In this paper, we prove the convergence of the current defined from the Schrödinger-Poisson system with the presence of a strong magnetic field toward a dissipative solution of the Euler equations.

In this paper, we prove the convergence of the current defined from the Schrödinger-Poisson system with the presence of a strong magnetic field toward a dissipative solution of the Euler equations.

This paper reviews popular acceleration techniques to converge the non-linear self-consistent field equations appearing in quantum chemistry calculations with localized basis sets. The different methodologies, as well as their advantages and limitations are discussed within the same framework. Several illustrative examples of calculations are presented. This paper attempts to describe recent achievements and remaining challenges in this field.

We correct an error in the normalizing constant of resonant states.

We introduce a method for construction of a covariant differential calculus over a Hopf algebra $A$ from a quantized calculus $da=[D,a]$, $a\in A$, where $D$ is a candidate for a Dirac operator for $A$. We recover the method of construction of a bicovariant differential calculus given by T. Brzeziński and S. Majid created from a central element of the dual Hopf algebra $A^{\circ }$. We apply this method to the Dirac operator for the quantum $\mathrm{SL}\left(2\right)$ given by S. Majid. We find that the differential calculus obtained by our method is the...

Nonlinear Schrödinger equations (NLS)${}_a$ with strongly singular potential ${a\left|x\right|}^{-2}$ on a bounded domain $\Omega$ are considered. If $\Omega ={ℝ}^N$ and $a>-{(N-2)}^2/4$, then the global existence of weak solutions is confirmed by applying the energy methods established by N. Okazawa, T. Suzuki, T. Yokota (2012). Here $a=-{(N-2)}^2/4$ is excluded because $D\left(P_{a\left(N\right)}^{1/2}\right)$ is not equal to $H^1\left({ℝ}^N\right)$, where $P_{a\left(N\right)}:=-\Delta -{(N-2)}^2/{\left(4\right|x|}^2)$ is nonnegative and selfadjoint in $L^2\left({ℝ}^N\right)$. On the other hand, if $\Omega$ is a smooth and bounded domain with $0\in \Omega$, the Hardy-Poincaré inequality is proved in J. L. Vazquez, E. Zuazua (2000)....

In this paper, we develop the crystal basis theory for the quantum queer superalgebra $U_q\left(𝔮\left(n\right)\right)$. We define the notion of crystal bases and prove the tensor product rule for $U_q\left(𝔮\left(n\right)\right)$-modules in the category ${𝒪}_{\mathrm{int}}^{\ge 0}$. Our main theorem shows that every $U_q\left(𝔮\left(n\right)\right)$-module in the category ${𝒪}_{\mathrm{int}}^{\ge 0}$ has a unique crystal basis.

We prove estimates for the sectional curvature of hyperkähler quotients and give applications to moduli spaces of solutions to Nahm’s equations and Hitchin’s equations.