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Hyperlinear and sofic groups: a brief guide
 Bull. Symbolic Logic
"... Relatively recently, two new classes of (discrete, countable) groups have been isolated: hyperlinear groups and sofic groups. They come from different corners of mathematics (operator algebras and symbolic dynamics, respectively), and were introduced independently from each other, but are closely re ..."
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Relatively recently, two new classes of (discrete, countable) groups have been isolated: hyperlinear groups and sofic groups. They come from different corners of mathematics (operator algebras and symbolic dynamics, respectively), and were introduced independently from each other, but are closely related nevertheless.
On the order of countable graphs
 European J. Comb
, 2003
"... sets, Rigid graphs, Universal graphs. The authors wish to thank the MittagLeffler Institute, for its ..."
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sets, Rigid graphs, Universal graphs. The authors wish to thank the MittagLeffler Institute, for its
Determinacy for Infinite Games with More Than Two Players with Preferences
 ILLC Publication Series PP2003
, 2003
"... We discuss infinite zerosum perfectinformation games with more than two players. They are not determined in the traditional sense, but as soon as you fix a preference function for the players and assume common knowledge of rationality and this preference function among the players, you get determi ..."
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We discuss infinite zerosum perfectinformation games with more than two players. They are not determined in the traditional sense, but as soon as you fix a preference function for the players and assume common knowledge of rationality and this preference function among the players, you get determinacy for open and closed payo# sets.
A simple inductive measure analysis for cardinals under the Axiom of Determinacy, submitted to
 the Proceedings of the North Texas Logic Conference; ILLC Publication Series
"... Abstract. In this paper, we give a thorough and basic introduction to the main techniques dealing with computation of cardinals under the Axiom of Determinacy by measure analyses. As an application, we give a simple inductive measure analysis (without invoking Jackson’s “description theory”) that al ..."
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Abstract. In this paper, we give a thorough and basic introduction to the main techniques dealing with computation of cardinals under the Axiom of Determinacy by measure analyses. As an application, we give a simple inductive measure analysis (without invoking Jackson’s “description theory”) that allows the computation of further Jónsson cardinals. The Axiom of Determinacy AD is a gametheoretic statement expressing that all infinite twoplayer perfect information games with a countable set of possible moves are determined, i.e., admit a winning strategy for one of the players. The restriction to countable sets of possible moves makes AD essentially a statement about real numbers and sets of real numbers, and traditionally it has been investigated by descriptive set theorists. As a consequence, it comes as a surprise to see that AD has strikingly peculiar consequences for the combinatorics on uncountable cardinals such as ℵω: for example, the Axiom of Determinacy implies that every algebra on ℵω has a proper subalgebra of cardinality ℵω. How can an axiom that talks about the existence of
The Simulation Technique and its Applications to Infinitary Combinatorics under the Axiom of Blackwell Determinacy
 Journal of Mathematics (ILLC Publications PP200318
"... motivated by games used in statistics. It is known that ..."
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motivated by games used in statistics. It is known that
The Pointwise View Of Determinacy: Arboreal Forcings, Measurability, and Weak Measurability
, 2003
"... We prove that for all standard arboreal forcing notions P there is a counterexample for the implication "If A is determined, then A is Pmeasurable". Moreover, we investigate for which forcing notions this is extendible to "weakly Pmeasurable". ..."
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We prove that for all standard arboreal forcing notions P there is a counterexample for the implication "If A is determined, then A is Pmeasurable". Moreover, we investigate for which forcing notions this is extendible to "weakly Pmeasurable".
HIGHER MOMENTS OF BANACH SPACE VALUED RANDOM VARIABLES
"... Abstract. We define the k:th moment of a Banach space valued random variable as the expectation of its k:th tensor power; thus the moment (if it exists) is an element of a tensor power of the original Banach space. We study both the projective and injective tensor products, and their relation. Moreo ..."
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Abstract. We define the k:th moment of a Banach space valued random variable as the expectation of its k:th tensor power; thus the moment (if it exists) is an element of a tensor power of the original Banach space. We study both the projective and injective tensor products, and their relation. Moreover, in order to be general and flexible, we study three different types of expectations: Bochner integrals, Pettis integrals and Dunford integrals. One of the problems studied is whether two random variables with the same injective moments (of a given order) necessarily have the same projective moments; this is of interest in applications. We show that this holds if the Banach space has the approximation property, but not in general. Several sections are devoted to results in special Banach spaces, including Hilbert spaces, C pK q and Dr0, 1s. The latter space is nonseparable, which complicates the arguments, and we prove various preliminary results on e.g. measurability in Dr0, 1s that we need. One of the main motivations of this paper is the application to Zolotarev metrics and their use in the contraction method. This is sketched in an appendix. 1.
Small Locally Compact Linearly Lindelöf Spaces ∗
, 2008
"... There is a locally compact Hausdorff space of weight ℵω which is linearly Lindelöf and not Lindelöf. We shall prove: Theorem 1 There is a compact Hausdorff space X and a point p in X such that: 1. χ(p, X) = w(X) = ℵω. 2. For all regular κ> ω, no κsequence of points distinct from p converges to p. ..."
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There is a locally compact Hausdorff space of weight ℵω which is linearly Lindelöf and not Lindelöf. We shall prove: Theorem 1 There is a compact Hausdorff space X and a point p in X such that: 1. χ(p, X) = w(X) = ℵω. 2. For all regular κ> ω, no κsequence of points distinct from p converges to p. As usual, χ(p, X), the character of p in X, is the least size of a local base at p, and w(X), the weight of X, is the least size of a base for X. This theorem with “ℶω ” replacing “ℵω ” was proved in [11]. Arhangel’skii and Buzyakova [1] point out that if X, p satisfy (2) of the theorem, then the space X\{p} is linearly Lindelöf and locally compact; if in addition χ(p, X)> ℵ0, then X\{p} is not Lindelöf. (2) requires cf(χ(p, X)) = ω, because there must be a sequence of type cf(χ(p, X)) converging to p. Thus, in (1) of the theorem, ℵω is the smallest possible uncountable value for χ(p, X) and w(X). As in [11], the X of the theorem will be constructed as an inverse limit, using the following terminology: