Number Of Representations Research Articles

Limits of real numbers in the binary signed digit representation

We extract verified algorithms for exact real number computation from constructive proofs. To this end we use a coinductive representation of reals as streams of binary signed digits. The main objective of this paper is the formalisation of a constructive proof that real numbers are closed with respect to limits. All the proofs of the main theorem and the first application are implemented in the Minlog proof system and the extracted terms are further translated into Haskell. We compare two approaches. The first approach is a direct proof. In the second approach we make use of the representation of reals by a Cauchy-sequence of rationals. Utilizing translations between the two represenation and using the completeness of the Cauchy-reals, the proof is very short. In both cases we use Minlog's program extraction mechanism to automatically extract a formally verified program that transforms a converging sequence of reals, i.e.~a sequence of streams of binary signed digits together with a modulus of convergence, into the binary signed digit representation of its limit. The correctness of the extracted terms follows directly from the soundness theorem of program extraction. As a first application we use the extracted algorithms together with Heron's method to construct an algorithm that computes square roots with respect to the binary signed digit representation. In a second application we use the convergence theorem to show that the signed digit representation of real numbers is closed under multiplication.

Teaching redundant residue number system for electronics and computer students

This paper describes a study for teaching number system in a computer arithmetic course and addresses the existing gap in the current course by focusing on the characteristics of applications. A redundant residue number system (RRNS) is an efficient and innovative number system that inherits the interesting properties of both residue number system (RNS) and redundant number systems. It breaks the operands into residues by considering a moduli-set and then signifies the residues by a redundant representation. So, by limiting the carry propagation chain inside and outside the moduli, RRNS proposes more efficient arithmetic units which could be employed in digital signal processing (DSP) applications. Despite the applicability of RRNS, there is not well-organized teaching on RRNS covering the recent achievements. Besides, there is an important question for students and researchers what is the most appropriate number representation for each application category? In this paper, we present a step-by-step education process for RRNS that includes basic concepts of RNS and the redundant number system. Then we address characteristics of different applications that RRNS is suitable to be employed for them. So, the method gives a well-organized proceeding to computer arithmetic designers and students.

ULog: a software-based approximate logarithmic number system for computations on SIMD processors

This paper presents a new number representation based on logarithmic number system (LNS) called unsigned logarithmic number system (ulog), as an alternative to the conventional floating-point (FP) number format, to use in approximate computing applications. ulog is tailored for software implementation on commercial general-purpose processors, and uses the same dynamic range as conventional IEEE Standard FP formats to prevent overflow and underflow. ulog converts FP numbers to fixed-point numbers and uses integer operations for all computations. Moreover, vectorization and approximate logarithmic addition have been used to increase the performance of the software implementation of ulog. Then, we used different BLAS benchmarks to evaluate the performance of the proposed format than IEEE standard formats. 16- and 32-bit ulog improve the runtime than double-precision at most 70.26% and 46.36%, respectively. Besides, accuracy analysis of the ulog based on different logarithm bases showed that base 4 has the lowest error in most cases.

Combining attention with spectrum to handle missing values on time series data without imputation

In the development of predictive models, the problem of missing data is a critical issue that traditionally requires a two-step analysis. Data scientists analyze the patterns of missing values, select variables, impute missing values on the basis of domain knowledge, and then train a model. Models typically have their input sizes hardcoded, and have limitations in handling data with high missing rates or changes in available variables. We propose an attention-based neural network combined with a novel real number representation, which requires little work on manually selecting variables, and in which missing data can be overlooked, making imputation unnecessary. In this proposed model, data analysis can be one step, omitting the first step of imputing missing values. The study included data on 32,709 intensive care unit (ICU) admissions and 60 healthcare variables from the Medical Information Mart for Intensive Care (MIMIC)-IV. The proposed algorithm yielded an area under the receiver operating characteristic curve (AUC) of 0.842 (95% CIs: 0.828–0.856) when predicting prolonged length of stay in the ICU, outperforming current approaches using imputation methods. The proposed algorithm can be applied to a range of problems in data science, as it addresses the issue of incomplete data with automatic variable selection.

Children's use of egocentric reference frames in spatial language is related to their numerical magnitude understanding.

