scholarly journals Application of Parallel Computing to Obtain all Real Solutions of a High Degree Univariate Polynomial Equation

2016 ◽  
Author(s):  
Liying Wang
Keyword(s):  
Parallel Computing ◽  
Polynomial Equation ◽  
Real Solutions ◽  
Univariate Polynomial ◽  
High Degree

Polynomial Solution to the Five-Position Synthesis of Spatial CC Dyads via Dialytic Elimination

2000 ◽  
Author(s):  
Chintien Huang ◽  
Yu-Jui Chang
Keyword(s):  
Polynomial Solution ◽  
Polynomial Equation ◽  
Elimination Method ◽  
Real Solutions ◽  
Numerical Example ◽  
Univariate Polynomial ◽  
Position Synthesis

Abstract This paper presents a polynomial solution to the five-position synthesis of spatial cylindrical-cylindrical dyads. The solution procedures start with the simplification of the synthesis equations derived by Tsai and Roth. The simplified equations are solved by Sylvester’s dialytic elimination method to obtain a univariate polynomial equation of degree six, which gives at most 6 CC dyads for the five-position synthesis. A numerical example with six real solutions is provided.

On the Solution of Five-Precision-Points Path Synthesis of Planar Four-Bar Linkages Using Algebraic Method

2016 ◽  
Author(s):  
Feng Wei ◽  
Shimin Wei ◽  
Ying Zhang ◽  
Qizheng Liao
Keyword(s):  
Algebraic Method ◽  
Polynomial Equation ◽  
Type I ◽  
Type Iv ◽  
Univariate Polynomial ◽  
Degenerate Solutions ◽  
Path Synthesis ◽  
Four Bar Linkage ◽  
High Degree ◽  
Resultant Matrix

The problem of five precision points path synthesis of planar four-bar linkage can be divided into four types in term of the input parameters. A unified formulation for the four types is built based on the planar displacement matrix. Next, the corresponding resultant matrix is constructed based on Groebner bases generated by applying the new term ordering (the groups graded reverse lexicographic ordering, <ggrevlex) for four types. Then, a high-degree univariate polynomial equation is accordingly obtained. At last, several examples are provided to validate the algorithm and the solutions are verified in the software SAM. And it is concluded that type I has 36 solutions, type II has 64 including 16 degenerate solutions, type III has 92 solutions and type IV has 82 solutions including 16 degenerate solutions.

scholarly journals Formulation and Mathematical Optimization of Controlled Release Calcium Alginate Micro Pellets of Frusemide

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Amitava Ghosh ◽  
Prithviraj Chakraborty
Keyword(s):  
Sodium Alginate ◽  
Calcium Alginate ◽  
Mathematical Optimization ◽  
Extended Period ◽  
Cost Effective ◽  
Polynomial Equation ◽  
Chronic Hypertension ◽  
Order Polynomial ◽  
High Degree

Objective. Frusemide loaded calcium alginate micropellets, an oral microparticulate delivery system, was statistically optimized exhibiting prolonged therapeutic action minimizing its adverse effects.Methods. Ionotropic Gelation technique was adopted employing 32Factorial designs and keeping the entire process free from organic solvents. Physicochemical and the release characteristics of the prepared formulations were studied, keeping variations only in sodium alginate (primary polymer) and Acrycoat E30D (copolymer) dispersion.Result. Sodium alginate was predominant over Acrycoat E30D in all batches. Nonadditives or interaction was observed to be insignificant. Multiple regressions produced second-order polynomial equation, and the predictive results obtained were validated with high degree of correlation. Thein vivostudy applauded that optimized calcium alginate micropellets of frusemide can produce a much greater diuretic effect over an extended period of 24 hours.Conclusion. This study reveals that the potential of a single dose of the mathematically optimized micro pellets of frusemide formulation is sufficient in the management of peripheral edema and ascites in congestive heart failure and as well in the treatment of chronic hypertension, leading to better patient compliance, and can be produced with minimum experimentation and time, proving far more cost-effective formulation than the conventional methods of formulating dosage forms.

