The growth of the area of a transcendental analytic set. II

1964 ◽  
Vol 156 (2) ◽  
pp. 144-170 ◽  
Author(s):  
Wilhelm Stoll
Keyword(s):  
Analytic Set

Remark on Irreducible Factorizations of Polynomials in Several Variables*

1981 ◽  
Vol 24 (4) ◽  
pp. 493-496
Author(s):  
Leiba Rodman
Keyword(s):  
Complex Variables ◽  
Several Complex Variables ◽  
Analytic Set ◽  
Several Variables ◽  
Set Of Points

AbstractIt is proved that a polynomial on several complex variables, whose coefficients depend analytically on a parameter ε, admits a factorization which is irreducible for every value of the parameter, with the possible exception of an analytic set of points. Moreover, the coefficients of the irreducible factors can be chosen to depend analytically on ε in a neighborhood of every point not belonging to this analytic set.

scholarly journals Tameness of Complex Dimension in a Real Analytic Set

2013 ◽  
Vol 65 (4) ◽  
pp. 721-739
Author(s):  
Janusz Adamus ◽  
Serge Randriambololona ◽  
Rasul Shafikov
Keyword(s):  
Positive Integer ◽  
Complex Manifold ◽  
Analytic Set ◽  
Complex Dimension ◽  
Real Analytic ◽  
Set Of Points

AbstractGiven a real analytic set X in a complex manifold and a positive integer d, denote by Ad the set of points p in X at which there exists a germ of a complex analytic set of dimension d contained in X. It is proved that Ad is a closed semianalytic subset of X.

scholarly journals Joint spectra and analytic set-valued functions

1986 ◽  
Vol 294 (1) ◽  
pp. 187-187 ◽  
Author(s):  
M. Klimek
Keyword(s):  
Analytic Set

scholarly journals Iteration of Analytic Multifunctions

2001 ◽  
Vol 162 ◽  
pp. 19-40 ◽  
Author(s):  
Maciej Klimek
Keyword(s):  
Julia Sets ◽  
Finite Family ◽  
Analytic Set ◽  
Polynomial Mappings ◽  
Markov's Inequality ◽  
Analytic Multifunctions ◽  
Markov’S Inequality

It is shown that iteration of analytic set-valued functions can be used to generate composite Julia sets in CN. Then it is shown that the composite Julia sets generated by a finite family of regular polynomial mappings of degree at least 2 in CN, depend analytically on the generating polynomials, in the sense of the theory of analytic set-valued functions. It is also proved that every pluriregular set can be approximated by composite Julia sets. Finally, iteration of infinitely many polynomial mappings is used to give examples of pluriregular sets which are not composite Julia sets and on which Markov’s inequality fails.

Every analytic set is Ramsey

1970 ◽  
Vol 35 (1) ◽  
pp. 60-64 ◽  
Author(s):  
Jack Silver
Keyword(s):  
Measurable Cardinal ◽  
Infinite Subset ◽  
The Other ◽  
Natural Numbers ◽  
Ramsey Property ◽  
Analytic Subset ◽  
Borel Set ◽  
Analytic Set ◽  
Principal Theorem ◽  
Countably Infinite

If X is a set, [Χ]ω will denote the set of countably infinite subsets of X. ω is the set of natural numbers. If S is a subset of [ω]ω, we shall say that S is Ramsey if there is some infinite subset X of ω such that either [Χ]ω ⊆ S or [Χ]ω ∩ S = 0. Dana Scott (unpublished) has asked which sets, in terms of logical complexity, are Ramsey.The principal theorem of this paper is: Every Σ11 (i.e., analytic) subset of [ω]ω is Ramsey (for the Σ, Π notations, see Addison [1]). This improves a result of Galvin-Prikry [2] to the effect that every Borel set is Ramsey. Our theorem is essentially optimal because, if the axiom of constructibility is true, then Gödel's Σ21 Π21 well-ordering of the set of reals [3], having the convenient property that the set of ω-sequences of reals enumerating initial segments is also Σ21 ∩ Π21, rather directly gives a Σ21 ∩ Π21 set which is not Ramsey. On the other hand, from the assumption that there is a measurable cardinal we shall derive the conclusion that every Σ21 (i.e., PCA) is Ramsey. Also, we shall explore the connection between Martin's axiom and the Ramsey property.

Strong and weak duality principles for fractal dimension in Euclidean space

1995 ◽  
Vol 118 (3) ◽  
pp. 393-410 ◽  
Author(s):  
Colleen D. Cutler
Keyword(s):  
Strong Duality ◽  
The Other ◽  
Duality Principle ◽  
Weak Duality ◽  
Pointwise Dimension ◽  
Analytic Subset ◽  
Analytic Set ◽  
Borel Measures ◽  
Dual Representations ◽  
Packing Dimensions

AbstractTricot [27] provided apparently dual representations of the Hausdorff and packing dimensions of any analytic subset of Euclidean d-space in terms of, respectively, the lower and upper pointwise dimension maps of the finite Borel measures on ℝd. In this paper we show that Tricot's two representations, while similar in appearance, are in fact not duals of each other, but rather the duals of two other ‘missing’ representations. The key to obtaining these missing representations lies in extended Frostman and antiFrostman lemmas, both of which we develop in this paper. This leads to the formulation of two distinct characterizations of dim (A) and Dim (A), one which we call the weak duality principle and the other the strong duality principle. In particular, the strong duality principle is concerned with the existence, for each analytic set A, of measures on A that are (almost) of the same exact dimension (Hausdorff or packing) as A. The connection with Rényi (or information) dimension and a variational principle of Cutler and Olsen[12] is also established.

scholarly journals Note on the Effective Choice of a Point in the Complement of an Analytic Set.

1970 ◽  
Vol 35 (1) ◽  
pp. 146
Author(s):  
M. R. Krom ◽  
Yoemon Sampei
Keyword(s):  
Analytic Set ◽  
Effective Choice

scholarly journals The saturation ofk-analytic rings and topological equivalence of associated analytic set germs

1982 ◽  
Vol 100 (2) ◽  
pp. 271-285
Author(s):  
Ulrich Daepp
Keyword(s):  
Topological Equivalence ◽  
Analytic Set
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