Representation Of Quantities Research Articles

Fundamental representation of the amount of substance fraction of carbon dioxide in nitrogen according to the dynamic volumetric principle

Abstract The new Austrian primary standard for the amount of substance fraction of carbon dioxide in nitrogen is presented. The primary standard is working according to the dynamic volumetric principle and is a fundamental representation of the measurement quantity. The direct traceability to the basic SI units is thereby thermodynamically realized without an artefact. A complete measurement uncertainty budget is calculated and verified with a bilateral inter-laboratory comparison.

Three-phase lag model for thermal conductivity of a thermo-viscoelastic porous medium

This paper investigates the propagation of waves and the thermal behavior in a thermo-viscoelastic porous medium using the three-phase-lag (TPL) model which accounts for phase lags in the heat flux vector, temperature gradient and thermal displacement gradient. The study aims to capture the interactions between thermal, mechanical, and structural properties of the thermo-viscoelastic porous, isotropic, homogeneous medium. The governing equations, incorporating the TPL heat conduction law and constitutive relations for the viscoelastic porous material, are derived and analyzed. The impact of relaxation times, porosity, and viscoelastic parameters on heat propagation is investigated through analytical solutions by normal modal analysis. Analytical representations of various physical quantities (stresses components, displacements, temperature,…) in the material's domain are derived. These expressions are then assessed numerically for a specific material, with the outcomes depicted graphically. A comparison is made between the predictions of the three-phase-lag (TPL), the dual-phase-lag (DPL) and Lord–Shulman (L–S) theories in the absence and presence of voids.

Profiles of preschoolers' numerical abilities across quantity representations.

Children's early numerical abilities shape their trajectories for math learning throughout schooling, and task quantity representation (e.g., nonsymbolic vs. symbolic) affects their reasoning about numerical concepts. The role of quantity representation in early numerical ability has typically been studied using variable-centered approaches. The present study builds on past work by using latent profile analysis as a person-centered approach to investigate heterogeneity in U.S. preschoolers' (N = 200, Mage = 4 years, 6 months) numerical abilities across nonsymbolic, verbal symbolic, and written symbolic representation types. The aim was to determine whether numerical ability indicators across the three representation types would result in empirically distinct ability profiles and whether preschool ability profile would predict variance on a standardized assessment of math achievement a year later, in kindergarten. We found evidence of four distinct preschool number ability profiles: (a) consistently low; (b) consistently high; (c) intermediate, with an advantage on nonsymbolic items; and (d) intermediate, with an advantage on verbal symbolic items. Although children in the consistently low and consistently high profiles performed reliably lower and higher, respectively, on the assessment of kindergarten math achievement, the two intermediate profiles performed similarly. The results reveal heterogeneity in preschool numerical ability across quantity representations and suggest that proficiency with either nonsymbolic or verbal representations may be a helpful foundation for building future math achievement. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Supplemental Material for Profiles of Preschoolers’ Numerical Abilities Across Quantity Representations

Supplemental Material for Profiles of Preschoolers’ Numerical Abilities Across Quantity Representations

Towards Better Quantity Representations for Solving Math Word Problems

Solving a math word problem requires selecting quantities in it and performing appropriate arithmetic operations to obtain the answer. For deep learning-based methods, it is vital to obtain good quantity representations, i.e., to selectively and emphatically aggregate information in the context of quantities. However, existing works have not paid much attention to this aspect. Many works simply encode quantities as ordinary tokens, or use some implicit or rule-based methods to select information in their context. This leads to poor results when dealing with linguistic variations and confounding quantities. This article proposes a novel method to identify question-related distinguishing features of quantities by contrasting their context with the question and the context of other quantities, thereby enhancing the representation of quantities. Our method not only considers the contrastive relationship between quantities but also considers multiple relationships jointly. Besides, we propose two auxiliary tasks to further guide the representation learning of quantities: (1) predicting whether a quantity is used in the question and (2) predicting the relations (operators) between quantities given the question. Experimental results show that our method outperforms previous methods on SVAMP and ASDiv-A under similar settings, even some newly released strong baselines. Supplementary experiments further confirm that our method indeed improves the performance of quantity selection by improving the representation of both quantities and questions.

Short-term number sense training recapitulates long-term neurodevelopmental changes from childhood to adolescence.

