Number System Research Articles

A geometric interpretation of the Peano Axioms as a didactic strategy for teaching and learning natural numbers

Peano’s axioms are a set of propositions that allow a formal definition of the set of natural numbers. In these axioms, Peano rescues the essence of counting objects and materializes it by assuming the existence of a set, of a function in this set (which he calls successor) and an axiom of induction. In this work a geometrical representation and interpretation of each of these axioms is made, which allows understanding the independence and necessity of each one of them, which facilitates giving the conceptual rigor of them by making use of mathematical language. Finally, using Euclidean geometry, a geometric construction of a system of natural numbers on a straight line is made.

Peculiarities of Applying Partial Convolutions to the Computation of Reduced Numerical Convolutions

A method is proposed that reduces the computation of the reduced digital convolution operation to a set of independent convolutions computed in Galois fields. The reduced digital convolution is understood as a modified convolution operation whose result is a function taking discrete values in the same discrete scale as the original functions. The method is based on the use of partial convolutions, reduced to computing a modulo integer q0, which is the product of several prime numbers: q0=p1p2…pn. It is shown that it is appropriate to use the expansion of the number q0, to q=p0p1p2…pn, where p0 is an additional prime number, to compute the reduced digital convolution. This corresponds to the use of additional digits in the number system used to convert to partial convolutions. The inverse procedure, i.e., reducing the result of calculations modulo q to the result corresponding to calculations modulo q0, is provided by the formula that used only integers proved in this paper. The possibilities of practical application of the obtained results are discussed.

It’s mathematics all the way down: revealing mathematical activity in non-formal learning spaces

We offer this synthesized framework as a tool to reveal mathematical activity in a non-formal making-space. In particular, we connect research at different grain sizes to illustrate and explain how mathematics plays a crucial, if often implicit, role in making activities. We begin by describing the Approximate Number System and the Ratio-Processing System, and explaining how those systems connect to both embodied cognition and Thompson’s (1994) conceptualization of quantities. Then, we examine how prediction and anticipation relate, with a particular emphasis on how social feedback guided the emergent mathematical activity. We offer this framework as a way to perceive, account for, and understand the multi-layered nature of experiencing mathematical activities.

The role of motor effort on the sensorimotor number system.

The integration of numerical information with motor processes has emerged as a fascinating area of investigation in both animal and human cognition. The interest in a sensorimotor number system has recently generated neurophysiological and psychophysical evidence which combine to highlight the importance of motor functions in the encoding of numerical information. Nevertheless, several key questions remain, such as the influence of non-numerical motor parameters over numerical perception. Here we tested the role of physical effort, a parameter positively correlated with the number of actions, in modulating the link between hand-actions and visual numerosity perception. Effort was manipulated during sensorimotor adaptation as well as during a new actions-estimation paradigm. The results of Experiment 1 shows that physical effort in the absence of actions (passive effort) is not sufficient to activate the sensorimotor number system, indicating that self-produced actions are instead necessary. Further experiments demonstrated that effort is marginally integrated during motor adaptation (Experiment 2) but discarded when estimating the number of self-produced hand actions (Experiment 3). Overall, the results indicate that the sensorimotor number system is largely fed by the number of discrete actions rather than the amount of effort but also indicates that effort (under specific circumstances) might be integrated. These findings provide novel insights into the sensorimotor numerical integration, paving the way for future investigations, such as on its functional role.

Hazard Identification and Risk Assessment for Sustainable Shipyard Floating Dock Operations: An Integrated Spherical Fuzzy Analytical Hierarchy Process and Fuzzy CoCoSo Approach

Background: This study investigated the process of selecting sustainable safety protocols for floating dock operations in shipyards by identifying potential workplace risks in emergency situations. Thirteen occupational hazards for shipyard floating dock operations were identified through a literature review and expert discussions. Methods: We incorporated four risk elements (consequence: C, frequency: F, probability: P, and number of people at risk: NP) from the Fine–Kinney and Hazard Rating Number System (HRNS) approaches as the risk assessment criteria. We obtained the importance weights of the risk assessment criteria via the Spherical Fuzzy Analytical Hierarchy Process (SF-AHP) and extended the Combined Compromise Solution (CoCoSo) method within the fuzzy framework to prioritize occupational hazards. This study demonstrated the practicality and efficiency of the proposed emergency risk assessment model for shipyard floating dock operations through a case example of occupational risk assessment. Results: The analysis results show that H4 is the occupational hazard with the highest priority, with a score of 3.553. H4 represents the hazard associated with insufficient access to the entire pool area. The second and third most important hazards are the inability of cranes to move freely in and out of the berthing dock and the inability to dispatch emergency teams. These hazards, denoted H1 and H12, follow closely behind with scores of 3.391 and 3.344, respectively. H10 is deemed the least concerning hazard, with a score of 1.802. Conclusions: Professionals can handle complex and uncertain risk assessment data more flexibly using the proposed system, which excels in accurately organizing occupational hazards.

