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A comparison and analysis of the use of the developed software implementation of the class MontgomeryArithmetic for computing modular exponentiation are conducted. The computation speed of the developed Montgomery modular multiplication is compared to the regular modular multiplication for calculating the modular exponentiation based on the right-to-left binary elevation method for a fixed basis with a preliminary calculation of a reduced set of remainders. The obtained results of performing modular exponentiation computations with parallelization based on multithreading on general-purpose computers speed up the computations by an average of 1.5 times using the developed modular Montgomery multiplication compared to the modular exponentiation functions of the MPIR, OpenSSL, and Crypto++ software libraries. [ABSTRACT FROM AUTHOR]

Universal presentations and interpretations of the history of mathematics have now been challenged, identifying the need of resurrecting hitherto marginalized contributions. In such studies, cognitive factors influencing the related development trajectories play a crucial role. Different societies would have had different mathematical cognitions contributing to the development of distinct forms of mathematics. This study attempts to surface the related cognitive contents of using large numbers. The principal source used for the study—the Buddhist Commentaries—determines the period and region covered. As such, it explores the conditions prevailed in 5th century CE (approximately) in Sri Lanka. 2000 Mathematics Subject Classification (MSC 2000) code: 01A07 [ABSTRACT FROM AUTHOR]

Very large numbers words such as "hundred," "thousand," "million," "billion," and "trillion" pose a learning problem for children because they are sparse in everyday speech and children's experience with extremely large quantities is scarce. In this study, we examine when children acquire the relative ordering of very large number words as a first step toward understanding their acquisition. In Study 1, a hundred and twenty‐five 5–8‐year‐olds participated in a verbal number comparison task involving very large number words. We found that children can judge which of two very large numbers is more as early as age 6, prior to entering first grade. In Study 2, we provided a descriptive analysis on the usage of very large number words using the CHILDES database. We found that the relative frequency of large number words does not change across the years, with "hundred" uttered more frequently than others by an order of magnitude. We also found that adults were more likely to use large number words to reference units of quantification for money, weight, and time, than for discrete, physical entities. Together, these results show that children construct a numerical scale for large number words prior to learning their precise cardinal meanings, and highlight how frequency and context may support their acquisition. Our results have pedagogical implications and highlight a need to investigate how children acquire meanings for number words that reference quantities beyond our everyday experience. [ABSTRACT FROM AUTHOR]

We provide various tools that give an answer to the following question: given two distinct finite tower powers with entries at least 1 and (possibly) different heights, determine which power tower is greater than the other one. [ABSTRACT FROM AUTHOR]

In recent years, hardware sorters have been an attracted topic for researchers. Since hardware sorters play a crucial role in embedded systems, several attempts have been made to efficiently design and implement these sorters. Previous state-of-the-art hardware sorters are not suitable for embedded edge computing devices because they (1) consume high power, (2) occupy high area, (3) work for limited data-width numbers, (4) require many memory resources, and (5) finally, their architecture is not scalable with the number of input records. This paper proposes a hardware sorter for edge devices with limited hardware resources. The proposed hardware sorter, called Edge-Sorter, processes 4 bits of input records at each clock cycle. Edge-Sorter utilizes the unary processing in its main processing core. Edge-Sorter has valuable attributes compared to previous state-of-the-art techniques, including low power consumption, low area occupation, sorting numbers without storing their indices, sorting numbers with arbitrary data-width, and scalable with the number of input records. The proposed approach is evaluated and compared with previous state-of-the-art techniques with two different implementation and synthesis environments: Xilinx Vivado FPGA-based and Synopsys Design Compiler 45-nm ASIC-based. The Synthesis results of both environments indicate that both Edge-Sorter techniques reduces area and power consumption on average by 80% and 90%, respectively compared to previous techniques. [ABSTRACT FROM AUTHOR]

Underdetermined blind source separation (UBSS) is an important issue for sparse signals, and a novel two-step approach for UBSS based on the law of large numbers and minimum intersection angle rule (LM method) is presented. In the first step, an estimation of the mixed matrix is obtained by using the law of large numbers, and the number of source signals is displayed graphically. In the second step, a method of estimating the source signals by the minimum intersection angle rule is proposed. The significance of this step is that the minimum intersection rule is better than the shortest path method, and the decomposition components can be found optimally by the former. The simulation results illustrate the effectiveness of the LM method. It has a simple principle, has good transplantation capability and may be widely applied in various fields of digital signal processing. [ABSTRACT FROM AUTHOR]

