Contour lines totally fix the structure of leftcontinuous t-norms. For each t-norm, these contour lines are determined by the corresponding residual implicator. Most properties involving residual implicators can now easily be translated into properties involving contour lines. As the portation law expresses associativity, we dispose of a powerful tool for constructing left-continuous t-norms. I...

We prove duality results for adjoint operators and product norms in the framework of Euclidean spaces. We show how these results can be used to derive condition numbers especially when perturbations on data are measured componentwise relatively to the original data. We apply this technique to obtain formulas for componentwise and mixed condition numbers for a linear function of a linear least s...

The goal of this paper is to introduce and study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets Ω of hyperbolic or spherical spaces. At least at a formal level, there are striking similarities among the three cases: Euclidean, spherical and hyperbolic. We start by defining non-Euclidean analogues of the Euclidean Funk weak metric and we give three distinct representa...

Comparison results are obtained between infinity subharmonic and infinity superharmonic functions defined on unbounded domains. The primary new tool employed is an approximation of infinity subharmonic functions that allows one to assume that gradients are bounded away from zero. This approximation also demystifies the proof in the case of a bounded domain. A second, quite different, topic is a...

We consider the empirical risk minimization problem for linear supervised learning, with regularization by structured sparsity-inducing norms. These are defined as sums of Euclidean norms on certain subsets of variables, extending the usual l1-norm and the group l1-norm by allowing the subsets to overlap. This leads to a specific set of allowed nonzero patterns for the solutions of such problem...

We developed gradient magnetic-field topography (GMFT) for magnetoencephalography (MEG). We plotted the Euclidean norms of gradient magnetic fields occurring at the centers of 102 sensors onto 49-point grids and projected these norms onto the MRI brain surface of a twelve-year-old boy who presented with neocortical epilepsy secondary to a left temporal tumor. The peak gradient magnetic field lo...

In this paper we introduce a method for the fuzzy model reconstruction and a method for measurements reduction on the basis of test signals by maximization a posteriori possibility. It ensures the maximum accuracy of the measurements reduction. It is used the model of measurement errors with fuzzy constraints on its Euclidean norms.

We extend basic properties of the diameter and the thickness of a convex body from Euclidean space to Minkowski spaces (i.e., real finite-dimensional Banach spaces), present purely “Minkowskian properties” and new analytical representations of these two quantities, and discuss their evaluation in the particular case of polyhedral norms.

The paper presents a competitive prediction-style upper bound on the square loss of the Aggregating Algorithm for Regression with Changing Dependencies in the linear case. The algorithm is able to compete with a sequence of linear predictors provided the sum of squared Euclidean norms of differences of regression coefficient vectors grows at a sublinear rate.

We consider the least-square regression problem with regularization by a block 1-norm, i.e., a sum of Euclidean norms over spaces of dimensions larger than one. This problem, referred to as the group Lasso, extends the usual regularization by the 1-norm where all spaces have dimension one, where it is commonly referred to as the Lasso. In this paper, we study the asymptotic model consistency of...