REGULAR SOLUTIONS FOR SCHRODINGER EQUATION ON UNBOUNDED DOMAINS

The authors study a class of solutions,namely, regular solutions of the Schrodinger equation (1/2 Delta + q)u = 0 on unbounded domains. They definite the regular solutions in terms of sample path properties of Brownian motion and then characterize them by analytic method. In Section 4, they discuss the regular solution to the stochastic Dirichlet problem for the equation (1/2 Delta + q)u = 0 having limit alpha at infinity.

The Regularity of Solutions to Mixed Boundary Value Problems of Second-Order Elliptic Equations with Small Angles

This paper considers the regularity of solutions to mixed boundary value problems in small-angle regions for elliptic equations. By constructing a specific barrier function, we proved that under the assumption of sufficient regularity of boundary conditions and coefficients, as long as the angle is sufficiently small, the regularity of the solution to the mixed boundary value problem of the second-order elliptic equation can reach any order.

On Blow-up of Regular Solutions to the Isentropic Euler and Euler-Boltzmann Equations with Vacuum

In this paper,the authors study the Cauchy problem of n-dimensional isentropic Euler equations and Euler-Boltzmann equations with vacuum in the whole space.They show that if the initial velocity satisfies some condition on the integral J in the"isolated mass group"(see(1.13)),then there will be finite time blow-up of regular solutions to the Euler system with J≤0(n≥1)and to the Euler-Boltzmann system with J<0(n≥1)and J=0(n≥2),no matter how small and smooth the initial data are.It is worth mentioning that these blow-up results imply the following:The radiation is not strong enough to prevent the formation of singularities caused by the appearance of vacuum,with the only possible exception in the case J=0 and n=1 since the radiation behaves differently on this occasion.

THE REGULAR SOLUTIONS OF THE ISENTROPIC EULER EQUATIONS WITH DEGENERATE LINEAR DAMPING

The regular solutions of the isentropic Euler equations with degenerate linear damping for a perfect gas are studied in this paper. And a critical degenerate linear damping coefficient is found, such that if the degenerate linear damping coefficient is larger than it and the gas lies in a compact domain initially, then the regular solution will blow up in finite time; if the degenerate linear damping coefficient is less than it, then under some hvpotheses on the initial data. the regular solution exists globally.

THE REGULARIZED SOLUTION APPROXIMATION OF FORWARD/BACKWARD PROBLEMS FOR A FRACTIONAL PSEUDO-PARABOLIC EQUATION WITH RANDOM NOISE

This paper deals with the forward and backward problems for the nonlinear fractional pseudo-parabolic equation ut+(-Δ)^(s1)ut+β(-Δ)^(s2)u=F(u,x,t)subject o random Gaussian white noise for initial and final data.Under the suitable assumptions s1,s2andβ,we first show the ill-posedness of mild solutions for forward and backward problems in the sense of Hadamard,which are mainly driven by random noise.Moreover,we propose the Fourier truncation method for stabilizing the above ill-posed problems.We derive an error estimate between the exact solution and its regularized solution in an E‖·‖Hs22norm,and give some numerical examples illustrating the effect of above method.

Inversion and III-Posed Problem Solutions in Atmospheric Remote Sensing

With the swift advances in earth observation,satellite remote sensing and application of atmospheric radiation theory have been developed in the past decades,atmospheric sensing inversion with its algorithms is getting more and more importance.It is known that since a remote sensing equation falls into an integral equation of the first kind,thus leading to the fact that it is ill-posed and particularly the solution is unsteady,tremendous difficulties arise from the retrieval.This paper will present a simple review on the inversion techniques with some necessary remarks,before introducing the successful efforts with respect to such equations and the encouraging solutions achieved in recent decades by researchers of the world.

Calculation of Activity Coefficient from Immiscible Binary Alloy Phase Diagram by Means of Modified Sub-regular Solution Model

The modified sub regular solution model was used for a calculation of the activity coefficient of immiscible binary alloy systems. The parameters needed for the calculation are the interaction parameters, λ 1 and λ 2, which are represented as a linear function of temperature, T . The molar excess Gibbs free energy, G m E, can be written in the form G m E= x A x B[( λ 11 + λ 12 T )+( λ 21 + λ 22 T ) x B ] The calculation is carried out numerically for three immiscible binary alloy systems, Al Pb, Cu Tl and In V. The agreement between the calculated and experimentally determined values of activity coefficient is excellent.

A New Method for Achieving an Initial Regular Solution of a Linear Programming

A method is provided for finding an initial regular solution of a linear programming in this paper. The key to this method is to solve an auxiliary linear programming instead of to introduce any artificial variable or constraint. Compared with the traditional method of achieving the regular solution by introducing an artificial constraint, it has advantages of saving the memories and little computational efforts.

Unbiased Estimation Formula of Unit Weight Standard Deviation in Regularization Solution

Regularization method is an effective method for solving ill\|posed equation. In this paper the unbiased estimation formula of unit weight standard deviation in the regularization solution is derived and the formula is verified with numerical case of 1 000 sample data by use of the typical ill\|posed equation, i.e. the Fredholm integration equation of the first kind.

