bivariate polynomials
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2021 ◽  
Vol 32 (1) ◽  
Author(s):  
Johanna Lercher ◽  
Hans-Peter Schröcker

AbstractWe consider bivariate polynomials over the skew field of quaternions, where the indeterminates commute with all coefficients and with each other. We analyze existence of univariate factorizations, that is, factorizations with univariate linear factors. A necessary condition for existence of univariate factorizations is factorization of the norm polynomial into a product of univariate polynomials. This condition is, however, not sufficient. Our central result states that univariate factorizations exist after multiplication with a suitable univariate real polynomial as long as the necessary factorization condition is fulfilled. We present an algorithm for computing this real polynomial and a corresponding univariate factorization. If a univariate factorization of the original polynomial exists, a suitable input of the algorithm produces a constant multiplication factor, thus giving an a posteriori condition for existence of univariate factorizations. Some factorizations obtained in this way are of interest in mechanism science. We present an example of a curious closed-loop mechanism with eight revolute joints.


2021 ◽  
Author(s):  
Kabutakapua Kakanda ◽  
Zhaolong Han ◽  
Bao Yan ◽  
Narakorn Srinil ◽  
Dai Zhou

Abstract The mechanics of offshore mooring lines are described by a set of nonlinear equations of motion which have typically been solved through a numerical finite element or finite difference method (FEM or FDM), and through the lumped mass method (LMM). The mooring line nonlinearities are associated with the distributed drag forces depending on the relative velocities of the environmental flow and the structure, as well as the axial dynamic strain-displacement relationship given by the geometric compatibility condition of the flexible mooring line. In this study, a semi analytical-numerical novel approach based on the power series method (PSM) is presented and applied to the analysis of offshore mooring lines for renewable energy and oil and gas applications. This PSM enables the construction of analytical solutions for ordinary and partial differential equations (ODEs and PDEs) by using series of polynomials whose coefficients are determined, depending on initial and boundary conditions. We introduce the mooring spatial response as a vector in the Lagrangian coordinate, whose components are infinite bivariate polynomials. For case studies, a two-dimensional mooring line with fixed-fixed ends and subject to nonlinear drag, buoyancy and gravity forces is considered. The introduced boundary and initial conditions enable the analysis of an equilibrium or steady-state of a catenary-like mooring line configuration with variable slenderness and flexibility. Polynomials’ coefficients computation is performed with the aid of a MATLAB package. Numerical results of mooring line configurations and resultant tensions are presented for deep-water applications, and compared with those obtained from a semi-analytical and finite element model. The PSM applied to the mooring line in the present study is efficient and more computationally robust than traditional numerical methods. The PSM can be directly applied to the dynamic analysis of mooring lines.


2021 ◽  
Vol 27 (2) ◽  
pp. 70-78
Author(s):  
Renata Passos Machado Vieira ◽  
Milena Carolina dos Santos Mangueira ◽  
Francisco Regis Vieira Alves ◽  
Paula Maria Machado Cruz Catarino

In this article, a study is carried out around the Perrin sequence, these numbers marked by their applicability and similarity with Padovan’s numbers. With that, we will present the recurrence for Perrin’s polynomials and also the definition of Perrin’s complex bivariate polynomials. From this, the recurrence of these numbers, their generating function, generating matrix and Binet formula are defined.


2021 ◽  
Vol 47 (2) ◽  
Author(s):  
Costanza Conti ◽  
Mariantonia Cotronei ◽  
Demetrio Labate ◽  
Wilfredo Molina

AbstractWe present a new method for the stable reconstruction of a class of binary images from a small number of measurements. The images we consider are characteristic functions of algebraic domains, that is, domains defined as zero loci of bivariate polynomials, and we assume to know only a finite set of uniform samples for each image. The solution to such a problem can be set up in terms of linear equations associated to a set of image moments. However, the sensitivity of the moments to noise makes the numerical solution highly unstable. To derive a robust image recovery algorithm, we represent algebraic polynomials and the corresponding image moments in terms of bivariate Bernstein polynomials and apply polynomial-generating, refinable sampling kernels. This approach is robust to noise, computationally fast and simple to implement. We illustrate the performance of our reconstruction algorithm from noisy samples through extensive numerical experiments. Our code is released open source and freely available.


