Parallel (operator) (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Parallel (operator)" in English language version.

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Van Droogenbroeck, Frans J., 'Coalescence, unlocking insights in the intricacies of merging independent probability density functions' (2025).

archive.org

Kersey (the elder), John (1673). "Chapter I: Concerning the Scope of this fourth Book and the Signification of Characters, Abbreviations and Citations used therein". The Elements of that Mathematical Art, commonly called Algebra. Vol. Book IV - The Elements of the Algebraical Arts. London: Thomas Passinger, Three-Bibles, London-Bridge. pp. 177–178. Archived from the original on 2020-08-05. Retrieved 2019-08-09.
Cajori, Florian (1993) [September 1928]. "§ 184, § 359, § 368". A History of Mathematical Notations – Notations in Elementary Mathematics. Vol. 1 (two volumes in one unaltered reprint ed.). Chicago, US: Open court publishing company. pp. 193, 402–403, 411–412. ISBN 0-486-67766-4. LCCN 93-29211. Retrieved 2019-07-22. pp. 402–403, 411–412: §359. […] ∥ for parallel occurs in Oughtred's Opuscula mathematica hactenus inedita (1677) [p. 197], a posthumous work (§ 184) […] §368. Signs for parallel lines. […] when Recorde's sign of equality won its way upon the Continent, vertical lines came to be used for parallelism. We find ∥ for "parallel" in Kersey,^[A] Caswell, Jones,^[B] Wilson,^[C] Emerson,^[D] Kambly,^[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^[F] use "par^[F] or ∥" for parallel […] [A] John Kersey, Algebra (London, 1673), Book IV, p. 177. [B] W. Jones, Synopsis palmarioum matheseos (London, 1706). [C] John Wilson, Trigonometry (Edinburgh, 1714), characters explained. [D] W. Emerson, Elements of Geometry (London, 1763), p. 4. [E] L. Kambly [de], Die Elementar-Mathematik, Part 2: Planimetrie, 43. edition (Breslau, 1876), p. 8. […] [F] H. S. Hall and F. H. Stevens, Euclid's Elements, Parts I and II (London, 1889), p. 10. […] [3]

berkeley.edu

inst.eecs.berkeley.edu

Ranade, Gireeja; Stojanovic, Vladimir, eds. (Fall 2018). "Chapter 15.7.2 Parallel Resistors" (PDF). EECS 16A Designing Information Devices and Systems I (PDF) (lecture notes). University of California, Berkeley. p. 12. Note 15. Archived (PDF) from the original on 2018-12-27. Retrieved 2018-12-28. p. 12: […] This mathematical relationship comes up often enough that it actually has a name: the "parallel operator", denoted ∥. When we say x∥y, it means ${\frac {xy}{x+y}}$ . Note that this is a mathematical operator and does not say anything about the actual configuration. In the case of resistors the parallel operator is used for parallel resistors, but for other components (like capacitors) this is not the case. […] (16 pages)

books.google.com

Duffin, Richard James (1971) [1970, 1969]. "Network Models". Written at Durham, North Carolina, USA. In Wilf, Herbert Saul; Hararay, Frank (eds.). Mathematical Aspects of Electrical Network Analysis. 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. Vol. III of SIAM-AMS Proceedings (illustrated ed.). Providence, Rhode Island: American Mathematical Society (AMS) / Society for Industrial and Applied Mathematics (SIAM). pp. 65–92 [68]. ISBN 0-8218-1322-6. ISSN 0080-5084. LCCN 79-167683. ISBN 978-0-8218-1322-5. Report 69-21. Retrieved 2019-08-05. pp. 68–69: […] 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 and associative. Moreover, multiplication is 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 + B as "A series B" and A : B as "A parallel B" then it is clear that the expression […] is the joint resistance of the network […] [1] [2] (206 pages)
Bober, William; Stevens, Andrew (2016). "Chapter 7.6. Laplace Transforms Applied to Circuits". Numerical and Analytical Methods with MATLAB for Electrical Engineers. Applied and Computational Mechanics (1 ed.). CRC Press. p. 224. ISBN 978-1-46657607-0. ISBN 1-46657607-3. (388 pages)
Ellerman, David Patterson (1995-03-21). "Chapter 12: Parallel Addition, Series-Parallel Duality, and Financial Mathematics". Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics (PDF). The worldly philosophy: studies in intersection of philosophy and economics (illustrated ed.). Rowman & Littlefield Publishers, Inc. pp. 237–268. ISBN 0-8476-7932-2. Archived (PDF) from the original on 2016-03-05. Retrieved 2019-08-09. p. 237: […] When resistors with resistance a and b are placed in series, their compound resistance is the usual sum (hereafter the series sum) of the resistances a + b. If the resistances are placed in parallel, their compound resistance is the parallel sum of the resistances, which is denoted by the full colon […] [4] (271 pages)
Basso, Christophe P. (2016). "Chapter 1.1.2 The Current Divider". Linear Circuit Transfer Functions: An Introduction to Fast Analytical Techniques (1 ed.). Chichester, West Sussex, New Jersey, USA: John Wiley & Sons Ltd. p. 12. ISBN 978-1-11923637-5. LCCN 2015047967. Retrieved 2018-12-28. (464 pages)
Georg, Otfried (2013) [1999]. "Chapter 2.11.4.3: Aufstellen der Differentialgleichung aus der komplexen Darstellung - MATHCAD Anwendung 2.11-6: Benutzerdefinierte Operatoren". Elektromagnetische Felder und Netzwerke: Anwendungen in Mathcad und PSpice. Springer-Lehrbuch (in German) (1 ed.). Springer-Verlag. pp. 246–248. doi:10.1007/978-3-642-58420-6. ISBN 978-3-642-58420-6. ISBN 3-642-58420-9. Retrieved 2019-08-04. (728 pages)
Mitra, Sujit Kumar; Bhimasankaram, Pochiraju; Malik, Saroj B. (2010). Matrix Partial Orders, Shorted Operators and Applications. Series in Algebra. Vol. 10 (illustrated 1st ed.). World Scientific Publishing Co. Pte. Ltd. ISBN 978-981-283-844-5. ISBN 981-283-844-9. Retrieved 2019-08-19. (446 pages)
McWane, John W. (1981-05-01). Introduction to Electronics and Instrumentation (illustrated ed.). North Scituate, Massachusetts, USA: Breton Publishers, Wadsworth, Inc. pp. 78, 96–98, 100, 104. ISBN 0-53400938-7. ISBN 978-0-53400938-0. Retrieved 2019-08-04. p. xiii, 96–98, 100: […] Bruce D. Wedlock […] was the principle contributing author to Part I, BASIC CIRCUIT NETWORKS including the design of the companion examples. […] Most of the development of the IEI program was undertaken as part of the Technical Curriculum Research and Development Project of the MIT Center of Advanced Engineering Study. […] shorthand notation […] shorthand symbol ∥ […] (xiii+545 pages) (NB. In 1981, a 216-pages laboratory manual accompanying this book existed as well. The work grew out of an MIT course program "The MIT Technical Curriculum Development Project - Introduction to Electronics and Instrumentation" developed between 1974 and 1979. In 1986, a second edition of this book was published under the title "Introduction to Electronics Technology".)
Paul, Steffen; Paul, Reinhold (2014-10-24). "Chapter 2.3.2: Zusammenschaltungen linearer resistiver Zweipole – Parallelschaltung". Grundlagen der Elektrotechnik und Elektronik 1: Gleichstromnetzwerke und ihre Anwendungen (in German). Vol. 1 (5 ed.). Springer-Verlag. p. 78. ISBN 978-3-64253948-0. ISBN 3-64253948-3. Retrieved 2019-08-04. p. 78: […] Bei abgekürzter Schreibweise achte man sorgfältig auf die Anwendung von Klammern. […] Das Parallelzeichen ∥ der Kurzschreibweise hat die gleiche Bedeutung wie ein Multiplikationszeichen. Deshalb können Klammern entfallen. (446 pages)

