The general framework of Legendre transformation is extended to the case of symplectic groupoids, using an appropriate generalization of the notion of generating function (of a Lagrangian submanifold). 1 Tulczyjew triple and its generalization The general framework of Legendre transformation was introduced by Tulczyjew [1]. It consists in recognition of the following structure, which we call th...

We introduce a version of the calculus of variations on time scales, which includes as special cases the classical calculus of variations and the discrete calculus of variations. Necessary conditions for weak local minima are established, among them the Euler condition, the Legendre condition, the strengthened Legendre condition, and the Jacobi condition. AMS (MOS) Subject Classification. 39A10.

We analyze the Legendre and Chebyshev spectral Galerkin semidiscretizations of a one dimensional homogeneous parabolic problem with nonconstant coefficients. We present error estimates for both smooth and nonsmooth data. In the Chebyshev case a limit in the order of approximation is established. On the contrary, in the Legendre case we find an arbitrary high order of convegence.

We present a novel nonnegative tensor decomposition method, called Legendre decomposition, which factorizes an input tensor into a multiplicative combination of parameters. Thanks to the well-developed theory of information geometry, the reconstructed tensor is unique and alwaysminimizes the KL divergence from an input tensor. We empirically show that Legendre decomposition can more accurately ...

This paper describes a new framework for white matter tractography in high angular resolution diffusion data. A direction-dependent local cost is defined based on the diffusion data for every direction on the unit sphere. Minimum cost curves are determined by solving the Hamilton-Jacobi-Bellman using an efficient algorithm. Classical costs based on the diffusion tensor field can be seen as a sp...

We present a continuous model for structural brain connectivity based on the Poisson point process. The model treats each stream-line curve in a tractography as an observed event in connectome space, here a product space of cortical white matter boundaries. We approximate the model parameter via kernel density estimation. To deal with the heavy computational burden, we develop a fast parameter ...