Error Estimates and Superconvergence of Mixed Finite Element Methods for Optimal Control Problems with Low Regularity

In this paper, we investigate the error estimates and superconvergence property of mixed finite element methods for elliptic optimal control problems. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. We derive $L^2$ and $L^\infty$-error estimates for the control variable. Moreover, using a recovery operator, we also derive some superconvergence results for the control variable. Finally, a numerical example is given to demonstrate the theoretical results.     

A Posteriori Error Estimates of Triangular Mixed Finite Element Methods for Semilinear Optimal Control Problems

In this paper, we present an a posteriori error estimates of semilinear quadratic constrained optimal control problems using triangular mixed finite element methods. The state and co-state are approximated by the order $k\leq 1$ Raviart- Thomas mixed finite element spaces and the control is approximated by piecewise constant element. We derive a posteriori error estimates for the coupled state and control approximations. A numerical example is presented in confirmation of the theory.     

On Error Estimates for the Trotter–Kato Product Formula

We study the error bound in the operator-norm topology for the Trotter exponential product formula as well as for its generalization à la Kato. Within the framework of an abstract setting, we give a simple proof of error estimates which improve some recent results in this direction.     

A Priori Error Estimates of Finite Element Methods for Linear Parabolic Integro-Differential Optimal Control Problems

In this paper, we study the mathematical formulation for an optimal control problem governed by a linear parabolic integro-differential equation and present the optimality conditions. We then set up its weak formulation and the finite element approximation scheme. Based on these we derive the a priori error estimates for its finite element approximation both in $H^1$ and $L^2$ norms. Furthermore, some numerical tests are presented to verify the theoretical results.     

The Pullback Asymptotic Behavior of the Solutions for 2D Nonautonomous G-Navier-Stokes Equations

The pullback asymptotic behavior of the solutions for 2D Nonautonomous G-Navier-Stokes equations is studied, and the existence of its $L^2$-pullback attractors on some bounded domains with Dirichlet boundary conditions is investigated by using the measure of noncompactness. Then the estimation of the fractal dimensions for the 2D G-Navier-Stokes equations is given.     

A Posteriori Error Estimates of a Combined Mixed Finite Element and Discontinuous Galerkin Method for a Kind of Compressible Miscible Displacement Problems

A kind of compressible miscible displacement problems which include molecular diffusion and dispersion in porous media are investigated. The mixed finite element method is applied to the flow equation, and the transport one is solved by the symmetric interior penalty discontinuous Galerkin method. Based on a duality argument, employing projection estimates and approximation properties, a posteriori residual-type $hp$ error estimates for the coupled system are presented, which is often used for guiding adaptivity. Comparing with the error analysis carried out by Yang (Int. J. Numer. Meth. Fluids, 65(7) (2011), pp. 781-797), the current work is more complicated and challenging.     

与铷（5^2S1／2）原子碰撞产生的铷7^2D态精细结构转移截面

本文报道使用蒸汽泡和两光子两步激发方法,测量原子激发态敏化荧光I_(3/2)~1和直接荧光I_(3/2)~2其与温度的关系,得到与基态(5~2S_(1/2)铷原子碰撞产生的铷7~2D_(5/2)→7~2D_(3/2)和7~2D_(3/2)→7~2D_(5/2)精细结构转移截面分别为:σ_(fs)=4.7×10~(-13)cm~2、σ_(fs)~’=7.0×10~(-13)cm~2;碰撞转移出7~2D双态的转移截面σ_(tr)(5/2)=0.62×10~(-13)cm~2.由计算的7~2D态几何截面σ_(geom)能够相对很好地描述σ_(fs)和σ_(fs)~’的数量级.     

