scholarly journals Ordered Gyrovector Spaces

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1041 ◽  
Author(s):  
Sejong Kim
Keyword(s):  
Vector Space ◽  
Partial Order ◽  
Convex Cone ◽  
Construction Scheme ◽  
Proper Convex

The well-known construction scheme to define a partial order on a vector space is to use a proper convex cone. Applying this idea to the gyrovector space we construct the partial order, called a gyro-order. We also give several inequalities of gyrolines and cogyrolines in terms of the gyro-order.

scholarly journals The Joly topology and the Mosco-Beer topology revisited

1993 ◽  
Vol 48 (3) ◽  
pp. 353-363 ◽  
Author(s):  
Dominique Azé ◽  
Jean-Paul Penot
Keyword(s):  
Vector Space ◽  
Convex Functions ◽  
Normed Vector Space ◽  
Fenchel Transform ◽  
Proper Convex ◽  
Normed Vector

Some extensions to the non reflexive case of continuity results for the Legendre-Fenchel transform are presented following an approach due to J.-L. Joly. We compare the topology introduced by J.-L. Joly and the Mosco-Beer topology introduced by G. Beer. In particular, in the case of the space of closed proper convex functions defined on the dual of a normed vector space they coincide.

scholarly journals Optimality conditions for a cone-convex programming problem

1979 ◽  
Vol 27 (2) ◽  
pp. 141-162 ◽  
Author(s):  
Hélène M. Massam
Keyword(s):  
Vector Space ◽  
Optimality Conditions ◽  
Programming Problem ◽  
Convex Programming ◽  
Convex Cone ◽  
Constraint Qualifications ◽  
Closed Convex Cone ◽  
Convex Programming Problem ◽  
Nonnegative Orthant ◽  
Locally Convex

AbstractOptimality conditions without constraint qualifications are given for the convex programming problem: Maximize f(x) such that g(x) ∈ B, where f maps X into R and is concave, g maps X into Rm and is B-concave, X is a locally convex topological vector space and B is a closed convex cone containing no line. In the case when B is the nonnegative orthant, the results reduce to some of those obtained recently by Ben-Israel, Ben-Tal and Zlobec.

Partial orders on linear transformation semigroups

2005 ◽  
Vol 135 (2) ◽  
pp. 413-437 ◽  
Author(s):  
R. P. Sullivan
Keyword(s):  
Vector Space ◽  
Regular Semigroup ◽  
Partial Order ◽  
Linear Transformation ◽  
Natural Partial Order ◽  
Transformation Semigroups ◽  
Linear Transformations ◽  
Partially Ordered ◽  
Partial Linear ◽  
A Chain

Let V be any vector space and P(V) the set of all partial linear transformations defined on V, that is, all linear α: A → B, where A, B are subspaces of V. Then P(V) is a semigroup under composition, which is partially ordered by ⊆ (that is, α ⊆ β if and only if dom α ⊆ dom β and α = β | dom α). We compare this order with the so-called 'natural partial order' ≤ on P(V) and we determine their meet and join. We also describe all elements of P(V) that are minimal (or maximal) with respect to each of these four orders, and we characterize all elements that are 'compatible' with them. In addition, we answer similar questions for the semigroup T(V) consisting of all α ∈ P(V) whose domain equals V. Other orders have been defined by Petrich on any regular semigroup: three of them form a chain below ≤, and we show that two of these are equal on the semigroup P(V) and on the ring T(V). We also consider questions for these orders that are similar to those already mentioned

Rank and dimension functions

2015 ◽  
Vol 29 ◽  
pp. 144-155
Author(s):  
K. Prasad ◽  
Nupur Nandini ◽  
Divya Shenoy
Keyword(s):  
Vector Space ◽  
Partial Order ◽  
Commutative Ring ◽  
Free Module ◽  
Projective Modules ◽  
Finitely Generated ◽  
The Matrix ◽  
Dimension Functions ◽  
Rank Of A Matrix ◽  
Minus Partial Order

In this paper, we invoke theory of generalized inverses and minus partial order on regular matrices over a commutative ring to define rank–function for regular matrices and dimension–function for finitely generated projective modules which are direct summands of a free module. Some properties held by the rank of a matrix and the dimension of a vector space over a field are generalized. Also, a generalization of rank-nullity theorem has been established when the matrix given is regular.

