scholarly journals On Maximal Vector Spaces of Finite Noncooperative Games

2017 ◽  
Vol 19 (02) ◽  
pp. 1750003
Author(s):  
Victoria Kreps
Keyword(s):  
Vector Space ◽  
Equilibrium Points ◽  
Negative Answer ◽  
Vector Spaces ◽  
Mixed Strategies ◽  
Necessary And Sufficient Condition ◽  
Nash Equilibrium Points ◽  
Pure Strategies ◽  
Necessary And Sufficient ◽  
Unit Formula

We consider finite noncooperative [Formula: see text] person games with fixed numbers [Formula: see text], [Formula: see text], of pure strategies of Player [Formula: see text]. We propose the following question: is it possible to extend the vector space of finite noncooperative [Formula: see text]-games in mixed strategies such that all games of a broader vector space of noncooperative [Formula: see text] person games on the product of unit [Formula: see text]-dimensional simplices have Nash equilibrium points? We get a necessary and sufficient condition for the negative answer. This condition consists of a relation between the numbers of pure strategies of the players. For two-person games the condition is that the numbers of pure strategies of the both players are equal.

scholarly journals Four-dimensional vector multiplets in arbitrary signature (I)

2020 ◽  
Vol 17 (10) ◽  
pp. 2050150 ◽  
Author(s):  
V. Cortés ◽  
L. Gall ◽  
T. Mohaupt
Keyword(s):  
Vector Space ◽  
Classification Problem ◽  
Lie Superalgebras ◽  
Symmetry Groups ◽  
Sufficient Condition ◽  
Dimensional Vector ◽  
Necessary And Sufficient Condition ◽  
Arbitrary Signature ◽  
Necessary And Sufficient ◽  
R Symmetry

We derive a necessary and sufficient condition for Poincaré Lie superalgebras in any dimension and signature to be isomorphic. This reduces the classification problem, up to certain discrete operations, to classifying the orbits of the Schur group on the vector space of superbrackets. We then classify four-dimensional [Formula: see text] supersymmetry algebras, which are found to be unique in Euclidean and in neutral signature, while in Lorentz signature there exist two algebras with R-symmetry groups [Formula: see text] and [Formula: see text], respectively.

On the Vanishing of a (G, σ) Product in a (G, σ) Space

1970 ◽  
Vol 22 (2) ◽  
pp. 363-371 ◽  
Author(s):  
K. Singh
Keyword(s):  
Vector Space ◽  
Multiplicative Group ◽  
Cartesian Product ◽  
Arbitrary Field ◽  
Necessary And Sufficient Condition ◽  
Tensor Space ◽  
Linear Character ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Necessary And Sufficient

In this paper, we shall construct a vector space, called the (G, σ) space, which generalizes the tensor space, the Grassman space, and the symmetric space. Then we shall determine a necessary and sufficient condition that the (G, σ) product of the vectors x1, x2, …, xn is zero.1. Let G be a permutation group on I = {1, 2, …, n} and F, an arbitrary field. Let σ be a linear character of G, i.e., σ is a homomorphism of G into the multiplicative group F* of F.For each i ∈ I, let Vi be a finite-dimensional vector space over F. Consider the Cartesian product W = V1 × V2 × … × Vn.1.1. Definition. W is called a G-set if and only if Vi = Vg(i) for all i ∊ I, and for all g ∊ G.

Equicontinuity of Families of Convex and Concave-Convex Operators

1984 ◽  
Vol 36 (5) ◽  
pp. 883-898 ◽  
Author(s):  
Mohamed Jouak ◽  
Lionel Thibault
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Convex Functions ◽  
Positive Cone ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Convex Operator ◽  
Necessary And Sufficient ◽  
Convex Operators

