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https://en.wikipedia.org/wiki/2000%20Emperor%27s%20Cup	Statistics of Emperor's Cup in the 2000 season. Overview It was contested by 80 teams, and Kashima Antlers won the championship. Results First round Nirasaki Astros 0–4 Hosei University Fukuoka University 4–1 Shimizu Commercial High School Gunma Fortuna 0–2 Consadole Sapporo Kusatsu Higashi High School 3–1 Yamagata Chuo High School Teihens F.C. 1–5 Albirex Niigata Ritsumeikan University 2–4 Kunimi High School Gifu Technical High School 1–7 Sagan Tosu Kanagawa Teachers 1–0 YKK AP SC Ehime FC 1–3 Yokohama Aichi Gakuin University 1–0 Fukui KSC Iwami 0–4 Shonan Bellmare Kochi University 1–1 (PK 4–5) Kibi International University Hatsushiba Hashimoto High School 0–5 Oita Trinita Sanfrecce Hiroshima Youth 0–1 FC Ueda Gentian Tenri University 0–6 Honda FC Nippon Steel Corporation Oita FC 4–2 Nippon Steel Kamaishi FC FC Primeiro 1–6 Denso Juntendo University 3–2 TDK Tottori 0–3 Omiya Ardija Tokuyama University 1–0 Apple Sports College Kwansei Gakuin University 2–1 Vegalta Sendai Tochigi SC 3–0 Hachinohe University Matsushita Electric Iga 0–4 Ventforet Kofu Sony Sendai 2–3 Doto University Saga Kita High School 0–12 Montedio Yamagata NIFS Kanoya 1–4 Jatco SC Kaiho Bank 0–3 Mito HollyHock Yokogawa Electric 1–1 (PK 4–5) Hannan University Saitama SC 0–2 Urawa Red Diamonds NTT Kumamoto 0–2 Honda Lock Tsukuba University 4–1 Tokai University Sun Life FC 0–5 Otsuka Pharmaceuticals Second round Hosei University 2–3 Fukuoka University Consadole Sapporo 6–0 Kusatsu Higashi High School Albirex Niigata 2–0 Kunimi High School Sagan Tosu 6–0 Kanagawa Teachers Yokohama FC 1–2 Aichi Gakuin University Shonan Bellmare 3–0 Kibi International University Oita Trinita 7–0 FC Ueda Gentian Honda 2–0 Nippon Steel Corporation Oita FC Denso 2–1 Juntendo University Omiya Ardija 4–1 Tokuyama University Kwansei Gakuin University 0–1 Tochigi Ventforet Kofu 2–1 Doto University Montedio Yamagata 1–2 Jatco SC Mito HollyHock 2–0 Hannan University Urawa Red Diamonds 9–0 Honda Lock Tsukuba University 1–1 (PK 2–4) Otsuka Pharmaceuticals Third round Yokohama F. Marinos 2–0 Fukuoka University Kyoto Purple Sanga 0–1 Consadole Sapporo Verdy Kawasaki 2–1 Albirex Niigata Kashima Antlers 2–1 Sagan Tosu Júbilo Iwata 5–0 Aichi Gakuin University Nagoya Grampus Eight 3–2 Shonan Bellmare Gamba Osaka 4–1 Oita Trinita Kashiwa Reysol 2–1 Honda Shimizu S-Pulse 3–0 Denso Avispa Fukuoka 4–2 Omiya Ardija JEF United Ichihara 1–0 Tochigi SC Tokyo 0–1 Ventforet Kofu Vissel Kobe 2–1 Jatco SC Sanfrecce Hiroshima 7–0 Mito HollyHock Kawasaki Frontale 0–2 Urawa Red Diamonds Cerezo Osaka 2–1 Otsuka Pharmaceuticals Fourth round Yokohama F. Marinos 2–1 Consadole Sapporo Verdy Kawasaki 0–2 Kashima Antlers Júbilo Iwata 2–0 Nagoya Grampus Eight Gamba Osaka 1–1 (PK 10–9) Kashiwa Reysol Shimizu S-Pulse 1–0 Avispa Fukuoka JEF United Ichihara 3–1 Ventforet Kofu Vissel Kobe 1–0 Sanfrecce Hiroshima Urawa Red Diamonds 1–4 Cerezo Osaka Quarter finals Yokohama F. Marinos 1–1 (PK 1–4) Kashima Antlers Júbilo Iwata 0–1 Gamba Osaka
https://en.wikipedia.org/wiki/2001%20Emperor%27s%20Cup	Statistics of Emperor's Cup in the 2001 season. Overview It was contested by 80 teams, and Shimizu S-Pulse won the championship. Results First round Ohara Gakuen JaSRA 1–3 Komazawa University Juntendo University 3–4 Sagawa Express Sun Life FC 0–4 Kunimi High School Saga Nanyo FC 1–0 Kibi International University Tokai University 2–0 Omiya Ardija Okinawa Kariyushi FC 1–2 Oita Trinita Nagasaki University 1–3 Nippon Steel Corporation Oita FC Tottori 1–0 Apple Sports College Saitama SC 0–14 Yokohama FC Muchz FC 0–7 Shonan Bellmare NTT Kumamoto 3–1 Yamagata FC Sony Sendai 0–1 Nara Sangyo University Kwansei Gakuin University 0–5 Kawasaki Frontale Fukuoka University 1–1(PK 1–3) Sagan Tosu Hosei University 1–0 Honda Lock Iwami FC 0–4 Jatco Gifu Technical High School 1–7 Mito HollyHock Osaka University of Health and Sport Sciences 0–4 Ventforet Kofu Denso 4–2 Akita Shogyo High School ALO's Hokuriku 2–1 Doto University Mind House Yokkaichi 1–3 Albirex Niigata Volca Kagoshima 2–3 Montedio Yamagata Tochigi SC 3–1 Fukui University of Technology Ehime FC 3–2 Fukuyama University FC Primeiro 0–4 Vegalta Sendai Nirasaki Astros 0–4 Kyoto Purple Sanga FC Kyoken Kyoto 4–0 Morioka Zebra Gunma Fortuna 6–1 Aster Aomori Kochi University 0–1 Honda FC Ryutsu Keizai University 3–1 Hannan University Kainan FC 1–1 (PK 4–2) Teihens FC Yamaguchi Teachers 0–8 Otsuka Pharmaceuticals Second round Komazawa University 4–1 Kunimi High School Sagawa Express 10–0 Saga Nanyo FC Tokai University 4–3 Nippon Steel Corporation Oita FC Oita Trinita 2–1 Gainare Tottori Yokohama FC 5–0 NTT Kumamoto Shonan Bellmare 0–0(PK 3–4) Nara Sangyo University Kawasaki Frontale 1–0 Hosei University Sagan Tosu 1–0 Jatco SC Mito HollyHock 1–0 Denso Ventforet Kofu 1–0 ALO's Hokuriku Albirex Niigata 2–1 Tochigi SC Montedio Yamagata 1–0 Ehime FC Vegalta Sendai 4–1 FC Kyoken Kyoto Kyoto Purple Sanga 3–0 Gunma Fortuna Honda FC 6–0 Kainan FC Ryutsu Keizai University 2–3 Otsuka Pharmaceuticals Third round Júbilo Iwata 3–2 Komazawa University Nagoya Grampus Eight 0–4 Sagawa Express Tokyo Verdy 2–0 Tokai University Cerezo Osaka 3–2 Oita Trinita Tokyo 0–1 Yokohama Kashima Antlers 6–0 Nara Sangyo University Consadole Sapporo 2–3 Kawasaki Frontale Kashiwa Reysol 1–2 Sagan Tosu Gamba Osaka 5–0 Mito HollyHock Urawa Red Diamonds 2–0 Ventforet Kofu Avispa Fukuoka 2–3 Albirex Niigata Vissel Kobe 1–0 Montedio Yamagata Sanfrecce Hiroshima 1–0 Vegalta Sendai Yokohama F. Marinos 0–1 Kyoto Purple Sanga Shimizu S-Pulse 2–1 Honda JEF United Ichihara 5–0 Otsuka Pharmaceuticals Fourth round Júbilo Iwata 1–3 Tokyo Verdy Sagawa Express 0–2 Cerezo Osaka Yokohama 1–3 Kawasaki Frontale Kashima Antlers 6–0 Sagan Tosu Gamba Osaka 1–0 Albirex Niigata Urawa Red Diamonds 4–1 Vissel Kobe Sanfrecce Hiroshima 0–4 Shimizu S-Pulse Kyoto Purple Sanga 0–4 JEF United Ichihara Quarter finals Tokyo Verdy 0–3 Kawasaki Frontale Cerezo Osaka 4–2 Kashima Antlers Gamba Osaka 0–2 Shimizu S-Pulse Urawa Red Diamonds 2–1 JEF United Ichihara Semi finals
https://en.wikipedia.org/wiki/2002%20Emperor%27s%20Cup	The 82nd Emperor's Cup Statistics of Emperor's Cup in the 2002 season. Overview It was contested by 80 teams, and Kyoto Purple Sanga won the cup. Results First round West Kagawa High School 0–1 Kokushikan University YSCC 1–7 Otsuka Pharmaceuticals Kunimi High School 3–0 Yamagata Central High School Centro de Futebol Edu 3–1 Volca Kagoshima International Budo University 1–2 Oita Trinita Teihens FC 0–6 Cerezo Osaka Iwami FC 2–1 Fukui KSC Tokuyama University 0–2 Sagawa Express Osaka Tsukuba University 1–3 Ventforet Kofu Ohara Gakuen JaSRA 0–1 Avispa Fukuoka Fukuoka University of Education 1–2 Alouette Kumamoto Tochigi SC 3–1 Iwate University Kyushu INAX 0–5 Mito HollyHock Takada 0–3 Yokohama FC Sagawa Printing 1–0 TDK Denso 1–1(PK 3–1) Sapporo University Gunma Horikoshi 2–1 Montedio Yamagata Kwansei Gakuin University 2–5 Sagan Tosu Tokyo Gakugei University 5–1 Hachinohe University Saitama 2–1 YKK AP SC Fukushima University 1–5 Omiya Ardija Nangoku Kochi 1–4 Shonan Bellmare Sony Sendai 2–3 Japan Soccer College Ehime FC 6–1 Mitsubishi Nagasaki SC Muchz FC 0–7 Kawasaki Frontale Kihoku Shukyudan 0–9 Albirex Niigata Tottori 2–1 Alex SC Profesor Miyazaki 4–3 Kibi International University Okinawa Kariyushi 2–3 Honda FC Kakamihara High School 0–9 Komazawa University Oita Trinita U-18 1–1(PK 5–3) Sanfrecce Hiroshima Youth Nirasaki Astros 0–2 Hamamatsu University Second round Kokushikan University 1–2 Kunimi High School Otsuka Pharmaceuticals 3–0 Centro de Futebol Edu Oita Trinita 3–0 Iwami FC Cerezo Osaka 4–1 Sagawa Express Osaka Ventforet Kofu 2–0 Alouette Kumamoto Avispa Fukuoka 2–1 Tochigi SC Mito HollyHock 4–0 Sagawa Printing Yokohama 3–2 Denso Gunma Horikoshi 0–1 Tokyo Gakugei University Sagan Tosu 7–0 Saitama Omiya Ardija 4–0 Japan Soccer College Shonan Bellmare 1–0 Ehime Kawasaki Frontale 6–0 Tottori Albirex Niigata 1–0 Profesor Miyazaki Honda 3–0 Oita Trinita U-18 Komazawa University 1–0 Hamamatsu University Third round Júbilo Iwata 2–0 Kunimi High School Nagoya Grampus Eight 2–0 Otsuka Pharmaceuticals Consadole Sapporo 0–5 Oita Trinita Kashiwa Reysol 1–2 Cerezo Osaka Vegalta Sendai 1–0 Ventforet Kofu Urawa Red Diamonds 1–2 Avispa Fukuoka JEF United Ichihara 4–0 Mito HollyHock Kyoto Purple Sanga 4–0 Yokohama Kashima Antlers 4–0 Tokyo Gakugei University Shimizu S-Pulse 4–2 Sagan Tosu Tokyo Verdy 0–2 Omiya Ardija Tokyo 3–4 Shonan Bellmare Vissel Kobe 1–3 Kawasaki Frontale Sanfrecce Hiroshima 2–0 Albirex Niigata Gamba Osaka 3–1 Honda FC Yokohama F. Marinos 3–0 Komazawa University Fourth round Júbilo Iwata 2–0 Oita Trinita Nagoya Grampus Eight 5–2 Cerezo Osaka Vegalta Sendai 1–2 JEF United Ichihara Avispa Fukuoka 0–1 Kyoto Purple Sanga Kashima Antlers 1–0 Omiya Ardija Shimizu S-Pulse 3–2 Shonan Bellmare Kawasaki Frontale 1–0 Gamba Osaka Sanfrecce Hiroshima 2–1 Yokohama F. Marinos Quarter finals Júbilo Iwata 0–1 JEF United Ichihara Nagoya Grampus Eight 0–1 Kyoto Purple Sanga Kashima Antlers 1–0 Kawasaki Frontale Shimizu S-Pulse 1–3 Sanfrecce Hirosh
https://en.wikipedia.org/wiki/2003%20Emperor%27s%20Cup	The 83rd Emperor's Cup Statistics of Emperor's Cup in the 2003 season. Overview It was contested by 80 teams, and Júbilo Iwata won the cup for the first time. Results First round Hannan University 2–0 Wakayama Kihoku Shukyudan Funabashi Municipal High School 1–0 Thespa Kusatsu Kanazawa 4–1 Yamaguchi Teachers Sanfrecce Hiroshima 1–0 Kwansei Gakuin University Omiya Ardija 5–0 Matsuzaka University Tochigi SC 5–2 Aomori Yamada High School Avispa Fukuoka 7–0 Tokai University Maruyasu Okazaki 1–0 Kibi International University Okinawa Kariyushi 1–0 Sagan Tosu ALO's Hokuriku 3–2 Doto University Consadole Sapporo 4–0 Jinsei Gakuen High School Shizuoka Sangyo University 4–2 Shiga Yasu High School Montedio Yamagata 4–3 Gifu Technical High School Ritsumeikan University 4–1 Tsuruoka Higashi High School Honda FC 5–0 Nirasaki Astros Tsukuba University 5–0 Matsucho Gakuen High School Otsuka Pharmaceuticals 5–0 Iwami FC TDK 3–1 Alouette Kumamoto Kawasaki Frontale 2–0 Juntendo University Kunimi High School 4–0 Panasonic Energy Tokushima Mito HollyHock 1–0 Sanfrecce Hiroshima Youth Gainare Tottori 1–0 Nippon Bunri University Shonan Bellmare 6–1 Tenri University Ehime FC 4–1 Japan Soccer College Yokohama 8–1 Viancone Fukushima Fukuoka University of Education 1–1(PK 6–5) Sony Sendai Ventforet Kofu 3–0 Fukui University of Technology Momoyama Gakuin University 6–1 Profesor Miyazaki Albirex Niigata 5–0 Volca Kagoshima Kochi University 3–1 Morioka Zebra Honda Luminoso Sayama 1–0 Komazawa University Sagawa Express Tokyo 4–0 Kyushu INAX Second round Funabashi Municipal High School 1–0 Hannan University Kawasaki Frontale 7–1 Kunimi High School Honda FC 2–1 Tsukuba University Avispa Fukuoka 3–0 Maruyasu Okazaki Yokohama 5–0 Fukuoka University of Education Albirex Niigata 2–0 Kochi University Omiya Ardija 4–0 Tochigi SC Shonan Bellmare 2–1 Ehime FC Mito HollyHock 4–1 Gainare Tottori Sanfrecce Hiroshima 3–0 Kanazawa ALO's Hokuriku 2–1 Okinawa Kariyushi Consadole Sapporo 3–2 Shizuoka Sangyo University Montedio Yamagata 6–1 Ritsumeikan University Sagawa Express Tokyo 2–1 Honda Luminoso Sayama Otsuka Pharmaceuticals 6–0 TDK Ventforet Kofu 2–1 Momoyama Gakuin University Third round Yokohama F. Marinos 2–2(PK 4–1) Funabashi Municipal High School Gamba Osaka 3–1 Consadole Sapporo Júbilo Iwata 2–0 Sagawa Express Tokyo Tokyo Verdy 2–1 Ventforet Kofu Cerezo Osaka 4–1 ALO's Hokuriku Nagoya Grampus Eight 1–0 Yokohama Kashiwa Reysol 1–0 Omiya Ardija Sanfrecce Hiroshima 2–0 Kyoto Purple Sanga Shimizu S-Pulse 2–0 Mito HollyHock Kashima Antlers 3–2 Avispa Fukuoka Kawasaki Frontale 3–0 Oita Trinita Albirex Niigata 2–1 Vegalta Sendai JEF United Ichihara 5–0 Otsuka Pharmaceuticals Shonan Bellmare 2–1 Urawa Red Diamonds Vissel Kobe 3–0 Montedio Yamagata FC Tokyo 2–2 Honda FC Fourth round Kashima Antlers 3–2 Kashiwa Reysol Yokohama F. Marinos 2–1 Sanfrecce Hiroshima JEF United Ichihara 5–0 Kawasaki Frontale Júbilo Iwata 4–0 Albirex Niigata Cerezo Osaka 3–2 Gamba Osaka Shimizu S-Pulse 2–1 Sh
https://en.wikipedia.org/wiki/2004%20Emperor%27s%20Cup	The 84th Emperor's Cup Statistics of Emperor's Cup in the 2004 season. Overview It was contested by 80 teams, and Tokyo Verdy won the cup for the fifth time. Results First round Hiroshima University of Economics 4–0 Sanyo Electric Tokushima Gainare Tottori 3–2 TDK Mitsubishi Motors Mizushima 3–2 Renaiss Gakuen Kōga Alouette Kumamoto 5–2 Mitsubishi Heavy Industries Nagasaki Oita Trinita U-18 1–0 Fuji University Hachinohe University 3–1 Central Kobe Saga University 2–0 Haguro High School FC Ryukyu 3–0 Yamaguchi Teachers Kochi University 2–1 Hannan University Honda Lock 4–1 Teikyo Daisan High School Sony Sendai 6–1 Maruoka High School Japan Soccer College 1–0 Chukyo High School Second round Sony Sendai 5–0 Yumoto High School Sagawa Printing 2–1 Mitsubishi Motors Mizushima Momoyama Gakuin University 2–0 National Institute of Fitness and Sports in Kanoya Tochigi SC 1–0 Gainare Tottori FC Ryukyu 1–0 Fukuoka University Tenri University 3–1 Saga University Ehime FC 3–0 Kochi University Tokai University 3–2 Otsuka Pharmaceuticals Sagawa Express Tokyo SC 2–0 Ryutsu Keizai University Honda FC 3–2 Sapporo University Gunma Horikoshi 2–1 Luminoso Sayama Hachinohe University 8–3 Yokkaichi University Thespa Kusatsu 6–0 Mattō FC Oita Trinita U-18 2–1 Kihoku Shūkyūdan Honda Lock 4–1 Nagano Elsa ALO's Hokuriku 2–0 Hiroshima University of Economics Japan Soccer College 0–0 (PK 4–2) JEF United Ichihara Amateur Chukyo University 2–1 Komazawa University Alouette Kumamoto 2–0 Takamatsu Kita High School FC Central Chūgoku 1–1 (PK 5–4) Shizuoka FC Third round Thespa Kusatsu 1–0 Momoyama Gakuin University Shonan Bellmare 1–0 Hachinohe University Ventforet Kofu 1–0 Sony Sendai FC Omiya Ardija 2–1 ALO's Hokuriku Gunma Horikoshi 2–1 Tokai University Yokohama FC 4–0 Oita Trinita U-18 Avispa Fukuoka 9–1 Tenri University Montedio Yamagata 3–2 FC Ryukyu Sagan Tosu 2–0 Tochigi SC Consadole Sapporo 2–1 Honda Lock Vegalta Sendai 2–0 Sagawa Printing Kyoto Purple Sanga 11–2 Japan Soccer College Kawasaki Frontale 4–0 Ehime FC Mito HollyHock 4–0 Alouette Kumamoto Honda FC 8–0 FC Central Chūgoku Sagawa Express Tokyo SC 3–2 Chukyo University Fourth round Yokohama F. Marinos 2–1 Montedio Yamagata Gamba Osaka 3–1 Sagan Tosu Gunma Horikoshi 1–0 Kashiwa Reysol Omiya Ardija 1–0 Shimizu S-Pulse Júbilo Iwata 3–2 Sagawa Express Tokyo SC FC Tokyo 1–0 Vegalta Sendai Kashima Antlers 1–0 Mito HollyHock Tokyo Verdy 2–1 Kyoto Purple Sanga Shonan Bellmare 3–2 Albirex Niigata Avispa Fukuoka 1–3 Urawa Red Diamonds Thespa Kusatsu 2–1 Cerezo Osaka Consadole Sapporo 2–1 JEF United Ichihara Kawasaki Frontale 3–2 Vissel Kobe Yokohama FC 1–0 Sanfrecce Hiroshima Oita Trinita 2–1 Ventforet Kofu Nagoya Grampus Eight 3–0 Honda FC Fifth round Júbilo Iwata 2–1 Gunma Horikoshi Tokyo Verdy 2–1 Nagoya Grampus Eight Kashima Antlers 3–2 Kawasaki Frontale Consadole Sapporo 1–0 Oita Trinita Gamba Osaka 5–0 Yokohama FC FC Tokyo 6–3 Omiya Ardija Yokohama F. Marinos 1–2 Thespa Kusatsu Urawa Red Diamonds 3–0 Shonan Bell
https://en.wikipedia.org/wiki/Double%20limit%20theorem	In hyperbolic geometry, Thurston's double limit theorem gives condition for a sequence of quasi-Fuchsian groups to have a convergent subsequence. It was introduced in and is a major step in Thurston's proof of the hyperbolization theorem for the case of manifolds that fiber over the circle. Statement By Bers's theorem, quasi-Fuchsian groups (of some fixed genus) are parameterized by points in T×T, where T is Teichmüller space of the same genus. Suppose that there is a sequence of quasi-Fuchsian groups corresponding to points (gi, hi) in T×T. Also suppose that the sequences gi, hi converge to points μ,μ in the Thurston boundary of Teichmüller space of projective measured laminations. If the points μ,μ have the property that any nonzero measured lamination has positive intersection number with at least one of them, then the sequence of quasi-Fuchsian groups has a subsequence that converges algebraically. References Translated into English as Kleinian groups Hyperbolic geometry Theorems in geometry
https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Tur%C3%A1n%20conjecture%20on%20additive%20bases	The Erdős–Turán conjecture is an old unsolved problem in additive number theory (not to be confused with Erdős conjecture on arithmetic progressions) posed by Paul Erdős and Pál Turán in 1941. The question concerns subsets of the natural numbers, typically denoted by , called additive bases. A subset is called an (asymptotic) additive basis of finite order if there is some positive integer such that every sufficiently large natural number can be written as the sum of at most elements of . For example, the natural numbers are themselves an additive basis of order 1, since every natural number is trivially a sum of at most one natural number. Lagrange's four-square theorem says that the set of positive square numbers is an additive basis of order 4. Another highly non-trivial and celebrated result along these lines is Vinogradov's theorem. One is naturally inclined to ask whether these results are optimal. It turns out that Lagrange's four-square theorem cannot be improved, as there are infinitely many positive integers which are not the sum of three squares. This is because no positive integer which is the sum of three squares can leave a remainder of 7 when divided by 8. However, one should perhaps expect that a set which is about as sparse as the squares (meaning that in a given interval , roughly of the integers in lie in ) which does not have this obvious deficit should have the property that every sufficiently large positive integer is the sum of three elements from . This follows from the following probabilistic model: suppose that is a positive integer, and are 'randomly' selected from . Then the probability of a given element from being chosen is roughly . One can then estimate the expected value, which in this case will be quite large. Thus, we 'expect' that there are many representations of as a sum of three elements from , unless there is some arithmetic obstruction (which means that is somehow quite different than a 'typical' set of the same density), like with the squares. Therefore, one should expect that the squares are quite inefficient at representing positive integers as the sum of four elements, since there should already be lots of representations as sums of three elements for those positive integers that passed the arithmetic obstruction. Examining Vinogradov's theorem quickly reveals that the primes are also very inefficient at representing positive integers as the sum of four primes, for instance. This begets the question: suppose that , unlike the squares or the prime numbers, is very efficient at representing positive integers as a sum of elements of . How efficient can it be? The best possibility is that we can find a positive integer and a set such that every positive integer is the sum of at most elements of in exactly one way. Failing that, perhaps we can find a such that every positive integer is the sum of at most elements of in at least one way and at most ways, where is a function of
https://en.wikipedia.org/wiki/Plancherel%20measure	In mathematics, Plancherel measure is a measure defined on the set of irreducible unitary representations of a locally compact group , that describes how the regular representation breaks up into irreducible unitary representations. In some cases the term Plancherel measure is applied specifically in the context of the group being the finite symmetric group – see below. It is named after the Swiss mathematician Michel Plancherel for his work in representation theory. Definition for finite groups Let be a finite group, we denote the set of its irreducible representations by . The corresponding Plancherel measure over the set is defined by where , and denotes the dimension of the irreducible representation . Definition on the symmetric group An important special case is the case of the finite symmetric group , where is a positive integer. For this group, the set of irreducible representations is in natural bijection with the set of integer partitions of . For an irreducible representation associated with an integer partition , its dimension is known to be equal to , the number of standard Young tableaux of shape , so in this case Plancherel measure is often thought of as a measure on the set of integer partitions of given order n, given by The fact that those probabilities sum up to 1 follows from the combinatorial identity which corresponds to the bijective nature of the Robinson–Schensted correspondence. Application Plancherel measure appears naturally in combinatorial and probabilistic problems, especially in the study of longest increasing subsequence of a random permutation . As a result of its importance in that area, in many current research papers the term Plancherel measure almost exclusively refers to the case of the symmetric group . Connection to longest increasing subsequence Let denote the length of a longest increasing subsequence of a random permutation in chosen according to the uniform distribution. Let denote the shape of the corresponding Young tableaux related to by the Robinson–Schensted correspondence. Then the following identity holds: where denotes the length of the first row of . Furthermore, from the fact that the Robinson–Schensted correspondence is bijective it follows that the distribution of is exactly the Plancherel measure on . So, to understand the behavior of , it is natural to look at with chosen according to the Plancherel measure in , since these two random variables have the same probability distribution. Poissonized Plancherel measure Plancherel measure is defined on for each integer . In various studies of the asymptotic behavior of as , it has proved useful to extend the measure to a measure, called the Poissonized Plancherel measure, on the set of all integer partitions. For any , the Poissonized Plancherel measure with parameter on the set is defined by for all . Plancherel growth process The Plancherel growth process is a random sequence of Young diagrams su
