On Self-Inverse Binary Matrices Over the Binary Galois Field

Rushdi, None

doi:10.3844/jmssp.2013.238.248

Cited by 10 publications

(4 citation statements)

References 26 publications

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“…This feature should be studied with the hope of simplifying the algorithm that extracts all the prime implicants. A pertinent question in this respect is whether a linear representation of a switching function (e.g., Rushdi and Ghaleb, 2013;Rushdi and Alsogati, 2013) could provide anyadvantage over the current sum-of-products representation.…”

Section: Resultsmentioning

confidence: 99%

Switching-Algebraic Analysis of Relational Databases

Rushdi¹

2014

Journal of Mathematics and Statistics

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There is an established equivalence between relational database Functional Dependencies (FDs) and a fragment of switching algebra that is built typically of Horn clauses. This equivalence pertains to both concepts and procedures of the FD relational database domain and the switching algebraic domain. This study is an exposition of the use of switching-algebraic tools in solving problems typically encountered in the analysis and design of relational databases. The switching-algebraic tools utilized include purely-algebraic techniques, purely-visual techniques employing the Karnaugh map and intermediary techniques employing the variable-entered Karnaugh map. The problems handled include; (a) the derivation of the closure of a Dependency Set (DS), (b) the derivation of the closure of a set of attributes, (c) the determination of all candidate keys and (d) the derivation of irredundant dependency sets equivalent to a given DS and consequently the determination of the minimal cover of such a set. A relatively large example illustrates the switching-algebraic approach and demonstrates its pedagogical and computational merits over the traditional approach.

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Section: Resultsmentioning

confidence: 99%

Switching-Algebraic Analysis of Relational Databases

Rushdi¹

2014

Journal of Mathematics and Statistics

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“…Our definitions of duality and transposition mean that each conditional probability has a dual , a transpose or inverse , and a dual of its transpose or inverse (a transpose of its dual) . Note that both the duality and transposition operators are involutary or self-inverse operators, i.e., each of them satisfies 'the law of involution' (applying any of them twice to a specific conditional probability leaves it intact) [54][55][56]. Table 1 defines the four possible sets { , , , } pertaining to the set of four direct indicators of diagnostic testing.…”

Section: On Diagnostic Testing and Its Basic Measuresmentioning

confidence: 99%

Has the Pandemic Triggered a ‘Paperdemic’? Towards an Assessment of Diagnostic Indicators for COVID-19

Rushdi

Serag

2021

IJPR

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This paper is a preliminary step towards the assessment of an alarming widespread belief that victims of the novel coronavirus SARS-CoV-2 include the quality and accuracy of scientific publications about it. Our initial results suggest that this belief cannot be readily ignored, denied, dismissed or refuted, since some genuine supporting evidence can be forwarded for it. This evidence includes an obvious increase in retractions of papers published about the COVID-19 pandemic plus an extra-ordinary phenomenon of inconsistency that we report herein. In fact, we provide a novel method for validating any purported set of the four most prominent indicators of diagnostic testing (Sensitivity, Specificity, Positive Predictive Value, and Negative Predictive Value), by observing that these indicators constitute three rather than four independent quantities. This observation has virtually been unheard of in the open medical literature, and hence researchers have not taken it into consideration. We define two functions, which serve as consistency criteria, since each of them checks consistency for any set of four numerical values (naturally belonging to the interval [0.0,1.0]) claimed to be the four basic diagnostic indicators. Most of the data we came across in various international journals met our criteria for consistency, but in a few cases, there were obvious unexplained blunders. We explored the same consistency problem for some diagnostic data published in 2020 concerning the ongoing COVID-19 pandemic and observed that the afore-mentioned unexplained blunders tended to be on the rise. A systematic extensive statistical assessment of this presumed tendency is warranted.

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“…In this perspective, the equation f (X) 0 = can be viewed as a set of premises in a logic-deduction process, while the equation CS(f (X)) is thought of as a set of consequents in this deduction. An interesting topic for further research is whether a linear (Reed-Muller) representation of the pertinent Boolean function (see, e.g., Rushdi and Ghaleb, 2013;Rushdi and Alsogati, 2013) could provide a new alternative for solving the corresponding Boolean equation.…”

Section: Jmssmentioning

confidence: 99%

Construction of General Subsumptive Solutions of Boolean Equations via Complete-Sum Derivation

Rushdi¹,

Albarakati²

2014

Journal of Mathematics and Statistics

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Boolean-equation solving permeates many diverse areas of modern science. To solve a system of Boolean equations, one usually combines them into an equivalent single Boolean equation {f (X) 0} = whose set of solutions is exactly the same as that of the original system of equations. One of the general classes of solutions for Boolean equations is the subsumptive general solution, in which each variable is expressed as an interval decided by a double inequality in terms of the succeeding variables. The solution validity depends on the satisfaction of a required consistency condition. In this study, we introduce a novel method (henceforth called the CS method) for producing subsumptive Boolean-equation solutions based on deriving the complete sum (CS(f (X)) of the pertinent Boolean function f (X). The complete sum CS(f (X)) is a disjunction of all prime implicants of f (X) and nothing else. It explicitly shows all information about f (X) in the most compact form. We demonstrate the proposed CS solutions in terms of four examples, covering Boolean algebras of different sizes and using two prominent methods for deriving CS(f (X)). Occasionally, the consistency condition results in a collapse of the underlying Boolean algebra into a smaller subalgebra. We also illustrate how an expansion tree (typically reduced to an acyclic graph) can be used to deduce a complete list of all particular solutions from the subsumptive solution. The present CS method yields correct solutions, since it fits into the frame of the most general subsumptive solution. Among competing subsumptive methods, the CS method strikes a reasonable tradeoff between the conflicting requirements of less computational cost and more compact form for the solution obtained. In fact, it is the second best known method from both criteria of efficiency and compactness of solution.

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On Self-Inverse Binary Matrices Over the Binary Galois Field

Cited by 10 publications

References 26 publications

Switching-Algebraic Analysis of Relational Databases

Switching-Algebraic Analysis of Relational Databases

Has the Pandemic Triggered a ‘Paperdemic’? Towards an Assessment of Diagnostic Indicators for COVID-19

Construction of General Subsumptive Solutions of Boolean Equations via Complete-Sum Derivation

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