In Bayesian inference‎, ‎the acquisition of prior distributions plays a fundamental role‎. ‎While authorized priors need not conform to traditional probability densities and may be improper priors‎, ‎obtaining proper prior densities remains a challenge in the Bayesian literature‎. ‎This article explores a set of conditions that enable the establishment of specific assumptions‎, ‎ensuring that maximum entropy priors and restricted reference priors become proper and transform into probability density priors‎. ‎By examining these conditions‎, ‎this study contributes to the advancement of proper prior acquisition in Bayesian analysis.

In the linear system Ax=b the points x are sometimes Constrained to lie in a given subspace S of column space of A. Drazin inverse for any singular or nonsingular matrix, exist and is unique. In this paper, the singular consistent or inconsistent Constrained linear systems are introduced and the effect of Drazin inverse in solving such systems is investigated. Constrained linear system arise in electrical network theory.

In this paper, optimization of a linear objective func-tion with fuzzy relational inequality constraints is in-vestigated where the feasible region is formed as the in-tersection of two inequality fuzzy systems and Dombi family of t-norms is considered as fuzzy composition. Dombi family of t-norms includes a parametric family of continuous strict t-norms, whose members are increas-ing functions of the parameter. This family of t-norms covers the whole spectrum of t-norms when the param-eter is changed from zero to in nity. The resolution of the feasible region of the problem is  rstly investigated when it is de ned with max-Dombi composition. Based on some theoretical results, a necessary and su cient condition and three other necessary conditions are de-rived for determining the feasibility. Moreover, in order to simplify the problem, some procedures are presented.

Introduction In industrial designing and manufacturing, it is often required to generate a smooth function approximating a given set of data which preserves certain shape properties of the data such as positivity, monotonicity, or convexity, that is, a smooth shape preserving approximation. ...