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  • 1
    Electronic Resource
    Electronic Resource
    Conjugation for polynomial mappings (1995)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 46 (1995), S. 872-882 
    Details
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We consider Keller's functions, namely polynomial functionsf:C n →C n with detf(x)=1 at allx εC n. Keller conjectured that they are all bijective and have polynomial inverses. The problem is still open. Without loss of generality assumef(0)=0 andf'(0)=I. We study the existence of certain mappingsh λ, λ 〉 1, defined by power series in a ball with center at the origin, such thath′λ(0)=I andh λ(λf(x))=λh λ(x). So eachh λ conjugates λf to its linear part λI in a ball where it is injective. We conjecture that for Keller's functionsf of the homogeneous formf(x)=x +g(x),g(sx)=s dg(x),g′(x)n=0,xεC n,sεC the conjugationh λ for λf is anentire function.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00917874
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Global inversion of functions: an introduction (1994)
    Springer
    Nonlinear differential equations and applications 1 (1994), S. 229-248 
    Details
    ISSN: 1420-9004
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This is an exposition of some basic ideas in the realm of Global Inverse Function theorems. We address ourselves mainly to readers who are interested in the applications to Differential Equations. But we do not deal with those applications and we give a ‘self-contained” elementary exposition. The first part is devoted to the celebrated Hadamard-Caccioppoli theorem on proper local homeomorphisms treated in the framework of the Hausdorff spaces. In the proof, the concept of ‘ω-limit set’ is used in a crucial way and this is perhaps the novelty of our approach. In the second part we deal with open sets in Banach spaces. The concept of ‘attraction basin’ here is the main tool of our exposition which also shows a few recent results, here extended from finite dimensional to general Banach spaces, together with the classical theorem of Hadamard-Levy which assumes that the operator norm of the inverse of the derivative does not grow too fast (roughly at most linearly).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01197748
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Local extrema of analytic functions (1996)
    Springer
    Nonlinear differential equations and applications 3 (1996), S. 287-303 
    Details
    ISSN: 1420-9004
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We give a complete answer to the problem of the finite decidability of the local extremality character of a real analytic function at a given point, a problem that found partial answers in some works by Severi and Łojasiewicz. Consider a real analytic functionf defined in a neighbourhood of a pointx 0∈R n . Restrictf to the spherical surface centered inx 0 and with radiusr≥0 and take its infimumm(r) and its supremumM(r). We establish some properties ofm(r) andM(r) for smallr〉0. In particular, we prove that they have asymptotic expansions of the formf(x 0)+c·(r α+o(r α)) asr→0 for a realc and a rational α≥1 (of course the parameters will usually be different form andM).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01194068
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    On the Jacobian conjecture for global asymptotic stability (1992)
    Springer
    Journal of dynamics and differential equations 4 (1992), S. 43-55 
    Details
    ISSN: 1572-9222
    Keywords: Jacobian conditions ; global injectivity ; global stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An old conjecture says that, for the two-dimensional system of ordinary differential equationsx=f(x), wheref: ℝ2 → ℝ2,f εC 1, andf(0)=0 the originx=0 should beglobally asymptotically stable (i.e., a stable equilibrium and all trajectoriesx(t) converge to it ast → +∞) whenever the following conditions on the Jacobian matrixJ(x) off hold: trJ(x) 〈 0, detJ(x) 〉 0, ∀x ε ℝ2 It is known that if such anf is globallyone-to-one as a mapping of the plane into itself, then the origin is a globally asymptotically stable equilibrium point for the systemx =f(x). In this paper we outline a new strategy to tackle the injectivity off, based on anauxiliary boundary value problem. The strategy is shown to be successful if the norm of the matrixJ(x) T J(x)t/det J(x) is bounded or, at least, grows slowly (for instance, linearly) as ¦x¦ → t∞.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01048154
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    Paper (German National Licenses)
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