Showing papers on "Coprime integers published in 1978"

On a generalized Szegö- Levinson realization algorithm for optimal linear predictors based on a network synthesis approach

Abstract: We show the correspondence between the following: i) the Darlington synthesis procedure, well known in the network theory literature; ii) J -coprime (MFD) representations of transfer functions; iii) the Szego-Levinson theory of orthogonal polynomials on the unit circle; iv) the theory of generalized Schur indices; and v) the theory of NevanlinnaPick approximations. More specifically: i) and ii) are equivalent, ii) and iv) are also equivalent for a subclass of i) and ii), while v) provides a nice framework for the study of convergence properties. The paper produces also an exact and an approximate construction procedure of prediction and modeling filters of a general (ARMA) type, for stationary processes, and shows its convergence.

On the use of right-coprime factorizations in distributed feedback systems containing unstable subsystems

Abstract: In this paper, we prove that every rational transfer function matrix has a right-coprime factorization in {\cal Q}^{n \times n} , and that a right-coprime factorization contains all the information about the domain and range of an unstable operator. We also derive a general necessary and sufficient condition for feedback stability that is applicable even to nonlinear systems, and show that right-coprime factorizations arise naturally when this general condition is applied to linear time-invariant systems.

Vector groups and the equality problem for vector addition systems

Abstract: Our purpose is to demonstrate that results concerning the equality problem for vector addition systems, may be used to establish the decidabilityand undecidability of decision problems associated with the class of HNN extensions of the infinite cyclic group. We call these groups 'vector groups.' By vector groups we understand the HNN groups G(p1, q1, l , Pk, qk) given by (I) ~(a,, **.,. a, b; a-lb 1P lal =b bq 1 , ak1 Pk It = bq k), where the exponent pairs pi, qi occurring in (I) are positive and relatively prime. In [2] the conjugacy problem for vector groups was reduced to the reachability problem for self-dual vector addition systems, and subsequently an algebraic solution of both problems was presented in [3]. Here we demonstrate that recent results concerning the equality problem for vector addition systems, having surprising algebraic consequences for vector groups. Let G be a vector group and call m a conjugate power of 1 in G when b' = xblx -1, x in G and a positive conjugate power if in addition x is given by a positive word in the generators a1, ... , ak, b of G (i.e., one which involves no negative exponents). The set of (positive) conjugate powers of 1 in G is called a (positive) conjugate power set. Here we consider the question as to whether the (positive) conjugate powers of 1 in G1 and G2 coincide where G1 is given in (I) and G2 arises from G1 by removing a particular defining relation, a-1' a = bpi b. We will prove: THEOREM A. It is decidable whether the removal of a particular defining relation from a vector group changes a conjugate power set. THEOREM B. It is undecidable whether the removal of a particular defining relation from a vector group changes a positive conjugate power set. For G = G(p1, q1, ... , Pk, qk) let CP(l, G) and PCP(I, G) denote respectively the conjugate and positive conjugate powers of 1 on G. From [1, pp. 22-23], it follows that m lies in CP(l, G) if and only if there is a sequence 1 = 11, ..., In = m such that li+ 1 = lipjlqj) or li+ 1 = li(qjlp1). Similarly, m lies in PCP(I, G) if and only if l = 11, .n. , In = m where li 1 = li(plqi). These observations allow us to restrict our attention to those positive integers 1 whose prime divisors are among the prime divisors of the exponents of G. Received March 17, 1977. AMS (MOS) subject classifications (1970). Primary 20E05, 20F05; Secondary 02G05, 68A30. Copyright ? 1978, American Mathematical Society

A Short Proof of Quadratic Reciprocity

Abstract: Let p = 2p'+ 1 and q = 2q'+ 1 be odd primes, p #q. For a and q relatively prime, we have aq 1 (mod q) by Fermat's theorem, so aq'_ ? 1 (mod q), and we define the Legendre symbol (a/q) to be + 1 or 1 in the two cases. If a_x2 (mod q), then a -=1 (mod q), so the q' squares, or "quadratic residues" (mod q) satisfy (a/ q) = 1 and the q' non-squares satisfy (a/ q) = 1. The famous quadratic reciprocity theorem, discovered by Euler and Legendre and first proved by Gauss (Werke II, p. 3-8), states that for odd primes p, q,