Large dimensionless numbers, arising out of ratios of various physical constants, intrigued many scientists, especially Dirac. Relying on the coincidence of large numbers, Dirac arrived at the revolutionary hypothesis that the gravitational constant G should vary inversely at the cosmic time t. This hypothesis of Dirac, known as the Large Number Hypothesis (LNH), sparked off many speculations, arguments and new ideas in terms of applications. Works done by several authors with LNH as their basic platform are extensively reviewed in this work. Relationship between some of those works are pointed out here elaborately. Possibility of time variations of physical constants other than G as well as large numbers in various realm of physical and biological sciences are also discussed. [ABSTRACT FROM AUTHOR]

This paper aims at complementing the cross-disciplinary intellectual portrait of Jean des Murs presented in this special issue from the particular perspective of his arithmetical and geometrical works. If looked at from this angle, the general question immediately implies new ones: what are the connections between Jean's arithmetical or geometrical works and those in the other fields of the quadrivium ? What is their importance in the global intellectual picture of Jean des Murs as a quadrivial author of the fourteenth century? In order to explore these different issues, I have chosen to focus on a specific question: to what extent do Jean des Murs's mathematical works reflect a persistent effort throughout his career to address the various issues related to small amounts? Focusing on this particularly significant problem will give me the opportunity to cover a representative sample of Jean des Murs's quadrivial works from a specific perspective. Music and astronomy address small amounts in different contexts and with different mathematical tools. However, some deeper similarities appear. They concern mainly the question of measure and incommensurability as well as the relation between mathematical and physical reasoning. Overall, they reflect the intellectual coherence between quadrivial sciences as attested by Jean des Murs's works. [ABSTRACT FROM AUTHOR]

This investigation examined the thinking of 128 pre-service teachers about the magnitude of large numbers and the relationships between large numbers. These pre-service teachers responded to a written task asking them to place one billion on a number line that included zero and one trillion and to provide an explanation for their placement. One billion was placed proportionately to zero and one trillion on the number line in only 20% of responses. The data revealed nine distinct strategies that emerged and different conceptions the pre-service teachers used for making the placements. The authors discuss possible reasons for the different conceptions. [ABSTRACT FROM PUBLISHER]

Literature and environment professor Scott Slovic, and his father, psychologist Paul Slovic, editors of this collection of essays and interviews, describe and demonstrate the psychological effects which hamper our ability to comprehend and respond appropriately to large numerical data. The collection then offers a brief survey of art works which, by first appealing to viewers' emotions, can potentially move the viewer to a better understanding of numbers. [ABSTRACT FROM AUTHOR]

The ability to discriminate between different quantities has important ecological relevance for animals when engaging in behaviours such as forming groups, foraging or trying to avoid predators. Quantity discrimination has been shown in a diversity of human and nonhuman animal species. In angelfish this discrimination ability has been investigated using dichotomous choice tests when the numerically different stimulus groups (shoals) of conspecifics were fully visible to the test fish. Here, using a new procedure we investigated whether test fish were able to discriminate between the contrasting shoals using their memory. After a period of full visual access to the contrasted shoals on the two sides of their test tank, the test fish was required to make a choice while being able to see only a single member of the stimulus shoals on each side. With this cognitively more demanding procedure we tested discrimination between numerically large shoals (≥ four fish per stimulus shoal). As in our previous studies, we found that angelfish consistently chose the larger of the two shoals when the shoals differed by a 2:1 or higher ratio, but not those that differed by a 3:2 or 4:3 ratio. The results followed Weber's law in that performance became poorer as the ratio between the two stimulus shoals approached one. In addition, when we kept the absolute difference between the contrasted shoals constant, discrimination was less accurate as the shoal sizes increased. This pattern of results lends support for the analogue magnitude representational system in the angelfish, a nonverbal approximation system believed to be employed by a diversity of human and nonhuman animal species. Furthermore, our results also demonstrate that angelfish remember the different shoals presented to them, i.e. they make their choice based upon mental representation of the different quantities. [ABSTRACT FROM AUTHOR]

Scott Slovic and Paul Slovic (Eds.). Numbers and Nerves: Information, Emotion, and Meaning in a World of Data (Corvallis, OR: Oregon State University Press, 2015). 272 pp. ISBN 978-0-87071-776-5. Literature and environment professor Scott Slovic, and his father, psychologist Paul Slovic, editors of this collection of essays and interviews, describe and demonstrate the psychological effects which hamper our ability to comprehend and respond appropriately to large numerical data. The collection then offers a brief survey of art works which, by first appealing to viewers’ emotions, can potentially move the viewer to a better understanding of numbers.

