Generalized regular points of a C^1 map between Banach spaces

Let f be a C^1 map between two Banach spaces E and F. It has been proved that the concept of generalized regular points of f, which is a generalization of the notion of regular points of f, has some crucial applications in nonlinearity and global analysis. We characterize the generalized regular points of f using the three integer-valued （or infinite） indices M（x0）, Mc（x0） and Mr（x0） at x0 ∈ E generated by f and by analyzing generalized inverses of bounded linear operators on Banach spaces, that is, iff ＇（x0） has a generalized inverse in the Banach space B（E, F） of all bounded linear operators on E into F and at least one of the indices M（x0）, Mc（x0） and Mr（x0） is finite, then xo is a generalized regular point off if and only if the multi-index （M（x）, Me（x）, Mr（x）） is continuous at X0.

A remark on regular points of Ricci limit spaces

Let Y be a Gromov-Hausdorff limit of complete Riemannian nmanifolds with Ricci curvature bounded from below. A point in Y is called k-regular, if its tangent is unique and is isometric to a k-dimensional Euclidean space. By Cheeger-Colding and Colding-Naber, there is k 〉 0 such that the set of all k-regular point ：Rk has a full renormalized measure. An open problem is if Rl = 0 for all l 〈 k？ The main result in this paper asserts that if R1 ≠ 0, then Y is a one-dimensional topological manifold. Our result improves Honda＇s result that under the assumption that 1 ≤ dimH（Y） 〈 2.

HAMILTON-JACOBI EQUATIONS FOR A REGULAR CONTROLLED HAMILTONIAN SYSTEM AND ITS REDUCED SYSTEMS

In this paper,we give the geometric constraint conditions of a canonical symplectic form and regular reduced symplectic forms for the dynamical vector fields of a regular controlled Hamiltonian(RCH)system and its regular reduced systems,which are called the Type I and Type II Hamilton-Jacobi equations.First,we prove two types of Hamilton-Jacobi theorems for an RCH system on the cotangent bundle of a configuration manifold by using the canonical symplectic form and its dynamical vector field.Second,we generalize the above results for a regular reducible RCH system with symmetry and a momentum map,and derive precisely two types of Hamilton-Jacobi equations for the regular point reduced RCH system and the regular orbit reduced RCH system.Third,we prove that the RCH-equivalence for the RCH system,and the RpCH-equivalence and RoCH-equivalence for the regular reducible RCH systems with symmetries,leave the solutions of corresponding Hamilton-Jacobi equations invariant.Finally,as an application of the theoretical results,we show the Type I and Type II Hamilton-Jacobi equations for the Rp-reduced controlled rigid body-rotor system and the Rp-reduced controlled heavy top-rotor system on the generalizations of the rotation group SO(3)and the Euclidean group SE(3),respectively.This work reveals the deeply internal relationships of the geometrical structures of phase spaces,the dynamical vector fields and the controls of the RCH system.

Robust elastic impedance inversion using L1-norm misfit function and constraint regularization

The classical elastic impedance （EI） inversion method, however, is based on the L2-norm misfit function and considerably sensitive to outliers, assuming the noise of the seismic data to be the Guassian-distribution. So we have developed a more robust elastic impedance inversion based on the Ll-norm misfit function, and the noise is assumed to be non-Gaussian. Meanwhile, some regularization methods including the sparse constraint regularization and elastic impedance point constraint regularization are incorporated to improve the ill-posed characteristics of the seismic inversion problem. Firstly, we create the Ll-norm misfit objective function of pre-stack inversion problem based on the Bayesian scheme within the sparse constraint regularization and elastic impedance point constraint regularization. And then, we obtain more robust elastic impedances of different angles which are less sensitive to outliers in seismic data by using the IRLS strategy. Finally, we extract the P-wave and S-wave velocity and density by using the more stable parameter extraction method. Tests on synthetic data show that the P-wave and S-wave velocity and density parameters are still estimated reasonable with moderate noise. A test on the real data set shows that compared to the results of the classical elastic impedance inversion method, the estimated results using the proposed method can get better lateral continuity and more distinct show of the gas, verifying the feasibility and stability of the method.

