A result of Ambrosetti–Prodi type for first-order ODEs with cubic non-linearities. Part I

In this paper, we prove a result of Ambrosetti–Prodi type for the problem x′=f(t,x)+λx, where f(t,x) is T-periodic in t, f(t,0)≡0 and f(t,x) has “cubic nonlinearities”. Received: February 4, 2000?Published online: April 14, 2003 RID="*" ID="*"This paper was partially supported by CDCHT, Universidad de los Andes.     

Picard values of <Emphasis Type="Italic">p</Emphasis>-adic meromorphic functions

We investigate Picard-Hayman behavior of derivatives of meromorphic functions on an algebraically closed field K, complete with respect to a non-trivial ultrametric absolute value. We present an analogue of the well-known Hayman’s alternative theorem both in K and in any open disk. Here the main hypothesis is based on the behaviour of |f|(r) when r tends to +∞ on properties of special values and quasi-exceptional values.We apply this study to give some sufficient conditions on meromorphic functions so that they satisfy Hayman’s conjectures for n = 1and for n = 2. Given a meromorphic transcendental function f, at least one of the two functions f′f and f′f ² assumes all non-zero values infinitely often. Further, we establish that if the sequence of residues at simple poles of a meromorphic transcendental function on K admits no infinite stationary subsequence, then either f′ + af ² has infinitely many zeros that are not zeros of f for every a ∈ K* or both f′ + bf ³ and f′ + bf ⁴ have infinitely many zeros that are not zeros of f for all b ∈ K*. Most of results have a similar version for unbounded meromorphic functions inside an open disk.     

A result of Ambrosetti–Prodi type for first-order ODEs with cubic non-linearities. Part II

In this paper we study the scalar equation x′=f(t,x), where f(t,x) has cubic non-linearities in x and we prove that this equation has at most three bounded separate solutions. We say that λ∈ℝ is a critical value of the equation x′=f(t,x)+λx if this equation has a degenerate bounded solution and we exhibit two classes of functions f such that the above equation has a unique critical value. Received: February 4, 2000; in final form: March 19, 2002?Published online: April 14, 2003 RID="*" ID="*"This paper was partially supported by CDCHT, Universidad de los Andes.     

Differentiable functions defined in closed sets. A problem of Whitney

In 1934, Whitney raised the question of how to recognize whether a function f defined on a closed subset X of ℝ ⁿ is the restriction of a function of class 𝒞 ^p . A necessary and sufficient criterion was given in the case n=1 by Whitney, using limits of finite differences, and in the case p=1 by Glaeser (1958), using limits of secants. We introduce a necessary geometric criterion, for general n and p, involving limits of finite differences, that we conjecture is sufficient at least if X has a “tame topology”. We prove that, if X is a compact subanalytic set, then there exists q=q _X (p) such that the criterion of order q implies that f is 𝒞 ^p . The result gives a new approach to higher-order tangent bundles (or bundles of differential operators) on singular spaces. Oblatum 21-XI-2001 & 3-VII-2002?Published online: 8 November 2002 RID="*" ID="*"Research partially supported by the following grants: E.B. – NSERC OGP0009070, P.M. – NSERC OGP0008949 and the Killam Foundation, W.P. – KBN 5 PO3A 005 21.     

Nonuniform hyperbolicity for C1-generic diffeomorphisms

We study the ergodic theory of non-conservative C ¹-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C ¹-generic diffeomorphisms are non-uniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set Λ of any C ¹-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set Λ.     

Feuilletages holomorphes locaux et analyticité partielle d'applications C ∞

 Let f : M → M′ be a smooth CR mapping between a generic real analytic submanifold M ⊂ ℂ ⁿ , n > 1, and a real analytic subset M′ ⊂ ℂ ^n′ . We prove that if M is minimal and if M′ does not contain any complex curves, then f is analytic on a dense open subset of M. More generally, we establish an upper estimate of the partial analyticity of f, which depends on the maximal dimension of local holomorphic foliations contained in M ^′ . Received: 7 August 2001 Mathematics Subject Classification (2000): 32V25, 32V40, 32H99     

Une procédure de Calderón-Zygmund pour le problème de la racine <Emphasis Type="Italic">k</Emphasis>-ième

Soit f une fonction définie sur un ouvert de ℝ ^d , à valeurs positives ou nulles, de classe C ^k , où k est un entier ≥2. Un résultat de Colombini–Jannelli–Spagnolo assure que ∇(f ^{1/ k} ) appartient à L ¹ _loc. Nous démontrons ici que ∇(f ^{1/ k} )∈L _w ^{k / k −2}, ce qui fournit une amélioration optimale du résultat ci-dessus et une généralisation d’un théorème classique de Glaeser sur la racine carrée. La méthode de preuve requiert l’usage d’une décomposition de Calderón–Zygmund de la fonction f qui nous permet de nous ramener à des formes normales de manipulation simple. Nous montrons également que la régularité C ^k est essentiellement nécessaire au résultat. Nous donnons quelques applications à des problèmes d’équations aux dérivées partielles faiblement hyperboliques. L’adaptabilité des méthodes de décomposition de Calderón-Zygmund, utilisées en analyse harmonique et pseudo-différentielle, trouve dans ces résultats une nouvelle illustration.

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Received: December 13, 2001?Published online: April 14, 2003     

Preimages of CR submanifolds with generic holomorphic maps

We investigate the notion of CR transversality of a generic holomorphic map f: ℂ ⁿ → ℂ ^m to a smooth CR submanifold M of ℂ ^m . We construct a stratification of the set of non-CR transversal points in the preimage M′ = f ⁻¹ (M) by smooth submanifolds, consisting of points where the CR dimension of M′ is constant. We show the existence of a Whitney stratification for sets which are locally diffeomorphic to the product of an open set and an analytic set. Work on this paper was supported by ARRS, Republic of Slovenia.     

Attracting basins in ℙ2

We prove the existence of infinitely many attracting basins for some holomorphic mappings in ℙ². We also show that if a family of mappings has a complex generic homoclinic tangency, then some of the mappings in the family have an attractive periodic fixed point.     

Simple and immune relations on countable structures

 Let 𝒜 be a computable structure and let R be a new relation on its domain. We establish a necessary and sufficient condition for the existence of a copy ℬ of 𝒜 in which the image of R (?R, resp.) is simple (immune, resp.) relative to ℬ. We also establish, under certain effectiveness conditions on 𝒜 and R, a necessary and sufficient condition for the existence of a computable copy ℬ of 𝒜 in which the image of R (?R, resp.) is simple (immune, resp.). Received: 4 February 2001 Published online: 5 November 2002 RID="*" ID="*" The first three authors gratefully acknowledge support of the NFS Binational Grant DMS-0075899. RID="*" ID="*" The first three authors gratefully acknowledge support of the NFS Binational Grant DMS-0075899. RID="*" ID="*" The first three authors gratefully acknowledge support of the NFS Binational Grant DMS-0075899.     

Codimension 2 reversible heteroclinic bifurcations with inclination flips

In this paper, the heteroclinic bifurcation problem with real eigenvalues and two inclination-flips is investigated in a four-dimensional reversible system. We perform a detailed study of this case by using the method originally established in the papers “Problems in Homoclinic Bifurcation with Higher Dimensions” and “Bifurcation of Heteroclinic Loops,” and obtain fruitful results, such as the existence and coexistence of R-symmetric homoclinic orbit and R-symmetric heteroclinic loops, R-symmetric homoclinic orbit and R-symmetric periodic orbit. The double R-symmetric homoclinic bifurcation (i.e., two-fold R-symmetric homoclinic bifurcation) for reversible heteroclinic loops is found, and the existence of infinitely many R-symmetric periodic orbits accumulating onto a homoclinic orbit is demonstrated. The relevant bifurcation surfaces and the existence regions are also located. This work was supported by National Natural Science Foundation of China (Grant No. 10671069)     

Van Kampen’s embedding obstruction for discrete groups

We give a lower bound to the dimension of a contractible manifold on which a given group can act properly discontinuously. In particular, we show that the n-fold product of nonabelian free groups cannot act properly discontinuously on ℝ^{2 n −1}. Oblatum 19-I-2001 & 29-V-2002?Published online: 5 September 2002 RID="*" ID="*"All three authors gratefully acknowledge the support by the National Science Foundation.     

Complete rank theorem of advanced calculus and singularities of bounded linear operators

Let E and F be Banach spaces, f: U ⊂ E → F be a map of C ^r (r ⩾ 1), x ₀ ∈ U, and ft (x ₀) denote the FréLechet differential of f at x ₀. Suppose that f′(x ₀) is double split, Rank(f′(x ₀)) = ∞, dimN(f′(_{x 0})) > 0 and codimR(f′(x ₀)) s> 0. The rank theorem in advanced calculus asks to answer what properties of f ensure that f(x) is conjugate to f′(x ₀) near x ₀. We have proved that the conclusion of the theorem is equivalent to one kind of singularities for bounded linear operators, i.e., x ₀ is a locally fine point for f′(x) or generalized regular point of f(x); so, a complete rank theorem in advanced calculus is established, i.e., a sufficient and necessary condition such that the conclusion of the theorem to be held is given.      

Fonction maximale centrée de Hardy–Littlewood sur les espaces hyperboliques

On montre que la fonction maximale centrée de Hardy–Littlewood, M, sur les espaces hyperboliques réels , satisfait l’inégalité de type faible pour toute f∈L ¹(ℍ ⁿ ), où A>0 est une constante indépendante de la dimension n.     

Multi-self-similar Markov processes on ℝ<Subscript>+</Subscript><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> and their Lamperti representations

 A classical result, due to Lamperti, establishes a one-to-one correspondence between a class of strictly positive Markov processes that are self-similar, and the class of one-dimensional Lévy processes. This correspondence is obtained by suitably time-changing the exponential of the Lévy process. In this paper we generalise Lamperti's result to processes in n dimensions. For the representation we obtain, it is essential that the same time-change be applied to all coordinates of the processes involved. Also for the statement of the main result we need the proper concept of self-similarity in higher dimensions, referred to as multi-self-similarity in the paper. The special case where the Lévy process ξ is standard Brownian motion in n dimensions is studied in detail. There are also specific comments on the case where ξ is an n-dimensional compound Poisson process with drift. Finally, we present some results concerning moment sequences, obtained by studying the multi-self-similar processes that correspond to n-dimensional subordinators. Received: 22 August 2002 / Revised version: 10 February 2003 Published online: 15 April 2003 RID="*" ID="*" MaPhySto – Centre for Mathematical Physics and Stochastics, funded by a grant from the Danish National Research Foundation Mathematics Subject Classification (2000): 60G18, 60G51, 60J25, 60J60, 60J75 Key words or phrases: Lévy process – Self-similarity – Time-change – Exponential functional – Brownian motion – Bessel process – Piecewise deterministic Markov process – Moment sequence     

On Critical Values of Polynomials with Real Critical Points

Let f be a polynomial of degree at least 2 with f(0)=0 and f′(0)=1. Suppose that all the zeros of f′ are real. We show that there is a zero ζ of f′ such that |f(ζ)/ζ|≤2/3, and that this inequality can be taken to be strict unless f is of the form f(z)=z+cz ³.     

Periodic solutions for some second order differential equations with singularity

In this paper, we will prove the existence of infinitely many harmonic and subharmonic solutions for the second order differential equation ẍ + g(x) = f(t, x) using the phase plane analysis methods and Poincaré–Birkhoff Theorem, where the nonlinear restoring field g exhibits superlinear conditions near the infinity and strong singularity at the origin, and f(t, x) = a(t)x ^γ + b(t, x) where 0 ≤ γ ≤ 1 and b(t, x) is bounded.     

Sur l'extension dans le domaine complémentaire d'une variété

Résumé étant donnés une variétéM, un ensemble compactN enM, des entiersr, n avec 1≤r≤n, un espace euclidienE àn dimensions, uner-variétéA enE et une transformation continuef: N→E−A, alors c'est un problème bien connu s'il existe une extensionf′ def satisfaisantf′(M)∈E−A. Ce problème est discuté.

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Riassunto SianoM una varietà,N un insieme compatto inM, 1≤r≤n degli interi,E uno spazio euclideo adn dimensioni,A unar-varietà inE edf:N→E−A una trasformazione continua. Allora è trattato il problema ben noto dell'esistenza di una estensionef′ dif conf′ (M)∈E−A.

     

Differential substitutions of the miura transformation type

The existence of a nontrivial integral and a sufficiently large set of symmetries with respect to one of the characteristics of a hyperbolic equation implies the existence of nontrivial integrals and symmetries with respect to the other characteristic. We classify differential substitutions of the form ω=P(x, u, u _x ) relating families of evolution equations that depend on an arbitrary function. Among such substitutions, there are infinitely many pairwise-nonequivalent ones. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 116, No. 3, pp. 336–348, September, 1998.     

Periodic solutions for some second order differential equations with singularity

In this paper, we will prove the existence of infinitely many harmonic and subharmonic solutions for the second order differential equation ẍ + g(x) = f(t, x) using the phase plane analysis methods and Poincaré–Birkhoff Theorem, where the nonlinear restoring field g exhibits superlinear conditions near the infinity and strong singularity at the origin, and f(t, x) = a(t)x ^γ + b(t, x) where 0 ≤ γ ≤ 1 and b(t, x) is bounded. This project was supported by the Program for New Century Excellent Talents of Ministry of Education of China and the National Natural Science Foundation of China (Grant No. 10671020 and 10301006).