Parallel (operator): Difference between revisions

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<ref name="Cajori_1928">{{cite book |author-first=Florian |author-last=Cajori |author-link=Florian Cajori |title=A History of Mathematical Notations – Notations in Elementary Mathematics |chapter=§ 184, § 359, § 368 |volume=1 |orig-date=September 1928 |publisher=[[Open court publishing company]] |___location=Chicago, US |date=1993 |edition=two volumes in one unaltered reprint |pages=[https://archive.org/details/historyofmathema00cajo_0/page/193 193, 402–403, 411–412] |isbn=0-486-67766-4 |lccn=93-29211 |url=https://archive.org/details/historyofmathema00cajo_0/page/193 |access-date=2019-07-22 |quote-pages=402–403, 411–412 |quote=§359. […] ∥ for parallel occurs in [[William Oughtred|Oughtred]]'s ''Opuscula mathematica hactenus inedita'' (1677) [p. 197], a posthumous work (§ 184) […] §368. Signs for parallel lines. […] when [[Robert Recorde|Recorde]]'s sign of equality won its way upon [[the Continent]], vertical lines came to be used for parallelism. We find ∥ for "parallel" in [[John Kersey the elder|Kersey]],{{citeref|A|ref=FC-A}} [[John Caswell|Caswell]], [[William Jones (mathematician)|Jones]],{{citeref|B|ref=FC-B}} Wilson,{{citeref|C|ref=FC-C}} [[William Emerson (mathematician)|Emerson]],{{citeref|D|ref=FC-D}} Kambly,{{citeref|E|ref=FC-E}} and the writers of the last fifty years who have been already quoted in connection with other pictographs. Before about 1875 it does not occur as often […] Hall and Stevens{{citeref|F|ref=FC-F}} use "par{{citeref|F|ref=FC-F}} or ∥" for parallel […] {{anchor|FC-A}}[A] [[John Kersey the elder|John Kersey]], ''{{citeref|Kersey (the elder)|1673|Algebra|style=plain}}'' (London, 1673), Book IV, p. 177. {{anchor|FC-B}}[B] [[William Jones (mathematician)|W. Jones]], ''Synopsis palmarioum matheseos'' (London, 1706). {{anchor|FC-C}}[C] John Wilson, ''Trigonometry'' (Edinburgh, 1714), characters explained. {{anchor|FC-D}}[D] [[William Emerson (mathematician)|W. Emerson]], ''Elements of Geometry'' (London, 1763), p. 4. {{anchor|FC-E}}[E] {{ill|Ludwig Kambly{{!}}L.<!-- Ludwig --> Kambly|de|Ludwig Kambly}}, ''Die Elementar-Mathematik'', Part 2: ''Planimetrie'', 43. edition (Breslau, 1876), p. 8. […] {{anchor|FC-F}}[F] H. S.<!-- Henry Sinclair --> Hall and F. H.<!-- Frederick Haller --> Stevens, ''Euclid's Elements'', Parts I and II (London, 1889), p. 10. […]}} [https://monoskop.org/images/2/21/Cajori_Florian_A_History_of_Mathematical_Notations_2_Vols.pdf]</ref>

<ref name="Seshu_1956">{{cite journal |title=On Electrical Circuits and Switching Circuits |date=September 1956 |author-first=Sundaram |author-last=Seshu |journal=[[IRE Transactions on Circuit Theory]] |volume=CT-3 |issue=3 |publisher=[[Institute of Radio Engineers]] (IRE) |pages=172–178 |doi=10.1109/TCT.1956.1086310 |url=https://www.researchgate.net/publication/3440524_On_Electrical_Circuits_and_Switching_Circuits}} (7 pages) (NB. See {{citeref|Seshu|Gould|1957|errata|style=plain}}.)</ref>