Author(s):  
SOHAM BASU

Abstract Without resorting to complex numbers or any advanced topological arguments, we show that any real polynomial of degree greater than two always has a real quadratic polynomial factor, which is equivalent to the fundamental theorem of algebra. The proof uses interlacing of bivariate polynomials similar to Gauss's first proof of the fundamental theorem of algebra using complex numbers, but in a different context of division residues of strictly real polynomials. This shows the sufficiency of basic real analysis as the minimal platform to prove the fundamental theorem of algebra.


2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Chingfang Hsu ◽  
Lein Harn ◽  
Shan Wu ◽  
Lulu Ke

Secret sharing (SS) schemes have been widely used in secure computer communications systems. Recently, a new type of SS scheme, called the secure secret reconstruction scheme (SSRS), was proposed, which ensures that the secret can only be recovered by participants who present valid shares. In other words, if any outside adversary participated in the secret reconstruction without knowing any valid share, the secret cannot be recovered by anyone including the adversary. However, the proposed SSRS can only prevent an active attacker from obtaining the recovered secret, but cannot prevent a passive attacker from obtaining the secret since exchange information among participants is unprotected. In this paper, based on bivariate polynomials, we propose a novel design for the SSRS that can prevent both active and passive attackers. Furthermore, we propose a verification scheme which can verify all shares at once, i.e., it allows all shareholders to efficiently verify that their shares obtained from the dealer are generated consistently without revealing their shares and the secret. The proposed scheme is really attractive for efficient and secure secret reconstruction in communications systems.


2019 ◽  
Vol 16 (2) ◽  
pp. 1
Author(s):  
Shamsatun Nahar Ahmad ◽  
Nor’ Aini Aris ◽  
Azlina Jumadi

Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated with the entries of the resultant matrix of the system under consideration. This paper presents some conditions for the homogeneous coordinates of certain system of bivariate polynomials through the construction and implementation of the Sylvester-Bèzout hybrid resultant matrix formulation. This basis of the implementation of the Bèzout block applies a combinatorial approach on a set of linear inequalities, named 5-rule. The inequalities involved the set of exponent vectors of the monomials of the system and the entries of the matrix are determined from the coefficients of facets variable known as brackets. The approach can determine the homogeneous coordinates of the given system and the entries of the Bèzout block. Conditions for determining the homogeneous coordinates are also given and proven.   


2019 ◽  
Vol 16 (2) ◽  
pp. 1
Author(s):  
Shamsatun Nahar Ahmad ◽  
Nor’Aini Aris ◽  
Azlina Jumadi

Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated with the entries of the resultant matrix of the system under consideration. This paper presents some conditions for the homogeneous coordinates of a certain system of bivariate polynomials through the construction and implementation of the Sylvester-Bèzout hybrid resultant matrix formulation. This basis of the implementation of the Bèzout block applies a combinatorial approach on a set of linear inequalities, named 5-rule. The inequalities involved the set of exponent vectors of the monomials of the system and the entries of the matrix are determined from the coefficients of facets variable known as brackets. The approach can determine the homogeneous coordinates of the given system and the entries of the Bèzout block. Conditions for determining the homogeneous coordinates are also given and proven.


2019 ◽  
Vol 52 (1) ◽  
pp. 451-466
Author(s):  
Ilija Jegdić ◽  
Plamen Simeonov ◽  
Vasilis Zafiris

AbstractWe introduce the (q, h)-blossom of bivariate polynomials, and we define the bivariate (q, h)-Bernstein polynomials and (q, h)-Bézier surfaces on rectangular domains using the tensor product. Using the (q, h)-blossom, we construct recursive evaluation algorithms for (q, h)-Bézier surfaces and we derive a dual functional property, a Marsden identity, and a number of other properties for bivariate (q, h)-Bernstein polynomials and (q, h)-Bézier surfaces. We develop a subdivision algorithm for (q, h)-Bézier surfaces with a geometric rate of convergence. Recursive evaluation algorithms for quantum (q, h)-partial derivatives of bivariate polynomials are also derived.


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