Tamper Detection and Self-recovery in a Visual Secret Sharing based Security Mechanism for Medical Records.

Medical records contain highly sensitive patient information. These medical records are significant for better research, diagnosis, and treatment. However, ensuring secure medical records storage is paramount to protect patient confidentiality, integrity, and privacy. Conventional methods involve encrypting and storing medical records in third-party clouds. Such storage enables convenient access and remote consultation. This cloud storage poses single-point attack risks and may lead to erroneous diagnoses and treatment. To address this, a novel (n,n)VSS scheme is proposed with data embedding, permutation ordered binary number system, tamper detection, and self-recovery mechanism. This approach enables the reconstruction of medical records even in the case of tampering. The tamper detection algorithm ensures data integrity. Simulation results demonstrate the superiority of proposed method in terms of security and reconstruction quality. Here, security analysis is done by considering attacks such as brute force, differential, and tampering attacks. Similarly, the reconstruction quality is evaluated using various human visual system parameters. The results show that the proposed technique provides a lower bit error rate ( ≈ 0), high average peak signal-to-noise ratio ( ≈ 35 dB), high structured similarity ( ≈ 1), high text embedding rate ( ≈ 0.7 BPP), and lossless reconstruction in the case of attacks.

About Absolute Convergence of Fourier Series of Almost Periodic Functions

The current stage of development of the theory of almost periodic functions is characterized by a desire for analysis and processing of a huge amount of accumulated scientific and practical material. The theory of almost periodic functions arose in the 20-30 s of the twentieth century; currently, extensive literature has accumulated on various issues of this theory. Long before the creation of the general theory of almost periodic functions, the outstanding Riga mathematician P. Bol drew attention to such functions. For functions of many variables f(x<sub>1</sub>, x<sub>2,</sub>...x<sub>p</sub>), Bol considered the corresponding multiple Fourier series and, in p-dimensional Euclidean space, a straight line passing through the origin: x<sub>1</sub>=a<sub>1</sub> t, x<sub>2</sub>=a<sub>2</sub> t,..., x<sub>p</sub>=a<sub>p</sub>t, where a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>p</sub> - some real, non-zero numbers. Considering the value of the function f(x<sub>1</sub>, x<sub>2,</sub>...x<sub>p</sub>) on this line, he obtains a function of one variable φ(t) = f(a<sub>1</sub> t, a<sub>2</sub> t,...a<sub>p</sub> t) and proves that this function is almost periodic. With some choice of numbers a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>p</sub> - it may happen that this function is periodic. However, if the numbers a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>p</sub> are linearly independent, then you can easily make sure that the function will not be a periodic function. Further development of the problem was carried out by the French mathematician E. Escalangon. However, the main significant drawback of the results of Bol and Escalangon was that from the very beginning, starting with the definition of almost-periodic functions, they introduced into consideration a fixed system of numbers a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>p</sub> associated with the almost-period (τ). This drawback was eliminated by the Danish mathematician G. Bohr, who developed in general terms the theory of continuous almost-periodic functions. Bohr's research in its methods was closely related to Bohl's research. However, Bohr did not impose restrictions such as Bohl’s inequality in advance for the almost period. The results obtained by Bol and Bohr were based on the deep connection between almost periodic functions and periodic functions of many variables. The article examines the question of sufficient conditions for the absolute and uniform convergence of Fourier series of uniform almost periodic functions in the case when the Fourier exponents have a single limit point at zero, i.e. λ<sub>k</sub>→0 (k→∞). In this case, the Laplace transform is used for the first time as a structural characteristic.

