F. J. MacWilliams proved that Hamming isometries between linear codes extend to monomial transformations. This theorem has recently been generalized by J. Wood who proved it for Frobenius rings using character theoretic methods. The present paper provides a combinatorial approach: First we extend I. Constantinescu's concept of homogeneous weights on arbitrary finite rings and prove MacWilliams' equivalence theorem to hold with respect to these weights for all finite Frobenius rings. As a central tool we then establish a general inversion principle for real functions on finite modules that involves Mo bius inversion on partially ordered sets. An application of the latter yields the aforementioned result of Wood. Academic Press

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We present the synthesis, structure, magnetic properties, as well as the Mössbauer and electron paramagnetic resonance studies of a ring-shaped [FeLn(Htea)(μ-N)(N)(piv)] (Ln = Y 1, Gd 2, Tb 3, Dy 4, Ho 5, Er, 6) coordination cluster. The Dy, Tb, and Ho analogues show blocking of the magnetization at low temperatures without applied fields. The anisotropy of the 3d ion and the exchange interaction between 3d and 4f ions in FeLn complexes are unambiguously determined by high-field/high-frequency electron paramagnetic resonance measurements at low temperature. Ferromagnetic exchange interaction J is found which decreases upon variation of the Ln ions to larger atomic numbers. This dependence is similar to the behavior shown in the effective barrier values of complexes 3-5. Further information about the anisotropy of the Ln ions was gathered with Fe Mössbauer spectroscopy, and the combination of these methods provides detailed information regarding the electronic structure of these complexes.

The aim of this study was to identify and independently validate a novel gene signature predicting locoregional tumor control (LRC) for treatment individualization of patients with locally advanced HPV-negative head and neck squamous cell carcinomas (HNSCC) who are treated with postoperative radio(chemo)therapy (PORT-C). Gene expression analyses were performed using NanoString technology on a multicenter training cohort of 130 patients and an independent validation cohort of 121 patients. The analyzed gene set was composed of genes with a previously reported association with radio(chemo)sensitivity or resistance to radio(chemo)therapy. Gene selection and model building were performed comparing several machine-learning algorithms. We identified a 7-gene signature consisting of the three individual genes , and one metagene combining the highly correlated genes, and The 7-gene signature was used, in combination with clinical parameters, to fit a multivariable Cox model to the training data (concordance index, ci = 0.82), which was successfully validated (ci = 0.71). The signature showed improved performance compared with clinical parameters alone (ci = 0.66) and with a previously published model including hypoxia-associated genes and cancer stem cell markers (ci = 0.65). It was used to stratify patients into groups with low and high risk of recurrence, leading to significant differences in LRC in training and validation ( < 0.001). We have identified and validated the first hypothesis-based gene signature for HPV-negative HNSCC treated by PORT-C including genes related to several radiobiological aspects. A prospective validation is planned in an ongoing prospective clinical trial before potential application in clinical trials for patient stratification. .