Preduals of quadratic Campanato spaces associated to operators with heat kernel bounds

Let $L$ be a nonnegative, self-adjoint operator on $L^2(\mathbb{R}^n)$ with the Gaussian upper bound on its heat kernel. As a generalization of the square Campanato space $\mathcal{L}^{2,\lambda}_{-\Delta}(\mathbb R^n)$, in \cite{DXY} the quadratic Campanato space $\mathcal{L}_L^{2,\lambda}(\mathbb{R}^n)$ is defined by a variant of the maximal function associated with the semigroup $\{e^{-tL}\}_{t\geq 0}$. On the basis of \cite{DX} and \cite{YY} this paper addresses the preduality of $\mathcal{L}_L^{2,\lambda}(\mathbb{R}^n)$ through an induced atom (or molecular) decomposition. Even in the case $L=-\Delta$ the discovered predual result is new and natural.Comment: 19 page

ZMCintegral-v5.1: Support for Multi-function Integrations on GPUs

In this new version of ZMCintegral, we have added the functionality of multi-function integrations, i.e. the ability to integrate more than $10^{3}$ different functions on GPUs. The Python API remains the similar as the previous versions. For integrands less than 5 dimensions, it usually takes less than 10 minutes to finish the evaluation of $10^{3}$ integrations on one Tesla v100 card. The performance scales linearly with the increasing of the GPUs.Comment: 5pages, 1figur

Bis(μ-biphenyl-2,2′-dicarboxyl­ato)bis­[aqua­(4,4′-dimethyl-2,2′-bipyridine-κ2 N,N′)copper(II)]

The mol­ecule of the title binuclear copper(II) complex, [Cu2(C14H8O4)2(C12H12N2)2(H2O)2], is bis­ected by a crystallographic twofold axis. Each CuII atom is coordinated in a distorted octa­hedral geometry by three O atoms from two biphen­yl-2,2′-dicarboxyl­ate anions, one aqua O atom and two N atoms of a 4,4′-dimethyl-2,2′-bipyridine ligand. Intramolecular O—H⋯O hydrogen bonds between the coordinated water molecules and the carboxylate O atoms are also present

All-optical control of thermal conduction in waveguide QED

We investigate the heat conduction between two one-dimension waveguides intermediated by a Laser-driving atom. The Laser provides the optical control on the heat conduction. The tunable asymmetric conduction of the heat against the temperature gradient is realized. Assisted by the modulated Laser, the heat conduction from either waveguide to the other waveguide can be suppressed. Meanwhile, the conduction towards the direction opposite to the suppressed one is gained. The heat currents can be significantly amplified by the energy flow of the Laser. Moveover, the scheme can act like a heat engine