Number Of Representations Research Articles

Differential processing of "small" and "large" multidigit numbers.

Little is known about the mental representation of large multidigit numbers that are usually beyond our personal experience. The present study explored the processing mechanisms of these numbers in a series of experiments, using the numerical comparison task. Experiment 1 included within and between-scale comparisons of multidigit numbers varying in their left digits (e.g., 8,000,000), with one group comparing small numbers (tens, hundreds, and thousands) and the other large ones (millions, billions, and trillions). In Experiment 2, comparisons of small (tens, hundreds) and large (millions, billions) multidigit numbers that varied in their left and right digits (e.g., 8,000,003) were presented in separate blocks. Experiment 3 presented small and large multidigit numbers (from tens to trillions) that varied in their left digits in the same block. We found novel compatibility effects between the left digit and scale components, as well as between the left digit, right digit, and scale components, and extended the previously reported unit-decade compatibility effect to larger scales. We also obtained global and scale distance effects for all scales in most conditions. Both compatibility and distance effects showed context dependency in large, but not small, multidigit numbers. Overall, these results demonstrate that small and large multidigit numbers are processed differently. We discuss these differences and propose a processing model that accounts for them. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Precision of bit slicing with in-memory computing based on analog phase-change memory crossbars

Abstract In-memory computing is a promising non-von Neumann approach to perform certain computational tasks efficiently within memory devices by exploiting their physical attributes. However, the computational accuracy achieved with this approach has been rather low, owing to significant inter-device variability and inhomogeneity across an array as well as intra-device variability and randomness from the analog memory devices. Bit slicing, a technique for constructing a high precision processor from several modules of lower precision, is a promising approach for overcoming this accuracy limitation. However, a systematic study to assess the precision ultimately achieved by bit slicing with analog in-memory computing has so far been lacking. In this work, we assess the computational error from bit slicing when performing in-memory matrix-vector multiplications. Using accurate models of phase-change memory crossbar arrays, we demonstrate that unlike in digital processors where bit slicing is used to extend the dynamic range of the number representation, bit slicing with in-memory computing should aim at minimizing the error from the analog matrix representation through averaging within a given dynamic range. The results are validated using a prototype phase-change memory chip and the impact on the neural network inference accuracy on CIFAR-10 and ImageNet benchmarks is evaluated.

Epistemic and Mediational Suitability of Tasks Designed in a Mobile Learning Context

The aim of this research is to propose indicators of epistemic and mediational suitability to assess tasks designed in a mobile learning context, using theoretical and methodological elements of the onto-semiotic approach (OSA) and the SAMR model (substitution, augmentation, modification, and redefinition). With these indicators, 110 tasks designed by two Mexican teachers are evaluated, for the teaching of the natural number in primary education. Some results indicate that the situation-problems addressed the meanings of numerical sequence, operational, ordinal, and measure, and that 56% of the tasks were located at the substitution level. Some implications derived from the research are that the mathematical content can be complemented by including other representations of the natural number and that the design of tasks that are positioned at the modification and redefinition levels should be encouraged. The foregoing serves as a basis for the design and assessment of mobile content for the natural number concept.

Exact Number Concepts Are Limited to the Verbal Count Range

Previous findings suggest that mentally representing exact numbers larger than four depends on a verbal count routine (e.g., “one, two, three .”). However, these findings are controversial because they rely on comparisons across radically different languages and cultures. We tested the role of language in number concepts within a single population—the Tsimane’ of Bolivia—in which knowledge of number words varies across individual adults. We used a novel data-analysis model to quantify the point at which participants (N = 30) switched from exact to approximate number representations during a simple numerical matching task. The results show that these behavioral switch points were bounded by participants’ verbal count ranges; their representations of exact cardinalities were limited to the number words they knew. Beyond that range, they resorted to numerical approximation. These results resolve competing accounts of previous findings and provide unambiguous evidence that large exact number concepts are enabled by language.

