Abstract. The purpose of this paper is to prove that the spectrum of the non-self-adjoint oneparticle Hamiltonian proposed by J. Feinberg and A. Zee (Phys. Rev. E 59 (1999), 6433-6443) has interior points. We do this by first recalling that the spectrum of this random operator is the union of the set of ∞ eigenvalues of all infinite matrices with the same structure. We then construct an infinite matrix of this structure for which every point of the open unit disk is an ∞ eigenvalue, this following from the fact that the components of the eigenvector are polynomials in the spectral parameter whose non-zero coefficients are ±1 's, forming the pattern of an infinite discrete Sierpinski triangle.Mathematics subject classification (2010): Primary 47B80; Secondary 47A10, 47B36.
Abstract. In this paper we develop and apply methods for the spectral analysis of non-selfadjoint tridiagonal infinite and finite random matrices, and for the spectral analysis of analogous deterministic matrices which are pseudo-ergodic in the sense of E. B. Davies (Commun. Math. Phys. 216 (2001), 687-704). As a major application to illustrate our methods we focus on the "hopping sign model" introduced by J. Feinberg and A. Zee (Phys. Rev. E 59 (1999), 6433-6443), in which the main objects of study are random tridiagonal matrices which have zeros on the main diagonal and random ±1 's as the other entries. We explore the relationship between spectral sets in the finite and infinite matrix cases, and between the semi-infinite and bi-infinite matrix cases, for example showing that the numerical range and p -norm ε -pseudospectra ( ε > 0 , p ∈ [1,∞] ) of the random finite matrices converge almost surely to their infinite matrix counterparts, and that the finite matrix spectra are contained in the infinite matrix spectrum Σ . We also propose a sequence of inclusion sets for Σ which we show is convergent to Σ , with the n th element of the sequence computable by calculating smallest singular values of (large numbers of) n × n matrices. We propose similar convergent approximations for the 2-norm ε -pseudospectra of the infinite random matrices, these approximations sandwiching the infinite matrix pseudospectra from above and below.Mathematics subject classification (2010): Primary 47B80; Secondary 47A10, 47B36.
This study has two parts: phase I designed activities to support all students’ learning preferences, and phase II used open-ended questions and activities based on these preferences to develop students’ mathematical critical thinking skills in polynomials at all performance levels (i.e., high-achieving, fair-achieving, and low-achieving students). This research used an embedded mixed-method design. The subjects selected were 28 out of 98 seventh graders at a boys’ junior high school in Bangkok, Thailand, who were chosen by cluster random sampling technique. The instruments, which were validated by five experts, included a questionnaire, lesson plans, exit tickets, interview protocols, and tests of critical thinking skills in polynomials. The content validity was assessed via expert judgment, and reliability was assessed by item analysis. The quality and effectiveness of the instruments were acceptable. The research results showed the following: (1) most students at all performance levels prefer activities in which they can learn from participating in classroom activities, such as games, activities with real-life applications, and activities involving listening instead of reading and writing, and (2) critical thinking skills in high-achieving and fair-achieving students were at the fair level, while those of low-achieving students were poor. Analysis was the highest critical thinking subskill among high-achieving and low-achieving students, while interpretation was the highest subskill in among fair-achieving students. Open-ended questions and activities based on students’ preferences appear to be practical for developing critical thinking skills among students of all achievement levels.
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