RECONSTRUCTION STABILITY OF NEARFIELD ACOUSTIC HOLOGRAPHY

The distributed source boundary point method （DSBPM） is used as the spatial transform algorithm for realizing nearfield acoustic holography （NAH）, the sensitivity of the reconstructed solution to the measurement errors is analyzed, and the regularization method is proposed to stabilize the reconstruction process, control the influence of the measurement errors and get a better approximate solution. An oscillating sphere is investigated as a numerical example, the influence of the measurement errors on the reconstruction solution is demonstrated, and the feasibility and validity of the regularization method are validated. Key words： Acoustic holography Boundary point method Inverse problem Regularization

A CLASS OF REACTION-DIFFUSION EQUATIONS WITH HYSTERESIS DIFFERENTIAL OPERATOR

In this paper, the classical and weak derivatives with respect to spatial variable of a class of hysteresis functional are discussed. Some conclusions about solutions of a class of reaction-diffusion equations with hysteresis differential operator are given.

Frobenius Method for Solving Second-Order Ordinary Differential Equations

As we know that the power series method is a very effective method for solving the Ordinary differential equations (ODEs) which have variable coefficient, so in this paper we have studied how to solve second-order ordinary differential equation with variable coefficient at a singular point t = 0 and determined the form of second linearly independent solution. Based on the roots of initial equation there are real and complex cases. When the roots of initial equation are real then there are three kinds of second linearly independent solutions. If the roots of the initial equation are distinct complex numbers, then the solution is complex-valued.

On the Spectra of General Ordinary Quasi-Differential Operators and Their L2w-Solutions

In this paper, we consider the general ordinary quasi-differential expression τ of order n with complex coefficients and its formal adjoint τ+ on the interval [a,b). We shall show in the case of one singular end-point and under suitable conditions that all solutions of a general ordinary quasi-differential equation are in the weighted Hilbert space provided that all solutions of the equations and its adjoint are in . Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions may be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while the others are new.

Existence theory for Rosseland equation and its homogenized equation

The global boundness and existence are presented for the kind of the Rosseland equation with a general growth condition. A linearized map in a closed convex set is defined. The image set is precompact, and thus a fixed point exists. A multi-scale expansion method is used to obtain the homogenized equation. This equation satisfies a similar growth condition.

Symmetric Reduction and Hamilton-Jacobi Equations of the Controlled Underwater Vehicle-Rotor System

.As an application of the theoretical results,in this paper,we study the symmetric reduction and Hamilton-Jacobi theory for the underwater ve-hicle with two internal rotors as a regular point reducible RCH system,in the cases of coincident and non-coincident centers of the buoyancy and the gravity.At first,we give the regular point reduction and the two types of Hamilton-Jacobi equations for a regular controlled Hamiltonian(RCH)system with sym-metry and a momentum map on the generalization of a semidirect product Lie group.Next,we derive precisely the geometric constraint conditions of the reduced symplectic forms for the dynamical vector fields of the regular point reducible controlled underwater vehicle-rotor system,that is,the two types of Hamilton-Jacobi equations for the reduced controlled underwater vehicle-rotor system,by calculations in detail.These work reveal the deeply internal relationships of the geometrical structures of the phase spaces,the dynamical vector fields and the controls of the system.

INTERFACE PROBLEMS FOR ELLIPTIC DIFFERENTIAL EQUATIONS

A new approach is given to analyse the regularity of solutions near singular points for the interface problems of second order elliptic partial differential equations. For general equations with nonsymmetric dominant terms and discontinuous piecewise smooth coefficients, it is proved that solutions in H 1 can be docomposed into two parts, one of which is a finite sum of particular solutions to the corresponding homogeneous equations with piecewise constant coefficients, and the other one of which is the regular part. Moreover a priori estimations are proven.

DIFFERENTIABILITY OF A CLASS OF HYSTERESIS FUNCTIONAL

In this paper, the differentiability of a class of hysteresis functional is discussed, and its expression formulas are given.