cmu.edu

kilthub.cmu.edu

Duffin, Richard James (1971) [1970, 1969]. "Network Models". Written at Durham, North Carolina, USA. In Wilf, Herbert Saul; Hararay, Frank (eds.). Mathematical Aspects of Electrical Network Analysis. 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. Vol. III of SIAM-AMS Proceedings (illustrated ed.). Providence, Rhode Island: American Mathematical Society (AMS) / Society for Industrial and Applied Mathematics (SIAM). pp. 65–92 [68]. ISBN 0-8218-1322-6. ISSN 0080-5084. LCCN 79-167683. ISBN 978-0-8218-1322-5. Report 69-21. Retrieved 2019-08-05. pp. 68–69: […] 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 and associative. Moreover, multiplication is 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 + B as "A series B" and A : B as "A parallel B" then it is clear that the expression […] is the joint resistance of the network […] [1] [2] (206 pages)
Anderson, Jr., William Niles; Duffin, Richard James (1969) [1968-05-27]. "Series and parallel addition of matrices". Journal of Mathematical Analysis and Applications. 26 (3). Academic Press, Inc.: 576–594. doi:10.1016/0022-247X(69)90200-5. p. 576: […] we define the parallel sum of A and B by the formula A(A + B)⁺B and denote it by A : B. If A and B are nonsingular this reduces to A : B = (A⁻¹ + B⁻¹)⁻¹ which is the well known electrical formula for addition of resistors in parallel. Then it is shown that the Hermitian semi-definite matrices form a commutative partially ordered semigroup under the parallel sum operation. […] [6]

core.ac.uk

Mitra, Sujit Kumar; Puri, Madan Lal (October 1973). "On Parallel Sum and Difference of Matrices" (PDF). Journal of Mathematical Analysis and Applications. 44 (1). Academic Press, Inc.: 92–97. doi:10.1016/0022-247X(73)90027-9. Archived from the original (PDF) on 2019-04-13.

doi.org

Georg, Otfried (2013) [1999]. "Chapter 2.11.4.3: Aufstellen der Differentialgleichung aus der komplexen Darstellung - MATHCAD Anwendung 2.11-6: Benutzerdefinierte Operatoren". Elektromagnetische Felder und Netzwerke: Anwendungen in Mathcad und PSpice. Springer-Lehrbuch (in German) (1 ed.). Springer-Verlag. pp. 246–248. doi:10.1007/978-3-642-58420-6. ISBN 978-3-642-58420-6. ISBN 3-642-58420-9. Retrieved 2019-08-04. (728 pages)
Mitra, Sujit Kumar (February 1970). "A Matrix Operation for Analyzing Series-parallel Multiports". Journal of the Franklin Institute. Brief Communication. 289 (2). Franklin Institute: 167–169. doi:10.1016/0016-0032(70)90302-9. p. 167: The purpose of this communication is to extend the concept of the scalar operation Reduced Sum introduced by Seshu […] and later elaborated by Erickson […] 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 networks. Let A and B be two non-singular square matrices having inverses, A⁻¹ and B⁻¹ respectively. We define the operation ∙ as A ∙ B = (A⁻¹ + B⁻¹)⁻¹ and the operation ⊙ as A ⊙ B = A ∙ (−B). The operation ∙ is commutative and associative and is also distributive with respect to multiplication. […] (3 pages)
Duffin, Richard James; Hazony, Dov; Morrison, Norman Alexander (March 1966) [1965-04-12, 1964-08-25]. "Network synthesis through hybrid matrices". SIAM Journal on Applied Mathematics. 14 (2). Society for Industrial and Applied Mathematics (SIAM): 390–413. doi:10.1137/0114032. JSTOR 2946272. (24 pages)
Anderson, Jr., William Niles; Duffin, Richard James (1969) [1968-05-27]. "Series and parallel addition of matrices". Journal of Mathematical Analysis and Applications. 26 (3). Academic Press, Inc.: 576–594. doi:10.1016/0022-247X(69)90200-5. p. 576: […] we define the parallel sum of A and B by the formula A(A + B)⁺B and denote it by A : B. If A and B are nonsingular this reduces to A : B = (A⁻¹ + B⁻¹)⁻¹ which is the well known electrical formula for addition of resistors in parallel. Then it is shown that the Hermitian semi-definite matrices form a commutative partially ordered semigroup under the parallel sum operation. […] [6]
Mitra, Sujit Kumar; Puri, Madan Lal (October 1973). "On Parallel Sum and Difference of Matrices" (PDF). Journal of Mathematical Analysis and Applications. 44 (1). Academic Press, Inc.: 92–97. doi:10.1016/0022-247X(73)90027-9. Archived from the original (PDF) on 2019-04-13.
Eriksson-Bique, Sirkka-Liisa Anneli [in Finnish]; Leutwiler, Heinz (February 1989) [1989-01-10]. "A generalization of parallel addition" (PDF). Aequationes Mathematicae. 38 (1). Birkhäuser Verlag: 99–110. doi:10.1007/BF01839498. Archived (PDF) from the original on 2020-08-20. Retrieved 2020-08-20.
Seshu, Sundaram (September 1956). "On Electrical Circuits and Switching Circuits". IRE Transactions on Circuit Theory. CT-3 (3). Institute of Radio Engineers (IRE): 172–178. doi:10.1109/TCT.1956.1086310. (7 pages) (NB. See errata.)
Seshu, Sundaram; Gould, Roderick (September 1957). "Correction to 'On Electrical Circuits and Switching Circuits'". IRE Transactions on Circuit Theory. Correction. CT-4 (3). Institute of Radio Engineers (IRE): 284. doi:10.1109/TCT.1957.1086390. (1 page) (NB. Refers to previous reference.)
Erickson, Kent E. (March 1959). "A New Operation for Analyzing Series-Parallel Networks". IRE Transactions on Circuit Theory. CT-6 (1). Institute of Radio Engineers (IRE): 124–126. doi:10.1109/TCT.1959.1086519. p. 124: […] The operation ∗ is defined as A ∗ B = AB/A + B. The symbol ∗ has algebraic properties which simplify the formal solution of many series-parallel network problems. If the operation ∗ were included as a subroutine in a digital computer, it could simplify the programming of certain network calculations. […] (3 pages) (NB. See comment.)
Kaufman, Howard (June 1963). "Remark on a New Operation for Analyzing Series-Parallel Networks". IEEE Transactions on Circuit Theory. CT-10 (2). Institute of Electrical and Electronics Engineers (IEEE): 283. doi:10.1109/TCT.1963.1082126. p. 283: […] Comments on the operation ∗ […] a∗b = ab/(a+b) […] (1 page) (NB. Refers to previous reference.)