https://en.wikipedia.org/wiki/Teichm%C3%BCller%20modular%20form	In mathematics, a Teichmüller modular form is an analogue of a Siegel modular form on Teichmüller space. References Automorphic forms
https://en.wikipedia.org/wiki/1996%20J.League%20Cup	Statistics of J. League Cup, officially the '96 J.League Yamazaki Nabisco Cup, in the 1996 season. Overview It was contested by 16 teams, and Shimizu S-Pulse won the championship. Results Group A Group B Semifinals Kashiwa Reysol 1–2 Verdy Kawasaki Shimizu S-Pulse 5–0 Bellmare Hiratsuka Final Verdy Kawasaki 3–3 (PK 4–5) Shimizu S-Pulse Shimizu S-Pulse won the cup References rsssf J. League J.League Cup Lea
https://en.wikipedia.org/wiki/1997%20J.League%20Cup	Statistics of J. League Cup, officially the '97 J.League Yamazaki Nabisco Cup, in the 1997 season. Overview It was contested by 20 teams, and Kashima Antlers won the cup. Results Group A Group B Group C Group D Group E Quarterfinals Nagoya Grampus Eight 4–0 ; 1–1 JEF United Ichihara Consadole Sapporo 1–2 ; 0–7 Kashima Antlers Urawa Red Diamonds 0–0 ; 2–3 Jubilo Iwata Yokohama Flugels 0–1 ; 3–0 Kashiwa Reysol Semifinals Kashima Antlers 1–0 ; 0–0 Nagoya Grampus Eight Yokohama Flugels 1–0 ; 0–2 Jubilo Iwata Final Jubilo Iwata 1–2 ; 1–5 Kashima Antlers Kashima Antlers won the cup. References rsssf J. League J.League Cup Lea
https://en.wikipedia.org/wiki/1998%20J.League%20Cup	Statistics of J. League Cup, officially the '1998 J.League Yamazaki Nabisco Cup, in the 1998 season. Overview It was contested by 20 teams, and Jubilo Iwata won the cup. Results Group A Group B Group C Group D Semifinals Shimizu S-Pulse 0–2 Jubilo Iwata JEF United Ichihara 3–2 Kashima Antlers Final Jubilo Iwata 4–0 JEF United Ichihara Jubilo Iwata won the cup. References rsssf J. League J.League Cup Lea
https://en.wikipedia.org/wiki/1999%20J.League%20Cup	Statistics of J. League Cup, officially the '99 J.League Yamazaki Nabisco Cup, in the 1999 season. Overview It was contested by 26 teams, and Kashiwa Reysol won the championship. Results 1st round Consadole Sapporo 1–0 ; 0–3 Avispa Fukuoka Albirex Niigata 0–3 ; 0–2 Kashiwa Reysol Vegalta Sendai 1–2 ; 1–4 Sanfrecce Hiroshima Montedio Yamagata 0–5 ; 1–4 Kyoto Purple Sanga Omiya Ardija 1–1 ; 0–3 Yokohama F. Marinos FC Tokyo 1–1 ; 2–1 Vissel Kobe Kawasaki Frontale 1–3 ; 1–0 Gamba Osaka Ventforet Kofu 0–2 ; 1–1 Verdy Kawasaki Sagan Tosu 0–3 ; 0–2 Cerezo Osaka Oita Trinita 2–1 ; 0–0 Bellmare Hiratsuka 2nd round Avispa Fukuoka 1–1 ; 0–1 Jubilo Iwata Cerezo Osaka 0–2 ; 2–1 Kashiwa Reysol Kyoto Purple Sanga 1–0 ; 0–2 Shimizu S-Pulse Verdy Kawasaki 0–3 ; 2–4 Nagoya Grampus Eight FC Tokyo 1–2 ; 4–1 JEF United Ichihara Sanfrecce Hiroshima 2–3 ; 0–1 Yokohama F. Marinos Gamba Osaka 1–1 ; 0–1 Kashima Antlers Oita Trinita 1–0 ; 1–3 Urawa Red Diamonds Quarterfinals Kashiwa Reysol 1–1 ; 2–0 Jubilo Iwata Nagoya Grampus Eight 3–2 ; 0–0 Shimizu S-Pulse Yokohama F. Marinos 0–3 ; 2–0 FC Tokyo Urawa Red Diamonds 2–0 ; 0–3 Kashima Antlers Semifinals Nagoya Grampus Eight 1–3 ; 2–1 Kashiwa Reysol Kashima Antlers 2–0 ; 1–1 FC Tokyo Final Kashiwa Reysol 2–2 (PK 5–4) Kashima Antlers Kashiwa Reysol won the championship. References rsssf J. League J.League Cup Lea
https://en.wikipedia.org/wiki/2000%20J.League%20Cup	Statistics of J. League Cup, officially the 2000 J.League Yamazaki Nabisco Cup, in the 2000 season. Overview It was contested by 27 teams, and Kashima Antlers won the cup. There were 22 teams played the first round, with 5 teams getting byes to the second round. Results 1st round Montedio Yamagata 0–3 ; 1–0 Sanfrecce Hiroshima Omiya Ardija 0–4 ; 0–2 Vissel Kobe Kawasaki Frontale 3–0 ; 1–2 Urawa Red Diamonds Shonan Bellmare 2–3 ; 0–0 Avispa Fukuoka Ventforet Kofu 0–2 ; 1–5 Yokohama F. Marinos Albirex Niigata 0–1 ; 1–3 Kyoto Purple Sanga Cerezo Osaka 2–0 ; 1–0 Vegalta Sendai Sagan Tosu 0–1 ; 1–2 Verdy Kawasaki Oita Trinita 2–2 ; 1–3 JEF United Ichihara Gamba Osaka 2–1 ; 1–0 Consadole Sapporo Shimizu S-Pulse 4–1 ; 3–1 Mito HollyHock 2nd round Yokohama F. Marinos 4–1 ; 0–1 Sanfrecce Hiroshima Kawasaki Frontale 1–0 ; 1–1 Kashiwa Reysol Verdy Kawasaki 1–0 ; 1–0 Cerezo Osaka Kyoto Purple Sanga 1–1 ; 0–1 FC Tokyo Gamba Osaka 0–1 ; 2–1 Jubilo Iwata Avispa Fukuoka 1–1 ; 2–3 Kashima Antlers JEF United Ichihara 1–1 ; 1–2 Nagoya Grampus Eight Vissel Kobe 2–0 ; 0–4 Shimizu S-Pulse Quarterfinals Verdy Kawasaki 0–0 ; 0–2 Kawasaki Frontale Jubilo Iwata 1–1 ; 1–2 Kyoto Purple Sanga Yokohama F. Marinos 1–2 ; 1–1 Kashima Antlers Shimizu S-Pulse 4–6 ; 0–0 Nagoya Grampus Eight Semifinals Kyoto Purple Sanga 0–2 ; 2–1 Kawasaki Frontale Nagoya Grampus Eight 1–3 ; 2–3 Kashima Antlers Final Kashima Antlers 2–0 Kawasaki Frontale Kashima Antlers won the cup. References rsssf J. League J.League Cup Lea
https://en.wikipedia.org/wiki/2001%20J.League%20Cup	Statistics of J. League Cup, officially the 2001 J.League Yamazaki Nabisco Cup, in the 2001 season. Overview It was contested by 28 teams, and Yokohama F. Marinos won the championship. Results 1st round The first legs were played on 4 April, and the second legs were played on 18 April. 12 teams from the Division 1 and all 12 teams from the Division 2 entered this round. \|} 1st Leg 2nd Leg 2nd round The first legs were played on 13 June, and the second legs were played on 20 June. The 4 remaining teams from the Division 1 entered this round. \|} 1st Leg 2nd Leg Quarterfinals The first legs were played on 8 August, and the second legs were played from 22 to 29 August. \|} 1st Leg 2nd Leg Semifinals The first legs were played on 26 September, and the second legs were played on 10 October. \|} 1st Leg 2nd Leg Final Yokohama F. Marinos won the championship. References rsssf J. League J.League Cup League Cup
https://en.wikipedia.org/wiki/2002%20J.League%20Cup	Statistics of J. League Cup, officially the 2002 J.League Yamazaki Nabisco Cup, in the 2002 season. Overview It was contested by 16 teams, and Kashima Antlers won the championship. Results Group A Group B Group C Group D Quarterfinals Semifinals Final Kashima Antlers won the championship. References rsssf J. League J.League Cup Lea
https://en.wikipedia.org/wiki/1976%20JSL%20Cup	Statistics of JSL Cup in the 1976 season. Overview It was contested by 20 teams, and Hitachi won the championship. Results East-A East-B West-A West-B Quarterfinals Mitsubishi Motors 2-0 Honda Eidai 2-1 Fujita Industries Hitachi 2-1 Toyota Motors Toyo Industries 1-3 Furukawa Electric Semifinals Mitsubishi Motors 0-1 Eidai Hitachi 1-1 (PK 2–0) Furukawa Electric Final Eidai 0-1 Hitachi Hitachi won the championship References JSL Cup League Cup
https://en.wikipedia.org/wiki/1978%20JSL%20Cup	Statistics of JSL Cup in the 1978 season. Overview It was contested by 20 teams, and Mitsubishi Motors won the championship. Results Group A Group B Group C Group D Quarterfinals Fujita Industries 2-1 Honda Nippon Steel 1-0 Furukawa Electric Mitsubishi Motors 2-0 Yanmar Diesel Nippon Kokan 1-2 Yomiuri Semifinals Fujita Industries 3-2 Nippon Steel Mitsubishi Motors 2-0 Yomiuri Final Fujita Industries 1-2 Mitsubishi Motors Mitsubishi Motors won the championship References JSL Cup League Cup
https://en.wikipedia.org/wiki/1979%20JSL%20Cup	Statistics of JSL Cup in the 1979 season. Overview It was contested by 20 teams, and Yomiuri won the championship. Results 1st round Yanmar Diesel 5-0 Toshiba Horikawa Nippon Steel 2-1 Kofu Mitsubishi Motors 2-1 Toyo Industries Furukawa Electric 2-0 Nippon Kokan 2nd round Yamaha Motors 3-1 Nissan Motors Yanmar Diesel 1-2 Yomiuri Toyota Motors 1-1 (PK 5–4) Nippon Steel Teijin Matsuyama 1-2 Fujitsu Fujita Industries 2-0 Sumitomo Metals Mitsubishi Motors 1-0 Yanmar Club Honda 0-1 Furukawa Electric Hitachi 4-0 Tanabe Pharmaceuticals Quarterfinals Yamaha Motors 2-4 Yomiuri Toyota Motors 1-2 Fujitsu Fujita Industries 0-3 Mitsubishi Motors Furukawa Electric 2-1 Hitachi Semifinals Yomiuri 3-3 (PK 4–3) Fujitsu Mitsubishi Motors 2-2 (PK 5–4) Furukawa Electric Final Yomiuri 3-2 Furukawa Electric Yomiuri won the championship References JSL Cup League Cup
https://en.wikipedia.org/wiki/1980%20JSL%20Cup	Statistics of JSL Cup in the 1980 season. Overview It was contested by 20 teams, and Nippon Kokan won the championship. Results 1st round Yomiuri 3-2 Yamaha Motors Furukawa Electric 3-2 Fujita Industries Honda 3-0 Tanabe Pharmaceuticals Yanmar Diesel 2-0 Fujitsu 2nd round Hitachi 5-1 Sumitomo Metals Yomiuri 1-3 Toyo Industries Mitsubishi Motors 0-1 Furukawa Electric Nissan Motors 2-0 Daikyo Oil Nippon Steel 2-4 Nippon Kokan Honda 0-1 Toshiba Kofu 0-3 Yanmar Diesel Teijin Matsuyama 4-2 Toyota Motors Quarterfinals Hitachi 4-1 Toyo Industries Furukawa Electric 1-1 (PK 4–5) Nissan Motors Nippon Kokan 1-0 Toshiba Yanmar Diesel 1-2 Teijin Matsuyama Semifinals Hitachi 2-0 Nissan Motors Nippon Kokan 2-1 Teijin Matsuyama Final Hitachi 1-3 Nippon Kokan Nippon Kokan won the championship References JSL Cup League Cup
https://en.wikipedia.org/wiki/1981%20JSL%20Cup	Statistics of JSL Cup in the 1981 season. Overview It was contested by 20 teams, and Toshiba and Mitsubishi Motors won the championship. Results 1st round Honda 3-0 Tanabe Pharmaceuticals Toshiba 2-1 Sumitomo Metals Yomiuri 1-2 Nippon Kokan Mitsubishi Motors 4-0 Teijin Matsuyama 2nd round Kofu 3-2 Nagoya Honda 0-3 Furukawa Electric Yamaha Motors 2-4 Toshiba Nissan Motors 3-0 Nippon Steel Hitachi 5-1 Toyota Motors Nippon Kokan 1-2 Fujita Industries Yanmar Diesel 3-5 Mitsubishi Motors Mazda 0-2 Fujitsu Quarterfinals Kofu 1-6 Furukawa Electric Toshiba 3-1 Nissan Motors Hitachi 3-5 Fujita Industries Mitsubishi Motors 1-1 (PK 5–3) Fujitsu Semifinals Furukawa Electric 2-2 (PK 2–3) Toshiba Fujita Industries 2-4 Mitsubishi Motors Final Toshiba 4-4 Mitsubishi Motors Toshiba and Mitsubishi Motors won the championship References JSL Cup League Cup
https://en.wikipedia.org/wiki/1982%20JSL%20Cup	Statistics of JSL Cup in the 1982 season. Overview It was contested by 20 teams, and Furukawa Electric won the championship. Results 1st round Furukawa Electric 3-1 Yomiuri Teijin 2-1 Saitama Teachers Hitachi 0-1 Nippon Steel Fujita Industries 10-0 Kofu 2nd round Mitsubishi Motors 4-1 Toyota Motors Furukawa Electric 6-1 Nissan Motors Sumitomo Metals 1-1 (PK 4–1) Teijin Nippon Kokan 1-1 (PK 5–4) Fujitsu Yamaha Motors 1-0 Tanabe Pharmaceuticals Nippon Steel 1-1 (PK 3–4) Toshiba Yanmar Diesel 1-1 (PK 7–6) Fujita Industries Mazda 1-3 Honda Quarterfinals Mitsubishi Motors 3-4 Furukawa Electric Sumitomo Metals 0-6 Nippon Kokan Yamaha Motors 1-0 Toshiba Yanmar Diesel 3-0 Honda Semifinals Furukawa Electric 1-1 (PK 4–3) Nippon Kokan Yamaha Motors 0-0 (PK 2–4) Yanmar Diesel Final Furukawa Electric 3-2 Yanmar Diesel Furukawa Electric won the championship References JSL Cup League Cup
https://en.wikipedia.org/wiki/1983%20JSL%20Cup	Statistics of JSL Cup in the 1983 season. Overview It was contested by 20 teams, and Yanmar Diesel won the championship. Results 1st round Yomiuri 1-1 (PK 3–1) Nippon Steel Kofu 1-1 (PK 4–1) Toho Titanium Furukawa Electric 0-1 Nippon Kokan Tanabe Pharmaceuticals 1-0 Saitama Teachers 2nd round Yanmar Diesel 5-1 Mitsubishi Motors Yomiuri 4-2 Fujitsu Toyota Motors 3-0 Kofu Fujita Industries 1-0 Mazda Toshiba 2-2 (PK 4–2) Yamaha Motors Nippon Kokan 2-1 Sumitomo Metals Nissan Motors 3-1 Tanabe Pharmaceuticals Honda 3-1 Hitachi Quarterfinals Yanmar Diesel 1-0 Yomiuri Toyota Motors 1-3 Fujita Industries Toshiba 1-3 Nippon Kokan Nissan Motors 2-1 Honda Semifinals Yanmar Diesel 1-1 (PK 3–2) Fujita Industries Nippon Kokan 3-3 (PK 3–4) Nissan Motors Final Yanmar Diesel 1-0 Nissan Motors Yanmar Diesel won the championship References JSL Cup League Cup
https://en.wikipedia.org/wiki/1984%20JSL%20Cup	Statistics of JSL Cup in the 1984 season. Overview It was contested by 20 teams, and Yanmar Diesel won the championship. Results 1st round Nippon Kokan 2-0 All Nippon Airways Yomiuri 4-2 Fujitsu Nissan Motors 1-0 Toyota Motors Furukawa Electric 1-1 (PK 3–2) Nippon Steel 2nd round Yamaha Motors 0-1 Matsushita Electric Nippon Kokan 0-2 Hitachi Mazda 4-1 Yomiuri Yanmar Diesel 3-1 Sumitomo Metals Honda 3-0 Kofu Nissan Motors 0-0 (PK 4–5) Mitsubishi Motors Fujita Industries 0-2 Furukawa Electric Toshiba 2-1 Tanabe Pharmaceuticals Quarterfinals Matsushita Electric 1-2 Hitachi Mazda 0-3 Yanmar Diesel Honda 2-0 Mitsubishi Motors Furukawa Electric 0-1 Toshiba Semifinals Hitachi 2-2 (PK 3–4) Yanmar Diesel Honda 0-1 Toshiba Final Yanmar Diesel 3-0 Toshiba Yanmar Diesel won the cup References JSL Cup League Cup
https://en.wikipedia.org/wiki/1985%20JSL%20Cup	Statistics of JSL Cup in the 1985 season. Overview It was contested by 20 teams, and Yomiuri won the championship. Results 1st round Yanmar Diesel 3-1 Kofu Nippon Kokan 5-0 Kyoto Police Sumitomo Metals 1-2 Matsushita Electric Furukawa Electric 5-1 Tanabe Pharmaceuticals Hitachi 0-1 Toshiba Mazda 1-1 (PK 5–6) Toyota Motors Honda 6-0 TDK Fujita Industries 2-0 Nippon Steel 2nd round Nissan Motors 5-0 Yanmar Diesel Nippon Kokan 1-2 Mitsubishi Motors Seino Transportations 0-2 Matsushita Electric Furukawa Electric 4-1 Osaka Gas Yomiuri 5-2 Toshiba Toyota Motors 0-1 Yamaha Motors Fujitsu 0-3 Honda Fujita Industries 1-1 (PK 5–4) All Nippon Airways Quarterfinals Nissan Motors 4-0 Mitsubishi Motors Matsushita Electric 0-1 Furukawa Electric Yomiuri 3-1 Yamaha Motors Honda 1-1 (PK 5–4) Fujita Industries Semifinals Nissan Motors 1-0 Furukawa Electric Yomiuri 3-1 Honda Final Nissan Motors 0-2 Yomiuri Yomiuri won the championship References JSL Cup League Cup
https://en.wikipedia.org/wiki/1986%20JSL%20Cup	Statistics of JSL Cup in the 1986 season. Overview It was contested by 27 teams, and Furukawa Electric won the championship. Results 1st round Furukawa Electric 4-1 Tanabe Pharmaceuticals Fujita Industries 4-0 Osaka Gas Yamaha Motors 3-1 Seino Transportations Honda 9-0 Kyoto Police Mitsubishi Motors 3-2 Mazda Fujitsu 2-2 (PK 4–3) Toho Titanium Nippon Kokan 1-1 (PK 3–0) Matsushita Electric Kawasaki Steel 2-0 TDK Toshiba 2-1 Nippon Steel Sumitomo Metals 3-1 Toyota Motors Yomiuri 0-2 Nissan Motors 2nd round NTT Kansai 0-10 Furukawa Electric Fujita Industries 3-0 Yamaha Motors Honda 1-0 Mitsubishi Motors Kofu 2-1 Cosmo Oil Yanmar Diesel 3-0 Fujitsu Nippon Kokan 2-0 Kawasaki Steel Toshiba 2-0 Sumitomo Metals Nissan Motors 1-0 Hitachi Quarterfinals Furukawa Electric 1-0 Fujita Industries Honda 2-2 (PK 5–4) Kofu Yanmar Diesel 2-1 Nippon Kokan Toshiba 0-3 Nissan Motors Semifinals Furukawa Electric 0-0 (PK 4–3) Honda Yanmar Diesel 1-1 Nissan Motors (PK 3–4) Final Furukawa Electric 4-0 Nissan Motors Furukawa Electric won the championship References JSL Cup League Cup
https://en.wikipedia.org/wiki/1987%20JSL%20Cup	Statistics of JSL Cup in the 1987 season. Overview It was contested by 28 teams, and Nippon Kokan won the championship. Results 1st round Fujitsu 2-0 Kawasaki Steel Mitsubishi Motors 1-1 (PK 4–3) Toyota Motors Toshiba 4-0 Kofu Yamaha Motors 3-0 Toho Titanium Nippon Kokan 4-0 Seino Transportations Fujita Industries 4-0 Tanabe Pharmaceuticals All Nippon Airways 3-2 NTT Kansai Matsushita Electric 3-0 Osaka Gas Yomiuri 1-3 Sumitomo Metals Mazda Auto Hiroshima 2-2 (PK 5–4) NTT Kanto Hitachi 3-0 Cosmo Oil 2nd round Honda 3-1 Fujitsu Mitsubishi Motors 1-0 Mazda Toshiba 1-0 Yamaha Motors Nippon Kokan 1-1 (PK 5–4) Nissan Motors Furukawa Electric 1-0 Fujita Industries All Nippon Airways 1-0 Matsushita Electric Sumitomo Metals 7-1 Mazda Auto Hiroshima Hitachi 0-4 Yanmar Diesel Quarterfinals Honda 3-0 Mitsubishi Motors Toshiba 0-3 Nippon Kokan Furukawa Electric 1-1 (PK 7–8) All Nippon Airways Sumitomo Metals 2-1 Yanmar Diesel Semifinals Honda 0-3 Nippon Kokan All Nippon Airways 0-0 (PK 3–4) Sumitomo Metals Final Nippon Kokan 3-1 Sumitomo Metals Nippon Kokan won the championship References JSL Cup League Cup
https://en.wikipedia.org/wiki/1988%20JSL%20Cup	Statistics of JSL Cup in the 1988 season. Overview It was contested by 28 teams, and Nissan Motors won the championship. Results 1st round Nissan Motors 7-0 Osaka Gas Yomiuri 5-0 Cosmo Oil Hitachi 3-1 Kofu Mitsubishi Motors 4-1 Kawasaki Steel Fujita Industries 3-0 Fujitsu Yamaha Motors 3-0 Teijin Mazda 2-1 NTT Kansai Toshiba 1-0 Tanabe Pharmaceuticals Furukawa Electric 1-0 Nippon Steel Yanmar Diesel 4-0 Fujieda City Office Matsushita Electric 0-0 (PK 3–4) NTT Kanto Toyota Motors 5-0 Toho Titanium 2nd round NKK 1-3 Nissan Motors Yomiuri 1-0 Hitachi Mitsubishi Motors 2-1 Fujita Industries Yamaha Motors 0-2 All Nippon Airways Honda 1-1 (PK 3–4) Mazda Toshiba 3-2 Furukawa Electric Yanmar Diesel 1-1 (PK 5–4) NTT Kanto Toyota Motors 2-3 Sumitomo Metals Quarterfinals Nissan Motors 1-0 Yomiuri Mitsubishi Motors 2-1 All Nippon Airways Mazda 0-0 (PK 1–4) Toshiba Yanmar Diesel 5-1 Sumitomo Metals Semifinals Nissan Motors 2-0 Mitsubishi Motors Toshiba 1-0 Yanmar Diesel Final Nissan Motors 3-0 Toshiba Nissan Motors won the championship References JSL Cup League Cup
https://en.wikipedia.org/wiki/1989%20JSL%20Cup	Statistics of JSL Cup in the 1989 season. Overview It was contested by 28 teams, and Nissan Motors won the championship. Results 1st round Kawasaki Steel 3-2 Mazda Auto Hiroshima All Nippon Airways 2-1 Kofu NKK 4-0 Osaka Gas Fujita Industries 1-1 Kyoto Shiko Furukawa Electric 1-0 Fujitsu Matsushita Electric 0-0 (PK 1–3) Cosmo Oil Yomiuri 2-0 Nippon Steel Toyota Motors 3-0 Toho Titanium NTT Kanto 3-3 (PK 5–4) Tanabe Pharmaceuticals Yamaha Motors 3-0 Sumitomo Metals Hitachi 1-1 (PK 2–4) Mazda Honda 6-0 Teijin 2nd round Nissan Motors 3-2 Kawasaki Steel All Nippon Airways 2-1 NKK Fujita Industries 4-2 Furukawa Electric Cosmo Oil 1-4 Yanmar Diesel Mitsubishi Motors 0-2 Yomiuri Toyota Motors 1-1 (PK 5–4) NTT Kanto Yamaha Motors 1-0 Mazda Honda 1-1 Toshiba Quarterfinals Nissan Motors 2-0 All Nippon Airways Fujita Industries 1-0 Yanmar Diesel Yomiuri 4-2 Toyota Motors Yamaha Motors 2-1 Toshiba Semifinals Nissan Motors 2-1 Fujita Industries Yomiuri 0-1 Yamaha Motors Final Nissan Motors 1-0 Yamaha Motors Nissan Motors won the championship References JSL Cup League Cup
https://en.wikipedia.org/wiki/1990%20JSL%20Cup	Statistics of JSL Cup in the 1990 season. Overview It was contested by 28 teams, and Nissan Motors won the championship. Results 1st round Toyota Motors 2-0 Kawasaki Steel NKK 3-0 Toho Titanium Mazda 2-2 (PK 4–3) Kyoto Shiko All Nippon Airways 4-0 Nippon Steel Matsushita Electric 1-2 Fujitsu Hitachi 1-1 (PK 3–2) Sumitomo Metals Toshiba 3-0 Tanabe Pharmaceuticals Furukawa Electric 4-1 Kofu Yanmar Diesel 1-0 Mitsubishi Motors NTT Kanto 2-0 Osaka Gas Honda 2-1 Yomiuri Juniors Cosmo Oil 2-0 Otsuka Pharmaceutical 2nd round Nissan Motors 3-0 Toyota Motors NKK 0-3 Mazda All Nippon Airways 1-1 (PK 4–2) Fujitsu Hitachi 0-4 Yomiuri Yamaha Motors 1-0 Toshiba Furukawa Electric 1-1 (PK 4–1) Yanmar Diesel NTT Kanto 0-2 Honda Cosmo Oil 3-1 Fujita Industries Quarterfinals Nissan Motors 1-0 Mazda All Nippon Airways 1-0 Yomiuri Yamaha Motors 1-1 (PK 4–5) Furukawa Electric Honda 1-0 Cosmo Oil Semifinals Nissan Motors 1-0 All Nippon Airways Furukawa Electric 0-0 (PK 4–3) Honda Final Nissan Motors 3-1 Furukawa Electric Nissan Motors won the championship References JSL Cup Lea
https://en.wikipedia.org/wiki/1991%20JSL%20Cup	Statistics of JSL Cup in the 1991 season. Overview It was contested by 28 teams, and Yomiuri won the championship. Results 1st round Tanabe Pharmaceuticals 2-1 Cosmo Oil Toshiba 2-0 Otsuka Pharmaceutical Hitachi 8-0 Toho Titanium NKK 1-0 Kyoto Shiko Matsushita Electric 1-0 Tokyo Gas Toyota Motors 3-0 Yomiuri Juniors Yomiuri 6-0 Chuo Bohan Mitsubishi Motors 2-4 Sumitomo Metals Yamaha Motors 1-3 Fujitsu Mazda 3-2 NTT Kanto Yanmar Diesel 1-4 Kofu Fujita Industries 2-0 Kawasaki Steel 2nd round Honda 7-1 Tanabe Pharmaceuticals Toshiba 1-1 (PK 4–5) Hitachi NKK 1-2 Matsushita Electric Toyota Motors 3-3 (PK 4–5) Nissan Motors All Nippon Airways 1-3 Yomiuri Sumitomo Metals 0-3 Fujitsu Mazda 3-1 Kofu Fujita Industries 1-0 Furukawa Electric Quarterfinals Honda 2-2 (PK 6–5) Hitachi Matsushita Electric 0-1 Nissan Motors Yomiuri 4-1 Fujitsu Mazda 1-1 (PK 4–2) Fujita Industries Semifinals Honda 0-0 (PK 4–2) Nissan Motors Yomiuri 1-0 Mazda Final Honda 3-4 Yomiuri Yomiuri won the championship References JSL Cup Lea
https://en.wikipedia.org/wiki/Lutz%20Gerresheim	Lutz Gerresheim (born 19 September 1958 – 10 March 1980) was a German footballer who played as a midfielder. Career statistics References External links 1958 births 1980 deaths German men's footballers Footballers from Bochum Men's association football midfielders Bundesliga players 2. Bundesliga players SC Westfalia Herne players VfL Bochum players
