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.

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

scholarly journals Distribution of rational points on the real line

1973 ◽  
Vol 15 (2) ◽  
pp. 243-256 ◽  
Author(s):  
T. K. Sheng
Keyword(s):  
Algebraic Number ◽  
Rational Number ◽  
Real Line ◽  
Rational Points ◽  
The Other ◽  
Liouville Numbers ◽  
The Real ◽  
Other Hand ◽  
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It is well known that no rational number is approximable to order higher than 1. Roth [3] showed that an algebraic number is not approximable to order greater than 2. On the other hand it is easy to construct numbers, the Liouville numbers, which are approximable to any order (see [2], p. 162). We are led to the question, “Let Nn(α, β) denote the number of distinct rational points with denominators ≦ n contained in an interval (α, β). What is the behaviour of Nn(α, + 1/n) as α varies on the real line?” We shall prove that and that there are “compressions” and “rarefactions” of rational points on the real line.

The Range of a Gap Series

1975 ◽  
Vol 18 (5) ◽  
pp. 753-754
Author(s):  
J. S. Hwang
Keyword(s):  
Natural Number ◽  
Gap Series ◽  
Image Position

Theorem. Letbe a function holomorphic in the disk, wherep is a natural number andIfthen then f(z) assumes every complex value infinitely often in every sector.The purpose of this note is to prove the above result. To do this, we first observe that from the condition a<∞, we can easily show that the derivative f′(z) satisfying

scholarly journals The determination of Fibonacci groups

1974 ◽  
Vol 11 (1) ◽  
pp. 11-14 ◽  
Author(s):  
A.M. Brunner
Keyword(s):  
Natural Number ◽  
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Fibonacci groups are the groupswhere r is a natural number. The groups F(2, 8) and F(2, 10) are shown to he infinite, thus leaving F(2, 9) as the only Fibonacci group whose finiteness or infiniteness has not been determined.

Register machine proof of the theorem on exponential diophantine representation of enumerable sets

1984 ◽  
Vol 49 (3) ◽  
pp. 818-829 ◽  
Author(s):  
J. P. Jones ◽  
Y. V. Matijasevič
Keyword(s):  
Natural Number ◽  
Polynomial Time ◽  
Simple Proof ◽  
Diophantine Equations ◽  
Polynomial Equation ◽  
Exponential Diophantine Equations ◽  
Recursively Enumerable ◽  
Image Position ◽  
The Right ◽  
Exponential Relation

The purpose of the present paper is to give a new, simple proof of the theorem of M. Davis, H. Putnam and J. Robinson [1961], which states that every recursively enumerable relation A(a1, …, an) is exponential diophantine, i.e. can be represented in the formwhere a1 …, an, x1, …, xm range over natural numbers and R and S are functions built up from these variables and natural number constants by the operations of addition, A + B, multiplication, AB, and exponentiation, AB. We refer to the variables a1,…,an as parameters and the variables x1 …, xm as unknowns.Historically, the Davis, Putnam and Robinson theorem was one of the important steps in the eventual solution of Hilbert's tenth problem by the second author [1970], who proved that the exponential relation, a = bc, is diophantine, and hence that the right side of (1) can be replaced by a polynomial equation. But this part will not be reproved here. Readers wishing to read about the proof of that are directed to the papers of Y. Matijasevič [1971a], M. Davis [1973], Y. Matijasevič and J. Robinson [1975] or C. Smoryński [1972]. We concern ourselves here for the most part only with exponential diophantine equations until §5 where we mention a few consequences for the class NP of sets computable in nondeterministic polynomial time.

scholarly journals Feature Recognition for Manufacturability Analysis

1994 ◽  
Author(s):  
William C. Regli ◽  
Satyandra K. Gupta ◽  
Dana S. Nau
Keyword(s):  
Time Complexity ◽  
Material Removal ◽  
Feature Recognition ◽  
Solid Modeling ◽  
Cad Model ◽  
Worst Case ◽  
Machining Features ◽  
Automated Recognition ◽  
Wide Range ◽  
Complete Set

Abstract While automated recognition of features has been attempted for a wide range of applications, no single existing approach possesses the functionality required to perform manufacturability analysis. In this paper, we present a methodology for taking a CAD model of a part and extracting a set of machinable features that contains the complete set of alternative interpretations of the part as collections of MRSEVs (Material Removal Shape Element Volumes, a STEP-based library of machining features). The approach handles a variety of features including those describing holes, pockets, slots, and chamfering and filleting operations. In addition, the approach considers accessibility constraints for these features, has an worst-case algorithmic time complexity quadratic in the number of solid modeling operations, and modifies features recognized to account for available tooling and produce more realistic volumes for manufacturability analysis.

