Maximal function and Multiplier Theorem for Weighted Space on the Unit Sphere

For a family of weight functions invariant under a finite reflection group, the boundedness of a maximal function on the unit sphere is established and used to prove a multiplier theorem for the orthogonal expansions with respect to the weight function on the unit sphere. Similar results are also established for the weighted space on the unit ball and on the standard simplex.Comment: 24 pages, to appear in J. Funct. Analysi

Polynomial Approximation in Sobolev Spaces on the Unit Sphere and the Unit Ball

This work is a continuation of the recent study by the authors on approximation theory over the sphere and the ball. The main results define new Sobolev spaces on these domains and study polynomial approximations for functions in these spaces, including simultaneous approximation by polynomials and relation between best approximation to a function and to its derivatives.Comment: 16 page

Moduli of Smoothness and Approximation on the Unit Sphere and the Unit Ball

A new modulus of smoothness based on the Euler angles is introduced on the unit sphere and is shown to satisfy all the usual characteristic properties of moduli of smoothness, including direct and inverse theorem for the best approximation by polynomials and its equivalence to a $K$-functional, defined via partial derivatives in Euler angles. The set of results on the moduli on the sphere serves as a basis for defining new moduli of smoothness and their corresponding $K$-functionals on the unit ball, which are used to characterize the best approximation by polynomials on the ball.Comment: 63 pages, to appear in Advances in Mat

Adsorption of molecular oxygen on doped graphene: atomic, electronic and magnetic properties

Adsorption of molecular oxygen on B-, N-, Al-, Si-, P-, Cr- and Mn-doped graphene is theoretically studied using density functional theory in order to clarify if O2 can change the possibility of using doped graphene for gas sensors, electronic and spintronic devices. O2 is physisorbed on B-, and Ndoped graphene with small adsorption energy and long distance from the graphene plane, indicating the oxidation will not happen; chemisorption is observed on Al-, Si-, P-, Cr- and Mn-doped graphene. The local curvature caused by the large bond length of X-C (X represents the dopants) relative to CC bond plays a very important role in this chemisorption. The chemisorption of O2 induces dramatic changes of electronic structures and localized spin polarization of doped graphene, and in particular, chemisorption of O2 on Cr-doped graphene is antiferromagnetic. The analysis of electronic density of states shows the contribution of the hybridization between O and dopants is mainly from the p or d orbitals. Furthermore, spin density shows that the magnetization locates mainly around the doped atoms, which may be responsible for the Kondo effect. These special properties supply a good choice to control the electronic properties and spin polarization in the field of graphene engineering.Comment: 7 pages, 10 figure

Dynamically Spontaneous Symmetry Breaking and Masses of Lightest Nonet Scalar Mesons as Composite Higgs Bosons

Based on the (approximate) chiral symmetry of QCD Lagrangian and the bound state assumption of effective meson fields, a nonlinearly realized effective chiral Lagrangian for meson fields is obtained from integrating out the quark fields by using the new finite regularization method. As the new method preserves the symmetry principles of the original theory and meanwhile keeps the finite quadratic term given by a physically meaningful characteristic energy scale $M_c$, it then leads to a dynamically spontaneous symmetry breaking in the effective chiral field theory. The gap equations are obtained as the conditions of minimal effective potential in the effective theory. The instanton effects are included via the induced interactions discovered by 't Hooft and found to play an important role in obtaining the physical solutions for the gap equations. The lightest nonet scalar mesons($\sigma$, $f_0$, $a_0$ and $\kappa$) appearing as the chiral partners of the nonet pseudoscalar mesons are found to be composite Higgs bosons with masses below the chiral symmetry breaking scale $\Lambda_{\chi} \sim 1.2$ GeV. In particular, the mass of the singlet scalar (or the $\sigma$) is found to be $m_{\sigma} \simeq 677$ MeV.Comment: 15 pages, Revtex, published version, Eur. Phys. J. C (2004) (DOI) 10.1140/epjcd/s2004-01-001-