ed.gov

files.eric.ed.gov

Wolf, Lawrence J. (1977) [1976, 1974]. "Section 4. Instructional Materials – 4.3. The MIT Technical Curriculum Development Project – Introduction to Electronics and Instrumentation". In Aldridge, Bill G.; Mowery, Donald R.; Wolf, Lawrence J.; Dixon, Peggy (eds.). Science and Engineering Technology – Curriculum Guide: A Guide to a Two-Year Associate Degree Curriculum (PDF). Saint Louis Community College–Florissant Valley, St. Louis, Missouri, USA: National Science Teachers Association, Washington DC, USA. pp. 21, 77. Archived (PDF) from the original on 2017-02-15. Retrieved 2019-08-08. p. 21: […] Introduction to Electronics and Instrumentation is a new and contemporary approach to the introductory electronics course. Designed for students with no prior experience with electronics, it develops the skills and knowledge necessary to use and understand modern electronic systems. […] John W. McWane […] (NB. The SET Project was a two-year post-secondary curriculum developed between 1974 and 1977 preparing technicians to use electronic instruments.)

ellerman.org

Ellerman, David Patterson (1995-03-21). "Chapter 12: Parallel Addition, Series-Parallel Duality, and Financial Mathematics". Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics (PDF). The worldly philosophy: studies in intersection of philosophy and economics (illustrated ed.). Rowman & Littlefield Publishers, Inc. pp. 237–268. ISBN 0-8476-7932-2. Archived (PDF) from the original on 2016-03-05. Retrieved 2019-08-09. p. 237: […] When resistors with resistance a and b are placed in series, their compound resistance is the usual sum (hereafter the series sum) of the resistances a + b. If the resistances are placed in parallel, their compound resistance is the parallel sum of the resistances, which is denoted by the full colon […] [4] (271 pages)
Ellerman, David Patterson (May 2004) [1995-03-21]. "Introduction to Series-Parallel Duality" (PDF). University of California at Riverside. CiteSeerX 10.1.1.90.3666. Archived from the original on 2019-08-10. Retrieved 2019-08-09. The parallel sum of two positive real numbers x:y = [(1/x) + (1/y)]⁻¹ arises in electrical circuit theory as the resistance resulting from hooking two resistances x and y in parallel. There is a duality between the usual (series) sum and the parallel sum. […] [5] (24 pages)

gitlab.com

Bonin, Walter (2020) [2015]. WP 43S Owner's Manual (PDF). 0.16 (draft ed.). p. 119. ISBN 978-1-72950098-9. ISBN 1-72950098-6. Archived (PDF) from the original on 2022-07-21. Retrieved 2020-08-20. [10] [11] (328 pages)
Bonin, Walter (2020) [2015]. WP 43S Reference Manual (PDF). 0.16 (draft ed.). p. 127. ISBN 978-1-72950106-1. ISBN 1-72950106-0. Archived (PDF) from the original on 2022-07-21. Retrieved 2020-08-20. [12] [13] (315 pages)

hpmuseum.org

Dowrick, Nigel (2015-05-03) [2015-03-16]. "Complex Lock mode for WP-34s". HP Museum. Archived from the original on 2019-04-03. Retrieved 2019-08-07.

ieee.org

ieeexplore.ieee.org

Senturia, Stephen D. [at Wikidata]; Wedlock, Bruce D. (1975) [August 1974]. "Part A. Learning the Language, Chapter 3. Linear Resistive Networks, 3.2 Basic Network Configurations, 3.2.3. Resistors in Parallel". Written at Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. Electronic Circuits and Applications (1 ed.). New York, London, Sydney, Toronto: John Wiley & Sons, Inc. pp. viii–ix, 44–46 [45]. ISBN 0-471-77630-0. LCCN 74-7404. S2CID 61070327. pp. viii, ix, 45: This textbook evolved from a one-semester introductory electronics course taught by the authors at the Massachusetts Institute of Technology. […] The course is used by many freshmen as a precursor to the MIT Electrical Engineering Core Program. […] The preparation of a book of this size has drawn on the contribution of many people. The concept of teaching network theory and electronics as a single unified subject derives from Professor Campbell Searle, who taught the introductory electronics course when one of us (S.D.S.) was a first-year physics graduate student trying to learn electronics. In addition, Professor Searle has provided invaluable constructive criticism throughout the writing of this text. Several members of the MIT faculty and nearly 40 graduate technical assistants have participated in the teaching of this material over the past five years, many of whom have made important contributions through their suggestions and examples. Among these, we especially wish to thank O. R. Mitchell, Irvin Englander, George Lewis, Ernest Vincent, David James, Kenway Wong, Gim Hom, Tom Davis, James Kirtley, and Robert Donaghey. The chairman of the MIT Department of Electrical Engineering, Professor Louis D. Smullin, has provided support and encouragement during this project, as have many colleagues throughout the department. […] The first result […] states that the total voltage across the parallel combination of R₁ and R₂ is the same as that which occurs across a single resistance of value R₁ R₂ (R₁ + R₂). Because this expression for parallel resistance occurs so often, it is given a special notation (R₁∥R₂). That is, when R₁ and R₂ are in parallel, the equivalent resistance is $(R_{1}\parallel R_{2})={\frac {R_{1}R_{2}}{R_{1}+R_{2}}}$ […] (xii+623+5 pages) (NB. A teacher's manual was available as well. Early print runs contains a considerable number of typographical errors. See also: Wedlock's 1978 book.) [7]

