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 with positive curvature, we also obtain many more different solutions. In particular, we obtain spiral wave solutions with sign-changing curvature and with negative curvature.

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.

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 shown that, under some assumptions on $p$ and $f$, the solution for the discrete problem has the second order accuracy $O(h^2)$, where $h$ denotes the maximum mesh size. Finally, observations are given.

We propose a method with raised speed of convergence for simultaneous extraction of a part of all roots of polynomials. The method is efficient for the polynomials which have well separated real roots. The proof of local convergence is shown and numerical results are given.

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.

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.

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 other models of equations describing the magnetization switching by spin-polarized current and show that our method can be used to solve them.

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.