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Introduction and Mathematical Preliminaries
Published in L. Prasad, S. S. Iyengar, WAVELET ANALYSIS with Applications to IMAGE PROCESSING, 2020
Definition 1.10If S is any set, a nonempty system A of some subsets of S is called an algebra of sets over S if ∀A, B ∈ A, the following conditions hold:A ∪ B ∈ A andAc = S\A ∈ A.
Preliminaries
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
Set Relations—Algebra of Sets The set operations described in the previous paragraph can be used to construct a sort of algebra of sets that is governed by a number of basic laws. We list several of these as follows:
BSDEs driven by fractional Brownian motion with time-delayed generators
Published in Applicable Analysis, 2023
Let Ω be a non-empty set, a algebra of sets of Ω and a probability measure defined on . The triplet defines a probability space and the mathematical expectation with respect to the probability measure . Let us recall that, for , a fractional Brownian motion with Hurst parameter H is a continuous and centered Gaussian process with covariance Denote . Let ξ and η be measurable functions on . Define and . Note that, for any , is a Hilbert scalar product. Let be the completion of the set of measurable functions such that .
Averaging principle for BSDEs driven by two mutually independent fractional Brownian motions
Published in Applicable Analysis, 2023
Sadibou Aidara, Yaya Sagna, Ibrahima Faye
Let Ω be a non-empty set, a σ-algebra of sets Ω, a probability measure defined on and a σ-algebra generated by both fractional Brownian motions.
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
Cylindric set algebras are algebras whose elements are relations of a certain pre-assigned arity, endowed with set-theoretic operations that utilise the form of elements of the algebra as sets of sequences. For a set V, denotes the Boolean set algebra . Let U be a set and α an ordinal; α will be the dimension of the algebra. For and let and The algebra is called the full cylindric set algebra of dimension α with unit (or greatest element). Any subalgebra of the latter is called a set algebra of dimension α. Examples of subalgebras of such set algebras arise naturally from models of first-order theories. Indeed, if is a first-order structure in a first-order signature L with α many variables, then one manufactures a cylindric set algebra based on as follows, cf. Henkin et al. (1985, §4.3). Let (here means that s satisfies φ in ), then the set is a cylindric set algebra of dimension α, where denotes the set of the first-order formulas taken in the signature L. To see why, we have: Following Henkin et al. (1985), denotes the class of all subalgebras of full set algebras of dimension α. The (equationally defined) class is obtained from cylindric set algebras by a process of abstraction and is defined by a finite schema of equations given in Henkin et al. (1985, Definition 1.1.1) that holds of course in the more concrete set algebras.