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Based on the matrix expression of general nonlinear numerical analogues presented by the present author, this paper proposes a novel philosophy of nonlinear computation and analysis. The nonlinear problems are considered an ill-posed linear…
We propose a boundary regularity condition for the $M_n(\mathbb{C})$-valued subordination functions in free probability to prove the local limit theorem and delocalization of eigenvectors for polynomials in two random matrices. We prove…
Consider the eigenvalues $\lambda_i(M_n)$ (in increasing order) of a random Hermitian matrix $M_n$ whose upper-triangular entries are independent with mean zero and variance one, and are exponentially decaying. By Wigner's semicircular law,…
For a finite dimensional unital complex simple Jordan superalgebra $J$, the Tits-Kantor-Koecher construction yields a 3-graded Lie superalgebra $\mathfrak g_\flat\cong \mathfrak g_\flat(-1)\oplus\mathfrak g_\flat(0)\oplus\mathfrak…
This paper is a tutorial for eigenvalue and generalized eigenvalue problems. We first introduce eigenvalue problem, eigen-decomposition (spectral decomposition), and generalized eigenvalue problem. Then, we mention the optimization problems…
In this paper we study Steklov eigenvalues for the Lam\'e operator which arise in the theory of linear elasticity. In this eigenproblem the spectral parameter appears in a Robin boundary condition, linking the traction and the displacement.…
Given $\Omega(\subseteq\;R^{1+m})$, a smooth bounded domain and a nonnegative measurable function $f$ defined on $\Omega$ with suitable summability. In this paper, we will study the existence and regularity of solutions to the quasilinear…
We study the eigenvalue problem for a superlinear convolution operator in the special case of bilinear constitutive laws and establish the existence and uniqueness of a one-parameter family of nonlinear eigenfunctions under a topological…
We describe a novel algorithm for solving general parametric (nonlinear) eigenvalue problems. Our method has two steps: first, high-accuracy solutions of non-parametric versions of the problem are gathered at some values of the parameters;…
Given an optimization problem, the Hessian matrix and its eigenspectrum can be used in many ways, ranging from designing more efficient second-order algorithms to performing model analysis and regression diagnostics. When nonlinear models…
We derive some localization and perturbation results for coneigenvalues of quaternion matrices. In localization results, we derive Ger\v{s}gorin type theorems for right and left coneigenvalues of quaternion matrices. We prove that certain…
We prove a single-value version of Reshetnyak's theorem. Namely, if a non-constant map $f \in W^{1,n}_{\text{loc}}(\Omega, \mathbb{R}^n)$ from a domain $\Omega \subset \mathbb{R}^n$ satisfies the estimate $\lvert Df(x) \rvert^n \leq K…
We consider radial complex scaling/perfectly matched layer methods for scalar resonance problems in homogeneous exterior domains. We introduce a new abstract framework to analyze the convergence of domain truncations and discretizations.…
We consider the bifurcation problem $u'' + \lambda u = N(u)$ with two point boundary conditions where $N(u)$ is a general nonlinear term which may also depend on the eigenvalue $\lambda$. We give a variational characterization of the…
This paper introduces a fast and numerically stable algorithm for the solution of fourth-order linear boundary value problems on an interval. This type of equation arises in a variety of settings in physics and signal processing. Our method…
We present a novel integral-equation algorithm for evaluation of Zaremba eigenvalues and eigenfunctions}, that is, eigenvalues and eigenfunctions of the Laplace operator with mixed Dirichlet-Neumann boundary conditions; of course, (slight…
We study singular solutions to the fractional Laplace equation and, more generally, to nonlocal linear equations with measurable kernels. We establish B\^ocher type results that characterize the behavior of singular solutions near the…
In matrix theory and numerical analysis there are two very famous and important results. One is Gersgorin circle theorem, the other is strictly diagonally dominant theorem. They have important application and research value, and have been…
We present a Chebyshev collocation method for linear ODE and DDE problems. We first give a posteriori estimates for the accuracy of the approximate solution of a scalar ODE initial value problem. Examples of the success of the estimate are…
In this paper, we analyze an eigenvalue problem for nonlinear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We prove bifurcation results from trivial solutions and from infinity for…