The Laplace-Beltrami eigenvalue equation $H\Phi = \lambda \Phi $ on the $n$-sphere is studied, with an added vector potential term motivated by the differential equations for the polynomial ...Lauricella functions $F_A $. The operator $H$ is self adjoint with respect to the natural inner product induced on the sphere and, in certain special coordinates, it admits a spectral decomposition with eigenspaces composed entirely of polynomials. The eigenvalues are degenerate but the degeneracy can be broken through use of the possible separable coordinate systems on the $n$-sphere. Then a basis for each eigenspace can be selected in terms of the simultaneous eigenfunctions of a family of commuting second-order differential operators that also commute with $H$. The results provide a multiplicity of $n$-variable orthogonal and biorthogonal families of polynomials that generalize classical results for one and two variable families of Jacobi polynomials on intervals, disks, and paraboloids.
This paper presents the results of a research study on quality of life and migration intentions from Croatian rural areas. The goal of the study was to investigate the satisfaction of rural ...population with quality of life in their communities and to determine the decisive factors in resolving the dilemma: "to stay or to leave the rural area?" The research study was conducted using survey method on the sample of 914 respondents from Croatian rural areas (all 20 counties were included in the sample proportionally). The rural communities and respondents in the age group between 24 to 45 years were selected randomly. The study results indicate that the major hardships of rural life are of economic nature, lack of employment opportunities, inadequate choice of profession and lower income compared to a city. About 20% of the respondents are not satisfied with conditions of rural life and intend to leave. Further depopulation of Croatian rural areas may have unfavourable effects such as excessive urbanization of large cities, uneven development of Croatia and insufficient utilization of spatial, production and human resources.PUBLICATION ABSTRACT
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