We study certain function algebras and their operator algebra completions on r-discrete abelian groupoids, the corresponding conditional expectations, maximal abelian subalgebras (masa) and eigen-functionals.

The Computational theoretical techniques using first principle which make use of the density functional theory as implemented by quantum expresso as well as inter – atomic force constant (IFC) techniques (Born-von Jarman) were used to investigate the phonon dispersion curves of Silver (Ag) and Gold (Au) The results obtained show that for Au, the extension to 1 – 6^th neighbour gave slightly close agreement with the experimental phonon dispersions when compared to the 1 – 5^th neighbor at W_T and L_L symmetry points. The percentage error for Au at W_T are 2.5% and 2.9% ; at point L_L we obtained 0.6% and 0.7% respectively. The Local density Approximation (LDA) results for Ag underestimates the lattice with about 1.6% while the Generalized Gradient Approximation (GGA) overestimates it by 1.9%.

Newton’s law of universal gravitation applies between two point-masses. True gravitation of solid Earth is volume integration of gravitation of all point-masses inside the solid Earth on a point-mass in atmosphere. However, in meteorology the Earth “shrinks” into a point-mass located at Earth center with entire Earth mass to identify the Earth gravitation (untrue). Combination of untrue gravitational and centrifugal accelerations gives effective gravity (geff). Combination of true gravitational and centrifugal accelerations leads to true gravity (g). The true gravity g minus the effective gravity geff is the gravity disturbance vector, δg = g – geff. With the true gravity g used in the basic equations, seven non-dimensional numbers are proposed to identify the importance of δg versus traditional forcing terms such as horizontal pressure gradient force and Coriolis force. These non-dimensional numbers are calculated from two publicly available datasets with the geoid undulation (N) from the static gravity field model EIGEN-6C4 and long-term mean geopotential height (Z), wind velocity (u, v), and temperature (Ta) at 12 pressure levels in troposphere from the NCEP/NCAR reanalyzed climatology. The results demonstrate δg nonnegligible in hydrostatic equilibrium, geostrophic wind, geostrophic vorticity, Ekman pumping, Q vector, and Omega equation, but negligible in thermal wind relation.

The connection of two phenomena - non-conservative friction forces and dissipation-induced instability can lead to many interesting engineering problems. The paper studies general material-dependent damping influence on dynamical instability of disc brake systems leading to brake squeal. The effect of general damping is demonstrated on a minimal and complex model of a disc brake. A complex system including material-dependent damping is defined in the commercial finite element software. The finite element model validated by experimental data on the brake-disc test bench is used to compute the influence of a pad and a disc damping variations on system stability by complex eigenvalue analysis. Analyzes show a significant sensitivity of the experimentally verified unstable mode of the system to the ratio of the damping between the disc and the friction material components.

Clifford algebra is unified language and efficient tool for geometry and physics. In this paper, we introduce this algebra to derive the integrable conditions for Dirac and Pauli equations. This algebra shows the standard operation procedure and deep insights into the structure of the equations. Usually, the integrable condition is related to the special symmetry of transformation group, which involves some advanced mathematical tools and its availability is limited. In this paper, the integrable conditions are only regarded as algebraic conditions. The commutators expressed by Clifford algebra have a neat and covariant form, which are naturally valid in curvilinear coordinate system and curved space-time. For Pauli and Schr\"odinger equation, many solutions in axisymmetric potential and magnetic field are also integrable. We get the scalar eigen equation in dipole magnetic field. By the virtue of Clifford algebra, the physical researches may be greatly promoted.

The effectiveness of a recently proposed methodology in the identification of damage in planar, multi-storey, Reinforced Concrete (RC) moment-frames, which develop a plastic-yield mechanism on their beams, is showcased here via the examining of a group of such existing multi-storey frames with three or more unequal spans. According to the methodology, the diagram of the instantaneous Eigen-Frequencies of the frame in the nonlinear regime is drawn as a function of the inelastic seismic roof displacement by performing a sequence of pushover and instantaneous modal analyses with gradually increasing target displacement. Using this key-diagram, the locations of severe seismic damage in an existing moment-frame can be evaluated if the instantaneous fundamental eigen-frequency of the damaged frame, at an analysis step within the nonlinear area, is known in advance by “the monitoring and the identification of frequencies” using a local network of uniaxial accelerometers. This is a hybrid technique because both procedures, the instrumental Monitoring of the structure and the Pushover analysis on the frame (M and P technique), are combined. Moreover, the damage image of the planar multi-storey moment-frame is illustrated, and the lateral stiffness matrix of the damaged frame is calculated with high accuracy.