A birational mapping with a strange attractor: Post critical set and covariant curves

We consider some two-dimensional birational transformations. One of them is a birational deformation of the H\'enon map. For some of these birational mappings, the post critical set (i.e. the iterates of the critical set) is infinite and we show that this gives straightforwardly the algebraic covariant curves of the transformation when they exist. These covariant curves are used to build the preserved meromorphic two-form. One may have also an infinite post critical set yielding a covariant curve which is not algebraic (transcendent). For two of the birational mappings considered, the post critical set is not infinite and we claim that there is no algebraic covariant curve and no preserved meromorphic two-form. For these two mappings with non infinite post critical sets, attracting sets occur and we show that they pass the usual tests (Lyapunov exponents and the fractal dimension) for being strange attractors. The strange attractor of one of these two mappings is unbounded.Comment: 26 pages, 11 figure

Post-critical set and non existence of preserved meromorphic two-forms

We present a family of birational transformations in $CP_2$ depending on two, or three, parameters which does not, generically, preserve meromorphic two-forms. With the introduction of the orbit of the critical set (vanishing condition of the Jacobian), also called ``post-critical set'', we get some new structures, some "non-analytic" two-form which reduce to meromorphic two-forms for particular subvarieties in the parameter space. On these subvarieties, the iterates of the critical set have a polynomial growth in the \emph{degrees of the parameters}, while one has an exponential growth out of these subspaces. The analysis of our birational transformation in $CP_2$ is first carried out using Diller-Favre criterion in order to find the complexity reduction of the mapping. The integrable cases are found. The identification between the complexity growth and the topological entropy is, one more time, verified. We perform plots of the post-critical set, as well as calculations of Lyapunov exponents for many orbits, confirming that generically no meromorphic two-form can be preserved for this mapping. These birational transformations in $CP_2$, which, generically, do not preserve any meromorphic two-form, are extremely similar to other birational transformations we previously studied, which do preserve meromorphic two-forms. We note that these two sets of birational transformations exhibit totally similar results as far as topological complexity is concerned, but drastically different results as far as a more ``probabilistic'' approach of dynamical systems is concerned (Lyapunov exponents). With these examples we see that the existence of a preserved meromorphic two-form explains most of the (numerical) discrepancy between the topological and probabilistic approach of dynamical systems.Comment: 34 pages, 7 figure

The impact of age on major orthopaedic trauma: an analysis of the United Kingdom Trauma Audit Research Network database.

AIMS: To compare the early management and mortality of older patients sustaining major orthopaedic trauma with that of a younger population with similar injuries. PATIENTS AND METHODS: The Trauma Audit Research Network database was reviewed to identify eligible patients admitted between April 2012 and June 2015. Distribution and severity of injury, interventions, comorbidity, critical care episodes and mortality were recorded. The population was divided into young (64 years or younger) and older (65 years and older) patients. RESULTS: Of 142 765 adults sustaining major trauma, 72 942 (51.09 %) had long bone or pelvic fractures and 45.81% of these were > 65 years old. Road traffic collision was the most common mechanism in the young (40.4%) and, in older people, fall from standing height (80.4%) predominated. The 30 day mortality in older patients with fractures is greater (6.8% versus2.5%), although critical care episodes are more common in the young (18.2%versus9.7%). Older people are less likely to be admitted to critical care beds and are often managed in isolation by surgeons. Orthopaedic surgery is the most common admitting and operating specialty and, in older people, fracture surgery accounted for 82.1% of procedures. CONCLUSION: Orthopaedic trauma in older people is associated with mortality that is significantly greater than for similar fractures in the young. As with the hip fracture population, major trauma in the elderly is a growing concern which highlights the need for a review of admission pathways and shared orthogeriatric care models

Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals

Lattice statistical mechanics, often provides a natural (holonomic) framework to perform singularity analysis with several complex variables that would, in a general mathematical framework, be too complex, or could not be defined. Considering several Picard-Fuchs systems of two-variables "above" Calabi-Yau ODEs, associated with double hypergeometric series, we show that holonomic functions are actually a good framework for actually finding the singular manifolds. We, then, analyse the singular algebraic varieties of the n-fold integrals $\chi^{(n)}$, corresponding to the decomposition of the magnetic susceptibility of the anisotropic square Ising model. We revisit a set of Nickelian singularities that turns out to be a two-parameter family of elliptic curves. We then find a first set of non-Nickelian singularities for $\chi^{(3)}$ and $\chi^{(4)}$, that also turns out to be rational or ellipic curves. We underline the fact that these singular curves depend on the anisotropy of the Ising model. We address, from a birational viewpoint, the emergence of families of elliptic curves, and of Calabi-Yau manifolds on such problems. We discuss the accumulation of these singular curves for the non-holonomic anisotropic full susceptibility.Comment: 36 page

Using Abbreviated Injury Scale (AIS) codes to classify Computed Tomography (CT) features in the Marshall System

<p>Abstract</p> <p>Background</p> <p>The purpose of Abbreviated Injury Scale (AIS) is to code various types of Traumatic Brain Injuries (TBI) based on their anatomical location and severity. The Marshall CT Classification is used to identify those subgroups of brain injured patients at higher risk of deterioration or mortality. The purpose of this study is to determine whether and how AIS coding can be translated to the Marshall Classification</p> <p>Methods</p> <p>Initially, a Marshall Class was allocated to each AIS code through cross-tabulation. This was agreed upon through several discussion meetings with experts from both fields (clinicians and AIS coders). Furthermore, in order to make this translation possible, some necessary assumptions with regards to coding and classification of mass lesions and brain swelling were essential which were all approved and made explicit.</p> <p>Results</p> <p>The proposed method involves two stages: firstly to determine all possible Marshall Classes which a given patient can attract based on allocated AIS codes; via cross-tabulation and secondly to assign one Marshall Class to each patient through an algorithm.</p> <p>Conclusion</p> <p>This method can be easily programmed in computer softwares and it would enable future important TBI research programs using trauma registry data.</p

Survival Trends After Surgery for Acute Subdural Hematoma in Adults Over a 20-year Period.

OBJECTIVE: We sought to determine 30-day survival trends and prognostic factors following surgery for acute subdural hematomas (ASDHs) in England and Wales over a 20-year period. SUMMARY OF BACKGROUND DATA: ASDHs are still considered the most lethal type of traumatic brain injury. It remains unclear whether the adjusted odds of survival have improved significantly over time. METHODS: Using the Trauma Audit and Research Network (TARN) database, we analyzed ASDH cases in the adult population (>16 yrs) treated surgically between 1994 and 2013. Two thousand four hundred ninety-eight eligible cases were identified. Univariable and multiple logistic regression analyses were performed, using multiple imputation for missing data. RESULTS: The cohort was 74% male with a median age of 48.9 years. Over half of patients were comatose at presentation (53%). Mechanism of injury was due to a fall (2 m 24%), road traffic collision (25%), and other (17%). Thirty-six per cent of patients presented with polytrauma. Gross survival increased from 59% in 1994 to 1998 to 73% in 2009 to 2013. Under multivariable analysis, variables independently associated with survival were year of injury, Glasgow Coma Scale, Injury Severity Score, age, and pupil reactivity. The time interval from injury to craniotomy and direct admission to a neurosurgical unit were not found to be significant prognostic factors. CONCLUSIONS: A significant improvement in survival over the last 20 years was observed after controlling for multiple prognostic factors. Prospective trials and cohort studies are expected to elucidate the distribution of functional outcome in survivors.AGK is supported by a Royal College of Surgeons of England Research Fellowship, a National Institute for Health Research (NIHR) Academic Clinical Fellowship, and a Raymond and Beverly Sackler Studentship. PJH is supported by a NIHR Research Professorship and the NIHR Cambridge Biomedical Research Centre.This is the author accepted manuscript. It is currently under an indefinite embargo pending publication by Wolters Kluwer

Improved outcomes for hepatic trauma in England and Wales over a decade of trauma and hepatobiliary surgery centralisation

