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

Decomposition of splitting invariants in split real groups

To a maximal torus in a quasi-split semi-simple simply-connected group over a local field of characteristic 0, Langlands and Shelstad construct a cohomological invariant called the splitting invariant, which is an important component of their endoscopic transfer factors. We study this invariant in the case of a split real group and prove a decomposition theorem which expresses this invariant for a general torus as a product of the corresponding invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants between different tori in the given real group.Comment: 22 page

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

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

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

Lowest dimensional example on non-universality of generalized In\"on\"u-Wigner contractions

We prove that there exists just one pair of complex four-dimensional Lie algebras such that a well-defined contraction among them is not equivalent to a generalized IW-contraction (or to a one-parametric subgroup degeneration in conventional algebraic terms). Over the field of real numbers, this pair of algebras is split into two pairs with the same contracted algebra. The example we constructed demonstrates that even in the dimension four generalized IW-contractions are not sufficient for realizing all possible contractions, and this is the lowest dimension in which generalized IW-contractions are not universal. Moreover, this is also the first example of nonexistence of generalized IW-contraction for the case when the contracted algebra is not characteristically nilpotent and, therefore, admits nontrivial diagonal derivations. The lower bound (equal to three) of nonnegative integer parameter exponents which are sufficient to realize all generalized IW-contractions of four-dimensional Lie algebras is also found.Comment: 15 pages, extended versio