scholarly journals Addition and subtraction of integers in codes operands with negative zero

2020 ◽  
Vol 69 (2) ◽  
Author(s):  
O. V. Samoshchenko ◽  
◽  
O. A. Zolotukhina
Keyword(s):  
Initial Data ◽  
Input Data ◽  
Analytical Description ◽  
Negative Numbers ◽  
Source Data ◽  
Computer Representation ◽  
Addition And Subtraction ◽  
Encoding Method ◽  
Positive Properties ◽  
True Values

Code on the outputs of adder binary numbers described as the remainder of the sum the initial data on the adder module is equal to output carry weight. An original technique for synthesizing a way of operands representation in the addition and subtraction schemes of integers in a code with a negative zero was developed, which is based on the representation the source data in the form a remainder on the adder module. A method of computer representation for integer numbers is proposed, in which the codes of posi-tive and negative numbers are formed by the same procedure. The property of duality the addition and sub-traction operations on the initial data in the code with a negative zero have justified analytically. Areas of allowable results values for the correct input data addition and subtraction operations are determined. It is identified combination of the adder output signals, which determine the presence and polarity the adder bit grid overflow. It is shown that designed fixing scheme bit grid overflow of adder outputs invariant with re-spect to operations of addition and subtraction of source data with a negative zero code. For the analytical description of arithmetic operations on integer numbers represented with the proposed encoding method, a technique of calculating the sum and difference of numbers using the biased supplementary code has been proposed. Analytically substantiated, that the technique makes the scheme of the operational adder homoge-neous. The rules for establishing the correctness of the addition and subtraction operations of the integers given in the proposed encoding form are given. For true values of the initial arguments, the sums and the differences codes ranges are obtained, and the rules for positive and negative overflows identification are proposed. The original usage of a common numerical bias during the operands encoding, that evinces itself in the advantages of basic computer operations technical implementation, predetermines positive properties in practical implementations of more complex arithmetical actions.

scholarly journals A Study of Errors and Misconceptions in the Learning of Addition and Subtraction of Directed Numbers in Grade 8

SAGE Open ◽  
2016 ◽  
Vol 6 (4) ◽  
pp. 215824401667137 ◽  
Author(s):  
Judah Paul Makonye ◽  
Josiah Fakude
Keyword(s):  
Conceptual Understanding ◽  
Teaching And Learning ◽  
Study Data ◽  
Negative Numbers ◽  
Teaching And Learning Mathematics ◽  
Learning Mathematics ◽  
Number Lines ◽  
Addition And Subtraction ◽  
Logical Errors ◽  
Grade 8

The study focused on the errors and misconceptions that learners manifest in the addition and subtraction of directed numbers. Skemp’s notions of relational and instrumental understanding of mathematics and Sfard’s participation and acquisition metaphors of learning mathematics informed the study. Data were collected from 35 Grade 8 learners’ exercise book responses to directed numbers tasks as well as through interviews. Content analysis was based on Kilpatrick et al.’s strands of mathematical proficiency. The findings were as follows: 83.3% of learners have misconceptions, 16.7% have procedural errors, 67% have strategic errors, and 28.6% have logical errors on addition and subtraction of directed numbers. The sources of the errors seemed to be lack of reference to mediating artifacts such as number lines or other real contextual situations when learning to deal with directed numbers. Learners seemed obsessed with positive numbers and addition operation frames—the first number ideas they encountered in school. They could not easily accommodate negative numbers or the subtraction operation involving negative integers. Another stumbling block seemed to be poor proficiency in English, which is the language of teaching and learning mathematics. The study recommends that building conceptual understanding on directed numbers and operations on them must be encouraged through use of multirepresentations and other contexts meaningful to learners. For that reason, we urge delayed use of calculators.

scholarly journals About stability of mixed integer optimization problems under perturbations of input data of vector criterion

2021 ◽  
pp. 7-11
Author(s):  
Tetiana Lebedeva ◽  
Natalia Semenova ◽  
Tetiana Sergienko
Keyword(s):  
Initial Data ◽  
Input Data ◽  
Optimization Problems ◽  
Point Of View ◽  
Mixed Integer ◽  
Stability Of Solutions ◽  
Integer Optimization ◽  
Mixed Integer Optimization ◽  
Small Perturbations ◽  
Vector Criterion

The article is devoted to the study of the influence of uncertainty in initial data on the solutions of mixed integer optimization vector problems. In the optimization problems, including problems with vector criterion, small perturbations in initial data can result in solutions strongly different from the true ones. The problem of stability of the indicated tasks is studied from the point of view of direct coupled with her question in relation to stability of solutions belonging to some subsets of feasible set.

scholarly journals Low-precision Logarithmic Number Systems

2021 ◽  
Vol 18 (4) ◽  
pp. 1-25
Author(s):  
Syed Asad Alam ◽  
James Garland ◽  
David Gregg
Keyword(s):  
Input Data ◽  
Global Optimum ◽  
Average Error ◽  
Conversion Error ◽  
Real Numbers ◽  
Number Systems ◽  
External Sources ◽  
Addition And Subtraction ◽  
Base Selection ◽  
Logarithmic Number

