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...

In this paper, we prove that 2 dimensional transversal small perturbations of d-dimensional Euclidean planes under the skew mean curvature flow lead to global solutions which converge unperturbed in suitable norms. And clarify long time behaviors Sobolev spaces.

This paper is concerned with the problem of nonlinear best simultaneous approximations in K?the Bochner function spaces respect to Minkowski? norms Euclidean spaces. Characterization results generalized approximation are established. These considered a generalization concerning Lebesgue and Orlicz

The assumption that voters’ preferences share some common structure is a standard way to circumvent NP-hardness results in social choice problems. While the Kemeny ranking problem NP-hard general case, it known become easy if are 1-dimensional Euclidean. In this note, we prove remains for k-dimensional Euclidean with k≥2 under norms ℓ1, ℓ2 and ℓ∞, by showing any weighted tournament (resp. bipar...

In this article we establish optimal estimates for the first eigenvalue of Schrödinger operators on the d-dimensional unit sphere. These estimates depend on Lp norms of the potential, or of its inverse, and are equivalent to interpolation inequalities on the sphere. We also characterize a semi-classical asymptotic regime and discuss how our estimates on the sphere differ from those on the Eucli...

A John function is a continuously differentiable function whose gradient is bounded by the reciprocal of the Euclidean distance to the boundary of the domain. Here we construct John functions whose gradient norms have positive lower bounds for o-minimal domains. We prove their definability and give explicit estimates for number of John functions required for a given set to obtain uniform positi...

Recently Antilla, Ball and Perissinaki proved that the squares of coordinate functions in ln p are negatively correlated. This paper extends their results to balls in generalized Orlicz norms on Rn. From this, the concentration of the Euclidean norm and a form of the Central Limit Theorem for the generalized Orlicz balls is deduced. Also, a counterexample for the square negative correlation hyp...

1. Calculus on spheres 2. Spherical Laplacian from Euclidean 3. Eigenvectors for the spherical Laplacian 4. Invariant integrals on spheres 5. L spectral decompositions on spheres 6. Sup-norms of spherical harmonics on Sn−1 7. Pointwise convergence of Fourier-Laplace series 8. Irreducibility of representation spaces for O(n) 9. Hecke’s identity • Appendix: Bernstein’s proof of Weierstraß approxi...

The approximation of road distances by the weighted lp distance measure has been studied in a series of papers ([5, 6], etc.) and it is argued with empirical study that lp distances weighted by an inflation factor tailored to given regions can better describe the irregularity in the transportation networks such as hill, bends, and are therefore superior to the weighted rectangular and Euclidean...

The issue of Cα-degree reduction of triangular Bézier surfaces is exposed. It is anticipated that both triangular Bézier surfaces are Cα-continuous at the vertices. The Euclidean norm as well as the L2−norm is used. The final solutions are given in terms of the matrix of degree raising, the Gram matrix, and the Bézier points. Moreover, it is shown that the solutions using both norms are equival...