Bright and dark solitons in a quasi-1D Bose-Einstein condensates modelled by 1D Gross-Pitaevskii equation with time-dependent parameters

We investigate the exact bright and dark solitary wave solutions of an effective 1D Bose-Einstein condensate (BEC) by assuming that the interaction energy is much less than the kinetic energy in the transverse direction. In particular, following the earlier works in the literature Pérez-García et al. (2004) [50], Serkin et al. (2007) [51], Gurses (2007) [52] and Kundu (2009) [53], we point out that the effective 1D equation resulting from the Gross-Pitaevskii (GP) equation can be transformed into the standard soliton (bright/dark) possessing, completely integrable 1D nonlinear Schrödinger (NLS) equation by effecting a change of variables of the coordinates and the wave function. We consider both confining and expulsive harmonic trap potentials separately and treat the atomic scattering length, gain/loss term and trap frequency as the experimental control parameters by modulating them as a function of time. In the case when the trap frequency is kept constant, we show the existence of different kinds of soliton solutions, such as the periodic oscillating solitons, collapse and revival of condensate, snake-like solitons, stable solitons, soliton growth and decay and formation of two-soliton bound state, as the atomic scattering length and gain/loss term are varied. However, when the trap frequency is also modulated, we show the phenomena of collapse and revival of two-soliton like bound state formation of the condensate for double modulated periodic potential and bright and dark solitons for step-wise modulated potentials.     

Asymptotic behavior of fluctuations for the 1D Ising model in zero-temperature limit

Fluctuation of the average spin for one-dimensional Ising spins with nearest neighbor interactions are studied. The distribution function for the average spin is calculated for a finite volume, finite temperature, and finite magnetic field. As the volume increases and the temperature diminishes at zero magnetic field, there are two limits in which the probability distribution shows quite different behaviors: in the thermodynamic limit as the volume goes to infinity for finite temperature, small deviations of the fluctuations are described by a Gaussian distribution, and in the limit as the temperature vanishes for a finite volume, the ground states are realized with probability one. The crossover between these limits is analyzed via a ratio of the correlation length to the volume. The helix-coil transition in a polypeptide is discussed as an application.     

Posteriori Error Estimation for an Interior Penalty Discontinuous Galerkin Method for Maxwell's Equations in Cold Plasma

In this paper, we develop a residual-based a posteriori error estimator for the time-dependent Maxwell's equations in the cold plasma. Here we consider a semi-discrete interior penalty discontinuous Galerkin (DG) method for solving the governing equations. We provide both the upper bound and lower bound analysis for the error estimator. This is the first posteriori error analysis carried out for the Maxwell's equations in dispersive media.     

Equivalent a Posteriori Error Estimator of Spectral Approximation for Control Problems with Integral Control-State Constraints in One Dimension

In this paper, we investigate the Galerkin spectral approximation for elliptic control problems with integral control and state constraints. Firstly, an a posteriori error estimator is established, which can be acted as the equivalent indicator with explicit expression. Secondly, appropriate base functions of the discrete spaces make it probable to solve the discrete system. Numerical test indicates the reliability and efficiency of the estimator, and shows the proposed method is competitive for this class of control problems. These discussions can certainly be extended to two- and three-dimensional cases.     

Anderson transition in 1D systems with spatial disorder

A simple Kronig-Penney model for 1D mesoscopic systems with δ peak potentials is used to study numerically the influence of spatial disorder on conductance fluctuations and distribution at different regimes. The Lévy laws are used to investigate the statistical properties of the eigenstates. It is found that an Anderson transition occurs even in 1D meaning that the disorder can also provide constructive quantum interferences. The critical disorder W_c for this transition is estimated. In these 1D systems, the metallic phase is well characterized by a Gaussian conductance distribution. Indeed, the results relative to conductance distribution are in good agreement with the previous works in 2D and 3D systems for other models. At this transition, the conductance probability distribution has a system size independent shape with large fluctuations in good agreement with previous works.     

A Spectral Method for Second Order Volterra Integro-Differential Equation with Pantograph Delay

In this paper, a Legendre-collocation spectral method is developed for the second order Volterra integro-differential equation with pantograph delay. We provide a rigorous error analysis for the proposed method. The spectral rate of convergence for the proposed method is established in both $L^2$-norm and $L^∞$-norm.     

