SummaryComputational analyses of gradient‐elasticity often require the trial solution to be C1 yet constructing simple C1 finite elements is not trivial. This article develops two 48‐dof 4‐node tetrahedral elements for 3D gradient‐elasticity analyses by generalizing the discrete Kirchhoff method and a relaxed hybrid‐stress method. Displacement and displacement‐gradient are the only nodal dofs. Both methods start with the derivation of a C0 quadratic‐complete displacement interpolation from which the strain is derived. In the first element, displacement‐gradient at the mid‐edge points are approximated and then interpolated together with those at the nodes whilst the strain‐gradient is derived from the displacement‐gradient interpolation. In the second element, the assumed constant double‐stress modes are employed to enforce the continuity of the normal derivative of the displacement. The whole formulation can be viewed as if the strain‐gradient matrix derived from the displacement interpolation matrix is refined by a constant matrix. Both elements are validated by the individual element patch test and other numerical benchmark tests. To the best knowledge of the authors, the proposed elements are probably the first nonmixed/penalty 3D elements which employ only displacement and displacement‐gradient as the nodal dofs for gradient‐elasticity analyses.

In this article, we obtain an equation for the high-dimensional limit measure of eigenvalues of generalized Wishart processes, and the results is extended to random particle systems that generalize SDEs of eigenvalues. We also introduce a new set of conditions on the coefficient matrices for the existence and uniqueness of a strong solution for the SDEs of eigenvalues. The equation of the limit measure is further discussed assuming self-similarity on the eigenvalues.

In this article, we obtain an equation for the high-dimensional limit measure of eigenvalues of generalized Wishart processes, and the results is extended to random particle systems that generalize SDEs of eigenvalues. We also introduce a new set of conditions on the coefficient matrices for the existence and uniqueness of a strong solution for the SDEs of eigenvalues. The equation of the limit measure is further discussed assuming self-similarity on the eigenvalues.

In this article, we establish a limiting distribution for eigenvalues of a class of auto-covariance matrices. The same distribution has been found in the literature for a regularized version of these auto-covariance matrices. The original non-regularized autocovariance matrices are non invertible which introduce supplementary difficulties for the study of their eigenvalues through Girko's Hermitization scheme. The key result in this paper is a new polynomial lower bound for the least singular value of the resolvent matrices associated to a rank-defective quadratic function of a random matrix with independent and identically distributed entries.

For real symmetric and complex Hermitian Gaussian processes whose values are d × d matrices, we characterize the conditions under which the probability that at least k eigenvalues collide is positive for 2 ≤ k ≤ d, and we obtain the Hausdorff dimension of the set of collision times.