Background: Over the last decade trauma services have undergone a reconfiguration in England and Wales. The objective is to describe the epidemiology, management and outcomes for liver trauma over this period and examine factors predicting survival. Methods: Patients sustaining hepatic trauma were identified using the Trauma Audit and Research Network database. Demographics, management and outcomes were assessed between January 2005 and December 2014 and analysed over five, 2-year study periods. Independent predictor variables for the outcome of liver trauma were analysed using multiple logistic regression. Results: 4368 Patients sustained hepatic trauma (with known outcome) between January 2005 and December 2014. Median age was 34 years (interquartile range 23–49). 81% were due to blunt and 19% to penetrating trauma. Road traffic collisions were the main mechanism of injury (58.2%). 241 patients (5.5%) underwent liver-specific surgery. The overall 30-day mortality rate was 16.4%. Improvements were seen in early consultant input, frequency and timing of computed tomography (CT) scanning, use of tranexamic acid and 30-day mortality over the five time periods. Being treated in a unit with an on-site HPB service increased the odds of survival (odds ratio 3.5, 95% confidence intervals 2.7–4.5). Conclusions: Our study has shown that being treated in a unit with an on-site HPB service increased the odds of survival. Further evaluation of the benefits of trauma and HPB surgery centralisation is warranted

Epidemiology and predictors of spinal injury in adult major trauma patients: European cohort study

This is a European cohort study on predictors of spinal injury in adult (≥16 years) major trauma patients, using prospectively collected data of the Trauma Audit and Research Network from 1988 to 2009. Predictors for spinal fractures/dislocations or spinal cord injury were determined using univariate and multivariate logistic regression analysis. 250,584 patients were analysed. 24,000 patients (9.6%) sustained spinal fractures/dislocations alone and 4,489 (1.8%) sustained spinal cord injury with or without fractures/dislocations. Spinal injury patients had a median age of 44.5 years (IQR = 28.8–64.0) and Injury Severity Score of 9 (IQR = 4–17). 64.9% were male. 45% of patients suffered associated injuries to other body regions. Age <45 years (≥45 years OR 0.83–0.94), Glasgow Coma Score (GCS) 3–8 (OR 1.10, 95% CI 1.02–1.19), falls >2 m (OR 4.17, 95% CI 3.98–4.37), sports injuries (OR 2.79, 95% CI 2.41–3.23) and road traffic collisions (RTCs) (OR 1.91, 95% CI 1.83–2.00) were predictors for spinal fractures/dislocations. Age <45 years (≥45 years OR 0.78–0.90), male gender (female OR 0.78, 95% CI 0.72–0.85), GCS <15 (OR 1.36–1.93), associated chest injury (OR 1.10, 95% CI 1.01–1.20), sports injuries (OR 3.98, 95% CI 3.04–5.21), falls >2 m (OR 3.60, 95% CI 3.21–4.04), RTCs (OR 2.20, 95% CI 1.96–2.46) and shooting (OR 1.91, 95% CI 1.21–3.00) were predictors for spinal cord injury. Multilevel injury was found in 10.4% of fractures/dislocations and in 1.3% of cord injury patients. As spinal trauma occurred in >10% of major trauma patients, aggressive evaluation of the spine is warranted, especially, in males, patients <45 years, with a GCS <15, concomitant chest injury and/or dangerous injury mechanisms (falls >2 m, sports injuries, RTCs and shooting). Diagnostic imaging of the whole spine and a diligent search for associated injuries are substantial

Intracranial bleeding in patients with traumatic brain injury: A prognostic study

