Computer Progress in Czechoslovakia

Increasing the speed of computer structures is associated with the involvement of new principles based on the use of the number system of residual classes. When performing a number of computational operations using a non-positional number system of residual classes, the problem of direct conversion of numbers from the positional number system to the number system of residual classes and the reverse conversion of numbers from the number system of residual classes to the positional one is solved. system. A number of works contain methods for implementing these operations of varying complexity, but their solution is far from perfect. The results of the study indicate the possibility of obtaining a more efficient solution that simplifies the practical implementation of operations for direct conversion of numbers from the positional number system to the number system of residual classes and inverse conversion of numbers. The aim of the study is to analyze the implementation of the operations of direct conversion of numbers from the positional number system to the number system of residual classes and the reverse conversion of numbers. The tools of the research methodology are system analysis, number theory and the Chinese remainder theorem. The research methodology consists in the analysis of special tables of residues by modules. An algorithm for performing complex operations based on the use of a polyadic code of a number presented in the number system of residual classes is proposed. A simple and efficient approach is proposed using iterative table operations with residuals. For each value of the given residuals, the constants are obtained by a simple selection from the corresponding table.The method underlying the implementation of these complex operations is to sum the chosen constants to the desired value, while the corresponding constants are subtracted from the values of the known remainders until the latter vanishes. The method is algorithmically simple and uncomplicated for circuit implementation, and it should be considered as one of the areas of research to improve the efficiency of computations in the system of residual classes.

The report considers correction codes with mutually non-prime bases (CMNPB). To date, almost no one has been engaged in a deep study of the corrective properties of systems of residual classes (SRC), the bases of which are not mutually prime numbers. Such a system has certain corrective properties, which makes it necessary to assess the possibility and feasibility of using CMNPB to increase the reliability of computing in computer systems (CS). For the purpose of further research and development of CMNPB, we will consider a number of scientific statements, the use of the results of which will allow us to more fully study the corrective properties of codes with non-prime bases. Algorithms for monitoring and correcting errors in the SRC with mutually non-prime in pairs bases have been developed. Using these algorithms makes it relatively easy to implement a procedure for detecting and correcting one-time errors. Note that by the simplicity of the design of the decoder circuits, CMNPB have no analogues in the positional number systems. This is achieved by limiting the class of possible correctable errors, by introducing additional hardware redundancy in the representation of code words.

We introduce a novel method for complex number representation. The proposed, redundant complex binary number system (RCBNS) is developed by combining a redundant binary number and a complex number in base (-1+j). Donald (1960) and Walter Penny (1964, 1965) represented complex numbers using base -j and (-1+j) in the classified algorithmic models. A redundant complex binary number system consists of both real and imaginary-radix number systems that form a redundant integer digit set. This system is formed by using complex radix of (-1+j) and a digit set of /spl alpha/=3, where /spl alpha/ assumes a value of -3, -2, -1, 0, 1, 2, 3. The arithmetic operations of complex numbers with this system treat the real and imaginary parts as one unit. The carry-free addition has the advantage of redundancy in number representation in the arithmetic operations. Results of the arithmetic operations are in the RCBNS form. The two methods for conversion from the RCBNS form to the standard binary number form have been presented. The RCBNS reduces the number of steps required to perform complex number arithmetic operations, thus enhancing the speed.

A number system uses a set of digits and sign to represent all the numbers. Different number systems use different amount of digits and sign. To compare the leanness of a number system, the cardinality of the complete set of digits and sign are employed in this work. The 01 binary number system is used by almost all the modern computers. It can represent all the real numbers by two digits 0, 1 and a sign (-), which means its cardinality is 3. Thus, the 01 binary system is not a true binary system. After reviewing all the existing number systems it is found that no true binary system exists for real numbers. All the current binary number systems either don’t have radix 2, or have cardinality 3. In this paper, a true binary system for real numbers, with both radix and cardinality 2, is invented based on the principle of Yin-Yang. Surprisingly, this binary system can not only represent all real numbers without using a sign, but also perform arithmetic and Boolean operations without any problem. Furthermore, due to its radix being 2, this true binary system is compatible with the 01 binary system and many operations are interchangeable between the two systems. It is expected that this true binary system will be applied in the fields of number representation, computer science and cryptography.

In order to effectively plan and develop projects involving microcomputers, the industrial engineer requires background information on number systems, binary arithmetic, and binary logic. In this tutorial article, a basic introduction to these topics is provided. Specific topics include: positional number systems, conversion between number systems, addition and subtraction of binary numbers, the two's complement of negative numbers, the ASCII and Gray codes, the basic logic functions (AND, OR, NOT, NAND, NOR, XOR), and some applications of basic logic functions, including a decoder, an adder, and a binary latch. This is the first of an integrated series of articles dealing with concepts and techniques for interfacing microcomputers to industrial devices.

Nowadays it is strongly believed that in a large number of spiral galaxies there are regular magnetic fields. Their existence is proved by measurements of the Faraday rotation of polarization plane of electromagnetic waves which are received on modern radio telescopes. The theoretical description of magnetic fields generation is connected with the mean field dynamo mechanism. It is based on joint action of the ?-effect and differential rotation. At the first stage of evolution, magnetic fields enlarge exponentially with the growth rate described by the largest eigenvalue of the corresponding differential operator. If the field becomes comparable with the equipartition value, the nonlinear terms become more important, and they are connected with saturation of the field growth. The non-linear system of equations has several stationary points, and some of them are stable. According to the contrast structures theory for a quite small turbulent viscosity and some initial conditions, in different regions there will be magnetic fields of opposite directions, and they will be divided by narrow transition layers. Such features, for example, characterize the magnetic field of the Milky Way. Most of the existing works describe reversals for quite moderate distances from the galaxy center. However, it was shown previously that the magnetic field can principally be generated also in outer parts of galaxies, which are situated at distances of more than 10 kpc from the rotation axis. It is interesting to describe the appearance of reversals in such parts of galaxies. In this work we have studied the dynamo action in outer parts of the Milky-Way-like galaxies. It is shown that it is possible to generate opposite magnetic fields there. We have used random initial conditions to obtain spatial reversals. Because of a large volume of computations, they were carried out by using graphics processing units (GPUs).

