Univalent Foundations and the UniMath Library

We give a concise presentation of the Univalent Foundations of mathematics outlining the main ideas, followed by a discussion of the UniMath library of formalized mathematics implementing the ideas of the Univalent Foundations (section 1), and the challenges one faces in attempting to design a large-scale library of formalized mathematics (section 2). This leads us to a general discussion about the links between architecture and mathematics where a meeting of minds is revealed between architects and mathematicians (section 3). On the way our odyssey from the foundations to the "horizon" of mathematics will lead us to meet the mathematicians David Hilbert and Nicolas Bourbaki as well as the architect Christopher Alexander

Clinical trial simulation to evaluate power to compare the antiviral effectiveness of two hepatitis C protease inhibitors using nonlinear mixed effect models: a viral kinetic approach.

International audienceBACKGROUND: Models of hepatitis C virus (HCV) kinetics are increasingly used to estimate and to compare in vivo drug's antiviral effectiveness of new potent anti-HCV agents. Viral kinetic parameters can be estimated using non-linear mixed effect models (NLMEM). Here we aimed to evaluate the performance of this approach to precisely estimate the parameters and to evaluate the type I errors and the power of the Wald test to compare the antiviral effectiveness between two treatment groups when data are sparse and/or a large proportion of viral load (VL) are below the limit of detection (BLD). METHODS: We performed a clinical trial simulation assuming two treatment groups with different levels of antiviral effectiveness. We evaluated the precision and the accuracy of parameter estimates obtained on 500 replication of this trial using the stochastic approximation expectation-approximation algorithm which appropriately handles BLD data. Next we evaluated the type I error and the power of the Wald test to assess a difference of antiviral effectiveness between the two groups. Standard error of the parameters and Wald test property were evaluated according to the number of patients, the number of samples per patient and the expected difference in antiviral effectiveness. RESULTS: NLMEM provided precise and accurate estimates for both the fixed effects and the inter-individual variance parameters even with sparse data and large proportion of BLD data. However Wald test with small number of patients and lack of information due to BLD resulted in an inflation of the type I error as compared to the results obtained when no limit of detection of VL was considered. The corrected power of the test was very high and largely outperformed what can be obtained with empirical comparison of the mean VL decline using Wilcoxon test. CONCLUSION: This simulation study shows the benefit of viral kinetic models analyzed with NLMEM over empirical approaches used in most clinical studies. When designing a viral kinetic study, our results indicate that the enrollment of a large number of patients is to be preferred to small population sample with frequent assessments of VL

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

Sur les exposants de Lyapounov des applications meromorphes

Let f be a dominating meromorphic self-map of a compact Kahler manifold. We give an inequality for the Lyapounov exponents of some ergodic measures of f using the metric entropy and the dynamical degrees of f. We deduce the hyperbolicity of some measures.Comment: 27 pages, paper in french, final version: to appear in Inventiones Mat

Crises and collective socio-economic phenomena: simple models and challenges

Financial and economic history is strewn with bubbles and crashes, booms and busts, crises and upheavals of all sorts. Understanding the origin of these events is arguably one of the most important problems in economic theory. In this paper, we review recent efforts to include heterogeneities and interactions in models of decision. We argue that the Random Field Ising model (RFIM) indeed provides a unifying framework to account for many collective socio-economic phenomena that lead to sudden ruptures and crises. We discuss different models that can capture potentially destabilising self-referential feedback loops, induced either by herding, i.e. reference to peers, or trending, i.e. reference to the past, and account for some of the phenomenology missing in the standard models. We discuss some empirically testable predictions of these models, for example robust signatures of RFIM-like herding effects, or the logarithmic decay of spatial correlations of voting patterns. One of the most striking result, inspired by statistical physics methods, is that Adam Smith's invisible hand can badly fail at solving simple coordination problems. We also insist on the issue of time-scales, that can be extremely long in some cases, and prevent socially optimal equilibria to be reached. As a theoretical challenge, the study of so-called "detailed-balance" violating decision rules is needed to decide whether conclusions based on current models (that all assume detailed-balance) are indeed robust and generic.Comment: Review paper accepted for a special issue of J Stat Phys; several minor improvements along reviewers' comment

Visual versus semi-quantitative analysis of 18F-FDG-PET in amnestic MCI. An European Alzheimer\u27s Disease Consortium (EADC) project

