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Summation Kernels for Orthogonal Polynomial Systems
Published in George Anastassiou, Handbook of Analytic-Computational Methods in Applied Mathematics, 2019
Frank Filbir, Rupert Lasser, Josef Obermaier
Let K be a non-void set equipped with the discrete topology. A Borel measure m on K is completely determined by the values m(x) = m({x}) for all x ∈ K. Hence one may take Borel measures on K as functions. Denote by M(K) the Banach space of all complex Borel measures on K and by Mc1(K) the subset of probability measures with finite support. We write ϵx, x ∈ K, for the dirac measure or function with ϵ(x) = 1, and ϵ(y) = 0 if y ≠ x. For μ ∈ M(K) there is a unique representation () μ=∑x∈Iaxϵx,
Topology and Logic Programming
Published in Pascal Hitzler, Anthony Seda, Mathematical Aspects of Logic Programming Semantics, 2016
Because Q is a product topology, it is easy to describe the basic open sets of I(X, T) in Q as follows (the nature of X is actually irrelevant, although it is being taken here to be BP,J). First, given any truth value t ∈ T, the singleton set {t} is open in T, since T is endowed with the discrete topology. Therefore, see Section A.5, the basic open sets here are of the form πi1−1(ti1 )∩…∩πi1−1(tin). They therefore can be written in the form G(Ai1,…, Ain ; ti1,…, tin ) = {I ∈ I(X, T) | I(Aij) = tij for j = 1,…, n}, where Ai1,…, Ain are arbitrary, but fixed, elements of X.
Existence of Periodic Solutions for First-Order Difference Equations Subjected to Allee Effects
Published in Hemen Dutta, Mathematical Methods in Engineering and Applied Sciences, 2020
A map f : Z(−∞, ∞) × [0, ∞) → [0, ∞) is continuous if it is continuous as a map of the topological space Z(−∞, ∞) × [0, ∞) onto the topological space [0, ∞). Throughout the chapter, the topology on Z(−∞, ∞) is the discrete topology. One may refer [1] for basic definitions and results of difference equations.
Constructing condensed memories in functorial time
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2023
is a sheaf on.We consider the discrete topology on . Both sheaf conditions are guaranteed by the functorial nature of the morphisms between presheaves nested in , i.e. by Eqs. (10) and (11). The entropic typing is, in particular, maximal because is profinite.□
Bratteli diagrams for bounded topological speedups
Published in Dynamical Systems, 2023
Drew D. Ash, Andrew Dykstra, Michelle LeMasurier
An important type of Cantor system takes the form , where is a finite alphabet and is the shift map defined as follows: for a doubly-infinite sequence , is the sequence where . Here, is given the product of the discrete topology on . The system is called the full shift on . A subshift is a system where X is a closed and shift-invariant subset of some .
Optimal controller applied to robotic systems using covariant control equations
Published in International Journal of Control, 2022
Juan Antonio Rojas-Quintero, Juan Antonio Rojas-Estrada, Jorge Villalobos-Chin, Victor Santibañez, Eusebio Bugarin
It is possible to show uniform global asymptotic stability for systems of the form (37) if there exist a function and three class-K functions and η such that for each is continuously differentiable and satisfies the following inequalities: To show stability of our switched system, first, consider the index set . The set , together with the discrete topology forms a topological space Λ. Let each element of Λ correspond to one of the phases described above. Define , then, the switched system is given by The switching signal is given by which allows us to define and .