ABSTRACT We study the relativistic self-gravitating, hydrostatic spheres with a polytropic equation of state, considering structures with the polytropic indices n = 1(0.5)3 and illustrate the results for the relativistic parameters σ = 0 − 0.75. We determine the critical relativistic parameter at which the mass of the polytrope has a maximum value and represents the first mode of radial instability. For n = 1(0.5)2.5, stable relativistic polytropes occur for σ less than the critical values 0.42,0.20,0.10, and 0.04, respectively, while unstable relativistic polytropes are obtained when σ is greater than the same values. When n = 3.0 and σ > 0.5, energetically unstable solutions occur. The results of critical values are in full agreement with those evaluated by several authors. Comparisons between analytical and numerical solutions of the given relativistic functions provide a maximum relative error of order 10−3.

RESUMEN Estudiamos esferas hidrostáticas, autogravitantes y relativistas, con una ecuación de estado politrópica, considerando estructuras con índices politrópicos n = 1(0.5)3, e ilustramos los resultados para parámetros relativistas σ = 0 − 0.75. Determinamos el parámetro relativista crítico, para el cual la masa del politropo alcanza un valor máximo y representa el primer modo de inestabilidad radial. Con n = 1(0.5)2.5 encontramos politropos relativistas estables para σ menor que los valores críticos 0.42,0.20,0.10, y 0.04, respectivamente. Se obtienen politropos relativistas inestables para valores mayores de σ. Cuando n = 3.0 y σ > 0.5 encontramos soluciones energéticamente inestables. Los resutados sobre los valores críticos concuerdan muy bien con los de otros autores. Al comparar las soluciones analíticas y numéricas de las funciones relativistas estudiadas se encuentran errores relativos máximos del orden de 10−3.

Abstract The production of milk-clotting enzyme (MCE) by free and immobilized Bacillus circulans 25 cells was investigated. The investigation evaluates cell immobilization through entrapment and covalent binding to different carriers. B. circulans 25 cells immobilized covalently via glutaraldehyde (cross-linker) to natural wool exhibited higher effectiveness factor (0.76) compared to other carriers. Immobilized cells produce maximum level of MCE and highest ratio of milk-clotting activity/proteolytic activity (MCA/PA) after 24 h using 0.4g wet weight cells/g wool. In batch operation, decreasing biomass loading to 0.2 g enhanced the MCA/PA ratio and specific productivity by 5.94 and 59.88%, respectively. Immobilized cells on natural wool as an effective and suitable carrier were able to produce the same level of MCA, productivity and MCA/PA ratio for 9 repeated cycles (216 h). Under continuous operation immobilized cells in a packed-bed bioreactor were able to keep producing MCA with MCA/PA ratio for about 7 days at the same level. Moreover, continuous operation demonstrates a very good productivity of 0.385 KU/L/h, which is higher than the other production systems by 7.0-11.7 fold. Immobilized B. circulans 25 cells proved to be fully capable of continuous MCE production in a packed-bed bioreactor.