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Locally Convex Topological Vector Spaces
Published in Kenneth Kuttler, Modern Analysis, 2017
or else the two simplicies do not intersect. The collection of simplicies is said to be locally finite if, for every point, there exists a ball containing that point which also intersects only finitely many of the simplicies in the collection. We leave it to the reader to verify that for each ϵ > 0, there exists a locally finite tiling of ℝn which is composed of simplicies which have diameters less than ϵ.
On continuous variations of trajectories of differential inclusion with unbounded right-hand side
Published in Optimization, 2020
A (proper) open covering of the metric space is a collection of non-empty open sets of such that and for any . An open covering of a separable metric space is called locally finite if for any point there exists a neighbourhood such that only for a finite collection of indices .
Atom-canonicity in varieties of cylindric algebras with applications to omitting types in multi-modal logic
Published in Journal of Applied Non-Classical Logics, 2020
The varieties of representable algebras of dimension α, α an ordinal is defined as via which turns out to be a variety, that is to say, closed under , as well. An algebra is locally finite, if the dimension set of every element is finite. The dimension set of x, or for short, is the set Locally finite algebras correspond to Tarski–Lindenbaum algebras of (first-order) formulas; in such algebras the dimension set of (an equivalence class of) a formula reflects the number of (finite) set of free variables in this formula. Tarski proved that every locally finite ω-dimensional cylindric algebra is representable, i.e. isomorphic to a subdirect product of set algebra each of dimension ω. Let denote the class of locally finite cylindric algebras. Let stand for the class of isomorphic copies of subdirect products of set algebras each of dimension ω, or briefly, the class of ω dimensional representable cylindric algebras. Then Tarski's theorem reads . This representation theorem is non-trivial; in fact it is equivalent to Gödel's celebrated Completeness Theorem (Henkin et al., 1985, § 4.3), cf. Theorem 3.7. Completeness in the general case is a huge subject that has provoked extensive research.