https://en.wikipedia.org/wiki/Amorphous%20set	In set theory, an amorphous set is an infinite set which is not the disjoint union of two infinite subsets. Existence Amorphous sets cannot exist if the axiom of choice is assumed. Fraenkel constructed a permutation model of Zermelo–Fraenkel with Atoms in which the set of atoms is an amorphous set. After Cohen's initial work on forcing in 1963, proofs of the consistency of amorphous sets with Zermelo–Fraenkel were obtained. Additional properties Every amorphous set is Dedekind-finite, meaning that it has no bijection to a proper subset of itself. To see this, suppose that is a set that does have a bijection to a proper subset. For each natural number define to be the set of elements that belong to the image of the -fold composition of with itself but not to the image of the -fold composition. Then each is non-empty, so the union of the sets with even indices would be an infinite set whose complement in is also infinite, showing that cannot be amorphous. However, the converse is not necessarily true: it is consistent for there to exist infinite Dedekind-finite sets that are not amorphous. No amorphous set can be linearly ordered. Because the image of an amorphous set is itself either amorphous or finite, it follows that every function from an amorphous set to a linearly ordered set has only a finite image. The cofinite filter on an amorphous set is an ultrafilter. This is because the complement of each infinite subset must not be infinite, so every subset is either finite or cofinite. Variations If is a partition of an amorphous set into finite subsets, then there must be exactly one integer such that has infinitely many subsets of size ; for, if every size was used finitely many times, or if more than one size was used infinitely many times, this information could be used to coarsen the partition and split into two infinite subsets. If an amorphous set has the additional property that, for every partition , , then it is called strictly amorphous or strongly amorphous, and if there is a finite upper bound on then the set is called bounded amorphous. It is consistent with ZF that amorphous sets exist and are all bounded, or that they exist and are all unbounded. References Axiom of choice Cardinal numbers
https://en.wikipedia.org/wiki/Unfolding	Unfolding may refer to: Mathematics Unfolding (functions), of a manifold Unfolding (geometry), of a polyhedron Deconvolution Other uses Unfolding (DSP implementation) Unfolding (music), in Schenkerian analysis Unfolding (sculpture), by Bernhard Heiliger located near Milwaukee, Wisconsin, United States Equilibrium unfolding, in biochemistry See also Unfold (disambiguation) Unfoldment (disambiguation)
https://en.wikipedia.org/wiki/Dieter%20Zorc	Dieter Zorc (born 17 October 1939 – 16 October 2007) was a German footballer who played as a defender. He was the father of Michael Zorc. Career statistics References External links 1939 births 2007 deaths German men's footballers Bundesliga players VfL Bochum players Men's association football defenders People from Lünen Footballers from Arnsberg (region) West German men's footballers
https://en.wikipedia.org/wiki/Martine%20Nida-R%C3%BCmelin	Martine Nida-Rümelin (born 1957 in Munich) is a philosopher. Biography Nida-Rümelin studied philosophy, psychology, mathematics and political science at the University of Munich. In her doctoral thesis, she discusses the knowledge argument, by the Australian philosopher Frank Jackson, which is directed against a materialist conception of phenomenal consciousness. In it she presents one of the most important arguments, which is based on qualia, i.e., individual instances of subjective, conscious experience. Her transformed version of the Mary's room thought-experiment has been much discussed and coined the "Nida-Rümelin room" by John Perry. In her habilitation she developed a non-reductionist view about the identity of conscious individuals. In 2019, she won the Jean Nicod Prize. She is the daughter of the sculptor Rolf Nida-Rümelin, the granddaughter of the sculptor Wilhelm Nida-Rümelin and the sister of the philosopher and politician Julian Nida-Rümelin. Academic career Since 1999, she has been a professor at the University of Fribourg in Switzerland. Her main areas of interest are philosophy of mind, theory of knowledge and philosophy of language. The major part of her published work is concerned with the special status of conscious individuals and aims at developing a non-materialist account which avoids the weaknesses of traditional dualism. Phenomenal consciousness, identity of conscious beings through time and across possible worlds, and the active role of the subject in its doings are central themes of her research. Rational intuitions and phenomenological reflexion play a prominent role in her philosophical approach. Between 2019 and 2022 she was Visiting Professor at the University of Italian Switzerland. Bibliography Farben und phänomenales Wissen. Eine Kritik materialistischer Theorien des Geistes, Conceptus Sonderband, Academia, St. Augustin 1993. Der besondere Status von Personen: Eine Anomalie für die Theorie praktischer Rationalität. in: Julian Nida-Rümelin und Ulrike Wessels (Hg.): Praktische Rationalität, de Gruyter, Berlin, 1993, S.143-166. Was Mary nicht wissen konnte. Phänomenale Zustände als Gegenstand von Überzeugungen. In Thomas Metzinger (Hg.): Bewußtsein, Beiträge aus der Gegenwartsphilosophie, Schöningh, Paderborn, 1995, S.259-282. Pseudonormal Vision and Color Qualia, in Stuart Hameroff, Alfred Kaszniak und David Chalmers (Hg.) "Toward a Science of Consciousness III, The Third Tucson Discussions and Debates", MIT Press, 1999, S.75-84. Grasping Phenomenal Properties, in Torin Alter & Sven Walter (Hrg.), Phenomenal Belief and Phenomenal Concepts, Oxford University Press, 2004. Der Blick von innen. Zur transtemporalen Identität bewusstseinsfähiger Wesen. Suhrkamp, Frankfurt am Main, 2006. Dualist Emergentism, in Jonathan Cohen & Brian McLaughlin, Contemporary Debates in Philosophy of Mind, Blackwell Publishing. Die Person als Autor ihres Tuns. Bemerkungen zur Deutung neurobiologischer Daten, in Adrian Hold
https://en.wikipedia.org/wiki/Duffin%E2%80%93Schaeffer%20theorem	The Duffin–Schaeffer theorem is a theorem in mathematics, specifically, the Diophantine approximation proposed by R. J. Duffin and A. C. Schaeffer in 1941. It states that if is a real-valued function taking on positive values, then for almost all (with respect to Lebesgue measure), the inequality has infinitely many solutions in coprime integers with if and only if where is Euler's totient function. In 2019, the Duffin–Schaeffer conjecture was proved by Dimitris Koukoulopoulos and James Maynard. Introduction That existence of the rational approximations implies divergence of the series follows from the Borel–Cantelli lemma. The converse implication is the crux of the conjecture. There have been many partial results of the Duffin–Schaeffer conjecture established to date. Paul Erdős established in 1970 that the conjecture holds if there exists a constant such that for every integer we have either or . This was strengthened by Jeffrey Vaaler in 1978 to the case . More recently, this was strengthened to the conjecture being true whenever there exists some such that the series This was done by Haynes, Pollington, and Velani. In 2006, Beresnevich and Velani proved that a Hausdorff measure analogue of the Duffin–Schaeffer conjecture is equivalent to the original Duffin–Schaeffer conjecture, which is a priori weaker. This result was published in the Annals of Mathematics. Proof In July 2019, Dimitris Koukoulopoulos and James Maynard announced a proof of the conjecture. In July 2020, the proof was published in the Annals of Mathematics. Related problems A higher-dimensional analogue of this conjecture was resolved by Vaughan and Pollington in 1990. See also Khinchin's theorem Notes References External links Quanta magazine article about Duffin-Schaeffer conjecture. Numberphile interview with James Maynard about the proof. Conjectures Conjectures that have been proved Diophantine approximation
https://en.wikipedia.org/wiki/Thurston%20boundary	In mathematics, the Thurston boundary of Teichmüller space of a surface is obtained as the boundary of its closure in the projective space of functionals on simple closed curves on the surface. The Thurston boundary can be interpreted as the space of projective measured foliations on the surface. The Thurston boundary of the Teichmüller space of a closed surface of genus is homeomorphic to a sphere of dimension . The action of the mapping class group on the Teichmüller space extends continuously over the union with the boundary. Measured foliations on surfaces Let be a closed surface. A measured foliation on is a foliation on which may admit isolated singularities, together with a transverse measure , i.e. a function which to each arc transverse to the foliation associates a positive real number . The foliation and the measure must be compatible in the sense that the measure is invariant if the arc is deformed with endpoints staying in the same leaf. Let be the space of isotopy classes of closed simple curves on . A measured foliation can be used to define a function as follows: if is any curve let where the supremum is taken over all collections of disjoint arcs which are transverse to (in particular if is a closed leaf of ). Then if the intersection number is defined by: . Two measured foliations are said to be equivalent if they define the same function on (there is a topological criterion for this equivalence via Whitehead moves). The space of projective measured laminations is the image of the set of measured laminations in the projective space via the embedding . If the genus of is at least 2, the space is homeomorphic to the -dimensional sphere (in the case of the torus it is the 2-sphere; there are no measured foliations on the sphere). Compactification of Teichmüller space Embedding in the space of functionals Let be a closed surface. Recall that a point in the Teichmüller space is a pair where is a hyperbolic surface (a Riemannian manifold with sectional curvatures all equal to ) and a homeomorphism, up to a natural equivalence relation. The Teichmüller space can be realised as a space of functionals on the set of isotopy classes of simple closed curves on as follows. If and then is defined to be the length of the unique closed geodesic on in the isotopy class . The map is an embedding of into , which can be used to give the Teichmüller space a topology (the right-hand side being given the product topology). In fact, the map to the projective space is still an embedding: let denote the image of there. Since this space is compact, the closure is compact: it is called the Thurston compactification of the Teichmüller space. The Thurston boundary The boundary is equal to the subset of . The proof also implies that the Thurston compactfification is homeomorphic to the -dimensional closed ball. Applications Pseudo-Anosov diffeomorphisms A diffeomorphism is called pseudo-Anosov if the
https://en.wikipedia.org/wiki/Simultaneous%20uniformization%20theorem	In mathematics, the simultaneous uniformization theorem, proved by , states that it is possible to simultaneously uniformize two different Riemann surfaces of the same genus using a quasi-Fuchsian group of the first kind. The quasi-Fuchsian group is essentially uniquely determined by the two Riemann surfaces, so the space of marked quasi-Fuchsian group of the first kind of some fixed genus g can be identified with the product of two copies of Teichmüller space of the same genus. References Kleinian groups Riemann surfaces
https://en.wikipedia.org/wiki/Theodor%20Sch%C3%B6nemann	Theodor Schönemann, also written Schoenemann (4 April 181216 January 1868), was a German mathematician who obtained several important results in number theory concerning the theory of congruences, which can be found in several publications in Crelle's journal, volumes 17 to 40. Notably, he obtained Hensel's lemma before Hensel, Scholz's reciprocity law before Scholz, and formulated Eisenstein's criterion before Eisenstein. He also studied, under the form of integer polynomials modulo both a prime number and an irreducible polynomial (remaining irreducible modulo that prime number), what can nowadays be recognized as finite fields (more general than those of prime order). He was educated in Königsberg and Berlin, where among his teachers were Jakob Steiner and Carl Gustav Jacob Jacobi. He obtained his doctorate in 1842, after which he became Gymnasialoberlehrer (professor at a gymnasium) in Brandenburg an der Havel. Apart from the mentioned mathematical papers, he also published, mainly after 1850, in mechanics and physical technique. Works Ueber die Bewegung veränderlicher ebener Figuren, welche während der Bewegung sich ähnlich bleiben in ihrer Ebene. 1862 digital References Biography (in German) H. L. Dorwart, Irreducibility of polynomials, American Mathematical Monthly 42 Vol 6 (1935), 369–381, . Reference to Schoenemann on page 370. 1812 births 1868 deaths People from Drezdenko People from the Province of Brandenburg 19th-century German mathematicians Number theorists
https://en.wikipedia.org/wiki/Otto%20Schilling	Otto Franz Georg Schilling (3 November 1911 – 20 June 1973) was a German-American mathematician known as one of the leading algebraists of his time. He was born in Apolda and studied in the 1930s at the Universität Jena and the Universität Göttingen under Emmy Noether. After Noether was forced to leave Germany by the Nazis, he found a new advisor in Helmut Hasse, and obtained his Ph.D. from Marburg University in 1934 on the thesis Über gewisse Beziehungen zwischen der Arithmetik hyperkomplexer Zahlsysteme und algebraischer Zahlkörper. He then was post doc at Trinity College, Cambridge before moving to Institute for Advanced Study 1935–37 and the Johns Hopkins University 1937–39. He became an instructor with the University of Chicago in 1939, promoted to assistant professor 1943, associate 1945 and full professor in 1958. In 1961 he moved to Purdue University. He died in Highland Park, Illinois. His students were, among others, the game theorist Anatol Rapoport and the mathematician Harley Flanders. Articles (typo in Schilling's name) with Saunders Mac Lane: with Saunders Mac Lane: with Saunders Mac Lane: with Saunders Mac Lane: with Irving Kaplansky: Books References 20th-century German mathematicians Algebraists People from Apolda University of Marburg alumni Purdue University faculty University of Chicago faculty 1911 births 1973 deaths 20th-century American mathematicians Emigrants from Nazi Germany to the United States
https://en.wikipedia.org/wiki/Circular%20law	In probability theory, more specifically the study of random matrices, the circular law concerns the distribution of eigenvalues of an random matrix with independent and identically distributed entries in the limit . It asserts that for any sequence of random matrices whose entries are independent and identically distributed random variables, all with mean zero and variance equal to , the limiting spectral distribution is the uniform distribution over the unit disc. Precise statement Let be a sequence of matrix ensembles whose entries are i.i.d. copies of a complex random variable with mean 0 and variance 1. Let denote the eigenvalues of . Define the empirical spectral measure of as With these definitions in mind, the circular law asserts that almost surely (i.e. with probability one), the sequence of measures converges in distribution to the uniform measure on the unit disk. History For random matrices with Gaussian distribution of entries (the Ginibre ensembles), the circular law was established in the 1960s by Jean Ginibre. In the 1980s, Vyacheslav Girko introduced an approach which allowed to establish the circular law for more general distributions. Further progress was made by Zhidong Bai, who established the circular law under certain smoothness assumptions on the distribution. The assumptions were further relaxed in the works of Terence Tao and Van H. Vu, Guangming Pan and Wang Zhou, and Friedrich Götze and Alexander Tikhomirov. Finally, in 2010 Tao and Vu proved the circular law under the minimal assumptions stated above. The circular law result was extended in 1988 by Sommers, Crisanti, Sompolinsky and Stein to an elliptical law for ensembles of matrices with arbitrary correlations. The elliptic and circular laws were further generalized by Aceituno, Rogers and Schomerus to the hypotrochoid law which includes higher order correlations. See also Wigner semicircle distribution References Random matrices
https://en.wikipedia.org/wiki/Gary%20R.%20Mar	Gary R. Mar is an American philosopher and logician specializing in logic, the philosophy of logic, the philosophy of mathematics, analytic philosophy, philosophy of language and linguistics, philosophy of science, computational philosophy, the philosophy of religion, and Asian American philosophy. Professor Mar is a member of the Philosophy Department at Stony Brook University. Gary Mar was the last student to have a Ph.D. directed by Alonzo Church. He is co-author with Donald Kalish and Richard Montague of the second edition of Logic: Techniques of Formal Reasoning. He is also co-author of The Philosophical Computer with Patrick Grim and Paul St. Denis, which uses computer modelling to explore fractal images and chaos in semantic paradoxes. This research was featured in a column by Ian Stewart ('A Partly True Story,' in Scientific American (Feb. 1993, 110–112). This research was also presented at the Kurt Gödel Centenary Symposium, Horizons of Truth at the University of Vienna in April 2006. Gary Mar is the founding director of the Stony Brook Philosophy Department Logic Lab at Stony Brook and the founding director of the Asian American Center at Stony Brook, after being the catalyst for the donation of the Charles B. Wang Asian American Center at Stony Brook University, which at that time was the largest donation in the history of the public education system in New York State. In 2003 he hosted a graduate seminar with Noam Chomsky through the President's Rotating Stars Program, and in 2005 he was instrumental in the awarding of an honorary doctorate to documentary filmmaker Loni Ding, whose landmark series Ancestors in the Americas is an in-depth documentary on the history of Asians in the Americas. Gary Mar has been the recipient of the Chancellor's and President's Award for Excellence in Teaching (1993), the Chancellor's Award for Excellence in University Service (2015), the Alumni Association Outstanding Professor Award (1995), a Pew Foundation Research Fellowship (1995–1996), and was a charter member Stony Brook's Academy of Scholar-Teachers (1996). As a graduate student, Gary Mar was a co-winner of the Rudolf Carnap Dissertation prize (UCLA, 1985). Publications Books Forthcoming: Thinking Matters -- Module I: Critical Thinking as Creative Problem Solving (World Scientific) Forthcoming: Gödel’s Ontological Dreams: Excursions in Logic, Part I (World Scientific) Logic: Techniques of Formal Reasoning(second edition), co-authored with Donald Kalish and Richard Montague (HBJ, 1980, republished by Oxford University Press, 2000) The Philosophical Computer: Exploratory Essays in Philosophical Computer Modeling co-authored with Patrick Grim and Paul St. Denis (M.I.T. Press, 1998) Selected articles include 'Gödel's Ontological Dreams,' in Space, Time and the Limits of Human Understanding, eds. S. Wuppuluri and G. Ghirardi, Springer, (2017), 461–76. 'Hao Wang's Logical Journey,' Journal of Chinese Philosophy, Thematic Issue: Time,
https://en.wikipedia.org/wiki/Muszaki%20Abu%20Bakar	Muszaki Abu Bakar (born 15 March 1989, in Selangor) is a Malaysian footballer who plays for Malaysia Premier League club, Perlis FA as a right-back. External links Muszaki Statistics Malaysian men's footballers Living people 1989 births Footballers from Selangor Perlis F.A. players Negeri Sembilan FC players Men's association football fullbacks
https://en.wikipedia.org/wiki/Meyerhoff%20manifold	In hyperbolic geometry, the Meyerhoff manifold is the arithmetic hyperbolic 3-manifold obtained by surgery on the figure-8 knot complement. It was introduced by as a possible candidate for the hyperbolic 3-manifold of smallest volume, but the Weeks manifold turned out to have slightly smaller volume. It has the second smallest volume of orientable arithmetic hyperbolic 3-manifolds, where is the zeta function of the quartic field of discriminant . Alternatively, where is the polylogarithm and is the absolute value of the complex root (with positive imaginary part) of the quartic . showed that this manifold is arithmetic. See also Gieseking manifold Weeks manifold References 3-manifolds Hyperbolic geometry
https://en.wikipedia.org/wiki/Sun%20Yeneng	Sun Yeneng is Goh Keng Swee Professor of Economics and Professor of Mathematics at the National University of Singapore (NUS). Sun received his B.S. from University of Science and Technology of China in 1983 and his M.S. and Ph.D from University of Illinois at Urbana-Champaign respectively in 1985 and 1989. He joined NUS as a lecturer at the Department of Mathematics in 1989 and was promoted to professor in 2002. He was Raffles Professor of Social Sciences at the Department of Economics in NUS from 2009 to 2015. Sun was formerly appointed as acting head and head in the Department of Economics in 2008 to 2012. He has been Dean of Faculty of Science from 1 July 2020. Research Interests Sun's research interests include mathematical economics, analysis and probability theory. Selected works Awards and honors Sun has been a Economic Theory Fellow of the Society for the Advancement of Economic Theory since 2011, and a Fellow of the Singapore National Academy of Science since 2014. References External links Homepage at NUS Singaporean mathematicians Singaporean economists Academic staff of the National University of Singapore National University of Singapore deans Living people University of Science and Technology of China alumni University of Illinois Urbana-Champaign alumni Year of birth missing (living people)
