TensorFlowとは?特徴やメリットと活用事例を解説 最終更新日:2024/02/07 0017132人目の素数さん2024/04/12(金) 14:53:18.48ID:aptMDkCS これいいね https://math.stackexchange.com/questions/2030558/what-is-the-history-of-the-term-tensor What is the history of the term "tensor"? asked Nov 25, 2016 at 19:08 Yahya Abdal-Aziz 0018132人目の素数さん2024/04/12(金) 15:08:15.82ID:l/8lry6C マラテンソルってなんでしょうか? 0019132人目の素数さん2024/04/13(土) 05:31:46.65ID:QqlPnDNV tension fieldの強さを測るものではないか 0020132人目の素数さん2024/04/13(土) 19:08:18.13ID:NLkm48qb 物理と微分幾何で扱うテンソルは同じ。 微分幾何のテンソルはベクトル空間のテンソルで代数は可換環のテンソルで気にする点がちょっと違う。 0021132人目の素数さん2024/04/13(土) 20:42:12.89ID:AkaTH9ql ありがとうございます そういえば、雪江 代数学3に”テンソル代数”がありましたね (私も書棚のこやしですが)
https://kconrad.math.uconn.edu/blurbs/ Expository papers KEITH CONRAD Linear/Multilinear algebra (下記以外にTensor products II、Fields and Galois theoryでSeparable extensions and tensor products、Splitting fields and tensor products のpdfがあります) https://kconrad.math.uconn.edu/blurbs/linmultialg/tensorprod.pdf TENSOR PRODUCTS I KEITH CONRAD P2 Here is a brief history of tensors and tensor products. Tensor comes from the Latin tendere, which means “to stretch.” In 1822 Cauchy introduced the Cauchy stress tensor in continuum mechanics, and in 1861 Riemann created the Riemann curvature tensor in geometry, but they did not use those names. In 1884, Gibbs [7, Chap. 3] introduced tensor products of vectors in R3 with the label “indeterminate product”*3 and applied it to study strain on a body. He extended the indeterminate product to n dimensions in 1886 [8]. Voigt used tensors to describe stress and strain on crystals in 1898 [25], and the term tensor first appeared with its modern physical meaning there.*4 In geometry Ricci used tensors in the late 1800s and his 1901 paper [22] with Levi-Civita (in English in [15]) was crucial in Einstein’s work on general relativity. Wide use of the term “tensor” in physics and math is due to Einstein; Ricci and Levi-Civita called tensors by the bland name “systems”. The notation ⊗ is due to Murray and von Neumann in 1936 [17, Chap. II] for tensor products (they wrote “direct products”) of Hilbert spaces.*5 The tensor product of abelian groups A and B, with that name but written as A◦B instead of A⊗Z B, is due to Whitney [27] in 1938. Tensor products of modules over a commutative ring are due to Bourbaki [2] in 1948.
脚注 *3 Gibbs chose that label since this product was, in his words, “the most general form of product of two vectors,” as it is subject to no laws except bilinearity, which must be satisfied by any operation deserving to be called a product. In 1844, Grassmann created a special tensor called an “open product” [20, Chap. 3].
*4 Writing i, j, and k for the standard basis of R3, Gibbs called a sum ai⊗i+bj⊗j+ck⊗k with positive a, b, and c a right tensor [7, p. 57], but I don’t know if this had an influence on Voigt’s terminology.
