Complete Set of Invariants of a 4th Order Tensor: The 12 Tasks of HARDI from Ternary Quartics

International audienceInvariants play a crucial role in Diffusion MRI. In DTI (2 nd order tensors), invariant scalars (FA, MD) have been successfully used in clinical applications. But DTI has limitations and HARDI models (e.g. 4 th order tensors) have been proposed instead. These, however, lack invariant features and computing them systematically is challenging. We present a simple and systematic method to compute a functionally complete set of invariants of a non-negative 3D 4 th order tensor with re-spect to SO3. Intuitively, this transforms the tensor's non-unique ternary quartic (TQ) decomposition (from Hilbert's theorem) to a unique canon-ical representation independent of orientation – the invariants. The method consists of two steps. In the first, we reduce the 18 degrees-of-freedom (DOF) of a TQ representation by 3-DOFs via an orthogonal transformation. This transformation is designed to enhance a rotation-invariant property of choice of the 3D 4 th order tensor. In the second, we further reduce 3-DOFs via a 3D rotation transformation of coordinates to arrive at a canonical set of invariants to SO3 of the tensor. The resulting invariants are, by construction, (i) functionally complete, (ii) functionally irreducible (if desired), (iii) computationally efficient and (iv) reversible (mappable to the TQ coefficients or shape); which is the novelty of our contribution in comparison to prior work. Results from synthetic and real data experiments validate the method and indicate its importance

Harnessing plant biomass for biofuels and biomaterials

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/72519/1/j.1365-313X.2008.03512.x.pd

Nucleosomes in gene regulation: theoretical approaches

This work reviews current theoretical approaches of biophysics and bioinformatics for the description of nucleosome arrangements in chromatin and transcription factor binding to nucleosomal organized DNA. The role of nucleosomes in gene regulation is discussed from molecular-mechanistic and biological point of view. In addition to classical problems of this field, actual questions of epigenetic regulation are discussed. The authors selected for discussion what seem to be the most interesting concepts and hypotheses. Mathematical approaches are described in a simplified language to attract attention to the most important directions of this field

Preliminary results on high polymorphism of glycosyltransferase VvGt7 in Grapevine (Vitis vinifera)

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