1.
On the Steadily Rotating Spirals - Guo, Jong-Shenq; Nakamura, Ken-Ichi; Ogiwara, Toshiko; Tsai, Je-Chiang
We study an autonomous system of two first order ordinary differential equations. This system arises from a model for steadily rotating spiral waves in excitable media. The sharply located spiral wave fronts are modeled as planar curves. Their normal velocity is assumed to depend affine linearly on curvature. The spiral tip rotates along a circle with a constant positive rotation frequency. The tip neither grows nor retracts tangentially to the curve. With rotation frequency as a parameter, we obtain the complete classification of solutions of this system. Besides providing another approach to derive the results obtained by Fiedler-Guo-Tsai for spirals...
2.
A Note on Discrete Convexity and
Local Optimality - Ui, Takashi
One of the most important properties of a convex function is that a local optimum is also a global optimum. This paper explores the discrete analogue of this property. We consider arbitrary locality in a discrete space and the corresponding local optimum of a function over the discrete space. We introduce the corresponding notion of discrete convexity and show that the local optimum of a function satisfying the discrete convexity is also a global optimum. The special cases include discretely-convex, integrally-convex, M-convex, $\text{M}^\natural$-convex, L-convex, and $\text{L}^\natural$-convex functions.
3.
A Mathematical Theory for Numerical Treatment
of Nonlinear Two-Point Boundary Value Problems - Yamamoto, Tetsuro; Oishi, Shin'ichi
This paper gives a unified mathematical theory for numerical treatment of two--point boundary value problems of the form $-(p(x)u')'+f(x,u,u')=0,\ a\le x \le b,\ \alpha_0 u(a)-\alpha_1 u'(a)=\alpha,\ \beta_0 u(b)+\beta_1 u'(b)=\beta,\ \alpha_0,\alpha_1,\beta_0,\beta_1 \ge 0,\ \alpha_0+\alpha_1>0,\ \beta_0+\beta_1>0,\ \alpha_0 +\beta_0 >0$. Firstly, a unique existence of solution is shown with the use of the Schauder fixed point theorem, which improves Keller's result \cite{Keller}. Next, a new discrete boundary value problem with arbitrary nodes is proposed. The unique existence of solution for the problem is also proved by using the Brouwer theorem, which extends some results in Keller \cite{Keller} and Ortega--Rheinboldt \cite{Ortega}. Furthermore, it is...
5.
Determination of the Babuska-Aziz Constant
for the Linear Triangular Finite Element - Kikuchi, Fumio; Liu, Xuefeng
We explicitly determine the Babuska-Aziz constant, which plays an essential role in the interpolation error estimation of the linear triangular finite element. The equation for determination is the transcendental equation $t + \tan t = 0$, so that the solution can be numerically obtained with desired accuracy and verification. Such highly accurate approximate values for the constant can be widely used for a priori and a posteriori error estimations in adaptive computation and/or numerical verification.
6.
Traveling Curved Fronts of Anisotropic Curvature Flows - Marutani, Yoshiko; Ninomiya, Hirokazu; Weidenfeld, Rémi
In this paper, the anisotropic curvature flows with driving force are considered. The existence of traveling curved fronts is shown by constructing supersolutions and subsolutions. By the advantage of this method, their global stability is also proved. In the last section the profiles of the traveling fronts are discussed when the anisotropy becomes strong and converges to a non-smooth function.
7.
On a Model of Magnetization Switching by Spin-Polarized Current - Hamdache, K.; Hamroun, D.; Tilioua, M.
This paper is concerned with global existence of weak solutions to a model equations of magnetization reversal by spin-polarized current in a layer introduced in [19]. The local magnetization of the ferromagnet satisfies the usual Landau-Lifshitz equation which is coupled to the nonlinear heat equation satisfied by the spin accumulation field defined in all the layer. The coupling is due to the contact interaction energy. We use an hyperbolic regularization method with penalization of the saturation constraint satisfied by the local magnetization to prove global existence result, in any finite time interval, of weak solutions with finite energy. We present...
8.
A Sweep-Line Algorithm for the Inclusion Hierarchy among Circles - Kim, Deok-Soo; Lee, Byunghoon; Sugihara, Kokichi
Suppose that there are a number of circles in a plane and some of them may contain several smaller circles. In this case, it is necessary to find the inclusion hierarchy among circles for the various applications such as the simulation of emulsion and diameter estimation for wire bundles. In this paper, we present a plane-sweep algorithm that can identify the inclusion hierarchy among the circles in $O(n\log n)$ time in the worst-case. Also, the proposed algorithm uses the sweep-line method and a red-black tree for the efficient
computation.
9.
Comparison Between Passive and Active Control of a Non-Linear Dynamical System - El-Serafi, S. A.; Eissa, M. H.; El-Sherbiny, H. M.; El-Ghareeb, T. H.
Vibrations and dynamic chaos should be controlled in structures and machines. The wing of the airplane should be free from vibrations or it should be kept minimum. To do so, two main strategies are used. They are passive and active control methods. In this paper we present a mathematical study of passive and active control in some non-linear differential equations describing the vibration of the wing. Firstly, non-linear differential equation representing the wing system subjected to multi-excitation force is considered and solved using the method of multiple scale perturbation. Secondly, a tuned mass absorber (TMA) is applied to the system...
10.
A Finite Element Scheme for Two-Layer Viscous Shallow-Water Equations - Kanayama, Hiroshi; Dan, Hiroshi
In this paper, the two-layer viscous shallow-water equations are derived from the threedimensional Navier-Stokes equations under the hydrostatic assumption. It is noted that the combination of upper and lower equations in the two-layer model produces the classical one-layer equations if the density of each layer is the same. The two-layer equations are approximated by a finite element method which follows our numerical scheme established for the one-layer model in 1978. Finally, it is numerically demonstrated that the interfacial instability generated when the densities are the same can be eliminated by providing a sufficient density difference.
11.
Numerical Green's Function Method Based on the DE Transformation - Mori, Masatake; Echigo, Toshihiko
A method for numerical solution of boundary value problems with ordinary differential equation based on the method of Green's function incorporated with the double exponential transformation is presented. The method proposed does not require solving a system of linear equations and gives an approximate solution of very high accuracy with a small number of function evaluations. The error of the method is $O\left(\exp\left(-C_1N/\log(C_2N)\right)\right)$ where $N$ is a parameter representing the number of function evaluations and $C_1$ and $C_2$ are some positive constants. Numerical examples also prove the high efficiency of the method. An alternative method via an integral equation is...
12.
Mathematical Modeling Analyses for Obtaining an Optimal Railway Track Maintenance Schedule - Oyama, Tatsuo; Miwa, Masashi
Railway track irregularities need to be kept at a satisfactory level by taking appropriate maintenance activities. This paper aims at obtaining an optimal maintenance schedule for improving the railway track irregularities using all-integer linear programming (AILP) optimization model analyses.
Firstly, we try to predict a change of surface irregularities by investigating the transition process through degradation and restoration model analyses. Then we develop an AILP model for obtaining an optimal schedule of multiple tie tamper (MTT) operation. The model takes both maintenance costs and the level of surface irregularities that reflects riding quality and safety into account, then finally gives an...
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