Decision Making and Games with Vector Outcomes

2019 ◽  
Vol 20 (1) ◽  
Author(s):  
Jaeok Park
Keyword(s):  
Decision Making ◽  
Vector Space ◽  
Topological Vector Space ◽  
Convex Cone ◽  
General Framework ◽  
Equilibrium Selection ◽  
Utility Representation ◽  
Decision Making Problem ◽  
Real Topological Vector Space ◽  
Continuity Axiom

AbstractIn this paper, we study decision making and games with vector outcomes. We provide a general framework where outcomes lie in a real topological vector space and the decision maker’s preferences over outcomes are described by a preference cone, which is defined as a convex cone satisfying a continuity axiom. Further, we define a notion of utility representation and introduce a duality between outcomes and utilities. We provide conditions under which a preference cone is represented by a utility and is the dual of a set of utilities. We formulate a decision-making problem with vector outcomes and study optimal choices. We also consider games with vector outcomes and characterize the set of equilibria. Lastly, we discuss the problem of equilibrium selection based on our characterization.

scholarly journals SPECIAL VINBERG CONES

2021 ◽  
Author(s):  
D. V. ALEKSEEVSKY ◽  
V. CORTÉS
Keyword(s):  
Vector Space ◽  
Convex Cone ◽  
Euclidean Jordan Algebra ◽  
Jordan Algebras ◽  
Euclidean Case ◽  
Euclidean Metric ◽  
Euclidean Vector Space ◽  
Lorentz Cone ◽  
Homogeneous Convex ◽  
Vinberg Algebras

AbstractThe paper is devoted to the generalization of the Vinberg theory of homogeneous convex cones. Such a cone is described as the set of “positive definite matrices” in the Vinberg commutative algebra ℋn of Hermitian T-matrices. These algebras are a generalization of Euclidean Jordan algebras and consist of n × n matrices A = (aij), where aii ∈ ℝ, the entry aij for i < j belongs to some Euclidean vector space (Vij ; 𝔤) and $$ {a}_{ji}={a}_{ij}^{\ast }=\mathfrak{g}\left({a}_{ij},\cdot \right)\in {V}_{ij}^{\ast } $$ a ji = a ij ∗ = g a ij ⋅ ∈ V ij ∗ belongs to the dual space $$ {V}_{ij}^{\ast }. $$ V ij ∗ . The multiplication of T-Hermitian matrices is defined by a system of “isometric” bilinear maps Vij × Vjk → Vij ; i < j < k, such that |aij ⋅ ajk| = |aij| ⋅ |aik|, alm ∈ Vlm. For n = 2, the Hermitian T-algebra ℋn= ℋ2 (V) is determined by a Euclidean vector space V and is isomorphic to a Euclidean Jordan algebra called the spin factor algebra and the associated homogeneous convex cone is the Lorentz cone of timelike future directed vectors in the Minkowski vector space ℝ1,1⊕ V . A special Vinberg Hermitian T-algebra is a rank 3 matrix algebra ℋ3(V; S) associated to a Clifford Cl(V )-module S together with an “admissible” Euclidean metric 𝔤S.We generalize the construction of rank 2 Vinberg algebras ℋ2(V ) and special Vinberg algebras ℋ3(V; S) to the pseudo-Euclidean case, when V is a pseudo-Euclidean vector space and S = S0 ⊕ S1 is a ℤ2-graded Clifford Cl(V )-module with an admissible pseudo-Euclidean metric. The associated cone 𝒱 is a homogeneous, but not convex cone in ℋm; m = 2; 3. We calculate the characteristic function of Koszul-Vinberg for this cone and write down the associated cubic polynomial. We extend Baez’ quantum-mechanical interpretation of the Vinberg cone 𝒱2 ⊂ ℋ2(V ) to the special rank 3 case.

Sign in / Sign up

Export Citation Format

Share Document