Approximate solutions of problems involving normal operators

AbstractThe spectral theory for unbounded normal operators is used to develop a systematic method of approximating functions of operators with other, more easily computable functions, leading to a priori error estimates in the operator norm. In particular, polynomial approximations are obtained for resolvents and semigroups in terms of inverses and resolvents, respectively

Visual perception in dyslexia is limited by sub-optimal scale selection

Readers with dyslexia are purported to have a selective visual impairment but the underlying nature of the deficit remains elusive. Here, we used a combination of behavioural psychophysics and biologically-motivated computational modeling to investigate if this deficit extends to object segmentation, a process implicated in visual word form recognition. Thirty-eight adults with a wide range of reading abilities were shown random-dot displays spatially divided into horizontal segments. Adjacent segments contained either local motion signals in opposing directions or analogous static form cues depicting orthogonal orientations. Participants had to discriminate these segmented patterns from stimuli containing identical motion or form cues that were spatially intermingled. Results showed participants were unable to perform the motion or form task reliably when segment size was smaller than a spatial resolution (acuity) limit that was independent of reading skill. Coherence thresholds decreased as segment size increased, but for the motion task the rate of improvement was shallower for readers with dyslexia and the segment size where performance became asymptotic was larger. This suggests that segmentation is impaired in readers with dyslexia but only on tasks containing motion information. We interpret these findings within a novel framework in which the mechanisms underlying scale selection are impaired in developmental dyslexia

Encoding of rapid time-varying information is impaired in poor readers

A characteristic set of eye movements and fixations are made during reading, so the position of words on the retinae is constantly being updated. Effective decoding of print requires this temporal stream of visual information to be segmented or parsed into its constituent units (e.g., letters or words). Poor readers' difficulties with word recognition could arise at the point of segmenting time-varying visual information, but the mechanisms underlying this process are little understood. Here, we used random-dot displays to explore the effects of reading ability on temporal segmentation. Thirty-eight adult readers viewed test stimuli that were temporally segmented by constraining either local motions or analogous form cues to oscillate back and fourth at each of a range of rates. Participants had to discriminate these segmented patterns from comparison stimuli containing the same motion and form cues but these were temporally intermingled. Results showed that the motion and form tasks could not be performed reliably when segment duration was shorter than a temporal resolution (acuity) limit. The acuity limits for both tasks were significantly and negatively correlated with reading scores. Importantly, the minimum segment duration needed to detect the temporally segmented stimuli was longer in relatively poor readers than relatively good readers. This demonstrates that adult poor readers have difficulty segmenting temporally changing visual input particularly at short segment durations. These results are consistent with evidence suggesting that precise encoding of rapid time-varying information is impaired in developmental dyslexia

Why is the processing of global motion impaired in adults with developmental dyslexia?

Individuals with dyslexia are purported to have a selective dorsal stream impairment that manifests as a deficit in perceiving visual global motion relative to global form. However, the underlying nature of the visual deficit in readers with dyslexia remains unclear. It may be indicative of a difficulty with motion detection, temporal processing, or any task that necessitates integration of local visual information across multiple dimensions (i.e. both across space and over time). To disentangle these possibilities we administered four diagnostic global motion and global form tasks to a large sample of adult readers (N = 106) to characterise their perceptual abilities. Two sets of analyses were conducted. First, to investigate if general reading ability is associated with performance on the visual tasks across the entire sample, a composite reading score was calculated and entered into a series of continuous regression analyses. Next, to investigate if the performance of readers with dyslexia differs from that of good readers on the visual tasks we identified a group of forty-three individuals for whom phonological decoding was specifically impaired, consistent with the dyslexic profile, and compared their performance with that of good readers who did not exhibit a phonemic deficit. Both analyses yielded a similar pattern of results. Consistent with previous research, coherence thresholds of poor readers were elevated on a random-dot global motion task and a spatially one-dimensional (1-D) global motion task, but no difference was found on a static global form task. However, our results extend those of previous studies by demonstrating that poor readers exhibited impaired performance on a temporally-defined global form task, a finding that is difficult to reconcile with the dorsal stream vulnerability hypothesis. This suggests that the visual deficit in developmental dyslexia does not reflect an impairment detecting motion per se. It is better characterised as a difficulty processing temporal information, which is exacerbated when local visual cues have to be integrated across multiple (>2) dimensions