instructure.com

utah.instructure.com

Cotter, Neil E., ed. (2015-10-12) [2014-09-20]. "ECE1250 Cookbook – Nodes, Series, Parallel" (lecture notes). Cookbooks. University of Utah. Archived (PDF) from the original on 2020-08-20. Retrieved 2019-08-11. […] One convenient way to indicate two resistors are in parallel is to put a ∥ between them. […]

jstor.org

Duffin, Richard James; Hazony, Dov; Morrison, Norman Alexander (March 1966) [1965-04-12, 1964-08-25]. "Network synthesis through hybrid matrices". SIAM Journal on Applied Mathematics. 14 (2). Society for Industrial and Applied Mathematics (SIAM): 390–413. doi:10.1137/0114032. JSTOR 2946272. (24 pages)

loc.gov

lccn.loc.gov

Duffin, Richard James (1971) [1970, 1969]. "Network Models". Written at Durham, North Carolina, USA. In Wilf, Herbert Saul; Hararay, Frank (eds.). Mathematical Aspects of Electrical Network Analysis. 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. Vol. III of SIAM-AMS Proceedings (illustrated ed.). Providence, Rhode Island: American Mathematical Society (AMS) / Society for Industrial and Applied Mathematics (SIAM). pp. 65–92 [68]. ISBN 0-8218-1322-6. ISSN 0080-5084. LCCN 79-167683. ISBN 978-0-8218-1322-5. Report 69-21. Retrieved 2019-08-05. pp. 68–69: […] 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 and associative. Moreover, multiplication is 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 + B as "A series B" and A : B as "A parallel B" then it is clear that the expression […] is the joint resistance of the network […] [1] [2] (206 pages)
Cajori, Florian (1993) [September 1928]. "§ 184, § 359, § 368". A History of Mathematical Notations – Notations in Elementary Mathematics. Vol. 1 (two volumes in one unaltered reprint ed.). Chicago, US: Open court publishing company. pp. 193, 402–403, 411–412. ISBN 0-486-67766-4. LCCN 93-29211. Retrieved 2019-07-22. pp. 402–403, 411–412: §359. […] ∥ for parallel occurs in Oughtred's Opuscula mathematica hactenus inedita (1677) [p. 197], a posthumous work (§ 184) […] §368. Signs for parallel lines. […] when Recorde's sign of equality won its way upon the Continent, vertical lines came to be used for parallelism. We find ∥ for "parallel" in Kersey,^[A] Caswell, Jones,^[B] Wilson,^[C] Emerson,^[D] Kambly,^[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^[F] use "par^[F] or ∥" for parallel […] [A] John Kersey, Algebra (London, 1673), Book IV, p. 177. [B] W. Jones, Synopsis palmarioum matheseos (London, 1706). [C] John Wilson, Trigonometry (Edinburgh, 1714), characters explained. [D] W. Emerson, Elements of Geometry (London, 1763), p. 4. [E] L. Kambly [de], Die Elementar-Mathematik, Part 2: Planimetrie, 43. edition (Breslau, 1876), p. 8. […] [F] H. S. Hall and F. H. Stevens, Euclid's Elements, Parts I and II (London, 1889), p. 10. […] [3]
Basso, Christophe P. (2016). "Chapter 1.1.2 The Current Divider". Linear Circuit Transfer Functions: An Introduction to Fast Analytical Techniques (1 ed.). Chichester, West Sussex, New Jersey, USA: John Wiley & Sons Ltd. p. 12. ISBN 978-1-11923637-5. LCCN 2015047967. Retrieved 2018-12-28. (464 pages)
Senturia, Stephen D. [at Wikidata]; Wedlock, Bruce D. (1975) [August 1974]. "Part A. Learning the Language, Chapter 3. Linear Resistive Networks, 3.2 Basic Network Configurations, 3.2.3. Resistors in Parallel". Written at Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. Electronic Circuits and Applications (1 ed.). New York, London, Sydney, Toronto: John Wiley & Sons, Inc. pp. viii–ix, 44–46 [45]. ISBN 0-471-77630-0. LCCN 74-7404. S2CID 61070327. pp. viii, ix, 45: This textbook evolved from a one-semester introductory electronics course taught by the authors at the Massachusetts Institute of Technology. […] The course is used by many freshmen as a precursor to the MIT Electrical Engineering Core Program. […] The preparation of a book of this size has drawn on the contribution of many people. The concept of teaching network theory and electronics as a single unified subject derives from Professor Campbell Searle, who taught the introductory electronics course when one of us (S.D.S.) was a first-year physics graduate student trying to learn electronics. In addition, Professor Searle has provided invaluable constructive criticism throughout the writing of this text. Several members of the MIT faculty and nearly 40 graduate technical assistants have participated in the teaching of this material over the past five years, many of whom have made important contributions through their suggestions and examples. Among these, we especially wish to thank O. R. Mitchell, Irvin Englander, George Lewis, Ernest Vincent, David James, Kenway Wong, Gim Hom, Tom Davis, James Kirtley, and Robert Donaghey. The chairman of the MIT Department of Electrical Engineering, Professor Louis D. Smullin, has provided support and encouragement during this project, as have many colleagues throughout the department. […] The first result […] states that the total voltage across the parallel combination of R₁ and R₂ is the same as that which occurs across a single resistance of value R₁ R₂ (R₁ + R₂). Because this expression for parallel resistance occurs so often, it is given a special notation (R₁∥R₂). That is, when R₁ and R₂ are in parallel, the equivalent resistance is $(R_{1}\parallel R_{2})={\frac {R_{1}R_{2}}{R_{1}+R_{2}}}$ […] (xii+623+5 pages) (NB. A teacher's manual was available as well. Early print runs contains a considerable number of typographical errors. See also: Wedlock's 1978 book.) [7]

mit.edu

libraries.mit.edu

Wiesner, Jerome Bert; Johnson, Howard Wesley; Killian, Jr., James Rhyne, eds. (1978-04-11). "School of Engineering – Center for Advanced Engineering Study (C.A.E.S.) – Research and Development – Technical Curriculum Research and Development Project". Report of the President and the Chancellor 1977–78 – Massachusetts Institute of Technology (PDF). Massachusetts Institute of Technology (MIT). pp. 249, 252–253. Archived (PDF) from the original on 2015-09-10. Retrieved 2019-08-08. pp. 249, 252–253: […] The Technical Curriculum Research and Development Program, sponsored by the Imperial Organization of Social Services [fa] of Iran, is entering the fourth year of a five-year contract. Curriculum development in electronics and mechanical engineering continues. […] Administered jointly by C.A.E.S. and the Department of Materials Science and Engineering, the Project is under the supervision of Professor Merton C. Flemings. It is directed by Dr. John W. McWane. […] Curriculum Materials Development. This is the principal activity of the project and is concerned with the development of innovative, state-of-the-art course materials in needed areas of engineering technology […] new introductory course in electronics […] is entitled Introduction to Electronics and Instrumentation and consists of eight […] modules […] dc Current, Voltage, and Resistance; Basic Circuit Networks; Time Varying Signals; Operational Amplifiers; Power Supplies; ac Current, Voltage, and Impedance; Digital Circuits; and Electronic Measurement and Control. This course represents a major change and updating of the way in which electronics is introduced, and should be of great value to STI as well as to many US programs. […]

