Showing papers on "Coprime integers published in 1983"

Structure, Smith-MacMillan form and coprime MFDs of a rational matrix inside a region P =ω∪{∞}

Abstract: The structure of the Smith-MacMillan form of a rational matrix T(s) inside a region P=ω∪{∞} (where ω is asymmetric with respect to the real axis subset of the finite complex plane C) is determined. Algorithmic procedures based on elementary row and column operations over the euclidean ring R P(s) consisting of all rational functions with no poles in P are given. Coprimeness in P of a pair of rational matrices is studied in detail. These results lead to constructive procedures for determining the coprime in P matrix fraction descriptions of T(a).

A carry-free algorithm for finding the greatest common divisor of two integers

Abstract: We investigate a variant of the so-called “binary” algorithm for finding the GCD (greatest common divisor) of two numbers which requires no comparisons We show that when implemented with carry-save hardware, it can be used to find the modulo B inverse of an n-bit binary integer in a time proportional to n, using only registers of length proportional to n Such a hardware implementation of this algorithm set up for finding inverses with respect to a 336 bit modulus B would have applications in the currently expanding field of secure data transmission and storage In such an implementation, multiplication in linear time-both modulo B and ordinary—would come along as a by-product because multiplication can be achieved by a sequence of nine inversions, some additions and negations

When the cartesian product of two directed cycles is hyperhamiltonian

Abstract: We show that the Cartesian product Za × Zb of two directed cycles is hypo-Hamiltonian (Hamiltonian) if and only if there is a pair of relatively prime positive integers m and n with ma + nb = ab - 1 (ma + nb = ab). The result for hypo-Hamiltonian is new; that for Hamiltonian is known. These are special cases of the fact that there is a simple circuit of length p in Za × Zb if and only if there is a pair of relatively prime non-negative integers m and n with ma + nb = p ≤ ab.

Threshold-Bounded Interval Orders and a Theory of Picycles

Abstract: Given $1\leqq m\leqq n$ with m and n relatively prime if $m\geqq 2$, let $\mathcal{P} [ m,n ]$ be the class of finite partially ordered sets $( A,P )$ whose points $a,b, \cdots $ can be mapped into closed intervals with lengths in $[ m,n ]$ such that, for all $a,b \in A,aPb$ if and only if a’s interval lies completely to the right of b’s interval. A theory of picycles based on a mixture of algebraic and combinatorial ideas leads to the conclusion that each $\mathcal{P} [ m,n ]$ is axiomatizable by a universal sentence of first-order logic. Necessary and sufficient conditions for membership in $\mathcal{P} [ m,n ]$ are specified.The present results lie in sharp contrast to an earlier conclusion that the class $\mathcal{P}_n $ of finite interval orders which can be represented using no more than n interval lengths is not axiomatizable by a universal sentence when $n\geqq 2$.

On genus 2 Heegaard diagrams for the 3-sphere

Abstract: Let D be any genus 2 Heegaard diagram for the 3-sphere and (a,, a2; ?I, r2) be the cyclically reduced presentation associated with D. We shall show that ?I contains ?2 or Pi' as a subword in cyclic sense if {(P, ?2} #4 {a -', a '} holds, and that, using this property, (a1, a 2; r1, r2) can be transformed to the trivial one (al, a2; a ", a 1). By the recent positive solution of genus 2 Poincare conjec- ture, our result implies the purely algebraic, algorithmic solution to the decision problem; whether a given 3-manifold with a genus 2 Heegaard splitting is simply connected or not, equivalently, is homeomorphic to the 3-sphere or not. 1. Introduction. This is the continued work of (5) related to the experimental discovery due to Homma and Ochiai (cf. (5, 4)) which indicates the possibility of the existence of an elegant and practical algorithm for simplifying the presentations of the fundamental group associated with genus 2 Heegaard diagrams for the 3-sphere S3 by mutual substitutions (Definition 1). It is similar to Euclidean algorithm applied to relatively prime integers. In this paper, we shall establish it in complete manner (Theorem 2). The recent result announced by Thurston, Bass, Shalen, Meeks-Yau, Gordon-Litherland and others implies the positive solution of Poincare conjecture in case of Heegaard genus 2. Then our result implies the solution to the isomorphism problem with respect to the trivial group among all the presentations associated with genus 2 Heegaard diagrams. In other words, it gives a simple algorithm to decide whether a given 3-manifold with a genus 2 Heegaard splitting is homeomorphic to the 3-sphere or not. In order to prove Theorem 2, we shall show a key theorem (Theorem 1) which assures the existence of the substitution realized by some Heegaard diagram for S3. Our proof of Theorem 1 is based on the new concept of fake Heegaard diagrams (Definition 4), the surgery on them (Definition 5) and the result of (4). In the next section, we shall state our results precisely and prove them in the subsequent ??3, 4. In the last ?5, we shall show some examples for supplemental remarks related to our results.

