)∫ィイD1a-0ィエD1gィイD1-1ィエD1(s)ds = ∞ or ∫ィイD2a+0ィエD2gィイD1-1ィエD1(s)ds = ∞ for any a : g(a) = 0. Then We obtain that any continuous viscosity solution is the radial solution and it is unique. If the assumption 3) is not satisfied there exist examples such that the continuous viscosity solutions are not unique. Here, the domain is a bounded boall in n-dimension space. We next state the existence and uniqueness of the continuous unbounded viscosity solution in RィイD12ィエD1. We use the order of the infinite neighborhood of the solution as the boundary condition. We know that the existence or nonexistence of the solution are dependent on a kind of the order of the solution. Moreover, we get the results which the uniqueness or non-uniqueness are also dependent on a kind of the order of the solution. In case, we assume that g, f is sufficiently smooth. We now show the existence of a continuous viscosity solution to quasi-semilinear degenerate elliptic problem. Here, g(|x|, u), f(|x|, u) are continuous and f is strictly monotone for u. Moreover, we assume there exists an implicite function of f = 0 and the implicite function holds some smoothness. Then we can prove the existence of the continuous viscosity solution. But it is difficult to prove the uniqueness of the solution.