source
stringlengths 16
98
| text
stringlengths 40
168k
|
|---|---|
Wikipedia:The Nine Chapters on the Mathematical Art#0
|
The Nine Chapters on the Mathematical Art is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 1st century CE. This book is one of the earliest surviving mathematical texts from China, the others being the Suan shu shu (202 BCE – 186 BCE) and Zhoubi Suanjing (compiled throughout the Han until the late 2nd century CE). It lays out an approach to mathematics that centres on finding the most general methods of solving problems, which may be contrasted with the approach common to ancient Greek mathematicians, who tended to deduce propositions from an initial set of axioms. Entries in the book usually take the form of a statement of a problem, followed by the statement of the solution and an explanation of the procedure that led to the solution. These were commented on by Liu Hui in the 3rd century. The book was later included in the early Tang collection, the Ten Computational Canons. == History == === Original book === The full title of The Nine Chapters on the Mathematical Art appears on two bronze standard measures which are dated to 179 CE, but there is speculation that the same book existed beforehand under different titles. The title is also mentioned in volume 24 of the Book of the Later Han as one of the books studied by Ma Xu (馬續). Based on this known knowledge, his younger brother Ma Rong (馬融) places the date of composition to no later than 93 CE. Most scholars believe that Chinese mathematics and the mathematics of the ancient Mediterranean world had developed more or less independently up to the time when The Nine Chapters reached its final form. The method of chapter 7 was not found in Europe until the 13th century, and the method of chapter 8 uses Gaussian elimination before Carl Friedrich Gauss (1777–1855). There is also the mathematical proof given in the treatise for the Pythagorean theorem. The influence of The Nine Chapters greatly assisted the development of ancient mathematics in the regions of Korea and Japan. Its influence on mathematical thought in China persisted until the Qing dynasty era. Liu Hui wrote a detailed commentary in 263. He analyses the procedures of The Nine Chapters step by step, in a manner which is clearly designed to give the reader confidence that they are reliable, although he is not concerned to provide formal proofs in the Euclidean manner. Liu's commentary is of great mathematical interest in its own right. Liu credits the earlier mathematicians Zhang Cang (fl. 165 BCE – d. 142 BCE) and Geng Shouchang (fl. 75 BCE – 49 BCE) (see armillary sphere) with the initial arrangement and commentary on the book, yet Han dynasty records do not indicate the names of any authors of commentary, as they are not mentioned until the 3rd century The Nine Chapters is an anonymous work, and its origins are not clear. Until recent years, there was no substantial evidence of related mathematical writing that might have preceded it, with the exception of mathematical work by those such as Jing Fang (78–37 BCE), Liu Xin (d. 23), and Zhang Heng (78–139) and the geometry clauses of the Mozi of the 4th century BCE. This is no longer the case. The Suàn shù shū (算數書) or Writings on Reckonings is an ancient Chinese text on mathematics approximately seven thousand characters in length, written on 190 bamboo strips. It was discovered together with other writings in 1983 when archaeologists opened a tomb in Hubei province. It is among the corpus of texts known as the Zhangjiashan Han bamboo texts. From documentary evidence this tomb is known to have been closed in 186 BCE, early in the Western Han dynasty. While its relationship to The Nine Chapters is still under discussion by scholars, some of its contents are clearly paralleled there. The text of the Suàn shù shū is however much less systematic than The Nine Chapters; and appears to consist of a number of more or less independent short sections of text drawn from a number of sources. The Zhoubi Suanjing, a mathematics and astronomy text, was also compiled during the Han, and was even mentioned as a school of mathematics in and around 180 CE by Cai Yong. === Western translations === The title of the book has been translated in a wide variety of ways. In 1852, Alexander Wylie referred to it as Arithmetical Rules of the Nine Sections. With only a slight variation, the Japanese historian of mathematics Yoshio Mikami shortened the title to Arithmetic in Nine Sections. David Eugene Smith, in his History of Mathematics (Smith 1923), followed the convention used by Yoshio Mikami. Several years later, George Sarton took note of the book, but only with limited attention and only mentioning the usage of red and black rods for positive and negative numbers. In 1959, Joseph Needham and Wang Ling (historian) translated Jiu Zhang Suan shu as The Nine Chapters on the Mathematical Art for the first time. Later in 1994, Lam Lay Yong used this title in her overview of the book, as did other mathematicians including John N. Crossley and Anthony W.-C Lun in their translation of Li Yan and Du Shiran's Chinese Mathematics: A Concise History (Li and Du 1987). Afterwards, the name The Nine Chapters on the Mathematical Art stuck and became the standard English title for the book. == Table of contents == Contents of The Nine Chapters are as follows: == Major contributions == === Real number system === The Nine Chapters on the Mathematical Art does not discuss natural numbers, that is, positive integers and their operations, but they are widely used and written on the basis of natural numbers. Although it is not a book on fractions, the meaning, nature, and four operations of fractions are fully discussed. For example: combined division (addition), subtraction (subtraction), multiplication (multiplication), warp division (division), division (comparison size), reduction (simplified fraction), and bisector (average). The concept of negative numbers also appears in "Nine Chapters of Arithmetic". In order to cooperate with the algorithm of equations, the rules of addition and subtraction of positive and negative numbers are given. The subtraction is "divide by the same name, benefit by different names. The addition is "divide by different names, benefit from each other by the same name. Among them, "division" is subtraction, "benefit" is addition, and "no entry" means that there is no counter-party, but multiplication and division are not recorded. The Nine Chapters on the Mathematical Art gives a certain discussion on natural numbers, fractions, positive and negative numbers, and some special irrationality. Generally speaking, it has the prototype of the real number system used in modern mathematics. === Gou Gu (Pythagorean) Theorem === The geometric figures included in The Nine Chapters on the Mathematical Art are mostly straight and circular figures because of its focus on the applications onto the agricultural fields. In addition, due to the needs of civil architecture, The Nine Chapters on the Mathematical Art also discusses volumetric algorithms of linear and circular 3 dimensional solids. The arrangement of these volumetric algorithms ranges from simple to complex, forming a unique mathematical system. Regarding the direct application of the Gou Gu Theorem, which is precisely the Chinese version of the Pythagorean Theorem, the book divides it into four main categories: Gou Gu mutual seeking, Gou Gu integer, Gou Gu dual capacity, Gou Gu similar. Gou Gu mutual seeking discusses the algorithm of finding the length of a side of the right triangle while knowing the other two. Gou Gu integer is precisely the finding of some significant integer Pythagorean numbers, including famously the triple 3,4,5. Gou Gu dual capacity discusses algorithms for calculating the areas of the inscribed rectangles and other polygons in the circle, which also serves an algorithm to calculate the value of pi. Lastly, Gou Gu similars provide algorithms of calculating heights and lengths of buildings on the mathematical basis of similar right triangles. === Completing of squares and solutions of system of equations === The methods of completing the squares and cubes as well as solving simultaneous linear equations listed in The Nine Chapters on the Mathematical Art can be regarded one of the major content of ancient Chinese mathematics. The discussion of these algorithms in The Nine Chapters on the Mathematical Art are very detailed. Through these discussions, one can understand the achievements of the development of ancient Chinese mathematics. Completing the squaring and cubes can not only solve systems of two linear equations with two unknowns, but also general quadratic and cubic equations. It is the basis for solving higher-order equations in ancient China, and it also plays an important role in the development of mathematics. The "equations" discussed in the Fang Cheng chapter are equivalent to today's simultaneous linear equations. The solution method called "Fang Cheng Shi" is best known today as Gaussian elimination. Among the eighteen problems listed in the Fang Cheng chapter, some are equivalent to simultaneous linear equations with two unknowns, some are equivalent to simultaneous linear equations with 3 unknowns, and the most complex example analyzes the solution to a system of linear equations with up to 5 unknowns. == Significance == The word jiu, or "9", means more than just a digit in ancient Chinese. In fact, since it is the largest digit, it often refers to something of a grand scale or a supreme authority. Further, the word zhang, or "chapter", also has more connotations than simply being the "chapter". It may refer to a section, several parts of an article, or an entire treatise. In this light, many scholars of the history of Chinese mathematics compare the significance of The Nine Chapters on the Mathematical Art on the development of Eastern mathematical traditions to that of Euclid's Elements on the Western mathematical traditions. However, the influence of The Nine Chapters on the Mathematical Art stops short at the advancement of modern mathematics due to its focus on practical problems and inductive proof methods as opposed to the deductive, axiomatic tradition that Euclid's Elements establishes. However, it is dismissive to say that The Nine Chapters on the Mathematical Art has no impact at all on modern mathematics. The style and structure of The Nine Chapters on the Mathematical Art can be best concluded as "problem, formula, and computation". This process of solving applied mathematical problems can now be considered the standard approach in the field of applied mathematics. == Notable translations == Abridged English translation: Yoshio Mikami: "Arithmetic in Nine Sections", in The Development of Mathematics in China and Japan, 1913. Highly Abridged English translation: Florian Cajori: "Arithmetic in Nine Sections", in A History of Mathematics, Second Edition, 1919 (possibly copied or paraphrased from Mikami). Abridged English translation: Lam Lay Yong: Jiu Zhang Suanshu: An Overview, Archive for History of Exact Sciences, Springer Verlag, 1994. A full translation and study of the Nine Chapters and Liu Hui's commentary is available in Kangshen Shen, The Nine Chapters on the Mathematical Art, Oxford University Press, 1999. ISBN 0-19-853936-3 A French translation with detailed scholarly addenda and a critical edition of the Chinese text of both the book and its commentary by Karine Chemla and Shuchun Guo is Les neuf chapitres: le classique mathématique de la Chine ancienne et ses commentaires. Paris: Dunod, 2004. ISBN 978-2-10-049589-4. German translation: Kurt Vogel, Neun Bücher Arithmetischer Technik, Friedrich Vieweg und Sohn Braunsweig, 1968. Russian translation: E. I Beriozkina, Математика в девяти книгах (Mathematika V Devyati Knigah), Moscow: GITTL, 1957. == See also == Haidao Suanjing History of mathematics History of geometry == References == === Bibliography === == External links == Full text of the book (Chinese)
|
Wikipedia:The Quarterly Journal of Pure and Applied Mathematics#0
|
The Quarterly Journal of Pure and Applied Mathematics was a mathematics journal that first appeared as such in 1855, but as the continuation of The Cambridge Mathematical Journal that had been launched in 1836 and had run in four volumes before changing its title to The Cambridge and Dublin Mathematical Journal for a further nine volumes (these latter volumes carried dual numbering). Papers in the first issue, which carried a preface dated April, 1855, and promised further issues on a quarterly schedule in June, September, December and March, have dates going back to November, 1854; the first volume carried a further preface dated January, 1857. From the outset, keeping the journal up and running was to prove a challenging task. It was edited under the new title by James Joseph Sylvester and Norman Macleod Ferrers, assisted by George G. Stokes and Arthur Cayley, with Charles Hermite as corresponding editor in Paris, an arrangement that remained stable for the first fifteen volumes. With the sixteenth volume in 1879, the new editorial line-up was N. M. Ferrers, A. Cayley, and J. W. L. Glaisher. Andrew Forsyth was recruited to the board for the twentieth volume in 1885. However, the following decade saw an attrition in the editorship in short succession: Ferrers continued up to the twenty-fifth volume in 1891; Cayley the following volume in 1893; and Forsyth two volumes later, in 1895. Thus, by the twenty-eighth volume in 1896, Glaisher, who had edited Messenger of Mathematics single-handedly since its inception in May, 1870, was also left as the sole editor of the Quarterly. It came to be felt that these periodicals had become so identified with Glaisher that it would be awkward to attempt to continue them after his death (in 1928). In the mid-1920s, this led G. H. Hardy to push for two new titles, the Journal of the London Mathematical Society and The Quarterly Journal (Oxford Series); Hardy was then secretary of the London Mathematical Society and Savilian Professor of Geometry at the University of Oxford. == References == == External links == Cambridge Mathematical Journal digitized by GDZ (volumes 1 to 4). Cambridge and Dublin Mathematical Journal digitized by GDZ (volumes 5 to 13 of the Cambridge Mathematical Journal, or 1 to 9 of the Cambridge and Dublin Mathematical Journal). Cambridge and Dublin Mathematical Journal digitized by HathiTrust (volumes 1-13) The Quarterly Journal of Pure and Applied Mathematics digitized by GDZ (volumes 1 (1857) to 31 (1900) only) The Quarterly Journal of Pure and Applied Mathematics digitized by HathiTrust (volumes 1-50) Messenger of Mathematics digitized by GDZ (volumes 1 (1871) to 30 (1901); published on May to April schedule, with title page of bound volumes bearing date of later year, so vol. 30 runs from May, 1900 to April, 1901).
|
Wikipedia:Theodorus Dekker#0
|
Theodorus Jozef Dekker (Dirk Dekker, 1 March 1927 - 25 November 2021) was a Dutch mathematician. Dekker completed his Ph.D. degree from the University of Amsterdam in 1958. His thesis was titled "Paradoxical Decompositions of Sets and Spaces". Dekker invented an algorithm that allows two processes to share a single-use resource without conflict, using only shared memory for communication, named Dekker's algorithm. == References == Prof. dr. T.J. Dekker, 1927 - at the University of Amsterdam Album Academicum website == External links == Theodorus (Dirk) Jozef Dekker at the Mathematics Genealogy Project
|
Wikipedia:Theodosius' Spherics#0
|
The Spherics (Greek: τὰ σφαιρικά, tà sphairiká) is a three-volume treatise on spherical geometry written by the Hellenistic mathematician Theodosius of Bithynia in the 2nd or 1st century BC. Book I and the first half of Book II establish basic geometric constructions needed for spherical geometry using the tools of Euclidean solid geometry, while the second half of Book II and Book III contain propositions relevant to astronomy as modeled by the celestial sphere. Primarily consisting of theorems which were known at least informally a couple centuries earlier, the Spherics was a foundational treatise for geometers and astronomers from its origin until the 19th century. It was continuously studied and copied in Greek manuscript for more than a millennium. It was translated into Arabic in the 9th century during the Islamic Golden Age, and thence translated into Latin in 12th century Iberia, though the text and diagrams were somewhat corrupted. In the 16th century printed editions in Greek were published along with better translations into Latin. == History == Several of the definitions and theorems in the Spherics were used without mention in Euclid's Phenomena and two extant works by Autolycus concerning motions of the celestial sphere, all written about two centuries before Theodosius. It has been speculated that this tradition of Greek "spherics" – founded in the axiomatic system and using the methods of proof of solid geometry exemplified by Euclid's Elements but extended with additional definitions relevant to the sphere – may have originated in a now-unknown work by Eudoxus, who probably established a two-sphere model of the cosmos (spherical Earth and celestial sphere) sometime between 370–340 BC. The Spherics is a supplement to the Elements, and takes its content for granted as a prerequisite. The Spherics follows the general presentation style of the Elements, with definitions followed by a list of theorems (propositions), each of which is first stated abstractly as prose, then restated with points lettered for the proof. It analyses spherical circles as flat circles lying in planes intersecting the sphere and provides geometric constructions for various configurations of spherical circles. Spherical distances and radii are treated as Euclidean distances in the surrounding 3-dimensional space. The relationship between planes is described in terms of dihedral angle. As in the Elements, there is no concept of angle measure or trigonometry per se. This approach differs from other quantitative methods of Greek astronomy such as the analemma (orthographic projection), stereographic projection, or trigonometry (a fledgling subject introduced by Theodosius' contemporary Hipparchus). It also differs from the approach taken in Menelaus' Spherics, a treatise of the same title written 3 centuries later, which treats the geometry of the sphere intrinsically, analyzing the inherent structure of the spherical surface and circles drawn on it rather than primarily treating it as a surface embedded in three-dimensional space. In late antiquity, the Spherics was part of a collection of treatises now called the Little Astronomy, an assortment of shorter works on geometry and astronomy building on Euclid's Elements. Other works in the collection included Aristarchus' On the Sizes and Distances, Autolycus' On Rising and Settings and On the Moving Sphere, Euclid's Catoptrics, Data, Optics, and Phenomena, Hypsicles' On Ascensions, Theodosius' On Geographic Places and On Days and Nights, and Menelaus' Spherics. Often several of these were bound together in a single volume. During the Islamic Golden Age, the books in the collection were translated into Arabic, and with the addition of a few new works, were known as the Middle Books, intended to fit between the Elements and Ptolemy's Almagest. Authoritative critical editions of the Greek text, compiled from several manuscripts, were made by Heiberg (1927) and Czinczenheim (2000). Sidoli & Thomas (2023) is an English translation by modern scholars. == Editions and translations == partial edition in: Valla, Giorgio, ed. (1501). "De sphaericis (book XII, chapter V)". De fugiendis et expetendis rebus (in Latin). Vol. 1. Venetiis: Aldus Manutius. Sphera mundi noviter recognita cum commentariis et authoribus in hoc volumine contentis, videlicet [...] Theodosii de Spheris [...] (in Latin). Venetiis. 1518. Vögelin, Johannes, ed. (1529). Theodosii de Sphaericis libri tres (in Latin). Vienna: Joannes Singrenius. Maurolico, Francesco, ed. (1558). Theodosii sphaericorum elementorum libri III, ex traditione Maurolyci Messanensis mathematici (in Latin). Messina: Petrus Spira mense Augusto. Péna, Jean, ed. (1558). Theodosij Tripolitae Sphaericorum, libri tres (in Greek and Latin). Paris: Andreas Wechelus. Dasypodius, Conrad, ed. (1573). Sphaericae doctrinae propositiones (in Latin and Greek). Argentorati: Excudebat Christianus Mylius. Clavius, Christopher, ed. (1586). Theodosii Tripolitae Sphaericorum Libri III (in Latin). Rome: Ex Typographia Dominici Basae. Henrion, Denis, ed. (1615). Les trois livres des Élémens spériques de Théodose Tripolitain (in French). Paris: Chez Abraham Pacard. Hérigone, Pierre, ed. (1637). "Theodosii Sphaerica = Sphériques de Théodose". Cursus mathematicus = Cours mathématique (in Latin and French). Vol. 5. Parisiis: Henry le Gras. pp. 218–329. Barrow, Isaac, ed. (1675). Theodosii Sphaerica: Methodo Nova Illustrata, & Succinctè Demonstrata (in Latin). London: Guil. Godbid. Hunt, Joseph, ed. (1707). Theodosiou Sphairikōn biblia 3. Theodosii Sphaericorum libri tres (in Greek and Latin). Oxford: H. Clements. Stone, Edmund, ed. (1721). Clavius's Commentary on the Sphericks of Theodosius Tripolitae: or, Spherical Elements. London: J. Senex. Nizze, Ernst, ed. (1826). Die Sphärik des Theodosios (in German). Stralsund. Nizze, Ernst, ed. (1852). Theodosii Tripolitae Sphaericorum Libros Tres (in Greek and Latin). Berlin: Georgii Reimer. Heiberg, Johan Ludvig, ed. (1927). Theodosius. Sphaerica (in Greek and Latin). Berlin: Weidmannsche Buchhandlung. Ver Eecke, Paul, ed. (1927). Les sphériques de Théodose de Tripoli (in French). Bruges: Brouwer et Cie. Czwalina, Arthur, ed. (1931). Autolykos: Rotierende kugel und Aufgang und untergang der gestirne. Theodosios von Tripolis: Sphaerik. Übersetzt und mit anmerkungen versehen (in German). Leipzig: Akademische verlagsgesellschaft m. b. h. Naṣīr al-Dīn al-Ṭūsī, ed. (1939). Kitāb al-ukar li-Thāʾudhūsiyūs: Taḥrīr al-alāma al-faylasūf al-H̱awāǧah Naṣīr al-Dīn Muḥammad ibn Muḥammad ibn al-Ḥasan al-Ṭūsī كتاب الاكر لثاوذوسيوس: تحرير العلامة الفيلسوف الخوا نصير الدين محمد بن محمد بن الحسن الطوصي (in Arabic). Ḥaydarʾābād: Dāʾirat al-maʿārif al-ʿuthmānīyyah. Martin, Thomas J., ed. (1975). The Arabic Translation of Theodosius's Sphaerica (PhD thesis) (in Arabic and English). University of St. Andrews. Czinczenheim, Claire, ed. (2000). Édition, traduction et commentaire des Sphériques de Théodose (PhD thesis) (in Greek and French). Université de Paris IV, Paris-Sorbonne. Spandagos, Vangelēs, ed. (2000). Ta Sphairika tu Theodosiu tu Tripolitu Τα Σφαιρικα του Θεοδοσιου του Τριπολιτου (in Greek). Athens: Aithra. ISBN 9789607007889. Kunitzsch, Paul; Lorch, Richard, eds. (2010). Theodosius, "Sphaerica": Arabic and Medieval Latin Translations (in Arabic, Latin, and English). Stuttgart: Franz Steiner. ISBN 9783515092883. Sidoli, Nathan; Thomas, Robert Spencer David, eds. (2023). The Spherics of Theodosios. London: Routledge. doi:10.4324/9781003142164. ISBN 9780367557300. == Notes == == References == Lorch, Richard (1996). "The transmission of Theodosius' Sphaerica". In Folkerts, Menso (ed.). Mathematische Probleme im Mittelalter: Der lateinische und arabische Sprachbereich. Wiesbaden: Harrassowitz. pp. 159–184. Malpangotto, Michela (2010). "Graphical Choices and Geometrical Thought in the Transmission of Theodosius' Spherics from Antiquity to the Renaissance". Archive for History of Exact Sciences. 64 (1): 75–112. doi:10.1007/s00407-009-0054-1. JSTOR 41342412. Sidoli, Nathan; Saito, Ken (2009). "The role of geometrical construction in Theodosius's Spherics" (PDF). Archive for History of Exact Sciences. 63 (6): 581–609. doi:10.1007/s00407-009-0045-2. JSTOR 41134325. Sidoli, Nathan; Kusuba, Takanori (2017). "Naṣīr al-Dīn al-Ṭūsī's revision of Theodosius's Spherics" (PDF). In Iqbal, Muzaffar (ed.). New Perspectives on the History of Islamic Science. Vol. 3. Routledge. pp. 355–392. doi:10.4324/9781315248011-18. Thomas, Robert S.D. (2013). "Acts of Geometrical Construction in the Spherics of Theodosios". From Alexandria, Through Baghdad. Springer. pp. 227–237. doi:10.1007/978-3-642-36736-6_11. Thomas, Robert S.D. (2018). "The definitions and theorems of The Spherics of Theodosios". In Sidoli, Nathan; Brummelen, Glen Van (eds.). Research in History and Philosophy of Mathematics. CSHPM Annual Meeting, Toronto, Ontario, May 28–30 2017. Springer. pp. 1–21. doi:10.1007/978-3-642-36736-6_11. Thomas, Robert S.D. (2018). "An Appreciation of the First Book of Spherics". Mathematics Magazine. 91 (1): 3–15. doi:10.1080/0025570X.2017.1404798. JSTOR 48664899.
|
Wikipedia:Theorem of transition#0
|
In algebra, the theorem of transition is said to hold between commutative rings A ⊂ B {\displaystyle A\subset B} if B {\displaystyle B} dominates A {\displaystyle A} ; i.e., for each proper ideal I of A, I B {\displaystyle IB} is proper and for each maximal ideal n {\displaystyle {\mathfrak {n}}} of B, n ∩ A {\displaystyle {\mathfrak {n}}\cap A} is maximal for each maximal ideal m {\displaystyle {\mathfrak {m}}} and m {\displaystyle {\mathfrak {m}}} -primary ideal Q {\displaystyle Q} of A {\displaystyle A} , length B ( B / Q B ) {\displaystyle \operatorname {length} _{B}(B/QB)} is finite and moreover length B ( B / Q B ) = length B ( B / m B ) length A ( A / Q ) . {\displaystyle \operatorname {length} _{B}(B/QB)=\operatorname {length} _{B}(B/{\mathfrak {m}}B)\operatorname {length} _{A}(A/Q).} Given commutative rings A ⊂ B {\displaystyle A\subset B} such that B {\displaystyle B} dominates A {\displaystyle A} and for each maximal ideal m {\displaystyle {\mathfrak {m}}} of A {\displaystyle A} such that length B ( B / m B ) {\displaystyle \operatorname {length} _{B}(B/{\mathfrak {m}}B)} is finite, the natural inclusion A → B {\displaystyle A\to B} is a faithfully flat ring homomorphism if and only if the theorem of transition holds between A ⊂ B {\displaystyle A\subset B} . == Notes == == References == Nagata, M. (1975). Local Rings. Interscience tracts in pure and applied mathematics. Krieger. ISBN 978-0-88275-228-0. Matsumura, Hideyuki (1986). Commutative ring theory. Cambridge Studies in Advanced Mathematics. Vol. 8. Cambridge University Press. ISBN 0-521-36764-6. MR 0879273. Zbl 0603.13001.
|
Wikipedia:Theory of equations#0
|
In algebra, the theory of equations is the study of algebraic equations (also called "polynomial equations"), which are equations defined by a polynomial. The main problem of the theory of equations was to know when an algebraic equation has an algebraic solution. This problem was completely solved in 1830 by Évariste Galois, by introducing what is now called Galois theory. Before Galois, there was no clear distinction between the "theory of equations" and "algebra". Since then algebra has been dramatically enlarged to include many new subareas, and the theory of algebraic equations receives much less attention. Thus, the term "theory of equations" is mainly used in the context of the history of mathematics, to avoid confusion between old and new meanings of "algebra". == History == Until the end of the 19th century, "theory of equations" was almost synonymous with "algebra". For a long time, the main problem was to find the solutions of a single non-linear polynomial equation in a single unknown. The fact that a complex solution always exists is the fundamental theorem of algebra, which was proved only at the beginning of the 19th century and does not have a purely algebraic proof. Nevertheless, the main concern of the algebraists was to solve in terms of radicals, that is to express the solutions by a formula which is built with the four operations of arithmetics and with nth roots. This was done up to degree four during the 16th century. Scipione del Ferro and Niccolò Fontana Tartaglia discovered solutions for cubic equations. Gerolamo Cardano published them in his 1545 book Ars Magna, together with a solution for the quartic equations, discovered by his student Lodovico Ferrari. In 1572 Rafael Bombelli published his L'Algebra in which he showed how to deal with the imaginary quantities that could appear in Cardano's formula for solving cubic equations. The case of higher degrees remained open until the 19th century, when Paolo Ruffini gave an incomplete proof in 1799 that some fifth degree equations cannot be solved in radicals followed by Niels Henrik Abel's complete proof in 1824 (now known as the Abel–Ruffini theorem). Évariste Galois later introduced a theory (presently called Galois theory) to decide which equations are solvable by radicals. == Further problems == Other classical problems of the theory of equations are the following: Linear equations: this problem was solved during antiquity. Simultaneous linear equations: The general theoretical solution was provided by Gabriel Cramer in 1750. However devising efficient methods (algorithms) to solve these systems remains an active subject of research now called linear algebra. Finding the integer solutions of an equation or of a system of equations. These problems are now called Diophantine equations, which are considered a part of number theory (see also integer programming). Systems of polynomial equations: Because of their difficulty, these systems, with few exceptions, have been studied only since the second part of the 19th century. They have led to the development of algebraic geometry. == See also == Root-finding algorithm Properties of polynomial roots Quintic function == References == https://www.britannica.com/science/mathematics/Theory-of-equations == Further reading == Uspensky, James Victor, Theory of Equations (McGraw-Hill), 1963 Dickson, Leonard E., Elementary Theory of Equations (Internet Archive), originally 1914 [1]
|
Wikipedia:Theory of functional connections#0
|
The Theory of Functional Connections (TFC) is a mathematical framework designed for functional interpolation. It introduces a method to derive a functional— a function that operates on another function—capable of transforming constrained optimization problems into equivalent unconstrained problems. This transformation enables the application of TFC to various mathematical challenges, including the solution of differential equations. Functional interpolation, in this context, refers to constructing functionals that always satisfy given constraints, regardless of the expression of the internal (free) function. == From interpolation to functional interpolation == To provide a general context for the TFC, consider a generic interpolation problem involving n {\displaystyle n} constraints, such as a differential equation subject to a boundary value problem (BVP). Regardless of the differential equation, these constraints may be consistent or inconsistent. For instance, in a problem over the domain D : ( 0 , 1 ) ∪ ( 0 , 1 ) {\displaystyle {\mathcal {D}}:(0,1)\cup (0,1)} , the constraints f 1 ( x , 0 ) = 1 + x {\displaystyle f_{1}(x,0)=1+x} and f 2 ( 0 , y ) = 2 − y {\displaystyle f_{2}(0,y)=2-y} are inconsistent, as they yield different values at the shared point ( 0 , 0 ) {\displaystyle (0,0)} . If the n {\displaystyle n} constraints are consistent, a function interpolating these constraints can be constructed by selecting n {\displaystyle n} linearly independent support functions, such as monomials, { 1 , x , x 2 , ⋯ , x n − 1 } {\displaystyle \{1,x,x^{2},\cdots ,x^{n-1}\}} . The chosen set of support functions may or may not be consistent with the given constraints. For instance, the constraints y ( − 1 ) = y ( + 1 ) = 0 {\displaystyle y(-1)=y(+1)=0} and d y d x | x = 0 = 1 {\displaystyle {\dfrac {dy}{dx}}{\bigg |}_{x=0}=1} are inconsistent with the support functions, { 1 , x , x 2 } {\displaystyle \{1,x,x^{2}\}} , as can be easily verified. If the support functions are consistent with the constraints, the interpolation problem can be solved, yielding an interpolant—a function that satisfies all constraints. Choosing a different set of support functions would result in a different interpolant. When an interpolation problem is solved and an initial interpolant is determined, all possible interpolants can, in principle, be generated by performing the interpolation process with every distinct set of linearly independent support functions consistent with the constraints. However, this method is impractical, as the number of possible sets of support functions is infinite. This challenge was addressed through the development of the TFC, an analytical framework for performing functional interpolation introduced by Daniele Mortari at Texas A&M University. The approach involves constructing a functional f ( x , g ( x ) ) {\displaystyle f{\big (}\mathbf {x} ,g(\mathbf {x} ){\big )}} that satisfies the given constraints for any arbitrary expression of g ( x ) {\displaystyle g(\mathbf {x} )} , referred to as the free function. This functional, known as the constrained functional, provides a complete representation of all possible interpolants. By varying g ( x ) {\displaystyle g(\mathbf {x} )} , it is possible to generate the entire set of interpolants, including those that are discontinuous or partially defined. Function interpolation produces a single interpolating function, while functional interpolation generates a family of interpolating functions represented through a functional. This functional defines the subspace of functions that inherently satisfy the given constraints, effectively reducing the solution space to the region where solutions to the constrained optimization problem are located. By employing these functionals, constrained optimization problems can be reformulated as unconstrained problems. This reformulation allows for simpler and more efficient solution methods, often improving accuracy, robustness, and reliability. Within this context, the Theory of Functional Connections (TFC) provides a systematic framework for transforming constrained problems into unconstrained ones, thereby streamlining the solution process. TFC addresses univariate constraints involving points, derivatives, integrals, and any linear combination of these. The theory is also extended to accommodate infinite and multivariate constraints and applied to solving ordinary, partial, and integro-differential equations. The consistency problem, which pertains to constraints, interpolation, and functional interpolation, is comprehensively addressed in. This includes the consistency challenges associated with boundary conditions that involve shear and mixed derivatives. The univariate version of TFC can be expressed in one of the following two forms: { f ( x , g ( x ) ) = g ( x ) + ∑ j = 1 n η j ( x , g ( x ) ) s j ( x ) f ( x , g ( x ) ) = g ( x ) + ∑ j = 1 n ϕ j ( x , s ( x ) ) ρ j ( x , g ( x ) ) , {\displaystyle {\begin{cases}f{\big (}x,g(x){\big )}=g(x)+\displaystyle \sum _{j=1}^{n}\eta _{j}{\big (}x,g(x){\big )}\,s_{j}(x)\\f{\big (}x,g(x){\big )}=g(x)+\displaystyle \sum _{j=1}^{n}\phi _{j}{\big (}x,\mathbf {s} (x){\big )}\,\rho _{j}{\big (}x,g(x){\big )},\end{cases}}} where n {\displaystyle n} represents the number of linear constraints, g ( x ) {\displaystyle g(x)} is the free function, and s j ( x ) {\displaystyle s_{j}(x)} are n {\displaystyle n} user-defined, linearly independent support functions. The terms η j ( x , g ( x ) ) {\displaystyle \eta _{j}(x,g(x))} are the coefficient functionals, ϕ j ( x ) {\displaystyle \phi _{j}(x)} are switching functions (which take a value of 1 when evaluated at their respective constraint and 0 at other constraints), and ρ j ( x , g ( x ) ) {\displaystyle \rho _{j}{\big (}x,g(x){\big )}} are projection functionals that express the constraints in terms of the free function. == A rational example == To show how TFC generalizes interpolation, consider the constraints, y ˙ ( x 1 ) = y ˙ 1 {\displaystyle {\dot {y}}(x_{1})={\dot {y}}_{1}} and y ˙ ( x 2 ) = y ˙ 2 {\displaystyle {\dot {y}}(x_{2})={\dot {y}}_{2}} . An interpolating function satisfying these constraints is, f a ( x ) = x ( 2 x 2 − x ) 2 ( x 2 − x 1 ) y ˙ 1 + x ( x − 2 x 1 ) 2 ( x 2 − x 1 ) y ˙ 2 , {\displaystyle f_{a}(x)={\dfrac {x(2x_{2}-x)}{2(x_{2}-x_{1})}}\,{\dot {y}}_{1}+{\dfrac {x(x-2x_{1})}{2(x_{2}-x_{1})}}\,{\dot {y}}_{2},} as can be easily verified. Because of this interpolation property, the derivative of the function, δ ( x ) = g ( x ) − x ( 2 x 2 − x ) 2 ( x 2 − x 1 ) g ˙ ( x 1 ) − x ( x − 2 x 1 ) 2 ( x 2 − x 1 ) g ˙ ( x 2 ) , {\displaystyle \delta (x)=g(x)-{\dfrac {x(2x_{2}-x)}{2(x_{2}-x_{1})}}\,{\dot {g}}(x_{1})-{\dfrac {x(x-2x_{1})}{2(x_{2}-x_{1})}}\,{\dot {g}}(x_{2}),} vanishes at x 1 {\displaystyle x_{1}} and x 2 {\displaystyle x_{2}} , for \textit{any} function, g ( x ) {\displaystyle g(x)} . Therefore, by adding δ ( x ) {\displaystyle \delta (x)} to f a ( x ) {\displaystyle f_{a}(x)} , a functional is obtained that still satisfies the constraints, f ( x , g ( x ) ) = f a ( x ) + δ ( x ) = x ( 2 x 2 − x ) 2 ( x 2 − x 1 ) y ˙ 1 + x ( x − 2 x 1 ) 2 ( x 2 − x 1 ) y ˙ 2 + g ( x ) − x ( 2 x 2 − x ) 2 ( x 2 − x 1 ) g ˙ ( x 1 ) − x ( x − 2 x 1 ) 2 ( x 2 − x 1 ) g ˙ ( x 2 ) , {\displaystyle f{\big (}x,g(x){\big )}=f_{a}(x)+\delta (x)={\dfrac {x(2x_{2}-x)}{2(x_{2}-x_{1})}}\,{\dot {y}}_{1}+{\dfrac {x(x-2x_{1})}{2(x_{2}-x_{1})}}\,{\dot {y}}_{2}+g(x)-{\dfrac {x(2x_{2}-x)}{2(x_{2}-x_{1})}}\,{\dot {g}}(x_{1})-{\dfrac {x(x-2x_{1})}{2(x_{2}-x_{1})}}\,{\dot {g}}(x_{2}),} no matter what g ( x ) {\displaystyle g(x)} is. Due to this property, this functional is referred to as constrained functional. The key requirement for the functional f ( x , g ( x ) ) {\displaystyle f{\big (}x,g(x){\big )}} to work as intended is that the terms g ˙ ( x 1 ) {\displaystyle {\dot {g}}(x_{1})} and g ˙ ( x 2 ) {\displaystyle {\dot {g}}(x_{2})} are defined. Once this condition is met, the functional f ( x , g ( x ) ) {\displaystyle f{\big (}x,g(x){\big )}} is free to take on any arbitrary values beyond the specified constraints, thanks to the infinite flexibility provided by g ( x ) {\displaystyle g(x)} . Importantly, this flexibility is not limited to the specific constraints chosen in this example. Instead, it applies universally to any set of constraints. This universality illustrates how TFC performs functional interpolation: it constructs a function that satisfies the given constraints while simultaneously allowing complete freedom in behavior elsewhere through the choice of g ( x ) {\displaystyle g(x)} . In essence, this example demonstrates that the constrained functional f ( x , g ( x ) ) {\displaystyle f{\big (}x,g(x){\big )}} captures all possible functions that meet the given constraints, showcasing the power and generality of TFC in handling a wide variety of interpolation problems. == Applications of TFC == TFC has been extended and employed in various applications, including its use in shear-type and mixed derivative problems, the analysis of fractional operators, the determination of geodesics for BVP in curved spaces, and in continuation methods. Additionally, TFC has been applied to indirect optimal control, the modeling of stiff chemical kinetics, and the study of epidemiological dynamics. TFC extends into astrodynamics [1], where Lambert's problem is efficiently solved. It has also demonstrated potential in nonlinear programming and structural mechanics and radiative transfer, among other areas. An efficient, free Python TFC toolbox is available at https://github.com/leakec/tfc. Of particular note is the application of TFC in neural networks, where it has shown exceptional efficiency, especially addressing high-dimensional problems and in enhancing the performance of physics-informed neural networks by effectively eliminating constraints from the optimization process, a challenge that traditional neural networks often struggle to address. This capability significantly improves computational efficiency and accuracy, enabling the resolution of complex problems with greater ease, as proved by the University of Arizona. TFC has been employed with physics-informed neural networks and symbolic regression techniques for physics discovery of dynamical systems. == Difference with spectral methods == At first glance, TFC and spectral methods may appear similar in their approach to solving constrained optimization problems. However, there are two fundamental distinctions between them: Representation of solutions: Spectral methods represent the solution as a sum of basis functions, whereas TFC represents the free function as a sum of basis functions. This distinction allows TFC to analytically satisfy the constraints, while spectral methods treat constraints as additional data, approximating them with an accuracy dependent on the residuals. Computational approach in BVP: In linear BVPs, the computational strategies of the two methods differ significantly. Spectral methods typically employ iterative techniques, such as the shooting method, to reformulate the BVP as an initial value problem, which is simpler to solve. Conversely, TFC directly addresses these problems through linear least-squares techniques, avoiding the need for iterative procedures. Both methods can perform optimization using either the Galerkin method, which ensures the residual vector is orthogonal to the chosen basis functions, or the Collocation method, which minimizes the norm of the residual vector. == Difference with Lagrange multipliers technique == The Lagrange multipliers method is a widely used approach for imposing constraints in an optimization problem. This technique introduces additional variables, known as multipliers, which must be computed to enforce the constraints. While the computation of these multipliers is straightforward in some cases, it can be challenging or even practically infeasible in others, thereby adding significant complexity to the problem. In contrast, TFC doesn't add new variables and enables the derivation of constrained functionals without encountering insurmountable difficulties. However, it is important to note that the Lagrange multiplier method has the advantage of handling inequality constraints, a capability that TFC currently lacks. A notable limitation of both approaches is their propensity to produce solutions that correspond to local optima rather than guaranteed global optima, particularly in the context of non-convex problems. Consequently, supplementary verification procedures or alternative methods may be required to assess and confirm the quality and global validity of the obtained solution. In summary, while TFC does not entirely replace the Lagrange multipliers method, it serves as a powerful alternative in cases where the computation of multipliers becomes excessively complex or infeasible, provided the constraints are limited to equalities. == References ==
|
Wikipedia:Theta operator#0
|
In mathematics, the theta operator is a differential operator defined by θ = z d d z . {\displaystyle \theta =z{d \over dz}.} This is sometimes also called the homogeneity operator, because its eigenfunctions are the monomials in z: θ ( z k ) = k z k , k = 0 , 1 , 2 , … {\displaystyle \theta (z^{k})=kz^{k},\quad k=0,1,2,\dots } In n variables the homogeneity operator is given by θ = ∑ k = 1 n x k ∂ ∂ x k . {\displaystyle \theta =\sum _{k=1}^{n}x_{k}{\frac {\partial }{\partial x_{k}}}.} As in one variable, the eigenspaces of θ are the spaces of homogeneous functions. (Euler's homogeneous function theorem) == See also == Difference operator Delta operator Elliptic operator Fractional calculus Invariant differential operator Differential calculus over commutative algebras == References == == Further reading == Watson, G.N. (1995). A treatise on the theory of Bessel functions (Cambridge mathematical library ed., [Nachdr. der] 2. ed.). Cambridge: Univ. Press. ISBN 0521483913.
|
Wikipedia:Thiele's interpolation formula#0
|
In mathematics, Thiele's interpolation formula is a formula that defines a rational function f ( x ) {\displaystyle f(x)} from a finite set of inputs x i {\displaystyle x_{i}} and their function values f ( x i ) {\displaystyle f(x_{i})} . The problem of generating a function whose graph passes through a given set of function values is called interpolation. This interpolation formula is named after the Danish mathematician Thorvald N. Thiele. It is expressed as a continued fraction, where ρ represents the reciprocal difference: f ( x ) = f ( x 1 ) + x − x 1 ρ ( x 1 , x 2 ) + x − x 2 ρ 2 ( x 1 , x 2 , x 3 ) − f ( x 1 ) + x − x 3 ρ 3 ( x 1 , x 2 , x 3 , x 4 ) − ρ ( x 1 , x 2 ) + ⋯ {\displaystyle f(x)=f(x_{1})+{\cfrac {x-x_{1}}{\rho (x_{1},x_{2})+{\cfrac {x-x_{2}}{\rho _{2}(x_{1},x_{2},x_{3})-f(x_{1})+{\cfrac {x-x_{3}}{\rho _{3}(x_{1},x_{2},x_{3},x_{4})-\rho (x_{1},x_{2})+\cdots }}}}}}} Note that the n {\displaystyle n} -th level in Thiele's interpolation formula is ρ n ( x 1 , x 2 , ⋯ , x n + 1 ) − ρ n − 2 ( x 1 , x 2 , ⋯ , x n − 1 ) + x − x n + 1 ρ n + 1 ( x 1 , x 2 , ⋯ , x n + 2 ) − ρ n − 1 ( x 1 , x 2 , ⋯ , x n ) + ⋯ , {\displaystyle \rho _{n}(x_{1},x_{2},\cdots ,x_{n+1})-\rho _{n-2}(x_{1},x_{2},\cdots ,x_{n-1})+{\cfrac {x-x_{n+1}}{\rho _{n+1}(x_{1},x_{2},\cdots ,x_{n+2})-\rho _{n-1}(x_{1},x_{2},\cdots ,x_{n})+\cdots }},} while the n {\displaystyle n} -th reciprocal difference is defined to be ρ n ( x 1 , x 2 , … , x n + 1 ) = x 1 − x n + 1 ρ n − 1 ( x 1 , x 2 , … , x n ) − ρ n − 1 ( x 2 , x 3 , … , x n + 1 ) + ρ n − 2 ( x 2 , … , x n ) {\displaystyle \rho _{n}(x_{1},x_{2},\ldots ,x_{n+1})={\frac {x_{1}-x_{n+1}}{\rho _{n-1}(x_{1},x_{2},\ldots ,x_{n})-\rho _{n-1}(x_{2},x_{3},\ldots ,x_{n+1})}}+\rho _{n-2}(x_{2},\ldots ,x_{n})} . The two ρ n − 2 {\displaystyle \rho _{n-2}} terms are different and can not be cancelled. == References == Weisstein, Eric W. "Thiele's Interpolation Formula". MathWorld.
|
Wikipedia:Thierry Goudon#0
|
Thierry Goudon (born January 1969 in Aix-en-Provence, France) is a French mathematician. He works in applied mathematics, with interest in the study of Partial Differential Equations motivated from physics. He has made contributions on kinetic theory, which corresponds to a description of matter in terms of statistical physics. The Boltzmann equation for gas dynamics is a typical example of this activity. The kinetic framework also arises in many other fields: neutron transport, radiative transfer, and biology. He is interested in asymptotic analysis, including the study of hydrodynamic regimes and homogenization theory, establishing relationships between microscopic and macroscopic descriptions. He also works on fluid mechanics, both as regards the analysis of the equations and also the design of numerical methods for computing the solutions. Currently he holds a Senior INRIA Researcher (Directeur de recherche) position at Sophia Antipolis; he is the head of the team COFFEE devoted to Complex Flows For Energy and Environment. == Biography == Thierry Goudon completed his undergraduate studies in Aix-en-Provence and in Bordeaux, where he attended the Matmeca program. He obtained the PhD degree under supervision of Kamal Hamdache in 1997 at University Bordeaux 1. He joined the University of Nice as Assistant Professor (Maitre de conferences). He obtain the Habilitation to conduct research in 2001 and he became Full Professor at the University of Lille in 2003. Since 2008, he has held the pots of a Senior INRIA Researcher, in Lille until 2011, then in Sophia Antipolis. In 2008, he was awarded the Robert Dautray prize, jointly with Jean-Francois Clouet from the French Atomic Commission: this exceptional prize, funded by the French Society of Applied and Industrial Mathematics in the honor of R. Dautray's 80th birthday, honours remarkable works on radiative transfer theory and its applications. == References == == External links == Thierry Goudon's professional website Thierry Goudon at the Mathematics Genealogy Project Movie: Avis de recherche
|
Wikipedia:Thomas A. Scott Professorship of Mathematics#0
|
The Thomas A. Scott Professorship of Mathematics is an academic grant made to the University of Pennsylvania. It was established in 1881 by the railroad executive and financier Thomas Alexander Scott. == Recipients == Ezra Otis Kendall, 1881–1899 Edwin Schofield Crawley, 1899–1933 George Hervey Hallett, 1933–1941 John Robert Kline, 1941–1955 Hans A. Rademacher, 1956–1962 Eugenio Calabi, 1967–1993 Shmuel Weinberger, 1994–1996 Herbert S. Wilf, 1998–2006 Charles Epstein, 2008–present == See also == Thomas A. Scott Fellowship in Hygiene == External links == University of Pennsylvania Mathematics Department page about the Professorship == References ==
|
Wikipedia:Thomas Bartholin#0
|
Thomas Bartholin (; Latinized as Thomas Bartholinus; 20 October 1616 – 4 December 1680) was a Danish physician, mathematician, and theologian. He discovered the lymphatic system in humans and advanced the theory of refrigeration anesthesia, being the first to describe it scientifically. Thomas Bartholin came from a family that has become famous for its pioneering scientists, twelve of whom became professors at the University of Copenhagen. Three generations of the Bartholin family made significant contributions to anatomical science and medicine in the 17th and 18th centuries: Thomas Bartholin's father, Caspar Bartholin the Elder (1585–1629), his brother Rasmus Bartholin (1625–1698), and his son Caspar Bartholin the Younger (1655–1738). Thomas Bartholin's son Thomas Bartholin the Younger (1659–1690) became a professor of history at the University of Copenhagen and was later appointed royal antiquarian and secretary to the Royal Archives. == Personal life == Thomas Bartholin was the second of the six sons of Caspar Bartholin the Elder, a physician born in Malmø, Scania, and his spouse Anne Fincke. Bartholin the Elder published the first collected anatomical work in 1611. This work was later augmented, illustrated and revised by Thomas Bartholin, becoming the standard reference on anatomy; the son notably added updates on William Harvey's theory of blood circulation and on the lymphatic system. Bartholin visited the Italian botanist Pietro Castelli at Messina in 1644. In 1663 Bartholin bought Hagestedgård, which burned down in 1670 including his library, with the loss of many manuscripts. King Christian V of Denmark appointed Bartholin as his physician with a substantial salary and freed the farm from taxation as recompense for the loss. In 1680 Bartholin's health failed, the farm was sold, and he moved back to Copenhagen, where he died. He was buried in Vor Frue Kirke (Church of Our Lady). The Bartholinsgade, a street in Copenhagen, is named for the family. Nearby is the Bartholin Institute (Bartholin Institutet). One of the buildings of the University of Aarhus is named after him. == Contributions to medical research == In December 1652, Bartholin published the first full description of the human lymphatic system. Jean Pecquet had previously noted the lymphatic system in animals in 1651, and Pecquet's discovery of the thoracic duct and its entry into the veins made him the first person to describe the correct route of the lymphatic fluid into the blood. Shortly after the publication of Pecquet's and Bartholin's findings, a similar discovery of the human lymphatic system was published by Olof Rudbeck in 1653, although Rudbeck had presented his findings at the court of Queen Christina of Sweden in April–May 1652, before Bartholin, but delayed in writing about it until 1653 (after Bartholin). As a result, an intense priority dispute ensued. Niels Stensen or Steno became Bartholin's most famous pupil. Thomas' publication De nivis usu medico observationes variae Chapter XXII, contains the first known mention of refrigeration anaesthesia, a technique whose invention Thomas Bartholin credits to the Italian Marco Aurelio Severino of Naples. According to Bartholin, Severino was the first to present the use of freezing mixtures of snow and ice (1646), and Thomas Bartholin initially learnt about the technique from him during a visit to Naples. Bartholin–Patau syndrome, a congenital syndrome of multiple abnormalities produced by trisomy 13, was first described by Bartholin in 1656. Caspar Bartholin the Elder, Thomas Bartholin's father; his brother Rasmus Bartholin; and his son Caspar Bartholin the Younger (who first described "Bartholin's glands"), all contributed to the practice of modern medicine through their discoveries of important anatomical structures and phenomena. Bartholin the Elder started his tenure as professor at Copenhagen University in 1613, and over the next 125 years, the scientific accomplishments of the Bartholins while serving on the medical faculty of the University of Copenhagen won international acclaim and contributed to the reputation of the institution. == Selected works == Historiarum anatomicarum rariorum [...] (Case histories of unusual anatomical and clinical structures, including descriptions and illustrations of anomalies and normal structures) ... centuria I et II at Google Books, Amsterdam, 1654. ... centuria III et IV at Google Books. The Hague: Vlacq, 1657. ... centuria V et VI at Google Books, Copenhagen: P. Haubold, 1661 (with Mantissa anatomica, by Johannes Rodius). De unicornu. Padua, 1645. De Angina Puerorum Campaniae Siciliaeque Epidemica Exercitationes. Paris, 1646. De lacteis thoracicis in homine brutisque nuperrime observatis historia anatomica at Google Books, Copenhagen: M. Martzan, 1652 (Bartholin's discovery of the thoracic duct). Vasa lymphatica nuper Hafniae in animalibus inventa et hepatis exsequiae. Hafniae (Copenhagen), Petrus Hakius, 1653. Vasa lymphatica in homine nuper inventa. Hafniae (Copenhagen), 1654. Historarium anatomicarum rariorum centuria I-VI. Copenhagen, 1654–1661. Anatomia. The Hague. Ex typographia Adriani Vlacq, 1655. Dispensarium hafniense. Copenhagen, 1658. De nivis usu medico observationes variae. Accessit D. Erasmi Bartholini de figura nivis dissertatio. With a book by Rasmus Bartholin. Copenhagen: Typis Matthiase Godichii, sumptibus Petri Haubold, 1661. (Contains the first known mention of refrigeration anaesthesia) Cista medica hafniensis. Copenhagen, 1662. De pulmonum substantia et motu. Copenhagen, 1663. De insolitis partus humani viis. Copenhagen, 1664. De medicina danorum domestica. Copenhagen, 1666. De flammula cordis epistola. Copenhagen, 1667. Orationes et dissertationes omnino varii argumenti. Copenhagen, 1668. Carmina varii argumenti. Copenhagen, 1669. De medicis poetis dissertatio. Hafinae, apud D. Paulli, 1669. De bibliothecae incendio. Copenhagen, 1670. De morbis biblicis miscellanea medica. Francofurti, D. Paulli, 1672. De cruce Christi hypomnemata IV, Typis Andreae ab Hoogenhuysen, Vesaliae (Wesel), 1673. Acta medica et philosophica. 1673–1680. == References == == External links == View digitized titles by Thomas Bartholin in Botanicus.org Thomas Bartholin in Whonamedit.com Bartholin's (1647) De luce animalium – digital facsimile at the Linda Hall Library MyNDIR (My Norse Digital Image Repository) Illustrations by Thomas Bartholin from manuscripts and early print books.
