町頭 義朗

コンパスと定規による,正17角形の作図法はガウスにより発見されたものである。この発見を記念して,ガウスの墓には,正17角形が描かれているという。この方法がガロア理論へと結びついたのであるが,今では,正17角形の作図にはガロア理論に基づく方法を述べた本が多い。本論文では,ガロア理論という言葉を用いずに,高校生にもわかる正17角形の描き方を述べる。A construction by using only a ruler without scale and a compass is very interesting. There have been a lot of construction problems from ancient Greece era. In 1796, C. F. Gauss discovered how to construct a regular heptadecagon. His idea had joined to the Galois theory. Now, almost books introduce the construction of a regular heptadecagon by using the Galois theory. In this paper, we study how to construct a regular heptadecagon without using the word "Galois theory".

In the second author's previous work, we considered an approximation of a catenoid constructed from even truncated cones that maintains minimality in a certain sense. In this paper, we consider such an approximation consisting of odd truncated cones that maintains minimality in the same sense. Through this procedure, we obtain a discrete curve approximating a catenary by exploiting the fact that it is the function that generates a catenoid. In this investigation, the theory of the Gauss hypergeometric functions plays an important role.

空間内で,半径が等しく,お互いに平行で,お互いの中心を結ぶ直線がどちらにも直交しているような2つの円を考える。正の実数voに対し,その2つの円を境界とし,内部の体積がvoに等しいような曲面の族の中で,表面積が極小であるものは,アンデュロイドと呼ばれる曲面である。本論文では,アンデュロイドを複数個の円錐台で,"体積を保ったまま"かつ"面積極小のまま"近似する方法を調べる。An unduloid is a surface of revolution whose area is minimizing in the surfaces conserving the constant volume enclosed inside them. In this paper, we consider an approximation of an unduloid keeping the minimality by a combination of some truncated cones.

平面上に4点を与えたとき,その4点を通る放物線が存在するための必要十分条件は,4点が凸四角形の頂点となることである。また一般に,その時2本の放物線が存在する。放物線はコンパスと定規だけでは作図出来ないが,4点が与えられたときに,その4点を通る放物線の焦点と準線をコンパスと定規で求めることは可能である。本論文では,その求め方を論じ,幾何学ソフトウェアKSEG で,4点を通る放物線を描くやり方を述べる。Let A, B, C, D be four points in E2. In general, there are two parabolas through the four points, if and only if the four points are the vertices of a convex quadrilateral. In this paper, we study how to draw the parabolas through the given 4 points by a geometric software"KSEG".

鎖の両端を固定し,鎖自体が重力により吊り下がった時になす曲線を懸垂線といい,懸垂線を回転させてできる曲面を懸垂面という。懸垂面は回転面であるような極小曲面として知られており,吊り橋やアーチの建設に役立てられている。本論文では,懸垂面を複数個の円錐台で,"面積極小のまま"近似する方法を調べる。It is well-known that a catenoid is a minimal surface and the shape which a soap film between two parallel circles forms. In this article,we consider the approximation of a catenoid by combinations of some truncated cones keeping the minimality in a certain sense.

The catenary is the curve which a hanging chain forms, that is, mathematically, the graph of the function t ↦ c cosh t/c for a constant c > 0. The study of catenaries is applied to the design of arches and suspension bridges. The surface of revolution generated by a catenary is called a catenoid. It is well-known that a catenoid is a minimal surface and the shape which a soap film between two parallel circles forms. In this article, we consider the approximation of a catenoid by combinations of some truncated cones keeping the minimality in a certain sense. In investigating the minimal combinations, the theory of the Gauss hypergeometric functions plays an important role.

The study of triangles or ellipses in a plane is very interesting and we have been studying a lot of character of them since B.C.. On the other hand, geodesic triangles in a Riemannian manifold plays an important role on the study of the relation between the topology and the curvature of the manifold. In this paper, we study some properties of ellipses in a non-positive curved Riemannian surface. Moreover by using the properties, we prove that a well-known theorem for triangles in a plane holds for them in a non-positive curved Riemmanian surface.平面上の三角形や楕円などは非常に面白い性質を持ち,古代ギリシャの時代から研究されている。一方, リーマン多様体の研究は19世紀ぐらいから始められ,リーマン多様体上の測地三角形は,リーマン多様体の曲率とその位相の関係を調べる上で,非常に重要な役割を果たしている。本論文は, 非正曲率曲面上の三角形や楕円の性質を調べ,平面幾何で知られている定理を,非正曲率曲面上の定理に拡張することを目的とする。

空間内に,半径の等しい2つの平行な円で,2円の中心を結ぶ直線がそれぞれの円と直交しているものを考える。その2つの円を境界とする極小曲面は(あれば)懸垂面であることが知られている。しかし,lを円の間の距離の半分,aを円の半径とするとき,もしも a/l=c_0 ならば,そういう懸垂面は存在しない。ただし,c_0はcosh(1/x)-(1/x)sinh(1/x) という関数の零点である。 本論文では,a/l = c_0 の時の懸垂面の面積の振る舞いについて調べ,その時の懸垂面は(平均曲率が0という意味での)極小曲面ではあるが,面積が極小となっていないことを実証する。Let us consider two circles in R^3 such that the radii are the same and the line joining two centers is perpendicular to the circles. The minimal surfaces bounded by the two circles are catenoids. However, if we put l the half of the distance between two circles and a the radii of the circles, there does not exist a catenoid bounded by the circles in the case where a/l < c_0, where c_0 is the zero point of the function cosh(1/x)-(1/x)sinh(1/x).In this paper, we study the behavior of the area of the catenoid in the case where a/l = c_0.

We prove the compactness of the imbedding of the Sobolev space W-0(1,2) (Ohm )into L-2 (Ohm) for any relatively compact open subset Ohm of an Alexandrov space. As a corollary, the generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be approximated by the Laplacian induced from the DC-structure on the Alexandrov space. We also prove the existence of the locally Holder continuous heat kernel.

We study the space of directions on a length space and examine examples having particular spaces of directions. Then we generalize the notion of total excess on length spaces satisfying some suitable conditions, which we call good surfaces. For good surfaces we generalize the Euler characteristic, and prove the generalized Gauss-Bonnet Theorem and other relations between the total excess and the Euler characteristic. Furthermore, we see that the Gaussian curvature can be defined almost everywhere on a good surface with non-positive total excess. © Springer-Verlag 2001.

We generalize the Gauss-Bonnet theorem for Alexandrov surfaces and show that we can define the Gaussian curvature almost everywhere on an Alexandrov surface.

We generalize the Toponogov hinge theorem and the Alexandrov convexity to the context of radial curvature, and study complete open Riemannian manifolds of non-negative radial curvature. © 1994 by Pacific Journal of Mathematics.

We generalize Toponogov’s theorem to the context of radial curvature and obtain corresponding generalizations of classical sphere theorems. © 1993 American Mathematical Society.