A competitive neural model of small number detection

The numerical distance effect (inverse relationship between numerical distance and reaction time in relative number comparison tasks) has frequently been used to characterize the mental representation of number. The size of the distance effect decreases over developmental time. However, it is unclear whether this reduction simply reflects developmental changes in domain-general speed of processing and whether it is specific to numerical compared with non-numerical magnitude. To examine these open questions, we conducted a cross-sectional study with 6-, 7-, and 8-year-old children as well as adult college students. Participants performed comparisons on Arabic numerals, arrays of squares, squares of varying luminance and bars of varying height. To control for general age-related changes in reaction time, a measure of speed of processing was used as a covariate in the analysis. A significant developmental decrease in the distance effect was found across numerical and non-numerical comparison tasks over and above general changes in processing speed. However, this change was not found to differ as a function of format. These data suggest that developmental changes in the distance effect are reflective of changes in a domain-general comparison process, rather than domain-specific developmental changes in number representations. However, analysis of overall reaction times revealed significantly greater developmental changes for numerical relative to non-numerical comparison tasks. These findings highlight the importance of taking multiple measures into account when characterizing developmental changes in numerical magnitude processing. Implications for theories of numerical cognition and its development are discussed.

The reliability of and the relation between non-symbolic numerical distance effects in comparison, same-different judgments and priming

For well over one-hundred years, several key factors have been well established in the study of number comparison, including mental number line, numerical distance effect, and effect of sensory representation on number processing. The purpose of this article is to put some of these studies together to discuss design parameters and research questions addressed in the mental number comparison studies. Most of the studies discuss sensory representation and abstract number representation as well as degree of their interaction. In order to give the different views on a particular research question, the author classified studies under the related research questions. For example, Stroop and size congruity effect studies are addressed under this title chronologically. It was very clear that the design parameters and research question might change the interpretation of a task. It may be time to shift attention from the question of the interaction degree of sensory representation and abstract representation to a larger scope. The larger scope would be to understand the differences and similarities between different groups using a universal approach.

BackgroundA dominant view in numerical cognition is that numerical comparisons operate on a notation independent representation (Dehaene, 1992). Although previous human neurophysiological studies using scalp-recorded event-related potentials (ERPs) on the numerical distance effect have been interpreted as supporting this idea, differences in the electrophysiological correlates of the numerical distance effect in symbolic notations (e.g. Arabic numerals) and non-symbolic notations (e.g. a set of visually presented dots of a certain number) are not entirely consistent with this view.Methods and resultsTwo experiments were conducted to resolve these discrepancies. In Experiment 1, participants performed a symbolic and a non-symbolic numerical comparison task ("smaller or larger than 5?") with numerical values 1–4 and 6–9 while ERPs were recorded. Consistent with a previous report (Temple & Posner, 1998), in the symbolic condition the amplitude of the P2p ERP component (210–250 ms post-stimulus) was larger for values near to the standard than for values far from the standard whereas this pattern was reversed in the non-symbolic condition. However, closer analysis indicated that the reversal in polarity was likely due to the presence of a confounding stimulus effect on the early sensory ERP components for small versus larger numerical values in the non-symbolic condition. In Experiment 2 exclusively large numerosities (8–30) were used, thereby rendering sensory differences negligible, and with this control in place the numerical distance effect in the non-symbolic condition mirrored the symbolic condition of Experiment 1.ConclusionCollectively, the results support the claim of an abstract semantic processing stage for numerical comparisons that is independent of input notation.

Previous studies have suggested that numerical processing relates to mathematical performance, but it seems that such relationship is more evident for intentional than for automatic numerical processing. In the present study we assessed the relationship between the two types of numerical processing and specific mathematical abilities in a sample of 109 children in grades 1–6. Participants were tested in an ample range of mathematical tests and also performed both a numerical and a size comparison task. The results showed that numerical processing related to mathematical performance only when inhibitory control was involved in the comparison tasks. Concretely, we found that intentional numerical processing, as indexed by the numerical distance effect in the numerical comparison task, was related to mathematical reasoning skills only when the task-irrelevant dimension (the physical size) was incongruent; whereas automatic numerical processing, indexed by the congruency effect in the size comparison task, was related to mathematical calculation skills only when digits were separated by small distance. The observed double dissociation highlights the relevance of both intentional and automatic numerical processing in mathematical skills, but when inhibitory control is also involved.

