We are interested in lower and upper bounds of asymptotic irrationality measures for certain simple continued fractions. A general procedure for estimation of real numbers by rational numbers is described. It is applied for simple continued fractions with quotients satisfying some asymptotic density conditions.
Murata and Umehara gave a classification of complete flat fronts in the Euclidean 3-space and proved their orientability. Here, a flat front is a flat surface (i.e., a surface where one of the principal curvatures is identically zero) with admissible singularities. In this paper,we investigate wave fronts where one of the principal curvatures is a non-zero constant. Although they are orientable in the regular surface case, there exist non-orientable examples. We classify weakly complete ones and derive their orientability.
In the book [To2], Totaro determined Chow rings CH∗(BG)/p of classifying spaces BG for all p-groups G of order |G|≤ p4. In this paper, we compute CH∗(BG)/2 for G = 2+1+4 = D8 · D8, which has nilpotent elements.
In this paper, we characterize irreducible symmetric cones among homogeneous cones Ωof rank r by the fact that the basic relative invariants for Ωand for Ω∗ (the dual cone of Ω) both have the degrees 1, 2,... , r, up to permutations.
We introduce several topics for sequences in the unit disc of the complex plane and the space of bounded analytic functions. We characterize sequences that are the union of one interpolating subsequence and one zero subsequence. We also require the interpolation of the derivative and that it vanishes where the function does.
A harmonic, Kähler Hadamard manifold (M2m, g), m ≥ 2, with Ricci curvature Ric =−(m + 1)/2 and volume entropy ρ(M, g)= m, is biholomorphically isometric to a complex hyperbolic space of holomorphic sectional curvature −1, provided (M, g) is of hypergeometric type. A similar characterization of the real hyperbolic space and the quaternionic hyperbolic space is also obtained in terms of Ricci curvature and volume entropy, without hypergeometric assumption.
Erokhin showed that the Siegel theta series associated with the even unimodular 32-dimensional extremal lattices of degree up to three is unique. Later, Salvati Manni showed that the difference of the Siegel theta series of degree four associated with the two even unimodular 32-dimensional extremal lattices is a constant multiple of the square J2 of the Schottky modular form J, which is a Siegel cusp form of degree four and weight eight. In the present paper we show that the Fourier coefficients of the Siegel theta series associated with the even unimodular 32-dimensional extremal lattices of degrees two and three can be computed explicitly, and the Fourier coefficients of the Siegel theta series of degree four for those lattices are computed almost explicitly.
The systems of differential equations associated with the classical hypergeometric functions and the hypergeometric functions on the space of point configurations are investigated from the viewpoint of the twisted de Rham theory. In each case, it is proved that the integral of a certain multivalued function over an arbitrary twisted (or loaded) cycle satisfies the system of differential equations in question. The classical hypergeometric functions studied here include Appell's hypergeometric functions F1, F2, F3, F4, Lauricella's hypergeometric functions FA, FB, FC, FD, and the generalized hypergeometric function n+1Fn.
Under mean radius of curvature flow, a closed convex surface in Euclidean space is known to expand exponentially to infinity. In the three-dimensional case we prove that the oriented normals to the flowing surface converge to the oriented normals of a round sphere whose centre is the Steiner point of the initial surface, which remains constant under the flow.
To prove this we show that the oriented normal lines, considered as a surface in the space of all oriented lines, evolve by a parabolic flow which preserves the Lagrangian condition.Moreover, this flow converges to a holomorphic Lagrangian section, which forms the set of oriented lines through a point.
The coordinates of the Steiner point are projections of the support function into the first non-zero eigenspace of the spherical Laplacian and are given by explicit integrals of initial surface data.
In the present paper, multiscale systems of polynomial wavelets on an n-dimensional sphere are constructed. Scaling functions and wavelets are investigated,and their reproducing and localization properties and positive definiteness are examined. Decomposition and reconstruction algorithms for the wavelet transform are presented. Formulae for variances in space and momentum domain, as well as for the uncertainty product, of zonal functions over n-dimensional spheres are derived and applied to the scaling functions.
We give a fundamental set of solutions to the generalized hypergeometrice quation n+1En in terms of integrals of Euler type and explicitly determine the matrix elements of the circuit matrices with respect to this set of solutions.