Two-Dimensional Parabolic Inverse Source Problem with Final Overdetermination in Reproducing Kernel Space

A new method of the reproducing kernel Hilbert space is applied to a twodimensional parabolic inverse source problem with the final overdetermination. The exact and approximate solutions are both obtained in a reproducing kernel space. The approximate solution and its partial derivatives are proved to converge to the exact solution and its partial derivatives, respectively. A technique is proposed to improve some existing methods. Numerical results show that the method is of high precision, and confirm the robustness of our method for reconstructing source parameter.

Optimal Load Balancing Leveling Method for Multi-leg Flexible Platforms

The working platforms supported with multiple extensible legs must be leveled before they come into operation.Although the supporting stiffness and reliability of the platform are improved with the increasing number of the supporting legs,the increased overdetermination of the multi-leg platform systems leads to leveling coupling problem among legs and virtual leg problem in which some of the supporting legs bear zero or quasi zero loads.These problems make it quite complex and time consuming to level such a multi-leg platform.Based on rigid body kinematics,an approximate equation is formulated to rapidly calculate the leg extension for leveling a rigid platform,then a proportional speed control strategy is proposed to reduce the unexpected platform distortion and leveling coupling between supporting legs.Taking both the load coupling between supporting legs and the elastic flexibility of the working platform into consideration,an optimal balancing legs’ loads(OBLL) model is firstly put forward to deal with the traditional virtual leg problem.By taking advantage of the concept of supporting stiffness matrix,a coupling extension method(CEM) is developed to solve this OBLL problem for multi-leg flexible platform.At the end,with the concept of supporting stiffness matrix and static transmissibility matrix,an optimal load balancing leveling method is proposed to achieve geometric leveling and legs’ loads balancing simultaneously.Three numerical examples are given out to illustrate the performance of proposed methods.This paper proposes a method which can effectively quantify all of the legs’ extension at the same time,achieve geometric leveling and legs’ loads balancing simultaneously.By using the proposed methods,the stability,precision and efficiency of auto-leveling control process can be improved.

Investigation of the Metabolic Flux Analysis Method for Overdetermined System

With the xanthan synthesis in Xanthomoaas campestris as an example, two methods for metabolic flux analysis of overdetermined system, the experimental data error minimization method and the equation error minimization method, are compared from their mathematical basis, rationality of the results and the easiness of computation. The results show that the experimental data error minimization method is appropriate in metabolic flux analysis of overdetermined system.

Some Symmetry Results for the A-Laplacian Equation via the Moving Planes Method

In this paper, we are concerned with a positive solution of the non-homogeneous A-Laplacian equation in an open bounded connected domain. We use moving planes method to prove that the domain is a ball and the solution is radially symmetric.

Processing Human Colonic Pressure Signals by Using Overdetermined ICA

Independent component analysis （ICA） is a widely used method for blind source separation （BSS）. The mature ICA model has a restriction that the number of the sources must equal to that of the sensors used to collect data, which is hard to meet in most practical cases. In this paper, an overdetermined ICA method is proposed and successfully used in the analysis of human colonic pressure signals. Using principal component analysis （PCA）, the method estimates the number of the sources firstly and reduces the dimensions of the observed signals to the same with that of the sources; and then, Fast- ICA is used to estimate all the sources. From 26 groups of colonic pressure recordings, several colonic motor patterns are extracted, which riot only prove the effectiveness of this method, but also greatly facilitate further medical researches.

Some Steiner Symmetry Results in Overdetermined Boundary Value Problem

In this paper, we use the moving planes method to prove that the domain Ω and the solution u are Steiner symmetric if u is a positive solution of the overdetermined boundary value problem in Ω.