|
Wikipedia:Thomas Bloom#0
|
Thomas F. Bloom is a mathematician, who is a Royal Society University Research Fellow at the University of Manchester. He works in arithmetic combinatorics and analytic number theory. == Education and career == Thomas did his undergraduate degree in Mathematics and Philosophy at Merton College, Oxford. He then went on to do his PhD in mathematics at the University of Bristol under the supervision of Trevor Wooley. After finishing his PhD, he was a Heilbronn Research Fellow at the University of Bristol. In 2018, he became a postdoctoral research fellow at the University of Cambridge with Timothy Gowers. In 2021, he joined the University of Oxford as a Research Fellow. Then, in 2024, he moved to the University of Manchester, where he also took on a Research Fellow position. == Research == In July 2020, Bloom and Sisask proved that any set such that ∑ n ∈ A 1 n {\displaystyle \sum _{n\in A}{\frac {1}{n}}} diverges must contain arithmetic progressions of length 3. This is the first non-trivial case of a conjecture of Erdős postulating that any such set must in fact contain arbitrarily long arithmetic progressions. In November 2020, in joint work with James Maynard, he improved the best-known bound for square-difference-free sets, showing that a set A ⊂ [ N ] {\displaystyle A\subset [N]} with no square difference has size at most N ( log N ) c log log log N {\displaystyle {\frac {N}{(\log N)^{c\log \log \log N}}}} for some c > 0 {\displaystyle c>0} . In December 2021, he proved that any set A ⊂ N {\displaystyle A\subset \mathbb {N} } of positive upper density contains a finite S ⊂ A {\displaystyle S\subset A} such that ∑ n ∈ S 1 n = 1 {\displaystyle \sum _{n\in S}{\frac {1}{n}}=1} . This answered a question of Erdős and Graham. == References ==
|
Wikipedia:Thomas F. Coleman#0
|
Thomas F. Coleman (1950–2021) was a Canadian mathematician and computer scientist who is a Professor in the Department of Combinatorics and Optimization at the University of Waterloo, where he held the Ophelia Lazaridis University Research Chair. In addition, Coleman was the director of WatRISQ, an institute composed of quantitative and computational finance researchers spanning several Faculties at the University of Waterloo. His research focused on mathematical optimization. == Education == Coleman earned his PhD from University of Waterloo in 1979 with the dissertation A Superlinear Penalty Function Method to Solve the Nonlinear Programming Problem supervised by Andrew Conn. He followed that up with a two-year postdoctoral appointment in the Applied Mathematics Division at Argonne National Laboratory. == Career == From 1981 to 2005, Coleman was a professor of computer science at Cornell University. From 1998 to 2005 he served as the director of Cornell Theory Center, now Cornell University Center for Advanced Computing From 2005 to 2010, Coleman served as the dean of the Faculty of Mathematics at the University of Waterloo. During his tenure as Cornell Theory Center director, Coleman founded and directed a computational finance academic-industry-government venture located at 55 Broad Street in New York, which shaped into Cornell Financial Engineering Manhattan. He died of cancer on April 20, 2021. == Awards and honors == Coleman was selected a SIAM Fellow in 2016 "for his contributions to financial optimization, sparse numerical optimization and leadership in mathematical education and industry engagement". == References == == External links == Thomas F. Coleman publications indexed by Google Scholar
|
Wikipedia:Thomas Fantet de Lagny#0
|
Thomas Fantet de Lagny (7 November 1660 – 11 April 1734) was a French mathematician, well known for his contributions to computational mathematics, and for calculating π to 112 correct decimal places. == Biography == Thomas Fantet de Lagny was son of Pierre Fantet, a royal official in Grenoble, and Jeanne d'Azy, the daughter of a physician from Montpellier. He entered a Jesuit College in Lyon, where he became passionate about mathematics, as he studied some mathematical texts such as Euclid by Georges Fournier and an algebra text by Jacques Pelletier du Mans. Then he studied three years in the Faculty of Law in Toulouse. In 1686, he went to Paris and became a mathematics tutor to the Noailles family. He collaborated with de l'Hospital under the name of de Lagny, and at that time he started publishing his first mathematical papers. He came back to Lyon when, on 11 December 1695, he was named an associate of the Académie Royale des Sciences. Then, in 1697, he became professor of hydrography at Rochefort for 16 years. De Lagny returned to Paris in 1714, and became a librarian at the Bibliothèque du roi, and a deputy director of the Banque Générale between 1716 and 1718. On 7 July 1719, he was awarded a pension by the Académie Royale des Sciences, finally earning his living from science. In 1723, he became a pensionnaire at the academy, replacing Pierre Varignon who died in 1722, but had to retire in 1733. De Lagny died on 11 April 1734. While he was dying, someone asked him: "What is the square of 12?" and he answered immediately: "144." == Computing π == In 1719, de Lagny calculated π to 127 decimal places, using Gregory's series for arctangent, but only 112 decimals were correct. This remained the record until 1789, when Jurij Vega calculated 126 correct digits of π. == Bibliography == Méthode nouvelle infiniment générale et infiniment abrégée pour l’extraction des racines quarrées, cubiques... (Paris, 1691) Méthodes nouvelles et abrégées pour l’extraction et l’approximation des racines (Paris, 1692) Nouveaux élémens d’arithmétique et d’algébre ou introduction aux mathématiques (Paris, 1697) Trignonmétrie française ou reformée (Rochefort, 1703) De la cubature de la sphére où l’on démontr une infinité de portions de sphére égales à des pyramides rectilignes (La Rochelle, 1705) Analyse générale ou Méthodes nouvelles pour résoudre les probémes de tous les genres et de tous degrés à l’infini, M. Richer, ed. (Paris, 1733) == References == O'Connor, John J.; Robertson, Edmund F., "Thomas Fantet de Lagny", MacTutor History of Mathematics Archive, University of St Andrews Lagny, Thomas Fantet de, Encyclopedia.com
|
Wikipedia:Thomas Fincke#0
|
Thomas Fincke (6 January 1561 – 24 April 1656) was a Danish mathematician and physicist, and a professor at the University of Copenhagen for more than 60 years. == Biography == Thomas Jacobsen Fincke was born in Flensburg in Schleswig. Fincke was the son of Councillor Jacob Fincke and Anna Thorsmede. He completed his primary schooling at Flensburg. From 1577, he studied mathematics, rhetoric and other philosophical studies for five years at the University of Strasbourg. Fincke's lasting achievement is found in his book Geometria rotundi (1583), in which he introduced the modern names of the trigonometric functions tangent and secant. In 1590, he became professor of mathematics at the University of Copenhagen. In 1603 he also obtained a professorship in medicine. == Personal life == He was married to Ivaria Jungesdatter Ivers (1574–1614). His son Jacob Fincke (1592–1663) was a professor of physics. His daughters married scientist Caspar Bartholin the Elder (1585–1629), botanist Jørgen Fuiren (1581–1628), historian Ole Worm (1588–1654) and theologian Hans Brochmand (1594–1630). Fincke died at Copenhagen and was buried at Vor Frue Kirke. == References == == External links == O'Connor, John J.; Robertson, Edmund F., "Thomas Fincke", MacTutor History of Mathematics Archive, University of St Andrews
|
Wikipedia:Thomas Hou#0
|
Thomas Yizhao Hou (born 1962) is a Chinese-American mathematician who is the Charles Lee Powell Professor of Applied and Computational Mathematics in the Department of Computing and Mathematical Sciences at the California Institute of Technology. He is known for his work in numerical analysis and mathematical analysis. == Academic biography == Hou studied at the South China University of Technology, where he received a B.S. in mathematics in 1982. He completed his Ph.D. in mathematics at the University of California, Los Angeles in 1987 under the supervision of Björn Engquist. His dissertation was titled Convergence of Particle Methods for Euler and Boltzmann Equations with Oscillatory Solutions. From 1989 to 1993, he taught at the Courant Institute of Mathematical Sciences at New York University. He has been on the faculty of the California Institute of Technology since 1993. He became the Charles Lee Powell Professor of Applied and Computational Mathematics in 2004. == Research == Hou is known for his research on multiscale analysis and singularity formation of the three-dimensional incompressible Euler and Navier-Stokes equations. He is an author of the monograph Multiscale finite element methods. The multiscale finite element method developed by Hou and his former postdoc, Xiao-Hui Wu, was one of the earliest multiscale methods and has found many applications from the engineering community. A variant of his method has been adopted by several major oil companies in their new generation of flow simulators. Hou has worked extensively on computational and analytical aspects of the Euler and Navier-Stokes equations. In 2014, Hou and his former postdoc, Guo Luo, presented convincing numerical evidence that the axisymmetric Euler equations develop finite time singularity from smooth initial data. In 2022, Hou and his former Ph.D. student, Jiajie Chen, made a breakthrough by proving the finite time singularity of the axisymmetric Euler equations with smooth data and boundary (the so-called Hou-Luo blowup scenario). Hou’s recent work on the potentially singular behavior of the three-dimensional Navier-Stokes equations has also generated a lot of interests. Hou is also known for his work in computational fluid dynamics. His early work on the convergence of the point vortex method for incompressible Euler equations was unexpected and considered a breakthrough. The level set method developed by Hou and co-workers was the first level set method for multiphase flows and has found many applications. The Small-Scale Decomposition method developed by Hou-Lowengrub-Shelley was considered a tour de force for fluid interface problems and has been used widely in computational fluid dynamics, materials science, and biology. Hou was founder of "SIAM Journal on Multiscale Modeling and Simulation", and he served as the editor-in-chief from 2002 to 2007. He was also cofounder of Advances in Adaptive Data Analysis. == Awards and honors == Hou has won several major awards. He received an Alfred P. Sloan Research Fellowship in 1990. He was awarded the Feng Kang Prize in Scientific Computing in 1997 and the Francois Frenkiel Award from the American Physical Society in 1998. He received the James H. Wilkinson Prize in Numerical Analysis and Scientific Computing from the Society for Industrial and Applied Mathematics (SIAM) in 2001, the J. Tinsley Oden Medal from the United States Association of Computational Mechanics in 2005, the Outstanding Paper Prize from SIAM in 2018, the Ralph E. Kleinman Prize from SIAM in 2023, and the William Benter Prize in Applied Mathematics in 2024. He was an invited speaker at the 1998 International Congress of Mathematicians in Berlin, and he was a plenary speaker at the 2003 International Congress on Industrial and Applied Mathematics in Sydney. Hou is a Member of the National Academy of Sciences and a Fellow of the American Academy of Arts and Sciences, the Society for Industrial and Applied Mathematics, and the American Mathematical Society. == References == == External links == Thomas Y. Hou professional home page
|
Wikipedia:Thomas Jakobsen#0
|
Thomas Jakobsen is a mathematician, cryptographer, and computer programmer, formerly an assistant professor at the Technical University of Denmark (DTU) and head of research and development at IO Interactive. His notable work includes designing the physics engine and 3-D pathfinder algorithms for Hitman: Codename 47, and the cryptanalysis of a number of block ciphers. Jakobsen earned an M.Sc. in engineering and Ph.D. in mathematics, both from DTU. == External links == Publications by Thomas Jakobsen at ResearchGate
|
Wikipedia:Thomas Joannes Stieltjes#0
|
Thomas Joannes Stieltjes ( STEEL-chəz, Dutch: [ˈtoːmɑ ˈstiltɕəs]; 29 December 1856 – 31 December 1894) was a Dutch mathematician. He was a pioneer in the field of moment problems and contributed to the study of continued fractions. The Thomas Stieltjes Institute for Mathematics at Leiden University, dissolved in 2011, was named after him, as is the Riemann–Stieltjes integral. == Biography == Stieltjes was born in Zwolle on 29 December 1856. His father (who had the same first names) was a civil engineer and politician. Stieltjes Sr. was responsible for the construction of various harbours around Rotterdam, and also seated in the Dutch parliament. Stieltjes Jr. went to university at the Polytechnical School in Delft in 1873. Instead of attending lectures, he spent his student years reading the works of Gauss and Jacobi — the consequence of this being he failed his examinations. There were two further failures (in 1875 and 1876), and his father despaired. His father was friends with H. G. van de Sande Bakhuyzen (who was the director of Leiden University), and Stieltjes Jr. was able to get a job as an assistant at Leiden Observatory. Soon afterwards, Stieltjes began a correspondence with Charles Hermite which lasted for the rest of his life. He originally wrote to Hermite concerning celestial mechanics, but the subject quickly turned to mathematics and he began to devote his spare time to mathematical research. The director of Leiden Observatory, van de Sande-Bakhuyzen, responded quickly to Stieltjes' request on 1 January 1883 to stop his observational work to allow him to work more on mathematical topics. In 1883, he also married Elizabeth Intveld in May. She also encouraged him to move from astronomy to mathematics. And in September, Stieltjes was asked to substitute at University of Delft for F.J. van den Berg. From then until December of that year, he lectured on analytical geometry and on descriptive geometry. He resigned his post at the observatory at the end of that year. In 1884, Stieltjes applied for a chair in Groningen. He was initially accepted, but in the end turned down by the Department of Education, since he lacked the required diplomas. In 1884, Hermite and professor David Bierens de Haan arranged for an honorary doctorate to be granted to Stieltjes by Leiden University, enabling him to become a professor. In 1885, he was appointed as member of the Royal Dutch Academy of Sciences (Koninklijke Nederlandse Akademie van Wetenschappen, KNAW), and the next year he became a foreign member. In 1889, he was appointed professor of differential and integral calculus at Toulouse University. Stieltjes died on 31 December 1894 in Toulouse, France. He was buried in Terre-Cabade cemetery on 2 January 1895. == Research == Stieltjes worked on almost all branches of analysis, continued fractions and number theory. For his work, he is sometimes referred to as "the father of the analytic theory of continued fractions". His work is also seen as important as a first step towards the theory of Hilbert spaces. Other important contributions to mathematics that he made involved discontinuous functions and divergent series, differential equations, interpolation, the gamma function and elliptic functions. He became known internationally because of the Riemann–Stieltjes integral. In 1886, Henri Poincaré and Stieltjes, simultaneously and independently, provided a definition of an asymptotic approximation and illustrated its use and practicality. Their papers were published in Acta Math and Annales scientifiques de l'Ecole normale supérieure, respectively. == Awards and Honours == Stieltjes' work on continued fractions earned him the Ormoy Prize (Prix Ormoy) of the Académie des Sciences in 1893. In 1884 the University of Leiden awarded him an honorary doctorate, and in 1885 he was elected to membership in the Royal Academy of Sciences of Amsterdam. In honour of Stieltjes, since 1996, the Stieltjes Prize (Stieltjesprijs) has been awarded annually for the best PhD thesis in mathematics to a student of any Dutch university. All mathematics institutes and departments of Dutch universities are asked for an overview of the PhDs that have taken place in the academic year. The list thus obtained forms the list of candidates for the prize. The award consists of a certificate and an amount of 1200 Euros. == See also == Annales de la Faculté des Sciences de Toulouse co-founded by Stieltjes Chebyshev–Markov–Stieltjes inequalities Heine–Stieltjes polynomials Laplace–Stieltjes transform Lebesgue–Stieltjes integral Montel's theorem Riemann–Stieltjes integral Stieltjes constants Stieltjes matrix Stieltjes moment problem Stieltjes polynomials Stieltjes transformation (and Stieltjes inversion formula) Stieltjes–Wigert polynomials == References == == External links == Media related to Thomas Joannes Stieltjes jr. at Wikimedia Commons O'Connor, John J.; Robertson, Edmund F., "Thomas Joannes Stieltjes", MacTutor History of Mathematics Archive, University of St Andrews Thomas Joannes Stieltjes at the Mathematics Genealogy Project Œuvres complètes de Thomas Jan Stieltjes, pub. par les soins de la Société mathématique d'Amsterdam. (Groningen: P. Noordhoff, 1914–18) (PDF copy at UMDL, text in Dutch, French and German)
|
Wikipedia:Thomas Klein#0
|
Thomas Klein (born April 14, 1948) is a German civil rights activist, historian, and politician. == Life == Klein was born in Berlin on April 14, 1948. He trained as an electrical mechanic before enrolling at Humboldt University to study mathematics, where he received his doctorate in 1976. Meanwhile, Klein had begun working at the Central Institute for Economic Sciences, part of the Academy of Sciences of the GDR, in 1973. During the 1970s, Klein began to associate with opposition groups in the German Democratic Republic. As a result, he was arrested by the Stasi in September 1979, and taken to Berlin-Hohenschönhausen Prison. Klein was convicted of "unlawful contact" ("ungesetzlicher Verbindungsaufnahme") under section 219 of the East German criminal code. During his trial, Gregor Gysi served as his lawyer. He was held in the Bautzen II prison until his release in December 1980. Following his release, Klein was banned from working in the sciences, and was assigned to work at a state-owned furniture company. In 1987, Klein was a founding member of the opposition Gegenstimmen Group. The group brought together leftist opponents of the ruling Socialist Unity Party, and included Marxists, Titoists, Trotskyites, and members of the Christian left. In a report from 1989, the Stasi counted Klein among the most ardent members of the opposition. In 1989, Klein was one of the founding members of the United Left, and a co-author of the party's "Böhlener Platform". In the 1990 East German election, he was the United Left's lead candidate, and was the only member of the party elected to the Volkskammer. When Germany reunited, Klein then became a member of the Bundestag until a new federal election was held on December 2, 1990. The United Left began to disintegrate following German reunification, and during the 2000s Klein joined Die Linke. Following his time as a member of the Bundestag, Klein then spent two years as an employee of the Bundestag. From 1996 to 2009 he worked at the Centre for Contemporary History in Potsdam as a historian specializing in the history of the German Democratic Republic and its opposition movements. In the course of this work, he has authored five books on East German political history. == References ==
|
Wikipedia:Thomas L. Saaty#0
|
Thomas L. Saaty (July 18, 1926 – August 14, 2017) was a Distinguished University Professor at the University of Pittsburgh, where he taught in the Joseph M. Katz Graduate School of Business. He is the inventor, architect, and primary theoretician of the Analytic Hierarchy Process (AHP), a decision-making framework used for large-scale, multiparty, multi-criteria decision analysis, and of the Analytic Network Process (ANP), its generalization to decisions with dependence and feedback. Later on, he generalized the mathematics of the ANP to the Neural Network Process (NNP) with application to neural firing and synthesis but none of them gain such popularity as AHP. He died on the 14th of August 2017 after a year-long battle with cancer. Prior to coming to the University of Pittsburgh, Saaty was professor of statistics and operations research at the Wharton School of the University of Pennsylvania (1969–79). Before that, he spent fifteen years working for U.S. government agencies and for companies doing government-sponsored research. His employers at that time included the Operations Evaluation Group of MIT at the Pentagon, the Office of Naval Research, and the Arms Control and Disarmament Agency at the U.S. State Department. == Contributions == Saaty was a Distinguished University Professor at the University of Pittsburgh. He has made contributions in the fields of operations research (parametric linear programming, epidemics and the spread of biological agents, queuing theory, and behavioral mathematics as it relates to operations), arms control and disarmament, and urban design. He has written more than 35 books and 350 papers on mathematics, operations research, and decision making. Their subjects include graph theory and its applications, nonlinear mathematics, analytical planning, and game theory and conflict resolution. According to the Mathematics Genealogy Project, he has had 14 doctoral students. Saaty himself was a student of Einar Hille at Yale. In line with his long-time interest in peace and conflict resolution, in 1983 Saaty proposed that an International Center for Conflict Resolution needs to be established that would have branches in many countries and would be staffed by retired diplomats, negotiators and conflict analysts. This idea was first published in an article "Center for Conflict Resolution," in the March 1984 issue of the Bulletin of the Atomic Scientists, and it later appeared as an appendix in his 1989 book on Conflict Resolution co-authored with J.M. Alexander. A current revised version of this proposal is posted here with his University of Pittsburgh vita. A 2002 article listing the most important contributions to operations research from 1954 to date listed four from Saaty: "Parametric Programming" (1954, with S. I. Gass), "Mathematical Methods of Operations Research" (1959), "Elements of Queueing Theory" (1961), and "The Analytic Hierarchy Process" (1980). The book on operations research was the first to summarize the formal mathematical methods in the field of Operations Research and was translated to Russian and Japanese. The comprehensive work on queueing theory was reviewed by D.G. Kendall of Oxford University in Mathematical Reviews who wrote that this book is "a substantial encyclopedia of queueing theory whose scope is indicated by the 910 items in the bibliography at the end of the book." The book "Mathematical Methods of Arms Control and Disarmament" was reviewed in Management Science in April 1969, "This fascinating book is an important contribution to the present task of discovering some valid underlying mathematical structures in politics...highly recommended both because of its numerous fascinating models and because of the deadly importance of its subject." The Analytic Hierarchy Process itself anticipates the PageRank algorithm by more than 20 years, with the same basic idea of using the eigenvector corresponding to the largest eigenvalue of a suitable matrix. Saaty has been elected to the National Academy of Engineering (2005), and the Real Academia de Ciencias Exactas, Físicas y Naturales (Spanish Royal Academy of Sciences, 1971). In 1973, he received the Lester R. Ford Award from the Mathematical Association of America for expository excellence in his paper "Thirteen Colorful Variations on Guthrie's Four-color Conjecture" on the four color problem, and in 2000 he was awarded the gold medal of the International Society on Multi-criteria Decision Making. He is the 2007 recipient of the Akao Prize Archived 2007-08-21 at the Wayback Machine of the QFD Institute. In 2008, he received the INFORMS Impact Prize for his development of the Analytic Hierarchy Process. The Impact Prize is awarded every two years to recognize contributions that have had a broad impact on the fields of operations research and the management sciences. Emphasis is placed on the breadth of the impact of an idea or body of research. In 2011 he was awarded, in a ceremony with ancient roots on YouTube, the Doktor Honoris Causa degree by Jagiellonian University in Kraków, Poland. In December 2011 he received the Herbert Simon Award for Outstanding Contribution in Information Technology and Decision Making for the paper "The Possibility Of Group Welfare Functions" coauthored with Professor Luis G. Vargas, published in the International Journal of Information Technology & Decision Making (IJITDM), Most of the university scholars are working on the basis he provided, Anum Bakhtyar. Saaty died at the age of 91 on August 14, 2017, 14 months after a cancer diagnosis. == Education == PhD, Mathematics, Yale University, 1953 (thesis, under Einar Carl Hille: "On the Bessel Tricomi Equation"). Post-graduate study, University of Paris, 1952–53. MA, Mathematics, Yale University, 1951. MS, Physics, The Catholic University of America, 1949. BA, Columbia Union College, 1948. == Bibliography == === Analytic hierarchy process (AHP) === 1980 The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation, ISBN 0-07-054371-2, McGraw-Hill 1982 Decision Making for Leaders: The Analytical Hierarchy Process for Decisions in a Complex World, ISBN 0-534-97959-9, Wadsworth. 1988, Paperback, ISBN 0-9620317-0-4, RWS 1982 The Logic of Priorities: Applications in Business, Energy, Health, and Transportation, with Luis G. Vargas, ISBN 0-89838-071-5 (Hardcover) ISBN 0-89838-078-2 (Paperback), Kluwer-Nijhoff 1985 Analytical Planning: The Organization of Systems, with Kevin P. Kearns, ISBN 0-08-032599-8, Pergamon 1989 Conflict Resolution: The Analytic Hierarchy Process, with Joyce Alexander, ISBN 0-275-93229-X, Praeger 1991 Prediction, Projection and Forecasting: Applications of the Analytic Hierarchy Process in Economics, Finance, Politics, Games and Sports, with Luis G. Vargas, ISBN 0-7923-9104-7, Kluwer Academic 1992 The Hierarchon: A Dictionary of Hierarchies, with Ernest H. Forman, ISBN 0-9620317-5-5, RWS 1994 Fundamentals of Decision Making and Priority Theory with the Analytic Hierarchy Process, ISBN 0-9620317-6-3, RWS 1994 Decision Making in Economic, Social and Technological Environments, with Luis G. Vargas, ISBN 0-9620317-7-1, RWS 1996 Vol. III and IV of the Analytic Hierarchy Process Series, ISBN 1-888603-07-0 RWS 2001 Models, Methods, Concepts & Applications of the Analytic Hierarchy Process, with Luis G. Vargas, ISBN 0-7923-7267-0, Kluwer Academic 2007 Group Decision Making: Drawing Out and Reconciling Differences, with Kirti Peniwati, ISBN 1-888603-08-9, RWS 2008 Decision making with the analytic hierarchy process, Int. J. Services Sciences, Vol. 1, No. 1, 2008 (http://www.colorado.edu/geography/leyk/geog_5113/readings/saaty_2008.pdf Archived 2017-09-18 at the Wayback Machine) - includes a statement of priority scales which measure intangibles in relative terms. === Analytic network process (ANP) === 1996 Decision Making with Dependence and Feedback: The Analytic Network Process, ISBN 0-9620317-9-8, RWS 2005 Theory and Applications of the Analytic Network Process: Decision Making with Benefits, Opportunities, Costs and Risks, ISBN 1-888603-06-2, RWS 2005 The Encyclicon, A Dictionary of Decisions with Dependence and Feedback based on the Analytic Network Process, with Müjgan S. Özdemir, ISBN 1-888603-05-4, RWS 2006 Decision Making with the Analytic Network Process: Economic, Political, Social and Technological Applications with Benefits, Opportunities, Costs and Risks, with Luis G. Vargas, ISBN 0-387-33859-4, Springer 2008 The Encyclicon, Volume 2: A Dictionary of Complex Decisions using the Analytic Network Process, with Brady Cillo, ISBN 1-888603-09-7, RWS 2008 The analytic hierarchy and analytic network measurement processes: Applications to decisions under Risk', European Journal of Pure and Applied Mathematics, 1 (1), 122–196, (2008) === Neural network process (NNP) === 1999 The Brain: Unraveling the Mystery of How it Works, The Neural Network Process, ISBN 1-888603-02-X, RWS 2009 Principia Mathematica Decernendi: Mathematical Principles of Decision Making - Generalization of the Analytic Network Process to Neural Firing and Synthesis, ISBN 1-888603-10-0, RWS === Operations research === 1959 Mathematical Methods of Operations Research, no ISBN (translated into Japanese and Russian), McGraw-Hill. 1988 Extended edition, ISBN 0-486-65703-5, Dover (paperback) 1961 Elements of Queueing Theory with Applications, no ISBN (translated into Russian, Spanish and German), McGraw-Hill === Mathematics === 1964 Nonlinear Mathematics, with J. Bram, no ISBN, McGraw-Hill. 1981 Reprinted as ISBN 0-486-64233-X, Dover (paperback) 1964–1965 Lectures on Modern Mathematics, Volumes I, II, III (Thomas L. Saaty, Editor), no ISBN (translated into Japanese), John Wiley 1965 Finite Graphs and Networks, with R. Busacker, no ISBN (translated into Japanese, Russian, German and Hungarian), McGraw-Hill 1967 Modern Nonlinear Equations, no ISBN, McGraw-Hill. 1981, reprinted as ISBN 0-486-64232-1, Dover (paperback) 1969 The Spirit and Uses of the Mathematical Sciences, (Thomas L. Saaty, Editor, with F.J. Weyl), no ISBN, McGraw-Hill 1970 Optimization in Integers and Related Extremal Problems, no ISBN (translated into Russian), McGraw-Hill 1977 The Four-Color Problem; Assaults and Conquest, ISBN 0-07-054382-8, with Paul C. Kainen, McGraw-Hill. 1986 Revised edition, ISBN 0-486-65092-8, Dover (paperback) === Applied mathematics === 1968 Mathematical Models of Arms Control and Disarmament, ISBN 0-471-74810-2, John Wiley 1973 Topics in Behavioral Mathematics, no ISBN, Mathematical Association of America 1981 Thinking with Models: Mathematical Models in the Physical, Biological, and Social Sciences, with Joyce Alexander, hardback ISBN 0-08-026475-1, paperback ISBN 978-0-08-026474-5, Pergamon === Other topics === 1973 Compact City, with George B. Dantzig, hardback ISBN 0-7167-0784-5, paperback ISBN 0-7167-0794-2 (translated into Japanese and Russian), W.H. Freeman 1990 Embracing the Future, with Larry W. Boone, ISBN 0-275-93573-6, Praeger 2001 Creative Thinking, Problem Solving & Decision Making, ISBN 1-888603-03-8, RWS 2013 Compact City: The Next Urban Evolution in Response to Climate Change, ISBN 1-888603-12-7, RWS == References == == External links == University of Pittsburgh faculty biography of Thomas L. Saaty Decision Lens Board of Advisors Real Academia de Ciencias, Relación de Académicos Biography of Thomas L. Saaty from the Institute for Operations Research and the Management Sciences
|
Wikipedia:Thomas R. Kane#0
|
Thomas Reif Kane (March 23, 1924 – February 16, 2019) was a professor emeritus of applied mechanics at Stanford University. == Early life == Kane was born in Vienna, Austria. He immigrated to the United States with his parents in 1938 after Austria fell to Nazi Germany. In 1943, he enlisted in the United States Army and was stationed in the South Pacific as a combat photographer. From 1946 to 1953 he attended Columbia University during which he earned two BS degrees in mathematics and civil engineering, as well as an MS in civil engineering and a PhD in applied mechanics. == Career == In 1953, Dr. Kane joined the engineering faculty at the University of Pennsylvania as an assistant professor of mechanical engineering and three years later was promoted to associate professor. While at Penn, he served as a research engineer and on the committee whose focus was investigating the question of sabbatical leave. In the 1960s, Kane devised a method for formulating equations of motion for complex mechanical systems that requires less labor and leads to simpler equations than the classical approaches, while avoiding the vagueness of virtual quantities. The method is based on the use of partial angular velocities and partial velocities. == References ==
|
Wikipedia:Thomas Simpson#0
|
Thomas Simpson FRS (20 August 1710 – 14 May 1761) was a British mathematician and inventor known for the eponymous Simpson's rule to approximate definite integrals. The attribution, as often in mathematics, can be debated: this rule had been found 100 years earlier by Johannes Kepler, and in German it is called Keplersche Fassregel, or roughly "Kepler's Barrel Rule". == Biography == Simpson was born in Sutton Cheney, Leicestershire. The son of a weaver, Simpson taught himself mathematics. At the age of nineteen, he married a fifty-year old widow with two children. As a youth, he became interested in astrology after seeing a solar eclipse. He also dabbled in divination and caused fits in a girl after 'raising a devil' from her. After this incident, he and his wife fled to Derby. He moved with his wife and children to London at age twenty-five, where he supported his family by weaving during the day and teaching mathematics at night. From 1743, he taught mathematics at the Royal Military Academy, Woolwich. Simpson was a fellow of the Royal Society. In 1758, Simpson was elected a foreign member of the Royal Swedish Academy of Sciences. He died in Market Bosworth, and was laid to rest in Sutton Cheney. A plaque inside the church commemorates him. == Early work == Simpson's treatise entitled The Nature and Laws of Chance and The Doctrine of Annuities and Reversions were based on the work of De Moivre and were attempts at making the same material more brief and understandable. Simpson stated this clearly in The Nature and Laws of Chance, referring to Abraham De Moivre's The Doctrine of Chances: "tho' it neither wants Matter nor Elegance to recommend it, yet the Price must, I am sensible, have put it out of the Power of many to purchase it". In both works, Simpson cited De Moivre's work and did not claim originality beyond the presentation of some more accurate data. While he and De Moivre initially got along, De Moivre eventually felt that his income was threatened by Simpson's work and in his second edition of Annuities upon Lives, wrote in the preface: "After the pains I have taken to perfect this Second Edition, it may happen, that a certain Person, whom I need not name, out of Compassion to the Public, will publish a Second Edition of his Book on the same Subject, which he will afford at a very moderate Price, not regarding whether he mutilates my Propositions, obscures what is clear, makes a Shew of new Rules, and works by mine; in short, confounds, in his usual way, every thing with a croud of useless Symbols; if this be the Case, I must forgive the indigent Author, and his disappointed Bookseller." == Work == The method commonly called Simpson's Rule was known and used earlier by Bonaventura Cavalieri (a student of Galileo) in 1639, and later by James Gregory; still, the long popularity of Simpson's textbooks invites this association with his name, in that many readers would have learnt it from them. In the context of disputes surrounding methods advanced by René Descartes, Pierre de Fermat proposed the challenge to find a point D such that the sum of the distances to three given points, A, B and C is least, a challenge popularised in Italy by Marin Mersenne in the early 1640s. Simpson treats the problem in the first part of Doctrine and Application of Fluxions (1750), on pp. 26–28, by the description of circular arcs at which the edges of the triangle ABC subtend an angle of pi/3; in the second part of the book, on pp. 505–506 he extends this geometrical method, in effect, to weighted sums of the distances. Several of Simpson's books contain selections of optimisation problems treated by simple geometrical considerations in similar manner, as (for Simpson) an illuminating counterpart to possible treatment by fluxional (calculus) methods. But Simpson does not treat the problem in the essay on geometrical problems of maxima and minima appended to his textbook on Geometry of 1747, although it does appear in the considerably reworked edition of 1760. Comparative attention might, however, usefully be drawn to a paper in English from eighty years earlier as suggesting that the underlying ideas were already recognised then: J. Collins A Solution, Given by Mr. John Collins of a Chorographical Probleme, Proposed by Richard Townley Esq. Who Doubtless Hath Solved the Same Otherwise, Philosophical Transactions of the Royal Society of London, 6 (1671), pp. 2093–2096. Of further related interest are problems posed in the early 1750s by J. Orchard, in The British Palladium, and by T. Moss, in The Ladies' Diary; or Woman's Almanack (at that period not yet edited by Simpson). == Simpson-Weber triangle problem == This type of generalisation was later popularised by Alfred Weber in 1909. The Simpson-Weber triangle problem consists in locating a point D with respect to three points A, B, and C in such a way that the sum of the transportation costs between D and each of the three other points is minimised. In 1971, Luc-Normand Tellier found the first direct (non iterative) numerical solution of the Fermat and Simpson-Weber triangle problems. Long before Von Thünen's contributions, which go back to 1818, the Fermat point problem can be seen as the very beginning of space economy. In 1985, Luc-Normand Tellier formulated an all-new problem called the “attraction-repulsion problem”, which constitutes a generalisation of both the Fermat and Simpson-Weber problems. In its simplest version, the attraction-repulsion problem consists in locating a point D with respect to three points A1, A2 and R in such a way that the attractive forces exerted by points A1 and A2, and the repulsive force exerted by point R cancel each other out. In the same book, Tellier solved that problem for the first time in the triangle case, and he reinterpreted the space economy theory, especially, the theory of land rent, in the light of the concepts of attractive and repulsive forces stemming from the attraction-repulsion problem. That problem was later further analysed by mathematicians like Chen, Hansen, Jaumard and Tuy (1992), and Jalal and Krarup (2003). The attraction-repulsion problem is seen by Ottaviano and Thisse (2005) as a prelude to the New Economic Geography that developed in the 1990s, and earned Paul Krugman a Nobel Memorial Prize in Economic Sciences in 2008. == Publications == Treatise of Fluxions (1737) The Nature and Laws of Chance (1740) Essays on several curious and useful subjects, in speculative and mix'd mathematicks. London: John Nourse. 1740. The Doctrine of Annuities and Reversions (1742) Mathematical dissertations on a variety of physical and analytical subjects. London: Thomas Woodward. 1743. A Treatise of Algebra (1745) Elements of Plane Geometry. To which are added, An Essay on the Maxima and Minima of Geometrical Quantities, And a brief Treatise of regular Solids; Also, the Mensuration of both Superficies and Solids, together with the Construction of a large Variety of Geometrical Problems (Printed for the Author; Samuel Farrer; and John Turner, London, 1747) [The book is described as being Designed for the Use of Schools and the main body of text is Simpson's reworking of the early books of The Elements of Euclid. Simpson is designated Professor of Geometry in the Royal Academy at Woolwich.] Trigonometry, Plane and Spherical (1748) Doctrine and Application of Fluxions. Containing (besides what is common on the subject) a Number of New Improvements on the Theory. And the Solution of a Variety of New, and very Interesting, Problems in different Branches of the Mathematicks (two parts bound in one volume; J. Nourse, London, 1750) Select Exercises in Mathematics (1752) Miscellaneous tracts on some curious, and very interesting subjects in mechanics, physical-astronomy, and speculative mathematics. London: John Nourse. 1757. Miscellaneous tracts on some curious and very interesting subjects in mechanics, physical-astronomy and speculative mathematics. London: John Nourse. 1768. == See also == Probability Series multisection Simpson's rules (ship stability) == References == == External links == Thomas Simpson and his Work on Maxima and Minima at Convergence "Simpson, Thomas" . Encyclopædia Britannica. Vol. 25 (11th ed.). 1911. pp. 135–136. O'Connor, John J.; Robertson, Edmund F., "Thomas Simpson", MacTutor History of Mathematics Archive, University of St Andrews
|
Wikipedia:Thomas William Körner#0
|
Thomas William Körner (born 17 February 1946) is a British pure mathematician and the author of three books on popular mathematics. He is titular Professor of Fourier Analysis in the University of Cambridge and a Fellow of Trinity Hall. He is the son of the philosopher Stephan Körner and of Edith Körner. He studied at Trinity Hall, Cambridge, and wrote his PhD thesis Some Results on Kronecker, Dirichlet and Helson Sets there in 1971, studying under Nicholas Varopoulos. In 1972 he won the Salem Prize. He has written academic mathematics books aimed at undergraduates: Fourier Analysis Exercises for Fourier Analysis A Companion to Analysis Vectors, Pure and Applied Calculus for the Ambitious He has also written three books aimed at secondary school students, the popular 1996 title The Pleasures of Counting, Naive Decision Making (published 2008) on probability, statistics and game theory, and Where Do Numbers Come From? (published October 2019). == References == == External links == Professor Körner's website
|
Wikipedia:Thoralf Skolem#0
|
Thoralf Albert Skolem (Norwegian: [ˈtûːrɑɫf ˈskûːlɛm]; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory. == Life == Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skolem attended secondary school in Kristiania (later renamed Oslo), passing the university entrance examinations in 1905. He then entered Det Kongelige Frederiks Universitet to study mathematics, also taking courses in physics, chemistry, zoology and botany. In 1909, he began working as an assistant to the physicist Kristian Birkeland, known for bombarding magnetized spheres with electrons and obtaining aurora-like effects; thus Skolem's first publications were physics papers written jointly with Birkeland. In 1913, Skolem passed the state examinations with distinction, and completed a dissertation titled Investigations on the Algebra of Logic. He also traveled with Birkeland to the Sudan to observe the zodiacal light. He spent the winter semester of 1915 at the University of Göttingen, at the time the leading research center in mathematical logic, metamathematics, and abstract algebra, fields in which Skolem eventually excelled. In 1916 he was appointed a research fellow at Det Kongelige Frederiks Universitet. In 1918, he became a Docent in Mathematics and was elected to the Norwegian Academy of Science and Letters. Skolem did not at first formally enroll as a Ph.D. candidate, believing that the Ph.D. was unnecessary in Norway. He later changed his mind and submitted a thesis in 1926, titled Some theorems about integral solutions to certain algebraic equations and inequalities. His notional thesis advisor was Axel Thue, even though Thue had died in 1922. In 1927, he married Edith Wilhelmine Hasvold. Skolem continued to teach at Det kongelige Frederiks Universitet (renamed the University of Oslo in 1939) until 1930 when he became a Research Associate in Chr. Michelsen Institute in Bergen. This senior post allowed Skolem to conduct research free of administrative and teaching duties. However, the position also required that he reside in Bergen, a city which then lacked a university and hence had no research library, so that he was unable to keep abreast of the mathematical literature. In 1938, he returned to Oslo to assume the Professorship of Mathematics at the university. There he taught the graduate courses in algebra and number theory, and only occasionally on mathematical logic. Skolem's Ph.D. student Øystein Ore went on to a career in the USA. Skolem served as president of the Norwegian Mathematical Society, and edited the Norsk Matematisk Tidsskrift ("The Norwegian Mathematical Journal") for many years. He was also the founding editor of Mathematica Scandinavica. After his 1957 retirement, he made several trips to the United States, speaking and teaching at universities there. He remained intellectually active until his sudden and unexpected death. For more on Skolem's academic life, see Fenstad (1970). == Mathematics == Skolem published around 180 papers on Diophantine equations, group theory, lattice theory, and most of all, set theory and mathematical logic. He mostly published in Norwegian journals with limited international circulation, so that his results were occasionally rediscovered by others. An example is the Skolem–Noether theorem, characterizing the automorphisms of simple algebras. Skolem published a proof in 1927, but Emmy Noether independently rediscovered it a few years later. Skolem was among the first to write on lattices. In 1912, he was the first to describe a free distributive lattice generated by n elements. In 1919, he showed that every implicative lattice (now also called a Skolem lattice) is distributive and, as a partial converse, that every finite distributive lattice is implicative. After these results were rediscovered by others, Skolem published a 1936 paper in German, "Über gewisse 'Verbände' oder 'Lattices'", surveying his earlier work in lattice theory. Skolem was a pioneer model theorist. In 1920, he greatly simplified the proof of a theorem Leopold Löwenheim first proved in 1915, resulting in the Löwenheim–Skolem theorem, which states that if a countable first-order theory has an infinite model, then it has a countable model. His 1920 proof employed the axiom of choice, but he later (1922 and 1928) gave proofs using Kőnig's lemma in place of that axiom. It is notable that Skolem, like Löwenheim, wrote on mathematical logic and set theory employing the notation of his fellow pioneering model theorists Charles Sanders Peirce and Ernst Schröder, including Π, Σ as variable-binding quantifiers, in contrast to the notations of Peano, Principia Mathematica, and Principles of Mathematical Logic. Skolem (1934) pioneered the construction of non-standard models of arithmetic and set theory. Skolem (1922) refined Zermelo's axioms for set theory by replacing Zermelo's vague notion of a "definite" property with any property that can be coded in first-order logic. The resulting axiom is now part of the standard axioms of set theory. Skolem also pointed out that a consequence of the Löwenheim–Skolem theorem is what is now known as Skolem's paradox: If Zermelo's axioms are consistent, then they must be satisfiable within a countable domain, even though they prove the existence of uncountable sets. == Completeness == The completeness of first-order logic is a corollary of results Skolem proved in the early 1920s and discussed in Skolem (1928), but he failed to note this fact, perhaps because mathematicians and logicians did not become fully aware of completeness as a fundamental metamathematical problem until the 1928 first edition of Hilbert and Ackermann's Principles of Mathematical Logic clearly articulated it. In any event, Kurt Gödel first proved this completeness in 1930. Skolem distrusted the completed infinite and was one of the founders of finitism in mathematics. Skolem (1923) sets out his primitive recursive arithmetic, a very early contribution to the theory of computable functions, as a means of avoiding the so-called paradoxes of the infinite. Here he developed the arithmetic of the natural numbers by first defining objects by primitive recursion, then devising another system to prove properties of the objects defined by the first system. These two systems enabled him to define prime numbers and to set out a considerable amount of number theory. If the first of these systems can be considered as a programming language for defining objects, and the second as a programming logic for proving properties about the objects, Skolem can be seen as an unwitting pioneer of theoretical computer science. In 1929, Presburger proved that Peano arithmetic without multiplication was consistent, complete, and decidable. The following year, Skolem proved that the same was true of Peano arithmetic without addition, a system named Skolem arithmetic in his honor. Gödel's famous 1931 result is that Peano arithmetic itself (with both addition and multiplication) is incompletable and hence a posteriori undecidable. Hao Wang praised Skolem's work as follows: Skolem tends to treat general problems by concrete examples. He often seemed to present proofs in the same order as he came to discover them. This results in a fresh informality as well as a certain inconclusiveness. Many of his papers strike one as progress reports. Yet his ideas are often pregnant and potentially capable of wide application. He was very much a 'free spirit': he did not belong to any school, he did not found a school of his own, he did not usually make heavy use of known results... he was very much an innovator and most of his papers can be read and understood by those without much specialized knowledge. It seems quite likely that if he were young today, logic... would not have appealed to him. (Skolem 1970: 17-18) For more on Skolem's accomplishments, see Hao Wang (1970). == See also == Leopold Löwenheim Model theory Skolem arithmetic Skolem normal form Skolem's paradox Skolem problem Skolem sequence Skolem–Mahler–Lech theorem == References == === Primary === Skolem, Thoralf (1934). "Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschließlich Zahlenvariablen" (PDF). Fundamenta Mathematicae (in German). 23 (1): 150–161. doi:10.4064/fm-23-1-150-161. Skolem, T. A., 1970. Selected works in logic, Fenstad, J. E., ed. Oslo: Scandinavian University Books. Contains 22 articles in German, 26 in English, 2 in French, 1 English translation of an article originally published in Norwegian, and a complete bibliography. Skolem, Thoralf (23 April 2018). Zach, Richard (ed.). "Skolem's 1920, 1923 Papers". richardzach.org. Retrieved 4 January 2024. === Writings in English translation === Jean van Heijenoort, 1967. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press. 1920. "Logico-combinatorial investigations on the satisfiability or provability of mathematical propositions: A simplified proof of a theorem by Löwenheim," 252–263. 1922. "Some remarks on axiomatized set theory," 290-301. 1923. "The foundations of elementary arithmetic," 302-33. 1928. "On mathematical logic," 508–524. === Secondary === Brady, Geraldine, 2000. From Peirce to Skolem. North Holland. Fenstad, Jens Erik, 1970, "Thoralf Albert Skolem in Memoriam" in Skolem (1970: 9–16). Hao Wang, 1970, "A survey of Skolem's work in logic" in Skolem (1970: 17–52). == External links == O'Connor, John J.; Robertson, Edmund F., "Thoralf Skolem", MacTutor History of Mathematics Archive, University of St Andrews Thoralf Skolem at the Mathematics Genealogy Project Fenstad, Jens Erik, 1996, "Thoralf Albert Skolem 1887-1963: A Biographical Sketch," Nordic Journal of Philosophical Logic 1: 99-106.
|
Wikipedia:Thue–Morse sequence#0
|
In mathematics, the Thue–Morse or Prouhet–Thue–Morse sequence is the binary sequence (an infinite sequence of 0s and 1s) that can be obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far. It is sometimes called the fair share sequence because of its applications to fair division or parity sequence. The first few steps of this procedure yield the strings 0, 01, 0110, 01101001, 0110100110010110, and so on, which are the prefixes of the Thue–Morse sequence. The full sequence begins: 01101001100101101001011001101001.... The sequence is named after Axel Thue, Marston Morse and (in its extended form) Eugène Prouhet. == Definition == There are several equivalent ways of defining the Thue–Morse sequence. === Direct definition === To compute the nth element tn, write the number n in binary. If the number of ones in this binary expansion is odd then tn = 1, if even then tn = 0. That is, tn is the even parity bit for n. John H. Conway et al. deemed numbers n satisfying tn = 1 to be odious (intended to be similar to odd) numbers, and numbers for which tn = 0 to be evil (similar to even) numbers. === Fast sequence generation === This method leads to a fast method for computing the Thue–Morse sequence: start with t0 = 0, and then, for each n, find the highest-order bit in the binary representation of n that is different from the same bit in the representation of n − 1. If this bit is at an even index, tn differs from tn − 1, and otherwise it is the same as tn − 1. In Python: The resulting algorithm takes constant time to generate each sequence element, using only a logarithmic number of bits (constant number of words) of memory. === Recurrence relation === The Thue–Morse sequence is the sequence tn satisfying the recurrence relation t 0 = 0 , t 2 n = t n , t 2 n + 1 = 1 − t n , {\displaystyle {\begin{aligned}t_{0}&=0,\\t_{2n}&=t_{n},\\t_{2n+1}&=1-t_{n},\end{aligned}}} for all non-negative integers n. === L-system === The Thue–Morse sequence is a morphic word: it is the output of the following Lindenmayer system: === Characterization using bitwise negation === The Thue–Morse sequence in the form given above, as a sequence of bits, can be defined recursively using the operation of bitwise negation. So, the first element is 0. Then once the first 2n elements have been specified, forming a string s, then the next 2n elements must form the bitwise negation of s. Now we have defined the first 2n+1 elements, and we recurse. Spelling out the first few steps in detail: We start with 0. The bitwise negation of 0 is 1. Combining these, the first 2 elements are 01. The bitwise negation of 01 is 10. Combining these, the first 4 elements are 0110. The bitwise negation of 0110 is 1001. Combining these, the first 8 elements are 01101001. And so on. So T0 = 0. T1 = 01. T2 = 0110. T3 = 01101001. T4 = 0110100110010110. T5 = 01101001100101101001011001101001. T6 = 0110100110010110100101100110100110010110011010010110100110010110. And so on. In Python: Which can then be converted to a (reversed) string as follows: === Infinite product === The sequence can also be defined by: ∏ i = 0 ∞ ( 1 − x 2 i ) = ∑ j = 0 ∞ ( − 1 ) t j x j , {\displaystyle \prod _{i=0}^{\infty }\left(1-x^{2^{i}}\right)=\sum _{j=0}^{\infty }(-1)^{t_{j}}x^{j},} where tj is the jth element if we start at j = 0. == Properties == The Thue–Morse sequence contains many squares: instances of the string X X {\displaystyle XX} , where X {\displaystyle X} denotes the string A {\displaystyle A} , A ¯ {\displaystyle {\overline {A}}} , A A ¯ A {\displaystyle A{\overline {A}}A} , or A ¯ A A ¯ {\displaystyle {\overline {A}}A{\overline {A}}} , where A = T k {\displaystyle A=T_{k}} for some k ≥ 0 {\displaystyle k\geq 0} and A ¯ {\displaystyle {\overline {A}}} is the bitwise negation of A {\displaystyle A} . For instance, if k = 0 {\displaystyle k=0} , then A = T 0 = 0 {\displaystyle A=T_{0}=0} . The square A A ¯ A A A ¯ A = 010010 {\displaystyle A{\overline {A}}AA{\overline {A}}A=010010} appears in T {\displaystyle T} starting at the 16th bit. Since all squares in T {\displaystyle T} are obtained by repeating one of these 4 strings, they all have length 2 n {\displaystyle 2^{n}} or 3 ⋅ 2 n {\displaystyle 3\cdot 2^{n}} for some n ≥ 0 {\displaystyle n\geq 0} . T {\displaystyle T} contains no cubes: instances of X X X {\displaystyle XXX} . There are also no overlapping squares: instances of 0 X 0 X 0 {\displaystyle 0X0X0} or 1 X 1 X 1 {\displaystyle 1X1X1} . The critical exponent of T {\displaystyle T} is 2. The Thue–Morse sequence is a uniformly recurrent word: given any finite string X in the sequence, there is some length nX (often much longer than the length of X) such that X appears in every block of length nX. Notably, the Thue–Morse sequence is uniformly recurrent without being either periodic or eventually periodic (i.e., periodic after some initial nonperiodic segment). The sequence T2n is a palindrome for any n. Furthermore, let qn be a word obtained by counting the ones between consecutive zeros in T2n . For instance, q1 = 2 and q2 = 2102012. Since Tn does not contain overlapping squares, the words qn are palindromic squarefree words. The Thue–Morse morphism μ is defined on alphabet {0,1} by the substitution map μ(0) = 01, μ(1) = 10: every 0 in a sequence is replaced with 01 and every 1 with 10. If T is the Thue–Morse sequence, then μ(T) is also T. Thus, T is a fixed point of μ. The morphism μ is a prolongable morphism on the free monoid {0,1}∗ with T as fixed point: T is essentially the only fixed point of μ; the only other fixed point is the bitwise negation of T, which is simply the Thue–Morse sequence on (1,0) instead of on (0,1). This property may be generalized to the concept of an automatic sequence. The generating series of T over the binary field is the formal power series t ( Z ) = ∑ n = 0 ∞ T ( n ) Z n . {\displaystyle t(Z)=\sum _{n=0}^{\infty }T(n)Z^{n}\ .} This power series is algebraic over the field of rational functions, satisfying the equation Z + ( 1 + Z ) 2 t + ( 1 + Z ) 3 t 2 = 0 {\displaystyle Z+(1+Z)^{2}t+(1+Z)^{3}t^{2}=0} === In combinatorial game theory === The set of evil numbers (numbers n {\displaystyle n} with t n = 0 {\displaystyle t_{n}=0} ) forms a subspace of the nonnegative integers under nim-addition (bitwise exclusive or). For the game of Kayles, evil nim-values occur for few (finitely many) positions in the game, with all remaining positions having odious nim-values. === The Prouhet–Tarry–Escott problem === The Prouhet–Tarry–Escott problem can be defined as: given a positive integer N and a non-negative integer k, partition the set S = { 0, 1, ..., N-1 } into two disjoint subsets S0 and S1 that have equal sums of powers up to k, that is: ∑ x ∈ S 0 x i = ∑ x ∈ S 1 x i {\displaystyle \sum _{x\in S_{0}}x^{i}=\sum _{x\in S_{1}}x^{i}} for all integers i from 1 to k. This has a solution if N is a multiple of 2k+1, given by: S0 consists of the integers n in S for which tn = 0, S1 consists of the integers n in S for which tn = 1. For example, for N = 8 and k = 2, 0 + 3 + 5 + 6 = 1 + 2 + 4 + 7, 02 + 32 + 52 + 62 = 12 + 22 + 42 + 72. The condition requiring that N be a multiple of 2k+1 is not strictly necessary: there are some further cases for which a solution exists. However, it guarantees a stronger property: if the condition is satisfied, then the set of kth powers of any set of N numbers in arithmetic progression can be partitioned into two sets with equal sums. This follows directly from the expansion given by the binomial theorem applied to the binomial representing the nth element of an arithmetic progression. For generalizations of the Thue–Morse sequence and the Prouhet–Tarry–Escott problem to partitions into more than two parts, see Bolker, Offner, Richman and Zara, "The Prouhet–Tarry–Escott problem and generalized Thue–Morse sequences". === Fractals and turtle graphics === Using turtle graphics, a curve can be generated if an automaton is programmed with a sequence. When Thue–Morse sequence members are used in order to select program states: If t(n) = 0, move ahead by one unit, If t(n) = 1, rotate by an angle of π/3 radians (60°) The resulting curve converges to the Koch curve, a fractal curve of infinite length containing a finite area. This illustrates the fractal nature of the Thue–Morse Sequence. It is also possible to draw the curve precisely using the following instructions: If t(n) = 0, rotate by an angle of π radians (180°), If t(n) = 1, move ahead by one unit, then rotate by an angle of π/3 radians. === Equitable sequencing === In their book on the problem of fair division, Steven Brams and Alan Taylor invoked the Thue–Morse sequence but did not identify it as such. When allocating a contested pile of items between two parties who agree on the items' relative values, Brams and Taylor suggested a method they called balanced alternation, or taking turns taking turns taking turns . . . , as a way to circumvent the favoritism inherent when one party chooses before the other. An example showed how a divorcing couple might reach a fair settlement in the distribution of jointly-owned items. The parties would take turns to be the first chooser at different points in the selection process: Ann chooses one item, then Ben does, then Ben chooses one item, then Ann does. Lionel Levine and Katherine E. Stange, in their discussion of how to fairly apportion a shared meal such as an Ethiopian dinner, proposed the Thue–Morse sequence as a way to reduce the advantage of moving first. They suggested that “it would be interesting to quantify the intuition that the Thue–Morse order tends to produce a fair outcome.” Robert Richman addressed this problem, but he too did not identify the Thue–Morse sequence as such at the time of publication. He presented the sequences Tn as step functions on the interval [0,1] and described their relationship to the Walsh and Rademacher functions. He showed that the nth derivative can be expressed in terms of Tn. As a consequence, the step function arising from Tn is orthogonal to polynomials of order n − 1. A consequence of this result is that a resource whose value is expressed as a monotonically decreasing continuous function is most fairly allocated using a sequence that converges to Thue–Morse as the function becomes flatter. An example showed how to pour cups of coffee of equal strength from a carafe with a nonlinear concentration gradient, prompting a whimsical article in the popular press. Joshua Cooper and Aaron Dutle showed why the Thue–Morse order provides a fair outcome for discrete events. They considered the fairest way to stage a Galois duel, in which each of the shooters has equally poor shooting skills. Cooper and Dutle postulated that each dueler would demand a chance to fire as soon as the other's a priori probability of winning exceeded their own. They proved that, as the duelers’ hitting probability approaches zero, the firing sequence converges to the Thue–Morse sequence. In so doing, they demonstrated that the Thue–Morse order produces a fair outcome not only for sequences Tn of length 2n, but for sequences of any length. Thus the mathematics supports using the Thue–Morse sequence instead of alternating turns when the goal is fairness but earlier turns differ monotonically from later turns in some meaningful quality, whether that quality varies continuously or discretely. Sports competitions form an important class of equitable sequencing problems, because strict alternation often gives an unfair advantage to one team. Ignacio Palacios-Huerta proposed changing the sequential order to Thue–Morse to improve the ex post fairness of various tournament competitions, such as the kicking sequence of a penalty shoot-out in soccer. He did a set of field experiments with pro players and found that the team kicking first won 60% of games using ABAB (or T1), 54% using ABBA (or T2), and 51% using full Thue–Morse (or Tn). As a result, ABBA is undergoing extensive trials in FIFA (European and World Championships) and English Federation professional soccer (EFL Cup). An ABBA serving pattern has also been found to improve the fairness of tennis tie-breaks. In competitive rowing, T2 is the only arrangement of port- and starboard-rowing crew members that eliminates transverse forces (and hence sideways wiggle) on a four-membered coxless racing boat, while T3 is one of only four rigs to avoid wiggle on an eight-membered boat. Fairness is especially important in player drafts. Many professional sports leagues attempt to achieve competitive parity by giving earlier selections in each round to weaker teams. By contrast, fantasy football leagues have no pre-existing imbalance to correct, so they often use a “snake” draft (forward, backward, etc.; or T1). Ian Allan argued that a “third-round reversal” (forward, backward, backward, forward, etc.; or T2) would be even more fair. Richman suggested that the fairest way for “captain A” and “captain B” to choose sides for a pick-up game of basketball mirrors T3: captain A has the first, fourth, sixth, and seventh choices, while captain B has the second, third, fifth, and eighth choices. === Hash collisions === The initial 2k bits of the Thue–Morse sequence are mapped to 0 by a wide class of polynomial hash functions modulo a power of two, which can lead to hash collisions. === Riemann zeta function === Certain linear combinations of Dirichlet series whose coefficients are terms of the Thue–Morse sequence give rise to identities involving the Riemann Zeta function (Tóth, 2022 ). For instance: ∑ n ≥ 1 5 t n − 1 + 3 t n n 2 = 4 ζ ( 2 ) = 2 π 2 3 , ∑ n ≥ 1 9 t n − 1 + 7 t n n 3 = 8 ζ ( 3 ) , {\displaystyle {\begin{aligned}\sum _{n\geq 1}{\frac {5t_{n-1}+3t_{n}}{n^{2}}}&=4\zeta (2)={\frac {2\pi ^{2}}{3}},\\\sum _{n\geq 1}{\frac {9t_{n-1}+7t_{n}}{n^{3}}}&=8\zeta (3),\end{aligned}}} where ( t n ) n ≥ 0 {\displaystyle (t_{n})_{n\geq 0}} is the n t h {\displaystyle n^{\rm {th}}} term of the Thue–Morse sequence. In fact, for all s {\displaystyle s} with real part greater than 1 {\displaystyle 1} , we have ( 2 s + 1 ) ∑ n ≥ 1 t n − 1 n s + ( 2 s − 1 ) ∑ n ≥ 1 t n n s = 2 s ζ ( s ) . {\displaystyle (2^{s}+1)\sum _{n\geq 1}{\frac {t_{n-1}}{n^{s}}}+(2^{s}-1)\sum _{n\geq 1}{\frac {t_{n}}{n^{s}}}=2^{s}\zeta (s).} == History == The Thue–Morse sequence was first studied by Eugène Prouhet in 1851, who applied it to number theory. However, Prouhet did not mention the sequence explicitly; this was left to Axel Thue in 1906, who used it to found the study of combinatorics on words. The sequence was only brought to worldwide attention with the work of Marston Morse in 1921, when he applied it to differential geometry. The sequence has been discovered independently many times, not always by professional research mathematicians; for example, Max Euwe, a chess grandmaster and mathematics teacher, discovered it in 1929 in an application to chess: by using its cube-free property (see above), he showed how to circumvent the threefold repetition rule aimed at preventing infinitely protracted games by declaring repetition of moves a draw. At the time, consecutive identical board states were required to trigger the rule; the rule was later amended to the same board position reoccurring three times at any point, as the sequence shows that the consecutive criterion can be evaded forever. == See also == Dejean's theorem Fabius function First difference of the Thue–Morse sequence Gray code Komornik–Loreti constant Prouhet–Thue–Morse constant == Notes == == References == == Further reading == Bugeaud, Yann (2012). Distribution modulo one and Diophantine approximation. Cambridge Tracts in Mathematics. Vol. 193. Cambridge: Cambridge University Press. ISBN 978-0-521-11169-0. Zbl 1260.11001. Lothaire, M. (2005). Applied combinatorics on words. Encyclopedia of Mathematics and Its Applications. Vol. 105. A collective work by Jean Berstel, Dominique Perrin, Maxime Crochemore, Eric Laporte, Mehryar Mohri, Nadia Pisanti, Marie-France Sagot, Gesine Reinert, Sophie Schbath, Michael Waterman, Philippe Jacquet, Wojciech Szpankowski, Dominique Poulalhon, Gilles Schaeffer, Roman Kolpakov, Gregory Koucherov, Jean-Paul Allouche and Valérie Berthé. Cambridge: Cambridge University Press. ISBN 978-0-521-84802-2. Zbl 1133.68067. == External links == "Thue-Morse sequence", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Weisstein, Eric W. "Thue-Morse Sequence". MathWorld. Allouche, J.-P.; Shallit, J. O. The Ubiquitous Prouhet-Thue-Morse Sequence. (contains many applications and some history) Thue–Morse Sequence over (1,2) (sequence A001285 in the OEIS) OEIS sequence A000069 (Odious numbers: numbers with an odd number of 1's in their binary expansion) OEIS sequence A001969 (Evil numbers: numbers with an even number of 1's in their binary expansion) Reducing the influence of DC offset drift in analog IPs using the Thue-Morse Sequence. A technical application of the Thue–Morse Sequence MusiNum - The Music in the Numbers. Freeware to generate self-similar music based on the Thue–Morse Sequence and related number sequences. Parker, Matt. "The Fairest Sharing Sequence Ever" (video). standupmaths. Retrieved 20 January 2016.
|
Wikipedia:Thyra Eibe#0
|
Thyra Eibe (3 November 1866 – 4 January 1955) was a Danish mathematician and translator, the first woman to earn a mathematics degree from the University of Copenhagen. She is known for her translation of Euclid's Elements into the Danish Language. == Education and career == Eibe was one of ten children of a Copenhagen bookseller. After completing a degree in historical linguistics in 1889 from N. Zahle's School (then a girls' school), Eibe studied mathematics at the University of Copenhagen, and earned a cand.mag. there in 1895. She returned to Zahle's School as a teacher, also teaching boys at Slomann's School and becoming the first woman to become an advanced mathematics teacher for boys in Denmark. In 1898 she moved to H. Adler Community College, later to become the Sortedam Gymnasium, where she remained until 1934, serving as principal for a year in 1929–1930. == Contributions == In undertaking her translation of Euclid, Eibe was motivated by the earlier work of Danish historian Johan Ludvig Heiberg, who published an edition of Euclid's Elements in its original Greek, with translations into Latin. As well as her translations, Eibe wrote several widely used Danish mathematics textbooks. == Recognition == In 1942, she was given the Tagea Brandt Rejselegat, an award for Danish woman who have made a significant contribution in science, literature or art. == References ==
|
Wikipedia:Théodore Moutard#0
|
Théodore Florentin Moutard (27 July 1827 – 13 March 1901) was a French mining engineer who worked at the École des Mines and contributed to mathematical geometry. The Moutard transformation in inverse geometry is named after him. Moutard was born in Soultz, Haut-Rhin, to Florentin and Elisabeth BERNOU. He was educated at the École Polytechnique and graduated in 1846 and entered the École des Mines and after graduating in 1849 he joined the Mining corps but was discharged in 1852 as he refused to take the oath required following the overthrow of Napoleon III. He joined back in 1870 and became a professor of mechanics at the École des Mines in 1875. He was also an examiner for the École Polytechnique from 1883. Moutard contributed to the La grande encyclopédie and his mathematical work was on algebraic surfaces and differential geometry. He collaborated with Victor Poncelet on elliptic functions. He was made Commander of the Legion of Honor in 1899. Moutard married twice and had two sons and two daughters. One of his sons was Douard Julien Moutard (1877-1948). One daughter Berthe married the mathematician Hermann Laurent. A daughter from his second marriage, Elisabeth married André Bujeaud, a politician and photographer, in 1868. == References ==
|
Wikipedia:Théophile Lepage#0
|
Théophile Henri Joseph Lepage (24 March 1901 – 1 April 1991) was a Belgian mathematician. == Biography == Théophile Lepage was born in Limburg on March 24, 1901. Together with Alfred Errera he founded the seminar for mathematical analysis at the ULB. This seminar played an important role in the flourishing of the department of mathematics at this university. He was professor of mathematics at the University of Liège from 1928 till 1930. He taught differential and integral calculus at the ULB from 1931 till 1956 and higher analysis from 1956 till 1971. For 43 years he was a member of the Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique. On June 5, 1948, he was nominated a corresponding member and on June 9, 1956 an effective member of the Académie. In 1963 he became president of the Académie and director of the Klasse Wetenschappen. He was also active in the Belgisch Wiskundig Genootschap. He died in Verviers on April 1, 1991. == Mathematical work == At the ULB, the ideas and the enthusiasm of Théophile de Donder formed the foundation of a flourishing mathematical tradition. Thanks to student Théophile Lepage, external differential calculus acquired one of the most helpful methods introduced in mathematics during the 20th century, and one for which De Donder was a pioneer, presenting new applications in the resolution of a classical problem—the partial differential equation of Monge-Ampère—and in the synthesis of the methods of Théophile de Donder, Hermann Weyl and Constantin Carathéodory into a calculus of variations of multipal integrals. Thanks to the use of differential geometry, it is possible to avoid long and boring calculations. The results of Lepage were named in reference works. His methods are still inspiring contemporary mathematicians: Boener and Sniatycki talked about the congruence of Lepage; not so long ago, Demeter Krupka, introduced—beside the eulerian forms which correspond to the classical equations of the calculus of variations of Euler—the so-called lepagian forms or equivalents of Lepage in equations of variations on fiber spaces. We also have Lepage to thank for interesting results concerning linear representations of the symplectic group, and more specifically Lepage's dissolution of an outer potency of the product of an even number of duplicates of a complex surface. == References == == External links == Théophile Lepage at the Mathematics Genealogy Project
|
Wikipedia:Théophile Pépin#0
|
Jean François Théophile Pépin (14 May 1826 – 3 April 1904) was a French mathematician. Born in Cluses, Haute-Savoie, he became a Jesuit in 1846, and from 1850 to 1856 and from 1862 to 1871 he was Professor of Mathematics at various Jesuit colleges. He was appointed Professor of Canon Law in 1873, moving to Rome in 1880. He died in Lyon at the age of 77. His work centred on number theory. In 1876 he found a new proof of Fermat's Last Theorem for n = 7, and in 1880 he published the first general solution to Frénicle de Bessy's problem x2 + y2 = z2, x2 = u2 + v2, x − y = u − v. He also gave his name to Pépin's test, a test of primality for Fermat numbers. == References == == Bibliography == Franz Lemmermeyer. "A Note on Pépin's counter examples to the Hasse principle for curves of genus 1." Abh. Math. Sem. Hamburg 69 (1999), 335–345.
|
Wikipedia:Théophile de Donder#0
|
Théophile Ernest de Donder (French: [də dɔ̃dɛʁ]; 19 August 1872 – 11 May 1957) was a Belgian mathematician, physicist and chemist famous for his work (published in 1923) in developing correlations between the Newtonian concept of chemical affinity and the Gibbsian concept of free energy. == Education == He received his doctorate in physics and mathematics from the Université Libre de Bruxelles in 1899, for a thesis entitled Sur la Théorie des Invariants Intégraux (On the Theory of Integral Invariants). == Career == He was professor between 1911 and 1942, at the Université Libre de Bruxelles. Initially he continued the work of Henri Poincaré and Élie Cartan. From 1914 on, he was influenced by the work of Albert Einstein and was an enthusiastic proponent of the theory of relativity. He gained significant reputation in 1923, when he developed his definition of chemical affinity. He pointed out a connection between the chemical affinity and the Gibbs free energy. He is considered the father of thermodynamics of irreversible processes. De Donder's work was later developed further by Ilya Prigogine. De Donder was an associate and friend of Albert Einstein. He was in 1927, one of the participants of the fifth Solvay Conference on Physics, that took place at the International Solvay Institute for Physics in Belgium. == Books by De Donder == Thermodynamic Theory of Affinity: A Book of Principles. Oxford, England: Oxford University Press (1936) The Mathematical Theory of Relativity. Cambridge, MA: MIT (1927) Sur la théorie des invariants intégraux (thesis) (1899). Théorie du champ électromagnétique de Maxwell-Lorentz et du champ gravifique d'Einstein (1917) La gravifique Einsteinienne (1921) Introduction à la gravifique einsteinienne (1925) Théorie mathématique de l'électricité (1925) Théorie des champs gravifiques (1926) Application de la gravifique einsteinienne (1930) Théorie invariantive du calcul des variations (1931) == See also == Klein–Gordon equation Schrödinger equation == References == == External links == Theophile de Donder - Science World at Wolfram.com Prigogine on de Donder De Donder's math genealogy De Donder's academic tree
|
Wikipedia:Tian yuan shu#0
|
Tian yuan shu (simplified Chinese: 天元术; traditional Chinese: 天元術; pinyin: tiān yuán shù) is a Chinese system of algebra for polynomial equations. Some of the earliest existing writings were created in the 13th century during the Yuan dynasty. However, the tianyuanshu method was known much earlier, in the Song dynasty and possibly before. == History == The Tianyuanshu was explained in the writings of Zhu Shijie (Jade Mirror of the Four Unknowns) and Li Zhi (Ceyuan haijing), two Chinese mathematicians during the Mongol Yuan dynasty. However, after the Ming overthrew the Mongol Yuan, Zhu and Li's mathematical works went into disuse as the Ming literati became suspicious of knowledge imported from Mongol Yuan times. Only recently, with the advent of modern mathematics in China, has the tianyuanshu been re-deciphered. Meanwhile, tian yuan shu arrived in Japan, where it is called tengen-jutsu. Zhu's text Suanxue qimeng was deciphered and was important in the development of Japanese mathematics (wasan) in the 17th and 18th centuries. == Description == Tian yuan shu means "method of the heavenly element" or "technique of the celestial unknown". The "heavenly element" is the unknown variable, usually written x in modern notation. It is a positional system of rod numerals to represent polynomial equations. For example, 2x2 + 18x − 316 = 0 is represented as , which in Arabic numerals is The 元 (yuan) denotes the unknown x, so the numerals on that line mean 18x. The line below is the constant term (-316) and the line above is the coefficient of the quadratic (x2) term. The system accommodates arbitrarily high exponents of the unknown by adding more lines on top and negative exponents by adding lines below the constant term. Decimals can also be represented. In later writings of Li Zhi and Zhu Shijie, the line order was reversed so that the first line is the lowest exponent. == See also == Yigu yanduan Ceyuan haijing == References == === Bibliography === Martzloff, Jean-Claude (2006). A History of Chinese Mathematics. trans. Stephen S. Wilson. Springer. pp. 258–272. ISBN 3-540-33782-2. Retrieved 2009-12-28. Murata, Tamotsu (2003). "Indigenous Japanese mathematics, Wasan". In Ivor Grattan-Guinness (ed.). Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. Vol. 1. JHU Press. pp. 105–106. ISBN 0-8018-7396-7. Retrieved 2009-12-28.
|
Wikipedia:Tianxin Cai#0
|
Cai Tianxin (Chinese: 蔡天新, born March 3, 1963, in Taizhou, Zhejiang) is a Chinese mathematician, poet and essayist noted for his books Mathematical Legends, A Brief History of Mathematics, Mathematics an Arts, A Modern Introduction to Classical Number Theory, Little memory: my Childhood in Mao’s Time, etc. He is a professor in the Mathematical School of Zhejiang University. == Early life and education == Cai was born in Taizhou, Zhejiang Province. He spent his childhood around 7 villages and one small town in southeastern China. He gained bachelor (1982), master (1984) and doctorate (1987) degrees at Shandong University, and his doctoral advisor was Pan Chengdong (潘承洞), whose supervisor Ming Shihe got Ph.D. in University of Oxford under the direct of E. C. Titchmarch. Cai became full professor in Hangzhou University in 1994, and full professor in Zhejiang University since 1998. == Research interests == Additive and multiplicative number theory, perfect numbers, congruence modulo integer power, Witten zeta values; history of mathematics, history of arts. == Writing and Publications == Cai has published more than 30 books of poetry, essays, travels, photograph, autobiography, popular mathematics and number theory. His work has been translated into more than 20 languages, and he has published more than 20 books worldwide. He has translated or edited 8 volumes of modern world poetry. He was selected by Herinrich and Jane Ledig-Rowohlt Foundation as a resident writer at the Chateau de Lavigny, Switzerland in 2007, a guest of the Arabic Capital of Culture in Baghdad, Iraq in 2014, and participated the International Writing Program in Iowa, USA in 2018. == Books == === English === Song of the quiet life, translated by Robert Berold and Cai Tianxin, Deep South, South Africa, 2005. Every Cloud Has Its Own Name, translated by Robert Berold and Cai Tianxin, 1-plus, San Francisco, 2017. The Book of Numbers, World Scientific, Singapore, 2018. A Modern Introduction to Classical Number Theory, World Scientific, Singapore, 2018. Perfect Numbers and Fibonacci Sequences, translated by Tyer Ross, World Scientific, Singapore, 2022. A Brief History of Mathematics, translated by Tyer Ross, to appear in Springer Nature, New York. == Awards and honors == Naji Naaman Literary Prize for Poetry, Beirut, 2013 China’s National Science and Technology Award, Beijing, 2018 Kathak Literary Award for Poetry, Dakar, 2019 Dang Dang Award for Influential Writer, Beijing, 2022 == References ==
|
Wikipedia:Tianyi Zheng#0
|
Tianyi Zheng is a Chinese-American mathematician specializing in geometric group theory and probability theory, including the theory of random walks and harmonic functions on groups. She is a professor of mathematics at the University of California, San Diego. == Education and career == Zheng was an undergraduate mathematics student at Tsinghua University, graduating in 2008. She completed a Ph.D. in 2013 at Cornell University, with the dissertation Random Walks On Some Classes Of Solvable Groups advised by Laurent Saloff-Coste. She became a postdoctoral Szegő Assistant Professor at Stanford University from 2013 to 2016 before obtaining a regular-rank faculty position as assistant professor at the University of California, San Diego in 2016. == Recognition == Zheng was named a Sloan Research Fellow in 2019. She was an invited speaker in mathematical analysis at the 2022 (virtual) International Congress of Mathematicians. In 2024, she was a recipient of the Rollo Davidson Prize, given "for her deep results and resolution of long-standing conjectures on random walks on groups". == References == == External links == Home page Tianyi Zheng publications indexed by Google Scholar
|
Wikipedia:Tibor Szele#0
|
Tibor Szele (21 June 1918 – 5 April 1955) Hungarian mathematician, working in combinatorics and abstract algebra. Szele was born in Debrecen. After graduating at the Debrecen University, he became a researcher at the Szeged University in 1946, then he went back at the Debrecen University in 1948 where he became full professor in 1952. He worked especially in the theory of Abelian groups and ring theory. He generalized Hajós's theorem. He founded the Hungarian school of algebra. Tibor Szele received the Kossuth Prize in 1952. He died in Szeged. == References == A panorama of Hungarian Mathematics in the Twentieth Century, p. 601. == External links == "Szele Grave". 2019-08-13. Archived from the original on 2019-08-13. Retrieved 2024-01-10. O'Connor, John J.; Robertson, Edmund F., "Tibor Szele", MacTutor History of Mathematics Archive, University of St Andrews
|
Wikipedia:Tibor Šalát#0
|
Tibor Šalát ((1926-05-13)May 13, 1926 – (2005-05-14)May 14, 2005) was a Slovak mathematician, professor of mathematics, and Doctor of Mathematics who specialized in number theory and real analysis. He was the author and co-author of undergraduate and graduate textbooks in mathematics, mostly in Slovak. And most of his scholarly papers have been published in various scientific journals. == Life == Originally from Žitava by the southern region of Slovakia, he studied at the Faculty of Natural Sciences of Charles University in Prague, where in 1952 he defended a dissertation entitled Príspevok k teorii súčtov a nekonečných radov s reálnými členami and supervised by Miloš Kössler and Vojtěch Jarník. In 1952 he went to work at the Faculty of Natural Sciences of Comenius University in Bratislava, where he became an assistant professor in 1962. He was appointed to a full professorship position in 1972. And in 1974, he earned a Ph.D. in Mathematics from the same institution. He specialized in Cantor's expansions, uniform distribution, statistical convergence, summation methods and theory of numbers. He wrote several undergraduate and graduate textbooks. == Academic papers == Juraj Činčura; Tibor Šalát; Martin Sleziak; Vladimír Toma (2005). "Sets of statistical cluster points and ℐ-cluster points". Real Analysis Exchange. 30 (2): 565–580. doi:10.14321/realanalexch.30.2.0565. MR 2177419. Pavel Kostyrko; Władysław Wilczyński; Tibor Šalát (2001). "I-Convergence". Real Analysis Exchange. 26 (2): 669–686. MR 1844385. Tibor Šalát (1980). "On statistically convergent sequences of real numbers". Mathematica Slovaca. 30 (2): 139–150. MR 0587239. Tibor Šalát; S. James Taylor; János T. Tóth (1998). "Radii of Convergence of Power Series". Real Analysis Exchange. 24 (1): 263–274. doi:10.2307/44152953. JSTOR 44152953. MR 1691750. Ján Borsík; Jaroslav Červeňanský; Tibor Šalát (1995). "Remarks on functions preserving convergence of infinite series". Real Analysis Exchange. 21 (2): 725–731. doi:10.2307/44152683. JSTOR 44152683. MR 1407285. M. Dindoš; T. Šalát; V. Toma (2003). "Statistical Convergence of Infinite Series". Czechoslovak Mathematical Journal. 53 (4): 989–1000. doi:10.1023/B:CMAJ.0000024535.89828.e8. hdl:10338.dmlcz/127854. MR 2018844. S2CID 121987409. Tibor Šalát (2000). "Remarks on Steinhaus' property and ratio sets of sets of positive integers". Czechoslovak Mathematical Journal. 50 (1): 175–183. doi:10.1023/A:1022457724187. hdl:10338.dmlcz/127559. MR 1745470. S2CID 117275867. Tibor Šalát (2000). "On uniform distribution of sequences". Czechoslovak Mathematical Journal. 50 (2): 331–340. doi:10.1023/A:1022422919181. hdl:10338.dmlcz/127572. MR 1761390. S2CID 117677265. == References == P. Kostyrko, O. Strauch: Professor Tibor Šalát (1926-2005), Tatra Mt. Math. Publ. 31 (2005), 1-16 [1]
|
Wikipedia:Tikhon Moiseev#0
|
Tikhon Yevgenyevich Moiseev (Russian: Ти́хон Евге́ньевич Моисе́ев; born 1978) is a Russian mathematician, Professor, Dr.Sc., a professor at the Faculty of Computer Science at the Moscow State University. Corresponding Member of the Russian Academy of Sciences. He defended the thesis "On the solvability of boundary value problems for the Lavrent'ev-Bitsadze equation with mixed boundary conditions" for the degree of Doctor of Physical and Mathematical Sciences (2013). Author more than 50 scientific articles. He is the son of famed mathematician Evgeny Moiseev. == References == == Bibliography == Evgeny Grigoriev (2010). Faculty of Computational Mathematics and Cybernetics: History and Modernity: A Biographical Directory (1 500 экз ed.). Moscow: Publishing house of Moscow University. pp. 131–132. ISBN 978-5-211-05838-5. == External links == Russian Academy of Sciences(in Russian) MSU CMC(in Russian) Scientific works of Tikhon Moiseev Scientific works of Tikhon Moiseev(in English)
|
Wikipedia:Tikhonov regularization#0
|
Ridge regression (also known as Tikhonov regularization, named for Andrey Tikhonov) is a method of estimating the coefficients of multiple-regression models in scenarios where the independent variables are highly correlated. It has been used in many fields including econometrics, chemistry, and engineering. It is a method of regularization of ill-posed problems. It is particularly useful to mitigate the problem of multicollinearity in linear regression, which commonly occurs in models with large numbers of parameters. In general, the method provides improved efficiency in parameter estimation problems in exchange for a tolerable amount of bias (see bias–variance tradeoff). The theory was first introduced by Hoerl and Kennard in 1970 in their Technometrics papers "Ridge regressions: biased estimation of nonorthogonal problems" and "Ridge regressions: applications in nonorthogonal problems". Ridge regression was developed as a possible solution to the imprecision of least square estimators when linear regression models have some multicollinear (highly correlated) independent variables—by creating a ridge regression estimator (RR). This provides a more precise ridge parameters estimate, as its variance and mean square estimator are often smaller than the least square estimators previously derived. == Overview == In the simplest case, the problem of a near-singular moment matrix X T X {\displaystyle \mathbf {X} ^{\mathsf {T}}\mathbf {X} } is alleviated by adding positive elements to the diagonals, thereby decreasing its condition number. Analogous to the ordinary least squares estimator, the simple ridge estimator is then given by β ^ R = ( X T X + λ I ) − 1 X T y {\displaystyle {\hat {\boldsymbol {\beta }}}_{R}=\left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} +\lambda \mathbf {I} \right)^{-1}\mathbf {X} ^{\mathsf {T}}\mathbf {y} } where y {\displaystyle \mathbf {y} } is the regressand, X {\displaystyle \mathbf {X} } is the design matrix, I {\displaystyle \mathbf {I} } is the identity matrix, and the ridge parameter λ ≥ 0 {\displaystyle \lambda \geq 0} serves as the constant shifting the diagonals of the moment matrix. It can be shown that this estimator is the solution to the least squares problem subject to the constraint β T β = c {\displaystyle {\boldsymbol {\beta }}^{\mathsf {T}}{\boldsymbol {\beta }}=c} , which can be expressed as a Lagrangian minimization: β ^ R = argmin β ( y − X β ) T ( y − X β ) + λ ( β T β − c ) {\displaystyle {\hat {\boldsymbol {\beta }}}_{R}={\text{argmin}}_{\boldsymbol {\beta }}\,\left(\mathbf {y} -\mathbf {X} {\boldsymbol {\beta }}\right)^{\mathsf {T}}\left(\mathbf {y} -\mathbf {X} {\boldsymbol {\beta }}\right)+\lambda \left({\boldsymbol {\beta }}^{\mathsf {T}}{\boldsymbol {\beta }}-c\right)} which shows that λ {\displaystyle \lambda } is nothing but the Lagrange multiplier of the constraint. In fact, there is a one-to-one relationship between c {\displaystyle c} and β {\displaystyle \beta } and since, in practice, we do not know c {\displaystyle c} , we define λ {\displaystyle \lambda } heuristically or find it via additional data-fitting strategies, see Determination of the Tikhonov factor. Note that, when λ = 0 {\displaystyle \lambda =0} , in which case the constraint is non-binding, the ridge estimator reduces to ordinary least squares. A more general approach to Tikhonov regularization is discussed below. == History == Tikhonov regularization was invented independently in many different contexts. It became widely known through its application to integral equations in the works of Andrey Tikhonov and David L. Phillips. Some authors use the term Tikhonov–Phillips regularization. The finite-dimensional case was expounded by Arthur E. Hoerl, who took a statistical approach, and by Manus Foster, who interpreted this method as a Wiener–Kolmogorov (Kriging) filter. Following Hoerl, it is known in the statistical literature as ridge regression, named after ridge analysis ("ridge" refers to the path from the constrained maximum). == Tikhonov regularization == Suppose that for a known real matrix A {\displaystyle A} and vector b {\displaystyle \mathbf {b} } , we wish to find a vector x {\displaystyle \mathbf {x} } such that A x = b , {\displaystyle A\mathbf {x} =\mathbf {b} ,} where x {\displaystyle \mathbf {x} } and b {\displaystyle \mathbf {b} } may be of different sizes and A {\displaystyle A} may be non-square. The standard approach is ordinary least squares linear regression. However, if no x {\displaystyle \mathbf {x} } satisfies the equation or more than one x {\displaystyle \mathbf {x} } does—that is, the solution is not unique—the problem is said to be ill posed. In such cases, ordinary least squares estimation leads to an overdetermined, or more often an underdetermined system of equations. Most real-world phenomena have the effect of low-pass filters in the forward direction where A {\displaystyle A} maps x {\displaystyle \mathbf {x} } to b {\displaystyle \mathbf {b} } . Therefore, in solving the inverse-problem, the inverse mapping operates as a high-pass filter that has the undesirable tendency of amplifying noise (eigenvalues / singular values are largest in the reverse mapping where they were smallest in the forward mapping). In addition, ordinary least squares implicitly nullifies every element of the reconstructed version of x {\displaystyle \mathbf {x} } that is in the null-space of A {\displaystyle A} , rather than allowing for a model to be used as a prior for x {\displaystyle \mathbf {x} } . Ordinary least squares seeks to minimize the sum of squared residuals, which can be compactly written as ‖ A x − b ‖ 2 2 , {\displaystyle \left\|A\mathbf {x} -\mathbf {b} \right\|_{2}^{2},} where ‖ ⋅ ‖ 2 {\displaystyle \|\cdot \|_{2}} is the Euclidean norm. In order to give preference to a particular solution with desirable properties, a regularization term can be included in this minimization: ‖ A x − b ‖ 2 2 + ‖ Γ x ‖ 2 2 = ‖ ( A Γ ) x − ( b 0 ) ‖ 2 2 {\displaystyle \left\|A\mathbf {x} -\mathbf {b} \right\|_{2}^{2}+\left\|\Gamma \mathbf {x} \right\|_{2}^{2}=\left\|{\begin{pmatrix}A\\\Gamma \end{pmatrix}}\mathbf {x} -{\begin{pmatrix}\mathbf {b} \\{\boldsymbol {0}}\end{pmatrix}}\right\|_{2}^{2}} for some suitably chosen Tikhonov matrix Γ {\displaystyle \Gamma } . In many cases, this matrix is chosen as a scalar multiple of the identity matrix ( Γ = α I {\displaystyle \Gamma =\alpha I} ), giving preference to solutions with smaller norms; this is known as L2 regularization. In other cases, high-pass operators (e.g., a difference operator or a weighted Fourier operator) may be used to enforce smoothness if the underlying vector is believed to be mostly continuous. This regularization improves the conditioning of the problem, thus enabling a direct numerical solution. An explicit solution, denoted by x ^ {\displaystyle {\hat {\mathbf {x} }}} , is given by x ^ = ( A T A + Γ T Γ ) − 1 A T b = ( ( A Γ ) T ( A Γ ) ) − 1 ( A Γ ) T ( b 0 ) . {\displaystyle {\hat {\mathbf {x} }}=\left(A^{\mathsf {T}}A+\Gamma ^{\mathsf {T}}\Gamma \right)^{-1}A^{\mathsf {T}}\mathbf {b} =\left({\begin{pmatrix}A\\\Gamma \end{pmatrix}}^{\mathsf {T}}{\begin{pmatrix}A\\\Gamma \end{pmatrix}}\right)^{-1}{\begin{pmatrix}A\\\Gamma \end{pmatrix}}^{\mathsf {T}}{\begin{pmatrix}\mathbf {b} \\{\boldsymbol {0}}\end{pmatrix}}.} The effect of regularization may be varied by the scale of matrix Γ {\displaystyle \Gamma } . For Γ = 0 {\displaystyle \Gamma =0} this reduces to the unregularized least-squares solution, provided that (ATA)−1 exists. Note that in case of a complex matrix A {\displaystyle A} , as usual the transpose A T {\displaystyle A^{\mathsf {T}}} has to be replaced by the Hermitian transpose A H {\displaystyle A^{\mathsf {H}}} . L2 regularization is used in many contexts aside from linear regression, such as classification with logistic regression or support vector machines, and matrix factorization. === Application to existing fit results === Since Tikhonov Regularization simply adds a quadratic term to the objective function in optimization problems, it is possible to do so after the unregularised optimisation has taken place. E.g., if the above problem with Γ = 0 {\displaystyle \Gamma =0} yields the solution x ^ 0 {\displaystyle {\hat {\mathbf {x} }}_{0}} , the solution in the presence of Γ ≠ 0 {\displaystyle \Gamma \neq 0} can be expressed as: x ^ = B x ^ 0 , {\displaystyle {\hat {\mathbf {x} }}=B{\hat {\mathbf {x} }}_{0},} with the "regularisation matrix" B = ( A T A + Γ T Γ ) − 1 A T A {\displaystyle B=\left(A^{\mathsf {T}}A+\Gamma ^{\mathsf {T}}\Gamma \right)^{-1}A^{\mathsf {T}}A} . If the parameter fit comes with a covariance matrix of the estimated parameter uncertainties V 0 {\displaystyle V_{0}} , then the regularisation matrix will be B = ( V 0 − 1 + Γ T Γ ) − 1 V 0 − 1 , {\displaystyle B=(V_{0}^{-1}+\Gamma ^{\mathsf {T}}\Gamma )^{-1}V_{0}^{-1},} and the regularised result will have a new covariance V = B V 0 B T . {\displaystyle V=BV_{0}B^{\mathsf {T}}.} In the context of arbitrary likelihood fits, this is valid, as long as the quadratic approximation of the likelihood function is valid. This means that, as long as the perturbation from the unregularised result is small, one can regularise any result that is presented as a best fit point with a covariance matrix. No detailed knowledge of the underlying likelihood function is needed. === Generalized Tikhonov regularization === For general multivariate normal distributions for x {\displaystyle \mathbf {x} } and the data error, one can apply a transformation of the variables to reduce to the case above. Equivalently, one can seek an x {\displaystyle \mathbf {x} } to minimize ‖ A x − b ‖ P 2 + ‖ x − x 0 ‖ Q 2 , {\displaystyle \left\|A\mathbf {x} -\mathbf {b} \right\|_{P}^{2}+\left\|\mathbf {x} -\mathbf {x} _{0}\right\|_{Q}^{2},} where we have used ‖ x ‖ Q 2 {\displaystyle \left\|\mathbf {x} \right\|_{Q}^{2}} to stand for the weighted norm squared x T Q x {\displaystyle \mathbf {x} ^{\mathsf {T}}Q\mathbf {x} } (compare with the Mahalanobis distance). In the Bayesian interpretation P {\displaystyle P} is the inverse covariance matrix of b {\displaystyle \mathbf {b} } , x 0 {\displaystyle \mathbf {x} _{0}} is the expected value of x {\displaystyle \mathbf {x} } , and Q {\displaystyle Q} is the inverse covariance matrix of x {\displaystyle \mathbf {x} } . The Tikhonov matrix is then given as a factorization of the matrix Q = Γ T Γ {\displaystyle Q=\Gamma ^{\mathsf {T}}\Gamma } (e.g. the Cholesky factorization) and is considered a whitening filter. This generalized problem has an optimal solution x ∗ {\displaystyle \mathbf {x} ^{*}} which can be written explicitly using the formula x ∗ = ( A T P A + Q ) − 1 ( A T P b + Q x 0 ) , {\displaystyle \mathbf {x} ^{*}=\left(A^{\mathsf {T}}PA+Q\right)^{-1}\left(A^{\mathsf {T}}P\mathbf {b} +Q\mathbf {x} _{0}\right),} or equivalently, when Q is not a null matrix: x ∗ = x 0 + ( A T P A + Q ) − 1 ( A T P ( b − A x 0 ) ) . {\displaystyle \mathbf {x} ^{*}=\mathbf {x} _{0}+\left(A^{\mathsf {T}}PA+Q\right)^{-1}\left(A^{\mathsf {T}}P\left(\mathbf {b} -A\mathbf {x} _{0}\right)\right).} == Lavrentyev regularization == In some situations, one can avoid using the transpose A T {\displaystyle A^{\mathsf {T}}} , as proposed by Mikhail Lavrentyev. For example, if A {\displaystyle A} is symmetric positive definite, i.e. A = A T > 0 {\displaystyle A=A^{\mathsf {T}}>0} , so is its inverse A − 1 {\displaystyle A^{-1}} , which can thus be used to set up the weighted norm squared ‖ x ‖ P 2 = x T A − 1 x {\displaystyle \left\|\mathbf {x} \right\|_{P}^{2}=\mathbf {x} ^{\mathsf {T}}A^{-1}\mathbf {x} } in the generalized Tikhonov regularization, leading to minimizing ‖ A x − b ‖ A − 1 2 + ‖ x − x 0 ‖ Q 2 {\displaystyle \left\|A\mathbf {x} -\mathbf {b} \right\|_{A^{-1}}^{2}+\left\|\mathbf {x} -\mathbf {x} _{0}\right\|_{Q}^{2}} or, equivalently up to a constant term, x T ( A + Q ) x − 2 x T ( b + Q x 0 ) . {\displaystyle \mathbf {x} ^{\mathsf {T}}\left(A+Q\right)\mathbf {x} -2\mathbf {x} ^{\mathsf {T}}\left(\mathbf {b} +Q\mathbf {x} _{0}\right).} This minimization problem has an optimal solution x ∗ {\displaystyle \mathbf {x} ^{*}} which can be written explicitly using the formula x ∗ = ( A + Q ) − 1 ( b + Q x 0 ) , {\displaystyle \mathbf {x} ^{*}=\left(A+Q\right)^{-1}\left(\mathbf {b} +Q\mathbf {x} _{0}\right),} which is nothing but the solution of the generalized Tikhonov problem where A = A T = P − 1 . {\displaystyle A=A^{\mathsf {T}}=P^{-1}.} The Lavrentyev regularization, if applicable, is advantageous to the original Tikhonov regularization, since the Lavrentyev matrix A + Q {\displaystyle A+Q} can be better conditioned, i.e., have a smaller condition number, compared to the Tikhonov matrix A T A + Γ T Γ . {\displaystyle A^{\mathsf {T}}A+\Gamma ^{\mathsf {T}}\Gamma .} == Regularization in Hilbert space == Typically discrete linear ill-conditioned problems result from discretization of integral equations, and one can formulate a Tikhonov regularization in the original infinite-dimensional context. In the above we can interpret A {\displaystyle A} as a compact operator on Hilbert spaces, and x {\displaystyle x} and b {\displaystyle b} as elements in the domain and range of A {\displaystyle A} . The operator A ∗ A + Γ T Γ {\displaystyle A^{*}A+\Gamma ^{\mathsf {T}}\Gamma } is then a self-adjoint bounded invertible operator. == Relation to singular-value decomposition and Wiener filter == With Γ = α I {\displaystyle \Gamma =\alpha I} , this least-squares solution can be analyzed in a special way using the singular-value decomposition. Given the singular value decomposition A = U Σ V T {\displaystyle A=U\Sigma V^{\mathsf {T}}} with singular values σ i {\displaystyle \sigma _{i}} , the Tikhonov regularized solution can be expressed as x ^ = V D U T b , {\displaystyle {\hat {x}}=VDU^{\mathsf {T}}b,} where D {\displaystyle D} has diagonal values D i i = σ i σ i 2 + α 2 {\displaystyle D_{ii}={\frac {\sigma _{i}}{\sigma _{i}^{2}+\alpha ^{2}}}} and is zero elsewhere. This demonstrates the effect of the Tikhonov parameter on the condition number of the regularized problem. For the generalized case, a similar representation can be derived using a generalized singular-value decomposition. Finally, it is related to the Wiener filter: x ^ = ∑ i = 1 q f i u i T b σ i v i , {\displaystyle {\hat {x}}=\sum _{i=1}^{q}f_{i}{\frac {u_{i}^{\mathsf {T}}b}{\sigma _{i}}}v_{i},} where the Wiener weights are f i = σ i 2 σ i 2 + α 2 {\displaystyle f_{i}={\frac {\sigma _{i}^{2}}{\sigma _{i}^{2}+\alpha ^{2}}}} and q {\displaystyle q} is the rank of A {\displaystyle A} . == Determination of the Tikhonov factor == The optimal regularization parameter α {\displaystyle \alpha } is usually unknown and often in practical problems is determined by an ad hoc method. A possible approach relies on the Bayesian interpretation described below. Other approaches include the discrepancy principle, cross-validation, L-curve method, restricted maximum likelihood and unbiased predictive risk estimator. Grace Wahba proved that the optimal parameter, in the sense of leave-one-out cross-validation minimizes G = RSS τ 2 = ‖ X β ^ − y ‖ 2 [ Tr ( I − X ( X T X + α 2 I ) − 1 X T ) ] 2 , {\displaystyle G={\frac {\operatorname {RSS} }{\tau ^{2}}}={\frac {\left\|X{\hat {\beta }}-y\right\|^{2}}{\left[\operatorname {Tr} \left(I-X\left(X^{\mathsf {T}}X+\alpha ^{2}I\right)^{-1}X^{\mathsf {T}}\right)\right]^{2}}},} where RSS {\displaystyle \operatorname {RSS} } is the residual sum of squares, and τ {\displaystyle \tau } is the effective number of degrees of freedom. Using the previous SVD decomposition, we can simplify the above expression: RSS = ‖ y − ∑ i = 1 q ( u i ′ b ) u i ‖ 2 + ‖ ∑ i = 1 q α 2 σ i 2 + α 2 ( u i ′ b ) u i ‖ 2 , {\displaystyle \operatorname {RSS} =\left\|y-\sum _{i=1}^{q}(u_{i}'b)u_{i}\right\|^{2}+\left\|\sum _{i=1}^{q}{\frac {\alpha ^{2}}{\sigma _{i}^{2}+\alpha ^{2}}}(u_{i}'b)u_{i}\right\|^{2},} RSS = RSS 0 + ‖ ∑ i = 1 q α 2 σ i 2 + α 2 ( u i ′ b ) u i ‖ 2 , {\displaystyle \operatorname {RSS} =\operatorname {RSS} _{0}+\left\|\sum _{i=1}^{q}{\frac {\alpha ^{2}}{\sigma _{i}^{2}+\alpha ^{2}}}(u_{i}'b)u_{i}\right\|^{2},} and τ = m − ∑ i = 1 q σ i 2 σ i 2 + α 2 = m − q + ∑ i = 1 q α 2 σ i 2 + α 2 . {\displaystyle \tau =m-\sum _{i=1}^{q}{\frac {\sigma _{i}^{2}}{\sigma _{i}^{2}+\alpha ^{2}}}=m-q+\sum _{i=1}^{q}{\frac {\alpha ^{2}}{\sigma _{i}^{2}+\alpha ^{2}}}.} == Relation to probabilistic formulation == The probabilistic formulation of an inverse problem introduces (when all uncertainties are Gaussian) a covariance matrix C M {\displaystyle C_{M}} representing the a priori uncertainties on the model parameters, and a covariance matrix C D {\displaystyle C_{D}} representing the uncertainties on the observed parameters. In the special case when these two matrices are diagonal and isotropic, C M = σ M 2 I {\displaystyle C_{M}=\sigma _{M}^{2}I} and C D = σ D 2 I {\displaystyle C_{D}=\sigma _{D}^{2}I} , and, in this case, the equations of inverse theory reduce to the equations above, with α = σ D / σ M {\displaystyle \alpha ={\sigma _{D}}/{\sigma _{M}}} . == Bayesian interpretation == Although at first the choice of the solution to this regularized problem may look artificial, and indeed the matrix Γ {\displaystyle \Gamma } seems rather arbitrary, the process can be justified from a Bayesian point of view. Note that for an ill-posed problem one must necessarily introduce some additional assumptions in order to get a unique solution. Statistically, the prior probability distribution of x {\displaystyle x} is sometimes taken to be a multivariate normal distribution. For simplicity here, the following assumptions are made: the means are zero; their components are independent; the components have the same standard deviation σ x {\displaystyle \sigma _{x}} . The data are also subject to errors, and the errors in b {\displaystyle b} are also assumed to be independent with zero mean and standard deviation σ b {\displaystyle \sigma _{b}} . Under these assumptions the Tikhonov-regularized solution is the most probable solution given the data and the a priori distribution of x {\displaystyle x} , according to Bayes' theorem. If the assumption of normality is replaced by assumptions of homoscedasticity and uncorrelatedness of errors, and if one still assumes zero mean, then the Gauss–Markov theorem entails that the solution is the minimal unbiased linear estimator. == See also == LASSO estimator is another regularization method in statistics. Elastic net regularization Matrix regularization == Notes == == References == == Further reading == Gruber, Marvin (1998). Improving Efficiency by Shrinkage: The James–Stein and Ridge Regression Estimators. Boca Raton: CRC Press. ISBN 0-8247-0156-9. Kress, Rainer (1998). "Tikhonov Regularization". Numerical Analysis. New York: Springer. pp. 86–90. ISBN 0-387-98408-9. Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007). "Section 19.5. Linear Regularization Methods". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8. Saleh, A. K. Md. Ehsanes; Arashi, Mohammad; Kibria, B. M. Golam (2019). Theory of Ridge Regression Estimation with Applications. New York: John Wiley & Sons. ISBN 978-1-118-64461-4. Taddy, Matt (2019). "Regularization". Business Data Science: Combining Machine Learning and Economics to Optimize, Automate, and Accelerate Business Decisions. New York: McGraw-Hill. pp. 69–104. ISBN 978-1-260-45277-8.
