scholarly journals On some properties of the Łojasiewicz exponent

2021 ◽  
Vol 359 (8) ◽  
pp. 991-997
Author(s):  
Christophe Eyral ◽  
Tadeusz Mostowski ◽  
Piotr Pragacz
Keyword(s):  
Łojasiewicz Exponent ◽  
Lojasiewicz Exponent

scholarly journals On the Łojasiewicz Exponent near the Fibre of a Polynomial

2004 ◽  
Vol 52 (3) ◽  
pp. 231-236 ◽  
Author(s):  
Grzegorz Skalski
Keyword(s):  
Łojasiewicz Exponent ◽  
Lojasiewicz Exponent

scholarly journals Multiplicity and the Łojasiewicz exponent

2000 ◽  
Vol 73 (3) ◽  
pp. 257-267 ◽  
Author(s):  
S. Spodzieja
Keyword(s):  
Łojasiewicz Exponent ◽  
Lojasiewicz Exponent

scholarly journals The Lojasiewicz exponent at infinity for overdetermined polynomial mappings

2002 ◽  
Vol 78 (1) ◽  
pp. 1-10 ◽  
Author(s):  
S. Spodzieja
Keyword(s):  
Polynomial Mappings ◽  
Łojasiewicz Exponent ◽  
Lojasiewicz Exponent

scholarly journals Multiplicity and the Lojasiewicz exponent

1988 ◽  
Vol 20 (1) ◽  
pp. 353-364 ◽  
Author(s):  
Arkadiusz Płoski
Keyword(s):  
Łojasiewicz Exponent ◽  
Lojasiewicz Exponent

Sum of squares and the Łojasiewicz exponent at infinity

2014 ◽  
Vol 112 (3) ◽  
pp. 223-237
Author(s):  
Krzysztof Kurdyka ◽  
Beata Osińska-Ulrych ◽  
Grzegorz Skalski ◽  
Stanisław Spodzieja
Keyword(s):  
Sum Of Squares ◽  
Łojasiewicz Exponent ◽  
Lojasiewicz Exponent

scholarly journals On the Łojasiewicz exponent and Newton polyhedron

2005 ◽  
Vol 28 (1) ◽  
pp. 106-110 ◽  
Author(s):  
Ould M. Abderrahmane
Keyword(s):  
Newton Polyhedron ◽  
Łojasiewicz Exponent ◽  
Lojasiewicz Exponent

scholarly journals Extensions of real regular mappings and the Łojasiewicz exponent at infinity

2013 ◽  
Vol 137 (6) ◽  
pp. 718-729
Author(s):  
Beata Osińska-Ulrych ◽  
Grzegorz Skalski ◽  
Stanisław Spodzieja
Keyword(s):  
Łojasiewicz Exponent ◽  
Lojasiewicz Exponent

THE ŁOJASIEWICZ EXPONENT AT AN EQUILIBRIUM POINT OF A STANDARD CNN IS 1/2

2006 ◽  
Vol 16 (08) ◽  
pp. 2191-2205 ◽  
Author(s):  
MAURO FORTI ◽  
ALBERTO TESI
Keyword(s):  
Vector Field ◽  
Equilibrium Point ◽  
Critical Point ◽  
Vector Fields ◽  
Equilibrium Points ◽  
Łojasiewicz Inequality ◽  
Łojasiewicz Exponent ◽  
Lojasiewicz Inequality ◽  
Output Trajectory ◽  
Lojasiewicz Exponent

In the sixties, Łojasiewicz proved a fundamental inequality for vector fields defined by the gradient of an analytic function, which gives a lower bound on the norm of the gradient in a neighborhood of a (possibly) non-isolated critical point. The inequality involves a number belonging to (0, 1), which depends on the critical point, and is known as the Łojasiewicz exponent. In this paper, a class of vector fields which are defined on a hypercube of ℝn, is considered. Each vector field is the gradient of a quadratic function in the interior of the hypercube, however it is discontinuous on the boundary of the hypercube. An extended Łojasiewicz inequality for this class of vector fields is proved, and it is also shown that the Łojasiewicz exponent at each point where a vector field vanishes is equal to 1/2. The considered fields include a class of vector fields which describe the dynamics of the output trajectories of a standard Cellular Neural Network (CNN) with a symmetric neuron interconnection matrix. By applying the extended Łojasiewicz inequality, it is shown that each output trajectory of a symmetric CNN has finite length, and as a consequence it converges to an equilibrium point. Furthermore, since the Łojasiewicz exponent at each equilibrium point of a symmetric CNN is equal to 1/2, it follows that each (state) trajectory, and each output trajectory, is exponentially convergent toward an equilibrium point, and this is true even in the most general case where the CNN possesses infinitely many nonisolated equilibrium points. In essence, the obtained results mean that standard symmetric CNNs enjoy the property of absolute stability of exponential convergence.

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