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Consider a linear and continuous operator T between Banach function spaces. We prove that under certain requirements an integral inequality for T is equivalent to a factorization of T through a specific kernel operator: in other words, the operator T has what we call a Maharam-type kernel representation. In the case that the inequality provides a domination involving trigonometric functions, a special factorization through the Fourier operator is given. We apply this result to study the problem that motivates the paper: the approximation of functions in $$L^{2}[0,1]$$ by means of trigonometric series whose Fourier coefficients are given by weighted trigonometric integrals. [ABSTRACT FROM AUTHOR]

Let ν(2 n) be the number of antipodal bicolored necklaces with 2 n pearls. In this note, we find the first two terms of the asymptotic expansion of ν(2 n). As a byproduct of this result, we also show that the sequence ( ν(2 n)) is non-holonomic, i.e., it satisfies no linear recurrence of a fixed finite order k with polynomial coefficients. [ABSTRACT FROM AUTHOR]

A k-colouring of a hypergraph is an assignment of no more than k colours to the vertices of the hypergraph in such a way that the coloured hypergraph has no monochromatic edges. In this paper a polymatroid is associated with a hypergraph. It is shown that the number of k-colourings of the hypergraph is determined by an evaluation of the characteristic polynomial of this polymatroid. Hypergraph colouring is then related to an extension of the critical problem developed by the author. It is shown that, if q is a prime power, then a hypergraph is q-colourable if and only if the critical exponent over GF(q) of its associated polymatroid is less than or equal to k. [ABSTRACT FROM AUTHOR]

Poincaré characteristic for reflexive relations (oriented graphs) is defined in terms of homology and is not invariant under transitive closure. Formulas for the Poincaré characteristic of products, joins, and bounded products are given. Euler's definition of characteristic extends to certain filtrations of reflexive relations which exist iff there are no oriented loops. Euler characteristic is independent of filtration, agrees with Poincaré characteristic, and is unique. One-sided Möbius characteristic is defined as the sum of all values of a one-sided inverse of the zeta function. Such one-sided inverses exist iff there are no local oriented loops (although there may be global oriented loops). One-sided Möbius characteristic need not be Poincaré characteristic, but it is when a one-sided local transitivity condition is satisfied. A two-sided local transitivity condition insures the existence of the Möbius function but Möbius inversion fails for non-posets. [ABSTRACT FROM AUTHOR]

This is the second part of the bibliography published in Aequationes Math. 1, 152-191 (1968). For the classification and other explanations and remarks see pp. 152-155 and 190-191 of the first part, loc. cit. [ABSTRACT FROM AUTHOR]