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# Two-element Boolean algebra

About: Two-element Boolean algebra is a research topic. Over the lifetime, 2585 publications have been published within this topic receiving 43758 citations.

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01 Jan 1993TL;DR: The fifth edition of this hugely successful textbook retains the quality of earlier editions while at the same time seeing numerous minor improvements and major additions as mentioned in this paper, including a new chapter on singular values and singular vectors, including ways to analyze a matrix of data.

Abstract: Linear algebra is something all mathematics undergraduates and many other students, in subjects ranging from engineering to economics, have to learn. The fifth edition of this hugely successful textbook retains the quality of earlier editions while at the same time seeing numerous minor improvements and major additions. The latter include: a new chapter on singular values and singular vectors, including ways to analyze a matrix of data; a revised chapter on computing in linear algebra, with professional-level algorithms and code that can be downloaded for a variety of languages; a new section on linear algebra and cryptography; and a new chapter on linear algebra in probability and statistics. A dedicated and active website also offers solutions to exercises as well as new exercises from many different sources (e.g. practice problems, exams, development of textbook examples), plus codes in MATLAB, Julia, and Python.

1,913 citations

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^{1}TL;DR: It is proved the validity of an algorithm whose running time goes up slightly faster than the square of d, the running times of which increase-other things being equal-as the cube of d.

Abstract: Given two boolean matrices A arid B, we define the boolean product A A B as that matrix whose (i, j)th entry is vk(a~/, A bki). We define tile boolean sum A V B as that matrix whose (i, j)th entry is a ij V b~j. The use of boolean matrices to represent program topology (Presser [1], and Marimont [2], t'or example) has led to interest in algorithms for transforming the d × d boolean matrix M to the d × d boolean matrix M' given by: d M' = v M s where we defineM ~ = MandM ~+I = M ~AM. 4=1 ne convenience of describing the transformation as a boolean sum of boolean products has apparently l suggested the corresponding algorithms, the running times of which increase-other things being equal-as the cube of d. While refraining from comment on the area of utility of such matrices, we prove the validity of an algorithm whose running time goes up slightly faster than the square of d. T,~EoeE~z. Given a square (d × d) matrix M each of whose elements m~5 is 0 or 1. Define M' by m,{~ = 1 if and only if either mii= 1 or there exist integers 1. Set i = 1. 2. (Va3 :my* = 1)(V£) set. rnj* =mik V mik. We assert M* = M'. PROOF. Trivially, m~*j = 1 ~ m~i = 1. For, either m~s was unity initially (m,4j = J)-in which case m~i is surely unity-or m~*-was set to unity in step two. That is, there were, at the L0th application of step two, m~L0 = m~0~\" = 1. 1 Presser, op. cir. In his definition of Boolean sum and product, Presser uses \"V\" for product and \"/k\" for sum. This is apparently a typographicM error, for his subsequent usage is consistent with ours. This definition of M' is trivially equivalent to the previous one. a This definition by construction is equivalent to the recursive definition: 0. (mo)~ = mij ; 1.

1,684 citations

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TL;DR: Boolean algebras are those mathematical systems first developed by George Boole in the treatment of logic by symbolic methods and since extensively investigated by other students of logic, including Schröder, Whitehead, Sheffer, Bernstein, and Huntington as mentioned in this paper.

Abstract: Boolean algebras are those mathematical systems first developed by George Boole in the treatment of logic by symbolic methodsf and since extensively investigated by other students of logic, including Schröder, Whitehead, Sheffer, Bernstein, and Huntington.J Since they embody in abstract form the principal algebraic rules governing the manipulation of classes or aggregates, these systems are of technical interest to the mathematician quite as much as to the logician. It is thus natural to suppose that a study of Boolean algebras by the methods of modern algebra will prove fruitful of important and useful results. Indeed, if one reflects upon various algebraic phenomena occurring in group theory, in ideal theory, and even in analysis, one is easily convinced that a systematic investigation of Boolean algebras, together with still more general systems, is probably essential to further progress in these theories.! The writer's interest in the subject, for example, arose in connection with the spectral theory of symmetric transformations in Hubert space and certain related properties of abstract integrals. In the actual development of the proposed theory of Boolean algebras, there emerged some extremely close connections with general topology which led at once to results of sufficient importance to confirm our a priori views of the probable value of such a theory.|| In the present paper, which is one of a projected series, we shall be concerned primarily with the problem of determining the representation of a

870 citations

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08 Apr 2011TL;DR: A new matrix product, called semi-tensor product of matrices, is used, which can covert the Boolean networks into discrete-time linear dynamic systems and the controllability of Boolean control networks is considered in the paper as an application.

Abstract: A Boolean network is a logical dynamic system, which has been used to describe cellular networks. Using a new matrix product, called semi-tensor product of matrices, a logical function can be expressed as an algebraic function. This expression can covert the Boolean networks into discrete-time linear dynamic systems. Similarly, the Boolean control networks can also be converted into discrete time bilinear dynamic systems. Under these forms the standard matrix analysis can be used to consider the structure and the control problems of Boolean (control) networks. After the detailed description of this new approach, the controllability of Boolean control networks is considered in the paper as an application.

834 citations