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21
Choosing roots of polynomials smoothly
 Israel J. Math
, 1998
"... Abstract. We clarify the question whether for a smooth curve of polynomials one can choose the roots smoothly and related questions. Applications to perturbation theory of operators are given. Table of contents 1. Introduction........................... ..."
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Cited by 22 (12 self)
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Abstract. We clarify the question whether for a smooth curve of polynomials one can choose the roots smoothly and related questions. Applications to perturbation theory of operators are given. Table of contents 1. Introduction...........................
The Mahler measure of algebraic numbers: a survey.” Conference Proceedings
 University of Bristol
, 2008
"... Abstract. A survey of results for Mahler measure of algebraic numbers, and onevariable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (‘house’) of an algebraic integer are also discussed. Some generalisations are also mentioned, though n ..."
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Abstract. A survey of results for Mahler measure of algebraic numbers, and onevariable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (‘house’) of an algebraic integer are also discussed. Some generalisations are also mentioned, though not to Mahler measure of polynomials in more than one variable. 1.
Chebyshev’s bias for composite numbers with restricted prime divisors
 Math. Comp
, 2005
"... Abstract. Let π(x; d, a) denote the number of primes p ≤ x with p ≡ a(mod d). Chebyshev’s bias is the phenomenon for which “more often” π(x; d, n)>π(x; d, r), than the other way around, where n is a quadratic nonresidue mod d and r is a quadratic residue mod d. Ifπ(x; d, n) ≥ π(x; d, r) for every x ..."
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Abstract. Let π(x; d, a) denote the number of primes p ≤ x with p ≡ a(mod d). Chebyshev’s bias is the phenomenon for which “more often” π(x; d, n)>π(x; d, r), than the other way around, where n is a quadratic nonresidue mod d and r is a quadratic residue mod d. Ifπ(x; d, n) ≥ π(x; d, r) for every x up to some large number, then one expects that N(x; d, n) ≥ N(x; d, r) for every x. Here N(x; d, a) denotes the number of integers n ≤ x such that every prime divisor p of n satisfies p ≡ a(mod d). In this paper we develop some tools to deal with this type of problem and apply them to show that, for example, N(x;4, 3) ≥ N(x;4, 1) for every x. In the process we express the socalled second order LandauRamanujan constant as an infinite series and show that the same type of formula holds for a much larger class of constants. 1.
Countable versus uncountable ranks in infinite semigroups of transformations and relations
"... and relations ..."
CONVEXITY ACCORDING TO THE GEOMETRIC MEAN
"... (communicated by Zs. Páles) Abstract. We develop a parallel theory to the classical theory of convex functions, based on a change of variable formula, by replacing the arithmetic mean by the geometric one. It is shown that many interesting functions such as exp � sinh � cosh � sec � csc � arc sin � ..."
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(communicated by Zs. Páles) Abstract. We develop a parallel theory to the classical theory of convex functions, based on a change of variable formula, by replacing the arithmetic mean by the geometric one. It is shown that many interesting functions such as exp � sinh � cosh � sec � csc � arc sin � Γ etc illustrate the multiplicative version of convexity when restricted to appropriate subintervals of (0 � 1).Asa consequence, we are not only able to improve on a number of classical elementary inequalities but also to discover new ones. 1.
Differentiable perturbation of unbounded operators
 Math. Ann
, 2003
"... Abstract. If A(t) is a C 1,αcurve of unbounded selfadjoint operators with compact resolvents and common domain of definition, then the eigenvalues can be parameterized C 1 in t. If A is C ∞ then the eigenvalues can be parameterized twice differentiably. Theorem. Let t ↦ → A(t) for t ∈ R be a curve ..."
