On the Spectra and Pseudospectra of a Class of Non-Self-Adjoint Random Matrices and Operators

In this paper we develop and apply methods for the spectral analysis of non-self-adjoint 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 $\pm 1$'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 \eps-pseudospectra (\eps>0, $p\in [1,\infty]$) 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 $\Sigma$. We also propose a sequence of inclusion sets for $\Sigma$ which we show is convergent to $\Sigma$, with the $n$th element of the sequence computable by calculating smallest singular values of (large numbers of) $n\times n$ matrices. We propose similar convergent approximations for the 2-norm \eps-pseudospectra of the infinite random matrices, these approximations sandwiching the infinite matrix pseudospectra from above and below

On Spectral Inclusion Sets and Computing the Spectra and Pseudospectra of Bounded Linear Operators

In this paper we derive novel families of inclusion sets for the spectrum and pseudospectrum of large classes of bounded linear operators, and establish convergence of particular sequences of these inclusion sets to the spectrum or pseudospectrum, as appropriate. Our results apply, in particular, to bounded linear operators on a separable Hilbert space that, with respect to some orthonormal basis, have a representation as a bi-infinite matrix that is banded or band-dominated. More generally, our results apply in cases where the matrix entries themselves are bounded linear operators on some Banach space. In the scalar matrix entry case we show that our methods, given the input information we assume, lead to a sequence of approximations to the spectrum, each element of which can be computed in finitely many arithmetic operations, so that, with our assumed inputs, the problem of determining the spectrum of a band-dominated operator has solvability complexity index one, in the sense of Ben-Artzi et al. (C. R. Acad. Sci. Paris, Ser. I 353 (2015), 931-936). As a concrete and substantial application, we apply our methods to the determination of the spectra of non-self-adjoint bi-infinite tridiagonal matrices that are pseudoergodic in the sense of Davies (Commun. Math. Phys. 216 (2001) 687-704)

Identifying common errors in polynomials of eighth grade students

This research aims to study and classify errors in polynomials made by secondary school students. The data for error identification was collected from exercise books of 72 eighth grade students. Three types of errors were examined: careless, computational, and conceptual errors. The errors were considered according to four topics in polynomials: similar terms of monomials; addition of polynomials; subtraction of polynomials; and multiplication of polynomials. It is found that students made the highest computational errors in identifying monomials’ similarity, which accounts for 17.86%. They have the highest percentage of making computational errors in the addition and subtraction of polynomials, which account for 10.88% and 12.04%, respectively. Lastly, they have the highest percentage of making careless errors in the multiplication of polynomials, which accounts for 14.44%. Furthermore, it can be seen that the source of errors is learners’ carelessness when writing the question and its answer. In addition, the basic knowledge of computing addition, subtraction, and multiplication of integers is the most crucial factor that leads to incorrect answers. Nevertheless, most students understand the principle of polynomials, but frequently make errors on other issues

Broad clinical involvement in a family affected by the fragile X premutation

The mutations in the FMR1 gene have been described as a family of disorders called fragile X-associated disorders including fragile X syndrome, fragile X-associated tremor/ataxia syndrome, primary ovarian insufficiency, and other problems associated with the premutation, such as hypothyroidism, hypertension, neuropathy, anxiety, depression, attention-deficit hyperactivity disorders, and autism spectrum disorders. The premutation is relatively common in the general population affecting 1 of 130 to 250 female individuals and 1 of 250 to 800 male individuals. Therefore, to provide appropriate treatment and genetic counseling for all of the carriers and affected individuals in a family, a detailed family history that reviews many of the disorders that are related to both the premutation and the full mutation should be carried out as exemplified in these cases. To facilitate the integration of this knowledge into clinical practice, this is the first case report that demonstrates only premutation involvement across 3 generations

Sertraline May Improve Language Developmental Trajectory in Young Children with Fragile X Syndrome: A Retrospective Chart Review

