Computing SL(2,C) Central Functions with Spin Networks

Let G=SL(2,C) and F_r be a rank r free group. Given an admissible weight in N^{3r-3}, there exists a class function defined on Hom(F_r,G) called a central function. We show that these functions admit a combinatorial description in terms of graphs called trace diagrams. We then describe two algorithms (implemented in Mathematica) to compute these functions.Comment: to appear in Geometriae Dedicat

Childhood lymphoma: Diagnostic accuracy of chest radiography for severe pulmonary complications

Introduction: We sought to determine whether chest radiography can be reliably used to distinguish persistent or relapsing pulmonary lymphoma from a variety of infectious and noninfectious pulmonary conditions that can occur in children receiving treatment for lymphoma. Methods: We studied chest radiographs of 37 patients (30 with non-Hodgkin's lymphoma, and seven with Hodgkin's disease) who died of paediatric lymphoma or of treatment complications. Pulmonary findings at autopsy comprised lung tumour ( n=14), pleural tumour ( n=12), pneumonia ( n=22), adult respiratory distress syndrome (ARDS; n=16), haemorrhage ( n=27), and infarction ( n=13). Using a 4-point scale and without knowledge of autopsy findings, three radiologists independently rated antemortem radiographs for the presence of pulmonary tumour, pleural tumour, pneumonia in general, pneumonia caused by viral, bacterial, fungal, and protozoan pathogens, ARDS, pulmonary haemorrhage, and pulmonary infarction. Diagnostic accuracy was defined by the area under the receiveroperating-characteristic curve (A z). Results: Diagnostic accuracy was good for pulmonary tumour (A z, 0.71±0.6), protozoan pneumonia (A z, 0.77±0.06), and ARDS (A z, 0.86±0.07) but poor for all other conditions. The absence of both pleural effusions and mediastinal/right hilar lymphadenopathy was significantly associated ( P≤0.04) with the absence of lung tumour. Discussion: The pulmonary processes in these patients can all demonstrate diffuse airspace opacification, and many patients had multiple lung abnormalities at autopsy. The radiologist-readers were unable to identify which pulmonary conditions were responsible for radiographic findings in most patients. The readers were able to identify patients who did not have pulmonary lymphoma. If pulmonary involvement with lymphoma is unlikely, bronchoscopy with bronchoalveolar lavage may be sufficient to establish a diagnosis. When pulmonary lymphoma is a clinical consideration, open lung biopsy is usually required for diagnosis