Numbers are omnipresent in our daily life. They are used to denote the date or time, the value of products, or to indicate the speed at which you drive your car. Indeed, numbers are part of our everyday life and as adults we use and manipulate them seemingly without any effort. But how do we do this? How is the meaning of numbers acquired and represented? And what brain mechanisms subserve the representation of number and mental operations involving numbers that guide our actions? The burgeoning field of “Numerical Cognition” seeks to provide answers to these kinds of questions. Researchers in this field seek to understand the representation and the neural correlates of basic numerical processes and how these basic processes relate to higher-level mental operations of these foundational representations, such as mental arithmetic. It should be clear that the relation between the most elementary numerical skill, such as the extraction of the numerosity of a set of elements, and solving mathematical equations is quite complex and rather difficult to investigate. In fact, the many faces of numbers further complicate the matter. A number is a highly abstract symbol that can represent different things. It can symbolize a magnitude (cardinality; “4” could indicate the number of children you have) or the rank (ordinality; “4” could also refer to your youngest child when placed in chronological order). The same number can also be written in many forms: 4 is equal to IV, to “Four,” and to ••••. Furthermore, in mathematics, numbers can also be positive or negative, small or large, natural, or decimal numbers. Numerical cognition research aims at understanding how these different conceptions of numbers relate to each other and how we develop an understanding their meaning. Over the last few decades these issues have been addressed by studying numerical skills in different animal species, human infants, and adults. However, the result of these years of focused research has not provided us with definite answers. Although some models of numerical cognition have dominated the research for many years, the basic representation of numbers remains debated until today. Even the mechanisms behind the most basic numerical skills, such as the comparison of two numbers, lack agreement in the research community. Indeed, the true nature of most of the commonly observed effects in simple numerical tasks, e.g., subitizing, the distance effect, the size congruity effect, or the SNARC-effect, remains obscure. On the other hand, the repeated observation of these effects in numerous studies indicates that they are genuine and that they could hold the key toward a proper understanding of number processing. It appears that more agreement is reached on the brain areas related to basic number processing. Many brain imaging experiments in humans and monkeys, and studies on patients with brain lesions have shown a strong involvement of the horizontal segment of the intraparietal sulcus (hIPS) in number processing. However, despite the apparent agreement of a central role of the hIPS in number processing, the specificity of this area for numbers and the exact role of the hIPS in a more elaborate number processing brain network is still under investigation. This Research Topic for Frontiers in Human Neuroscience covers a wide range of the remaining issues in numerical cognition. It discusses how numerical quantity is extracted from small sets of stimuli (Hyde, 2011), how the extraction of numerosity can be related to a system that extracts information from continuous dimensions (Henik et al., 2012) or to sensory-motor experiences (Ranzini et al., 2011). Furthermore, the causes of developmental dyscalculia are evaluated and the role played by symbolic (i.e., Arabic numerals) and non-symbolic (i.e., arrays of dots) in understanding this specific difficulty in mathematical processing is discussed (Noel and Rousselle, 2011). Research in the field of numerical cognition has grappled for to understand exactly how numerical and spatial processing are related. A number of contributions in this research topic shed further light on the association between number and space (Koten et al., 2011; Priftis et al., 2012; Van Dijck et al., 2012; Zorzi et al., 2012). Another question concerns the differences between automatic and intentional processing of number and the brain processes that mediate these different levels of number processing (Cohen Kadosh et al., 2012). Indeed, rather than confining number processing to a single area of the brain, a more elaborate network might be involved when a more elaborate number processing is needed, as in the processing of negative numbers (Blair et al., 2012), or doing mathematics (Grabner et al., 2011).

In the paper entitled ‘Asymptotic formula in a generalized Waring's problem’, I established an asymptotic formula for the number of representations of a large natural number N in the formwhere x1, x2, …, x7 and k are natural numbers with k ≥ 4 (see (2) Theorem 2).

