Some Properties of Reduced Polynomial Equations

1983 ◽  
pp. 869-884
Author(s):  
A. M. Ostrowski
Keyword(s):  
Polynomial Equations

Computation of Self-Intersections of Offsets of Be´zier Surface Patches

1997 ◽  
Vol 119 (2) ◽  
pp. 275-283 ◽  
Author(s):  
Takashi Maekawa ◽  
Wonjoon Cho ◽  
Nicholas M. Patrikalakis
Keyword(s):  
Differential Geometry ◽  
Distance Function ◽  
Polynomial Equations ◽  
Intersection Curve ◽  
Parameter Domain ◽  
Surface Patches ◽  
Intersection Curves

Self-intersection of offsets of regular Be´zier surface patches due to local differential geometry and global distance function properties is investigated. The problem of computing starting points for tracing self-intersection curves of offsets is formulated in terms of a system of nonlinear polynomial equations and solved robustly by the interval projected polyhedron algorithm. Trivial solutions are excluded by evaluating the normal bounding pyramids of the surface subpatches mapped from the parameter boxes computed by the polynomial solver with a coarse tolerance. A technique to detect and trace self-intersection curve loops in the parameter domain is also discussed. The method has been successfully tested in tracing complex self-intersection curves of offsets of Be´zier surface patches. Examples illustrate the principal features and robustness characteristics of the method.

Construction of systems of polynomial equations with exact p-Divisibility via the covering method

2014 ◽  
Vol 13 (06) ◽  
pp. 1450013 ◽  
Author(s):  
Francis N. Castro ◽  
Ivelisse M. Rubio
Keyword(s):  
Exponential Sums ◽  
Polynomial Equations ◽  
Prime Field ◽  
Elementary Method ◽  
Covering Method ◽  
Systems Of Polynomial Equations

We present an elementary method to compute the exact p-divisibility of exponential sums of systems of polynomial equations over the prime field. Our results extend results by Carlitz and provide concrete and simple conditions to construct families of polynomial equations that are solvable over the prime field.

Geometric Design of Symmetric 3-RRS Constrained Parallel Platforms

2004 ◽  
Author(s):  
Eric Wolbrecht ◽  
Hai-Jun Su ◽  
Alba Perez ◽  
J. Michael McCarthy
Keyword(s):  
Geometric Design ◽  
Homotopy Continuation ◽  
Total Degree ◽  
Polynomial Equations ◽  
Dimensional Synthesis ◽  
Kinematic Synthesis ◽  
Real Solutions ◽  
Parallel Platform ◽  
Three Degree Of Freedom ◽  
Serial Chains

The paper presents the kinematic synthesis of a symmetric parallel platform supported by three RRS serial chains. The dimensional synthesis of this three degree-of-freedom system is obtained using design equations for each of three RRS chains obtained by requiring that they reach a specified set of task positions. The result is 10 polynomial equations in 10 unknowns, which is solved using polynomial homotopy continuation. An example is provided in which the direction of the first revolute joint (2 parameters) and the z component of the base and platform are specified as well as the two task positions. The system of polynomials has a total degree of 4096 which means that in theory it can have as many solutions. Our example has 70 real solutions that define 70 different symmetric platforms that can reach the specified positions.

SOLVING THE INVERSE KINEMATICAL PROBLEM OF A ROBOT ARM BY USING GROEBNER BASIS

2021 ◽  
Vol 13 (1) ◽  
pp. 15
Author(s):  
Alisya Masturoh ◽  
Bambang Hendriya Guswanto ◽  
Triyani Triyani
Keyword(s):  
Robot Hand ◽  
Polynomial Equations ◽  
Robot Arm ◽  
Groebner Basis

The inverse kinematical problem of a robot arm is a problem to find some appropriate joint configurations for a pair of position and direction of a robot hand which is represented by a polynomial equations system. The system is solved by employing Groebner basis notion. Thus, the appropriate joint configurations for a pair of position and direction of the robot hand are obtained.

COMPOSITION SERIES OF THE SOLVABLE ABELIAN FACTOR GROUP SOURCE OF EQUATION OF ALL POLYNOMIAL EQUATIONS

2021 ◽  
Vol 5 (2) ◽  
pp. 462-469
Author(s):  
Bernard Alechenu ◽  
Babayo Muhammed Abdullahi ◽  
Daniel Eneojo Emmanuel
Keyword(s):  
Abelian Group ◽  
Complex Analysis ◽  
Factor Group ◽  
Roots Of Unity ◽  
Composition Series ◽  
Isomorphism Theorem ◽  
Polynomial Equations ◽  
Canonical Map ◽  
The Third ◽  
Recursive Set

This work penciled down the Composition Series of Factor Abelian Group over the source of all polynomial equations gleaned through  the nth roots of unity regular gons on a unit circle, a circle of radius one and centered at zero. To get the composition series, the third isomorphism theorem has to be passed through. But, the third isomorphism theorem itself gleaned via the first which is a deduction of the naturally existing canonical map. The solution of the source atom of the equation of all equation of polynomials are solvable by the intertwine of the Euler’s Formula and the De Moivre’s Theorem which after the inter-math, they become within the domain of complex analysis. For the source root of the equations, there is a recursive set of homomorphisms and ontoness of the mappings geneting the sequential terms in the composition series.    

Discovery of polynomial equations for regression

2004 ◽  
Vol 1 (1) ◽  
pp. 131-142
Author(s):  
Ljupčo Todorovski ◽  
Sašo Džeroski ◽  
Peter Ljubič
Keyword(s):  
Efficient Method ◽  
Regression Models ◽  
Predictive Accuracy ◽  
State Of The Art ◽  
Numerical Data ◽  
Predictive Performance ◽  
Polynomial Equations ◽  
Regression Methods ◽  
Piecewise Regression ◽  
Standard Regression

Both equation discovery and regression methods aim at inducing models of numerical data. While the equation discovery methods are usually evaluated in terms of comprehensibility of the induced model, the emphasis of the regression methods evaluation is on their predictive accuracy. In this paper, we present Ciper, an efficient method for discovery of polynomial equations and empirically evaluate its predictive performance on standard regression tasks. The evaluation shows that polynomials compare favorably to linear and piecewise regression models, induced by the existing state-of-the-art regression methods, in terms of degree of fit and complexity.

Solving the Kinematics of Planar Mechanisms

1998 ◽  
Author(s):  
Charles W. Wampler
Keyword(s):  
Eigenvalue Problem ◽  
Complex Plane ◽  
System Of Equations ◽  
Polynomial Equations ◽  
Input Output ◽  
Planar Mechanisms ◽  
General Method

Abstract This paper presents a general method for the analysis of planar mechanisms consisting of rigid links connected by rotational and/or translational joints. After describing the links as vectors in the complex plane, a simple recipe is outlined for formulating a set of polynomial equations which determine the locations of the links when the mechanism is assembled. It is then shown how to reduce this system of equations to a standard eigenvalue problem, or if preferred, a single resultant polynomial. Both input/output problems and tracing-curve equations are treated.

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