We investigate properties of harmonic Gauss maps and their applications to Lawson-Osserman s problem, to the rigidity of space-like submanifolds in a pseudo-Euclidean space and to the mean curvature flow.

For a class of data often arising in engineering, we have developed an approach to compute the lower confidence limit for structure reliability with a given confidence level. Especially, in a case with no failure and a case with only one failure, the concrete computational methods are presented.

This work deals with the numerical differentiation for an unknown smooth function whose data on a given set are available. The numerical differentiation is an ill-posed problem. In this work, the first and second derivatives of the smooth function are approximated by using the Tikhonov regularization method. It is proved that the approximate function can be chosen as a minimizer to a cost functional. The existence and uniqueness theory of the minimizer is established. Errors in the derivatives between the smooth unknown function and the approximate function are obtained, which depend on the mesh size of the grid and the noise level in the data. The numerical results are provided to support the theoretical analysis of this work.

A new multiplier method for solving the linear complementarity problem LCP(q,M) is proposed. By introducing a Lagrangian of LCP(q,M), a new smooth merit function θ(χ, λ) for LCP(q,M) is constructed. Based on it, a simple damped Newton-type algorithm with multiplier self-adjusting step is presented. When M is a P-matrix, the sequence {θ(χ^k, λ^k)} (where {(χ^k, λ^k)} is generated by the algorithm) is globally linearly convergent to zero and convergent in a finite number of iterations if the solution is degenerate. Numerical results suggest that the method is highly effcient and promising.

The main aim of this paper is to discuss the problem concerning the analyticity of the solutions of analytic non-linear elliptic boundary value problems. It is proved that if the corresponding first variation is regular in Lopatinskiïi sense, then the solution is analytic up to the boundary. The method of proof really covers the case that the corresponding first variation is regularly elliptic in the sense of Douglis-Nirenberg-Volevich, and hence completely generalize the previous result of C. B. Morrey. The author also discusses linear elliptic boundary value problems for systems of elliptic partial differential equations where the boundary operators are allowed to have singular integral operators as their coeffcients. Combining the standard Fourier transform technique with analytic continuation argument, the author constructs the Poisson and Green s kernel matrices related to the problems discussed and hence obtain some representation formulae to the solutions. Some a priori estimates of Schauder type and L_p type are obtained.

A rounding error analysis for the symplectic Lanczos method is given to solve the large-scale sparse Hamiltonian eigenvalue problem. If no breakdown occurs in the method, then it can be shown that the Hamiltonian structure preserving requirement does not destroy the essential feature of the nonsymmetric Lanczos algorithm. The relationship between the loss of J-orthogonality among the symplectic Lanczos vectors and the convergence of the Ritz values in the symmetric Lanczos algorithm is discussed. It is demonstrated that under certain assumptions the computed J-orthogonal Lanczos vectors lose the J-orthogonality when some Ritz values begin to converge. Our analysis closely follows the recent works of Bai and Fabbender.

Chinese remainder codes are constructed by applying weak block designs and the Chinese remainder theorem of ring theory. The new type of linear codes take the congruence class in the congruence class ring R/I₁ ∩ I₂ ∩ • • • ∩ I_n for the information bit, embed R/J_i into R/I₁ ∩ I₂ ∩ • • • ∩ I_n, and assign the cosets of R/J_i as the subring of R/I₁ ∩ I₂ ∩ • • • ∩ I_n and the cosets of R/J_i in R/I₁∩I₂∩• • •∩I_n as check lines. Many code classes exist in the Chinese remainder codes that have high code rates. Chinese remainder codes are the essential generalization of Sun Zi codes.

The purpose of this paper is to generalize the deformation theory of special Lagrangian submanifolds developed by Mclean and Hitchin to special Lagrangian suborbifolds.

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