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Binary Game
In mathematics, the binary game is a topological game introduced by Stanislaw Ulam in 1935 in an addendum to problem 43 of the Scottish book as a variation of the Banach–Mazur game In general topology, set theory and game theory, a Banach– Mazur game is a topological game played by two players, trying to pin down elements in a set (space). The concept of a Banach–Mazur game is closely related to the concept of Baire s .... In the binary game, one is given a fixed subset ''X'' of the set ''N'' of all sequences of 0s and 1s. The players take it in turn to choose a digit 0 or 1, and the first player wins if the sequence they form lies in the set ''X''. Another way to represent this game is to pick a subset X of the interval ,2/math> on the real line, then the players alternatively choose binary digits x_0, x_1, x_2, .... Player I wins the game if and only if the binary number (x_0.x_1x_2x_3...)_2 \inX, that is, \Sigma^_\frac\inX. See, page 237. The binary game is sometimes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Game
In mathematics, a topological game is an infinite game of perfect information played between two players on a topological space. Players choose objects with topological properties such as points, open sets, closed sets and open coverings. Time is generally discrete, but the plays may have transfinite lengths, and extensions to continuum time have been put forth. The conditions for a player to win can involve notions like topological closure and convergence. It turns out that some fundamental topological constructions have a natural counterpart in topological games; examples of these are the Baire property, Baire spaces, completeness and convergence properties, separation properties, covering and base properties, continuous images, Suslin sets, and singular spaces. At the same time, some topological properties that arise naturally in topological games can be generalized beyond a game-theoretic context: by virtue of this duality, topological games have been widely used to describ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stanislaw Ulam
Stanisław Marcin Ulam (; 13 April 1909 – 13 May 1984) was a Polish-American scientist in the fields of mathematics and nuclear physics. He participated in the Manhattan Project, originated the Teller–Ulam design of thermonuclear weapons, discovered the concept of the cellular automaton, invented the Monte Carlo method of computation, and suggested nuclear pulse propulsion. In pure and applied mathematics, he proved some theorems and proposed several conjectures. Born into a wealthy Polish Jewish family, Ulam studied mathematics at the Lwów Polytechnic Institute, where he earned his PhD in 1933 under the supervision of Kazimierz Kuratowski and Włodzimierz Stożek. In 1935, John von Neumann, whom Ulam had met in Warsaw, invited him to come to the Institute for Advanced Study in Princeton, New Jersey, for a few months. From 1936 to 1939, he spent summers in Poland and academic years at Harvard University in Cambridge, Massachusetts, where he worked to establish import ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach–Mazur Game
In general topology, set theory and game theory, a Banach– Mazur game is a topological game played by two players, trying to pin down elements in a set (space). The concept of a Banach–Mazur game is closely related to the concept of Baire spaces. This game was the first infinite positional game of perfect information to be studied. It was introduced by Stanisław Mazur as problem 43 in the Scottish book, and Mazur's questions about it were answered by Banach. Definition Let Y be a non-empty topological space, X a fixed subset of Y and \mathcal a family of subsets of Y that have the following properties: * Each member of \mathcal has non-empty interior. * Each non-empty open subset of Y contains a member of \mathcal. Players, P_1 and P_2 alternately choose elements from \mathcal to form a sequence W_0 \supseteq W_1 \supseteq \cdots. P_1 wins if and only if :X \cap \left (\bigcap_ W_n \right ) \neq \emptyset. Otherwise, P_2 wins. This is called a general Banach–Mazur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rényi–Ulam Game
Ulam's game, or the Rényi–Ulam game, is a mathematical game similar to the popular game of twenty questions. In Ulam's game, a player attempts to guess an unnamed object or number by asking yes–no questions of another, but ''one'' of the answers given may be a lie. introduced the game in a 1961 paper, based on Hungary's Bar Kokhba game, but the paper was overlooked for many years. rediscovered the game, presenting the idea that there are a million objects and the answer to one question can be wrong, and considered the minimum number of questions required, and the strategy that should be adopted. gave a survey of similar games and their relation to information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a .... See also * Knights and Knaves References * * * { ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