|
Wikipedia:Tilted large deviation principle#0
|
In mathematics — specifically, in large deviations theory — the tilted large deviation principle is a result that allows one to generate a new large deviation principle from an old one by exponential tilting, i.e. integration against an exponential functional. It can be seen as an alternative formulation of Varadhan's lemma. == Statement of the theorem == Let X be a Polish space (i.e., a separable, completely metrizable topological space), and let (με)ε>0 be a family of probability measures on X that satisfies the large deviation principle with rate function I : X → [0, +∞]. Let F : X → R be a continuous function that is bounded from above. For each Borel set S ⊆ X, let J ε ( S ) = ∫ S e F ( x ) / ε d μ ε ( x ) {\displaystyle J_{\varepsilon }(S)=\int _{S}e^{F(x)/\varepsilon }\,\mathrm {d} \mu _{\varepsilon }(x)} and define a new family of probability measures (νε)ε>0 on X by ν ε ( S ) = J ε ( S ) J ε ( X ) . {\displaystyle \nu _{\varepsilon }(S)={\frac {J_{\varepsilon }(S)}{J_{\varepsilon }(X)}}.} Then (νε)ε>0 satisfies the large deviation principle on X with rate function IF : X → [0, +∞] given by I F ( x ) = sup y ∈ X [ F ( y ) − I ( y ) ] − [ F ( x ) − I ( x ) ] . {\displaystyle I^{F}(x)=\sup _{y\in X}{\big [}F(y)-I(y){\big ]}-{\big [}F(x)-I(x){\big ]}.} == References == den Hollander, Frank (2000). Large deviations. Fields Institute Monographs 14. Providence, RI: American Mathematical Society. pp. x+143. ISBN 0-8218-1989-5. MR1739680
|
Wikipedia:Timeline of algebra#0
|
The following is a timeline of key developments of algebra: == See also == Mathematics portal History of algebra – Historical development of algebra == References == Boyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc. ISBN 0-471-54397-7. Hayashi, Takao (2005). "Indian Mathematics". In Flood, Gavin (ed.). The Blackwell Companion to Hinduism. Oxford: Basil Blackwell. pp. 360–375. ISBN 978-1-4051-3251-0.
|
Wikipedia:Timeline of calculus and mathematical analysis#0
|
A timeline of calculus and mathematical analysis. == 500BC to 1600 == 5th century BC - The Zeno's paradoxes, 5th century BC - Antiphon attempts to square the circle, 5th century BC - Democritus finds the volume of cone is 1/3 of volume of cylinder, 4th century BC - Eudoxus of Cnidus develops the method of exhaustion, 3rd century BC - Archimedes displays geometric series in The Quadrature of the Parabola. He further develops the method of exhaustion. 3rd century BC - Archimedes develops a concept of the indivisibles—a precursor to infinitesimals—allowing him to solve several problems using methods now termed as integral calculus. Archimedes also derives several formulae for determining the area and volume of various solids including sphere, cone, paraboloid and hyperboloid. Before 50 BC - Babylonian cuneiform tablets show use of the Trapezoid rule to calculate of the position of Jupiter. 3rd century - Liu Hui rediscovers the method of exhaustion in order to find the area of a circle. 4th century - The Pappus's centroid theorem, 5th century - Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere. 600 - Liu Zhuo is the first person to use second-order interpolation for computing the positions of the sun and the moon. 665 - Brahmagupta discovers a second order Newton-Stirling interpolation for sin ( x + ϵ ) {\displaystyle \sin(x+\epsilon )} , 862 - The Banu Musa brothers write the "Book on the Measurement of Plane and Spherical Figures", 9th century - Thābit ibn Qurra discusses the quadrature of the parabola and the volume of different types of conic sections. 12th century - Bhāskara II discovers a rule equivalent to Rolle's theorem for sin x {\displaystyle \sin x} , 14th century - Nicole Oresme proves of the divergence of the harmonic series, 14th century - Madhava discovers the power series expansion for sin x {\displaystyle \sin x} , cos x {\displaystyle \cos x} , arctan x {\displaystyle \arctan x} and π / 4 {\displaystyle \pi /4} This theory is now well known in the Western world as the Taylor series or infinite series. 14th century - Parameshvara discovers a third order Taylor interpolation for sin ( x + ϵ ) {\displaystyle \sin(x+\epsilon )} , 1445 - Nicholas of Cusa attempts to square the circle, 1501 - Nilakantha Somayaji writes the Tantrasamgraha, which contains the Madhava's discoveries, 1548 - Francesco Maurolico attempted to calculate the barycenter of various bodies (pyramid, paraboloid, etc.), 1550 - Jyeshtadeva writes the Yuktibhāṣā, a commentary to Nilakantha's Tantrasamgraha, 1560 - Sankara Variar writes the Kriyakramakari, 1565 - Federico Commandino publishes De centro Gravitati, 1588 - Commandino's translation of Pappus' Collectio gets published, 1593 - François Viète discovers the first infinite product in the history of mathematics, == 17th century == 1606 - Luca Valerio applies methods of Archimedes to find volumes and centres of gravity of solid bodies, 1609 - Johannes Kepler computes the integral ∫ 0 θ sin x d x = 1 − cos θ {\displaystyle \int _{0}^{\theta }\sin x\ dx=1-\cos \theta } , 1611 - Thomas Harriot discovers an interpolation formula similar to Newton's interpolation formula, 1615 - Johannes Kepler publishes Nova stereometria doliorum, 1620 - Grégoire de Saint-Vincent discovers that the area under a hyperbola represented a logarithm, 1624 - Henry Briggs publishes Arithmetica Logarithmica, 1629 - Pierre de Fermat discovers his method of maxima and minima, precursor of the derivative concept, 1634 - Gilles de Roberval shows that the area under a cycloid is three times the area of its generating circle, 1635 - Bonaventura Cavalieri publishes Geometria Indivisibilibus, 1637 - René Descartes publishes La Géométrie, 1638 - Galileo Galilei publishes Two New Sciences, 1644 - Evangelista Torricelli publishes Opera geometrica, 1644 - Fermat's methods of maxima and minima published by Pierre Hérigone, 1647 - Cavalieri computes the integral ∫ 0 a x n d x = 1 n + 1 a n + 1 {\displaystyle \int _{0}^{a}x^{n}\ dx={\frac {1}{n+1}}a^{n+1}} , 1647 - Grégoire de Saint-Vincent publishes Opus Geometricum, 1650 - Pietro Mengoli proves of the divergence of the harmonic series, 1654 - Johannes Hudde discovers the power series expansion for ln ( 1 + x ) {\displaystyle \ln(1+x)} , 1656 - John Wallis publishes Arithmetica Infinitorum, 1658 - Christopher Wren shows that the length of a cycloid is four times the diameter of its generating circle, 1659 - Second edition of Van Schooten's Latin translation of Descartes' Geometry with appendices by Hudde and Heuraet, 1665 - Isaac Newton discovers the generalized binomial theorem and develops his version of infinitesimal calculus, 1667 - James Gregory publishes Vera circuli et hyperbolae quadratura, 1668 - Nicholas Mercator publishes Logarithmotechnia, 1668 - James Gregory computes the integral of the secant function, 1669 - Newton invents a Newton's method for the computation of roots of a function, 1670 - Newton rediscovers the power series expansion for sin x {\displaystyle \sin x} and cos x {\displaystyle \cos x} (originally discovered by Madhava), 1670 - Isaac Barrow publishes Lectiones Geometricae, 1671 - James Gregory rediscovers the power series expansion for arctan x {\displaystyle \arctan x} and π / 4 {\displaystyle \pi /4} (originally discovered by Madhava), 1672 - René-François de Sluse publishes A Method of Drawing Tangents to All Geometrical Curves, 1673 - Gottfried Leibniz also develops his version of infinitesimal calculus, 1675 - Leibniz uses the modern notation for an integral for the first time, 1677 - Leibniz discovers the rules for differentiating products, quotients, and the function of a function. 1683 - Jacob Bernoulli discovers the number e, 1684 - Leibniz publishes his first paper on calculus, 1685 - Newon formulates and solves Newton's minimal resistance problem, giving birth to the field of calculus of variations, 1686 - The first appearance in print of the ∫ {\displaystyle \int } notation for integrals, 1687 - Isaac Newton publishes Philosophiæ Naturalis Principia Mathematica, 1691 - The first proof of Rolle's theorem is given by Michel Rolle, 1691 - Leibniz discovers the technique of separation of variables for ordinary differential equations, 1694 - Johann Bernoulli discovers the L'Hôpital's rule, 1696 - Guillaume de L'Hôpital publishes Analyse des Infiniment Petits, the first calculus textbook, 1696 - Jakob Bernoulli and Johann Bernoulli solve the brachistochrone problem. == 18th century == 1711 - Isaac Newton publishes De analysi per aequationes numero terminorum infinitas, 1712 - Brook Taylor develops Taylor series, 1722 - Roger Cotes computes the derivative of sine function in his Harmonia Mensurarum, 1730 - James Stirling publishes The Differential Method, 1734 - George Berkeley publishes The Analyst, 1734 - Leonhard Euler introduces the integrating factor technique for solving first-order ordinary differential equations, 1735 - Leonhard Euler solves the Basel problem, relating an infinite series to π, 1736 - Newton's Method of Fluxions posthumously published, 1737 - Thomas Simpson publishes Treatise of Fluxions, 1739 - Leonhard Euler solves the general homogeneous linear ordinary differential equation with constant coefficients, 1742 - Modern definion of logarithm by William Gardiner, 1742 - Colin Maclaurin publishes Treatise on Fluxions, 1748 - Euler publishes Introductio in analysin infinitorum, 1748 - Maria Gaetana Agnesi discusses analysis in Instituzioni Analitiche ad Uso della Gioventu Italiana, 1762 - Joseph Louis Lagrange discovers the divergence theorem, 1797 - Lagrange publishes Théorie des fonctions analytiques, == 19th century == 1807 - Joseph Fourier announces his discoveries about the trigonometric decomposition of functions, 1811 - Carl Friedrich Gauss discusses the meaning of integrals with complex limits and briefly examines the dependence of such integrals on the chosen path of integration, 1815 - Siméon Denis Poisson carries out integrations along paths in the complex plane, 1817 - Bernard Bolzano presents the intermediate value theorem — a continuous function which is negative at one point and positive at another point must be zero for at least one point in between, 1822 - Augustin-Louis Cauchy presents the Cauchy integral theorem for integration around the boundary of a rectangle in the complex plane, 1825 - Augustin-Louis Cauchy presents the Cauchy integral theorem for general integration paths—he assumes the function being integrated has a continuous derivative, and he introduces the theory of residues in complex analysis, 1825 - André-Marie Ampère discovers Stokes' theorem, 1828 - George Green introduces Green's theorem, 1831 - Mikhail Vasilievich Ostrogradsky rediscovers and gives the first proof of the divergence theorem earlier described by Lagrange, Gauss and Green, 1841 - Karl Weierstrass discovers but does not publish the Laurent expansion theorem, 1843 - Pierre-Alphonse Laurent discovers and presents the Laurent expansion theorem, 1850 - Victor Alexandre Puiseux distinguishes between poles and branch points and introduces the concept of essential singular points, 1850 - George Gabriel Stokes rediscovers and proves Stokes' theorem, 1861 - Karl Weierstrass starts to use the language of epsilons and deltas, 1873 - Georg Frobenius presents his method for finding series solutions to linear differential equations with regular singular points, == 20th century == 1908 - Josip Plemelj solves the Riemann problem about the existence of a differential equation with a given monodromic group and uses Sokhotsky - Plemelj formulae, 1966 - Abraham Robinson presents non-standard analysis. 1985 - Louis de Branges de Bourcia proves the Bieberbach conjecture, == See also == Timeline of ancient Greek mathematicians Timeline of geometry – Notable events in the history of geometry Timeline of mathematical logic Timeline of mathematics == References ==
|
Wikipedia:Timeline of geometry#0
|
The following is a timeline of key developments of geometry: == Before 1000 BC == ca. 2000 BC – Scotland, carved stone balls exhibit a variety of symmetries including all of the symmetries of Platonic solids. 1800 BC – Moscow Mathematical Papyrus, findings volume of a frustum 1800 BC – Plimpton 322 contains the oldest reference to the Pythagorean triplets. 1650 BC – Rhind Mathematical Papyrus, copy of a lost scroll from around 1850 BC, the scribe Ahmes presents one of the first known approximate values of π at 3.16, the first attempt at squaring the circle, earliest known use of a sort of cotangent, and knowledge of solving first order linear equations == 1st millennium BC == 800 BC – Baudhayana, author of the Baudhayana Sulba Sutra, a Vedic Sanskrit geometric text, contains quadratic equations, and calculates the square root of 2 correct to five decimal places ca. 600 BC – the other Vedic "Sulba Sutras" ("rule of chords" in Sanskrit) use Pythagorean triples, contain a number of geometrical proofs, and approximate π at 3.16 5th century BC – Hippocrates of Chios utilizes lunes in an attempt to square the circle 5th century BC – Apastamba, author of the Apastamba Sulba Sutra, another Vedic Sanskrit geometric text, makes an attempt at squaring the circle and also calculates the square root of 2 correct to five decimal places 530 BC – Pythagoras studies propositional geometry and vibrating lyre strings; his group also discover the irrationality of the square root of two, 370 BC – Eudoxus states the method of exhaustion for area determination 300 BC – Euclid in his Elements studies geometry as an axiomatic system, proves the infinitude of prime numbers and presents the Euclidean algorithm; he states the law of reflection in Catoptrics, and he proves the fundamental theorem of arithmetic 260 BC – Archimedes proved that the value of π lies between 3 + 1/7 (approx. 3.1429) and 3 + 10/71 (approx. 3.1408), that the area of a circle was equal to π multiplied by the square of the radius of the circle and that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. He also gave a very accurate estimate of the value of the square root of 3. 225 BC – Apollonius of Perga writes On Conic Sections and names the ellipse, parabola, and hyperbola, 150 BC – Jain mathematicians in India write the "Sthananga Sutra", which contains work on the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations 140 BC – Hipparchus develops the bases of trigonometry. == 1st millennium == ca 340 – Pappus of Alexandria states his hexagon theorem and his centroid theorem 50 – Aryabhata writes the "Aryabhata-Siddhanta", which first introduces the trigonometric functions and methods of calculating their approximate numerical values. It defines the concepts of sine and cosine, and also contains the earliest tables of sine and cosine values (in 3.75-degree intervals from 0 to 90 degrees) 7th century – Bhaskara I gives a rational approximation of the sine function 8th century – Virasena gives explicit rules for the Fibonacci sequence, gives the derivation of the volume of a frustum using an infinite procedure. 8th century – Shridhara gives the rule for finding the volume of a sphere and also the formula for solving quadratic equations 820 – Al-Mahani conceived the idea of reducing geometrical problems such as doubling the cube to problems in algebra. ca. 900 – Abu Kamil of Egypt had begun to understand what we would write in symbols as x n ⋅ x m = x m + n {\displaystyle x^{n}\cdot x^{m}=x^{m+n}} 975 – Al-Batani – Extended the Indian concepts of sine and cosine to other trigonometrical ratios, like tangent, secant and their inverse functions. Derived the formula: sin α = tan α / 1 + tan 2 α {\displaystyle \sin \alpha =\tan \alpha /{\sqrt {1+\tan ^{2}\alpha }}} and cos α = 1 / 1 + tan 2 α {\displaystyle \cos \alpha =1/{\sqrt {1+\tan ^{2}\alpha }}} . == 1000–1500 == ca. 1000 – Law of sines is discovered by Muslim mathematicians, but it is uncertain who discovers it first between Abu-Mahmud al-Khujandi, Abu Nasr Mansur, and Abu al-Wafa. ca. 1100 – Omar Khayyám "gave a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections." He became the first to find general geometric solutions of cubic equations and laid the foundations for the development of analytic geometry and non-Euclidean geometry. He also extracted roots using the decimal system (Hindu–Arabic numeral system). 1135 – Sharafeddin Tusi followed al-Khayyam's application of algebra to geometry, and wrote a treatise on cubic equations which "represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry." ca. 1250 – Nasir Al-Din Al-Tusi attempts to develop a form of non-Euclidean geometry. 15th century – Nilakantha Somayaji, a Kerala school mathematician, writes the "Aryabhatiya Bhasya", which contains work on infinite-series expansions, problems of algebra, and spherical geometry == 17th century == 17th century – Putumana Somayaji writes the "Paddhati", which presents a detailed discussion of various trigonometric series 1619 – Johannes Kepler discovers two of the Kepler-Poinsot polyhedra. 1637 - René Descartes publishes La Géométrie which introduces analytic geometry, which involves reducing geometry to a form of arithmetic and algebra and translating geometric shapes into algebraic equations. == 18th century == 1722 – Abraham de Moivre states de Moivre's formula connecting trigonometric functions and complex numbers, 1733 – Giovanni Gerolamo Saccheri studies what geometry would be like if Euclid's fifth postulate were false, 1796 – Carl Friedrich Gauss proves that the regular 17-gon can be constructed using only a compass and straightedge 1797 – Caspar Wessel associates vectors with complex numbers and studies complex number operations in geometrical terms, 1799 – Gaspard Monge publishes Géométrie descriptive, in which he introduces descriptive geometry. == 19th century == 1806 – Louis Poinsot discovers the two remaining Kepler-Poinsot polyhedra. 1829 – Bolyai, Gauss, and Lobachevsky invent hyperbolic non-Euclidean geometry, 1837 – Pierre Wantzel proves that doubling the cube and trisecting the angle are impossible with only a compass and straightedge, as well as the full completion of the problem of constructibility of regular polygons 1843 – William Hamilton discovers the calculus of quaternions and deduces that they are non-commutative, 1854 – Bernhard Riemann introduces Riemannian geometry, 1854 – Arthur Cayley shows that quaternions can be used to represent rotations in four-dimensional space, 1858 – August Ferdinand Möbius invents the Möbius strip, 1870 – Felix Klein constructs an analytic geometry for Lobachevski's geometry thereby establishing its self-consistency and the logical independence of Euclid's fifth postulate, 1873 – Charles Hermite proves that e is transcendental, 1878 – Charles Hermite solves the general quintic equation by means of elliptic and modular functions 1882 – Ferdinand von Lindemann proves that π is transcendental and that therefore the circle cannot be squared with a compass and straightedge, 1882 – Felix Klein discovers the Klein bottle, 1899 – David Hilbert presents a set of self-consistent geometric axioms in Foundations of Geometry == 20th century == 1901 – Élie Cartan develops the exterior derivative, 1912 – Luitzen Egbertus Jan Brouwer presents the Brouwer fixed-point theorem, 1916 – Einstein's theory of general relativity. 1930 – Casimir Kuratowski shows that the three-cottage problem has no solution, 1931 – Georges de Rham develops theorems in cohomology and characteristic classes, 1933 – Karol Borsuk and Stanislaw Ulam present the Borsuk-Ulam antipodal-point theorem, 1955 – H. S. M. Coxeter et al. publish the complete list of uniform polyhedron, 1975 – Benoit Mandelbrot, fractals theory, 1981 – Mikhail Gromov develops the theory of hyperbolic groups, revolutionizing both infinite group theory and global differential geometry, 1983 – the classification of finite simple groups, a collaborative work involving some hundred mathematicians and spanning thirty years, is completed, 1991 – Alain Connes and John Lott develop non-commutative geometry, 1998 – Thomas Callister Hales proves the Kepler conjecture, == 21st century == 2003 – Grigori Perelman proves the Poincaré conjecture, 2007 – a team of researchers throughout North America and Europe used networks of computers to map E8 (mathematics). == See also == History of geometry – Historical development of geometry Timeline of ancient Greek mathematicians Timeline of mathematical logic Timeline of mathematics == References ==
|
Wikipedia:Timeline of women in mathematics in the United States#0
|
There is a long history of women in mathematics in the United States. All women mentioned here are American unless otherwise noted. == Timeline == === 19th Century === 1829: The first public examination of an American girl in geometry was held. 1886: Winifred Edgerton Merrill became the first American woman to earn a PhD in mathematics, which she earned from Columbia University. 1891: Charlotte Angas Scott of Britain became the first woman to join the American Mathematical Society, then called the New York Mathematical Society. 1894: Charlotte Angas Scott of Britain became the first woman on the first Council of the American Mathematical Society. === 20th Century === 1913: Mildred Sanderson earned her PhD for a thesis that included an important theorem about modular invariants. 1927: Anna Pell-Wheeler became the first woman to present a lecture at the American Mathematical Society Colloquium. 1943: Euphemia Haynes became the first African-American woman to earn a Ph.D. in mathematics, which she earned from Catholic University of America. 1949: Gertrude Mary Cox became the first woman elected into the International Statistical Institute. 1956: Gladys West began collecting data from satellites at the Naval Surface Warfare Center Dahlgren Division. Her calculations directly impacted the development of accurate GPS systems. 1962: Mina Rees became the first person to receive the Award for Distinguished Service to Mathematics from the Mathematical Association of America. 1966: Mary L. Boas published Mathematical Methods in the Physical Sciences, which was still widely used in college classrooms as of 1999. ==== 1970s ==== 1970: Mina Rees became the first female president of the American Association for the Advancement of Science. 1971: Mary Ellen Rudin constructed the first Dowker space. The Association for Women in Mathematics (AWM) was founded. It is a professional society whose mission is to encourage women and girls to study and to have active careers in the mathematical sciences, and to promote equal opportunity for and the equal treatment of women and girls in the mathematical sciences. It is incorporated in the state of Massachusetts. The American Mathematical Society established its Joint Committee on Women in the Mathematical Sciences (JCW), which later became a joint committee of multiple scholarly societies. 1973: Jean Taylor published her dissertation on "Regularity of the Singular Set of Two-Dimensional Area-Minimizing Flat Chains Modulo 3 in R3" which solved a long-standing problem about length and smoothness of soap-film triple function curves. 1974: Joan Birman published the book Braids, Links, and Mapping Class Groups. It has become a standard introduction, with many of today's researchers having learned the subject through it. 1975: Julia Robinson became the first female mathematician elected to the National Academy of Sciences. 1976-1977: Marjorie Rice, an amateur mathematician, discovered four new types of tessellating pentagons in 1976 and 1977. 1979: Dorothy Lewis Bernstein became the first female president of the Mathematical Association of America. Mary Ellen Rudin became the first woman to present the Mathematical Association of America’s Earle Raymond Hedrick Lectures, intended to showcase skilled expositors and enrich the understanding of instructors of college-level mathematics. ==== 1980s ==== 1981: Canadian-American mathematician Cathleen Morawetz became the first woman to give the Gibbs Lecture of the American Mathematical Society. 1981: Doris Schattschneider became the first female editor of Mathematics Magazine, a refereed bimonthly publication of the Mathematical Association of America. 1983: Julia Robinson became the first female president of the American Mathematical Society, and the first female mathematician to be awarded a MacArthur Fellowship. 1987: Eileen Poiani became the first female president of Pi Mu Epsilon. 1988: Doris Schattschneider became the first woman to present the Mathematical Association of America’s J. Sutherland Frame Lectures. ==== 1990s ==== 1992: Gloria Gilmer became the first woman to deliver a major National Association of Mathematicians lecture (it was the Cox-Talbot address). 1995: Margaret Wright became the first female president of the Society for Industrial and Applied Mathematics. 1996: Joan Birman became the first woman to receive the Mathematical Association of America’s Chauvenet Prize, an annual award for expository articles. 1998: Melanie Wood became the first female American to make the U.S. International Math Olympiad Team. She won silver medals in the 1998 and 1999 International Mathematical Olympiads. === 21st Century === 2002: Melanie Wood became the first American woman and second woman overall to be named a Putnam Fellow in 2002. Putnam Fellows are the top five (or six, in case of a tie) scorers on William Lowell Putnam Mathematical Competition. 2004: Melanie Wood became the first woman to win the Frank and Brennie Morgan Prize for Outstanding Research in Mathematics by an Undergraduate Student. It is an annual award given to an undergraduate student in the US, Canada, or Mexico who demonstrates superior mathematics research. Alison Miller became the first female gold medal winner on the U.S. International Mathematical Olympiad Team. 2006: Stefanie Petermichl, a German mathematical analyst then at the University of Texas at Austin, became the first woman to win the Salem Prize, an annual award given to young mathematicians who have worked in Raphael Salem's field of interest, chiefly topics in analysis related to Fourier series. She shared the prize with Artur Avila. 2019: Karen Uhlenbeck became the first woman to win the Abel Prize, with the award committee citing "the fundamental impact of her work on analysis, geometry and mathematical physics." Marissa Kawehi Loving became the first Native Hawaiian woman to earn a PhD in mathematics when she graduated from the University of Illinois Urbana-Champaign in 2019. In addition to being Native Hawaiian, she is also black, Japanese, and Puerto Rican. 2020: Lisa Piccirillo published a mathematical proof in the journal Annals of Mathematics determining that the Conway knot is not a smoothly slice knot, answering an unsolved problem in knot theory first proposed over fifty years prior by English mathematician John Horton Conway. == See also == Timeline of women in mathematics == References == == Further reading == A Brief History of the Association for Women in Mathematics: The Presidents' Perspectives, by Lenore Blum (1991)
|
Wikipedia:Timofei Osipovsky#0
|
Timofei Fyodorovich Osipovsky (Russian: Тимофей Федорович Осиповский; February 2, 1766, Osipovo – June 24, 1832, Moscow) was a Russian Imperial mathematician, physicist, astronomer, and philosopher. Timofei Osipovsky graduated from the St Petersburg Teachers Seminary. He became a teacher at Imperial Kharkov University, in 1805, the year it was founded. The city of Kharkov, thanks to its educational establishments, became one of the most important cultural and educational centres of Russian Empire. In 1813 he became rector of the university. However, in 1820, Osipovsky was suspended from his post on religious grounds. Osipovsky's most famous work was the three volume book A Course of Mathematics (1801–1823), which soon became a standard university text and was used in universities for many years. == References == B. A. Rozenfeld, A History of Non-Euclidean Geometry : Evolution of the Concept of a Geometric Space (Springer, 1988). E. Ya. Bahmutskaya, Timofei Fedorovich Osipovsky and his 'Course of mathematics' (Russian), Istor.-Mat. Issled. 5 (1952), 28–74. U. I. Frankfurt, "On the question of the critical analysis of Newton's teachings of space and time in the 18th century. From Leibniz to Lomonosov" (Russian), in Mechanics and physics in the second half of the 18th century (Russian) (Nauka, Moscow, 1978), 148–190. T. S. Polyakova, "Russian paternalistic traditions in mathematics education in the 18th century and the first half of the 19th century" (Russian), Istor.-Mat. Issled. (2) 5 (40) (2000), 174–191; 383. V. E. Prudnikov, "Supplementary information on T F Osipovsky" (Russian), Istor.-Mat. Issled. 5 (1952), 75–83. G. F. Rybkin, "Materialistic features of the Weltanschauung of M V Ostrogradskii and his teacher T F Osipovsky" (Russian), Uspekhi Matem. Nauk (N.S.) 7 2(48) (1952), 123–144. A. P. Yushkevich, "The French Revolution and the development of mathematics in Russia" (Russian), Priroda 1989 (7) (1989), 91–97. == External links == O'Connor, John J.; Robertson, Edmund F., "Timofei Osipovsky", MacTutor History of Mathematics Archive, University of St Andrews
|
Wikipedia:Timothy Trudgian#0
|
Timothy Trudgian is an Australian mathematician specializing in number theory and related fields. He is known for his work on Riemann zeta function, analytic number theory, and distribution of primes. He currently is a Professor at the University of New South Wales (Canberra). == Education and Career == Trudgian completed his BSc (Hons) at the Australian National University in December 2005, then his Ph.D. from the University of Oxford in June 2010 under the supervision of Roger Heath-Brown. His dissertation was titled Further results on Gram's Law. == Research == Trudgian has made significant contributions to the field of (analytic) number theory. His research includes work on Riemann zeta function, distribution of primes, and primitive root modulo n. One of his notable achievements is proving that the Riemann hypothesis is true up to 3 × 1012. In 2024, together with Terence Tao and Andrew Yang, Trudgian published an on-going database of known theorems for various exponents appearing in analytic number theory, named Analytic Number Theory Exponent Database (ANTEDB), which could be used in the future for Lean formalization. == Recognition == Trudgian is a Fellow of the Australian Mathematical Society, elected in 2023. == Personal life == Trudgian is married and he has two son. == References ==
|
Wikipedia:Titu's lemma#0
|
In mathematics, the following inequality is known as Titu's lemma, Bergström's inequality, Engel's form or Sedrakyan's inequality, respectively, referring to the article About the applications of one useful inequality of Nairi Sedrakyan published in 1997, to the book Problem-solving strategies of Arthur Engel published in 1998 and to the book Mathematical Olympiad Treasures of Titu Andreescu published in 2003. It is a direct consequence of Cauchy–Bunyakovsky–Schwarz inequality. Nevertheless, in his article (1997) Sedrakyan has noticed that written in this form this inequality can be used as a proof technique and it has very useful new applications. In the book Algebraic Inequalities (Sedrakyan) several generalizations of this inequality are provided. == Statement of the inequality == For any real numbers a 1 , a 2 , a 3 , … , a n {\displaystyle a_{1},a_{2},a_{3},\ldots ,a_{n}} and positive reals b 1 , b 2 , b 3 , … , b n , {\displaystyle b_{1},b_{2},b_{3},\ldots ,b_{n},} we have a 1 2 b 1 + a 2 2 b 2 + ⋯ + a n 2 b n ≥ ( a 1 + a 2 + ⋯ + a n ) 2 b 1 + b 2 + ⋯ + b n . {\displaystyle {\frac {a_{1}^{2}}{b_{1}}}+{\frac {a_{2}^{2}}{b_{2}}}+\cdots +{\frac {a_{n}^{2}}{b_{n}}}\geq {\frac {\left(a_{1}+a_{2}+\cdots +a_{n}\right)^{2}}{b_{1}+b_{2}+\cdots +b_{n}}}.} (Nairi Sedrakyan (1997), Arthur Engel (1998), Titu Andreescu (2003)) === Probabilistic statement === Similarly to the Cauchy–Schwarz inequality, one can generalize Sedrakyan's inequality to random variables. In this formulation let X {\displaystyle X} be a real random variable, and let Y {\displaystyle Y} be a positive random variable. X and Y need not be independent, but we assume E [ | X | ] {\displaystyle E[|X|]} and E [ Y ] {\displaystyle E[Y]} are both defined. Then E [ X 2 / Y ] ≥ E [ | X | ] 2 / E [ Y ] ≥ E [ X ] 2 / E [ Y ] . {\displaystyle \operatorname {E} [X^{2}/Y]\geq \operatorname {E} [|X|]^{2}/\operatorname {E} [Y]\geq \operatorname {E} [X]^{2}/\operatorname {E} [Y].} == Direct applications == Example 1. Nesbitt's inequality. For positive real numbers a , b , c : {\displaystyle a,b,c:} a b + c + b a + c + c a + b ≥ 3 2 . {\displaystyle {\frac {a}{b+c}}+{\frac {b}{a+c}}+{\frac {c}{a+b}}\geq {\frac {3}{2}}.} Example 2. International Mathematical Olympiad (IMO) 1995. For positive real numbers a , b , c {\displaystyle a,b,c} , where a b c = 1 {\displaystyle abc=1} we have that 1 a 3 ( b + c ) + 1 b 3 ( a + c ) + 1 c 3 ( a + b ) ≥ 3 2 . {\displaystyle {\frac {1}{a^{3}(b+c)}}+{\frac {1}{b^{3}(a+c)}}+{\frac {1}{c^{3}(a+b)}}\geq {\frac {3}{2}}.} Example 3. For positive real numbers a , b {\displaystyle a,b} we have that 8 ( a 4 + b 4 ) ≥ ( a + b ) 4 . {\displaystyle 8(a^{4}+b^{4})\geq (a+b)^{4}.} Example 4. For positive real numbers a , b , c {\displaystyle a,b,c} we have that 1 a + b + 1 b + c + 1 a + c ≥ 9 2 ( a + b + c ) . {\displaystyle {\frac {1}{a+b}}+{\frac {1}{b+c}}+{\frac {1}{a+c}}\geq {\frac {9}{2(a+b+c)}}.} === Proofs === Example 1. Proof: Use n = 3 , {\displaystyle n=3,} ( a 1 , a 2 , a 3 ) := ( a , b , c ) , {\displaystyle \left(a_{1},a_{2},a_{3}\right):=(a,b,c),} and ( b 1 , b 2 , b 3 ) := ( a ( b + c ) , b ( c + a ) , c ( a + b ) ) {\displaystyle \left(b_{1},b_{2},b_{3}\right):=(a(b+c),b(c+a),c(a+b))} to conclude: a 2 a ( b + c ) + b 2 b ( c + a ) + c 2 c ( a + b ) ≥ ( a + b + c ) 2 a ( b + c ) + b ( c + a ) + c ( a + b ) = a 2 + b 2 + c 2 + 2 ( a b + b c + c a ) 2 ( a b + b c + c a ) = a 2 + b 2 + c 2 2 ( a b + b c + c a ) + 1 ≥ 1 2 ( 1 ) + 1 = 3 2 . ◼ {\displaystyle {\frac {a^{2}}{a(b+c)}}+{\frac {b^{2}}{b(c+a)}}+{\frac {c^{2}}{c(a+b)}}\geq {\frac {(a+b+c)^{2}}{a(b+c)+b(c+a)+c(a+b)}}={\frac {a^{2}+b^{2}+c^{2}+2(ab+bc+ca)}{2(ab+bc+ca)}}={\frac {a^{2}+b^{2}+c^{2}}{2(ab+bc+ca)}}+1\geq {\frac {1}{2}}(1)+1={\frac {3}{2}}.\blacksquare } Example 2. We have that ( 1 a ) 2 a ( b + c ) + ( 1 b ) 2 b ( a + c ) + ( 1 c ) 2 c ( a + b ) ≥ ( 1 a + 1 b + 1 c ) 2 2 ( a b + b c + a c ) = a b + b c + a c 2 a 2 b 2 c 2 ≥ 3 a 2 b 2 c 2 3 2 a 2 b 2 c 2 = 3 2 . {\displaystyle {\frac {{\Big (}{\frac {1}{a}}{\Big )}^{2}}{a(b+c)}}+{\frac {{\Big (}{\frac {1}{b}}{\Big )}^{2}}{b(a+c)}}+{\frac {{\Big (}{\frac {1}{c}}{\Big )}^{2}}{c(a+b)}}\geq {\frac {{\Big (}{\frac {1}{a}}+{\frac {1}{b}}+{\frac {1}{c}}{\Big )}^{2}}{2(ab+bc+ac)}}={\frac {ab+bc+ac}{2a^{2}b^{2}c^{2}}}\geq {\frac {3{\sqrt[{3}]{a^{2}b^{2}c^{2}}}}{2a^{2}b^{2}c^{2}}}={\frac {3}{2}}.} Example 3. We have a 2 1 + b 2 1 ≥ ( a + b ) 2 2 {\displaystyle {\frac {a^{2}}{1}}+{\frac {b^{2}}{1}}\geq {\frac {(a+b)^{2}}{2}}} so that a 4 + b 4 = ( a 2 ) 2 1 + ( b 2 ) 2 1 ≥ ( a 2 + b 2 ) 2 2 ≥ ( ( a + b ) 2 2 ) 2 2 = ( a + b ) 4 8 . {\displaystyle a^{4}+b^{4}={\frac {\left(a^{2}\right)^{2}}{1}}+{\frac {\left(b^{2}\right)^{2}}{1}}\geq {\frac {\left(a^{2}+b^{2}\right)^{2}}{2}}\geq {\frac {\left({\frac {(a+b)^{2}}{2}}\right)^{2}}{2}}={\frac {(a+b)^{4}}{8}}.} Example 4. We have that 1 a + b + 1 b + c + 1 a + c ≥ ( 1 + 1 + 1 ) 2 2 ( a + b + c ) = 9 2 ( a + b + c ) . {\displaystyle {\frac {1}{a+b}}+{\frac {1}{b+c}}+{\frac {1}{a+c}}\geq {\frac {(1+1+1)^{2}}{2(a+b+c)}}={\frac {9}{2(a+b+c)}}.} == References ==
|
Wikipedia:Tjalling Koopmans#0
|
Tjalling Charles Koopmans (August 28, 1910 – February 26, 1985) was a Dutch-American mathematician and economist. He was the joint winner with Leonid Kantorovich of the 1975 Nobel Memorial Prize in Economic Sciences for his work on the theory of the optimum allocation of resources. Koopmans showed that on the basis of certain efficiency criteria, it is possible to make important deductions concerning optimum price systems. == Biography == Koopmans was born in 's-Graveland, Netherlands. He began his university education at the Utrecht University at seventeen, specializing in mathematics. Three years later, in 1930, he switched to theoretical physics. In 1933, he met Jan Tinbergen, the winner of the 1969 Nobel Memorial Prize in Economics and moved to Amsterdam to study mathematical economics under him. In addition to mathematical economics, Koopmans extended his explorations to econometrics and statistics. In 1936, he graduated from Leiden University with a PhD, under the direction of Hendrik Kramers. The title of the thesis was "Linear regression analysis of economic time series". He also worked for the Economic and Financial Organization of the League of Nations.: 28 Koopmans moved to the United States in 1940. There, he worked for a while for a government body in Washington, D.C., where he published on the economics of transportation focusing on optimal routing, then moved to Chicago where he joined a research body, the Cowles Commission for Research in Economics, affiliated with the University of Chicago. In 1946, he became a naturalized citizen of the United States, and in 1948, director of the Cowles Commission. Also in 1948, he was elected as a Fellow of the American Statistical Association. In 1950, he became a corresponding member of the Royal Netherlands Academy of Arts and Sciences. Rising hostile opposition to the Cowles Commission by the department of economics at University of Chicago during the 1950s led Koopmans to convince the Cowles family to move it to Yale University in 1955 (where it was renamed the Cowles Foundation). He continued to publish, on the economics of optimal growth and activity analysis. Koopmans's early works on the Hartree–Fock theory are associated with the Koopmans' theorem, which is very well known in quantum chemistry. Koopmans was awarded his Nobel memorial prize (jointly with Leonid Kantorovich) for his contributions to the field of resource allocation, specifically the theory of optimal use of resources. The work for which the prize was awarded focused on activity analysis, the study of interactions between the inputs and outputs of production, and their relationship to economic efficiency and prices. Finally, the importance of the article by Koopmans (1942) deriving the distribution of the serial correlation coefficient was recognized by John von Neumann, and it later influenced the optimal tests for a unit root by John Denis Sargan and Alok Bhargava (Sargan and Bhargava, 1983). === Family and name === Tjalling Charles Koopmans was a son of Sjoerd Koopmans and Wytske van der Zee; his middle name Charles was probably derived from his patronymic "Sjoerds". One of Sjoerd Koopmans's sisters, Gatske Koopmans, and her husband Symon van der Meer were the paternal grandparents of Nobel Prize winner Simon van der Meer. Tjalling Koopmans and Simon van der Meer were therefore first cousins once removed. Tjalling had two brothers, one of whom was theologian Rev. Dr Jan Koopmans, who in 1940, early during the German occupation of the Netherlands, wrote the widely distributed pamphlet "Bijna te laat" ("Almost too late", 30,000 copies), warning about the future of the Jews under the Nazi regime. In 1945, towards the end of the war, he witnessed an execution of hostages in Amsterdam from behind a window and was mortally wounded by a stray bullet. Koopmans married Truus Wanningen in October 1936. The couple had three children - a son, Henry, and two daughters, Anne and Helen. == Selected works == Koopmans, Tjalling C. (March 1942). "Serial correlation and quadratic forms in normal variables". Annals of Mathematical Statistics. 13 (1). Institute of Mathematical Statistics: 14–33. doi:10.1214/aoms/1177731639. JSTOR 2236158. Koopmans, Tjalling C.; Montias, J.M. (1971). "On the Description and Comparison of Economic Systems". {{cite journal}}: Cite journal requires |journal= (help) Cowles Foundation Paper No. 357. Koopmans, Tjalling C. (December 11, 1975). Nobel Memorial Lecture: Concepts of optimality and their uses (PDF). Koopmans, Tjalling C.; Debreu, Gérard (December 1982). "Additively decomposed quasiconvex functions" (PDF). Mathematical Programming. 24 (1). Springer: 1–38. doi:10.1007/BF01585092. S2CID 206799604. == Further reading == Hughes Hallett, Andrew J. (1989). "Econometrics and the Theory of Economic Policy: The Tinbergen–Theil Contributions 40 Years On". Oxford Economic Papers. 41 (1): 189–214. doi:10.1093/oxfordjournals.oep.a041892. JSTOR 2663189. Sargan, J. D.; Bhargava, Alok (1983). "Testing residuals from least squares regressions for being generated by the Gaussian random walk". Econometrica. 51 (1): 153–174. doi:10.2307/1912252. JSTOR 1912252. == See also == Linear programming Transportation theory (mathematics) Koopmans' Theorem (in quantum chemistry) == References == == External links == Tjalling Koopmans on Nobelprize.org Scarf, Herbert E., "Tjalling Charles Koopmans: August 28, 1910 – February 26, 1985", National Academy of Science IDEAS/RePEc Tjalling Koopmans at the Mathematics Genealogy Project Tjalling Charles Koopmans (1910–1985). Library of Economics and Liberty (2nd ed.). Liberty Fund. 2008. {{cite book}}: |work= ignored (help) Biography of Tjalling Koopmans from the Institute for Operations Research and the Management Sciences Tjalling Charles Koopmans Papers (MS 1439). Manuscripts and Archives, Yale University Library.
|
Wikipedia:Tlalcuahuitl#0
|
Tlalcuahuitl [t͡ɬaɬˈkʷawit͡ɬ] or land rod also known as a cuahuitl [ˈkʷawit͡ɬ] was an Aztec unit of measuring distance that was approximately 2.5 m (8.2 ft), 6 ft (1.8 m) to 8 ft (2.4 m) or 7.5 ft (2.3 m) long. The abbreviation used for tlalcuahuitl is (T) and the unit square of a tlalcuahuitl is (T²). == Subdivisions of tlalcuahuitl == == Acolhua Congruence Arithmetic == Using their knowledge of tlalcuahuitl, Barbara J. Williams of the Department of Geology at the University of Wisconsin and María del Carmen Jorge y Jorge of the Research Institute for Applied Mathematics and FENOMEC Systems at the National Autonomous University of Mexico believe the Aztecs used a special type of arithmetic. This arithmetic (tlapōhuallōtl [t͡ɬapoːˈwalːoːt͡ɬ]) the researchers called Acolhua [aˈkolwa] Congruence Arithmetic and it was used to calculate the area of Aztec people's land as demonstrated below: == See also == meter feet == References ==
|
Wikipedia:Toby Gee#0
|
Toby Stephen Gee (born 2 January 1980) is a British mathematician working in number theory and arithmetic aspects of the Langlands Program. He specialises in algebraic number theory. Gee was awarded the Whitehead Prize in 2012, the Leverhulme Prize in 2012, and was elected as a Fellow of the American Mathematical Society in 2014 and of the Royal Society in 2024. == Career == Gee read mathematics at Trinity College, Cambridge, where he was Senior Wrangler in 2000. After completing his PhD with Kevin Buzzard at Imperial College in 2004, he was a Benjamin Peirce Assistant Professor at Harvard University until 2010. From 2010 to 2011 Gee was an assistant professor at Northwestern University, at which point he moved to Imperial College London, where he has been a professor since 2013. With Mark Kisin, he proved the Breuil–Mézard conjecture for potentially Barsotti–Tate representations, and with Thomas Barnet-Lamb and David Geraghty, he proved the Sato–Tate conjecture for Hilbert modular forms. One of his most influential ideas has been the introduction of a general 'philosophy of weights', which has clarified some aspects of the emerging mod p Langlands philosophy. == References == == External links == Toby Gee's Professional Webpage Toby Gee's Curriculum Vitae Toby Gee's results at International Mathematical Olympiad
|
Wikipedia:Toka Diagana#0
|
Toka Diagana (also published as Tocka Diagana) is a Mauritanian-American mathematician, a professor of mathematics and chair of mathematics at the University of Alabama in Huntsville, and the editor-in-chief of the journal Nonautonomous Dynamical Systems. The topics of his research include functional analysis, stochastic processes, differential equations, dynamical systems, and operator theory. == Education and career == Diagana is originally from Kaédi, in southern Mauritania, a country in Northwest Africa. After attending a top high school in Kaédi, he went north to Tunisia for undergraduate studies at the Faculté des sciences de Tunis of Tunis El Manar University. There, his interests in mathematical analysis and topology were sparked by Abdenabi Achour and Said Zarati. At the suggestion of Achour, he traveled to France for doctoral study in mathematics, at Claude Bernard University Lyon 1. He earned a diploma of advanced studies (master's degree) there in 1995, and completed his Ph.D. in 1999, under the direction of Jean-Bernard Baillon, also working in Lyon with Étienne Ghys. After a brief stint as a secondary school teacher in Thoissey, France, Diagana came to Howard University in the US as a lecturer in 2000, and obtained an assistant professorship there in 2002. He was quickly promoted to associate professor in 2004, and then to full professor in 2007. He moved to his present position as professor and chair at the University of Alabama in Huntsville in 2018. == Contributions == Diagana founded the African Diaspora Journal of Mathematics, and is the editor-in-chief of the journal Nonautonomous Dynamical Systems. His 13 authored and edited books include both mathematical monographs and multiple compilations of mathematics from researchers of the African diaspora. Despite many academic publications, in a profile on Mathematically Gifted & Black, he describes his greatest accomplishment as his mentorship of eight African Americans to doctorates in mathematics, including two women, countering the historic underrepresentation of people from these groups in mathematics. == Recognition == Diagana is a 2006 recipient of the Prix Chinguitt, a Mauritanian national prize given annually for excellence in science research. He is a Fellow of the African Academy of Sciences, elected in 2009. == References == == External links == Home page Toka Diagana publications indexed by Google Scholar
|
Wikipedia:Tom Bohman#0
|
Tom Bohman is an American mathematician who is a former head of the Department of Mathematical Sciences and is a Alexander M. Knaster Professor at Carnegie Mellon University. == References ==
|
Wikipedia:Tomasz Łuczak#0
|
Tomasz Łuczak (born 13 March 1963 in Poznań) is a Polish mathematician and professor at Adam Mickiewicz University in Poznań and Emory University. His main field of research is combinatorics, specifically discrete structures, such as random graphs, and their chromatic number. Under supervision of Michał Karoński, Łuczak earned his doctorate at Adam Mickiewicz University in Poznań in 1987. In 1991, he received the Kuratowski Prize and the following year, he was awarded the EMS Prize. In 1997, he won the prestigious Prize of the Foundation for Polish Science for his work on the theory of random discrete structures. == References == == External links == Website at Emory University Archived 2011-05-25 at the Wayback Machine Website at Adam Mickiewicz University
|
Wikipedia:Tomer Schlank#0
|
Tomer Moshe Schlank (Hebrew: תומר משה שלנק; born 1982) is an Israeli mathematician and a professor at The University of Chicago. Previously, he was a professor at Hebrew University of Jerusalem. He primarily works in homotopy theory, algebraic geometry, and number theory. In 2022 he won the Erdős prize in mathematics and in 2023 he was awarded a European Research Council consolidator grant. He is an editor for the Israel Journal of Mathematics. == Biography == Schlank was born on July 29, 1982, in Jerusalem, Israel. He graduated with a bachelor's degree from Tel Aviv University in 2001 and a master's degree from Tel Aviv University in 2008. He received his PhD from Hebrew University of Jerusalem in January, 2013, working under the supervision of Ehud de Shalit. His education was also influenced by the close proximity of David Kazhdan and Emmanuel Dror Farjoun. After completing his PhD, Schlank was hired as a Simons postdoctoral fellow at MIT. Afterwards he moved back to the Hebrew University in Jerusalem. Schlank is the great-grandson of the scientist Maria Pogonowska. == Research == Schlank is primarily known for his work on chromatic homotopy theory. Together with Robert Burklund, Jeremy Hahn, and Ishan Levy, he disproved the telescope conjecture for all heights greater than 1 and for all primes. This was the last outstanding conjecture among Ravenel's conjectures. The disproof made use of his work on ambidexterity of the T(n)-local category and cyclotomic extensions of the T(n)-local sphere with Ben-Moshe, Carmeli, and Yanovski. With Barthel, Stapleton, and Weinstein, he calculated the homotopy groups of the rationalization of the K(n)-local sphere. With Burklund and Yuan, Schlank proved the "chromatic nullstellensatz", a version of Hilbert's nullstellensatz for the T(n)-local category in which Morava E-theories play the role of algebraically closed fields. This work resolved the Ausoni—Rognes redshift conjecture for E ∞ {\displaystyle E_{\infty }} -ring spectra and also produced E ∞ {\displaystyle E_{\infty }} -orientations of Morava E-theory. Schlank's early work was a synthesis of homotopy theory and number theory. With Harpaz, he developed homotopy obstructions to the existence of rational points on smooth varieties over number fields and related these homotopy obstructions to the Manin obstruction. He wrote his thesis, titled "Applications of homotopy theory to the study of obstructions to existence of rational points", on this topic. Schlank is known for the breadth of his work and for bringing together seemingly unrelated concepts from different fields to solve problems. In mathematics, he has published papers in algebraic geometry, algebraic topology, category theory, combinatorics, dynamical systems, geometric topology, number theory, and representation theory. == References ==
|
Wikipedia:Tomoyuki Arakawa#0
|
Tomoyuki Arakawa (Japanese: 荒川 知幸; born 22 May 1968) is a Japanese mathematician, mathematical physicist, and a professor at the RIMS of the Kyoto University. His research interests are representation theory and vertex algebras and he is known especially for the work in W-algebras. He obtained his PhD from the University of Nagoya in 1999. In 2018 he was invited speaker at the International Congress of Mathematicians in Rio de Janeiro. He won the MSJ Autumn Prize in 2017 for his work on representation theory of W-algebras. == Selected publications == Arakawa, Tomoyuki (1 August 2007). "Representation theory of W-algebras". Inventiones Mathematicae. 169 (2): 219–320. doi:10.1007/s00222-007-0046-1. hdl:10935/1423. ISSN 1432-1297. S2CID 122047968. Arakawa, Tomoyuki (2015). "Rationality of W-algebras: principal nilpotent cases". Annals of Mathematics. 182 (2): 565–604. arXiv:1211.7124. doi:10.4007/annals.2015.182.2.4. ISSN 0003-486X. JSTOR 24523343. S2CID 53477362. Arakawa, Tomoyuki; Creutzig, Thomas; Linshaw, Andrew R. (1 October 2019). "W-algebras as coset vertex algebras". Inventiones Mathematicae. 218 (1): 145–195. arXiv:1801.03822. Bibcode:2019InMat.218..145A. doi:10.1007/s00222-019-00884-3. ISSN 1432-1297. S2CID 253745764. == References == == External links == Personal website
|
Wikipedia:Tonelli–Hobson test#0
|
In mathematics, the Tonelli–Hobson test gives sufficient criteria for a function ƒ on R2 to be an integrable function. It is often used to establish that Fubini's theorem may be applied to ƒ. It is named for Leonida Tonelli and E. W. Hobson. More precisely, the Tonelli–Hobson test states that if ƒ is a real-valued measurable function on R2, and either of the two iterated integrals ∫ R ( ∫ R | f ( x , y ) | d x ) d y {\displaystyle \int _{\mathbb {R} }\left(\int _{\mathbb {R} }|f(x,y)|\,dx\right)\,dy} or ∫ R ( ∫ R | f ( x , y ) | d y ) d x {\displaystyle \int _{\mathbb {R} }\left(\int _{\mathbb {R} }|f(x,y)|\,dy\right)\,dx} is finite, then ƒ is Lebesgue-integrable on R2. == References ==
|
Wikipedia:Tony Tan#0
|
Tony Tan Keng Yam (Chinese: 陈庆炎; pinyin: Chén Qìngyán; Pe̍h-ōe-jī: Tân Khèng-iām; born 7 February 1940) is a Singaporean banker and politician who served as the seventh president of Singapore between 2011 to 2017. Prior to entering politics, Tan was a general manager at OCBC Bank. He made his political debut in the 1979 by-elections as a PAP candidate contesting in Sembawang SMC and won. He also served as Deputy Prime Minister of Singapore between 1995 and 2005. Tan resigned from the Cabinet in 2005 and was appointed deputy chairman and executive director of GIC, the country's sovereign wealth fund, chairman of the National Research Foundation and Chairman of SPH. He resigned from all of his positions in 2010 before contesting in the 2011 presidential election as an independent candidate. Tan won the 2011 presidential election in a four-cornered fight and served as the president of Singapore until 2017. He did not seek for a re-election in the 2017 presidential election, which was reserved for Malay candidates after a constitutional amendment. His presidential term ended and he officially retired on 1 September 2017. He was succeeded by Halimah Yacob on 14 September 2017. == Education == Tan was educated at St Patrick's School and St Joseph's Institution before topping his class and graduating from the University of Singapore (now the National University of Singapore) with a Bachelor of Science with first class honours degree in physics, under a scholarship conferred by the Singapore Government. He subsequently went on to complete a Master of Science degree in operations research at the Massachusetts Institute of Technology, under the Asia Foundation Scholarship. He also completed a Doctor of Philosophy in applied mathematics at the University of Adelaide, and went on to teach mathematics at the University of Singapore. == Career == In 1969, Tan left the University of Singapore and joined OCBC Bank, where he became the general manager, before leaving the bank in 1979 to enter politics. In December 1991, Tan stepped down from the Cabinet to return to the private sector, where he rejoined OCBC Bank as the chairman and chief executive officer from 1992 to 1995, while retaining his seat in the Parliament as the Member of Parliament for Sembawang GRC. Tan was appointed as Deputy Chairman and Executive Director of GIC, the country's sovereign wealth fund, following his second retirement from Cabinet in 2005. He was also appointed as chairman of the National Research Foundation, Deputy Chairman of the Research, Innovation and Enterprise Council, and Chairman of Singapore Press Holdings concurrently. Tan's tenure at GIC coincided with moves toward greater disclosure in the investment fund's activities amid mounting concerns about the secretive fund's influence after high-profile investments in UBS and Citigroup. === Political career === A former member of the governing People's Action Party (PAP), Tan was elected as the Member of Parliament (MP) for Sembawang GRC after his electoral victory in the 1979 by-elections. He was subsequently appointed as Senior Minister of State for Education in 1979. ==== Minister for Education (1980–1981, 1985–1991) ==== He joined the Cabinet in 1980, serving as Minister for Education. As the Minister for Education, Tan scrapped a policy that favoured children of more well-educated mothers ahead of children of less-educated mothers in primary school placement in response to popular discontent and public criticism of the policy which saw PAP receiving the lowest votes since independence during the 1984 general election. He also introduced the independent schools system, allowing established educational institutions in Singapore to charge its own fees and have control over their governance and teaching staff, though this was criticised by parents as being "elitist" and made top-ranked schools increasingly out of reach to poorer families due to subsequent fee hikes. ==== Minister for Trade and Industry (1981–1986) ==== Tan took on the role of Minister for Trade and Industry from 1981 to 1986. He was also appointed as Minister for Finance from 1983 to 1985, and Minister for Health from 1985 to 1986. Tan espoused a cut in the Central Provident Fund (CPF) in the 1980s, which Prime Minister Lee Kuan Yew had said would not be allowed except "in an economic crisis". Tan was also known to have opposed the shipping industry strike in January 1986, the first for about a decade in Singapore, which was sanctioned by fellow Cabinet minister, Ong Teng Cheong, who is also Secretary-General of the National Trades Union Congress, felt the strike was necessary. As Minister for Trade and Industry, Tan was concerned about investors' reactions to a perceived deterioration of labour relations and the impact on foreign direct investment. In his analysis, historian Michael Barr explains that older [grassroots] union leaders bore "increasing disquiet" at their exclusion from consultation in NTUC's policies, which were effectively managed by "technocrats" in the government. Unlike the previous NTUC secretary-general Lim Chee Onn, Lee Kuan Yew's protégé Ong Teng Cheong in 1983 had an "implicit pact" with the trade unions—involving grassroots leaders in top decisions and "working actively and forcefully" in the interests of the unions "in a way Lim had never seen to do"—in exchange for the unions' continued "cooperation on the government's core industrial relations strategies". (In 1969, the NTUC had adopted "a cooperative, rather than a confrontational policy towards employers".) Although striking was prohibited and trade unions were barred from negotiating such matters as promotion, transfer, employment, dismissal, retrenchment, and reinstatement, issues that "accounted for most earlier labour disputes", the government provided measures for workers' safety and welfare, and serious union disputes with employers were almost always handled through the Industrial Arbitration Court, which had powers of both binding arbitration and voluntary mediation. However, Ong felt these measures did not prevent "management [from] taking advantage of the workers", recalling in a 2000 interview in Asiaweek: "Some of them were angry with me about that... the minister for trade and industry [Tan] was very angry, his officers were upset. They had calls from America, asking what happened to Singapore?" However the fact that the strike only lasted two days before "all the issues were settled" was cited by Ong in a 2000 interview with Asiaweek as proof that "management was just trying to pull a fast one". Tan initially opposed the timing of building the Mass Rapid Transit (MRT) in 1981 when it was raised by Ong. Tan held the view that the local construction industry was overheated at the time, and public housing should take priority. ==== Deputy Prime Minister (1995–2005) ==== After Ong Teng Cheong and Lee Hsien Loong were diagnosed with cancer in 1992 and 1993 respectively, Tan was asked to return to Cabinet in August 1995 as Deputy Prime Minister and Minister for Defence. It was reported that he declined an offer of make-up pay, which compensate ministers for a loss in salary when they leave the private sector. Tan declared that "the interests of Singapore must take precedence over that of a bank and my own personal considerations". In August 2003, he relinquished the portfolio of Minister for Defence and became Coordinating Minister for Security and Defence, while retaining the portfolio of Deputy Prime Minister. He later persuaded Minister for National Development Mah Bow Tan to abandon plans to demolish an old mosque in his constituency of Sembawang. Dubbed the "Last Kampung Mosque in Singapore", it was later designated a heritage site. Tan joined other dissenting colleagues in opposing the implementation of Integrated Resorts (IRs) with their attached casinos to Singapore. Commenting on a survey of gambling habits conducted by the Ministry of Community Development, Youth and Sports, Tan had said he was "appalled" that a newspaper headline dismissed the number of likely problem gamblers—55,000 as insignificant: "I don't think it's insignificant. Every Singaporean is important. Every Singaporean that gets into trouble means one family that is destroyed. It cannot be a matter of small concern to the Government." Prime Minister Lee Kuan Yew picked Tan to succeed him as prime minister, but Tan declined. Prime Minister Lee once praised Tan for his quick mind and decisiveness. "He would say 'yes or no' and he would stick to it," said Prime Minister Lee. As deputy prime minister, Tan was instrumental in the establishment of the Singapore Management University (SMU) and shaped its direction and early history. In 1997, the Singapore Government raised the idea of a third university for Singapore. Tan believed that the new university should differentiate itself from the two established institutions—the National University of Singapore (NUS) and the Nanyang Technological University (NTU), as the government wanted SMU to be an experiment in diversity. Tan believed that the third university should follow the American example which concentrated on management, business and economics. He made trips to universities in the United States to know more about them and search for potential partnerships. He helped to make the third university happen, reaching out to veteran businessman and current chairman of the SMU board of trustees Ho Kwon Ping to help in its establishment. Tan, having begun to look after the university education in the 1990s, was the driving force behind SMU, which in 2000 was set up as the country's first publicly-funded autonomous university. Tan stepped down as Deputy Prime Minister and Coordinating Minister for Security and Defence on 1 September 2005. == 2011 presidential election == On 22 December 2010, Tan announced that he would step down from his government-linked positions at GIC and SPH to run for the office of President of Singapore. Tan's campaign stressed his independence and his divergent views from the PAP government in specific policies, citing a remark made by MP Tan Soo Khoon in 2005: "It is probably the first time that I have heard Cabinet ministers, starting with no less than the deputy prime minister, Dr. Tony Tan, expressing divergent views [on the Integrated Resorts question]." However, competing presidential candidates and former PAP members Tan Kin Lian and Tan Cheng Bock questioned Tan's independence from the party. On 7 July 2011, Tony Tan submitted his presidential eligibility forms. In July 2011, Tony Tan stepped down from his positions at GIC and SPH to contest in the presidential election. Tan subsequently won 35.20% of the vote. On 29 July 2011, Tan responded to online allegations that his son Patrick Tan had received preferential treatment during his National Service (NS). "My sons all completed their National Service obligations fully and I have never intervened in their postings," he said. Tan also noted that he had served as Minister for Defence from 1995 to 2003, while Patrick Tan said that it was in 1988 that he been permitted by the Ministry of Defence (MINDEF) to disrupt his NS for premedical studies in Harvard University, where he graduated with a Bachelor of Science degree in biology and chemistry, and an MD-PhD program at Stanford University under the President's Scholarship and Loke Cheng Kim Scholarship. MINDEF clarified that, prior to 1992, disruptions were allowed for overseas medical studies, and longer periods of disruption were granted for those admitted to universities in the United States, where medicine is a graduate course. American medical students are required to complete a "pre-medical component for a general undergraduate degree" before applying to medical school. In response to a question in Parliament on the subject of deferments, Minister for Defence Ng Eng Hen stated on 20 October 2011 that Patrick Tan had not been given any special treatment. === Campaign platform === Describing himself as "Tested, Trusted, True", Tan said his past experiences will help him steer Singapore through the financial uncertainty lying ahead. During the Nomination Day on 17 August 2011, Tan unveiled his election symbol—a pair of black glasses which resembles the trademark spectacles he has steadfastly worn for years. His campaign materials, which included caps, postcards and fridge magnets also carried the symbol. About 9,400 posters and 200 banners were printed. === Campaign endorsements === Tan's presidential bid was endorsed by the 10,000-strong Federation of Tan Clan Associations on 7 August 2011. By 13 August 2011, the leaders of 19 NTUC-affiliated unions—which have 128,000 members, had endorsed his bid. On 14 August, the leadership of the Singapore Federation of Chinese Clan Associations (SFCCA) and the Singapore Chinese Chamber of Commerce & Industry (SCCCI) also endorsed his bid. The leadership of another four unions from the construction and real estate sector, which represent more than 50,000 members, endorsed Tan's bid on 16 August. Nine Teochew clan associations also supported Tan. Union leaders in three sectors—Transport and Logistics, Marine and Machinery-engineering, and Infocomm and Media—endorsed Tan on 17 August. They together represent 112,000 workers. The Singapore Malay Chamber of Commerce and Industry (SMCCI) endorsed Tan's presidential candidacy on 18 August 2011. It is also was the first Malay organisation to do so. === Campaign proceedings === After a closed door meeting with the Singapore Malay Chamber of Commerce and Industry on 11 August 2011, Tan remarked that it is "not too early" for the government to have contingency plans in case an economic crisis hits Singapore, noting that "with his background and knowledge", he added that he was in a position to provide "a steady hand". Speaking to reporters after a dialogue with the Singapore Manufacturers' Federation the following day, Tan remarked that it would be a "grave mistake" to phase out manufacturing in Singapore, which has been transitioning to a service economy and an information economy since the 1980s. He then went on to describe manufacturing as a "key pillar of Singapore's economy". Without the sector, he feels Singapore's economy will be "less resilient, less diversified" and there will be "fewer options for our young people and Singapore will lose." On 15 August 2011, following the National Day Rally speech by Prime Minister Lee Hsien Loong, Tan said that one point he found particularly interesting in Lee's address was whether Singapore would remain pragmatic in its policy making, or if it would turn populist. He added that the temptation to make populist decisions was affecting the presidential election, "with some candidates appealing to the public in ways that could go beyond the parameters of the Singapore's Constitution". On 17 August 2011, crowds booed at Tan and his son as he delivered his two-minute Nomination Day speech. According to The Straits Times, the jeers came from a vocal group of people who mostly supported another presidential candidate Tan Jee Say. At a press conference later that day, Tony Tan said that while different points of view were to be expected in a campaign, it was disappointing to have people who would not even listen, and hoped that Singaporeans would listen to the views of all the candidates. He said, "I don't think that jeering or heckling is the right way to go about the campaign, particularly in a campaign for the president, which has to be conducted with decorum and dignity." During the first presidential candidate broadcast on 18 August 2011, while other candidates made promises in their first presidential candidate broadcasts on Thursday night, Tan refrained from making promises during the broadcast and focused on the role of the president instead. Speaking in English, Chinese and Malay, Tan said, "Some people argue that the president must take a public stand on current issues. I hear and share the concerns of Singaporeans. But policies are debated in Parliament and implemented by the government. Others have said that the president must oppose the government. That is a job for the opposition. People interested in such roles should run for Parliament in the next general election." == Presidency (2011–2017) == Tan sought to distinguish his presidency by promoting a more active civil society, believing that Singapore needed to build up its "social reserves" to complement the substantial financial reserves the city state had accumulated over time. An example of this, he said, was the way that he had expanded Singapore's President's Challenge charity event to go beyond fund-raising to promote volunteerism and social entrepreneurship. On 8 November 2016, Tan announced that he would not be standing in 2017 presidential election, which was reserved for Malay candidates after a constitutional amendment on 9 November 2016. Tan left office on 31 August 2017. He was succeeded by Halimah Yacob who became president after a walkover of the presidential elections, as no other candidates were deemed eligible. == Other appointments == From 1980 to 1981, Tan served as Vice-Chancellor of the National University of Singapore. Tan was subsequently ex officio appointed Chancellor of the National University of Singapore and the Nanyang Technological University when he was elected as the president of Singapore in 2017. Tan had served as patron of many organisations, including the Singapore Dance Theatre, Singapore Computer Society, SJI International, Duke–NUS Medical School, and the MIT Club of Singapore. He was also named as the first patron of Dover Park Hospice in May 2011. On 21 November 2017, GIC, the country's sovereign wealth fund, announced that Tan will be appointed Director and Special Advisor from 1 January 2018. == Personal life == Tan's paternal grandfather was Tan Cheng Siong, the former general manager of the Overseas Chinese Bank, one of three precursor banks to OCBC Bank. His uncle was former chairman of OCBC Bank, Tan Chin Tuan. Through his maternal grandmother Annie Tan Sun Neo, he is also a great-great-great grandson of philanthropist Tan Kim Seng. When Tan was a first year physics student in 1959 at the University of Malaya—the predecessor of the National University of Singapore—at Bukit Timah campus, he met an arts undergraduate whom he fell in love and would marry five years later. Tan married Mary Chee Bee Kiang in 1964 and they have five children together; Peter Tan Boon Huan, Sharon Tan Shu Lin, Patricia Tan Shu Ming, Patrick Tan Boon Ooi and Philip Tan Boon Yew. Tan's son-in-law is NUS vice provost and dean Simon Chesterman. == Honours == In 2005, Tan was presented the NUS Eminent Alumni Award in recognition of his role as a visionary architect of Singapore's university sector. In 2010, he was presented the inaugural Distinguished Australian Alumnus Award by the Australian Alumni Singapore (AAS) at its 55th anniversary dinner in recognition of his distinguished career, and his significant contribution to society and to the Australian alumni community. Tan was awarded a medal from the Foreign Policy Association in 2011 for "outstanding leadership and service". In 2014, Tan was conferred an honorary doctorate by his alma mater, the University of Adelaide, for his "long record of outstanding achievements both as a leader in the Singapore government and in the business sectors. He was also made a Knight Grand Cross of the Order of the Bath. In 2017, Tan received the "Key to the City" for the Czech city of Prague, Czech Republic, during his state visit. In 2018, Tan received the top honour of the Order of Temasek (First Class) during Singapore's National Day Awards. In 2018, Tan was conferred an honorary degree of Doctor of Letters (D.Litt) from NTU. In 2022, Tan was conferred an honorary Doctor of Laws (LL.D) from SMU. == References ==
|
Wikipedia:Topological module#0
|
In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous. == Examples == A topological vector space is a topological module over a topological field. An abelian topological group can be considered as a topological module over Z , {\displaystyle \mathbb {Z} ,} where Z {\displaystyle \mathbb {Z} } is the ring of integers with the discrete topology. A topological ring is a topological module over each of its subrings. A more complicated example is the I {\displaystyle I} -adic topology on a ring and its modules. Let I {\displaystyle I} be an ideal of a ring R . {\displaystyle R.} The sets of the form x + I n {\displaystyle x+I^{n}} for all x ∈ R {\displaystyle x\in R} and all positive integers n , {\displaystyle n,} form a base for a topology on R {\displaystyle R} that makes R {\displaystyle R} into a topological ring. Then for any left R {\displaystyle R} -module M , {\displaystyle M,} the sets of the form x + I n M , {\displaystyle x+I^{n}M,} for all x ∈ M {\displaystyle x\in M} and all positive integers n , {\displaystyle n,} form a base for a topology on M {\displaystyle M} that makes M {\displaystyle M} into a topological module over the topological ring R . {\displaystyle R.} == See also == == References == Atiyah, Michael Francis; MacDonald, I.G. (1969). Introduction to Commutative Algebra. Westview Press. ISBN 978-0-201-40751-8. Kuz'min, L. V. (1993). "Topological modules". In Hazewinkel, M. (ed.). Encyclopedia of Mathematics. Vol. 9. Kluwer Academic Publishers.