scholarly journals An estimate for trigonometrical sums over square-free integers with a constant number of prime factors

1967 ◽  
Vol 15 (4) ◽  
pp. 249-255
Author(s):  
Sean Mc Donagh
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Prime Number ◽  
Rational Number ◽  
Large Integer ◽  
Constant Number ◽  
The Real ◽  
Prime Factors ◽  
Squarefree Number ◽  
Image Position

1. In deriving an expression for the number of representations of a sufficiently large integer N in the formwhere k: is a positive integer, s(k) a suitably large function of k and pi is a prime number, i = 1, 2, …, s(k), by Vinogradov's method it is necessary to obtain estimates for trigonometrical sums of the typewhere ω = l/k and the real number a satisfies 0 ≦ α ≦ 1 and is “near” a rational number a/q, (a, q) = 1, with “large” denominator q. See Estermann (1), Chapter 3, for the case k = 1 or Hua (2), for the general case. The meaning of “near” and “arge” is made clear below—Lemma 4—as it is necessary for us to quote Hua's estimate. In this paper, in Theorem 1, an estimate is obtained for the trigonometrical sumwhere α satisfies the same conditions as above and where π denotes a squarefree number with r prime factors. This estimate enables one to derive expressions for the number of representations of a sufficiently large integer N in the formwhere s(k) has the same meaning as above and where πri, i = 1, 2, …, s(k), denotes a square-free integer with ri prime factors.

Planning the Shortest Path in Cluttered Environments: A Review and a Planar Convex Hull-Based Approach

2019 ◽  
Vol 19 (4) ◽  
Author(s):  
Nafiseh Masoudi ◽  
Georges M. Fadel ◽  
Margaret M. Wiecek
Keyword(s):  
Path Planning ◽  
Shortest Path ◽  
Time Complexity ◽  
Optimal Path ◽  
Optimal Solution ◽  
Free Graph ◽  
Worst Case ◽  
Cluttered Environments ◽  
Planar Convex ◽  
Polyhedral Obstacles

Abstract Routing or path-planning is the problem of finding a collision-free and preferably shortest path in an environment usually scattered with polygonal or polyhedral obstacles. The geometric algorithms oftentimes tackle the problem by modeling the environment as a collision-free graph. Search algorithms such as Dijkstra’s can then be applied to find an optimal path on the created graph. Previously developed methods to construct the collision-free graph, without loss of generality, explore the entire workspace of the problem. For the single-source single-destination planning problems, this results in generating some unnecessary information that has little value and could increase the time complexity of the algorithm. In this paper, first a comprehensive review of the previous studies on the path-planning subject is presented. Next, an approach to address the planar problem based on the notion of convex hulls is introduced and its efficiency is tested on sample planar problems. The proposed algorithm focuses only on a portion of the workspace interacting with the straight line connecting the start and goal points. Hence, we are able to reduce the size of the roadmap while generating the exact globally optimal solution. Considering the worst case that all the obstacles in a planar workspace are intersecting, the algorithm yields a time complexity of O(n log(n/f)), with n being the total number of vertices and f being the number of obstacles. The computational complexity of the algorithm outperforms the previous attempts in reducing the size of the graph yet generates the exact solution.

The subnormal structure of general linear groups

1986 ◽  
Vol 99 (3) ◽  
pp. 425-431 ◽  
Author(s):  
L. N. Vaserstein
Keyword(s):  
Normal Subgroup ◽  
Natural Number ◽  
Associative Ring ◽  
Inverse Image ◽  
General Linear ◽  
Linear Groups ◽  
Canonical Homomorphism ◽  
General Linear Groups ◽  
Subnormal Structure ◽  
Image Position

Let A be an associative ring with 1. For any natural number n, let GLnA denote the group of invertible n by n matrices over A, and let EnA be the subgroup generated by all elementary matrices ai, j, where aεA and 1 ≤ i ≡ j ≤ n. For any (two-sided) ideal B of A, let GLnB be the kernel of the canonical homomorphism GLnA→GLn(A/B) and Gn(A, B) the inverse image of the centre of GLn(A/B) (when n > 1, the centre consists of scalar matrices over the centre of the ring A/B). Let EnB denote the subgroup of GLnB generated by its elementary matrices, and let En(A, B) be the normal subgroup of EnA generated by EnB (when n > 2, the group GLn(A, B) is generated by matrices of the form ai, jbi, j(−a)i, j with aA, b in B, i ≡ j, see [7]). In particuler,is the centre of GLnA

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