scholarly journals Bounds on the number of solutions of polynomial systems and the Betti numbers of real piecewise algebraic hypersurfaces

2018 ◽  
Vol 329 ◽  
pp. 147-163 ◽  
Author(s):  
Yisheng Lai ◽  
Weiping Du ◽  
Dexin Duan ◽  
Xiaoke Fang
Keyword(s):  
Betti Numbers ◽  
Polynomial Systems ◽  
Number Of Solutions ◽  
Algebraic Hypersurfaces

scholarly journals ON SOLUTIONS TO SOME POLYNOMIAL CONGRUENCES IN SMALL BOXES

2013 ◽  
Vol 89 (2) ◽  
pp. 300-307
Author(s):  
IGOR E. SHPARLINSKI
Keyword(s):  
Exponential Sums ◽  
Polynomial Systems ◽  
Side Length ◽  
Character Sum ◽  
Number Of Solutions ◽  
Polynomial Congruences

AbstractWe use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime $p$. In particular, we obtain nontrivial results about the number of solutions in boxes with the side length below ${p}^{1/ 2} $, which seems to be the limit of more general methods based on the bounds of exponential sums along varieties.

On the number of solutions of multivariable polynomial systems

1983 ◽  
Vol 28 (2) ◽  
pp. 224-227 ◽  
Author(s):  
A. Benallou ◽  
D. Mellichamp ◽  
D. Seborg
Keyword(s):  
Polynomial Systems ◽  
Number Of Solutions ◽  
Multivariable Polynomial

Synthesis of Watt-type Timed Curve Generators and Selection from Continuous Cognate Spaces

2021 ◽  
pp. 1-11
Author(s):  
Aravind Baskar ◽  
Mark Plecnik
Keyword(s):  
Recent Work ◽  
Polynomial Systems ◽  
Number Of Solutions ◽  
End Effector ◽  
Continuous Space ◽  
Trade Offs ◽  
Synthesis Of Mechanisms ◽  
Free Parameters ◽  
Selection Of

Abstract Following recent work on Stephenson-type mechanisms, the synthesis equations of Watt six-bar mechanisms that act as timed curve generators are formulated and systematically solved. Four variations of the problem arise by assigning the actuator and end effector onto different links. The approach produces exact synthesis of mechanisms up to eight precision points. Polynomial systems are formulated and their maximum number of solutions is estimated using the algorithm of random monodromy loops. Certain variations of Watt timed curve generators possess free parameters that do not affect the output motion, indicating a continuous space of cognate mechanisms. Packaging compactness, clearance, and dimensional sensitivity are characterized across the cognate space to illustrate trade-offs and aid in selection of a final mechanism.

scholarly journals Solving polynomial systems via homotopy continuation and monodromy

2018 ◽  
Vol 39 (3) ◽  
pp. 1421-1446 ◽  
Author(s):  
Timothy Duff ◽  
Cvetelina Hill ◽  
Anders Jensen ◽  
Kisun Lee ◽  
Anton Leykin ◽  
...  
Keyword(s):  
State Of The Art ◽  
Polynomial Systems ◽  
Solution Set ◽  
Software Implementation ◽  
Homotopy Continuation ◽  
Expected Number ◽  
Number Of Solutions ◽  
Study Methods ◽  
Software Packages ◽  
Generic System

Abstract We study methods for finding the solution set of a generic system in a family of polynomial systems with parametric coefficients. We present a framework for describing monodromy-based solvers in terms of decorated graphs. Under the theoretical . that monodromy actions are generated uniformly, we show that the expected number of homotopy paths tracked by an algorithm following this framework is linear in the number of solutions. We demonstrate that our software implementation is competitive with the existing state-of-the-art methods implemented in other software packages.

The Convex Polytopes and Homogeneous Coordinate Rings of Bivariate Polynomials

2019 ◽  
Vol 16 (2) ◽  
pp. 1
Author(s):  
Shamsatun Nahar Ahmad ◽  
Nor’Aini Aris ◽  
Azlina Jumadi
Keyword(s):  
Convex Polytopes ◽  
Polynomial Systems ◽  
Matrix Formulation ◽  
Bivariate Polynomials ◽  
Homogeneous Coordinates ◽  
The Matrix ◽  
Projective Vector ◽  
Coordinate Rings ◽  
Resultant Matrix ◽  
The Given

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.

Researching the factors influencing the satisfaction of master students at VNU School of Interdisciplinary Studies

2019 ◽  
Vol 35 (3) ◽  
Author(s):  
Do Huy Thuong ◽  
Nguyen Thi Phuong Hong
Keyword(s):  
Service Industry ◽  
Interdisciplinary Studies ◽  
Number Of Solutions ◽  
Factors Affecting ◽  
Training Institutions ◽  
Market Needs ◽  
Market Strategies ◽  
Students Satisfaction ◽  
Training Service

Improving the quality in order to keep up with the trend in the world is the vital task of training institutions today. Training institutions need to grasp market needs and satisfy the requirements of customers - learners. Nadiri, H., Kandampully, J & Hussain, K. (2009) argue that the managers in education need to apply market strategies that are being used by manufacturing and business enterprises and need to be aware that the role of training institutions is a service industry which is responsible for satisfying learner needs (Elliott & Shin, 2002). Currently, there have been many researches on students’ satisfaction. However, each research has its own objectives and is conducted on different scales. This study is implemented to provide information about the factors affecting master students’ satisfaction with the training service at VNU School of Interdisciplinary Studies (VNU SIS). Through it, the research offers a number of solutions to improving the satisfaction level of the master students at VNU SIS in the coming time.

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