Polarizations and differential calculus in affine spaces

Within the framework of mappings between affine spaces, the notion of $n$-th polarization of a function will lead to an intrinsic characterization of polynomial functions. We prove that the characteristic features of derivations, such as linearity, iterability, Leibniz and chain rules, are shared -- at the finite level -- by the polarization operators. We give these results by means of explicit general formulae, which are valid at any order $n$, and are based on combinatorial identities. The infinitesimal limits of the $n$-th polarizations of a function will yield its $n$-th derivatives (without resorting to the usual recursive definition), and the above mentioned properties will be recovered directly in the limit. Polynomial functions will allow us to produce a coordinate free version of Taylor's formula

Overexpression of the Cytokine BAFF and Autoimmunity Risk

$\textbf{BACKGROUND}$: Genomewide association studies of autoimmune diseases have mapped hundreds of susceptibility regions in the genome. However, only for a few association signals has the causal gene been identified, and for even fewer have the causal variant and underlying mechanism been defined. Coincident associations of DNA variants affecting both the risk of autoimmune disease and quantitative immune variables provide an informative route to explore disease mechanisms and drug-targetable pathways. $\textbf{METHODS}$: Using case-control samples from Sardinia, Italy, we performed a genomewide association study in multiple sclerosis followed by TNFSF13B locus-specific association testing in systemic lupus erythematosus (SLE). Extensive phenotyping of quantitative immune variables, sequence-based fine mapping, cross-population and cross-phenotype analyses, and gene-expression studies were used to identify the causal variant and elucidate its mechanism of action. Signatures of positive selection were also investigated. $\textbf{RESULTS}$: A variant in TNFSF13B, encoding the cytokine and drug target B-cell activating factor (BAFF), was associated with multiple sclerosis as well as SLE. The disease-risk allele was also associated with up-regulated humoral immunity through increased levels of soluble BAFF, B lymphocytes, and immunoglobulins. The causal variant was identified: an insertion-deletion variant, GCTGT→A (in which A is the risk allele), yielded a shorter transcript that escaped microRNA inhibition and increased production of soluble BAFF, which in turn up-regulated humoral immunity. Population genetic signatures indicated that this autoimmunity variant has been evolutionarily advantageous, most likely by augmenting resistance to malaria. $\textbf{CONCLUSIONS}$: A TNFSF13B variant was associated with multiple sclerosis and SLE, and its effects were clarified at the population, cellular, and molecular levels. (Funded by the Italian Foundation for Multiple Sclerosis and others.).Supported by grants (2011/R/13 and 2015/R/09, to Dr. Cucca) from the Italian Foundation for Multiple Sclerosis; contracts (N01-AG-1-2109 and HHSN271201100005C, to Dr. Cucca) from the Intramural Research Program of the National Institute on Aging, National Institutes of Health (NIH); a grant (FaReBio2011 “Farmaci e Reti Biotecnologiche di Qualità,” to Dr. Cucca) from the Italian Ministry of Economy and Finance; a grant (633964, to Dr. Cucca) from the Horizon 2020 Research and Innovation Program of the European Union; a grant (U1301.2015/AI.1157.BE Prat. 2015-1651, to Dr. Cucca) from Fondazione di Sardegna; grants (“Centro per la ricerca di nuovi farmaci per malattie rare, trascurate e della povertà” and “Progetto collezione di composti chimici ed attività di screening,” to Dr. Cucca) from Ministero dell’Istruzione, dell’Università e della Ricerca; grants (HG005581, HG005552, HG006513, and HG007022, to Dr. Abecasis) from the National Human Genome Research Institute; a grant (9-2011-253, to Dr. Todd) from JDRF; a grant (091157, to Dr. Todd) from the Wellcome Trust; a grant (to Dr. Todd) from the National Institute for Health Research (NIHR); and the NIHR Cambridge Biomedical Research Centre. Dr. Idda was a recipient of a Master and Back fellowship from the Autonomous Region of Sardinia

Modeling and estimation of signal-dependent and correlated noise

The additive white Gaussian noise (AWGN) model is ubiquitous in signal processing. This model is often justified by central-limit theorem (CLT) arguments. However, whereas the CLT may support a Gaussian distribution for the random errors, it does not provide any justification for the assumed additivity and whiteness. As a matter of fact, data acquired in real applications can seldom be described with good approximation by the AWGN model, especially because errors are typically correlated and not additive. Failure to model accurately the noise leads to inaccurate analysis, ineffective filtering, and distortion or even failure in the estimation. This chapter provides an introduction to both signal-dependent and correlated noise and to the relevant models and basic methods for the analysis and estimation of these types of noise. Generic one-parameter families of distributions are used as the essential mathematical setting for the observed signals. The distribution families covered as leading examples include Poisson, mixed Poisson–Gaussian, various forms of signal-dependent Gaussian noise (including multiplicative families and approximations of the Poisson family), as well as doubly censored heteroskedastic Gaussian distributions. We also consider various forms of noise correlation, encompassing pixel and readout cross-talk, fixed-pattern noise, column/row noise, etc., as well as related issues like photo-response and gain nonuniformity. The introduced models and methods are applicable to several important imaging scenarios and technologies, such as raw data from digital camera sensors, various types of radiation imaging relevant to security and to biomedical imaging.acceptedVersionPeer reviewe

Manifolds of mappings for continuum mechanics

This is an overview article. After an introduction to convenient calculus in infinite dimensions, the foundational material for manifolds of mappings is presented. The central character is the smooth convenient manifold $C^{\infty}(M,N)$ of all smooth mappings from a finite dimensional Whitney manifold germ $M$ into a smooth manifold $N$. A Whitney manifold germ is a smooth (in the interior) manifold with a very general boundary, but still admitting a continuous Whitney extension operator. This notion is developed here for the needs of geometric continuum mechanics.Comment: 66 pages. arXiv admin note: substantial text overlap with arXiv:1505.02359, arXiv:math/9202208. Last version: some misprintscorrecte