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https://en.wikipedia.org/wiki/1977%20Japan%20Soccer%20League | Statistics of Japan Soccer League for the 1977 season. This was the inaugural season of the Regional League promotion series, which replaced the Senior Cup as the source of the clubs promoted from the regional Japanese football leagues.
First Division
Promotion/relegation Series
Yomiuri promoted, Toyota Motors relegated.
Second Division
Promotion/relegation Series
Toshiba promoted, Furukawa Chiba relegated.
References
Japan - List of final tables (RSSSF)
1977
1
Jap
Jap |
https://en.wikipedia.org/wiki/1978%20Japan%20Soccer%20League | Statistics of Japan Soccer League for the 1978 season.
First Division
By winning the 1978 Emperor's Cup and 1978 Japan Soccer League Cup along with the title, Mitsubishi completed the first Japanese treble ever.
Promotion/relegation Series
Nissan promoted, Fujitsu relegated.
Second Division
Promotion/relegation Series
Yamaha promoted, Kyoto Shiko relegated.
References
Japan - List of final tables (RSSSF)
1978
1
Jap
Jap |
https://en.wikipedia.org/wiki/1979%20Japan%20Soccer%20League | Statistics of Japan Soccer League for the 1979 season.
First Division
Promotion/relegation Series
Yamaha promoted, NKK relegated.
Second Division
Promotion/relegation Series
No relegations. Due to withdrawal of Yanmar Club, Yanmar Diesel's B-squad, Daikyo was promoted.
References
Japan - List of final tables (RSSSF)
1979
1
Jap
Jap |
https://en.wikipedia.org/wiki/1980%20Japan%20Soccer%20League | Statistics of Japan Soccer League for the 1980 season. For the first time ever, automatic promotion and relegation was introduced for the first and last places of the Second Division, which means that the last place in the First Division went down.
First Division
Yanmar Diesel won the title for a fourth time.
Nissan, who had saved itself from relegation in the playout the previous season, went down after the bottom place was granted automatic relegation, while Yamaha saved itself by beating Fujitsu in the playout.
Promotion/relegation Series
Second Division
Honda was finally promoted on the second attempt after the 1978 debacle.
Kofu Club saved itself from relegation by defeating Furukawa Electric Chiba, Furukawa's B-team. Cosmo Oil Yokkaichi fell through and went back to the Tokai regional league.
Promotion/relegation Series
References
Japan - List of final tables (RSSSF)
Japan Soccer League seasons
1
Jap
Jap |
https://en.wikipedia.org/wiki/1981%20Japan%20Soccer%20League | Statistics of Japan Soccer League for the 1981 season.
First Division
Fujita Industries won their third League title.
Nippon Steel, one of eight inaugural member of the First Division in 1965 as Yawata Steel, was defeated by Second Division runner-up Nissan in the playout and relegated, never to play top flight football again. Yamaha Motors was relegated in bottom place, having won only two matches.
Promotion/relegation Series
Second Division
NKK and Nissan returned after two years in the second tier, NKK also grabbing the Emperor's Cup.
Kofu Club saved itself from relegation yet again by defeating NTT West Japan Kyoto, who were looking to regain their League place. Nagoya Soccer Club, an amateur outfit who never looked like League material, went back to the Tokai regional league after a single attempt.
Promotion/relegation Series
References
Japan - List of final tables (RSSSF)
Japan Soccer League seasons
1
Jap
Jap |
https://en.wikipedia.org/wiki/1983%20Japan%20Soccer%20League | Statistics of Japan Soccer League for the 1983 season.
First Division
Yomiuri, the football club became one of big names of earlier years of J.League as Verdy Kawasaki, and currently known as Tokyo Verdy, won its first of seven League championships, fully riding in the wave of its parent company's funds and prestige.
Mazda, five-time First Division champions in the 1960s, was relegated for the first time. Hitachi saved itself by defeating Sumitomo in the playout.
Promotion/relegation Series
Second Division
NKK returned to the top flight at the first time of asking.
Saitama Teachers went back to the Kantō regional league, and Toho Titanium followed when they lost the playout to Matsushita, a rising club at the time based in Nara which would eventually become Gamba Osaka.
Promotion/relegation Series
References
Japan - List of final tables (RSSSF)
Japan Soccer League seasons
1
Jap
Jap |
https://en.wikipedia.org/wiki/1984%20Japan%20Soccer%20League | Statistics of Japan Soccer League for the 1984 season. No promotion/relegation series for both division were held due to expansion of both divisions in the following season.
First Division
No relegation took place due to expansion to 12 clubs.
Yomiuri was invited to the revived Asian Club Championship, but withdrew.
Second Division
No relegation took place due to expansion to 12 clubs.
References
Japan - List of final tables (RSSSF)
Japan Soccer League seasons
1
Jap
Jap |
https://en.wikipedia.org/wiki/1985%E2%80%9386%20Japan%20Soccer%20League | Statistics of Japan Soccer League for the 1985–86 season.
First Division
Led by their star player Yasuhiko Okudera, who had returned to the club after successful periods in Europe, Furukawa Electric won their second title. Okudera became the first widely recognized professional Japanese player.
Sumitomo and ANA Yokohama were relegated after one season in the top division.
Second Division
No relegation took place for a second wave of expansion that would bring the division's number of clubs to 16.
First stage
East
West
Second stage
Promotion Group
Relegation Group
East
West
7-12 Playoff
References
Japan - List of final tables (RSSSF)
Japan Soccer League seasons
1985 in Japanese football
1986 in Japanese football
Japan Soccer League |
https://en.wikipedia.org/wiki/1986%E2%80%9387%20Japan%20Soccer%20League | Statistics of Japan Soccer League for the 1986–87 season.
First Division
Despite moving to Kashiwa, Chiba and a soccer-specific stadium of their own, Hitachi did not adjust well and were relegated in bottom place, the first drop for the former champions. Matsushita, despite having more victories than relegation rivals Yamaha, had more losses as well and thus joined Hitachi.
Second Division
Sumitomo returned to the top flight at the first time of asking, followed by Toyota Motors, who had been struggling since their 1977 relegation and came close to dropping out of the League. TDK and the Kyoto Police Dept. team went back to the regional divisions; TDK would not return to the second tier until 2021.
First stage
East
West
Second stage
Promotion Group
Relegation Group
East
West
9th-16th Place Playoff
References
Japan - List of final tables (RSSSF)
Japan Soccer League seasons
1987 in Japanese football
1986 in Japanese football
Japan Soccer League |
https://en.wikipedia.org/wiki/1987%E2%80%9388%20Japan%20Soccer%20League | Statistics of Japan Soccer League for the 1987–88 season.
First Division
Second Division
First stage
East
West
Second stage
Promotion Group
Relegation Group
East
West
9th-16th Places Playoff
References
Japan - List of final tables (RSSSF)
Japan Soccer League seasons
1987 in Japanese football
1988 in Japanese football
Japan Soccer League |
https://en.wikipedia.org/wiki/1988%E2%80%9389%20Japan%20Soccer%20League | Statistics of Japan Soccer League for the 1988–89 season.
First Division
For the first time, the format of three points for a win was adopted, but only for the First Division. Nissan won their first title.
Four-time champion Mitsubishi was relegated for the first time, along with struggling Sumitomo.
Second Division
This was the last season in which the second tier was contested in an East-and-West format. Toshiba won a second championship, but this time their promotion was automatic. Fallen giant Hitachi, still adjusting to the change in town, joined them. Regional outfits Fujieda Municipal and NTT Kansai went back to their regional leagues.
First stage
East
West
Second stage
Promotion Group
Relegation Group
East
West
9th-16th Places Playoff
References
Japan - List of final tables (RSSSF)
Japan Soccer League seasons
1989 in Japanese football leagues
1988 in Japanese football
Japan Soccer League |
https://en.wikipedia.org/wiki/1983%E2%80%9384%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League for the 1983–84 season.
Overview
It was contested by 14 teams, and Al Arabi Kuwait won the championship.
League standings
References
Kuwait - List of final tables (RSSSF)
1984
1983–84 in Asian association football leagues
1 |
https://en.wikipedia.org/wiki/1984%E2%80%9385%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League for the 1984–85 season.
Overview
It was contested by 14 teams, and Al Arabi Kuwait won the championship.
League standings
References
Kuwait - List of final tables (RSSSF)
1985
1984–85 in Asian association football leagues
1 |
https://en.wikipedia.org/wiki/1985%E2%80%9386%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League for the 1985–86 season.