BACKGROUND: Intracranial bleeding (IB) is a common and serious consequence of traumatic brain injury (TBI). IB can be classified according to the location into: epidural haemorrhage (EDH) subdural haemorrhage (SDH) intraparenchymal haemorrhage (IPH) and subarachnoid haemorrhage (SAH). Studies involving repeated CT scanning of TBI patients have found that IB can develop or expand in the 48 hours after injury. If IB enlarges after hospital admission and larger bleeds have a worse prognosis, this would provide a therapeutic rationale for treatments to prevent increase in the extent of bleeding. We analysed data from the Trauma Audit & Research Network (TARN), a large European trauma registry, to evaluate the association between the size of IB and mortality in patients with TBI. METHODS: We analysed 13,962 patients presenting to TARN participating hospitals between 2001 and 2008 with a Glasgow Coma Score (GCS) less than 15 at presentation or any head injury with Abbreviated Injury Scale (AIS) severity code 3 and above. The extent of intracranial bleeding was determined by the AIS code. Potential confounders were age, presenting Glasgow Coma Score, mechanism of injury, presence and nature of other brain injuries, and presence of extra-cranial injuries. The outcomes were in-hospital mortality and haematoma evacuation. We conducted a multivariable logistic regression analysis to evaluate the independent effect of large and small size of IB, in comparison with no bleeding, on patient outcomes. We also conducted a multivariable logistic regression analysis to assess the independent effect on mortality of large IB in comparison with small IB. RESULTS: Almost 46% of patients had at some type of IB. Subdural haemorrhages were present in 30% of the patients, with epidural and intraparenchymal present in approximately 22% each. After adjusting for potential confounders, we found that large IB, wherever located, was associated with increased mortality in comparison with no bleeding. We also found that large IB was associated with an increased risk of mortality in comparison with small IB. The odds ratio for mortality for large SDH, IPH and EDH, in comparison with small bleeds, were: 3.41 (95% CI: 2.684.33), 3.47 (95% CI: 2.265.33) and 2.86 (95% CI: 1.864.38) respectively. CONCLUSION: Large EDH, SDH and IPH are associated with a substantially higher probability of hospital mortality in comparison with small IB. However, the limitations of our data, such as the large proportion of missing data and lack of data on other confounding factors, such as localization of the bleeding, make the results of this report only explanatory. Future studies should also evaluate the effect of IB size on functional outcomes

Extra-corporeal membrane oxygenation for refractory cardiogenic shock after adult cardiac surgery:a systematic review and meta-analysis

Background - Postcardiotomy cardiogenic shock (PCCS) refractory to inotropic support and intra-aortic balloon pump (IABP) occurs rarely but is almost universally fatal without mechanical circulatory support. In this systematic review and meta-analysis we looked at the evidence behind the use of veno-arterial extra-corporeal membrane oxygenation (VA ECMO) in refractory PCCS from a patient survival rate and determinants of outcome viewpoint. Methods - A systematic review was performed in January 2017 using PubMed (with no defined time period) using the keywords “postcardiotomy”, “cardiogenic shock”, “extracorporeal membrane oxygenation” and “cardiac surgery”. We excluded papers pertaining to ECMO following paediatric cardiac surgery, medical causes of cardiogenic shock, as well as case reports, review articles, expert opinions, and letters to the editor. Once the studies were collated, a meta-analysis was performed on the proportion of survivors in those papers that met the inclusion criteria. Meta-regression was performed for the most commonly reported adverse prognostic indicators (API). Results - We identified 24 studies and a cumulative pool of 1926 patients from 1992 to 2016. We tabulated the demographic data, including the strengths and weaknesses for each of the studies, outcomes of VA ECMO for refractory PCCS, complications, and APIs. All the studies were retrospective cohort studies. Meta-analysis of the moderately heterogeneous data (95% CI 0.29 to 0.34, p 70 years, 95% CI −0.057 to 0.001, P = 0.058), and long ECMO support (95% CI −0.068 to 0.166, P = 0.412). Postoperative renal failure, high EuroSCORE (>20%), diabetes mellitus, obesity, rising lactate whilst on ECMO, gastrointestinal complications had also been reported. Conclusion - Haemodynamic support with VA ECMO provides a survival benefit with reasonable intermediate and long-term outcomes. Many studies had reported advanced age, renal failure and prolonged VA ECMO support as the most likely APIs for VA ECMO in PCCS. EuroSCORE can be utilized to anticipate the need for prophylactic perioperative VA ECMO in the high-risk category. APIs can be used to aid decision-making regarding both the institution and weaning of ECMO for refractory PCCS