We give a classification of dual functions, they are m -al functions. We call a function m-al with respect to an operator if the operator lives any function unchanged after m times of using the operator. And 2 ≤ m ≤ k. Functions with different m have very different properties. We give theoretical results for clones of self-dual (m = 2) and selfk -al (m = k) functions in k -valued logic at k ≤ 3...

Boosting combines weak learners into a predictor with low empirical risk. Its dual constructs a high entropy distribution upon which weak learners and training labels are uncorrelated. This manuscript studies this primal-dual relationship under a broad family of losses, including the exponential loss of AdaBoost and the logistic loss, revealing: • Weak learnability aids the whole loss family: f...

This paper approaches implementation evaluation as an integral part of the public administration organizational action. Drawing on contributions from organization theory, we assign a dual role to evaluation: valuable cognitive resource and accountability tool for the policymakers. This exploratory case study, supported by a review of secondary data sources, seeks to enrich the literature on imp...

Abstract. In this paper a duality of transformation functions in the interior point method is treated. A dual pair of convex or linear programming problems is considered and the primal problem is transformed by the parametrized transformation function of a more general form than logarithmic is. The construction of the parametrized transformation function for the dual problem is carried out so t...

We introduce the Hopf algebra of uniform block permutations and show that it is self-dual, free, and cofree. These results are closely related to the fact that uniform block permutations form a factorizable inverse monoid. This Hopf algebra contains the Hopf algebra of permutations of Malvenuto and Reutenauer and the Hopf algebra of symmetric functions in non-commuting variables of Gebhard, Ros...

a highly efficient and simple method for the preparation of α,α´-bis(arylidene)cycloalkanones under solvent-free conditions has been developed. the crossed-aldol condensation of cycloalkanones (1 eq.) with arylaldehydes (2 eq.) in the presence of 2.5 mol% of ionic liquid 1,3-disulfonic acid imidazolium hydrogen sulfate {[dsim]hso4} afforded the title compounds in high yields and short reaction ...