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On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system (ANS), that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique – the arguments from congruency, confounds, and imprecision – and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes for number, such as “numerosities” or “quanticals,” as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind(s) of number being represented. In response, we propose that the ANS represents not only natural numbers (e.g., 7), but also non-natural rational numbers (e.g., 3.5). It does not represent irrational numbers (e.g., √2), however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research.

It is widely accepted that human and nonhuman species possess a specialized system to process large approximate numerosities. The theory of an evolutionarily ancient approximate number system (ANS) has received con- verging support from developmental studies, comparative experiments, neuroimaging, and computational modelling, and it is one of the most dominant and influential theories in numerical cognition. The existence of an ANS system is significant, as it is believed to be the building block of numerical development in general. The acuity of the ANS is related to future arithmetic achievements, and intervention strategies therefore aim to improve the ANS. Here we critically review current evidence supporting the existence of an ANS. We show that important shortcomings and confounds exist in the empirical studies on human and non-human animals as well as the logic used to build computational models that support the ANS theory. We conclude that rather than taking the ANS theory for granted, a more comprehensive explanation might be provided by a sensory-integration system that compares or estimates large approximate numerosities by integrating the different sensory cues comprising number stimuli.

Big data becomes the key for ubiquitous computing and intelligence, and Distributed Storage Systems (DSS) are widely used in large-scale data centers or in the cloud for efficient data management. However, the data on stored are likely to be unavailable due to hardware failures and cyberattacks, e.g. DDoS. Maximum Distance Separable (MDS) codes are commonly used for the recovery of faulty storage nodes or unavailable data. However, the recovery of data nodes usually involves access to multiple nodes, which introduces significant communication overheads to the DSS. In this paper, a new DSS based on the Redundant Residue Number System (RRNS) is proposed, where efficient recovery is enabled by applying the second version of Chinese Remainder Theorem (CRT-II). The complexity and network traffic of the proposed data protection scheme is analyzed theoretically and compared with that of traditional MDS based DSSs. Experimental results show that the proposed DSS achieves lower encoding complexity, lower recovery complexity and lower network traffic than the MDS based schemes. Although the proposed data protection scheme introduces computation overheads for the case on which there are no failing nodes, its complexity is still lower for scenarios with frequent data updates. In addition, the proposed scheme introduces additional advantages in terms of security and storage flexibility.

Helping readers numerically solve optimization problems, this book focuses on the fundamental principles and applications of PSO and QPSO algorithms. The authors develop their novel QPSO algorithm, a PSO variant motivated from quantum mechanics, and show how to implement it in real-world applications, including inverse problems, digital filter d.

How do numerical symbols, such as number words, acquire semantic meaning? This question, also referred to as the \"symbol-grounding problem,\" is a central problem in the field of numerical cognition. Present theories suggest that symbols acquire their meaning by being mapped onto an approximate system for the nonsymbolic representation of number (Approximate Number System or ANS). In the present literature review, we first asked to which extent current behavioural and neuroimaging data support this theory, and second, to which extent the ANS, upon which symbolic numbers are assumed to be grounded, is numerical in nature. We conclude that (a) current evidence that has examined the association between the ANS and number symbols does not support the notion that number symbols are grounded in the ANS and (b) given the strong correlation between numerosity and continuous variables in nonsymbolic number processing tasks, it is next to impossible to measure the pure association between symbolic and nonsymbolic numerosity. Instead, it is clear that significant cognitive control resources are required to disambiguate numerical from continuous variables during nonsymbolic number processing. Thus, if there exists any mapping between the ANS and symbolic number, then this process of association must be mediated by cognitive control. Taken together, we suggest that studying the role of both cognitive control and continuous variables in numerosity comparison tasks will provide a more complete picture of the symbol-grounding problem. Comment les symboles numériques, tels que des mots exprimant un nombre, acquièrent-ils leur sens? Cette question, ou « problème du fondement des symboles », est au cœur du domaine de la cognition numérique. Les théories actuelles suggèrent que les symboles acquièrent leur signification lorsqu'ils sont intégrés dans un système approximatif pour la représentation non symbolique du nombre (système du nombre approximatif; SNA). Dans la présente revue de littérature, on s'est demandé, premièrement, dans quelle mesure les données actuelles sur le comportement et les données de neuroimagerie appuient cette théorie, et deuxièmement, dans quelle mesure le SNA, dans le cadre duquel il est assumé que les nombres symboliques sont ancrés, est de nature numérique. Nous concluons que a) les preuves actuelles du rapport entre le SNA et les symboles numériques n'appuient pas la notion que ceux-ci sont ancrés dans le SNA et, 2) que vu la forte corrélation entre la numérosité et les variables continues dans la tâche de traitement de nombres non symboliques, il est à peu près impossible de mesurer l'association nette entre la numérosité symbolique et non symbolique. Toutefois, il est clair que d'importantes ressources cognitives sont requises pour distinguer les variables numériques des variables continues durant le traitement de nombres non symboliques. Ainsi, s'il existe une mise en correspondance entre le SNA et le nombre symbolique, ce processus doit être géré par un contrôle cognitif. Nous suggérons que l'étude à la fois du rôle du contrôle cognitif et des variables continues dans les tâches de comparaison de numérosité permettra de dresser un portrait plus complet du problème du fondement des symboles.

