Sufficient Condition for Global Observability Decomposition of Polynomial Systems

2015 ◽  
Vol 8 (3) ◽  
pp. 228-233
Author(s):  
Yu KAWANO ◽  
Toshiyuki OHTSUKA
Keyword(s):  
Polynomial Systems ◽  
Sufficient Condition

On global and local observability of nonlinear polynomial systems: a decidable criterion

2020 ◽  
Vol 68 (6) ◽  
pp. 395-409 ◽  
Author(s):  
Daniel Gerbet ◽  
Klaus Röbenack
Keyword(s):  
Algebraic Geometry ◽  
Nonlinear Systems ◽  
Polynomial Systems ◽  
Sufficient Condition ◽  
Rank Condition ◽  
Observability Analysis ◽  
Global And Local

AbstractIt is very difficult to check the observability of nonlinear systems. Even for local observability, the observability rank condition provides only a sufficient condition. Much more difficult is the verification of global observability. This paper deals with the local and global observability analysis of polynomial systems based on algebraic geometry. In particular, we derive a decidable criterion for the verification of global observability of polynomial systems. Our framework can also be employed for local observability analysis.

scholarly journals An optimal $ Z $-eigenvalue inclusion interval for a sixth-order tensor and its an application

2021 ◽  
Vol 7 (1) ◽  
pp. 967-985
Author(s):  
Tinglan Yao ◽  
Keyword(s):  
Homogeneous Polynomial ◽  
Symmetric Tensor ◽  
Positive Definiteness ◽  
Polynomial Systems ◽  
Sixth Order ◽  
Invariant Polynomial ◽  
Sufficient Condition ◽  
Order Tensor ◽  
Inclusion Interval ◽  
Time Invariant

<abstract><p>An optimal $ Z $-eigenvalue inclusion interval for a sixth-order tensor is presented. As an application, a sufficient condition for the positive definiteness of a sixth-order real symmetric tensor (also a homogeneous polynomial form) is obtained, which is used to judge the asymptotically stability of time-invariant polynomial systems.</p></abstract>

scholarly journals Sylvester-type matrices for sparse resultants

2014 ◽  
Vol 6 (1) ◽  
Author(s):  
Shamsatun Nahar Ahmad ◽  
Nor’aini Aris
Keyword(s):  
Polynomial System ◽  
Polynomial Systems ◽  
Special Structure ◽  
Incremental Algorithm ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Newton Polytopes ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Resultant Matrix

The resultant matrix of a polynomial system depends on the geometry of its input Newton polytopes. Therefore for sparse inputs, the matrix is lower in dimension. The aim of the study is to infer conditions on the class of polynomial systems that can give a resultant matrix whose size is minimized, that is an optimal or Sylvester-type sparse resultant matrix. From the work of Emiris, the ‘incremental algorithm’ has been claimed to produce optimal matrices for the class of multi-homogeneous (or multigraded) systems of special structure. Cyclic polynomial systems for n-root problems also fall under this classification. We have applied the Maple multires package to obtain Sylvester-type matrices for some examples. The ultimate aim of the study is to verify whether the multigraded systems constitute to the only class of polynomial systems that can give sparse resultant optimal matrix; hence giving a necessary and sufficient condition for producing exact sparse resultants.

The Current Situation in Chemical Stabilization of Biological Materials

1972 ◽  
Vol 30 ◽  
pp. 132-133
Author(s):  
John H. Luft
Keyword(s):  
Electron Microscopy ◽  
Electron Microscope ◽  
Osmium Tetroxide ◽  
Molecular Level ◽  
Cross Linking ◽  
Chemical Stabilization ◽  
Specimen Preparation ◽  
Sufficient Condition ◽  
Radio Telescopes

With information processing devices such as radio telescopes, microscopes or hi-fi systems, the quality of the output often is limited by distortion or noise introduced at the input stage of the device. This analogy can be extended usefully to specimen preparation for the electron microscope; fixation, which initiates the processing sequence, is the single most important step and, unfortunately, is the least well understood. Although there is an abundance of fixation mixtures recommended in the light microscopy literature, osmium tetroxide and glutaraldehyde are favored for electron microscopy. These fixatives react vigorously with proteins at the molecular level. There is clear evidence for the cross-linking of proteins both by osmium tetroxide and glutaraldehyde and cross-linking may be a necessary if not sufficient condition to define fixatives as a class.

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.

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