The Spectra of Large Toeplitz Band Matrices with a Randomly Perturbed Entry

This report is concerned with the union $sp_{\Omega}^{(j,k)}T_{n}(a)$ of all possible spectra that may emerge when perturbing a large $n \times n$ Toeplitz band matrix $T_{n}(a)$ in the $(j,k)$ site by a number randomly chosen from some set $\Omega$. The main results give descriptive bounds and, in several interesting situations, even provide complete identifications of the limit of $sp_{\Omega}^{(j,k)}T_{n}(a)$ as $n \to \infty$. Also discussed are the cases of small and large sets $\Omega$ as well as the "discontinuity of the infinite volume case", which means that in general $sp_{\Omega}^{(j,k)}T_{n}(a)$ does not converge to something close to $sp_{\Omega}^{(j,k)}T_{n}(a)$ as $n \to \infty$, where $T(a)$ is the corresponding infinite Toeplitz matrix. Illustrations are provided for tridiagonal Toeplitz matrices, a notable special case. \ud \ud The second author was supported by UK Enginering and Physical Sciences Research Council Grant GR/M1241

Prevalence of Suicide Attempts in a Deaf Population with Co-Occurring Substance Use Disorder

The Deaf Off Drugs & Alcohol (DODA) Program provides culturally appropriate recovery services via e-therapy to Deaf and hard of hearing (HH) individuals with substance use disorder (SUD). In the first three years DODA was providing services,149 consumers (107 Deaf, 42 HH) received treatment. A retrospective secondary data analysis sought to examine the lifetime prevalence of suicidal behavior in Deaf individuals receiving alcohol and drug treatment services from the DODA program. The prevalence of self-reported lifetime suicide attempts in the Deaf sample was 42.1%, higher than rates reported for other subgroups with coexisting conditions. Suicidal ideation was reported by 50.5% of Deaf consumers and by 65.1% of Deaf women. Variables significantly associated with suicide attempts included past mental health diagnosis. Possible explanations and future study are discussed

Spectral approximation of banded Laurent matrices with localized random perturbations

This paper explores the relationship between the spectra of perturbed infinite banded Laurent matrices $L(a)+K$ and their approximations by perturbed circulant matrices $C_{n}(a)+P_{n}KP_{n}$ for large $n$. The entries $K_{jk}$ of the perturbation matrices assume values in prescribed sets $\Omega_{jk}$ at the sites $(j,k)$ of a fixed set $E$, and are zero at the sites $(j,k)$ outside $E$. With ${\cal K}_{\Omega}^{E}$ denoting the ensemble of these perturbation matrices, it is shown that \ud \displaystyle\lim_{n\to\infty} \ud \displaystyle\bigcup_{K\in{\cal K}_{\Omega}^{E}}\ud sp(C_{n}(a)+P_{n}KP_{n})=\ud \displaystyle\bigcup_{K\in{\cal K}_{\Omega}^{E}}\ud sp(L(a)=K)\ud under several fairly general assumptions on $E$ and $\Omega$

On large Toeplitz band matrices with an uncertain block

This report investigates the possible spectra of large, finite dimensional Toeplitz band matrices with perturbations (impurities, uncertainties) in the upper-left m x m block. The main result shows that the asymptotic spectrum of such a matrix is not affected by these perturbations, provided they have sufficiently small norm. This follows from analysis of structured pseudospectra (structured spectral value sets). In contrast, for typical non-Hermitian Toeplitz matrices there exist certain rank-one perturbations of arbitrarily small norm that move an eigenvalue away from the asymptotic spectrum in the large-dimensional limit

Piecewise continuous Toeplitz matrices and operators: slow approach to infinity

The pseudospectra of banded finite dimensional Toeplitz matrices rapidly converge to the pseudospectra of the corresponding infinite dimensional operator. This exponential convergence makes a compelling case for analyzing pseudospectra of such Toeplitz matrices, not just eigenvalues. But what if the matrix is dense and its symbol has a jump discontinuity? The pseudospectra of the finite matrices still converge, but it is shown here that the rate is no longer exponential in the matrix dimension, only algebraic.\ud \ud The second and third authors were supported by UK Engineering and Physical Sciences Research Council Grant GR/M12414

Comparison of Forward, Backward, and Conventional Training in the Learning of a List of CVC Trigrams

Ss were instructed to learn a list of 10 CVC trigrams by either the conventional serial anticipation method, backward conditioning or forward conditioning. The F ratio failed to show a significant difference between the three experimental groups. The results contradict previous results which shown that forward training is more efficient than the other two methods