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For the scalar programming problem, some characterizations for optimal solutions are known. In these characterizations convexity properties play a very important role. In this work, we study characterizations for multiobjective programming problem solutions when functions belonging to the problem are differentiable. These characterizations need some conditions of convexity. In differentiable scalar programming problems the concept of invexity is very important. We prove that it is also necessary for the multiobjective programming problem and give some characterizations of multiobjective programming problem solutions under weaker conditions. We define analogous concepts to those of stationary points and to the conditions of Kuhn᎐Tucker and Fritz᎐John for the multiobjective programming problem. ᮊ

The response of parathyroid cells to serum calcium is regulated by a calcium-sensing receptor protein (CaR). In patients with chronic renal failure, hypocalcemia contributes to the parathyroid hyperplasia and increased parathyroid hormone secretion characteristic of secondary hyperparathyroidism (sHPT). However, patients with uremia also display reduced sensitivity to extracellular calcium; this seems to be owing to an alteration of the receptor mechanism. This study examined calcium receptor expression in the parathyroid tissue of patients with sHPT, using immunohistochemical techniques and comparison with normal tissue and parathyroid glands of patients with primary hyperparathyroidism. In all the glands studied, immunostaining was more intense in chief cells than in oxyphilic, transitional, and clear cells. The parathyroid glands of patients with sHPT displayed significantly reduced expression of CaR with respect to morphologically normal ones; a very similar reduction is reported in adenomas. Furthermore, in glands displaying multinodular hyperplasia, expression was less marked in nodule-forming cells than in internodular areas. The decreased expression of calcium receptors in the parathyroid tissue of uremic patients was thought to be owing to the different cell populations present; these parathyroid glands contained predominantly transitional, oxyphilic, and clear cells, which normally express fewer receptors than chief cells, which are more abundant in normal glands.

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