J. M. Borwein has given in [1] a practical necessary and sufficient condition for a convex operator to be continuous at some point. Indeed J. M. Borwein has proved in his paper that a convex operator with values in an order topological vector space F (with normal positive cone F+) is continuous at some point if and only if it is bounded from above by a mapping which is continuous at this point. This result extends a previous one by M. Valadier in [16] asserting that a convex operator is continuous at a point whenever it is bounded from above by an element in F on a neighbourhood of the concerned point. Note that Valadier's result is necessary if and only if the topological interior of F+ is nonempty. Obviously both results above are generalizations of the classical one about real-valued convex functions formulated in this context exactly as Valadier's result (see for example [5]).

scholarly journals Sifat-Sifat Representasi Indekomposabel The Properties of Indecomposable Representations

2019 ◽  
Vol 8 (3) ◽  
Author(s):  
Vika Yugi Kurniawan
Keyword(s):  
Vector Space ◽  
Directed Graph ◽  
Linear Mapping ◽  
Sufficient Condition ◽  
Paper Study ◽  
Necessary And Sufficient Condition ◽  
Direct Sum ◽  
Indecomposable Representations ◽  
Finite Set ◽  
Necessary And Sufficient

A directed graph is also called as a quiver  where  is a finite set of vertices,  is a set of arrows, and  are two maps from  to . A representation  of a quiver  is an assignment of a vector space  to each vertex  of  and a linear mapping  to each arrow.  We denote by  the direct sum of representasions  and  of a quiver  . A representation  is called indecomposable if  is not ishomorphic to a direct sum of non-zero representations. This paper study about the properties of indecomposable representations. These properties will be used to investigate the necessary and sufficient condition of indecomposable representations.

scholarly journals Elementary properties of vector space graphs

1984 ◽  
Vol 30 (3) ◽  
pp. 411-420
Author(s):  
Grace Orzech
Keyword(s):  
Vector Space ◽  
Direct Summand ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Special Basis ◽  
Necessary And Sufficient

Let SΓ be a vector space graph. A graphic subspace of SΓ need not be a direct summand with a graphic complement. A necessary and sufficient condition for the existence of a graphic complement is given. Also, it is shown that every graphic subspace possesses an o-special basis which extends to an o-special basis of SΓ.

scholarly journals Stability of Fuzzy Dynamical Systems via Lyapunov Functions

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Abdelati El Allaoui ◽  
Said Melliani ◽  
Lalla Saadia Chadli
Keyword(s):  
Dynamical Systems ◽  
Initial Value Problems ◽  
Lyapunov Functions ◽  
Equilibrium Points ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Initial Value ◽  
Necessary And Sufficient ◽  
Fuzzy Dynamical Systems

The purpose of this paper is to introduce the concept of fuzzy Lyapunov functions to study the notion of stability of equilibrium points for fuzzy dynamical systems associated with fuzzy initial value problems, through the principle of Zadeh. Our contribution consists in a qualitative characterization of stability by a study of the trajectories of fuzzy dynamical systems, using auxiliary functions, and they will be called fuzzy Lyapunov functions. And, among the main results that have been proven is that the existence of fuzzy Lyapunov functions is a necessary and sufficient condition for stability. Some examples are given to illustrate the obtained results.

Open decompositions on ordered convex spaces

1973 ◽  
Vol 74 (1) ◽  
pp. 49-59 ◽  
Author(s):  
Yau-Chuen Wong
Keyword(s):  
Vector Space ◽  
Banach Spaces ◽  
Convex Space ◽  
Locally Convex Space ◽  
Positive Cone ◽  
Vector Spaces ◽  
Topological Vector Spaces ◽  
Closed Cone ◽  
Necessary And Sufficient ◽  
Locally Convex

Let (E, ) be a topological vector space with a positive cone C. Jameson (3) says that C given an open decomposition on E if V ∩ C − V ∩ C is a -neighbourhood of 0 whenever V is a -neighbourhood of 0. The concept of open decompositions plays an important rôle in the theory of ordered topological vector spaces; see (3). It is clear that C is generating if C gives an open decomposition on E; the converse is true for Banach spaces with a closed cone, by Andô's theorem (cf. (1) or (9)). Therefore the following question arises naturally:(Q 1) Let (E, ) be a locally convex space with a positive cone C. What condition on is necessary and sufficient for the cone C to give an open decomposition on E?

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