scholarly journals Acceleration of Euclidean Algorithm and Rational Number Reconstruction

2003 ◽  
Vol 32 (2) ◽  
pp. 548-556 ◽  
Author(s):  
Xinmao Wang ◽  
Victor Y. Pan
Keyword(s):  
Rational Number ◽  
Euclidean Algorithm

Is the Euclidean Algorithm Optimal Among its Peers?

2004 ◽  
Vol 10 (3) ◽  
pp. 390-418 ◽  
Author(s):  
Lou Van Den Dries ◽  
Yiannis N. Moschovakis
Keyword(s):  
Natural Number ◽  
Rational Number ◽  
Time Complexity ◽  
Euclidean Algorithm ◽  
Worst Case ◽  
On Demand ◽  
Recursive Program ◽  
Remainder Function ◽  
Proper Formulation ◽  
Image Position

The Euclidean algorithm on the natural numbers ℕ = {0,1,…} can be specified succinctly by the recursive programwhere rem(a, b) is the remainder in the division of a by b, the unique natural number r such that for some natural number q,It is an algorithm from (relative to) the remainder function rem, meaning that in computing its time complexity function cε (a, b), we assume that the values rem(x, y) are provided on demand by some “oracle” in one “time unit”. It is easy to prove thatMuch more is known about cε(a, b), but this simple-to-prove upper bound suggests the proper formulation of the Euclidean's (worst case) optimality among its peers—algorithms from rem:Conjecture. If an algorithm α computes gcd (x,y) from rem with time complexity cα (x,y), then there is a rational number r > 0 such that for infinitely many pairs a > b > 1, cα (a,b) > r log2a.

Spontaneous focusing on quantitative relations as a predictor of the development of rational number conceptual knowledge.

2016 ◽  
Vol 108 (6) ◽  
pp. 857-868 ◽  
Author(s):  
Jake McMullen ◽  
Minna M. Hannula-Sormunen ◽  
Eero Laakkonen ◽  
Erno Lehtinen
Keyword(s):  
Rational Number ◽  
Conceptual Knowledge

STRUCTURAL AND LOGIC DIAGRAM OF RESEARCH ON THE MILITARY-ECONOMIC SUBSTANTIATION OF THE RATIONAL NUMBER OF ARMED FORCES OF UKRAINE IN THE PROCESS OF FORMING THE PROGRAM OF THEIR DEVELOPMENT

2019 ◽  
Vol 2 (12) ◽  
pp. 102-110
Author(s):  
V. Nazarkin ◽  
O. Semenenko ◽  
A. Efimenko ◽  
V. Ivanov
Keyword(s):  
Rational Number ◽  
Armed Forces ◽  
The State ◽  
Limiting Factors ◽  
Development Programs ◽  
Power Structures ◽  
Scientific Substantiation ◽  
Conducting Research ◽  
Excess Number ◽  
The Military

The task of choosing the rational number of power structures is always one of the main priorities of any political leadership of the state. An insufficient number of armed forces is a threat to the national security of the state; an excess number creates pressures on the development of the country's national economy. Today, when the development programs of the Armed Forces of Ukraine are being formed in the context of the practical application of their units and subunits to carry out combat missions, questions of choosing a priority approach to the formation (justification) of the rational size of the Armed Forces of Ukraine is an urgent issue. The article proposes a structure for conducting research on the development and implementation of the methodology of military-economic substantiation of the rational strength of the Armed Forces of Ukraine in the system of defense planning of Ukraine in the formation of programs for their development for the medium and long term. The main objectives of this methodology are: scientific substantiation of the range of the necessary strength of the Armed Forces of Ukraine for the period of the program of their development; the choice of the indicator of the rational size of the Armed Forces of Ukraine according to the years of the program from a certain range of its changes; military-economic substantiation of this number under the influence of various limiting factors. The development and implementation of such a methodology will increase the efficiency of the formation and implementation of development programs of the Armed Forces of Ukraine, as well as the efficiency of using public funds for the development of power structures.

