We show that any pair X, Y of independent non-compactly supported random variables on [0, ∞) satisfies $$\mathop {\lim \inf}\limits_{m \to \infty} \mathbb{P}(\min (X,Y) > m\,\left| {X + 2m} \right.) = 0.$$ .We conjecture multi-variate and weighted generalizations this result, prove them under the additional assumption are identically distributed.

In this study, microwave-assisted extraction was applied for pectin extraction from the dried orange peel and Box-Behnken response surface design was used to study and optimize the effects of processing variables (microwave power, irradiation time, pH and solid-liquid ratio) on the yield of pectin. The amount of pectin extracted increased with increasing microwave power, while it reduces as the...

abstractbackground and objective: microwave-assisted extraction (mae) is one of the novel, fast, and efficient methods for extraction of bioactive compounds. the aim of this study was optimization of mae of betalains from red beet (beta vulgaris) and evaluating antioxidant activity of its methanolic extract.materials and methods: the effects of independent variables namely, microwave power leve...