scholarly journals The use of parallel computing for the high-resolution determination of earthquake source parameters

2019 ◽  
pp. 68-75
Author(s):  
A. S. Fomochkina ◽  
V. G. Bukchin
Keyword(s):  
Parallel Computing ◽  
Focal Mechanism ◽  
Source Parameters ◽  
Nodal Plane ◽  
Earthquake Parameters ◽  
Earthquake Source Parameters ◽  
Detailed Search ◽  
High Degree ◽  
Special Case

Alongside the determination of the focal mechanism and source depth of an earthquake by direct examination of their probable values on a grid in the parameter space, also the resolution of these determinations can be estimated. However, this approach requires considerable time in the case of a detailed search. A special case of a shallow earthquake whose one nodal plane is subhorizontal is an example of the sources that require the use of a detailed grid. For studying these events based on the records of the long-period surface waves, the grids with high degree of detail in the angles of the focal mechanism are required. We discuss the application of the methods of parallel computing for speeding up the calculations of earthquake parameters and present the results of studying the strongest aftershock of the Tohoku, Japan, earthquake by this approach.

Forward Positional Analysis for the General 4–6 In-Parallel Platform

1995 ◽  
Vol 209 (1) ◽  
pp. 55-67 ◽  
Author(s):  
Q Liao ◽  
L D Seneviratne ◽  
S W E Earles
Keyword(s):  
Polynomial Equation ◽  
Unknown Variable ◽  
Real Solutions ◽  
Positional Analysis ◽  
Order Polynomial ◽  
Parallel Platform ◽  
New Equation ◽  
Published Paper ◽  
Special Case ◽  
Kinematic Solution

Presented is the forward positional (kinematic) solution for the general case of the 4–6 in-parallel platform mechanism; in particular, the spherical joints of the moving and base platforms are not restricted to lie in planes, but can be freely chosen. The forward positional analysis consists of 13 equations which are reduced to a single thirty-second order polynomial equation in one unknown variable. This new equation is numerically solved and validated by substituting the 32 roots into the 13 forward positional equations. The new analysis is also used to solve an example, with a set of known results from a previously published paper, in which a special case of the 4–6 in-parallel platform is considered; the results are in exact agreement. For a number of general platform and actuator inputs a maximum of 24 real solutions have been found. One example is illustrated.

The Position Analysis of Spherical Seven-Bar Mechanism

2011 ◽  
Vol 101-102 ◽  
pp. 193-196
Author(s):  
Zhao Feng Zhang ◽  
Zhi Huan Zhang
Keyword(s):  
Mathematical Model ◽  
Analytical Solutions ◽  
Polynomial Equation ◽  
Constraint Equations ◽  
Position Analysis ◽  
Numerical Example ◽  
Location Parameters ◽  
Sylvester Resultant ◽  
Univariate Polynomial

In this paper, we turn plane seven-bar mechanism into spherical seven-bar mechanism, using quaternion to construct mathematical model for spherical seven-bar mechanism. Three constraint equations are obtained according to the angles constraint. Using Sylvester resultant elimination by two steps, a 32 degree univariate polynomial equation can be obtained. A numerical example confirms that analytical solutions of spherical seven-bar mechanism are 32 and with the help of Mathematic software to solve the location parameters.

Forward Kinematics in Polynomial Form of the General Stewart Platform1

1998 ◽  
Vol 123 (2) ◽  
pp. 254-260 ◽  
Author(s):  
Carlo Innocenti
Keyword(s):  
Parallel Manipulator ◽  
Polynomial Equation ◽  
Polynomial Form ◽  
Floating Point ◽  
Forward Kinematics ◽  
Numerical Example ◽  
Univariate Polynomial ◽  
General Geometry ◽  
Floating Point Computation

The paper presents a new algorithm to solve, in polynomial form, the forward kinematics of the general-geometry 6-6 fully-parallel manipulator. The forty solutions that the problem at hand admits in the complex domain are found by determining the roots of a 40th-order univariate polynomial equation. Unlike the existing algorithm, the proposed one is suitable for implementation in a standard floating-point computation environment. A numerical example shows application of the new algorithm to a case study.