Overview
It was contested by 7 teams, and Kazma Sporting Club won the championship.
League standings
References
Kuwait - List of final tables (RSSSF)
1986
1985–86 in Asian association football leagues
1 |
https://en.wikipedia.org/wiki/1986%E2%80%9387%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League for the 1986–87 season.
Overview
It was contested by 8 teams, and Kazma Sporting Club won the championship.
League standings
References
Kuwait - List of final tables (RSSSF)
1987
1986–87 in Asian association football leagues
1 |
https://en.wikipedia.org/wiki/1987%E2%80%9388%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League for the 1987–88 season.
Overview
It was contested by 8 teams, and Al Arabi Kuwait won the championship.
League standings
References
Kuwait - List of final tables (RSSSF)
1988
1987–88 in Asian association football leagues
1 |
https://en.wikipedia.org/wiki/1988%E2%80%9389%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League for the 1988–89 season.
Overview
It was contested by 8 teams, and Al Arabi Kuwait won the championship.
League standings
References
Kuwait - List of final tables (RSSSF)
1989
1988–89 in Asian association football leagues
1 |
https://en.wikipedia.org/wiki/1989%E2%80%9390%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League for the 1989–90 season.
Overview
It was contested by 8 teams, and Al Jahra won the championship.
League standings
References
Kuwait - List of final tables (RSSSF)
1990
1989–90 in Asian association football leagues
1 |
https://en.wikipedia.org/wiki/1991%E2%80%9392%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League for the 1991–92 season.
Overview
It was contested by 14 teams, and Al Qadisiya Kuwait won the championship.
Group stage
Group A
Group B
Championship Playoffs
Semifinals
Al Salmiya Club 0-1 Al Yarmouk
Al Qadisiya Kuwait 1-0 Al Arabi Kuwait
Third place match
Al Salmiya Club 2-1 Al Arabi Kuwait
Final
Al Qadisiya Kuwait 2-0 Al Yarmouk
References
Kuwait - List of final tables (RSSSF)
1992
Kuw
1 |
https://en.wikipedia.org/wiki/1992%E2%80%9393%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League for the 1992–93 season.
Overview
It was contested by 8 teams, and Al Arabi Kuwait won the championship.
League standings
References
Kuwait - List of final tables (RSSSF)
1993
Kuw
1 |
https://en.wikipedia.org/wiki/1994%E2%80%9395%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League for the 1994–95 season.
Overview
It was contested by 14 teams, and Al Salmiya Club won the championship.
League standings
References
Kuwait - List of final tables (RSSSF)
1994–95
1
1994–95 in Asian association football leagues |
https://en.wikipedia.org/wiki/1995%E2%80%9396%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League for the 1995–96 season.
Overview
It was contested by 14 teams, and Kazma Sporting Club won the championship.
League standings
Championship playoff
References
Kuwait - List of final tables (RSSSF)
1995–96
1
1995–96 in Asian association football leagues |
https://en.wikipedia.org/wiki/1996%E2%80%9397%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League for the 1996–97 season.
Overview
It was contested by 13 teams, and Al Arabi Kuwait won the championship.
League standings
Semifinals
Al Arabi Kuwait 3-2 : 0-0 Al Salmiya Club
Kazma Sporting Club 1-1 : 0-3 Al Naser Sporting Club
Third place match
Kazma Sporting Club 1-0 Al Salmiya Club
Final
Al Arabi Kuwait 1-0 Al Naser Sporting Club
References
Kuwait - List of final tables (RSSSF)
1996–97
1
1996–97 in Asian association football leagues |
https://en.wikipedia.org/wiki/1997%E2%80%9398%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League for the 1997–98 season.
Overview
It was contested by 14 teams, and Al Salmiya Club won the championship.
League standings
References
Kuwait - List of final tables (RSSSF)
RSSSF
1997–98
1
1997–98 in Asian association football leagues |
https://en.wikipedia.org/wiki/1998%E2%80%9399%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League for the 1998–99 season.
Overview
It was contested by 14 teams, and Al Qadisiya Kuwait won the championship.
First stage
Second stage
Group 1
Group 2
Group 3
Championship play-offs
Quarterfinals
Semifinals
Third place match
Final
References
Kuwait – List of final tables (RSSSF)
1998-99
1
1998–99 in Asian association football leagues |
https://en.wikipedia.org/wiki/1999%E2%80%932000%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League for the 1999–2000 season.
Overview
It was contested by 14 teams, and Al Salmiya Club won the championship.
League standings
Championship playoff
References
Kuwait - List of final tables (RSSSF)
1999–2000
1999–2000 in Asian association football leagues
1 |
https://en.wikipedia.org/wiki/2000%E2%80%9301%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League for the 2000–01 season.
Overview
It was contested by 8 teams, and Al Kuwait Kaifan won the championship.
League standings
References
Kuwait - List of final tables (RSSSF)
2000
2000–01 in Asian association football leagues
1 |
https://en.wikipedia.org/wiki/2001%E2%80%9302%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League for the 2001–02 season.
Overview
It was competed between 8 teams, and Al Arabi Kuwait won the championship.
League standings
References
Kuwait - List of final tables (RSSSF)
2001–02
1
2001–02 in Asian association football leagues |
https://en.wikipedia.org/wiki/2002%E2%80%9303%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League for the 2002–03 season.
Overview
It was contested by 8 teams, and Al Qadisiya Kuwait won the championship.
League standings
Top scorers
References
Kuwait - List of final tables (RSSSF)
2002–03
1
2002–03 in Asian association football leagues |
https://en.wikipedia.org/wiki/2003%E2%80%9304%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League for the 2003–04 season.
Overview
It was contested by 14 teams, and Al Qadisiya Kuwait won the championship.
League standings
Places 5–8
Semifinals
Tadamon 1–2 : abd Al-Shabab
Kazma Sporting Club 4–1 : 1–2 Al Jahra
7th Place Match
Al Jahra 0–1 Tadamon
5th Place Match
Al-Shabab 1–2 Kazma Sporting Club
Places 1–4
Semifinals
Al Qadisiya Kuwait 3–0 : 0–1 Al Kuwait Kaifan
Al Arabi Kuwait 1–2 : 1–0 Al Salmiya Club
3rd Place Match
Al Kuwait Kaifan 1–3 Al Arabi Kuwait
Championship final
Al Qadisiya Kuwait 2–1Al Salmiya Club
References
Kuwait – List of final tables (RSSSF)
2003–04
1
2003–04 in Asian association football leagues |
https://en.wikipedia.org/wiki/2005%E2%80%9306%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League for the 2005–06 season. The bottom 6 teams of that season were relegated as the commencing season was the inaugurating season of the Kuwaiti second division.
Overview
It was contested by 14 teams, and Al Kuwait Kaifan won the championship.
League standings
References
Kuwait - List of final tables (RSSSF)
Kuwait Premier League seasons
1
Kuw |
https://en.wikipedia.org/wiki/2006%E2%80%9307%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League for the 2006–07 season.
Overview
It was contested by 8 teams, and Al Kuwait Kaifan won the championship.
League standings
Championship playoff
Al Kuwait Kaifan 2-0 Kazma Sporting Club
References
Kuwait - List of final tables (RSSSF)
Kuwait Premier League seasons
Kuwait
1 |
https://en.wikipedia.org/wiki/From%20Here%20to%20Infinity%20%28book%29 | From Here to Infinity: A Guide to Today's Mathematics, a 1996 book by mathematician and science popularizer Ian Stewart, is a guide to modern mathematics for the general reader. It aims to answer questions such as "What is mathematics?", "What is it for " and "What are mathematicians doing nowadays?". Author Simon Singh describes it as "An interesting and accessible account of current mathematical topics".
Summary
After an introductory chapter The Nature of Mathematics, Stewart devotes each of the following 18 chapters to an exposition of a particular problem that has given rise to new mathematics or an area of research in modern mathematics.
