Overview of (pro-)Lie group structures on Hopf algebra character groups

Character groups of Hopf algebras appear in a variety of mathematical and physical contexts. To name just a few, they arise in non-commutative geometry, renormalisation of quantum field theory, and numerical analysis. In the present article we review recent results on the structure of character groups of Hopf algebras as infinite-dimensional (pro-)Lie groups. It turns out that under mild assumptions on the Hopf algebra or the target algebra the character groups possess strong structural properties. Moreover, these properties are of interest in applications of these groups outside of Lie theory. We emphasise this point in the context of two main examples: The Butcher group from numerical analysis and character groups which arise from the Connes--Kreimer theory of renormalisation of quantum field theories.Comment: 31 pages, precursor and companion to arXiv:1704.01099, Workshop on "New Developments in Discrete Mechanics, Geometric Integration and Lie-Butcher Series", May 25-28, 2015, ICMAT, Madrid, Spai

Randomised controlled trial of cervical radiofrequency lesions as a treatment for cervicogenic headache [ISRCTN07444684]

BACKGROUND: Cervicogenic headache (CEH) is a unilateral headache localised in the neck or occipital region, projecting to the frontal and temporal regions. Since the pathogenesis of this syndrome appears to have an anatomical basis in the cervical region, several surgical procedures aimed at reducing the nociceptive input on the cervical level, have been tested. We developed a sequence of various cervical radiofrequency neurotomies (facet joint denervations eventually followed by upper dorsal root ganglion neurotomies) that proved successful in a prospective pilot trial with 15 CEH patients. To further evaluate this sequential treatment program we conducted a randomised controlled trial METHODS: 30 patients with cervicogenic headache according to the Sjaastad diagnostic criteria, were randomised. 15 patients received a sequence of radiofrequency treatments (cervical facet joint denervation, followed by cervical dorsal root ganglion lesions when necessary), and the other 15 patients underwent local injections with steroid and anaesthetic at the greater occipital nerve, followed by transcutaneous electrical nerve stimulation (TENS) when necessary. Visual analogue scores for pain, global perceived effects scores, quality of life scores were assessed at 8, 16, 24 and 48 weeks. Patients also kept a headache diary. RESULTS: There were no statistically significant differences between the two treatment groups at any time point in the trial. CONCLUSION: We did not find evidence that radiofrequency treatment of cervical facet joints and upper dorsal root ganglions is a better treatment than the infiltration of the greater occipital nerve, followed by TENS for patients fulfilling the clinical criteria of cervicogenic headache

Recovery 3 and 12 months after hysterectomy: epidemiology and predictors of chronic pain, physical functioning, and global surgical recovery

Chronic postsurgical pain (CPSP) is 1 important aspect of surgical recovery. To improve perioperative care and postoperative recovery knowledge on predictors of impaired recovery is essential. The aim of this study is to assess predictors and epidemiological data of CPSP, physical functioning (SF-36PF, 0–100), and global surgical recovery (global surgical recovery index, 0–100%) 3 and 12 months after hysterectomy for benign indication. A prospective multicenter cohort study was performed. Sociodemographic, somatic, and psychosocial data were assessed in the week before surgery, postoperatively up to day 4, and at 3- and 12-month follow-up. Generalized linear model (CPSP) and linear-mixed model analyses (SF-36PF and global surgical recovery index) were used. Baseline data of 468 patients were collected, 412 (88%) patients provided data for 3-month evaluation and 376 (80%) patients for 12-month evaluation. After 3 and 12 months, prevalence of CPSP (numeric rating scale ≥ 4, scale 0–10) was 10.2% and 9.0%, respectively, SF-36PF means (SD) were 83.5 (20.0) and 85.9 (20.2), global surgical recovery index 88.1% (15.6) and 93.3% (13.4). Neuropathic pain was reported by 20 (5.0%) patients at 3 months and 14 (3.9%) patients at 12 months. Preoperative pain, surgery-related worries, acute postsurgical pain on day 4, and surgery-related infection were significant predictors of CPSP. Baseline level, participating center, general psychological robustness, indication, acute postsurgical pain, and surgery-related infection were significant predictors of SF-36PF. Predictors of global surgical recovery were baseline expectations, surgery-related worries, American Society of Anesthesiologists classification, type of anesthesia, acute postsurgical pain, and surgery-related infection. Several predictors were identified for CPSP, physical functioning, and global surgical recovery. Some of the identified factors are modifiable and optimization of patients’ preoperative pain status and psychological condition as well as reduction of acute postsurgical pain and surgery-related infection may lead to improvement of outcome

Differences in sensory processing between chronic cervical zygapophysial joint pain patients with and without cervicogenic headache

BACKGROUND: It is not known why some patients with underlying chronic nociceptive sources in the neck develop cervicogenic headache (CEH) and why others do not. This quantitative sensory testing (QST) study systematically explores the differences in sensory pain processing in 17 CEH patients with underlying chronic cervical zygapophysial joint pain compared to 10 patients with chronic cervical zygapophysial joint pain but without CEH. METHODS: The QST protocol comprises pressure pain threshold testing, thermal detection threshold testing, electrical pain threshold testing and measurement of descending inhibitory modulation using the conditioned pain modulation (CPM) paradigm. RESULTS: The main difference between patients with or without CEH was the lateralization of pressure hyperalgesia to the painful side of the head of CEH patients, accompanied by cold as well as warm relative hyperesthesia on the painful side of the head and neck. DISCUSSION: From this hypothesis-generating study, our results suggest that rostral neuraxial spread of central sensitization, probably to the trigeminal spinal nucleus, plays a major role in the development of CEH