இணை இயக்கி

இணை இயக்கி (parallel operator $\|$ )(இது கொணர்கூட்டல் அல்லது இணை கூட்டுத்தொகை அல்லது இணை கூட்டல் எனவும் வழங்கும்) இங்கு $\|$ ( "இணை" எனும் சொல்,^[1] வடிவியலின் இணைகோடுகள் குறியீட்டைப் பின்பற்றி வழங்கும் சொல்லாகும்)^[2]^[3]) என்பது ஒரு சார்பு (கணிதம்) ஆகும்; இது குறிப்பாக மின் பொறியியலிலும்^[4]^[5]^[6]^{[nb 1]} இயக்கவியலிலும் பாய்ம இயக்கவியலிலும் நிதிசார் கணிதவியலிலும் சுருக்கெழுத்தாக பயன்படுத்தப்படுகிறது. .^[7]^[8] இது பெருக்கல் நேர்மாறைக் குறிப்பதோடு அதை வரையறுக்கிறது.

${\begin{array}{rlcl}\|:\ &{\overline {\mathbb {C} }}\times {\overline {\mathbb {C} }}&\to &{\overline {\mathbb {C} }}\\&(a,b)&\mapsto &a\|b={\frac {1}{{\frac {1}{a}}+{\frac {1}{b}}}}\end{array}}$

அதாவது, இது இரண்டு எண்களான a, b ஆகியவர்ரின் இசைச் சராசரியைக் காட்டுகின்றது.

குறிப்புகள்[தொகு]

↑ While the use of the symbol ∥ for "parallel" in geometry reaches as far back as 1673 in John Kersey the elder's work,வார்ப்புரு:Citeref this came into more use only since about 1875.வார்ப்புரு:Citeref 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,வார்ப்புரு:Citeref Kent E. Erickson proposed an asterisk (∗) to symbolize the operator in 1959,வார்ப்புரு:Citeref whilst Richard James Duffin and William Niles Anderson, Jr. used a colon (:) for the parallel addition since 1966.வார்ப்புரு:Citeref Sujit Kumar Mitra used a middot (∙) for it in 1970.வார்ப்புரு:Citeref The first usage of the parallel symbol (∥) for this operator in applied electronics is unknown, but might have originated from d (d) and Bruce D. Wedlock's 1974 book "Electronic Circuits and Applications",வார்ப்புரு:Citeref 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",வார்ப்புரு:Citeref which grew out of an identically-named MIT course developed as part of the influential வார்ப்புரு:Citeref 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.

மேற்கோள்கள்[தொகு]

↑ "Network Models". written at Durham, North Carolina, USA. 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. III of SIAM-AMS Proceedings (illustrated ). Providence, Rhode Island: American Mathematical Society (AMS) / Society for Industrial and Applied Mathematics (SIAM). 1971. பக். 65–92 [68]. ISBN 978-0-8218-1322-5. Report 69-21. பன்னாட்டுத் தரப்புத்தக எண்:0-8218-1322-6. https://books.google.com/books?id=j2vFhxA5K-UC&pg=PA65. பார்த்த நாள்: 2019-08-05. "[…] 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)
↑ "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. Book IV - The Elements of the Algebraical Arts. London: Thomas Passinger, Three-Bibles, London-Bridge. 1673. பக். 177–178. https://archive.org/details/b30333726_0002. பார்த்த நாள்: 2019-08-09.
↑ "§ 184, § 359, § 368". A History of Mathematical Notations – Notations in Elementary Mathematics. 1 (two volumes in one unaltered reprint ). Chicago, US: Open court publishing company. 1993. பக். 193, 402–403, 411–412. பன்னாட்டுத் தரப்புத்தக எண்:0-486-67766-4. https://archive.org/details/historyofmathema00cajo_0/page/193. பார்த்த நாள்: 2019-07-22. "§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,வார்ப்புரு:Citeref Caswell, Jones,வார்ப்புரு:Citeref Wilson,வார்ப்புரு:Citeref Emerson,வார்ப்புரு:Citeref Kambly,வார்ப்புரு:Citeref 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 Stevensவார்ப்புரு:Citeref use "parவார்ப்புரு:Citeref or ∥" for parallel […] [A] John Kersey, வார்ப்புரு:Citeref (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|Deutsch (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]
↑ "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. பார்க்கப்பட்ட நாள் 2019-07-13.
↑ "Chapter 7.6. Laplace Transforms Applied to Circuits". Numerical and Analytical Methods with MATLAB for Electrical Engineers. Applied and Computational Mechanics (1 ). CRC Press. 2016. பக். 224. ISBN 1-46657607-3. பன்னாட்டுத் தரப்புத்தக எண்:978-1-46657607-0. https://books.google.com/books?id=O2POBQAAQBAJ&pg=PA224. (388 pages)
↑ "Chapter 15.7.2 Parallel Resistors". EECS 16A Designing Information Devices and Systems I (lecture notes). University of California, Berkeley. Fall 2018. பக். 12. Note 15. https://inst.eecs.berkeley.edu/~ee16a/fa18/lectures/Note15.pdf#page=12. பார்த்த நாள்: 2018-12-28. "[…] 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)
↑ "Chapter 12: Parallel Addition, Series-Parallel Duality, and Financial Mathematics". Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics. G – Reference, Information and Interdisciplinary Subjects Series (illustrated ). Rowman & Littlefield Publishers, Inc.. 1995-03-21. பக். 237–268. பன்னாட்டுத் தரப்புத்தக எண்:0-8476-7932-2. https://books.google.com/books?id=NgJqXXk7zAAC&pg=PA237. பார்த்த நாள்: 2019-08-09. "[…] 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. பார்க்கப்பட்ட நாள் 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)