Number sense is fundamental to the development of numerical problem-solving skills. In early childhood, children establish associations between non-symbolic (e.g., a set of dots) and symbolic (e.g., Arabic numerals) representations of quantity. The developmental estrangement theory proposes that the relationship between non-symbolic and symbolic representations of quantity evolves with age, with increased dissociation across development. Consistent with this theory, recent research suggests that cross-format neural representational similarity (NRS) between non-symbolic and symbolic quantities is correlated with arithmetic fluency in children but not in adolescents. However, it is not known if short-term training (STT) can induce similar changes as long-term development. In this study, children aged 7-10 years underwent a theoretically motivated 4-week number sense training. Using multivariate neural pattern analysis, we investigated whether short-term learning could modify the relation between cross-format NRS and arithmetic skills. Our results revealed a significant correlation between cross-format NRS and arithmetic fluency in distributed brain regions, including the parietal and prefrontal cortices, prior to training. However, this association was no longer observed after training, and multivariate predictive models confirmed these findings. Our findings provide evidence that intensive STT during early childhood can promote behavioral improvements and neural plasticity that resemble and recapitulate long-term neurodevelopmental changes that occur from childhood to adolescence. More generally, our study contributes to our understanding of the malleability of number sense and highlights the potential for targeted interventions to shape neurodevelopmental trajectories in early childhood. RESEARCH HIGHLIGHTS: We tested the hypothesis that short-term number sense training induces the dissociation of symbolic numbers from non-symbolic representations of quantity in children. We leveraged a theoretically motivated intervention and multivariate pattern analysis to determine training-induced neurocognitive changes in the relation between number sense and arithmetic problem-solving skills. Neural representational similarity between non-symbolic and symbolic quantity representations was correlated with arithmetic skills before training but not after training. Short-term training recapitulates long-term neurodevelopmental changes associated with numerical problem-solving from childhood to adolescence.

A number sense as an emergent property of the manipulating brain

The ability to understand and manipulate numbers and quantities emerges during childhood, but the mechanism through which humans acquire and develop this ability is still poorly understood. We explore this question through a model, assuming that the learner is able to pick up and place small objects from, and to, locations of its choosing, and will spontaneously engage in such undirected manipulation. We further assume that the learner’s visual system will monitor the changing arrangements of objects in the scene and will learn to predict the effects of each action by comparing perception with a supervisory signal from the motor system. We model perception using standard deep networks for feature extraction and classification. Our main finding is that, from learning the task of action prediction, an unexpected image representation emerges exhibiting regularities that foreshadow the perception and representation of numbers and quantity. These include distinct categories for zero and the first few natural numbers, a strict ordering of the numbers, and a one-dimensional signal that correlates with numerical quantity. As a result, our model acquires the ability to estimate numerosity, i.e. the number of objects in the scene, as well as subitization, i.e. the ability to recognize at a glance the exact number of objects in small scenes. Remarkably, subitization and numerosity estimation extrapolate to scenes containing many objects, far beyond the three objects used during training. We conclude that important aspects of a facility with numbers and quantities may be learned with supervision from a simple pre-training task. Our observations suggest that cross-modal learning is a powerful learning mechanism that may be harnessed in artificial intelligence.

Human brain representations of internally generated outcomes of approximate calculation revealed by ultra-high-field brain imaging

Much of human culture’s advanced technology owes its existence to the ability to mentally manipulate quantities. Neuroscience has described the brain regions overall recruited by numerical tasks and the neuronal codes representing individual quantities during perceptual tasks. Nevertheless, it remains unknown how quantity representations are combined or transformed during mental computations and how specific quantities are coded in the brain when generated as the result of internal computations rather than evoked by a stimulus. Here, we imaged the brains of adult human subjects at 7 Tesla during an approximate calculation task designed to disentangle in- and outputs of the computation from the operation itself. While physically presented sample numerosities were distinguished in activity patterns along the dorsal visual pathway and within frontal and occipito-temporal regions, a representation of the internally generated result was most prominently detected in higher order regions such as angular gyrus and lateral prefrontal cortex. Behavioral precision in the task was related to cross-decoding performance between sample and result representations in medial IPS regions. This suggests the transformation of sample into result may be carried out within dorsal stream sensory-motor integration regions, and resulting outputs maintained for task purposes in higher-level regions in a format possibly detached from sensory-evoked inputs.