Chapter 5 - Computer Numbering Systems

Complex numbers play a very important role in various applications of electrical and computer engineering. These days, arithmetic operations dealing with these numbers rely on a divide-and-conquer technique wherein a complex number is broken into its real and imaginary parts and then, after representing each part in binary number system, operation is carried out on each part as if part of the real arithmetic. Thus, addition of two complex numbers requires two separate additions and their multiplication requires four individual multiplications, one subtraction and one addition. In an effort to reduce the number of arithmetic operations within the realm of complex arithmetic, binary number system with base (-1+j), called complex binary number system, has been proposed in the literature which allows a complex number to be represented as single-unit instead of two separate units as in the base-2 binary number system. In this paper, the effects of shift-left operations on complex binary numbers have been examined

Even though decimal arithmetic is pervasive in financial and commercial transactions, computers are still implementing almost all arithmetic calculations using binary arithmetic. As chip real estate becomes cheaper it is becoming likely that more computer manufacturers will provide processors with decimal arithmetic engines. Programming languages and databases are expanding the decimal data types available while there has been little change in the base hardware. As a result, each language and application is defining a different arithmetic and few have considered the efficiency of hardware implementations when setting requirements. We propose a decimal format which meets the requirements of existing standards for decimal arithmetic and is efficient for hardware implementation. We propose this specification in the hope that designers will consider providing decimal arithmetic in future microprocessors and that future decimal software specifications will consider hardware efficiencies.

The article describes the book of the Italian mathematician Francesco Brunetti "Arimmetica binomica...", published in 1746. Examples of numbers from the book, examples of addition, subtraction, multiplication, division and extraction of the square root in the binary system, as well as binary values of the logarithms of a number of numbers are given. The book also presents operations on fractions and mixed binary numbers. Attention is drawn to the original way of writing binary numbers proposed by Brunetti. According to this method, instead of the entire binary number, only the numbers (in the form of decimal digits) of those digits in which units are represented (with numbering starting from the right) are recorded. Ordinal numbers of digits greater than 9, Brunetti proposes to designate as their last digit (0, 1, ..., 9), writing a comma (prime) to the right of all such digits, and numbers greater than 19 — similarly with two commas, etc. With such a notation, four arithmetic operations on binary numbers are greatly facilitated (the article provides an example of adding the numbers written accordingly).

У статті пропонується до розгляду концепція розробки інформаційно-керуючих систем на основі біноміальних систем числення, в основі якої лежить нова інформаційна технологія обробки даних на базі біноміальних чисел. В рамках даної технології проводиться формування кодових об'єктів біноміальної структури, їх перетворення в двійкові біноміальні числа і зворотне перетворення результуючих кодових об'єктів у вихідні дані на основі двійкової системи числення. Наведені підходи до вирішення типових інформаційних задач з використанням біноміальних чисел.

Technologies and applications of the field-programmable gate array (FPGAs) and digital signal processing (DSP) require both new customizable number systems and new data formats. This paper introduces a new class of parameterized number systems, namely the generalized Phi number system (GPNS). By selecting appropriate parameters, the new system derives the traditional Phi number system, binary number system, beta encoder, and other commonly used number systems. GPNS also creates new opportunities for developing customized number systems, multimedia security systems, and image decomposition and enhancement systems. A new image enhancement algorithm is also developed by integrating the GPNS-based bit-plane decomposition with Parameterized Logarithmic Image Processing (PLIP) models. Simulation results are given to demonstrate the GPNS's performance.

The author has been teaching programming to Gomel schoolchildren for more than four decades. For the last twenty-five years, training has been focused primarily on preparing for programming competitions from school to international competitions (IOI). Since 1997, 15 students of the author have won a total of 26 medals at IOI 1997 - 2023. The article contains a description of the teaching methods and tools used by the author to teach the topic “Number Systems”. An essential technical basis for training is the instrumental distance learning system DL.GSU.BY, was created and developed under the leadership of the author from 1999 to the present time. In this article, the following questions are sequentially considered (with the solution of the corresponding original problems from the American and Russian Internet Olympiads): binary number system, binary counting, converting numbers from decimal to binary, and vice versa. Hexadecimal and octal number systems. Conversion of numbers from one number system to another.

Algorithms are developed to compute the number of active devices for residue number system (RNS) and binary number system (BNS) signal processing architectures. The precharged CMOS technology is used in order to build the filter architecture. The ROM lookup table and two's complement multiplier are used as the computational cells in the RNS and BNS, respectively. A direct implementation transversal systolic filter architecture is considered in RNS and BNS algorithms.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A new method to perform binary division and take square roots is presented. It produces the Gray Code representation of the quotient/root, by performing a sequence of shift, subtract and absolute value operations on the dividend and the divisor or the square and the root, and converts the result to the binary representation at the same time. The method has a simpler synchronous hardware implementation than the non-restoring algorithm[1]. The division method may be regarded as the digital equivalent of an A/D conversions method proposed earlier[2][3]. As the division and square rooting operations require rather similar shift-and-subtract logic circuits, they can be incorporated into a multi-function arithmetic unit along the lines of [4].