மேலும் படிக்க[தொகு]

"Parallel Addition and Parallel Subtraction of Operators". Mathematics of the USSR-Izvestiya (American Mathematical Society) 10 (2): 351–370. 1976-04-30. doi:10.1070/IM1976v010n02ABEH001694. Bibcode: 1976IzMat..10..351P. https://iopscience.iop.org/article/10.1070/IM1976v010n02ABEH001694/pdf.
"Almost Definite Operators and Electro-mechanical Systems". SIAM Journal on Applied Mathematics (Society for Industrial and Applied Mathematics (SIAM)) 35 (1): 21–30. July 1978. doi:10.1137/0135003. (10 pages)
"Parallel Summation, Maxwell's Principle and the Infimum of Projections". Journal of Mathematical Analysis and Applications (Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA) 70 (1): 33–41. July 1979. doi:10.1016/0022-247X(79)90073-8. https://core.ac.uk/download/pdf/82642040.pdf. பார்த்த நாள்: 2020-08-20.
"Direct and Inverse Addition in Convex Analysis and Applications". Journal of Mathematical Analysis and Applications (Department of Mathematics, University of Washington, Seattle, Washington, USA: Academic Press, Inc.) 148 (2): 317–349. May 1990. doi:10.1016/0022-247X(90)90004-Y. https://core.ac.uk/download/pdf/81183866.pdf. பார்த்த நாள்: 2020-08-20. (33 pages)
"Geometric characterization of series-parallel variable resistor networks". IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 41 (11): 686–698. 1994. doi:10.1109/81.331520. https://people.eecs.berkeley.edu/~tygar/papers/Geometric_characterization_of_series-parallel/Bryant_Journal_preprint.pdf.
"Bilateral Shorted Operators and Parallel Sums". Linear Algebra and Its Applications (La Plata, Argentina & Buenos Aires, Argentina) 414 (2–3): 570–588. April 2006. doi:10.1016/j.laa.2005.10.039. http://www.mate.unlp.edu.ar/~demetrio/Papers/Papers%20en%20pdf/17.%20Bilateral%20Shorted%20Operators%20and%20Parallel%20Sums.pdf. பார்த்த நாள்: 2020-08-20. [6] (19 pages)
"Mathematical aspects for electrical network connections". KKU Engineering Journal 43 (1): 47–54. February 2016. doi:10.14456/kkuenj.2016.8. https://www.researchgate.net/publication/307713517_Mathematical_aspects_for_electrical_network_connections. பார்த்த நாள்: 2020-08-20.
Besenyei, Ádám (2016-09-01). "The irresistible inequality of Milne" (PDF). Budapest: Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University. CIA2016. Archived (PDF) from the original on 2019-08-08. பார்க்கப்பட்ட நாள் 2019-08-11.
"7.5 Electrical Characteristics: VCC = 5 V / 7.6 Electrical Characteristics: VCC = 2.7 V / 9.1.2.1 Inverting Comparator with Hysteresis". TLV3201, TLV3202: TLV320x 40-ns, microPOWER, Push-Pull Output Comparators. Dallas, Texas, USA: Texas Instruments Incorporated. 2022-06-03. பக். 5, 6, 13–14 [13]. SBOS561B. https://www.ti.com/lit/ds/symlink/tlv3201.pdf?ts=1660718632803. பார்த்த நாள்: 2022-08-18. "PARAMETER […] TYP […] UNIT […] INPUT IMPEDANCE […] Common mode […] 10¹³ ∥ 2 […] Ω ∥ pF […] Differential […] 10¹³ ∥ 4 […] Ω ∥ pF […]" (37 pages) (NB. Unusual usage of ∥ for both values and units.)