Cortical quantity representations of visual numerosity and timing overlap increasingly into superior cortices but remain distinct

Many sensory brain areas are organized as topographic maps where neural response preferences change gradually across the cortical surface. Within association cortices, 7-Tesla fMRI and neural model-based analyses have also revealed many topographic maps for quantities like numerosity and event timing, often in similar locations. Numerical and temporal quantity estimations also show behavioral similarities and even interactions. For example, the duration of high-numerosity displays is perceived as longer than that of low-numerosity displays. Such interactions are often ascribed to a generalized magnitude system with shared neural responses across quantities. Anterior quantity responses are more closely linked to behavior. Here, we investigate whether common quantity representations hierarchically emerge by asking whether numerosity and timing maps become increasingly closely related in their overlap, response preferences, and topography. While the earliest quantity maps do not overlap, more superior maps overlap increasingly. In these overlapping areas, some intraparietal maps have consistently correlated numerosity and timing preferences, and some maps have consistent angles between the topographic progressions of numerosity and timing preferences. However, neither of these relationships increases hierarchically like the amount of overlap does. Therefore, responses to different quantities are initially derived separately, then progressively brought together, without generally becoming a common representation. Bringing together distinct responses to different quantities may underlie behavioral interactions and allow shared access to comparison and action planning systems.

Проблема разработки методов измерения точности репрезентации количества на всем протяжении школьного обучения

The relevance and social demand for developing methods for measuring the accuracy of representation of quantitative information are associated with the need to analyze specific cognitive functions that underlie individual differences in speed and quality of learning in school. Expressed in symbolic, non-symbolic and mixed formats, representation of quantity is one of the most important cognitive functions that determines the success of learning mathematics. The article presents the results of the development and adaptation of three tests – “Number Sense”, “Number Line” and “Dot Number Task” – that measure the accuracy of the representation of quantitative information presented as sets of objects, numbers, and their combinations. The total number of study participants involved in the process of test adaptation amounted to 1,751 students in grades 1–11 aged from 6.8 to 18.8 years, of which 48.8% were girls. For each test, an internal consistency analysis was carried out, descriptive statistics were calculated, and the distribution of indicators of quantity representation accuracy at various levels of general education was analyzed. The results of the analysis showed satisfactory psychometric characteristics of computerized tests, which indicates their reliability and makes them suitable for application at the primary, basic and full levels of general education.

Poster Session I: Criterion effects in maximum likelihood difference scaling: Similar is not always the opposite of different.

Maximum Likelihood Difference Scaling (MLDS) is an efficient method of estimating perceptual representations of suprathreshold physical quantities (Maloney & Yang, 2003), such as luminance contrast. In MLDS, observers can be instructed to judge which of two stimulus pairs are more similar to one another, or which of the two pairs are more different from one another. If the same physical attributes are used for both the similar and dissimilar tasks, the two criteria should produce the same perceptual scales. We estimated perceptual scales for suprathreshold achromatic square patches. Increments and decrements on the mid-gray background were estimated separately. Observers judged which pair of stimuli were more similar in half of the sessions, and more different in the other half sessions. For most observers, the two tasks produced the same perceptual scales: a decelerating curve for increment contrasts and a cubic curve for decremental contrasts (cf. Whittle, 1992). These scales predicted forced-choice contrast discrimination thresholds for both increments and decrements. However, for a subset of observers, the 'more different' judgments produced scales that accelerated with contrast for both increments and decrements; these scale shapes do not predict their discrimination thresholds. Our results suggest that, even with these simple stimuli, observers in an MLDS experiment may attend to different aspects of the stimulus depending on the assigned task.

Representation of physical quantities: From scalars, vectors, tensors and spinors to multivectors

Mathematical representations of physical variables and operators are of primary importance in developing a theory - the relationship among different relevant quantities of any physical process. A thorough account of the representations of different classes of physical variables is drawn up with a brief discussion of various related mathematical systems including quaternion and spinor. The present study is intended to facilitate a comprehensive introduction to the 'geometric algebra', which provides an immensely productive unification of these systems and promises more.

