Greatest common divisors with moving targets and consequences for linear recurrence sequences

In this correspondence, a new viewpoint is proposed for estimating an image from its distorted versions in presence of noise without the a priori knowledge of the distortion functions. In z-domain, the desired image can be regarded as the greatest common polynomial divisor among the distorted versions. With the assumption that the distortion filters are finite impulse response (FIR) and relatively coprime, in the absence of noise, this becomes a problem of taking the greatest common divisor (GCD) of two or more two-dimensional (2-D) polynomials. Exact GCD is not desirable because even extremely small variations due to quantization error or additive noise can destroy the integrity of the polynomial system and lead to a trivial solution. Our approach to this blind deconvolution approximation problem introduces a new robust interpolative 2-D GCD method based on a one-dimensional (1-D) Sylvester-type GCD algorithm. Experimental results with both synthetically blurred images and real motion-blurred pictures show that it is computationally efficient and moderately noise robust.

We present new methods for computing the greatest common right divisor of polynomial matrices. These methods involve the recently studied generalized Sylvester and generalized Bezoutian resultant matrices, which require no polynomial operations. They can provide a row proper greatest common right divisor, test for coprimeness and calculate dual dynamical indices. The generalized resultant matrices are developments of the scalar Sylvester and Bezoutian resultants and many of the familiar properties of these latter matrices are demonstrated to have analogs with the properties of the generalized resultant matrices for matrix polynomials.

Let $(x_n)_{n\geq 0}$ be a linear recurrence sequence of order $k\geq 2$ satisfying $$ \begin{align*}x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}\end{align*} $$ for all integers $n\geq k$ , where $a_1,\dots ,a_k,x_0,\dots , x_{k-1}\in \mathbb {Z},$ with $a_k\neq 0$ . In 2017, Sanna posed an open question to classify primes p for which the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb {Q}_p$ . In a recent paper, we showed that if the characteristic polynomial of the recurrence sequence has a root $\pm \alpha $ , where $\alpha $ is a Pisot number and if p is a prime such that the characteristic polynomial of the recurrence sequence is irreducible in $\mathbb {Q}_p$ , then the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb {Q}_p$ . In this article, we answer the problem for certain linear recurrence sequences whose characteristic polynomials are reducible over $\mathbb {Q}$ .

A matrix Euclidean algorithm induced by state space realization

SummaryThe implementation of cryptography algorithms is constantly observing threat from multiple techniques in breaking secure system. The current trend in breaking the secure system of cryptography is by power analysis technique. In order to break the secure key, the energy consumed in the digital circuit during computation is measured. This technique is commonly known as differential power analysis, explored by the hackers to break the secure systems. To circumvent these type of attacks, it is necessary to explore different designs, which dissipate less energy. Ideally, reversible circuits dissipate zero energy. We present a new reversible architecture for greatest common divisor (GCD) computation using modified Binary GCD algorithm. We present the generalized design methodology of reversible GCD computation unit. We compare the proposed GCD computation design with the existing design. The proposed reversible GCD architecture takes less number of iterations compared with the existing GCD architecture in the literature. The proposed design outperforms the existing GCD design in terms of Quantum Cost, Gate Count, and Ancilla Inputs. Copyright © 2016 John Wiley & Sons, Ltd.

We give a probabilistic algorithm for computing the greatest common divisor (GCD) of two polynomials over an algebraic number field. We can compute the GCD using O(llog5(l)) expected binary operations where l is the size of the GCD given by standard estimations. Since we require time Ω(l) just to write down the GCD, the algorithm is close to optimal.

Information systems are developing and improving very fast. The requirements for synchronization for such systems also increase. In such systems linear recurrent sequences are widely used for synchronization. In this paper filtration algorithms of linear recurrent sequences sequences are synthesized. The synthesized algorithms can be classified as quasi-noncoherent (subincoherent) algorithms, as opposed to quasi-coherent algorithms, which use phase estimate of the received signal, or to noncoherent algorithms that use averaging over the entire region of phase uncertainty. The results of computer simulation of developed algorithms for various signal-to-noise ratios and accumulation time are presented.

