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Probability and Distribution Theory for Radar Detection
Published in Graham V. Weinberg, Radar Detection Theory of Sliding Window Processes, 2017
The probability measure P assigns probabilities to events in F $ {\mathcal{F}} $ , and is hence a mapping from F $ {\mathcal{F}} $ to the unit interval [0, 1 ]. It is assumed to satisfy the three properties:P(Ω) = 1;P(A) ∊ [0, 1];P(A∪B)=P(A)+P(B)ifA∩B=Φ, $ P(A\mathop \cup \nolimits B) = P(A) + P(B)\,\,\,if\,\,A\mathop \cap \nolimits B = \iPhi , $
Structure and dimension of invariant subsets of expanding Markov maps and joint invariance
Published in Dynamical Systems, 2023
In our setting, we denote the unit interval where 0 and 1 are identified. Let be an expanding Markov map of the circle, i.e. there exist finitely many points such that , and a so that for all , on which f is one-sided differentiable. For such a map we consider to be the set of all compact and non-empty subsets of that are also invariant under the action of f. We endow the set with the Hausdorff metric . M. Urbanski in [17] and C.C. Conley in [3] present some the topological properties of this metric space for expanding Markov maps and for flows respectively. Motivated by that, we further study the topological structure of the metric space . More precisely, we will show that is a compact and totally disconnected metric space.
Continuous propositional modal logic
Published in Journal of Applied Non-Classical Logics, 2018
In this paper, we introduce a modal extension of Continuous Propositional Logic. See Ben Yaacov and Usvyatsov (2010) or Ben Yaacov and Pedersen (2010). Indeed, the system that we present can equally be regarded as a continuous extension (in the sense of Continuous Logic) of Propositional Modal Logic . See Blackburn, de Rijke, and Venema (2001). Throughout this paper, [0, 1] denotes the real unit interval.