|
Wikipedia:Topological semigroup#0
|
In mathematics, a topological semigroup is a semigroup that is simultaneously a topological space, and whose semigroup operation is continuous. Every topological group is a topological semigroup. == See also == == References ==
|
Wikipedia:Tord Ganelius#0
|
Tord Hjalmar Ganelius (born 23 May 1925 in Stockholm, dead 14 March 2016 in Stockholm) was a Swedish mathematician and professor emeritus. He served as Permanent Secretary of the Royal Swedish Academy of Sciences and was a board member of the Nobel Foundation from 1981 to 1989. His primary research interests were holomorphic functions and approximation theory. == Education and career == Ganelius completed his Ph.D. in 1953 at Stockholms Högskola (known as Stockholm University since 1960) by presenting his dissertation Sequences of Analytic Functions and Their Zeros. He was an associate professor at Lund University in 1953–56 and in 1957 he was appointed Professor of Mathematics at University of Gothenburg, where he served until 1981. During this time he was Dean of the Faculty of Science twice, in 1963–65 and 1977–80. In 1972, he was elected a member of the Royal Swedish Academy of Sciences and he was appointed the permanent secretary in 1981, a position he held until 1989. He was a board member of the Nobel Foundation from 1981 to 1989, and every October he announced the Nobel Laureates in Physics, Chemistry and Economic Sciences. Tord Ganelius has also been a visiting professor at the University of Washington in 1962, Cornell University in 1967–68 and the University of California, San Diego in 1972–73. In 1966, Ganelius published the Swedish-language mathematics textbook ”Introduktion till matematiken”, which since 2006 has been available on-line. == Family == Tord Ganelius is the son of Hjalmar and Ebba G. In 1951 he married Aggie Hemberg (born 1928). They have four children: Per (1952), Truls (1955), Svante (1957) and Aggie Öhman (1963). == References ==
|
Wikipedia:Toric ideal#0
|
In algebra, a toric ideal is an ideal generated by differences of two monomials. An affine or projective algebraic variety defined by a toric prime ideal or a homogeneous toric ideal is an affine or projective toric variety. == References ==
|
Wikipedia:Torsten Carleman#0
|
Torsten Carleman (8 July 1892, Visseltofta, Osby Municipality – 11 January 1949, Stockholm), born Tage Gillis Torsten Carleman, was a Swedish mathematician, known for his results in classical analysis and its applications. As the director of the Mittag-Leffler Institute for more than two decades, Carleman was the most influential mathematician in Sweden. == Work == The dissertation of Carleman under Erik Albert Holmgren, as well as his work in the early 1920s, was devoted to singular integral equations. He developed the spectral theory of integral operators with Carleman kernels, that is, kernels K(x, y) such that K(y, x) = K(x, y) for almost every (x, y), and ∫ | K ( x , y ) | 2 d y < ∞ {\displaystyle \int |K(x,y)|^{2}dy<\infty } for almost every x. In the mid-1920s, Carleman developed the theory of quasi-analytic functions. He proved the necessary and sufficient condition for quasi-analyticity, now called the Denjoy–Carleman theorem. As a corollary, he obtained a sufficient condition for the determinacy of the moment problem. As one of the steps in the proof of the Denjoy–Carleman theorem in Carleman (1926), he introduced the Carleman inequality ∑ n = 1 ∞ ( a 1 a 2 ⋯ a n ) 1 / n ≤ e ∑ n = 1 ∞ a n , {\displaystyle \sum _{n=1}^{\infty }\left(a_{1}a_{2}\cdots a_{n}\right)^{1/n}\leq e\sum _{n=1}^{\infty }a_{n},} valid for any sequence of non-negative real numbers ak. At about the same time, he established the Carleman formulae in complex analysis, which reconstruct an analytic function in a domain from its values on a subset of the boundary. He also proved a generalisation of Jensen's formula, now called the Jensen–Carleman formula. In the 1930s, independently of John von Neumann, he discovered the mean ergodic theorem. Later, he worked in the theory of partial differential equations, where he introduced the Carleman estimates, and found a way to study the spectral asymptotics of Schrödinger operators. In 1932, following the work of Henri Poincaré, Erik Ivar Fredholm, and Bernard Koopman, he devised the Carleman embedding (also called Carleman linearization), a way to embed a finite-dimensional system of nonlinear differential equations du⁄dt = P(u) for u: Rk → R, where the components of P are polynomials in u, into an infinite-dimensional system of linear differential equations. In 1933 Carleman published a short proof of what is now called the Denjoy–Carleman–Ahlfors theorem. This theorem states that the number of asymptotic values attained by an entire function of order ρ along curves in the complex plane going outwards toward infinite absolute value is less than or equal to 2ρ. In 1935, Torsten Carleman introduced a generalisation of Fourier transform, which foreshadowed the work of Mikio Sato on hyperfunctions; his notes were published in Carleman (1944). He considered the functions f of at most polynomial growth, and showed that every such function can be decomposed as f = f+ + f−, where f+ and f− are analytic in the upper and lower half planes, respectively, and that this representation is essentially unique. Then he defined the Fourier transform of (f+, f−) as another such pair (g+, g−). Though conceptually different, the definition coincides with the one given later by Laurent Schwartz for tempered distributions. Carleman's definition gave rise to numerous extensions. Returning to mathematical physics in the 1930s, Carleman gave the first proof of global existence for Boltzmann's equation in the kinetic theory of gases (his result applies to the space-homogeneous case). The results were published posthumously in Carleman (1957). Carleman supervised the Ph.D. theses of Ulf Hellsten, Karl Persson (Dagerholm), Åke Pleijel and (jointly with Fritz Carlson) of Hans Rådström. == Life == Carleman was born in Visseltofta to Alma Linnéa Jungbeck and Karl Johan Carleman, a school teacher. He studied at Växjö Cathedral School, graduating in 1910. He continued his studies at Uppsala University, being one of the active members of the Uppsala Mathematical Society. Kjellberg recalls: He was a genius! My older friends in Uppsala used to tell me about the wonderful years they had had when Carleman was there. He was the most active speaker in the Uppsala Mathematical Society and a well-trained gymnast. When people left the seminar crossing the Fyris River, he walked on his hands on the railing of the bridge. From 1917 he was docent at Uppsala University, and from 1923 — a full professor at Lund University. In 1924 he was appointed professor at Stockholm University. He was elected a member of the Royal Swedish Academy of Sciences in 1926 and of the Finnish Society of Sciences and Letters in 1934. From 1927, he was director of the Mittag-Leffler Institute and editor of Acta Mathematica. From 1929 to 1946 Carleman was married to Anna-Lisa Lemming (1885–1954), the half-sister of the athlete Eric Lemming who won four golden medals and three bronze at the Olympic Games. During this period he was also known as a recognized fascist, anti-semite and xenophobe. His interaction with William Feller before the former departure to the United States was not particularly pleasant, at some point being reported due to his opinion that "Jews and foreigners should be executed". Carlson remembers Carleman as: "secluded and taciturn, who looked at life and people with a bitter humour. In his heart, he was inclined to kindliness towards those around him, and strove to assist them swiftly." Towards the end of his life, he remarked to his students that "professors ought to be shot at the age of fifty." During the last decades of his life, Carleman abused alcohol, according to Norbert Wiener and William Feller. His final years were plagued by neuralgia. At the end of 1948, he developed the liver disease jaundice; he died from complications of the disease. == Selected publications == Carleman, T. (1926). Les fonctions quasi analytiques (in French). Paris: Gauthier-Villars. JFM 52.0255.02. Carleman, T. (1944). L'Intégrale de Fourier et Questions que s'y Rattachent (in French). Uppsala: Publications Scientifiques de l'Institut Mittag-Leffler. MR 0014165. Carleman, T. (1957). Problèmes mathématiques dans la théorie cinétique des gaz (in French). Uppsala: Publ. Sci. Inst. Mittag-Leffler. MR 0098477. Carleman, Torsten (1960), Pleijel, Ake; Lithner, Lars; Odhnoff, Jan (eds.), Edition Complete Des Articles De Torsten Carleman, Litos reprotryk and l'Institut mathematique Mittag-Leffler == Notes == == External links == Torsten Carleman at the Mathematics Genealogy Project
|
Wikipedia:Toshmuhammad Qori-Niyoziy#0
|
Toshmuhammad Qori-Niyoziy (Uzbek Cyrillic: Тошмуҳаммад Ниёзович Қори-Ниёзий, Russian: Ташмухамед Ниязович Кары-Ниязов, Tashmukhamed Niyazovich Kary-Niyazov; 2 September [O.S. 21 August] 1897 — 17 March 1970) was an Uzbek mathematician and historian who served as the first president of the Academy of Sciences of the Uzbek SSR. == Early life == Born in Khujand on 2 September [O.S. 21 August] 1897 to a shoemaker, he initially received schooling in a maktab, but attended for less than a year due to abuse from the teacher. His family went on to move to Skobelev (now Fergana), where he eventually attended a Russian school and graduated with excellent marks in the mid-1910s. In 1917, he became a teacher at a school he founded in Kokand, which quickly became a regional school. Initially having volunteered to serve as head of schools for the Skobelev district, he went on to serve as director of the Uzbek Pedagogical College in Kokand from 1920 to 1925. Several years later he graduated from the Faculty of Physics and Mathematics at Central Asian State University in Tashkent; he defended his thesis in Uzbek. His wife Oishakhon, who he married in 1920 in a Muslim ceremony, was one of the first women teachers in the Uzbek SSR. She frequently advised him on his philology work, including the first Uzbek dictionaries that they worked on together. == Career == Whilst a university student, he was tasked with teaching advanced math classes such as analytic geometry in the Uzbek language. After graduating he continued to teach university-level mathematics in the Uzbek language, becoming the first Uzbek to receive the title of professor in 1931. That year, he became a member of the Communist Party. From then to 1933 he served as a rector at the university, although he did not receive his doctorate of physics and mathematics until 1939. He then became the Deputy Chairman of the Committee of the Uzbek SSR for Science, Culture and Art, and worked on the transition of the Uzbek alphabet to a Cyrillic script. He also devoted a considerable amount of time to researching the history of Uzbekistan and historic academic works, with a special focus on astronomy and archaeology. As part of his research about early astronomy in what is present-day Uzbekistan, he had to read through numerous Arabic manuscripts. In addition to his academic work, he held various political offices, serving as a deputy in the Supreme Soviet of the USSR for the 1st and 2nd convocations. He also authored numerous textbooks and academic papers, including the first Uzbek-language math textbooks and papers about Uzbek culture and society. == World War II == In June 1941, he led alongside Mikhail Gerasimov a scientific expedition to examine the tomb of Timur in Samarkand. According to local legend, an inscription on the tomb threatened to bring about a catastrophe to whoever opened it, and shortly after it was opened, Nazi Germany began invading the Soviet Union. After the remains were reburied with Muslim rites in 1942, some in Uzbekistan credited the Soviet victory in the Battle of Stalingrad to the reburial. After the German invasion of the Soviet Union, his only son Shavkat applied to go to the frontlines with the Red Army. Being skilled in mathematics like his father, he was chosen for artillery school. After surviving the war, Shavkat went on to graduate from the F.E. Dzerzhinky Military Academy and follow in his father's footsteps with a career in mathematics, but specialized in ballistics and rocket technology. When the Academy of Sciences of the Uzbek SSR was established in 1943, Qori-Niyazov was made its first president and held the post until 1947. == Postwar == In 1946 Qori-Niyoziy became a professor at the Tashkent Institute of Engineers and Agricultural Mechanization. For his paper "Ulugbek's Astronomical School" he was awarded the Stalin Prize. In 1954 he became a member of the International Astronomical Union, in 1967 he became a corresponding member of the International Academy of the History of Science, and that same year on 1 September he was awarded the title Hero of Socialist Labour for his work promoting academics in the Uzbek SSR. His work included serving as editor-in-chief of the Uzbek science magazine Fan va turmush and deputy chairman of the board for preserving historic and cultural monuments of Uzbekistan. During the course of his work, he travelled to various foreign countries including Afghanistan, Bulgaria, India, Italy, and Japan. He died on 17 March 1970 and was buried in the Chigatoy Cemetery. == Awards and honors == Honoured Scientist of the Uzbek SSR (1939) Three Orders of Lenin (4 November 1944, 1 March 1965, 1 September 1967) Three Orders of the Red Banner of Labour (23 January 1946, 16 January 1950, 27 October 1953) Stalin Prize 3rd class (1952) Hero of Socialist Labour (1 September 1967) Beruniy State Prize (1970) Order of Outstanding Merit (23 August 2002) == Notes == == References ==
|
Wikipedia:Toshmuhammad Sarimsoqov#0
|
Toshmuhammad Sarimsoqov (Uzbek Cyrillic: Тошмуҳаммад Алиевич Саримсоқов, Russian: Ташмухамед Алиевич Сарымсаков, Tashmukhamed Alievich Sarymsakov; 10 September 1915 – 17 December 1995) was an Uzbek mathematician who served as president of the Academy of Sciences of the Uzbek SSR from 1947 to 1952. == Early life and education == Born on 10 September 1915 in Shahrixon to an Uzbek family, in 1931 he graduated from a Russian secondary school in Kokand; he subsequently enrolled in the Central Asia State University. There, he was one of the first students of Vsevolod Romanovsky. After graduating from the Faculty of Physics and Mathematics of the university in 1936, he remained at the university, where he attended graduate school. At the same time, he worked as assistant and associate professor. After briefly serving in the Red Army, he returned to the university in 1942 to defend his doctoral dissertation. That year he received his Doctor of Sciences degree. He became a member of the Communist party in 1944 and served as a deputy in the third convocation of the Supreme Soviet of the Soviet Union. == Career == In 1943 he became the rector of his university, and held that post until June 1944. When the Academy of Sciences of the Uzbek SSR was founded in 1943, he became its vice president. In 1947 he became its president, and held the post until 1952. He then returned to being the rector of Central Asia State University, where he remained until 1958. From 1959 to 1971 he served as the minister of higher education of the Uzbek SSR before again returning to being the rector of the university, which had been renamed to Tashkent State University in 1960. In 1983 he returned to working at the Academy of Sciences of the Uzbek SSR, and in 1988 he became advisor to the Presidium of the Academy of Sciences of the Uzbek SSR. His main areas of study were probability, statistics, and functional analysis. During his career he authored over 170 academic papers. His work on the theory of non-homogeneous Markov chains is cited in modern academic papers. For his work, he was awarded the title Hero of Socialist Labour on 3 April 1990. After Uzbekistan became independent he worked as an advisor to the President of the Academy of Sciences of Uzbekistan. He died in Tashkent on 18 December 1995. == Awards == Hero of Socialist Labour (3 April 1990) Four Orders of Lenin (23 January 1946, 16 January 1950, 15 September 1961, 3 April 1990) Three Orders of the Red Banner of Labour (4 November 1944, 11 January 1957, 9 September 1971) Order of the Badge of Honour (1 March 1965) Order of the October Revolution (3 October 1975) Medal "For Labour Valour" (6 November 1951) Order of Outstanding Merit (23 August 2002) Stalin Prize (1948) Biruni State Prize (1967) Honoured Worker of Science and Technology of the Uzbek SSR (1960) == References ==
|
Wikipedia:Tosun Terzioğlu#0
|
Nazım Terzioğlu (1912 – September 20, 1976) was a Turkish mathematician. He was one of the first mathematicians in Turkish academia. His son, Tosun Terzioğlu, was also a mathematician. == Early life == Nazım Terzioğlu completed his primary education in his place of birth, Kayseri. He started his secondary education in Istanbul and then continued in İzmir until his graduation from İzmir High School in 1930. At that time, some of Turkey's most qualified mathematics teachers worked at İzmir High School. Alumni of that school included mathematicians such as Cahit Arf (1910–1997) and Tevfik Oktay Kabakcıoğlu (1910–1971). In those years, the successful young people were sent abroad by the government to be trained as qualified workforce in various fields needed for the country. Terzioğlu passed the relevant exam and left for Germany to study mathematics on behalf of the Ministry of Education of Turkey. He pursued his higher education in the University of Göttingen and Munich University. He completed his Ph.D. under the supervision of the famous mathematician of that period, Constantin Carathéodory (1873–1950), who was a member of a Greek family in Fener, Istanbul. == Career == Upon completion of his education in Germany, Terzioğlu began to work as an assistant of Mathematical Mechanics and Advanced Geometry in the Institute of Mathematics of the Faculty of Science of Istanbul University in 1937. He became associate professor in 1942 and the following year, he was appointed to professorship in the newly established Institute of Mathematics of the Faculty of Science of Ankara University (1943). After spending two years in this faculty, he returned to Istanbul University as a professor (1944). At Istanbul University, he worked as the Dean of the Faculty of Science in 1950–1952. During the same period, Terzioğlu established some of the scientific institutions for which Turkey had felt the major need until those years. These are the Institute for Geophysics of Istanbul University, the Institute for Hydrobiology in Istanbul Baltalimanı and the Cosmic Ray Institute which Terzioğlu founded at Uludağ, Bursa in cooperation with Adnan Sokullu and Sait Akpınar. After his deanship in the Faculty of Science, he became the Chairman of the Analysis Division of the Institute of Mathematics in the same faculty (1953). In 1965–1967, Terzioğlu, in addition to his responsibilities in Istanbul University, worked first by proxy then acting as the principal founder-rector of Karadeniz Technical University (KTU). It is his honour to establish the first Faculty of Fundamental Sciences of Turkey in KTU. In 1967, Terzioğlu returned to his mission in the Faculty of Science of Istanbul University. In 1969 and 1971, he was elected as the rector of Istanbul University. He had maintained this position for two periods (28 October 1969 – 28 October 1971 and 28 October 1971 – 31 May 1974). In his first years as a rector, he restored the building of a historical soup kitchen which was assigned by Wakfs to the university as a part of the Sehzade Mosque. On 6 August 1971, by setting up a new printing system in it, he put the same building into service with the name of Research Institute for Mathematics of Faculty of Science. Terzioğlu also established a mathematics library within this institute with a capacity of 2000 books which he provided through donations and purchase from foreign countries. After his death, the institute was named on the proposal of the Faculty of Science as Nazim Terzioğlu Mathematics Research Institute. As an outcome of the negotiations with Silivri Municipality, Terzioğlu provided Istanbul University with 35 acres of land to be donated in Silivri. In a part of this land, 18 study rooms, 3 large conference halls, a library and a guest house to accommodate scientists coming from abroad were constructed in accord with his order. Terzioğlu considered graduate education very seriously. He believed that talented young people ought to be trained in a particular way. To provide such an environment, he invited foreign scientists and organized congresses, seminars, colloquia, summer and progress courses in Silivri facilities which was opened into use on September 3, 1973. Thanks to these activities, he made significant contributions to the education of young generations. The scientific meetings organized by Terzioğlu in Silivri facilities are: February 10–14, 1973: First National Meeting of Mathematicians; July 9–14, 1973: the preparatory course related to the Summer Seminar on International Display Theory of Finite Groups; July 15–28, 1973: Summer Seminar on International Display Theory of Finite Groups; August 20 – September 9, 1973: International Symposium on Functional Analysis; September 8–21, 1975: the preparatory course related to the International Symposium on Algebraic Number Theory; September 22–27, 1975: International Symposium on Algebraic Number Theory; April 23–26, 1976: Second National Meeting of Mathematicians; August 1976: Ultrasound Congress (joint with physicists); September 5–11, 1976: International Congress of Functional Analysis; September 20–25, 1976: Rolf Nevanlinna International Symposium. == Death == Terzioğlu died as a result of a heart attack in the morning of the opening day of the International Symposium organized to tribute Rolf Nevanlinna, who had been a teacher of Terzioğlu. Albeit his unexpected loss, the symposium was completed after some rearrangements were made in the program. The guest mathematicians also attended the funeral ceremony on September 22 and the symposium began on September 23. Terzioğlu was elected as the honorary guest of this symposium and the title doctoris honoris causa was awarded to Rolf Nevanlinna by Istanbul University. == Legacy == One of the contributions of Terzioğlu as the director of the Mathematics Research Institute to Turkey's mathematical culture and the history of science was the systematic scan of the Islamic literature relevant to mathematics and the presentation of the information related to conic sections in ancient mathematics to the scientific community. As a result of these efforts, the facsimile of two ancient texts of mathematics originally written in Arabic were realized. The first one is the preface of Mecmuatu'r-risail, the Arabic translation by Beni Musa b. Sakir (died in 873) of Conica, which is the work of Apollonius of Perga (BC 262–190) on the conic sections. This preface, published with the title Das Vorwort des Astronomen Bani Musa b. Sakir, describes how the Apollonius' Conica was acquired by the Islamic world. After that, Terzioğlu published the facsimile of the copy of the lost 8th book of Apollonius' Conica which was rewritten by Ibnu'l-Heysem (965–1039) with the help from other sources. In the introduction part of this book with the title Das Achte Buch zu den Conica des Apollonios von Perge, the following information is provided in summary: In ancient mathematics, the interest for conics starts with Menaechmus (BC IV. Century) and reaches the summit with Apollonius of Perga. Apollonius wrote his famous work Conica by processing previous information and adding up his own inventions. The first 7 volumes of this work consisting of 8 volumes in total are known whereas the 8th volume is missing. The Islamic and Western mathematicians working in this field took place in the reconstruction of the 8th volume. The most successful one of these works is that of Edmund Halley's (1656–1742) Apollonii Per-gaei conicorum (Oxoniae, 1710). The 8th book of Conica reconstructed by Ibn el-Heysem is the 4th manuscript with the name Makalatu'l-Hasan b.el-Hasan b.el Heysem fi el-kitabu'l-mahrutat in the Mecmu'atu'r-risail, which is recorded under no. 1796 in Manisa Library. The fact that Ibn el-Heysem completed this work nearly 700 years before Halley is interesting. Within the framework of this program, Terzioğlu was preparing for publication the first 7 books of Conica, which were translated into Arabic in 415/1024 AD by Ibnu'l-Heysem who had also examined the previous translations of his time. Terzioğlu's death coincides with the time when the facsimile of the manuscript located at No. 2762 of Suleymaniye Library, Ayasofya had been completed. As the part of the book he wanted to include related to the history of conics remained incomplete, it was removed from press and was published later with the title Kitab al-Mahrutat Das Buch der Kegelschnitte des Apollonios von Perge by the Research Institute for Mathematics. It includes a part in which the description of the manuscript and the direct translation of its preface are given in Turkish and German. One of the most important services of Terzioğlu to the Turkish history of science is to make translate into Turkish in Latin letters the published first two volumes and the third volume as a manuscript (see Istanbul University Library TY. 903, 904, 905 for copies of manuscripts) of Asar-i Bakiye} (Vol. I–II, Istanbul, 1329/1913) by Salih Zeki Bey (1863–1921) during his presidency of the Turkish Mathematics Association. His aim was to offer such an old source to the benefit of young generations. === Positions and awards === Terzioğlu, who had an important role in the revival of the Union of Balkan Mathematicians (French: Union Balkanique des Mathematiciens) which was founded before World War II, had been the president of that organization for two periods (1966–1971). He was also selected as the chairman of the IV. Congress of Balkan Mathematicians organized in Istanbul on August 29, 1972. Among his other international activities, the role he played in providing Turkey with the membership of the International Mathematical Union is an unforgettable service. In 1973, Terzioğlu was selected as a member of Hahnemann Medical Society of America. In 1974, he has been awarded the Medal of Merit of Federal Republic of Germany by the German President on his endeavor for the development of Turkish–German relations. He also has two medals given by the Charles University in Prague and the Finland University of Jyväskylä. Nazim Terzioğlu has been awarded on December 2, 1982 the TÜBİTAK Service Award thanks to his contributions to the development of mathematics in our country. His family established a Mathematics Research Award on behalf of Terzioğlu who gave efforts during his life for the development of mathematics and the creation of a research potential. For the first time, this award had been given to three young mathematicians in a ceremony at the Faculty of Science of Istanbul University on September 20, 1981, which is the fifth year of his death. The second award in 1982 was given to a young mathematician in the opening ceremony of the International Symposium on Mathematics that was held on 14–24 September 1982 at the Karadeniz Technical University where Terzioğlu served as the founder-rector. His son, Tosun Terzioğlu (1942–2016) was a Turkish mathematician and academic administrator. == Books == The books written by Terzioğlu, who has many published articles in his own field, are: Über Finslersche Raume (Doktorarbeit), München, 1936 (On Finsler Spaces (Ph.D. Thesis), Munich, 1936.) Fonksiyonlar Teorisine Baslangic. Fonksiyonlar Teorisi. 2 Cilt. (Konrad Knopp'dan ceviri), Istanbul, 1938–1939. (Introduction to the Theory of Functions. Theory of Functions by Konrad Knopp, 2 Volumes (translated), Istanbul, 1938–1939.) Finsler Uzay\i nda Gauss–Bonnet Teoremi, Istanbul 1948. (Gauss–Bonnet theorem in Finsler Spaces, Istanbul 1948.) Lise Fen Kolu Icin Modern Geometri: Konikler, (Ahmet Nazmi Ilker ile), Istanbul, 1960. (Modern Geometry for the Science Sections of High Schools: Conics, (with Ahmet Nazmi Ilker), Istanbul, 1960.) Liseler Icin Cebir Temrinleri (P. Aubert ve G. Papelier'den ceviri), Istanbul, 1960. (Exercises in Algebra for High Schools by P. Aubert and G. Papelier (translated), Istanbul, 1960.) Diferansiyel ve Integral Hesap, (Edmund Landau'dan ceviri), Istanbul, 1961. (Differential and Integral Calculus by Edmund Landau (translated), Istanbul, 1961.) Lise Fen Kolu Icin Modern Geometri. Fasikül I-Kesenler; Fasikül II-Harmonik Bolme, Harmonik Demet, Daireye Göre Kuvvet; Fasikül III-Daireye Göre Kutup ve Kutup Dogrusu (G. Papelier'den ceviri), Istanbul, 1968. (Modern Geometry for the Science Sections of High Schools. Fascicle I: Secants; Fascicle II: Harmonic Division, Harmonic Pencil, Power with respect to the sphere; Fascicle III: Pole and polar line with respect to the sphere by G. Papelier (translated), Istanbul, 1968.) Analiz Problemleri, Istanbul, 1973. (Problems in Analysis, Istanbul, 1973.) Das Vorwort des Astronomen Bani Musa b. Sakir zu den Conica des Apollonios von Perge, Istanbul, 1974. (The foreword of the Astronomer Bani Musa b. Sakir to the Conics of Apollonius of Perga, Istanbul, 1974.) Das achte Buch zu den Conica des Apollonios von Perge re-konstruiert von Ibn al-Haysam, Istanbul, 1974. (The Eighth Book to the Conics of Apollonius of Perga Reconstructed by Ibn-Haysam, Istanbul, 1974.) Kitab al-Mahrutat. Das Buch der Kegelschnitte des Apollonios von Perge, Istanbul, 1981. (Kitab al-Mahrutat. The Book of Conic Sections of Apollonius of Perga, Istanbul, 1981.) == References == == Further reading == Nazim Terzioğlu, History of Science (Monthly Journal), February 1993, Number 16, 11–19. International Symposium on Analysis and Theory of Functions, ATF2009 (Dedicated to Nazim Terzioğlu) Abstract Book.
|
Wikipedia:Total algebra#0
|
In abstract algebra, the total algebra of a monoid is a generalization of the monoid ring that allows for infinite sums of elements of a ring. Suppose that S is a monoid with the property that, for all s ∈ S {\displaystyle s\in S} , there exist only finitely many ordered pairs ( t , u ) ∈ S × S {\displaystyle (t,u)\in S\times S} for which t u = s {\displaystyle tu=s} . Let R be a ring. Then the total algebra of S over R is the set R S {\displaystyle R^{S}} of all functions α : S → R {\displaystyle \alpha :S\to R} with the addition law given by the (pointwise) operation: ( α + β ) ( s ) = α ( s ) + β ( s ) {\displaystyle (\alpha +\beta )(s)=\alpha (s)+\beta (s)} and with the multiplication law given by: ( α ⋅ β ) ( s ) = ∑ t u = s α ( t ) β ( u ) . {\displaystyle (\alpha \cdot \beta )(s)=\sum _{tu=s}\alpha (t)\beta (u).} The sum on the right-hand side has finite support, and so is well-defined in R. These operations turn R S {\displaystyle R^{S}} into a ring. There is an embedding of R into R S {\displaystyle R^{S}} , given by the constant functions, which turns R S {\displaystyle R^{S}} into an R-algebra. An example is the ring of formal power series, where the monoid S is the natural numbers. The product is then the Cauchy product. == References == Nicolas Bourbaki (1989), Algebra, Springer: §III.2
|
Wikipedia:Total set#0
|
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation ≤ {\displaystyle \leq } on some set X {\displaystyle X} , which satisfies the following for all a , b {\displaystyle a,b} and c {\displaystyle c} in X {\displaystyle X} : a ≤ a {\displaystyle a\leq a} (reflexive). If a ≤ b {\displaystyle a\leq b} and b ≤ c {\displaystyle b\leq c} then a ≤ c {\displaystyle a\leq c} (transitive). If a ≤ b {\displaystyle a\leq b} and b ≤ a {\displaystyle b\leq a} then a = b {\displaystyle a=b} (antisymmetric). a ≤ b {\displaystyle a\leq b} or b ≤ a {\displaystyle b\leq a} (strongly connected, formerly called totality). Requirements 1. to 3. just make up the definition of a partial order. Reflexivity (1.) already follows from strong connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders. Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, toset and loset are also used. The term chain is sometimes defined as a synonym of totally ordered set, but generally refers to a totally ordered subset of a given partially ordered set. An extension of a given partial order to a total order is called a linear extension of that partial order. == Strict and non-strict total orders == For delimitation purposes, a total order as defined above is sometimes called non-strict order. For each (non-strict) total order ≤ {\displaystyle \leq } there is an associated relation < {\displaystyle <} , called the strict total order associated with ≤ {\displaystyle \leq } that can be defined in two equivalent ways: a < b {\displaystyle a<b} if a ≤ b {\displaystyle a\leq b} and a ≠ b {\displaystyle a\neq b} (reflexive reduction). a < b {\displaystyle a<b} if not b ≤ a {\displaystyle b\leq a} (i.e., < {\displaystyle <} is the complement of the converse of ≤ {\displaystyle \leq } ). Conversely, the reflexive closure of a strict total order < {\displaystyle <} is a (non-strict) total order. Thus, a strict total order on a set X {\displaystyle X} is a strict partial order on X {\displaystyle X} in which any two distinct elements are comparable. That is, a strict total order is a binary relation < {\displaystyle <} on some set X {\displaystyle X} , which satisfies the following for all a , b {\displaystyle a,b} and c {\displaystyle c} in X {\displaystyle X} : Not a < a {\displaystyle a<a} (irreflexive). If a < b {\displaystyle a<b} then not b < a {\displaystyle b<a} (asymmetric). If a < b {\displaystyle a<b} and b < c {\displaystyle b<c} then a < c {\displaystyle a<c} (transitive). If a ≠ b {\displaystyle a\neq b} , then a < b {\displaystyle a<b} or b < a {\displaystyle b<a} (connected). Asymmetry follows from transitivity and irreflexivity; moreover, irreflexivity follows from asymmetry. == Examples == Any subset of a totally ordered set X is totally ordered for the restriction of the order on X. The unique order on the empty set, ∅, is a total order. Any set of cardinal numbers or ordinal numbers (more strongly, these are well-orders). If X is any set and f an injective function from X to a totally ordered set then f induces a total ordering on X by setting x1 ≤ x2 if and only if f(x1) ≤ f(x2). The lexicographical order on the Cartesian product of a family of totally ordered sets, indexed by a well ordered set, is itself a total order. The set of real numbers ordered by the usual "less than or equal to" (≤) or "greater than or equal to" (≥) relations is totally ordered. Hence each subset of the real numbers is totally ordered, such as the natural numbers, integers, and rational numbers. Each of these can be shown to be the unique (up to an order isomorphism) "initial example" of a totally ordered set with a certain property, (here, a total order A is initial for a property, if, whenever B has the property, there is an order isomorphism from A to a subset of B): The natural numbers form an initial non-empty totally ordered set with no upper bound. The integers form an initial non-empty totally ordered set with neither an upper nor a lower bound. The rational numbers form an initial totally ordered set which is dense in the real numbers. Moreover, the reflexive reduction < is a dense order on the rational numbers. The real numbers form an initial unbounded totally ordered set that is connected in the order topology (defined below). Ordered fields are totally ordered by definition. They include the rational numbers and the real numbers. Every ordered field contains an ordered subfield that is isomorphic to the rational numbers. Any Dedekind-complete ordered field is isomorphic to the real numbers. The letters of the alphabet ordered by the standard dictionary order, e.g., A < B < C etc., is a strict total order. == Chains == The term chain is sometimes defined as a synonym for a totally ordered set, but it is generally used for referring to a subset of a partially ordered set that is totally ordered for the induced order. Typically, the partially ordered set is a set of subsets of a given set that is ordered by inclusion, and the term is used for stating properties of the set of the chains. This high number of nested levels of sets explains the usefulness of the term. A common example of the use of chain for referring to totally ordered subsets is Zorn's lemma which asserts that, if every chain in a partially ordered set X has an upper bound in X, then X contains at least one maximal element. Zorn's lemma is commonly used with X being a set of subsets; in this case, the upper bound is obtained by proving that the union of the elements of a chain in X is in X. This is the way that is generally used to prove that a vector space has Hamel bases and that a ring has maximal ideals. In some contexts, the chains that are considered are order isomorphic to the natural numbers with their usual order or its opposite order. In this case, a chain can be identified with a monotone sequence, and is called an ascending chain or a descending chain, depending whether the sequence is increasing or decreasing. A partially ordered set has the descending chain condition if every descending chain eventually stabilizes. For example, an order is well founded if it has the descending chain condition. Similarly, the ascending chain condition means that every ascending chain eventually stabilizes. For example, a Noetherian ring is a ring whose ideals satisfy the ascending chain condition. In other contexts, only chains that are finite sets are considered. In this case, one talks of a finite chain, often shortened as a chain. In this case, the length of a chain is the number of inequalities (or set inclusions) between consecutive elements of the chain; that is, the number minus one of elements in the chain. Thus a singleton set is a chain of length zero, and an ordered pair is a chain of length one. The dimension of a space is often defined or characterized as the maximal length of chains of subspaces. For example, the dimension of a vector space is the maximal length of chains of linear subspaces, and the Krull dimension of a commutative ring is the maximal length of chains of prime ideals. "Chain" may also be used for some totally ordered subsets of structures that are not partially ordered sets. An example is given by regular chains of polynomials. Another example is the use of "chain" as a synonym for a walk in a graph. == Further concepts == === Lattice theory === One may define a totally ordered set as a particular kind of lattice, namely one in which we have { a ∨ b , a ∧ b } = { a , b } {\displaystyle \{a\vee b,a\wedge b\}=\{a,b\}} for all a, b. We then write a ≤ b if and only if a = a ∧ b {\displaystyle a=a\wedge b} . Hence a totally ordered set is a distributive lattice. === Finite total orders === A simple counting argument will verify that any non-empty finite totally ordered set (and hence any non-empty subset thereof) has a least element. Thus every finite total order is in fact a well order. Either by direct proof or by observing that every well order is order isomorphic to an ordinal one may show that every finite total order is order isomorphic to an initial segment of the natural numbers ordered by <. In other words, a total order on a set with k elements induces a bijection with the first k natural numbers. Hence it is common to index finite total orders or well orders with order type ω by natural numbers in a fashion which respects the ordering (either starting with zero or with one). === Category theory === Totally ordered sets form a full subcategory of the category of partially ordered sets, with the morphisms being maps which respect the orders, i.e. maps f such that if a ≤ b then f(a) ≤ f(b). A bijective map between two totally ordered sets that respects the two orders is an isomorphism in this category. === Order topology === For any totally ordered set X we can define the open intervals (a, b) = {x | a < x and x < b}, (−∞, b) = {x | x < b}, (a, ∞) = {x | a < x}, and (−∞, ∞) = X. We can use these open intervals to define a topology on any ordered set, the order topology. When more than one order is being used on a set one talks about the order topology induced by a particular order. For instance if N is the natural numbers, < is less than and > greater than we might refer to the order topology on N induced by < and the order topology on N induced by > (in this case they happen to be identical but will not in general). The order topology induced by a total order may be shown to be hereditarily normal. === Completeness === A totally ordered set is said to be complete if every nonempty subset that has an upper bound, has a least upper bound. For example, the set of real numbers R is complete but the set of rational numbers Q is not. In other words, the various concepts of completeness (not to be confused with being "total") do not carry over to restrictions. For example, over the real numbers a property of the relation ≤ is that every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R. However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation ≤ to the rational numbers. There are a number of results relating properties of the order topology to the completeness of X: If the order topology on X is connected, X is complete. X is connected under the order topology if and only if it is complete and there is no gap in X (a gap is two points a and b in X with a < b such that no c satisfies a < c < b.) X is complete if and only if every bounded set that is closed in the order topology is compact. A totally ordered set (with its order topology) which is a complete lattice is compact. Examples are the closed intervals of real numbers, e.g. the unit interval [0,1], and the affinely extended real number system (extended real number line). There are order-preserving homeomorphisms between these examples. === Sums of orders === For any two disjoint total orders ( A 1 , ≤ 1 ) {\displaystyle (A_{1},\leq _{1})} and ( A 2 , ≤ 2 ) {\displaystyle (A_{2},\leq _{2})} , there is a natural order ≤ + {\displaystyle \leq _{+}} on the set A 1 ∪ A 2 {\displaystyle A_{1}\cup A_{2}} , which is called the sum of the two orders or sometimes just A 1 + A 2 {\displaystyle A_{1}+A_{2}} : For x , y ∈ A 1 ∪ A 2 {\displaystyle x,y\in A_{1}\cup A_{2}} , x ≤ + y {\displaystyle x\leq _{+}y} holds if and only if one of the following holds: x , y ∈ A 1 {\displaystyle x,y\in A_{1}} and x ≤ 1 y {\displaystyle x\leq _{1}y} x , y ∈ A 2 {\displaystyle x,y\in A_{2}} and x ≤ 2 y {\displaystyle x\leq _{2}y} x ∈ A 1 {\displaystyle x\in A_{1}} and y ∈ A 2 {\displaystyle y\in A_{2}} Intuitively, this means that the elements of the second set are added on top of the elements of the first set. More generally, if ( I , ≤ ) {\displaystyle (I,\leq )} is a totally ordered index set, and for each i ∈ I {\displaystyle i\in I} the structure ( A i , ≤ i ) {\displaystyle (A_{i},\leq _{i})} is a linear order, where the sets A i {\displaystyle A_{i}} are pairwise disjoint, then the natural total order on ⋃ i A i {\displaystyle \bigcup _{i}A_{i}} is defined by For x , y ∈ ⋃ i ∈ I A i {\displaystyle x,y\in \bigcup _{i\in I}A_{i}} , x ≤ y {\displaystyle x\leq y} holds if: Either there is some i ∈ I {\displaystyle i\in I} with x ≤ i y {\displaystyle x\leq _{i}y} or there are some i < j {\displaystyle i<j} in I {\displaystyle I} with x ∈ A i {\displaystyle x\in A_{i}} , y ∈ A j {\displaystyle y\in A_{j}} === Decidability === The first-order theory of total orders is decidable, i.e. there is an algorithm for deciding which first-order statements hold for all total orders. Using interpretability in S2S, the monadic second-order theory of countable total orders is also decidable. == Orders on the Cartesian product of totally ordered sets == There are several ways to take two totally ordered sets and extend to an order on the Cartesian product, though the resulting order may only be partial. Here are three of these possible orders, listed such that each order is stronger than the next: Lexicographical order: (a,b) ≤ (c,d) if and only if a < c or (a = c and b ≤ d). This is a total order. (a,b) ≤ (c,d) if and only if a ≤ c and b ≤ d (the product order). This is a partial order. (a,b) ≤ (c,d) if and only if (a < c and b < d) or (a = c and b = d) (the reflexive closure of the direct product of the corresponding strict total orders). This is also a partial order. Each of these orders extends the next in the sense that if we have x ≤ y in the product order, this relation also holds in the lexicographic order, and so on. All three can similarly be defined for the Cartesian product of more than two sets. Applied to the vector space Rn, each of these make it an ordered vector space. See also examples of partially ordered sets. A real function of n real variables defined on a subset of Rn defines a strict weak order and a corresponding total preorder on that subset. == Related structures == A binary relation that is antisymmetric, transitive, and reflexive (but not necessarily total) is a partial order. A group with a compatible total order is a totally ordered group. There are only a few nontrivial structures that are (interdefinable as) reducts of a total order. Forgetting the orientation results in a betweenness relation. Forgetting the location of the ends results in a cyclic order. Forgetting both data results use of point-pair separation to distinguish, on a circle, the two intervals determined by a point-pair. == See also == == Notes == == References == Birkhoff, Garrett (1967). Lattice Theory. Colloquium Publications. Vol. 25. Providence: Am. Math. Soc. Davey, Brian A.; Priestley, Hilary Ann (1990). Introduction to Lattices and Order. Cambridge Mathematical Textbooks. Cambridge University Press. ISBN 0-521-36766-2. LCCN 89009753. Fuchs, L (1963). Partially Ordered Algebraic Systems. Pergamon Press. George Grätzer (1971). Lattice theory: first concepts and distributive lattices. W. H. Freeman and Co. ISBN 0-7167-0442-0 Halmos, Paul R. (1968). Naive Set Theory. Princeton: Nostrand. John G. Hocking and Gail S. Young (1961). Topology. Corrected reprint, Dover, 1988. ISBN 0-486-65676-4 Rosenstein, Joseph G. (1982). Linear orderings. New York: Academic Press. Schmidt, Gunther; Ströhlein, Thomas (1993). Relations and Graphs: Discrete Mathematics for Computer Scientists. Berlin: Springer-Verlag. ISBN 978-3-642-77970-1. == External links == "Totally ordered set", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
|
Wikipedia:Total variation#0
|
In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function f, defined on an interval [a, b] ⊂ R, its total variation on the interval of definition is a measure of the one-dimensional arclength of the curve with parametric equation x ↦ f(x), for x ∈ [a, b]. Functions whose total variation is finite are called functions of bounded variation. == Historical note == The concept of total variation for functions of one real variable was first introduced by Camille Jordan in the paper (Jordan 1881). He used the new concept in order to prove a convergence theorem for Fourier series of discontinuous periodic functions whose variation is bounded. The extension of the concept to functions of more than one variable however is not simple for various reasons. == Definitions == === Total variation for functions of one real variable === Definition 1.1. The total variation of a real-valued (or more generally complex-valued) function f {\displaystyle f} , defined on an interval [ a , b ] ⊂ R {\displaystyle [a,b]\subset \mathbb {R} } is the quantity V a b ( f ) = sup P ∑ i = 0 n P − 1 | f ( x i + 1 ) − f ( x i ) | , {\displaystyle V_{a}^{b}(f)=\sup _{\mathcal {P}}\sum _{i=0}^{n_{P}-1}|f(x_{i+1})-f(x_{i})|,} where the supremum runs over the set of all partitions P = { P = { x 0 , … , x n P } ∣ P is a partition of [ a , b ] } {\displaystyle {\mathcal {P}}=\left\{P=\{x_{0},\dots ,x_{n_{P}}\}\mid P{\text{ is a partition of }}[a,b]\right\}} of the given interval. Which means that a = x 0 < x 1 < . . . < x n P = b {\displaystyle a=x_{0}<x_{1}<...<x_{n_{P}}=b} . === Total variation for functions of n > 1 real variables === Definition 1.2. Let Ω be an open subset of Rn. Given a function f belonging to L1(Ω), the total variation of f in Ω is defined as V ( f , Ω ) := sup { ∫ Ω f ( x ) div ϕ ( x ) d x : ϕ ∈ C c 1 ( Ω , R n ) , ‖ ϕ ‖ L ∞ ( Ω ) ≤ 1 } , {\displaystyle V(f,\Omega ):=\sup \left\{\int _{\Omega }f(x)\operatorname {div} \phi (x)\,\mathrm {d} x\colon \phi \in C_{c}^{1}(\Omega ,\mathbb {R} ^{n}),\ \Vert \phi \Vert _{L^{\infty }(\Omega )}\leq 1\right\},} where C c 1 ( Ω , R n ) {\displaystyle C_{c}^{1}(\Omega ,\mathbb {R} ^{n})} is the set of continuously differentiable vector functions of compact support contained in Ω {\displaystyle \Omega } , ‖ ‖ L ∞ ( Ω ) {\displaystyle \Vert \;\Vert _{L^{\infty }(\Omega )}} is the essential supremum norm, and div {\displaystyle \operatorname {div} } is the divergence operator. This definition does not require that the domain Ω ⊆ R n {\displaystyle \Omega \subseteq \mathbb {R} ^{n}} of the given function be a bounded set. === Total variation in measure theory === ==== Classical total variation definition ==== Following Saks (1937, p. 10), consider a signed measure μ {\displaystyle \mu } on a measurable space ( X , Σ ) {\displaystyle (X,\Sigma )} : then it is possible to define two set functions W ¯ ( μ , ⋅ ) {\displaystyle {\overline {\mathrm {W} }}(\mu ,\cdot )} and W _ ( μ , ⋅ ) {\displaystyle {\underline {\mathrm {W} }}(\mu ,\cdot )} , respectively called upper variation and lower variation, as follows W ¯ ( μ , E ) = sup { μ ( A ) ∣ A ∈ Σ and A ⊂ E } ∀ E ∈ Σ {\displaystyle {\overline {\mathrm {W} }}(\mu ,E)=\sup \left\{\mu (A)\mid A\in \Sigma {\text{ and }}A\subset E\right\}\qquad \forall E\in \Sigma } W _ ( μ , E ) = inf { μ ( A ) ∣ A ∈ Σ and A ⊂ E } ∀ E ∈ Σ {\displaystyle {\underline {\mathrm {W} }}(\mu ,E)=\inf \left\{\mu (A)\mid A\in \Sigma {\text{ and }}A\subset E\right\}\qquad \forall E\in \Sigma } clearly W ¯ ( μ , E ) ≥ 0 ≥ W _ ( μ , E ) ∀ E ∈ Σ {\displaystyle {\overline {\mathrm {W} }}(\mu ,E)\geq 0\geq {\underline {\mathrm {W} }}(\mu ,E)\qquad \forall E\in \Sigma } Definition 1.3. The variation (also called absolute variation) of the signed measure μ {\displaystyle \mu } is the set function | μ | ( E ) = W ¯ ( μ , E ) + | W _ ( μ , E ) | ∀ E ∈ Σ {\displaystyle |\mu |(E)={\overline {\mathrm {W} }}(\mu ,E)+\left|{\underline {\mathrm {W} }}(\mu ,E)\right|\qquad \forall E\in \Sigma } and its total variation is defined as the value of this measure on the whole space of definition, i.e. ‖ μ ‖ = | μ | ( X ) {\displaystyle \|\mu \|=|\mu |(X)} ==== Modern definition of total variation norm ==== Saks (1937, p. 11) uses upper and lower variations to prove the Hahn–Jordan decomposition: according to his version of this theorem, the upper and lower variation are respectively a non-negative and a non-positive measure. Using a more modern notation, define μ + ( ⋅ ) = W ¯ ( μ , ⋅ ) , {\displaystyle \mu ^{+}(\cdot )={\overline {\mathrm {W} }}(\mu ,\cdot )\,,} μ − ( ⋅ ) = − W _ ( μ , ⋅ ) , {\displaystyle \mu ^{-}(\cdot )=-{\underline {\mathrm {W} }}(\mu ,\cdot )\,,} Then μ + {\displaystyle \mu ^{+}} and μ − {\displaystyle \mu ^{-}} are two non-negative measures such that μ = μ + − μ − {\displaystyle \mu =\mu ^{+}-\mu ^{-}} | μ | = μ + + μ − {\displaystyle |\mu |=\mu ^{+}+\mu ^{-}} The last measure is sometimes called, by abuse of notation, total variation measure. ==== Total variation norm of complex measures ==== If the measure μ {\displaystyle \mu } is complex-valued i.e. is a complex measure, its upper and lower variation cannot be defined and the Hahn–Jordan decomposition theorem can only be applied to its real and imaginary parts. However, it is possible to follow Rudin (1966, pp. 137–139) and define the total variation of the complex-valued measure μ {\displaystyle \mu } as follows Definition 1.4. The variation of the complex-valued measure μ {\displaystyle \mu } is the set function | μ | ( E ) = sup π ∑ A ∈ π | μ ( A ) | ∀ E ∈ Σ {\displaystyle |\mu |(E)=\sup _{\pi }\sum _{A\in \pi }|\mu (A)|\qquad \forall E\in \Sigma } where the supremum is taken over all partitions π {\displaystyle \pi } of a measurable set E {\displaystyle E} into a countable number of disjoint measurable subsets. This definition coincides with the above definition | μ | = μ + + μ − {\displaystyle |\mu |=\mu ^{+}+\mu ^{-}} for the case of real-valued signed measures. ==== Total variation norm of vector-valued measures ==== The variation so defined is a positive measure (see Rudin (1966, p. 139)) and coincides with the one defined by 1.3 when μ {\displaystyle \mu } is a signed measure: its total variation is defined as above. This definition works also if μ {\displaystyle \mu } is a vector measure: the variation is then defined by the following formula | μ | ( E ) = sup π ∑ A ∈ π ‖ μ ( A ) ‖ ∀ E ∈ Σ {\displaystyle |\mu |(E)=\sup _{\pi }\sum _{A\in \pi }\|\mu (A)\|\qquad \forall E\in \Sigma } where the supremum is as above. This definition is slightly more general than the one given by Rudin (1966, p. 138) since it requires only to consider finite partitions of the space X {\displaystyle X} : this implies that it can be used also to define the total variation on finite-additive measures. ==== Total variation of probability measures ==== The total variation of any probability measure is exactly one, therefore it is not interesting as a means of investigating the properties of such measures. However, when μ and ν are probability measures, the total variation distance of probability measures can be defined as ‖ μ − ν ‖ {\displaystyle \|\mu -\nu \|} where the norm is the total variation norm of signed measures. Using the property that ( μ − ν ) ( X ) = 0 {\displaystyle (\mu -\nu )(X)=0} , we eventually arrive at the equivalent definition ‖ μ − ν ‖ = | μ − ν | ( X ) = 2 sup { | μ ( A ) − ν ( A ) | : A ∈ Σ } {\displaystyle \|\mu -\nu \|=|\mu -\nu |(X)=2\sup \left\{\,\left|\mu (A)-\nu (A)\right|:A\in \Sigma \,\right\}} and its values are non-trivial. The factor 2 {\displaystyle 2} above is usually dropped (as is the convention in the article total variation distance of probability measures). Informally, this is the largest possible difference between the probabilities that the two probability distributions can assign to the same event. For a categorical distribution it is possible to write the total variation distance as follows δ ( μ , ν ) = ∑ x | μ ( x ) − ν ( x ) | . {\displaystyle \delta (\mu ,\nu )=\sum _{x}\left|\mu (x)-\nu (x)\right|\;.} It may also be normalized to values in [ 0 , 1 ] {\displaystyle [0,1]} by halving the previous definition as follows δ ( μ , ν ) = 1 2 ∑ x | μ ( x ) − ν ( x ) | {\displaystyle \delta (\mu ,\nu )={\frac {1}{2}}\sum _{x}\left|\mu (x)-\nu (x)\right|} == Basic properties == === Total variation of differentiable functions === The total variation of a C 1 ( Ω ¯ ) {\displaystyle C^{1}({\overline {\Omega }})} function f {\displaystyle f} can be expressed as an integral involving the given function instead of as the supremum of the functionals of definitions 1.1 and 1.2. ==== The form of the total variation of a differentiable function of one variable ==== Theorem 1. The total variation of a differentiable function f {\displaystyle f} , defined on an interval [ a , b ] ⊂ R {\displaystyle [a,b]\subset \mathbb {R} } , has the following expression if f ′ {\displaystyle f'} is Riemann integrable V a b ( f ) = ∫ a b | f ′ ( x ) | d x {\displaystyle V_{a}^{b}(f)=\int _{a}^{b}|f'(x)|\mathrm {d} x} If f {\displaystyle f} is differentiable and monotonic, then the above simplifies to V a b ( f ) = | f ( a ) − f ( b ) | {\displaystyle V_{a}^{b}(f)=|f(a)-f(b)|} For any differentiable function f {\displaystyle f} , we can decompose the domain interval [ a , b ] {\displaystyle [a,b]} , into subintervals [ a , a 1 ] , [ a 1 , a 2 ] , … , [ a N , b ] {\displaystyle [a,a_{1}],[a_{1},a_{2}],\dots ,[a_{N},b]} (with a < a 1 < a 2 < ⋯ < a N < b {\displaystyle a<a_{1}<a_{2}<\cdots <a_{N}<b} ) in which f {\displaystyle f} is locally monotonic, then the total variation of f {\displaystyle f} over [ a , b ] {\displaystyle [a,b]} can be written as the sum of local variations on those subintervals: V a b ( f ) = V a a 1 ( f ) + V a 1 a 2 ( f ) + ⋯ + V a N b ( f ) = | f ( a ) − f ( a 1 ) | + | f ( a 1 ) − f ( a 2 ) | + ⋯ + | f ( a N ) − f ( b ) | {\displaystyle {\begin{aligned}V_{a}^{b}(f)&=V_{a}^{a_{1}}(f)+V_{a_{1}}^{a_{2}}(f)+\,\cdots \,+V_{a_{N}}^{b}(f)\\[0.3em]&=|f(a)-f(a_{1})|+|f(a_{1})-f(a_{2})|+\,\cdots \,+|f(a_{N})-f(b)|\end{aligned}}} ==== The form of the total variation of a differentiable function of several variables ==== Theorem 2. Given a C 1 ( Ω ¯ ) {\displaystyle C^{1}({\overline {\Omega }})} function f {\displaystyle f} defined on a bounded open set Ω ⊆ R n {\displaystyle \Omega \subseteq \mathbb {R} ^{n}} , with ∂ Ω {\displaystyle \partial \Omega } of class C 1 {\displaystyle C^{1}} , the total variation of f {\displaystyle f} has the following expression V ( f , Ω ) = ∫ Ω | ∇ f ( x ) | d x {\displaystyle V(f,\Omega )=\int _{\Omega }\left|\nabla f(x)\right|\mathrm {d} x} . ===== Proof ===== The first step in the proof is to first prove an equality which follows from the Gauss–Ostrogradsky theorem. ===== Lemma ===== Under the conditions of the theorem, the following equality holds: ∫ Ω f div φ = − ∫ Ω ∇ f ⋅ φ {\displaystyle \int _{\Omega }f\operatorname {div} \varphi =-\int _{\Omega }\nabla f\cdot \varphi } ====== Proof of the lemma ====== From the Gauss–Ostrogradsky theorem: ∫ Ω div R = ∫ ∂ Ω R ⋅ n {\displaystyle \int _{\Omega }\operatorname {div} \mathbf {R} =\int _{\partial \Omega }\mathbf {R} \cdot \mathbf {n} } by substituting R := f φ {\displaystyle \mathbf {R} :=f\mathbf {\varphi } } , we have: ∫ Ω div ( f φ ) = ∫ ∂ Ω ( f φ ) ⋅ n {\displaystyle \int _{\Omega }\operatorname {div} \left(f\mathbf {\varphi } \right)=\int _{\partial \Omega }\left(f\mathbf {\varphi } \right)\cdot \mathbf {n} } where φ {\displaystyle \mathbf {\varphi } } is zero on the border of Ω {\displaystyle \Omega } by definition: ∫ Ω div ( f φ ) = 0 {\displaystyle \int _{\Omega }\operatorname {div} \left(f\mathbf {\varphi } \right)=0} ∫ Ω ∂ x i ( f φ i ) = 0 {\displaystyle \int _{\Omega }\partial _{x_{i}}\left(f\mathbf {\varphi } _{i}\right)=0} ∫ Ω φ i ∂ x i f + f ∂ x i φ i = 0 {\displaystyle \int _{\Omega }\mathbf {\varphi } _{i}\partial _{x_{i}}f+f\partial _{x_{i}}\mathbf {\varphi } _{i}=0} ∫ Ω f ∂ x i φ i = − ∫ Ω φ i ∂ x i f {\displaystyle \int _{\Omega }f\partial _{x_{i}}\mathbf {\varphi } _{i}=-\int _{\Omega }\mathbf {\varphi } _{i}\partial _{x_{i}}f} ∫ Ω f div φ = − ∫ Ω φ ⋅ ∇ f {\displaystyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } =-\int _{\Omega }\mathbf {\varphi } \cdot \nabla f} ===== Proof of the equality ===== Under the conditions of the theorem, from the lemma we have: ∫ Ω f div φ = − ∫ Ω φ ⋅ ∇ f ≤ | ∫ Ω φ ⋅ ∇ f | ≤ ∫ Ω | φ | ⋅ | ∇ f | ≤ ∫ Ω | ∇ f | {\displaystyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } =-\int _{\Omega }\mathbf {\varphi } \cdot \nabla f\leq \left|\int _{\Omega }\mathbf {\varphi } \cdot \nabla f\right|\leq \int _{\Omega }\left|\mathbf {\varphi } \right|\cdot \left|\nabla f\right|\leq \int _{\Omega }\left|\nabla f\right|} in the last part φ {\displaystyle \mathbf {\varphi } } could be omitted, because by definition its essential supremum is at most one. On the other hand, we consider θ N := − I [ − N , N ] I { ∇ f ≠ 0 } ∇ f | ∇ f | {\displaystyle \theta _{N}:=-\mathbb {I} _{\left[-N,N\right]}\mathbb {I} _{\{\nabla f\neq 0\}}{\frac {\nabla f}{\left|\nabla f\right|}}} and θ N ∗ {\displaystyle \theta _{N}^{*}} which is the up to ε {\displaystyle \varepsilon } approximation of θ N {\displaystyle \theta _{N}} in C c 1 {\displaystyle C_{c}^{1}} with the same integral. We can do this since C c 1 {\displaystyle C_{c}^{1}} is dense in L 1 {\displaystyle L^{1}} . Now again substituting into the lemma: lim N → ∞ ∫ Ω f div θ N ∗ = lim N → ∞ ∫ { ∇ f ≠ 0 } I [ − N , N ] ∇ f ⋅ ∇ f | ∇ f | = lim N → ∞ ∫ [ − N , N ] ∩ { ∇ f ≠ 0 } ∇ f ⋅ ∇ f | ∇ f | = ∫ Ω | ∇ f | {\displaystyle {\begin{aligned}&\lim _{N\to \infty }\int _{\Omega }f\operatorname {div} \theta _{N}^{*}\\[4pt]&=\lim _{N\to \infty }\int _{\{\nabla f\neq 0\}}\mathbb {I} _{\left[-N,N\right]}\nabla f\cdot {\frac {\nabla f}{\left|\nabla f\right|}}\\[4pt]&=\lim _{N\to \infty }\int _{\left[-N,N\right]\cap {\{\nabla f\neq 0\}}}\nabla f\cdot {\frac {\nabla f}{\left|\nabla f\right|}}\\[4pt]&=\int _{\Omega }\left|\nabla f\right|\end{aligned}}} This means we have a convergent sequence of ∫ Ω f div φ {\textstyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } } that tends to ∫ Ω | ∇ f | {\textstyle \int _{\Omega }\left|\nabla f\right|} as well as we know that ∫ Ω f div φ ≤ ∫ Ω | ∇ f | {\textstyle \int _{\Omega }f\operatorname {div} \mathbf {\varphi } \leq \int _{\Omega }\left|\nabla f\right|} . Q.E.D. It can be seen from the proof that the supremum is attained when φ → − ∇ f | ∇ f | . {\displaystyle \varphi \to {\frac {-\nabla f}{\left|\nabla f\right|}}.} The function f {\displaystyle f} is said to be of bounded variation precisely if its total variation is finite. === Total variation of a measure === The total variation is a norm defined on the space of measures of bounded variation. The space of measures on a σ-algebra of sets is a Banach space, called the ca space, relative to this norm. It is contained in the larger Banach space, called the ba space, consisting of finitely additive (as opposed to countably additive) measures, also with the same norm. The distance function associated to the norm gives rise to the total variation distance between two measures μ and ν. For finite measures on R, the link between the total variation of a measure μ and the total variation of a function, as described above, goes as follows. Given μ, define a function φ : R → R {\displaystyle \varphi \colon \mathbb {R} \to \mathbb {R} } by φ ( t ) = μ ( ( − ∞ , t ] ) . {\displaystyle \varphi (t)=\mu ((-\infty ,t])~.} Then, the total variation of the signed measure μ is equal to the total variation, in the above sense, of the function φ {\displaystyle \varphi } . In general, the total variation of a signed measure can be defined using Jordan's decomposition theorem by ‖ μ ‖ T V = μ + ( X ) + μ − ( X ) , {\displaystyle \|\mu \|_{TV}=\mu _{+}(X)+\mu _{-}(X)~,} for any signed measure μ on a measurable space ( X , Σ ) {\displaystyle (X,\Sigma )} . == Applications == Total variation can be seen as a non-negative real-valued functional defined on the space of real-valued functions (for the case of functions of one variable) or on the space of integrable functions (for the case of functions of several variables). As a functional, total variation finds applications in several branches of mathematics and engineering, like optimal control, numerical analysis, and calculus of variations, where the solution to a certain problem has to minimize its value. As an example, use of the total variation functional is common in the following two kind of problems Numerical analysis of differential equations: it is the science of finding approximate solutions to differential equations. Applications of total variation to these problems are detailed in the article "total variation diminishing" Image denoising: in image processing, denoising is a collection of methods used to reduce the noise in an image reconstructed from data obtained by electronic means, for example data transmission or sensing. "Total variation denoising" is the name for the application of total variation to image noise reduction; further details can be found in the papers of (Rudin, Osher & Fatemi 1992) and (Caselles, Chambolle & Novaga 2007). A sensible extension of this model to colour images, called Colour TV, can be found in (Blomgren & Chan 1998). == See also == Bounded variation p-variation Total variation diminishing Total variation denoising Quadratic variation Total variation distance of probability measures Kolmogorov–Smirnov test Anisotropic diffusion == Notes == == Historical references == Arzelà, Cesare (7 May 1905), "Sulle funzioni di due variabili a variazione limitata (On functions of two variables of bounded variation)", Rendiconto delle Sessioni della Reale Accademia delle Scienze dell'Istituto di Bologna, Nuova serie (in Italian), IX (4): 100–107, JFM 36.0491.02, archived from the original on 2007-08-07. Golubov, Boris I. (2001) [1994], "Arzelà variation", Encyclopedia of Mathematics, EMS Press. Golubov, Boris I. (2001) [1994], "Fréchet variation", Encyclopedia of Mathematics, EMS Press. Golubov, Boris I. (2001) [1994], "Hardy variation", Encyclopedia of Mathematics, EMS Press. Golubov, Boris I. (2001) [1994], "Pierpont variation", Encyclopedia of Mathematics, EMS Press. Golubov, Boris I. (2001) [1994], "Vitali variation", Encyclopedia of Mathematics, EMS Press. Golubov, Boris I. (2001) [1994], "Tonelli plane variation", Encyclopedia of Mathematics, EMS Press. Golubov, Boris I.; Vitushkin, Anatoli G. (2001) [1994], "Variation of a function", Encyclopedia of Mathematics, EMS Press Jordan, Camille (1881), "Sur la série de Fourier", Comptes rendus hebdomadaires des séances de l'Académie des sciences (in French), 92: 228–230, JFM 13.0184.01 (available at Gallica). This is, according to Boris Golubov, the first paper on functions of bounded variation. Hahn, Hans (1921), Theorie der reellen Funktionen (in German), Berlin: Springer Verlag, pp. VII+600, JFM 48.0261.09. Vitali, Giuseppe (1908) [17 dicembre 1907], "Sui gruppi di punti e sulle funzioni di variabili reali (On groups of points and functions of real variables)", Atti dell'Accademia delle Scienze di Torino (in Italian), 43: 75–92, JFM 39.0101.05, archived from the original on 2009-03-31. The paper containing the first proof of Vitali covering theorem. == References == Adams, C. Raymond; Clarkson, James A. (1933), "On definitions of bounded variation for functions of two variables", Transactions of the American Mathematical Society, 35 (4): 824–854, doi:10.1090/S0002-9947-1933-1501718-2, JFM 59.0285.01, MR 1501718, Zbl 0008.00602. Cesari, Lamberto (1936), "Sulle funzioni a variazione limitata (On the functions of bounded variation)", Annali della Scuola Normale Superiore, II (in Italian), 5 (3–4): 299–313, JFM 62.0247.03, MR 1556778, Zbl 0014.29605. Available at Numdam. Leoni, Giovanni (2017), A First Course in Sobolev Spaces: Second Edition, Graduate Studies in Mathematics, American Mathematical Society, pp. xxii+734, ISBN 978-1-4704-2921-8. Saks, Stanisław (1937). Theory of the Integral. Monografie Matematyczne. Vol. 7 (2nd ed.). Warszawa–Lwów: G.E. Stechert & Co. pp. VI+347. JFM 63.0183.05. Zbl 0017.30004.. (available at the Polish Virtual Library of Science). English translation from the original French by Laurence Chisholm Young, with two additional notes by Stefan Banach. Rudin, Walter (1966), Real and Complex Analysis, McGraw-Hill Series in Higher Mathematics (1st ed.), New York: McGraw-Hill, pp. xi+412, MR 0210528, Zbl 0142.01701. == External links == One variable "Total variation" on PlanetMath. One and more variables Function of bounded variation at Encyclopedia of Mathematics Measure theory Rowland, Todd. "Total Variation". MathWorld.. Jordan decomposition at PlanetMath.. Jordan decomposition at Encyclopedia of Mathematics === Applications === Caselles, Vicent; Chambolle, Antonin; Novaga, Matteo (2007), The discontinuity set of solutions of the TV denoising problem and some extensions, SIAM, Multiscale Modeling and Simulation, vol. 6 n. 3, archived from the original on 2011-09-27 (a work dealing with total variation application in denoising problems for image processing). Rudin, Leonid I.; Osher, Stanley; Fatemi, Emad (1992), "Nonlinear total variation based noise removal algorithms", Physica D: Nonlinear Phenomena, 60 (1–4), Physica D: Nonlinear Phenomena 60.1: 259-268: 259–268, Bibcode:1992PhyD...60..259R, doi:10.1016/0167-2789(92)90242-F. Blomgren, Peter; Chan, Tony F. (1998), "Color TV: total variation methods for restoration of vector-valued images", IEEE Transactions on Image Processing, 7 (3), Image Processing, IEEE Transactions on, vol. 7, no. 3: 304-309: 304–309, Bibcode:1998ITIP....7..304B, doi:10.1109/83.661180, PMID 18276250. Tony F. Chan and Jackie (Jianhong) Shen (2005), Image Processing and Analysis - Variational, PDE, Wavelet, and Stochastic Methods, SIAM, ISBN 0-89871-589-X (with in-depth coverage and extensive applications of Total Variations in modern image processing, as started by Rudin, Osher, and Fatemi).