Robust Parametric Modeling of Speech in Additive White Gaussian Noise

In estimating the linear prediction coefficients for an autoregressive spectral model, the concept of using the Yule-Walker equations is often invoked. In case of additive white Gaussian noise (AWGN), a typical parameter compensation method involves using a minimal set of Yule-Walker equation evaluations and removing a noise variance estimate from the principal diagonal of the autocorrelation matrix. Due to a potential over-subtraction of the noise variance, however, this method may not retain the symmetric Toeplitz structure of the autocorrelation matrix and thereby may not guarantee a positive-definite matrix estimate. As a result, a significant decrease in estimation performance may occur. To counteract this problem, a parametric modelling of speech contaminated by AWGN, assuming that the noise variance can be estimated, is herein presented. It is shown that by combining a suitable noise variance estimator with an efficient iterative scheme, a significant improvement in modelling performance can be achieved. The noise variance is estimated from the least squares analysis of an overdetermined set of p lower-order Yule-Walker equations. Simulation results indicate that the proposed method provides better parameter estimates in comparison to the standard Least Mean Squares (LMS) technique which uses a minimal set of evaluations for determining the spectral parameters.

Serrin-Type Overdetermined Problem in H^(n)

In this paper,we prove the symmetry of the solution to overdetermined problem for the equationσ_(k)(D^(2)u-ul)=C^(k)_(n)in hyperbolic space.Our approach is based on establishing a Rellich-Pohozaev type identity and using a P function.Our result generalizes the overdetermined problem for Hessian equation in Euclidean space.

Method of characteristics and first integrals for systems of quasi-linear partial differential equations of first order

Given a set of independent vector fields on a smooth manifold, we discuss how to find a function whose zero-level set is invariant under the flows of the vector fields. As an application, we study the solvability of overdetermined partial differential equations: Given a system of quasi-linear PDEs of first order for one unknown function we find a necessary and sufficient condition for the existence of solutions in terms of the second jet of the coefficients. This generalizes to certain quasi-linear systems of first order for several unknown functions.

PSEUDO-SOLUTION ON GEODETIC OVERDETERMINED BOUNDARY VALUE PROBLEMS

A higher step of study on the GOBVPs (geodetic overdetermined boundary value problems) is reached in this paper, which covers the proposal of new concept of pseudo-solutionon the GOBVPs, its strict definition of mathematics and solving principle. The so-called pseudosolution is a harmonical function having the property of optimum approximating the given boundary values in the sense of a relevant norm. Analytical expressions of the pseudo-solutions of two typical OBVPs for biboundary surfaces in physical geodesy, the problems S—D and S—N, are obtained, which is elegant and concise in form and convenient for practice, by using the derived formulas of norms of fractional exponential Sobolev's spaces in the case of spherical biboundary. The pseudo-solution is composed of two parts: the major is the solution of classical Stokes' problem, playing control role in field representation; the minor is correction term, serving the function of synergist and precision of the gravity field. Besides, a general case of the GOBVP is also dealt with.

Arbitrariness of the general solution of the partial differential equations and its applications

Using the framework of formal theory of partial differential equations, we consider a method of computation of the bi-Hilbert polynomial (i.e. Hilbert polynomial in two variables). Furthermore, present an approach to compute the number of arbitrary functions of positive differential order in the general solution. Then, under the "AC=BD" model for mathematics mechanization developed by Hong-qing ZHANG, we present a method to reduce an overdetermined system to a well-determined one. As applications, the Maxwell equations and weakly overdetermined equations are considered.

Overdetermined Systems of First Order Partial Differential Equations with Singular Solution

We study overdetermined systems of first order partial differential equations with singular solutions.The main result gives a characterization of such systems and asserts that the singular solution is equal to the contact singular set.

CONVERGENCE OF NEWTON＇S METHOD FOR SYSTEMS OF EQUATIONS WITH CONSTANT RANK DERIVATIVES

The convergence properties of Newton＇s method for systems of equations with constant rank derivatives are studied under the hypothesis that the derivatives satisfy some weak Lipschitz conditions. The unified convergence results, which include Kantorovich type theorems and Smale＇s point estimate theorems as special cases, are obtained.

Convergent Overdetermined-RBF-MLPG for Solving Second Order Elliptic PDEs

This paper deals with the solvability and the convergence of a class of unsymmetric Meshless Local Petrov-Galerkin(MLPG)method with radial basis function(RBF)kernels generated trial spaces.Local weak-form testings are done with stepfunctions.It is proved that subject to sufficiently many appropriate testings,solvability of the unsymmetric RBF-MLPG resultant systems can be guaranteed.Moreover,an error analysis shows that this numerical approximation converges at the same rate as found in RBF interpolation.Numerical results(in double precision)give good agreement with the provided theory.