Numerical magnitude information is assumed to be spatially represented in the form of a mental number line defined with respect to a body-centred, egocentric frame of reference. In this context, spatial language skills such as mastery of verbal descriptions of spatial position (e.g., in front of, behind, to the right/left) have been proposed to be relevant for grasping spatial relations between numerical magnitudes on the mental number line. We examined 4- to 5-year-old’s spatial language skills in tasks that allow responses in egocentric and allocentric frames of reference, as well as their relative understanding of numerical magnitude (assessed by a number word comparison task). In addition, we evaluated influences of children’s absolute understanding of numerical magnitude assessed by their number word comprehension (montring different numbers using their fingers) and of their knowledge on numerical sequences (determining predecessors and successors as well as identifying missing dice patterns of a series). Results indicated that when considering responses that corresponded to the egocentric perspective, children’s spatial language was associated significantly with their relative numerical magnitude understanding, even after controlling for covariates, such as children’s SES, mental rotation skills, and also absolute magnitude understanding or knowledge on numerical sequences. This suggests that the use of egocentric reference frames in spatial language may facilitate spatial representation of numbers along a mental number line and thus seem important for preschoolers’ relative understanding of numerical magnitude.

Statistically characterized subgroups of the circle (II): Continued fractions

In this note, we continue the investigation of the new version of characterized subgroups of the circle group T, namely, “statistically characterized subgroups” (shortly, “s-characterized subgroups”) recently introduced in [12]. We primarily investigate these subgroups for sequences arising out of continued fraction representation of irrational numbers α in line of [22] and [20] (followed by [3–5]) comparing their main results for this new notion and show that these Borel subgroups are strictly larger in size (so nontrivial) than the corresponding characterized subgroups, always having cardinality c and containing the subgroup 〈α〉 and in the process answer the Open Question 6.4 [12] which had asked about the size of the s-characterized subgroup for the celebrated Fibonacci sequence.

What is worthy of investigation?

We describe in dialogue form a possible way of discovering and investigating 10-adic numbers starting from the naive question about a “largest natural number”. Among the topics we pursue are possibilities of extensions to transfinite 10-adic numbers, 10-adic representations of rational numbers, zero divisors, square roots and 10-adic roots of higher degree of natural numbers, and applications of 10-adic number representation in computer arithmetic. The participants of the dialogue are idealized embodiments of different philosophical attitudes towards mathematics. The article aims at illustrating how these attitudes interact, in both jarring and stimulating ways, and how they impact mathematical development. Moreover, the article demonstrates how different attitudes translate into mathematical heuristics. In our opinion this is also relevant for teaching mathematics as it raises consciousness for reflections on how mathematical investigations work.

Pengembangan Model dan Penggunaan Konteks dalam Mendukung Pemahaman Siswa pada Topik Penjumlahan yang Melibatkan Bilangan Negatif

This study aims to develop learning activities to contribute to the theory based on empirical activities in learning the topic of summation involving negative numbers. This research also develops learning activities that combine context and models that students can apply to understand the topic. The type of research used to attain the objectives in this study is design research, each cycle consists of three steps, namely preliminary study, classroom experiment and retrospective method. The subjects of the study were grade 4 students and teachers at SD Islam Athirah Makassar. The results showed that to help students understand the concept of summation involving negative numbers, students must understand the concept of negative numbers and use the number line model to count. This can be supported by the context of equilibrium and the number line as a representation of the negative numbers. In classroom learning, the use of context stimulates students to think informally, furthermore, by indicating a state using numbers and with the help of a number line model, students can solve mathematical problems at a formal level.

Syntactic chunking reveals a core syntactic representation of multi-digit numbers, which is generative and automatic

Representing the base-10 structure of numbers is a challenging cognitive ability, unique to humans, but it is yet unknown how precisely this is done. Here, we examined whether and how literate adults represent a number’s full syntactic structure. In 5 experiments, participants repeated number-word sequences and we systematically varied the order of words within each sequence. Repetition on grammatical sequences (e.g., two hundred ninety-seven) was better than on non-grammatical ones (hundred seven two ninety). We conclude that the participants represented the number’s full syntactic structure and used it to merge number words into chunks in short-term memory. Accuracy monotonously improved for sequences with increasingly longer grammatical segments, up to a limit of ~ 4 words per segment, irrespectively of the number of digits, and worsened thereafter. Namely, short chunks improved memorization, whereas oversized chunks disrupted memorization. This chunk size limit suggests that the chunks are not based on predefined structures, whose size limit is not expected to be so low, but are created ad hoc by a generative process, such as the hierarchical syntactic representation hypothesized in Michael McCloskey’s number-processing model. Chunking occurred even when it disrupted performance, as in the oversized chunks, and even when external cues for chunking were controlled for or were removed. We conclude that the above generative process operates automatically rather than voluntarily. To date, this is the most detailed account of the core representation of the syntactic structure of numbers—a critical aspect of numerical literacy and of the ability to read and write numbers.

