Resonance theory of decoherence and thermalization

We present a rigorous analysis of the phenomenon of decoherence for general N-level systems coupled to reservoirs. The latter are described by free massless bosonic fields. We apply our general results to the specific cases of the qubit and the quantum register. We compare our results with the explicitly solvable case of systems whose interaction with the environment does not allow for energy exchange (non-demolition, or energy conserving interactions). We suggest a new approach which applies to a wide variety of systems which are not explicitly solvable.     

Stochastic Liouville equation studies of FT-EPR spectra of singlet-triplet mixing photo-chemically generated radical pair system

A stochastic Liouville equation (SLE) was numerically solved to obtain pulsed Fourier-transform (FT) EPR spectra on a radical pair system created in a photo-induced chemical reaction. Numerical calculations were applied to the photo-chemical reaction of deuterated acetone and 2-propanol at low temperatures. In this reaction system, the antiphase structures of the EPR signals, so called spin-correlated radical pair (SCRP) signals of two identical isopropyl ketyl radicals and spin-polarized free isopropyl ketyl radicals were observed by FT-EPR and continuous wave time-resolved (CW TR) EPR techniques. In the present work, FT-EPR spectra of the antiphase structure signals of the radical pair themselves as well as spin-polarized free radical signals were simulated. Additionally, rising behavior of free radical signals polarized by the radical pair mechanism (RPM) was also clarified. Furthermore two-dimensional (2D) FT-EPR nutation spectra were simulated in the both cases with and without the radical pairs by the use of SLE. In these simulations, strong DC components in the nutation frequency dimension, were well reproduced as was obtained in experiments. It was shown that relaxation during the microwave pulse was essential for the appearance of the DC components.     

New stationary distributions of the Vlasov-Maxwell-Fokker-Planck's system

In this paper we investigate the nonlinear Vlasov-Maxwell-Fokker-Plank (VMFP) system. We propose new families of distributions for the stationary VMFP equation, which is reduced to certain systems of nonlinear elliptic equations depending on arbitrary parameters. We consider the cases when these elliptic systems allow the reduction to the nonlinear PDEs of fourth order, or to the well-known Liouville equation.     

黎曼流形上一类指数调和型热方程的梯度估计

本文我们推到了黎曼流形上指数调和型热方程的一个Hamilton-Souplet-Zhang型梯度估计. 利用这个估计, 我们得到了一个Liouville型定理.     

Properties of fractional calculus with respect to a function and Bernstein type polynomials

This paper is divided into two stages. In the first stage, we investigated a new approach for the $\psi$-Riemann–Liouville fractional integral and the Faa di Bruno formula for the $\psi$-Hilfer fractional derivative. In addition, we discussed other properties involving the $\psi$-Hilfer fractional derivative and the $\psi$-Riemann–Liouville fractional integral. In the second stage, Bernstein polynomials involving the $\psi (·)$ function are investigated and the $\psi$-Riemann–Liouville fractional integral and $\psi$-Hilfer fractional derivative from the Bernstein polynomials are evaluated. We also discussed the relationship between the $\psi$-Hilfer fractional derivative with Laguerre polynomials and hypergeometric functions, and a version of the fractional mean value theorem with respect to a function. Motivated by the Bernstein polynomials, the second stage uses the Bernstein polynomials to approximate the solution of a fractional integro-differential equation with Hilfer fractional derivative and concluding with a numerical approach with its respective graph.     

四元Heisenberg群上的一个半线性问题

本文研究了四元Heisenberg群上的一个半线性方程问题,通过把对应的方程问题化为积分进行估计,证明了其对应的半线性方程的非负双椭圆解只有唯一的零解,推广了相应Heisenberg群上的定理.     

On fractional semidiscrete Dirac operators of Lévy–Leblond type

In this paper, we introduce a wide class of space-fractional and time-fractional semidiscrete Dirac operators of Lévy–Leblond type on the semidiscrete space-time lattice $h{\mathbb{Z}}^n\times [0,\infty )$ ($h>0$), resembling to fractional semidiscrete counterparts of the so-called parabolic Dirac operators. The methods adopted here are fairly operational, relying mostly on the algebraic manipulations involving Clifford algebras, discrete Fourier analysis techniques as well as standard properties of the analytic fractional semidiscrete semigroup ${\left\{\exp (-t{e}^{i\theta }{(-{\mathrm{\Delta }}_h)}^{\alpha })\right\}}_{t\ge 0}$, carrying the parameter constraints and $|\theta |\le \frac{\alpha \pi }{2}$. The results obtained involve the study of Cauchy problems on $h{\mathbb{Z}}^n\times [0,\infty )$.     

Liouville Type Theorems for Nonlinear p-Laplacian Equation on Complete Noncompact Riemannian Manifolds

In this paper,the authors study the gradient estimates for positive weak solutions to the following p-Laplacian equation △_pu+au^σ=0 on complete noncompact Riemannian manifold,where a,σ are two nonzero real constants with p≠2.Using the gradient estimate,they can get the corresponding Lionville theorem.On the other hand,by virtue of the Poincare inequality,they also obtain a Liouville theorem under some integral conditions with respect to positive weak solutions.