THE NATURAL BOUNDARY OF SOME RANDOM POWER SERIES

Suppose that {X(n)(omega)} are independent random complex variable sequence, E(X(n)) = 0 and [GRAPHICS] (V(X(n) = sigma(n)2). If reversed capital E-epsilon > 0 such that for all P (H) > 1-epsilon, we have [GRAPHICS] Then the circle {\Z\ = rho} is almost surely a natural boundary of the random series [GRAPHICS]

The Best Natural Boundary:a New Concept Developed by Combining the Sequence Boundary with the GSSP

Through a detailed study of sequence boundaries, it is concluded that sequence stratigraphy is an independent regional and transitional stratigraphic system between local lithostratigraphy and global chronostratigraphy. Therefore, a new tripartite stratigraphic classification scheme has been proposed. By combining organically the concept of sequence boundaries with the GSSP, it is suggested that the GSSP should be chosen in a conformale portion of a related sequence boundary, and the boundary established in light of this concept is defined as the Best Natural Boundary (BNB). The definition of the BNB points out the working area and stratigraphic level for the GSSP. By referring to a case study of the Permian Guadalupian/Lopingian boundary, the concept of the BNB has been elaborated in detail, and it is proposed that the BNB of the Guadalupian and the Lopingian lies between the Mesogondolella granti Zone and the Ctarkina postbitteri Zone, which is also the sequence chronostratigraphic boundary between the Transpecos composite sequence (chronozone)and the UAA- 1 composite sequence (chronozone).

THE UNSTABLE MODES OF NATURAL CONVECTION BOUNDARY LAYER

The instability of natural convection boundary layer around a vertical heated flat plate is analyzed theoretically in this paper. The results illustrate that the 'loop' in the neutral curve is not a real loop but a twist of the curve in the frequency-wave number-Grashof number space, and there is only one unstable mode at small Prandtl numbers. Specially, when the Prandtl number is large enough two unstable modes will be found in the 'loop' region. Along the amplifying surface intersection the two unstable modes have the same Grashof number, wave number and frequency but different amplifying rates. Their instability characteristics are analyzed and the criterion for determining the existence of the multi-unstable modes is also discussed.

Sequence Boundary:Its Significance in Stratigraphic Subdivision,Correlation and Classification

Sequence boundary is studied in detail in reference to the Permian Guadalupian Lopingian boundary, based on high resolution correlation of conodont zones. A conceptual synchronous stratigraphic boundary, corresponding to the age of the sequence boundary dated in the conformable portion of the sequence boundary, is defined as the sequence chronostratigraphic boundary (SCB). The sequence boundary is proved to be dual in nature in regards to tin-synchrony and diachroneity of stratigraphic boundaries. The merits of the sequence boundary in stratigraphic subdivision, correlation and classification are elucidated in comparison with the lithostratigraphic boundary, the biostratigraphic boundary and the traditional approach of the global stratotype section and point (GSSP ). By integrating the virtues of the sequence boundary and the GSSP, it is proposed that the GSSP should be established in the conformable portion of the related sequence boundary. The boundary established in light of this approach is defined as the best natural boundary. The analysis of the sequence boundary leads to the conclusion that sequence stratigraphy is a regional and transitional stratigraphic system between local lithostratigraphy and global chronostratigraphy. Therefore, a new tripartite stratigraphic classification scheme is tentatively proposed.

On the Coupled of NBEM and CFEM for an Anisotropic Quasilinear Problem in an Unbounded Domain with a Concave Angle

In this paper, based on the Kirchhoff transformation and the natural boundary element method, a coupled natural boundary element and curved edge finite element is applied to solve the anisotropic quasi-linear problem in an unbounded domain with a concave angle. By using the principle of the natural boundary reduction, we obtain the natural integral equation on the artificial boundary of circular arc boundary, and get the coupled variational problem and its numerical method. Then the error and convergence of coupling solution are analyzed. Finally, some numerical examples are verified to show the feasibility of our method.

NATURAL BOUNDARY ELEMENT METHOD FOR THREE DIMENSIONAL EXTERIOR HARMONIC PROBLEM WITH AN INNER PROLATE SPHEROID BOUNDARY

In this paper, we study natural boundary reduction for Laplace equation with Dirichlet or Neumann boundary condition in a three-dimensional unbounded domain, which is the outside domain of a prolate spheroid. We express the Poisson integral formula and natural integral operator in a series form explicitly. Thus the original problem is reduced to a boundary integral equation on a prolate spheroid. The variational formula for the reduced problem and its well-posedness are discussed. Boundary element approximation for the variational problem and its error estimates, which have relation to the mesh size and the terms after the series is truncated, are also presented. Two numerical examples are presented to demonstrate the effectiveness and error estimates of this method.

THE COUPLING OF NATURAL BOUNDARY ELEMENT AND FINITE ELEMENT METHOD FOR 2D HYPERBOLIC EQUATIONS

In this paper, we investigate the coupling of natural boundary element and finite element methods of exterior initial boundary value problems for hyperbolic equations. The governing equation is first discretized in time, leading to a time-step scheme, where an exterior elliptic problem has to be solved in each time step. Second, a circular artificial boundary TR consisting of a circle of radius R is introduced, the original problem in an unbounded domain is transformed into the nonlocal boundary value problem in a bounded subdomain. And the natural integral equation and the Poisson integral formula are obtained in the infinite domain Ω2 outside circle of radius R. The coupled variational formulation is given. Only the function itself, not its normal derivative at artificial boundary TR, appears in the variational equation, so that the unknown numbers are reduced and the boundary element stiffness matrix has a few different elements. Such a coupled method is superior to the one based on direct boundary element method. This paper discusses finite element discretization for variational problem and its corresponding numerical technique, and the convergence for the numerical solutions. Finally, the numerical example is presented to illustrate feasibility and efficiency of this method.