Young children with fragile X syndrome (FXS) often experience anxiety, irritability, and hyperactivity related to sensory hyperarousal. However, there are no medication recommendations with documented efficacy for children under 5 years old of age with FXS. We examined data through a chart review for 45 children with FXS, 12-50 months old, using the Mullen Scales of Early Learning (MSEL) for baseline and longitudinal assessments. All children had clinical level of anxiety, language delays based on MSEL scores, and similar early learning composite (ELC) scores at their first visit to our clinic. Incidence of autism spectrum disorder (ASD) was similar in both groups. There were 11 children who were treated with sertraline, and these patients were retrospectively compared to 34 children who were not treated with sertraline by chart review. The baseline assessments were done at ages ranging from 18 to 44 months (mean 26.9, SD 7.99) and from 12 to 50 months (mean 29.94, SD 8.64) for treated and not treated groups, respectively. Mean rate of improvement in both expressive and receptive language development was significantly higher in the group who was treated with sertraline (P < 0.0001 and P = 0.0071, resp.). This data supports the need for a controlled trial of sertraline treatment in young children with FXS

Autism phenotype in fragile X premutation males is not associated with FMR1 expression: a preliminary evaluation

To explore the association between autism phenotype and FMR1 protein (FMRP), FMR1 mRNA and CGG repeat length in 31 male FMR1 premutation carriers aged 3.0 to 27.9 years old (mean 13.0 ± SD 6.5) using the ADOS communication, social interactive and total scores. FMRP levels were determined using the sandwich Enzyme-linked Immunosorbent Assay (ELISA) method, FMR1 mRNA expression levels were measured by qRT-PCR, and CGG repeat size was determined using Southern blot and PCR analyses. There was no significant difference in FMRP, CGG repeat length, and FMR1 mRNA between fifteen subjects without (ASD / PDDNOS / autism and sixteen subjects with ASD / PDDNOS / autism. ADOS scores were not significantly associated with either FMRP or FMR1 mRNA, This preliminary evaluation found that autism phenotype is not associated with the level of expression of either FMR1 mRNA or FMRP. However, CGG was significantly negative associated with both ADOS communication score (p= 0.0173) and ADOS total score (p= 0.0358).This work was supported by National Institute of Health grants HD036071, HD02274, DE019583, DA024854, AG032119, AG032115, MH77554; National Center for Advancing Translational Sciences grant UL1RR024146 and UL1TR000153; and Health and Human Services Administration of Developmental Disabilities grant 90DD05969

Autoimmune disease in mothers with the FMR1 premutation is associated with seizures in their children with fragile X syndrome

An increased prevalence of autoimmune diseases in family members of children with autism spectrum disorders (ASD) has been previously reported. ASD is also a common problem co-occurring in children with fragile X syndrome (FXS). Why ASD occurs in some individuals with FXS, but not all, is largely unknown. Furthermore, in premutation carrier mothers, there is an increased risk for autoimmune diseases. This study compared the rate of ASD and other neurodevelopmental/behavioral problems in 61 children with FXS born to 41 carrier mothers who had autoimmune disease and in 97 children with FXS of 78 carrier mothers who did not have autoimmune disease. There were no significant differences in the mean age (9.61 ± 5.59 vs. 9.41 ± 6.31, P = 0.836), cognitive and adaptive functioning in children of mothers with and without autoimmune disease. Among children whose mothers had autoimmune disease, the odds ratio (OR) for ASD was 1.27 (95% CI 0.62–2.61, P = 0.5115). Interestingly, the OR for seizures and tics was 3.81 (95% CI 1.13–12.86, P = 0.031) and 2.94 (95% CI 1.19–7.24, P = 0.019), respectively, in children of mothers with autoimmune disease compared to children of mothers without autoimmune disease. In conclusion, autoimmune disease in carrier mothers was not associated with the presence of ASD in their children. However, seizures and tics were significantly increased in children of mothers with autoimmune disease. This suggests a potential new mechanism of seizure and tic exacerbation in FXS related to an intergenerational influence from autoimmunity in the carrier mother

Television viewing in Thai infants and toddlers: impacts to language development and parental perceptions

