Multipole expansions depend on the coordinate system, so that coefficients of multipole moments can be set equal to zero by an appropriate choice of coordinates. Therefore, it is meaningless to say that a physical system has a nonvanishing quadrupole moment, say, without specifying which coordinate system is used. (Except if this moment is the lowest non-vanishing one.) This result is demonstra...

We review Maxwell’s multipole vectors, and elucidate some of their mathematical properties, with emphasis on the application of this tool to the cosmic microwave background (CMB). In particular, for a completely random function on the sphere (corresponding to the statistically isotropic Gaussian model of the CMB), we derive the full probability density function of the multipole vectors. This fu...

This paper concerns numerical simulations of time-dependent problems in computational solid mechanics. A perturbation method, with the time as parameter, is proposed to solve two classical problems: an elastic bar excited by end force and dynamic buckling a cylindrical panel. Specific quadratic recast equations nonlinear problems. Numerical results show that asymptotic expansions robust, effici...

The influences of mechanically activated fly ash on the microstructure and thermal behaviour atmospheric-cured concrete panels have been evaluated. For this purpose, comparative analyses concretes prepared with without addition raw milled ashes were performed. was an attrition mill for 1 h. Cylindrical specimens 20% substitution large same compositions prepared. Mechanical properties measured a...

We measured the nondipole parameters for the spin-orbit doublets Xe 4d5/2 and Xe 4d3/2 over a photon-energy range from 100 eV to 250 eV at beamline 8.0.1.3 of the Advanced Light Source at the Lawrence Berkeley National Laboratory. Significant nondipole effects are found at relatively low energies as a result of Cooper minima in dipole channels and interchannel coupling in quadrupole channels. M...