Chapter 2 - The Price of Primality - primality tests and integer factorisation
Chapter 3 - Marginal Interest - Fermat's Last Theorem
Chapter 4 - Parallel Thinking - non-Euclidean geometry
Chapter 5 - The Miraculous Jar - Cantor's theorem and cardinal numbers
Chapter 6 - Ghosts of Departed Quantities - calculus and non-standard analysis
Chapter 7 - The Duellist and the Monster - the classification of finite simple groups
Chapter 8 - The Purple Wallflower - the four colour theorem
Chapter 9 - Much Ado About Knotting - topology and the Poincaré conjecture
Chapter 10 - More Ado About Knotting - knot polynomials
Chapter 11 - Squarerooting the Unsquarerootable - complex numbers and the Riemann hypothesis
Chapter 12 - Squaring the Unsquarable - the Banach-Tarski paradox
Chapter 13 - Strumpet Fortune - probability and random walks
Chapter 14 - The Mathematics of Nature - the stability of the Solar System
Chapter 15 - The Patterns of Chaos - chaos theory and strange attractors
Chapter 16 - The Two-and-a-halfth Dimension - fractals
Chapter 17 - Dixit Algorizmi - algorithms and NP-complete problems
Chapter 18 - The Limits of Computability - Turing machines and computable numbers
Chapter 19 - The Ultimate in Technology Transfer - experimental mathematics and the relationship between mathematics and science
Editions
Important advances in mathematics necessitated revisions of the book. For example, when the 1st edition came out, Fermat's Last Theorem was still an open problem. By the 3rd edition, it has been solved by Andrew Wiles. Other revised topics include Tarski's circle-squaring problem, Carmichael numbers, and the Kepler Problem.
1st edition (1987): published under the title The Problems of Mathematics
2nd edition (1992)
retitled/revised edition (1996)
References
Books by Ian Stewart (mathematician)
Popular mathematics books |
https://en.wikipedia.org/wiki/Fermat%27s%20Last%20Theorem | In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been known since antiquity to have infinitely many solutions.
The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of Arithmetica. Fermat added that he had a proof that was too large to fit in the margin. Although other statements claimed by Fermat without proof were subsequently proven by others and credited as theorems of Fermat (for example, Fermat's theorem on sums of two squares), Fermat's Last Theorem resisted proof, leading to doubt that Fermat ever had a correct proof. Consequently the proposition became known as a conjecture rather than a theorem. After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles and formally published in 1995. It was described as a "stunning advance" in the citation for Wiles's Abel Prize award in 2016. It also proved much of the Taniyama–Shimura conjecture, subsequently known as the modularity theorem, and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques.
The unsolved problem stimulated the development of algebraic number theory in the 19th and 20th centuries. It is among the most notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Records as the "most difficult mathematical problem", in part because the theorem has the largest number of unsuccessful proofs.
Overview
Pythagorean origins
The Pythagorean equation, , has an infinite number of positive integer solutions for x, y, and z; these solutions are known as Pythagorean triples (with the simplest example 3,4,5). Around 1637, Fermat wrote in the margin of a book that the more general equation had no solutions in positive integers if n is an integer greater than 2. Although he claimed to have a general proof of his conjecture, Fermat left no details of his proof, and no proof by him has ever been found. His claim was discovered some 30 years later, after his death. This claim, which came to be known as Fermat's Last Theorem, stood unsolved for the next three and a half centuries.
The claim eventually became one of the most notable unsolved problems of mathematics. Attempts to prove it prompted substantial development in number theory, and over time Fermat's Last Theorem gained prominence as an unsolved problem in mathematics.
Subsequent developments and solution
The special case , proved by Fermat himself, is sufficient to establish that if the theorem is false for some exponent n that is not a prime number, it must also be false for some smaller n, so only prime values of n need further investigation. Over the next two centuries (1637–1839), the conjecture was proved for only the primes 3, 5, and 7, a |
https://en.wikipedia.org/wiki/Superstrong | In mathematics, superstrong may refer to:
Superstrong cardinal in set theory
Superstrong approximation in algebraic group theory |
https://en.wikipedia.org/wiki/DIPS | DIPS may refer to:
Defense independent pitching statistics (baseball)
Dip (exercise)
Division of International Protection Services, under the United Nations High Commissioner for Refugees
Washington Diplomats, a defunct professional soccer team
Nickname of Bollywood actress, Deepika Padukone
DIPS (Digital Image Processing with Sound)
Dips (TV series), Swedish comedy series
See also
DIP (disambiguation) |
https://en.wikipedia.org/wiki/Markov%20spectrum | In mathematics, the Markov spectrum devised by Andrey Markov is a complicated set of real numbers arising in Markov Diophantine equation and also in the theory of Diophantine approximation.
Quadratic form characterization
Consider a quadratic form given by f(x,y) = ax2 + bxy + cy2 and suppose that its discriminant is fixed, say equal to −1/4. In other words, b2 − 4ac = 1.
One can ask for the minimal value achieved by when it is evaluated at non-zero vectors of the grid , and if this minimum does not exist, for the infimum.
The Markov spectrum M is the set obtained by repeating this search with different quadratic forms with discriminant fixed to −1/4:
Lagrange spectrum
Starting from Hurwitz's theorem on Diophantine approximation, that any real number has a sequence of rational approximations m/n tending to it with
it is possible to ask for each value of 1/c with 1/c ≥ about the existence of some for which
for such a sequence, for which c is the best possible (maximal) value. Such 1/c make up the Lagrange spectrum L, a set of real numbers at least (which is the smallest value of the spectrum). The formulation with the reciprocal is awkward, but the traditional definition invites it; looking at the set of c instead allows a definition instead by means of an inferior limit. For that, consider
where m is chosen as an integer function of n to make the difference minimal. This is a function of , and the reciprocal of the Lagrange spectrum is the range of values it takes on irrational numbers.
Relation with Markov spectrum
The initial part of the Lagrange spectrum, namely the part lying in the interval , is equal to the Markov spectrum. The first few values are , , /5, /13, ... and the nth number of this sequence (that is, the nth Lagrange number) can be calculated from the nth Markov number by the formulaFreiman's constant is the name given to the end of the last gap in the Lagrange spectrum, namely:
.
Real numbers greater than F are also members of the Markov spectrum. Moreover, it is possible to prove that L is strictly contained in M.
Geometry of Markov and Lagrange spectrum
On one hand, the initial part of the Markov and Lagrange spectrum lying in the interval [, 3) are both equal and they are a discrete set. On the other hand, the final part of these sets lying after Freiman's constant are also equal, but a continuous set. The geometry of the part between the initial part and final part has a fractal structure, and can be seen as a geometric transition between the discrete initial part and the continuous final part. This is stated precisely in the next theorem:
See also
Markov number
References
Further reading
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 188–189, 1996.
Cusick, T. W. and Flahive, M. E. The Markov and Lagrange Spectra. Providence, RI: Amer. Math. Soc., 1989.
External links
Diophantine approximation
Quadratic forms
Combinatorics |
https://en.wikipedia.org/wiki/Absolute%20probability%20judgement | Absolute probability judgement is a technique used in the field of human reliability assessment (HRA), for the purposes of evaluating the probability of a human error occurring throughout the completion of a specific task. From such analyses measures can then be taken to reduce the likelihood of errors occurring within a system and therefore lead to an improvement in the overall levels of safety. There exist three primary reasons for conducting an HRA; error identification, error quantification and error reduction. As there exist a number of techniques used for such purposes, they can be split into one of two classifications; first generation techniques and second generation techniques. First generation techniques work on the basis of the simple dichotomy of 'fits/doesn't fit' in the matching of the error situation in context with related error identification and quantification and second generation techniques are more theory based in their assessment and quantification of errors. 'HRA techniques have been utilised in a range of industries including healthcare, engineering, nuclear, transportation and business sector; each technique has varying uses within different disciplines.
Absolute probability judgement, which is also known as direct numerical estimation, is based on the quantification of human error probabilities (HEPs). It is grounded on the premise that people cannot recall or are unable to estimate with certainty, the probability of a given event occurring. Expert judgement is typically desirable for utilisation in the technique when there is little or no data with which to calculate HEPs, or when the data is unsuitable or difficult to understand. In theory, qualitative knowledge built through the experts' experience can be translated into quantitative data such as HEPs.
Required of the experts is a good level of both substantive experience (i.e. the expert must have a suitable level of knowledge of the problem domain) and normative experience (i.e. it must be possible for the expert, perhaps with the aid of a facilitator, to translate this knowledge explicitly into probabilities). If experts possess the required substantive knowledge but lack knowledge which is normative in nature, the experts may be trained or assisted in ensuring that the knowledge and expertise requiring to be captured is translated into the correct probabilities i.e. to ensure that it is an accurate representation of the experts' judgements.
