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1. CJM 2005 (vol 57 pp. 897)

Berezhnoĭ, Evgenii I.; Maligranda, Lech
 Representation of Banach Ideal Spaces and Factorization of Operators Representation theorems are proved for Banach ideal spaces with the Fatou property which are built by the Calder{\'o}n--Lozanovski\u\i\ construction. Factorization theorems for operators in spaces more general than the Lebesgue $L^{p}$ spaces are investigated. It is natural to extend the Gagliardo theorem on the Schur test and the Rubio de~Francia theorem on factorization of the Muckenhoupt $A_{p}$ weights to reflexive Orlicz spaces. However, it turns out that for the scales far from $L^{p}$-spaces this is impossible. For the concrete integral operators it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces are not valid. Representation theorems for the Calder{\'o}n--Lozanovski\u\i\ construction are involved in the proofs. Keywords:Banach ideal spaces, weighted spaces, weight functions,, CalderÃ³n--Lozanovski\u\i\ spaces, Orlicz spaces, representation of, spaces, uniqueness problem, positive linear operators, positive sublinear, operators, Schur test, factorization of operators, fCategories:46E30, 46B42, 46B70

2. CJM 2001 (vol 53 pp. 758)

Goulden, I. P.; Jackson, D. M.; Latour, F. G.
 Inequivalent Transitive Factorizations into Transpositions The question of counting minimal factorizations of permutations into transpositions that act transitively on a set has been studied extensively in the geometrical setting of ramified coverings of the sphere and in the algebraic setting of symmetric functions. It is natural, however, from a combinatorial point of view to ask how such results are affected by counting up to equivalence of factorizations, where two factorizations are equivalent if they differ only by the interchange of adjacent factors that commute. We obtain an explicit and elegant result for the number of such factorizations of permutations with precisely two factors. The approach used is a combinatorial one that rests on two constructions. We believe that this approach, and the combinatorial primitives that have been developed for the cut and join'' analysis, will also assist with the general case. Keywords:transitive, transposition, factorization, commutation, cut-and-joinCategories:05C38, 15A15, 05A15, 15A18