Logarithmic number systems (LNS) are used to represent real numbers in many applications using a constant base raised to a fixed-point exponent making its distribution exponential. This greatly simplifies hardware multiply, divide, and square root. LNS with base-2 is most common, but in this article, we show that for low-precision LNS the choice of base has a significant impact. We make four main contributions. First, LNS is not closed under addition and subtraction, so the result is approximate. We show that choosing a suitable base can manipulate the distribution to reduce the average error. Second, we show that low-precision LNS addition and subtraction can be implemented efficiently in logic rather than commonly used ROM lookup tables, the complexity of which can be reduced by an appropriate choice of base. A similar effect is shown where the result of arithmetic has greater precision than the input. Third, where input data from external sources is not expected to be in LNS, we can reduce the conversion error by selecting a LNS base to match the expected distribution of the input. Thus, there is no one base that gives the global optimum, and base selection is a trade-off between different factors. Fourth, we show that circuits realized in LNS require lower area and power consumption for short word lengths.

Subtraction of Positive and Negative Numbers: The Difference and Completion Approaches with Chips

2008 ◽  
Vol 14 (1) ◽  
pp. 21-23
Author(s):  
Alfinio Flores
Keyword(s):  
Basic Principle ◽  
Negative Numbers ◽  
Addition And Subtraction ◽  
The Difference

Diverse contexts such as take away, comparison, and completion give rise to subtraction problems. The take-away interpretation of subtraction has been explored using two-colored chips to help students understand addition and subtraction of integers (Bennett and Musser 1976). This article will show how the difference and completion interpretations of subtraction may be illustrated with colored chips to help students develop a better understanding of the subtraction of negative and positive numbers. Black chips will represent positive numbers, and white chips will represent negative numbers. The basic principle is that chips of different colors will cancel each other when joined together; this pair of chips is called a zero-pair. Several zero-pairs together will still have a zero value. Each collection in figure 1 represents zero.

scholarly journals The calculation of the boundaries of the product market in antitrust investigation

2020 ◽  
Vol 14 (79) ◽  
pp. 5-27
Author(s):  
V. A. Brodsky ◽  
I. S. Kozhukhovskiy
Keyword(s):  
Mathematical Model ◽  
Initial Data ◽  
Standard Method ◽  
Product Market ◽  
Commodity Market ◽  
Source Data

A method is proposed for calculating the product and geographical boundaries of the commodity market based on the initial data obtained by the Antimonopoly authority from sellers and buyers participating in the survey as part of the Antimonopoly investigation.The method includes special terminology, forms of representation of source data, and an economic and mathematical model of the commodity market designed to calculate the boundaries and composition of market participants.The method provides full computerization of processing of initial data received by the Antimonopoly authority from the survey participants.The main differences between the proposed method and the standard method of SSNIP (hypo thetical monopolist Test) are considered.

Hot & cold cubes

1977 ◽  
Vol 24 (1) ◽  
pp. 70-71 ◽  
Author(s):  
Stanley M. Jencks ◽  
Doald M. Peck
Keyword(s):  
Simple Model ◽  
Clear Picture ◽  
Negative Numbers ◽  
Addition And Subtraction

A simple model that portrays both addition and subtraction of positive and negative numbers is hard to find. A few years ago the authors ran across one that has been so successful it must be passed on for other teachers to use. This model lies in the realm of fanta sy yet provides an easily imaginable and crystal-clear picture of how positive and negative numbers behave when they are added or subtracted.

MATHEMATICAL METHODS IN ABŪ AL-WAFĀʾ'S ALMAGEST AND THE QIBLA DETERMINATIONS

2011 ◽  
Vol 21 (1) ◽  
pp. 1-56 ◽  
Author(s):  
Ali Moussa
Keyword(s):  
Input Data ◽  
Analytical Description ◽  
High Accuracy ◽  
Trigonometric Functions ◽  
Scientific Culture ◽  
Spherical Trigonometry ◽  
Mathematical Methods ◽  
Medieval Islam ◽  
The Given

AbstractThe problem of the Qibla was one of the central issues in the scientific culture of Medieval Islam, and to solve it properly, one needed mathematics and observation. The mathematics consisted of two parts: plane trigonometry (to construct the trigonometric tables) and spherical trigonometry (as the problem belongs to spherical astronomy). Observation and its instruments were needed to find the geographical coordinates of Mecca and the given location; these coordinates (latitude, longitude) will be the input data in the formulas of the Qibla. In his Almagest, Abū al-Wafāʾ produced a brilliant work to solve the problem. He worked on both mathematics and observation, and reached accurate and easy “modern” solutions. In plane trigonometry, he introduced the trigonometric functions with new definitions, proved the formulas for sines, approximated the sine of degree one, and thus constructed the tables of sines and tangents with high accuracy. In spherical trigonometry, he proved four new spherical theorems, including the tangent rule (which was based on the new definitions and this rule allowed him to work out the easiest solution, as will be shown). In observation, he described three instruments which he used over several years in Baghdad. This paper is a detailed technical and analytical description of Abū al-Wafāʾ's mathematical methods and the Qibla determinations, supplemented with many important original Arabic texts with translation and commentary.