monoskop.org

Cajori, Florian (1993) [September 1928]. "§ 184, § 359, § 368". A History of Mathematical Notations – Notations in Elementary Mathematics. Vol. 1 (two volumes in one unaltered reprint ed.). Chicago, US: Open court publishing company. pp. 193, 402–403, 411–412. ISBN 0-486-67766-4. LCCN 93-29211. Retrieved 2019-07-22. pp. 402–403, 411–412: §359. […] ∥ for parallel occurs in Oughtred's Opuscula mathematica hactenus inedita (1677) [p. 197], a posthumous work (§ 184) […] §368. Signs for parallel lines. […] when Recorde's sign of equality won its way upon the Continent, vertical lines came to be used for parallelism. We find ∥ for "parallel" in Kersey,^[A] Caswell, Jones,^[B] Wilson,^[C] Emerson,^[D] Kambly,^[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^[F] use "par^[F] or ∥" for parallel […] [A] John Kersey, Algebra (London, 1673), Book IV, p. 177. [B] W. Jones, Synopsis palmarioum matheseos (London, 1706). [C] John Wilson, Trigonometry (Edinburgh, 1714), characters explained. [D] W. Emerson, Elements of Geometry (London, 1763), p. 4. [E] L. Kambly [de], Die Elementar-Mathematik, Part 2: Planimetrie, 43. edition (Breslau, 1876), p. 8. […] [F] H. S. Hall and F. H. Stevens, Euclid's Elements, Parts I and II (London, 1889), p. 10. […] [3]

psu.edu

citeseerx.ist.psu.edu

Ellerman, David Patterson (May 2004) [1995-03-21]. "Introduction to Series-Parallel Duality" (PDF). University of California at Riverside. CiteSeerX 10.1.1.90.3666. Archived from the original on 2019-08-10. Retrieved 2019-08-09. The parallel sum of two positive real numbers x:y = [(1/x) + (1/y)]⁻¹ arises in electrical circuit theory as the resistance resulting from hooking two resistances x and y in parallel. There is a duality between the usual (series) sum and the parallel sum. […] [5] (24 pages)

researchgate.net

Mitra, Sujit Kumar (February 1970). "A Matrix Operation for Analyzing Series-parallel Multiports". Journal of the Franklin Institute. Brief Communication. 289 (2). Franklin Institute: 167–169. doi:10.1016/0016-0032(70)90302-9. p. 167: The purpose of this communication is to extend the concept of the scalar operation Reduced Sum introduced by Seshu […] and later elaborated by Erickson […] 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 networks. Let A and B be two non-singular square matrices having inverses, A⁻¹ and B⁻¹ respectively. We define the operation ∙ as A ∙ B = (A⁻¹ + B⁻¹)⁻¹ and the operation ⊙ as A ⊙ B = A ∙ (−B). The operation ∙ is commutative and associative and is also distributive with respect to multiplication. […] (3 pages)
Seshu, Sundaram (September 1956). "On Electrical Circuits and Switching Circuits". IRE Transactions on Circuit Theory. CT-3 (3). Institute of Radio Engineers (IRE): 172–178. doi:10.1109/TCT.1956.1086310. (7 pages) (NB. See errata.)
Seshu, Sundaram; Gould, Roderick (September 1957). "Correction to 'On Electrical Circuits and Switching Circuits'". IRE Transactions on Circuit Theory. Correction. CT-4 (3). Institute of Radio Engineers (IRE): 284. doi:10.1109/TCT.1957.1086390. (1 page) (NB. Refers to previous reference.)
Erickson, Kent E. (March 1959). "A New Operation for Analyzing Series-Parallel Networks". IRE Transactions on Circuit Theory. CT-6 (1). Institute of Radio Engineers (IRE): 124–126. doi:10.1109/TCT.1959.1086519. p. 124: […] The operation ∗ is defined as A ∗ B = AB/A + B. The symbol ∗ has algebraic properties which simplify the formal solution of many series-parallel network problems. If the operation ∗ were included as a subroutine in a digital computer, it could simplify the programming of certain network calculations. […] (3 pages) (NB. See comment.)

semanticscholar.org

api.semanticscholar.org

Senturia, Stephen D. [at Wikidata]; Wedlock, Bruce D. (1975) [August 1974]. "Part A. Learning the Language, Chapter 3. Linear Resistive Networks, 3.2 Basic Network Configurations, 3.2.3. Resistors in Parallel". Written at Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. Electronic Circuits and Applications (1 ed.). New York, London, Sydney, Toronto: John Wiley & Sons, Inc. pp. viii–ix, 44–46 [45]. ISBN 0-471-77630-0. LCCN 74-7404. S2CID 61070327. pp. viii, ix, 45: This textbook evolved from a one-semester introductory electronics course taught by the authors at the Massachusetts Institute of Technology. […] The course is used by many freshmen as a precursor to the MIT Electrical Engineering Core Program. […] The preparation of a book of this size has drawn on the contribution of many people. The concept of teaching network theory and electronics as a single unified subject derives from Professor Campbell Searle, who taught the introductory electronics course when one of us (S.D.S.) was a first-year physics graduate student trying to learn electronics. In addition, Professor Searle has provided invaluable constructive criticism throughout the writing of this text. Several members of the MIT faculty and nearly 40 graduate technical assistants have participated in the teaching of this material over the past five years, many of whom have made important contributions through their suggestions and examples. Among these, we especially wish to thank O. R. Mitchell, Irvin Englander, George Lewis, Ernest Vincent, David James, Kenway Wong, Gim Hom, Tom Davis, James Kirtley, and Robert Donaghey. The chairman of the MIT Department of Electrical Engineering, Professor Louis D. Smullin, has provided support and encouragement during this project, as have many colleagues throughout the department. […] The first result […] states that the total voltage across the parallel combination of R₁ and R₂ is the same as that which occurs across a single resistance of value R₁ R₂ (R₁ + R₂). Because this expression for parallel resistance occurs so often, it is given a special notation (R₁∥R₂). That is, when R₁ and R₂ are in parallel, the equivalent resistance is $(R_{1}\parallel R_{2})={\frac {R_{1}R_{2}}{R_{1}+R_{2}}}$ […] (xii+623+5 pages) (NB. A teacher's manual was available as well. Early print runs contains a considerable number of typographical errors. See also: Wedlock's 1978 book.) [7]