ON THE REGULARITY OF THE MINIMUM SOLUTION OF THE RESTRAINED VARIATIONAL PROBLEMS

This paper is concerned with the regularity of minimum solution u of the following functional L(u) = integral(Omega) a alpha(beta)(x)g(ij)(u)D alpha u(i)D(beta)upsilon(i)dx on the restraint E = {u is an element of W-0(1,2) (Omega, R(N))\parallel to u parallel to L(D) = 1}. Under appropriate conditions, the bounded minimum solution u of the above functional is proved to be nothing but Holder continuous.

REGULARIZATION METHOD FOR IMPROVING OPTIMAL CONVERGENCE RATE OF THE REGULARIZED SOLUTION OF ILL-POSED PROBLEMS

This paper presents anew regularization method for solving operator equations of the first kind; the convergence rate of the regularized solution is improved, as compared with the ordinary Tikhonov regularization.

一类三维趋化–斯托克斯方程组正则化问题解的先验估计

本文研究了一类带流体耦合的三维趋化–斯托克斯方程,该模型刻画了流体环境中的生物趋化现象。大量的生物实验观测表明,在趋化流体模型中,重力对细胞运动的影响和趋化本身对流体的交互作用都应该放在方程中同时考虑。本文将在 的条件下利用权函数的方法建立正则化问题解的先验估计,为进一步研究解的定性理论做好准备。

Non-Global Existence of Regular Solution to Initial Value Problem of Relativistic Euler Equations in R^(N)

In fluid mechanics and astrophysics,relativistic Euler equations can be used to describe the effects of special relativity which are an extension of the classical Euler equations.In this paper,we will consider the initial value problem of relativistic Euler equations in an initial bounded region of R N.If the initial velocity satisfies max→x 0∈∂Ω(0)N∑i=1 v_(i)^(2)(0,→x 0)<c^(2)A_(1)/2,where A 1 is a positive constant depend on some sufficiently large T^(*),then we can get the non-global existence of the regular solution for the N-dimensional relativistic Euler equations.

Two non-perturbativeα'or loop corrected string cosmological solutions

In this paper,we present two non-perturbative string cosmological solutions without curvature singularities for the bosonic gravi-dilaton system.These solutions are general in that they can straightforwardly match the perturbative solution to arbitrarily high orders in the perturbative region.The first solution includes non-perturbative α’corrections based on Hohm-Zwiebach action.We then use the simple phenomenological map between the α’ and loop corrected theories in string cosmology to construct a non-perturbative loop corrected non-singular solution.Both solutions are non-singular everywhere.Therefore,the pre-and post-big-bangs are smoothly connected by these solutions.

A FUNDAMENTAL SOLUTION FOR THE LAPLACE OPERATOR ON THE QUATERNIONIC HEISENBERG GROUP

In this paper, the author studies the Laplace operator on the quaternionic Heisenberg group, construct a fundamental solution for it and use this solution to prove the L-p-boundedness and the weak (1-1) boundedness of certain singular convolution operators on the quaternionic Heisenberg group.

Nested regular polygon solutions for planar 2N-body problems

In this paper we study the necessary conditions for the masses of the nested regular polygon solutions of the planar 2N-body problem.We prove that the masses at the vertices of each regular polygon must be equal to each other.

An iterative algorithm for solving ill-conditioned linear least squares problems

Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics and geosciences, where regularization algorithms are employed to seek optimal solutions. For many problems, even with the use of regularization algorithms it may be impossible to obtain an accurate solution. Riley and Golub suggested an iterative scheme for solving LLS problems. For the early iteration algorithm, it is difficult to improve the well-conditioned perturbed matrix and accelerate the convergence at the same time. Aiming at this problem, self-adaptive iteration algorithm（SAIA） is proposed in this paper for solving severe ill-conditioned LLS problems. The algorithm is different from other popular algorithms proposed in recent references. It avoids matrix inverse by using Cholesky decomposition, and tunes the perturbation parameter according to the rate of residual error decline in the iterative process. Example shows that the algorithm can greatly reduce iteration times, accelerate the convergence,and also greatly enhance the computation accuracy.

Thermodynamic analysis of β → α(α_2) phase transformation temperature in a Ti-Al-H alloy system

The T0 face equation of a Ti-Al-H alloy system was set up by the regular solution model,and the relationship between the β phase stabilizing parameter of hydrogen and the equilibrium phase compositions was attained.According to the T0 face equation and the thermodynamic parameters from literature,the effect of hydrogen on the β→α（α2） transformation temperature was evaluated.The calculated results were in a better consistence with the measured ones.

ON SPINODAL BOUNDARIES OF Al-Li ALLOYS

The spinodal decomposition can occur in Al-Li alloys containing 5.8-14.2 at.% Li at room temperature. The modutated structure wavelength is approximately 3.1 nm for com mercial Al-LI alloys. The limit composition of the miscibility gap is 3.66 -16.06 at.%Li at 298 K. The highest temperature of the miscibility gap is 377 K.

COMPENSATED COMPACTNESS APPLIED TO PERTURBED FOURTH AND SIXTH ORDER P.D.E

In the paper, by using the methods of compensated compactness and energy estimate, the convergence of class of fourth and sixth orders singular perturbed, partial differential equations is obtained, and furthermore, the regularity of solutions are improved.