We aimed to investigate the accuracy of FDG-PET to detect the Alzheimer\u27s disease (AD) brain glucose hypometabolic pattern in 142 patients with amnestic mild cognitive impairment (aMCI) and 109 healthy controls. aMCI patients were followed for at least two years or until conversion to dementia. Images were evaluated by means of visual read by either moderately-skilled or expert readers, and by means of a summary metric of AD-like hypometabolism (PALZ score). Seventy-seven patients converted to AD-dementia after 28.6?19.3 months of follow-up. Expert reading was the most accurate tool to detect these MCI converters from healthy controls (sensitivity 89.6%, specificity 89.0%, accuracy 89.2%) while two moderately-skilled readers were less (p < 0.05) specific (sensitivity 85.7%, specificity 79.8%, accuracy 82.3%) and PALZ scorewas less (p < 0.001) sensitive (sensitivity 62.3%, specificity 91.7%, accuracy 79.6%). Among the remaining 67 aMCI patients, 50 were confirmed as aMCI after an average of 42.3 months, 12 developed other dementia, and 3 reverted to normalcy. In 30/50 persistent MCI patients, the expert recognized the AD hypometabolic pattern. In 13/50 aMCI, both the expert and PALZ score were negative while in 7/50, only the PALZ score was positive due to sparse hypometabolic clusters mainly in frontal lobes. Visual FDG-PET reads by an expert is the most accurate method but an automated, validated system may be particularly helpful to moderately-skilled readers because of high specificity, and should be mandatory when even a moderately-skilled reader is unavailable

Modeling the dynamics of biomarkers during primary HIV infection taking into account the uncertainty of infection date

During primary HIV infection, the kinetics of plasma virus concentrations and CD4+ cell counts is very complex. Parametric and nonparametric models have been suggested for fitting repeated measurements of these markers. Alternatively, mechanistic approaches based on ordinary differential equations have also been proposed. These latter models are constructed according to biological knowledge and take into account the complex nonlinear interactions between viruses and cells. However, estimating the parameters of these models is difficult. A main difficulty in the context of primary HIV infection is that the date of infection is generally unknown. For some patients, the date of last negative HIV test is available in addition to the date of first positive HIV test (seroconverters). In this paper we propose a likelihood-based method for estimating the parameters of dynamical models using a population approach and taking into account the uncertainty of the infection date. We applied this method to a sample of 761 HIV-infected patients from the Concerted Action on SeroConversion to AIDS and Death in Europe (CASCADE).Comment: Published in at http://dx.doi.org/10.1214/10-AOAS364 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Melanocortin-1 Receptor, Skin Cancer and Phenotypic Characteristics (M-SKIP) Project: Study Design and Methods for Pooling Results of Genetic Epidemiological Studies

Background: For complex diseases like cancer, pooled-analysis of individual data represents a powerful tool to investigate the joint contribution of genetic, phenotypic and environmental factors to the development of a disease. Pooled-analysis of epidemiological studies has many advantages over meta-analysis, and preliminary results may be obtained faster and with lower costs than with prospective consortia. Design and methods: Based on our experience with the study design of the Melanocortin-1 receptor (MC1R) gene, SKin cancer and Phenotypic characteristics (M-SKIP) project, we describe the most important steps in planning and conducting a pooled-analysis of genetic epidemiological studies. We then present the statistical analysis plan that we are going to apply, giving particular attention to methods of analysis recently proposed to account for between-study heterogeneity and to explore the joint contribution of genetic, phenotypic and environmental factors in the development of a disease. Within the M-SKIP project, data on 10,959 skin cancer cases and 14,785 controls from 31 international investigators were checked for quality and recoded for standardization. We first proposed to fit the aggregated data with random-effects logistic regression models. However, for the M-SKIP project, a two-stage analysis will be preferred to overcome the problem regarding the availability of different study covariates. The joint contribution of MC1R variants and phenotypic characteristics to skin cancer development will be studied via logic regression modeling. Discussion: Methodological guidelines to correctly design and conduct pooled-analyses are needed to facilitate application of such methods, thus providing a better summary of the actual findings on specific fields

Using C. elegans to decipher the cellular and molecular mechanisms underlying neurodevelopmental disorders

Prova tipográfica (uncorrected proof)Neurodevelopmental disorders such as epilepsy, intellectual disability (ID), and autism spectrum disorders (ASDs) occur in over 2 % of the population, as the result of genetic mutations, environmental factors, or combination of both. In the last years, use of large-scale genomic techniques allowed important advances in the identification of genes/loci associated with these disorders. Nevertheless, following association of novel genes with a given disease, interpretation of findings is often difficult due to lack of information on gene function and effect of a given mutation in the corresponding protein. This brings the need to validate genetic associations from a functional perspective in model systems in a relatively fast but effective manner. In this context, the small nematode, Caenorhabditis elegans, presents a good compromise between the simplicity of cell models and the complexity of rodent nervous systems. In this article, we review the features that make C. elegans a good model for the study of neurodevelopmental diseases. We discuss its nervous system architecture and function as well as the molecular basis of behaviors that seem important in the context of different neurodevelopmental disorders. We review methodologies used to assess memory, learning, and social behavior as well as susceptibility to seizures in this organism. We will also discuss technological progresses applied in C. elegans neurobiology research, such as use of microfluidics and optogenetic tools. Finally, we will present some interesting examples of the functional analysis of genes associated with human neurodevelopmental disorders and how we can move from genes to therapies using this simple model organism.The authors would like to acknowledge Fundação para a Ciência e Tecnologia (FCT) (PTDC/SAU-GMG/112577/2009). AJR and CB are recipients of FCT fellowships: SFRH/BPD/33611/2009 and SFRH/BPD/74452/2010, respectively