Infinite -games with imperfect information
D. Blackwell (1969)
Applicationes Mathematicae
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D. Blackwell (1969)
Applicationes Mathematicae
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Tapani Hyttinen, Saharon Shelah, Jouko Vaananen (2002)
Fundamenta Mathematicae
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By results of [9] there are models and for which the Ehrenfeucht-Fraïssé game of length ω₁, , is non-determined, but it is consistent relative to the consistency of a measurable cardinal that no such models have cardinality ≤ ℵ₂. We now improve the work of [9] in two ways. Firstly, we prove that the consistency strength of the statement “CH and is determined for all models and of cardinality ℵ₂” is that of a weakly compact cardinal. On the other hand, we show that if , T is a countable...
C. A. Pensavalle, G. Pieri (2010)
Bollettino dell'Unione Matematica Italiana
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Consider a M-player game in strategic form where the set is a closed interval of real numbers and the payoff function is concave and differentiable with respect to the variable , for any . The aim of this paper is to find appropriate conditions on the payoff functions under the well-posedness with respect to the related variational inequality is equivalent to the formulation of the Tykhonov well-posedness in a game context. The idea of the proof is to appeal to a third equivalence,...
Marion Scheepers, Franklin D. Tall (2010)
Fundamenta Mathematicae
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Arhangel’skii proved that if a first countable Hausdorff space is Lindelöf, then its cardinality is at most . Such a clean upper bound for Lindelöf spaces in the larger class of spaces whose points are has been more elusive. In this paper we continue the agenda started by the second author, [Topology Appl. 63 (1995)], of considering the cardinality problem for spaces satisfying stronger versions of the Lindelöf property. Infinite games and selection principles, especially the Rothberger...
Santi Spadaro (2016)
Fundamenta Mathematicae
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We apply the theory of infinite two-person games to two well-known problems in topology: Suslin’s Problem and Arhangel’skii’s problem on the weak Lindelöf number of the topology on a compact space. More specifically, we prove results of which the following two are special cases: 1) every linearly ordered topological space satisfying the game-theoretic version of the countable chain condition is separable, and 2) in every compact space satisfying the game-theoretic version of the weak...
Shunsuke Nakamura, Ryo Hanafusa, Wataru Ogasa, Takeru Kitagawa, Ryohei Miyadera (2013)
Visual Mathematics
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Krzysztof Ciesielski, Andrés Millán, Janusz Pawlikowski (2003)
Fundamenta Mathematicae
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We formulate a Covering Property Axiom , which holds in the iterated perfect set model, and show that it implies the existence of uncountable strong γ-sets in ℝ (which are strongly meager) as well as uncountable γ-sets in ℝ which are not strongly meager. These sets must be of cardinality ω₁ < , since every γ-set is universally null, while implies that every universally null has cardinality less than = ω₂. We also show that implies the existence of a partition of ℝ into ω₁ null...
Vladimir Vladimirovich Tkachuk (1995)
Commentationes Mathematicae Universitatis Carolinae
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The old game is the point-open one discovered independently by F. Galvin [7] and R. Telgársky [17]. Recall that it is played on a topological space as follows: at the -th move the first player picks a point and the second responds with choosing an open . The game stops after moves and the first player wins if . Otherwise the victory is ascribed to the second player. In this paper we introduce and study the games and . In the moves are made exactly as in the point-open game,...
Jan Hauke, Charles R. Johnson, Tadeusz Ostrowski (2015)
Discussiones Mathematicae - General Algebra and Applications
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For a given square matrix and the vector of ones denote by (A,e) the matrix ⎡ A e ⎤ ⎣ 0 ⎦ This is often called the saddle point matrix and it plays a significant role in several branches of mathematics. Here we show some applications of it in: game theory and analysis. An application of specific saddle point matrices that are hollow, symmetric, and nonnegative is likewise shown in geometry as a generalization of Heron’s formula to give the volume of a general simplex, as well as...
Ljubiša D. R. Kočinac, Marion Scheepers (2003)
Fundamenta Mathematicae
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We use Ramseyan partition relations to characterize: ∙ the classical covering property of Hurewicz; ∙ the covering property of Gerlits and Nagy; ∙ the combinatorial cardinal numbers and add(ℳ ). Let X be a -space. In [9] we showed that has countable strong fan tightness as well as the Reznichenko property if, and only if, all finite powers of X have the Gerlits-Nagy covering property. Now we show that the following are equivalent: 1. has countable fan tightness and the Reznichenko...
Nathan Keller (2012)
Journal of the European Mathematical Society
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The well-known Impossibility Theorem of Arrow asserts that any generalized social welfare function (GSWF) with at least three alternatives, which satisfies Independence of Irrelevant Alternatives (IIA) and Unanimity and is not a dictatorship, is necessarily non-transitive. In 2002, Kalai asked whether one can obtain the following quantitative version of the theorem: For any , there exists such that if a GSWF on three alternatives satisfies the IIA condition and its probability of...
Ahmad Al-Omari, Takashi Noiri (2017)
Archivum Mathematicum
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A topological space is said to be -Lindelöf [1] if every cover of by cozero sets of admits a countable subcover. In this paper, we obtain new characterizations and preservation theorems of -Lindelöf spaces.
Salvador García-Ferreira, Paul J. Szeptycki (2007)
Commentationes Mathematicae Universitatis Carolinae
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The Katětov ordering of two maximal almost disjoint (MAD) families and is defined as follows: We say that if there is a function such that for every . In [Garcia-Ferreira S., Hrušák M., Ordering MAD families a la Katětov, J. Symbolic Logic 68 (2003), 1337–1353] a MAD family is called -uniform if for every , we have that . We prove that CH implies that for every -uniform MAD family there is a -point of such that the set of all Rudin-Keisler predecessors of is dense...
José F. Fernando, José Manuel Gamboa, Jesús M. Ruiz (2014)
Journal of the European Mathematical Society
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We study here several finiteness problems concerning affine Nash manifolds and Nash subsets . Three main results are: (i) A Nash function on a semialgebraic subset of has a Nash extension to an open semialgebraic neighborhood of in , (ii) A Nash set that has only normal crossings in can be covered by finitely many open semialgebraic sets equipped with Nash diffeomorphisms such that , (iii) Every affine Nash manifold with corners is a closed subset of an affine Nash...
Lúcia R. Junqueira, Franklin D. Tall (2003)
Fundamenta Mathematicae
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We consider the question of when , where is the elementary submodel topology on X ∩ M, especially in the case when is compact.