<ref name="Seshu_1957">{{cite journal |title=Correction to 'On Electrical Circuits and Switching Circuits' |series=Correction |date=September 1957 |author-first1=Sundaram |author-last1=Seshu |author-first2=Roderick |author-last2=Gould |journal=[[IRE Transactions on Circuit Theory]] |volume=CT-4 |issue=3 |publisher=[[Institute of Radio Engineers]] (IRE) |page=284 |doi=10.1109/TCT.1957.1086390 |url=https://www.researchgate.net/publication/3440591_Correction_to_'On_Electric_Circuits_and_Switching_Circuits'|doi-access=free }} (1 page) (NB. Refers to {{citeref|Seshu|1956|previous|style=plain}} reference.)</ref>

<ref name="Erickson_1959">{{cite journal |author-first=Kent E. |author-last=Erickson |title=A New Operation for Analyzing Series-Parallel Networks |journal=[[IRE Transactions on Circuit Theory]] |volume=CT-6 |issue=1 |date=March 1959 |publisher=[[Institute of Radio Engineers]] (IRE) |doi=10.1109/TCT.1959.1086519 |pages=124–126 |url=https://www.researchgate.net/publication/3440722_A_New_Operation_for_Analyzing_Series-Parallel_Networks |quote-page=124 |quote=[…] The operation&nbsp;∗ is defined as A&nbsp;∗ B = AB/A&nbsp;+ B. The symbol&nbsp;∗ has algebraic properties which simplify the formal solution of many series-parallel network problems. If the operation&nbsp;∗ were included as a subroutine in a [[digital computer]], it could simplify the programming of certain network calculations. […]}} (3 pages) (NB. See {{citeref|Kaufman|1963|comment|style=plain}}.)</ref>

<ref name="Kaufman_1963">{{cite journal |author-first=Howard |author-last=Kaufman |title=Remark on a New Operation for Analyzing Series-Parallel Networks |journal=[[IEEE Transactions on Circuit Theory]] |volume=CT-10 |issue=2 |date=June 1963 |publisher=[[Institute of Electrical and Electronics Engineers]] (IEEE) |doi=10.1109/TCT.1963.1082126 |page=283 |quote-page=283 |quote=[…] Comments on the operation&nbsp;∗ […] a∗b&nbsp;= ab/(a+b) […]}} (1 page) (NB. Refers to {{citeref|Erickson|1959|previous|style=plain}} reference.)</ref>

<ref name="Duffin_1966">{{cite journal |title=Network synthesis through hybrid matrices |author-last1=Duffin |author-first1=Richard James |author-link1=Richard James Duffin |author-first2=Dov |author-last2=Hazony |author-first3=Norman Alexander |author-last3=Morrison |journal=[[SIAM Journal on Applied Mathematics]] |publisher=[[Society for Industrial and Applied Mathematics]] (SIAM) |volume=14 |number=2 |date=March 1966 |orig-date=1965-04-12, 1964-08-25 |pages=390–413 |jstor=2946272 |doi=10.1137/0114032}} (24 pages)</ref>

<ref name="Anderson-Duffin_1969">{{cite journal |author-last1=Anderson, Jr. |author-first1=William Niles<!-- https://web.archive.org/web/20110723004431/http://www.tomstrong.org/kappa/03/b_ind_0399.html --> |author-last2=Duffin |author-first2=Richard James |author-link2=Richard James Duffin |title=Series and parallel addition of matrices |journal=[[Journal of Mathematical Analysis and Applications]] |publisher=[[Academic Press, Inc.]] |volume=26 |issue=3 |pages=576–594 |date=1969 |orig-date=1968-05-27 |doi=10.1016/0022-247X(69)90200-5 |quote-page=576 |quote=[…] we define the parallel sum of A and B by the formula A(A&nbsp;+ B)⁺B and denote it by A&nbsp;: B. If A and B are nonsingular this reduces to A&nbsp;: B&nbsp;= (A⁻¹&nbsp;+ B⁻¹)⁻¹ which is the well known electrical formula for addition of resistors in parallel. Then it is shown that the [[Hermitian matrix|Hermitian]] semi-definite matrices form a [[commutative law|commutative]] partially ordered semigroup under the parallel sum operation. […]|doi-access=free }} [https://kilthub.cmu.edu/articles/Series_and_parallel_addition_of_matrices/6479477/1]</ref>