The defect technique for partially absorbing and reflecting boundaries: Application to the Ornstein–Uhlenbeck process

A particle roaming in a potential in the presence of spatial heterogeneities such as traps or physical barriers is a mathematical representation of a vast number of processes across multiple disciplines, ranging from the dynamics of chemical reactions and the movement of foraging animals to the performance of a financial asset. Among the approaches to tackle such problems, a powerful one is the so-called defect technique. Within a continuous space–time formulation, the technique has been successfully used to analyze diffusion problems with partially absorbing boundaries, but has not been employed to study other types of boundaries. Here we exploit the continuity equation to extend the defect technique to when partially reflecting barriers are present. For each boundary type, we apply the defect formalism to the Ornstein–Uhlenbeck process, recovering known analytical results and presenting new ones.

Fuzzy Concept Evaluation Based on Prospect Theory and Heterogeneous Evaluation Information

Abstract Concept evaluation is the core stage of new product development and has a significant impact on the downstream process of product development. Because of the uncertainty and ambiguity of early design information, the concept evaluation process not only relies on the semantic terms of decision makers (DMs) but also includes uncertain criteria values (such as crisp numbers and interval numbers). In addition, DMs will have psychological preference errors when evaluating concepts owing to the risks taken in the evaluation. To address these drawbacks, a fuzzy concept evaluation approach based on prospect theory and heterogeneous evaluation information is proposed. Initially, based on the definition of intuitionistic fuzzy numbers (IFNs), a numerical model is developed to unify the representation of crisp numbers, interval numbers, and fuzzy numbers, and a normalized decision matrix is constructed. Second, a weight distribution method of DMs is proposed, which introduces hesitation and similarity, and the weighted intuitionistic fuzzy evaluation information is transformed into interval IFNs. Third, the weight of the evaluation criteria is determined using the decision-making trial and evaluation laboratory model (DEMATEL), and an evaluation information correction model based on the prospect theory is established to select the optimal concepts. Finally, the feasibility of the proposed approach is demonstrated using a bar-peeling machine, and a comparison and sensitivity analysis is conducted to verify the robustness of the decision results.

An Efficient Parallel Reverse Conversion of Residue Code to Mixed-Radix Representation Based on the Chinese Remainder Theorem.

In this paper, we deal with the critical problems in residue arithmetic. The reverse conversion from a Residue Number System (RNS) to positional notation is a main non-modular operation, and it constitutes a basis of other non-modular procedures used to implement various computational algorithms. We present a novel approach to the parallel reverse conversion from the residue code into a weighted number representation in the Mixed-Radix System (MRS). In our proposed method, the calculation of mixed-radix digits reduces to a parallel summation of the small word-length residues in the independent modular channels corresponding to the primary RNS moduli. The computational complexity of the developed method concerning both required modular addition operations and one-input lookup tables is estimated as , where k equals the number of used moduli. The time complexity is modular clock cycles. In pipeline mode, the throughput rate of the proposed algorithm is one reverse conversion in one modular clock cycle.

Replication study challenges and new number formats for chaotic pseudo random number generators

AbstractChaotic Pseudo Random Number Generators have been seen as a promising candidate for secure random number generation. Using the logistic map as state transition function, we perform number generation experiments that illustrate the challenges when trying to do a replication study. Those challenges range from uncertainties about the rounding mode in arithmetic hardware over chosen number representations for variables to compiler or programmer decisions on evaluation order for arithmetic expressions. We find that different decisions lead to different streams with different security properties, where we focus on period length. However, descriptions in articles often are not detailed enough to deduce all decisions unambiguously. To address similar problems in other replication studies for security applications, we propose recommendations for descriptions of numerical experiments on security applications to avoid the above challenges. Moreover, we use the results to propose the use of higher-radix and mixed-radix representations to trade storage size for period length, and investigate if exploiting the symmetry of the logistic map function for number representation is advantageous.