There is often the need to update an installed intrusion detection system (IDS) due to new attack methods or upgraded computing environments. Since many current IDSs are constructed by manual encoding of expert knowledge, changes to IDSs are expensive and slow. We describe a data mining framework for adaptively building Intrusion Detection (ID) models. The central idea is to utilize auditing programs to extract an extensive set of features that describe each network connection or host session, and apply data mining programs to learn rules that accurately capture the behavior of intrusions and normal activities. These rules can then be used for misuse detection and anomaly detection. New detection models are incorporated into an existing IDS through a meta-learning (or co-operative learning) process, which produces a meta detection model that combines evidence from multiple models. We discuss the strengths of our data mining programs, namely, classification, meta-learning, association rules, and frequent episodes. We report on the results of applying these programs to the extensively gathered network audit data for the 1998 DARPA Intrusion Detection Evaluation Program.

Formulas are obtained for the number of representations of positive integers by quadratic forms .

Numerical distance effect in patients with schizophrenia

We develop the concept of minimum weight binary continued fraction representation of a rational number as an extension of minimum weight binary radix representation of an integer. The relation of these representations to the attainment of optimum efficiency in the shift and add or subtract model of binary computer arithmetic is discussed.

The substrata factor theory of reading explains that reading consists of multiple systems, including perceptual, linguistic, cognitive, motivational, affective, and physiological, that interact within the reader and between the reader and the text. An obvious implication is that any theory of reading that is based on only one of these systems must be inadequate. In this chapter it will be shown that this implication applies to Mattingly’s chapter’s linguistically based speculations on the process of reading and learning to read. Further, it will be shown that, in contrast, more adequate explanations have come from other linguists (Gleitman & Rozin, 1977) and from some psychologists (Gough, 1976; Adams & Collins, 1977). Furthermore, it will be shown how multiple systems within the reader must be drawn upon in learning to read by providing a brief description of instructional processes and strategies in teaching reading. Finally, two of Mattingly’s claims will be tested. First, although phonological processes may be bypassed when deaf subjects learn to read, hearing subjects only seem to bypass them as they develop automaticity in reading. Second, there seems to be no evidence to support the hypothesis that there is a critical age for learning to read. Our conclusion is that, although teachers may draw upon linguistic processes in teaching reading, to date there is no valid evidence that linguistic awareness is necessary for learning to read.KeywordsSkilled ReaderPhonological ProcessOral LanguageWord IdentificationReading InstructionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Topological properties of real number representations

ABSTRACTRecent research has suggested an important role for spatial attention in reading, however the nature of this influence is unclear. Further, it has been shown that intention to engage in linguistic processing can enhance lexical processing. Therefore, we developed a novel paradigm to examine the impact of spatial attention on primed and unprimed lexical processing. In a lexical decision task, targets were preceded by a cue to lexical access and spatial location simultaneously. Results indicated that on spatially valid trials, the lexical cuing effect was large and word processing was enhanced, suggesting that spatial attention influences lexical access. On spatially invalid trials, lexical cuing effects were significantly decreased, resulting in a spatial cuing by lexical cuing interaction. Results from this experiment provide evidence that spatial attention interacts with lexical access during word processing. This novel paradigm provides a new avenue for exploring the relations between spatial attention and reading processes.

Efficient representation of data is a fundamental prerequisite for addressing computational problems effectively using computers. The continual improvement in methods for representing numbers in computers serves as a critical step in expanding the scope and capabilities of computing systems. In this research, we conduct a comprehensive review of both fundamental and advanced techniques for representing numbers in computers. Additionally, we propose a novel model capable of representing rational numbers with absolute precision, catering to specific high precision applications. Specifically, we adopt fractional positional notation coupled with explicit codification of the periodic parts, thereby accommodating the entire rational number set without any loss of accuracy. We elucidate the properties and hardware representation of this proposed format and provide the results of extensive experiments to demonstrate its expressiveness and minimal codification error when compared to other real number representation formats. This research contributes to the advancement of numerical representation in computer systems, empowering them to handle complex computations with heightened accuracy, making them more reliable and versatile in a wide range of applications.