A Computational Algorithm for a Spatial Serial Robot

2011 ◽  
Vol 217-218 ◽  
pp. 233-237
Author(s):  
Xi Guang Huang
Keyword(s):  
Automatic Control ◽  
Inverse Kinematics ◽  
Computational Algorithm ◽  
Polynomial Equation ◽  
Central Problem ◽  
Serial Robots ◽  
Symbolic Calculations ◽  
Inverse Kinematics Problem ◽  
Serial Robot ◽  
Univariate Polynomial

The inverse kinematics of serial robots is a central problem in the automatic control of robot manipulators. The aim of this paper is to obtain a computational algorithm to compute the inverse kinematics problem of a spatial serial robot. We use a series of algebraic and numeric transformations to reduce the problem to a univariate polynomial equation. The results can be directly applied to symbolic calculations and decreased considerably the calculation time.

A New Algebraic Solution to Inverse Static Force Analysis of a Special Planar Three-Spring System

2016 ◽  
Author(s):  
Ying Zhang ◽  
Qizheng Liao ◽  
Shimin Wei ◽  
Feng Wei ◽  
Duanling Li
Keyword(s):  
Polynomial Equation ◽  
Algebraic Solution ◽  
Static Force ◽  
Force Analysis ◽  
Elimination Algorithm ◽  
Equilibrium Configurations ◽  
Spring System ◽  
Univariate Polynomial ◽  
Variable Substitution ◽  
Resultant Matrix

In this paper, we present a new algebraic elimination algorithm for the inverse static force analysis of a special planar three-spring system. The system consists of three linear springs joined to the ground at the two fixed pivots and connected to the two moving pivots at the platform. When exerted by specified static force, the goal of inverse static analysis is to determine all the equilibrium configurations. First of all, a system of seven polynomial equations in seven variables is established based on the geometric constraint and static force balancing. Then, four basic constraint equations in four variables are obtained by variable substitution. Next, a 20 by 20 resultant matrix is reduced by means of three consecutive Sylvester elimination process. Finally, a 54th-degree univariate polynomial equation is directly derived without extraneous roots in the computer algebra system Mathematica 9.0. At last, a numerical example is given to verify the elimination procedure.

Research on Position Analysis of a Kind of Nine-Link Barranov Truss

2007 ◽  
Vol 130 (1) ◽  
Author(s):  
Pin Wang ◽  
Qizheng Liao ◽  
Yufeng Zhuang ◽  
Shimin Wei
Keyword(s):  
Gaussian Elimination ◽  
Constraint Equation ◽  
Polynomial Equation ◽  
Vector Method ◽  
Constraint Equations ◽  
Position Analysis ◽  
Sylvester Resultant ◽  
Univariate Polynomial ◽  
New Equation ◽  
First Time

The position analysis of a nine-link Barranov truss is finished by using Dixon resultants together with Sylvester resultants. Above all, using vector method in complex plane to construct four constraint equations and transform them into complex exponential form, then three constraint equations are used to construct a 6×6 Dixon matrix, which contains two variables to be eliminated. We extract the greatest common divisor (GCD) of two columns of Dixon matrix and compute its determinant to obtain a new equation. This equation together with the fourth constraint equation can be used to construct a Sylvester resultant. A 50deg univariate polynomial equation is obtained from the determinant of Sylvester resultant. Other variables can be computed by Euclidean algorithm and Gaussian elimination. Lastly, a numerical example confirms that the analytical solution number of the Barranov truss is 50. It is the first time to complete analytical solutions of this kind of Barranov truss.

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