|
Wikipedia:Toufik Mansour#0
|
Toufik Mansour (Arabic: توفيق منصور, Hebrew: תאופיק מנסור) is an Israeli mathematician working in algebraic combinatorics. He is a member of the Druze community and is the first Israeli Druze to become a professional mathematician. Mansour obtained his Ph.D. in mathematics from the University of Haifa in 2001 under Alek Vainshtein. As of 2007, he is a professor of mathematics at the University of Haifa. He served as chair of the department from 2015 to 2017. He has previously been a faculty member of the Center for Combinatorics at Nankai University from 2004 to 2007, and at The John Knopfmacher Center for Applicable Analysis and Number Theory at the University of the Witwatersrand. Mansour is an expert on Discrete Mathematics and its applications. In particular, he is interested in permutation patterns, colored permutations, set partitions, combinatorics on words, and compositions. He has written more than 260 research papers, which means that he publishes a paper roughly every 20 days, or that he produces one publication page roughly every day. == Books == Heubach, Silvia; Mansour, Toufik (2010), Combinatorics of Compositions and Words, Discrete Mathematics and its Applications, Boca Raton, Florida: CRC Press, ISBN 978-1-4200-7267-9, MR 2531482. Mansour, Toufik (2013), Combinatorics of Set Partitions, Discrete Mathematics and its Applications, Boca Raton, Florida: CRC Press, ISBN 978-1-4398-6333-6, MR 2953184. Mansour, Toufik; Schork, Matthias (2015), Commutation Relations, Normal Ordering, and Stirling Numbers, Discrete Mathematics and its Applications, Boca Raton, Florida: CRC Press, ISBN 978-1-4665-7988-0. == See also == List of Israeli Druze Schröder–Hipparchus number == References == == External links == Home page and list of publications Google Scholar profile ORCID profile
|
Wikipedia:Trace (linear algebra)#0
|
In linear algebra, the trace of a square matrix A, denoted tr(A), is the sum of the elements on its main diagonal, a 11 + a 22 + ⋯ + a n n {\displaystyle a_{11}+a_{22}+\dots +a_{nn}} . It is only defined for a square matrix (n × n). The trace of a matrix is the sum of its eigenvalues (counted with multiplicities). Also, tr(AB) = tr(BA) for any matrices A and B of the same size. Thus, similar matrices have the same trace. As a consequence, one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). == Definition == The trace of an n × n square matrix A is defined as: 34 tr ( A ) = ∑ i = 1 n a i i = a 11 + a 22 + ⋯ + a n n {\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i=1}^{n}a_{ii}=a_{11}+a_{22}+\dots +a_{nn}} where aii denotes the entry on the i th row and i th column of A. The entries of A can be real numbers, complex numbers, or more generally elements of a field F. The trace is not defined for non-square matrices. == Example == Let A be a matrix, with A = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) = ( 1 0 3 11 5 2 6 12 − 5 ) {\displaystyle \mathbf {A} ={\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}}={\begin{pmatrix}1&0&3\\11&5&2\\6&12&-5\end{pmatrix}}} Then tr ( A ) = ∑ i = 1 3 a i i = a 11 + a 22 + a 33 = 1 + 5 + ( − 5 ) = 1 {\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i=1}^{3}a_{ii}=a_{11}+a_{22}+a_{33}=1+5+(-5)=1} == Properties == === Basic properties === The trace is a linear mapping. That is, tr ( A + B ) = tr ( A ) + tr ( B ) tr ( c A ) = c tr ( A ) {\displaystyle {\begin{aligned}\operatorname {tr} (\mathbf {A} +\mathbf {B} )&=\operatorname {tr} (\mathbf {A} )+\operatorname {tr} (\mathbf {B} )\\\operatorname {tr} (c\mathbf {A} )&=c\operatorname {tr} (\mathbf {A} )\end{aligned}}} for all square matrices A and B, and all scalars c.: 34 A matrix and its transpose have the same trace:: 34 tr ( A ) = tr ( A T ) . {\displaystyle \operatorname {tr} (\mathbf {A} )=\operatorname {tr} \left(\mathbf {A} ^{\mathsf {T}}\right).} This follows immediately from the fact that transposing a square matrix does not affect elements along the main diagonal. === Trace of a product === The trace of a square matrix which is the product of two matrices can be rewritten as the sum of entry-wise products of their elements, i.e. as the sum of all elements of their Hadamard product. Phrased directly, if A and B are two m × n matrices, then: tr ( A T B ) = tr ( A B T ) = tr ( B T A ) = tr ( B A T ) = ∑ i = 1 m ∑ j = 1 n a i j b i j . {\displaystyle \operatorname {tr} \left(\mathbf {A} ^{\mathsf {T}}\mathbf {B} \right)=\operatorname {tr} \left(\mathbf {A} \mathbf {B} ^{\mathsf {T}}\right)=\operatorname {tr} \left(\mathbf {B} ^{\mathsf {T}}\mathbf {A} \right)=\operatorname {tr} \left(\mathbf {B} \mathbf {A} ^{\mathsf {T}}\right)=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}b_{ij}\;.} If one views any real m × n matrix as a vector of length mn (an operation called vectorization) then the above operation on A and B coincides with the standard dot product. According to the above expression, tr(A⊤A) is a sum of squares and hence is nonnegative, equal to zero if and only if A is zero.: 7 Furthermore, as noted in the above formula, tr(A⊤B) = tr(B⊤A). These demonstrate the positive-definiteness and symmetry required of an inner product; it is common to call tr(A⊤B) the Frobenius inner product of A and B. This is a natural inner product on the vector space of all real matrices of fixed dimensions. The norm derived from this inner product is called the Frobenius norm, and it satisfies a submultiplicative property, as can be proven with the Cauchy–Schwarz inequality: 0 ≤ [ tr ( A B ) ] 2 ≤ tr ( A T A ) tr ( B T B ) , {\displaystyle 0\leq \left[\operatorname {tr} (\mathbf {A} \mathbf {B} )\right]^{2}\leq \operatorname {tr} \left(\mathbf {A} ^{\mathsf {T}}\mathbf {A} \right)\operatorname {tr} \left(\mathbf {B} ^{\mathsf {T}}\mathbf {B} \right),} if A and B are real matrices such that A B is a square matrix. The Frobenius inner product and norm arise frequently in matrix calculus and statistics. The Frobenius inner product may be extended to a hermitian inner product on the complex vector space of all complex matrices of a fixed size, by replacing B by its complex conjugate. The symmetry of the Frobenius inner product may be phrased more directly as follows: the matrices in the trace of a product can be switched without changing the result. If A and B are m × n and n × m real or complex matrices, respectively, then: 34 This is notable both for the fact that AB does not usually equal BA, and also since the trace of either does not usually equal tr(A)tr(B). The similarity-invariance of the trace, meaning that tr(A) = tr(P−1AP) for any square matrix A and any invertible matrix P of the same dimensions, is a fundamental consequence. This is proved by tr ( P − 1 ( A P ) ) = tr ( ( A P ) P − 1 ) = tr ( A ) . {\displaystyle \operatorname {tr} \left(\mathbf {P} ^{-1}(\mathbf {A} \mathbf {P} )\right)=\operatorname {tr} \left((\mathbf {A} \mathbf {P} )\mathbf {P} ^{-1}\right)=\operatorname {tr} (\mathbf {A} ).} Similarity invariance is the crucial property of the trace in order to discuss traces of linear transformations as below. Additionally, for real column vectors a ∈ R n {\displaystyle \mathbf {a} \in \mathbb {R} ^{n}} and b ∈ R n {\displaystyle \mathbf {b} \in \mathbb {R} ^{n}} , the trace of the outer product is equivalent to the inner product: === Cyclic property === More generally, the trace is invariant under circular shifts, that is, This is known as the cyclic property. Arbitrary permutations are not allowed: in general, tr ( A B C D ) ≠ tr ( A C B D ) . {\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} \mathbf {C} \mathbf {D} )\neq \operatorname {tr} (\mathbf {A} \mathbf {C} \mathbf {B} \mathbf {D} )~.} However, if products of three symmetric matrices are considered, any permutation is allowed, since: tr ( A B C ) = tr ( ( A B C ) T ) = tr ( C B A ) = tr ( A C B ) , {\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} \mathbf {C} )=\operatorname {tr} \left(\left(\mathbf {A} \mathbf {B} \mathbf {C} \right)^{\mathsf {T}}\right)=\operatorname {tr} (\mathbf {C} \mathbf {B} \mathbf {A} )=\operatorname {tr} (\mathbf {A} \mathbf {C} \mathbf {B} ),} where the first equality is because the traces of a matrix and its transpose are equal. Note that this is not true in general for more than three factors. === Trace of a Kronecker product === The trace of the Kronecker product of two matrices is the product of their traces: tr ( A ⊗ B ) = tr ( A ) tr ( B ) . {\displaystyle \operatorname {tr} (\mathbf {A} \otimes \mathbf {B} )=\operatorname {tr} (\mathbf {A} )\operatorname {tr} (\mathbf {B} ).} === Characterization of the trace === The following three properties: tr ( A + B ) = tr ( A ) + tr ( B ) , tr ( c A ) = c tr ( A ) , tr ( A B ) = tr ( B A ) , {\displaystyle {\begin{aligned}\operatorname {tr} (\mathbf {A} +\mathbf {B} )&=\operatorname {tr} (\mathbf {A} )+\operatorname {tr} (\mathbf {B} ),\\\operatorname {tr} (c\mathbf {A} )&=c\operatorname {tr} (\mathbf {A} ),\\\operatorname {tr} (\mathbf {A} \mathbf {B} )&=\operatorname {tr} (\mathbf {B} \mathbf {A} ),\end{aligned}}} characterize the trace up to a scalar multiple in the following sense: If f {\displaystyle f} is a linear functional on the space of square matrices that satisfies f ( x y ) = f ( y x ) , {\displaystyle f(xy)=f(yx),} then f {\displaystyle f} and tr {\displaystyle \operatorname {tr} } are proportional. For n × n {\displaystyle n\times n} matrices, imposing the normalization f ( I ) = n {\displaystyle f(\mathbf {I} )=n} makes f {\displaystyle f} equal to the trace. === Trace as the sum of eigenvalues === Given any n × n matrix A, there is where λ1, ..., λn are the eigenvalues of A counted with multiplicity. This holds true even if A is a real matrix and some (or all) of the eigenvalues are complex numbers. This may be regarded as a consequence of the existence of the Jordan canonical form, together with the similarity-invariance of the trace discussed above. === Trace of commutator === When both A and B are n × n matrices, the trace of the (ring-theoretic) commutator of A and B vanishes: tr([A, B]) = 0, because tr(AB) = tr(BA) and tr is linear. One can state this as "the trace is a map of Lie algebras gln → k from operators to scalars", as the commutator of scalars is trivial (it is an Abelian Lie algebra). In particular, using similarity invariance, it follows that the identity matrix is never similar to the commutator of any pair of matrices. Conversely, any square matrix with zero trace is a linear combination of the commutators of pairs of matrices. Moreover, any square matrix with zero trace is unitarily equivalent to a square matrix with diagonal consisting of all zeros. === Traces of special kinds of matrices === === Relationship to the characteristic polynomial === The trace of an n × n {\displaystyle n\times n} matrix A {\displaystyle A} is the coefficient of t n − 1 {\displaystyle t^{n-1}} in the characteristic polynomial, possibly changed of sign, according to the convention in the definition of the characteristic polynomial. == Relationship to eigenvalues == If A is a linear operator represented by a square matrix with real or complex entries and if λ1, ..., λn are the eigenvalues of A (listed according to their algebraic multiplicities), then This follows from the fact that A is always similar to its Jordan form, an upper triangular matrix having λ1, ..., λn on the main diagonal. In contrast, the determinant of A is the product of its eigenvalues; that is, det ( A ) = ∏ i λ i . {\displaystyle \det(\mathbf {A} )=\prod _{i}\lambda _{i}.} Everything in the present section applies as well to any square matrix with coefficients in an algebraically closed field. === Derivative relationships === If ΔA is a square matrix with small entries and I denotes the identity matrix, then we have approximately det ( I + Δ A ) ≈ 1 + tr ( Δ A ) . {\displaystyle \det(\mathbf {I} +\mathbf {\Delta A} )\approx 1+\operatorname {tr} (\mathbf {\Delta A} ).} Precisely this means that the trace is the derivative of the determinant function at the identity matrix. Jacobi's formula d det ( A ) = tr ( adj ( A ) ⋅ d A ) {\displaystyle d\det(\mathbf {A} )=\operatorname {tr} {\big (}\operatorname {adj} (\mathbf {A} )\cdot d\mathbf {A} {\big )}} is more general and describes the differential of the determinant at an arbitrary square matrix, in terms of the trace and the adjugate of the matrix. From this (or from the connection between the trace and the eigenvalues), one can derive a relation between the trace function, the matrix exponential function, and the determinant: det ( exp ( A ) ) = exp ( tr ( A ) ) . {\displaystyle \det(\exp(\mathbf {A} ))=\exp(\operatorname {tr} (\mathbf {A} )).} A related characterization of the trace applies to linear vector fields. Given a matrix A, define a vector field F on Rn by F(x) = Ax. The components of this vector field are linear functions (given by the rows of A). Its divergence div F is a constant function, whose value is equal to tr(A). By the divergence theorem, one can interpret this in terms of flows: if F(x) represents the velocity of a fluid at location x and U is a region in Rn, the net flow of the fluid out of U is given by tr(A) · vol(U), where vol(U) is the volume of U. The trace is a linear operator, hence it commutes with the derivative: d tr ( X ) = tr ( d X ) . {\displaystyle d\operatorname {tr} (\mathbf {X} )=\operatorname {tr} (d\mathbf {X} ).} == Trace of a linear operator == In general, given some linear map f : V → V (where V is a finite-dimensional vector space), we can define the trace of this map by considering the trace of a matrix representation of f, that is, choosing a basis for V and describing f as a matrix relative to this basis, and taking the trace of this square matrix. The result will not depend on the basis chosen, since different bases will give rise to similar matrices, allowing for the possibility of a basis-independent definition for the trace of a linear map. Such a definition can be given using the canonical isomorphism between the space End(V) of linear maps on V and V ⊗ V*, where V* is the dual space of V. Let v be in V and let g be in V*. Then the trace of the indecomposable element v ⊗ g is defined to be g(v); the trace of a general element is defined by linearity. The trace of a linear map f : V → V can then be defined as the trace, in the above sense, of the element of V ⊗ V* corresponding to f under the above mentioned canonical isomorphism. Using an explicit basis for V and the corresponding dual basis for V*, one can show that this gives the same definition of the trace as given above. == Numerical algorithms == === Stochastic estimator === The trace can be estimated unbiasedly by "Hutchinson's trick":Given any matrix W ∈ R n × n {\displaystyle {\boldsymbol {W}}\in \mathbb {R} ^{n\times n}} , and any random u ∈ R n {\displaystyle {\boldsymbol {u}}\in \mathbb {R} ^{n}} with E [ u u ⊺ ] = I {\displaystyle \mathbb {E} [{\boldsymbol {u}}{\boldsymbol {u}}^{\intercal }]=\mathbf {I} } , we have E [ u ⊺ W u ] = tr W {\displaystyle \mathbb {E} [{\boldsymbol {u}}^{\intercal }{\boldsymbol {W}}{\boldsymbol {u}}]=\operatorname {tr} {\boldsymbol {W}}} . For a proof expand the expectation directly. Usually, the random vector is sampled from N ( 0 , I ) {\displaystyle \operatorname {N} (\mathbf {0} ,\mathbf {I} )} (normal distribution) or { ± n − 1 / 2 } n {\displaystyle \{\pm n^{-1/2}\}^{n}} (Rademacher distribution). More sophisticated stochastic estimators of trace have been developed. == Applications == If a 2 x 2 real matrix has zero trace, its square is a diagonal matrix. The trace of a 2 × 2 complex matrix is used to classify Möbius transformations. First, the matrix is normalized to make its determinant equal to one. Then, if the square of the trace is 4, the corresponding transformation is parabolic. If the square is in the interval [0,4), it is elliptic. Finally, if the square is greater than 4, the transformation is loxodromic. See classification of Möbius transformations. The trace is used to define characters of group representations. Two representations A, B : G → GL(V) of a group G are equivalent (up to change of basis on V) if tr(A(g)) = tr(B(g)) for all g ∈ G. The trace also plays a central role in the distribution of quadratic forms. == Lie algebra == The trace is a map of Lie algebras tr : g l n → K {\displaystyle \operatorname {tr} :{\mathfrak {gl}}_{n}\to K} from the Lie algebra g l n {\displaystyle {\mathfrak {gl}}_{n}} of linear operators on an n-dimensional space (n × n matrices with entries in K {\displaystyle K} ) to the Lie algebra K of scalars; as K is Abelian (the Lie bracket vanishes), the fact that this is a map of Lie algebras is exactly the statement that the trace of a bracket vanishes: tr ( [ A , B ] ) = 0 for each A , B ∈ g l n . {\displaystyle \operatorname {tr} ([\mathbf {A} ,\mathbf {B} ])=0{\text{ for each }}\mathbf {A} ,\mathbf {B} \in {\mathfrak {gl}}_{n}.} The kernel of this map, a matrix whose trace is zero, is often said to be traceless or trace free, and these matrices form the simple Lie algebra s l n {\displaystyle {\mathfrak {sl}}_{n}} , which is the Lie algebra of the special linear group of matrices with determinant 1. The special linear group consists of the matrices which do not change volume, while the special linear Lie algebra is the matrices which do not alter volume of infinitesimal sets. In fact, there is an internal direct sum decomposition g l n = s l n ⊕ K {\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus K} of operators/matrices into traceless operators/matrices and scalars operators/matrices. The projection map onto scalar operators can be expressed in terms of the trace, concretely as: A ↦ 1 n tr ( A ) I . {\displaystyle \mathbf {A} \mapsto {\frac {1}{n}}\operatorname {tr} (\mathbf {A} )\mathbf {I} .} Formally, one can compose the trace (the counit map) with the unit map K → g l n {\displaystyle K\to {\mathfrak {gl}}_{n}} of "inclusion of scalars" to obtain a map g l n → g l n {\displaystyle {\mathfrak {gl}}_{n}\to {\mathfrak {gl}}_{n}} mapping onto scalars, and multiplying by n. Dividing by n makes this a projection, yielding the formula above. In terms of short exact sequences, one has 0 → s l n → g l n → tr K → 0 {\displaystyle 0\to {\mathfrak {sl}}_{n}\to {\mathfrak {gl}}_{n}{\overset {\operatorname {tr} }{\to }}K\to 0} which is analogous to 1 → SL n → GL n → det K ∗ → 1 {\displaystyle 1\to \operatorname {SL} _{n}\to \operatorname {GL} _{n}{\overset {\det }{\to }}K^{*}\to 1} (where K ∗ = K ∖ { 0 } {\displaystyle K^{*}=K\setminus \{0\}} ) for Lie groups. However, the trace splits naturally (via 1 / n {\displaystyle 1/n} times scalars) so g l n = s l n ⊕ K {\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus K} , but the splitting of the determinant would be as the nth root times scalars, and this does not in general define a function, so the determinant does not split and the general linear group does not decompose: GL n ≠ SL n × K ∗ . {\displaystyle \operatorname {GL} _{n}\neq \operatorname {SL} _{n}\times K^{*}.} === Bilinear forms === The bilinear form (where X, Y are square matrices) B ( X , Y ) = tr ( ad ( X ) ad ( Y ) ) {\displaystyle B(\mathbf {X} ,\mathbf {Y} )=\operatorname {tr} (\operatorname {ad} (\mathbf {X} )\operatorname {ad} (\mathbf {Y} ))} where ad ( X ) Y = [ X , Y ] = X Y − Y X {\displaystyle \operatorname {ad} (\mathbf {X} )\mathbf {Y} =[\mathbf {X} ,\mathbf {Y} ]=\mathbf {X} \mathbf {Y} -\mathbf {Y} \mathbf {X} } and for orientation, if det Y ≠ 0 {\displaystyle \operatorname {det} \mathbf {Y} \neq 0} then ad ( X ) = X − Y X Y − 1 . {\displaystyle \operatorname {ad} (\mathbf {X} )=\mathbf {X} -\mathbf {Y} \mathbf {X} \mathbf {Y} ^{-1}~.} B ( X , Y ) {\displaystyle B(\mathbf {X} ,\mathbf {Y} )} is called the Killing form; it is used to classify Lie algebras. The trace defines a bilinear form: ( X , Y ) ↦ tr ( X Y ) . {\displaystyle (\mathbf {X} ,\mathbf {Y} )\mapsto \operatorname {tr} (\mathbf {X} \mathbf {Y} )~.} The form is symmetric, non-degenerate and associative in the sense that: tr ( X [ Y , Z ] ) = tr ( [ X , Y ] Z ) . {\displaystyle \operatorname {tr} (\mathbf {X} [\mathbf {Y} ,\mathbf {Z} ])=\operatorname {tr} ([\mathbf {X} ,\mathbf {Y} ]\mathbf {Z} ).} For a complex simple Lie algebra (such as s l {\displaystyle {\mathfrak {sl}}} n), every such bilinear form is proportional to each other; in particular, to the Killing form. Two matrices X and Y are said to be trace orthogonal if tr ( X Y ) = 0. {\displaystyle \operatorname {tr} (\mathbf {X} \mathbf {Y} )=0.} There is a generalization to a general representation ( ρ , g , V ) {\displaystyle (\rho ,{\mathfrak {g}},V)} of a Lie algebra g {\displaystyle {\mathfrak {g}}} , such that ρ {\displaystyle \rho } is a homomorphism of Lie algebras ρ : g → End ( V ) . {\displaystyle \rho :{\mathfrak {g}}\rightarrow {\text{End}}(V).} The trace form tr V {\displaystyle {\text{tr}}_{V}} on End ( V ) {\displaystyle {\text{End}}(V)} is defined as above. The bilinear form ϕ ( X , Y ) = tr V ( ρ ( X ) ρ ( Y ) ) {\displaystyle \phi (\mathbf {X} ,\mathbf {Y} )={\text{tr}}_{V}(\rho (\mathbf {X} )\rho (\mathbf {Y} ))} is symmetric and invariant due to cyclicity. == Generalizations == The concept of trace of a matrix is generalized to the trace class of compact operators on Hilbert spaces, and the analog of the Frobenius norm is called the Hilbert–Schmidt norm. If K {\displaystyle K} is a trace-class operator, then for any orthonormal basis { e n } n = 1 {\displaystyle \{e_{n}\}_{n=1}} , the trace is given by tr ( K ) = ∑ n ⟨ e n , K e n ⟩ , {\displaystyle \operatorname {tr} (K)=\sum _{n}\left\langle e_{n},Ke_{n}\right\rangle ,} and is finite and independent of the orthonormal basis. The partial trace is another generalization of the trace that is operator-valued. The trace of a linear operator Z {\displaystyle Z} which lives on a product space A ⊗ B {\displaystyle A\otimes B} is equal to the partial traces over A {\displaystyle A} and B {\displaystyle B} : tr ( Z ) = tr A ( tr B ( Z ) ) = tr B ( tr A ( Z ) ) . {\displaystyle \operatorname {tr} (Z)=\operatorname {tr} _{A}\left(\operatorname {tr} _{B}(Z)\right)=\operatorname {tr} _{B}\left(\operatorname {tr} _{A}(Z)\right).} For more properties and a generalization of the partial trace, see traced monoidal categories. If A {\displaystyle A} is a general associative algebra over a field k {\displaystyle k} , then a trace on A {\displaystyle A} is often defined to be any functional tr : A → k {\displaystyle \operatorname {tr} :A\to k} which vanishes on commutators; tr ( [ a , b ] ) = 0 {\displaystyle \operatorname {tr} ([a,b])=0} for all a , b ∈ A {\displaystyle a,b\in A} . Such a trace is not uniquely defined; it can always at least be modified by multiplication by a nonzero scalar. A supertrace is the generalization of a trace to the setting of superalgebras. The operation of tensor contraction generalizes the trace to arbitrary tensors. Gomme and Klein (2011) define a matrix trace operator trm {\displaystyle \operatorname {trm} } that operates on block matrices and use it to compute second-order perturbation solutions to dynamic economic models without the need for tensor notation. == Traces in the language of tensor products == Given a vector space V, there is a natural bilinear map V × V∗ → F given by sending (v, φ) to the scalar φ(v). The universal property of the tensor product V ⊗ V∗ automatically implies that this bilinear map is induced by a linear functional on V ⊗ V∗. Similarly, there is a natural bilinear map V × V∗ → Hom(V, V) given by sending (v, φ) to the linear map w ↦ φ(w)v. The universal property of the tensor product, just as used previously, says that this bilinear map is induced by a linear map V ⊗ V∗ → Hom(V, V). If V is finite-dimensional, then this linear map is a linear isomorphism. This fundamental fact is a straightforward consequence of the existence of a (finite) basis of V, and can also be phrased as saying that any linear map V → V can be written as the sum of (finitely many) rank-one linear maps. Composing the inverse of the isomorphism with the linear functional obtained above results in a linear functional on Hom(V, V). This linear functional is exactly the same as the trace. Using the definition of trace as the sum of diagonal elements, the matrix formula tr(AB) = tr(BA) is straightforward to prove, and was given above. In the present perspective, one is considering linear maps S and T, and viewing them as sums of rank-one maps, so that there are linear functionals φi and ψj and nonzero vectors vi and wj such that S(u) = Σφi(u)vi and T(u) = Σψj(u)wj for any u in V. Then ( S ∘ T ) ( u ) = ∑ i φ i ( ∑ j ψ j ( u ) w j ) v i = ∑ i ∑ j ψ j ( u ) φ i ( w j ) v i {\displaystyle (S\circ T)(u)=\sum _{i}\varphi _{i}\left(\sum _{j}\psi _{j}(u)w_{j}\right)v_{i}=\sum _{i}\sum _{j}\psi _{j}(u)\varphi _{i}(w_{j})v_{i}} for any u in V. The rank-one linear map u ↦ ψj(u)φi(wj)vi has trace ψj(vi)φi(wj) and so tr ( S ∘ T ) = ∑ i ∑ j ψ j ( v i ) φ i ( w j ) = ∑ j ∑ i φ i ( w j ) ψ j ( v i ) . {\displaystyle \operatorname {tr} (S\circ T)=\sum _{i}\sum _{j}\psi _{j}(v_{i})\varphi _{i}(w_{j})=\sum _{j}\sum _{i}\varphi _{i}(w_{j})\psi _{j}(v_{i}).} Following the same procedure with S and T reversed, one finds exactly the same formula, proving that tr(S ∘ T) equals tr(T ∘ S). The above proof can be regarded as being based upon tensor products, given that the fundamental identity of End(V) with V ⊗ V∗ is equivalent to the expressibility of any linear map as the sum of rank-one linear maps. As such, the proof may be written in the notation of tensor products. Then one may consider the multilinear map V × V∗ × V × V∗ → V ⊗ V∗ given by sending (v, φ, w, ψ) to φ(w)v ⊗ ψ. Further composition with the trace map then results in φ(w)ψ(v), and this is unchanged if one were to have started with (w, ψ, v, φ) instead. One may also consider the bilinear map End(V) × End(V) → End(V) given by sending (f, g) to the composition f ∘ g, which is then induced by a linear map End(V) ⊗ End(V) → End(V). It can be seen that this coincides with the linear map V ⊗ V∗ ⊗ V ⊗ V∗ → V ⊗ V∗. The established symmetry upon composition with the trace map then establishes the equality of the two traces. For any finite dimensional vector space V, there is a natural linear map F → V ⊗ V'; in the language of linear maps, it assigns to a scalar c the linear map c⋅idV. Sometimes this is called coevaluation map, and the trace V ⊗ V' → F is called evaluation map. These structures can be axiomatized to define categorical traces in the abstract setting of category theory. == See also == Trace of a tensor with respect to a metric tensor Characteristic function Field trace Golden–Thompson inequality Singular trace Specht's theorem Trace class Trace identity Trace inequalities von Neumann's trace inequality == Notes == == References == == External links == "Trace of a square matrix", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
|
Wikipedia:Trace identity#0
|
In mathematics, a trace identity is any equation involving the trace of a matrix. == Properties == Trace identities are invariant under simultaneous conjugation. == Uses == They are frequently used in the invariant theory of n × n {\displaystyle n\times n} matrices to find the generators and relations of the ring of invariants, and therefore are useful in answering questions similar to that posed by Hilbert's fourteenth problem. == Examples == The Cayley–Hamilton theorem says that every square matrix satisfies its own characteristic polynomial. This also implies that all square matrices satisfy tr ( A n ) − c n − 1 tr ( A n − 1 ) + ⋯ + ( − 1 ) n n det ( A ) = 0 {\displaystyle \operatorname {tr} \left(A^{n}\right)-c_{n-1}\operatorname {tr} \left(A^{n-1}\right)+\cdots +(-1)^{n}n\det(A)=0\,} where the coefficients c i {\displaystyle c_{i}} are given by the elementary symmetric polynomials of the eigenvalues of A. All square matrices satisfy tr ( A ) = tr ( A T ) . {\displaystyle \operatorname {tr} (A)=\operatorname {tr} \left(A^{\mathsf {T}}\right).\,} == See also == Trace inequality – Concept in Hlibert spaces mathematics == References == Rowen, Louis Halle (2008), Graduate Algebra: Noncommutative View, Graduate Studies in Mathematics, vol. 2, American Mathematical Society, p. 412, ISBN 9780821841532.
|
Wikipedia:Traian Lalescu#0
|
Traian Lalescu (Romanian: [traˈjan laˈlesku]; 12 July 1882 – 15 June 1929) was a Romanian mathematician. His main focus was on integral equations and he contributed to work in the areas of functional equations, trigonometric series, mathematical physics, geometry, mechanics, algebra, and the history of mathematics. == Life == He was born in Bucharest. His father, also named Traian, was originally from Cornea, Caraș-Severin and worked as a superintendent at the Creditul Agricol Bank. Lalescu went to the Carol I High School in Craiova, continuing high school in Roman, and graduating from the Boarding High School in Iași. After entering the University of Iași, he completed his undergraduate studies in 1903 at the University of Bucharest. He earned his Ph.D. in Mathematics from the University of Paris in 1908. His dissertation, Sur les équations de Volterra, was written under the direction of Émile Picard. That same year, he presented his work at the International Congress of Mathematicians in Rome.: 30 In 1911, he published Introduction to the Theory of Integral Equations, the first book ever on the subject of integral equations. After returning to Romania in 1909, he first taught Mathematics at the Ion Maiorescu Gymnasium in Giurgiu. He then taught until 1912 at the Gheorghe Șincai High School and the Cantemir Vodă High School in Bucharest. From 1909 to 1910, he was a teaching assistant at the School of Bridges and Roads, in the department of graphic statistics. A year later, he was appointed full-time professor of analytical geometry, succeeding Spiru Haret; he lectured at the School (which would later become the Polytechnic University of Bucharest) until his death. In 1916, he became the first president of Sportul Studențesc, the university's football club. Also that year, he was appointed tenured professor of algebra and number theory at the University of Bucharest, a position he held until his death. In 1920, Lalescu became a professor and the inaugural rector of the Polytechnic University of Timișoara; for a year, he would commute by train for 20 hours between Timișoara and Bucharest to teach his classes. In 1921, he founded the football club Politehnica Timișoara. His wife, Ecaterina, was a former student of his; they had for children—two sons and two daughters: Nicolae, Mariana, Florica, and Traian. She died in childbirth in 1921, at age 28. In 1920, Lalescu was elected to the Parliament of Romania as deputy for Orșova, and then re-elected twice as deputy for Caransebeș. He presented in parliament a well-received report on the budget project for 1925. In the fall of 1927, he caught a double pneumonia; in 1928, he went for a vacation in Nice and for treatment in Paris, but he succumbed to the disease the next year, at age 46. In 1991, he was elected posthumously honorary member of the Romanian Academy. == The Lalescu sequence == In a 1900 issue of Gazeta Matematică, Lalescu proposed the study of the sequence L n = ( n + 1 ) ! n + 1 − n ! n {\displaystyle L_{n}={\sqrt[{n+1}]{(n+1)!}}-{\sqrt[{n}]{n!}}} . It turns out that the Lalescu sequence is decreasing and bounded below by 0, and thus is converging. Its limit is given by lim n → ∞ L n = 1 e {\displaystyle \lim _{n\to \infty }L_{n}={\frac {1}{e}}} . == Legacy == There are several institutions bearing his name, including Colegiul Național de Informatică Traian Lalescu in Hunedoara and Liceul Teoretic Traian Lalescu in Reșița. There are also streets named after him in Craiova, Oradea, Reșița, and Timișoara. The National Mathematics Contest Traian Lalescu for undergraduate students is also named after him. A statue of Lalescu, carved in 1930 by Cornel Medrea, is situated in front of the Faculty of Mechanical Engineering, in Timișoara and another statue of Lalescu is situated inside the University of Bucharest. == Work == T. Lalesco, Introduction à la théorie des équations intégrales. Avec une préface de É. Picard, Paris: A. Hermann et Fils, 1912. VII + 152 pp. JFM entry Traian Lalescu, Introducere la teoria ecuațiilor integrale, Editura Academiei Republicii Populare Romîne, 1956. 134 pp. (A reprint of the first edition [Bucharest, 1911], with a bibliography taken from the French translation [Paris, 1912]). MR0085450 == References == == External links == "Representative Figures of the Romanian Science and Technology" (in Romanian) "Traian Lalescu", from Colegiul Național de Informatică Traian Lalescu, Hunedoara (in Romanian) "Cine a fost Traian Lalescu?", from Liceul Teoretic Traian Lalescu, Reșița (in Romanian) "Monumentul lui Traian Lalescu (1930)", at infotim.ro A Class of Applications of AM-GM Inequality (From a 2004 Putnam Competition Problem to Lalescu’s Sequence) by Wladimir G. Boskoff and Bogdan Suceava, Australian Math. Society Gazette, 33 (2006), No.1, 51-56.
|
Wikipedia:Trairāśika#0
|
Trairāśika is the Sanskrit term used by Indian astronomers and mathematicians of the pre-modern era to denote what is known as the "rule of three" in elementary mathematics and algebra. In the contemporary mathematical literature, the term "rule of three" refers to the principle of cross-multiplication which states that if a b = c d {\displaystyle {\tfrac {a}{b}}={\tfrac {c}{d}}} then a d = b c {\displaystyle ad=bc} or a = b c d {\displaystyle a={\tfrac {bc}{d}}} . The antiquity of the term trairāśika is attested by its presence in the Bakhshali manuscript, a document believed to have been composed in the early centuries of the Common Era. == The trairāśika rule == Basically trairāśika is a rule which helps to solve the following problem: "If p {\displaystyle p} produces h {\displaystyle h} what would i {\displaystyle i} produce?" Here p {\displaystyle p} is referred to as pramāṇa ("argument"), h {\displaystyle h} as phala ("fruit") and i {\displaystyle i} as ichcā ("requisition"). The pramāṇa and icchā must be of the same denomination, that is, of the same kind or type like weights, money, time, or numbers of the same objects. Phala can be a of a different denomination. It is also assumed that phala increases in proportion to pramāṇa. The unknown quantity is called icchā-phala, that is, the phala corresponding to the icchā. Āryabhaṭa gives the following solution to the problem: "In trairāśika, the phala is multiplied by ichcā and then divided by pramāṇa. The result is icchā-phala." In modern mathematical notations, icchā-phala = phala × icchā pramāṇa . {\displaystyle {\text{icchā-phala }}={\tfrac {{\text{phala}}\times {\text{icchā}}}{\text{pramāṇa}}}.} The four quantities can be presented in a row like this: pramāṇa | phala | ichcā | icchā-phala (unknown) Then the rule to get icchā-phala can be stated thus: "Multiply the middle two and divide by the first." === Illustrative examples === 1. This example is taken from Bījagaṇita, a treatise on algebra by the Indian mathematician Bhāskara II (c. 1114–1185). Problem: "If two and a half pala-s (a unit of weight) of saffron be obtained for three-sevenths of a nishca (a unit of money); say instantly, best of merchants, how much is got for nine nishca-s?" Solution: pramāṇa = 3 7 {\displaystyle {\tfrac {3}{7}}} nishca, phala = 2 1 2 {\displaystyle 2{\tfrac {1}{2}}} pala-s of saffron, icchā = 9 {\displaystyle 9} nishca-s and we have to find the icchā-phala. icchā-phala = phala × icchā pramāṇa = ( 2 1 2 ) × 9 3 7 = 52 1 2 {\displaystyle {\text{icchā-phala }}={\tfrac {{\text{phala}}\times {\text{icchā}}}{\text{pramāṇa}}}={\tfrac {(2{\tfrac {1}{2}})\times 9}{\tfrac {3}{7}}}=52{\tfrac {1}{2}}} pala-s of safron. 2. This example is taken from Yuktibhāṣā, a work on mathematics and astronomy, composed by Jyesthadeva of the Kerala school of astronomy and mathematics around 1530. Problem: "When 5 measures of paddy is known to yield 2 measures of rice how many measures of rice will be obtained from 12 measures of paddy?" Solution: pramāṇa = 5 measures of paddy, phala = 2 measures of rice, icchā = 12 measures of rice and we have to find the icchā-phala. icchā-phala = phala × icchā pramāṇa = 2 × 12 5 = 24 5 {\displaystyle {\text{icchā-phala }}={\tfrac {{\text{phala}}\times {\text{icchā}}}{\text{pramāṇa}}}={\tfrac {2\times 12}{5}}={\tfrac {24}{5}}} measures of rice. == Vyasta-trairāśika: Inverse rule of three == The four quantities associated with trairāśika are presented in a row as follows: pramāṇa | phala | ichcā | icchā-phala (unknown) In trairāśika it was assumed that the phala increases with pramāṇa. If it is assumed that phala decreases with increases in pramāṇa, the rule for finding icchā-phala is called vyasta-trairāśika (or, viloma-trairāśika) or "inverse rule of three". In vyasta-trairāśika the rule for finding the icchā-phala may be stated as follows assuming that the relevant quantities are written in a row as indicated above. "In the three known quantities, multiply the middle term by the first and divide by the last." In modern mathematical notations we have, icchā-phala = phala × pramāṇa icchā . {\displaystyle {\text{icchā-phala }}={\tfrac {{\text{phala}}\times {\text{pramāṇa}}}{\text{icchā}}}.} === Illustrative example === This example is from Bījagaṇita: Problem: "If a female slave sixteen years of age, bring thirty-two nishca-s, what will one aged twenty cost?" Solution: pramāṇa = 16 years, phala 32 = nishca-s, ichcā = 20 years. It is assumed that phala decreases with pramāṇa. Hence icchā-phala = phala × pramāṇa icchā = 32 × 16 20 = 25 3 5 {\displaystyle {\text{icchā-phala }}={\tfrac {{\text{phala}}\times {\text{pramāṇa}}}{\text{icchā}}}={\tfrac {32\times 16}{20}}=25{\tfrac {3}{5}}} nishca-s. == Compound proportion == In trairāśika there is only one pramāṇa and the corresponding phala. We are required to find the phala corresponding to a given value of ichcā for the pramāṇa. The relevant quantities may also be represented in the following form: Indian mathematicians have generalized this problem to the case where there are more than one pramāṇa. Let there be n pramāṇa-s pramāṇa-1, pramāṇa-2, . . ., pramāṇa-n and the corresponding phala. Let the iccha-s corresponding to the pramāṇa-s be iccha-1, iccha-2, . . ., iccha-n. The problem is to find the phala corresponding to these iccha-s. This may be represented in the following tabular form: This is the problem of compound proportion. The ichcā-phala is given by ichcā-phala = ( ichcā-1 × ichcā-2 × ⋯ × ichcā-n ) × phala pramāṇa-1 × pramāṇa-2 × ⋯ × pramāṇa-n . {\displaystyle {\text{ ichcā-phala }}={\tfrac {({\text{ ichcā-1 }}\times {\text{ ichcā-2 }}\times \cdots \times {\text{ ichcā-n }})\times {\text{ phala }}}{{\text{ pramāṇa-1 }}\times {\text{ pramāṇa-2 }}\times \cdots \times {\text{ pramāṇa-n }}}}.} Since there are 2 n + 1 {\displaystyle 2n+1} quantities, the method for solving the problem may be called the "rule of 2 n + 1 {\displaystyle 2n+1} ". In his Bǐjagaṇita Bhāskara II has discussed some special cases of this general principle, like, "rule of five" (pañjarāśika), "rule of seven" (saptarāśika), "rule of nine" ("navarāśika") and "rule of eleven" (ekādaśarāśika). === Illustrative example === This example for rule of nine is taken from Bǐjagaṇita: Problem: If thirty benches, twelve fingers thick, square of four wide, and fourteen cubits long, cost a hundred [nishcas]; tell me, my friend, what price will fourteen benches fetch, which are four less in every dimension? Solution: The data is presented in the following tabular form: iccha-phala = ( 14 × 8 × 12 × 10 ) × 100 30 × 12 × 16 × 14 = 100 6 = 16 2 3 {\displaystyle {\tfrac {(14\times 8\times 12\times 10)\times 100}{30\times 12\times 16\times 14}}={\tfrac {100}{6}}=16{\tfrac {2}{3}}} . == Importance of the trairāśika == All Indian astronomers and mathematicians have placed the trairāśika principle on a high pedestal. For example, Bhaskara II in his Līlāvatī even compares the trairāśika to God himself! "As the being, who relieves the minds of his worshipers from suffering, and who is the sole cause of the production of this universe, pervades the whole, and does so with his various manifestations, as worlds, paradises, mountains, rivers, gods, demons, men, trees," and cities; so is all this collection of instructions for computations pervaded by the rule of three terms." == Additional reading == For advanced applications of trairāśika in astronomy, see: M. S. Sriram (2022). "Non-trivial Use of the "Trairāśika" (Proportionality Principle) in Indian Astronomy Texts". In Sita Sundar Ram; Ramakalyani V (eds.). History and Development of Mathematics in India (PDF). New Delhi: National Mission for Manuscripts. pp. 337–353. Retrieved 20 June 2024.. For a complete discussion on trairāśika, see: Bibhutibhushan Datta and Avadhesh Narayan Singh (1962). History of Hindu Mathematics: A Source Book Parts I and II. Mumbai: Asia Publishing House. pp. 203–218. Retrieved 21 June 2024. For applications of trairāśika in Indian architecture, see: P. Ramakrishnan (September 1998). Indian mathematics related to architecture and other areas with special reference to Kerala (PDF). Cochin, India: Cochin University of Science and Technology. pp. 72–92. Retrieved 21 June 2024. (Chapter V Trairāśika (Rule of Three) in Traditional Architecture) == References ==
|
Wikipedia:Transcendental function#0
|
In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable that can be written using only the basic operations of addition, subtraction, multiplication, and division (without the need of taking limits). This is in contrast to an algebraic function. Examples of transcendental functions include the exponential function, the logarithm function, the hyperbolic functions, and the trigonometric functions. Equations over these expressions are called transcendental equations. == Definition == Formally, an analytic function f {\displaystyle f} of one real or complex variable is transcendental if it is algebraically independent of that variable. This means the function does not satisfy any polynomial equation. For example, the function f {\displaystyle f} given by f ( x ) = a x + b c x + d {\displaystyle f(x)={\frac {ax+b}{cx+d}}} for all x {\displaystyle x} is not transcendental, but algebraic, because it satisfies the polynomial equation ( a x + b ) − ( c x + d ) f ( x ) = 0 {\displaystyle (ax+b)-(cx+d)f(x)=0} . Similarly, the function f {\displaystyle f} that satisfies the equation f ( x ) 5 + f ( x ) = x {\displaystyle f(x)^{5}+f(x)=x} for all x {\displaystyle x} is not transcendental, but algebraic, even though it cannot be written as a finite expression involving the basic arithmetic operations. This definition can be extended to functions of several variables. == History == The transcendental functions sine and cosine were tabulated from physical measurements in antiquity, as evidenced in Greece (Hipparchus) and India (jya and koti-jya). In describing Ptolemy's table of chords, an equivalent to a table of sines, Olaf Pedersen wrote: The mathematical notion of continuity as an explicit concept is unknown to Ptolemy. That he, in fact, treats these functions as continuous appears from his unspoken presumption that it is possible to determine a value of the dependent variable corresponding to any value of the independent variable by the simple process of linear interpolation. A revolutionary understanding of these circular functions occurred in the 17th century and was explicated by Leonhard Euler in 1748 in his Introduction to the Analysis of the Infinite. These ancient transcendental functions became known as continuous functions through quadrature of the rectangular hyperbola xy = 1 by Grégoire de Saint-Vincent in 1647, two millennia after Archimedes had produced The Quadrature of the Parabola. The area under the hyperbola was shown to have the scaling property of constant area for a constant ratio of bounds. The hyperbolic logarithm function so described was of limited service until 1748 when Leonhard Euler related it to functions where a constant is raised to a variable exponent, such as the exponential function where the constant base is e. By introducing these transcendental functions and noting the bijection property that implies an inverse function, some facility was provided for algebraic manipulations of the natural logarithm even if it is not an algebraic function. The exponential function is written exp ( x ) = e x {\displaystyle \exp(x)=e^{x}} . Euler identified it with the infinite series ∑ k = 0 ∞ x k / k ! {\textstyle \sum _{k=0}^{\infty }x^{k}/k!} , where k! denotes the factorial of k. The even and odd terms of this series provide sums denoting cosh(x) and sinh(x), so that e x = cosh x + sinh x . {\displaystyle e^{x}=\cosh x+\sinh x.} These transcendental hyperbolic functions can be converted into circular functions sine and cosine by introducing (−1)k into the series, resulting in alternating series. After Euler, mathematicians view the sine and cosine this way to relate the transcendence to logarithm and exponent functions, often through Euler's formula in complex number arithmetic. == Examples == The following functions are transcendental: f 1 ( x ) = x π f 2 ( x ) = e x f 3 ( x ) = log e x f 4 ( x ) = cosh x f 5 ( x ) = sinh x f 6 ( x ) = tanh x f 7 ( x ) = sinh − 1 x f 8 ( x ) = tanh − 1 x f 9 ( x ) = cos x f 10 ( x ) = sin x f 11 ( x ) = tan x f 12 ( x ) = sin − 1 x f 13 ( x ) = tan − 1 x f 14 ( x ) = x ! f 15 ( x ) = 1 / x ! f 16 ( x ) = x x {\displaystyle {\begin{aligned}f_{1}(x)&=x^{\pi }\\[2pt]f_{2}(x)&=e^{x}\\[2pt]f_{3}(x)&=\log _{e}{x}\\[2pt]f_{4}(x)&=\cosh {x}\\f_{5}(x)&=\sinh {x}\\f_{6}(x)&=\tanh {x}\\f_{7}(x)&=\sinh ^{-1}{x}\\[2pt]f_{8}(x)&=\tanh ^{-1}{x}\\[2pt]f_{9}(x)&=\cos {x}\\f_{10}(x)&=\sin {x}\\f_{11}(x)&=\tan {x}\\f_{12}(x)&=\sin ^{-1}{x}\\[2pt]f_{13}(x)&=\tan ^{-1}{x}\\[2pt]f_{14}(x)&=x!\\f_{15}(x)&=1/x!\\[2pt]f_{16}(x)&=x^{x}\\[2pt]\end{aligned}}} For the first function f 1 ( x ) {\displaystyle f_{1}(x)} , the exponent π {\displaystyle \pi } can be replaced by any other irrational number, and the function will remain transcendental. For the second and third functions f 2 ( x ) {\displaystyle f_{2}(x)} and f 3 ( x ) {\displaystyle f_{3}(x)} , the base e {\displaystyle e} can be replaced by any other positive real number base not equaling 1, and the functions will remain transcendental. Functions 4-8 denote the hyperbolic trigonometric functions, while functions 9-13 denote the circular trigonometric functions. The fourteenth function f 14 ( x ) {\displaystyle f_{14}(x)} denotes the analytic extension of the factorial function via the gamma function, and f 15 ( x ) {\displaystyle f_{15}(x)} is its reciprocal, an entire function. Finally, in the last function f 16 ( x ) {\displaystyle f_{16}(x)} , the exponent x {\displaystyle x} can be replaced by k x {\displaystyle kx} for any nonzero real k {\displaystyle k} , and the function will remain transcendental. == Algebraic and transcendental functions == The most familiar transcendental functions are the logarithm, the exponential (with any non-trivial base), the trigonometric, and the hyperbolic functions, and the inverses of all of these. Less familiar are the special functions of analysis, such as the gamma, elliptic, and zeta functions, all of which are transcendental. The generalized hypergeometric and Bessel functions are transcendental in general, but algebraic for some special parameter values. Transcendental functions cannot be defined using only the operations of addition, subtraction, multiplication, division, and n {\displaystyle n} th roots (where n {\displaystyle n} is any integer), without using some "limiting process". A function that is not transcendental is algebraic. Simple examples of algebraic functions are the rational functions and the square root function, but in general, algebraic functions cannot be defined as finite formulas of the elementary functions, as shown by the example above with f ( x ) 5 + f ( x ) = x {\displaystyle f(x)^{5}+f(x)=x} (see Abel–Ruffini theorem). The indefinite integral of many algebraic functions is transcendental. For example, the integral ∫ t = 1 x 1 t d t {\displaystyle \int _{t=1}^{x}{\frac {1}{t}}dt} turns out to equal the logarithm function l o g e ( x ) {\displaystyle log_{e}(x)} . Similarly, the limit or the infinite sum of many algebraic function sequences is transcendental. For example, lim n → ∞ ( 1 + x / n ) n {\displaystyle \lim _{n\to \infty }(1+x/n)^{n}} converges to the exponential function e x {\displaystyle e^{x}} , and the infinite sum ∑ n = 0 ∞ x 2 n ( 2 n ) ! {\displaystyle \sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}} turns out to equal the hyperbolic cosine function cosh x {\displaystyle \cosh x} . In fact, it is impossible to define any transcendental function in terms of algebraic functions without using some such "limiting procedure" (integrals, sequential limits, and infinite sums are just a few). Differential algebra examines how integration frequently creates functions that are algebraically independent of some class, such as when one takes polynomials with trigonometric functions as variables. == Transcendentally transcendental functions == Most familiar transcendental functions, including the special functions of mathematical physics, are solutions of algebraic differential equations. Those that are not, such as the gamma and the zeta functions, are called transcendentally transcendental or hypertranscendental functions. == Exceptional set == If f is an algebraic function and α {\displaystyle \alpha } is an algebraic number then f (α) is also an algebraic number. The converse is not true: there are entire transcendental functions f such that f (α) is an algebraic number for any algebraic α. For a given transcendental function the set of algebraic numbers giving algebraic results is called the exceptional set of that function. Formally it is defined by: E ( f ) = { α ∈ Q ¯ : f ( α ) ∈ Q ¯ } . {\displaystyle {\mathcal {E}}(f)=\left\{\alpha \in {\overline {\mathbb {Q} }}\,:\,f(\alpha )\in {\overline {\mathbb {Q} }}\right\}.} In many instances the exceptional set is fairly small. For example, E ( exp ) = { 0 } , {\displaystyle {\mathcal {E}}(\exp )=\{0\},} this was proved by Lindemann in 1882. In particular exp(1) = e is transcendental. Also, since exp(iπ) = −1 is algebraic we know that iπ cannot be algebraic. Since i is algebraic this implies that π is a transcendental number. In general, finding the exceptional set of a function is a difficult problem, but if it can be calculated then it can often lead to results in transcendental number theory. Here are some other known exceptional sets: Klein's j-invariant E ( j ) = { α ∈ H : [ Q ( α ) : Q ] = 2 } , {\displaystyle {\mathcal {E}}(j)=\left\{\alpha \in {\mathcal {H}}\,:\,[\mathbb {Q} (\alpha ):\mathbb {Q} ]=2\right\},} where H {\displaystyle {\mathcal {H}}} is the upper half-plane, and [ Q ( α ) : Q ] {\displaystyle [\mathbb {Q} (\alpha ):\mathbb {Q} ]} is the degree of the number field Q ( α ) . {\displaystyle \mathbb {Q} (\alpha ).} This result is due to Theodor Schneider. Exponential function in base 2: E ( 2 x ) = Q , {\displaystyle {\mathcal {E}}(2^{x})=\mathbb {Q} ,} This result is a corollary of the Gelfond–Schneider theorem, which states that if α ≠ 0 , 1 {\displaystyle \alpha \neq 0,1} is algebraic, and β {\displaystyle \beta } is algebraic and irrational then α β {\displaystyle \alpha ^{\beta }} is transcendental. Thus the function 2x could be replaced by cx for any algebraic c not equal to 0 or 1. Indeed, we have: E ( x x ) = E ( x 1 x ) = Q ∖ { 0 } . {\displaystyle {\mathcal {E}}(x^{x})={\mathcal {E}}\left(x^{\frac {1}{x}}\right)=\mathbb {Q} \setminus \{0\}.} A consequence of Schanuel's conjecture in transcendental number theory would be that E ( e e x ) = ∅ . {\displaystyle {\mathcal {E}}\left(e^{e^{x}}\right)=\emptyset .} A function with empty exceptional set that does not require assuming Schanuel's conjecture is f ( x ) = exp ( 1 + π x ) . {\displaystyle f(x)=\exp(1+\pi x).} While calculating the exceptional set for a given function is not easy, it is known that given any subset of the algebraic numbers, say A, there is a transcendental function whose exceptional set is A. The subset does not need to be proper, meaning that A can be the set of algebraic numbers. This directly implies that there exist transcendental functions that produce transcendental numbers only when given transcendental numbers. Alex Wilkie also proved that there exist transcendental functions for which first-order-logic proofs about their transcendence do not exist by providing an exemplary analytic function. == Dimensional analysis == In dimensional analysis, transcendental functions are notable because they make sense only when their argument is dimensionless (possibly after algebraic reduction). Because of this, transcendental functions can be an easy-to-spot source of dimensional errors. For example, log(5 metres) is a nonsensical expression, unlike log(5 metres / 3 metres) or log(3) metres. One could attempt to apply a logarithmic identity to get log(5) + log(metres), which highlights the problem: applying a non-algebraic operation to a dimension creates meaningless results. == See also == Complex function Function (mathematics) Generalized function List of special functions and eponyms List of types of functions Rational function Special functions == References == == External links == Definition of "Transcendental function" in the Encyclopedia of Math
|
Wikipedia:Transformation (function)#0
|
In mathematics, a transformation, transform, or self-map is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f: X → X. Examples include linear transformations of vector spaces and geometric transformations, which include projective transformations, affine transformations, and specific affine transformations, such as rotations, reflections and translations. == Partial transformations == While it is common to use the term transformation for any function of a set into itself (especially in terms like "transformation semigroup" and similar), there exists an alternative form of terminological convention in which the term "transformation" is reserved only for bijections. When such a narrow notion of transformation is generalized to partial functions, then a partial transformation is a function f: A → B, where both A and B are subsets of some set X. == Algebraic structures == The set of all transformations on a given base set, together with function composition, forms a regular semigroup. == Combinatorics == For a finite set of cardinality n, there are nn transformations and (n+1)n partial transformations. == See also == Coordinate transformation Data transformation (statistics) Geometric transformation Infinitesimal transformation Linear transformation List of transforms Rigid transformation Transformation geometry Transformation semigroup Transformation group Transformation matrix == References == == External links == Media related to Transformation (function) at Wikimedia Commons
|
Wikipedia:Transgression map#0
|
In algebraic topology, a transgression map is a way to transfer cohomology classes. It occurs, for example in the inflation-restriction exact sequence in group cohomology, and in integration in fibers. It also naturally arises in many spectral sequences; see spectral sequence#Edge maps and transgressions. == Inflation-restriction exact sequence == The transgression map appears in the inflation-restriction exact sequence, an exact sequence occurring in group cohomology. Let G be a group, N a normal subgroup, and A an abelian group which is equipped with an action of G, i.e., a homomorphism from G to the automorphism group of A. The quotient group G / N {\displaystyle G/N} acts on A N = { a ∈ A : n a = a for all n ∈ N } . {\displaystyle A^{N}=\{a\in A:na=a{\text{ for all }}n\in N\}.} Then the inflation-restriction exact sequence is: 0 → H 1 ( G / N , A N ) → H 1 ( G , A ) → H 1 ( N , A ) G / N → H 2 ( G / N , A N ) → H 2 ( G , A ) . {\displaystyle 0\to H^{1}(G/N,A^{N})\to H^{1}(G,A)\to H^{1}(N,A)^{G/N}\to H^{2}(G/N,A^{N})\to H^{2}(G,A).} The transgression map is the map H 1 ( N , A ) G / N → H 2 ( G / N , A N ) {\displaystyle H^{1}(N,A)^{G/N}\to H^{2}(G/N,A^{N})} . Transgression is defined for general n ∈ N {\displaystyle n\in \mathbb {N} } , H n ( N , A ) G / N → H n + 1 ( G / N , A N ) {\displaystyle H^{n}(N,A)^{G/N}\to H^{n+1}(G/N,A^{N})} , only if H i ( N , A ) G / N = 0 {\displaystyle H^{i}(N,A)^{G/N}=0} for i ≤ n − 1 {\displaystyle i\leq n-1} . == Notes == == References == Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics. Vol. 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001. Hazewinkel, Michiel (1995). Handbook of Algebra, Volume 1. Elsevier. p. 282. ISBN 0444822127. Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. Vol. 62 (2nd printing of 1st ed.). Springer-Verlag. ISBN 3-540-63003-1. Zbl 0819.11044. Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften. Vol. 323 (2nd ed.). Springer-Verlag. pp. 112–113. ISBN 978-3-540-37888-4. Zbl 1136.11001. Schmid, Peter (2007). The Solution of The K(GV) Problem. Advanced Texts in Mathematics. Vol. 4. Imperial College Press. p. 214. ISBN 978-1860949708. Serre, Jean-Pierre (1979). Local Fields. Graduate Texts in Mathematics. Vol. 67. Translated by Greenberg, Marvin Jay. Springer-Verlag. pp. 117–118. ISBN 0-387-90424-7. Zbl 0423.12016. == External links == transgression at the nLab
|
Wikipedia:Translation (geometry)#0
|
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is an isometry. == As a function == If v {\displaystyle \mathbf {v} } is a fixed vector, known as the translation vector, and p {\displaystyle \mathbf {p} } is the initial position of some object, then the translation function T v {\displaystyle T_{\mathbf {v} }} will work as T v ( p ) = p + v {\displaystyle T_{\mathbf {v} }(\mathbf {p} )=\mathbf {p} +\mathbf {v} } . If T {\displaystyle T} is a translation, then the image of a subset A {\displaystyle A} under the function T {\displaystyle T} is the translate of A {\displaystyle A} by T {\displaystyle T} . The translate of A {\displaystyle A} by T v {\displaystyle T_{\mathbf {v} }} is often written as A + v {\displaystyle A+\mathbf {v} } . === Application in classical physics === In classical physics, translational motion is movement that changes the position of an object, as opposed to rotation. For example, according to Whittaker: If a body is moved from one position to another, and if the lines joining the initial and final points of each of the points of the body are a set of parallel straight lines of length ℓ, so that the orientation of the body in space is unaltered, the displacement is called a translation parallel to the direction of the lines, through a distance ℓ. A translation is the operation changing the positions of all points ( x , y , z ) {\displaystyle (x,y,z)} of an object according to the formula ( x , y , z ) → ( x + Δ x , y + Δ y , z + Δ z ) {\displaystyle (x,y,z)\to (x+\Delta x,y+\Delta y,z+\Delta z)} where ( Δ x , Δ y , Δ z ) {\displaystyle (\Delta x,\ \Delta y,\ \Delta z)} is the same vector for each point of the object. The translation vector ( Δ x , Δ y , Δ z ) {\displaystyle (\Delta x,\ \Delta y,\ \Delta z)} common to all points of the object describes a particular type of displacement of the object, usually called a linear displacement to distinguish it from displacements involving rotation, called angular displacements. When considering spacetime, a change of time coordinate is considered to be a translation. == As an operator == The translation operator turns a function of the original position, f ( v ) {\displaystyle f(\mathbf {v} )} , into a function of the final position, f ( v + δ ) {\displaystyle f(\mathbf {v} +\mathbf {\delta } )} . In other words, T δ {\displaystyle T_{\mathbf {\delta } }} is defined such that T δ f ( v ) = f ( v + δ ) . {\displaystyle T_{\mathbf {\delta } }f(\mathbf {v} )=f(\mathbf {v} +\mathbf {\delta } ).} This operator is more abstract than a function, since T δ {\displaystyle T_{\mathbf {\delta } }} defines a relationship between two functions, rather than the underlying vectors themselves. The translation operator can act on many kinds of functions, such as when the translation operator acts on a wavefunction, which is studied in the field of quantum mechanics. == As a group == The set of all translations forms the translation group T {\displaystyle \mathbb {T} } , which is isomorphic to the space itself, and a normal subgroup of Euclidean group E ( n ) {\displaystyle E(n)} . The quotient group of E ( n ) {\displaystyle E(n)} by T {\displaystyle \mathbb {T} } is isomorphic to the group of rigid motions which fix a particular origin point, the orthogonal group O ( n ) {\displaystyle O(n)} : E ( n ) / T ≅ O ( n ) {\displaystyle E(n)/\mathbb {T} \cong O(n)} Because translation is commutative, the translation group is abelian. There are an infinite number of possible translations, so the translation group is an infinite group. In the theory of relativity, due to the treatment of space and time as a single spacetime, translations can also refer to changes in the time coordinate. For example, the Galilean group and the Poincaré group include translations with respect to time. === Lattice groups === One kind of subgroup of the three-dimensional translation group are the lattice groups, which are infinite groups, but unlike the translation groups, are finitely generated. That is, a finite generating set generates the entire group. == Matrix representation == A translation is an affine transformation with no fixed points. Matrix multiplications always have the origin as a fixed point. Nevertheless, there is a common workaround using homogeneous coordinates to represent a translation of a vector space with matrix multiplication: Write the 3-dimensional vector v = ( v x , v y , v z ) {\displaystyle \mathbf {v} =(v_{x},v_{y},v_{z})} using 4 homogeneous coordinates as v = ( v x , v y , v z , 1 ) {\displaystyle \mathbf {v} =(v_{x},v_{y},v_{z},1)} . To translate an object by a vector v {\displaystyle \mathbf {v} } , each homogeneous vector p {\displaystyle \mathbf {p} } (written in homogeneous coordinates) can be multiplied by this translation matrix: T v = [ 1 0 0 v x 0 1 0 v y 0 0 1 v z 0 0 0 1 ] {\displaystyle T_{\mathbf {v} }={\begin{bmatrix}1&0&0&v_{x}\\0&1&0&v_{y}\\0&0&1&v_{z}\\0&0&0&1\end{bmatrix}}} As shown below, the multiplication will give the expected result: T v p = [ 1 0 0 v x 0 1 0 v y 0 0 1 v z 0 0 0 1 ] [ p x p y p z 1 ] = [ p x + v x p y + v y p z + v z 1 ] = p + v {\displaystyle T_{\mathbf {v} }\mathbf {p} ={\begin{bmatrix}1&0&0&v_{x}\\0&1&0&v_{y}\\0&0&1&v_{z}\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}p_{x}+v_{x}\\p_{y}+v_{y}\\p_{z}+v_{z}\\1\end{bmatrix}}=\mathbf {p} +\mathbf {v} } The inverse of a translation matrix can be obtained by reversing the direction of the vector: T v − 1 = T − v . {\displaystyle T_{\mathbf {v} }^{-1}=T_{-\mathbf {v} }.\!} Similarly, the product of translation matrices is given by adding the vectors: T v T w = T v + w . {\displaystyle T_{\mathbf {v} }T_{\mathbf {w} }=T_{\mathbf {v} +\mathbf {w} }.\!} Because addition of vectors is commutative, multiplication of translation matrices is therefore also commutative (unlike multiplication of arbitrary matrices). == Translation of axes == While geometric translation is often viewed as an active transformation that changes the position of a geometric object, a similar result can be achieved by a passive transformation that moves the coordinate system itself but leaves the object fixed. The passive version of an active geometric translation is known as a translation of axes. == Translational symmetry == An object that looks the same before and after translation is said to have translational symmetry. A common example is a periodic function, which is an eigenfunction of a translation operator. == Translations of a graph == The graph of a real function f, the set of points ( x , f ( x ) ) {\displaystyle (x,f(x))} , is often pictured in the real coordinate plane with x as the horizontal coordinate and y = f ( x ) {\displaystyle y=f(x)} as the vertical coordinate. Starting from the graph of f, a horizontal translation means composing f with a function x ↦ x − a {\displaystyle x\mapsto x-a} , for some constant number a, resulting in a graph consisting of points ( x , f ( x − a ) ) {\displaystyle (x,f(x-a))} . Each point ( x , y ) {\displaystyle (x,y)} of the original graph corresponds to the point ( x + a , y ) {\displaystyle (x+a,y)} in the new graph, which pictorially results in a horizontal shift. A vertical translation means composing the function y ↦ y + b {\displaystyle y\mapsto y+b} with f, for some constant b, resulting in a graph consisting of the points ( x , f ( x ) + b ) {\displaystyle {\bigl (}x,f(x)+b{\bigr )}} . Each point ( x , y ) {\displaystyle (x,y)} of the original graph corresponds to the point ( x , y + b ) {\displaystyle (x,y+b)} in the new graph, which pictorially results in a vertical shift. For example, taking the quadratic function y = x 2 {\displaystyle y=x^{2}} , whose graph is a parabola with vertex at ( 0 , 0 ) {\displaystyle (0,0)} , a horizontal translation 5 units to the right would be the new function y = ( x − 5 ) 2 = x 2 − 10 x + 25 {\displaystyle y=(x-5)^{2}=x^{2}-10x+25} whose vertex has coordinates ( 5 , 0 ) {\displaystyle (5,0)} . A vertical translation 3 units upward would be the new function y = x 2 + 3 {\displaystyle y=x^{2}+3} whose vertex has coordinates ( 0 , 3 ) {\displaystyle (0,3)} . The antiderivatives of a function all differ from each other by a constant of integration and are therefore vertical translates of each other. == Applications == For describing vehicle dynamics (or movement of any rigid body), including ship dynamics and aircraft dynamics, it is common to use a mechanical model consisting of six degrees of freedom, which includes translations along three reference axes (as well as rotations about those three axes). These translations are often called surge, sway, and heave. == See also == == References == == Further reading == Zazkis, R., Liljedahl, P., & Gadowsky, K. Conceptions of function translation: obstacles, intuitions, and rerouting. Journal of Mathematical Behavior, 22, 437-450. Retrieved April 29, 2014, from www.elsevier.com/locate/jmathb Transformations of Graphs: Horizontal Translations. (2006, January 1). BioMath: Transformation of Graphs. Retrieved April 29, 2014 == External links == Translation Transform at cut-the-knot Geometric Translation (Interactive Animation) at Math Is Fun Understanding 2D Translation and Understanding 3D Translation by Roger Germundsson, The Wolfram Demonstrations Project.