scholarly journals On the integer part of the reciprocal of the Riemann zeta function tail at certain rational numbers in the critical strip

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
WonTae Hwang ◽  
Kyunghwan Song
Keyword(s):  
Zeta Function ◽  
Rational Number ◽  
Riemann Zeta Function ◽  
Rational Numbers ◽  
Critical Strip ◽  
Integer Part ◽  
Integral Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove that the integer part of the reciprocal of the tail of $\zeta (s)$ ζ ( s ) at a rational number $s=\frac{1}{p}$ s = 1 p for any integer with $p \geq 5$ p ≥ 5 or $s=\frac{2}{p}$ s = 2 p for any odd integer with $p \geq 5$ p ≥ 5 can be described essentially as the integer part of an explicit quantity corresponding to it. To deal with the case when $s=\frac{2}{p}$ s = 2 p , we use a result on the finiteness of integral points of certain curves over $\mathbb{Q}$ Q .

scholarly journals Public-key cryptosystem based on invariants of diagonalizable groups

2017 ◽  
Vol 9 (1) ◽  
Author(s):  
František Marko ◽  
Alexandr N. Zubkov ◽  
Martin Juráš
Keyword(s):  
Linear Algebra ◽  
Finite Fields ◽  
Number Fields ◽  
Euclidean Algorithm ◽  
Public Key ◽  
Public Key Cryptosystem ◽  
Finite Rings

AbstractWe develop a public-key cryptosystem based on invariants of diagonalizable groups and investigate properties of such a cryptosystem first over finite fields, then over number fields and finally over finite rings. We consider the security of these cryptosystem and show that it is necessary to restrict the set of parameters of the system to prevent various attacks (including linear algebra attacks and attacks based on the Euclidean algorithm).

Current situation and development direction of sports industry in colleges and universities based on euclidean algorithm

2021 ◽  
pp. 1-7
Author(s):  
Shunjiang Ma ◽  
Gaicheng Liu ◽  
Zhiwu Huang
Keyword(s):  
Computer Technology ◽  
Colleges And Universities ◽  
Status Quo ◽  
Euclidean Algorithm ◽  
Test Results ◽  
Sports Industry ◽  
Development Direction ◽  
The Status ◽  
Traditional Sports ◽  
And Function

With the development of sports in colleges and universities, the research on innovation reform of sports industry has been deepened. Therefore, based on the above situation, a study of the status quo and development direction of sports industry in colleges and universities based on the Euclid algorithm is proposed. In the research here, according to the traditional sports industry concept to sum up, and then according to the advantages of computer technology to deal with the relevant data. In order to realize good overlap between data, an application of Euclidean algorithm is proposed. In the test of Euclidean algorithm, the efficiency and function of the algorithm are tested comprehensively, and the test results show that the research is feasible.

scholarly journals ON -K2 FOR NORMAL SURFACE SINGULARITIES

1998 ◽  
Vol 09 (06) ◽  
pp. 653-668 ◽  
Author(s):  
HAO CHEN ◽  
SHIHOKO ISHII
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Rational Number ◽  
Accumulation Point ◽  
Normal Surface ◽  
Accumulation Points ◽  
Surface Singularities

In this paper we show the lower bound of the set of non-zero -K2 for normal surface singularities establishing that this set has no accumulation points from above. We also prove that every accumulation point from below is a rational number and every positive integer is an accumulation point. Every rational number can be an accumulation point modulo ℤ. We determine all accumulation points in [0, 1]. If we fix the value -K2, then the values of pg, pa, mult, embdim and the numerical indices are bounded, while the numbers of the exceptional curves are not bounded.

Call for Manuscripts for the 2014 Focus Issue: Rational Number Sense – October 2012

2012 ◽  
Vol 18 (3) ◽  
pp. 189
Keyword(s):  
Rational Number ◽  
Number Sense ◽  
Rational Numbers

This call for manuscripts is requesting articles that address how to make sense of rational numbers in their myriad forms, including as fractions, ratios, rates, percentages, and decimals.

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