Existence of Moduli for Bi-Lipschitz Equivalence of Analytic Functions

We show that the bi-Lipschitz equivalence of analytic function germs (², 0)(, 0) admits continuous moduli. More precisely, we propose an invariant of the bi-Lipschitz equivalence of such germs that varies continuously in many analytic families f _t : (², 0)(, 0). For a single germ f the invariant of f is given in terms of the leading coefficients of the asymptotic expansions of f along the branches of generic polar curve of f.     

On a global descent method for polynomials

Summary Given a complex polynomialp we determine a functionf _p : such that |p(f _p (z))||p(z)|,z withk<1. This result is used to introduce a global root-finding algorithm for polynomials.     

Invariant subrings of ℂ[X,Y,Z] which are complete intersections

It is known that for every finite subgroup G of SL(2,), the invariant subring [X,Y]^G is a hyper-surface. In this note we treat finite subgroups of SL(3,) and give complete classification of the finite subgroups of SL(3,) whose invariant subrings are complete intersections.     

Nicht endlich erzeugte Primideale in Steinschen Algebren

In this note we exhibit a closed prime idealF in the ring Ó(³) of all holomorphic functions on ³ which is not finitely generated.F is the ideal of a certain irreducible curve Y³, obtained as the image of a proper holomorphic map f³.

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Herrn Karl Stein gewidmet     

On the Monodromy of a Multivalued Function along Its Ramification Locus

We consider multivalued analytic functions in ⁿ) whose set of singular points occupies a very small part of ⁿ). Under a mapping of a topological space Y into ⁿ), such a function f can induce a multivalued function on Y. This is possible even if the image of Y entirely lies in the ramification set of f. We estimate the monodromy group of the induced function via the monodromy group of f.     

Surjectivity of quotient maps for algebraic (ℂ,+)-actions and polynomial maps with contractible fibers

In this paper we establish two results concerning algebraic (,+)-actions on ⁿ . First, let be an algebraic (,+)-action on ³. By a result of Miyanishi, its ring of invariants is isomorphic to [t ₁,t ₂]. Iff ₁,f ₂ generate this ring, the quotient map of is the mapF:³²,x(f ₁(x), f₂(x)). By using some topological arguments we prove thatF is always surjective. Secon, we are interested in dominant polynomial mapsF: ^{n n-1} whose connected components of their generic fibers are contractible. For such maps, we prove the existence of an algebraic (,+)-action on ⁿ for whichF is invariant. Moreover we give some conditions so thatF*([t ₁,...,t _n-1 ]) is the ring of invariants of .Dedicated to all my friends and my family     

Direct and inverse spectral problems for generalized strings

Let the functionQ be holomorphic in he upper half plane ⁺ and such that ImQ(z 0 and ImzQ(z) 0 ifz ⁺. A basic result of M.G. Krein states that these functionsQ are the principal Titchmarsh-Weyl coefficiens of a (regular or singular) stringS[L,m] with a (non-decreasing) mass distribution functionm on some interval [0,L) with a free left endpoint 0. This string corresponds to the eigenvalue problemdf + fdm = 0; f(0–) = 0. In this note we show that the set of functionsQ which are holomorphic in ⁺ and such that the kernel has negative squares of ⁺ and ImzQ(z) 0 ifz ⁺ is the principal Titchmarsh-Weyl coefficient of a generalized string, which is described by the eigenvalue problemdf +f dm + ² fdD = 0 on [0,L),f(0–) = 0. Here is the number of pointsx whereD increases or 0 >m(x + 0) –m(x – 0) –; outside of these pointsx the functionm is locally non-decreasing and the functionD is constant.To the memory of M.G. Krein with deep gratitude and affection.This author is supported by the Fonds zur Förderung der wissenschaftlichen Forschung of Austria, Project P 09832     

Common cyclic entire functions for partial differential operators

Let H(^N) denote the Fréchet space of all entire functions of N variables (N1). The purpose of this paper is to prove the existence of a dense set of functions f in H(^N) such that if L is any nonscalar linear differential operator with constant coefficients, then the set {p(L)fp(·) is a polynomial} is dense in H(^N).Research supported in part by an NSF grant     

K-theory of fine topological algebras, Chern character, and assembly

TheK-theory of the group algebra [] for a countable, discrete group is defined in terms of the simplicial ring of smooth simplices on [], where [] is given the fine topology with respect to its finite-dimensional, linear subspaces. The assembly map for this theory :K _* B K _* [] is studied and shown to be a rational injection. The proof uses the Connes-Karoubi Chern character fromK-theory of Banach algebras to cyclic homology, here generalized to any fine topological algebra, and proved to be multiplicative.     

A note on the classification of holomorphic harmonic morphisms

In this note we give a complete classification of those holomorphic maps :U ⁿ defined on open and connected subsets of ^m which are harmonic morphisms.The first author was supported by the Icelandic Science Fund.     

A spectral mapping theorem for scalar-type spectral operators in locally convex spaces

LetT be a continuous scalar-type spectral operator defined on a quasi-complete locally convex spaceX, that is,T=fdP whereP is an equicontinuous spectral measure inX andf is aP-integrable function. It is shown that (T) is precisely the closedP-essential range of the functionf or equivalently, that (T) is equal to the support of the (unique) equicontinuous spectral measureQ ^* defined on the Borel sets of the extended complex plane ^* such thatQ ^*({})=0 andT=zdQ _*(z). This result is then used to prove a spectral mapping theorem; namely, thatg((T))=(g(T)) for anyQ ^*-integrable functiong: ^{* *} which is continuous on (T). This is an improvement on previous results of this type since it covers the case wheng((T))/{} is an unbounded set in a phenomenon which occurs often for continuous operatorsT defined in non-normable spacesX.     

