Polyhedra and commensurability

This snapshot introduces the notion of commensurability of polyhedra. At its bottom, this concept can be developed from constructions with paper, scissors, and glue. Starting with an elementary example, we formalize it subsequently. Finally, we discuss intriguing connections with other fields of mathematics

Commensurability of hyperbolic Coxeter groups: theory and computation

For hyperbolic Coxeter groups of finite covolume we review and present new theoretical and computational aspects of wide commensurability. We discuss separately the arithmetic and the non-arithmetic cases. Some worked examples are added as well as a panoramic viewto hyperbolic Coxeter groups and their classification

Predicting calyceal access for percutaneous nephrolithotomy with computed tomography multiplanar reconstruction

OBJECTIVES: Objective parameters in computed tomography (CT) scans that could predict calyceal access during percutaneous nephrolithotomy have not been evaluated. These parameters could improve access planning for percutaneous nephrolithotomy. We aimed to determine which parameters extracted from a preoperative multiplanar reconstructed CT could predict renal calyceal access during a percutaneous nephrolithotomy. METHODS: From January 2009 through April 2011, 230 patients underwent 284 percutaneous nephrolithotomies at our institution. Sixteen patients presented with complete staghorn calculi, and 11 patients (13 renal units) were analyzed. Five parameters were extracted from a preoperative reconstructed CT and compared with the surgical results of percutaneous nephrolithotomy. RESULTS: Fifty-eight calyces were studied, with an average of 4.4 calyces per procedure. A rigid nephroscope was used to access a particular calyx, and a univariate analysis showed that the entrance calyx had a smaller length (2.7 vs. 3.98 cm, p = 0.018). The particular calyx to be accessed should have a smaller length (2.22 vs. 3.19 cm, p = 0.012), larger angles (117.6 vs. 67.96,

On commensurable hyperbolic Coxeter groups

For Coxeter groups acting non-cocompactly but with finite covolume on real hyperbolic space Hn, new methods are presented to distinguish them up to (wide) commensurability. We exploit these ideas and determine the commensurability classes of all hyperbolic Coxeter groups whose fundamental polyhedra are pyramids over a product of two simplices of positive dimensions

Commensurability of hyperbolic Coxeter groups: theory and computation (Geometry and Analysis of Discrete Groups and Hyperbolic Spaces)

"Geometry and Analysis of Discrete Groups and Hyperbolic Spaces". June 22～26, 2015. edited by Michihiko Fujii, Nariya Kawazumi and Ken'ichi Ohshika. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.For hyperbolic Coxeter groups of finite covolume we review and present new theoretical and computational aspects of wide commensurability. We discuss separately the arithmetic and the non-arithmetic cases. Some worked examples are added as well as a panoramic view to hyperbolic Coxeter groups and their classification

Polyèdres et commensurabilité

Le but de cet article est de présenter la notion de commensurabilité de polyèdres. Cette notion est basée uniquement sur l’utilisation de ciseaux et de colle. Les deux premières sections sont élémentaires et servent à présenter un exemple concret et sa formalisation. Dans la dernière section, certains liens avec d’autres domaines des mathématiques sont discutés. Cet article est une traduction abrégée du "Snapshot of Modern Mathematics MFO" [1] des mêmes auteurs

Robotic Salvage Lymph Node Dissection After Radical Prostatectomy

ABSTRACTIntroduction and objective:Radical prostatectomy is a first-line treatment for localized prostate cancer. However, in some cases, biochemical recurrence associated with imaging-detected nodal metastases may happen. Herein, we aim to present the surgical technique for salvage lymph node dissection after radical prostatectomy.Materials and Methods:A 70 year-old asymptomatic man presented with a prostate-specific antigen (PSA) of 7.45ng/ mL. Digital rectal examination was normal and trans-rectal prostate biopsy revealed a prostate adenocarcinoma Gleason 7 (3+4). Pre-operative computed tomography scan and bone scintigraphy showed no metastatic disease. In other service, the patient underwent a robotic-assisted radical prostatectomy plus obturador lymphadenectomy. Pathologic examination showed a pT3aN0 tumor. After 6 months of follow-up, serum PSA was 1.45ng/mL. Further investigation with 11C–Choline PET/CT revealed only a 2-cm lymph node close to the left internal iliac artery. The patient was counseled for salvage lymph node dissection.Results:Salvage lymph node dissection was uneventfully performed. Operative time was 1.5 hour, blood loss was minimal, and there were no intra- or postoperative complications. The patient was discharged from hospital in the 1st postoperative day. After 12 months of follow-up, his PSA was undetectable with no other adjuvant therapy.Conclusion:Robotic salvage pelvic lymph node dissection is an effective option for treatment of patients with biochemical recurrence after radical prostatectomy and only pelvic lymph node metastasis detected by C11-Choline PET/CT