Yusuke Sakane

Chebyshev blossoming in Müntz spaces: Toward shaping with Young diagrams

The notion of a blossom in extended Chebyshev spaces offers adequate generalizations and extra-utilities to the tools for free-form design schemes. Unfortunately, such advantages are often overshadowed by the complexity of the resulting algorithms. In this work, we show that for the case of Muntz spaces with integer exponents, the notion of a Chebyshev blossom leads to elegant algorithms whose complexities are embedded in the combinatorics of Schur functions. We express the blossom and the pseudo-affinity property in Muntz spaces in terms of Schur functions. We derive an explicit expression for the Chebyshev-Bernstein basis via an inductive argument on nested Muntz spaces. We also reveal a simple algorithm for dimension elevation. Free-form design schemes in Muntz spaces with Young diagrams as shape parameters are discussed.

Non-naturally reductive Einstein metrics on exceptional Lie groups

Abstract Given an exceptional compact simple Lie group G we describe new left-invariant Einstein metrics which are not naturally reductive. In particular, we consider fibrations of G over flag manifolds with a certain kind of isotropy representation and we construct the Einstein equation with respect to the induced left-invariant metrics. Then we apply a technique based on Grobner bases and classify the real solutions of the associated algebraic systems. For the Lie group G 2 we obtain the first known example of a left-invariant Einstein metric, which is not naturally reductive. Moreover, for the Lie groups E 7 and E 8 , we conclude that there exist non-isometric non-naturally reductive Einstein metrics, which are Ad ( K ) -invariant by different Lie subgroups K .

A Müntz type theorem for a family of corner cutting schemes

Dimension elevation process of Gelfond-Bezier curves generates a family of control polygons obtained through a sequence of corner cuttings. We give a Muntz type condition for the convergence of the generated control polygons to the underlying curve. The surprising emergence of the Muntz condition in the problem raises the question of a possible connection between the density questions of nested Chebyshev spaces and the convergence of the corresponding dimension elevation algorithms.

Gelfond-Bézier curves

We show that the generalized Bernstein bases in Muntz spaces defined by Hirschman and Widder (1949) and extended by Gelfond (1950) can be obtained as pointwise limits of the Chebyshev-Bernstein bases in Muntz spaces with respect to an interval [a,1] as the positive real number a converges to zero. Such a realization allows for concepts of curve design such as de Casteljau algorithm, blossom, dimension elevation to be transferred from the general theory of Chebyshev blossoms in Muntz spaces to these generalized Bernstein bases that we termed here as Gelfond-Bernstein bases. The advantage of working with Gelfond-Bernstein bases lies in the simplicity of the obtained concepts and algorithms as compared to their Chebyshev-Bernstein bases counterparts.

HOMOGENEOUS EINSTEIN METRICS ON GENERALIZED FLAG MANIFOLDS Sp(n)=(U(p) U(q) Sp(n p q))

We construct the Einstein equation for an invariant Riemannian metric on generalized flag manifolds Sp(n)/(U(p)×U(q)×Sp(n− p− q)). By computing a Gröbner basis for a system of polynomials on six variables, we prove that the generalized flag manifolds Sp(3)/(U(1) × U(1) × Sp(1)), Sp(4)/(U(1) × U(1) × Sp(2)) and Sp(4)/(U(2) × U(1) × Sp(1)) admit exactly three, six and two non-Kähler invariant Einstein metrics up to isometry, respectively.

HOMOGENEOUS EINSTEIN METRICS ON GENERALIZED FLAG MANIFOLDS WITH FIVE ISOTROPY SUMMANDS

We construct the homogeneous Einstein equation for generalized flag manifolds G/K of a compact simple Lie group G whose isotropy representation decomposes into five inequivalent irreducible Ad(K)-submodules. To this end, we apply a new technique which is based on a fibration of a flag manifold over another such space and the theory of Riemannian submersions. We classify all generalized flag manifolds with five isotropy summands, and we use Grobner bases to study the corresponding polynomial systems for the Einstein equation. For the generalized flag manifolds E6/(SU(4) × SU(2) × U(1) × U(1)) and E7/(U(1) × U(6)) we find explicitly all invariant Einstein metrics up to isometry. For the generalized flag manifolds SO(2l + 1)/(U(1) × U(p) × SO(2(l - p - 1) + 1)) and SO(2l)/(U(1) × U(p) × SO(2(l - p - 1))) we prove existence of at least two non-Kahler–Einstein metrics. For small values of l and p we give the precise number of invariant Einstein metrics.

On Compact Einstein Kähler Manifolds with Abundant Holomorphic Transformations

Let (M,J,g) be a compact connected Kahler manifold and let Ric(g) denote the Ricci tensor. A compact Kahler manifold (M,J,g) is said to be Einstein if Ric(g) = kg for some k ∈ R. If we denote by γ the Ricci form of (M,J,g) (γ(X,Y) = Ric(g)(X,JY)) and by ω the Kahler form, (M,J,g) is Einstein if and only if γ = kω (k ∈ R). Let H2 (M, ℝ) denote the 2nd cohomology group with the coefficients in R. It is known that the first Chern class c1 (M) of a compact Kahler manifold (M, J, g) is given by

Einstein hypersurfaces of Kählerian

{c_1}\left( M \right) = \frac{1}{{2\pi }}\left[ \gamma \right] \in {H^2}\left( {M,R} \right)