Convergence and gradient algorithm of a class of neural networks based on the polygonal fuzzy numbers representation

As a special case of general fuzzy numbers, the polygonal fuzzy number can describe a fuzzy object by means of an ordered representation of finite real numbers. Different from general fuzzy numbers, the polygonal fuzzy numbers overcome the shortcoming of complex operations based on Zadeh’s traditional expansion principle, and can maintain the closeness of arithmetic operation. Hence, it is feasible to use a polygonal fuzzy number to approximate a general fuzzy number. First, an extension theorem of continuous functions on a real compact set is given according to open set construction theorem. Then using Weierstrass approximation theorem and ordered representation of the polygonal fuzzy numbers, the convergence of a single hidden layer feedforward polygonal fuzzy neural network is proved. Secondly, the gradient vector of the approximation error function and the optimization parameter vector of the network are given by using the ordered representation of polygonal fuzzy numbers, and then the gradient descent algorithm is used to train the optimal parameters of the polygonal fuzzy neural network iteratively. Finally, two simulation examples are given to verify the approximation ability of the network. Simulation result shows that the proposed network and the gradient descent algorithm are effective, and the single hidden layer feedforward network have good abilities in learning and generalization.

Multicomponent hybrid numbers: On algebraic properties and matrix representations of hybrid-hyperbolic numbers

In this study, the hybrid-hyperbolic numbers are introduced. This number system is a more general form of the hybrid number system, which is an interesting number system, as well as a number system that includes multicomponent number systems (i.e., complex-hyperbolic, dual-hyperbolic and bihyperbolic numbers). In this paper, we give algebraic properties of hybrid-hyperbolic numbers. In addition, 2 × 2 and 4 × 4 hyperbolic matrix representations of hybrid-hyperbolic numbers are given and some properties of them are examined.

Extending ideas of numerical order beyond the count-list from kindergarten to first grade

Ordinal processing plays a fundamental role in both the representation and manipulation of symbolic numbers. As such, it is important to understand how children come to develop a sense of ordinality in the first place. The current study examines the role of the count-list in the development of ordinal knowledge through the investigation of two research questions: (1) Do K-1 children struggle to extend the notion of numerical order beyond the count-list, and if so (2) does this extension develop incrementally or manifest as a qualitative re-organization of how children recognize the ordinality of numerical sequences. Overall, we observed that although young children reliably identified adjacent ordered sequences (i.e., those that match the count-list; ‘2-3-4') as being in the correct ascending order, they performed significantly below chance on non-adjacent ordered trials (i.e., those that do not match the count-list but are in the correct order; ‘2-4-6’) from the beginning of kindergarten to the end of first grade. Further, both qualitative and quantitative analyses supported the conclusion that the ability to extend notions of ordinality beyond the count-list emerged as a conceptual shift in ordinal understanding rather than through incremental improvements. These findings are the first to suggest that the ability to extend notions of ordinality beyond the count-list to include non-adjacent numbers is non-trivial and reflects a significant developmental hurdle that most children must overcome in order to develop a mature sense of ordinality.

A novel normalization algorithm to facilitate pre-assessment of Covid-19 disease by improving accuracy of CNN and its FPGA implementation

COVID-19 is still a fatal disease, which has threatened all people by affecting the human lungs. Chest X-Ray or computed tomography imaging is commonly used to make a fast and reliable medical investigation to detect the COVID-19 virus. These medical images are remarkably challenging because it is a full-time job and prone to human errors. In this paper, a new normalization algorithm that consists of Mean–Variance-Softmax-Rescale (MVSR) processes respectively is proposed to provide facilitation pre-assessment and diagnosis Covid-19 disease. In order to show the effect of MVSR normalization technique, the algorithm of proposed method is applied to chest X-ray and Sars-Cov-2 computed tomography images dataset. The normalized X-ray images with MVSR are used to recognize Covid-19 virus via Convolutional Neural Network (CNN) model. At the implementation stage, the MVSR algorithm is executed on MATLAB environment, then all the arithmetic operations of the MVSR normalization are coded in VHDL with the help of fixed-point fractional number representation format on FPGA platform. The experimental platform consists of Zynq-7000 Development FPGA Board and VGA monitor to display the both original and MVSR normalized chest X-ray images. The CNN model is constructed and executed using Anaconda Navigator interface with python language. Based on the results of this study, infections of Covid-19 disease can be easily diagnosed with MVSR normalization technique. The proposed MVSR normalization technique increased the classification accuracy of the CNN model from 83.01, to 96.16% for binary class of chest X-ray images.

