Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations

Equazioni alle derivate parziali di tipo ellittico. By C. Miranda. Pp. viii, 222. DM 28.80. 1955, (Springer, Berlin)

This text has two parts. The first one is the essentially unmodified text of our 1973-74 seminar on integral dependence in complex analytic geometry at the Ecole Polytechnique with J-J. Risler's appendix on the Lojasiewicz exponents in the real-analytic framework. The second part is a short survey of more recent results directly related to the content of the seminar. The first part begins with the definition and elementary properties of the ν order function associated to an ideal I of a reduced analytic algebra A. Denoting by ν I (x) the largest power of I containing the element ∈ A, one defines ν I (x) = lim i→∞ ν I (x k )/k. The second paragraph is devoted to the equivalent definitions of the integral closure of an ideal in complex analytic geometry, one of them being I = {x ∈ A/ν I (x) ≥ 1}. The third paragraph describes the normalized blowing-up of an ideal and the fourth explains how to compute ν I (x) with the help of the normalized blowing-up of the ideal I. It contains the basic finiteness results of the seminar, such as the rationality of ν I (x) (which had been proved by Nagata in algebraic geometry, a fact of which we were not aware at the time), tlie definitions of the fractional powers of coherent sheaves of ideals and the proof of their coherency. Given a coherent sheaf I of O X -ideals on a reduced analytic space X one can define for each open set U of X and f ∈ T(U, O X ) the number v U I (f) as the infimum of the ν Iy (f y ) for y ∈ U. Then one defines for each positive real number v the sheaf I v (resp. Z v+ ) associated to the presheaf U → {f ∈ Γ(U,O X )/ν U I (f) ≥ ν} (resp. U → {f ∈ Γ(U,O X )/ν U I (f) > ν}). Finally one has the graded O X /I-algebra gr I O X = ⊕ I v /I v+ . One important result is then that this algebra is locally finitely generated and that locally there is a universal denominator q in the sense that all nonzero homogeneous components of the graded algebra have degree in 1 qN. In § 5 it is shown that one can compute ν using analytic arcs h: (C, 0) → (X, x), and §6 shows that Lojasiewicz exponents are the inverses of ν, which implies that they are rational. Risler's appendix shows how to use blowing-ups to compute Lojasiewicz exponents and prove their rationality in the real analytic case. The complements, added for this publication, point to some developments directly related to the subject of the seminar: The first one is the proof in the spirit of the seminar of the classical Lojasiewicz inequality |grad(/(z))| ≥ C 1 |f/(z)| θ with θ < 1. Then we point to litter work which shows that in fact given an ideal I and an element f ∈ A the rational number ν I (f) can be seen as the slope of one of the sides of a natural Newton polygon associated to I and f, which is in several ways a better indicator of the relations of the powers of f with the powers of I and has some useful incarnations. The third complement points to results of Izumi using P to characterize the Gabrielov rank condition for a morphism of analytic algebras, the fourth is a presentation of a generalization due to Ciuperca, Enescu and Spiroff of the rationality of v to the case of several ideals, where it becomes the rationality of a certain polyhedral cone. The fifth comment presents the connection of v with the type of ideals, which was introduced by D'Angelo in complex analysis and used recently by Heier for the proof of an effective Nullstellensatz. In the middle 1980's, A. Ploski, J. Chadzynski and T. Krasinski found methods of evaluation for the local and global Lojasiewiez exponents in inequalities of the form [P(z)| ≥ C|z| θ where either P = (P 1 ,..., P k ) is a collection of analytic functions on C" having an isolated zero at the origin and the inequality should be true for |z| small enough, or P is a collection of polynomials with finitely many common zeroes and the inequality should be true for |z| large enough. The results on the type are of the same nature, because it follows from the seminar that the type is in fact a Lojasiewicz exponent. The sixth comment points to results of Morales and others about the Hilbert function associated to the integrally closed powers I n of a primary ideal in an excellent local ring and the associated graded algebra. Finally we point to two different but not unrelated uses of what is in fact the main object of study in the seminar: the reduced graded ring gr I A defined and studied in § 4. In [T5] the second author uses the fact that for the local algebra 0 of a plane analytic branch the algebra gr m O is the algebra of the semigroup associated to the singularity and is a complete intersection (a result due to the first author) to revisit the local moduli problem. The key is that the local analytic algebra 0 of every plane branch in the same equisingularity class has the same gr m O because it has the same semigroup, so that the branch is a deformation of the monomial curve corresponding to that algebra. In [Kn], Allen Knutson uses the same specialization to the "balanced normal cone" corresponding to gr I A in intersection theory. Each paragraph has its own bibliography. Unfortunately at the time of the seminar we were unaware of the beautiful results of Samuel, Rees and Nagata (see [Sa], [N], [R.1], [R2], [R3] in the bibliography of the complements), of which it appears a posteriori that some parts of the seminar are translations into the complex analytic framework. The demand for this text over the years, however, and the fact that some mathematicians are led to rediscover some of its results, indicate that its publication is probably of some use.

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Clôture intégrale des idéaux et équisingularité

Isolated singularities defined by weighted homogeneous polynomials

This survey paper discusses five equivalent ways of defining a “critical point at infinity” for a complex polynomial of two variables.

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Five Definitions of Critical Point at Infinity

We prove a sharp regular separation condition for arbitrary projective algebraic sets. As a corollary an extension of Kollar's results on the Łojasiewicz exponent of polynomial mappings is obtained.

Separation of algebraic sets and the Łojasiewicz exponent of polynomial mappings

Mei-Chi Shaw) We give an explicit formula for the Lojasiewicz exponent of an isolated weighted homogeneous surface singularity in terms of its weights. From the formula we get that the Lojasiewicz exponent is a topological invariant of these singularities.

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The Lojasiewicz exponent of an isolated weighted homogeneous surface singularity

/pdf/on-the-lojasiewicz-exponent-at-infinity-for-polynomial-19bss3r4n1.pdf

On the Łojasiewicz exponent at infinity for polynomial mappings of ℂ² into ℂ² and components of polynomial automorphisms of ℂ²

/pdf/a-set-on-which-the-lojasiewicz-exponent-at-infinity-is-3os4fmhlg6.pdf

A set on which the Łojasiewicz exponent at infinity is attained

/pdf/exponent-of-growth-of-polynomial-mappings-of-c-2-into-c-2-qi3olc055n.pdf

Tadeusz Krasiński

Papers

Separation of algebraic sets and the Łojasiewicz exponent of polynomial mappings

The Lojasiewicz exponent of an isolated weighted homogeneous surface singularity

On the Łojasiewicz exponent at infinity for polynomial mappings of ℂ² into ℂ² and components of polynomial automorphisms of ℂ²

A set on which the Łojasiewicz exponent at infinity is attained

Exponent of growth of polynomial mappings of $C^2$ into $C^2$