Displaying similar documents to “Applications of saddle-point determinants”

On the Variational Inequality and Tykhonov Well-Posedness in Game Theory

C. A. Pensavalle, G. Pieri (2010)

Bollettino dell'Unione Matematica Italiana

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Consider a M-player game in strategic form G = ( X 1 , , X M , g 1 , , g M ) where the set X i is a closed interval of real numbers and the payoff function g i is concave and differentiable with respect to the variable x i X i , for any i = 1 , , M . 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,...

Some applications of the point-open subbase game

D. Guerrero Sánchez, Vladimir Vladimirovich Tkachuk (2017)

Commentationes Mathematicae Universitatis Carolinae

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Given a subbase 𝒮 of a space X , the game P O ( 𝒮 , X ) is defined for two players P and O who respectively pick, at the n -th move, a point x n X and a set U n 𝒮 such that x n U n . The game stops after the moves { x n , U n : n ø } have been made and the player P wins if n ø U n = X ; otherwise O is the winner. Since P O ( 𝒮 , X ) is an evident modification of the well-known point-open game P O ( X ) , the primary line of research is to describe the relationship between P O ( X ) and P O ( 𝒮 , X ) for a given subbase 𝒮 . It turns out that, for any subbase 𝒮 , the player P has a winning...

More on the Ehrenfeucht-Fraisse game of length ω₁

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, E F G ω ( , ) , 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 E F G ω ( , ) is determined for all models and of cardinality ℵ₂” is that of a weakly compact cardinal. On the other hand, we show that if 2 < 2 , T is a countable...

Topological games and product spaces

Salvador García-Ferreira, R. A. González-Silva, Artur Hideyuki Tomita (2002)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we deal with the product of spaces which are either 𝒢 -spaces or 𝒢 p -spaces, for some p ω * . These spaces are defined in terms of a two-person infinite game over a topological space. All countably compact spaces are 𝒢 -spaces, and every 𝒢 p -space is a 𝒢 -space, for every p ω * . We prove that if { X μ : μ < ω 1 } is a set of spaces whose product X = μ < ω 1 X μ is a 𝒢 -space, then there is A [ ω 1 ] ω such that X μ is countably compact for every μ ω 1 A . As a consequence, X ω 1 is a 𝒢 -space iff X ω 1 is countably compact, and if X 2 𝔠 is a 𝒢 -space,...

G-matrices, J -orthogonal matrices, and their sign patterns

Frank J. Hall, Miroslav Rozložník (2016)

Czechoslovak Mathematical Journal

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A real matrix A is a G-matrix if A is nonsingular and there exist nonsingular diagonal matrices D 1 and D 2 such that A - T = D 1 A D 2 , where A - T denotes the transpose of the inverse of A . Denote by J = diag ( ± 1 ) a diagonal (signature) matrix, each of whose diagonal entries is + 1 or - 1 . A nonsingular real matrix Q is called J -orthogonal if Q T J Q = J . Many connections are established between these matrices. In particular, a matrix A is a G-matrix if and only if A is diagonally (with positive diagonals) equivalent to a column permutation...

Linear preservers of rc-majorization on matrices

Mohammad Soleymani (2024)

Czechoslovak Mathematical Journal

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Let A , B be n × m matrices. The concept of matrix majorization means the j th column of A is majorized by the j th column of B and this is done for all j by a doubly stochastic matrix D . We define rc-majorization that extended matrix majorization to columns and rows of matrices. Also, the linear preservers of rc-majorization will be characterized.

Circulant matrices with orthogonal rows and off-diagonal entries of absolute value 1

Daniel Uzcátegui Contreras, Dardo Goyeneche, Ondřej Turek, Zuzana Václavíková (2021)

Communications in Mathematics

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It is known that a real symmetric circulant matrix with diagonal entries d 0 , off-diagonal entries ± 1 and orthogonal rows exists only of order 2 d + 2 (and trivially of order 1 ) [Turek and Goyeneche 2019]. In this paper we consider a complex Hermitian analogy of those matrices. That is, we study the existence and construction of Hermitian circulant matrices having orthogonal rows, diagonal entries d 0 and any complex entries of absolute value 1 off the diagonal. As a particular case, we consider...

Analytic aspects of the circulant Hadamard conjecture

Teodor Banica, Ion Nechita, Jean-Marc Schlenker (2014)

Annales mathématiques Blaise Pascal

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We investigate the problem of counting the real or complex Hadamard matrices which are circulant, by using analytic methods. Our main observation is the fact that for | q 0 | = ... = | q N - 1 | = 1 the quantity Φ = i + k = j + l q i q k q j q l satisfies Φ N 2 , with equality if and only if q = ( q i ) is the eigenvalue vector of a rescaled circulant complex Hadamard matrix. This suggests three analytic problems, namely: (1) the brute-force minimization of Φ , (2) the study of the critical points of Φ , and (3) the computation of the moments of Φ . We explore here...

Empirical approximation in Markov games under unbounded payoff: discounted and average criteria

Fernando Luque-Vásquez, J. Adolfo Minjárez-Sosa (2017)

Kybernetika

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This work deals with a class of discrete-time zero-sum Markov games whose state process x t evolves according to the equation x t + 1 = F ( x t , a t , b t , ξ t ) , where a t and b t represent the actions of player 1 and 2, respectively, and ξ t is a sequence of independent and identically distributed random variables with unknown distribution θ . Assuming possibly unbounded payoff, and using the empirical distribution to estimate θ , we introduce approximation schemes for the value of the game as well as for optimal strategies considering...

Norm continuity of pointwise quasi-continuous mappings

Alireza Kamel Mirmostafaee (2018)

Mathematica Bohemica

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Let X be a Baire space, Y be a compact Hausdorff space and ϕ : X C p ( Y ) be a quasi-continuous mapping. For a proximal subset H of Y × Y we will use topological games 𝒢 1 ( H ) and 𝒢 2 ( H ) on Y × Y between two players to prove that if the first player has a winning strategy in these games, then ϕ is norm continuous on a dense G δ subset of X . It follows that if Y is Valdivia compact, each quasi-continuous mapping from a Baire space X to C p ( Y ) is norm continuous on a dense G δ subset of X .

Linear preservers of row-dense matrices

Sara M. Motlaghian, Ali Armandnejad, Frank J. Hall (2016)

Czechoslovak Mathematical Journal

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Let 𝐌 m , n be the set of all m × n real matrices. A matrix A 𝐌 m , n is said to be row-dense if there are no zeros between two nonzero entries for every row of this matrix. We find the structure of linear functions T : 𝐌 m , n 𝐌 m , n that preserve or strongly preserve row-dense matrices, i.e., T ( A ) is row-dense whenever A is row-dense or T ( A ) is row-dense if and only if A is row-dense, respectively. Similarly, a matrix A 𝐌 n , m is called a column-dense matrix if every column of A is a column-dense vector. At the end, the structure...