Background
Absolute probability judgement is an expert judgement-based approach which involves using the beliefs of experts (e.g. front-line staff, process engineers etc.) to estimate HEPs. There are two primary forms of the technique; Group Methods and Single Expert Methods i.e. it can be done either as a group or as an individual exercise. Group methods tend to be the more popular and widely used as they are more robust and are less subject to bias. Moreover, within the context of use, it is unusual for a single individual |
https://en.wikipedia.org/wiki/Human%20cognitive%20reliability%20correlation | Human Cognitive Reliability Correlation (HCR) is a technique used in the field of Human reliability Assessment (HRA), for the purposes of evaluating the probability of a human error occurring throughout the completion of a specific task. From such analyses measures can then be taken to reduce the likelihood of errors occurring within a system and therefore lead to an improvement in the overall levels of safety. There exist three primary reasons for conducting an HRA; error identification, error quantification and error reduction. As there exist a number of techniques used for such purposes, they can be split into one of two classifications; first generation techniques and second generation techniques. First generation techniques work on the basis of the simple dichotomy of ‘fits/doesn’t fit’ in the matching of the error situation in context with related error identification and quantification and second generation techniques are more theory based in their assessment and quantification of errors. HRA techniques have been utilised in a range of industries including healthcare, engineering, nuclear, transportation and business sector; each technique has varying uses within different disciplines.
HCR is based on the premise that an operator’s likelihood of success or failure in a time-critical task is dependent on the cognitive process used to make the critical decisions that determine the outcome. Three Performance Shaping Factors (PSFs) – Operator Experience, Stress Level, and Quality of Operator/Plant Interface - also influence the average (median) time taken to perform the task. Combining these factors enables “response-time” curves to be calibrated and compared to the available time to perform the task. Using these curves, the analyst can then estimate the likelihood that an operator will take the correct action, as required by a given stimulus (e.g. pressure warning signal), within the available time window. The relationship between these normalised times and Human Error Probabilities (HEPs) is based on simulator experimental data.
Background
HCR is a psychology/cognitive modelling approach to HRA developed by Hannaman et al. in 1984. The method uses Rasmussen’s idea of rule-based, skill-based, and knowledge-based decision making to determine the likelihood of failing a given task, as well as considering the PSFs of operator experience, stress and interface quality. The database underpinning this methodology was originally developed through the use of nuclear power-plant simulations due to a requirement for a method by which nuclear operating reliability could be quantified.
HCR Methodology
The HCR methodology is broken down into a sequence of steps as given below:
The first step is for the analyst to determine the situation in need of a human reliability assessment. It is then determined whether this situation is governed by rule-based, skill-based or knowledge-based decision making.
From the relevant literature, the appropriate HCR math |
https://en.wikipedia.org/wiki/Vance%20Faber | Vance Faber (born December 1, 1944 in Buffalo, New York) is a mathematician, known for his work in combinatorics, applied linear algebra and image processing.
Faber received his Ph.D. in 1971 from Washington University in St. Louis. His advisor was Franklin Tepper Haimo.
Faber was a professor at University of Colorado at Denver during the 1970s. He spent parts of 3 years at the National Center for Atmospheric Research in Boulder on a NASA postdoctoral fellowship, where he wrote a second thesis on the numerical solution of the Shallow Water Equations under the direction of numerical analyst Paul Swarztrauber. In the 1980s and 1990s, he was on the staff of the Computer Research and Applications Group at Los Alamos National Laboratory. He was Group Leader from 1990 to 1995.
From 1998 to 2003 Faber was CTO and Head of Research for three different small companies building imaging software: LizardTech, Mapping Science and Cytoprint. He is currently a consultant.
In 1981, Gene Golub offered a US$500 prize for “the construction of a 3-term conjugate gradient like descent method for non-symmetric real matrices or a proof that there can be no such method”. Faber and his co-author Thomas A. Manteuffel won this prize for their 1984 paper, in which they gave conditions for the existence of such a method and showed that, in general, there can be no such method.
See also
Erdős–Faber–Lovász conjecture
References
External links
Some publications of Vance Faber in the field of computer science.
1944 births
20th-century American mathematicians
21st-century American mathematicians
Graph theorists
Living people
Washington University in St. Louis alumni
Washington University in St. Louis mathematicians
University of Colorado Denver faculty |
https://en.wikipedia.org/wiki/Grassmannian%20%28disambiguation%29 | In mathematics, a Grassmannian may refer to:
Affine Grassmannian
Affine Grassmannian (manifold)
Grassmannian, the classical parameter space for linear subspaces of a linear space or projective space
Lagrangian Grassmannian
See also
Grassmann algebra, or exterior algebra, a setting where the exterior product is defined.
Grassmann number, a construction for path integrals of fermionic fields in physics.
Grassmann integral, a method for integrating functions of Grassmann variables |
https://en.wikipedia.org/wiki/Prime%20manifold | In topology, a branch of mathematics, a prime manifold is an n-manifold that cannot be expressed as a non-trivial connected sum of two n-manifolds. Non-trivial means that neither of the two is an n-sphere.
A similar notion is that of an irreducible n-manifold, which is one in which any embedded (n − 1)-sphere bounds an embedded n-ball. Implicit in this definition is the use of a suitable category, such as the category of differentiable manifolds or the category of piecewise-linear manifolds.
The notions of irreducibility in algebra and manifold theory are related. An irreducible manifold is prime, although the converse does not hold. From an algebraist's perspective, prime manifolds should be called "irreducible"; however the topologist (in particular the 3-manifold topologist) finds the definition above more useful. The only compact, connected 3-manifolds that are prime but not irreducible are the trivial 2-sphere bundle over the circle S1 and the twisted 2-sphere bundle over S1.
According to a theorem of Hellmuth Kneser and John Milnor, every compact, orientable 3-manifold is the connected sum of a unique (up to homeomorphism) collection of prime 3-manifolds.
Definitions
Consider specifically 3-manifolds.
Irreducible manifold
A 3-manifold is if any smooth sphere bounds a ball. More rigorously, a differentiable connected 3-manifold is irreducible if every differentiable submanifold homeomorphic to a sphere bounds a subset (that is, ) which is homeomorphic to the closed ball
The assumption of differentiability of is not important, because every topological 3-manifold has a unique differentiable structure. The assumption that the sphere is smooth (that is, that it is a differentiable submanifold) is however important: indeed the sphere must have a tubular neighborhood.
A 3-manifold that is not irreducible is called .
Prime manifolds
A connected 3-manifold is prime if it cannot be expressed as a connected sum of two manifolds neither of which is the 3-sphere (or, equivalently, neither of which is homeomorphic to ).
Examples
Euclidean space
Three-dimensional Euclidean space is irreducible: all smooth 2-spheres in it bound balls.
On the other hand, Alexander's horned sphere is a non-smooth sphere in that does not bound a ball. Thus the stipulation that the sphere be smooth is necessary.
Sphere, lens spaces
The 3-sphere is irreducible. The product space is not irreducible, since any 2-sphere (where is some point of ) has a connected complement which is not a ball (it is the product of the 2-sphere and a line).
A lens space with (and thus not the same as ) is irreducible.
Prime manifolds and irreducible manifolds
A 3-manifold is irreducible if and only if it is prime, except for two cases: the product and the non-orientable fiber bundle of the 2-sphere over the circle are both prime but not irreducible.
From irreducible to prime
An irreducible manifold is prime. Indeed, if we express as a connected sum
then is |
https://en.wikipedia.org/wiki/Concentric%20objects | In geometry, two or more objects are said to be concentric when they share the same center. Any pair of (possibly unalike) objects with well-defined centers can be concentric, including circles, spheres, regular polygons, regular polyhedra, parallelograms, cones, conic sections, and quadrics.
Geometric objects are coaxial if they share the same axis (line of symmetry). Geometric objects with a well-defined axis include circles (any line through the center), spheres, cylinders, conic sections, and surfaces of revolution.
Concentric objects are often part of the broad category of whorled patterns, which also includes spirals (a curve which emanates from a point, moving farther away as it revolves around the point).
Geometric properties
In the Euclidean plane, two circles that are concentric necessarily have different radii from each other.
However, circles in three-dimensional space may be concentric, and have the same radius as each other, but nevertheless be different circles. For example, two different meridians of a terrestrial globe are concentric with each other and with the globe of the earth (approximated as a sphere). More generally, every two great circles on a sphere are concentric with each other and with the sphere.
By Euler's theorem in geometry on the distance between the circumcenter and incenter of a triangle, two concentric circles (with that distance being zero) are the circumcircle and incircle of a triangle if and only if the radius of one is twice the radius of the other, in which case the triangle is equilateral.
The circumcircle and the incircle of a regular n-gon, and the regular n-gon itself, are concentric. For the circumradius-to-inradius ratio for various n, see Bicentric polygon#Regular polygons. The same can be said of a regular polyhedron's insphere, midsphere and circumsphere.