*5 I thank Jim Casey for bringing [17] to my attention. 0026132人目の素数さん2024/04/14(日) 11:33:32.20ID:g/SCaNYS>>25
要約すると ・1822 Cauchy introduced the Cauchy stress tensor in continuum mechanics ・1861 Riemann created the Riemann curvature tensor in geometry, but they did not use those names( tensor使わず) ・1884, Gibbs [7, Chap. 3] introduced tensor products of vectors in R3 with the label “indeterminate product”*3 and applied it to study strain on a body. He extended the indeterminate product to n dimensions in 1886 [8]. ・1898 Voigt used tensors to describe stress and strain on crystals ・1800s and 1901 In geometry Ricci used tensors in the late 1800s and his 1901 paper [22] with Levi-Civita (in English in [15]) was crucial in Einstein’s work on general relativity. ・1913 Wide use of the term “tensor” in physics and math is due to Einstein; Ricci and Levi-Civita called tensors by the bland name “systems”. (この後の年表略す)
(参考) https://en.wikipedia.org/wiki/General_relativity A version of non-Euclidean geometry, called Riemannian geometry, enabled Einstein to develop general relativity by providing the key mathematical framework on which he fit his physical ideas of gravity.[6] This idea was pointed out by mathematician Marcel Grossmann and published by Grossmann and Einstein in 1913.[7]
7 Grossmann for the mathematical part and Einstein for the physical part (1913). Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation (Outline of a Generalized Theory of Relativity and of a Theory of Gravitation), Zeitschrift für Mathematik und Physik, 62, 225–261. English translate https://sites.pitt.edu/~jdnorton/teaching/GR&Grav_2007/pdf/Einstein_Entwurf_1913.pdf 0027132人目の素数さん2024/04/14(日) 15:40:44.67ID:g/SCaNYS Woldemar Voigtさん、2 September 1850 – 13 December 1919 German physicist(下記) 特殊相対性理論と関係しているのか だから、アインシュタインは Voigt notation つまり、用語"tensor"に詳しいのですね
https://en.wikipedia.org/wiki/Voigt_notation Voigt notation Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order.[1] There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig[2] of old ideas of Lord Kelvin. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application.
Applications The notation is named after physicist Woldemar Voigt & John Nye (scientist). It is useful, for example, in calculations involving constitutive models to simulate materials, such as the generalized Hooke's law, as well as finite element analysis,[4] and Diffusion MRI.[5]
https://en.wikipedia.org/wiki/Woldemar_Voigt Woldemar Voigt Woldemar Voigt (German: [foːkt] ⓘ; 2 September 1850 – 13 December 1919) was a German physicist. Voigt transformation Further information: History of Lorentz transformations
https://en.wikipedia.org/wiki/History_of_Lorentz_transformations History of Lorentz transformations Electrodynamics and special relativity Overview In physics, analogous transformations have been introduced by Voigt (1887) related to an incompressible medium, and by Heaviside (1888), Thomson (1889), Searle (1896) and Lorentz (1892, 1895) who analyzed Maxwell's equations. They were completed by Larmor (1897, 1900) and Lorentz (1899, 1904), and brought into their modern form by Poincaré (1905) who gave the transformation the name of Lorentz.[3] Eventually, Einstein (1905) showed in his development of special relativity that the transformations follow from the principle of relativity and constant light speed alone by modifying the traditional concepts of space and time, without requiring a mechanical aether in contradistinction to Lorentz and Poincaré.[4] Minkowski (1907–1908) used them to argue that space and time are inseparably connected as spacetime. Voigt (1887) Woldemar Voigt (1887)[R 1] developed a transformation in connection with the Doppler effect and an incompressible medium, being in modern notation:[5][6] 0028132人目の素数さん2024/04/14(日) 17:09:42.29ID:g/SCaNYS これも参考に貼っておきます
2 Answers Sorted by: 18 I think the OP referes to the modern meaning of the word, in which case, according to that website, it first appeared in german physicist Woldemar Voigt's paper Die fundamentalen physikalischen Eigenschaften der Krystalle in elementarer Darstellung published in 1898. (I do not have access to this paper, but probably this deals with deformation tensors in crystals). answered Sep 18, 2011 at 17:21 Thomas Sauvaget 3 This is the right answer. Voigt used the word tensor to describe stress and strain (i.e., things that stretched). In mechanics there is the stress tensor, strain tensor, and elasticity tensor. More precisely, it seems that what interested Voigt were special cases of symmetric tensors. See en.wikipedia.org/wiki/Woldemar_Voigt and en.wikipedia.org/wiki/Voigt_notation. – KConrad Sep 18, 2011 at 20:09
11 A tensor muscle is a muscle that stretches some part of the body, e.g. the tensor veli palatini or tensor tympani. The word ultimately derives from the Latin tendere meaning "to stretch", see Douglas Harper's etymonline. Hamilton first introduced the term to mathematics; see its entry in "Earliest Known Uses of Some of the Words of Mathematics". http://jeff560.tripod.com/t.html answered Sep 18, 2011 at 17:14 Charles 2 Hamilton's usage has been abandoned, however, and the word means something entirely different now. – Ryan Reich Sep 19, 2011 at 5:48 0029132人目の素数さん2024/04/14(日) 17:12:49.95ID:g/SCaNYS 追加 https://jeff560.tripod.com/t.html Earliest Known Uses of Some of the Words of Mathematics (T) Last revision: Apr. 13, 2019
TENSOR was one of the family of terms introduced by William Rowan Hamilton (1805-1865) in his study of QUATERNIONS. VECTOR and SCALAR and VERSOR were among the others. The tensor is for quaternions what the MODULUS is for complex numbers. The term derives from the Latin tendĕre to stretch. In 1846 Hamilton wrote in The London, Edinburgh, and Dublin Philosophical Magazine XXIX. 27: Since the square of a scalar is always positive, while the square of a vector is always negative, the algebraical excess of the former over the latter square is always a positive number; if then we make (TQ)2 = (SQ)2 — (VQ)2, and if we suppose TQ to be always a real and positive or absolute number, which we may call the tensor of the quaternion Q, we shall not thereby diminish the generality of that quaternion. This tensor is what was called in former articles the modulus. The passage is reproduced in Section 19 of “On Quaternions”. This ‘article’ is a compilation of 18 short papers published in the Philosophical Magazine between 1844 and 1850 made by the editors of Hamilton’s Mathematical Papers. The editors concatenated them to form a seamless whole, with no indication as to how the material was distributed into the individual papers. Tensor in Hamilton’s sense is no longer used. [Information for this article was provided by David Wilkins and Julio González Cabillón.]
