several tables have been given due to a − minimal sets. our main aim in this paper is tocomplete these tables by employing several examples.

1. The norm || • || in a Banach algebra A is said to be minimal [l ] if, given any other norm || •||1 in A (with respect to which A need not be complete), the condition ||a||iá||a|| for each oG^4 implies that ||a[|i = ||a||. We shall say that || •|| is absolutely minimal if, given any other norm ||-||i whatever in A, then ||a||iè||a|| for each aEA. An absolutely minimal norm is of course minima...

A model of traversable wormhole in an extended gravity theory has been proposed. The Casimir effect and Generalised Uncertainty Principle (GUP) arising out the concept minimal length have considered to obtain shape function, radial tangential pressure wormhole. effects GUP parameter on properties discussed. energy condition violation by exotic matter shown. Also we calculated content that viola...

Nowadays, vinegar is industrially produced by mesophilic acetic acid bacteria (AAB). However, temperature fluctuation during acetous fermentation is inevitable, and may cause process disturbances. This can be mostly avoided using thermo-tolerant AAB. The main purpose of the present study was to isolate thermo-tolerant AAB. Fermentation performances were then evaluated. Twenty-eight different is...

For a left and right Noetherian ring $R$, we give some equivalent characterizations for $\_RR$ satisfying the Auslander condition in terms of flat (resp. injective) dimensions minimal injective coresolution resolution) $R$-modules. Furthermore, prove that an artin algebra $R$ condition, is Gorenstein if only subcategory consisting finitely generated modules contravariantly finite. As applicatio...