https://en.wikipedia.org/wiki/Institute%20of%20Statistics%20%28Albania%29	The Institute of Statistics () is an independent public legal entity tasked with producing official statistics in the Republic of Albania. INSTAT is organized at the central level, with regional statistical offices at the local level that operate within its organizational structure, which is approved by a decision of the Assembly, in accordance with the provisions of the legislation in force for independent institutions. Overview The statistical service in the Republic of Albania is carried out by the Institute of Statistics. In 1924, a statistical office was created that kept various economic records at the Ministry of Public Works and Agriculture. The activity of this office was limited to agricultural inventories that included the number of farmers and the type and amount of land use with agricultural and livestock plants, as well as some detailed statistics on industry, trade, export-imports and prices. The statistical service was eventually institutionalized by Decree no. 121, dt. 8/04/1940. The statistical system that followed, was established by Decision no. 35, dt. 13.01.1945 with the creation of the Directorate of Statistics, a subordinate institution of the Council of Ministers. Later, this directorate came under the jurisdiction of the State Planning Commission. Today, pursuant to law no. 17, dated 05.04.2018 "On official statistics", this system consists of a number of institutions responsible for producing official statistics and other institutions, public or otherwise, which produce statistics in various fields for monitoring or implementing their developmental policies. The National Statistical System carries out its activities in accordance with the aforementioned law and the multi-year Statistics Program which is approved by a special decision of the Assembly. See also Demographics of Albania List of national and international statistical services Eurostat References Albania 1940 establishments in Albania
https://en.wikipedia.org/wiki/2011%20Puerto%20Rico%20Islanders%20season	The 2011 season is the Puerto Rico Islanders 8th season over all and their 1st season in the North American Soccer League. This article shows player statistics and all matches that the club have and will play during the 2011 season. Club Technical staff Kit Squad First Team squad As of September 13, 2011. Transfers In Out Match results North American Soccer League CFU Club championship CONCACAF Champions League Standings References 2011 American soccer clubs 2011 season 2011 North American Soccer League season Islanders
https://en.wikipedia.org/wiki/Rips%20machine	In geometric group theory, the Rips machine is a method of studying the action of groups on R-trees. It was introduced in unpublished work of Eliyahu Rips in about 1991. An R-tree is a uniquely arcwise-connected metric space in which every arc is isometric to some real interval. Rips proved the conjecture of Morgan and Shalen that any finitely generated group acting freely on an R-tree is a free product of free abelian and surface groups. Actions of surface groups on R-trees By Bass–Serre theory, a group acting freely on a simplicial tree is free. This is no longer true for R-trees, as Morgan and Shalen showed that the fundamental groups of surfaces of Euler characteristic less than −1 also act freely on a R-trees. They proved that the fundamental group of a connected closed surface S acts freely on an R-tree if and only if S is not one of the 3 nonorientable surfaces of Euler characteristic ≥−1. Applications The Rips machine assigns to a stable isometric action of a finitely generated group G a certain "normal form" approximation of that action by a stable action of G on a simplicial tree and hence a splitting of G in the sense of Bass–Serre theory. Group actions on real trees arise naturally in several contexts in geometric topology: for example as boundary points of the Teichmüller space (every point in the Thurston boundary of the Teichmüller space is represented by a measured geodesic lamination on the surface; this lamination lifts to the universal cover of the surface and a naturally dual object to that lift is an -tree endowed with an isometric action of the fundamental group of the surface), as Gromov-Hausdorff limits of, appropriately rescaled, Kleinian group actions, and so on. The use of -trees machinery provides substantial shortcuts in modern proofs of Thurston's Hyperbolization Theorem for Haken 3-manifolds. Similarly, -trees play a key role in the study of Culler-Vogtmann's Outer space as well as in other areas of geometric group theory; for example, asymptotic cones of groups often have a tree-like structure and give rise to group actions on real trees. The use of -trees, together with Bass–Serre theory, is a key tool in the work of Sela on solving the isomorphism problem for (torsion-free) word-hyperbolic groups, Sela's version of the JSJ-decomposition theory and the work of Sela on the Tarski Conjecture for free groups and the theory of limit groups. References Further reading Hyperbolic geometry Geometric group theory Trees (topology)
https://en.wikipedia.org/wiki/Yao%20graph	In computational geometry, the Yao graph, named after Andrew Yao, is a kind of geometric spanner, a weighted undirected graph connecting a set of geometric points with the property that, for every pair of points in the graph, their shortest path has a length that is within a constant factor of their Euclidean distance. The basic idea underlying the two-dimensional Yao graph is to surround each of the given points by equally spaced rays, partitioning the plane into sectors with equal angles, and to connect each point to its nearest neighbor in each of these sectors. Associated with a Yao graph is an integer parameter which is the number of rays and sectors described above; larger values of produce closer approximations to the Euclidean distance. The stretch factor is at most , where is the angle of the sectors. The same idea can be extended to point sets in more than two dimensions, but the number of sectors required grows exponentially with the dimension. Andrew Yao used these graphs to construct high-dimensional Euclidean minimum spanning trees. Software for drawing Yao graphs Cone-based Spanners in Computational Geometry Algorithms Library (CGAL) See also Theta graph Semi-Yao graph References Computational geometry Geometric graph theory
https://en.wikipedia.org/wiki/John%20R.%20Isbell	John Rolfe Isbell (October 27, 1930 – August 6, 2005) was an American mathematician, for many years a professor of mathematics at the University at Buffalo (SUNY). Biography Isbell was born in Portland, Oregon, the son of an army officer from Isbell, a town in Franklin County, Alabama. He attended several undergraduate institutions, including the University of Chicago, where professor Saunders Mac Lane was a source of inspiration. He began his graduate studies in mathematics at Chicago, briefly studied at Oklahoma A&M University and the University of Kansas, and eventually completed a Ph.D. in game theory at Princeton University in 1954 under the supervision of Albert W. Tucker. After graduation, Isbell was drafted into the U.S. Army, and stationed at the Aberdeen Proving Ground. In the late 1950s he worked at the Institute for Advanced Study in Princeton, New Jersey, from which he then moved to the University of Washington and Case Western Reserve University. He joined the University at Buffalo in 1969, and remained there until his retirement in 2002. Research Isbell published over 140 papers under his own name, and several others under pseudonyms. Isbell published the first paper by John Rainwater, a fictitious mathematician who had been invented by graduate students at the University of Washington in 1952. After Isbell's paper, other mathematicians have published papers using the name "Rainwater" and have acknowledged "Rainwater's assistance" in articles. Isbell published other articles using two additional pseudonyms, M. G. Stanley and H. C. Enos, publishing two under each. Many of his works involved topology and category theory: He was "the leading contributor to the theory of uniform spaces". Isbell duality is a form of duality arising when a mathematical object can be interpreted as a member of two different categories; a standard example is the Stone duality between sober spaces and complete Heyting algebras with sufficiently many points. Isbell was the first to study the category of metric spaces defined by metric spaces and the metric maps between them, and did early work on injective metric spaces and the tight span construction. In abstract algebra, Isbell found a rigorous formulation for the Pierce–Birkhoff conjecture on piecewise-polynomial functions. He also made important contributions to the theory of median algebras. In geometric graph theory, Isbell was the first to prove the bound χ ≤ 7 on the Hadwiger–Nelson problem, the question of how many colors are needed to color the points of the plane in such a way that no two points at unit distance from each other have the same color. See also Isbell conjugacy Isbell's zigzag theorem References External resources Mathematical Reviews Pseudonyms used by Isbell (and other mathematicians): 1930 births 2005 deaths 20th-century American mathematicians 21st-century American mathematicians Category theorists Game theorists Topologists University of Chicago alumni Princeton Un
https://en.wikipedia.org/wiki/Point%20Cloud%20Library	The Point Cloud Library (PCL) is an open-source library of algorithms for point cloud processing tasks and 3D geometry processing, such as occur in three-dimensional computer vision. The library contains algorithms for filtering, feature estimation, surface reconstruction, 3D registration, model fitting, object recognition, and segmentation. Each module is implemented as a smaller library that can be compiled separately (for example, libpcl_filters, libpcl_features, libpcl_surface, ...). PCL has its own data format for storing point clouds - PCD (Point Cloud Data), but also allows datasets to be loaded and saved in many other formats. It is written in C++ and released under the BSD license. These algorithms have been used, for example, for perception in robotics to filter outliers from noisy data, stitch 3D point clouds together, segment relevant parts of a scene, extract keypoints and compute descriptors to recognize objects in the world based on their geometric appearance, and create surfaces from point clouds and visualize them. PCL requires several third-party libraries to function, which must be installed. Most mathematical operations are implemented using the Eigen library. The visualization module for 3D point clouds is based on VTK. Boost is used for shared pointers and the FLANN library for quick k-nearest neighbor search. Additional libraries such as Qhull, OpenNI, or Qt are optional and extend PCL with additional features. PCL is cross-platform software that runs on the most commonly used operating systems: Linux, Windows, macOS and Android. The library is fully integrated with the Robot Operating System (ROS) and provides support for OpenMP and Intel Threading Building Blocks (TBB) libraries for multi-core parallelism. The library is constantly updated and expanded, and its use in various industries is constantly growing. For example, PCL participated in the Google Summer of Code 2020 initiative with three projects. One was the extension of PCL for use with Python using Pybind11. A large number of examples and tutorials are available on the PCL website, either as C++ source files or as tutorials with a detailed description and explanation of the individual steps. Applications Point cloud library is widely used in many different fields, here are some examples: stitching 3D point clouds together recognize 3D objects on their geometric appearance filtering and smoothing out noisy data create surfaces from point clouds aligning a previously captured model of an object to some newly captured data cluster recognition and 6DoF pose estimation point cloud streaming to mobile devices with real-time visualization 3rd party libraries PCL requires for its installation several third-party libraries, which are listed below. Some libraries are optional and extend PCL with additional features. The PCL library is built with the CMake build system (http://www.cmake.org/) at least in version 3.5.0. Mandatory libraries: Boost (http://
https://en.wikipedia.org/wiki/Stability%20postulate	In probability theory, to obtain a nondegenerate limiting distribution of the extreme value distribution, it is necessary to "reduce" the actual greatest value by applying a linear transformation with coefficients that depend on the sample size. If are independent random variables with common probability density function then the cumulative distribution function of is If there is a limiting distribution of interest, the stability postulate states the limiting distribution is some sequence of transformed "reduced" values, such as , where may depend on n but not on x. To distinguish the limiting cumulative distribution function from the "reduced" greatest value from F(x), we will denote it by G(x). It follows that G(x) must satisfy the functional equation This equation was obtained by Maurice René Fréchet and also by Ronald Fisher. Boris Vladimirovich Gnedenko has shown there are no other distributions satisfying the stability postulate other than the following: Gumbel distribution for the minimum stability postulate If and then where and In other words, Extreme value distribution for the maximum stability postulate If and then where and In other words, Fréchet distribution for the maximum stability postulate If and then where and In other words, References Stability (probability) Extreme value data
https://en.wikipedia.org/wiki/Naoto%20Yoshii	is a Japanese footballer. Club statistics Updated to 23 February 2016. References External links Profile at Kataller Toyama 1987 births Living people Osaka Gakuin University alumni Association football people from Hyōgo Prefecture Japanese men's footballers J2 League players J3 League players Kataller Toyama players FC Machida Zelvia players Men's association football defenders
https://en.wikipedia.org/wiki/Mehran%20Mesbahi	Mehran Mesbahi is an Iranian-American control theorist and aerospace engineer. He is a Professor of Aeronautics and Astronautics, and Adjunct Professor of Electrical Engineering and Mathematics at the University of Washington in Seattle. His research is on systems and control theory over networks, optimization, and aerospace controls. Mehran Mesbahi earned his Ph.D. from the University of Southern California in June 1996. From July 1996 until December 1999, he was with the Guidance and Control Analysis Group of the Jet Propulsion Laboratory at California Institute of Technology. During this time, he also had appointments in the Department of Electrical Engineering- Systems at USC (1997–1998) and in the Department of Control and Dynamical Systems at Caltech (1998–1999). From January 2000 to July 2002, he was an Assistant Professor of Aerospace Engineering and Mechanics at the University of Minnesota-Twin Cities. Honors and awards IEEE Fellow University of Washington College of Engineering Innovator Award, 2008 University of Washington Distinguished Teaching Award, 2005 NASA Space Act Award, 2004 National Science Foundation CAREER Award, 2001 Selected publications M. Mesbahi and Magnus Egerstedt, Graph-theoretic Methods in Multiagent Networks, Princeton University Press, 2010. A. Rahmani, M. Ji, M. Mesbahi, and M. Egerstedt. Controllability of multi-agent systems from a graph theoretic perspective, SIAM Journal on Control and Optimization, 48 (1): 162-186, 2009. M. Mesbahi. On state-dependent dynamic graphs and their controllability properties, IEEE Transactions on Automatic Control (50) 3: 387- 392, 2005. Y. Hatano and M. Mesbahi. Agreement over random networks, IEEE Transactions on Automatic Control, (50) 11: 1867-1872, 2005. M. Mesbahi and F. Y. Hadaegh. Formation flying of multiple spacecraft via graphs, matrix inequalities, and switching, AIAA Journal of Guidance, Control, and Dynamics, (24) 2: 369-377, 2001. References External links Homepage RainLab Mathematics Genealogy Project profile Control theorists Living people USC Viterbi School of Engineering alumni University of Minnesota faculty University of Washington faculty Fellow Members of the IEEE Year of birth missing (living people)
https://en.wikipedia.org/wiki/Tetsuya%20Funatsu	is a Japanese football player who plays for Thespakusatsu Gunma. Club statistics Updated to 23 February 2020. References External links Profile at Thespakusatsu Gunma 1982 births Living people Biwako Seikei Sport College alumni Association football people from Osaka Prefecture Japanese men's footballers J1 League players J2 League players J3 League players Kataller Toyama players Cerezo Osaka players Montedio Yamagata players Thespakusatsu Gunma players Men's association football defenders
https://en.wikipedia.org/wiki/Yuya%20Nagatomi	is a former Japanese football player. Club statistics References External links 1982 births Living people Chukyo University alumni Association football people from Fukuoka Prefecture Japanese men's footballers J2 League players Japan Football League players Ehime FC players Kataller Toyama players Men's association football forwards
https://en.wikipedia.org/wiki/Keisuke%20Kimoto	is a former Japanese football player, who played for all of his career with Kataller Toyama. Club career statistics Updated to 2 February 2018. References External links Profile at Kataller Toyama 1984 births Living people Kansai University alumni Association football people from Ōita Prefecture Japanese men's footballers J2 League players J3 League players Japan Football League players Kataller Toyama players Men's association football midfielders
https://en.wikipedia.org/wiki/Taijiro%20Mori	is a Japanese footballer who plays as a midfielder for Toyama Shinjo in Hokushinetsu Football League. Club statistics References External links Profile at Kataller 1991 births Living people Association football people from Toyama Prefecture Japanese men's footballers J2 League players J3 League players Japan Football League players Kataller Toyama players SP Kyoto FC players Men's association football midfielders
https://en.wikipedia.org/wiki/Rado%27s%20theorem	In mathematics, Rado's theorem or Radó's theorem may refer to: Tibor Radó's theorem (harmonic functions) Tibor Radó's theorem (Riemann surfaces) Richard Rado's theorem (Ramsey theory) Richard Rado's theorem (matroid theory)
https://en.wikipedia.org/wiki/Elementary%20group	In algebra, more specifically group theory, a p-elementary group is a direct product of a finite cyclic group of order relatively prime to p and a p-group. A finite group is an elementary group if it is p-elementary for some prime number p. An elementary group is nilpotent. Brauer's theorem on induced characters states that a character on a finite group is a linear combination with integer coefficients of characters induced from elementary subgroups. More generally, a finite group G is called a p-hyperelementary if it has the extension where is cyclic of order prime to p and P is a p-group. Not every hyperelementary group is elementary: for instance the non-abelian group of order 6 is 2-hyperelementary, but not 2-elementary. See also Elementary abelian group References Arthur Bartels, Wolfgang Lück, Induction Theorems and Isomorphism Conjectures for K- and L-Theory G. Segal, The representation-ring of a compact Lie group J.P. Serre, "Linear representations of finite groups". Graduate Texts in Mathematics, vol. 42, Springer-Verlag, New York, Heidelberg, Berlin, 1977, Properties of groups
https://en.wikipedia.org/wiki/Rad%C3%B3%27s%20theorem%20%28Riemann%20surfaces%29	In mathematical complex analysis, Radó's theorem, proved by , states that every connected Riemann surface is second-countable (has a countable base for its topology). The Prüfer surface is an example of a surface with no countable base for the topology, so cannot have the structure of a Riemann surface. The obvious analogue of Radó's theorem in higher dimensions is false: there are 2-dimensional connected complex manifolds that are not second-countable. References Riemann surfaces Theorems in complex analysis
https://en.wikipedia.org/wiki/Selberg%27s%20zeta%20function%20conjecture	In mathematics, the Selberg conjecture, named after Atle Selberg, is a theorem about the density of zeros of the Riemann zeta function ζ(1/2 + it). It is known that the function has infinitely many zeroes on this line in the complex plane: the point at issue is how densely they are clustered. Results on this can be formulated in terms of N(T), the function counting zeroes on the line for which the value of t satisfies 0 ≤ t ≤ T. Background In 1942 Atle Selberg investigated the problem of the Hardy–Littlewood conjecture 2; and he proved that for any there exist and such that for and the inequality holds true. In his turn, Selberg stated a conjecture relating to shorter intervals, namely that it is possible to decrease the value of the exponent a = 0.5 in Proof of the conjecture In 1984 Anatolii Karatsuba proved that for a fixed satisfying the condition a sufficiently large T and the interval in the ordinate t (T, T + H) contains at least cH ln T real zeros of the Riemann zeta function and thereby confirmed the Selberg conjecture. The estimates of Selberg and Karatsuba cannot be improved in respect of the order of growth as T → +∞. Further work In 1992 Karatsuba proved that an analog of the Selberg conjecture holds for "almost all" intervals (T, T + H], H = Tε, where ε is an arbitrarily small fixed positive number. The Karatsuba method permits one to investigate zeroes of the Riemann zeta-function on "supershort" intervals of the critical line, that is, on the intervals (T, T + H], the length H of which grows slower than any, even arbitrarily small degree T. In particular, he proved that for any given numbers ε, ε1 satisfying the conditions 0 < ε, ε1< 1 almost all intervals (T, T + H] for H ≥ exp[(ln T)ε] contain at least H (ln T)1 −ε1 zeros of the function ζ(1/2 + it). This estimate is quite close to the conditional result that follows from the Riemann hypothesis. References Zeta and L-functions Conjectures that have been proved
