On second-order, divergence-free tensors

On the Structure of Divergence-Free Tensors

Glyphs for Asymmetric Second-Order 2D Tensors

Tensors model a wide range of physical phenomena. While symmetric tensors are sufficient for some applications (such as diffusion), asymmetric tensors are required, for example, to describe differential properties of fluid flow. Glyphs permit inspecting individual tensor values, but existing tensor glyphs are fully defined only for symmetric tensors. We propose a glyph to visualize asymmetric s...

Hamiltonian Systems on Manifolds of Second-order Tensors *

Considered is the Schrödinger equation in a finite-dimensional space as an equation of mathematical physics derivable from the variational principle and treatable in terms of the Lagrange-Hamilton formalism. It provides an interesting example of " mechanics " with singular Lagrangians, effectively treatable within the framework of Dirac formalism. We discuss also some modified " Schrödinger " e...

Divergence-free positive symmetric tensors and fluid dynamics

We consider d× d tensors A(x) that are symmetric, positive semi-definite, and whose rowdivergence vanishes identically. We establish sharp inequalities for the integral of (detA) 1 d−1 . We apply them to models of compressible inviscid fluids: Euler equations, Euler–Fourier, relativistic Euler, Boltzman, BGK, etc... We deduce an a priori estimate for a new quantity, namely the space-time integr...

From Second to Higher Order Tensors in Diffusion-MRI

Diffusion MRI, which is sensitive to the Brownian motion of molecules, has become today an excellent medical tool for probing the tissue micro-structure of cerebral white matter in vivo and non-invasively. It makes it possible to reconstruct fiber pathways and segment major fiber bundles that reflect the structures in the brain which are not visible to other non-invasive imaging modalities. Sin...