The region of the plane between two concentric circles is an annulus, and analogously the region of space between two concentric spheres is a spherical shell.
For a given point c in the plane, the set of all circles having c as their center forms a pencil of circles. Each two circles in the pencil are concentric, and have different radii. Every point in the plane, except for the shared center, belongs to exactly one of the circles in the pencil. Every two disjoint circles, and every hyperbolic pencil of circles, may be transformed into a set of concentric circles by a Möbius transformation.
Applications and examples
The ripples formed by dropping a small object into still water naturally form an expanding system of concentric circles. Evenly spaced circles on the targets used in target archery or similar sports provide another familiar example of concentric circles.
Coaxial cable is a type of electrical cable in which the combined neutral and earth core completely surrounds the live core(s) in system of concentric cylindrical shells.
Johannes Kepler's Mysterium Cosmographicum envisioned a cosmological system formed by concentric re |
https://en.wikipedia.org/wiki/Leopoldo%20Nachbin | Leopoldo Nachbin (7 January 1922 – 3 April 1993) was a Jewish-Brazilian mathematician who dealt with topology, and harmonic analysis.
Nachbin was born in Recife, and is best known for Nachbin's theorem. He died, aged 71, in Rio de Janeiro.
Nachbin was a Ph.D. student of Laurent Schwartz.
His Ph.D. students include Francisco Antônio Dória and Seán Dineen.
He was an invited speaker at the International Congress of Mathematicians (ICM) of 1962 in Stockholm.
Bibliography
Topology and Order (Krieger Pub. Co., 1965)
Introdução à Álgebra (McGraw-Hill, 1971, in Portuguese)
References
Ralph A. Raimi - Leopoldo Nachbin, 1922-1993.
Candido Lima da Silva Dias, Chaim Samuel Hönig, Luis Adauto da Justa Medeiros - Leopoldo Nachbin.
J. Horváth. The life and works of Leopoldo Nachbin.
External links
Leopoldo Nachbin, 1922-1993
Os trabalhos de Leopoldo Nachbin (1922-1993), by Jorge Mujica, in Portuguese, free translation: "The Works of Leopoldo Nachbin (1922-1993)"
20th-century Brazilian mathematicians
1922 births
1993 deaths
Brazilian Jews
Topologists
Instituto Nacional de Matemática Pura e Aplicada researchers
Mathematical analysts |
https://en.wikipedia.org/wiki/Signature%20operator | In mathematics, the signature operator is an elliptic differential operator defined on a certain subspace of the space of differential forms on an even-dimensional compact Riemannian manifold, whose analytic index is the same as the topological signature of the manifold if the dimension of the manifold is a multiple of four. It is an instance of a Dirac-type operator.
Definition in the even-dimensional case
Let be a compact Riemannian manifold of even dimension . Let
be the exterior derivative on -th order differential forms on . The Riemannian metric on allows us to define the Hodge star operator and with it the inner product
on forms. Denote by
the adjoint operator of the exterior differential . This operator can be expressed purely in terms of the Hodge star operator as follows:
Now consider acting on the space of all forms .
One way to consider this as a graded operator is the following: Let be an involution on the space of all forms defined by:
It is verified that anti-commutes with and, consequently, switches the -eigenspaces of
Consequently,
Definition: The operator with the above grading respectively the above operator is called the signature operator of .
Definition in the odd-dimensional case
In the odd-dimensional case one defines the signature operator to be acting
on the even-dimensional forms of .
Hirzebruch Signature Theorem
If , so that the dimension of is a multiple of four, then Hodge theory implies that:
where the right hand side is the topological signature (i.e. the signature of a quadratic form on defined by the cup product).
The Heat Equation approach to the Atiyah-Singer index theorem can then be used to show that:
where is the Hirzebruch L-Polynomial, and the the Pontrjagin forms on .
Homotopy invariance of the higher indices
Kaminker and Miller proved that the higher indices of the signature operator are homotopy-invariant.
See also
Hirzebruch signature theorem
Pontryagin class
Friedrich Hirzebruch
Michael Atiyah
Isadore Singer
Notes
References
Elliptic partial differential equations |
https://en.wikipedia.org/wiki/Intersection%20form%20of%20a%204-manifold | In mathematics, the intersection form of an oriented compact 4-manifold is a special symmetric bilinear form on the 2nd (co)homology group of the 4-manifold. It reflects much of the topology of the 4-manifolds, including information on the existence of a smooth structure.
Definition using intersection
Let M be a closed 4-manifold (PL or smooth).
Take a triangulation T of M.
Denote by the dual cell subdivision.
Represent classes by 2-cycles A and B modulo 2 viewed as unions of 2-simplices of T and of , respectively.
Define the intersection form modulo 2
by the formula
This is well-defined because the intersection of a cycle and a boundary consists of an even number of points (by definition of a cycle and a boundary).
If M is oriented, analogously (i.e. counting intersections with signs) one defines the intersection form on the 2nd homology group
Using the notion of transversality, one can state the following results (which constitute an equivalent definition of the intersection form).
If classes are represented by closed surfaces (or 2-cycles modulo 2) A and B meeting transversely, then
If M is oriented and classes are represented by closed oriented surfaces (or 2-cycles) A and B meeting transversely, then every intersection point in has the sign +1 or −1 depending on the orientations, and is the sum of these signs.
Definition using cup product
Using the notion of the cup product , one can give a dual (and so an equivalent) definition as follows.
Let M be a closed oriented 4-manifold (PL or smooth).
Define the intersection form on the 2nd cohomology group
by the formula
The definition of a cup product is dual (and so is analogous) to the above definition of the intersection form on homology of a manifold, but is more abstract.
However, the definition of a cup product generalizes to complexes and topological manifolds.
This is an advantage for mathematicians who are interested in complexes and topological manifolds (not only in PL and smooth manifolds).
When the 4-manifold is smooth, then in de Rham cohomology, if a and b are represented by 2-forms and , then the intersection form can be expressed by the integral
where is the wedge product.
The definition using cup product has a simpler analogue modulo 2 (which works for non-orientable manifolds).
Of course one does not have this in de Rham cohomology.
Properties and applications
Poincare duality states that the intersection form is unimodular (up to torsion).
By Wu's formula, a spin 4-manifold must have even intersection form, i.e., is even for every x. For a simply-connected smooth 4-manifold (or more generally one with no 2-torsion residing in the first homology), the converse holds.
The signature of the intersection form is an important invariant. A 4-manifold bounds a 5-manifold if and only if it has zero signature. Van der Blij's lemma implies that a spin 4-manifold has signature a multiple of eight. In fact, Rokhlin's theorem implies th |
https://en.wikipedia.org/wiki/1996%20Kyrgyzstan%20League | Statistics of Kyrgyzstan League for the 1996 season.
Overview
It was contested by 12 teams, and Metallurg Kadamjay won the championship.
League standings
References
Kyrgyzstan - List of final tables (RSSSF)
Kyrgyzstan League seasons
1
Kyrgyzstan
Kyrgyzstan |
https://en.wikipedia.org/wiki/1997%20Kyrgyzstan%20League | Statistics for the Kyrgyzstan League for the 1997 season.
Overview
There were 10 teams. Dinamo Bishkek won the championship.
League standings
References
Kyrgyzstan - List of final tables (RSSSF)
Kyrgyzstan League seasons
1
Kyrgyzstan
Kyrgyzstan |
https://en.wikipedia.org/wiki/1998%20Kyrgyzstan%20League | Statistics of Kyrgyzstan League for the 1998 season.
Overview
It was contested by 20 teams, and CAG Dinamo MVD Bishkek won the championship.
First stage
Zone A
Zone B
Final stage
References
Kyrgyzstan – List of final tables (RSSSF)
Kyrgyzstan League seasons
1
Kyrgyzstan
Kyrgyzstan |
https://en.wikipedia.org/wiki/1999%20Kyrgyzstan%20League | Statistics of Kyrgyzstan League for the 1999 season.
Overview
It was contested by 12 teams, and Dinamo Bishkek won the championship.
League standings
References
Kyrgyzstan – List of final tables (RSSSF)
Kyrgyzstan League seasons
1
Kyrgyzstan
Kyrgyzstan |
https://en.wikipedia.org/wiki/2000%20Kyrgyzstan%20League | Statistics of Kyrgyzstan League for the 2000 season.
Overview
The football season of 2000 consisted of 12 competing teams, resulting in SKA PVO Bishkek winning the championship.