sourceforge.net

Dale, Paul; Bonin, Walter (2012-11-30) [2008-12-09]. WP 34S Owner's Manual (PDF) (3.1 ed.). pp. 1, 14, 32, 66, 116. Archived (PDF) from the original on 2019-07-09. Retrieved 2019-07-13. [8] (211 pages)
Bonin, Walter (2015) [2008-12-09]. WP 34S Owner's Manual (3.3 ed.). CreateSpace Independent Publishing Platform. ISBN 978-1-5078-9107-0. [9]
Bonin, Walter (2020) [2015]. WP 43S Owner's Manual (PDF). 0.16 (draft ed.). p. 119. ISBN 978-1-72950098-9. ISBN 1-72950098-6. Archived (PDF) from the original on 2022-07-21. Retrieved 2020-08-20. [10] [11] (328 pages)
Bonin, Walter (2020) [2015]. WP 43S Reference Manual (PDF). 0.16 (draft ed.). p. 127. ISBN 978-1-72950106-1. ISBN 1-72950106-0. Archived (PDF) from the original on 2022-07-21. Retrieved 2020-08-20. [12] [13] (315 pages)

springer.com

link.springer.com

Eriksson-Bique, Sirkka-Liisa Anneli [in Finnish]; Leutwiler, Heinz (February 1989) [1989-01-10]. "A generalization of parallel addition" (PDF). Aequationes Mathematicae. 38 (1). Birkhäuser Verlag: 99–110. doi:10.1007/BF01839498. Archived (PDF) from the original on 2020-08-20. Retrieved 2020-08-20.

st.com

"A7987: 61 V, 3 A asynchronous step-down switching regulator with adjustable current limitation for automotive" (PDF) (Datasheet). Revision 3. STMicroelectronics NV. 2020-09-22 [2019-03-19]. pp. 17, 18, 20. DS12928. Archived (PDF) from the original on 2022-07-18. Retrieved 2022-07-18. (36 pages)

thecalculatorstore.com

Dale, Paul; Bonin, Walter (2012-11-30) [2008-12-09]. WP 34S Owner's Manual (PDF) (3.1 ed.). pp. 1, 14, 32, 66, 116. Archived (PDF) from the original on 2019-07-09. Retrieved 2019-07-13. [8] (211 pages)

ti.com

"INA 326/INA 327 – Precision, Rail-to-Rail I/O Instrumentation Amplifier" (PDF). Burr-Brown / Texas Instruments. 2018 [November 2004, November 2001]. pp. 3, 9, 13. SBOS222D. Archived (PDF) from the original on 2019-07-13. Retrieved 2019-07-13.
"7.5.3 Selection of the External Resistance". TPL5110 Nano-Power System Timer for Power Gating (PDF) (Datasheet). Revision A. Texas Instruments Incorporated. September 2018 [January 2015]. pp. 13–14. SNAS650A. Archived (PDF) from the original on 2022-09-25. Retrieved 2022-09-25. (27 pages)

uni-paderborn.de

ei.uni-paderborn.de

Böcker, Joachim (2019-03-18) [April 2008]. "Grundlagen der Elektrotechnik Teil B" (PDF) (in German). Universität Paderborn. p. 12. Archived (PDF) from the original on 2018-04-17. Retrieved 2019-08-09. p. 12: Für die Berechnung des Ersatzwiderstands der Parallelschaltung wird […] gern die Kurzschreibweise ∥ benutzt.

web.archive.org

Kersey (the elder), John (1673). "Chapter I: Concerning the Scope of this fourth Book and the Signification of Characters, Abbreviations and Citations used therein". The Elements of that Mathematical Art, commonly called Algebra. Vol. Book IV - The Elements of the Algebraical Arts. London: Thomas Passinger, Three-Bibles, London-Bridge. pp. 177–178. Archived from the original on 2020-08-05. Retrieved 2019-08-09.
"INA 326/INA 327 – Precision, Rail-to-Rail I/O Instrumentation Amplifier" (PDF). Burr-Brown / Texas Instruments. 2018 [November 2004, November 2001]. pp. 3, 9, 13. SBOS222D. Archived (PDF) from the original on 2019-07-13. Retrieved 2019-07-13.
Ranade, Gireeja; Stojanovic, Vladimir, eds. (Fall 2018). "Chapter 15.7.2 Parallel Resistors" (PDF). EECS 16A Designing Information Devices and Systems I (PDF) (lecture notes). University of California, Berkeley. p. 12. Note 15. Archived (PDF) from the original on 2018-12-27. Retrieved 2018-12-28. p. 12: […] This mathematical relationship comes up often enough that it actually has a name: the "parallel operator", denoted ∥. When we say x∥y, it means ${\frac {xy}{x+y}}$ . Note that this is a mathematical operator and does not say anything about the actual configuration. In the case of resistors the parallel operator is used for parallel resistors, but for other components (like capacitors) this is not the case. […] (16 pages)
Ellerman, David Patterson (1995-03-21). "Chapter 12: Parallel Addition, Series-Parallel Duality, and Financial Mathematics". Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics (PDF). The worldly philosophy: studies in intersection of philosophy and economics (illustrated ed.). Rowman & Littlefield Publishers, Inc. pp. 237–268. ISBN 0-8476-7932-2. Archived (PDF) from the original on 2016-03-05. Retrieved 2019-08-09. p. 237: […] When resistors with resistance a and b are placed in series, their compound resistance is the usual sum (hereafter the series sum) of the resistances a + b. If the resistances are placed in parallel, their compound resistance is the parallel sum of the resistances, which is denoted by the full colon […] [4] (271 pages)
Ellerman, David Patterson (May 2004) [1995-03-21]. "Introduction to Series-Parallel Duality" (PDF). University of California at Riverside. CiteSeerX 10.1.1.90.3666. Archived from the original on 2019-08-10. Retrieved 2019-08-09. The parallel sum of two positive real numbers x:y = [(1/x) + (1/y)]⁻¹ arises in electrical circuit theory as the resistance resulting from hooking two resistances x and y in parallel. There is a duality between the usual (series) sum and the parallel sum. […] [5] (24 pages)
Cotter, Neil E., ed. (2015-10-12) [2014-09-20]. "ECE1250 Cookbook – Nodes, Series, Parallel" (lecture notes). Cookbooks. University of Utah. Archived (PDF) from the original on 2020-08-20. Retrieved 2019-08-11. […] One convenient way to indicate two resistors are in parallel is to put a ∥ between them. […]
Böcker, Joachim (2019-03-18) [April 2008]. "Grundlagen der Elektrotechnik Teil B" (PDF) (in German). Universität Paderborn. p. 12. Archived (PDF) from the original on 2018-04-17. Retrieved 2019-08-09. p. 12: Für die Berechnung des Ersatzwiderstands der Parallelschaltung wird […] gern die Kurzschreibweise ∥ benutzt.
Mitra, Sujit Kumar; Puri, Madan Lal (October 1973). "On Parallel Sum and Difference of Matrices" (PDF). Journal of Mathematical Analysis and Applications. 44 (1). Academic Press, Inc.: 92–97. doi:10.1016/0022-247X(73)90027-9. Archived from the original (PDF) on 2019-04-13.
Eriksson-Bique, Sirkka-Liisa Anneli [in Finnish]; Leutwiler, Heinz (February 1989) [1989-01-10]. "A generalization of parallel addition" (PDF). Aequationes Mathematicae. 38 (1). Birkhäuser Verlag: 99–110. doi:10.1007/BF01839498. Archived (PDF) from the original on 2020-08-20. Retrieved 2020-08-20.
Wolf, Lawrence J. (1977) [1976, 1974]. "Section 4. Instructional Materials – 4.3. The MIT Technical Curriculum Development Project – Introduction to Electronics and Instrumentation". In Aldridge, Bill G.; Mowery, Donald R.; Wolf, Lawrence J.; Dixon, Peggy (eds.). Science and Engineering Technology – Curriculum Guide: A Guide to a Two-Year Associate Degree Curriculum (PDF). Saint Louis Community College–Florissant Valley, St. Louis, Missouri, USA: National Science Teachers Association, Washington DC, USA. pp. 21, 77. Archived (PDF) from the original on 2017-02-15. Retrieved 2019-08-08. p. 21: […] Introduction to Electronics and Instrumentation is a new and contemporary approach to the introductory electronics course. Designed for students with no prior experience with electronics, it develops the skills and knowledge necessary to use and understand modern electronic systems. […] John W. McWane […] (NB. The SET Project was a two-year post-secondary curriculum developed between 1974 and 1977 preparing technicians to use electronic instruments.)
Wiesner, Jerome Bert; Johnson, Howard Wesley; Killian, Jr., James Rhyne, eds. (1978-04-11). "School of Engineering – Center for Advanced Engineering Study (C.A.E.S.) – Research and Development – Technical Curriculum Research and Development Project". Report of the President and the Chancellor 1977–78 – Massachusetts Institute of Technology (PDF). Massachusetts Institute of Technology (MIT). pp. 249, 252–253. Archived (PDF) from the original on 2015-09-10. Retrieved 2019-08-08. pp. 249, 252–253: […] The Technical Curriculum Research and Development Program, sponsored by the Imperial Organization of Social Services [fa] of Iran, is entering the fourth year of a five-year contract. Curriculum development in electronics and mechanical engineering continues. […] Administered jointly by C.A.E.S. and the Department of Materials Science and Engineering, the Project is under the supervision of Professor Merton C. Flemings. It is directed by Dr. John W. McWane. […] Curriculum Materials Development. This is the principal activity of the project and is concerned with the development of innovative, state-of-the-art course materials in needed areas of engineering technology […] new introductory course in electronics […] is entitled Introduction to Electronics and Instrumentation and consists of eight […] modules […] dc Current, Voltage, and Resistance; Basic Circuit Networks; Time Varying Signals; Operational Amplifiers; Power Supplies; ac Current, Voltage, and Impedance; Digital Circuits; and Electronic Measurement and Control. This course represents a major change and updating of the way in which electronics is introduced, and should be of great value to STI as well as to many US programs. […]
"7.5.3 Selection of the External Resistance". TPL5110 Nano-Power System Timer for Power Gating (PDF) (Datasheet). Revision A. Texas Instruments Incorporated. September 2018 [January 2015]. pp. 13–14. SNAS650A. Archived (PDF) from the original on 2022-09-25. Retrieved 2022-09-25. (27 pages)
"A7987: 61 V, 3 A asynchronous step-down switching regulator with adjustable current limitation for automotive" (PDF) (Datasheet). Revision 3. STMicroelectronics NV. 2020-09-22 [2019-03-19]. pp. 17, 18, 20. DS12928. Archived (PDF) from the original on 2022-07-18. Retrieved 2022-07-18. (36 pages)
Dale, Paul; Bonin, Walter (2012-11-30) [2008-12-09]. WP 34S Owner's Manual (PDF) (3.1 ed.). pp. 1, 14, 32, 66, 116. Archived (PDF) from the original on 2019-07-09. Retrieved 2019-07-13. [8] (211 pages)
Dowrick, Nigel (2015-05-03) [2015-03-16]. "Complex Lock mode for WP-34s". HP Museum. Archived from the original on 2019-04-03. Retrieved 2019-08-07.
Bonin, Walter (2020) [2015]. WP 43S Owner's Manual (PDF). 0.16 (draft ed.). p. 119. ISBN 978-1-72950098-9. ISBN 1-72950098-6. Archived (PDF) from the original on 2022-07-21. Retrieved 2020-08-20. [10] [11] (328 pages)
Bonin, Walter (2020) [2015]. WP 43S Reference Manual (PDF). 0.16 (draft ed.). p. 127. ISBN 978-1-72950106-1. ISBN 1-72950106-0. Archived (PDF) from the original on 2022-07-21. Retrieved 2020-08-20. [12] [13] (315 pages)