Abstract concepts: external influences, internal constraints, and methodological issues

There is a longstanding and widely held misconception about the relative remoteness of abstract concepts from concrete experiences. This review examines the current evidence for external influences and internal constraints on the processing, representation, and use of abstract concepts, like truth, friendship, and number. We highlight the theoretical benefit of distinguishing between grounded and embodied cognition and then ask which roles do perception, action, language, and social interaction play in acquiring, representing and using abstract concepts. By reviewing several studies, we show that they are, against the accepted definition, not detached from perception and action. Focussing on magnitude-related concepts, we also discuss evidence for cultural influences on abstract knowledge and explore how internal processes such as inner speech, metacognition, and inner bodily signals (interoception) influence the acquisition and retrieval of abstract knowledge. Finally, we discuss some methodological developments. Specifically, we focus on the importance of studies that investigate the time course of conceptual processing and we argue that, because of the paramount role of sociality for abstract concepts, new methods are necessary to study concepts in interactive situations. We conclude that bodily, linguistic, and social constraints provide important theoretical limitations for our theories of conceptual knowledge.

Interval-filling sequences of order $N$ and a representation of real numbers in canonical number systems

Interval-filling sequences of order $N$ and a representation of real numbers in canonical number systems

PERCIVAL: Open-Source Posit RISC-V Core With Quire Capability

The posit representation for real numbers is an alternative to the ubiquitous IEEE 754 floating-point standard. In this work, we present PERCIVAL, an application-level posit RISC-V core based on CVA6 that can execute all posit instructions, including the quire fused operations. This solves the obstacle encountered by previous works, which only included partial posit support or which had to emulate posits in software. In addition, Xposit, a RISC-V extension for posit instructions is incorporated into LLVM. Therefore, PERCIVAL is the first work that integrates the complete posit instruction set in hardware. These elements allow for the native execution of posit instructions as well as the standard floating-point ones, further permitting the comparison of these representations. FPGA and ASIC synthesis show the hardware cost of implementing 32-bit posits and highlight the significant overhead of including a quire accumulator. However, results show that the quire enables a more accurate execution of dot products. In general matrix multiplications, the accuracy error is reduced up to 4 orders of magnitude. Furthermore, performance comparisons show that these accuracy improvements do not hinder their execution, as posits run as fast as single-precision floats and exhibit better timing than double-precision floats, thus potentially providing an alternative representation.

Online construction of spatial representation of numbers: Evidence from the SNARC effect in number processing in interferential situations

Online construction of spatial representation of numbers: Evidence from the SNARC effect in number processing in interferential situations

The Laplace Microarchitecture for Tracking Data Uncertainty

This article presents Laplace, a microarchitecture for tracking machine representations of probability distributions paired with architectural state. Laplace uses in-processor distribution representations, which are approximations of probability distributions just as floating-point number representations are approximations of real-valued numbers. The article presents two sets of instruction set architecture (ISA) extensions to 1) provide a mechanism to initialize distributional information in the microarchitecture; and 2) to allow applications to query statistics of the distributional information without exposing the uncertainty representations above the ISA. Unlike existing methods for uncertainty tracking, which require software to be rewritten in a domain-specific language or extensive source-level changes, Laplace achieves all of these benefits while requiring no changes to existing binaries to track uncertainty through them. Compared to repeated Monte Carlo re-executions of applications on a conventional microarchitecture, Laplace achieves the same level of uncertainty tracking accuracy with 2,076× fewer executed instructions on average (up to 21,343× fewer).

Efficient congruencing in ellipsephic sets: the general case

In this paper, we bound the number of solutions to a general Vinogradov system of equations [Formula: see text] as well as other related systems, in which the variables are required to satisfy digital restrictions in a given base. Specifically, our sets of permitted digits have the property that there are few representations of a natural number as sums of elements of the digit set — the set of squares serving as a key example. We obtain better bounds using this additive structure than could be deduced purely from the size of the set of variables. In particular, when the digits are required to be squares, we obtain diagonal behavior with [Formula: see text] variables.

