On Spherically Symmetric Motions of the Atmosphere Surrounding a Planet Governed by the Compressible Euler Equations

We consider spherically symmetric motions of inviscid compressible gas surrounding a solid ball under the gravity of the core. Equilibria touch the vacuum with finite radii, and the linearized equation around one of the equilibria has time-periodic solutions. To justify the linearization, we should construct true solutions for which this time-periodic solution plus the equilibrium is the first approximation. But this leads us to difficulty caused by singularities at the free boundary touching the vacuum. We solve this problem by the Nash-Moser theorem.Comment: Finally revise

A Note on a Modified Catuskoti

The `catuskoti' or tetralemma in Buddhist logic is a problematic subject from the modern logical point of view. Recently a many-valued paraconsistent logic was proposed in order to formalize catuskoti adequately by G. Priest. On the other hand a slight modification of the formalization of catuskoti seems to allow an appropriate interpretation in the framework of the classical propositional calculus in the mathematical logic developed by Russell-Whitehead and Hilbert-Ackermann.Comment: Appendix has been slightly corrected. No substantial change

The Impact of Brand Image and Customer Commitment on Loyalty: An Empirical Study of Automobile Sector

Corporate brand image has been assessed as an important antecedent of customer commitment and loyalty. Corporate brands help companies achieving higher performance, such as sales. A poor brand image will lead to disaster, and lose customers. Marketing exists to deliver more value to satisfy customers as well as build a long-term and mutually profitability relationship with customers. If a firm’s products or services do not satisfy or meet the customer’s needs and wants, all the strategies are insufficient. With loyal customers, companies can have higher market share and reduce the operating cost. This initial study was from relevant literature, then set up research structure and hypotheses. Survey was employed, and respondents were collected from automobile sector in Taiwan. There were 170 usable questionnaires to analyze normality, convergent and discriminant validities, and SEM model by PASW 18 and AMOS 18.0. The research found that corporate brand image significantly affects customer commitment and loyalty, and customer commitment has strong impact on customer loyalty for the sample. Therefore, firms need to specifically focus on a long-term and mutually profitability relationship with a customer and create loyalty as competitive advantages in the markets

On Linear Adiabatic Perturbations of Spherically Symmetric Gaseous Stars Governed by the Euler-Poisson Equations

The linearized operator for non-radial oscillations of spherically symmetric self-gravitating gaseous stars is analyzed in view of the functional analysis. The evolution of the star is supposed to be governed by the Euler-Poisson equations under the equation of state of the ideal gas, and the motion is supposed to be adiabatic. We consider the case of not necessarily isentropic, that is, not barotropic motions. Basic theory of self-adjoint realization of the linearized operator is established. Some problems in the investigation of the concrete properties of the spectrum of the linearized operator are proposed. The existence of eigenvalues which accumulate to 0 is proved in a mathematically rigorous fashion.The absence of continuous spectra and the completeness of eigenfunctions for the operators reduced by spherical harmonics is discussed