Zbigniew Slodkowski | University of Illinois at Chicago (original) (raw)

Papers by Zbigniew Slodkowski

Rendiconti del Seminario Matematico della Università di Padova, 1986

L'accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova... more L'accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ A Generalization of Vesentini and Wermer's Theorems. ZBIGNIEW SLODKOWSKI Introduction. Applications of potential theory to spectral theory began in 1968 with the following theorem of E. Vesentini [11]. If z-~ Tz is an analytic operator valued function in Gee then where r (~) denotes the spectral radius, is subharmonic in G. J. Wermer [12] has obtained a similar result in uniform algebras: If f, g belong to a uniform algebra A then the function log

Annali della Scuola normale superiore di Pisa. Classe di scienze, Mar 9, 2017

Environment and Planning A, May 1, 1979

A mathematical model of urban spatial interaction based on the intervening-opportunities principl... more A mathematical model of urban spatial interaction based on the intervening-opportunities principle is discussed and its equilibria are studied. It is shown that, under natural assumptions, the number of equilibria is finite, and a mathematical criterion for distinguishing the equilibrium corresponding to reality is given.

Environment and Planning A, Feb 1, 1984

In this paper the equilibria of an urban retail model based on the principle of gravitation are s... more In this paper the equilibria of an urban retail model based on the principle of gravitation are studied. It is shown, among other things, that a positive equilibrium exists and is unique when the parameter a of the model is less than 1; for a equal to 1 a sufficient condition for uniqueness is given.

International Journal of Mathematics and Mathematical Sciences, 2014

We provide a new proof for the description of holomorphic and biholomorphic flows on multiply con... more We provide a new proof for the description of holomorphic and biholomorphic flows on multiply connected domains in the complex plane. In contrast to the original proof of Heins (1941) we do this by the means of operator theory and by utilizing the techniques of universal coverings of the underlying domains of holomorphic flows and their liftings on the corresponding universal coverings.

Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 1995

© Scuola Normale Superiore, Pisa, 1995, tous droits réservés. L’accès aux archives de la revue « ... more © Scuola Normale Superiore, Pisa, 1995, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

arXiv (Cornell University), Apr 6, 2019

In previous works, G. Tomassini and the authors studied and classified complex surfaces admitting... more In previous works, G. Tomassini and the authors studied and classified complex surfaces admitting a real-analytic plurisubharmonic exhaustion function; let X be such a surface and D ⊆ X a domain admitting a continuous plurisubharmonic exhaustion function: what can be said about the geometry of D? If the exhaustion of D is assumed to be smooth, the second author already answered this question; however, the continuous case is more difficult and requires different methods. In the present paper, we address such question by studying the local maximum sets contained in D and their interplay with the complex geometric structure of X; we conclude that, if D is not a modification of a Stein space, then it shares the same geometric features of X.

arXiv (Cornell University), Nov 17, 2016

Journal of Functional Analysis, May 1, 1996

Let M=[u=0] be a smooth hypersurface of a domain 0 in C 2. As it is well known M is Levi flat if ... more Let M=[u=0] be a smooth hypersurface of a domain 0 in C 2. As it is well known M is Levi flat if and only if 0 u z1 u z2 L(u)=&det \ u zÄ 1 u zÄ 1 z1 u zÄ 1 z2 + =0. u zÄ 2 u zÄ 2 z1 u zÄ 2 z2 In general, for a given M we introduce the function k L (M)=L(u) 3 Â| u|, thè`L evi curvature'' of M(| u| 2 =|u z1 | 2 + |u z2 | 2). |k L (M)| depends only on M and k L (M) 0 means that locally on [u=0], [u<0] is pseudoconvex [12]. L(u), viewed as a differential operator acting on u is called the Levi operator (for non Cartesian hypersurfaces); L(u) is an elliptic degenerate quasi-linear operator. In this paper we study for L(u) the Dirichlet problem (C): L(u)=k | u| 3 in 0, u= g on b0 where 0 is bounded and strictly pseudoconvex, k=k(z, t) is a real function on 0_R and g : b0 Ä R. The geometric counter part of this problem is the following: given a bounded domain 0 in C 2 and a family of hypersurfaces # c =[ g=c] of b0 find a family of level sets M c =[u=c] such that bM c =# c and k L (M c)=k(}, c). In particular, when k=0 then the level sets M c form a family of Levi flat hypersurfaces with article no.

$Research paper thumbnail of Evolution of subsets of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">C</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb {C}^2</annotation></semantics></math>C2 and parabolic problem for the Levi equation$

Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 1997

Proceedings of the American Mathematical Society, 1991

Let ^ bea maximum modulus algebra on X , and V a maximal open subset of X such that V has the str... more Let ^ bea maximum modulus algebra on X , and V a maximal open subset of X such that V has the structure of one-dimensional variety on which functions from A are analytic. Then, the restriction algebra Ax,v is again a maximum modulus algebra.

$Research paper thumbnail of Polynomial hulls in <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">C</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb {C}^2</annotation></semantics></math>C2 and quasicircles$

Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 1989

Cornell University - arXiv, Nov 10, 2022

Replying to three questions posed by N. Shcherbina, we show that a compact psudoconcave set can h... more Replying to three questions posed by N. Shcherbina, we show that a compact psudoconcave set can have the core smaller than itself, that the core of a compact set must be pseudoconcave, and that it can be decomposed into compact pseudoconcave sets on which all smooth plurisubharmonic functions are constant.

arXiv: Complex Variables, Apr 27, 2015

A weakly complete space is a complex space admitting a (smooth) plurisubharmonic exhaustion funct... more A weakly complete space is a complex space admitting a (smooth) plurisubharmonic exhaustion function. In this paper, we classify those weakly complete complex surfaces for which such exhaustion function can be chosen real analytic: they can be modifications of Stein spaces or proper over a non compact (possibly singular) complex curve or foliated with real analytic Levi-flat hypersurfaces which in turn are foliated by dense complex leaves (these we call surfaces of Grauert type). In the last case, we also show that such Levi-flat hypersurfaces are in fact level sets of a global proper pluriharmonic function, up to passing to a holomorphic double cover of the space. An example of Brunella shows that not every weakly complete surface can be endowed with a real analytic plurisubharmonic exhaustion function. Our method of proof is based on the careful analysis of the level sets of the given exhaustion function and their intersections with the minimal singular set, i.e the set where every plurisubharmonic exhaustion function has a degenerate Levi form.

Advances in Complex Geometry

We prove that the core of a complex manifold X is the union of pairwise disjoint pseudoconcave se... more We prove that the core of a complex manifold X is the union of pairwise disjoint pseudoconcave sets on which all uniformly bounded continuous plurisubharmonic functions on X are constant. Similarly, the minimal kernel of a weakly complete complex manifold decomposes into the union of compact pseudoconcave sets on which all continuous plurisub-harmonic functions are constant. Versions of these results for standard smoothness classes are obtained. Analogous facts are discussed in the context of Richberg's regularization of continuous strongly plurisubharmonic functions.

L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scien... more L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.