Modular Mappings and Data Distribution Independent Computations

1997 ◽  
Vol 07 (02) ◽  
pp. 169-180 ◽  
Author(s):  
Lee Hyuk-Jae ◽  
José A.B. Fortes
Keyword(s):  
Initial Data ◽  
Input Data ◽  
Systematic Approach ◽  
Matrix Multiplication ◽  
Data Distribution ◽  
Parallel Computers ◽  
Linearly Independent ◽  
Zero Entry ◽  
Pattern Distribution ◽  
Data Redistribution

This paper considers the problem of writing data distribution independent (DDI) programs in order to eliminate or reduce initial data redistribution overheads for distributed memory parallel computers. The functionality and execution time of DDI programs are independent of initial data distributions. Modular mappings, which can be used to derive many equally optimal and functionally equivalent programs, are briefly reviewed. Relations between modular mappings and input data distributions are then established. These relations are the basis of a systematic approach to the derivation of DDI programs which is illustrated for matrix-matrix multiplication (c = a × b). Conditions of data distributions for which it is possible to find a modular mapping that yields a programa as efficient as Cannon's algorithm are: (1) the first row of the inverse of pattern distribution of array 'a' should be equal to be equal to the second row of the inverse of pattern distribution of array 'b', (2) the second row of the inverse of pattern distribution of array 'a' should be linearly independent of the first row of the inverse of pattern distribution of array 'b', and (3) each pattern distribution of arrays 'a', 'b', and 'c' should have at least one zero entry, respectively.

scholarly journals The use of long short-term memory and gated recurrent unit for predicting the values of geomagnetic indices

2020 ◽  
pp. 110-121
Author(s):  
В.А. Мочалов ◽  
А.В. Мочалова
Keyword(s):  
Deep Learning ◽  
Initial Data ◽  
Short Term Memory ◽  
Short Term ◽  
Forecasting Accuracy ◽  
Term Memory ◽  
Geomagnetic Indices ◽  
Source Data ◽  
Long Short Term Memory ◽  
Gated Recurrent Unit

В работе с помощью глубокого обучения рассматривается прогнозирование значений следующих геомагнитных индексов (ГИ): Dst, Kp, AE и AP. Для прогнозирования используются архитектуры долгой краткосрочной памяти (LSTM) и управляемых рекуррентных блоков (GRU). Для различных ГИ индексов анализируется функция потерь в за-висимости от периодичности исходных данных. Установлено, что чем меньше периодичность исходных данных ГИ тем точнее осуществляется прогноз следующего значения ГИ. Для анализа использовались следую-щие периоды исходных данных ГИ: час, 3 часа, сутки. In this work, with the help of deep learning, predicting the values of the following geomagnetic indices (GI) is considered: Dst, Kp, AE and Ap. For forecasting we use the architectures are long short-term memory (LSTM) and gated recurrent unit (GRU). For various GI indices, the loss function is analyzed depending on the periodicity of the source data. It has been established that forecasting accuracy increases with decreasing periodicity of the initial data of geomagnetic indices. For the analysis, the following periods of the initial GI data were used: hour, 3 hours, day. For the analysis we used hour, 3 hours and day periods of the initial GI source data.

scholarly journals Stability kernel of vector optimization problems under perturbations of criterion functions

2021 ◽  
pp. 17-23
Author(s):  
Т.Т. Lebedeva ◽  
◽  
N.V. Semenova ◽  
T.I. Sergienko ◽  
◽  
...  
Keyword(s):  
Initial Data ◽  
Vector Optimization ◽  
Input Data ◽  
Optimization Problems ◽  
Vector Optimization Problem ◽  
Vector Optimization Problems ◽  
Optimal Solutions ◽  
Vector Criterion ◽  
Criterion Functions ◽  
Conditions Of Stability

The article is devoted to the study of the influence of uncertainty in initial data on the solutions of optimiza-tion multicriterial problems. In the optimization problems, including problems with vector criterion, small per-turbations in initial data can result in solutions strongly different from the true ones. The results of the con-ducted researches allow us to extend the known class of vector optimization problems, stable with respect to in-put data perturbations in vector criterion. We are talking about stability in the sense of Hausdorff lower semicontinuity for point-set mapping that characterizes the dependence of the set of optimal solutions on the input data of the vector optimization problem. The conditions of stability against input data perturbations in vector criterion for the problem of finding Pareto optimal solutions with continuous partial criterion func-tions and feasible set of arbitrary structure are obtained by studying the sets of points that are stable belonging and stable not belonging to the Pareto set.

Sign in / Sign up

Export Citation Format

Share Document