A and the vanishing topology of discriminants

Summary Suppose thatf: ⁿ , 0 ^p , 0 is finitely -determined withnp. We define a Milnor fiber for the discriminant off; it is the discriminant of a stabilization off. We prove that this discriminant Milnor fiber has the homotopy type of a wedge of spheres of dimensionp–1, whose number we denote byµ (f). One of the main theorems of the paper is a = type result: if (n, p) is in the range of nice dimensions in the sense of Mather, then -codium,with equality iff is weighted homogeneous. Outside the nice dimensions we obtain analogous formulae with correction terms measuring the presence of unstable but topologically stable germs in the stabilization. These results are further extended to nonlinear sections of free divisors.Oblatum 15-VIII-1990Partially supported by a grant from the National Science Foundation and a Fullbright Fellowship     

Toeplitz and Hankel matrices on ℂ n

It is known that the characterizations of the Toeplitz operatorT onH ² and also the Hankel operatorH onH ² by using the simple unilateral shiftT _z . Recently, some characterization of the normal Toeplitz matrix truncated on ⁿ is given by D.R.Farenick, M. Krupnik, N. Krupnik and W. Y. Lee [1] and, independently, by T.Ito [2]. In this paper, we shall give some characterizations of the Toeplitz matrix and also the Hankel matrix truncated on ⁿ .Dedicated to Professor Masanori Fukamiya on his 88th birthday     

Harmonic morphisms from complex projective spaces

In this paper we study harmonic morphisms :U P ^m N ² from open subsets of complex projective spaces to Riemann surfaces. We construct many new examples of such maps which are not holomorphic with respect to the standard Kähler structure on P ^m.The research leading to this paper was supported by the Icelandic Science Fund and the Danish National Science Fund.     

Differentiable structures with zero entropy on simply connected 4-manifolds

We show that a closed 4-dimensional simply connected topological manifoldM admits a differentiable structure with aC Riemannian metric whose geodesic flow has zero topological entropy if and only ifM is homeomorphic toS ⁴, ²,S ²×S ², or ²#².     

The degrees of polynomial self-mappings of ℂ2

Two results on the degrees of polynomial mappings ²² are obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 527–534, April, 1998.     

Entire functions of exponential type and model subspaces in HP

Let W ^p be the space of entire fonctions in of exponential type , whose restrictions to the real line belong to L^p(). For an inner function in the upper half- plane ₊ we denote by K ^p (p 1) the invariant subspace of the inverse shift operator (the model subspace) in H^p, generated by the function K ^p =H^p , where H^p=H^p(₊) is the Hardy class. In the paper it is shown that many known properties of the spaces W ^p (some imbedding and uniqueness theorems, the Logvinenko-Sereda theorem on equivalent norms, S. N. Bernshtein's differential inequality) remain valid for the spaces K ^p if and only if the derivative is bounded. The classical results for entire functions are obtained for (x)= e^ix.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 81–100, 1991.     

Milnor numbers and the topology of polynomial hypersurfaces

Summary LetF: n + 1 be a polynomial. The problem of determining the homology groupsH _q (F ^–1 (c)), c , in terms of the critical points ofF is considered. In the best case it is shown, for a certain generic class of polynomials (tame polynomials), that for allc,F ^–1 (c) has the homotopy type of a bouquet of - ^c n-spheres. Here is the sum of all the Milnor numbers ofF at critical points ofF and ^c is the corresponding sum for critical points lying onF ^–1 (c). A second best case is also discussed and the homology groupsH _q (F ^–1 (c)) are calculated for genericc. This case gives an example in which the critical points at infinity ofF must be considered in order to determine the homology groupsH _q (F ^–1 (c)).     

On fixed points of conformal pseudogroups

We show in this paper Theorem 2 that if (H, H ₁) is a pseudogroup generated by a finite numberH ₁ of germs of conformal diffeomorphisms of defined on a sufficiently small discD, which is not linearizable and such that the linear group (L,H ₁)={g(0)/g(H,H ₁)}* is dense in *, then the set of fixed points of the pseudogroup (H, H ₁) is dense inD. This implies the abundance of distinct homotopy classes of loops in leaves of foliations defined in ² by generic polynomial vector fields as well as for germs of holomorphic vector fields in ² beginning with generic jets, both of degree at least 2. These homotopy classes may be realized arbitrarily close to the line at infinity or to 0, respectively. This shows the genericity of polynomial vector fields with infinite Petrovsky-Landis genus ([5]).The idea of the proof is very simple. Ifg is a non-linear conformal diffeomorphism with multiplier =g'(0), then the map obtained by the composition ofg and the linear map with multiplier ^–1 will have at 0 a fixed point of multiplicity at least 2. Since we may approximate ^–1 by elementsh in the pseudogroup and the multiplicity of fixed points satisfy a law of conservation of number, we obtain thath o g has fixed points close to 0. These fixed points appear as a by product of the relative nonlinearity of the generators of the pseudogroup, since linearizable pseudogroups have 0 as an isolated fixed point. The fixed points obtained are not conjugate since they have distinct multipliers.The main technical tool is the angular derivative introduced in [8]. It allows one to split the search for fixed points into two parts: One is to obtain a contraction and the other is to return arbitrarily close to the starting point without modifying the property of contraction. This is carried out since the angular derivative is multiplicative for compositions and is identically 1 for linear maps.Supported by CONACYT-CNRS and CONACYT 3398-E9307.     

Quasisimilarity of model contractions with unequal defects

Let T and T be C₁₀ contractions with characteristic functions H (ⁿⁿ⁺¹), H (^mm+1). The fundamental result is: T and T are quasisimilar if and only if The paper contains an analysis of this condition; examples are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 24–37, 1986.