League standings
References
Kyrgyzstan - List of final tables (RSSSF)
Kyrgyzstan League seasons
1
Kyrgyzstan
Kyrgyzstan |
https://en.wikipedia.org/wiki/2001%20Kyrgyzstan%20League | Statistics of Kyrgyzstan League for the 2001 season.
Overview
It was contested by 8 teams, and SKA PVO Bishkek won the championship.
League standings
References
Kyrgyzstan - List of final tables (RSSSF)
Kyrgyzstan League seasons
1
Kyrgyzstan
Kyrgyzstan |
https://en.wikipedia.org/wiki/2002%20Kyrgyzstan%20League | Statistics of Kyrgyzstan League for the 2002 season.
The 2002 Kyrgyzstan League was contested by 10 teams with SKA PVO Bishkek winning the championship.
League Standings
Top scorers
References
Kyrgyzstan - List of final tables (RSSSF)
Kyrgyzstan League seasons
1
Kyrgyzstan
Kyrgyzstan |
https://en.wikipedia.org/wiki/2003%20Kyrgyzstan%20League | Statistics of Kyrgyzstan League for the 2003 season.
Overview
It was contested by 18 teams, and Zhashtyk Ak Altyn Kara-Suu won the championship.
First stage
Zone A
Zone B
League standings
References
Kyrgyzstan - List of final tables (RSSSF)
Kyrgyzstan League seasons
1
Kyrgyzstan
Kyrgyzstan |
https://en.wikipedia.org/wiki/2004%20Kyrgyzstan%20League | Statistics of Kyrgyzstan League for the 2004 season.
Overview
It was contested by 10 teams, and Dordoi-Dynamo Naryn won the championship.
League standings
References
Kyrgyzstan - List of final tables (RSSSF)
Kyrgyzstan League seasons
1
Kyrgyzstan
Kyrgyzstan |
https://en.wikipedia.org/wiki/2005%20Kyrgyzstan%20League | Statistics of Kyrgyzstan League for the 2005 season.
Overview
It was contested by 8 teams, and Dordoi-Dynamo Naryn won the championship. Al Fagir Aravan withdrew after playing 4 matches and started playing at the Kyrgyzstan League Second Level. Remaining matches of the first quarter of the season were awarded 0-3 defeats against them
League standings
Championship play off
Dordoi-Dynamo Naryn 1-1 (pen 4-2) Shoro SKA Bishkek
References
Kyrgyzstan - List of final tables (RSSSF)
Kyrgyzstan League seasons
1
Kyrgyzstan
Kyrgyzstan |
https://en.wikipedia.org/wiki/2006%20Kyrgyzstan%20League | Statistics of Kyrgyzstan League for the 2006 season.
Overview
It was contested by 11 teams, and Dordoi-Dynamo Naryn won the championship.
First stage
Group A
Group B
Final stage
References
Kyrgyzstan - List of final tables (RSSSF)
Kyrgyzstan League seasons
1
Kyrgyzstan
Kyrgyzstan |
https://en.wikipedia.org/wiki/2007%20Kyrgyzstan%20League | Statistics of Kyrgyzstan League for the 2007 season.
Overview
It was performed in 10 teams, and Dordoi-Dynamo Naryn won the championship.
League standings
References
Kyrgyzstan - List of final tables (RSSSF)
Kyrgyzstan League seasons
1
Kyrgyzstan
Kyrgyzstan |
https://en.wikipedia.org/wiki/2008%20Kyrgyzstan%20League | Statistics of Kyrgyzstan League for the 2008 season.
Overview
It was contested by 9 teams, and Dordoi-Dynamo Naryn won the championship after beating Abdish-Ata Kant in an end of season playoff final after both sides had the same number of points in the regular season.
Final league standings
References
Kyrgyzstan - List of final tables (RSSSF)
Kyrgyzstan League seasons
1
Kyrgyzstan
Kyrgyzstan |
https://en.wikipedia.org/wiki/2004%20Lao%20League | Statistics of Lao League for the 2004 season.
Overview
It was contested by 11 teams, and MCTPC won the championship.
League standings
Results
Relegation playoff
Prime Minister's Office FC and No-8 Road Construction FC were automatically promoted from Lao League 2. A play off was held between the third place team and the third bottom team in the top division.
References
Lao Premier League seasons
1
Laos
Laos |
https://en.wikipedia.org/wiki/2005%20Lao%20League | Statistics of Lao League for the 2005 season.
Overview
It was contested by 11 teams, and Vientiane FC won the championship.
League standings
Relegation playoff
Vilakone FC and Kavin College FC were automatically promoted from Lao League 2. A play off was held between the third place team and the third bottom team in the top division.
References
Lao Premier League seasons
1
Laos
Laos |
https://en.wikipedia.org/wiki/2008%20Lao%20League | Statistics of Lao League in the 2008 season.
Overview
Lao Army FC won the championship.
References
RSSSF
Lao Premier League seasons
1
Laos
Laos |
https://en.wikipedia.org/wiki/TESEO | Tecnica Empirica Stima Errori Operatori (TESEO) is a technique in the field of Human reliability Assessment (HRA), that evaluates the probability of a human error occurring throughout the completion of a specific task. From such analyses measures can then be taken to reduce the likelihood of errors occurring within a system and therefore lead to an improvement in the overall levels of safety. There exist three primary reasons for conducting an HRA; error identification, error quantification and error reduction. As there exist a number of techniques used for such purposes, they can be split into one of two classifications; first generation techniques and second generation techniques. First generation techniques work on the basis of the simple dichotomy of ‘fits/doesn’t fit’ in the matching of the error situation in context with related error identification and quantification and second generation techniques are more theory based in their assessment and quantification of errors. ‘HRA techniques have been utilised in a range of industries including healthcare, engineering, nuclear, transportation and business sector; each technique has varying uses within different disciplines.
This is a time based model that describes the probability of a system operator's failure as a multiplicative function of 5 main factors. These factors are as follows:
K1: The type of task to be executed
K2: The time available to the operator to complete the task
K3: The operator's level of experience/characteristics
K4: The operator's state of mind
K5: The environmental and ergonomic conditions prevalent
Using these figures, an overall Human Error Probability (HEP) can be calculated with the formulation provided below:
K1 x K2 x K3 x K4 x K5
The specific value of each of the above functions can be obtained by consulting standard tables that take account of the method in which the HEP is derived.
Background
Developed in 1980 by Bello and Colombari, TESEO created with the intention of using it for the purpose of conducting HRA of process industries. The methodology is relatively straightforward and is easy to use but is also limited; it is useful for quick overview HRA assessments, as opposed to highly detailed and in-depth assessments. Within the field of HRA, there is a lack of theoretical foundation for the technique, as is widely acknowledged throughout.
TESEO Methodology
When putting this technique into practice, it is necessary for the designated HRA assessor to thoroughly consider the task requiring assessment and therefore also consider the value for Kn that applies in the context. Once this value has been decided upon, the tables, previously mentioned, are then consulted from which a related value for each of the identified factors is found to allow the HEP to be calculated.
Worked Example
Provided below is an example of how TESEO methodology can be used in practice; each of the stages of the process described above are worked through in order.
Context
An |
https://en.wikipedia.org/wiki/Technique%20for%20human%20error-rate%20prediction | The technique for human error-rate prediction (THERP) is a technique used in the field of human reliability assessment (HRA), for the purposes of evaluating the probability of a human error occurring throughout the completion of a specific task. From such analyses measures can then be taken to reduce the likelihood of errors occurring within a system and therefore lead to an improvement in the overall levels of safety. There exist three primary reasons for conducting an HRA: error identification, error quantification and error reduction. As there exist a number of techniques used for such purposes, they can be split into one of two classifications: first-generation techniques and second-generation techniques. First-generation techniques work on the basis of the simple dichotomy of ‘fits/doesn’t fit’ in matching an error situation in context with related error identification and quantification. Second generation techniques are more theory-based in their assessment and quantification of errors. ‘HRA techniques have been utilised for various applications in a range of disciplines and industries including healthcare, engineering, nuclear, transportation and business.
THERP models human error probabilities (HEPs) using a fault-tree approach, in a similar way to an engineering risk assessment, but also accounts for performance shaping factors (PSFs) that may influence these probabilities. The probabilities for the human reliability analysis event tree (HRAET), which is the primary tool for assessment, are nominally calculated from the database developed by the authors Swain and Guttman; local data e.g. from simulators or accident reports may however be used instead. The resultant tree portrays a step by step account of the stages involved in a task, in a logical order. The technique is known as a total methodology as it simultaneously manages a number of different activities including task analysis, error identification, representation in form of HRAET and HEP quantification.