<p>Abstract</p> <p>Background</p> <p>Effects of television to language development in infants and toddlers, especially in the Asian children, are inconclusive. This study aimed to (a) study time spent on television in Thai infants and toddlers (age < 2 years), (b) investigate the association between time spent on television (as recommended by the American Academy of Paediatrics (AAP), < 2 hours per day) and language development in Thai 2-year-old children, and (c) explore parental perceptions on television toward their child's development.</p> <p>Methods</p> <p>Two hundred and sixty children and their parents were recruited into the study. Time spent on television and parental perceptions on television viewing toward their child's development were recorded during face-to-face and telephone interviews. Language development was assessed at the age of 2 years using the Clinical Linguistic Auditory Milestone Scale (CLAMS), and parents' report. Association between delayed language development and time spent on television viewing, as well as other various parameters such as gender, maternal education and family income, were analysed using a multivariate logistic regression model.</p> <p>Results</p> <p>Most Thai infants and toddlers watched television at the age of 6 months, 1 year and 2 years old (98.0, 95.3 and 96.7%, respectively). On average, 1-year-old children watched television 1.23 ± 1.42 hours per day. This increased to 1.69 ± 1.56 hours per day when they were 2 years old. However, watching television longer than 2 hours per day did not associate with delayed language development. On multivariate logistic regression analysis, gender (male) was the only significant factor associated with delayed language development (OR = 6.9, 95% CI = 1.5–31.3). Moreover, 75%, 71%, and 66% of Thai parents believed that television viewing yielded benefits to children's developments.</p> <p>Conclusion</p> <p>Thai children commenced watching television at an early age and the amount of television viewing time increased by age. Most parents had positive perceptions to television viewing. The study found no association between time spent on television viewing (≥ 2 hours per day) and delayed language development at the age of 2 years.</p> <p>Gender (male) was the only variable associated with delayed language development.</p

The Role of AGG Interruptions in the Transcription of FMR1 Premutation Alleles

Fragile X associated disorders are caused by a premutation allele in the fragile X mental retardation 1 gene (FMR1) and are hypothesized to result from the toxic effect of elevated levels of expanded FMR1 transcripts. Increased levels of FMR1 mRNA have indeed been reported in premutation carriers; however the mechanism by which expanded alleles lead to elevated levels of FMR1 mRNA in premutation carriers is unknown. Within the CGG repeat tract AGG interruptions are found, generally 1–3 present in normal/intermediate alleles (6–54 CGG repeats) and usually 0–1 in premutation alleles (55–200 CGG repeats). They are present at specific locations, generally occurring after 9 or 10 uninterrupted CGG repeats [(CGG)9AGG(CGG)9AGG(CGG)n]. We evaluated both the number of AGG interruptions and the resulting length of the uninterrupted 3′ CGG repeat pure tract in premutation alleles derived from two large cohorts of male and female carriers to determine whether the presence of AGG interruptions or the length of a pure stretch of CGG repeats influence the levels of FMR1 mRNA in blood. Our findings indicate that neither the number of AGG interruptions, nor their position along the CGG tract have a significant affect on mRNA levels in premutation carriers. We also, as expected based on previous findings, observed a highly significant correlation between CGG repeat number (as both total length and length of pure CGG stretch) and FMR1 mRNA expression levels, in both males and females. Importantly, we did not observe any significant difference in FMR1 mRNA levels in premutation carriers based on age

FMR1 premutation and full mutation molecular mechanisms related to autism

Fragile X syndrome (FXS) is caused by an expanded CGG repeat (>200 repeats) in the 5′ un-translated portion of the fragile X mental retardation 1 gene (FMR1) leading to a deficiency or absence of the FMR1 protein (FMRP). FMRP is an RNA-binding protein that regulates the translation of a number of other genes that are important for synaptic development and plasticity. Furthermore, many of these genes, when mutated, have been linked to autism in the general population, which may explain the high comorbidity that exists between FXS and autism spectrum disorders (ASD). Additionally, premutation repeat expansions (55 to 200 CGG repeats) may also give rise to ASD through a different molecular mechanism that involves a direct toxic effect of FMR1 mRNA. It is believed that RNA toxicity underlies much of the premutation-related involvement, including developmental concerns like autism, as well as neurodegenerative issues with aging such as the fragile X-associated tremor ataxia syndrome (FXTAS). RNA toxicity can also lead to mitochondrial dysfunction, which is common in older premutation carriers both with and without FXTAS. Many of the problems with cellular dysregulation in both premutation and full mutation neurons also parallel the cellular abnormalities that have been documented in idiopathic autism. Research regarding dysregulation of neurotransmitter systems caused by the lack of FMRP in FXS, including metabotropic glutamate receptor 1/5 (mGluR1/5) pathway and GABA pathways, has led to new targeted treatments for FXS. Preliminary evidence suggests that these new targeted treatments will also be beneficial in non-fragile X forms of autism