வெளி இணைப்புகள்[தொகு]

https://github.com/microsoftarchive/edx-platform-1/blob/master/common/lib/calc/calc/calc.py

[NB_Origin-7] While the use of the symbol ∥ for "parallel" in geometry reaches as far back as 1673 in John Kersey the elder's work,வார்ப்புரு:Citeref this came into more use only since about 1875.வார்ப்புரு:Citeref 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,வார்ப்புரு:Citeref Kent E. Erickson proposed an asterisk (∗) to symbolize the operator in 1959,வார்ப்புரு:Citeref whilst Richard James Duffin and William Niles Anderson, Jr. used a colon (:) for the parallel addition since 1966.வார்ப்புரு:Citeref Sujit Kumar Mitra used a middot (∙) for it in 1970.வார்ப்புரு:Citeref The first usage of the parallel symbol (∥) for this operator in applied electronics is unknown, but might have originated from d (d) and Bruce D. Wedlock's 1974 book "Electronic Circuits and Applications",வார்ப்புரு:Citeref 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",வார்ப்புரு:Citeref which grew out of an identically-named MIT course developed as part of the influential வார்ப்புரு:Citeref 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.

[Duffin_1971-1] "Network Models". written at Durham, North Carolina, USA. 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. III of SIAM-AMS Proceedings (illustrated ). Providence, Rhode Island: American Mathematical Society (AMS) / Society for Industrial and Applied Mathematics (SIAM). 1971. பக். 65–92 [68]. ISBN 978-0-8218-1322-5. Report 69-21. பன்னாட்டுத் தரப்புத்தக எண்:0-8218-1322-6. https://books.google.com/books?id=j2vFhxA5K-UC&pg=PA65. பார்த்த நாள்: 2019-08-05. "[…] 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)

[Kersey_1673-2] "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. Book IV - The Elements of the Algebraical Arts. London: Thomas Passinger, Three-Bibles, London-Bridge. 1673. பக். 177–178. https://archive.org/details/b30333726_0002. பார்த்த நாள்: 2019-08-09.

[Cajori_1928-3] "§ 184, § 359, § 368". A History of Mathematical Notations – Notations in Elementary Mathematics. 1 (two volumes in one unaltered reprint ). Chicago, US: Open court publishing company. 1993. பக். 193, 402–403, 411–412. பன்னாட்டுத் தரப்புத்தக எண்:0-486-67766-4. https://archive.org/details/historyofmathema00cajo_0/page/193. பார்த்த நாள்: 2019-07-22. "§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,வார்ப்புரு:Citeref Caswell, Jones,வார்ப்புரு:Citeref Wilson,வார்ப்புரு:Citeref Emerson,வார்ப்புரு:Citeref Kambly,வார்ப்புரு:Citeref 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 Stevensவார்ப்புரு:Citeref use "parவார்ப்புரு:Citeref or ∥" for parallel […] [A] John Kersey, வார்ப்புரு:Citeref (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|Deutsch (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]

[TI_2004-4] "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. பார்க்கப்பட்ட நாள் 2019-07-13.

[Bober_2016-5] "Chapter 7.6. Laplace Transforms Applied to Circuits". Numerical and Analytical Methods with MATLAB for Electrical Engineers. Applied and Computational Mechanics (1 ). CRC Press. 2016. பக். 224. ISBN 1-46657607-3. பன்னாட்டுத் தரப்புத்தக எண்:978-1-46657607-0. https://books.google.com/books?id=O2POBQAAQBAJ&pg=PA224. (388 pages)

[EECS_2018-6] "Chapter 15.7.2 Parallel Resistors". EECS 16A Designing Information Devices and Systems I (lecture notes). University of California, Berkeley. Fall 2018. பக். 12. Note 15. https://inst.eecs.berkeley.edu/~ee16a/fa18/lectures/Note15.pdf#page=12. பார்த்த நாள்: 2018-12-28. "[…] 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_1995-8] "Chapter 12: Parallel Addition, Series-Parallel Duality, and Financial Mathematics". Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics. G – Reference, Information and Interdisciplinary Subjects Series (illustrated ). Rowman & Littlefield Publishers, Inc.. 1995-03-21. பக். 237–268. பன்னாட்டுத் தரப்புத்தக எண்:0-8476-7932-2. https://books.google.com/books?id=NgJqXXk7zAAC&pg=PA237. பார்த்த நாள்: 2019-08-09. "[…] 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_2004-9] 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. பார்க்கப்பட்ட நாள் 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)

[1]

[2]

[3]

[4]

[5]

[6]

[nb 1]

[7]

[8]