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1 Notion We introduce some notions used in this supplementary material. For regression task, we define y max = max y |y|. We further denote the set S as S = B 0, y max λ −1/2 if L2 is used and λ ≤1 R D otherwise where B 0, y max λ −1/2 = w ∈ R D : w ≤ y max λ −1/2 and R D specifies the whole feature space. We introduce five types of loss functions that can be used in our proposed algorithm, nam...

This article is to contribute to the classification of finite-dimensional complex pointed Hopf algebras with Weyl groups of E8. Many papers are about the classification of finite dimensional pointed Hopf algebras, for example, [AS98, AS02, AS00, AS05, He06, AHS08, AG03, AFZ, AZ07, Gr00, Fa07, AF06, AF07, ZZC, ZC]. In these research ones need the centralizers and character tables of groups. In t...

for some fixed large N0; we shall call such weights admissible. Rubio de Francia [11] showed that for every w ∈ L(R) there is a nonnegative W ∈ L(R) such that ‖W‖2 ≤ Cλ‖w‖2, Cλ < ∞ if λ > 0, and the analogous weighted norm inequality for S t holds uniformly in t. He used methods related to factorization theory of operators and the proof gave no information on how to construct w from W . In [3] ...

This article is to contribute to the classification of finite-dimensional complex pointed Hopf algebras with Weyl groups of E6, E7, F4, G2. Many papers are about the classification of finite dimensional pointed Hopf algebras, for example, [AS98, AS02, AS00, AS05, He06, AHS08, AG03, AFZ, AZ07, Gr00, Fa07, AF06, AF07, ZZC, ZC]. In these research ones need the centralizers and character tables of ...

Let L be a special Lagrangian submanifold of R with boundary lying on the symplectic, codimension 2 submanifold W . It is shown how deformations of L which keep the boundary of L confined to W can be described by an elliptic boundary value problem, and two results about minimal Lagrangian submanifolds with boundary are derived using this fact. The first is that the space of minimal Lagrangian s...

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We investigate the conventional tight-binding model of L-electrons on a ring-shaped molecule of L atoms with nearest neighbor hopping. The hopping amplitudes, t(w), depend on the atomic spacings, w, with an associated distortion energy V (w). A Hubbard type on-site interaction as well as nearest-neighbor repulsive potentials can also be included. We prove that when L = 4k+2 the minimum energy E...