Analysis of buoyancy driven flow inside a vertical filter chamber

Designs based on scientific evidence play an essential role when engineering machinery. The current case study seeks an alternative way of filtering particles using a permeable surface. The filter is designed so that the effects of heat from a permeable bottom wall create a force that drives permeates out of the filter chamber. The model follows the mass conservation, incompressible Navier-Stokes and the energy conservation equations. Effects of parameters arising from flow dynamics and heat distribution are studied to better understand the theory behind the filtrations process. Graphical representations of skin friction and Nusselt number inside the filter chamber are plotted against filtration dynamics parameters. We found that more permeates are produced when the internal energy is low since the system loses internal temperature when less dense fluid, which possesses heat, is withdrawn from the chamber. In addition, the chamber volume increase permeates outflow since it enhances the pulling effects of the buoyancy force. It is found that increasing the Richardson number enhances the buoyancy forces and viscous effects; thus, decreasing permeates outflow velocity and eventually reduces wall drag coefficient. The wall drag coefficients on the left and right vertical walls have extrema towards the bottom of the filter chamber, in the region, 0 <y < 2 % H.

Self-Communicating Deep Reinforcement Learning Agents Develop External Number Representations

Symbolic numbers are a remarkable product of human cultural development. The developmental process involved the creation and progressive refinement of material representational tools, such as notched tallies, knotted strings, and counting boards. In this paper, we introduce a computational framework that allows the investigation of how material representations might support number processing in a deep reinforcement learning scenario. In this framework, agents can use an external, discrete state to communicate information to solve a simple numerical estimation task. We find that different perceptual and processing constraints result in different emergent representations, whose specific characteristics can facilitate the learning and communication of numbers.

How Does the Accuracy of Children's Number Representations Influence the Accuracy of Their Numerical Predictions?

Predictions begin with an extrapolation of the properties of their underlying representations to forecast a future state not presently in evidence. For numerical predictions, sets of numbers are summarized and the result forms the basis of and constrains numerical predictions. One open question is how the accuracy of underlying representations influences predictions, particularly numerical predictions. It is possible that inaccuracies in individual number representations are randomly distributed and averaged over during summarization (e.g., wisdom of crowds). It is also possible that inaccuracies are not random and lead to errors in predictions. We investigated this question by measuring the accuracy of individual number representations of 279 children ages 8–12 years, using a 0–1,000 number line, and numerical predictions, measured using a home run derby task. Consistent with prior research, our results from mixed random effects models evaluating percent absolute error (PAE; prediction error) demonstrated that third graders’ representations of individual numbers were less accurate, characterized by overestimation errors, and were associated with overpredictions (i.e., predictions above the set mean). Older children had more accurate individual number representations and a slight tendency to underpredict (i.e., predictions below the set mean). The results suggest that large, systematic inaccuracies appear to skew predictions while small, random errors appear to be averaged over during summarization. These findings add to our understanding of summarization and its role in numerical predictions.

Parity Properties of Configurations

In the paper, the crossing number of the join product G*+Dn for the disconnected graph G* consisting of two components isomorphic to K2 and K3 is given, where Dn consists of n isolated vertices. Presented proofs are completed with the help of the graph of configurations that is a graphical representation of minimum numbers of crossings between two different subgraphs whose edges do not cross the edges of G*. For the first time, multiple symmetry between configurations are presented as parity properties. We also determine crossing numbers of join products of G* with paths Pn and cycles Cn on n vertices by adding new edges joining vertices of Dn.

The Impact of Coding Levels of Magnitude and of Spatial-Direction on the Spatial-Numerical Association of Response Codes Effect of Negative Numbers.

Whether negative numbers have a fixed spatial–numerical association of response codes effect (SNARC effect), and (if they have) whether the spatial representation of negative numbers is associated with negative numbers’ absolute or signed values remains controversial. In this study, through three experiments, the coding level of the magnitude and the spatial-direction is manipulated. In the first experiment, participants are required to code the magnitude and spatial-direction explicitly by using a magnitude classification task. In the second experiment, participants are forced to code the magnitude implicitly as well as to code the spatial-direction explicitly by utilizing a cuing task. In the third experiment, participants are obliged to code the magnitude explicitly as well as to code the spatial-direction implicitly by adopting a magnitude and arrow-direction classification tasks with Go/No-Go responses. The results show that (1) the absolute value of negative numbers associates with space when the magnitude of negative numbers is explicitly coded, no matter employing the explicit or implicit spatial-direction; (2) the signed value of negative numbers associates with space under the condition of implicit magnitude as well as explicit spatial-direction. In conclusion, the current study indicates that the SNARC effect of negative numbers is variable in different conditions, and the type of SNARC effect about negative numbers is modulated by the joint coding level of the magnitude and spatial-direction.