A new axiomatic characterization with a minimum of conditions for entropy as a function on the set of states in quantum mechanics is presented. Traditionally unspoken assumptions are unveiled and replaced by proven consequences of the axioms. First the Boltzmann-Planck formula is derived. Building on this formula, using the Law of Large Numbers-a basic theorem of probability theory-the von Neumann formula is deduced. Axioms used in older theories on the foundations are now derived facts. [ABSTRACT FROM AUTHOR]

Highlights: [•] The 4- and 5-year-olds matched the notations (digit included) equally in the two transcoding directions. [•] The 4-year-olds matched better arrays and digits than number words and digits, with small numbers first. [•] These results highlight the developmental importance of the analogical code in the learning of digital code. [•] They suggest the utility of exploring children’s acquisition of digits in addition to number words. [Copyright &y& Elsevier]

Research about geologic time conceptions generally focuses on the placement of events on the geologic timescale, with few studies dealing with the duration of geologic processes or events. Those studies indicate that students often have very poor conceptions about temporal durations of geologic processes, but the reasons for that are relatively unexplored. Close connections between ideas about time and space over short time periods, as well as poor number sense for numbers in unfamiliar ranges have been repeatedly demonstrated. This study explored whether the conceptions geoscience novices hold about the temporal duration of geoscience processes across a variety of temporal scales are influenced by their ideas about space and large numbers. Seventeen undergraduates in an introductory geoscience course participated in task-based qualitative interviews. Students tended to equate spatial size with temporal duration over short and long time periods, sometimes modifying contradictory data to fit their interpretation. Confusion about the relative size of temporal periods up to 100,000,000 years was observed. They described durations operating on long temporal scales in largely qualitative, imprecise terms. Spatial compression of large temporal periods and expansion of short time periods were common. Students possessed few specific temporal markers for durations of task events. Specific pedagogical and curricular recommendations are discussed. [ABSTRACT FROM AUTHOR]

Human adults master sophisticated, abstract numerical calculations that are mostly based on symbolic language and thus inimitably human. Humans may nonetheless share a subset of non-verbal numerical skills, available soon after birth and considered the evolutionary foundation of more complex numerical reasoning, with other animals. These skills are thought to be based on the two systems: the object file system which processes small values (<3) and the analogue magnitude system which processes large magnitudes (>4). Infants' ability to discriminate 1 vs. 2, 1 vs. 3, 2 vs. 3, but not 1 vs. 4, seems to indicate that the two systems are independent, implying that the conception of a continuous number processing system is based on precursors that appear to be interrupted at or about the number four. The findings from the study being presented here indicating that chicks are able to make a series of discriminations regarding that borderline number (1 vs. 4, 1 vs. 5, 2 vs. 4) support the hypothesis that there is continuity in the number system which processes both small and large numerousness. [ABSTRACT FROM AUTHOR]

An understanding of geologic time is comprised of 2 facets. Events in Earth's history can be placed in relative and absolute temporal succession on a vast timescale. Rates of geologic processes vary widely, and some occur over time periods well outside human experience. Several factors likely contribute to an understanding of geologic time, one of which is an individual's ability to perceive the relative size of large time periods and to move multiplicatively through quantities that differ by many orders of magnitude. Thirty-five US students aged 13-24 years participated in task-based interviews to assess their understanding of large temporal periods. Fewer than half of the students performed well enough to indicate that their knowledge of large numbers was robust enough to enable them to understand processes in geologic time. Some students were confused about relationships between quantities in the thousands and millions, while others had difficulty showing proportional relationships among relatively small temporal units (up to 100 years). Students differed in their ability to perceive the entire scale upon which numbers were to be placed as well as broader problem-solving strategies. Spatial mapping of numbers was evident. Implications for future research are discussed. [ABSTRACT FROM AUTHOR]