https://en.wikipedia.org/wiki/Yonny%20Hern%C3%A1ndez%20%28motorcyclist%29	Yonny Hernández Vega (born 25 July 1988) is a Colombian motorcycle racer. He is the older brother of Santiago Hernández. Career statistics Grand Prix motorcycle racing By season By class Races by year (key) (Races in bold indicate pole position; races in italics indicate fastest lap) Superbike World Championship Races by year (key) (Races in bold indicate pole position; races in italics indicate fastest lap) References External links Moto2 World Championship riders Colombian motorcycle racers 1988 births Living people Avintia Racing MotoGP riders Sportspeople from Medellín Pramac Racing MotoGP riders Aspar Racing Team MotoGP riders Superbike World Championship riders MotoGP World Championship riders MotoE World Cup riders 21st-century Colombian people
https://en.wikipedia.org/wiki/Special%20case	In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of . A limiting case is a type of special case which is arrived at by taking some aspect of the concept to the extreme of what is permitted in the general case. If is true, one can immediately deduce that is true as well, and if is false, can also be immediately deduced to be false. A degenerate case is a special case which is in some way qualitatively different from almost all of the cases allowed. Examples Special case examples include the following: All squares are rectangles (but not all rectangles are squares); therefore the square is a special case of the rectangle. Fermat's Last Theorem, that has no solutions in positive integers with , is a special case of Beal's conjecture, that has no primitive solutions in positive integers with , , and all greater than 2, specifically, the case of . The unproven Riemann hypothesis is a special case of the generalized Riemann hypothesis, in the case that χ(n) = 1 for all n. Fermat's little theorem, which states "if is a prime number, then for any integer a, then " is a special case of Euler's theorem, which states "if n and a are coprime positive integers, and is Euler's totient function, then ", in the case that is a prime number. Euler's identity is a special case of Euler's formula which states "for any real number x: ", in the case that = . Mathematical logic
https://en.wikipedia.org/wiki/Network%20topology	Network topology is the arrangement of the elements (links, nodes, etc.) of a communication network. Network topology can be used to define or describe the arrangement of various types of telecommunication networks, including command and control radio networks, industrial fieldbusses and computer networks. Network topology is the topological structure of a network and may be depicted physically or logically. It is an application of graph theory wherein communicating devices are modeled as nodes and the connections between the devices are modeled as links or lines between the nodes. Physical topology is the placement of the various components of a network (e.g., device location and cable installation), while logical topology illustrates how data flows within a network. Distances between nodes, physical interconnections, transmission rates, or signal types may differ between two different networks, yet their logical topologies may be identical. A network’s physical topology is a particular concern of the physical layer of the OSI model. Examples of network topologies are found in local area networks (LAN), a common computer network installation. Any given node in the LAN has one or more physical links to other devices in the network; graphically mapping these links results in a geometric shape that can be used to describe the physical topology of the network. A wide variety of physical topologies have been used in LANs, including ring, bus, mesh and star. Conversely, mapping the data flow between the components determines the logical topology of the network. In comparison, Controller Area Networks, common in vehicles, are primarily distributed control system networks of one or more controllers interconnected with sensors and actuators over, invariably, a physical bus topology. Topologies Two basic categories of network topologies exist, physical topologies and logical topologies. The transmission medium layout used to link devices is the physical topology of the network. For conductive or fiber optical mediums, this refers to the layout of cabling, the locations of nodes, and the links between the nodes and the cabling. The physical topology of a network is determined by the capabilities of the network access devices and media, the level of control or fault tolerance desired, and the cost associated with cabling or telecommunication circuits. In contrast, logical topology is the way that the signals act on the network media, or the way that the data passes through the network from one device to the next without regard to the physical interconnection of the devices. A network's logical topology is not necessarily the same as its physical topology. For example, the original twisted pair Ethernet using repeater hubs was a logical bus topology carried on a physical star topology. Token Ring is a logical ring topology, but is wired as a physical star from the media access unit. Physically, AFDX can be a cascaded star topology of multiple dual r
https://en.wikipedia.org/wiki/Operation	Operation or Operations may refer to: Arts, entertainment and media Operation (game), a battery-operated board game that challenges dexterity Operation (music), a term used in musical set theory Operations (magazine), Multi-Man Publishing's house organ for articles and discussion about its wargaming products The Operation (film), a 1973 British television film The Operation (1990), a crime, drama, TV movie starring Joe Penny, Lisa Hartman, and Jason Beghe The Operation (1992–1998), a reality television series from TLC The Operation M.D., formerly The Operation, a Canadian garage rock band "Operation", a song by Relient K from The Creepy EP, 2001 Business Business operations, the harvesting of value from assets owned by a business Manufacturing operations, operation of a facility Operations management, an area of management concerned with designing and controlling the process of production Military and law enforcement Military operation, a military action (usually in a military campaign) using deployed forces Black operation, or "Black op", an operation that may be outside of standard military protocol or against the law Clandestine operation, an intelligence or military operation carried out so that the operation goes unnoticed Combined operations, operations by forces of two or more allied nations Covert operation, an operation which conceals the identity of the sponsor Psychological operations, an operation consisting of psychological manipulations, tactics, and warfare Special operations, military operations that are unconventional Operations (J3), third level of Nation Level Command Structure Operations (military staff), staff involved in planning operations Sting operation, an operation designed to catch a person committing a crime, by means of deception Science and technology Inference, a step in reasoning Information technology operations Operation (mathematics), a calculation from zero or more input values (called operands) to an output value Arity, number of arguments or operands that the function takes Binary operation, calculation that combines two elements of the set to produce another element of the set Graph operations, produce new graphs from initial ones Modulo operation, operation finds the remainder after division of one number by another Operations research, in British usage, application of advanced analytical methods to make better decisions Unary operation, an operation with only one operand Rail transport operations, the control of a rail system Scientific operation Surgical operation, or surgery, in medicine Unit operation, a basic step in a chemical engineering process Other uses Anomalous operation, in parapsychology, a term describing a broad category of purported paranormal effects Operation, a word which represents a grammatical relation (i.e., function) or instruction, rather than a term or name Operation of law, a legal term that indicates that a right or liability
https://en.wikipedia.org/wiki/Precision	Precision, precise or precisely may refer to: Science, and technology, and mathematics Mathematics and computing (general) Accuracy and precision, measurement deviation from true value and its scatter Significant figures, the number of digits that carry real information about a measurement Precision and recall, in information retrieval: the proportion of relevant documents returned Precision (computer science), a measure of the detail in which a quantity is expressed Precision (statistics), a model parameter or a quantification of precision Computing products Dell Precision, a line of Dell workstations Precision Architecture, former name for PA-RISC, a reduced instruction set architecture developed by Hewlett-Packard Ubuntu 12.04 "Precise Pangolin", Canonical's sixteenth release of Ubuntu Companies Precision Air, an airline based in Tanzania Precision Castparts Corp., a casting company based in Portland, Oregon, in the United States Precision Drilling, the largest drilling-rig contractor in Canada Precision Monolithics, an American company that produced linear semiconductors Precision Talent, a voice-over talent-management company F. E. Baker Ltd, maker of Precision motorcycle and cycle-car engines pre-WW1 Precisely (company), formerly Syncsort Other Precision Club, a bidding system in the game of contract bridge Precision (march), the official marching music of the Royal Military College of Canada Precision 15, a self-bailing dinghy Fender Precision Bass, by Fender Musical Instruments Corporation Precisely (sketch), a dramatic sketch by the English playwright Harold Pinter See also Precisionism, an artistic movement also known as Cubist Realism Precisionist (1981–2006), an American Hall of Fame Thoroughbred racehorse
https://en.wikipedia.org/wiki/Random%20number	In mathematics and statistics, a random number is either Pseudo-random or a number generated for, or part of, a set exhibiting statistical randomness. Algorithms and implementations A 1964-developed algorithm is popularly known as the Knuth shuffle or the Fisher–Yates shuffle (based on work they did in 1938). A real-world use for this is sampling water quality in a reservoir. In 1999, a new feature was added to the Pentium III: a hardware-based random number generator. It has been described as "several oscillators combine their outputs and that odd waveform is sampled asynchronously." These numbers, however, were only 32 bit, at a time when export controls were on 56 bits and higher, so they were not state of the art. Common understanding In common understanding, "1 2 3 4 5" is not as random as "3 5 2 1 4" and certainly not as random as "47 88 1 32 41" but "we can't say authoritavely that the first sequence is not random ... it could have been generated by chance." When a police officer claims to have done a "random .. door-to-door" search, there is a certain expectation that members of a jury will have. Real world consequences Flaws in randomness have real-world consequences. A 99.8% randomness was shown by researchers to negatively affect an estimated 27,000 customers of a large service and that the problem was not limited to just that situation. See also Algorithmically random sequence Quasi-random sequence Random number generation Random sequence Random variable Random variate Random real References Permutations
https://en.wikipedia.org/wiki/Cycloid	In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve. The cycloid, with the cusps pointing upward, is the curve of fastest descent under uniform gravity (the brachistochrone curve). It is also the form of a curve for which the period of an object in simple harmonic motion (rolling up and down repetitively) along the curve does not depend on the object's starting position (the tautochrone curve). In physics, when a charged particle at rest is put under a uniform electric and magnetic field perpendicular to one another, the particle’s trajectory draws out a cycloid. History The cycloid has been called "The Helen of Geometers" as, like Helen of Troy, it caused frequent quarrels among 17th-century mathematicians, while Sarah Hart sees it named as such "because the properties of this curve are so beautiful". Historians of mathematics have proposed several candidates for the discoverer of the cycloid. Mathematical historian Paul Tannery speculated that such a simple curve must have been known to the ancients, citing similar work by Carpus of Antioch described by Iamblichus. English mathematician John Wallis writing in 1679 attributed the discovery to Nicholas of Cusa, but subsequent scholarship indicates that either Wallis was mistaken or the evidence he used is now lost. Galileo Galilei's name was put forward at the end of the 19th century and at least one author reports credit being given to Marin Mersenne. Beginning with the work of Moritz Cantor and Siegmund Günther, scholars now assign priority to French mathematician Charles de Bovelles based on his description of the cycloid in his Introductio in geometriam, published in 1503. In this work, Bovelles mistakes the arch traced by a rolling wheel as part of a larger circle with a radius 120% larger than the smaller wheel. Galileo originated the term cycloid and was the first to make a serious study of the curve. According to his student Evangelista Torricelli, in 1599 Galileo attempted the quadrature of the cycloid (determining the area under the cycloid) with an unusually empirical approach that involved tracing both the generating circle and the resulting cycloid on sheet metal, cutting them out and weighing them. He discovered the ratio was roughly 3:1, which is the true value, but he incorrectly concluded the ratio was an irrational fraction, which would have made quadrature impossible. Around 1628, Gilles Persone de Roberval likely learned of the quadrature problem from Père Marin Mersenne and effected the quadrature in 1634 by using Cavalieri's Theorem. However, this work was not published until 1693 (in his Traité des Indivisibles). Constructing the tangent of the cycloid dates to August 1638 when Mersenne received unique methods from Roberval, Pierre de Fermat and René Descartes. Merse
https://en.wikipedia.org/wiki/Reliability	Reliability, reliable, or unreliable may refer to: Science, technology, and mathematics Computing Data reliability (disambiguation), a property of some disk arrays in computer storage Reliability (computer networking), a category used to describe protocols Reliability (semiconductor), outline of semiconductor device reliability drivers Other uses in science, technology, and mathematics Reliability (statistics), the overall consistency of a measure Reliability engineering, concerned with the ability of a system or component to perform its required functions under stated conditions for a specified time High reliability is informally reported in "nines" Human reliability in engineered systems Reliability theory, as a theoretical concept, to explain biological aging and species longevity Other uses Reliabilism, in philosophy and epistemology Unreliable narrator, whose credibility has been seriously compromised See also Reliant (disambiguation)
https://en.wikipedia.org/wiki/Skew	Skew may refer to: In mathematics Skew lines, neither parallel nor intersecting. Skew normal distribution, a probability distribution Skew field or division ring Skew-Hermitian matrix Skew lattice Skew polygon, whose vertices do not lie on a plane Infinite skew polyhedron Skew-symmetric graph Skew-symmetric matrix Skew tableau, a generalization of Young tableau Skewness, a measure of the asymmetry of a probability distribution Shear mapping In science and technology Skew, also synclinal or gauche in alkane stereochemistry Skew ray (optics), an optical path not in a plane of symmetry Skew arch, not at a right angle In computing Clock skew Transitive data skew, an issue of data synchronization In telecommunications Skew (fax), unstraightness Skew (antenna) a method to improve the horizontal radiation pattern Other uses Volatility skew, in finance, a downward-sloping volatility smile SKEW, the ticker symbol for the CBOE Skew Index See also SKU or Stock-keeping unit
https://en.wikipedia.org/wiki/Transposition	Transposition may refer to: Logic and mathematics Transposition (mathematics), a permutation which exchanges two elements and keeps all others fixed Transposition, producing the transpose of a matrix AT, which is computed by swapping columns for rows in the matrix A Transpose of a linear map Transposition (logic), a rule of replacement in philosophical logic Transpose relation, another name for converse relation Games Transposition (chess), different moves or a different move order leading to the same position, especially during the openings Transposition table, used in computer games to speed up the search of the game tree Biology Transposition (birth defect), a group of congenital defects involving an abnormal spatial arrangement of tissue or organ Transposition of the great vessels, cardiac transposition, a congenital heart defect with malformation of any of the major vessels Transposition of teeth Penoscrotal transposition Transposition (horizontal gene transfer), the transfer of genetic material between organisms other than by vertical gene transfer Transposons, or genetic transposition, a mutation in which a chromosomal segment is transferred to a new position on the same or another chromosome Other uses Transposition (law), the incorporation of the provisions of a European Union directive into a Member State's domestic law Transposition (music), moving a note or collection of notes up or down in pitch by a constant number of semitones Transposition (transmission lines), periodic swapping of positions of the conductors of a transmission line Transposition cipher, an elementary cryptographic operation Transposition, docking, and extraction an orbital maneuver performed on the Apollo lunar missions Transposition, sleight of hand (magic), a performer appears to make two different objects ([usually] coins or cards) switch places with one another faster than physically possible. Transpose, a database of academic journal preprinting policies maintained by ASAPbio
https://en.wikipedia.org/wiki/Unavailability	Unavailability, in mathematical terms, is the probability that an item will not operate correctly at a given time and under specified conditions. It opposes availability. Numerical values associated with the calculation of availability are often awkward, consisting of a series of 9s before reaching any significant numerical information (e.g. 0.9999999654). For this reason, it is more convenient to use the complement measure of availability, namely, unavailability. Expressed mathematically, unavailability is 1 minus the availability. Therefore, a system with availability 0.9999999654 is more concisely described as having an unavailability of 3.46 × 10−8. Calculations using unavailability Often fault trees and reliability block diagrams will use unavailability of the various components in the calculation of the top level failure rates through AND gates or parallel redundant components. Repairable model Unavailability (Q), using the repairable model, may be expressed mathematically by the equation: where MTTR is the mean time to repair, and MTBF is the mean time between failures of a repairable system. Alternatively, this can be written as: where λ is the failure rate and μ is the repair rate. When μ >> λ, the preceding formula is often approximated to: Non-Repairable model For the non-repairable model of unavailability (Q), the unreliability function (often F(t) the CDF of the exponential distribution) is used to approximate the worst-case-unavailability. If the rate of failure is constant the Poisson distribution and exponential distribution describe this rate. The unreliability function approximating the worst case unavailability is as follows: Q = 1 - e-λt Where t is the time at risk. Telecommunications In telecommunication, an unavailability is an expression of the degree to which a system, subsystem, or equipment is not operable and not in a committable state at the start of a mission, when the mission is called for at an unknown, i.e. random, time. The conditions determining operability and committability must be specified. Railway In the railway industry, the railway normally keeps operating for 24 hours a day 7 days per week all year round making the idea of mission time meaningless. Both the repairable model and non-repairable model are known to be used in railway. The repairable model is used for total system availability or unavailability and the non-repairable model is used for system safety. Safe down time is the time between when a wrong side failure happens and when it is detected and mitigated. Aerospace The aerospace industry often uses a mission time equal to the expected flight time given that certain pre-flight tests are performed. Space The mission time for space systems may be as long as a satellite or system in orbit. Space systems are exceedingly difficult to repair making mission time a consideration when evaluating unavailability. References Systems engineering