VLSI IMPLEMENTATION OF HIGH SPEED SINGLE PRECESSION FLOATING POINT UNIT USING VERILOG

Single-precision floating-point format is a computer number format that is used to represent a wide dynamic range of values. Floating point numbers representation has widespread dominance over fixed point numbers. Since the recent years, researchers are putting a lot of efforts in interfacing complex modules which are used in signal processing with processors for increasing the speed. In this work implementation of a floating point arithmetic unit which can perform addition, subtraction, multiplication, and division functions on 32-bit operands that use the IEEE 754-2008 standard is done using Verilog. The FPU of this work is a single precision IEEE754 compliant integrated unit. Pre-normalization of operands is employed for addition and subtraction, multiplication using bit pair recoding and division using non restoring division. It can handle not only basic floating point operations like addition, subtraction, multiplication and division but can also handle operations like transcendental functions like sine, cosine and tangential function. The logical method for Addition and Subtraction operation is expanded in order to decrease the no. of gates used.

Get in touch with numbers \u2013 an approximate number comparison task in the haptic modality

The Approximate Number System (ANS) is conceptualized as an innate cognitive system that allows humans to perceive numbers of objects or events (>4) in a fuzzy, imprecise manner. The representation of numbers is assumed to be abstract and not bound to a particular sense. In the present study, we test the assumption of a shared cross-sensory system. We investigated approximate number processing in the haptic modality and compared performance to that of the visual modality. We used a dot comparison task (DCT), in which participants compare two dot arrays and decide which one contains more dots. In the haptic DCT, 67 participants had to compare two simultaneously presented dot arrays with the palms of their hands; in the visual DCT, participants inspected and compared dot arrays on a screen. Tested ratios ranged from 2.0 (larger/smaller number) to 1.1. As expected, in both the haptic and the visual DCT responses similarly depended on the ratio of the numbers of dots in the two arrays. However, on an individual level, we found evidence against medium or stronger positive correlations between “ANS acuity” in the visual and haptic DCTs. A regression model furthermore revealed that besides number, spacing-related features of dot patterns (e.g., the pattern’s convex hull) contribute to the percept of numerosity in both modalities. Our results contradict the strong theory of the ANS solely processing number and being independent of a modality. According to our regression and response prediction model, our results rather point towards a modality-specific integration of number and number-related features.

Boltzmann Configurational Entropy Revisited in the Framework of Generalized Statistical Mechanics

As known, a method to introduce non-conventional statistics may be realized by modifying the number of possible combinations to put particles in a collection of single-particle states. In this paper, we assume that the weight factor of the possible configurations of a system of interacting particles can be obtained by generalizing opportunely the combinatorics, according to a certain analytical function of the actual number of particles present in every energy level. Following this approach, the configurational Boltzmann entropy is revisited in a very general manner starting from a continuous deformation of the multinomial coefficients depending on a set of deformation parameters . It is shown that, when is related to the solutions of a simple linear difference–differential equation, the emerging entropy is a scaled version, in the occupational number representation, of the entropy of degree known, in the framework of the information theory, as Sharma–Taneja–Mittal entropic form.

Strengthening Quality of Chaotic Bit Sequences

We discuss chaos and its quality as measured through the 0-1 test for chaos. When the 0-1 test indicates deteriorating quality of chaos, because of the finite precision representations of real numbers in digital implementations, then the process may eventually lead to a periodic sequence. A simple method for improving the quality of a chaotic signal is to mix the signal with another signal by using the XOR operation. In this paper, such mixing of weak chaotic signals is considered, yielding new signals with improved quality (with K values from the 0-1 test close to 1). In some sense, such a mixing of signals could be considered as a two-layer prevention strategy to maintain chaos. That fact may be important in those applications when the hardware resources are limited. The 0-1 test is used to show the improved chaotic behavior in the case when a continuous signal (for example, from the Chua, Rössler or Lorenz system) intermingles with a discrete signal (for example, from the logistic, Tinkerbell or Henon map). The analysis is presented for chaotic bit sequences. Our approach can further lead to hardware applications, and possibly, to improvements in the design of chaotic bit generators. Several illustrative examples are included.