In this short review we study the state of the art of the great problems in cosmology and their interrelationships. The reconciliation of these problems passes undoubtedly through the idea of a quantum universe. [ABSTRACT FROM AUTHOR]

Previous empirical studies using multiple-choice procedures have suggested that there are misconceptions about the scale of astronomical distances. The present study provides a quantitative estimate of the nature of this misconception among US university students by asking them, in an open-ended response format, to make estimates of the distances from the Earth to the Moon, to the Sun, to the nearest star, and to the nearest galaxy. The 83 participants were asked to give their estimates on a scale with the Earth the size of a baseball, using a familiar local landmark for its position, and asked to indicate the appropriate location of the other astronomical objects on this scale. These psychological estimates were then compared to the actual physical distances. The data showed that while there is great variation, a general pattern emerged that US undergraduate participants overestimated the distance from the Earth to the Moon, moderately underestimated the distance from the Earth to the Sun, and dramatically underestimated the distances to the nearest star and to the nearest galaxy. The results suggest that these distance misconceptions should receive direct instructional attention in science courses. [ABSTRACT FROM AUTHOR]

Adults increase the certainty of their inductive inferences by observing more diverse instances. However, most young children fail to do so. The present study tested the hypothesis that children's sensitivity to instance diversity is determined by three variables: ability to discriminate among instances (Discrimination); an intuition that large numbers of instances increase the strength of conclusion (Monotonicity); ability to detect subcategories and evaluate numerical differences between the subcategories, or Extraction. A total of 219 Chinese children aged 6 to 11 were tested for sensitivity to diversity by means of Discrimination, Monotonicity, and Extraction. The results indicated that children at all ages were able to discriminate instances and attend to set size. However, only 9- and 11-year-olds demonstrated Extraction and sensitivity to diversity. Furthermore, among all children diversity scores increased linearly with the level of Extraction. These results suggest that the law of large numbers plays a role in children's diversity-based reasoning. [ABSTRACT FROM AUTHOR]

The large number coincidences that fascinated theorists such as Eddington and Dirac are shown here to be a specific example of a general set of scaling factors defining universes in which fundamental forces are equated. The numbers have prescriptive power and they are therefore correct and exact a priori. The universes thus defined exhibit a fractal structure centred on the Planck/Stoney scale with some formal resemblance to black holes and with properties analogous to Hawking radiation. The problematic case of emerging and evaporating universes is briefly considered in the context of quantum gravity. Historically, the large numbers are associated with the mass of a charged particle and the mass of the universe. This paper demonstrates that the numbers are properly understood in the context of four masses including a non-zero mass derived from Hubble's Constant and the Planck or Stoney mass. [ABSTRACT FROM AUTHOR]

In many problems, modular exponentiation |xb|m is a basic computation, often responsible for the overall time performance, as in some cryptosystems, since its implementation requires a large number of multiplications.It is known that |xb|m=|x|b|ϕ(m)|m for any x in [1,m−1] if m is prime; in this case the number of multiplications depends on ϕ(m) instead of depending on b. It was also stated that previous relation holds in the case m=pq, with p and q prime; this case occurs in the RSA method.In this paper it is proved that such a relation holds in general for any x in [1,m−1] when m is a product of any number n of distinct primes and that it does not hold in the other cases for the whole range [1,m−1].Moreover, a general method is given to compute |xb|m without any hypothesis on m, for any x in [1,m−1], with a number of modular multiplications not exceeding those required when m is a product of primes.Next, it is shown that representing x in a residue number system (RNS) with proper moduli mi allows to compute |xb|m by n modular exponentiations |xib|mi in parallel and, in turn, to replace b by |b|ϕ(mi) in the worst case, thus executing a very low number of multiplications, namely ⌈log2mi⌉ for each residue digit.A general architecture is also proposed and evaluated, as a possible implementation of the proposed method for the modular exponentiation. [Copyright &y& Elsevier]

An electromagnetic Mach principle is introduced, considering the zero-point radiation field as being produced by all visible matter in the universe, and at the same time being responsible for the quantum fluctuations of matter. An order-of-magnitude estimate allows thus to derive a relation between Planck's constant (as a measure of the strength of the field fluctuations) and cosmological constants, which is well satisfied by the accepted values of the intervening quantities. [ABSTRACT FROM AUTHOR]