Analytical Approximate Solution of SchrSdinger Equation in D Dimensions with Quadratic Exponential-Type Potential for Arbitrary/-State

We present the bound state solution of Schr6dinger equation in D dimensions for quadratic exponential-type potential for arbitrary l-state. We use generalized parametric Nikiforov-Uvarov method to obtain the energy levels and the corresponding eigenfunction in dosed form. We also compute the energy eigenvalues numerically.     

Collective Excitations and Nonlinear Dynamics of 1D BEC with Two- and Three-body Interactions in Anharmonic Traps

The collective excitations of low-dimensional Bose-Einstein condensates with two- and three-body interactions in anharmonic potentials are investigated. Using the standard variational approach, the governing equations of motions for the low-energy excitations are obtained by solving time-dependent Gross-Pitaevskii-Ginzburg equation, and the excitation spectrums are calculated in small amplitude limit. The frequency shift and nonlinear mode coupling induced by the anharmonic distortion （adding cubic, quartic, or quintic terra to a harmonic trap） are studied.     

Finite size corrections within the continuum limit for quantum spins: two-magnon bound states in 1D Heisenberg ferromagnet

A continuum medium approach is proposed to describe the finite size dependent effects for the 1D isotropic Heisenberg ferromagnet. The results are compared to the exact Bethe ansatz solution for the finite chain. The approach is shown to adequately account for the behaviour of the eigenfunctions and eigenenergies. The continuum is obtained by integration in Fourier space via introduction of cut-offs at the integration limits and analytical continuation from the discrete lattice to the continuous medium. It offers a new perspective on the instability of bound states, and reveals the linear behaviour of the amplitude in the critical region and other features of the model in an analytical way. We further apply this approach to investigate the long wavelength expansion of the master equation and to show the route of constructing reliable approximations valid for more complicated models. It is concluded that the approach can be useful to study mesoscopic spin systems. Received 28 May 2000 and Received in final form 6 April 2001     

Triplet superconductivity in a 1D itinerant electron system with transverse spin anisotropy

In this paper we study the ground state phase diagram of a one-dimensional t-J-U model away from half-filling. In the large-bandwidth limit and for ferromagnetic exchange with easy-plane anisotropy a phase with gapless charge and massive spin excitations, characterized by the coexistence of triplet superconducting and spin density wave instabilities is realized in the ground state. With increasing ferromagnetic exchange transitions into a ferrometallic and then a spin gapped triplet superconducting phase take place.     

A rubidium-stabilized ring-cavity resonator for optical frequency metrology: precise measurement of the D1 line in 133Cs

We have developed a ring-cavity resonator that can be used to measure the absolute frequencies of optical transitions with an uncertainty below 40 kHz. The length of the resonator is calibrated against a reference laser locked to the D₂ line of ⁸⁷Rb, the frequency of which is known with 6 kHz accuracy. We demonstrate the power of this technique by measuring the absolute frequencies of various hyperfine transitions in the D₁ line of ¹³³Cs. Our results agree with earlier measurements using the frequency-comb technique, and have similar accuracy. Measurement of the D₁-line frequency could lead to a more precise determination of the fine-structure constant. We also report a precise value of A=291.918(8) MHz for the hyperfine constant in the 6P_1/2 state.     

The Ansatz Method for Analyzing Schrödinger’s Equation with Three Anharmonic Potentials in D Dimensions

In this letter, by applying a suitable ansatz to the wave functions, the solutions of the D-dimensional radial Schrödinger equation with some anharmonic potentials are obtained.     

Multiple mobility edges in a 1D Aubry chain with Hubbard interaction in presence of electric field: Controlled electron transport

Electronic behavior of a 1D Aubry chain with Hubbard interaction is critically analyzed in presence of electric field. Multiple energy bands are generated as a result of Hubbard correlation and Aubry potential, and, within these bands localized states are developed under the application of electric field. Within a tight-binding framework we compute electronic transmission probability and average density of states using Green's function approach where the interaction parameter is treated under Hartree–Fock mean field scheme. From our analysis we find that selective transmission can be obtained by tuning injecting electron energy, and thus, the present model can be utilized as a controlled switching device.