Tile Number and Space-Efficient Knot Mosaics

In this paper we introduce the concept of a space-efficient knot mosaic. That is, we seek to determine how to create knot mosaics using the least number of non-blank tiles necessary to depict the knot. This least number is called the tile number of the knot. We determine strict bounds for the tile number of a knot in terms of the mosaic number of the knot. In particular, if $t$ is the tile number of a prime knot with mosaic number $m$, then $5m-8 \leq t \leq m^2-4$ if $m$ is even and $5m-8 \leq t \leq m^2-8$ if $m$ is odd. We also determine the tile number of several knots and provide space-efficient knot mosaics for each of them.Comment: The original version of this article was split into two articles during refereein

Spontaneous Mapping of Number and Space in Adults and Young Children

Mature representations of space and number are connected to one another in ways suggestive of a ‘mental number line,’ but this mapping could either be a cultural construction or a reflection of a more fundamental link between the domains of number and geometry. Using a manual bisection paradigm, we tested for number line representations in adults, young school children, and preschool children. Non-symbolic numerical displays systematically distorted localization of the midpoint of a horizontal line at all three ages. Numerical and spatial representations therefore are linked prior to the onset of formal instruction, in a manner that suggests a privileged relation between spatial and numerical cognition.Psycholog

Broad-band X-ray analysis of local mid-infrared selected Compton-thick AGN candidates

The estimate of the number and space density of obscured AGN over cosmic time still represents an open issue. While the obscured AGN population is a key ingredient of the X-ray background synthesis models and is needed to reproduce its shape, a complete census of obscured AGN is still missing. Here we test the selection of obscured sources among the local 12-micron sample of Seyfert galaxies. Our selection is based on a difference up to three orders of magnitude in the ratio between the AGN bolometric luminosity, derived from the spectral energy distribution (SED) decomposition, and the same quantity obtained by the published XMM-Newton 2-10 keV luminosity. The selected sources are UGC05101, NGC1194 and NGC3079 for which the available X-ray wide bandpass, from Chandra and XMM-Newton plus NuSTAR data, extending to energies up to ~30-45 keV, allows us an accurate determination of the column density, and hence of the true intrinsic power. The newly derived NH values clearly indicate heavy obscuration (about 1.2, 2.1 and 2.4 x10^{24} cm-2 for UGC05101, NGC1194 and NGC3079, respectively) and are consistent with the prominent silicate absorption feature observed in the Spitzer-IRS spectra of these sources (at 9.7 micron rest frame). We finally checked that the resulting X-ray luminosities in the 2-10 keV band are in good agreement with those derived from the mid-IR band through empirical L_MIR-L_X relations.Comment: 14 pages, 6 figures, accepted for publication in MNRA

An experimental investigation of the spatial-numerical association in preschool children

openThis thesis presents an experimental investigation aimed at exploring the association between number and space in preschool children. The study employs computerized tests within a number line bisection task to assess the understanding of spatial numerical association

Development of a Possible General Magnitude System for Number and Space

There is strong evidence for a link between numerical and spatial processing. However, whether this association is based on a common general magnitude system is far from conclusive and the impact of development is not yet known. Hence, the present study aimed to investigate the association between discrete non-symbolic number processing (comparison of dot arrays) and continuous spatial processing (comparison of angle sizes) in children between the third and sixth grade (N = 367). Present findings suggest that the processing of comparisons of number of dots or angle are related to each other, but with angle processing developing earlier and being more easily comparable than discrete number representations for children of this age range. Accordingly, results favor the existence of a more complex underlying magnitude system consisting of dissociated but closely interacting representations for continuous and discrete magnitudes

Space-Efficient Prime Knot 7-Mosaics

The concepts of tile number and space-efficiency for knot mosaics were first explored by Heap and Knowles (arXiv:1702.06462), where they determined the possible tile numbers and space-efficient layouts for every prime knot with mosaic number 6 or less. In this paper, we extend those results to prime knots with mosaic number 7

Variability in the Alignment of Number and Space Across Languages and Tasks

While the domains of space and number appear to be linked in human brains and minds, their conceptualization still differs across languages and cultures. For instance, frames of reference for spatial descriptions vary according to task, context, and cultural background, and the features of the mental number line depend on formal education and writing direction. To shed more light on the influence of culture/language and task on such conceptualizations, we conducted a large-scale survey with speakers of five languages that differ in writing systems, preferences for spatial and temporal representations, and/or composition of number words. Here, we report data obtained from tasks on ordered arrangements, including numbers, letters, and written text. Comparing these data across tasks, domains, and languages indicates that, even within a single domain, representations may differ depending on task characteristics, and that the degree of cross-domain alignment varies with domains and culture

Registered Replication Report on Fischer, Castel, Dodd, and Pratt (2003)

The attentional spatial-numerical association of response codes (Att-SNARC) effect (Fischer, Castel, Dodd, & Pratt, 2003)—the finding that participants are quicker to detect left-side targets when the targets are preceded by small numbers and quicker to detect right-side targets when they are preceded by large numbers—has been used as evidence for embodied number representations and to support strong claims about the link between number and space (e.g., a mental number line). We attempted to replicate Experiment 2 of Fischer et al. by collecting data from 1,105 participants at 17 labs. Across all 1,105 participants and four interstimulus-interval conditions, the proportion of times the effect we observed was positive (i.e., directionally consistent with the original effect) was 50. Further, the effects we observed both within and across labs were minuscule and incompatible with those observed by Fischer et al. Given this, we conclude that we failed to replicate the effect reported by Fischer et al. In addition, our analysis of several participant-level moderators (finger-counting habits, reading and writing direction, handedness, and mathematics fluency and mathematics anxiety) revealed no substantial moderating effects. Our results indicate that the Att-SNARC effect cannot be used as evidence to support strong claims about the link between number and space

Registered Replication Report on Fischer, Castel, Dodd, and Pratt (2003)

The attentional spatial-numerical association of response codes (Att-SNARC) effect (Fischer, Castel, Dodd, & Pratt, 2003)—the finding that participants are quicker to detect left-side targets when the targets are preceded by small numbers and quicker to detect right-side targets when they are preceded by large numbers—has been used as evidence for embodied number representations and to support strong claims about the link between number and space (e.g., a mental number line). We attempted to replicate Experiment 2 of Fischer et al. by collecting data from 1,105 participants at 17 labs. Across all 1,105 participants and four interstimulus-interval conditions, the proportion of times the effect we observed was positive (i.e., directionally consistent with the original effect) was .50. Further, the effects we observed both within and across labs were minuscule and incompatible with those observed by Fischer et al. Given this, we conclude that we failed to replicate the effect reported by Fischer et al. In addition, our analysis of several participant-level moderators (finger-counting habits, reading and writing direction, handedness, and mathematics fluency and mathematics anxiety) revealed no substantial moderating effects. Our results indicate that the Att-SNARC effect cannot be used as evidence to support strong claims about the link between number and space