https://en.wikipedia.org/wiki/Felix%20Klein	Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group theory. His 1872 Erlangen program, classifying geometries by their basic symmetry groups, was an influential synthesis of much of the mathematics of the time. Life Felix Klein was born on 25 April 1849 in Düsseldorf, to Prussian parents. His father, Caspar Klein (1809–1889), was a Prussian government official's secretary stationed in the Rhine Province. His mother was Sophie Elise Klein (1819–1890, née Kayser). He attended the Gymnasium in Düsseldorf, then studied mathematics and physics at the University of Bonn, 1865–1866, intending to become a physicist. At that time, Julius Plücker had Bonn's professorship of mathematics and experimental physics, but by the time Klein became his assistant, in 1866, Plücker's interest was mainly geometry. Klein received his doctorate, supervised by Plücker, from the University of Bonn in 1868. Plücker died in 1868, leaving his book concerning the basis of line geometry incomplete. Klein was the obvious person to complete the second part of Plücker's Neue Geometrie des Raumes, and thus became acquainted with Alfred Clebsch, who had relocated to Göttingen in 1868. Klein visited Clebsch the next year, along with visits to Berlin and Paris. In July 1870, at the beginning of the Franco-Prussian War, he was in Paris and had to leave the country. For a brief time he served as a medical orderly in the Prussian army before being appointed lecturer at Göttingen in early 1871. Erlangen appointed Klein professor in 1872, when he was only 23 years old. For this, he was endorsed by Clebsch, who regarded him as likely to become the best mathematician of his time. Klein did not wish to remain in Erlangen, where there were very few students, and was pleased to be offered a professorship at the Technische Hochschule München in 1875. There he and Alexander von Brill taught advanced courses to many excellent students, including Adolf Hurwitz, Walther von Dyck, Karl Rohn, Carl Runge, Max Planck, Luigi Bianchi, and Gregorio Ricci-Curbastro. In 1875, Klein married Anne Hegel, granddaughter of the philosopher Georg Wilhelm Friedrich Hegel. After spending five years at the Technische Hochschule, Klein was appointed to a chair of geometry at Leipzig. There his colleagues included Walther von Dyck, Rohn, Eduard Study and Friedrich Engel. Klein's years at Leipzig, 1880 to 1886, fundamentally changed his life. In 1882, his health collapsed; in 1883–1884, he was afflicted with depression. Nevertheless, his research continued; his seminal work on hyperelliptic sigma functions, published between 1886 and 1888, dates from around this period. Klein accepted a professorship at the University of Göttingen in 1886. From then on, until his 1913 retirement, he sought to re-establish Göttingen as the world's
https://en.wikipedia.org/wiki/Group%20theory	In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography. The early history of group theory dates from the 19th century. One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 2004, that culminated in a complete classification of finite simple groups. History Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry. The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields. Early results about permutation groups were obtained by Lagrange, Ruffini, and Abel in their quest for general solutions of polynomial equations of high degree. Évariste Galois coined the term "group" and established a connection, now known as Galois theory, between the nascent theory of groups and field theory. In geometry, groups first became important in projective geometry and, later, non-Euclidean geometry. Felix Klein's Erlangen program proclaimed group theory to be the organizing principle of geometry. Galois, in the 1830s, was the first to employ groups to determine the solvability of polynomial equations. Arthur Cayley and Augustin Louis Cauchy pushed these investigations further by creating the theory of permutation groups. The second historical source for groups stems from geometrical situations. In an attempt to come to grips with possible geometries (such as euclidean, hyperbolic or projective geometry) using group theory, Felix Klein initiated the Erlangen programme. Sophus Lie, in 1884, started using groups (now called Lie groups) attached to analytic problems. Thirdly, groups were, at first implicitly and later explicitly, used in algebraic number theory. The different scope of these early sources resulted in different notions of groups. The theory of groups was unified starting around 1880. Since then, the impact of group theory has been ever growing, giving rise to the birth of abstract algebra in the early 20th c
https://en.wikipedia.org/wiki/Klein%20four-group	In mathematics, the Klein four-group is an abelian group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one. It can be described as the symmetry group of a non-square rectangle (with the three non-identity elements being horizontal reflection, vertical reflection and 180-degree rotation), as the group of bitwise exclusive or operations on two-bit binary values, or more abstractly as , the direct product of two copies of the cyclic group of order 2. It was named Vierergruppe (meaning four-group) by Felix Klein in 1884. It is also called the Klein group, and is often symbolized by the letter V or as K4. The Klein four-group, with four elements, is the smallest group that is not a cyclic group. There is only one other group of order four, up to isomorphism, the cyclic group of order 4. Both are abelian groups. Presentations The Klein group's Cayley table is given by: The Klein four-group is also defined by the group presentation All non-identity elements of the Klein group have order 2, thus any two non-identity elements can serve as generators in the above presentation. The Klein four-group is the smallest non-cyclic group. It is however an abelian group, and isomorphic to the dihedral group of order (cardinality) 4, i.e. D4 (or D2, using the geometric convention); other than the group of order 2, it is the only dihedral group that is abelian. The Klein four-group is also isomorphic to the direct sum , so that it can be represented as the pairs under component-wise addition modulo 2 (or equivalently the bit strings under bitwise XOR); with (0,0) being the group's identity element. The Klein four-group is thus an example of an elementary abelian 2-group, which is also called a Boolean group. The Klein four-group is thus also the group generated by the symmetric difference as the binary operation on the subsets of a powerset of a set with two elements, i.e. over a field of sets with four elements, e.g. ; the empty set is the group's identity element in this case. Another numerical construction of the Klein four-group is the set with the operation being multiplication modulo 8. Here a is 3, b is 5, and is . The Klein four-group has a representation as 2×2 real matrices with the operation being matrix multiplication: On a Rubik's Cube the "4 dots" pattern can be made in three ways, depending on the pair of faces that are left blank; these three positions together with the "identity" or home position form an example of the Klein group. Geometry Geometrically, in two dimensions the Klein four-group is the symmetry group of a rhombus and of rectangles that are not squares, the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180 degree rotation. In three dimensions there are three different symmetry groups that are algebraically the Klein four-group V: on
https://en.wikipedia.org/wiki/Bernhard%20Riemann	Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis. His 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as a foundational paper of analytic number theory. Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity. He is considered by many to be one of the greatest mathematicians of all time. Biography Early years Riemann was born on 17 September 1826 in Breselenz, a village near Dannenberg in the Kingdom of Hanover. His father, Friedrich Bernhard Riemann, was a poor Lutheran pastor in Breselenz who fought in the Napoleonic Wars. His mother, Charlotte Ebell, died before her children had reached adulthood. Riemann was the second of six children, shy and suffering from numerous nervous breakdowns. Riemann exhibited exceptional mathematical talent, such as calculation abilities, from an early age but suffered from timidity and a fear of speaking in public. Education During 1840, Riemann went to Hanover to live with his grandmother and attend lyceum (middle school years). After the death of his grandmother in 1842, he attended high school at the Johanneum Lüneburg. In high school, Riemann studied the Bible intensively, but he was often distracted by mathematics. His teachers were amazed by his ability to perform complicated mathematical operations, in which he often outstripped his instructor's knowledge. In 1846, at the age of 19, he started studying philology and Christian theology in order to become a pastor and help with his family's finances. During the spring of 1846, his father, after gathering enough money, sent Riemann to the University of Göttingen, where he planned to study towards a degree in theology. However, once there, he began studying mathematics under Carl Friedrich Gauss (specifically his lectures on the method of least squares). Gauss recommended that Riemann give up his theological work and enter the mathematical field; after getting his father's approval, Riemann transferred to the University of Berlin in 1847. During his time of study, Carl Gustav Jacob Jacobi, Peter Gustav Lejeune Dirichlet, Jakob Steiner, and Gotthold Eisenstein were teaching. He stayed in Berlin for two years and returned to Göttingen in 1849. Academia Riemann held his first lectures in 1854, which founded the field of Riemannian geometry and thereby set the stage for Albert Einstein's general theory of relativity. In 1857, there was an attempt to promote Riema
https://en.wikipedia.org/wiki/Dual%20number	In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy with . Dual numbers can be added component-wise, and multiplied by the formula which follows from the property and the fact that multiplication is a bilinear operation. The dual numbers form a commutative algebra of dimension two over the reals, and also an Artinian local ring. They are one of the simplest examples of a ring that has nonzero nilpotent elements. History Dual numbers were introduced in 1873 by William Clifford, and were used at the beginning of the twentieth century by the German mathematician Eduard Study, who used them to represent the dual angle which measures the relative position of two skew lines in space. Study defined a dual angle as , where is the angle between the directions of two lines in three-dimensional space and is a distance between them. The -dimensional generalization, the Grassmann number, was introduced by Hermann Grassmann in the late 19th century. Modern definition In modern algebra, the algebra of dual numbers is often defined as the quotient of a polynomial ring over the real numbers by the principal ideal generated by the square of the indeterminate, that is It may also be defined as the exterior algebra of a one-dimensional vector space with as its basis element. Division Division of dual numbers is defined when the real part of the denominator is non-zero. The division process is analogous to complex division in that the denominator is multiplied by its conjugate in order to cancel the non-real parts. Therefore, to divide an equation of the form we multiply the numerator and denominator by the conjugate of the denominator: which is defined when is non-zero. If, on the other hand, is zero while is not, then the equation has no solution if is nonzero is otherwise solved by any dual number of the form . This means that the non-real part of the "quotient" is arbitrary and division is therefore not defined for purely nonreal dual numbers. Indeed, they are (trivially) zero divisors and clearly form an ideal of the associative algebra (and thus ring) of the dual numbers. Matrix representation The dual number can be represented by the square matrix . In this representation the matrix squares to the zero matrix, corresponding to the dual number . There are other ways to represent dual numbers as square matrices. They consist of representing the dual number by the identity matrix, and by any matrix whose square is the zero matrix; that is, in the case of matrices, any nonzero matrix of the form with Differentiation One application of dual numbers is automatic differentiation. Any polynomial with real coefficients can be extended to a function of a dual-number-valued argument, where is the derivative of More generally, any (analytic) real function can be extended to the dual nu
https://en.wikipedia.org/wiki/Data%20mining	Data mining is the process of extracting and discovering patterns in large data sets involving methods at the intersection of machine learning, statistics, and database systems. Data mining is an interdisciplinary subfield of computer science and statistics with an overall goal of extracting information (with intelligent methods) from a data set and transforming the information into a comprehensible structure for further use. Data mining is the analysis step of the "knowledge discovery in databases" process, or KDD. Aside from the raw analysis step, it also involves database and data management aspects, data pre-processing, model and inference considerations, interestingness metrics, complexity considerations, post-processing of discovered structures, visualization, and online updating. The term "data mining" is a misnomer because the goal is the extraction of patterns and knowledge from large amounts of data, not the extraction (mining) of data itself. It also is a buzzword and is frequently applied to any form of large-scale data or information processing (collection, extraction, warehousing, analysis, and statistics) as well as any application of computer decision support system, including artificial intelligence (e.g., machine learning) and business intelligence. The book Data Mining: Practical Machine Learning Tools and Techniques with Java (which covers mostly machine learning material) was originally to be named Practical Machine Learning, and the term data mining was only added for marketing reasons. Often the more general terms (large scale) data analysis and analytics—or, when referring to actual methods, artificial intelligence and machine learning—are more appropriate. The actual data mining task is the semi-automatic or automatic analysis of large quantities of data to extract previously unknown, interesting patterns such as groups of data records (cluster analysis), unusual records (anomaly detection), and dependencies (association rule mining, sequential pattern mining). This usually involves using database techniques such as spatial indices. These patterns can then be seen as a kind of summary of the input data, and may be used in further analysis or, for example, in machine learning and predictive analytics. For example, the data mining step might identify multiple groups in the data, which can then be used to obtain more accurate prediction results by a decision support system. Neither the data collection, data preparation, nor result interpretation and reporting is part of the data mining step, although they do belong to the overall KDD process as additional steps. The difference between data analysis and data mining is that data analysis is used to test models and hypotheses on the dataset, e.g., analyzing the effectiveness of a marketing campaign, regardless of the amount of data. In contrast, data mining uses machine learning and statistical models to uncover clandestine or hidden patterns in a large volume of data. The
https://en.wikipedia.org/wiki/Arity	In logic, mathematics, and computer science, arity () is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank, but this word can have many other meanings. In logic and philosophy, arity may also be called adicity and degree. In linguistics, it is usually named valency. Examples In general, functions or operators with a given arity follow the naming conventions of n-based numeral systems, such as binary and hexadecimal. A Latin prefix is combined with the -ary suffix. For example: A nullary function takes no arguments. Example: A unary function takes one argument. Example: A binary function takes two arguments. Example: A ternary function takes three arguments. Example: An n-ary function takes n arguments. Example: Nullary A constant can be considered an operation of arity 0, called a nullary. Also, outside of functional programming, a function without arguments can be meaningful and not necessarily constant (due to side effects). Such functions may have some hidden input, such as global variables or the whole state of the system (time, free memory, etc.). Unary Examples of unary operators in mathematics and in programming include the unary minus and plus, the increment and decrement operators in C-style languages (not in logical languages), and the successor, factorial, reciprocal, floor, ceiling, fractional part, sign, absolute value, square root (the principal square root), complex conjugate (unary of "one" complex number, that however has two parts at a lower level of abstraction), and norm functions in mathematics. In programming the two's complement, address reference, and the logical NOT operators are examples of unary operators. All functions in lambda calculus and in some functional programming languages (especially those descended from ML) are technically unary, but see n-ary below. According to Quine, the Latin distributives being singuli, bini, terni, and so forth, the term "singulary" is the correct adjective, rather than "unary". Abraham Robinson follows Quine's usage. In philosophy, the adjective monadic is sometimes used to describe a one-place relation such as 'is square-shaped' as opposed to a two-place relation such as 'is the sister of'. Binary Most operators encountered in programming and mathematics are of the binary form. For both programming and mathematics, these include the multiplication operator, the radix operator, the often omitted exponentiation operator, the logarithm operator, the addition operator, and the division operator. Logical predicates such as OR, XOR, AND, IMP are typically used as binary operators with two distinct operands. In CISC architectures, it is common to have two source operands (and store result in one of them). Ternary The computer programming language C and its various descendants (including C++, C#, Java, Julia, Perl, and others) provide the ternary conditional operator ?:. The first operand
https://en.wikipedia.org/wiki/Closure%20%28topology%29	In topology, the closure of a subset of points in a topological space consists of all points in together with all limit points of . The closure of may equivalently be defined as the union of and its boundary, and also as the intersection of all closed sets containing . Intuitively, the closure can be thought of as all the points that are either in or "very near" . A point which is in the closure of is a point of closure of . The notion of closure is in many ways dual to the notion of interior. Definitions Point of closure For as a subset of a Euclidean space, is a point of closure of if every open ball centered at contains a point of (this point can be itself). This definition generalizes to any subset of a metric space Fully expressed, for as a metric space with metric is a point of closure of if for every there exists some such that the distance ( is allowed). Another way to express this is to say that is a point of closure of if the distance where is the infimum. This definition generalizes to topological spaces by replacing "open ball" or "ball" with "neighbourhood". Let be a subset of a topological space Then is a or of if every neighbourhood of contains a point of (again, for is allowed). Note that this definition does not depend upon whether neighbourhoods are required to be open. Limit point The definition of a point of closure of a set is closely related to the definition of a limit point of a set. The difference between the two definitions is subtle but important – namely, in the definition of a limit point of a set , every neighbourhood of must contain a point of , i.e., each neighbourhood of obviously has but it also must have a point of that is not equal to in order for to be a limit point of . A limit point of has more strict condition than a point of closure of in the definitions. The set of all limit points of a set is called the . A limit point of a set is also called cluster point or accumulation point of the set. Thus, every limit point is a point of closure, but not every point of closure is a limit point. A point of closure which is not a limit point is an isolated point. In other words, a point is an isolated point of if it is an element of and there is a neighbourhood of which contains no other points of than itself. For a given set and point is a point of closure of if and only if is an element of or is a limit point of (or both). Closure of a set The of a subset of a topological space denoted by or possibly by (if is understood), where if both and are clear from context then it may also be denoted by or (Moreover, is sometimes capitalized to .) can be defined using any of the following equivalent definitions: is the set of all points of closure of is the set together with all of its limit points. (Each point of is a point of closure of , and each limit point of is also a point of closure of .) is the intersection of all closed se
