100 Collaborative Products and their Uses

Collaborative technologies, applications and uses

Copulae and Their Uses

This short survey of copulas presents their properties, tries to stress their relevance for statistics and their connection with Markov processes and conditional expectations. 1 What is a copula? Copul were introduced by Sklar(1959). The reader is referred to Schweizer and Sklar (1983), Schweizer (1991), Sklar (1973), Sklar (1996), Nelsen (1999). A copula is a function C : [0; 1] ]0; 1[! ]0; 1[...

Nuclei and their uses

⊇ Basic ⊇ Congruence ⊇ Division of sets of intervals, those sets closed under taking (a) similar intervals, (b) sub-intervals, (c) abutting intervals, (d) suprema, respectively. We let B(Λ) and D(Λ) be the family of all basic sets and division sets, respectively. Abstract and basic sets are not particularly important, but the do help with the organization. However, observe that the family B(Λ) ...

Catalan Numbers, Their Generalization, and Their Uses

Probably the most prominent among the special integers that arise in combinatorial contexts are the binomial coefficients (~). These have many uses and, often, fascinating interpretations [9]. We would like to stress one particular interpretation in terms of paths on the integral lattice in the coordinate plane, and discuss the celebrated ballot problem using this interpretation. A path is a se...

Activated mono-, di-, and polysaccharides reaction products thereof, their preparation and uses

Reaction at the interface of an organic solution containing an acidic reactant and an aqueous alkaline solution containing nonreducing carbohydrates such as sucrose, sugar alcohols, cyclodextrins, and polysaccharides imparts a specificity to the reaction for one or more of the primary alcohol groups of the carbohydrate reactant. The resulting activated, nonreducing carbohydrate intermediate can...