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Cited by 10 (5 self)
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Abstract. If A(t) is a C 1,αcurve of unbounded selfadjoint operators with compact resolvents and common domain of definition, then the eigenvalues can be parameterized C 1 in t. If A is C ∞ then the eigenvalues can be parameterized twice differentiably. Theorem. Let t ↦ → A(t) for t ∈ R be a curve of unbounded selfadjoint operators in a Hilbert space with common domain of definition and with compact resolvent. (A) If A(t) is real analytic in t ∈ R, then the eigenvalues and the eigenvectors of A(t) may be parameterized real analytically in t. (B) If A(t) is C ∞ in t ∈ R and if no two unequal continuously parameterized eigenvalues meet of infinite order at any t ∈ R, then the eigenvalues and the eigenvectors can be parameterized smoothly in t, on the whole parameter domain. (C) If A is C ∞ , then the eigenvalues of A(t) may be parameterized twice differentiably in t. (D) If A(t) is C 1,α for some α> 0 in t ∈ R, then the eigenvalues of A(t) may be parameterized in a C 1 way in t. Part (A) is due to Rellich [10] in 1940, see also [2] and [6], VII, 3.9. Part (B) has been proved in [1], 7.8, see also [8], 50.16, in 1997; there we gave also a different proof of (A). The purpose of this paper is to prove parts (C) and (D). Both results cannot be improved to obtain a C 1,βparameterization of the eigenvalues for some β> 0, by the first example below. In our proof of (D) the assumption C 1,α cannot be weakened to C 1, see the second example. For finite dimensional Hilbert spaces part (D) has been proved under the assumption of C 1 by Rellich [11], with a small inaccuracy in the auxiliary theorem on p. 48: Condition (4) must be more restrictive, otherwise the induction argument on p. 50 is not valid, since the proof on p. 52 relies on the fact that all values coincide at the point in question. A proof can also be found in [6], II, 6.8. We need a strengthened version of this result, thus our proof covers it also. We thank T. and M. HoffmannOstenhof and T. Kappeler for their interest and hints.
Ultraproducts in Analysis
 IN ANALYSIS AND LOGIC, VOLUME 262 OF LONDON MATHEMATICAL SOCIETY LECTURE NOTES
, 2002
"... ..."
The Mathematical Import Of Zermelo's WellOrdering Theorem
 Bull. Symbolic Logic
, 1997
"... this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs ..."
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Cited by 5 (1 self)
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this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his wellknown paradox, an early expression of our motif. The motif becomes fully manifest through the study of functions f :
The Lehmer Constants Of An Annulus
 J. Th. Nombres Bordeaux
, 2001
"... Introduction and results. Let V be an open subset of C . In 1985, Langevin [La1] introduced the following function, the Lehmer constant y of V (see [La3]), defined as L (V ) = inf M (ff) 1= deg(ff) , where the infimum is taken over every nonzero noncyclotomic algebraic number ff lying with it ..."
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Cited by 4 (2 self)
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Introduction and results. Let V be an open subset of C . In 1985, Langevin [La1] introduced the following function, the Lehmer constant y of V (see [La3]), defined as L (V ) = inf M (ff) 1= deg(ff) , where the infimum is taken over every nonzero noncyclotomic algebraic number ff lying with its conjugates in C nV . Here M (ff) is the Mahler measure of ff: M (ff) = ja 0 j d Q i=1 max (1; jff i j) , where ff = ff 1 has minimal polynomial a 0 z d + :::
MANY PARAMETER LIPSCHITZ PERTURBATION OF UNBOUNDED OPERATORS
, 2007
"... If u ↦ A(u) is a C 1,αmapping having as values unbounded selfadjoint operators with compact resolvents and common domain of definition, parametrized by u in an (even infinite dimensional) space then any continuous arrangement of the eigenvalues u ↦ → λi(u) is C 0,1 in u. If u ↦ → A(u) is C 0,1, th ..."
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Cited by 4 (3 self)
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If u ↦ A(u) is a C 1,αmapping having as values unbounded selfadjoint operators with compact resolvents and common domain of definition, parametrized by u in an (even infinite dimensional) space then any continuous arrangement of the eigenvalues u ↦ → λi(u) is C 0,1 in u. If u ↦ → A(u) is C 0,1, then the eigenvalues may be chosen C 0,1/N (even C 0,1 if N = 2), locally in u, where N is locally the maximal multiplicity of the eigenvalues.