Background
The technique for human error rate prediction (THERP) is a first generation methodology, which means that its procedures follow the way conventional reliability analysis models a machine. The technique was developed in the Sandia Laboratories for the US Nuclear Regulatory Commission. Its primary author is Swain, who developed the THERP methodology gradually over a lengthy period of time. THERP relies on a large human reliability database that contains HEPs, and is based upon both plant data and expert judgments. The technique was the first approach in HRA to come into broad use and is still widely used in a range of applications even beyond its original nuclear setting.
THERP methodology
The methodology for the THERP technique is broken down into 5 main stages:
1. Define the system failures of interest
These failures include functions of the system where human error has a greater likelihood of influencing the probability of a fault, and those of interest to the ris |
https://en.wikipedia.org/wiki/Heather%20Gordon | Heather Gordon (born 1967) is an American contemporary visual artist.
Career
Gordon creates large-scale paintings and immersive art projects, using numbers, algorithms, and geometry in her creative process.
In November 2017 Gordon's installation And Then the Sun Swallowed Me was exhibited at the Contemporary Art Museum of Raleigh.
Her piece Cinnabar was featured in the North Carolina Museum of Art's exhibit titled You Are Here: Light, Color, and Sound Experiences from April 7, 2018, until July 2, 2018. Prior to the exhibit, her work was featured as part of the museum's Matrons of the Arts initiative, highlighting female-identified artists from around the world. She received a North Carolina Artists Fellowship in 2014.
Her collaborate works with dancer and choreographer Justin Tornow, titled Echo and SHOW, were shown at 21c Durham Museum Hotel and The Durham Fruit. In 2017 Gordon and Tornow collaborated to create No.19/Modulations, which was shown at the CCB Plaza in downtown Durham, North Carolina.
In August 2018 her work titled DOUBLE EDGED: Geometric Abstraction Then and Now was shown at the Weatherspoon Art Museum. Also in 2018, she debuted Steel, a tape installation, at The Dillon in Raleigh, North Carolina.
Her work has also been shown at the Ackland Art Museum, Waterworks, The Carrack Modern Art Museum, and the North Carolina School of Science and Math. She is part of Mural Durham, an art project in Durham.
In 2019 Gordon worked with the David M Rubenstein Rare Book & Manuscript Library and the Duke University Archives to research documents related to the Duke Forest for her work titled Forest for the Trees.
Personal life
Gordon was the only child of an accountant and engineer. Her father was a United States Air Force officer, and grew up primarily on military bases around the United States. Godron is lesbian, and said she knew when she was eight years old.
Gordon earned a Bachelor of Fine Arts degree from the University of Florida in 1990 and a Master of Fine Arts degree from New Mexico State University in 1995. She lives in Durham. Gordon has a son named Henry.
References
Living people
1967 births
21st-century American painters
21st-century American women artists
American conceptual artists
American contemporary painters
American digital artists
American women installation artists
American installation artists
American multimedia artists
New Mexico State University alumni
People from Durham, North Carolina
University of Florida alumni
Women conceptual artists
Women multimedia artists
American lesbian artists |
https://en.wikipedia.org/wiki/Human%20error%20assessment%20and%20reduction%20technique | Human error assessment and reduction technique (HEART) is a technique used in the field of human reliability assessment (HRA), for the purposes of evaluating the probability of a human error occurring throughout the completion of a specific task. From such analyses measures can then be taken to reduce the likelihood of errors occurring within a system and therefore lead to an improvement in the overall levels of safety. There exist three primary reasons for conducting an HRA: error identification, error quantification, and error reduction. As there exist a number of techniques used for such purposes, they can be split into one of two classifications: first-generation techniques and second generation techniques. First generation techniques work on the basis of the simple dichotomy of 'fits/doesn't fit' in the matching of the error situation in context with related error identification and quantification and second generation techniques are more theory based in their assessment and quantification of errors. HRA techniques have been used in a range of industries including healthcare, engineering, nuclear, transportation, and business sectors. Each technique has varying uses within different disciplines.
HEART method is based upon the principle that every time a task is performed there is a possibility of failure and that the probability of this is affected by one or more Error Producing Conditions (EPCs) – for instance: distraction, tiredness, cramped conditions etc. – to varying degrees. Factors which have a significant effect on performance are of greatest interest. These conditions can then be applied to a "best-case-scenario" estimate of the failure probability under ideal conditions to then obtain a final error chance. This figure assists in communication of error chances with the wider risk analysis or safety case. By forcing consideration of the EPCs potentially affecting a given procedure, HEART also has the indirect effect of providing a range of suggestions as to how the reliability may therefore be improved (from an ergonomic standpoint) and hence minimising risk.
Background
HEART was developed by Williams in 1986. It is a first generation HRA technique, yet it is dissimilar to many of its contemporaries in that it remains to be widely used throughout the UK. The method essentially takes into consideration all factors which may negatively affect performance of a task in which human reliability is considered to be dependent, and each of these factors is then independently quantified to obtain an overall Human Error Probability (HEP), the collective product of the factors.
HEART methodology
1. The first stage of the process is to identify the full range of sub-tasks that a system operator would be required to complete within a given task.
2. Once this task description has been constructed a nominal human unreliability score for the particular task is then determined, usually by consulting local experts. Based around this c |
https://en.wikipedia.org/wiki/1996%20Djurg%C3%A5rdens%20IF%20season |
Player statistics
Appearances for competitive matches only
|}
Goals
Allsvenskan
Svenska Cupen
Intertoto Cup
Friendlies
Competitions
Overall
Allsvenskan
League table
Matches
Svenska Cupen
1995–96
Group stage
1996–97
Preliminary rounds
UEFA Intertoto Cup
Group stage
Friendlies
References
Djurgårdens IF Fotboll seasons
Djurgarden |
https://en.wikipedia.org/wiki/Success%20likelihood%20index%20method | Success Likelihood Index Method (SLIM) is a technique used in the field of Human reliability Assessment (HRA), for the purposes of evaluating the probability of a human error occurring throughout the completion of a specific task. From such analyses measures can then be taken to reduce the likelihood of errors occurring within a system and therefore lead to an improvement in the overall levels of safety. There exist three primary reasons for conducting an HRA; error identification, error quantification and error reduction. As there exist a number of techniques used for such purposes, they can be split into one of two classifications; first generation techniques and second generation techniques. First generation techniques work on the basis of the simple dichotomy of ‘fits/doesn’t fit’ in the matching of the error situation in context with related error identification and quantification and second generation techniques are more theory based in their assessment and quantification of errors. ‘HRA techniques have been utilised in a range of industries including healthcare, engineering, nuclear, transportation and business sector; each technique has varying uses within different disciplines.
SLIM is a decision-analytic approach to HRA which uses expert judgement to quantify Performance Shaping Factors (PSFs); factors concerning the individuals, environment or task, which have the potential to either positively or negatively affect performance e.g. available task time. Such factors are used to derive a Success Likelihood Index (SLI), a form of preference index, which is calibrated against existing data to derive a final Human Error Probability (HEP). The PSF's which require to be considered are chosen by experts and are namely those factors which are regarded as most significant in relation to the context in question.
The technique consists of two modules: MAUD (multi-attribute utility decomposition) which scales the relative success likelihood in performing a range of tasks, given the PSFs probable to affect human performance; and SARAH (Systematic Approach to the Reliability Assessment of Humans) which calibrates these success scores with tasks with known HEP values, to provide an overall figure.
Background
SLIM was developed by Embrey et al. [1] for use within the US nuclear industry. By use of this method, relative success likelihoods are established for a range of tasks, and then calibrated using a logarithmic transformation.
SLIM methodology
The SLIM methodology breaks down into ten steps of which steps 1-7 are involved in SLIM-MAUD and 8-10 are SLIM-SARAH.
Definition of situations and subsets
Upon selection of a relevant panel of experts who will carry out the assessment, these individuals are provided with as fully detailed a task description as possible with regards to the individual designated to perform each task and further factors which are likely to influence the success of each of these. An in depth description is a cri |
https://en.wikipedia.org/wiki/Systoles%20of%20surfaces | In mathematics, systolic inequalities for curves on surfaces were first studied by Charles Loewner in 1949 (unpublished; see remark at end of P. M. Pu's paper in '52). Given a closed surface, its systole, denoted sys, is defined to be the least length of a loop that cannot be contracted to a point on the surface. The systolic area of a metric is defined to be the ratio area/sys2. The systolic ratio SR is the reciprocal quantity sys2/area. See also Introduction to systolic geometry.
