A nullstellensatz for sequences over F_p

Let p be a prime and let A=(a_1,...,a_l) be a sequence of nonzero elements in F_p. In this paper, we study the set of all 0-1 solutions to the equation a_1 x_1 + ... + a_l x_l = 0. We prove that whenever l >= p, this set actually characterizes A up to a nonzero multiplicative constant, which is no longer true for l < p. The critical case l=p is of particular interest. In this context, we prove that whenever l=p and A is nonconstant, the above equation has at least p-1 minimal 0-1 solutions, thus refining a theorem of Olson. The subcritical case l=p-1 is studied in detail also. Our approach is algebraic in nature and relies on the Combinatorial Nullstellensatz as well as on a Vosper type theorem.Comment: 23 page

A rigorous method for multigenic families' functional annotation: the peptidyl arginine deiminase (PADs) proteins family example

BACKGROUND: large scale and reliable proteins' functional annotation is a major challenge in modern biology. Phylogenetic analyses have been shown to be important for such tasks. However, up to now, phylogenetic annotation did not take into account expression data (i.e. ESTs, Microarrays, SAGE, ...). Therefore, integrating such data, like ESTs in phylogenetic annotation could be a major advance in post genomic analyses. We developed an approach enabling the combination of expression data and phylogenetic analysis. To illustrate our method, we used an example protein family, the peptidyl arginine deiminases (PADs), probably implied in Rheumatoid Arthritis. RESULTS: the analysis was performed as follows: we built a phylogeny of PAD proteins from the NCBI's NR protein database. We completed the phylogenetic reconstruction of PADs using an enlarged sequence database containing translations of ESTs contigs. We then extracted all corresponding expression data contained in EST database This analysis allowed us 1/To extend the spectrum of homologs-containing species and to improve the reconstruction of genes' evolutionary history. 2/To deduce an accurate gene expression pattern for each member of this protein family. 3/To show a correlation between paralogous sequences' evolution rate and pattern of tissular expression. CONCLUSION: coupling phylogenetic reconstruction and expression data is a promising way of analysis that could be applied to all multigenic families to investigate the relationship between molecular and transcriptional evolution and to improve functional annotation

Modelling avalanches in martensites

Solids subject to continuous changes of temperature or mechanical load often exhibit discontinuous avalanche-like responses. For instance, avalanche dynamics have been observed during plastic deformation, fracture, domain switching in ferroic materials or martensitic transformations. The statistical analysis of avalanches reveals a very complex scenario with a distinctive lack of characteristic scales. Much effort has been devoted in the last decades to understand the origin and ubiquity of scale-free behaviour in solids and many other systems. This chapter reviews some efforts to understand the characteristics of avalanches in martensites through mathematical modelling.Comment: Chapter in the book "Avalanches in Functional Materials and Geophysics", edited by E. K. H. Salje, A. Saxena, and A. Planes. The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-45612-6_

A note on linear Sperner families

In an earlier work we described Gr\"obner bases of the ideal of polynomials over a field, which vanish on the set of characteristic vectors $\mathbf{v} \in \{0,1\}^n$ of the complete $d$ unifom set family over the ground set $[n]$. In particular, it turns out that the standard monomials of the above ideal are {\em ballot monomials}. We give here a partial extension of the latter fact. We prove that the lexicographic standard monomials for linear Sperner systems are also ballot monomials. A set family is a linear Sperner system if the characteristic vectors satisfy a linear equation $a_1v_1+\cdots +a_nv_n=k$, where $0<a_q\leq a_2\leq \cdots \leq a_n$ and $k$ are integers. As an application, we confirm a conjecture of Frankl for linear Sperner systems.Comment: 12 page

Observational analytic studies in multiple sclerosis: controlling bias through study design and conduct. The Australian Multicentre Study of Environment and Immune Function

Rising multiple sclerosis incidence over the last 50 years and geographic patterns of occurrence suggest an environmental role in the causation of this multifactorial disease. Design options for epidemiological studies of environmental causes of multiple sclerosis are limited by the low incidence of the disease, possible diagnostic delay and budgetary constraints. We describe scientific and methodological issues considered in the development of the Australian Multicentre Study of Environment and Immune Function (the Ausimmune Study), which seeks, in particular, to better understand the causes of the well-known MS positive latitudinal gradient. A multicentre, case-control design down the eastern seaboard of Australia allows the recruitment of sufficient cases for adequate study power and provides data on environmental exposures that vary by latitude. Cases are persons with an incident first demyelinating event (rather than prevalent multiple sclerosis), sourced from a population base using a two tier notification system. Controls, matched on sex, age (within two years) and region of residence, are recruited from the general population. Biases common in case-control studies, eg, prevalence-incidence bias, admission-rate bias, non-respondent bias, observer bias and recall bias, as well as confounding have been carefully considered in the study design and conduct of the Ausimmune Study