Plane Wave Reflection in Nonlocal Semiconducting Rotating Media with Extended Model of Three-Phase-Lag Memory-Dependent Derivative

The present paper examines the plane wave’s reflection in a semiconducting magneto-thermoelastic rotating nonlocal half-space medium, which is stress-free, thermally insulated, and with a diffusion transport process at the boundaries. The novel mathematical model of extended three-phase-lag (TPL) heat transfer law with memory-dependent derivative (MDD) and two temperatures (2T) is used to model this problem. The investigated 2D model shows that a longitudinal wave, when striking the surface z = 0, produces four reflected waves. The characteristics of the plane wave such as phase velocity, amplitude ratios, penetration depth, attenuation-coefficients-specific loss, and energy ratios of various reflected waves are obtained. The symmetric and asymmetric tensor representations of all physical quantities are used. The effects of various theories of thermoelasticity, two temperatures, and rotation on wave characteristics are illustrated graphically using MATLAB software.

Representation of the RG-Invariant Quantities in Perturbative QCD through Powers of the Conformal Anomaly

In this work we consider the possibility of representing the perturbative series for renormalization group invariant quantities in QCD in the form of their decomposition in powers of the conformal anomaly {{beta ({{alpha }_{s}})} mathord{left/ {vphantom {{beta ({{alpha }_{s}})} {{{alpha }_{s}}}}} right. kern-0em} {{{alpha }_{s}}}} in the overline {{text{MS}}} -scheme. We remind that such expansion is possible for the Adler function of the process of {{e}^{ + }}{{e}^{ - }} annihilation into hadrons and the coefficient function of the Bjorken polarized sum rule for the deep-inelastic electron-nucleon scattering, which are both related by the CBK relation. In addition, we study the discussed decomposition for the static quark-antiquark Coulomb-like potential, its relation with the quantity defined by the cusp anomalous dimension and the coefficient function of the Bjorken unpolarized sum rule of neutrino-nucleon scattering. In conclusion we also present the formal results of applying this approach to the non-renormalization invariant ratio between the pole and overline {{text{MS}}} -scheme running mass of heavy quark in QCD and compare them with those already known in the literature. The arguments in favor of the validity of the considered representation in powers of {{beta ({{alpha }_{s}})} mathord{left/ {vphantom {{beta ({{alpha }_{s}})} {{{alpha }_{s}}}}} right. kern-0em} {{{alpha }_{s}}}} for all mentioned renorm-invariant perturbative quantities are discussed.

The componential nature of arithmetical cognition: some important questions.

Research on typically developing children and adults and people with developmental and acquired dyscalculia converges in indicating that arithmetical ability is not unitary but is made up of many different components. Categories of components include non-symbolic quantity representation and processing; symbolic quantity representation and processing; counting procedures and principles; arithmetic operations; arithmetical knowledge and understanding; multiple forms and applications of conceptual knowledge of arithmetic; and domain-general abilities such as attention, executive functions and working memory. There is much evidence that different components can and often do show considerable functional independence, not only in developmental and acquired dyscalculia, but in typically achieving children and adults. At the same time, it is possible to find complex interactions and bidirectional relationships between the different components, including between domain-specific and apparently domain-general abilities. There is a great deal that still needs to be discovered. In particular, we need to learn more about the origins in infancy of subitizing and approximate magnitude comparison, the extent to which these interact, the extent to which they may be further divisible, and the extent and ways in which they themselves may develop with age and the extent to which they may influence later-developing components. There also needs to be a lot more research on exactly how domain-general and domain-specific abilities contribute to mathematical development, and how they interact with one another.

Modular Representation of Rational Fractions

The computer is designed to perform calculations, therefore it is a device for automatically processing machine data formats that display structured information. In machine data formats, the digit grid of typical and specialized computers displays numerical values on which calculations are performed in various number systems. A new modular packed representation of numerical values by rational fractions with fixed denominators, previously unknown, is introduced. For such modular representations of numerical quantities, a basic algorithm for the formation of vector components of types of representations is given with an estimate of complexity. A method of optimal approximation by packed modular representations of arbitrary irregular fractions has been found. The features of algorithms for performing additive and multiplicative operations for packed representations in a modular processor are analyzed.