This paper presents a preliminary report on a new algorithm for computing the Greatest Common Divisor (GCD) of two multivariate polynomials over the integers. The algorithm is strongly influenced by the method used for factoring multivariate polynomials over the integers. It uses an extension of the Hensel lemma approach originally suggested by Zassenhaus for factoring univariate polynomials over the integers. We point out that the cost of the Modular GCD algorithm applied to sparse multivariate polynomials grows at least exponentially in the number of variables appearing in the GCD. This growth is largely independent of the number of terms in the GCD. The new algorithm, called the EZ (Extended Zassenhaus) GCD Algorithm, appears to have a computing bound which in most cases is a polynomial function of the number of terms in the original polynomials and the sum of the degrees of the variables in them. Especially difficult cases for the EZ GCD Algorithm are described. Applications of the algorithm to the computation of contents and square-free decompositions of polynomials are indicated.

The study of the Least Common Multiple (LCM) and the Greatest Common Divisor (GCD) explores the fundamental field of number theory. In addition to being essential to theoretical number theory, these mathematical ideas have extensive applications in computer science, algebra, and real-world settings like network routing and scheduling. Using techniques like the Euclidean algorithm for GCD calculation and prime factorization for LCM, the study starts by examining the fundamental concepts and notations of GCD and LCM. Then the author goes further into the detail about how these ideas are used to polynomial equations, highlighting how they may be used to solve challenging issues and simplify algebraic expressions. The studys practical applications, which demonstrate how GCD and LCM optimize scheduling tasks and boost network routing efficiency, are presented in the conclusion. The results of this study highlight the broad impact and adaptability of LCM and GCD in theoretical and applied mathematics, providing valuable insights for further research in these domains.

We propose an algorithm to compute a greatest common divisor (GCD) of univariate polynomials with large integer coefficients on Graphics Processing Units (GPUs). At the highest level, our algorithm relies on modular techniques to decompose the problem into subproblems that can be solved separately. Next, we employ resultant-based or matrix algebra methods to compute a GCD of each modular image in parallel. Our approach exhibits block structure to distribute the computation of a single modular GCD over several thread blocks, and thus to remove any hardware limitations on the maximal size of polynomials that can be handled. To "combine" all modular results, we have adopted Mixed-Radix Conversion (MRC) algorithm running on the GPU. Our approach shows a significant speed-up over host-based GCD algorithm from Maple 13.

Polynomial matrices play an important part in linear system calculations. New computational procedures are given for calculation of the Smith normal form and the greatest common right divisor of polynomial matrices. It is shown how suitable transformation matrices can be determined for the calculation of the Smith normal form, and how a set of polynomial matrix multipliers can be calculated for the greatest common right divisor problem. Neither of these algorithms relies on explicit calculation of the greate3t common divisor of polynomials. Limited numerical experience has shown that the3e algorithms are both fast and accurate.

The best known linear recurrence sequence is the Fibonacci Sequence 1,1,2,3,5,8,..., where each term (after the first two terms) is a sum of the two preceding terms. We may extend it to become a “doubly infinite” sequence $$ ..., - 8,5, - 3,2, - 1,1,0,1,1,2,3,5,8,... . $$ .

We investigate the decidability of termination, reachability, coverability and deadlock-freeness of Petri nets endowed with a hierarchy of places, and with inhibitor arcs, reset arcs and transfer arcs that respect this hierarchy. We also investigate what happens when we have a mix of these special arcs, some of which respect the hierarchy, while others do not. We settle the decidability status of the above four problems for all combinations of hierarchy, inhibitor, reset and transfer arcs, except the termination problem for two combinations. For both these combinations, we show that deciding termination is as hard as deciding the positivity problem on linear recurrence sequences -- a long-standing open problem.

The objective of this is to show the application of Euclid's Lemma to calculate the Greatest Common Divisor in Elementary Education and evaluate the receptivity that the students had the method. We studied and wrote about the division in integers, divisibility and properties involved, the theorem euclidian division, the greatest common divisor, Euclid's Lemma, the euclidean algorithm, the Fundamental Theorem of Arithmetic and also on the Diophantine Equations they are an interesting application of the Greatest Common Divisor. We use a short course on Greatest Common Divisor in two elementary school classes and present some results obtained.

We investigate the decidability of the monadic second-order (MSO) theory of the structure (N; <, P1,…,Pk), for various unary predicates P1,…,Pk ⊆ N. We focus in particular on 'arithmetic' predicates arising in the study of linear recurrence sequences, such as fixed-base powers Powk = {kn : n ∈ N}, k-th powers Nk = {nk : n ∈ N}, and the set of terms of the Fibonacci sequence Fib = {0, 1, 2, 3, 5, 8, 13,…} (and similarly for other linear recurrence sequences having a single, non-repeated, dominant characteristic root). We obtain several new unconditional and conditional decidability results, a select sample of which are the following: