Multiple Error Correction in Redundant Residue Number Systems: A Modified Modular Projection Method with Maximum Likelihood Decoding

Error detection and correction codes based on redundant residue number systems are powerful tools to control and correct arithmetic processing and data transmission errors. Decoding the magnitude and location of a multiple error is a complex computational problem: it requires verifying a huge number of different possible combinations of erroneous residual digit positions in the error localization stage. This paper proposes a modified correcting method based on calculating the approximate weighted characteristics of modular projections. The new procedure for correcting errors and restoring numbers in a weighted number system involves the Chinese Remainder Theorem with fractions. This approach calculates the rank of each modular projection efficiently. The ranks are used to calculate the Hamming distances. The new method speeds up the procedure for correcting multiple errors and restoring numbers in weighted form by an average of 18% compared to state-of-the-art analogs.

The summary lesson on the theme "number systems" in the web quest format

The theme "Number systems" is one of the basic in the school informatics course. The material of this theme is present in the tasks of the Main State Examination and the Unified State Examination, therefore it is very important to timely and efficiently consolidate the basic algorithms to convert numbers from one number system to another. The article provides a variant of such a summary lesson on the theme "Number systems" for the 8th grade in the format of web quest. The quest was developed in the Thinglink environment and contains several locations, each of which is associated with the convertion of numbers between different number systems. The tools for solving the quest tasks are Google Sheets and the LearningApps service, in which the author of the article has developed his own educational resources, as well as the Geogebra environment for performing constructions in one of the tasks. The presented quest can also be carried out in a computer-less form

John Horton Conway. 26 December 1937—11 April 2020

John Conway was without doubt one of the most celebrated British mathematicians of the last half century. He first gained international recognition in 1968 when he constructed the automorphism group of the then recently-discovered Leech lattice, and in so doing discovered three new sporadic simple groups. At around the same time he invented The Game of Life, which brought him to the attention of a much wider audience and led to a cult following of Lifers. He also combined the methods of Cantor and Dedekind for extending number systems to construct what Donald Knuth (ForMemRS 2003) called ‘surreal numbers’, the achievement of which Conway was probably most proud. Throughout his life he continued to make significant contributions to many branches of mathematics, including number theory, logic, algebra, combinatorics and geometry, and in his later years he teamed up with Simon Kochen to produce the Free Will theorem, which asserts that if humans have free will then, in a certain sense, so do elementary particles. In this biographical memoir I attempt to give some idea of the depth and breadth of Conway's contribution to mathematics.

Method of multiscale coincidence of pulses for measuring temporary loads from moving vehicles in bridges and overpasses

Modern development of road transport is characterized by constant updating of the nomenclature of cars (including heavy-duty vehicles), by increasing of intensity and speed of their movement, especially in large metropolises and on roads of national and local significance. This complicates the operating conditions of spans of highway bridges to some extent, leads to their damage and early wear. The deterioration of the technical condition of the structures of the spans is largely due to the increased dynamic impact of the cars on the bridges. The purpose of this article is to measure the amplitude of oscillations of a bridge structure from temporary loads of rolling stock with non-periodic long-term loading by optical methods. In accordance with this goal, it is necessary to develop a technique and device for measuring the amplitude of oscillations of the bridge structure by converting the amplitude of oscillations in time intervals and their subsequent measurement by multi-scale pulse coincidence. In a number of works, digital deployment systems that use number systems with different bases, represented by a family of deployment functions, are considered in detail. Such digital systems in general, are described by the corresponding operating scheme. According to this operating scheme, a method of multiscale matching for measuring time intervals was developed, which formed the basis for measuring the vibration amplitude of the bridge structure. The given mathematical model of calculation of the number of measurement channels was used in the synthesis of velocity transducers and linear displacements based on optical (laser) deployment systems, information conversion devices of radar velocity sensors based on the Doppler effect. Similarly, a frequency comparison device, a digital frequency meter, a digital phase meter, and a number of other frequency-time group devices were synthesized.

Typology of number systems in languages of Western and Central Siberia

This paper investigates the linguistic expression of number in seven languages from Western and Central Siberia. In a first step the number system of each language is described in detail, and afterwards the most relevant convergences and divergences of the languages are dealt with. Three particularly interesting phenomena are discussed in more detail: First, it is shown that the concept of general number, denoting noun forms underspecified for number, is able to account for a range of related phenomena (unmarked noun forms after numerals, nouns denoting paired objects). Second, singulatives in Selkup, Ket and partly Eastern Khanty are analyzed, whereby it is argued that their similar morphosyntactic and grammaticalization patterns allow for analyzing them as a contact phenomenon. Third, two splits on the animacy hierarchy between the first and second person in Dolgan as well as Chulym Turkic are presented. Finally, the results are evaluated against a broader areal-typological background, whereby it is shown that the category of number does not support any larger areal groupings within Western and Central Siberia, but that the analyzed languages rather adhere to patterns of number marking present all over Northern Eurasia.

Toolset for Supporting the Research of Lattice Based Number Expansions

The world of generalized number systems contains many challenging areas. Computer experiments often support the theoretical research. In this paper we introduce a toolset that helps to analyze some properties of lattice based number expansions. The toolset is able to (1) analyze the expansions, (2) decide the number system property, (3) classify and visualize the periodic points. The toolset is implemented in Python, published alongside with a database that stores plenty of special expansions, and is able to store the custom properties like signature, operator eigenvalues, etc. Researchers can connect to the server and request/upload data, or perform experiments on them. We present an introductory usage of the toolset and detail some results that has been observed by the toolset. The toolset can be downloaded from http://numsys.info domain.

Binomial Number System

This paper presents and first scientifically substantiates the generalized theory of binomial number systems (BNS) and the method of their formation for reliable digital signal processing (DSP), transmission, and data storage. The method is obtained based on the general theory of positional number systems (PNS) with conditions and number functions for converting BNS with a binary alphabet, also allowing to generate matrix BNS, linear-cyclic, and multivalued number systems. Generated by BNS, binomial numbers possess the error detection property. A characteristic property of binomial numbers is the ability, on their basis, to form various combinatorial configurations based on the binomial coefficients, e.g., compositions or constant-weight (CW) codes. The theory of positional binary BNS construction and generation of binary binomial numbers are proposed. The basic properties and possible areas of application of BNS researched, particularly for the formation and numbering of combinatorial objects, are indicated. The CW binomial code is designed based on binary binomial numbers with variable code lengths. BNS is efficiently used to develop error detection digital devices and has the property of compressing information.

Counting and the ontogenetic origins of exact equality

Humans are unique in their capacity to both represent number exactly and to express these representations symbolically. This correlation has prompted debate regarding whether symbolic number systems are necessary to represent exact number. Previous work addressing this question in innumerate adults and semi-numerate children has been limited by conflicting results and differing methodologies, and has not yielded a clear answer. We address this debate by adapting methods used with innumerate populations (a “set-matching” task) for 3- to 5-year-old US children at varying stages of symbolic number acquisition. In five studies we find that children’s ability to match sets exactly is related not simply to knowing the meanings of a few number words, but also to understanding how counting is used to generate sets (i.e., the cardinal principle). However, while children were more likely to match sets after acquiring the cardinal principle, they nevertheless demonstrated failures, compatible with the hypothesis that the ability to reason about exact equality emerges sometime later. These findings provide important data on the origin of exact number concepts, and point to knowledge of a counting system, rather than number language in general, as a key ingredient in the ability to reason about large exact number.