<ref name="Mitra_1970">{{cite journal |title=A Matrix Operation for Analyzing Series-parallel Multiports |journal=[[Journal of the Franklin Institute]] |publisher=[[Franklin Institute]] |series=Brief Communication |volume=289 |issue=2 |pages=167–169 |date=February 1970 |doi=10.1016/0016-0032(70)90302-9 |author-first=Sujit Kumar |author-last=Mitra<!-- 1932–2004 http://www.iisc.ernet.in/currsci/aug102004/395.pdf --> |url=https://www.researchgate.net/publication/239368282_A_matrix_operation_for_analyzing_series-parallel_multiports |quote-page=167 |quote=The purpose of this communication is to extend the concept of the [[scalar (mathematics)|scalar]] operation Reduced Sum introduced by {{citeref|Seshu|1956|Seshu|style=plain}} […] and later elaborated by {{citeref|Erickson|1959|Erickson|style=plain}} […] to matrices, to outline some interesting properties of this new matrix operation, and to apply the matrix operation in the analysis of series and parallel [[n-port network|''n''-port network]]s. Let A and B be two non-singular [[square matrices]] having [[inverse matrix|inverses]], A⁻¹ and B⁻¹ respectively. We define the operation&nbsp;∙ as A&nbsp;∙ B&nbsp;= (A⁻¹&nbsp;+ B⁻¹)⁻¹ and the operation&nbsp;⊙ as A&nbsp;⊙ B&nbsp;= A&nbsp;∙ (−B). The operation&nbsp;∙ is [[commutative law|commutative]] and [[associative law|associative]] and is also [[distributive law|distributive]] with respect to multiplication. […]}} (3 pages)</ref>

<ref name="Duffin_1971">{{cite book |title=Mathematical Aspects of Electrical Network Analysis |series=Proceedings of a Symposium in Applied Mathematics of the American Mathematical Society and the Society for Industrial and Applied Mathematics held in New York City, 1969-04-02/03 |chapter=Network Models |pages=65–92 [68] |volume=III of SIAM-AMS Proceedings |date=1971 |orig-date=1970, 1969 |publisher=[[American Mathematical Society]] (AMS) / [[Society for Industrial and Applied Mathematics]] (SIAM) |edition=illustrated |publication-place=Providence, Rhode Island |editor-first1=Herbert Saul |editor-last1=Wilf |editor-link1=Herbert Saul Wilf |editor-first2=Frank |editor-last2=Hararay |editor-link2=Frank Harary |author-first=Richard James |author-last=Duffin |author-link=Richard James Duffin |___location=Durham, North Carolina, USA |issn=0080-5084 |isbn=0-8218-1322-6 |id={{ISBN|978-0-8218-1322-5}}. Report 69-21 |lccn=79-167683 |chapter-url=https://books.google.com/books?id=j2vFhxA5K-UC&pg=PA65 |url=https://books.google.com/books?id=j2vFhxA5K-UC |access-date=2019-08-05 |quote-pages=68–69 |quote=[…] To have a convenient short notation for the joint resistance of resistors connected in parallel let […] A:B = AB/(A+B) […] A:B may be regarded as a new operation termed ''parallel addition'' […] Parallel addition is defined for any nonnegative numbers. The network model shows that parallel addition is [[commutative law|commutative]] and [[associative law|associative]]. Moreover, multiplication is [[distributive law|distributive]] over this operation. Consider now an algebraic expression in the operations (+) and (:) operating on positive numbers A, B, C, etc. […] To give a network interpretation of such a polynomial read ''A&nbsp;+ B'' as "A series B" and ''A&nbsp;: B'' as "A parallel B" then it is clear that the expression […] is the joint resistance of the network […]}} [https://kilthub.cmu.edu/articles/Network_models/6478514] [https://kilthub.cmu.edu/ndownloader/files/11915081] (206 pages)</ref>