|
Wikipedia:Translation of axes#0
|
In mathematics, a translation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x'y'-Cartesian coordinate system in which the x' axis is parallel to the x axis and k units away, and the y' axis is parallel to the y axis and h units away. This means that the origin O' of the new coordinate system has coordinates (h, k) in the original system. The positive x' and y' directions are taken to be the same as the positive x and y directions. A point P has coordinates (x, y) with respect to the original system and coordinates (x', y') with respect to the new system, where or equivalently In the new coordinate system, the point P will appear to have been translated in the opposite direction. For example, if the xy-system is translated a distance h to the right and a distance k upward, then P will appear to have been translated a distance h to the left and a distance k downward in the x'y'-system . A translation of axes in more than two dimensions is defined similarly. A translation of axes is a rigid transformation, but not a linear map. (See Affine transformation.) == Motivation == Coordinate systems are essential for studying the equations of curves using the methods of analytic geometry. To use the method of coordinate geometry, the axes are placed at a convenient position with respect to the curve under consideration. For example, to study the equations of ellipses and hyperbolas, the foci are usually located on one of the axes and are situated symmetrically with respect to the origin. If the curve (hyperbola, parabola, ellipse, etc.) is not situated conveniently with respect to the axes, the coordinate system should be changed to place the curve at a convenient and familiar location and orientation. The process of making this change is called a transformation of coordinates. The solutions to many problems can be simplified by translating the coordinate axes to obtain new axes parallel to the original ones. == Translation of conic sections == Through a change of coordinates, the equation of a conic section can be put into a standard form, which is usually easier to work with. For the most general equation of the second degree, which takes the form it is always possible to perform a rotation of axes in such a way that in the new system the equation takes the form that is, eliminating the xy term. Next, a translation of axes can reduce an equation of the form (3) to an equation of the same form but with new variables (x', y') as coordinates, and with D and E both equal to zero (with certain exceptions—for example, parabolas). The principal tool in this process is "completing the square." In the examples that follow, it is assumed that a rotation of axes has already been performed. === Example 1 === Given the equation 9 x 2 + 25 y 2 + 18 x − 100 y − 116 = 0 , {\displaystyle 9x^{2}+25y^{2}+18x-100y-116=0,} by using a translation of axes, determine whether the locus of the equation is a parabola, ellipse, or hyperbola. Determine foci (or focus), vertices (or vertex), and eccentricity. Solution: To complete the square in x and y, write the equation in the form 9 ( x 2 + 2 x ) + 25 ( y 2 − 4 y ) = 116. {\displaystyle 9(x^{2}+2x\qquad )+25(y^{2}-4y\qquad )=116.} Complete the squares and obtain 9 ( x 2 + 2 x + 1 ) + 25 ( y 2 − 4 y + 4 ) = 116 + 9 + 100 {\displaystyle 9(x^{2}+2x+1)+25(y^{2}-4y+4)=116+9+100} ⇔ 9 ( x + 1 ) 2 + 25 ( y − 2 ) 2 = 225. {\displaystyle \Leftrightarrow 9(x+1)^{2}+25(y-2)^{2}=225.} Define x ′ = x + 1 {\displaystyle x'=x+1} and y ′ = y − 2. {\displaystyle y'=y-2.} That is, the translation in equations (2) is made with h = − 1 , k = 2. {\displaystyle h=-1,k=2.} The equation in the new coordinate system is Divide equation (5) by 225 to obtain x ′ 2 25 + y ′ 2 9 = 1 , {\displaystyle {\frac {x'^{2}}{25}}+{\frac {y'^{2}}{9}}=1,} which is recognizable as an ellipse with a = 5 , b = 3 , c 2 = a 2 − b 2 = 16 , c = 4 , e = 4 5 . {\displaystyle a=5,b=3,c^{2}=a^{2}-b^{2}=16,c=4,e={\tfrac {4}{5}}.} In the x'y'-system, we have: center ( 0 , 0 ) {\displaystyle (0,0)} ; vertices ( ± 5 , 0 ) {\displaystyle (\pm 5,0)} ; foci ( ± 4 , 0 ) . {\displaystyle (\pm 4,0).} In the xy-system, use the relations x = x ′ − 1 , y = y ′ + 2 {\displaystyle x=x'-1,y=y'+2} to obtain: center ( − 1 , 2 ) {\displaystyle (-1,2)} ; vertices ( 4 , 2 ) , ( − 6 , 2 ) {\displaystyle (4,2),(-6,2)} ; foci ( 3 , 2 ) , ( − 5 , 2 ) {\displaystyle (3,2),(-5,2)} ; eccentricity 4 5 . {\displaystyle {\tfrac {4}{5}}.} == Generalization to several dimensions == For an xyz-Cartesian coordinate system in three dimensions, suppose that a second Cartesian coordinate system is introduced, with axes x', y' and z' so located that the x' axis is parallel to the x axis and h units from it, the y' axis is parallel to the y axis and k units from it, and the z' axis is parallel to the z axis and l units from it. A point P in space will have coordinates in both systems. If its coordinates are (x, y, z) in the original system and (x', y', z') in the second system, the equations hold. Equations (6) define a translation of axes in three dimensions where (h, k, l) are the xyz-coordinates of the new origin. A translation of axes in any finite number of dimensions is defined similarly. == Translation of quadric surfaces == In three-space, the most general equation of the second degree in x, y and z has the form where the quantities A , B , C , … , J {\displaystyle A,B,C,\ldots ,J} are positive or negative numbers or zero. The points in space satisfying such an equation all lie on a surface. Any second-degree equation which does not reduce to a cylinder, plane, line, or point corresponds to a surface which is called quadric. As in the case of plane analytic geometry, the method of translation of axes may be used to simplify second-degree equations, thereby making evident the nature of certain quadric surfaces. The principal tool in this process is "completing the square." === Example 2 === Use a translation of coordinates to identify the quadric surface x 2 + 4 y 2 + 3 z 2 + 2 x − 8 y + 9 z = 10. {\displaystyle x^{2}+4y^{2}+3z^{2}+2x-8y+9z=10.} Solution: Write the equation in the form x 2 + 2 x + 4 ( y 2 − 2 y ) + 3 ( z 2 + 3 z ) = 10. {\displaystyle x^{2}+2x\qquad +4(y^{2}-2y\qquad )+3(z^{2}+3z\qquad )=10.} Complete the square to obtain ( x + 1 ) 2 + 4 ( y − 1 ) 2 + 3 ( z + 3 2 ) 2 = 10 + 1 + 4 + 27 4 . {\displaystyle (x+1)^{2}+4(y-1)^{2}+3(z+{\tfrac {3}{2}})^{2}=10+1+4+{\tfrac {27}{4}}.} Introduce the translation of coordinates x ′ = x + 1 , y ′ = y − 1 , z ′ = z + 3 2 . {\displaystyle x'=x+1,\qquad y'=y-1,\qquad z'=z+{\tfrac {3}{2}}.} The equation of the surface takes the form x ′ 2 + 4 y ′ 2 + 3 z ′ 2 = 87 4 , {\displaystyle x'^{2}+4y'^{2}+3z'^{2}={\tfrac {87}{4}},} which is recognizable as the equation of an ellipsoid. == See also == Change of basis Translation (geometry) == Notes == == References == Anton, Howard (1987), Elementary Linear Algebra (5th ed.), New York: Wiley, ISBN 0-471-84819-0 Protter, Murray H.; Morrey, Charles B. Jr. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley, LCCN 76087042
|
Wikipedia:Transpose#0
|
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing another matrix, often denoted by AT (among other notations). The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley. == Transpose of a matrix == === Definition === The transpose of a matrix A, denoted by AT, ⊤A, A⊤, A ⊺ {\displaystyle A^{\intercal }} , A′, Atr, tA or At, may be constructed by any one of the following methods: Reflect A over its main diagonal (which runs from top-left to bottom-right) to obtain AT Write the rows of A as the columns of AT Write the columns of A as the rows of AT Formally, the i-th row, j-th column element of AT is the j-th row, i-th column element of A: [ A T ] i j = [ A ] j i . {\displaystyle \left[\mathbf {A} ^{\operatorname {T} }\right]_{ij}=\left[\mathbf {A} \right]_{ji}.} If A is an m × n matrix, then AT is an n × m matrix. In the case of square matrices, AT may also denote the Tth power of the matrix A. For avoiding a possible confusion, many authors use left upperscripts, that is, they denote the transpose as TA. An advantage of this notation is that no parentheses are needed when exponents are involved: as (TA)n = T(An), notation TAn is not ambiguous. In this article, this confusion is avoided by never using the symbol T as a variable name. ==== Matrix definitions involving transposition ==== A square matrix whose transpose is equal to itself is called a symmetric matrix; that is, A is symmetric if A T = A . {\displaystyle \mathbf {A} ^{\operatorname {T} }=\mathbf {A} .} A square matrix whose transpose is equal to its negative is called a skew-symmetric matrix; that is, A is skew-symmetric if A T = − A . {\displaystyle \mathbf {A} ^{\operatorname {T} }=-\mathbf {A} .} A square complex matrix whose transpose is equal to the matrix with every entry replaced by its complex conjugate (denoted here with an overline) is called a Hermitian matrix (equivalent to the matrix being equal to its conjugate transpose); that is, A is Hermitian if A T = A ¯ . {\displaystyle \mathbf {A} ^{\operatorname {T} }={\overline {\mathbf {A} }}.} A square complex matrix whose transpose is equal to the negation of its complex conjugate is called a skew-Hermitian matrix; that is, A is skew-Hermitian if A T = − A ¯ . {\displaystyle \mathbf {A} ^{\operatorname {T} }=-{\overline {\mathbf {A} }}.} A square matrix whose transpose is equal to its inverse is called an orthogonal matrix; that is, A is orthogonal if A T = A − 1 . {\displaystyle \mathbf {A} ^{\operatorname {T} }=\mathbf {A} ^{-1}.} A square complex matrix whose transpose is equal to its conjugate inverse is called a unitary matrix; that is, A is unitary if A T = A − 1 ¯ . {\displaystyle \mathbf {A} ^{\operatorname {T} }={\overline {\mathbf {A} ^{-1}}}.} === Examples === [ 1 2 ] T = [ 1 2 ] {\displaystyle {\begin{bmatrix}1&2\end{bmatrix}}^{\operatorname {T} }=\,{\begin{bmatrix}1\\2\end{bmatrix}}} [ 1 2 3 4 ] T = [ 1 3 2 4 ] {\displaystyle {\begin{bmatrix}1&2\\3&4\end{bmatrix}}^{\operatorname {T} }={\begin{bmatrix}1&3\\2&4\end{bmatrix}}} [ 1 2 3 4 5 6 ] T = [ 1 3 5 2 4 6 ] {\displaystyle {\begin{bmatrix}1&2\\3&4\\5&6\end{bmatrix}}^{\operatorname {T} }={\begin{bmatrix}1&3&5\\2&4&6\end{bmatrix}}} === Properties === Let A and B be matrices and c be a scalar. ( A T ) T = A . {\displaystyle \left(\mathbf {A} ^{\operatorname {T} }\right)^{\operatorname {T} }=\mathbf {A} .} The operation of taking the transpose is an involution (self-inverse). ( A + B ) T = A T + B T . {\displaystyle \left(\mathbf {A} +\mathbf {B} \right)^{\operatorname {T} }=\mathbf {A} ^{\operatorname {T} }+\mathbf {B} ^{\operatorname {T} }.} The transpose respects addition. ( c A ) T = c ( A T ) . {\displaystyle \left(c\mathbf {A} \right)^{\operatorname {T} }=c(\mathbf {A} ^{\operatorname {T} }).} The transpose of a scalar is the same scalar. Together with the preceding property, this implies that the transpose is a linear map from the space of m × n matrices to the space of the n × m matrices. ( A B ) T = B T A T . {\displaystyle \left(\mathbf {AB} \right)^{\operatorname {T} }=\mathbf {B} ^{\operatorname {T} }\mathbf {A} ^{\operatorname {T} }.} The order of the factors reverses. By induction, this result extends to the general case of multiple matrices, so (A1A2...Ak−1Ak)T = AkTAk−1T…A2TA1T. det ( A T ) = det ( A ) . {\displaystyle \det \left(\mathbf {A} ^{\operatorname {T} }\right)=\det(\mathbf {A} ).} The determinant of a square matrix is the same as the determinant of its transpose. The dot product of two column vectors a and b can be computed as the single entry of the matrix product a ⋅ b = a T b . {\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} ^{\operatorname {T} }\mathbf {b} .} If A has only real entries, then ATA is a positive-semidefinite matrix. ( A T ) − 1 = ( A − 1 ) T . {\displaystyle \left(\mathbf {A} ^{\operatorname {T} }\right)^{-1}=\left(\mathbf {A} ^{-1}\right)^{\operatorname {T} }.} The transpose of an invertible matrix is also invertible, and its inverse is the transpose of the inverse of the original matrix.The notation A−T is sometimes used to represent either of these equivalent expressions. If A is a square matrix, then its eigenvalues are equal to the eigenvalues of its transpose, since they share the same characteristic polynomial. ( A a ) ⋅ b = a ⋅ ( A T b ) {\displaystyle \left(\mathbf {A} \mathbf {a} \right)\cdot \mathbf {b} =\mathbf {a} \cdot \mathbf {\left(A^{T}\mathbf {b} \right)} } for two column vectors a , b {\displaystyle \mathbf {a} ,\mathbf {b} } and the standard dot product. Over any field k {\displaystyle k} , a square matrix A {\displaystyle \mathbf {A} } is similar to A T {\displaystyle \mathbf {A} ^{\operatorname {T} }} . This implies that A {\displaystyle \mathbf {A} } and A T {\displaystyle \mathbf {A} ^{\operatorname {T} }} have the same invariant factors, which implies they share the same minimal polynomial, characteristic polynomial, and eigenvalues, among other properties. A proof of this property uses the following two observations. Let A {\displaystyle \mathbf {A} } and B {\displaystyle \mathbf {B} } be n × n {\displaystyle n\times n} matrices over some base field k {\displaystyle k} and let L {\displaystyle L} be a field extension of k {\displaystyle k} . If A {\displaystyle \mathbf {A} } and B {\displaystyle \mathbf {B} } are similar as matrices over L {\displaystyle L} , then they are similar over k {\displaystyle k} . In particular this applies when L {\displaystyle L} is the algebraic closure of k {\displaystyle k} . If A {\displaystyle \mathbf {A} } is a matrix over an algebraically closed field in Jordan normal form with respect to some basis, then A {\displaystyle \mathbf {A} } is similar to A T {\displaystyle \mathbf {A} ^{\operatorname {T} }} . This further reduces to proving the same fact when A {\displaystyle \mathbf {A} } is a single Jordan block, which is a straightforward exercise. === Products === If A is an m × n matrix and AT is its transpose, then the result of matrix multiplication with these two matrices gives two square matrices: A AT is m × m and AT A is n × n. Furthermore, these products are symmetric matrices. Indeed, the matrix product A AT has entries that are the inner product of a row of A with a column of AT. But the columns of AT are the rows of A, so the entry corresponds to the inner product of two rows of A. If pi j is the entry of the product, it is obtained from rows i and j in A. The entry pj i is also obtained from these rows, thus pi j = pj i, and the product matrix (pi j) is symmetric. Similarly, the product AT A is a symmetric matrix. A quick proof of the symmetry of A AT results from the fact that it is its own transpose: ( A A T ) T = ( A T ) T A T = A A T . {\displaystyle \left(\mathbf {A} \mathbf {A} ^{\operatorname {T} }\right)^{\operatorname {T} }=\left(\mathbf {A} ^{\operatorname {T} }\right)^{\operatorname {T} }\mathbf {A} ^{\operatorname {T} }=\mathbf {A} \mathbf {A} ^{\operatorname {T} }.} === Implementation of matrix transposition on computers === On a computer, one can often avoid explicitly transposing a matrix in memory by simply accessing the same data in a different order. For example, software libraries for linear algebra, such as BLAS, typically provide options to specify that certain matrices are to be interpreted in transposed order to avoid the necessity of data movement. However, there remain a number of circumstances in which it is necessary or desirable to physically reorder a matrix in memory to its transposed ordering. For example, with a matrix stored in row-major order, the rows of the matrix are contiguous in memory and the columns are discontiguous. If repeated operations need to be performed on the columns, for example in a fast Fourier transform algorithm, transposing the matrix in memory (to make the columns contiguous) may improve performance by increasing memory locality. Ideally, one might hope to transpose a matrix with minimal additional storage. This leads to the problem of transposing an n × m matrix in-place, with O(1) additional storage or at most storage much less than mn. For n ≠ m, this involves a complicated permutation of the data elements that is non-trivial to implement in-place. Therefore, efficient in-place matrix transposition has been the subject of numerous research publications in computer science, starting in the late 1950s, and several algorithms have been developed. == Transposes of linear maps and bilinear forms == As the main use of matrices is to represent linear maps between finite-dimensional vector spaces, the transpose is an operation on matrices that may be seen as the representation of some operation on linear maps. This leads to a much more general definition of the transpose that works on every linear map, even when linear maps cannot be represented by matrices (such as in the case of infinite dimensional vector spaces). In the finite dimensional case, the matrix representing the transpose of a linear map is the transpose of the matrix representing the linear map, independently of the basis choice. === Transpose of a linear map === Let X# denote the algebraic dual space of an R-module X. Let X and Y be R-modules. If u : X → Y is a linear map, then its algebraic adjoint or dual, is the map u# : Y# → X# defined by f ↦ f ∘ u. The resulting functional u#(f) is called the pullback of f by u. The following relation characterizes the algebraic adjoint of u ⟨u#(f), x⟩ = ⟨f, u(x)⟩ for all f ∈ Y# and x ∈ X where ⟨•, •⟩ is the natural pairing (i.e. defined by ⟨h, z⟩ := h(z)). This definition also applies unchanged to left modules and to vector spaces. The definition of the transpose may be seen to be independent of any bilinear form on the modules, unlike the adjoint (below). The continuous dual space of a topological vector space (TVS) X is denoted by X'. If X and Y are TVSs then a linear map u : X → Y is weakly continuous if and only if u#(Y') ⊆ X', in which case we let tu : Y' → X' denote the restriction of u# to Y'. The map tu is called the transpose of u. If the matrix A describes a linear map with respect to bases of V and W, then the matrix AT describes the transpose of that linear map with respect to the dual bases. === Transpose of a bilinear form === Every linear map to the dual space u : X → X# defines a bilinear form B : X × X → F, with the relation B(x, y) = u(x)(y). By defining the transpose of this bilinear form as the bilinear form tB defined by the transpose tu : X## → X# i.e. tB(y, x) = tu(Ψ(y))(x), we find that B(x, y) = tB(y, x). Here, Ψ is the natural homomorphism X → X## into the double dual. === Adjoint === If the vector spaces X and Y have respectively nondegenerate bilinear forms BX and BY, a concept known as the adjoint, which is closely related to the transpose, may be defined: If u : X → Y is a linear map between vector spaces X and Y, we define g as the adjoint of u if g : Y → X satisfies B X ( x , g ( y ) ) = B Y ( u ( x ) , y ) {\displaystyle B_{X}{\big (}x,g(y){\big )}=B_{Y}{\big (}u(x),y{\big )}} for all x ∈ X and y ∈ Y. These bilinear forms define an isomorphism between X and X#, and between Y and Y#, resulting in an isomorphism between the transpose and adjoint of u. The matrix of the adjoint of a map is the transposed matrix only if the bases are orthonormal with respect to their bilinear forms. In this context, many authors however, use the term transpose to refer to the adjoint as defined here. The adjoint allows us to consider whether g : Y → X is equal to u −1 : Y → X. In particular, this allows the orthogonal group over a vector space X with a quadratic form to be defined without reference to matrices (nor the components thereof) as the set of all linear maps X → X for which the adjoint equals the inverse. Over a complex vector space, one often works with sesquilinear forms (conjugate-linear in one argument) instead of bilinear forms. The Hermitian adjoint of a map between such spaces is defined similarly, and the matrix of the Hermitian adjoint is given by the conjugate transpose matrix if the bases are orthonormal. == See also == Adjugate matrix, the transpose of the cofactor matrix Conjugate transpose Converse relation Moore–Penrose pseudoinverse Projection (linear algebra) == References == == Further reading == Bourbaki, Nicolas (1989) [1970]. Algebra I Chapters 1-3 [Algèbre: Chapitres 1 à 3] (PDF). Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64243-5. OCLC 18588156. Halmos, Paul (1974), Finite dimensional vector spaces, Springer, ISBN 978-0-387-90093-3. Maruskin, Jared M. (2012). Essential Linear Algebra. San José: Solar Crest. pp. 122–132. ISBN 978-0-9850627-3-6. Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. Schwartz, Jacob T. (2001). Introduction to Matrices and Vectors. Mineola: Dover. pp. 126–132. ISBN 0-486-42000-0. == External links == Gilbert Strang (Spring 2010) Linear Algebra from MIT Open Courseware
|
Wikipedia:Transpose of a linear map#0
|
In linear algebra, the transpose of a linear map between two vector spaces, defined over the same field, is an induced map between the dual spaces of the two vector spaces. The transpose or algebraic adjoint of a linear map is often used to study the original linear map. This concept is generalised by adjoint functors. == Definition == Let X # {\displaystyle X^{\#}} denote the algebraic dual space of a vector space X . {\displaystyle X.} Let X {\displaystyle X} and Y {\displaystyle Y} be vector spaces over the same field K . {\displaystyle {\mathcal {K}}.} If u : X → Y {\displaystyle u:X\to Y} is a linear map, then its algebraic adjoint or dual, is the map # u : Y # → X # {\displaystyle {}^{\#}u:Y^{\#}\to X^{\#}} defined by f ↦ f ∘ u . {\displaystyle f\mapsto f\circ u.} The resulting functional # u ( f ) := f ∘ u {\displaystyle {}^{\#}u(f):=f\circ u} is called the pullback of f {\displaystyle f} by u . {\displaystyle u.} The continuous dual space of a topological vector space (TVS) X {\displaystyle X} is denoted by X ′ . {\displaystyle X^{\prime }.} If X {\displaystyle X} and Y {\displaystyle Y} are TVSs then a linear map u : X → Y {\displaystyle u:X\to Y} is weakly continuous if and only if # u ( Y ′ ) ⊆ X ′ , {\displaystyle {}^{\#}u\left(Y^{\prime }\right)\subseteq X^{\prime },} in which case we let t u : Y ′ → X ′ {\displaystyle {}^{t}u:Y^{\prime }\to X^{\prime }} denote the restriction of # u {\displaystyle {}^{\#}u} to Y ′ . {\displaystyle Y^{\prime }.} The map t u {\displaystyle {}^{t}u} is called the transpose or algebraic adjoint of u . {\displaystyle u.} The following identity characterizes the transpose of u {\displaystyle u} : ⟨ t u ( f ) , x ⟩ = ⟨ f , u ( x ) ⟩ for all f ∈ Y ′ and x ∈ X , {\displaystyle \left\langle {}^{t}u(f),x\right\rangle =\left\langle f,u(x)\right\rangle \quad {\text{ for all }}f\in Y^{\prime }{\text{ and }}x\in X,} where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \left\langle \cdot ,\cdot \right\rangle } is the natural pairing defined by ⟨ z , h ⟩ := z ( h ) . {\displaystyle \left\langle z,h\right\rangle :=z(h).} == Properties == The assignment u ↦ t u {\displaystyle u\mapsto {}^{t}u} produces an injective linear map between the space of linear operators from X {\displaystyle X} to Y {\displaystyle Y} and the space of linear operators from Y # {\displaystyle Y^{\#}} to X # . {\displaystyle X^{\#}.} If X = Y {\displaystyle X=Y} then the space of linear maps is an algebra under composition of maps, and the assignment is then an antihomomorphism of algebras, meaning that t ( u v ) = t v t u . {\displaystyle {}^{t}(uv)={}^{t}v{}^{t}u.} In the language of category theory, taking the dual of vector spaces and the transpose of linear maps is therefore a contravariant functor from the category of vector spaces over K {\displaystyle {\mathcal {K}}} to itself. One can identify t ( t u ) {\displaystyle {}^{t}\left({}^{t}u\right)} with u {\displaystyle u} using the natural injection into the double dual. If u : X → Y {\displaystyle u:X\to Y} and v : Y → Z {\displaystyle v:Y\to Z} are linear maps then t ( v ∘ u ) = t u ∘ t v {\displaystyle {}^{t}(v\circ u)={}^{t}u\circ {}^{t}v} If u : X → Y {\displaystyle u:X\to Y} is a (surjective) vector space isomorphism then so is the transpose t u : Y ′ → X ′ . {\displaystyle {}^{t}u:Y^{\prime }\to X^{\prime }.} If X {\displaystyle X} and Y {\displaystyle Y} are normed spaces then ‖ x ‖ = sup ‖ x ′ ‖ ≤ 1 | x ′ ( x ) | for each x ∈ X {\displaystyle \|x\|=\sup _{\|x^{\prime }\|\leq 1}\left|x^{\prime }(x)\right|\quad {\text{ for each }}x\in X} and if the linear operator u : X → Y {\displaystyle u:X\to Y} is bounded then the operator norm of t u {\displaystyle {}^{t}u} is equal to the norm of u {\displaystyle u} ; that is ‖ u ‖ = ‖ t u ‖ , {\displaystyle \|u\|=\left\|{}^{t}u\right\|,} and moreover, ‖ u ‖ = sup { | y ′ ( u x ) | : ‖ x ‖ ≤ 1 , ‖ y ∗ ‖ ≤ 1 where x ∈ X , y ′ ∈ Y ′ } . {\displaystyle \|u\|=\sup \left\{\left|y^{\prime }(ux)\right|:\|x\|\leq 1,\left\|y^{*}\right\|\leq 1{\text{ where }}x\in X,y^{\prime }\in Y^{\prime }\right\}.} === Polars === Suppose now that u : X → Y {\displaystyle u:X\to Y} is a weakly continuous linear operator between topological vector spaces X {\displaystyle X} and Y {\displaystyle Y} with continuous dual spaces X ′ {\displaystyle X^{\prime }} and Y ′ , {\displaystyle Y^{\prime },} respectively. Let ⟨ ⋅ , ⋅ ⟩ : X × X ′ → C {\displaystyle \langle \cdot ,\cdot \rangle :X\times X^{\prime }\to \mathbb {C} } denote the canonical dual system, defined by ⟨ x , x ′ ⟩ = x ′ x {\displaystyle \left\langle x,x^{\prime }\right\rangle =x^{\prime }x} where x {\displaystyle x} and x ′ {\displaystyle x^{\prime }} are said to be orthogonal if ⟨ x , x ′ ⟩ = x ′ x = 0. {\displaystyle \left\langle x,x^{\prime }\right\rangle =x^{\prime }x=0.} For any subsets A ⊆ X {\displaystyle A\subseteq X} and S ′ ⊆ X ′ , {\displaystyle S^{\prime }\subseteq X^{\prime },} let A ∘ = { x ′ ∈ X ′ : sup a ∈ A | x ′ ( a ) | ≤ 1 } and S ∘ = { x ∈ X : sup s ′ ∈ S ′ | s ′ ( x ) | ≤ 1 } {\displaystyle A^{\circ }=\left\{x^{\prime }\in X^{\prime }:\sup _{a\in A}\left|x^{\prime }(a)\right|\leq 1\right\}\qquad {\text{ and }}\qquad S^{\circ }=\left\{x\in X:\sup _{s^{\prime }\in S^{\prime }}\left|s^{\prime }(x)\right|\leq 1\right\}} denote the (absolute) polar of A {\displaystyle A} in X ′ {\displaystyle X^{\prime }} (resp. of S ′ {\displaystyle S^{\prime }} in X {\displaystyle X} ). If A ⊆ X {\displaystyle A\subseteq X} and B ⊆ Y {\displaystyle B\subseteq Y} are convex, weakly closed sets containing the origin then t u ( B ∘ ) ⊆ A ∘ {\displaystyle {}^{t}u\left(B^{\circ }\right)\subseteq A^{\circ }} implies u ( A ) ⊆ B . {\displaystyle u(A)\subseteq B.} If A ⊆ X {\displaystyle A\subseteq X} and B ⊆ Y {\displaystyle B\subseteq Y} then [ u ( A ) ] ∘ = ( t u ) − 1 ( A ∘ ) {\displaystyle [u(A)]^{\circ }=\left({}^{t}u\right)^{-1}\left(A^{\circ }\right)} and u ( A ) ⊆ B implies t u ( B ∘ ) ⊆ A ∘ . {\displaystyle u(A)\subseteq B\quad {\text{ implies }}\quad {}^{t}u\left(B^{\circ }\right)\subseteq A^{\circ }.} If X {\displaystyle X} and Y {\displaystyle Y} are locally convex then ker t u = ( Im u ) ∘ . {\displaystyle \operatorname {ker} {}^{t}u=\left(\operatorname {Im} u\right)^{\circ }.} === Annihilators === Suppose X {\displaystyle X} and Y {\displaystyle Y} are topological vector spaces and u : X → Y {\displaystyle u:X\to Y} is a weakly continuous linear operator (so ( t u ) ( Y ′ ) ⊆ X ′ {\displaystyle \left({}^{t}u\right)\left(Y^{\prime }\right)\subseteq X^{\prime }} ). Given subsets M ⊆ X {\displaystyle M\subseteq X} and N ⊆ X ′ , {\displaystyle N\subseteq X^{\prime },} define their annihilators (with respect to the canonical dual system) by M ⊥ : = { x ′ ∈ X ′ : ⟨ m , x ′ ⟩ = 0 for all m ∈ M } = { x ′ ∈ X ′ : x ′ ( M ) = { 0 } } where x ′ ( M ) := { x ′ ( m ) : m ∈ M } {\displaystyle {\begin{alignedat}{4}M^{\bot }:&=\left\{x^{\prime }\in X^{\prime }:\left\langle m,x^{\prime }\right\rangle =0{\text{ for all }}m\in M\right\}\\&=\left\{x^{\prime }\in X^{\prime }:x^{\prime }(M)=\{0\}\right\}\qquad {\text{ where }}x^{\prime }(M):=\left\{x^{\prime }(m):m\in M\right\}\end{alignedat}}} and ⊥ N : = { x ∈ X : ⟨ x , n ′ ⟩ = 0 for all n ′ ∈ N } = { x ∈ X : N ( x ) = { 0 } } where N ( x ) := { n ′ ( x ) : n ′ ∈ N } {\displaystyle {\begin{alignedat}{4}{}^{\bot }N:&=\left\{x\in X:\left\langle x,n^{\prime }\right\rangle =0{\text{ for all }}n^{\prime }\in N\right\}\\&=\left\{x\in X:N(x)=\{0\}\right\}\qquad {\text{ where }}N(x):=\left\{n^{\prime }(x):n^{\prime }\in N\right\}\\\end{alignedat}}} The kernel of t u {\displaystyle {}^{t}u} is the subspace of Y ′ {\displaystyle Y^{\prime }} orthogonal to the image of u {\displaystyle u} : ker t u = ( Im u ) ⊥ {\displaystyle \ker {}^{t}u=(\operatorname {Im} u)^{\bot }} The linear map u {\displaystyle u} is injective if and only if its image is a weakly dense subset of Y {\displaystyle Y} (that is, the image of u {\displaystyle u} is dense in Y {\displaystyle Y} when Y {\displaystyle Y} is given the weak topology induced by ker t u {\displaystyle \operatorname {ker} {}^{t}u} ). The transpose t u : Y ′ → X ′ {\displaystyle {}^{t}u:Y^{\prime }\to X^{\prime }} is continuous when both X ′ {\displaystyle X^{\prime }} and Y ′ {\displaystyle Y^{\prime }} are endowed with the weak-* topology (resp. both endowed with the strong dual topology, both endowed with the topology of uniform convergence on compact convex subsets, both endowed with the topology of uniform convergence on compact subsets). (Surjection of Fréchet spaces): If X {\displaystyle X} and Y {\displaystyle Y} are Fréchet spaces then the continuous linear operator u : X → Y {\displaystyle u:X\to Y} is surjective if and only if (1) the transpose t u : Y ′ → X ′ {\displaystyle {}^{t}u:Y^{\prime }\to X^{\prime }} is injective, and (2) the image of the transpose of u {\displaystyle u} is a weakly closed (i.e. weak-* closed) subset of X ′ . {\displaystyle X^{\prime }.} === Duals of quotient spaces === Let M {\displaystyle M} be a closed vector subspace of a Hausdorff locally convex space X {\displaystyle X} and denote the canonical quotient map by π : X → X / M where π ( x ) := x + M . {\displaystyle \pi :X\to X/M\quad {\text{ where }}\quad \pi (x):=x+M.} Assume X / M {\displaystyle X/M} is endowed with the quotient topology induced by the quotient map π : X → X / M . {\displaystyle \pi :X\to X/M.} Then the transpose of the quotient map is valued in M ⊥ {\displaystyle M^{\bot }} and t π : ( X / M ) ′ → M ⊥ ⊆ X ′ {\displaystyle {}^{t}\pi :(X/M)^{\prime }\to M^{\bot }\subseteq X^{\prime }} is a TVS-isomorphism onto M ⊥ . {\displaystyle M^{\bot }.} If X {\displaystyle X} is a Banach space then t π : ( X / M ) ′ → M ⊥ {\displaystyle {}^{t}\pi :(X/M)^{\prime }\to M^{\bot }} is also an isometry. Using this transpose, every continuous linear functional on the quotient space X / M {\displaystyle X/M} is canonically identified with a continuous linear functional in the annihilator M ⊥ {\displaystyle M^{\bot }} of M . {\displaystyle M.} === Duals of vector subspaces === Let M {\displaystyle M} be a closed vector subspace of a Hausdorff locally convex space X . {\displaystyle X.} If m ′ ∈ M ′ {\displaystyle m^{\prime }\in M^{\prime }} and if x ′ ∈ X ′ {\displaystyle x^{\prime }\in X^{\prime }} is a continuous linear extension of m ′ {\displaystyle m^{\prime }} to X {\displaystyle X} then the assignment m ′ ↦ x ′ + M ⊥ {\displaystyle m^{\prime }\mapsto x^{\prime }+M^{\bot }} induces a vector space isomorphism M ′ → X ′ / ( M ⊥ ) , {\displaystyle M^{\prime }\to X^{\prime }/\left(M^{\bot }\right),} which is an isometry if X {\displaystyle X} is a Banach space. Denote the inclusion map by In : M → X where In ( m ) := m for all m ∈ M . {\displaystyle \operatorname {In} :M\to X\quad {\text{ where }}\quad \operatorname {In} (m):=m\quad {\text{ for all }}m\in M.} The transpose of the inclusion map is t In : X ′ → M ′ {\displaystyle {}^{t}\operatorname {In} :X^{\prime }\to M^{\prime }} whose kernel is the annihilator M ⊥ = { x ′ ∈ X ′ : ⟨ m , x ′ ⟩ = 0 for all m ∈ M } {\displaystyle M^{\bot }=\left\{x^{\prime }\in X^{\prime }:\left\langle m,x^{\prime }\right\rangle =0{\text{ for all }}m\in M\right\}} and which is surjective by the Hahn–Banach theorem. This map induces an isomorphism of vector spaces X ′ / ( M ⊥ ) → M ′ . {\displaystyle X^{\prime }/\left(M^{\bot }\right)\to M^{\prime }.} == Representation as a matrix == If the linear map u {\displaystyle u} is represented by the matrix A {\displaystyle A} with respect to two bases of X {\displaystyle X} and Y , {\displaystyle Y,} then t u {\displaystyle {}^{t}u} is represented by the transpose matrix A T {\displaystyle A^{T}} with respect to the dual bases of Y ′ {\displaystyle Y^{\prime }} and X ′ , {\displaystyle X^{\prime },} hence the name. Alternatively, as u {\displaystyle u} is represented by A {\displaystyle A} acting to the right on column vectors, t u {\displaystyle {}^{t}u} is represented by the same matrix acting to the left on row vectors. These points of view are related by the canonical inner product on R n , {\displaystyle \mathbb {R} ^{n},} which identifies the space of column vectors with the dual space of row vectors. == Relation to the Hermitian adjoint == The identity that characterizes the transpose, that is, [ u ∗ ( f ) , x ] = [ f , u ( x ) ] , {\displaystyle \left[u^{*}(f),x\right]=[f,u(x)],} is formally similar to the definition of the Hermitian adjoint, however, the transpose and the Hermitian adjoint are not the same map. The transpose is a map Y ′ → X ′ {\displaystyle Y^{\prime }\to X^{\prime }} and is defined for linear maps between any vector spaces X {\displaystyle X} and Y , {\displaystyle Y,} without requiring any additional structure. The Hermitian adjoint maps Y → X {\displaystyle Y\to X} and is only defined for linear maps between Hilbert spaces, as it is defined in terms of the inner product on the Hilbert space. The Hermitian adjoint therefore requires more mathematical structure than the transpose. However, the transpose is often used in contexts where the vector spaces are both equipped with a nondegenerate bilinear form such as the Euclidean dot product or another real inner product. In this case, the nondegenerate bilinear form is often used implicitly to map between the vector spaces and their duals, to express the transposed map as a map Y → X . {\displaystyle Y\to X.} For a complex Hilbert space, the inner product is sesquilinear and not bilinear, and these conversions change the transpose into the adjoint map. More precisely: if X {\displaystyle X} and Y {\displaystyle Y} are Hilbert spaces and u : X → Y {\displaystyle u:X\to Y} is a linear map then the transpose of u {\displaystyle u} and the Hermitian adjoint of u , {\displaystyle u,} which we will denote respectively by t u {\displaystyle {}^{t}u} and u ∗ , {\displaystyle u^{*},} are related. Denote by I : X → X ∗ {\displaystyle I:X\to X^{*}} and J : Y → Y ∗ {\displaystyle J:Y\to Y^{*}} the canonical antilinear isometries of the Hilbert spaces X {\displaystyle X} and Y {\displaystyle Y} onto their duals. Then u ∗ {\displaystyle u^{*}} is the following composition of maps: Y ⟶ J Y ∗ ⟶ t u X ∗ ⟶ I − 1 X {\displaystyle Y{\overset {J}{\longrightarrow }}Y^{*}{\overset {{}^{\text{t}}u}{\longrightarrow }}X^{*}{\overset {I^{-1}}{\longrightarrow }}X} == Applications to functional analysis == Suppose that X {\displaystyle X} and Y {\displaystyle Y} are topological vector spaces and that u : X → Y {\displaystyle u:X\to Y} is a linear map, then many of u {\displaystyle u} 's properties are reflected in t u . {\displaystyle {}^{t}u.} If A ⊆ X {\displaystyle A\subseteq X} and B ⊆ Y {\displaystyle B\subseteq Y} are weakly closed, convex sets containing the origin, then t u ( B ∘ ) ⊆ A ∘ {\displaystyle {}^{t}u\left(B^{\circ }\right)\subseteq A^{\circ }} implies u ( A ) ⊆ B . {\displaystyle u(A)\subseteq B.} The null space of t u {\displaystyle {}^{t}u} is the subspace of Y ′ {\displaystyle Y^{\prime }} orthogonal to the range u ( X ) {\displaystyle u(X)} of u . {\displaystyle u.} t u {\displaystyle {}^{t}u} is injective if and only if the range u ( X ) {\displaystyle u(X)} of u {\displaystyle u} is weakly closed. == See also == Adjoint functors – Relationship between two functors abstracting many common constructions Composition operator – Linear operator in mathematics Hermitian adjoint – Conjugate transpose of an operator in infinite dimensions Riesz representation theorem – Theorem about the dual of a Hilbert space Dual space § Transpose of a linear map Transpose § Transpose of a linear map == References == == Bibliography == Halmos, Paul (1974), Finite-dimensional Vector Spaces, Springer, ISBN 0-387-90093-4 Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
|
Wikipedia:Transseries#0
|
In mathematics, the field T L E {\displaystyle \mathbb {T} ^{LE}} of logarithmic-exponential transseries is a non-Archimedean ordered differential field which extends comparability of asymptotic growth rates of elementary nontrigonometric functions to a much broader class of objects. Each log-exp transseries represents a formal asymptotic behavior, and it can be manipulated formally, and when it converges (or in every case if using special semantics such as through infinite surreal numbers), corresponds to actual behavior. Transseries can also be convenient for representing functions. Through their inclusion of exponentiation and logarithms, transseries are a strong generalization of the power series at infinity ( ∑ n = 0 ∞ a n x n {\textstyle \sum _{n=0}^{\infty }{\frac {a_{n}}{x^{n}}}} ) and other similar asymptotic expansions. The field T L E {\displaystyle \mathbb {T} ^{LE}} was introduced independently by Dahn-Göring and Ecalle in the respective contexts of model theory or exponential fields and of the study of analytic singularity and proof by Ecalle of the Dulac conjectures. It constitutes a formal object, extending the field of exp-log functions of Hardy and the field of accelerando-summable series of Ecalle. The field T L E {\displaystyle \mathbb {T} ^{LE}} enjoys a rich structure: an ordered field with a notion of generalized series and sums, with a compatible derivation with distinguished antiderivation, compatible exponential and logarithm functions and a notion of formal composition of series. == Examples and counter-examples == Informally speaking, exp-log transseries are well-based (i.e. reverse well-ordered) formal Hahn series of real powers of the positive infinite indeterminate x {\displaystyle x} , exponentials, logarithms and their compositions, with real coefficients. Two important additional conditions are that the exponential and logarithmic depth of an exp-log transseries f , {\displaystyle f,} that is the maximal numbers of iterations of exp and log occurring in f , {\displaystyle f,} must be finite. The following formal series are log-exp transseries: ∑ n = 1 ∞ e x 1 n n ! + x 3 + log x + log log x + ∑ n = 0 ∞ x − n + ∑ i = 1 ∞ e − ∑ j = 1 ∞ e i x 2 − j x . {\displaystyle \sum _{n=1}^{\infty }{\frac {e^{x^{\frac {1}{n}}}}{n!}}+x^{3}+\log x+\log \log x+\sum _{n=0}^{\infty }x^{-n}+\sum _{i=1}^{\infty }e^{-\sum _{j=1}^{\infty }e^{ix^{2}-jx}}.} ∑ m , n ∈ N x 1 m + 1 e − ( log x ) n . {\displaystyle \sum _{m,n\in \mathbb {N} }x^{\frac {1}{m+1}}e^{-(\log x)^{n}}.} The following formal series are not log-exp transseries: ∑ n ∈ N x n {\displaystyle \sum _{n\in \mathbb {N} }x^{n}} — this series is not well-based. log x + log log x + log log log x + ⋯ {\displaystyle \log x+\log \log x+\log \log \log x+\cdots } — the logarithmic depth of this series is infinite 1 2 x + e 1 2 log x + e e 1 2 log log x + ⋯ {\displaystyle {\frac {1}{2}}x+e^{{\frac {1}{2}}\log x}+e^{e^{{\frac {1}{2}}\log \log x}}+\cdots } — the exponential and logarithmic depths of this series are infinite It is possible to define differential fields of transseries containing the two last series; they belong respectively to T E L {\displaystyle \mathbb {T} ^{EL}} and R ⟨ ⟨ ω ⟩ ⟩ {\displaystyle \mathbb {R} \langle \langle \omega \rangle \rangle } (see the paragraph Using surreal numbers below). == Introduction == A remarkable fact is that asymptotic growth rates of elementary nontrigonometric functions and even all functions definable in the model theoretic structure ( R , + , × , < , exp ) {\displaystyle (\mathbb {R} ,+,\times ,<,\exp )} of the ordered exponential field of real numbers are all comparable: For all such f {\displaystyle f} and g {\displaystyle g} , we have f ≤ ∞ g {\displaystyle f\leq _{\infty }g} or g ≤ ∞ f {\displaystyle g\leq _{\infty }f} , where f ≤ ∞ g {\displaystyle f\leq _{\infty }g} means ∃ x . ∀ y > x . f ( y ) ≤ g ( y ) {\displaystyle \exists x.\forall y>x.f(y)\leq g(y)} . The equivalence class of f {\displaystyle f} under the relation f ≤ ∞ g ∧ g ≤ ∞ f {\displaystyle f\leq _{\infty }g\wedge g\leq _{\infty }f} is the asymptotic behavior of f {\displaystyle f} , also called the germ of f {\displaystyle f} (or the germ of f {\displaystyle f} at infinity). The field of transseries can be intuitively viewed as a formal generalization of these growth rates: In addition to the elementary operations, transseries are closed under "limits" for appropriate sequences with bounded exponential and logarithmic depth. However, a complication is that growth rates are non-Archimedean and hence do not have the least upper bound property. We can address this by associating a sequence with the least upper bound of minimal complexity, analogously to construction of surreal numbers. For example, ( ∑ k = 0 n x − k ) n ∈ N {\textstyle (\sum _{k=0}^{n}x^{-k})_{n\in \mathbb {N} }} is associated with ∑ k = 0 ∞ x − k {\textstyle \sum _{k=0}^{\infty }x^{-k}} rather than ∑ k = 0 ∞ x − k − e − x {\textstyle \sum _{k=0}^{\infty }x^{-k}-e^{-x}} because e − x {\displaystyle e^{-x}} decays too quickly, and if we identify fast decay with complexity, it has greater complexity than necessary (also, because we care only about asymptotic behavior, pointwise convergence is not dispositive). Because of the comparability, transseries do not include oscillatory growth rates (such as sin x {\displaystyle \sin x} ). On the other hand, there are transseries such as ∑ k ∈ N k ! e x − k k + 1 {\textstyle \sum _{k\in \mathbb {N} }k!e^{x^{-{\frac {k}{k+1}}}}} that do not directly correspond to convergent series or real valued functions. Another limitation of transseries is that each of them is bounded by a tower of exponentials, i.e. a finite iteration e e . . . e x {\displaystyle e^{e^{.^{.^{.^{e^{x}}}}}}} of e x {\displaystyle e^{x}} , thereby excluding tetration and other transexponential functions, i.e. functions which grow faster than any tower of exponentials. There are ways to construct fields of generalized transseries including formal transexponential terms, for instance formal solutions e ω {\displaystyle e_{\omega }} of the Abel equation e e ω ( x ) = e ω ( x + 1 ) {\displaystyle e^{e_{\omega }(x)}=e_{\omega }(x+1)} . == Formal construction == Transseries can be defined as formal (potentially infinite) expressions, with rules defining which expressions are valid, comparison of transseries, arithmetic operations, and even differentiation. Appropriate transseries can then be assigned to corresponding functions or germs, but there are subtleties involving convergence. Even transseries that diverge can often be meaningfully (and uniquely) assigned actual growth rates (that agree with the formal operations on transseries) using accelero-summation, which is a generalization of Borel summation. Transseries can be formalized in several equivalent ways; we use one of the simplest ones here. A transseries is a well-based sum, ∑ a i m i , {\displaystyle \sum a_{i}m_{i},} with finite exponential depth, where each a i {\displaystyle a_{i}} is a nonzero real number and m i {\displaystyle m_{i}} is a monic transmonomial ( a i m i {\displaystyle a_{i}m_{i}} is a transmonomial but is not monic unless the coefficient a i = 1 {\displaystyle a_{i}=1} ; each m i {\displaystyle m_{i}} is different; the order of the summands is irrelevant). The sum might be infinite or transfinite; it is usually written in the order of decreasing m i {\displaystyle m_{i}} . Here, well-based means that there is no infinite ascending sequence m i 1 < m i 2 < m i 3 < ⋯ {\displaystyle m_{i_{1}}<m_{i_{2}}<m_{i_{3}}<\cdots } (see well-ordering). A monic transmonomial is one of 1, x, log x, log log x, ..., epurely_large_transseries. Note: Because x n = e n log x {\displaystyle x^{n}=e^{n\log x}} , we do not include it as a primitive, but many authors do; log-free transseries do not include log {\displaystyle \log } but x n e ⋯ {\displaystyle x^{n}e^{\cdots }} is permitted. Also, circularity in the definition is avoided because the purely_large_transseries (above) will have lower exponential depth; the definition works by recursion on the exponential depth. See "Log-exp transseries as iterated Hahn series" (below) for a construction that uses x a e ⋯ {\displaystyle x^{a}e^{\cdots }} and explicitly separates different stages. A purely large transseries is a nonempty transseries ∑ a i m i {\textstyle \sum a_{i}m_{i}} with every m i > 1 {\displaystyle m_{i}>1} . Transseries have finite exponential depth, where each level of nesting of e or log increases depth by 1 (so we cannot have x + log x + log log x + ...). Addition of transseries is termwise: ∑ a i m i + ∑ b i m i = ∑ ( a i + b i ) m i {\textstyle \sum a_{i}m_{i}+\sum b_{i}m_{i}=\sum (a_{i}+b_{i})m_{i}} (absence of a term is equated with a zero coefficient). Comparison: The most significant term of ∑ a i m i {\textstyle \sum a_{i}m_{i}} is a i m i {\displaystyle a_{i}m_{i}} for the largest m i {\displaystyle m_{i}} (because the sum is well-based, this exists for nonzero transseries). ∑ a i m i {\textstyle \sum a_{i}m_{i}} is positive iff the coefficient of the most significant term is positive (this is why we used 'purely large' above). X > Y iff X − Y is positive. Comparison of monic transmonomials: x = e log x , log x = e log log x , … {\displaystyle x=e^{\log x},\log x=e^{\log \log x},\ldots } – these are the only equalities in our construction. x > log x > log log x > ⋯ > 1 > 0. {\displaystyle x>\log x>\log \log x>\cdots >1>0.} e a < e b {\displaystyle e^{a}<e^{b}} iff a < b {\displaystyle a<b} (also e 0 = 1 {\displaystyle e^{0}=1} ). Multiplication: e a e b = e a + b {\displaystyle e^{a}e^{b}=e^{a+b}} ( ∑ a i x i ) ( ∑ b j y j ) = ∑ k ( ∑ i , j : z k = x i y j a i b j ) z k . {\displaystyle \left(\sum a_{i}x_{i}\right)\left(\sum b_{j}y_{j}\right)=\sum _{k}\left(\sum _{i,j\,:\,z_{k}=x_{i}y_{j}}a_{i}b_{j}\right)z_{k}.} This essentially applies the distributive law to the product; because the series is well-based, the inner sum is always finite. Differentiation: ( ∑ a i x i ) ′ = ∑ a i x i ′ {\displaystyle \left(\sum a_{i}x_{i}\right)'=\sum a_{i}x_{i}'} 1 ′ = 0 , x ′ = 1 {\displaystyle 1'=0,x'=1} ( e y ) ′ = y ′ e y {\displaystyle (e^{y})'=y'e^{y}} ( log y ) ′ = y ′ / y {\displaystyle (\log y)'=y'/y} (division is defined using multiplication). With these definitions, transseries is an ordered differential field. Transseries is also a valued field, with the valuation ν {\displaystyle \nu } given by the leading monic transmonomial, and the corresponding asymptotic relation defined for 0 ≠ f , g ∈ T L E {\displaystyle 0\neq f,g\in \mathbb {T} ^{LE}} by f ≺ g {\displaystyle f\prec g} if ∀ 0 < r ∈ R , | f | < r | g | {\displaystyle \forall 0<r\in \mathbb {R} ,|f|<r|g|} (where | f | = max ( f , − f ) {\displaystyle |f|=\max(f,-f)} is the absolute value). == Other constructions == === Log-exp transseries as iterated Hahn series === ==== Log-free transseries ==== We first define the subfield T E {\displaystyle \mathbb {T} ^{E}} of T L E {\displaystyle \mathbb {T} ^{LE}} of so-called log-free transseries. Those are transseries which exclude any logarithmic term. Inductive definition: For n ∈ N , {\displaystyle n\in \mathbb {N} ,} we will define a linearly ordered multiplicative group of monomials M n {\displaystyle {\mathfrak {M}}_{n}} . We then let T n E {\displaystyle \mathbb {T} _{n}^{E}} denote the field of well-based series R [ [ M n ] ] {\displaystyle \mathbb {R} [[{\mathfrak {M}}_{n}]]} . This is the set of maps R → M n {\displaystyle \mathbb {R} \to {\mathfrak {M}}_{n}} with well-based (i.e. reverse well-ordered) support, equipped with pointwise sum and Cauchy product (see Hahn series). In T n E {\displaystyle \mathbb {T} _{n}^{E}} , we distinguish the (non-unital) subring T n , ≻ E {\displaystyle \mathbb {T} _{n,\succ }^{E}} of purely large transseries, which are series whose support contains only monomials lying strictly above 1 {\displaystyle 1} . We start with M 0 = x R {\displaystyle {\mathfrak {M}}_{0}=x^{\mathbb {R} }} equipped with the product x a x b := x a + b {\displaystyle x^{a}x^{b}:=x^{a+b}} and the order x a ≺ x b ↔ a < b {\displaystyle x^{a}\prec x^{b}\leftrightarrow a<b} . If n ∈ N {\displaystyle n\in \mathbb {N} } is such that M n {\displaystyle {\mathfrak {M}}_{n}} , and thus T n E {\displaystyle \mathbb {T} _{n}^{E}} and T n , ≻ E {\displaystyle \mathbb {T} _{n,\succ }^{E}} are defined, we let M n + 1 {\displaystyle {\mathfrak {M}}_{n+1}} denote the set of formal expressions x a e θ {\displaystyle x^{a}e^{\theta }} where a ∈ R {\displaystyle a\in \mathbb {R} } and θ ∈ T n , ≻ E {\displaystyle \theta \in \mathbb {T} _{n,\succ }^{E}} . This forms a linearly ordered commutative group under the product ( x a e θ ) ( x a ′ e θ ′ ) = ( x a + a ′ ) e θ + θ ′ {\displaystyle (x^{a}e^{\theta })(x^{a'}e^{\theta '})=(x^{a+a'})e^{\theta +\theta '}} and the lexicographic order x a e θ ≺ x a ′ e θ ′ {\displaystyle x^{a}e^{\theta }\prec x^{a'}e^{\theta '}} if and only if θ < θ ′ {\displaystyle \theta <\theta '} or ( θ = θ ′ {\displaystyle \theta =\theta '} and a < a ′ {\displaystyle a<a'} ). The natural inclusion of M 0 {\displaystyle {\mathfrak {M}}_{0}} into M 1 {\displaystyle {\mathfrak {M}}_{1}} given by identifying x a {\displaystyle x^{a}} and x a e 0 {\displaystyle x^{a}e^{0}} inductively provides a natural embedding of M n {\displaystyle {\mathfrak {M}}_{n}} into M n + 1 {\displaystyle {\mathfrak {M}}_{n+1}} , and thus a natural embedding of T n E {\displaystyle \mathbb {T} _{n}^{E}} into T n + 1 E {\displaystyle \mathbb {T} _{n+1}^{E}} . We may then define the linearly ordered commutative group M = ⋃ n ∈ N M n {\textstyle {\mathfrak {M}}=\bigcup _{n\in \mathbb {N} }{\mathfrak {M}}_{n}} and the ordered field T E = ⋃ n ∈ N T n E {\textstyle \mathbb {T} ^{E}=\bigcup _{n\in \mathbb {N} }\mathbb {T} _{n}^{E}} which is the field of log-free transseries. The field T E {\displaystyle \mathbb {T} ^{E}} is a proper subfield of the field R [ [ M ] ] {\displaystyle \mathbb {R} [[{\mathfrak {M}}]]} of well-based series with real coefficients and monomials in M {\displaystyle {\mathfrak {M}}} . Indeed, every series f {\displaystyle f} in T E {\displaystyle \mathbb {T} ^{E}} has a bounded exponential depth, i.e. the least positive integer n {\displaystyle n} such that f ∈ T n E {\displaystyle f\in \mathbb {T} _{n}^{E}} , whereas the series e − x + e − e x + e − e e x + ⋯ ∈ R [ [ M ] ] {\displaystyle e^{-x}+e^{-e^{x}}+e^{-e^{e^{x}}}+\cdots \in \mathbb {R} [[{\mathfrak {M}}]]} has no such bound. Exponentiation on T E {\displaystyle \mathbb {T} ^{E}} : The field of log-free transseries is equipped with an exponential function which is a specific morphism exp : ( T E , + ) → ( T E , > , × ) {\displaystyle \exp :(\mathbb {T} ^{E},+)\to (\mathbb {T} ^{E,>},\times )} . Let f {\displaystyle f} be a log-free transseries and let n ∈ N {\displaystyle n\in \mathbb {N} } be the exponential depth of f {\displaystyle f} , so f ∈ T n E {\displaystyle f\in \mathbb {T} _{n}^{E}} . Write f {\displaystyle f} as the sum f = θ + r + ε {\displaystyle f=\theta +r+\varepsilon } in T n E , {\displaystyle \mathbb {T} _{n}^{E},} where θ ∈ T n , ≻ E {\displaystyle \theta \in \mathbb {T} _{n,\succ }^{E}} , r {\displaystyle r} is a real number and ε {\displaystyle \varepsilon } is infinitesimal (any of them could be zero). Then the formal Hahn sum E ( ε ) := ∑ k ∈ N ε k k ! {\displaystyle E(\varepsilon ):=\sum _{k\in \mathbb {N} }{\frac {\varepsilon ^{k}}{k!}}} converges in T n E {\displaystyle \mathbb {T} _{n}^{E}} , and we define exp ( f ) = e θ exp ( r ) E ( ε ) ∈ T n + 1 E {\displaystyle \exp(f)=e^{\theta }\exp(r)E(\varepsilon )\in \mathbb {T} _{n+1}^{E}} where exp ( r ) {\displaystyle \exp(r)} is the value of the real exponential function at r {\displaystyle r} . Right-composition with e x {\displaystyle e^{x}} : A right composition ∘ e x {\displaystyle \circ _{e^{x}}} with the series e x {\displaystyle e^{x}} can be defined by induction on the exponential depth by ( ∑ f m m ) ∘ e x := ∑ f m ( m ∘ e x ) , {\displaystyle \left(\sum f_{\mathfrak {m}}{\mathfrak {m}}\right)\circ e^{x}:=\sum f_{\mathfrak {m}}({\mathfrak {m}}\circ e^{x}),} with x r ∘ e x := e r x {\displaystyle x^{r}\circ e^{x}:=e^{rx}} . It follows inductively that monomials are preserved by ∘ e x , {\displaystyle \circ _{e^{x}},} so at each inductive step the sums are well-based and thus well defined. ==== Log-exp transseries ==== Definition: The function exp {\displaystyle \exp } defined above is not onto T E , > {\displaystyle \mathbb {T} ^{E,>}} so the logarithm is only partially defined on T E {\displaystyle \mathbb {T} ^{E}} : for instance the series x {\displaystyle x} has no logarithm. Moreover, every positive infinite log-free transseries is greater than some positive power of x {\displaystyle x} . In order to move from T E {\displaystyle \mathbb {T} ^{E}} to T L E {\displaystyle \mathbb {T} ^{LE}} , one can simply "plug" into the variable x {\displaystyle x} of series formal iterated logarithms ℓ n , n ∈ N {\displaystyle \ell _{n},n\in \mathbb {N} } which will behave like the formal reciprocal of the n {\displaystyle n} -fold iterated exponential term denoted e n {\displaystyle e_{n}} . For m , n ∈ N , {\displaystyle m,n\in \mathbb {N} ,} let M m , n {\displaystyle {\mathfrak {M}}_{m,n}} denote the set of formal expressions u ∘ ℓ n {\displaystyle {\mathfrak {u}}\circ \ell _{n}} where u ∈ M m {\displaystyle {\mathfrak {u}}\in {\mathfrak {M}}_{m}} . We turn this into an ordered group by defining ( u ∘ ℓ n ) ( v ∘ ℓ n ( x ) ) := ( u v ) ∘ ℓ n {\displaystyle ({\mathfrak {u}}\circ \ell _{n})({\mathfrak {v}}\circ \ell _{n}(x)):=({\mathfrak {u}}{\mathfrak {v}})\circ \ell _{n}} , and defining u ∘ ℓ n ≺ v ∘ ℓ n {\displaystyle {\mathfrak {u}}\circ \ell _{n}\prec {\mathfrak {v}}\circ \ell _{n}} when u ≺ v {\displaystyle {\mathfrak {u}}\prec {\mathfrak {v}}} . We define T m , n L E := R [ [ M m , n ] ] {\displaystyle \mathbb {T} _{m,n}^{LE}:=\mathbb {R} [[{\mathfrak {M}}_{m,n}]]} . If n ′ > n {\displaystyle n'>n} and m ′ ≥ m + ( n ′ − n ) , {\displaystyle m'\geq m+(n'-n),} we embed M m , n {\displaystyle {\mathfrak {M}}_{m,n}} into M m ′ , n ′ {\displaystyle {\mathfrak {M}}_{m',n'}} by identifying an element u ∘ ℓ n {\displaystyle {\mathfrak {u}}\circ \ell _{n}} with the term ( u ∘ e x ∘ ⋯ ∘ e x ⏞ n ′ − n ) ∘ ℓ n ′ . {\displaystyle \left({\mathfrak {u}}\circ \overbrace {e^{x}\circ \cdots \circ e^{x}} ^{n'-n}\right)\circ \ell _{n'}.} We then obtain T L E {\displaystyle \mathbb {T} ^{LE}} as the directed union T L E = ⋃ m , n ∈ N T m , n L E . {\displaystyle \mathbb {T} ^{LE}=\bigcup _{m,n\in \mathbb {N} }\mathbb {T} _{m,n}^{LE}.} On T L E , {\displaystyle \mathbb {T} ^{LE},} the right-composition ∘ ℓ {\displaystyle \circ _{\ell }} with ℓ {\displaystyle \ell } is naturally defined by T m , n L E ∋ ( ∑ f m ∘ ℓ n m ∘ ℓ n ) ∘ ℓ := ∑ f m ∘ ℓ n m ∘ ℓ n + 1 ∈ T m , n + 1 L E . {\displaystyle \mathbb {T} _{m,n}^{LE}\ni \left(\sum f_{{\mathfrak {m}}\circ \ell _{n}}{\mathfrak {m}}\circ \ell _{n}\right)\circ \ell :=\sum f_{{\mathfrak {m}}\circ \ell _{n}}{\mathfrak {m}}\circ \ell _{n+1}\in \mathbb {T} _{m,n+1}^{LE}.} Exponential and logarithm: Exponentiation can be defined on T L E {\displaystyle \mathbb {T} ^{LE}} in a similar way as for log-free transseries, but here also exp {\displaystyle \exp } has a reciprocal log {\displaystyle \log } on T L E , > {\displaystyle \mathbb {T} ^{LE,>}} . Indeed, for a strictly positive series f ∈ T m , n L E , > {\displaystyle f\in \mathbb {T} _{m,n}^{LE,>}} , write f = m r ( 1 + ε ) {\displaystyle f={\mathfrak {m}}r(1+\varepsilon )} where m {\displaystyle {\mathfrak {m}}} is the dominant monomial of f {\displaystyle f} (largest element of its support), r {\displaystyle r} is the corresponding positive real coefficient, and ε := f m r − 1 {\displaystyle \varepsilon :={\frac {f}{{\mathfrak {m}}r}}-1} is infinitesimal. The formal Hahn sum L ( 1 + ε ) := ∑ k ∈ N ( − ε ) k k + 1 {\displaystyle L(1+\varepsilon ):=\sum _{k\in \mathbb {N} }{\frac {(-\varepsilon )^{k}}{k+1}}} converges in T m , n L E {\displaystyle \mathbb {T} _{m,n}^{LE}} . Write m = u ∘ ℓ n {\displaystyle {\mathfrak {m}}={\mathfrak {u}}\circ \ell _{n}} where u ∈ M m {\displaystyle {\mathfrak {u}}\in {\mathfrak {M}}_{m}} itself has the form u = x a e θ {\displaystyle {\mathfrak {u}}=x^{a}e^{\theta }} where θ ∈ T m , ≻ E {\displaystyle \theta \in \mathbb {T} _{m,\succ }^{E}} and a ∈ R {\displaystyle a\in \mathbb {R} } . We define ℓ ( m ) := a ℓ n + 1 + θ ∘ ℓ n {\displaystyle \ell ({\mathfrak {m}}):=a\ell _{n+1}+\theta \circ \ell _{n}} . We finally set log ( f ) := ℓ ( m ) + log ( c ) + L ( 1 + ε ) ∈ T m , n + 1 L E . {\displaystyle \log(f):=\ell ({\mathfrak {m}})+\log(c)+L(1+\varepsilon )\in \mathbb {T} _{m,n+1}^{LE}.} === Using surreal numbers === ==== Direct construction of log-exp transseries ==== One may also define the field of log-exp transseries as a subfield of the ordered field N o {\displaystyle \mathbf {No} } of surreal numbers. The field N o {\displaystyle \mathbf {No} } is equipped with Gonshor-Kruskal's exponential and logarithm functions and with its natural structure of field of well-based series under Conway normal form. Define F 0 L E = R ( ω ) {\displaystyle F_{0}^{LE}=\mathbb {R} (\omega )} , the subfield of N o {\displaystyle \mathbf {No} } generated by R {\displaystyle \mathbb {R} } and the simplest positive infinite surreal number ω {\displaystyle \omega } (which corresponds naturally to the ordinal ω {\displaystyle \omega } , and as a transseries to the series x {\displaystyle x} ). Then, for n ∈ N {\displaystyle n\in \mathbb {N} } , define F n + 1 L E {\displaystyle F_{n+1}^{LE}} as the field generated by F n L E {\displaystyle F_{n}^{LE}} , exponentials of elements of F n L E {\displaystyle F_{n}^{LE}} and logarithms of strictly positive elements of F n L E {\displaystyle F_{n}^{LE}} , as well as (Hahn) sums of summable families in F n L E {\displaystyle F_{n}^{LE}} . The union F ω L E = ⋃ n ∈ N F n L E {\textstyle F_{\omega }^{LE}=\bigcup _{n\in \mathbb {N} }F_{n}^{LE}} is naturally isomorphic to T L E {\displaystyle \mathbb {T} ^{LE}} . In fact, there is a unique such isomorphism which sends ω {\displaystyle \omega } to x {\displaystyle x} and commutes with exponentiation and sums of summable families in F ω L E {\displaystyle F_{\omega }^{LE}} lying in F ω {\displaystyle F_{\omega }} . ==== Other fields of transseries ==== Continuing this process by transfinite induction on O r d {\displaystyle \mathbf {Ord} } beyond F ω L E {\displaystyle F_{\omega }^{LE}} , taking unions at limit ordinals, one obtains a proper class-sized field R ⟨ ⟨ ω ⟩ ⟩ {\displaystyle \mathbb {R} \langle \langle \omega \rangle \rangle } canonically equipped with a derivation and a composition extending that of T L E {\displaystyle \mathbb {T} ^{LE}} (see Operations on transseries below). If instead of F 0 L E {\displaystyle F_{0}^{LE}} one starts with the subfield F 0 E L := R ( ω , log ω , log log ω , … ) {\displaystyle F_{0}^{EL}:=\mathbb {R} (\omega ,\log \omega ,\log \log \omega ,\ldots )} generated by R {\displaystyle \mathbb {R} } and all finite iterates of log {\displaystyle \log } at ω {\displaystyle \omega } , and for n ∈ N , F n + 1 E L {\displaystyle n\in \mathbb {N} ,F_{n+1}^{EL}} is the subfield generated by F n E L {\displaystyle F_{n}^{EL}} , exponentials of elements of F n E L {\displaystyle F_{n}^{EL}} and sums of summable families in F n E L {\displaystyle F_{n}^{EL}} , then one obtains an isomorphic copy the field T E L {\displaystyle \mathbb {T} ^{EL}} of exponential-logarithmic transseries, which is a proper extension of T L E {\displaystyle \mathbb {T} ^{LE}} equipped with a total exponential function. The Berarducci-Mantova derivation on N o {\displaystyle \mathbf {No} } coincides on T L E {\displaystyle \mathbb {T} ^{LE}} with its natural derivation, and is unique to satisfy compatibility relations with the exponential ordered field structure and generalized series field structure of T E L {\displaystyle \mathbb {T} ^{EL}} and R ⟨ ⟨ ω ⟩ ⟩ . {\displaystyle \mathbb {R} \langle \langle \omega \rangle \rangle .} Contrary to T L E , {\displaystyle \mathbb {T} ^{LE},} the derivation in T E L {\displaystyle \mathbb {T} ^{EL}} and R ⟨ ⟨ ω ⟩ ⟩ {\displaystyle \mathbb {R} \langle \langle \omega \rangle \rangle } is not surjective: for instance the series 1 ω log ω log log ω ⋯ := exp ( − ( log ω + log log ω + log log log ω + ⋯ ) ) ∈ T E L {\displaystyle {\frac {1}{\omega \log \omega \log \log \omega \cdots }}:=\exp(-(\log \omega +\log \log \omega +\log \log \log \omega +\cdots ))\in \mathbb {T} ^{EL}} doesn't have an antiderivative in T E L {\displaystyle \mathbb {T} ^{EL}} or R ⟨ ⟨ ω ⟩ ⟩ {\displaystyle \mathbb {R} \langle \langle \omega \rangle \rangle } (this is linked to the fact that those fields contain no transexponential function). == Additional properties == === Operations on transseries === ==== Operations on the differential exponential ordered field ==== Transseries have very strong closure properties, and many operations can be defined on transseries: Log-exp transseries form an exponentially closed ordered field: the exponential and logarithmic functions are total. For example: exp ( x − 1 ) = ∑ n = 0 ∞ 1 n ! x − n and log ( x + ℓ ) = ℓ + ∑ n = 0 ∞ ( x − 1 ℓ ) n n + 1 . {\displaystyle \exp(x^{-1})=\sum _{n=0}^{\infty }{\frac {1}{n!}}x^{-n}\quad {\text{and}}\quad \log(x+\ell )=\ell +\sum _{n=0}^{\infty }{\frac {(x^{-1}\ell )^{n}}{n+1}}.} Logarithm is defined for positive arguments. Log-exp transseries are real-closed. Integration: every log-exp transseries f {\displaystyle f} has a unique antiderivative with zero constant term F ∈ T L E {\displaystyle F\in \mathbb {T} ^{LE}} , F ′ = f {\displaystyle F'=f} and F 1 = 0 {\displaystyle F_{1}=0} . Logarithmic antiderivative: for f ∈ T L E {\displaystyle f\in \mathbb {T} ^{LE}} , there is h ∈ T L E {\displaystyle h\in \mathbb {T} ^{LE}} with f ′ = f h ′ {\displaystyle f'=fh'} . Note 1. The last two properties mean that T L E {\displaystyle \mathbb {T} ^{LE}} is Liouville closed. Note 2. Just like an elementary nontrigonometric function, each positive infinite transseries f {\displaystyle f} has integral exponentiality, even in this strong sense: ∃ k , n ∈ N : ℓ n − k − 1 ≤ ℓ n ∘ f ≤ ℓ n − k + 1. {\displaystyle \exists k,n\in \mathbb {N} :\quad \ell _{n-k}-1\leq \ell _{n}\circ f\leq \ell _{n-k}+1.} The number k {\displaystyle k} is unique, it is called the exponentiality of f {\displaystyle f} . ==== Composition of transseries ==== An original property of T L E {\displaystyle \mathbb {T} ^{LE}} is that it admits a composition ∘ : T L E × T L E , > , ≻ → T L E {\displaystyle \circ :\mathbb {T} ^{LE}\times \mathbb {T} ^{LE,>,\succ }\to \mathbb {T} ^{LE}} (where T L E , > , ≻ {\displaystyle \mathbb {T} ^{LE,>,\succ }} is the set of positive infinite log-exp transseries) which enables us to see each log-exp transseries f {\displaystyle f} as a function on T L E , > , ≻ {\displaystyle \mathbb {T} ^{LE,>,\succ }} . Informally speaking, for g ∈ T L E , > , ≻ {\displaystyle g\in \mathbb {T} ^{LE,>,\succ }} and f ∈ T L E {\displaystyle f\in \mathbb {T} ^{LE}} , the series f ∘ g {\displaystyle f\circ g} is obtained by replacing each occurrence of the variable x {\displaystyle x} in f {\displaystyle f} by g {\displaystyle g} . ===== Properties ===== Associativity: for f ∈ T L E {\displaystyle f\in \mathbb {T} ^{LE}} and g , h ∈ T L E , > , ≻ {\displaystyle g,h\in \mathbb {T} ^{LE,>,\succ }} , we have g ∘ h ∈ T L E , > , ≻ {\displaystyle g\circ h\in \mathbb {T} ^{LE,>,\succ }} and f ∘ ( g ∘ h ) = ( f ∘ g ) ∘ h {\displaystyle f\circ (g\circ h)=(f\circ g)\circ h} . Compatibility of right-compositions: For g ∈ T L E , > , ≻ {\displaystyle g\in \mathbb {T} ^{LE,>,\succ }} , the function ∘ g : f ↦ f ∘ g {\displaystyle \circ _{g}:f\mapsto f\circ g} is a field automorphism of T L E {\displaystyle \mathbb {T} ^{LE}} which commutes with formal sums, sends x {\displaystyle x} onto g {\displaystyle g} , e x {\displaystyle e^{x}} onto exp ( g ) {\displaystyle \exp(g)} and ℓ {\displaystyle \ell } onto log ( g ) {\displaystyle \log(g)} . We also have ∘ x = id T L E {\displaystyle \circ _{x}=\operatorname {id} _{\mathbb {T} ^{LE}}} . Unicity: the composition is unique to satisfy the two previous properties. Monotonicity: for f ∈ T L E {\displaystyle f\in \mathbb {T} ^{LE}} , the function g ↦ f ∘ g {\displaystyle g\mapsto f\circ g} is constant or strictly monotonous on T L E , > , ≻ {\displaystyle \mathbb {T} ^{LE,>,\succ }} . The monotony depends on the sign of f ′ {\displaystyle f'} . Chain rule: for f ∈ T L E × {\displaystyle f\in \mathbb {T} ^{LE}\times } and g ∈ T L E , > , ≻ {\displaystyle g\in \mathbb {T} ^{LE,>,\succ }} , we have ( f ∘ g ) ′ = g ′ f ′ ∘ g {\displaystyle (f\circ g)'=g'f'\circ g} . Functional inverse: for g ∈ T L E , > , ≻ {\displaystyle g\in \mathbb {T} ^{LE,>,\succ }} , there is a unique series h ∈ T L E , > , ≻ {\displaystyle h\in \mathbb {T} ^{LE,>,\succ }} with g ∘ h = h ∘ g = x {\displaystyle g\circ h=h\circ g=x} . Taylor expansions: each log-exp transseries f {\displaystyle f} has a Taylor expansion around every point in the sense that for every g ∈ T L E , > , ≻ {\displaystyle g\in \mathbb {T} ^{LE,>,\succ }} and for sufficiently small ε ∈ T L E {\displaystyle \varepsilon \in \mathbb {T} ^{LE}} , we have f ∘ ( g + ε ) = ∑ k ∈ N f ( k ) ∘ g k ! ε k {\displaystyle f\circ (g+\varepsilon )=\sum _{k\in \mathbb {N} }{\frac {f^{(k)}\circ g}{k!}}\varepsilon ^{k}} where the sum is a formal Hahn sum of a summable family. Fractional iteration: for f ∈ T L E , > , ≻ {\displaystyle f\in \mathbb {T} ^{LE,>,\succ }} with exponentiality 0 {\displaystyle 0} and any real number a {\displaystyle a} , the fractional iterate f a {\displaystyle f^{a}} of f {\displaystyle f} is defined. === Decidability and model theory === ==== Theory of differential ordered valued differential field ==== The ⟨ + , × , ∂ , < , ≺ ⟩ {\displaystyle \left\langle +,\times ,\partial ,<,\prec \right\rangle } theory of T L E {\displaystyle \mathbb {T} ^{LE}} is decidable and can be axiomatized as follows (this is Theorem 2.2 of Aschenbrenner et al.): T L E {\displaystyle \mathbb {T} ^{LE}} is an ordered valued differential field. f > 0 ∧ f ≻ 1 ⟹ f ′ > 0 {\displaystyle f>0\wedge f\succ 1\Longrightarrow f'>0} f ≺ 1 ⟹ f ′ ≺ 1 {\displaystyle f\prec 1\Longrightarrow f'\prec 1} ∀ f ∃ g : g ′ = f {\displaystyle \forall f\exists g:\quad g'=f} ∀ f ∃ h : h ′ = f h {\displaystyle \forall f\exists h:\quad h'=fh} Intermediate value property (IVP): P ( f ) < 0 ∧ P ( g ) > 0 ⟹ ∃ h : P ( h ) = 0 , {\displaystyle P(f)<0\wedge P(g)>0\Longrightarrow \exists h:\quad P(h)=0,} where P is a differential polynomial, i.e. a polynomial in f , f ′ , f ″ , … , f ( k ) . {\displaystyle f,f',f'',\ldots ,f^{(k)}.} In this theory, exponentiation is essentially defined for functions (using differentiation) but not constants; in fact, every definable subset of R n {\displaystyle \mathbb {R} ^{n}} is semialgebraic. ==== Theory of ordered exponential field ==== The ⟨ + , × , exp , < ⟩ {\displaystyle \langle +,\times ,\exp ,<\rangle } theory of T L E {\displaystyle \mathbb {T} ^{LE}} is that of the exponential real ordered exponential field ( R , + , × , exp , < ) {\displaystyle (\mathbb {R} ,+,\times ,\exp ,<)} , which is model complete by Wilkie's theorem. === Hardy fields === T a s {\displaystyle \mathbb {T} _{\mathrm {as} }} is the field of accelero-summable transseries, and using accelero-summation, we have the corresponding Hardy field, which is conjectured to be the maximal Hardy field corresponding to a subfield of T {\displaystyle \mathbb {T} } . (This conjecture is informal since we have not defined which isomorphisms of Hardy fields into differential subfields of T {\displaystyle \mathbb {T} } are permitted.) T a s {\displaystyle \mathbb {T} _{\mathrm {as} }} is conjectured to satisfy the above axioms of T {\displaystyle \mathbb {T} } . Without defining accelero-summation, we note that when operations on convergent transseries produce a divergent one while the same operations on the corresponding germs produce a valid germ, we can then associate the divergent transseries with that germ. A Hardy field is said maximal if it is properly contained in no Hardy field. By an application of Zorn's lemma, every Hardy field is contained in a maximal Hardy field. It is conjectured that all maximal Hardy fields are elementary equivalent as differential fields, and indeed have the same first order theory as T L E {\displaystyle \mathbb {T} ^{LE}} . Logarithmic-transseries do not themselves correspond to a maximal Hardy field for not every transseries corresponds to a real function, and maximal Hardy fields always contain transexponential functions. == See also == Formal power series Hahn series Exponentially closed field Hardy field == References == Edgar, G. A. (2010), "Transseries for beginners", Real Analysis Exchange, 35 (2): 253–310, arXiv:0801.4877, doi:10.14321/realanalexch.35.2.0253, S2CID 14290638. Aschenbrenner, Matthias; Dries, Lou van den; Hoeven, Joris van der (2017), On Numbers, Germs, and Transseries, arXiv:1711.06936, Bibcode:2017arXiv171106936A.
