ŁOJASIEWICZ INEQUALITY ON NON-COMPACT DOMAINS AND SINGULARITIES AT INFINITY

2013 ◽  
Vol 24 (10) ◽  
pp. 1350079 ◽  
Author(s):  
DINH SI TIEP ◽  
KRZYSZTOF KURDYKA ◽  
OLIVIER LE GAL
Keyword(s):  
Sufficient Condition ◽  
Polynomial Functions ◽  
The Real ◽  
Polar Set ◽  
Several Variables ◽  
Łojasiewicz Inequality ◽  
Real Polynomials ◽  
Lojasiewicz Inequality

We give a version of the Łojasiewicz inequality for the real polynomials on non-compact domains. The inequality takes in account not only distance to a fiber, but also distance to a polar set. It improves the recent results of [D. S. Tiep, H. H. Vui and N. T. Thao, Łojasiewicz inequality for polynomial functions on non-compact domains, Int. J. Math.23(4) (2012), Article ID: 1250033, 28 pp., doi:10.1142/S0129167X12500334], since we consider a distance to a smaller set. Then we use this new version of the inequality to obtain a sufficient condition for the existence of a vanishing component at infinity for real polynomials in several variables.

ŁOJASIEWICZ INEQUALITY FOR POLYNOMIAL FUNCTIONS ON NON-COMPACT DOMAINS

2012 ◽  
Vol 23 (04) ◽  
pp. 1250033 ◽  
Author(s):  
DINH SI TIEP ◽  
HA HUY VUI ◽  
NGUYEN THI THAO
Keyword(s):  
Polynomial Functions ◽  
Łojasiewicz Inequality ◽  
Lojasiewicz Inequality

In this paper we give some versions of the Łojasiewicz inequality on non-compact domains for polynomial functions. We also point out some relations between the existence Łojasiewicz inequality and the phenomenon of singularities at infinity.

Level curves of open polynomial functions on the real plane

1994 ◽  
Vol 22 (14) ◽  
pp. 5973-5981
Author(s):  
J. Ferrera ◽  
M.J. de la Puente
Keyword(s):  
Real Plane ◽  
Polynomial Functions ◽  
The Real ◽  
Level Curves

Moral responsibility and the concept of agency

2011 ◽  
Author(s):  
HELEN STEWARD
Keyword(s):  
Moral Responsibility ◽  
Sufficient Condition ◽  
The Real

This chapter argues for the incompatibility of moral responsibility and determinism. The real reason why determinism and moral responsibility are inconsistent is not moral, but metaphysical. The real reason is that determinism is inconsistent with agency, which is a necessary (though not, of course, a sufficient) condition of moral responsibility.

Matrix representation of formal polynomials over max-plus algebra

2020 ◽  
pp. 2150216
Author(s):  
Cailu Wang ◽  
Yuegang Tao
Keyword(s):  
Matrix Method ◽  
Polynomial Algorithm ◽  
Polynomial Function ◽  
Matrix Representation ◽  
Sufficient Condition ◽  
Polynomial Functions ◽  
Canonical Forms ◽  
Necessary And Sufficient Condition ◽  
The Matrix ◽  
Necessary And Sufficient

This paper proposes the matrix representation of formal polynomials over max-plus algebra and obtains the maximum and minimum canonical forms of a polynomial function by standardizing this representation into a canonical form. A necessary and sufficient condition for two formal polynomials corresponding to the same polynomial function is derived. Such a matrix method is constructive and intuitive, and leads to a polynomial algorithm for factorization of polynomial functions. Some illustrative examples are presented to demonstrate the results.

Łojasiewicz inequality for a pair of semialgebraic functions

2021 ◽  
Vol 166 ◽  
pp. 102927
Author(s):  
Beata Osińska-Ulrych ◽  
Grzegorz Skalski ◽  
Anna Szlachcińska
Keyword(s):  
Łojasiewicz Inequality ◽  
Lojasiewicz Inequality

The Inverse Eigenvalue Problem for Symmetric Doubly Arrow Matrices

2013 ◽  
Vol 444-445 ◽  
pp. 625-627
Author(s):  
Kan Ming Wang ◽  
Zhi Bing Liu ◽  
Xu Yun Fei
Keyword(s):  
Eigenvalue Problem ◽  
Special Kind ◽  
Inverse Eigenvalue Problem ◽  
Symmetric Matrices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Inverse Eigenvalue ◽  
Necessary And Sufficient

In this paper we present a special kind of real symmetric matrices: the real symmetric doubly arrow matrices. That is, matrices which look like two arrow matrices, forward and backward, with heads against each other at the station, . We study a kind of inverse eigenvalue problem and give a necessary and sufficient condition for the existence of such matrices.

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