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Let k be a field and R a standard graded k-algebra. We denote by H R the homology algebra of the Koszul complex on a minimal set of generators of the irrelevant ideal of R. We discuss the relationship between the multiplicative structure of H R and the property that R is a Koszul algebra. More generally, we work in the setting of local rings and we show that certain conditions on the multiplicative structure of Koszul homology imply strong homological properties, such as existence of certain Golod homomorphisms, leading to explicit computations of Poincaré series. As an application, we show that the Poincaré series of all finitely generated modules over a stretched Cohen-Macaulay local ring are rational, sharing a common denominator.

It is proved that the minimal free resolution of a module M over a Gorenstein local ring R is eventually periodic if, and only if, the class of M is torsion in a certain Z[t ±1 ]-module associated to R. This module, denoted J(R), is the free Z[t ±1 ]-module on the isomorphism classes of finitely generated R-modules modulo relations reminiscent of those defining the Grothendieck group of R. The main result is a structure theorem for J(R) when R is a complete Gorenstein local ring; the link between periodicity and torsion stated above is a corollary.

Infants born with congenital anomalies demand individualized nutritional evaluations and recommendations. The anatomical changes of neonatal surgical diseases create specific physiological constraints. This article reviews several nutrition-centered options to aid the medical provider caring for babies with common surgical diseases. The anatomical basis of these diseases, the objective of surgical repairs, and the nutritional constraints associated with both the disease and the repair are presented. Specific nutritional interventions designed to work with these constraints can favorably alter physiology to meet individualized needs. In this way, surgically informed nutritional therapy improves one's chances for successful postsurgical results.

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