TRIANGLE CENTERS DEFINED BY QUADRATIC POLYNOMIALS

We consider a family of triangle centers whose barycentric coordinates are given by quadratic polynomials, and determine the lines that contain an infinite number of such triangle centers. We show that for a given quadratic triangle center, there exist in general four principal lines through this center. These four principal lines possess an intimate connection with the Nagel line

Decomposability of polynomial valued 2-forms

We give a characterization of decomposable polynomial valued 2-forms in terms of their components. Such 2-forms must satisfy some cubic condition in addition to Plücker's quadratic relation. Several GL(n, K)×GL(m, K)-invariant varieties naturally appear during this characterization, and we state the mutual relation of these varieties and study their geometric properties in detail

An algorithm to calculate plethysms of Schur functions : Dedicated to Professor Motoyoshi Sakuma on his 70th birthday

We present an algorithm to calculate plethysms of Schur functions which is fitted for computers, and give the decomposition table of plethysms {λ}⨂{μ} up to total degree 12

An example of convex heptagon with Heesch number one

We give an example of convex heptagon whose Heesch number is just equal to one, and among fourteen kinds of edge-to-edge coronas of this tile we present some of them, one of which admits a family of continuous deformations

Invariant subvarieties of the 3-tensor space C^2⨂C^2⨂C^2

We classify G-invariant subvarieties of the 3-tensor space C^2⨂C^2⨂C^2 that are defined by polynomials with degree≤6,where G=GL(2,C)×GL(2,C)×GL(2,C). We also calculate the character fo S^p(C^2⨂C^2⨂C^2), determine the generators of each irreducible component of S^p(C^2⨂C^2⨂C^2), and obtain some curious identities between them that play a fundamental role in classifying invariant subvarieties

SU (3)/SO (3) のガウス方程式の解と概解

We give new solutions and almost solutions of the Gauss equation of the Riemannian symmetric spaces SU (3)/SO (3) and its non-compact dual in codimensions 4 and 5,which improve the previously known estimates on the codimension. We also give experimental estimates on the infimum of the norm ∥γ_r (α) ±R∥ for each codimension r, where R is the curvature of SU (3)/SO (3), and α runs all over the space of second fundamental forms

外積代数におけるガウス方程式について

We define the Gauss equation in the exterior algebra, and state a relation to the original Gauss equation appearing in the theory of Riemannian submanifolds. We also state several necessary (and sufficient) conditions in order that this equation admits a solution mainly in the case codimension=1 and 2

On the Curvature of the Homogeneous Space U (n+1)/U (n)

We determine all torsion free invariant affine connections on the homogeneous space S^=U(n+1)/U(n), and characterize their curvatures in terms of the polynomials of their components in the space of curvature-like tensors. The essential difference between the case n=1 and n≥2 is explained in detail from the standpoint of flat affine geometry

Tilings of the 2-dimensional sphere by congruent right triangles

We classify all spherical tilings consisting of congruent right triangles. There exist five sporadic types and four series of such tilings. We also exhibit the figures of these tilings

合同な四角形による球面のタイリングの例

We give examples of monohedral tilings of the 2-dimensional sphere by quadrangles, three of whose edges have the same length. We show that to classify monohedral tilings by quadrangles with this property, we must consider a condition between four angles, in addition to combinatorial consideration, which we developed in [8] for the case of triangles