Approximate square-free part and decomposition

Finding the greatest common divisor (GCD) of two given polynomials is a basic problem in algebraic computing. The problem is usually stated as follows: given the (real or complex) coefficients of two polynomials, compute the coefficients of their greatest common divisor.

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.

A matrix Euclidean algorithm induced by state space realization

Computing the greatest common divisor of a set of polynomials is a problem which plays an important role in different fields, such as linear system, control, and network theory. In practice, the polynomials are obtained through measurements and computations, so that their coefficients are inexact. This poses the problem of computing an approximate common factor. We propose an improvement and a generalization of the method recently proposed in Guglielmi et al. (SIAM J. Numer. Anal. 55, 1456–1482, 2017), which restates the problem as a (structured) distance to singularity of the Sylvester matrix. We generalize the algorithm in order to work with more than 2 polynomials and to compute an Approximate GCD (Greatest Common Divisor) of degree k ≥ 1; moreover, we show that the algorithm becomes faster by replacing the eigenvalues by the singular values.

Fully homomorphic encryption (FHE) is a "holy grail" of cryptography. However, it is not yet adopted in practice because no known scheme is efficient. In this paper, we mainly focus on how to reduce the public key sizes in FHE. Based on Dijk et al.'s FHE scheme (DGHV) and Gentry's fully homomorhpic technology, we propose two schemes with shorter public keys by encrypting with a combination of the public key elements to replace a linear form in DGHV scheme and a quadratic form in Coron et al.'s scheme. Compared with DGHV scheme and Coron et al.'s scheme, our schemes can greatly reduce the public key sizes, which make our schemes more efficient. Furthermore, we prove the security of our schemes based on difficulties of the approximate greatest common divisor (AGCD) problem and the sparse subset sum problem.

Learning traffic signal phase and timing information from low-sampling rate taxi GPS trajectories

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.

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 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.

Based on the Lyapunov stability, an adaptive sliding mode controller for uncertain time-delay systems is presented Sliding mode motion of the closed-loop system from any initial states can asymptotically approach to zero in the finite time, which avoids need of the upper bound of perturbations. By applying free-weighting Matrices technique and delay transformation model, a delay-dependent sufficient condition is derived to guarantee the asymptotic stability of the closed-loop system. A numerical example is given to illustrate the feasibility of the proposed control methodology.

This paper addresses the motion control problem of autonomous underwater vehicles (AUVs) perturbed by unknown hydrodynamic coefficients, unmodelled dynamics and environmental disturbances. The proposed algorithm consists of an adaptive subcontroller to tackle the parametric uncertainties and a robust term to vanish the effects of unstructured uncertainties and disturbances. The resulting robust adaptive controller ensures the convergence of tracking error, without any assumption on the upper bound of perturbations in the design procedure. The closed loop stability is shown, using the Lyapunov stability theorem, and verified by various simulations.

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.

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.

This paper proposes an adaptive integral-type terminal sliding mode approach for the attitude and position tracking control of a quadrotor UAV subject to model uncertainties and external disturbances. First, an integral-type terminal sliding tracker is designed to attain the quadrotor UAV tracking performance in finite time when the upper bound of perturbations and uncertainties are known. Next, an adaptation law is proposed and a modified parameter-tuning integral-type terminal sliding mode tracking control scheme is designed to compensate of the model uncertainties and external disturbances. The stability and finite time convergence of the proposed approach is verified using the Lyapunov theory. Its performance is assessed using a simulation study encompassing various scenarios. Low chattering dynamics, fast convergence rate, and absence of singularities are the main features of the proposed approach.