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 வடிவவியல் 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 தொடர் இணைப்பு மற்றும் பக்க இணைப்பு மின்வழிகள்s originates from network theory in மின்பொறியியல். Sundaram Seshu introduced a reduced sum operator in 1956,^[C] Kent E. Erickson proposed an உடுக்குறி (∗) to symbolize the operator in 1959,^[D] whilst Richard James Duffin and William Niles Anderson, Jr. used a முக்கால்புள்ளி (:) 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 d (d) and Bruce D. Wedlock's 1974 book "Electronic Circuits and Applications",^[G] which evolved from their introductory electronics course at மாசாச்சூசெட்சு தொழில்நுட்பக் கழகம் (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 பெருக்கல் (கணிதம்) and division in some contexts.

மேற்கோள்கள்[தொகு]

↑ 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]. பன்னாட்டுத் தரப்புத்தக எண் 0-8218-1322-6. பன்னாட்டுத் தர தொடர் எண் 0080-5084. LCCN 79-167683. பன்னாட்டுத் தரப்புத்தக எண் 978-0-8218-1322-5. Report 69-21. பார்க்கப்பட்ட நாள் 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)
↑ 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. பார்க்கப்பட்ட நாள் 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. பன்னாட்டுத் தரப்புத்தக எண் 0-486-67766-4. LCCN 93-29211. பார்க்கப்பட்ட நாள் 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|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.
↑ 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. பன்னாட்டுத் தரப்புத்தக எண் 978-1-46657607-0. பன்னாட்டுத் தரப்புத்தக எண் 1-46657607-3. (388 pages)
↑ 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). கலிபோர்னியா பல்கலைக்கழகம் (பெர்க்லி). p. 12. Note 15. Archived (PDF) from the original on 2018-12-27. பார்க்கப்பட்ட நாள் 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). G – Reference, Information and Interdisciplinary Subjects Series (illustrated ed.). Rowman & Littlefield Publishers, Inc. pp. 237–268. பன்னாட்டுத் தரப்புத்தக எண் 0-8476-7932-2. Archived (PDF) from the original on 2016-03-05. பார்க்கப்பட்ட நாள் 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 கூட்டல்) 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 முக்கால்புள்ளி […] {{cite book}}: |work= ignored (help) [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)

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 வடிவவியல் 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 தொடர் இணைப்பு மற்றும் பக்க இணைப்பு மின்வழிகள்s originates from network theory in மின்பொறியியல். Sundaram Seshu introduced a reduced sum operator in 1956,^[C] Kent E. Erickson proposed an உடுக்குறி (∗) to symbolize the operator in 1959,^[D] whilst Richard James Duffin and William Niles Anderson, Jr. used a முக்கால்புள்ளி (:) 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 d (d) and Bruce D. Wedlock's 1974 book "Electronic Circuits and Applications",^[G] which evolved from their introductory electronics course at மாசாச்சூசெட்சு தொழில்நுட்பக் கழகம் (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 பெருக்கல் (கணிதம்) and division in some contexts.

[Duffin_1971-1] 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]. பன்னாட்டுத் தரப்புத்தக எண் 0-8218-1322-6. பன்னாட்டுத் தர தொடர் எண் 0080-5084. LCCN 79-167683. பன்னாட்டுத் தரப்புத்தக எண் 978-0-8218-1322-5. Report 69-21. பார்க்கப்பட்ட நாள் 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)

[Kersey_1673-2] 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. பார்க்கப்பட்ட நாள் 2019-08-09.

[Cajori_1928-3] 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. பன்னாட்டுத் தரப்புத்தக எண் 0-486-67766-4. LCCN 93-29211. பார்க்கப்பட்ட நாள் 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|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] 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. பன்னாட்டுத் தரப்புத்தக எண் 978-1-46657607-0. பன்னாட்டுத் தரப்புத்தக எண் 1-46657607-3. (388 pages)

[EECS_2018-6] 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). கலிபோர்னியா பல்கலைக்கழகம் (பெர்க்லி). p. 12. Note 15. Archived (PDF) from the original on 2018-12-27. பார்க்கப்பட்ட நாள் 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_1995-8] 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). G – Reference, Information and Interdisciplinary Subjects Series (illustrated ed.). Rowman & Littlefield Publishers, Inc. pp. 237–268. பன்னாட்டுத் தரப்புத்தக எண் 0-8476-7932-2. Archived (PDF) from the original on 2016-03-05. பார்க்கப்பட்ட நாள் 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 கூட்டல்) 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 முக்கால்புள்ளி […] {{cite book}}: |work= ignored (help) [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]