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"Number systems."
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Multiparameter eigenvalue problems : Sturm-Liouville theory
\"With special attention to the Sturm-Liouville theory, this book discusses the full multiparameter theory as applied to second-order linear equations. It considers the spectral theory of these multiparameter problems in detail for both the regular and singular cases. The text covers eignencurves, the essential spectrum, eigenfunctions, oscillation theorems, the distribution of eigencurves, the limit point, limit circle theory, and more. This text is the culmination of more than two decades of research by F.V. Atkinson, one of the masters in the field, and his successors, who continued his work after he passed away in 2002\"-- Provided by publisher.
Book
The number sense represents (rational) numbers
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.
Journal Article
The math book
Applying the Big Ideas Simply Explained series' trademark combination of authoritative, accessible text and bold graphics to chart the development of math through history, The Math Book explores and explains subjects ranging from ancient mathematical ideas and inventions, such as prehistoric tally bones and Sumerian multiplication tables, through the developments in mathematics during medieval and Renaissance Europe, to the more recent rise of game and group theory. Tracing math through the scientific revolution to its 21st-century use in computers, the internet, and AI, The Math Book uses an innovative graphic-led approach to make the subject accessible to everyone.
BOOK
Sensory-integration system rather than approximate number system underlies numerosity processing: A critical review
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.
Journal Article
The green computing book : tackling energy efficiency at large scale /edited by Wu-chun Feng
\"Driven by the newly placed importance and growing search for ways to make computing greener and more efficient, this reference is the first research-level book devoted to green computing and large-scale energy efficiency. With contributions from leading experts in the field, the book presents current research and developments in hardware, systems software, run-time systems, programming languages, data center management, and applications. It also covers the emerging green movement in computing, including the Green Grid and the Green 500 list, as well as important programs in grassroots organizations and government agencies\"-- Provided by publisher.
Book
Enhancing data protection with a distributed storage system based on the redundant residue number system
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.
Journal Article
Particle Swarm Optimisation
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.
eBook
Feature selection in intrusion detection systems: a new hybrid fusion of Bat algorithm and Residue Number System
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.
Journal Article
Fast RNS Implementation of Elliptic Curve Point Multiplication on FPGAs
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
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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.
Journal Article
An Efficient Method for Comparing Numbers and Determining the Sign of a Number in RNS for Even Ranges
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.
Journal Article