https://en.wikipedia.org/wiki/Topological%20group	In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other. Topological groups have been studied extensively in the period of 1925 to 1940. Haar and Weil (respectively in 1933 and 1940) showed that the integrals and Fourier series are special cases of a very wide class of topological groups. Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics. In functional analysis, every topological vector space is an additive topological group with the additional property that scalar multiplication is continuous; consequently, many results from the theory of topological groups can be applied to functional analysis. Formal definition A topological group, , is a topological space that is also a group such that the group operation (in this case product): , and the inversion map: , are continuous. Here is viewed as a topological space with the product topology. Such a topology is said to be compatible with the group operations and is called a group topology. Checking continuity The product map is continuous if and only if for any and any neighborhood of in , there exist neighborhoods of and of in such that , where }. The inversion map is continuous if and only if for any and any neighborhood of in , there exists a neighborhood of in such that , where }. To show that a topology is compatible with the group operations, it suffices to check that the map , is continuous. Explicitly, this means that for any and any neighborhood in of , there exist neighborhoods of and of in such that . Additive notation This definition used notation for multiplicative groups; the equivalent for additive groups would be that the following two operations are continuous: , , . Hausdorffness Although not part of this definition, many authors require that the topology on be Hausdorff. One reason for this is that any topological group can be canonically associated with a Hausdorff topological group by taking an appropriate canonical quotient; this however, often still requires working with the original non-Hausdorff topological group. Other reasons, and some equivalent conditions, are discussed below. This article will not assume that topological groups are necessarily Hausdorff. Category In the language of category theory, topological groups can be defined concisely as group objects in the category of topological spaces, in the same way that ordinary groups are group objects in the category of sets. Note that the axioms are given in terms of the maps (binary product, unary inverse, and nullary identity), hence are categorical definitions. Homomorphisms A homomorphism of topological groups mea
https://en.wikipedia.org/wiki/Upper%20and%20lower%20bounds	In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an element of that is less than or equal to every element of . A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound. The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds. Examples For example, is a lower bound for the set (as a subset of the integers or of the real numbers, etc.), and so is . On the other hand, is not a lower bound for since it is not smaller than every element in . The set has as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that . Every subset of the natural numbers has a lower bound since the natural numbers have a least element (0 or 1, depending on convention). An infinite subset of the natural numbers cannot be bounded from above. An infinite subset of the integers may be bounded from below or bounded from above, but not both. An infinite subset of the rational numbers may or may not be bounded from below, and may or may not be bounded from above. Every finite subset of a non-empty totally ordered set has both upper and lower bounds. Bounds of functions The definitions can be generalized to functions and even to sets of functions. Given a function with domain and a preordered set as codomain, an element of is an upper bound of if for each in . The upper bound is called sharp if equality holds for at least one value of . It indicates that the constraint is optimal, and thus cannot be further reduced without invalidating the inequality. Similarly, a function defined on domain and having the same codomain is an upper bound of , if for each in . The function is further said to be an upper bound of a set of functions, if it is an upper bound of each function in that set. The notion of lower bound for (sets of) functions is defined analogously, by replacing ≥ with ≤. Tight bounds An upper bound is said to be a tight upper bound, a least upper bound, or a supremum, if no smaller value is an upper bound. Similarly, a lower bound is said to be a tight lower bound, a greatest lower bound, or an infimum, if no greater value is a lower bound. Exact upper bounds An upper bound of a subset of a preordered set is said to be an exact upper bound for if every element of that is strictly majorized by is also majorized by some element of . Exact upper bounds of reduced products of linear orders play an important role in PCF theory. See also Greatest element and least element Infimum and supremum Maximal and minimal elements References Mathematical terminology Order theory Real analysis de:Schranke (Mathematik) pl:Kresy
https://en.wikipedia.org/wiki/Trace%20%28linear%20algebra%29	In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proven that the trace of a matrix is the sum of its (complex) eigenvalues (counted with multiplicities). It can also be proven that for any two matrices and of appropriate sizes. This implies that 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 square matrix is defined as where denotes the entry on the th row and th column of . The entries of 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 be a matrix, with Then Properties Basic properties The trace is a linear mapping. That is, for all square matrices and , and all scalars . A matrix and its transpose have the same trace: 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 real 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 and are two real matrices, then: If one views any real matrix as a vector of length (an operation called vectorization) then the above operation on and coincides with the standard dot product. According to the above expression, is a sum of squares and hence is nonnegative, equal to zero if and only if is zero. Furthermore, as noted in the above formula, . These demonstrate the positive-definiteness and symmetry required of an inner product; it is common to call the Frobenius inner product of and . 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: if and are real positive semi-definite matrices of the same size. 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 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 and are and real or complex matrices, respectively, then This is notable both for the fact that does not usually eq
https://en.wikipedia.org/wiki/List%20of%20mail%20server%20software	This is a list of mail server software: mail transfer agents, mail delivery agents, and other computer software which provide e-mail. Product statistics All such figures are necessarily estimates because data about mail server share is difficult to obtain; there are few reliable primary sources—and no agreed methodologies for its collection. Surveys probing Internet-exposed systems typically attempt to identify systems via their banner, or other identifying features; and report Postfix and exim as overwhelming leaders in March 2021, with greater than 92% share between them. While such methods are effective at identifying mail server share for receiving systems, most large-scale sending environments are not listening for traffic on the public internet and will not be counted using such methodologies. SMTP POP/IMAP Mail filtering Mail server packages Mail-in-a-Box iRedmail Modoboa Mailcow Poste.io docker-mailserver Mailu See also Comparison of mail servers Message transfer agent References Message transfer agents Mail servers
https://en.wikipedia.org/wiki/Probability%20space	In probability theory, a probability space or a probability triple is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a . A probability space consists of three elements: A sample space, , which is the set of all possible outcomes. An event space, which is a set of events, , an event being a set of outcomes in the sample space. A probability function, , which assigns each event in the event space a probability, which is a number between 0 and 1. In order to provide a sensible model of probability, these elements must satisfy a number of axioms, detailed in this article. In the example of the throw of a standard die, we would take the sample space to be . For the event space, we could simply use the set of all subsets of the sample space, which would then contain simple events such as ("the die lands on 5"), as well as complex events such as ("the die lands on an even number"). Finally, for the probability function, we would map each event to the number of outcomes in that event divided by 6 – so for example, would be mapped to , and would be mapped to . When an experiment is conducted, we imagine that "nature" "selects" a single outcome, , from the sample space . All the events in the event space that contain the selected outcome are said to "have occurred". This "selection" happens in such a way that if the experiment were repeated many times, the number of occurrences of each event, as a fraction of the total number of experiments, would most likely tend towards the probability assigned to that event by the probability function . The Soviet mathematician Andrey Kolmogorov introduced the notion of probability space, together with other axioms of probability, in the 1930s. In modern probability theory there are a number of alternative approaches for axiomatization – for example, algebra of random variables. Introduction A probability space is a mathematical triplet that presents a model for a particular class of real-world situations. As with other models, its author ultimately defines which elements , , and will contain. The sample space is the set of all possible outcomes. An outcome is the result of a single execution of the model. Outcomes may be states of nature, possibilities, experimental results and the like. Every instance of the real-world situation (or run of the experiment) must produce exactly one outcome. If outcomes of different runs of an experiment differ in any way that matters, they are distinct outcomes. Which differences matter depends on the kind of analysis we want to do. This leads to different choices of sample space. The σ-algebra is a collection of all the events we would like to consider. This collection may or may not include each of the elementary events. Here, an "event" is a set of zero or more outcomes; that is, a subset of the sample space. An event is considered to have
https://en.wikipedia.org/wiki/Borel%20set	In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel. For a topological space X, the collection of all Borel sets on X forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on X is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets). Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a Borel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory. In some contexts, Borel sets are defined to be generated by the compact sets of the topological space, rather than the open sets. The two definitions are equivalent for many well-behaved spaces, including all Hausdorff σ-compact spaces, but can be different in more pathological spaces. Generating the Borel algebra In the case that X is a metric space, the Borel algebra in the first sense may be described generatively as follows. For a collection T of subsets of X (that is, for any subset of the power set P(X) of X), let be all countable unions of elements of T be all countable intersections of elements of T Now define by transfinite induction a sequence Gm, where m is an ordinal number, in the following manner: For the base case of the definition, let be the collection of open subsets of X. If i is not a limit ordinal, then i has an immediately preceding ordinal i − 1. Let If i is a limit ordinal, set The claim is that the Borel algebra is Gω1, where ω1 is the first uncountable ordinal number. That is, the Borel algebra can be generated from the class of open sets by iterating the operation to the first uncountable ordinal. To prove this claim, any open set in a metric space is the union of an increasing sequence of closed sets. In particular, complementation of sets maps Gm into itself for any limit ordinal m; moreover if m is an uncountable limit ordinal, Gm is closed under countable unions. For each Borel set B, there is some countable ordinal αB such that B can be obtained by iterating the operation over αB. However, as B varies over all Borel sets, αB will vary over all the countable ordinals, and thus the first ordinal at which all the Borel sets are obtained is ω1, the first uncountable ordinal. Example An important example, especially in the theory of probability, is the Borel algebra on the set of real numbers. It is the algebra on which the Borel measure is defined. Given a real random variable defined on a probability space, its probability distribution is by definition also a measure on the Borel algebra. The Borel algebra on the reals is the smallest σ-alge
https://en.wikipedia.org/wiki/Measurable%20space	In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set and a σ-algebra on Then the tuple is called a measurable space. Note that in contrast to a measure space, no measure is needed for a measurable space. Example Look at the set: One possible -algebra would be: Then is a measurable space. Another possible -algebra would be the power set on : With this, a second measurable space on the set is given by Common measurable spaces If is finite or countably infinite, the -algebra is most often the power set on so This leads to the measurable space If is a topological space, the -algebra is most commonly the Borel -algebra so This leads to the measurable space that is common for all topological spaces such as the real numbers Ambiguity with Borel spaces The term Borel space is used for different types of measurable spaces. It can refer to any measurable space, so it is a synonym for a measurable space as defined above a measurable space that is Borel isomorphic to a measurable subset of the real numbers (again with the Borel -algebra) See also References Measure theory Space (mathematics)
https://en.wikipedia.org/wiki/Probability%20density%20function	In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would be equal to that sample. Probability density is the probability per unit length, in other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. This probability is given by the integral of this variable's PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and the area under the entire curve is equal to 1. The terms probability distribution function and probability function have also sometimes been used to denote the probability density function. However, this use is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over general sets of values or it may refer to the cumulative distribution function, or it may be a probability mass function (PMF) rather than the density. "Density function" itself is also used for the probability mass function, leading to further confusion. In general though, the PMF is used in the context of discrete random variables (random variables that take values on a countable set), while the PDF is used in the context of continuous random variables. Example Suppose bacteria of a certain species typically live 4 to 6 hours. The probability that a bacterium lives 5 hours is equal to zero. A lot of bacteria live for approximately 5 hours, but there is no chance that any given bacterium dies at exactly 5.00... hours. However, the probability that the bacterium dies between 5 hours and 5.01 hours is quantifiable. Suppose the answer is 0.02 (i.e., 2%). Then, the probability that the bacterium dies between 5 hours and 5.001 hours should be about 0.002, since this time interval is one-tenth as long as the previous. The probability that the bacterium dies between 5 hours and 5.0001 hours should be about 0.0002, and so on. In this example, the ratio (probability of dying during an interval) / (duration of the interval) is approximately constant, and equal to
https://en.wikipedia.org/wiki/Flux	Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport phenomena, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In vector calculus flux is a scalar quantity, defined as the surface integral of the perpendicular component of a vector field over a surface. Terminology The word flux comes from Latin: fluxus means "flow", and fluere is "to flow". As fluxion, this term was introduced into differential calculus by Isaac Newton. The concept of heat flux was a key contribution of Joseph Fourier, in the analysis of heat transfer phenomena. His seminal treatise Théorie analytique de la chaleur (The Analytical Theory of Heat), defines fluxion as a central quantity and proceeds to derive the now well-known expressions of flux in terms of temperature differences across a slab, and then more generally in terms of temperature gradients or differentials of temperature, across other geometries. One could argue, based on the work of James Clerk Maxwell, that the transport definition precedes the definition of flux used in electromagnetism. The specific quote from Maxwell is: According to the transport definition, flux may be a single vector, or it may be a vector field / function of position. In the latter case flux can readily be integrated over a surface. By contrast, according to the electromagnetism definition, flux is the integral over a surface; it makes no sense to integrate a second-definition flux for one would be integrating over a surface twice. Thus, Maxwell's quote only makes sense if "flux" is being used according to the transport definition (and furthermore is a vector field rather than single vector). This is ironic because Maxwell was one of the major developers of what we now call "electric flux" and "magnetic flux" according to the electromagnetism definition. Their names in accordance with the quote (and transport definition) would be "surface integral of electric flux" and "surface integral of magnetic flux", in which case "electric flux" would instead be defined as "electric field" and "magnetic flux" defined as "magnetic field". This implies that Maxwell conceived of these fields as flows/fluxes of some sort. Given a flux according to the electromagnetism definition, the corresponding flux density, if that term is used, refers to its derivative along the surface that was integrated. By the Fundamental theorem of calculus, the corresponding flux density is a flux according to the transport definition. Given a current such as electric current—charge per time, current density would also be a flux according to the transport definition—charge per time per area. Due to the conflicting definitions of flux, and the interchangeability of flux, flow, and current in nontechnical English, all of the terms u
https://en.wikipedia.org/wiki/Sill	Sill may refer to: Sill (dock), a weir at the low water mark retaining water within a dock Sill (geology), a subhorizontal sheet intrusion of molten or solidified magma Sill (geostatistics) Sill (river), a river in Austria Sill plate, a construction element Window sill, a more specific construction element than above Automotive sill, also known as a rocker panel; see Glossary of automotive design#R Fort Sill, a United States Army post near Lawton, Oklahoma Mount Sill, a California mountain Aquatic sill, a shoal near the mouth of a fjord, remnant of an extinct glacier's terminal moraine People Sill Anna Peck Sill (1816-1889), American educator Edward Rowland Sill (1841–1887), American poet and educator George G. Sill (1829–1907), American politician from Connecticut Joel Sill (born 1946), American music producer John M. B. Sill (1831–1901), American diplomat Joshua W. Sill (1831–1862), American Civil War brigadier general Judee Sill (1944–1979), American singer and songwriter Lester Sill (1918–1994), American record label executive Thomas Hale Sill (1783–1856), American politician from Pennsylvania Zach Sill (born 1988), Canadian ice hockey player Sills Beverly Sills (1929–2007), American operatic soprano David Sills (judge) (1938–2011), American jurist David Sills (American football) (born 1996), American football player Douglas Sills (born 1960), American actor Eileen Sills (born 1962), British chief nurse and NHS national guardian Josh Sills (born 1998), American football player Kenneth C.M. Sills (1879–1954), American educator Milton Sills (1882–1930), American stage and film actor Paul Sills (1927–2008), American director and improvisation teacher Saskia Sills (born 1996), British windsurfer and sailor Steven Sills (), American screenwriter and film producer Tim Sills (born 1979), English footballer See also Cill (disambiguation) Still (disambiguation)
https://en.wikipedia.org/wiki/Autonomous%20system	Autonomous system may refer to: Autonomous system (Internet), a collection of IP networks and routers under the control of one entity Autonomous system (mathematics), a system of ordinary differential equations which does not depend on the independent variable Autonomous robot, robots which can perform desired tasks in unstructured environments without continuous human guidance Autonomous underwater vehicle, a system that travels underwater without requiring input from an operator.
