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Published in Rowan Garnier, John Taylor, Discrete Mathematics, 2020
We have seen that a partially ordered set may have several minimal and/or maximal elements. It can, however, have at most one greatest element and at most one least element. That is, if a poset A has a greatest element α, then α is unique; and similarly for a least element β. (We have, in fact, been tacitly assuming this by referring to the greatest and least elements.) It is easy to see, for example, that A has at most one least element: suppose β and β' are two least elements. Then β R β', since β is a least element, and β' R β, since β' is a least element. Therefore β = β' (by anti-symmetry), so there is only one least element. The same kind of argument clearly works for the greatest element as well.
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
Algorithms and Data Structures for Exact Computation of Marginals
Published in Marloes Maathuis, Mathias Drton, Steffen Lauritzen, Martin Wainwright, Handbook of Graphical Models, 2018
An additional benefit of a junction tree, and the elimination process, is that it can be used for all marginals. In earlier sections, we discussed the case where we wish only for one marginal p(xA) $ p(x_{A}) $ but the algorithm presented in this section will optionally produce a marginal for all C∈C $ C \in \mathcal C $ . The desired marginal p(xA) $ p(x_{A}) $ will be a subset of one of the junction tree clusters (so A⊆C $ A \subseteq C $ for some C∈C $ C \in \mathcal C $ ), but we know it will not be a superset since, from the discussion at the end of Section 1.2, the variables comprising the desired marginal (those having index A) have been completed in the graph before any triangulation process begins. Therefore, A is a subset of some maximal clique in the triangulated graph. The cost of producing all of these marginals is O(rmaxC∈C|C|) $ O(r^{\max _{C \in \mathcal C } |C|}) $ , i.e., it is exponential in the largest maximal clique size in the triangulated graph used to form the junction tree. This is the same cost, ignoring constants w.r.t. the clique sizes, as what it would take to produce only one marginal p(xA) $ p(x_{A}) $ (assuming the same triangulated graph).
The extension of the linear inequality method for generalized rational Chebyshev approximation to approximation by general quasilinear functions
Published in Optimization, 2022
Vinesha Peiris, Nadezda Sukhorukova
The linear inequality method [23] for solving rational and generalized rational problems includes the following steps. Identify upper and lower bounds ( and ) for the maximal deviation. In particular, zero is always a lower bound, while can be used as an upper bound.Set . Check if the following system of inequalities has a feasible solution: If (5)–(7) has a feasible solution, assign , otherwise . If , go to Step 1.
Weighted upper metric mean dimension for amenable group actions
Published in Dynamical Systems, 2020
Dingxuan Tang, Haiyan Wu, Zhiming Li
Denote by the maximal cardinality of an -separated subset of , by the smallest cardinality of an -spanning subset of . Suppose that is a nonempty subset, . For any and the Følner sequence , we put For any sequence of positive real numbers , we denote by For and the Følner sequence in G, let
Minimal convex majorants of functions and Demyanov–Rubinov exhaustive super(sub)differentials
Published in Optimization, 2019
A. The Fenchel-Moreau subdifferential. Let be a lower semicontinuous convex function and let . Then the exhaustive DR-subdifferential of the function f at is nonempty and coincides with the Fenchel–Moreau subdifferential of f at in the sense of convex analysis. This conclusion follows from the fact that due to the Hahn-Banach theorem each maximal concave minorant of a continuous sublinear function is, in fact, a continuous linear one. As for the exhaustive DR-superdifferential of the function f at it is an one-element family consisting only of the continuous sublinear function