Abstract. Given a set N with n elements and a family F of subsets, we show how to partition N into k such subsets in 2 n n O(1) time. We also consider variations of this problem where the subsets may overlap or are weighted, and we solve the decision, counting, summation, and optimisation versions of these problems. Our algorithms are based on the principle of inclusion-exclusion and the zeta transform.In effect we get exact algorithms in 2 n n O(1) time for several well-studied partition problems including Domatic Number, Chromatic Number, Maximum k-Cut, Bin Packing, List Colouring, and the Chromatic Polynomial. We also have applications to Bayesian learning with decision graphs and to model-based data clustering.If only polynomial space is available, our algorithms run in time 3 n n O(1) if membership in F can be decided in polynomial time. We solve Chromatic Number in O(2.2461 n ) time and Domatic Number in O(2.8718 n ) time.Finally, we present a family of polynomial space approximation algorithms that find a number between χ(G) and (1 + )χ(G) in time O(1.2209 n + 2.2461 e − n ).

Aim: To study the macronutrient and energy contents of human milk fractions during the first 6 mo of lactation. Study design: A total of 483 milk samples, including 52 pairs of fore‐ and hindmilk samples from 20 mothers, 253 samples from 53 donor mothers and 126 samples from 36 mothers of preterm infants, were collected longitudinally, starting at 1 wk postpartum and continuing monthly up to 6 mo. Protein, lactose and fat contents were measured and energy density estimated. Results: The protein content was significantly lower in fore‐ and hindmilk than in donor or preterm milk during the first months of lactation. In donor and preterm milk, the protein content declined consistently from 2.0 g/100 ml at 1 wk to half of that at 6 mo, and a similar trend was observed in fore‐ and hindmilk. Lactose content showed no significant changes between the groups or in the course of lactation. The fat content was highest in hindmilk, being approximately two‐ to threefold that of foremilk. Accordingly, hindmilk included 25–35 kcal/100 ml more energy on average than foremilk. Conclusions: The fat content of human milk increases in relation to breast emptying, while the other macronutritients of milk show only slight changes. When enteral feeding with high‐energy human milk is preferred, as in the case of very preterm infants, hind milk, with its higher fat content, would be a natural choice.

Abstract. We present randomized algorithms for some well-studied, hard combinatorial problems: the k-path problem, the p-packing of qsets problem, and the q-dimensional p-matching problem. Our algorithms solve these problems with high probability in time exponential only in the parameter (k, p, q) and using polynomial space; the constant bases of the exponentials are significantly smaller than in previous works. For example, for the k-path problem the improvement is from 2 to 1.66. We also show how to detect if a d-regular graph admits an edge coloring with d colors in time within a polynomial factor of O(2 (d−1)n/2 ).Our techniques build upon and generalize some recently published ideas by I.

An association between DiGeorge's syndrome and an unbalanced chromosomal rearrangement leading to trisomy 20pter leads to 20q11 and monosomy 22pter leads to 22q11 was found in four individuals belongings to one family. These and other data from the literature are interpreted to suggest that DiGeorge's syndrome can be caused by deletion of a gene located in chromosome 22, probably in band 22q11.