Showing papers on "Coprime integers published in 1987"

On Thue's equation

Abstract: and more precisely for the number of primitive solutions to (1.1), that is, solutions in coprime integers x, y. The first important results on the number of solutions of Thue's equation were obtained by C.L. Siegel, in the case r = 3 and in the binomial case F ( x , y ) = a x r b y r. In view of his results and comments, we should attribute to Siegel the problem to decide whether the number of primitive solutions to (1.1), for irreducible F of degree r > 3, could be bounded by a function depending only on r and h, but otherwise independent of F. In 1983, Siegel's question was answered in the affirmative by J.-H. Evertse [2]. As a special case of his results (he also treats equations in number fields), Evertse obtains the bound

On codes with spectral nulls at rational submultiples of the symbol frequency

Abstract: In digital data transmission (respectively, storage systems), line codes (respectively, recording codes) are used to tailor the spectrum of the encoded sequences to satisfy constraints imposed by the channel transfer characteristics or other system requirements. For instance, pilot tone insertion requires codes with zero mean and zero spectral density at tone frequencies. Embedded tracking/focus servo signals produce similar needs. Codes are studied with spectral nulls at frequencies f=kf_{s}/n , where f , is the symbol frequency and k, n are relatively prime integers with k \leq n; in other words, nulls at rational submultiples of the symbol frequency. A necessary and sufficient condition is given for a null at f in the form of a finite discrete Fourier transform (DFT) running sum condition. A corollary of the result is the algebraic characterization of spectral nulls which can be simultaneously realized. Specializing to binary sequences, we describe canonical Mealy-type state diagrams (directed graphs with edges labeled by binary symbols) for each set of realizable spectral nulls. Using the canonical diagrams, we obtain a frequency domain characterization of the spectral null systems obtained by the technique of time domain interleaving.

An efficient solution of the congruence x^2 + ky^2 = mpmod{n}

Abstract: The equation of the title arose in the proposed signature scheme of Ong-Schnorr-Shamir. The large integers n, k and m are given and we are asked to find any solution x, y . It was believed that this task was of similar difficulty to that of factoring the modulus n; we show that, on the contrary, a solution can easily be found if k and m are relatively prime to n . Under the assumption of the generalized Riemann hypothesis, a solution can be found by a probabilistic algorithm in O(\log n)^{2}|\log\log|k||) arithmetical steps on O(\log n) -bit integers. The algorithm can be extended to solve the equation X^{2} + KY^{2} = M \pmod{n} for quadratic integers K, M \in {\bf Z}[\sqrt{d}] and to solve in integers the equation x^{3} + ky_{3} + k^{2}z^{3} - 3kxyz = m \pmod{n} .

A connection between normalized coprime factorizations and linear quadratic regulator theory

Abstract: Given a transfer matrix described by a minimal state-space triple, a method is given for computing state-space realizations for the numerator and denominator of a normalized, stable, right coprime factorization for the transfer matrix. The method involves the solution of an algebraic Riccati equation. It allows the use of existing computational state-space algorithms in finding normalized stable right coprime factorizations, and avoids explicit calculations of spectral factors.

Sublinear parallel algorithm for computing the greatest common divisor of two integers

Abstract: The paper presents a sublinear time parallel algorithm for computing the greatest common divisor of two integers. Its running time on two n bit integers is $O({{n\log \log n} / {\log n}})$ using the weak concurrent read concurrent write model.

Finitely generated ideals in certain algebras of transfer functions for infinite-dimensional systems†

Abstract: In this note we show that in certain algebras of stable transfer functions for infinite-dimensional systems there exist finitely generated ideals that are not principal. As a consequence there exist unstable transfer functions that have no coprime factorizations.

A new result on difference sets with -1 as multiplier

Abstract: In this paper we study finite abelian groups admitting a difference set with multiplier -1. In these groups we have that each integer, which is relatively prime to the group order, is a multiplier (see [1] and Section 1 of this paper).

Multiple-burst error-correcting cyclic product codes (Corresp.)

Abstract: Let C be the cyclic product code of p single parity check codes of relatively prime lengths n_{1}, n_{2},\cdots , n_{p} (n_{1} . It is proven that C can correct 2^{P-2}+2^{p-3}-1 bursts of length n_{1} , and lfloor(\max\{p+1, \min\{2^{p-s}+s-1,2^{p-s}+2^{p-s-1}\}\}-1)/2\rfloor bursts of length n_{1}n_{2} \cdots n_{s} (2\leq s \leq p-2) . For p=3 this means that C is double-burst- n_{1} -correcting. An efficient decoding algorithm is presented for this code.

Coprime fraction computation of 2-D rational matrices

Abstract: This note presents a numerical method of computing a coprime fraction of a two-dimensional (2-D) rational matrix, not necessarily proper. It is achieved by searching the primary linearly dependent rows, in order from top to bottom, of the two generalized resultants. The procedure can be extended to the three- or higher dimensional case and the result can also be used to compute the greatest common divisor (GCD) of 2-D polynomial matrices without employing primitive factorizations which does not exist in the three- or higher dimensional case.

An efficient algorithm for reducing transfer functions to coprime forms

Abstract: A very simple and efficient algorithm is formulated for reducing any transfer function to its coprime form without long divisions.

Boundary Control of Undamped Beam using Square-Root Velocity

Abstract: The coprime factorization approach is used to design a linear time-invariant feedback controller to stabilize an undamped beam in the bounded-input-bounded-output (BIBO) sense Transfer functions are factored into coprime ratios of elements from a ring of stable causal transfer functions that includes transcendental functions of ?S We consider a pinned-free beam with a control torque at the pinned end It is shown that feedback involving "square-root velocity" (= displacement × ?S) leads to BIBO-stability However, this stable system is "ill-posed" in the sense that it can be destabilized by the introduction of an arbitrarily small delay in the feedback loop

A factorization approach to control synthesis of distributed linear systems

Abstract: The research was concerned on the task of feedback controller synthesis for a class of linear distributed systems. More specifically, optimal frequency domain compensator designs for sensitivity reduction and for robust stabilization were sought. The optimization was formulated through coprime factorizations of a given system transfer function. Coprime factorization was reduced to lumped stabilizing compensator design. Lumped compensators were shown to exist for all strictly proper distributed systems and were determined by computing stabilizing compensators for lumped systems which are close to the given distributed systems when measured by the gap metric. The use of frequency domain minimax criterion for the optimization results in a (Nehari's) best approximation problem. A closed form solution, however, is difficult to obtain for the class of systems considered, especially when the systems are unstable. This difficulty was overcome by solving a finite dimensional optimization problem, which was derived from the original one via a generalized Fourier expansion. It was shown that the resulting design through the above approach is asymptotically optimal. An explicit procedure was given for obtaining a satisfactory near optimal solution using some of the existing numerically reliable algorithms.

Sufficient conditions for BIBO robust stabilization : given by the gap metric

Abstract: A relation between coprlme fractions and the gap metric is presented. Using this result we provide some sufficient conditions for BIBO robust stabilization for a very wide class of systems. These conditions allow the plant and compensator to be disturbed simultaneously. Keywords: Robust stabllization; Gap metric; Coprime fraction.