THE COUPLING METHOD FOR TORSION PROBLEM OF THE SQUARE CROSS-SECTION BAR WITH CRACKS

By coupling natural boundary element method (NBEM) with FEM based on domain decomposition, the torsion problem of the square cross-sections bar with cracks have been studied, the stresses of the nodes of the cross-sections and the stress intensity factors have been calculated, and some distribution pictures of the stresses have been drawn. During computing, the effect of the relaxed factors to the convergence speed of the iterative method has been discussed. The results of the computation have confirmed the advantages of the NBEM and its coupling with the FEM.

Natural boundary of the random power series

We investigate convergence properties of random Taylor series whose coefficients are ψ-mixing random variables. In particular, we give sufficient conditions such that the circle of convergence of the series forms almost surely a natural boundary.

NON-SYMMETRICAL BENDING PROBLEMS OF INFINITE ANNULAR PLATES SUPPORTED ON INNER EDGE

For non-asymmetrical bending problems of elastic annular plates, the exact solutions are not fond. To bending problems of infinite annular plate with two different boundary conditions, based on the boundary integral formula,the natural boundary integral equation for the boundary value problems of the biharmonic equation and the condition of bending moment in infinity,bending solutions under non-symmetrical loads are gained by the Fourier series and convolution formulae. The formula for the solutions has nicer convergence velocity and high computational accuracy, and the calculating process is simpler. Solutions of the given examples are compared with the finite element method. The textual solutions of moments near the loads are better than the finite element method to the fact that near the concentrative loads the inners forces trend to infinite.

A PRECONDITIONER FOR COUPLING SYSTEM OF NATURAL BOUNDARY ELEMENT AND COMPOSITE GRID FINITE ELEMENT

Discusses a coupling boundary element methods with finite element method for the Dirichlet exterior problems. Information on the natural boundary reduction; Discussion on the composite grid discretization; Details on a preconditioner for the discrete system.

Boundary Integral Equations and A Posteriori Error Estimates

Adaptive methods have been rapidly developed and applied in many fields of scientific and engi- neering computing. Reliable and efficient a posteriori error estimates play key roles for both adaptive finite element and boundary element methods. The aim of this paper is to develop a posteriori error estimates for boundary element methods. The standard a posteriori error estimates for boundary element methods are obtained from the classical boundary integral equations. This paper presents hyper-singular a posteriori er- ror estimates based on the hyper-singular integral equations. Three kinds of residuals are used as the esti- mates for boundary element errors. The theoretical analysis and numerical examples show that the hyper- singular residuals are good a posteriori error indicators in many adaptive boundary element computations.

THE ELLIPSOID ARTIFICIAL BOUNDARY METHOD FOR THREE-DIMENSIONAL UNBOUNDED DOMAINS

The artificial boundary method is applied to solve three-dimensional exterior problems. Two kind of rotating ellipsoids are chosen as the artificial boundaries and the exact artificial boundary conditions are derived explicitly in terms of an infinite series. Then the well-posedness of the coupled variational problem is obtained. It is found that error estimates derived depend on the mesh size, truncation term and the location of the artificial boundary. Three numerical examples are presented to demonstrate the effectiveness and accuracy of the proposed method.

THE OVERLAPPING DOMAIN DECOMPOSITION METHODFOR HARMONIC EQUATION OVER EXTERIORTHREE-DIMENSIONAL DOMAIN

Presents a study which applied the overlapping domain decomposition method based on the natural boundary reduction to solve the boundary value problem of harmonic equation over domain. Methods to solve boundary value problems; Contraction factor for the domain; Results.

Orientation Field Guided Texture Synthesis

We present a framework for example-based texture synthesis with feature directions aligned to vector fields with 2-way rotational symmetry, also known as orientation fields. Through a simple variational formulation, the framework allows the user to design the orientation field with intuitive controls, by interactively manipulating singularities and field directions. The resulting field is then used to guide a parallel synthesis. Our contribution is twofold： a design tool for orientation fields with a natural boundary condition, and a parallel texture synthesis adapted specifically for using such fields in feature alignment. We demonstrate the advantages of the procedure through examples on planar and curved patches with trivial topology.

DOMAIN DECOMPOSITION WITH NONMATCHING GRIDS FOR EXTERIOR TRANSMISSION PROBLEMS VIA FEM AND DTN MAPPING

In this paper, we are concerned with a non-overlapping domain decomposition method （DDM） for exterior transmission problems in the plane. Based on the natural boundary integral operator, we combine the DDM with a Dirichlet-to-Neumann （DtN） mapping and provide the numerical analysis with nonmatching grids. The weak continuity of the approximation solutions on the interface is imposed by a dual basis multiplier. We show that this multiplier space can generate optimal error estimate and obtain the corresponding rate of convergence. Finally, several numerical examples confirm the theoretical results.