ŁOJASIEWICZ INEQUALITY IN P-MINIMAL STRUCTURES

Th purpose of this paper is to extend the Łojasiewicz inequality for functions definable in some subclass of P-minimal structures. More precisely, we prove that the Łojasiewicz inequality holds for functions definable in poptimal expansions of Qp. It is also shown that the Łojasiewicz exponent is a rational number in such p-optimal expansions.

Download Full-text

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127406016008 ◽

2006 ◽

Vol 16 (08) ◽

pp. 2191-2205 ◽

Cited By ~ 17

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.

Download Full-text

Computation of the Łojasiewicz exponent of nonnegative and nondegenerate analytic functions

International Journal of Mathematics ◽

10.1142/s0129167x1450092x ◽

2014 ◽

Vol 25 (10) ◽

pp. 1450092 ◽

Cited By ~ 3

Author(s):

Nguyễn Thao Nguyên Búi ◽

Tiến Son Phạm

Keyword(s):

Analytic Function ◽

Analytic Functions ◽

Newton Polyhedron ◽

Łojasiewicz Inequality ◽

Łojasiewicz Exponent ◽

Lojasiewicz Inequality ◽

Lojasiewicz Exponent

Let f : (ℝn, 0) → (ℝ, 0) be a nonconstant analytic function defined in a neighborhood of the origin 0 ∈ ℝn. The classical Łojasiewicz inequality states that there exist positive constants δ, c and l such that |f(x)| ≥ cd(x, f-1(0))l for ‖x‖ ≤ δ, where d(x, f-1(0)) denotes the distance from x to the set f-1(0). The Łojasiewicz exponent of f at the origin 0 ∈ ℝn, denoted by [Formula: see text], is the infimum of the exponents l satisfying the Łojasiewicz inequality. In this paper, we establish a formula for computing the Łojasiewicz exponent [Formula: see text] of f in terms of the Newton polyhedron of f in the case where f is nonnegative and nondegenerate.

Download Full-text