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Statistical Analysis and Optimization of Nonlinear Filters
Published in Jaakko Astola, Pauli Kuosmanen, Fundamentals of Nonlinear Digital Filtering, 2020
Jaakko Astola, Pauli Kuosmanen
We recall some basic lattice terminology [94]. A partially ordered set (poset) is a set with a binary relation which is reflexive, antisymmetric and transitive. For example, BN= {x = (x1x2, …, xN) : xi ∈ {0, 1}} and the relation ≤ x≤yifandonlyifxn≤ynforalln.
Vector spaces
Published in Qingwen Hu, Concise Introduction to Linear Algebra, 2017
Zorn’s lemma in set theory claims that if every totally ordered subset X of a partially ordered set Y has a upper bound, then Y has a maximal element. If we take union to obtain the upper bound for every totally ordered subcollection such as X, Zorn’s lemma applies to our current situation and asserts the existence of a maximal element in Y, which is the maximal linearly independent set of vectors, namely, a basis of V. We arrive at
Logical object structure and system implementation for BIM database in civil infrastructures
Published in Architectural Engineering and Design Management, 2023
In Equations (22) and (23), the relation is a partial order relation. and are partially ordered sets, i.e. 〈〉. According to the set theory, the partially ordered set can be transformed as the Hasse diagram (Brüggemann & Patil, 2011). According to Notations 1 and 3, we have Then, based on Equations (22)–(24), we can get the Hasse diagram of and (See Figure 10).
Multi-dimensional ρ-almost periodic type functions and applications
Published in Applicable Analysis, 2022
M. Fečkan, M. T. Khalladi, M. Kostić, A. Rahmani
The use of binary relation ρ in Definition 2.1 suggests a very general way of approaching to many known classes of almost periodic functions; before we go any further, we would like to note that this general approach has some obvious unpleasant consequences because, under the general requirements of Definition 2.1, any continuous function is always Bohr -almost periodic provided that (in particular, it is very redundant to assume any kind of boundedness of function in Definition 2.1; see [2, Proposition 2.2] and [15, Proposition 2.8] for some particular results obtained in this direction). Therefore, given two sets , satisfying and a continuous function , it is natural to introduce the following non-empty sets: and Clearly, we have ; further on, the assumptions () and imply (). The set (), equipped with the relation of set inclusion, becomes a partially ordered set; for the sake of brevity, we will not consider the minimal elements and the least elements (if exist) of these partially ordered sets here. The interested reader may try to construct some examples concerning this issue.