Solution of the string equations for asymmetric potentials

We consider the large $N$ expansion of the partition function for the Hermitian one-matrix model. It is well known that the coefficients of this expansion are generating functions $F^{(g)}$ for a certain kind of graph embedded in a Riemann surface. Other authors have made a simplifying assumption that the potential $V$ is an even function. We present a method for computing $F^{(g)}$ in the case that $V$ is not an even function. Our method is based on the string equations, and yields "valence independent" formulas which do not depend explicitly on the potential. We introduce a family of differential operators, the "string polynomials", which make clear the valence independent nature of the string equations.Comment: 32 pages, 1 figur

Valence independent formula for the equilibrium measure

We derive a new formula for the equilibrium measure for eigenvalues of random matrices sampled from polynomial perturbations of the GUE, valid in the one-cut case. The virtue of our formula is that it depends on the potential only implicitly through the endpoints of support of the equilibrium measure. Our motivation is the problem of computing explicit formulas for generating functions which enumerate graphs embedded in a Riemann surface. To demonstrate the utility of our formula for the equilibrium measure, we derive a formula for the generating function $e_1$ enumerating maps on the torus. This formula is "valence independent" in the sense that it holds regardless of what numbers of edges are allowed to meet at vertices; furthermore it subsumes formulas for $e_1$ given by other authors as special cases.Comment: 17 page

A PDE Approach to the Combinatorics of the Full Map Enumeration Problem: Exact Solutions and their Universal Character

Maps are polygonal cellular networks on Riemann surfaces. This paper completes a program of constructing closed form general representations for the enumerative generating functions associated to maps of fixed but arbitrary genus. These closed form expressions have a universal character in the sense that they are independent of the explicit valence distribution of the tiling polygons. Nevertheless the valence distributions may be recovered from the closed form generating functions by a remarkable {\it unwinding identity} in terms of the Appell polynomials generated by Bessel functions. Our treatment, based on random matrix theory and Riemann-Hilbert problems for orthogonal polynomials reveals the generating functions to be solutions of nonlinear conservation laws and their prolongations. This characterization enables one to gain insights that go beyond more traditional methods that are purely combinatorial. Universality results are connected to stability results for characteristic singularities of conservation laws that were studied by Caflisch, Ercolani, Hou and Landis as well as directly related to universality results for random matrix spectra as described by Deift, Kriecherbauer, McLaughlin, Venakides and Zhou

Reporting performance of prognostic models in cancer: a review

<p>Abstract</p> <p>Background</p> <p>Appropriate choice and use of prognostic models in clinical practice require the use of good methods for both model development, and for developing prognostic indices and risk groups from the models. In order to assess reliability and generalizability for use, models need to have been validated and measures of model performance reported. We reviewed published articles to assess the methods and reporting used to develop and evaluate performance of prognostic indices and risk groups from prognostic models.</p> <p>Methods</p> <p>We developed a systematic search string and identified articles from PubMed. Forty-seven articles were included that satisfied the following inclusion criteria: published in 2005; aiming to predict patient outcome; presenting new prognostic models in cancer with outcome time to an event and including a combination of at least two separate variables; and analysing data using multivariable analysis suitable for time to event data.</p> <p>Results</p> <p>In 47 studies, Cox models were used in 94% (44), but the coefficients or hazard ratios for the variables in the final model were reported in only 72% (34). The reproducibility of the derived model was assessed in only 11% (5) of the articles. A prognostic index was developed from the model in 81% (38) of the articles, but researchers derived the prognostic index from the final prognostic model in only 34% (13) of the studies; different coefficients or variables from those in the final model were used in 50% (19) of models and the methods used were unclear in 16% (6) of the articles. Methods used to derive prognostic groups were also poor, with researchers not reporting the methods used in 39% (14 of 36) of the studies and data derived methods likely to bias estimates of differences between risk groups being used in 28% (10) of the studies. Validation of their models was reported in only 34% (16) of the studies. In 15 studies validation used data from the same population and in five studies from a different population. Including reports of validation with external data from publications up to four years following model development, external validation was attempted for only 21% (10) of models. Insufficient information was provided on the performance of models in terms of discrimination and calibration.</p> <p>Conclusions</p> <p>Many published prognostic models have been developed using poor methods and many with poor reporting, both of which compromise the reliability and clinical relevance of models, prognostic indices and risk groups derived from them.</p