Periodic points and topological entropy of maps of the circle

Abstract: Let f be a continuous map from the circle to itself, let P(f ) denote the set of integers nl for which f has a periodic point of period ni. In this paper it is shown that the two smallest numbers in P( f ) are either coprime or one is twice the other.

Reducibilities among decision problems for HNN groups, vector addition systems and subsystems of Peano arithmetic

Abstract: Our purpose is to exhibit reducibilities among decision problems for conjugate powers in HNN groups, reachability sets of vector addition systems and sentences in subsystems of Peano arithmetic, and show that although these problems are not primitive recursively decidable, they do admit decision procedures which are primitive recursive in the Ackermann function. By the class of vector groups VA we understand the HNN groups G( P 1, ql, . p4, qk\.) given by (1) (al...,akl b; a'bPI'a= b '...,ak = b where the exponent pairs p,, q, occurring in (I) are positive and relatively prime. (For concepts and results of a group-theoretic nature not explicitly discussed here the reader should consult Lyndon and Schupp [5].) Let G be a vector group, m a positive conjugate power of I in G when b' xb'x -, where x in G is given by a positive word in the generators al,. . . ,ak, b of G (i.e. one which involves no negative exponents). The set of positive conjugate powers of I in G, or positive conjugate power set is denoted PCP(l, G). By the equality problem for positive conjugate power sets, we mean the question of determining for any integers 1, 12 and vector groups G1, G2, whether PCP(I1, G1) = PCP(12, G2). The special equality problem is to decide the equality problem in those cases where l1 = 12 and G2 arises from G1 by removing a particular generating symbol a, and its corresponding defining relation a'bP a = bq, from the presentation of G1 as in (I). The finite special equality problem is to decide the special equality problem in those cases where PCP(11, G) is finite. We identify a decision problem with the set of Godel numbers of its instances and use this identification to discuss the complexity of the problem. A function g is primitive recursive in a function f iff g is in the class obtained by primitive recursion and composition from f together with the usual initial functions. It is shown in Anshel [1] that the special equality problem for vector groups is undecidable. In contrast, we will prove THEOREM 1. The finite special equality problem for vector groups is (i) decidable but not primitive recursive, (ii) primitive recursive in the Ackermann function. Received by the editors June 27, 1981. 1980 Mathematics Subject Classification. Primary 20F10, 03D40, 03F30; Secondary 68C99. 01983 American Mathematical Society 0002-9939/83 $1.00 + $.25 per page

On the symmetry of the Smith normal form for (υ, k, λ) designs

Abstract: It is shown in this paper (Theorems 5, 6, and 7) that ifS1,S2,Sv are the invariant factors of the incidence matrix for a (υ, k, λ) design with k and λ relatively prime, then

On sums of distinct integers belonging to certain sequences

Abstract: Many papers have been devoted to the problem of representing integers as sums of distinct numbers belonging to fixed sequences. We mention for example the paper of Birch [1], in which he proves the conjecture of Erd6s concerning the representability of all large integers as sum of distinct terms of the type p,qb where p, q are coprime, the very general results of Cassels [2], which contain Birch's result, and the paper of Erd6s [3]. Here we give conditions, different from thoseof Cassels and Erd6s, on a sequence guaranteeing that all large natural numbers are sums of its distinct terms. We introduce some notation: ~r is said to be of type 6 if, for every large x, we have (x, ( l + 6 ) x ) ~ d ~ O . Let d and ~ denote sequences of natural numbers; we say that the product d ~ = {ab, aCd, bC~} is direct if ab=a'b" implies a=a', b=b'. Further, let S(~r {m, rn=al+... +av, al>a2>... >av, a j~d} , and, as usual, set d ( x ) = 2 1 . aEa~' a~x Our results are the following

On the solvability of x2 = b(mod m)

Abstract: Euler's criterion is the usual procedure used for assessing the solvability of the equation x2 = b(mod m). This criterion is applicable only when b and m are relatively prime and must be applied to each b (0≤ b < m). Described in this paper is an alternative criterion which characterizes and counts, for a general modulus m, all numbers b for which x2 = b(mod m) is solvable.