Abstract
This paper presents a formulation of the notion of monotonicity on homogeneous spaces. We review the general theory of invariant cone fields on homogeneous spaces and provide a list of examples involving spaces that arise in applications in information engineering and applied mathematics. Invariant cone fields associate a cone with the tangent space at each point in a way that is invariant with respect to the group actions that define the homogeneous space. We argue that invariance of conal structures induces orders that are tractable for use in analysis and propose invariant differential positivity as a natural generalization of monotonicity on such spaces.
Highlights
Monotonicity is the property of dynamical systems or maps that preserve a partial order, which is defined as a binary relation that is reflexive, antisymmetric, and transitive
We review the general theory of invariant cone fields on homogeneous spaces and provide a list of examples involving spaces that arise in applications in information engineering and applied mathematics
We argue that invariance of conal structures induces orders that are tractable for use in analysis and propose invariant differential positivity as a natural generalization of monotonicity on such spaces
Summary
Monotonicity is the property of dynamical systems or maps that preserve a partial order, which is defined as a binary relation that is reflexive, antisymmetric, and transitive. A monotone dynamical system is characterized by the property that any two points that are ordered at one instant in time will remain ordered at all subsequent times as the system evolves with the flow. Monotone flows and their discrete-time analogues, order-preserving maps, play an important role in the theory of dynamical systems and find applications to many biological, physical, chemical, and economic.
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