|
Wikipedia:Treatise on Analysis#0
|
Treatise on Analysis is a translation by Ian G. Macdonald of the nine-volume work Éléments d'analyse on mathematical analysis by Jean Dieudonné, and is an expansion of his textbook Foundations of Modern Analysis. It is a successor to the various Cours d'Analyse by Augustin-Louis Cauchy, Camille Jordan, and Édouard Goursat. == Contents and publication history == === Volume I === The first volume was originally a stand-alone graduate textbook with a different title. It was first written in English and later translated into French, unlike the other volumes which were first written in French. It has been republished several times and is much more common than the later volumes of the series. The contents include Chapter I: Sets Chapter II Real numbers Chapter III Metric spaces Chapter IV The real line Chapter V Normed spaces Chapter VI Hilbert spaces Chapter VII Spaces of continuous functions Chapter VIII Differential calculus (This uses the Cauchy integral rather than the more common Riemann integral of functions.) Chapter IX Analytic functions (of a complex variable) Chapter X Existence theorems (for ordinary differential equations) Chapter XI Elementary spectral theory Dieudonné, J. (1960), Foundations of modern analysis, Pure and Applied Mathematics, vol. X, New York-London: Academic Press, MR 0120319 Dieudonné, J. (1963), Éléments d'analyse. Tome I: Fondements de l'analyse moderne, Cahiers Scientifiques, vol. XXVIII, Paris: Gauthier-Villars, MR 0161945 Dieudonné, J. (1968), Éléments d'analyse. Tome I: Fondements de l'analyse moderne, Cahiers Scientifiques, vol. XXVIII (2nd ed.), Paris: Gauthier-Villars, MR 0235945 Dieudonné, J. (1969), Foundations of modern analysis., Pure and Applied Mathematics, vol. 10-I (2nd ed.), New York-London: Academic Press, ISBN 978-0122155505, MR 0349288 === Volume II === The second volume includes Chapter XII Topology and topological algebra Chapter XIII Integration Chapter XIV Integration in locally compact groups Chapter XV Normed algebras and spectral theory Dieudonné, J. (1968), Éléments d'analyse. Tome II: Chapitres XII à XV, Cahiers Scientifiques, vol. XXXI, Paris: Gauthier-Villars, MR 0235946 Dieudonné, J. (1970), Treatise on analysis. Vol. II, Pure and Applied Mathematics, vol. 10-II, New York-London: Academic Press, MR 0258551 Dieudonné, J. (1976), Treatise on analysis. Vol. II, Pure and Applied Mathematics, vol. 10-II (2nd ed.), New York-London: Academic Press, ISBN 0-12-215502-5, MR 0530406 === Volume III === The third volume includes chapter XVI on differential manifolds and chapter XVII on distributions and differential operators. === Volume IV === The fourth volume includes Chapter XVIII Differential systems Chapter XIX Lie groups Chapter XX Riemannian geometry === Volume V === Volume V consists of chapter XXI on compact Lie groups. === Volume VI === Volume VI consists of chapter XXII on harmonic analysis (mostly on locally compact groups) === Volume VII === Volume VII consists of the first part of chapter XXIII on linear functional equations. This chapter is considerably more advanced than most of the other chapters. === Volume VIII === Volume VIII consists of the second part of chapter XXIII on linear functional equations. === Volume IX === Volume IX contains chapter XXIV on elementary differential topology. Unlike the earlier volumes there is no English translation of it. Dieudonné, J. (1982), Éléments d'analyse. Tome IX. Chapitre XXIV, Cahiers Scientifiques, vol. XL11, Paris: Gauthier-Villars, ISBN 2-04-011499-8, MR 0658305 === Volume X === Dieudonne planned a final volume containing chapter XXV on nonlinear problems, but this was never published. == References == Nachbin, Leopoldo (1961), "Review: J. Dieudonné, Foundations of Modern Analysis", Bull. Amer. Math. Soc., 67 (3): 246–250, doi:10.1090/s0002-9904-1961-10566-1 Frank, Peter (1960), "Book reviews: Foundations of Modern Analysis. J. Dieudonné. Academic Press, New York, 1960", Science, 132 (3441): 1759, doi:10.1126/science.132.3441.1759-a Marsden, Jerrold E. (1980), "Review: Jean Dieudonné, Treatise on analysis", Bull. Amer. Math. Soc. (N.S.), 3 (1): 719–724, doi:10.1090/s0273-0979-1980-14804-1
|
Wikipedia:Trevor Evans (mathematician)#0
|
Trevor Evans (1925–1991) was a mathematician specializing in abstract algebra, finite geometry, and the word problem. Originally British, he worked for many years in the US. == Early life and education == Evans was born on December 22, 1925, in Wolverhampton. He read mathematics at the University of Oxford, receiving a bachelor's degree in 1946. After a 1948 master's degree from the University of Manchester, he returned to Oxford for a master's degree in 1950 and a doctorate (D.Sc.) in 1960. His doctorate was granted based on 12 previous publications rather than on a dissertation; it was jointly advised by Graham Higman and Philip Hall. == Career and later life == From 1946 to 1950, Evans was an assistant lecturer at the University of Manchester. He moved to the US in 1950. After becoming an instructor at the University of Wisconsin–Madison in 1950, he took a faculty position at Emory University in 1951. He remained at Emory for the rest of his career, despite several other visiting positions: he joined the Institute for Advanced Study in 1952, and became a research associate at the University of Chicago in 1953. Later he visited the University of Nebraska, the California Institute of Technology, and the Darmstadt University of Technology in Germany. He headed the mathematics department at Emory from 1963 until 1978, and was given the Fuller E. Callaway Professorship of Mathematics in 1980. His notable students included Etta Zuber Falconer, who completed her doctorate under his supervision in 1969. == Awards == Emory University gave Evans their 1972 Emory Williams Distinguished Teaching Award for Graduate Education. He was also a recipient of the distinguished service award of the Southeastern Section of the Mathematical Association of America. An undergraduate award, the Trevor Evans Award, is given in memory of Evans by the Emory University Department of Mathematics. Another award with the same name, the Trevor Evans Award, is given annually by the Mathematical Association of America for an outstanding publication in its undergraduate mathematics magazine, Math Horizons. == Selected publications == === Research articles === Evans, Trevor (1951), "On multiplicative systems defined by generators and relations, I: Normal form theorems", Proceedings of the Cambridge Philosophical Society, 47: 637–649, doi:10.1017/S0305004100027092, MR 0043764 Evans, Trevor (1960), "Embedding incomplete latin squares", American Mathematical Monthly, 67 (10): 958–961, doi:10.2307/2309221, JSTOR 2309221, MR 0122728 Evans, Trevor (1969), "Some connections between residual finiteness, finite embeddability and the word problem", Journal of the London Mathematical Society, Second Series, 1: 399–403, doi:10.1112/jlms/s2-1.1.399, MR 0249344 Evans, Trevor (1971), "The lattice of semigroup varieties", Semigroup Forum, 2 (1): 1–43, doi:10.1007/BF02572269, MR 0284528 === Surveys === Evans, Trevor (1978), "Word problems", Bulletin of the American Mathematical Society, 84 (5): 789–802, doi:10.1090/S0002-9904-1978-14516-9, MR 0498063 === Books === Evans, Trevor (1959), Fundamentals Of Mathematics, Prentice-Hall Lindner, Charles C.; Evans, Trevor (1977), Finite embedding theorems for partial designs and algebras, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 56, Les Presses de l'Université de Montréal, MR 0460213 == References ==
|
Wikipedia:Triangle inequality#0
|
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If a, b, and c are the lengths of the sides of a triangle then the triangle inequality states that c ≤ a + b , {\displaystyle c\leq a+b,} with equality only in the degenerate case of a triangle with zero area. In Euclidean geometry and some other geometries, the triangle inequality is a theorem about vectors and vector lengths (norms): ‖ u + v ‖ ≤ ‖ u ‖ + ‖ v ‖ , {\displaystyle \|\mathbf {u} +\mathbf {v} \|\leq \|\mathbf {u} \|+\|\mathbf {v} \|,} where the length of the third side has been replaced by the length of the vector sum u + v. When u and v are real numbers, they can be viewed as vectors in R 1 {\displaystyle \mathbb {R} ^{1}} , and the triangle inequality expresses a relationship between absolute values. In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proved without these theorems. The inequality can be viewed intuitively in either R 2 {\displaystyle \mathbb {R} ^{2}} or R 3 {\displaystyle \mathbb {R} ^{3}} . The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a 180° angle and two 0° angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line. In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in [0, π]) with those endpoints. The triangle inequality is a defining property of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces (p ≥ 1), and inner product spaces. == Euclidean geometry == The triangle inequality theorem is stated in Euclid's Elements, Book 1, Proposition 20: […] in the triangle ABC the sum of any two sides is greater than the remaining one, that is, the sum of BA and AC is greater than BC, the sum of AB and BC is greater than AC, and the sum of BC and CA is greater than AB. Euclid proved the triangle inequality for distances in plane geometry using the construction in the figure. Beginning with triangle ABC, an isosceles triangle is constructed with one side taken as BC and the other equal leg BD along the extension of side AB. It then is argued that angle β has larger measure than angle α, so side AD is longer than side AC. However: A D ¯ = A B ¯ + B D ¯ = A B ¯ + B C ¯ , {\displaystyle {\overline {AD}}={\overline {AB}}+{\overline {BD}}={\overline {AB}}+{\overline {BC}},} so the sum of the lengths of sides AB and BC is larger than the length of AC. This proof appears in Euclid's Elements, Book 1, Proposition 20. === Mathematical expression of the constraint on the sides of a triangle === For a proper triangle, the triangle inequality, as stated in words, literally translates into three inequalities (given that a proper triangle has side lengths a, b, c that are all positive and excludes the degenerate case of zero area): a + b > c , b + c > a , c + a > b . {\displaystyle a+b>c,\quad b+c>a,\quad c+a>b.} A more succinct form of this inequality system can be shown to be | a − b | < c < a + b . {\displaystyle |a-b|<c<a+b.} Another way to state it is max ( a , b , c ) < a + b + c − max ( a , b , c ) {\displaystyle \max(a,b,c)<a+b+c-\max(a,b,c)} implying 2 max ( a , b , c ) < a + b + c {\displaystyle 2\max(a,b,c)<a+b+c} and thus that the longest side length is less than the semiperimeter. A mathematically equivalent formulation is that the area of a triangle with sides a, b, c must be a real number greater than zero. Heron's formula for the area is 4 ⋅ area = ( a + b + c ) ( − a + b + c ) ( a − b + c ) ( a + b − c ) = − a 4 − b 4 − c 4 + 2 a 2 b 2 + 2 a 2 c 2 + 2 b 2 c 2 . {\displaystyle {\begin{aligned}4\cdot {\text{area}}&={\sqrt {(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}\\&={\sqrt {-a^{4}-b^{4}-c^{4}+2a^{2}b^{2}+2a^{2}c^{2}+2b^{2}c^{2}}}.\end{aligned}}} In terms of either area expression, the triangle inequality imposed on all sides is equivalent to the condition that the expression under the square root sign be real and greater than zero (so the area expression is real and greater than zero). The triangle inequality provides two more interesting constraints for triangles whose sides are a, b, c, where a ≥ b ≥ c and ϕ {\displaystyle \phi } is the golden ratio, as 1 < a + c b < 3 {\displaystyle 1<{\frac {a+c}{b}}<3} 1 ≤ min ( a b , b c ) < ϕ . {\displaystyle 1\leq \min \left({\frac {a}{b}},{\frac {b}{c}}\right)<\phi .} === Right triangle === In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides and less than their sum. The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle ADC. An isosceles triangle ABC is constructed with equal sides AB = AC. From the triangle postulate, the angles in the right triangle ADC satisfy: α + γ = π / 2 . {\displaystyle \alpha +\gamma =\pi /2\ .} Likewise, in the isosceles triangle ABC, the angles satisfy: 2 β + γ = π . {\displaystyle 2\beta +\gamma =\pi \ .} Therefore, α = π / 2 − γ , w h i l e β = π / 2 − γ / 2 , {\displaystyle \alpha =\pi /2-\gamma ,\ \mathrm {while} \ \beta =\pi /2-\gamma /2\ ,} and so, in particular, α < β . {\displaystyle \alpha <\beta \ .} That means side AD, which is opposite to angle α, is shorter than side AB, which is opposite to the larger angle β. But AB = AC. Hence: A C ¯ > A D ¯ . {\displaystyle {\overline {AC}}>{\overline {AD}}\ .} A similar construction shows AC > DC, establishing the theorem. An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point B: (i) as depicted (which is to be proved), or (ii) B coincident with D (which would mean the isosceles triangle had two right angles as base angles plus the vertex angle γ, which would violate the triangle postulate), or lastly, (iii) B interior to the right triangle between points A and D (in which case angle ABC is an exterior angle of a right triangle BDC and therefore larger than π/2, meaning the other base angle of the isosceles triangle also is greater than π/2 and their sum exceeds π in violation of the triangle postulate). This theorem establishing inequalities is sharpened by Pythagoras' theorem to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides. === Examples of use === Consider a triangle whose sides are in an arithmetic progression and let the sides be a, a + d, a + 2d. Then the triangle inequality requires that 0 < a < 2 a + 3 d , 0 < a + d < 2 a + 2 d , 0 < a + 2 d < 2 a + d . {\displaystyle {\begin{array}{rcccl}0&<&a&<&2a+3d,\\0&<&a+d&<&2a+2d,\\0&<&a+2d&<&2a+d.\end{array}}} To satisfy all these inequalities requires a > 0 and − a 3 < d < a . {\displaystyle a>0{\text{ and }}-{\frac {a}{3}}<d<a.} When d is chosen such that d = a/3, it generates a right triangle that is always similar to the Pythagorean triple with sides 3, 4, 5. Now consider a triangle whose sides are in a geometric progression and let the sides be a, ar, ar2. Then the triangle inequality requires that 0 < a < a r + a r 2 , 0 < a r < a + a r 2 , 0 < a r 2 < a + a r . {\displaystyle {\begin{array}{rcccl}0&<&a&<&ar+ar^{2},\\0&<&ar&<&a+ar^{2},\\0&<&\!ar^{2}&<&a+ar.\end{array}}} The first inequality requires a > 0; consequently it can be divided through and eliminated. With a > 0, the middle inequality only requires r > 0. This now leaves the first and third inequalities needing to satisfy r 2 + r − 1 > 0 r 2 − r − 1 < 0. {\displaystyle {\begin{aligned}r^{2}+r-1&{}>0\\r^{2}-r-1&{}<0.\end{aligned}}} The first of these quadratic inequalities requires r to range in the region beyond the value of the positive root of the quadratic equation r2 + r − 1 = 0, i.e. r > φ − 1 where φ is the golden ratio. The second quadratic inequality requires r to range between 0 and the positive root of the quadratic equation r2 − r − 1 = 0, i.e. 0 < r < φ. The combined requirements result in r being confined to the range φ − 1 < r < φ and a > 0. {\displaystyle \varphi -1<r<\varphi \,{\text{ and }}a>0.} When r the common ratio is chosen such that r = √φ it generates a right triangle that is always similar to the Kepler triangle. === Generalization to any polygon === The triangle inequality can be extended by mathematical induction to arbitrary polygonal paths, showing that the total length of such a path is no less than the length of the straight line between its endpoints. Consequently, the length of any polygon side is always less than the sum of the other polygon side lengths. ==== Example of the generalized polygon inequality for a quadrilateral ==== Consider a quadrilateral whose sides are in a geometric progression and let the sides be a, ar, ar2, ar3. Then the generalized polygon inequality requires that 0 < a < a r + a r 2 + a r 3 0 < a r < a + a r 2 + a r 3 0 < a r 2 < a + a r + a r 3 0 < a r 3 < a + a r + a r 2 . {\displaystyle {\begin{array}{rcccl}0&<&a&<&ar+ar^{2}+ar^{3}\\0&<&ar&<&a+ar^{2}+ar^{3}\\0&<&ar^{2}&<&a+ar+ar^{3}\\0&<&ar^{3}&<&a+ar+ar^{2}.\end{array}}} These inequalities for a > 0 reduce to the following r 3 + r 2 + r − 1 > 0 {\displaystyle r^{3}+r^{2}+r-1>0} r 3 − r 2 − r − 1 < 0. {\displaystyle r^{3}-r^{2}-r-1<0.} The left-hand side polynomials of these two inequalities have roots that are the tribonacci constant and its reciprocal. Consequently, r is limited to the range 1/t < r < t where t is the tribonacci constant. ==== Relationship with shortest paths ==== This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line. No polygonal path between two points is shorter than the line between them. This implies that no curve can have an arc length less than the distance between its endpoints. By definition, the arc length of a curve is the least upper bound of the lengths of all polygonal approximations of the curve. The result for polygonal paths shows that the straight line between the endpoints is the shortest of all the polygonal approximations. Because the arc length of the curve is greater than or equal to the length of every polygonal approximation, the curve itself cannot be shorter than the straight line path. === Converse === The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (that is, with zero area) with these numbers as its side lengths. In either case, if the side lengths are a, b, c we can attempt to place a triangle in the Euclidean plane as shown in the diagram. We need to prove that there exists a real number h consistent with the values a, b, and c, in which case this triangle exists. By the Pythagorean theorem we have b2 = h2 + d2 and a2 = h2 + (c − d)2 according to the figure at the right. Subtracting these yields a2 − b2 = c2 − 2cd. This equation allows us to express d in terms of the sides of the triangle: d = − a 2 + b 2 + c 2 2 c . {\displaystyle d={\frac {-a^{2}+b^{2}+c^{2}}{2c}}.} For the height of the triangle we have that h2 = b2 − d2. By replacing d with the formula given above, we have h 2 = b 2 − ( − a 2 + b 2 + c 2 2 c ) 2 . {\displaystyle h^{2}=b^{2}-\left({\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)^{2}.} For a real number h to satisfy this, h2 must be non-negative: 0 ≤ b 2 − ( − a 2 + b 2 + c 2 2 c ) 2 0 ≤ ( b − − a 2 + b 2 + c 2 2 c ) ( b + − a 2 + b 2 + c 2 2 c ) 0 ≤ ( a 2 − ( b − c ) 2 ) ( ( b + c ) 2 − a 2 ) 0 ≤ ( a + b − c ) ( a − b + c ) ( b + c + a ) ( b + c − a ) 0 ≤ ( a + b − c ) ( a + c − b ) ( b + c − a ) {\displaystyle {\begin{aligned}0&\leq b^{2}-\left({\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)^{2}\\[4pt]0&\leq \left(b-{\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)\left(b+{\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)\\[4pt]0&\leq \left(a^{2}-(b-c)^{2})((b+c)^{2}-a^{2}\right)\\[6pt]0&\leq (a+b-c)(a-b+c)(b+c+a)(b+c-a)\\[6pt]0&\leq (a+b-c)(a+c-b)(b+c-a)\end{aligned}}} which holds if the triangle inequality is satisfied for all sides. Therefore, there does exist a real number h {\displaystyle h} consistent with the sides a , b , c {\displaystyle a,b,c} , and the triangle exists. If each triangle inequality holds strictly, h > 0 {\displaystyle h>0} and the triangle is non-degenerate (has positive area); but if one of the inequalities holds with equality, so h = 0 {\displaystyle h=0} , the triangle is degenerate. === Generalization to higher dimensions === The area of a triangular face of a tetrahedron is less than or equal to the sum of the areas of the other three triangular faces. More generally, in Euclidean space the hypervolume of an (n − 1)-facet of an n-simplex is less than or equal to the sum of the hypervolumes of the other n facets. Much as the triangle inequality generalizes to a polygon inequality, the inequality for a simplex of any dimension generalizes to a polytope of any dimension: the hypervolume of any facet of a polytope is less than or equal to the sum of the hypervolumes of the remaining facets. In some cases the tetrahedral inequality is stronger than several applications of the triangle inequality. For example, the triangle inequality appears to allow the possibility of four points A, B, C, and Z in Euclidean space such that distances AB = BC = CA = 26 and AZ = BZ = CZ = 14. However, points with such distances cannot exist: the area of the 26–26–26 equilateral triangle ABC is 169 3 {\textstyle 169{\sqrt {3}}} , which is larger than three times 39 3 {\textstyle 39{\sqrt {3}}} , the area of a 26–14–14 isosceles triangle (all by Heron's formula), and so the arrangement is forbidden by the tetrahedral inequality. == Normed vector space == In a normed vector space V, one of the defining properties of the norm is the triangle inequality: ‖ u + v ‖ ≤ ‖ u ‖ + ‖ v ‖ ∀ u , v ∈ V {\displaystyle \|\mathbf {u} +\mathbf {v} \|\leq \|\mathbf {u} \|+\|\mathbf {v} \|\quad \forall \,\mathbf {u} ,\mathbf {v} \in V} That is, the norm of the sum of two vectors is at most as large as the sum of the norms of the two vectors. This is also referred to as subadditivity. For any proposed function to behave as a norm, it must satisfy this requirement. If the normed space is Euclidean, or, more generally, strictly convex, then ‖ u + v ‖ = ‖ u ‖ + ‖ v ‖ {\displaystyle \|\mathbf {u} +\mathbf {v} \|=\|\mathbf {u} \|+\|\mathbf {v} \|} if and only if the triangle formed by u, v, and u + v, is degenerate, that is, u and v are on the same ray, i.e., u = 0 or v = 0, or u = α v for some α > 0. This property characterizes strictly convex normed spaces such as the ℓp spaces with 1 < p < ∞. However, there are normed spaces in which this is not true. For instance, consider the plane with the ℓ1 norm (the Manhattan distance) and denote u = (1, 0) and v = (0, 1). Then the triangle formed by u, v, and u + v, is non-degenerate but ‖ u + v ‖ = ‖ ( 1 , 1 ) ‖ = | 1 | + | 1 | = 2 = ‖ u ‖ + ‖ v ‖ . {\displaystyle \|\mathbf {u} +\mathbf {v} \|=\|(1,1)\|=|1|+|1|=2=\|\mathbf {u} \|+\|\mathbf {v} \|.} === Example norms === The absolute value is a norm for the real line; as required, the absolute value satisfies the triangle inequality for any real numbers u and v. If u and v have the same sign or either of them is zero, then | u + v | = | u | + | v | . {\displaystyle |u+v|=|u|+|v|.} If u and v have opposite signs, then without loss of generality assume | u | > | v | . {\displaystyle |u|>|v|.} Then | u + v | = | u | − | v | < | u | + | v | . {\displaystyle |u+v|=|u|-|v|<|u|+|v|.} Combining these cases: | u + v | ≤ | u | + | v | . {\displaystyle |u+v|\leq |u|+|v|.} The triangle inequality is useful in mathematical analysis for determining the best upper estimate on the size of the sum of two numbers, in terms of the sizes of the individual numbers. There is also a lower estimate, which can be found using the reverse triangle inequality which states that for any real numbers u and v, | u − v | ≥ | | u | − | v | | . {\displaystyle |u-v|\geq {\bigl |}|u|-|v|{\bigr |}.} The taxicab norm or 1-norm is one generalization absolute value to higher dimensions. To find the norm of a vector v = ( v 1 , v 2 , … v n ) , {\displaystyle v=(v_{1},v_{2},\ldots v_{n}),} just add the absolute value of each component separately, ‖ v ‖ 1 = | v 1 | + | v 2 | + ⋯ + | v n | . {\displaystyle \|v\|_{1}=|v_{1}|+|v_{2}|+\dotsb +|v_{n}|.} The Euclidean norm or 2-norm defines the length of translation vectors in an n-dimensional Euclidean space in terms of a Cartesian coordinate system. For a vector v = ( v 1 , v 2 , … v n ) , {\displaystyle v=(v_{1},v_{2},\ldots v_{n}),} its length is defined using the n-dimensional Pythagorean theorem: ‖ v ‖ 2 = | v 1 | 2 + | v 2 | 2 + ⋯ + | v n | 2 . {\displaystyle \|v\|_{2}={\sqrt {|v_{1}|^{2}+|v_{2}|^{2}+\dotsb +|v_{n}|^{2}}}.} The inner product is norm in any inner product space, a generalization of Euclidean vector spaces including infinite-dimensional examples. The triangle inequality follows from the Cauchy–Schwarz inequality as follows: Given vectors u {\displaystyle u} and v {\displaystyle v} , and denoting the inner product as ⟨ u , v ⟩ {\displaystyle \langle u,v\rangle } : The Cauchy–Schwarz inequality turns into an equality if and only if u and v are linearly dependent. The inequality ⟨ u , v ⟩ + ⟨ v , u ⟩ ≤ 2 | ⟨ u , v ⟩ | {\displaystyle \langle u,v\rangle +\langle v,u\rangle \leq 2\left|\left\langle u,v\right\rangle \right|} turns into an equality for linearly dependent u {\displaystyle u} and v {\displaystyle v} if and only if one of the vectors u or v is a nonnegative scalar of the other. Taking the square root of the final result gives the triangle inequality. The p-norm is a generalization of taxicab and Euclidean norms, using an arbitrary positive integer exponent, ‖ v ‖ p = ( | v 1 | p + | v 2 | p + ⋯ + | v n | p ) 1 / p , {\displaystyle \|v\|_{p}={\bigl (}|v_{1}|^{p}+|v_{2}|^{p}+\dotsb +|v_{n}|^{p}{\bigr )}^{1/p},} where the vi are the components of vector v. Except for the case p = 2, the p-norm is not an inner product norm, because it does not satisfy the parallelogram law. The triangle inequality for general values of p is called Minkowski's inequality. It takes the form: ‖ u + v ‖ p ≤ ‖ u ‖ p + ‖ v ‖ p . {\displaystyle \|u+v\|_{p}\leq \|u\|_{p}+\|v\|_{p}\ .} == Metric space == In a metric space M with metric d, the triangle inequality is a requirement upon distance: d ( A , C ) ≤ d ( A , B ) + d ( B , C ) , {\displaystyle d(A,\ C)\leq d(A,\ B)+d(B,\ C)\ ,} for all points A, B, and C in M. That is, the distance from A to C is at most as large as the sum of the distance from A to B and the distance from B to C. The triangle inequality is responsible for most of the interesting structure on a metric space, namely, convergence. This is because the remaining requirements for a metric are rather simplistic in comparison. For example, the fact that any convergent sequence in a metric space is a Cauchy sequence is a direct consequence of the triangle inequality, because if we choose any xn and xm such that d(xn, x) < ε/2 and d(xm, x) < ε/2, where ε > 0 is given and arbitrary (as in the definition of a limit in a metric space), then by the triangle inequality, d(xn, xm) ≤ d(xn, x) + d(xm, x) < ε/2 + ε/2 = ε, so that the sequence {xn} is a Cauchy sequence, by definition. This version of the triangle inequality reduces to the one stated above in case of normed vector spaces where a metric is induced via d(u, v) ≔ ‖u − v‖, with u − v being the vector pointing from point v to u. == Reverse triangle inequality == The reverse triangle inequality is an equivalent alternative formulation of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry, the statement is: Any side of a triangle is greater than or equal to the difference between the other two sides. In the case of a normed vector space, the statement is: | ‖ u ‖ − ‖ v ‖ | ≤ ‖ u − v ‖ , {\displaystyle {\big |}\|u\|-\|v\|{\big |}\leq \|u-v\|,} or for metric spaces, | d ( A , C ) − d ( B , C ) | ≤ d ( A , B ) {\displaystyle |d(A,C)-d(B,C)|\leq d(A,B)} . This implies that the norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} as well as the distance-from- z {\displaystyle z} function d ( z , ⋅ ) {\displaystyle d(z,\cdot )} are Lipschitz continuous with Lipschitz constant 1, and therefore are in particular uniformly continuous. The proof of the reverse triangle inequality from the usual one uses ‖ v − u ‖ = ‖ − 1 ( u − v ) ‖ = | − 1 | ⋅ ‖ u − v ‖ = ‖ u − v ‖ {\displaystyle \|v-u\|=\|{-}1(u-v)\|=|{-}1|\cdot \|u-v\|=\|u-v\|} to find: ‖ u ‖ = ‖ ( u − v ) + v ‖ ≤ ‖ u − v ‖ + ‖ v ‖ ⇒ ‖ u ‖ − ‖ v ‖ ≤ ‖ u − v ‖ , {\displaystyle \|u\|=\|(u-v)+v\|\leq \|u-v\|+\|v\|\Rightarrow \|u\|-\|v\|\leq \|u-v\|,} ‖ v ‖ = ‖ ( v − u ) + u ‖ ≤ ‖ v − u ‖ + ‖ u ‖ ⇒ ‖ u ‖ − ‖ v ‖ ≥ − ‖ u − v ‖ , {\displaystyle \|v\|=\|(v-u)+u\|\leq \|v-u\|+\|u\|\Rightarrow \|u\|-\|v\|\geq -\|u-v\|,} Combining these two statements gives: − ‖ u − v ‖ ≤ ‖ u ‖ − ‖ v ‖ ≤ ‖ u − v ‖ ⇒ | ‖ u ‖ − ‖ v ‖ | ≤ ‖ u − v ‖ . {\displaystyle -\|u-v\|\leq \|u\|-\|v\|\leq \|u-v\|\Rightarrow {\big |}\|u\|-\|v\|{\big |}\leq \|u-v\|.} In the converse, the proof of the triangle inequality from the reverse triangle inequality works in two cases: If ‖ u + v ‖ − ‖ u ‖ ≥ 0 , {\displaystyle \|u+v\|-\|u\|\geq 0,} then by the reverse triangle inequality, ‖ u + v ‖ − ‖ u ‖ = | ‖ u + v ‖ − ‖ u ‖ | ≤ ‖ ( u + v ) − u ‖ = ‖ v ‖ ⇒ ‖ u + v ‖ ≤ ‖ u ‖ + ‖ v ‖ {\displaystyle \|u+v\|-\|u\|={\big |}\|u+v\|-\|u\|{\big |}\leq \|(u+v)-u\|=\|v\|\Rightarrow \|u+v\|\leq \|u\|+\|v\|} , and if ‖ u + v ‖ − ‖ u ‖ < 0 , {\displaystyle \|u+v\|-\|u\|<0,} then trivially ‖ u ‖ + ‖ v ‖ ≥ ‖ u ‖ > ‖ u + v ‖ {\displaystyle \|u\|+\|v\|\geq \|u\|>\|u+v\|} by the nonnegativity of the norm. Thus, in both cases, we find that ‖ u ‖ + ‖ v ‖ ≥ ‖ u + v ‖ {\displaystyle \|u\|+\|v\|\geq \|u+v\|} . For metric spaces, the proof of the reverse triangle inequality is found similarly by: d ( A , B ) + d ( B , C ) ≥ d ( A , C ) ⇒ d ( A , B ) ≥ d ( A , C ) − d ( B , C ) {\displaystyle d(A,B)+d(B,C)\geq d(A,C)\Rightarrow d(A,B)\geq d(A,C)-d(B,C)} d ( C , A ) + d ( A , B ) ≥ d ( C , B ) ⇒ d ( A , B ) ≥ d ( B , C ) − d ( A , C ) {\displaystyle d(C,A)+d(A,B)\geq d(C,B)\Rightarrow d(A,B)\geq d(B,C)-d(A,C)} Putting these equations together we find: d ( A , B ) ≥ | d ( A , C ) − d ( B , C ) | {\displaystyle d(A,B)\geq |d(A,C)-d(B,C)|} And in the converse, beginning from the reverse triangle inequality, we can again use two cases: If d ( A , C ) − d ( B , C ) ≥ 0 {\displaystyle d(A,C)-d(B,C)\geq 0} , then d ( A , B ) ≥ | d ( A , C ) − d ( B , C ) | = d ( A , C ) − d ( B , C ) ⇒ d ( A , B ) + d ( B , C ) ≥ d ( A , C ) {\displaystyle d(A,B)\geq |d(A,C)-d(B,C)|=d(A,C)-d(B,C)\Rightarrow d(A,B)+d(B,C)\geq d(A,C)} , and if d ( A , C ) − d ( B , C ) < 0 , {\displaystyle d(A,C)-d(B,C)<0,} then d ( A , B ) + d ( B , C ) ≥ d ( B , C ) > d ( A , C ) {\displaystyle d(A,B)+d(B,C)\geq d(B,C)>d(A,C)} again by the nonnegativity of the metric. Thus, in both cases, we find that d ( A , B ) + d ( B , C ) ≥ d ( A , C ) {\displaystyle d(A,B)+d(B,C)\geq d(A,C)} . == Triangle inequality for cosine similarity == By applying the cosine function to the triangle inequality and reverse triangle inequality for arc lengths and employing the angle addition and subtraction formulas for cosines, it follows immediately that sim ( u , w ) ≥ sim ( u , v ) ⋅ sim ( v , w ) − ( 1 − sim ( u , v ) 2 ) ⋅ ( 1 − sim ( v , w ) 2 ) {\displaystyle \operatorname {sim} (u,w)\geq \operatorname {sim} (u,v)\cdot \operatorname {sim} (v,w)-{\sqrt {\left(1-\operatorname {sim} (u,v)^{2}\right)\cdot \left(1-\operatorname {sim} (v,w)^{2}\right)}}} and sim ( u , w ) ≤ sim ( u , v ) ⋅ sim ( v , w ) + ( 1 − sim ( u , v ) 2 ) ⋅ ( 1 − sim ( v , w ) 2 ) . {\displaystyle \operatorname {sim} (u,w)\leq \operatorname {sim} (u,v)\cdot \operatorname {sim} (v,w)+{\sqrt {\left(1-\operatorname {sim} (u,v)^{2}\right)\cdot \left(1-\operatorname {sim} (v,w)^{2}\right)}}\,.} With these formulas, one needs to compute a square root for each triple of vectors {u, v, w} that is examined rather than arccos(sim(u,v)) for each pair of vectors {u, v} examined, and could be a performance improvement when the number of triples examined is less than the number of pairs examined. == Reversal in Minkowski space == The Minkowski space metric η μ ν {\displaystyle \eta _{\mu \nu }} is not positive-definite, which means that ‖ u ‖ 2 = η μ ν u μ u ν {\displaystyle \|u\|^{2}=\eta _{\mu \nu }u^{\mu }u^{\nu }} can have either sign or vanish, even if the vector u is non-zero. Moreover, if u and v are both timelike vectors lying in the future light cone, the triangle inequality is reversed: ‖ u + v ‖ ≥ ‖ u ‖ + ‖ v ‖ . {\displaystyle \|u+v\|\geq \|u\|+\|v\|.} A physical example of this inequality is the twin paradox in special relativity. The same reversed form of the inequality holds if both vectors lie in the past light cone, and if one or both are null vectors. The result holds in n + 1 {\displaystyle n+1} dimensions for any n ≥ 1 {\displaystyle n\geq 1} . If the plane defined by u {\displaystyle u} and v {\displaystyle v} is space-like (and therefore a Euclidean subspace) then the usual triangle inequality holds. == See also == Subadditivity Minkowski inequality Ptolemy's inequality == Notes == == References == Pedoe, Daniel (1988). Geometry: A comprehensive course. Dover. ISBN 0-486-65812-0. Rudin, Walter (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. ISBN 0-07-054235-X.
|
Wikipedia:Triangular decomposition#0
|
In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily similar to an upper triangular matrix whose diagonal elements are the eigenvalues of the original matrix. == Statement == The complex Schur decomposition reads as follows: if A is an n × n square matrix with complex entries, then A can be expressed as A = Q U Q − 1 {\displaystyle A=QUQ^{-1}} for some unitary matrix Q (so that the inverse Q−1 is also the conjugate transpose Q* of Q), and some upper triangular matrix U. This is called a Schur form of A. Since U is similar to A, it has the same spectrum, and since it is triangular, its eigenvalues are the diagonal entries of U. The Schur decomposition implies that there exists a nested sequence of A-invariant subspaces {0} = V0 ⊂ V1 ⊂ ⋯ ⊂ Vn = Cn, and that there exists an ordered orthonormal basis (for the standard Hermitian form of Cn) such that the first i basis vectors span Vi for each i occurring in the nested sequence. Phrased somewhat differently, the first part says that a linear operator J on a complex finite-dimensional vector space stabilizes a complete flag (V1, ..., Vn). There is also a real Schur decomposition. If A is an n × n square matrix with real entries, then A can be expressed as A = Q H Q − 1 {\displaystyle A=QHQ^{-1}} where Q is an orthogonal matrix and H is either upper or lower quasi-triangular. A quasi-triangular matrix is a matrix that when expressed as a block matrix of 2 × 2 and 1 × 1 blocks is triangular. This is a stronger property than being Hessenberg. Just as in the complex case, a family of commuting real matrices {Ai} may be simultaneously brought to quasi-triangular form by an orthogonal matrix. There exists an orthogonal matrix Q such that, for every Ai in the given family, H i = Q A i Q − 1 {\displaystyle H_{i}=QA_{i}Q^{-1}} is upper quasi-triangular. == Proof == A constructive proof for the Schur decomposition is as follows: every operator A on a complex finite-dimensional vector space has an eigenvalue λ, corresponding to some eigenspace Vλ. Let Vλ⊥ be its orthogonal complement. It is clear that, with respect to this orthogonal decomposition, A has matrix representation (one can pick here any orthonormal bases Z1 and Z2 spanning Vλ and Vλ⊥ respectively) [ Z 1 Z 2 ] ∗ A [ Z 1 Z 2 ] = [ λ I λ A 12 0 A 22 ] : V λ ⊕ V λ ⊥ → V λ ⊕ V λ ⊥ {\displaystyle {\begin{bmatrix}Z_{1}&Z_{2}\end{bmatrix}}^{*}A{\begin{bmatrix}Z_{1}&Z_{2}\end{bmatrix}}={\begin{bmatrix}\lambda \,I_{\lambda }&A_{12}\\0&A_{22}\end{bmatrix}}:{\begin{matrix}V_{\lambda }\\\oplus \\V_{\lambda }^{\perp }\end{matrix}}\rightarrow {\begin{matrix}V_{\lambda }\\\oplus \\V_{\lambda }^{\perp }\end{matrix}}} where Iλ is the identity operator on Vλ. The above matrix would be upper-triangular except for the A22 block. But exactly the same procedure can be applied to the sub-matrix A22, viewed as an operator on Vλ⊥, and its submatrices. Continue this way until the resulting matrix is upper triangular. Since each conjugation increases the dimension of the upper-triangular block by at least one, this process takes at most n steps. Thus the space Cn will be exhausted and the procedure has yielded the desired result. The above argument can be slightly restated as follows: let λ be an eigenvalue of A, corresponding to some eigenspace Vλ. A induces an operator T on the quotient space Cn/Vλ. This operator is precisely the A22 submatrix from above. As before, T would have an eigenspace, say Wμ ⊂ Cn modulo Vλ. Notice the preimage of Wμ under the quotient map is an invariant subspace of A that contains Vλ. Continue this way until the resulting quotient space has dimension 0. Then the successive preimages of the eigenspaces found at each step form a flag that A stabilizes. == Notes == Although every square matrix has a Schur decomposition, in general this decomposition is not unique. For example, the eigenspace Vλ can have dimension > 1, in which case any orthonormal basis for Vλ would lead to the desired result. Write the triangular matrix U as U = D + N, where D is diagonal and N is strictly upper triangular (and thus a nilpotent matrix). The diagonal matrix D contains the eigenvalues of A in arbitrary order (hence its Frobenius norm, squared, is the sum of the squared moduli of the eigenvalues of A, while the Frobenius norm of A, squared, is the sum of the squared singular values of A). The nilpotent part N is generally not unique either, but its Frobenius norm is uniquely determined by A (just because the Frobenius norm of A is equal to the Frobenius norm of U = D + N). It is clear that if A is a normal matrix, then U from its Schur decomposition must be a diagonal matrix and the column vectors of Q are the eigenvectors of A. Therefore, the Schur decomposition extends the spectral decomposition. In particular, if A is positive definite, the Schur decomposition of A, its spectral decomposition, and its singular value decomposition coincide. A commuting family {Ai} of matrices can be simultaneously triangularized, i.e. there exists a unitary matrix Q such that, for every Ai in the given family, Q Ai Q* is upper triangular. This can be readily deduced from the above proof. Take element A from {Ai} and again consider an eigenspace VA. Then VA is invariant under all matrices in {Ai}. Therefore, all matrices in {Ai} must share one common eigenvector in VA. Induction then proves the claim. As a corollary, we have that every commuting family of normal matrices can be simultaneously diagonalized. In the infinite dimensional setting, not every bounded operator on a Banach space has an invariant subspace. However, the upper-triangularization of an arbitrary square matrix does generalize to compact operators. Every compact operator on a complex Banach space has a nest of closed invariant subspaces. == Computation == The Schur decomposition of a given matrix is numerically computed by the QR algorithm or its variants. In other words, the roots of the characteristic polynomial corresponding to the matrix are not necessarily computed ahead in order to obtain its Schur decomposition. Conversely, the QR algorithm can be used to compute the roots of any given characteristic polynomial by finding the Schur decomposition of its companion matrix. Similarly, the QR algorithm is used to compute the eigenvalues of any given matrix, which are the diagonal entries of the upper triangular matrix of the Schur decomposition. Although the QR algorithm is formally an infinite sequence of operations, convergence to machine precision is practically achieved in O ( n 3 ) {\displaystyle {\mathcal {O}}(n^{3})} operations. See the Nonsymmetric Eigenproblems section in LAPACK Users' Guide. == Applications == Lie theory applications include: Every invertible operator is contained in a Borel group. Every operator fixes a point of the flag manifold. == Generalized Schur decomposition == Given square matrices A and B, the generalized Schur decomposition factorizes both matrices as A = Q S Z ∗ {\displaystyle A=QSZ^{*}} and B = Q T Z ∗ {\displaystyle B=QTZ^{*}} , where Q and Z are unitary, and S and T are upper triangular. The generalized Schur decomposition is also sometimes called the QZ decomposition.: 375 The generalized eigenvalues λ {\displaystyle \lambda } that solve the generalized eigenvalue problem A x = λ B x {\displaystyle A\mathbf {x} =\lambda B\mathbf {x} } (where x is an unknown nonzero vector) can be calculated as the ratio of the diagonal elements of S to those of T. That is, using subscripts to denote matrix elements, the ith generalized eigenvalue λ i {\displaystyle \lambda _{i}} satisfies λ i = S i i / T i i {\displaystyle \lambda _{i}=S_{ii}/T_{ii}} . == References ==
|
Wikipedia:Trichotomy theorem#0
|
In group theory, the trichotomy theorem divides the finite simple groups of characteristic 2 type and rank at least 3 into three classes. It was proved by Aschbacher (1981, 1983) for rank 3 and by Gorenstein & Lyons (1983) for rank at least 4. The three classes are groups of GF(2) type (classified by Timmesfeld and others), groups of "standard type" for some odd prime (classified by the Gilman–Griess theorem and work by several others), and groups of uniqueness type, where Aschbacher proved that there are no simple groups. == References == Aschbacher, Michael (1981), "Finite groups of rank 3. I", Inventiones Mathematicae, 63 (3): 357–402, Bibcode:1981InMat..63..357A, doi:10.1007/BF01389061, ISSN 0020-9910, MR 0620676 Aschbacher, Michael (1983), "Finite groups of rank 3. II", Inventiones Mathematicae, 71 (1): 51–163, Bibcode:1983InMat..71...51A, doi:10.1007/BF01393339, ISSN 0020-9910, MR 0688262 Gorenstein, D.; Lyons, Richard (1983), "The local structure of finite groups of characteristic 2 type", Memoirs of the American Mathematical Society, 42 (276): vii+731, doi:10.1090/memo/0276, ISBN 978-0-8218-2276-0, ISSN 0065-9266, MR 0690900
|
Wikipedia:Tricorn (mathematics)#0
|
In mathematics, the tricorn, sometimes called the Mandelbar set, is a fractal defined in a similar way to the Mandelbrot set, but using the mapping z ↦ z ¯ 2 + c {\displaystyle z\mapsto {\bar {z}}^{2}+c} instead of z ↦ z 2 + c {\displaystyle z\mapsto z^{2}+c} used for the Mandelbrot set. It was introduced by W. D. Crowe, R. Hasson, P. J. Rippon, and P. E. D. Strain-Clark. John Milnor found tricorn-like sets as a prototypical configuration in the parameter space of real cubic polynomials, and in various other families of rational maps. The characteristic three-cornered shape created by this fractal repeats with variations at different scales, showing the same sort of self-similarity as the Mandelbrot set. In addition to smaller tricorns, smaller versions of the Mandelbrot set are also contained within the tricorn fractal. == Formal definition == The tricorn T {\displaystyle T} is defined by a family of quadratic antiholomorphic polynomials f c : C → C {\displaystyle f_{c}:\mathbb {C} \to \mathbb {C} } given by f c : z ↦ z ¯ 2 + c , {\displaystyle f_{c}:z\mapsto {\bar {z}}^{2}+c,} where c {\displaystyle c} is a complex parameter. For each c {\displaystyle c} , one looks at the forward orbit ( 0 , f c ( 0 ) , f c ( f c ( 0 ) ) , f c ( f c ( f c ( 0 ) ) ) , … ) {\displaystyle (0,f_{c}(0),f_{c}(f_{c}(0)),f_{c}(f_{c}(f_{c}(0))),\ldots )} of the critical point 0 {\displaystyle 0} of the antiholomorphic polynomial p c {\displaystyle p_{c}} . In analogy with the Mandelbrot set, the tricorn is defined as the set of all parameters c {\displaystyle c} for which the forward orbit of the critical point is bounded. This is equivalent to saying that the tricorn is the connectedness locus of the family of quadratic antiholomorphic polynomials; i.e. the set of all parameters c {\displaystyle c} for which the Julia set J ( f c ) {\displaystyle J(f_{c})} is connected. The higher degree analogues of the tricorn are known as the multicorns. These are the connectedness loci of the family of antiholomorphic polynomials f c : z ↦ z ¯ d + c {\displaystyle f_{c}:z\mapsto {\bar {z}}^{d}+c} . == Basic properties == The tricorn is compact, and connected. In fact, Nakane modified Douady and Hubbard's proof of the connectedness of the Mandelbrot set to construct a dynamically defined real-analytic diffeomorphism from the exterior of the tricorn onto the exterior of the closed unit disc in the complex plane. One can define external parameter rays of the tricorn as the inverse images of radial lines under this diffeomorphism. Every hyperbolic component of the tricorn is simply connected. The boundary of every hyperbolic component of odd period of the tricorn contains real-analytic arcs consisting of quasi-conformally equivalent but conformally distinct parabolic parameters. Such an arc is called a parabolic arc of the tricorn. This is in stark contrast with the corresponding situation for the Mandelbrot set, where parabolic parameters of a given period are known to be isolated. The boundary of every odd period hyperbolic component consists only of parabolic parameters. More precisely, the boundary of every hyperbolic component of odd period of the tricorn is a simple closed curve consisting of exactly three parabolic cusp points as well as three parabolic arcs, each connecting two parabolic cusps. Every parabolic arc of period k has, at both ends, an interval of positive length across which bifurcation from a hyperbolic component of odd period k to a hyperbolic component of period 2k occurs. == Image gallery of various zooms == Much like the Mandelbrot set, the tricorn has many complex and intricate designs. Due to their similarity, they share many features. However, in the tricorn such features appear to be squeezed and stretched along its boundary. The following images are progressional zooms on a selected c {\displaystyle c} value where c = 0.48 + 0.58 i {\displaystyle c=0.48+0.58i} . The images are not stretched or altered, that is how they look on magnification. == Implementation == The below pseudocode implementation hardcodes the complex operations for Z. Consider implementing complex number operations to allow for more dynamic and reusable code. == Further topological properties == The tricorn is not path connected. Hubbard and Schleicher showed that there are hyperbolic components of odd period of the tricorn that cannot be connected to the hyperbolic component of period one by paths. A stronger statement to the effect that no two (non-real) odd period hyperbolic components of the tricorn can be connected by a path was proved by Inou and Mukherjee. It is well known that every rational parameter ray of the Mandelbrot set lands at a single parameter. On the other hand, the rational parameter rays at odd-periodic (except period one) angles of the tricorn accumulate on arcs of positive length consisting of parabolic parameters. Moreover, unlike the Mandelbrot set, the dynamically natural straightening map from a baby tricorn to the original tricorn is discontinuous at infinitely many parameters. == References ==
|
Wikipedia:Trilinear coordinates#0
|
In geometry, the trilinear coordinates x : y : z of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio x : y is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices A and B respectively; the ratio y : z is the ratio of the perpendicular distances from the point to the sidelines opposite vertices B and C respectively; and likewise for z : x and vertices C and A. In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances (a', b', c'), or equivalently in ratio form, ka' : kb' : kc' for any positive constant k. If a point is on a sideline of the reference triangle, its corresponding trilinear coordinate is 0. If an exterior point is on the opposite side of a sideline from the interior of the triangle, its trilinear coordinate associated with that sideline is negative. It is impossible for all three trilinear coordinates to be non-positive. == Notation == The ratio notation x : y : z {\displaystyle x:y:z} for trilinear coordinates is often used in preference to the ordered triple notation ( x , y , z ) , {\displaystyle (x,y,z),} with the latter reserved for triples of directed distances ( a ′ , b ′ , c ′ ) {\displaystyle (a',b',c')} relative to a specific triangle. The trilinear coordinates x : y : z , {\displaystyle x:y:z,} can be rescaled by any arbitrary value without affecting their ratio. The bracketed, comma-separated triple notation ( x , y , z ) {\displaystyle (x,y,z)} can cause confusion because conventionally this represents a different triple than e.g. ( 2 x , 2 y , 2 z ) , {\displaystyle (2x,2y,2z),} but these equivalent ratios x : y : z = {\displaystyle x:y:z={}\!} 2 x : 2 y : 2 z {\displaystyle 2x:2y:2z} represent the same point. == Examples == The trilinear coordinates of the incenter of a triangle △ABC are 1 : 1 : 1; that is, the (directed) distances from the incenter to the sidelines BC, CA, AB are proportional to the actual distances denoted by (r, r, r), where r is the inradius of △ABC. Given side lengths a, b, c we have: Note that, in general, the incenter is not the same as the centroid; the centroid has barycentric coordinates 1 : 1 : 1 (these being proportional to actual signed areas of the triangles △BGC, △CGA, △AGB, where G = centroid.) The midpoint of, for example, side BC has trilinear coordinates in actual sideline distances ( 0 , Δ b , Δ c ) {\displaystyle (0,{\tfrac {\Delta }{b}},{\tfrac {\Delta }{c}})} for triangle area Δ, which in arbitrarily specified relative distances simplifies to 0 : ca : ab. The coordinates in actual sideline distances of the foot of the altitude from A to BC are ( 0 , 2 Δ a cos C , 2 Δ a cos B ) , {\displaystyle (0,{\tfrac {2\Delta }{a}}\cos C,{\tfrac {2\Delta }{a}}\cos B),} which in purely relative distances simplifies to 0 : cos C : cos B.: p. 96 == Formulas == === Collinearities and concurrencies === Trilinear coordinates enable many algebraic methods in triangle geometry. For example, three points P = p : q : r U = u : v : w X = x : y : z {\displaystyle {\begin{aligned}P&=p:q:r\\U&=u:v:w\\X&=x:y:z\\\end{aligned}}} are collinear if and only if the determinant D = | p q r u v w x y z | {\displaystyle D={\begin{vmatrix}p&q&r\\u&v&w\\x&y&z\end{vmatrix}}} equals zero. Thus if x : y : z is a variable point, the equation of a line through the points P and U is D = 0.: p. 23 From this, every straight line has a linear equation homogeneous in x, y, z. Every equation of the form l x + m y + n z = 0 {\displaystyle lx+my+nz=0} in real coefficients is a real straight line of finite points unless l : m : n is proportional to a : b : c, the side lengths, in which case we have the locus of points at infinity.: p. 40 The dual of this proposition is that the lines p α + q β + r γ = 0 u α + v β + w γ = 0 x α + y β + z γ = 0 {\displaystyle {\begin{aligned}p\alpha +q\beta +r\gamma &=0\\u\alpha +v\beta +w\gamma &=0\\x\alpha +y\beta +z\gamma &=0\end{aligned}}} concur in a point (α, β, γ) if and only if D = 0.: p. 28 Also, if the actual directed distances are used when evaluating the determinant of D, then the area of triangle △PUX is KD, where K = − a b c 8 Δ 2 {\displaystyle K={\tfrac {-abc}{8\Delta ^{2}}}} (and where Δ is the area of triangle △ABC, as above) if triangle △PUX has the same orientation (clockwise or counterclockwise) as △ABC, and K = − a b c 8 Δ 2 {\displaystyle K={\tfrac {-abc}{8\Delta ^{2}}}} otherwise. === Parallel lines === Two lines with trilinear equations l x + m y + n z = 0 {\displaystyle lx+my+nz=0} and l ′ x + m ′ y + n ′ z = 0 {\displaystyle l'x+m'y+n'z=0} are parallel if and only if: p. 98, #xi | l m n l ′ m ′ n ′ a b c | = 0 , {\displaystyle {\begin{vmatrix}l&m&n\\l'&m'&n'\\a&b&c\end{vmatrix}}=0,} where a, b, c are the side lengths. === Angle between two lines === The tangents of the angles between two lines with trilinear equations l x + m y + n z = 0 {\displaystyle lx+my+nz=0} and l ′ x + m ′ y + n ′ z = 0 {\displaystyle l'x+m'y+n'z=0} are given by: Art. 48 ± ( m n ′ − m ′ n ) sin A + ( n l ′ − n ′ l ) sin B + ( l m ′ − l ′ m ) sin C l l ′ + m m ′ + n n ′ − ( m n ′ + m ′ n ) cos A − ( n l ′ + n ′ l ) cos B − ( l m ′ + l ′ m ) cos C . {\displaystyle \pm {\frac {(mn'-m'n)\sin A+(nl'-n'l)\sin B+(lm'-l'm)\sin C}{ll'+mm'+nn'-(mn'+m'n)\cos A-(nl'+n'l)\cos B-(lm'+l'm)\cos C}}.} Thus they are perpendicular if: Art. 49 l l ′ + m m ′ + n n ′ − ( m n ′ + m ′ n ) cos A − ( n l ′ + n ′ l ) cos B − ( l m ′ + l ′ m ) cos C = 0. {\displaystyle ll'+mm'+nn'-(mn'+m'n)\cos A-(nl'+n'l)\cos B-(lm'+l'm)\cos C=0.} === Altitude === The equation of the altitude from vertex A to side BC is: p.98, #x y cos B − z cos C = 0. {\displaystyle y\cos B-z\cos C=0.} === Line in terms of distances from vertices === The equation of a line with variable distances p, q, r from the vertices A, B, C whose opposite sides are a, b, c is: p. 97, #viii a p x + b q y + c r z = 0. {\displaystyle apx+bqy+crz=0.} === Actual-distance trilinear coordinates === The trilinears with the coordinate values a', b', c' being the actual perpendicular distances to the sides satisfy: p. 11 a a ′ + b b ′ + c c ′ = 2 Δ {\displaystyle aa'+bb'+cc'=2\Delta } for triangle sides a, b, c and area Δ. This can be seen in the figure at the top of this article, with interior point P partitioning triangle △ABC into three triangles △PBC, △PCA, △PAB with respective areas 1 2 a a ′ , 1 2 b b ′ , 1 2 c c ′ . {\displaystyle {\tfrac {1}{2}}aa',{\tfrac {1}{2}}bb',{\tfrac {1}{2}}cc'.} === Distance between two points === The distance d between two points with actual-distance trilinears ai : bi : ci is given by: p. 46 d 2 sin 2 C = ( a 1 − a 2 ) 2 + ( b 1 − b 2 ) 2 + 2 ( a 1 − a 2 ) ( b 1 − b 2 ) cos C {\displaystyle d^{2}\sin ^{2}C=(a_{1}-a_{2})^{2}+(b_{1}-b_{2})^{2}+2(a_{1}-a_{2})(b_{1}-b_{2})\cos C} or in a more symmetric way d 2 = a b c 4 Δ 2 ( a ( b 1 − b 2 ) ( c 2 − c 1 ) + b ( c 1 − c 2 ) ( a 2 − a 1 ) + c ( a 1 − a 2 ) ( b 2 − b 1 ) ) . {\displaystyle d^{2}={\frac {abc}{4\Delta ^{2}}}\left(a(b_{1}-b_{2})(c_{2}-c_{1})+b(c_{1}-c_{2})(a_{2}-a_{1})+c(a_{1}-a_{2})(b_{2}-b_{1})\right).} === Distance from a point to a line === The distance d from a point a' : b' : c' , in trilinear coordinates of actual distances, to a straight line l x + m y + n z = 0 {\displaystyle lx+my+nz=0} is: p. 48 d = l a ′ + m b ′ + n c ′ l 2 + m 2 + n 2 − 2 m n cos A − 2 n l cos B − 2 l m cos C . {\displaystyle d={\frac {la'+mb'+nc'}{\sqrt {l^{2}+m^{2}+n^{2}-2mn\cos A-2nl\cos B-2lm\cos C}}}.} === Quadratic curves === The equation of a conic section in the variable trilinear point x : y : z is: p.118 r x 2 + s y 2 + t z 2 + 2 u y z + 2 v z x + 2 w x y = 0. {\displaystyle rx^{2}+sy^{2}+tz^{2}+2uyz+2vzx+2wxy=0.} It has no linear terms and no constant term. The equation of a circle of radius r having center at actual-distance coordinates (a', b', c' ) is: p.287 ( x − a ′ ) 2 sin 2 A + ( y − b ′ ) 2 sin 2 B + ( z − c ′ ) 2 sin 2 C = 2 r 2 sin A sin B sin C . {\displaystyle (x-a')^{2}\sin 2A+(y-b')^{2}\sin 2B+(z-c')^{2}\sin 2C=2r^{2}\sin A\sin B\sin C.} ==== Circumconics ==== The equation in trilinear coordinates x, y, z of any circumconic of a triangle is: p. 192 l y z + m z x + n x y = 0. {\displaystyle lyz+mzx+nxy=0.} If the parameters l, m, n respectively equal the side lengths a, b, c (or the sines of the angles opposite them) then the equation gives the circumcircle.: p. 199 Each distinct circumconic has a center unique to itself. The equation in trilinear coordinates of the circumconic with center x' : y' : z' is: p. 203 y z ( x ′ − y ′ − z ′ ) + z x ( y ′ − z ′ − x ′ ) + x y ( z ′ − x ′ − y ′ ) = 0. {\displaystyle yz(x'-y'-z')+zx(y'-z'-x')+xy(z'-x'-y')=0.} ==== Inconics ==== Every conic section inscribed in a triangle has an equation in trilinear coordinates:: p. 208 l 2 x 2 + m 2 y 2 + n 2 z 2 ± 2 m n y z ± 2 n l z x ± 2 l m x y = 0 , {\displaystyle l^{2}x^{2}+m^{2}y^{2}+n^{2}z^{2}\pm 2mnyz\pm 2nlzx\pm 2lmxy=0,} with exactly one or three of the unspecified signs being negative. The equation of the incircle can be simplified to: p. 210, p.214 ± x cos A 2 ± y cos B 2 ± z cos C 2 = 0 , {\displaystyle \pm {\sqrt {x}}\cos {\frac {A}{2}}\pm {\sqrt {y}}\cos {\frac {B}{2}}\pm {\sqrt {z}}\cos {\frac {C}{2}}=0,} while the equation for, for example, the excircle adjacent to the side segment opposite vertex A can be written as: p. 215 ± − x cos A 2 ± y cos B 2 ± z cos C 2 = 0. {\displaystyle \pm {\sqrt {-x}}\cos {\frac {A}{2}}\pm {\sqrt {y}}\cos {\frac {B}{2}}\pm {\sqrt {z}}\cos {\frac {C}{2}}=0.} === Cubic curves === Many cubic curves are easily represented using trilinear coordinates. For example, the pivotal self-isoconjugate cubic Z(U, P), as the locus of a point X such that the P-isoconjugate of X is on the line UX is given by the determinant equation | x y z q r y z r p z x p q x y u v w | = 0. {\displaystyle {\begin{vmatrix}x&y&z\\qryz&rpzx&pqxy\\u&v&w\end{vmatrix}}=0.} Among named cubics Z(U, P) are the following: Thomson cubic: Z ( X ( 2 ) , X ( 1 ) ) {\displaystyle Z(X(2),X(1))} , where X ( 2 ) {\displaystyle X(2)} is centroid and X ( 1 ) {\displaystyle X(1)} is incenter Feuerbach cubic: Z ( X ( 5 ) , X ( 1 ) ) {\displaystyle Z(X(5),X(1))} , where X ( 5 ) {\displaystyle X(5)} is Feuerbach point Darboux cubic: Z ( X ( 20 ) , X ( 1 ) ) {\displaystyle Z(X(20),X(1))} , where X ( 20 ) {\displaystyle X(20)} is De Longchamps point Neuberg cubic: Z ( X ( 30 ) , X ( 1 ) ) {\displaystyle Z(X(30),X(1))} , where X ( 30 ) {\displaystyle X(30)} is Euler infinity point. == Conversions == === Between trilinear coordinates and distances from sidelines === For any choice of trilinear coordinates x : y : z to locate a point, the actual distances of the point from the sidelines are given by a' = kx, b' = ky, c' = kz where k can be determined by the formula k = 2 Δ a x + b y + c z {\displaystyle k={\tfrac {2\Delta }{ax+by+cz}}} in which a, b, c are the respective sidelengths BC, CA, AB, and ∆ is the area of △ABC. === Between barycentric and trilinear coordinates === A point with trilinear coordinates x : y : z has barycentric coordinates ax : by : cz where a, b, c are the sidelengths of the triangle. Conversely, a point with barycentrics α : β : γ has trilinear coordinates α a : β b : γ c . {\displaystyle {\tfrac {\alpha }{a}}:{\tfrac {\beta }{b}}:{\tfrac {\gamma }{c}}.} === Between Cartesian and trilinear coordinates === Given a reference triangle △ABC, express the position of the vertex B in terms of an ordered pair of Cartesian coordinates and represent this algebraically as a vector B → , {\displaystyle {\vec {B}},} using vertex C as the origin. Similarly define the position vector of vertex A as A → . {\displaystyle {\vec {A}}.} Then any point P associated with the reference triangle △ABC can be defined in a Cartesian system as a vector P → = k 1 A → + k 2 B → . {\displaystyle {\vec {P}}=k_{1}{\vec {A}}+k_{2}{\vec {B}}.} If this point P has trilinear coordinates x : y : z then the conversion formula from the coefficients k1 and k2 in the Cartesian representation to the trilinear coordinates is, for side lengths a, b, c opposite vertices A, B, C, x : y : z = k 1 a : k 2 b : 1 − k 1 − k 2 c , {\displaystyle x:y:z={\frac {k_{1}}{a}}:{\frac {k_{2}}{b}}:{\frac {1-k_{1}-k_{2}}{c}},} and the conversion formula from the trilinear coordinates to the coefficients in the Cartesian representation is k 1 = a x a x + b y + c z , k 2 = b y a x + b y + c z . {\displaystyle k_{1}={\frac {ax}{ax+by+cz}},\quad k_{2}={\frac {by}{ax+by+cz}}.} More generally, if an arbitrary origin is chosen where the Cartesian coordinates of the vertices are known and represented by the vectors A → , B → , C → {\displaystyle {\vec {A}},{\vec {B}},{\vec {C}}} and if the point P has trilinear coordinates x : y : z, then the Cartesian coordinates of P → {\displaystyle {\vec {P}}} are the weighted average of the Cartesian coordinates of these vertices using the barycentric coordinates ax, by, cz as the weights. Hence the conversion formula from the trilinear coordinates x, y, z to the vector of Cartesian coordinates P → {\displaystyle {\vec {P}}} of the point is given by P → = a x a x + b y + c z A → + b y a x + b y + c z B → + c z a x + b y + c z C → , {\displaystyle {\vec {P}}={\frac {ax}{ax+by+cz}}{\vec {A}}+{\frac {by}{ax+by+cz}}{\vec {B}}+{\frac {cz}{ax+by+cz}}{\vec {C}},} where the side lengths are | C → − B → | = a , | A → − C → | = b , | B → − A → | = c . {\displaystyle {\begin{aligned}&|{\vec {C}}-{\vec {B}}|=a,\\&|{\vec {A}}-{\vec {C}}|=b,\\&|{\vec {B}}-{\vec {A}}|=c.\end{aligned}}} == See also == Morley's triangles, giving examples of numerous points expressed in trilinear coordinates Ternary plot Viviani's theorem == References == == External links == Weisstein, Eric W. "Trilinear Coordinates". MathWorld. Encyclopedia of Triangle Centers - ETC by Clark Kimberling; has trilinear coordinates (and barycentric) for 64000 triangle centers.