Optimum harnessing of solar energy with proper selection of phase changing material using integrated fuzzy-COPRAS Model

ABSTRACT Thermal Energy Storage (TES) devices can assist in storing the energy during the sunlight hours which can be used later on non-availability of solar power. Implementation of Phase Changing Material (PCM) can enhance efficiency of TES devices. Selection of resourceful and efficient PCM plays an essential role such that it can be successfully implanted nearly with all solar thermal devices. Assortment of the material involves selection of certain characteristics and properties associated with the performance of material as well as their physical, mechanical, economic and environmental properties. This study deals with a methodological analysis by integrating techniques for finding optimum PCM among various selected materials. Hybridization involves integrating Triangular Fuzzy Number (TFN) with Complex Proportional Assessment (COPRAS) method. In general, Fuzzy set comprises of three or more numbers, denoting it as TFN, Gaussian Fuzzy numbers, Trapezoidal Fuzzy Numbers, and so on. Although alternatives are many to design a fuzzy set depending on elements count, the TFNs are the ones considered as actual representations of a Fuzzy number. COPRAS is a well-known robust MCDM technique chosen to carry out the work of selection. Eight PCM have been compared on the parameters like Latent Heat, Cost, Melting points and others. Calcium Chloride Hexahydrate is the optimum material found as PCM. The result of this study would help planners and designers for an optimum selection of PCM in efficient harnessing of solar energy.

Beyond categories: dynamic qualitative analysis of visuospatial representation in arithmetic

Visuospatial representations of numbers and their relationships are widely used in mathematics education. These include drawn images, models constructed with concrete manipulatives, enactive/embodied forms, computer graphics, and more. This paper addresses the analytical limitations and ethical implications of methodologies that use broad categorizations of representations and argues the benefits of dynamic qualitative analysis of arithmetical-representational strategy across multiple semi-independent aspects of display, calculation, and interaction. It proposes an alternative methodological approach combining the structured organization of classification with the detailed nuance of description and describes a systematic but flexible framework for analysing nonstandard visuospatial representations of early arithmetic. This approach is intended for use by researchers or practitioners, for interpretation of multimodal and nonstandard visuospatial representations, and for identification of small differences in learners’ developing arithmetical-representational strategies, including changes over time. Application is illustrated using selected data from a microanalytic study of struggling students’ multiplication and division in scenario tasks.

Hexadecagonal Fuzzy Numbers: Novel Ranking and Defuzzification Techniques for Fuzzy Matrix Game Problems

With the help of various different representations of a hexadecagonal fuzzy number, this paper investigates the uncertainty associated with ambiguity and imprecision in the results of 16-component game scenarios. Numerous membership functions for alpha-cuts are established in terms of symmetrical and asymmetrical situations. We look at the centroid methodology, as well as the mean of the alpha-cut technique, the mean of the bounded area removal method and the bounded area included by the fuzzy number. A new centroid-based technique is used to rank two hexadecagonal fuzzy numbers. Defuzzification approaches have also been used to numerical examples based on fuzzy game theory in order to illustrate the effectiveness of the techniques.

The traces left by the information designer. Data visualization and enunciation

A common understanding considers information design to be a clear and immediate transfer of information, in which the author disappears to make the data emerge with utmost clarity. This idea of infographics as a transparent and objective medium is questioned by several scholars and practitioners who consider visualization not just as a representation of numbers but as an interpretative device. In this essay, we will review these positions, with special regard to the use of the semiotic concept of enunciation, which is also beginning to be used in critical design theory and digital humanities. This concept allows us to detect the traces of the act of enunciation in the visual artefact. In particular, we will deal with the recognition of visualization as an act of interpretation, the visual calibration and distancing from one’s statement in journalism and scientific communication and the visual reference to the production process in graphic design.