Redundant Residue Number System (RRNS) Di-Base Table for SOLiD Sequencing

The next-generation sequencing (NGS) methodology, sequencing by oligonucleotide ligation and detection (SOLiD) uses a di-base table, a somewhat unusual method, to decode sequences. Its coding scheme is based on the binary number system. The di-base table is not connected to the genetic code, nor is the coding scheme structured in the space of an entire number system. Gamow also revealed the hidden attribute of a 4 × 4 code for the di-base table, supporting his proposal of a 4 × 4 × 4 codons for the genetic code. Consideration for digital applications has focused more on the Residue Number System (RNS) and Redundant Residue Number System (RRNS) lately. Consequently, an RRNS di-base table based on the number tree concept is designed. The designed RRNS di-base table deviates from the canonical di-base table but retains every attribute necessary for effective SOLiD decoding. It shares a close relationship with the RNS-Genetic code and this presents a compelling argument for creating a single instrument that possesses the capabilities of both the genetic code and the di-base table.

Elementary fractal geometry. 4. Automata-generated topological spaces

Finite automata were used to determine multiple addresses in number systems and to find topological properties of self-affine tiles and finite type fractals. We join these two lines of research by axiomatically defining automata which generate topological spaces. Simple examples show the potential of the concept. Spaces generated by automata are topologically self-similar. Two basic algorithms are outlined. The first one determines automata for all $k$-tuples of equivalent addresses from the automaton for double addresses. The second one constructs finite topological spaces which approximate the generated space. Finally, we discuss the realization of automata-generated spaces as self-similar sets.

Influence of atmospheric wind on the propagation of radio waves

Background. It is necessary to study the influence of the physical characteristics of the atmosphere, in particular, wind on atmospheric turbulence and, consequently, on the characteristics of the radio signal it is shown. Aim. The dependence of the time-spectral function of the radio signal energy flux on the wind speed in the troposphere in the antenna plane is found. Methods. A method of transition from the Cartesian coordinate system in the antenna plane to the polar coordinate system of wave numbers has been developed. Based on this method the relationship between the Fourier spectral function of the correlation moment and the representation of the Bessel function is found. For the Fourier spectral function of the correlation moment, the previously obtained solution of the differential equation for the fluctuations of the eikonal amplitude of electromagnetic wave in a turbulent atmosphere at the front of the electromagnetic wave at the coordinate of the receiving antenna is used. Using the inverse Fourier transform the relationship between the time-spectral function of the radio signal energy flux and the time-correlation function of this flux is found. Results. Based on the study of the time-correlation function of the radio signal energy flow its relationship with the two-point correlation moment characterizing the fluctuations of the eikonal amplitude of the radio signal is found. To analyze the effect of wind, a turbulence model was used, reflecting the inertial turbulence interval, in which the energy flow from larger turbulent vortices to smaller vortices is determined by the viscous dissipation of the smallest vortices. Conclusion. Numerical calculations have shown that such a wind in the antenna plane blows away turbulent vortices in this plane, bettering the quality of the received radio signal.

A blueprint for precise and fault-tolerant analog neural networks

Analog computing has reemerged as a promising avenue for accelerating deep neural networks (DNNs) to overcome the scalability challenges posed by traditional digital architectures. However, achieving high precision using analog technologies is challenging, as high-precision data converters are costly and impractical. In this work, we address this challenge by using the residue number system (RNS) and composing high-precision operations from multiple low-precision operations, thereby eliminating the need for high-precision data converters and information loss. Our study demonstrates that the RNS-based approach can achieve ≥99% FP32 accuracy with 6-bit integer arithmetic for DNN inference and 7-bit for DNN training. The reduced precision requirements imply that using RNS can achieve several orders of magnitude higher energy efficiency while maintaining the same throughput compared to conventional analog hardware with the same precision. We also present a fault-tolerant dataflow using redundant RNS to protect the computation against noise and errors inherent within analog hardware.

Reliablity and Security for Fog Computing Systems

Fog computing (FC) is a distributed architecture in which computing resources and services are placed on edge devices closer to data sources. This enables more efficient data processing, shorter latency times, and better performance. Fog computing was shown to be a promising solution for addressing the new computing requirements. However, there are still many challenges to overcome to utilize this new computing paradigm, in particular, reliability and security. Following this need, a systematic literature review was conducted to create a list of requirements. As a result, the following four key requirements were formulated: (1) low latency and response times; (2) scalability and resource management; (3) fault tolerance and redundancy; and (4) privacy and security. Low delay and response can be achieved through edge caching, edge real-time analyses and decision making, and mobile edge computing. Scalability and resource management can be enabled by edge federation, virtualization and containerization, and edge resource discovery and orchestration. Fault tolerance and redundancy can be enabled by backup and recovery mechanisms, data replication strategies, and disaster recovery plans, with a residual number system (RNS) being a promising solution. Data security and data privacy are manifested in strong authentication and authorization mechanisms, access control and authorization management, with fully homomorphic encryption (FHE) and the secret sharing system (SSS) being of particular interest.