|
Wikipedia:Trinity Mathematical Society#0
|
The Trinity Mathematical Society, abbreviated TMS, was founded in Trinity College, Cambridge in 1919 by G. H. Hardy to "promote the discussion of subjects of mathematical interest". It is the oldest mathematical university society in the United Kingdom and is believed to be the oldest existing subject society at any British university. Today, the society is one of the largest societies in Trinity College, with nearly 600 members, and each year holds an extensive range of talks, together with social events including an annual cricket match against the Adams Society of St John's College, Cambridge. The society has hosted a range of distinguished speakers, including: M.Atiyah, A.Baker; B.Birch; C.Birkar; B.Bollobás; M.Born; J.H.Conway; H.S.M.Coxeter; H.Davenport; P.Dirac; F.W.Dyson; O.R.Frisch; W.T.Gowers; G.H.Hardy; W.V.D.Hodge; P.Kaptiza; E.Landau; J.E.Littlewood; L.J.Mordell; R.Penrose; G.Polya; R.Rado; F.Ramsey; B.Russell; E.Rutherford; L.Susskind; P.Swinnerton-Dyer; J.J.Thomson; W.Thurston; F.Wilczek; L.Wittgenstein. The logo of the society is the minimal perfect squared square. == Significance of the apple == For historical reasons, the apple is very important symbolically to the society. An apple is dropped at the end of meetings to signify that the meeting is now social; the President bowls an apple as the first 'ball' at the annual cricket match; and, as outlined in the society's Standing Orders, an apple is part of the design of the Society tie. == See also == Apple (symbolism) Ulam spiral == References == == External links == Official website
|
Wikipedia:Triple product#0
|
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product. == Scalar triple product == The scalar triple product (also called the mixed product, box product, or triple scalar product) is defined as the dot product of one of the vectors with the cross product of the other two. === Geometric interpretation === Geometrically, the scalar triple product a ⋅ ( b × c ) {\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )} is the (signed) volume of the parallelepiped defined by the three vectors given. === Properties === The scalar triple product is unchanged under a circular shift of its three operands (a, b, c): a ⋅ ( b × c ) = b ⋅ ( c × a ) = c ⋅ ( a × b ) {\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )=\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} )} Swapping the positions of the operators without re-ordering the operands leaves the triple product unchanged. This follows from the preceding property and the commutative property of the dot product: a ⋅ ( b × c ) = ( a × b ) ⋅ c {\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=(\mathbf {a} \times \mathbf {b} )\cdot \mathbf {c} } Swapping any two of the three operands negates the triple product. This follows from the circular-shift property and the anticommutativity of the cross product: a ⋅ ( b × c ) = − a ⋅ ( c × b ) = − b ⋅ ( a × c ) = − c ⋅ ( b × a ) {\displaystyle {\begin{aligned}\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )&=-\mathbf {a} \cdot (\mathbf {c} \times \mathbf {b} )\\&=-\mathbf {b} \cdot (\mathbf {a} \times \mathbf {c} )\\&=-\mathbf {c} \cdot (\mathbf {b} \times \mathbf {a} )\end{aligned}}} The scalar triple product can also be understood as the determinant of the 3×3 matrix that has the three vectors either as its rows or its columns (a matrix has the same determinant as its transpose): a ⋅ ( b × c ) = det [ a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 ] = det [ a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 ] = det [ a b c ] . {\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\det {\begin{bmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\\\end{bmatrix}}=\det {\begin{bmatrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{bmatrix}}=\det {\begin{bmatrix}\mathbf {a} &\mathbf {b} &\mathbf {c} \end{bmatrix}}.} If the scalar triple product is equal to zero, then the three vectors a, b, and c are coplanar, since the parallelepiped defined by them would be flat and have no volume. If any two vectors in the scalar triple product are equal, then its value is zero: a ⋅ ( a × b ) = a ⋅ ( b × a ) = b ⋅ ( a × a ) = 0 {\displaystyle \mathbf {a} \cdot (\mathbf {a} \times \mathbf {b} )=\mathbf {a} \cdot (\mathbf {b} \times \mathbf {a} )=\mathbf {b} \cdot (\mathbf {a} \times \mathbf {a} )=0} Also: ( a ⋅ ( b × c ) ) a = ( a × b ) × ( a × c ) {\displaystyle (\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} ))\,\mathbf {a} =(\mathbf {a} \times \mathbf {b} )\times (\mathbf {a} \times \mathbf {c} )} The simple product of two triple products (or the square of a triple product), may be expanded in terms of dot products: ( ( a × b ) ⋅ c ) ( ( d × e ) ⋅ f ) = det [ a ⋅ d a ⋅ e a ⋅ f b ⋅ d b ⋅ e b ⋅ f c ⋅ d c ⋅ e c ⋅ f ] {\displaystyle ((\mathbf {a} \times \mathbf {b} )\cdot \mathbf {c} )\;((\mathbf {d} \times \mathbf {e} )\cdot \mathbf {f} )=\det {\begin{bmatrix}\mathbf {a} \cdot \mathbf {d} &\mathbf {a} \cdot \mathbf {e} &\mathbf {a} \cdot \mathbf {f} \\\mathbf {b} \cdot \mathbf {d} &\mathbf {b} \cdot \mathbf {e} &\mathbf {b} \cdot \mathbf {f} \\\mathbf {c} \cdot \mathbf {d} &\mathbf {c} \cdot \mathbf {e} &\mathbf {c} \cdot \mathbf {f} \end{bmatrix}}} This restates in vector notation that the product of the determinants of two 3 × 3 matrices equals the determinant of their matrix product. As a special case, the square of a triple product is a Gram determinant. Note that this determinant is well defined for vectors in Rm (m-dimensional Euclidean space) even when m ≠ 3; in particular, the absolute value of a triple product for three vectors in Rm can be computed from this formula for the square of a triple product by taking its square root: | ( a × b ) ⋅ c | = det [ a ⋅ a a ⋅ b a ⋅ c b ⋅ a b ⋅ b b ⋅ c c ⋅ a c ⋅ b c ⋅ c ] {\displaystyle |(\mathbf {a} \times \mathbf {b} )\cdot \mathbf {c} |={\sqrt {\det {\begin{bmatrix}\mathbf {a} \cdot \mathbf {a} &\mathbf {a} \cdot \mathbf {b} &\mathbf {a} \cdot \mathbf {c} \\\mathbf {b} \cdot \mathbf {a} &\mathbf {b} \cdot \mathbf {b} &\mathbf {b} \cdot \mathbf {c} \\\mathbf {c} \cdot \mathbf {a} &\mathbf {c} \cdot \mathbf {b} &\mathbf {c} \cdot \mathbf {c} \end{bmatrix}}}}} The ratio of the triple product and the product of the three vector norms is known as a polar sine: a ⋅ ( b × c ) ‖ a ‖ ‖ b ‖ ‖ c ‖ = psin ( a , b , c ) {\displaystyle {\frac {\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )}{\|{\mathbf {a} }\|\|{\mathbf {b} }\|\|{\mathbf {c} }\|}}=\operatorname {psin} (\mathbf {a} ,\mathbf {b} ,\mathbf {c} )} which ranges between −1 and 1. === The triple product is a scalar density === Strictly speaking, a scalar does not change at all under a coordinate transformation. (For example, the factor of 2 used for doubling a vector does not change if the vector is in spherical vs. rectangular coordinates.) However, if each vector is transformed by a matrix then the triple product ends up being multiplied by the determinant of the transformation matrix. That is, the triple product of covariant vectors is more properly described as a scalar density. T a ⋅ ( T b × T c ) = det ( T a T b T c ) = det ( T ( a b c ) ) = det ( T ) det ( a b c ) = det ( T ) ( a ⋅ ( b × c ) ) {\displaystyle {\begin{aligned}T\mathbf {a} \cdot (T\mathbf {b} \times T\mathbf {c} )&=\det \left({\begin{matrix}T\mathbf {a} &T\mathbf {b} &T\mathbf {c} \end{matrix}}\right)\\&=\det \left(T\left({\begin{matrix}\mathbf {a} &\mathbf {b} &\mathbf {c} \end{matrix}}\right)\right)\\&=\det(T)\det \left({\begin{matrix}\mathbf {a} &\mathbf {b} &\mathbf {c} \end{matrix}}\right)\\&=\det(T)(\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} ))\end{aligned}}} Some authors use "pseudoscalar" to describe an object that looks like a scalar but does not transform like one. Because the triple product transforms as a scalar density not as a scalar, it could be called a "pseudoscalar" by this broader definition. However, the triple product is not a "pseudoscalar density". When a transformation is an orientation-preserving rotation, its determinant is +1 and the triple product is unchanged. When a transformation is an orientation-reversing rotation then its determinant is −1 and the triple product is negated. An arbitrary transformation could have a determinant that is neither +1 nor −1. === As an exterior product === In exterior algebra and geometric algebra the exterior product of two vectors is a bivector, while the exterior product of three vectors is a trivector. A bivector is an oriented plane element and a trivector is an oriented volume element, in the same way that a vector is an oriented line element. Given vectors a, b and c, the product a ∧ b ∧ c {\displaystyle \mathbf {a} \wedge \mathbf {b} \wedge \mathbf {c} } is a trivector with magnitude equal to the scalar triple product, i.e. | a ∧ b ∧ c | = | a ⋅ ( b × c ) | {\displaystyle |\mathbf {a} \wedge \mathbf {b} \wedge \mathbf {c} |=|\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )|} , and is the Hodge dual of the scalar triple product. As the exterior product is associative brackets are not needed as it does not matter which of a ∧ b or b ∧ c is calculated first, though the order of the vectors in the product does matter. Geometrically the trivector a ∧ b ∧ c corresponds to the parallelepiped spanned by a, b, and c, with bivectors a ∧ b, b ∧ c and a ∧ c matching the parallelogram faces of the parallelepiped. === As a trilinear function === The triple product is identical to the volume form of the Euclidean 3-space applied to the vectors via interior product. It also can be expressed as a contraction of vectors with a rank-3 tensor equivalent to the form (or a pseudotensor equivalent to the volume pseudoform); see below. == Vector triple product == The vector triple product is defined as the cross product of one vector with the cross product of the other two. The following relationship holds: a × ( b × c ) = ( a ⋅ c ) b − ( a ⋅ b ) c {\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=(\mathbf {a} \cdot \mathbf {c} )\mathbf {b} -(\mathbf {a} \cdot \mathbf {b} )\mathbf {c} } . This is known as triple product expansion, or Lagrange's formula, although the latter name is also used for several other formulas. Its right hand side can be remembered by using the mnemonic "ACB − ABC", provided one keeps in mind which vectors are dotted together. A proof is provided below. Some textbooks write the identity as a × ( b × c ) = b ( a ⋅ c ) − c ( a ⋅ b ) {\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=\mathbf {b} (\mathbf {a} \cdot \mathbf {c} )-\mathbf {c} (\mathbf {a} \cdot \mathbf {b} )} such that a more familiar mnemonic "BAC − CAB" is obtained, as in “back of the cab”. Since the cross product is anticommutative, this formula may also be written (up to permutation of the letters) as: ( a × b ) × c = − c × ( a × b ) = − ( c ⋅ b ) a + ( c ⋅ a ) b {\displaystyle (\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =-\mathbf {c} \times (\mathbf {a} \times \mathbf {b} )=-(\mathbf {c} \cdot \mathbf {b} )\mathbf {a} +(\mathbf {c} \cdot \mathbf {a} )\mathbf {b} } From Lagrange's formula it follows that the vector triple product satisfies: a × ( b × c ) + b × ( c × a ) + c × ( a × b ) = 0 {\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )+\mathbf {b} \times (\mathbf {c} \times \mathbf {a} )+\mathbf {c} \times (\mathbf {a} \times \mathbf {b} )=\mathbf {0} } which is the Jacobi identity for the cross product. Another useful formula follows: ( a × b ) × c = a × ( b × c ) − b × ( a × c ) {\displaystyle (\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )-\mathbf {b} \times (\mathbf {a} \times \mathbf {c} )} These formulas are very useful in simplifying vector calculations in physics. A related identity regarding gradients and useful in vector calculus is Lagrange's formula of vector cross-product identity: ∇ × ( ∇ × A ) = ∇ ( ∇ ⋅ A ) − ( ∇ ⋅ ∇ ) A {\displaystyle {\boldsymbol {\nabla }}\times ({\boldsymbol {\nabla }}\times \mathbf {A} )={\boldsymbol {\nabla }}({\boldsymbol {\nabla }}\cdot \mathbf {A} )-({\boldsymbol {\nabla }}\cdot {\boldsymbol {\nabla }})\mathbf {A} } This can be also regarded as a special case of the more general Laplace–de Rham operator Δ = d δ + δ d {\displaystyle \Delta =d\delta +\delta d} . === Proof === The x {\displaystyle x} component of u × ( v × w ) {\displaystyle \mathbf {u} \times (\mathbf {v} \times \mathbf {w} )} is given by: ( u × ( v × w ) ) x = u y ( v x w y − v y w x ) − u z ( v z w x − v x w z ) = v x ( u y w y + u z w z ) − w x ( u y v y + u z v z ) = v x ( u y w y + u z w z ) − w x ( u y v y + u z v z ) + ( u x v x w x − u x v x w x ) = v x ( u x w x + u y w y + u z w z ) − w x ( u x v x + u y v y + u z v z ) = ( u ⋅ w ) v x − ( u ⋅ v ) w x {\displaystyle {\begin{aligned}(\mathbf {u} \times (\mathbf {v} \times \mathbf {w} ))_{x}&=\mathbf {u} _{y}(\mathbf {v} _{x}\mathbf {w} _{y}-\mathbf {v} _{y}\mathbf {w} _{x})-\mathbf {u} _{z}(\mathbf {v} _{z}\mathbf {w} _{x}-\mathbf {v} _{x}\mathbf {w} _{z})\\&=\mathbf {v} _{x}(\mathbf {u} _{y}\mathbf {w} _{y}+\mathbf {u} _{z}\mathbf {w} _{z})-\mathbf {w} _{x}(\mathbf {u} _{y}\mathbf {v} _{y}+\mathbf {u} _{z}\mathbf {v} _{z})\\&=\mathbf {v} _{x}(\mathbf {u} _{y}\mathbf {w} _{y}+\mathbf {u} _{z}\mathbf {w} _{z})-\mathbf {w} _{x}(\mathbf {u} _{y}\mathbf {v} _{y}+\mathbf {u} _{z}\mathbf {v} _{z})+(\mathbf {u} _{x}\mathbf {v} _{x}\mathbf {w} _{x}-\mathbf {u} _{x}\mathbf {v} _{x}\mathbf {w} _{x})\\&=\mathbf {v} _{x}(\mathbf {u} _{x}\mathbf {w} _{x}+\mathbf {u} _{y}\mathbf {w} _{y}+\mathbf {u} _{z}\mathbf {w} _{z})-\mathbf {w} _{x}(\mathbf {u} _{x}\mathbf {v} _{x}+\mathbf {u} _{y}\mathbf {v} _{y}+\mathbf {u} _{z}\mathbf {v} _{z})\\&=(\mathbf {u} \cdot \mathbf {w} )\mathbf {v} _{x}-(\mathbf {u} \cdot \mathbf {v} )\mathbf {w} _{x}\end{aligned}}} Similarly, the y {\displaystyle y} and z {\displaystyle z} components of u × ( v × w ) {\displaystyle \mathbf {u} \times (\mathbf {v} \times \mathbf {w} )} are given by: ( u × ( v × w ) ) y = ( u ⋅ w ) v y − ( u ⋅ v ) w y ( u × ( v × w ) ) z = ( u ⋅ w ) v z − ( u ⋅ v ) w z {\displaystyle {\begin{aligned}(\mathbf {u} \times (\mathbf {v} \times \mathbf {w} ))_{y}&=(\mathbf {u} \cdot \mathbf {w} )\mathbf {v} _{y}-(\mathbf {u} \cdot \mathbf {v} )\mathbf {w} _{y}\\(\mathbf {u} \times (\mathbf {v} \times \mathbf {w} ))_{z}&=(\mathbf {u} \cdot \mathbf {w} )\mathbf {v} _{z}-(\mathbf {u} \cdot \mathbf {v} )\mathbf {w} _{z}\end{aligned}}} By combining these three components we obtain: u × ( v × w ) = ( u ⋅ w ) v − ( u ⋅ v ) w {\displaystyle \mathbf {u} \times (\mathbf {v} \times \mathbf {w} )=(\mathbf {u} \cdot \mathbf {w} )\ \mathbf {v} -(\mathbf {u} \cdot \mathbf {v} )\ \mathbf {w} } === Using geometric algebra === If geometric algebra is used the cross product b × c of vectors is expressed as their exterior product b∧c, a bivector. The second cross product cannot be expressed as an exterior product, otherwise the scalar triple product would result. Instead a left contraction can be used, so the formula becomes − a ⌟ ( b ∧ c ) = b ∧ ( a ⌟ c ) − ( a ⌟ b ) ∧ c = ( a ⋅ c ) b − ( a ⋅ b ) c {\displaystyle {\begin{aligned}-\mathbf {a} \;{\big \lrcorner }\;(\mathbf {b} \wedge \mathbf {c} )&=\mathbf {b} \wedge (\mathbf {a} \;{\big \lrcorner }\;\mathbf {c} )-(\mathbf {a} \;{\big \lrcorner }\;\mathbf {b} )\wedge \mathbf {c} \\&=(\mathbf {a} \cdot \mathbf {c} )\mathbf {b} -(\mathbf {a} \cdot \mathbf {b} )\mathbf {c} \end{aligned}}} The proof follows from the properties of the contraction. The result is the same vector as calculated using a × (b × c). == Triple Bivector Product == In geometric algebra, three bivectors can also have a triple product. This product mimic the standard triple vector product. The antisymmetric product of three bivectors is. a ⇒ × ( b ⇒ × c ⇒ ) = − ( a ⇒ ⋅ c ⇒ ) b ⇒ + ( a ⇒ ⋅ b ⇒ ) c ⇒ {\displaystyle {\overset {\Rightarrow }{a}}\times ({\overset {\Rightarrow }{b}}\times {\overset {\Rightarrow }{c}})=-({\overset {\Rightarrow }{a}}\cdot {\overset {\Rightarrow }{c}}){\overset {\Rightarrow }{b}}+({\overset {\Rightarrow }{a}}\cdot {\overset {\Rightarrow }{b}}){\overset {\Rightarrow }{c}}} === Proof === This proof is made by taking dual of the geometric algebra version of the triple vector product until all vectors become bivectors. ( − a ⌟ ( b ∧ c ) ) ⋆ = − 1 2 ( a ⇒ ( b ∧ c ) − ( b ∧ c ) a ⇒ ) = − a ⇒ × ( b ∧ c ) ( − a ⇒ × ( b ∧ c ) ) ⋆ = − 1 2 ( a ⇒ 1 2 ( b ⇒ c − c b ⇒ ) − 1 2 ( b ⇒ c − c b ⇒ ) a ⇒ ) = − a ⇒ × ( b ⇒ ⋅ c ) ( − a ⇒ × ( b ⇒ ⋅ c ) ) ⋆ = − 1 2 ( a ⇒ 1 2 ( b ⇒ c ⇒ − c ⇒ b ⇒ ) − 1 2 ( b ⇒ c ⇒ − c ⇒ b ⇒ ) a ⇒ ) = a ⇒ × ( b ⇒ × c ⇒ ) {\displaystyle {\begin{aligned}(-\mathbf {a} \;{\big \lrcorner }\;(\mathbf {b} \wedge \mathbf {c} ))\star &=-{\frac {1}{2}}({\overset {\Rightarrow }{a}}(\mathbf {b} \wedge \mathbf {c} )-(\mathbf {b} \wedge \mathbf {c} ){\overset {\Rightarrow }{a}})=-{\overset {\Rightarrow }{a}}\times (\mathbf {b} \wedge \mathbf {c} )\\(-{\overset {\Rightarrow }{a}}\times (\mathbf {b} \wedge \mathbf {c} ))\star &=-{\frac {1}{2}}({\overset {\Rightarrow }{a}}{\frac {1}{2}}({\overset {\Rightarrow }{b}}\mathbf {c} -\mathbf {c} {\overset {\Rightarrow }{b}})-{\frac {1}{2}}({\overset {\Rightarrow }{b}}\mathbf {c} -\mathbf {c} {\overset {\Rightarrow }{b}}){\overset {\Rightarrow }{a}})=-{\overset {\Rightarrow }{a}}\times ({\overset {\Rightarrow }{b}}\cdot \mathbf {c} )\\(-{\overset {\Rightarrow }{a}}\times ({\overset {\Rightarrow }{b}}\cdot \mathbf {c} ))\star &=-{\frac {1}{2}}({\overset {\Rightarrow }{a}}{\frac {1}{2}}({\overset {\Rightarrow }{b}}{\overset {\Rightarrow }{c}}-{\overset {\Rightarrow }{c}}{\overset {\Rightarrow }{b}})-{\frac {1}{2}}({\overset {\Rightarrow }{b}}{\overset {\Rightarrow }{c}}-{\overset {\Rightarrow }{c}}{\overset {\Rightarrow }{b}}){\overset {\Rightarrow }{a}})={\overset {\Rightarrow }{a}}\times ({\overset {\Rightarrow }{b}}\times {\overset {\Rightarrow }{c}})\end{aligned}}} This was three duals. This must also be done to the left side. ( ( ( ( a ⋅ c ) b − ( a ⋅ b ) c ) ⋆ ) ⋆ ) ⋆ = ( 1 2 ( a ⇒ c ⇒ + c ⇒ a ⇒ ) ) b ⇒ − ( 1 2 ( a ⇒ b ⇒ + b ⇒ a ⇒ ) ) c ⇒ = ( a ⇒ ⋅ c ⇒ ) b ⇒ + ( a ⇒ ⋅ b ⇒ ) c ⇒ {\displaystyle {\begin{aligned}&((((\mathbf {a} \cdot \mathbf {c} )\mathbf {b} -(\mathbf {a} \cdot \mathbf {b} )\mathbf {c} )\star )\star )\star =\\&({\frac {1}{2}}({\overset {\Rightarrow }{a}}{\overset {\Rightarrow }{c}}+{\overset {\Rightarrow }{c}}{\overset {\Rightarrow }{a}})){\overset {\Rightarrow }{b}}-({\frac {1}{2}}({\overset {\Rightarrow }{a}}{\overset {\Rightarrow }{b}}+{\overset {\Rightarrow }{b}}{\overset {\Rightarrow }{a}})){\overset {\Rightarrow }{c}}=\\&({\overset {\Rightarrow }{a}}\cdot {\overset {\Rightarrow }{c}}){\overset {\Rightarrow }{b}}+({\overset {\Rightarrow }{a}}\cdot {\overset {\Rightarrow }{b}}){\overset {\Rightarrow }{c}}\end{aligned}}} By negating both side we obtain: a ⇒ × ( b ⇒ × c ⇒ ) = − ( a ⇒ ⋅ c ⇒ ) b ⇒ + ( a ⇒ ⋅ b ⇒ ) c ⇒ {\displaystyle {\overset {\Rightarrow }{a}}\times ({\overset {\Rightarrow }{b}}\times {\overset {\Rightarrow }{c}})=-({\overset {\Rightarrow }{a}}\cdot {\overset {\Rightarrow }{c}}){\overset {\Rightarrow }{b}}+({\overset {\Rightarrow }{a}}\cdot {\overset {\Rightarrow }{b}}){\overset {\Rightarrow }{c}}} == Interpretations == === Tensor calculus === In tensor notation, the triple product is expressed using the Levi-Civita symbol: a ⋅ [ b × c ] = ε i j k a i b j c k {\displaystyle \mathbf {a} \cdot [\mathbf {b} \times \mathbf {c} ]=\varepsilon _{ijk}a^{i}b^{j}c^{k}} and ( a × [ b × c ] ) i = ε i j k a j ε k ℓ m b ℓ c m = ε i j k ε k ℓ m a j b ℓ c m , {\displaystyle (\mathbf {a} \times [\mathbf {b} \times \mathbf {c} ])_{i}=\varepsilon _{ijk}a^{j}\varepsilon ^{k\ell m}b_{\ell }c_{m}=\varepsilon _{ijk}\varepsilon ^{k\ell m}a^{j}b_{\ell }c_{m},} referring to the i {\displaystyle i} -th component of the resulting vector. This can be simplified by performing a contraction on the Levi-Civita symbols, ε i j k ε k ℓ m = δ i j ℓ m = δ i ℓ δ j m − δ i m δ j ℓ , {\displaystyle \varepsilon _{ijk}\varepsilon ^{k\ell m}=\delta _{ij}^{\ell m}=\delta _{i}^{\ell }\delta _{j}^{m}-\delta _{i}^{m}\delta _{j}^{\ell }\,,} where δ j i {\displaystyle \delta _{j}^{i}} is the Kronecker delta function ( δ j i = 0 {\displaystyle \delta _{j}^{i}=0} when i ≠ j {\displaystyle i\neq j} and δ j i = 1 {\displaystyle \delta _{j}^{i}=1} when i = j {\displaystyle i=j} ) and δ i j ℓ m {\displaystyle \delta _{ij}^{\ell m}} is the generalized Kronecker delta function. We can reason out this identity by recognizing that the index k {\displaystyle k} will be summed out leaving only i {\displaystyle i} and j {\displaystyle j} . In the first term, we fix i = l {\displaystyle i=l} and thus j = m {\displaystyle j=m} . Likewise, in the second term, we fix i = m {\displaystyle i=m} and thus l = j {\displaystyle l=j} . Returning to the triple cross product, ( a × [ b × c ] ) i = ( δ i ℓ δ j m − δ i m δ j ℓ ) a j b ℓ c m = a j b i c j − a j b j c i = b i ( a ⋅ c ) − c i ( a ⋅ b ) . {\displaystyle (\mathbf {a} \times [\mathbf {b} \times \mathbf {c} ])_{i}=(\delta _{i}^{\ell }\delta _{j}^{m}-\delta _{i}^{m}\delta _{j}^{\ell })a^{j}b_{\ell }c_{m}=a^{j}b_{i}c_{j}-a^{j}b_{j}c_{i}=b_{i}(\mathbf {a} \cdot \mathbf {c} )-c_{i}(\mathbf {a} \cdot \mathbf {b} )\,.} === Vector calculus === Consider the flux integral of the vector field F {\displaystyle \mathbf {F} } across the parametrically-defined surface S = r ( u , v ) {\displaystyle S=\mathbf {r} (u,v)} : ∬ S F ⋅ n ^ d S {\textstyle \iint _{S}\mathbf {F} \cdot {\hat {\mathbf {n} }}\,dS} . The unit normal vector n ^ {\displaystyle {\hat {\mathbf {n} }}} to the surface is given by r u × r v | r u × r v | {\textstyle {\frac {\mathbf {r} _{u}\times \mathbf {r} _{v}}{|\mathbf {r} _{u}\times \mathbf {r} _{v}|}}} , so the integrand F ⋅ ( r u × r v ) | r u × r v | {\textstyle \mathbf {F} \cdot {\frac {(\mathbf {r} _{u}\times \mathbf {r} _{v})}{|\mathbf {r} _{u}\times \mathbf {r} _{v}|}}} is a scalar triple product. == See also == Quadruple product Vector algebra relations == Notes == == References == Lass, Harry (1950). Vector and Tensor Analysis. McGraw-Hill Book Company, Inc. pp. 23–25. == External links == Khan Academy video of the proof of the triple product expansion
|
Wikipedia:Triple product rule#0
|
The triple product rule, known variously as the cyclic chain rule, cyclic relation, cyclical rule or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables. The rule finds application in thermodynamics, where frequently three variables can be related by a function of the form f(x, y, z) = 0, so each variable is given as an implicit function of the other two variables. For example, an equation of state for a fluid relates temperature, pressure, and volume in this manner. The triple product rule for such interrelated variables x, y, and z comes from using a reciprocity relation on the result of the implicit function theorem, and is given by ( ∂ x ∂ y ) ( ∂ y ∂ z ) ( ∂ z ∂ x ) = − 1 , {\displaystyle \left({\frac {\partial x}{\partial y}}\right)\left({\frac {\partial y}{\partial z}}\right)\left({\frac {\partial z}{\partial x}}\right)=-1,} where each factor is a partial derivative of the variable in the numerator, considered to be a function of the other two. The advantage of the triple product rule is that by rearranging terms, one can derive a number of substitution identities which allow one to replace partial derivatives which are difficult to analytically evaluate, experimentally measure, or integrate with quotients of partial derivatives which are easier to work with. For example, ( ∂ x ∂ y ) = − ( ∂ z ∂ y ) ( ∂ z ∂ x ) {\displaystyle \left({\frac {\partial x}{\partial y}}\right)=-{\frac {\left({\frac {\partial z}{\partial y}}\right)}{\left({\frac {\partial z}{\partial x}}\right)}}} Various other forms of the rule are present in the literature; these can be derived by permuting the variables {x, y, z}. == Derivation == An informal derivation follows. Suppose that f(x, y, z) = 0. Write z as a function of x and y. Thus the total differential dz is d z = ( ∂ z ∂ x ) d x + ( ∂ z ∂ y ) d y {\displaystyle dz=\left({\frac {\partial z}{\partial x}}\right)dx+\left({\frac {\partial z}{\partial y}}\right)dy} Suppose that we move along a curve with dz = 0, where the curve is parameterized by x. Thus y can be written in terms of x, so on this curve d y = ( ∂ y ∂ x ) d x {\displaystyle dy=\left({\frac {\partial y}{\partial x}}\right)dx} Therefore, the equation for dz = 0 becomes 0 = ( ∂ z ∂ x ) d x + ( ∂ z ∂ y ) ( ∂ y ∂ x ) d x {\displaystyle 0=\left({\frac {\partial z}{\partial x}}\right)\,dx+\left({\frac {\partial z}{\partial y}}\right)\left({\frac {\partial y}{\partial x}}\right)\,dx} Since this must be true for all dx, rearranging terms gives ( ∂ z ∂ x ) = − ( ∂ z ∂ y ) ( ∂ y ∂ x ) {\displaystyle \left({\frac {\partial z}{\partial x}}\right)=-\left({\frac {\partial z}{\partial y}}\right)\left({\frac {\partial y}{\partial x}}\right)} Dividing by the derivatives on the right hand side gives the triple product rule ( ∂ x ∂ y ) ( ∂ y ∂ z ) ( ∂ z ∂ x ) = − 1 {\displaystyle \left({\frac {\partial x}{\partial y}}\right)\left({\frac {\partial y}{\partial z}}\right)\left({\frac {\partial z}{\partial x}}\right)=-1} Note that this proof makes many implicit assumptions regarding the existence of partial derivatives, the existence of the exact differential dz, the ability to construct a curve in some neighborhood with dz = 0, and the nonzero value of partial derivatives and their reciprocals. A formal proof based on mathematical analysis would eliminate these potential ambiguities. === Alternative derivation === Suppose a function f(x, y, z) = 0, where x, y, and z are functions of each other. Write the total differentials of the variables d x = ( ∂ x ∂ y ) d y + ( ∂ x ∂ z ) d z {\displaystyle dx=\left({\frac {\partial x}{\partial y}}\right)dy+\left({\frac {\partial x}{\partial z}}\right)dz} d y = ( ∂ y ∂ x ) d x + ( ∂ y ∂ z ) d z {\displaystyle dy=\left({\frac {\partial y}{\partial x}}\right)dx+\left({\frac {\partial y}{\partial z}}\right)dz} Substitute dy into dx d x = ( ∂ x ∂ y ) [ ( ∂ y ∂ x ) d x + ( ∂ y ∂ z ) d z ] + ( ∂ x ∂ z ) d z {\displaystyle dx=\left({\frac {\partial x}{\partial y}}\right)\left[\left({\frac {\partial y}{\partial x}}\right)dx+\left({\frac {\partial y}{\partial z}}\right)dz\right]+\left({\frac {\partial x}{\partial z}}\right)dz} By using the chain rule one can show the coefficient of dx on the right hand side is equal to one, thus the coefficient of dz must be zero ( ∂ x ∂ y ) ( ∂ y ∂ z ) + ( ∂ x ∂ z ) = 0 {\displaystyle \left({\frac {\partial x}{\partial y}}\right)\left({\frac {\partial y}{\partial z}}\right)+\left({\frac {\partial x}{\partial z}}\right)=0} Subtracting the second term and multiplying by its inverse gives the triple product rule ( ∂ x ∂ y ) ( ∂ y ∂ z ) ( ∂ z ∂ x ) = − 1. {\displaystyle \left({\frac {\partial x}{\partial y}}\right)\left({\frac {\partial y}{\partial z}}\right)\left({\frac {\partial z}{\partial x}}\right)=-1.} === Short derivation === This section is based on chapter 5 of Pippard. Suppose we are given four real variables ( x , y , z , w ) {\displaystyle (x,y,z,w)} , restricted to move on a 2-dimensional C 2 {\displaystyle C^{2}} surface in R 4 {\displaystyle \mathbb {R} ^{4}} . Then, if we know two of them, we can determine the other two uniquely (generically). In particular, we may take any two variables as the independent variables, and let the other two be the dependent variables, then we can take all these partial derivatives. Proposition: ( ∂ x ∂ y ) z ( ∂ y ∂ z ) x ( ∂ z ∂ x ) y = − 1 {\displaystyle \left({\frac {\partial x}{\partial y}}\right)_{z}\left({\frac {\partial y}{\partial z}}\right)_{x}\left({\frac {\partial z}{\partial x}}\right)_{y}=-1} Proof. We can ignore w {\displaystyle w} . Then locally the surface is just a x + b y + c z + d = 0 {\displaystyle ax+by+cz+d=0} . Then ( ∂ x ∂ y ) z = − b a {\displaystyle \left({\frac {\partial x}{\partial y}}\right)_{z}=-{\frac {b}{a}}} , etc. Now multiply them. == Applications == === Example: Ideal Gas Law === The ideal gas law relates the state variables of pressure (P), volume (V), and temperature (T) via P V = n R T {\displaystyle PV=nRT} which can be written as f ( P , V , T ) = P V − n R T = 0 {\displaystyle f(P,V,T)=PV-nRT=0} so each state variable can be written as an implicit function of the other state variables: P = P ( V , T ) = n R T V V = V ( P , T ) = n R T P T = T ( P , V ) = P V n R {\displaystyle {\begin{aligned}P&=P(V,T)={\frac {nRT}{V}}\\[1em]V&=V(P,T)={\frac {nRT}{P}}\\[1em]T&=T(P,V)={\frac {PV}{nR}}\end{aligned}}} From the above expressions, we have − 1 = ( ∂ P ∂ V ) ( ∂ V ∂ T ) ( ∂ T ∂ P ) = ( − n R T V 2 ) ( n R P ) ( V n R ) = ( − n R T P V ) = − P P = − 1 {\displaystyle {\begin{aligned}-1&=\left({\frac {\partial P}{\partial V}}\right)\left({\frac {\partial V}{\partial T}}\right)\left({\frac {\partial T}{\partial P}}\right)\\[1em]&=\left(-{\frac {nRT}{V^{2}}}\right)\left({\frac {nR}{P}}\right)\left({\frac {V}{nR}}\right)\\[1em]&=\left(-{\frac {nRT}{PV}}\right)\\[1em]&=-{\frac {P}{P}}=-1\end{aligned}}} === Geometric Realization === A geometric realization of the triple product rule can be found in its close ties to the velocity of a traveling wave ϕ ( x , t ) = A cos ( k x − ω t ) {\displaystyle \phi (x,t)=A\cos(kx-\omega t)} shown on the right at time t (solid blue line) and at a short time later t+Δt (dashed). The wave maintains its shape as it propagates, so that a point at position x at time t will correspond to a point at position x+Δx at time t+Δt, A cos ( k x − ω t ) = A cos ( k ( x + Δ x ) − ω ( t + Δ t ) ) . {\displaystyle A\cos(kx-\omega t)=A\cos(k(x+\Delta x)-\omega (t+\Delta t)).} This equation can only be satisfied for all x and t if k Δx − ω Δt = 0, resulting in the formula for the phase velocity v = Δ x Δ t = ω k . {\displaystyle v={\frac {\Delta x}{\Delta t}}={\frac {\omega }{k}}.} To elucidate the connection with the triple product rule, consider the point p1 at time t and its corresponding point (with the same height) p̄1 at t+Δt. Define p2 as the point at time t whose x-coordinate matches that of p̄1, and define p̄2 to be the corresponding point of p2 as shown in the figure on the right. The distance Δx between p1 and p̄1 is the same as the distance between p2 and p̄2 (green lines), and dividing this distance by Δt yields the speed of the wave. To compute Δx, consider the two partial derivatives computed at p2, ( ∂ ϕ ∂ t ) Δ t = rise from p 2 to p ¯ 1 in time Δ t (gold line) {\displaystyle \left({\frac {\partial \phi }{\partial t}}\right)\Delta t={\text{rise from }}p_{2}{\text{ to }}{\bar {p}}_{1}{\text{ in time }}\Delta t{\text{ (gold line)}}} ( ∂ ϕ ∂ x ) = slope of the wave (red line) at time t . {\displaystyle \left({\frac {\partial \phi }{\partial x}}\right)={\text{slope of the wave (red line) at time }}t.} Dividing these two partial derivatives and using the definition of the slope (rise divided by run) gives us the desired formula for Δ x = − ( ∂ ϕ ∂ t ) Δ t ( ∂ ϕ ∂ x ) , {\displaystyle \Delta x=-{\frac {\left({\frac {\partial \phi }{\partial t}}\right)\Delta t}{\left({\frac {\partial \phi }{\partial x}}\right)}},} where the negative sign accounts for the fact that p1 lies behind p2 relative to the wave's motion. Thus, the wave's velocity is given by v = Δ x Δ t = − ( ∂ ϕ ∂ t ) ( ∂ ϕ ∂ x ) . {\displaystyle v={\frac {\Delta x}{\Delta t}}=-{\frac {\left({\frac {\partial \phi }{\partial t}}\right)}{\left({\frac {\partial \phi }{\partial x}}\right)}}.} For infinitesimal Δt, Δ x Δ t = ( ∂ x ∂ t ) {\displaystyle {\frac {\Delta x}{\Delta t}}=\left({\frac {\partial x}{\partial t}}\right)} and we recover the triple product rule v = Δ x Δ t = − ( ∂ ϕ ∂ t ) ( ∂ ϕ ∂ x ) . {\displaystyle v={\frac {\Delta x}{\Delta t}}=-{\frac {\left({\frac {\partial \phi }{\partial t}}\right)}{\left({\frac {\partial \phi }{\partial x}}\right)}}.} == See also == Differentiation rules – Rules for computing derivatives of functions Exact differential – Type of infinitesimal in calculus (has another derivation of the triple product rule) Product rule – Formula for the derivative of a product Total derivative – Type of derivative in mathematics Triple product – Ternary operation on vectors and scalars. == References == Elliott, J. R.; Lira, C. T. (1999). Introductory Chemical Engineering Thermodynamics (1st ed.). Prentice Hall. p. 184. ISBN 0-13-011386-7. Carter, Ashley H. (2001). Classical and Statistical Thermodynamics. Prentice Hall. p. 392. ISBN 0-13-779208-5.
|
Wikipedia:Tristan Rivière#0
|
Tristan Rivière (born 1967) is a French mathematician, working on partial differential equations and the calculus of variations. == Biography == Rivière studied at the École Polytechnique and obtained his PhD in 1993 at the Pierre and Marie Curie University, under the supervision of Fabrice Bethuel, with a thesis on harmonic maps between manifolds. In 1992 he was appointed chargé de recherche at CNRS. In 1997 he received his habilitation at the University of Paris-Sud in Orsay. From 1999 to 2000 he was a visiting associate professor at the Courant Institute of Mathematical Sciences (New York University). Since 2003 he is full professor at ETH Zurich and since 2009 he is the Director of the Institute for Mathematical Research at ETH. == Research activity == His research interests include partial differential equations in physics (liquid crystals, Bose–Einstein condensates, micromagnetics, Ginzburg–Landau theory of superconductivity, gauge theory) and differential geometry (harmonic maps between manifolds, geometric flows, minimal surfaces, the Willmore functional and Yang–Mills fields). His work focuses in particular on non-linear phenomena, formation of vortices, energy quantization and regularity issues. == Awards and recognition == In 1996 he received the Bronze Medal of the CNRS, while in 2003 he was awarded the first Stampacchia Medal. In 2002 he was an invited speaker at the International Congress of Mathematicians in Beijing, where he gave a talk on bubbling, quantization and regularity issues in geometric non-linear analysis. == Selected publications == "Everywhere discontinuous Harmonic Maps into Spheres." Acta Mathematica, 175 (1995), 197-226 with F. Pacard: Linear and Nonlinear Aspects of Vortices. Birkhäuser 2000 "Conservation laws for conformally invariant variational problems". Inventiones Math., 168 (2007), 1-22 with R. Hardt: "Connecting rational homotopy type singularities of maps between manifolds". Acta Mathematica, 200 (2008), 15-83 "Analysis Aspects of Willmore Surfaces". Inventiones Math., 174 (2008), no. 1, 1-45 with G. Tian: "The singular set of 1-1 Integral currents". Annals of Mathematics, 169 (2009), no. 3, 741-794 with Y. Bernard: "Energy Quantization for Willmore Surfaces and Applications". Annals of Mathematics, 180 (2014), no. 1, 87-136 "A viscosity method in the min-max theory of minimal surfaces". Publications mathématiques de l'IHÉS, 126 (2017), no. 1, 177-246 == References == == External links == Homepage, ETH Zürich
|
Wikipedia:Trivial representation#0
|
In the mathematical field of representation theory, a trivial representation is a representation (V, φ) of a group G on which all elements of G act as the identity mapping of V. A trivial representation of an associative or Lie algebra is a (Lie) algebra representation for which all elements of the algebra act as the zero linear map (endomorphism) which sends every element of V to the zero vector. For any group or Lie algebra, an irreducible trivial representation always exists over any field, and is one-dimensional, hence unique up to isomorphism. The same is true for associative algebras unless one restricts attention to unital algebras and unital representations. Although the trivial representation is constructed in such a way as to make its properties seem tautologous, it is a fundamental object of the theory. A subrepresentation is equivalent to a trivial representation, for example, if it consists of invariant vectors; so that searching for such subrepresentations is the whole topic of invariant theory. The trivial character is the character that takes the value of one for all group elements. == References == Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103..
|
Wikipedia:Troels Jørgensen#0
|
Troels Jørgensen is a Danish mathematician at Columbia University working on hyperbolic geometry and complex analysis, who proved Jørgensen's inequality. He wrote his thesis in 1970 at the University of Copenhagen under the joint supervision of Werner Fenchel and Bent Fuglede. == Work == He is known for Jørgensen's inequality, and for his discovery of a hyperbolic structure on certain fibered 3-manifolds which were one of the inspirations for William Thurston's Geometrisation Conjecture. He is also credited with being one of the co-discoverers of the ordered structure of the set of volumes of hyperbolic 3-manifolds. == References == == External resources == [1] Troels Jørgensen at the Mathematics Genealogy Project
|
Wikipedia:Tropical compactification#0
|
In algebraic geometry, a tropical compactification is a compactification (projective completion) of a subvariety of an algebraic torus, introduced by Jenia Tevelev. Given an algebraic torus and a connected closed subvariety of that torus, a compactification of the subvariety is defined as a closure of it in a toric variety of the original torus. The concept of a tropical compactification arises when trying to make compactifications as "nice" as possible. For a torus T {\displaystyle T} and a toric variety P {\displaystyle \mathbb {P} } , the compactification X ¯ {\displaystyle {\bar {X}}} is tropical when the map Φ : T × X ¯ → P , ( t , x ) → t x {\displaystyle \Phi :T\times {\bar {X}}\to \mathbb {P} ,\ (t,x)\to tx} is faithfully flat and X ¯ {\displaystyle {\bar {X}}} is proper. == See also == Tropical geometry GIT quotient Chow quotient Toroidal embedding == References == Cavalieri, Renzo; Markwig, Hannah; Ranganathan, Dhruv (2017). "Tropical compactification and the Gromov–Witten theory of P 1 {\displaystyle \mathbb {P} ^{1}} ". Selecta Mathematica. 23: 1027–1060. arXiv:1410.2837. Bibcode:2014arXiv1410.2837C. doi:10.1007/s00029-016-0265-7.
|
Wikipedia:Trudinger's theorem#0
|
In mathematical analysis, Trudinger's theorem or the Trudinger inequality (also sometimes called the Moser–Trudinger inequality) is a result of functional analysis on Sobolev spaces. It is named after Neil Trudinger (and Jürgen Moser). It provides an inequality between a certain Sobolev space norm and an Orlicz space norm of a function. The inequality is a limiting case of Sobolev embedding and can be stated as the following theorem: Let Ω {\displaystyle \Omega } be a bounded domain in R n {\displaystyle \mathbb {R} ^{n}} satisfying the cone condition. Let m p = n {\displaystyle mp=n} and p > 1 {\displaystyle p>1} . Set A ( t ) = exp ( t n / ( n − m ) ) − 1. {\displaystyle A(t)=\exp \left(t^{n/(n-m)}\right)-1.} Then there exists the embedding W m , p ( Ω ) ↪ L A ( Ω ) {\displaystyle W^{m,p}(\Omega )\hookrightarrow L_{A}(\Omega )} where L A ( Ω ) = { u ∈ M f ( Ω ) : ‖ u ‖ A , Ω = inf { k > 0 : ∫ Ω A ( | u ( x ) | k ) d x ≤ 1 } < ∞ } . {\displaystyle L_{A}(\Omega )=\left\{u\in M_{f}(\Omega ):\|u\|_{A,\Omega }=\inf\{k>0:\int _{\Omega }A\left({\frac {|u(x)|}{k}}\right)~dx\leq 1\}<\infty \right\}.} The space L A ( Ω ) {\displaystyle L_{A}(\Omega )} is an example of an Orlicz space. == References == Moser, J. (1971), "A Sharp form of an Inequality by N. Trudinger", Indiana Univ. Math. J., 20 (11): 1077–1092, doi:10.1512/iumj.1971.20.20101. Trudinger, N. S. (1967), "On imbeddings into Orlicz spaces and some applications", J. Math. Mech., 17: 473–483.
|
Wikipedia:Trygve Haavelmo#0
|
Trygve Magnus Haavelmo (13 December 1911 – 28 July 1999), born in Skedsmo, Norway, was an economist whose research interests centered on econometrics. He received the Nobel Memorial Prize in Economic Sciences in 1989. == Biography == After attending Oslo Cathedral School, Haavelmo received a degree in economics from the University of Oslo in 1930 and eventually joined the Institute of Economics with the recommendation of Ragnar Frisch. Haavelmo was Frisch's assistant for a period of time until he was appointed as head of computations for the institute. In 1936, Haavelmo studied statistics at University College London while he subsequently traveled to Berlin, Geneva, and Oxford for additional studies. Haavelmo assumed a lecturing position at the University of Aarhus in 1938 for one year and then in the subsequent year was offered an academic scholarship to travel abroad and study in the United States. During World War II he worked with Nortraship in the Statistical Department in New York City. He received his PhD in 1946 for his work on The Probability Approach in Econometrics. He was a professor of economics and statistics at the University of Oslo between 1948–79 and was the trade department head of division from 1947 to 1948. Haavelmo acquired a prominent position in modern economics through his logical critique of a series of custom conceptions in mathematical analysis. In 1989, Haavelmo was awarded the Nobel Prize in Economics "for his clarification of the probability theory foundations of econometrics and his analyses of simultaneous economic structures." Haavelmo resided at Østerås in Bærum. He died on 28 July 1999 in Oslo. == Legacy == Judea Pearl wrote "Haavelmo was the first to recognize the capacity of economic models to guide policies" and "presented a mathematical procedure that takes an arbitrary model and produces quantitative answers to policy questions". According to Pearl, "Haavelmo's paper, 'The Statistical Implications of a System of Simultaneous Equations', marks a pivotal turning point, not in the statistical implications of econometric models, as historians typically presume, but in their causal counterparts." Haavelmo's idea that an economic model depicts a series of hypothetical experiments and that policies can be simulated by modifying equations in the model became the basis of all currently used formalisms of econometric causal inference. (The biostatistics and epidemiology literature on causal inference draws from different sources.) It was first operationalized by Robert H. Strotz and Herman Wold (1960) who advocated "wiping out" selected equations, and then translated into graphical models as "wiping out" incoming arrows. This operation has subsequently led to Pearl's "do"-calculus and to a mathematical theory of counterfactuals in econometric models. Pearl further speculates that the reason economists do not generally appreciate these revolutionary contributions of Haavelmo is because economists themselves have still not reached consensus of what an economic model stands for, as attested by profound disagreements among econometric textbooks. == References == == External links == List of publications Trygve Haavelmo on Nobelprize.org including the Nobel Lecture on 7 December 1989 Econometrics and the Welfare State Model Discovery and Trygve Haavelmo’s Legacy by David F. Hendry and Søren Johansen.] Trygve Haavelmo Growth Model by Elmer G. Wiens "Trygve Haavelmo (1911–1999)". The Concise Encyclopedia of Economics. Library of Economics and Liberty (2nd ed.). Liberty Fund. 2008. Pearl, Judea (2014). "Trygve Haavelmo and the Emergence of Causal Calculus" (PDF). Forthcoming, Econometric Theory, special issue on Haavelmo Centennial. UCLA Computer Science Department, Technical Report R-391. Chen, Bryant; Pearl, Judea (2013). "Regression and Causation: A Critical Examination of Six Econometrics Textbooks" (PDF). Real-World Economics Review. 65: 2–20.
|
Wikipedia:Trygve Nagell#0
|
In number theory, the Ramanujan–Nagell equation is an equation between a square number and a number that is seven less than a power of two. It is an example of an exponential Diophantine equation, an equation to be solved in integers where one of the variables appears as an exponent. The equation is named after Srinivasa Ramanujan, who conjectured that it has only five integer solutions, and after Trygve Nagell, who proved the conjecture. It implies non-existence of perfect binary codes with the minimum Hamming distance 5 or 6. == Equation and solution == The equation is 2 n − 7 = x 2 {\displaystyle 2^{n}-7=x^{2}\,} and solutions in natural numbers n and x exist just when n = 3, 4, 5, 7 and 15 (sequence A060728 in the OEIS). This was conjectured in 1913 by Indian mathematician Srinivasa Ramanujan, proposed independently in 1943 by the Norwegian mathematician Wilhelm Ljunggren, and proved in 1948 by the Norwegian mathematician Trygve Nagell. The values of n correspond to the values of x as:- x = 1, 3, 5, 11 and 181 (sequence A038198 in the OEIS). == Triangular Mersenne numbers == The problem of finding all numbers of the form 2b − 1 (Mersenne numbers) which are triangular is equivalent: 2 b − 1 = y ( y + 1 ) 2 ⟺ 8 ( 2 b − 1 ) = 4 y ( y + 1 ) ⟺ 2 b + 3 − 8 = 4 y 2 + 4 y ⟺ 2 b + 3 − 7 = 4 y 2 + 4 y + 1 ⟺ 2 b + 3 − 7 = ( 2 y + 1 ) 2 {\displaystyle {\begin{aligned}&\ 2^{b}-1={\frac {y(y+1)}{2}}\\[2pt]\Longleftrightarrow &\ 8(2^{b}-1)=4y(y+1)\\\Longleftrightarrow &\ 2^{b+3}-8=4y^{2}+4y\\\Longleftrightarrow &\ 2^{b+3}-7=4y^{2}+4y+1\\\Longleftrightarrow &\ 2^{b+3}-7=(2y+1)^{2}\end{aligned}}} The values of b are just those of n − 3, and the corresponding triangular Mersenne numbers (also known as Ramanujan–Nagell numbers) are: y ( y + 1 ) 2 = ( x − 1 ) ( x + 1 ) 8 {\displaystyle {\frac {y(y+1)}{2}}={\frac {(x-1)(x+1)}{8}}} for x = 1, 3, 5, 11 and 181, giving 0, 1, 3, 15, 4095 and no more (sequence A076046 in the OEIS). == Equations of Ramanujan–Nagell type == An equation of the form x 2 + D = A B n {\displaystyle x^{2}+D=AB^{n}} for fixed D, A, B and variable x, n is said to be of Ramanujan–Nagell type. The result of Siegel implies that the number of solutions in each case is finite. By representing n = 3 m + r {\displaystyle n=3m+r} with r ∈ { 0 , 1 , 2 } {\displaystyle r\in \{0,1,2\}} and B n = B r y 3 {\displaystyle B^{n}=B^{r}y^{3}} with y = B m {\displaystyle y=B^{m}} , the equation of Ramanujan–Nagell type is reduced to three Mordell curves (indexed by r {\displaystyle r} ), each of which has a finite number of integer solutions: r = 0 : ( A x ) 2 = ( A y ) 3 − A 2 D {\displaystyle r=0:\qquad (Ax)^{2}=(Ay)^{3}-A^{2}D} , r = 1 : ( A B x ) 2 = ( A B y ) 3 − A 2 B 2 D {\displaystyle r=1:\qquad (ABx)^{2}=(ABy)^{3}-A^{2}B^{2}D} , r = 2 : ( A B 2 x ) 2 = ( A B 2 y ) 3 − A 2 B 4 D {\displaystyle r=2:\qquad (AB^{2}x)^{2}=(AB^{2}y)^{3}-A^{2}B^{4}D} . The equation with A = 1 , B = 2 , D > 0 {\displaystyle A=1,\ B=2,\ D>0} has at most two solutions, except in the case D = 7 {\displaystyle D=7} corresponding to the Ramanujan–Nagell equation. This does not hold for D < 0 {\displaystyle D<0} , such as D = − 17 {\displaystyle D=-17} , where x 2 − 17 = 2 n {\displaystyle x^{2}-17=2^{n}} has the four solutions ( x , n ) = ( 5 , 3 ) , ( 7 , 5 ) , ( 9 , 6 ) , ( 23 , 9 ) {\displaystyle (x,n)=(5,3),(7,5),(9,6),(23,9)} . In general, if D = − ( 4 k − 3 ⋅ 2 k + 1 + 1 ) {\displaystyle D=-(4^{k}-3\cdot 2^{k+1}+1)} for an integer k ⩾ 3 {\displaystyle k\geqslant 3} there are at least the four solutions ( x , n ) = { ( 2 k − 3 , 3 ) ( 2 k − 1 , k + 2 ) ( 2 k + 1 , k + 3 ) ( 3 ⋅ 2 k − 1 , 2 k + 3 ) {\displaystyle (x,n)={\begin{cases}(2^{k}-3,3)\\(2^{k}-1,k+2)\\(2^{k}+1,k+3)\\(3\cdot 2^{k}-1,2k+3)\end{cases}}} and these are the only four if D > − 10 12 {\displaystyle D>-10^{12}} . There are infinitely many values of D for which there are exactly two solutions, including D = 2 m − 1 {\displaystyle D=2^{m}-1} . == Equations of Lebesgue–Nagell type == An equation of the form x 2 + D = A y n {\displaystyle x^{2}+D=Ay^{n}} for fixed D, A and variable x, y, n is said to be of Lebesgue–Nagell type. This is named after Victor-Amédée Lebesgue, who proved that the equation x 2 + 1 = y n {\displaystyle x^{2}+1=y^{n}} has no nontrivial solutions. Results of Shorey and Tijdeman imply that the number of solutions in each case is finite. Bugeaud, Mignotte and Siksek solved equations of this type with A = 1 and 1 ≤ D ≤ 100. In particular, the following generalization of the Ramanujan–Nagell equation: y n − 7 = x 2 {\displaystyle y^{n}-7=x^{2}\,} has positive integer solutions only when x = 1, 3, 5, 11, or 181. == See also == Pillai's conjecture Scientific equations named after people == Notes == == References == Beukers, F. (1981). "On the generalized Ramanujan-Nagell equation I" (PDF). Acta Arithmetica. 38 (4): 401–403. doi:10.4064/aa-38-4-389-410. Bugeaud, Y.; Mignotte, M.; Siksek, S. (2006). "Classical and modular approaches to exponential Diophantine equations II. The Lebesgue–Nagell equation". Compositio Mathematica. 142: 31–62. arXiv:math/0405220. doi:10.1112/S0010437X05001739. S2CID 18534268. Lebesgue (1850). "Sur l'impossibilité, en nombres entiers, de l'équation xm = y2 + 1". Nouv. Ann. Math. Série 1. 9: 178–181. Ljunggren, W. (1943). "Oppgave nr 2". Norsk Mat. Tidsskr. 25: 29. Nagell, T. (1948). "Løsning till oppgave nr 2". Norsk Mat. Tidsskr. 30: 62–64. Nagell, T. (1961). "The Diophantine equation x2 + 7 = 2n". Ark. Mat. 30 (2–3): 185–187. Bibcode:1961ArM.....4..185N. doi:10.1007/BF02592006. Ramanujan, S. (1913). "Question 464". J. Indian Math. Soc. 5: 130. Saradha, N.; Srinivasan, Anitha (2008). "Generalized Lebesgue–Ramanujan–Nagell equations". In Saradha, N. (ed.). Diophantine Equations. Narosa. pp. 207–223. ISBN 978-81-7319-898-4. Shorey, T. N.; Tijdeman, R. (1986). Exponential Diophantine equations. Cambridge Tracts in Mathematics. Vol. 87. Cambridge University Press. pp. 137–138. ISBN 0-521-26826-5. Zbl 0606.10011. Siegel, C. L. (1929). "Uber einige Anwendungen Diophantischer Approximationen". Abh. Preuss. Akad. Wiss. Phys. Math. Kl. 1: 41–69. == External links == "Values of X corresponding to N in the Ramanujan–Nagell Equation". Wolfram MathWorld. Retrieved 2012-05-08. Can N2 + N + 2 Be A Power Of 2?, Math Forum discussion
|
Wikipedia:Tsachik Gelander#0
|
Tsachik Gelander (Hebrew: צחיק גלנדר) is an Israeli mathematician working in the fields of Lie groups, topological groups, symmetric spaces, lattices and discrete subgroups (of Lie groups as well as general locally compact groups). He is a professor in Northwestern University. Gelander earned his PhD from the Hebrew University of Jerusalem in 2003, under the supervision of Shahar Mozes. His doctoral dissertation, Counting Manifolds and Tits Alternative, won the Haim Nessyahu Prize in Mathematics, awarded by the Israel Mathematical Union for the best annual doctoral dissertations in mathematics. After holding a Gibbs Assistant Professorship at Yale University, and faculty positions at the Hebrew University of Jerusalem and the Weizmann Institute of Science, Gelander joined Northwestern where he is currently a professor of mathematics. He contributed to the theory of lattices, Fuchsian groups and local rigidity, and the work on Chern's conjecture and the Derivation Problem. Among his well-known results is the solution to the Goldman conjecture, i.e. that the action of O u t ( F n ) {\displaystyle Out(F_{n})} on the deformation variety of a compact Lie group is ergodic when n {\displaystyle n} is at least 3 {\displaystyle 3} . He gave the distinguished Nachdiplom Lectures at ETH Zurich in 2011, and was an invited speaker at the 2018 International Congress of Mathematicians, giving a talk under the title of Asymptotic Invariants of Locally Symmetric Spaces. He was one of the recipients of the first call of the European Research Council (ERC) Starting Grant (2007), and in 2021 he won the ERC Advanced Grant. == Selected publications == Gelander, Tsachik (15 September 2004). "Homotopy type and volume of locally symmetric manifolds". Duke Mathematical Journal. 124 (3). Duke University Press. arXiv:math/0111165. doi:10.1215/s0012-7094-04-12432-7. ISSN 0012-7094. S2CID 14272953. Bader, Uri; Furman, Alex; Gelander, Tsachik; Monod, Nicolas (2007). "Property (T) and rigidity for actions on Banach spaces". Acta Mathematica. 198 (1). International Press of Boston: 57–105. arXiv:math/0506361. doi:10.1007/s11511-007-0013-0. ISSN 0001-5962. S2CID 5739931. Breuillard, Emmanuel; Gelander, Tsachik (1 September 2007). "A topological Tits alternative". Annals of Mathematics. 166 (2): 427–474. arXiv:math/0403043. doi:10.4007/annals.2007.166.427. ISSN 0003-486X. S2CID 14859975. Breuillard, E.; Gelander, T. (3 May 2008). "Uniform independence in linear groups". Inventiones Mathematicae. 173 (2). Springer Science and Business Media LLC: 225–263. arXiv:math/0611829. Bibcode:2008InMat.173..225B. doi:10.1007/s00222-007-0101-y. ISSN 0020-9910. S2CID 16029687. Belolipetsky, Mikhail; Gelander, Tsachik; Lubotzky, Alexander; Shalev, Aner (5 October 2010). "Counting arithmetic lattices and surfaces". Annals of Mathematics. 172 (3): 2197–2221. arXiv:0811.2482. doi:10.4007/annals.2010.172.2197. ISSN 0003-486X. S2CID 14172846. Bader, U.; Gelander, T.; Monod, N. (27 October 2011). "A fixed point theorem for L 1 spaces". Inventiones Mathematicae. 189 (1). Springer Science and Business Media LLC: 143–148. arXiv:1012.1488. doi:10.1007/s00222-011-0363-2. ISSN 0020-9910. S2CID 55594695. Abert, Miklos; Bergeron, Nicolas; Biringer, Ian; Gelander, Tsachik; Nikolav, Nikolay; Raimbault, Jean; Samet, Iddo (1 May 2017). "On the growth of $L^2$-invariants for sequences of lattices in Lie groups" (PDF). Annals of Mathematics. 185 (3). doi:10.4007/annals.2017.185.3.1. ISSN 0003-486X. S2CID 106398777. Gelander, Tsachik (2019). "A VIEW ON INVARIANT RANDOM SUBGROUPS AND LATTICES". Proceedings of the International Congress of Mathematicians (ICM 2018). WORLD SCIENTIFIC. pp. 1321–1344. arXiv:1807.06979. doi:10.1142/9789813272880_0099. ISBN 978-981-327-287-3. Fraczyk, Mikolaj; Gelander, Tsachik (January 2023). "Infinite volume and infinite injectivity radius". Annals of Mathematics. 197 (1): 389–421. arXiv:2101.00640. == References ==
|
Wikipedia:Tsinghua Bamboo Slips#0
|
The Tsinghua Bamboo Strips (simplified Chinese: 清华简; traditional Chinese: 清華簡; pinyin: Qīnghuá jiǎn) are a collection of Chinese texts dating to the Warring States period and written in ink on strips of bamboo, that were acquired in 2008 by Tsinghua University, China. The texts were obtained by illegal excavation, probably of a tomb in the area of Hubei or Hunan province, and were then acquired and donated to the university by an alumnus. The very large size of the collection and the significance of the texts for scholarship make it one of the most important discoveries of early Chinese texts to date. On 7 January 2014 the journal Nature announced that a portion of the Tsinghua Bamboo Strips represent "the world's oldest example" of a decimal multiplication table. == Discovery, conservation and publication == The Tsinghua Bamboo Strips (TBS) were donated to Tsinghua University in July 2008 by an alumnus of the university. The precise location(s) and date(s) of the illicit excavation that yielded the strips remain(s) unknown. An article in the Guangming Daily named the donor as Zhao Weiguo (赵伟国), and stated that the texts were purchased at "a foreign auction", Neither the name of the auction house, nor the location or sum involved in the transaction was mentioned. Li Xueqin, the director of the conservation and research project, has stated that the wishes of the alumnus to maintain his identity secret will be respected. Similarities with previous discoveries, such as the manuscripts from the Guodian tomb, indicate that the TBS came from a mid-to-late Warring States Period (480–221BC) tomb in the region of China culturally dominated at that time by the Chu state. A single radiocarbon date (305±30BC) and the style of ornament on the accompanying box are in keeping with this conclusion. By the time they reached the university, the strips were badly affected by mold. Conservation work on the strips was carried out, and a Center for Excavated Texts Research and Preservation was established at Tsinghua on April 25, 2009. There are 2388 strips altogether in the collection, including a number of fragments. A series of articles discussing the TBS, intended for an educated but non-specialist Chinese audience, appeared in the Guangming Daily during late 2008 and 2009. The first volume of texts (photographic reproductions, transcriptions, and commentary) was published by the Tsinghua team in 2010. The series is scheduled to have a total of 18 volumes, with the latest volume 13 forthcoming in December 2023. A series of studies and publications are appearing in the series The Tsinghua University Warring States Bamboo Manuscripts: Studies and Translations《清華大學藏戰國竹簡》研究与英译, edited by Huang Dekuang 黃德寬 and Edward Shaughnessy. == The texts == The Tsinghua manuscripts vary greatly in content. The collection caught attention because several of the TBS texts have connections to the received Shang Shu (尚書: Exalted Writings), a collection of texts dated to various periods from the first millennium BC to the 3rd century CE. Because of the important role the Shang Shu plays in Chinese culture, the discovery of Warring States manuscripts that bear on its formation attracted interest. For example, the Yin zhi 尹至 manuscript from volume one has a partial overlap with the "Tang shi" 湯誓 text in the Shang shu; volume 9 of the series includes a manuscript whose content largely overlaps the "Jin Teng" 金滕 text in the Shang shu, and accordingly the editors titled the manuscript *Jin teng. Several others "writings-style" manuscripts present in the excavated collection are not found in the received Exalted Writings, either never having been incorporated into the canonical text, or having been lost or removed in the process of transmission. Other content resembles that of annalistic histories (編年體史書), recording events from the beginning of the Western Zhou (mid-11th century BC) through to the early Warring States period (mid-5th century) is said to be similar in form and content to the received Bamboo Annals. Another text running across 14 strips recounts a celebratory gathering of the Zhou elite in the 8th year of the reign of King Wu of Zhou, prior to their conquest of the Shang dynasty. The gathering takes place in the ancestral temple of King Wen of Zhou, King Wu's father, and consisted of beer drinking and the recitation of hymns in the style of the received Shi Jing. == Texts by volume == === Volume one === The following texts were published in volume one: *Yin zhi 尹至, *Yin's arrival; *Yin gao 尹誥, *Yin's Announcement; Cheng wu 程寤; *Baoxun 保訓, *The Protective Instructions; *Qi ye 耆夜; *Jin teng 金縢;*Huangmen 皇門, *August gate; and Zhai Gong zhi gu ming 祭公之顧命 The Duke of Zhai's Retrospective Command; and Chu ju 楚居. *Yin zhi 尹至, *Yin's arrival and *Yin gao 尹誥, *Yin's Announcement were written by the same scribe, and were considered two texts in the "writings" style. *Baoxun 保訓, *The Protective Instructions. The text purports to be a record of a deathbed admonition by the Zhou king Wen Wang to his son and heir, Wu Wang. Although the team working on the text refers to it as "The Admonition of Protection" (or "Protector's Admonition", 保训), their transcription of the text refers to a "Precious Admonition" (Bao Xun) and that may be the more appropriate editorial title. The content of the king's speech revolves around a concept of The Middle (zhong 中) which seems to refer to an avoidance of extremes and an ability to consider multiple points of view. The king narrates a story of the sage-king Shun acquiring The Middle by living a modest, thoughtful life, and a more puzzling second tale which describes the Shang ancestor Wei (微) "borrowing The Middle from the River." A complete translation and study has been published as part of the series The Tsinghua University Warring States Bamboo Manuscripts: Studies and Translations《清華大學藏戰國竹簡》研究与英译. === Volume two === It includes one text only, the Xinian, probably composed c. 370 BCE. This text relates key events of Zhou history. It comprises 138 strips in a relatively well preserved condition. Among the contents they transmit is an account of the origin of Qin by supporters of the Shang dynasty, who were opposed to the Zhou conquest. === Volume three === It includes the Fu Yue zhi ming 傅說之命, Command to Fu Yue; the *Liang chen 良臣, the Zhu ci 祝辭, among others. === Volume seven === It includes "Zi Fan Zi Yu" 子犯子餘, "Jin Wen Gong ru yu Jin" 晉文公入於晉, "Zhao jianzi" 趙簡子, "Yue Gong qi shi" 越公其事. "Zi fan Zi Yu" 子犯子餘 records a dialogue between Zi Fan and Duke Mu of Qin. This takes place while Chong'er is in exile, traveling from state to state. "Jin Wen gong ru yu Jin" 晉文公入於晉 narrates the story of the Duke Wen of Jin (posthumous name of the aforementioned Chong'er) returning to his state after years of battle, and putting it in order. === Volume eight === It includes eight texts: *She ming 攝命, *Bang jia zhi zheng 邦家之政, *Bang jia chu wei" 邦家處位; *Xin shi wei zhong 心是謂中; *Tianxia zhi dao 天下之道; Ba qi wu wei wu si wu xing zhi shu 八氣五味五祀五行之屬, and Yu Xia Yin Shang zhi zhi 虞夏殷商之治. *She ming 攝命, *Command to She. Titled by the editors. 32 strips. It purports to be a royal command to a certain She. It is written in shu 書 style. The editors identified it as the "original" Shangshu chapter Jiong Ming 囧命. "Xin shi wei zhong" 心是謂中. *The heart is what is at the center, a short (8 strips) text of philosophical nature discussing the heart-mind (xin 心) as the central organ in charge of the body, but also the concept of "luck" and mandate (ming 命). It includes the statement that humans are in charge of their destiny, so far otherwise unattested. === Volume nine === Volume nine, published in 2019, presents five manuscripts: *Zhi zheng zhi dao 治政之道. This manuscript is of 43 strips, around 44 cm long by 0.6. Based on the incision cuts on the verso of the strips and the similarities in the writing, the editors realized that this manuscript was originally bound together with *Bang jia zhi zheng 邦家之政 from volume eight, and the two should be read together. Given that the manuscripts still present codicological differences, Jia Lianxian 賈連翔 identified them as an example of tong pian yi zhi 同篇異制, "one bundle with different configurations." Cheng ren 成人 *Nai ming (one and two) 迺命一二 (*Then he commanded). These are two manuscripts written by the same person, as noted by the editors. The first is of 12 strips; the strips are numbered from 1 to 11, with the last one being left blank after the conclusion of the text (signaled by a hook-shape mark). Similarly, the second manuscripts is of 16 strips, numbered from one to 15. These two texts record commands given by an unnamed person to a group of officials (in the first one), and to a group of males who are presumably serving the person who is speaking. Dao ci 禱辭 === Volume twelve === Volume twelve presents strip images, transcription and study of one manuscript, titled by the editors "San Bu Wei" 参不韋, the name of the person who talks in the manuscript. The manuscript is of 124 strips wavering 32.8 cm in length; they are numbered on the verso side in well-preserved conditions. The content is otherwise unattested. In the text, San Bu Wei admonishes Qi 啟 (founder of the Xia dynasty) on how to govern, revise punishments, and conduct rituals. === Volume thirteen === The volume contains 5 manuscripts: Daifu Shi Li 大夫食禮. Daifu Shi Li Ji 大夫食禮記. Wu yin tu 五音圖. One of the most striking features of this (and the following manuscript) is the size: the 35 extant strips (from 37, originally) of Wu Yin tu average around 19.3 cm, a length that is half of most of the manuscripts in the Tsinghua collection. The writing develops around the 5 edges of a star, which figures at the center. It has attracted a great deal of attention for being one of the few writings related to music that predate imperial times. Yue feng 樂風. Wei tian yong shen 畏天用身. The text is reproduced in full in a paper by Shi Xiaoli 石小力. The manuscript is of 17 strips, measuring 44.4 cm by 0.6 cm. The title was assigned by the editors, based on the two initial principles introduced by the manuscript: "being in awe of Heaven" 畏天 and "using one's abilities" 用身. The text articulates behavioral principles (pay attention to surroundings; speaking properly) and how even and the self/person differ. Shi Xiaoli notes several echoes with ideas known in the Xunzi 荀子. A first introduction to the manuscripts on music can be found in Jia Lianxian's 贾连翔 2023 article in Zhongguo shi yanjiu dongtai 中國史研究動態. === Volume fourteenth === Volume no. 14 presents three manuscripts: Cheng Hou 成后, a manuscript of 9 strips, some mildly damaged, and numbered on the verso side. The text is otherwise unattested. The title was assigned by the editor, on the basis of the content. Cheng Hou is identified as King Cheng of Zhou 周成王, who in the text reminiscences about the accomplishments of past kings, and goes on to describe his governing philosophies. Zhao Hou 昭后, which was originally bound together with Cheng Hou. This is another short manuscript, of 7 strips, also numbered on the back. The writing style also matches that of the scribe of Cheng Hou. The title, assigned by the editor, is likewise based on its content: King Zhao of Zhou 周昭王 is on the throne and, in a form of an address to an unnamed interlocutor, defines proper government. Both this and Cheng Hou are rhymed and are styled in four-character sentences. The language is reminiscent of styles used in bronze inscriptions and in some speeches collected in the Shangshu 尚書. A manuscript of 88 strips, 1 of which is lost, titled by the editors Liang Zhou 兩中. This is a dialogue between Qi 启, the second ruler of the legendary Xia 夏 dynasty, and two spirit-like figures, Gui Zhong 圭中 and Xiang Zhong 詳中. It begins by setting up the establishment of the dynasty, after which the dialogue ensues. The text can be divided into six units: strips 1 to 18, 19 to 29, 30 to 42, 43 to 63, 64 to 69, and 67 to 88. == Decimal multiplication table == Twenty-one bamboo strips of the Tsinghua Bamboo Strips, when assembled in the correct order, represent a decimal multiplication table that can be used to multiply numbers (any whole or half integer) up to 99.5. Joseph Dauben of the City University of New York called it "the earliest artefact of a decimal multiplication table in the world". According to Guo Shuchun, director of the Chinese Society of the History of Mathematics, those strips filled a historical gap for mathematical documents prior to the Qin dynasty. "It helps establish the place-value system, a crucial development in the history of math", as Professor Wen Xing of Dartmouth College explains. It is presumed that officials used the multiplication table to calculate land surface area, yields of crops and the amounts of taxes owed. == See also == Guodian Chu Strips Shuanggudui Yinqueshan Han Strips Zhangjiashan Han bamboo texts == References == === Citations === === Sources ===
|
Wikipedia:Tsit Yuen Lam#0
|
Tsit Yuen Lam (Chinese: 林節玄; Jyutping: lam4 zit3 jyun4; born 6 February 1942) is a Hong Kong-American mathematician specializing in algebra, especially ring theory and quadratic forms. == Academic career == Lam earned his bachelor's degree at the University of Hong Kong in 1963 and his Ph.D. at Columbia University in 1967 under Hyman Bass, with a thesis titled On Grothendieck Groups. Subsequently, he was an instructor at the University of Chicago and since 1968 he has been at the University of California, Berkeley, where he became assistant professor in 1969, associate professor in 1972, and full professor in 1976. He served as assistant department head several times. From 1995 to 1997 he was Deputy Director of the Mathematical Sciences Research Institute in Berkeley, California. Among his doctoral students is Richard Elman. == Awards and honors == From 1972 to 1974 he was a Sloan Fellow; in 1978–79 a Miller Research Professor; and in 1981–82 a Guggenheim Fellow. In 1982 he was awarded the Leroy P. Steele Prize for his textbooks. In 2012 he became a fellow of the American Mathematical Society. == Selected publications == Serre’s Conjecture. Lecture Notes in Mathematics, Springer, 1978 Serre’s Problem on Projective Modules. Springer 2006; 2nd printing, 2010 The Algebraic Theory of Quadratic Forms. Benjamin 1973, 1980; new version published as Introduction to Quadratic Forms over Fields, American Mathematical Society, 2005 A First Course in Non-Commutative Rings. Graduate Texts in Mathematics, Springer 1991, 2nd edition 2001, ISBN 0-387-95325-6 Lectures on Modules and Rings. Springer, Graduate Texts in Mathematics 1999, ISBN 978-0-387-98428-5 Sums of squares of real polynomials. (with Man-Duen Choi & Bruce Reznick), Proceedings of Symposia in Pure Mathematics 58, 103–126, 1995 Orderings, Valuations and Quadratic Forms. AMS 1983 Exercises in Classical Ring Theory. Springer 1985 Representations of Finite Groups: A Hundred Years. Part I, Part II. Notices of the AMS 1998. (pdf files) == References == == External links == Lam's homepage
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.