Modular arithmetic before C.F. Gauss: Systematizations and discussions on remainder problems in 18th-century Germany

Bullynck, Maarten

doi:10.1016/j.hm.2008.08.009

Cited by 13 publications

(6 citation statements)

References 8 publications

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“…according to Equation (3). We derive the minimum among these differences using the Bezout identity [6]. It states that if p i and p k are integers with greatest common divisor g i,k = gcd(p k , p i ), then the integers of the form j…”

Section: B Feasibility Stagementioning

confidence: 99%

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Control Performance Optimization for Application Integration on Automotive Architectures

Minaeva

Roy

Åkesson

et al. 2021

IEEE Trans. Comput.

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Automotive software implements different functionalities as multiple control applications sharing common platform resources. Although such applications are often developed independently, the control performance of the resulting system depends on how these applications are integrated. A key integration challenge is to efficiently schedule these applications on shared resources with minimal control performance degradation. We formulate this problem as that of scheduling multiple distributed periodic control tasks that communicate via messages with non-zero jitter. The optimization criterion used is a piecewise linear representation of the control performance degradation as a function of the end-to-end latency of the application. The three main contributions of this article are: 1) a constraint programming (CP) formulation to solve this integration problem optimally on time-triggered architectures, 2) an efficient heuristic called Flexi, and 3) an experimental evaluation of the scalability and efficiency of the proposed approaches. In contrast to the CP formulation, which for many real-life problems might have unacceptably long running times, Flexi returns nearly optimal results (0.5% loss in control performance compared to optimal) for most problems with more acceptable running times.

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Section: B Feasibility Stagementioning

confidence: 99%

“…The set R r is a union of time intervals, for which resource r is occupied by already scheduled activities. For the example in Figure 6, R ECU1 = {[0, 2], [4,6], [9,11], [14,15]}. It is introduced to reduce the computation time of the procedure.…”

Section: B Feasibility Stagementioning

confidence: 99%

Control Performance Optimization for Application Integration on Automotive Architectures

Minaeva

Roy

Åkesson

et al. 2021

IEEE Trans. Comput.

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“…As n, n 0 , m and m 0 are integers, it can be concluded that a is an integral multiple of g. Then we explore the possible values for a, before which we quote the famous Bézout's identity [29]:…”

Section: Analysis Of the Combinability Between Two Flowsmentioning

confidence: 99%

A Period-Aware Routing Method for IEEE 802.1Qbv TSN Networks

Huang

Jiang

et al. 2020

Electronics

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The IEEE 802.1Qbv standard provides deterministic delay and low jitter guarantee for time-critical communication using a precomputed cyclic transmission schedule. Computing such transmission schedule requires routing the flows first, which significantly affects the quality of the schedule. So far off-the-shelf algorithms like load-balanced routing, which minimize the maximum scheduled traffic load (MSTL), have been used to accommodate more time-triggered traffic. However, they do not consider that the bandwidth utilization of periodic flows is decentralized and their criteria for bottleneck of scheduling are imprecise. In this paper, we firstly explore the combinability among different periods of flows, which can measure their ability to share bandwidth without conflict. Then, we propose a novel period-aware routing algorithm to reduce the scheduling bottleneck, thus more flows can be accommodated. The experiment results show that the success rate of scheduling is significantly improved compared to shortest path routing and load balanced routing.

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“…In [28], Bullynck shows that the circulation of medieval problems gave rise to different traditions: some problems about remainders, of Chinese and Indian origin, were transmitted from the fifteenth to the seventeenth centuries in Italian algebra books, French and German calculus books, the works of La Coss, and later in the books about recreational mathematics, which were very successful in the seventeenth century. An important example is the collection of Problèmes plaisans et délectables qui se font par les nombres by Bachet mentioned above.…”

Section: Lagrange and Number Theory Seen In Their Contextmentioning

confidence: 99%

Lagrange and the four-square theorem

Boucard

2014

Lett Mat Int

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In this article we will first provide an overview of Lagrange's arithmetic work, written between 1768 and 1777. We then focus on the proof of the four-square theorem published in 1772, and shed light on the context of its implementation by relying on other memoirs by Lagrange and Leonhard Euler. By means of this analysis, our goal is to give a concrete vision of the arithmetic, algebraic and analytical methods and tools used by Lagrange in number theory and place his arithmetic practice in the context of the second half of the eighteenth century.

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Modular arithmetic before C.F. Gauss: Systematizations and discussions on remainder problems in 18th-century Germany

Cited by 13 publications

References 8 publications

Control Performance Optimization for Application Integration on Automotive Architectures

Control Performance Optimization for Application Integration on Automotive Architectures

A Period-Aware Routing Method for IEEE 802.1Qbv TSN Networks

Lagrange and the four-square theorem

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