TENSOR, TENSOR ANALYSIS, TENSOR CALCULUS, etc. are 20th century terms associated with the ABSOLUTE DIFFERENTIAL CALCULUS developed by Ricci-Curbastroin the 1880s and -90s on the basis of earlier work by Riemann, Christoffel, Bianchi and others. See Kline ch. 37 “The Differential Geometry of Gauss and Riemann” and ch. 48 “Tensor Analysis and Differential Geometry.” Ricci’s most influential publication was a substantial article written with his former student Levi-Civita. The article by the two Italians was written in French and appeared in the leading German mathematical journal: “Méthodes de calcul différentiel absolu et leurs applications,” Mathematische Annalen, 54 (1901), p. 125-201. The word tensor does not appear: Ricci and Levi-Civita write about systèmes. Tensor is due to the well-known Goettingen physicist Woldemar Voigt (1850-1919), who used it in his Die fundamentalen physikalischen Eigenschaften der Krystalle in elementarer Darstellung of 1898 (OED and Julio González Cabillón). Critically tensor was the term adopted by Einstein and Grossmann in their first publication on general relativity, Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation (Outline of a Generalized Theory of Relativity and of a Theory of Gravitation) (1913). Einstein made the subject fashionable. MacTutor relates that on a visit to Princeton in 1921 he commented on the large audience his lecture attracted: “I never realised that so many Americans were interested in tensor analysis.” See also MacTutor: General Relativity. The OED reports that tensor analysis is found in English in 1922 in H. L. Brose’s translation of Weyl’s Space-Time-Matter (Raum, Zeit, Materie): “Tensor analysis tells us how, by differentiating with respect to the space co-ordinates, a new tensor can be derived from the old one in a manner entirely independent of the co-ordinate system. This method, like tensor algebra, is of extreme simplicity.” The phrase tensor calculus appears in the same book. When Levi-Civita’s Lezioni di calcolo differenziale assoluto was translated into English in 1926, its title included an explanation: The Absolute Differential Calculus (Calculus of Tensors). [This entry was contributed by John Aldrich.] (引用終り) 以上 0031132人目の素数さん2024/04/14(日) 17:28:59.93ID:g/SCaNYS なので、ハミルトンの話を入れて、>>26を修正しておきます
・1822 Cauchy introduced the Cauchy stress tensor in continuum mechanics ・1846 ハミルトンはロンドン、エディンバラ、ダブリンの哲学雑誌XXIX の寄稿で 用語群で、VECTOR、SCALAR、VERSORなどと共に、用語tensorを導入したが その意味は、四元数の研究に特化した用語で、その後使われなくなった ・1861 Riemann created the Riemann curvature tensor in geometry, but they did not use those names( 用語tensor使わず) ・1884, Gibbs [7, Chap. 3] introduced tensor products of vectors in R3 with the label “indeterminate product”*3 and applied it to study strain on a body. He extended the indeterminate product to n dimensions in 1886 [8]. ・1898 Voigt used tensors to describe stress and strain on crystals ・1800s and 1901 In geometry Ricci used tensors in the late 1800s and his 1901 paper [22] with Levi-Civita (in English in [15]) was crucial in Einstein’s work on general relativity. ・1913 Wide use of the term “tensor” in physics and math is due to Einstein; Ricci and Levi-Civita called tensors by the bland name “systems”. (この後の年表略す)
