33 research outputs found

    Teaching Differential Equations with Graphics and without Linear Algebra

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    We present our approach to teaching the Method of Eigenvectors to solve linear systems of ODEs without assuming a prerequisite course in Linear Algebra. Rather we depend heavily on a graphical approach to systems in two dimensions to motivate the eigenvalue equation

    The developmental influence of primary memory capacity on working memory and academic achievement

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    In this study, we investigate the development of primary memory capacity among children. Children between the ages of 5-8 completed three novel tasks (split span, interleaved lists, and a modified free recall task) that measured primary memory by estimating the number of items in the focus of attention that could be spontaneously recalled in serial order. These tasks were calibrated against traditional measures of simple and complex span. Clear age- related changes in these primary memory estimates were observed. There were marked individual differences in primary memory capacity but each novel measure was predictive of simple span performance. Among older children, each measure shared variance with reading and mathematics performance, whereas for younger children the interleaved lists task was the strongest single predictor of academic ability. We argue that these novel tasks have considerable potential for the measurement of primary memory capacity and provide new, complementary ways of measuring the transient memory processes that predict academic performance. The interleaved lists task also shared features with interference control tasks, and our findings suggest that young children have a particular difficulty in resisting distraction, and that variance in the ability to resist distraction is also shared with measures of educational attainment

    A Cryptographic Attack: Finding the Discrete Logarithm on Elliptic Curves of Trace One

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    The crux of elliptic curve cryptography, a popular mechanism for securing data, is an asymmetric problem. The elliptic curve discrete logarithm problem, as it is called, is hoped to be generally hard in one direction but not the other, and it is this asymmetry that makes it secure. This paper describes the mathematics (and some of the computer science) necessary to understand and compute an attack on the elliptic curve discrete logarithm problem that works in a special case. The algorithm, proposed by Nigel Smart, renders the elliptic curve discrete logarithm problem easy in both directions for elliptic curves of so-called trace one. The implication is that these curves can never be used securely for cryptographic purposes. In addition, it calls for further investigation into whether or not the problem is hard in general

    What can we learn about immediate memory from the development of children's free recall?

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    We ask the question: Which aspects of immediate memory performance improve with age? In two studies, we reexamine the widely held view that primary memory capacity estimates derived from children's immediate free recall are age invariant. This was done by assessing children's immediate free-recall accuracy while also measuring the order in which they elected to recall items (Experiment 1) and by encouraging children to begin free recall with items from towards the end of the presented list (Experiment 2). Across samples aged between 5 and 8 years we replicated the previously reported age-related changes in free-recall serial position functions when aggregated across all trials of the standard task, including an absence of age differences in the recency portion of this curve. However, we also show that this does not reflect the fact that primary memory capacity is constant across age. Instead, when we incorporate order of report information, clear age differences are evident in the recall of list-final items that are output at the start of a participant's response. In addition, the total amount that individuals recalled varied little across different types of free-recall tasks. These findings have clear implications for the use of immediate free recall as a means of providing potential indices of primary memory capacity and in the study of the development of immediate memory

    A test of interference versus decay in working memory: Varying distraction within lists in a complex span task

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    We tested two competing explanations of the effect of processing on working memory. According to decay models, memory representations decay during processing and can be rehearsed or refreshed in the free time between processing steps. Alternatively, one interference-based model assumes that processing involves encoding of distractor representations in working memory, and free time is used to remove distractors. In several experiments the demand from distractor processing was varied within lists, such that one burst of processing following an item on the list was either particularly demanding or particularly undemanding. The exceptional distractor burst had its greatest effect on the list item that immediately preceded it (a local effect), and it affected items that had not yet been presented as well as preceding items. Both findings are predicted by a computational interference model of working memory, and together are problematic for the viewpoint that refreshing offsets decay.</p

    Modeling working memory: An interference model of complex span

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    This article introduces a new computational model for the complex-span task, the most popular task for studying working memory. SOB-CS is a two-layer neural network that associates distributed item representations with distributed, overlapping position markers. Memory capacity limits are explained by interference from a superposition of associations. Concurrent processing interferes with memory through involuntary encoding of distractors. Free time in-between distractors is used to remove irrelevant representations, thereby reducing interference. The model accounts for benchmark findings in four areas: (1) effects of processing pace, processing difficulty, and number of processing steps; (2) effects of serial position and error patterns; (3) effects of different kinds of item-distractor similarity; and (4) correlations between span tasks. The model makes several new predictions in these areas, which were confirmed experimentally

    Weierstrass points on cyclic covers of the projective line

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