wikibooks.org

en.wikibooks.org

Associative Composition Algebra/Homographies at Wikibooks

wikidata.org

While the use of the symbol ∥ for "parallel" in geometry reaches as far back as 1673 in John Kersey the elder's work,^[A] this came into more use only since about 1875.^[B] The usage of a mathematical operator for parallel circuits originates from network theory in electrical engineering. Sundaram Seshu introduced a reduced sum operator in 1956,^[C] Kent E. Erickson proposed an asterisk (∗) to symbolize the operator in 1959,^[D] whilst Richard James Duffin and William Niles Anderson, Jr. used a colon (:) for the parallel addition since 1966.^[E] Sujit Kumar Mitra used a middot (∙) for it in 1970.^[F] The first usage of the parallel symbol (∥) for this operator in applied electronics is unknown, but might have originated from Stephen D. Senturia [d] and Bruce D. Wedlock's 1974 book "Electronic Circuits and Applications",^[G] which evolved from their introductory electronics course at Massachusetts Institute of Technology (MIT) with concepts of teaching network theory and electronics derived from an earlier course taught by Campbell "Cam" Leach Searle. It was further popularized through John W. McWane's 1981 book "Introduction to Electronics and Instrumentation",^[H] which grew out of an identically-named MIT course developed as part of the influential Technical Curriculum Development Project between 1974 and 1979. This symbol was probably also introduced because the other used symbols could be easily confused with signs commonly used for multiplication and division in some contexts.
Senturia, Stephen D. [at Wikidata]; Wedlock, Bruce D. (1975) [August 1974]. "Part A. Learning the Language, Chapter 3. Linear Resistive Networks, 3.2 Basic Network Configurations, 3.2.3. Resistors in Parallel". Written at Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. Electronic Circuits and Applications (1 ed.). New York, London, Sydney, Toronto: John Wiley & Sons, Inc. pp. viii–ix, 44–46 [45]. ISBN 0-471-77630-0. LCCN 74-7404. S2CID 61070327. pp. viii, ix, 45: This textbook evolved from a one-semester introductory electronics course taught by the authors at the Massachusetts Institute of Technology. […] The course is used by many freshmen as a precursor to the MIT Electrical Engineering Core Program. […] The preparation of a book of this size has drawn on the contribution of many people. The concept of teaching network theory and electronics as a single unified subject derives from Professor Campbell Searle, who taught the introductory electronics course when one of us (S.D.S.) was a first-year physics graduate student trying to learn electronics. In addition, Professor Searle has provided invaluable constructive criticism throughout the writing of this text. Several members of the MIT faculty and nearly 40 graduate technical assistants have participated in the teaching of this material over the past five years, many of whom have made important contributions through their suggestions and examples. Among these, we especially wish to thank O. R. Mitchell, Irvin Englander, George Lewis, Ernest Vincent, David James, Kenway Wong, Gim Hom, Tom Davis, James Kirtley, and Robert Donaghey. The chairman of the MIT Department of Electrical Engineering, Professor Louis D. Smullin, has provided support and encouragement during this project, as have many colleagues throughout the department. […] The first result […] states that the total voltage across the parallel combination of R₁ and R₂ is the same as that which occurs across a single resistance of value R₁ R₂ (R₁ + R₂). Because this expression for parallel resistance occurs so often, it is given a special notation (R₁∥R₂). That is, when R₁ and R₂ are in parallel, the equivalent resistance is $(R_{1}\parallel R_{2})={\frac {R_{1}R_{2}}{R_{1}+R_{2}}}$ […] (xii+623+5 pages) (NB. A teacher's manual was available as well. Early print runs contains a considerable number of typographical errors. See also: Wedlock's 1978 book.) [7]

wikipedia.org

de.wikipedia.org

Cajori, Florian (1993) [September 1928]. "§ 184, § 359, § 368". A History of Mathematical Notations – Notations in Elementary Mathematics. Vol. 1 (two volumes in one unaltered reprint ed.). Chicago, US: Open court publishing company. pp. 193, 402–403, 411–412. ISBN 0-486-67766-4. LCCN 93-29211. Retrieved 2019-07-22. pp. 402–403, 411–412: §359. […] ∥ for parallel occurs in Oughtred's Opuscula mathematica hactenus inedita (1677) [p. 197], a posthumous work (§ 184) […] §368. Signs for parallel lines. […] when Recorde's sign of equality won its way upon the Continent, vertical lines came to be used for parallelism. We find ∥ for "parallel" in Kersey,^[A] Caswell, Jones,^[B] Wilson,^[C] Emerson,^[D] Kambly,^[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^[F] use "par^[F] or ∥" for parallel […] [A] John Kersey, Algebra (London, 1673), Book IV, p. 177. [B] W. Jones, Synopsis palmarioum matheseos (London, 1706). [C] John Wilson, Trigonometry (Edinburgh, 1714), characters explained. [D] W. Emerson, Elements of Geometry (London, 1763), p. 4. [E] L. Kambly [de], Die Elementar-Mathematik, Part 2: Planimetrie, 43. edition (Breslau, 1876), p. 8. […] [F] H. S. Hall and F. H. Stevens, Euclid's Elements, Parts I and II (London, 1889), p. 10. […] [3]

fi.wikipedia.org

Eriksson-Bique, Sirkka-Liisa Anneli [in Finnish]; Leutwiler, Heinz (February 1989) [1989-01-10]. "A generalization of parallel addition" (PDF). Aequationes Mathematicae. 38 (1). Birkhäuser Verlag: 99–110. doi:10.1007/BF01839498. Archived (PDF) from the original on 2020-08-20. Retrieved 2020-08-20.

fa.wikipedia.org

Wiesner, Jerome Bert; Johnson, Howard Wesley; Killian, Jr., James Rhyne, eds. (1978-04-11). "School of Engineering – Center for Advanced Engineering Study (C.A.E.S.) – Research and Development – Technical Curriculum Research and Development Project". Report of the President and the Chancellor 1977–78 – Massachusetts Institute of Technology (PDF). Massachusetts Institute of Technology (MIT). pp. 249, 252–253. Archived (PDF) from the original on 2015-09-10. Retrieved 2019-08-08. pp. 249, 252–253: […] The Technical Curriculum Research and Development Program, sponsored by the Imperial Organization of Social Services [fa] of Iran, is entering the fourth year of a five-year contract. Curriculum development in electronics and mechanical engineering continues. […] Administered jointly by C.A.E.S. and the Department of Materials Science and Engineering, the Project is under the supervision of Professor Merton C. Flemings. It is directed by Dr. John W. McWane. […] Curriculum Materials Development. This is the principal activity of the project and is concerned with the development of innovative, state-of-the-art course materials in needed areas of engineering technology […] new introductory course in electronics […] is entitled Introduction to Electronics and Instrumentation and consists of eight […] modules […] dc Current, Voltage, and Resistance; Basic Circuit Networks; Time Varying Signals; Operational Amplifiers; Power Supplies; ac Current, Voltage, and Impedance; Digital Circuits; and Electronic Measurement and Control. This course represents a major change and updating of the way in which electronics is introduced, and should be of great value to STI as well as to many US programs. […]

worldcat.org

search.worldcat.org

Duffin, Richard James (1971) [1970, 1969]. "Network Models". Written at Durham, North Carolina, USA. In Wilf, Herbert Saul; Hararay, Frank (eds.). Mathematical Aspects of Electrical Network Analysis. 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. Vol. III of SIAM-AMS Proceedings (illustrated ed.). Providence, Rhode Island: American Mathematical Society (AMS) / Society for Industrial and Applied Mathematics (SIAM). pp. 65–92 [68]. ISBN 0-8218-1322-6. ISSN 0080-5084. LCCN 79-167683. ISBN 978-0-8218-1322-5. Report 69-21. Retrieved 2019-08-05. pp. 68–69: […] 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 and associative. Moreover, multiplication is 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 + B as "A series B" and A : B as "A parallel B" then it is clear that the expression […] is the joint resistance of the network […] [1] [2] (206 pages)