https://en.wikipedia.org/wiki/Permutation	In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. For example, there are six permutations (orderings) of the set {1,2,3}: written as tuples, they are (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). Anagrams of a word whose letters are all different are also permutations: the letters are already ordered in the original word, and the anagram reorders them. The study of permutations of finite sets is an important topic in combinatorics and group theory. Permutations are used in almost every branch of mathematics and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences. The number of permutations of distinct objects is factorial, usually written as , which means the product of all positive integers less than or equal to . Formally, a permutation of a set is defined as a bijection from to itself. That is, it is a function from to for which every element occurs exactly once as an image value. Such a function is equivalent to the rearrangement of the elements of in which each element i is replaced by the corresponding . For example, the permutation (3,1,2) is described by the function defined as . The collection of all permutations of a set form a group called the symmetric group of the set. The group operation is the composition of functions (performing one rearrangement after the other), which results in another function (rearrangement). The properties of permutations do not depend on the nature of the elements being permuted, only on their number, so one often considers the standard set . In elementary combinatorics, the -permutations, or partial permutations, are the ordered arrangements of distinct elements selected from a set. When is equal to the size of the set, these are the permutations in the previous sense. History Permutations called hexagrams were used in China in the I Ching (Pinyin: Yi Jing) as early as 1000 BC. In Greece, Plutarch wrote that Xenocrates of Chalcedon (396–314 BC) discovered the number of different syllables possible in the Greek language. This would have been the first attempt on record to solve a difficult problem in permutations and combinations. Al-Khalil (717–786), an Arab mathematician and cryptographer, wrote the Book of Cryptographic Messages. It contains the first use of permutations and combinations, to list all possible Arabic words with and without vowels. The rule to determine the number of permutations of n objects was known in Indian culture around 1150 AD. The Lilavati by the Indian mathematician Bhaskara II contains a passage that translates as follows: The product of multiplication
https://en.wikipedia.org/wiki/Measurable%20function	In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable. Formal definition Let and be measurable spaces, meaning that and are sets equipped with respective -algebras and A function is said to be measurable if for every the pre-image of under is in ; that is, for all That is, where is the σ-algebra generated by f. If is a measurable function, one writes to emphasize the dependency on the -algebras and Term usage variations The choice of -algebras in the definition above is sometimes implicit and left up to the context. For example, for or other topological spaces, the Borel algebra (generated by all the open sets) is a common choice. Some authors define measurable functions as exclusively real-valued ones with respect to the Borel algebra. If the values of the function lie in an infinite-dimensional vector space, other non-equivalent definitions of measurability, such as weak measurability and Bochner measurability, exist. Notable classes of measurable functions Random variables are by definition measurable functions defined on probability spaces. If and are Borel spaces, a measurable function is also called a Borel function. Continuous functions are Borel functions but not all Borel functions are continuous. However, a measurable function is nearly a continuous function; see Luzin's theorem. If a Borel function happens to be a section of a map it is called a Borel section. A Lebesgue measurable function is a measurable function where is the -algebra of Lebesgue measurable sets, and is the Borel algebra on the complex numbers Lebesgue measurable functions are of interest in mathematical analysis because they can be integrated. In the case is Lebesgue measurable if and only if is measurable for all This is also equivalent to any of being measurable for all or the preimage of any open set being measurable. Continuous functions, monotone functions, step functions, semicontinuous functions, Riemann-integrable functions, and functions of bounded variation are all Lebesgue measurable. A function is measurable if and only if the real and imaginary parts are measurable. Properties of measurable functions The sum and product of two complex-valued measurable functions are measurable. So is the quotient, so long as there is no division by zero. If and are measurable functions, then so is their composition If and are measurable functions, their composition need not be
https://en.wikipedia.org/wiki/Sim%C3%A9on%20Denis%20Poisson	Baron Siméon Denis Poisson FRS FRSE (; 21 June 1781 – 25 April 1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electricity and magnetism, thermodynamics, elasticity, and fluid mechanics. Moreover, he predicted the Poisson spot in his attempt to disprove the wave theory of Augustin-Jean Fresnel, which was later confirmed. Biography Poisson was born in Pithiviers, Loiret district in France, the son of Siméon Poisson, an officer in the French army. In 1798, he entered the École Polytechnique in Paris as first in his year, and immediately began to attract the notice of the professors of the school, who left him free to make his own decisions as to what he would study. In his final year of study, less than two years after his entry, he published two memoirs, one on Étienne Bézout's method of elimination, the other on the number of integrals of a finite difference equation and this was so impressive that he was allowed to graduate in 1800 without taking the final examination,. The latter of the memoirs was examined by Sylvestre-François Lacroix and Adrien-Marie Legendre, who recommended that it should be published in the Recueil des savants étrangers, an unprecedented honor for a youth of eighteen. This success at once procured entry for Poisson into scientific circles. Joseph Louis Lagrange, whose lectures on the theory of functions he attended at the École Polytechnique, recognized his talent early on, and became his friend. Meanwhile, Pierre-Simon Laplace, in whose footsteps Poisson followed, regarded him almost as his son. The rest of his career, until his death in Sceaux near Paris, was occupied by the composition and publication of his many works and in fulfilling the duties of the numerous educational positions to which he was successively appointed. Immediately after finishing his studies at the École Polytechnique, he was appointed répétiteur (teaching assistant) there, a position which he had occupied as an amateur while still a pupil in the school; for his schoolmates had made a custom of visiting him in his room after an unusually difficult lecture to hear him repeat and explain it. He was made deputy professor (professeur suppléant) in 1802, and, in 1806 full professor succeeding Jean Baptiste Joseph Fourier, whom Napoleon had sent to Grenoble. In 1808 he became astronomer to the Bureau des Longitudes; and when the Faculté des sciences de Paris was instituted in 1809 he was appointed a professor of rational mechanics (professeur de mécanique rationelle). He went on to become a member of the Institute in 1812, examiner at the military school (École Militaire) at Saint-Cyr in 1815, graduation examiner at the École Polytechnique in 1816, councillor of the university in 1820, and geometer to the Bureau des Longitudes succeeding Pierre-Simon Laplace in 1827. In 1817, he married Nancy de Bardi and with her, he had four
https://en.wikipedia.org/wiki/Goldbach%27s%20conjecture	Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to hold for all integers less than , but remains unproven despite considerable effort. History On 7 June 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII), in which he proposed the following conjecture: Goldbach was following the now-abandoned convention of considering 1 to be a prime number, so that a sum of units would be a sum of primes. He then proposed a second conjecture in the margin of his letter, which implies the first: Euler replied in a letter dated 30 June 1742 and reminded Goldbach of an earlier conversation they had had (), in which Goldbach had remarked that the first of those two conjectures would follow from the statement This is in fact equivalent to his second, marginal conjecture. In the letter dated 30 June 1742, Euler stated: Each of the three conjectures above has a natural analog in terms of the modern definition of a prime, under which 1 is excluded. A modern version of the first conjecture is: A modern version of the marginal conjecture is: And a modern version of Goldbach's older conjecture of which Euler reminded him is: These modern versions might not be entirely equivalent to the corresponding original statements. For example, if there were an even integer larger than 4, for a prime, that could not be expressed as the sum of two primes in the modern sense, then it would be a counterexample to the modern version of the third conjecture (without being a counterexample to the original version). The modern version is thus probably stronger (but in order to confirm that, one would have to prove that the first version, freely applied to any positive even integer , could not possibly rule out the existence of such a specific counterexample ). In any case, the modern statements have the same relationships with each other as the older statements did. That is, the second and third modern statements are equivalent, and either implies the first modern statement. The third modern statement (equivalent to the second) is the form in which the conjecture is usually expressed today. It is also known as the "strong", "even", or "binary" Goldbach conjecture. A weaker form of the second modern statement, known as "Goldbach's weak conjecture", the "odd Goldbach conjecture", or the "ternary Goldbach conjecture", asserts that A proof for the weak conjecture was submitted in 2013 by Harald Helfgott to Annals of Mathematics Studies series. Although the article was accepted, Helfgott decided to undertake the major modifications suggested by the referee. Despite several revisions, Helfgott's proof has not yet appeared in a peer-reviewed publication. The weak conjecture is implied by the strong conjecture, as if is a sum of two primes, then is a
https://en.wikipedia.org/wiki/Complete%20measure	In mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a measure space (X, Σ, μ) is complete if and only if Motivation The need to consider questions of completeness can be illustrated by considering the problem of product spaces. Suppose that we have already constructed Lebesgue measure on the real line: denote this measure space by We now wish to construct some two-dimensional Lebesgue measure on the plane as a product measure. Naively, we would take the -algebra on to be the smallest -algebra containing all measurable "rectangles" for While this approach does define a measure space, it has a flaw. Since every singleton set has one-dimensional Lebesgue measure zero, for subset of However, suppose that is a non-measurable subset of the real line, such as the Vitali set. Then the -measure of is not defined but and this larger set does have -measure zero. So this "two-dimensional Lebesgue measure" as just defined is not complete, and some kind of completion procedure is required. Construction of a complete measure Given a (possibly incomplete) measure space (X, Σ, μ), there is an extension (X, Σ0, μ0) of this measure space that is complete. The smallest such extension (i.e. the smallest σ-algebra Σ0) is called the completion of the measure space. The completion can be constructed as follows: let Z be the set of all the subsets of the zero-μ-measure subsets of X (intuitively, those elements of Z that are not already in Σ are the ones preventing completeness from holding true); let Σ0 be the σ-algebra generated by Σ and Z (i.e. the smallest σ-algebra that contains every element of Σ and of Z); μ has an extension μ0 to Σ0 (which is unique if μ is σ-finite), called the outer measure of μ, given by the infimum Then (X, Σ0, μ0) is a complete measure space, and is the completion of (X, Σ, μ). In the above construction it can be shown that every member of Σ0 is of the form A ∪ B for some A ∈ Σ and some B ∈ Z, and Examples Borel measure as defined on the Borel σ-algebra generated by the open intervals of the real line is not complete, and so the above completion procedure must be used to define the complete Lebesgue measure. This is illustrated by the fact that the set of all Borel sets over the reals has the same cardinality as the reals. While the Cantor set is a Borel set, has measure zero, and its power set has cardinality strictly greater than that of the reals. Thus there is a subset of the Cantor set that is not contained in the Borel sets. Hence, the Borel measure is not complete. n-dimensional Lebesgue measure is the completion of the n-fold product of the one-dimensional Lebesgue space with itself. It is also the completion of the Borel measure, as in the one-dimensional case. Properties Maharam's theorem states that every complete measure space is decomposable into measures on cont
https://en.wikipedia.org/wiki/Complete%20lattice	In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a conditionally complete lattice. For comparison, in a general lattice, only pairs of elements need to have a supremum and an infimum. Every non-empty finite lattice is complete, but infinite lattices may be incomplete. Complete lattices appear in many applications in mathematics and computer science. Being a special instance of lattices, they are studied both in order theory and universal algebra. Complete lattices must not be confused with complete partial orders (cpos), which constitute a strictly more general class of partially ordered sets. More specific complete lattices are complete Boolean algebras and complete Heyting algebras (locales). Formal definition A partially ordered set (L, ≤) is a complete lattice if every subset A of L has both a greatest lower bound (the infimum, also called the meet) and a least upper bound (the supremum, also called the join) in (L, ≤). The meet is denoted by , and the join by . In the special case where A is the empty set, the meet of A will be the greatest element of L. Likewise, the join of the empty set yields the least element. Since the definition also assures the existence of binary meets and joins, complete lattices thus form a special class of bounded lattices. More implications of the above definition are discussed in the article on completeness properties in order theory. Complete semilattices In order theory, arbitrary meets can be expressed in terms of arbitrary joins and vice versa (for details, see completeness (order theory)). In effect, this means that it is sufficient to require the existence of either all meets or all joins to obtain the class of all complete lattices. As a consequence, some authors use the terms complete meet-semilattice or complete join-semilattice as another way to refer to complete lattices. Though similar on objects, the terms entail different notions of homomorphism, as will be explained in the below section on morphisms. On the other hand, some authors have no use for this distinction of morphisms (especially since the emerging concepts of "complete semilattice morphisms" can as well be specified in general terms). Consequently, complete meet-semilattices have also been defined as those meet-semilattices that are also complete partial orders. This concept is arguably the "most complete" notion of a meet-semilattice that is not yet a lattice (in fact, only the top element may be missing). This discussion is also found in the article on semilattices. Complete sublattices A sublattice M of a complete lattice L is called a complete sublattice of L if for every subset A of M the elements and , as defined in L, are actually in M. If the above requirement is lessened to require only non-empty meet and joins to be in M, the sublattice M is called a closed s
https://en.wikipedia.org/wiki/Limit%20inferior%20and%20limit%20superior	In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant. Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit. The limit inferior of a sequence is denoted by and the limit superior of a sequence is denoted by Definition for sequences The of a sequence (xn) is defined by or Similarly, the of (xn) is defined by or Alternatively, the notations and are sometimes used. The limits superior and inferior can equivalently be defined using the concept of subsequential limits of the sequence . An element of the extended real numbers is a subsequential limit of if there exists a strictly increasing sequence of natural numbers such that . If is the set of all subsequential limits of , then and If the terms in the sequence are real numbers, the limit superior and limit inferior always exist, as the real numbers together with ±∞ (i.e. the extended real number line) are complete. More generally, these definitions make sense in any partially ordered set, provided the suprema and infima exist, such as in a complete lattice. Whenever the ordinary limit exists, the limit inferior and limit superior are both equal to it; therefore, each can be considered a generalization of the ordinary limit which is primarily interesting in cases where the limit does not exist. Whenever lim inf xn and lim sup xn both exist, we have The limits inferior and superior are related to big-O notation in that they bound a sequence only "in the limit"; the sequence may exceed the bound. However, with big-O notation the sequence can only exceed the bound in a finite prefix of the sequence, whereas the limit superior of a sequence like e−n may actually be less than all elements of the sequence. The only promise made is that some tail of the sequence can be bounded above by the limit superior plus an arbitrarily small positive constant, and bounded below by the limit inferior minus an arbitrarily small positive constant. The limit superior and limit inferior of a sequence are a special case of those of a function (see below). The case of sequences of real numbers In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers. Since the supremum and infimum of an unbounded set of real numbers

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