Reporting methods in studies developing prognostic models in cancer: a review

Development of prognostic models enables identification of variables that are influential in predicting patient outcome and the use of these multiple risk factors in a systematic, reproducible way according to evidence based methods. The reliability of models depends on informed use of statistical methods, in combination with prior knowledge of disease. We reviewed published articles to assess reporting and methods used to develop new prognostic models in cancer.We developed a systematic search string and identified articles from PubMed. Forty-seven articles were included that satisfied the following inclusion criteria: published in 2005; aiming to predict patient outcome; presenting new prognostic models in cancer with outcome time to an event and including a combination of at least two separate variables; and analysing data using multivariable analysis suitable for time to event data.In 47 studies, prospective cohort or randomised controlled trial data were used for model development in only 33% (15) of studies. In 30% (14) of the studies insufficient data were available, having fewer than 10 events per variable (EPV) used in model development. EPV could not be calculated in a further 40% (19) of the studies. The coding of candidate variables was only reported in 68% (32) of the studies. Although use of continuous variables was reported in all studies, only one article reported using recommended methods of retaining all these variables as continuous without categorisation. Statistical methods for selection of variables in the multivariate modelling were often flawed. A method that is not recommended, namely, using statistical significance in univariate analysis as a pre-screening test to select variables for inclusion in the multivariate model, was applied in 48% (21) of the studies.We found that published prognostic models are often characterised by both use of inappropriate methods for development of multivariable models and poor reporting. In addition, models are limited by the lack of studies based on prospective data of sufficient sample size to avoid overfitting. The use of poor methods compromises the reliability of prognostic models developed to provide objective probability estimates to complement clinical intuition of the physician and guidelines

Antibodies against hypocretin receptor 2 are rare in narcolepsy

Recently, antibodies to the hypocretin receptor 2 (HCRTR2-Abs) were reported in a high proportion of narcolepsy patients who developed the disease following Pandemrix\uae vaccination. We tested a group of narcolepsy patients for the HCRTR2-Abs using a newly established cell-based assay

The importance of early immunotherapy in patients with faciobrachial dystonic seizures.

Faciobrachial dystonic seizures and limbic encephalitis closely associate with antibodies to leucine-rich glioma-inactivated 1 (LGI1). Here, we describe 103 consecutive patients with faciobrachial dystonic seizures and LGI1 antibodies to understand clinical, therapeutic and serological differences between those with and without cognitive impairment, and to determine whether cessation of faciobrachial dystonic seizures can prevent cognitive impairment. The 22/103 patients without cognitive impairment typically had normal brain MRI, EEGs and serum sodium levels (P &lt; 0.0001). Overall, cessation of faciobrachial dystonic seizures with antiepileptic drugs alone occurred in only 9/89 (10%) patients. By contrast, 51% showed cessation of faciobrachial dystonic seizures 30 days after addition of immunotherapy (P &lt; 0.0001), with earlier cessation in cognitively normal patients (P = 0.038). Indeed, expedited immunotherapy (P = 0.031) and normal cognition (P = 0.0014) also predicted reduced disability at 24 months. Furthermore, of 80 patients with faciobrachial dystonic seizures as their initial feature, 56% developed cognitive impairment after 90 days of active faciobrachial dystonic seizures. Whereas only one patient developed cognitive impairment after cessation of faciobrachial dystonic seizures (P &lt; 0.0001). All patients had IgG4-LGI1 antibodies, but those with cognitive impairment had higher proportions of complement-fixing IgG1 antibodies (P = 0.03). Both subclasses caused LGI1-ADAM22 complex internalization, a potential non-inflammatory epileptogenic mechanism. In summary, faciobrachial dystonic seizures show striking time-sensitive responses to immunotherapy, and their cessation can prevent the development of cognitive impairment.awx323media15681705685001