For H=H_b:x^2y , a nontrivial semisimple Hopf algebra of dimension 16 classified by Kashina, all simple Yetter–Drinfeld modules over H are determined, and all finite-dimensional Nichols algebras obeying the structural constraint ℬ(V)≅⊗ _i∈ Iℬ(V_i) are subsequently characterized, where V=⊕ _i∈ IV_i and each V_i constitutes a simple object in the category a_H^H𝒴𝒟 . Finally, several liftings of the corresponding Radford biproducts ℬ(V)♯ H are classified, which provide some classification results for finite-dimensional Hopf algebras with coradical isomorphic to H_b:x^2y .
In this note we first discuss the solution of the time-dependent Schrödinger equation for a Hamiltonian with a point interaction in dimension three. This case has a pedagogical character and it has the aim to show how to construct the solution for an initial datum in the form domain. In the second part we extend the method to a more complex situation, i.e., we study the same problem for a Hamiltonian describing a particle interacting through zero-range interaction with another particle subject to a harmonic potential.
The purpose of this paper is to investigate the existence of normalized solutions for the coupled elliptic system with quadratic nonlinearity. In the first scenario, under an explicit smallness assumption on the potential, we establish the existence of a mountain pass solution at a positive energy level. In the second scenario, if the mass is smaller and depends on the potential, we identify a local minimizer at a negative energy level and a mountain pass solution at a positive energy level.
In this paper, we establish weighted norm inequalities for the harmonic function in the upper half-space, and corresponding reverse inequalities are also obtained. These weighted Morrey-Lorentz estimates generalize the L^p -isometry of the Poisson semigroup in Stein-Weiss’s book [Introduction to Fourier analysis on Euclidean spaces, 1971]. The novelty of our results lies in the weight constructed from a product of two weights μ and ω . The former weight μ arises from the structure of the elliptic operator, while the latter ω is in the Muckenhoupt class with respect to μ . This is totally different from the two-weight theory. As applications, weighted norm inequalities and their reversed for the harmonic/caloric function related to the Neumann/Dirichlet problem are considered.
We apply Bochner’s technique combined with a mild Dirichlet integral condition to show that a complete constant mean curvature (CMC) spacelike hypersurface immersed with positive semi-definite modified second fundamental form in a Lorentzian product space ℝ_1× M^n , whose Riemannian base M^n has nonnegative sectional curvature, must be a slice {t}× M^n . In particular, we conclude that the spacelike hyperplanes of the Lorentz–Minkowski space 𝕃_1^n+1 are the only complete CMC spacelike hypersurfaces immersed in 𝕃_1^n+1 , whose modified second fundamental form is positive semi-definite and the height function with respect to a spacelike hyperplane satisfies such a Dirichlet integral condition. A version of the classical Calabi–Bernstein Theorem to the context of Lorentzian product spaces is also given.