CFD Uncertainty Quantification using PCE–HDMR: Exemplary Application to a Buoyancy-Driven Mixing Process

For the investigation of uncertainties in high dimensional spaces of computationally expensive engineering applications, reliable Uncertainty Quantification (UQ) methods are needed. These methods should provide accurate and efficient High-Dimensional Model Representations of stochastic results using a reasonable number of calculations. Therefore, the PCE–HDMR approach (Polynomial Chaos Expansion–High-Dimensional Model Representation) is utilized to qualify appropriate UQ methods for large-scale computations in the field of Computational Fluid Dynamics. This technique is a combination of Cut-HDMR, a hierarchical decomposition modeling approach, with PCE. To demonstrate its effectiveness, the PCE–HDMR methodology in conjunction with complementary modeling techniques is applied for the UQ analysis of a buoyancy-driven mixing process between two miscible fluids within the Differentially Heated Cavity of aspect ratio 4. The results include a thorough probabilistic representation of time-dependent response quantities that comprehensively describe the mixing process. The stochastic models are derived from Large Eddy Simulations using PCE–HDMR and the Sparse Grid Method, which serves as a reference for the results from PCE–HDMR. The results show that PCE–HDMR provides accurate statistics of the modeled time-dependent stochastic processes and shows good agreement with the reference results. Thus, PCE–HDMR indicates great potential for UQ of technical-scale computations due to its efficiency and flexibility in the construction of stochastic models.

Prevalence of Number, Number Relations, and Number Operations Indicators in State Early Learning Standards

AbstractState Early Learning Standards (ELS) are multi-function tools that inform early childhood instruction and practices. Using an established framework of early numerical development, this study assessed the prevalence of number, number relations, and number operations indicators in ELS, specifically indicators of counting, numeral knowledge, cardinality, relations among quantity and number, and operations. The type of quantity representation and the set sizes, or upper limits, assigned to counting, subitizing, and cardinality were also summarized. State ELS were organized into two groups: Group 1 (n = 23) included states with one set of indicators for preschool (e.g., ages three to five years) and Group 2 (n = 27) included states with at least two sets of indicators for preschool (e.g., 36 to 48 months and 48 to 60 months). Key findings include: (1) how states organize their ELS is associated with level of consensus for the upper limits and set sizes associated with counting and cardinality; (2) notable variability in the prevalence of early number indicators with gaps for indicators of advanced counting, cardinal principle knowledge, symbolic number relations, and ordering; and, (3) differences in how ELS address the type of quantity representation. Results are intended to highlight strengths and shortcomings of state ELS for number, number relations, and number operations and to offer considerations for future revisions.

Learning-induced reorganization of number neurons and emergence of numerical representations in a biologically inspired neural network

Number sense, the ability to decipher quantity, forms the foundation for mathematical cognition. How number sense emerges with learning is, however, not known. Here we use a biologically-inspired neural architecture comprising cortical layers V1, V2, V3, and intraparietal sulcus (IPS) to investigate how neural representations change with numerosity training. Learning dramatically reorganized neuronal tuning properties at both the single unit and population levels, resulting in the emergence of sharply-tuned representations of numerosity in the IPS layer. Ablation analysis revealed that spontaneous number neurons observed prior to learning were not critical to formation of number representations post-learning. Crucially, multidimensional scaling of population responses revealed the emergence of absolute and relative magnitude representations of quantity, including mid-point anchoring. These learnt representations may underlie changes from logarithmic to cyclic and linear mental number lines that are characteristic of number sense development in humans. Our findings elucidate mechanisms by which learning builds novel representations supporting number sense.

Non-linear adaptive control inspired by neuromuscular systems

Current paradigms for neuromorphic computing focus on internal computing mechanisms, for instance using spiking-neuron models. In this study, we propose to exploit what is known about neuro-mechanical control, exploiting the mechanisms of neural ensembles and recruitment, combined with the use of second-order overdamped impulse responses corresponding to the mechanical twitches of muscle-fiber groups. Such systems may be used for controlling any analog process, by realizing three aspects: Timing, output quantity representation and wave-shape approximation. We present an electronic based model implementing a single motor unit for twitch generation. Such units can be used to construct random ensembles, separately for an agonist and antagonist ‘muscle’. Adaptivity is realized by assuming a multi-state memristive system for determining time constants in the circuit. Using SPICE-based simulations, several control tasks were implemented which involved timing, amplitude and wave shape: The inverted pendulum task, the ‘whack-a-mole’ task and a handwriting simulation. The proposed model can be used for both electric-to-electronic as well as electric-to-mechanical tasks. In particular, the ensemble-based approach and local adaptivity may be of use in future multi-fiber polymer or multi-actuator pneumatic artificial muscles, allowing for robust control under varying conditions and fatigue, as is the case in biological muscles.