A note on ball-covering property of Banach spaces

By a ball-covering B of a Banach space X, we mean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere SX of X; and X is said to have the ball-covering property (BCP) provided it admits a ball-covering by countably many balls. In this note we give a natural example showing that the ball-covering property of a Banach space is not inherited by its subspaces; and we present a sharp quantitative version of the recent Fonf and Zanco renorming result saying that if the dual X^∗ of X is w^∗ separable, then for every ε>0 there exist a (1+ε)-equivalent norm on X, and an R>0 such that in this new norm SX admits a ball-covering by countably many balls of radius R. Namely, we show that R=R(ε) can be taken arbitrarily close to (1+ε)/ε, and that for X=?₁[0,1] the corresponding R cannot be equal to 1/ε. This gives the sharp order of magnitude for R(ε) as ε→0.     

Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations

In this paper, based on a multidimensional Riemann theta function, a lucid and straightforward generalization of the Hirota-Riemann method is presented to explicitly construct multiperiodic Riemann theta functions periodic wave solutions for nonlinear equations such as the Caudrey-Dodd-Gibbon-Sawada-Kotera equation and (2+1)-dimensional breaking soliton equation. Among these periodic waves, the one-periodic waves are well-known cnoidal waves, their surface pattern is one-dimensional, and often they are used as one-dimensional models of periodic waves. The two-periodic waves are a direct generalization of one-periodic waves, their surface pattern is two-dimensional so that they have two independent spatial periods in two independent horizontal directions. A limiting procedure is presented to analyze in detail, asymptotic behavior of the multiperiodic waves and the relations between the periodic wave solutions and soliton solutions are rigorously established. This generalized Hirota-Riemann method can also be demonstrated on a class variety of nonlinear difference equations such as Toeplitz lattice equation.     

Generalized Navier-Stokes equations with initial data in local Q-type spaces

In this paper, we establish a link between Leray mollified solutions of the three-dimensional generalized Navier-Stokes equations and mild solutions for initial data in the adherence of the test functions for the norm of . This result applies to the usual incompressible Navier-Stokes equations.     

Maximally resonant polygonal systems

A benzenoid system G is k-resonant if any set F of no more than k disjoint hexagons is a resonant pattern, i.e, G−F has a perfect matching. In 1990’s M. Zheng constructed the 3-resonant benzenoid systems and showed that they are maximally resonant, that is, they are k-resonant for all k≥1. Recently, the equivalence of 3-resonance and maximal resonance has been shown to be valid also for coronoid systems, carbon nanotubes, polyhexes in tori and Klein bottles, and fullerene graphs. So our main problem is to investigate the extent of graphs possessing this interesting property. In this paper, by replacing the above hexagons with even faces, we define k-resonance of graphs in surfaces, possibly with boundary, in a unified way. Some exceptions exist. For plane polygonal systems tessellated with polygons of even size at least six such that all inner vertices have the same degree three and the others have degree two or three, we show that such 3-resonant polygonal systems are indeed maximally resonant. They can be constructed by gluing and lapping operations on three types of basic graphs.     

Variations on a theme of Boole and Stein-Weiss

We give an alternative proof of a theorem of Stein and Weiss: The distribution function of the Hilbert transform of a characteristic function of a set E only depends on the Lebesgue measure |E| of such a set. We exploit a rational change of variable of the type used by George Boole in his paper “On the comparison of transcendents, with certain applications to the theory of definite integrals” together with the observation that if two functions f and g have the same Lp norm in a range of exponents p₁<p<p₂ then their distribution functions coincide.     

Weighted estimates for a class of multilinear fractional type operators

In this paper, a class of multiple fractional type weights is defined as

     

A note on the rank parity function

We provide some further theorems on the partitions generated by the rank parity function. New Bailey pairs are established, which are of independent interest.     

Degree conditions for group connectivity

Let G be a 2-edge-connected simple graph on n≥13 vertices and A an (additive) abelian group with |A|≥4. In this paper, we prove that if for every uv∉E(G), max{d(u),d(v)}≥n/4, then either G is A-connected or G can be reduced to one of K_2,3,C₄ and C₅ by repeatedly contracting proper A-connected subgraphs, where C_k is a cycle of length k. We also show that the bound n≥13 is the best possible.     

Convex Komlós sets in Banach function spaces

In 1967 Komlós proved that for any sequence n{fn} in L₁(μ), with ‖fn‖?M<∞ (where μ is a probability measure), there exists a subsequence n{gn} of n{fn} and a function g∈L₁(μ) such that for any further subsequence n{hn} of n{gn},

     

Filling polygonal holes with minimal energy surfaces on Powell-Sabin type triangulations

In this paper we present two different methods for filling in a hole in an explicit 3D surface, defined by a smooth function f in a part of a polygonal domain D⊂R². We obtain the final reconstructed surface over the whole domain D. We do the filling in two different ways: discontinuous and continuous. In the discontinuous case, we fill the hole with a function in a Powell-Sabin spline space that minimizes a linear combination of the usual seminorms in an adequate Sobolev space, and approximates (in the least squares sense) the values of f and those of its normal derivatives at an adequate set of points. In the continuous case, we will first replace f outside the hole by a smoothing bivariate spline s_f, and then we fill the hole also with a Powell-Sabin spline minimizing a linear combination of given seminorms. In both cases, we obtain existence and uniqueness of solutions and we present some graphical examples, and, in the continuous case, we also give a local convergence result.