Displaying similar documents to “Analysis of Algorithms: An Example of a Sort Algorithm”

Program Algebra over an Algebra

Grzegorz Bancerek (2012)

Formalized Mathematics

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We introduce an algebra with free variables, an algebra with undefined values, a program algebra over a term algebra, an algebra with integers, and an algebra with arrays. Program algebra is defined as universal algebra with assignments. Programs depend on the set of generators with supporting variables and supporting terms which determine the value of free variables in the next state. The execution of a program is changing state according to successor function using supporting terms. ...

Cellularity of free products of Boolean algebras (or topologies)

Saharon Shelah (2000)

Fundamenta Mathematicae

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The aim this paper is to present an answer to Problem 1 of Monk [10], [11]. We do this by proving in particular that if μ is a strong limit singular cardinal, θ = ( 2 c f ( μ ) ) + and 2 μ = μ + then there are Boolean algebras 𝔹 1 , 𝔹 2 such that c ( 𝔹 1 ) = μ , c ( 𝔹 2 ) < θ b u t c ( 𝔹 1 * 𝔹 2 ) = μ + . Further we improve this result, deal with the method and the necessity of the assumptions. In particular we prove that if 𝔹 is a ccc Boolean algebra and μ ℶ ω ≤ λ = c f ( λ ) ≤ 2 μ then 𝔹 satisfies the λ-Knaster condition (using the “revised GCH theorem”).

First Order Languages: Further Syntax and Semantics

Marco Caminati (2011)

Formalized Mathematics

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Third of a series of articles laying down the bases for classical first order model theory. Interpretation of a language in a universe set. Evaluation of a term in a universe. Truth evaluation of an atomic formula. Reassigning the value of a symbol in a given interpretation. Syntax and semantics of a non atomic formula are then defined concurrently (this point is explained in [16], 4.2.1). As a consequence, the evaluation of any w.f.f. string and the relation of logical implication are...

Prenormality of ideals and completeness of their quotient algebras

A. Morawiec, B. Węglorz (1993)

Colloquium Mathematicae

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It is well known that if a nontrivial ideal ℑ on κ is normal, its quotient Boolean algebra P(κ)/ℑ is κ + -complete. It is also known that such completeness of the quotient does not characterize normality, since P(κ)/ℑ turns out to be κ + -complete whenever ℑ is prenormal, i.e. whenever there exists a minimal ℑ-measurable function in κ κ . Recently, it has been established by Zrotowski (see [Z1], [CWZ] and [Z2]) that for non-Mahlo κ, not only is the above condition sufficient but also necessary...

Stability of the 4-2 Binary Addition Circuit Cells. Part I

Katsumi Wasaki (2008)

Formalized Mathematics

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To evaluate our formal verification method on a real-size calculation circuit, in this article, we continue to formalize the concept of the 4-2 Binary Addition Cell primitives (FTAs) to define the structures of calculation units for a very fast multiplication algorithm for VLSI implementation [11]. We define the circuit structure of four-types FTAs, TYPE-0 to TYPE-3, using the series constructions of the Generalized Full Adder Circuits (GFAs) that generalized adder to have for each positive...

Non-trivial derivations on commutative regular algebras.

A. F. Ber, Vladimir I. Chilin, Fyodor A. Sukochev (2006)

Extracta Mathematicae

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Necessary and sufficient conditions are given for a (complete) commutative algebra that is regular in the sense of von Neumann to have a non-zero derivation. In particular, it is shown that there exist non-zero derivations on the algebra L(M) of all measurable operators affiliated with a commutative von Neumann algebra M, whose Boolean algebra of projections is not atomic. Such derivations are not continuous with respect to measure convergence. In the classical setting of the algebra...

A new proof of Kelley's Theorem

S. Ng (1991)

Fundamenta Mathematicae

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Kelley's Theorem is a purely combinatorial characterization of measure algebras. We first apply linear programming to exhibit the duality between measures and this characterization for finite algebras. Then we give a new proof of the Theorem using methods from nonstandard analysis.