Fully Homomorphic Encryption (FHE) permits processing information in the form of ciphertexts without decryption. It can ensure the security of information in common technologies used today, such as cloud computing, the Internet of Things, and machine learning, among others. A primary disadvantage for its practical application is the low efficiency of sign and comparison operations. Several FHE schemes use the Residue Number System (RNS) to decrease the time complexity of these operations. Converting from the RNS to the positional number system and calculating the positional characteristic of a number are standard approaches for both operations in the RNS domain. In this paper, we propose a new method for comparing numbers and determining the sign of a number in RNS. We focus on the even ranges that are computationally simple due to their peculiarities. We compare the performance of several state-of-art algorithms based on an implementation in C++ and relatively simple moduli with a bit depth from 24 to 64 bits. The experimental analysis shows a better performance of our approach for all the test cases; it improves the sign detection between 1.93 and 15.3 times and the number comparison within 1.55–11.35 times with respect to all the methods and configurations.

Elliptic curve cryptography is the second most important public-key cryptography following RSA cryptography. The fundamental arithmetic of elliptic curve cryptography is a series of modular multiplications and modular additions. Usually, Montgomery algorithm is applied for modular multiplications over large integers to reduce the computational complexity. Targeting at fast elliptic curve point multiplication over prime fields a new approach in residue number system is proposed. Compared with other implementations that apply Montgomery ladder for parallel elliptic curve point multiplication, the proposed method uses a residue number system with a wide dynamic range, which supports continuous multiplications and needs only one RNS Montgomery multiplication to bring down the temporary results to valid range. Hardware implementation results demonstrate that the computation time for elliptic curve point multiplication over F p can be greatly reduced, and it takes about 0.677 ms to compute one time of elliptic curve point multiplication over 384-bit prime curves in Xilinx XC6VSX475t device, costing an area of 41409 slices, 676 DSPs and 138 Brams.

This research introduces innovative approaches to enhance intrusion detection systems (IDSs) by addressing critical challenges in existing methods. Various machine-learning techniques, including nature-inspired metaheuristics, Bayesian algorithms, and swarm intelligence, have been proposed in the past for attribute selection and IDS performance improvement. However, these methods have often fallen short in terms of detection accuracy, detection rate, precision, and F-score. To tackle these issues, the paper presents a novel hybrid feature selection approach combining the Bat metaheuristic algorithm with the Residue Number System (RNS). Initially, the Bat algorithm is utilized to partition training data and eliminate irrelevant attributes. Recognizing the Bat algorithm's slower training and testing times, RNS is incorporated to enhance processing speed. Additionally, principal component analysis (PCA) is employed for feature extraction. In a second phase, RNS is excluded for feature selection, allowing the Bat algorithm to perform this task while PCA handles feature extraction. Subsequently, classification is conducted using naive bayes, and k-Nearest Neighbors. Experimental results demonstrate the remarkable effectiveness of combining RNS with the Bat algorithm, achieving outstanding detection rates, accuracy, and F-scores. Notably, the fusion approach doubles processing speed. The findings are further validated through benchmarking against existing intrusion detection methods, establishing their competitiveness.

It has long been known that people have the ability to estimate numerical quantities without counting. A standard account is that people develop a sense of the size of symbolic numbers by learning to map symbolic numbers (e.g., 6) to their corresponding numerosities (e.g. :::) and concomitant approximate magnitude system (ANS) representations. However, we here demonstrate that adults are capable of extracting fractional numerical quantities from non-symbolic visual ratios (i.e., labeling a ratio of two circle areas with the appropriate symbolic fraction). Not only were adult participants able to perform this task, but they were remarkably accurate: linear regressions on median estimates yielded slopes near 1, and accounted for 97% of the variability. Participants also performed at least as well on line-estimation and ratio-estimation tasks using non-numeric circular stimuli as they did in earlier experiments using non-symbolic numerosities, which are frequently considered to be numeric stimuli. We discuss results as consistent with accounts suggesting that non-symbolic ratios have the potential to act as a reliable and stable ground for symbolic number, even when composed of non-numeric stimuli.