Valuations, completions, and hyperbolic actions of metabelian groups

AbstractActions on hyperbolic metric spaces are an important tool for studying groups, and so it is natural, but difficult, to attempt to classify all such actions of a fixed group. In this paper, we build strong connections between hyperbolic geometry and commutative algebra in order to classify the cobounded hyperbolic actions of numerous metabelian groups up to a coarse equivalence. In particular, we turn this classification problem into the problems of classifying ideals in the completions of certain rings and calculating invariant subspaces of matrices. We use this framework to classify the cobounded hyperbolic actions of many abelian‐by‐cyclic groups associated to expanding integer matrices. Each such action is equivalent to an action on a tree or on a Heintze group (a classically studied class of negatively curved Lie groups). Our investigations incorporate number systems, factorization in formal power series rings, completions, and valuations.

Characterization of quadratic ε−CNS polynomials

In this paper, we give characterization of quadratic ε-canonical number system (ε−CNS) polynomials for all values ε∈[0,1). Our characterization provides a unified view of the well-known characterizations of the classical quadratic CNS polynomials (ε=0) and quadratic SCNS polynomials (ε=1/2). This result is a consequence of our new characterization results of ε-shift radix systems (ε−SRS) in the two-dimensional case and their relation to quadratic ε−CNS polynomials.

Area–time–energy efficient architecture of CBNS‐based fast Fourier transform

AbstractA new approach for implementing the fast Fourier transform (FFT) using complex binary number system (CBNS) is presented in this paper. The advantage of using CBNS is that instead of processing the real and imaginary parts of a complex number separately, we can process both as a single entity. A total of four FFT architectures have been developed, which are 16‐, 64‐, 256‐, and 1024‐point FFT architectures. The proposed FFT is designed using Verilog‐HDL and implemented using a Virtex‐7 field‐programmable gate array (FPGA). The maximum clock rate achieved by the proposed FPGA‐based design is 376.1 MHz. The FPGA resource utilization of the FFT is less than the state‐of‐the‐art FFT designs. An application‐specific integrated circuit (ASIC) for the FFT is also designed. The normalized energy per FFT of the proposed ASIC is 0.14 nJ, and the normalized area per FFT of the ASIC is 1.08 mm2 which is less than other existing designs.

Fixed-point Encoding and Architecture Exploration for Residue Number Systems

Fixed-point Encoding and Architecture Exploration for Residue Number Systems

The Hyperreals enlargement sets and its application on compact topology

This paper presents a study of hyperreal numbers and their relations to real numbers. The hyperreals are a number system extension of the real number system. With this system, simpler and more intuitively natural definition (topological notions of special points and compactness), proofs (Heine-Borel theorem), and new concepts (limited, unlimited numbers, enlargement sets, and halos) of mathematical interest are offered.

Statistical learning and mathematics knowledge: the case of arithmetic principles

Statistical learning—an unconscious cognitive process used to extract regularities—is well-established as a fundamental mechanism underlying learning. Yet, despite the prominence of patterns in the number system and operations, little is known about the relation between statistical learning and mathematics knowledge. This study examined the associations among statistical learning, executive control, and arithmetic knowledge among first graders (N = 54). The relations varied by operation. For addition, children with greater statistical learning capacity responded more quickly to problems that were part of a principle (i.e., commutativity) pair than to unrelated problems, even after accounting for baseline performance, executive control, and age. For subtraction, results indicated an interaction between children's baseline subtraction performance and their statistical learning on accuracy. These findings provide an impetus for testing new models of mathematics learning that include statistical learning as a potentially important mechanism.

Over-the-Air Computation Based on Balanced Number Systems for Federated Edge Learning

Over-the-Air Computation Based on Balanced Number Systems for Federated Edge Learning

Low-Latency Preprocessing Architecture for Residue Number System via Flexible Barrett Reduction for Homomorphic Encryption

Low-Latency Preprocessing Architecture for Residue Number System via Flexible Barrett Reduction for Homomorphic Encryption