A modulus of smoothness on the unit sphere

For functions onS ^d−1 (the unit sphere inR ^d) and, in particular, forf∈L _p(S ^d−1), we define new simple moduli of smoothness. We relate different orders of these moduli, and we also relate these moduli to best approximation by spherical harmonics of order smaller thann. Our new moduli lead to sharper results than those now available for the known moduli onL _p(S ^d−1). Supported by NSERC Grant A4816 of Canada.     

Some properties of the reduced inner modulus

The research was financially supported within the program Universities of Russia in the direction Fundamental Problems of Mathematics and Mechanics (Code 1.1.33).     

On the modulus of disjointness preserving operators on complex vector lattices

Let L and M be Archimedean vector lattices such that and are complex vector lattices. We constructively and intrinsically prove that if is an order bounded disjointness preserving operator from into then the modulus

The reduced minimum modulus and the spectrum

We use the reduced minimum modulus to derive asymptotic formulae for the jumps of the minimum index of a semi-Fredholm operator on a Banach space. The results are then extended to the Riesz eigenvalues greater in absolute value than the essential spectral radius. This reveals a link between the two kinds of singularities.     

Some properties of the reduced modulus in space

We obtain upper bounds for the conformal modulus of a condenser with uniformly perfect plates and for the reduced modulus of a uniformly perfect set E at . For the reduced moduli of α-uniformly perfect sets we prove the continuity property with respect to the kernel convergence of the complements to these sets in the sense of Carathéodory.     

Interpolation on the complex Hilbert sphere

Let S∞ denote the unit sphere in some infinite dimensional complex Hilbert space (H,<·,·>)Let z1,z2,…,z1 be distinct points on S∞ This paper deals with interpolation of arbitrary data on the zj by a function in the linear span of the l functionswhen is a suitable function that operates on nonnegative definite matrices.Conditions for the strict positive definiteness of the kernel are obtained.     

Exact point-distributions over the complex sphere

We study point-distributions over the surface of the unit sphere in unitary space that generate quadrature rules which are exact for spherical polynomials up to a certain bi-degree. In this first stage, we explore several different characterizations for this type of point sets using standard tools such as, positive definiteness, reproducing kernel techniques, linearization formulas, etc. We find bounds on the cardinality of a point-distribution, without discussing the deeper question regarding best bounds. We include examples, construction methods and explain, via isometric embeddings from real to complex spheres, the proper connections with the so-called spherical designs.     

On holomorphic foliations in complex surfaces transverse to a sphere

We prove that a holomorphic foliation on a Stein surface transverse to the boundary of a 4-ball is conjugated inside the ball to the foliation generated by the holomorphic vector field , provided that the transversely holomorphic flow induced by on the boundary of the ball has a parabolic closed orbit. The proof contains a classification of transversely holomorphic flows on 3-manifolds with a parabolic closed orbit.     

Equivalence of K-functional and modulus of smoothness of functions on the sphere

In the present note certain fundamental estimates of the constructive theory of functions on the sphere Sⁿ Rⁿ⁺¹, n 1, are sharpened on the basis of the equivalence of the K-functional and the modulus of smoothness of functions. In particular a Bernshtein-type inequality for spherical polynomials is made more precise. The estimates obtained are applied to deduce a membership criterion for the function f L_p(Sⁿ), 1 p , to the space H_r H_r L_p(Sⁿ) depending on the growth of the norm of derivatives of best approximation polynomials of the function f, which is an analog of a result found by S. B. Stechkin related to continuous periodic functions.Translated from Matematicheskie Zametki, Vol. 52, No. 3, pp. 123–129, September, 1992.The author wishes to express his deep gratitude to Academician S. M. Nikol'skii and Professor P. I. Lizorkin for discussion of the results of the present note.     

The effective shear modulus for an n-layered composite sphere

This work presents the derivation of the effective shear modulus for a heterogeneous material composed of multi-layered composite spheres embedded in a linear elastic matrix. It is based on the composite spheres model known from the literature. In contrast to previous publications the effective shear modulus is obtained by equating the results of two models: In the first model, a heterogeneous sphere is embedded in an equivalent homogeneous material, whereas in the second model, the heterogeneous sphere is replaced by an equivalent homogeneous sphere. In the context of both, a shear stress approach and a shear deformation approach, this results into an overdetermined system of equations which is solved with the least squares method. In a numerical study our results are compared to effective moduli and bounds from the literature. Furthermore, a convincing agreement with experimental data for glass microspheres embedded in a polyester matrix is demonstrated. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the generalized modulus

The generalized modulus has numerous applications in geometric function theory and analytic number theory. In this paper, we study the monotonicity and convexity of the generalized modulus and obtain sharp functional inequalities for this function. Our results are the extensions of some well-known inequalities of the modulus of the Grötzsch ring.     

On the modulus of -convexity

In this paper, we prove that the moduli of W^*-convexity, introduced by Ji Gao [J. Gao, The W^*-convexity and normal structure in Banach spaces, Appl. Math. Lett. 17 (2004) 1381–1386], of a Banach space X and of the ultrapower of X itself coincide whenever X is super-reflexive. Moreover, we improve a sufficient condition for uniform normal structure of the space and its dual. This generalizes and strengthens the main results of [J. Gao, The W^*-convexity and normal structure in Banach spaces, Appl. Math. Lett. 17 (2004) 1381–1386].     

Neighborhoods of the Riemann sphere in complex surfaces

Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 27, No. 3, pp. 29–41, July–September, 1993.     

On embeddings in the sphere

We consider embeddings of a finite complex in a sphere. We give a homotopy-theoretic classification of such embeddings in a wide range.

     

Calculation of the complete elliptic integrals with complex modulus

Summary This is an Addendum to a preceding paper of Morita and Horiguchi [Numer. Math.20, 425–430 (1973)]. Attention is called to an error in the algol procedure given in that paper. A corrected procedure of calculating the complete elliptic integrals of the first and the second kind with complex modulusk is presented, in the form that is itself useful in the calculation of their analytic continuations over the branch cuts.     

Geodesic axes in the pants complex of the five-holed sphere

We study the synthetic geometry of the pants graph of the 5-holed sphere, establishing the existence of geodesics connecting any vertex or ideal point to any ideal point. We prove the existence of geodesic axes for sufficiently high powers of any pseudo-Anosov mapping class, and that large link hierarchies from Harvey?s curve graph all induce geodesic paths.     

A CR Poincaré inequality on the complex sphere

The main purpose of this paper is to prove a CR Poincaré inequality with sharp exponent on the sphere in complex space. We use the complex tangential gradient on the sphere instead of the usual Laplace-Beltrami gradient on the sphere.