要するに 最初は 1822 Cauchy stress tensor (但し、用語tensor使わず) 1846 ハミルトンが、四元数の研究で VECTOR、SCALAR、用語tensorを導入したが、用語tensorは現在とは別の意味でその後使われなくなった 1898 Voigt used tensors to describe stress and strain on crystals (これはCauchy stress tensorと同じ) 1913 Einsteinが一般相対性理論で Ricci and Levi-Civita called tensors を採用した が、実はRicci and Levi-Civita自身は 用語tensorを使っていなかった (ドイツの物理学者 Voigtの特殊相対性理論への貢献があって、Einsteinは用語tensorの使用を思いついたのでしょう) 0032132人目の素数さん2024/04/14(日) 17:59:08.67ID:raM0Fqyg 長文荒らしが来た 0033132人目の素数さん2024/04/14(日) 21:03:17.07ID:g/SCaNYS ぼく中学生? この程度で長文っていってたら、大学入試の国語問題は解けないよ 速読力を鍛えなさい! 0034132人目の素数さん2024/04/14(日) 21:25:05.66ID:N/pcrRF0 誰も読まない長文 0035132人目の素数さん2024/04/14(日) 22:12:29.23ID:g/SCaNYS だれも読まなくても、いいのです
https://en.wikipedia.org/wiki/A_History_of_Vector_Analysis A History of Vector Analysis 0039132人目の素数さん2024/04/15(月) 18:02:14.90ID:iZSyJwDR Tensorという雑誌は今もあるようだ 0040132人目の素数さん2024/04/15(月) 18:11:58.29ID:iZSyJwDR Tensor. New series the Tensor Society
English ed (1950)- 0041132人目の素数さん2024/04/15(月) 21:09:25.74ID:oySOdDfw>>39-40 ありがとうございます そういうウンチクを書けるのは、御大かな
【次ページ】翻訳技術の仕組み、「言葉をベクトル化する」とは? 0042132人目の素数さん2024/04/15(月) 21:16:53.52ID:BCQUcGPL The Journal of the Tensor Society (JTS) is the official organ of The Tensor Society and publishes original research articles in differential geometry, relativity, cosmology, and all interdisciplinary areas in mathematics that utilize differential geometric methods and structures. The following main areas are covered: differentiable manifolds, Finsler geometry, Lie groups, local and global differential geometry, General Relativity, and geometric theories of gravitation; cosmology, dark energy, dark matter, the accelerating universe, geometric models for particle physics; supergravity and supersymmetric field theories; classical and quantum field theory; gauge theories; topological field theories; and the geometry of chaos. In addition to original research, the Journal of the Tensor Society also publishes focused review articles that assess the state of the art, identify upcoming challenges, and propose promising solutions for the community. 0043132人目の素数さん2024/04/15(月) 22:59:10.56ID:oySOdDfw>>42 ありがとうございます 0044132人目の素数さん2024/04/16(火) 08:48:03.85ID:h9QdmK4e Uses of Killing and Killing-Yano Tensors Ulf Lindström, Özgür Sarıoğlu 0045132人目の素数さん2024/04/21(日) 17:27:00.60ID:LRSTOrnW けつも かおも あなるもそりませう アヌステンソルですうう 0046132人目の素数さん2024/04/21(日) 19:22:47.71ID:+2zd27AU 面白い ザブトン一枚 0047132人目の素数さん2024/04/21(日) 21:59:42.38ID:+2zd27AU AIのテンソル
概要 機械学習や数値解析、ニューラルネットワーク(ディープラーニング)に対応しており、GoogleとDeepMindの各種サービスなどでも広く活用されている。 0048132人目の素数さん2024/04/21(日) 21:59:59.41ID:+2zd27AUhttps://en.wikipedia.org/wiki/TensorFlow TensorFlow is a free and open-source software library for machine learning and artificial intelligence. It can be used across a range of tasks but has a particular focus on training and inference of deep neural networks.[3][4] 0049132人目の素数さん2024/04/26(金) 00:31:18.07ID:Nnj4aAHS 同じ数学といっても線形代数と微分幾何じゃベクトルの意味が違う 同類だけど 0050132人目の素数さん2024/04/26(金) 17:14:26.75ID:em70EpiX こちらにも転載しておきます ”ベクトルの概念を数学と物理で捕らえ方が異なる点を指摘して注意を喚起してくれた本です”