Torus
In 1949 Loewner proved his inequality for metrics on the torus T2, namely that the systolic ratio SR(T2) is bounded above by , with equality in the flat (constant curvature) case of the equilateral torus (see hexagonal lattice).
Real projective plane
A similar result is given by Pu's inequality for the real projective plane from 1952, due to Pao Ming Pu, with an upper bound of π/2 for the systolic ratio SR(RP2), also attained in the constant curvature case.
Klein bottle
For the Klein bottle K, Bavard (1986) obtained an optimal upper bound of for the systolic ratio:
based on work by Blatter from the 1960s.
Genus 2
An orientable surface of genus 2 satisfies Loewner's bound , see (Katz-Sabourau '06). It is unknown whether or not every surface of positive genus satisfies Loewner's bound. It is conjectured that they all do. The answer is affirmative for genus 20 and above by (Katz-Sabourau '05).
Arbitrary genus
For a closed surface of genus g, Hebda and Burago (1980) showed that the systolic ratio SR(g) is bounded above by the constant 2. Three years later, Mikhail Gromov found an upper bound for SR(g) given by a constant times
A similar lower bound (with a smaller constant) was obtained by Buser and Sarnak. Namely, they exhibited arithmetic hyperbolic Riemann surfaces with systole behaving as a constant times . Note that area is 4π(g-1) from the Gauss-Bonnet theorem, so that SR(g) behaves asymptotically as a constant times .
The study of the asymptotic behavior for large genus of the systole of hyperbolic surfaces reveals some interesting constants. Thus, Hurwitz surfaces defined by a tower of principal congruence subgroups of the (2,3,7) hyperbolic triangle group satisfy the bound
resulting from an analysis of the Hurwitz quaternion order. A similar bound holds for more general arithmetic Fuchsian groups. This 2007 result by Mikhail Katz, Mary Schaps, and Uzi Vishne improves an inequality due to Peter Buser and Peter Sarnak in the case of arithmetic groups defined over , from 1994, which contained a nonzero additive constant. For the Hurwitz surfaces of principal congruence type, the systolic ratio SR(g) is asymptotic to
Using Katok's entropy inequality, the following asymptotic upper bound for SR(g) was found in (Katz-Sabourau 2005):
see also (Katz 2007), p. 85. Combining the two estimates, one obtains tight bounds for the asymptotic behavior of the systolic ratio of surfaces.
Sphere
There is also a version of the inequality for metr |
https://en.wikipedia.org/wiki/Mixed-design%20analysis%20of%20variance | In statistics, a mixed-design analysis of variance model, also known as a split-plot ANOVA, is used to test for differences between two or more independent groups whilst subjecting participants to repeated measures. Thus, in a mixed-design ANOVA model, one factor (a fixed effects factor) is a between-subjects variable and the other (a random effects factor) is a within-subjects variable. Thus, overall, the model is a type of mixed-effects model.
A repeated measures design is used when multiple independent variables or measures exist in a data set, but all participants have been measured on each variable.
An example
Andy Field (2009) provided an example of a mixed-design ANOVA in which he wants to investigate whether personality or attractiveness is the most important quality for individuals seeking a partner. In his example, there is a speed dating event set up in which there are two sets of what he terms "stooge dates": a set of males and a set of females. The experimenter selects 18 individuals, 9 males and 9 females to play stooge dates. Stooge dates are individuals who are chosen by the experimenter and they vary in attractiveness and personality. For males and females, there are three highly attractive individuals, three moderately attractive individuals, and three highly unattractive individuals. Of each set of three, one individual has a highly charismatic personality, one is moderately charismatic and the third is extremely dull.
The participants are the individuals who sign up for the speed dating event and interact with each of the 9 individuals of the opposite sex. There are 10 males and 10 female participants. After each date, they rate on a scale of 0 to 100 how much they would like to have a date with that person, with a zero indicating "not at all" and 100 indicating "very much".
The random factors, or so-called repeated measures, are looks, which consists of three levels (very attractive, moderately attractive, and highly unattractive) and the personality, which again has three levels (highly charismatic, moderately charismatic, and extremely dull). The looks and personality have an overall random character because the precise level of each cannot be controlled by the experimenter (and indeed may be difficult to quantify); the 'blocking' into discrete categories is for convenience, and does not guarantee precisely the same level of looks or personality within a given block; and the experimenter is interested in making inferences on the general population of daters, not just the 18 'stooges' The fixed-effect factor, or so-called between-subjects measure, is gender because the participants making the ratings were either female or male, and precisely these statuses were designed by the experimenter.
ANOVA assumptions
When running an analysis of variance to analyse a data set, the data set should meet the following criteria:
Normality: scores for each condition should be sampled from a normally distributed population. |
https://en.wikipedia.org/wiki/Influence%20diagrams%20approach | Influence Diagrams Approach (IDA) is a technique used in the field of Human reliability Assessment (HRA), for the purposes of evaluating the probability of a human error occurring throughout the completion of a specific task. From such analyses measures can then be taken to reduce the likelihood of errors occurring within a system and therefore lead to an improvement in the overall levels of safety. There exist three primary reasons for conducting an HRA; error identification, error quantification and error reduction. As there exist a number of techniques used for such purposes, they can be split into one of two classifications; first generation techniques and second generation techniques. First generation techniques work on the basis of the simple dichotomy of ‘fits/doesn’t fit’ in the matching of the error situation in context with related error identification and quantification and second generation techniques are more theory based in their assessment and quantification of errors. ‘HRA techniques have been utilised in a range of industries including healthcare, engineering, nuclear, transportation and business sector; each technique has varying uses within different disciplines.
An Influence diagram(ID) is essentially a graphical representation of the probabilistic interdependence between Performance Shaping Factors (PSFs), factors which pose a likelihood of influencing the success or failure of the performance of a task. The approach originates from the field of decision analysis and uses expert judgement in its formulations. It is dependent upon the principal of human reliability and results from the combination of factors such as organisational and individual factors, which in turn combine to provide an overall influence. There exists a chain of influences in which each successive level affects the next. The role of the ID is to depict these influences and the nature of the interrelationships in a more comprehensible format. In this way, the diagram may be used to represent the shared beliefs of a group of experts on the outcome of a particular action and the factors that may or may not influence that outcome. For each of the identified influences quantitative values are calculated, which are then used to derive final Human Error Probability (HEP) estimates.
Background
IDA is a decision analysis based framework which is developed through eliciting expert judgement through group workshops. Unlike other first generation HRA, IDA explicitly considers the inter-dependency of operator and organisational PSFs. The IDA approach was first outlined by Howard and Matheson [1], and then developed specifically for the nuclear industry by Embrey et al. [2].
IDA Methodology
The IDA methodology is conducted in a series of 10 steps as follows:
1. Describe all relevant conditioning events
Experts who have sufficient knowledge of the situation under evaluation form a group; in depth knowledge is essential for the technique to be used to its o |
https://en.wikipedia.org/wiki/ATHEANA | A Technique for Human Event Analysis (ATHEANA) is a technique used in the field of human reliability assessment (HRA). The purpose of ATHEANA is to evaluate the probability of human error while performing a specific task. From such analyses, preventative measures can then be taken to reduce human errors within a system and therefore lead to improvements in the overall level of safety.
There exist three primary reasons for conducting a HRA; error identification, error quantification and error reduction. As there exist a number of techniques used for such purposes, they can be split into one of two classifications; first generation techniques and second generation techniques.
First generation techniques work on the basis of the simple dichotomy of ‘fits/doesn’t fit’ in the matching of the error situation in context with related error identification and quantification and second generation techniques are more theory based in their assessment and quantification of errors. ‘HRA techniques have been utilised in a range of industries including healthcare, engineering, nuclear, transportation and business sector; each technique has varying uses within different disciplines.
ATHEANA is used following the occurrence of an incident. The various drivers of an incident and the possible outcomes are categorised into one of the following groupings: organisational influences; performance shaping factors; error mechanisms; unsafe actions; human failure event; unacceptable outcome(s). The resultant model may indicate solutions to improve reliability, however there are no numerical aspects involved in the methodology used to construct the model. Due to this characteristic, the technique is thus not considered to be suitable for use in certain fields such as comparative design work or sensitivity analysis. The methodology of ATHEANA is not predictive but does serve as a diagnostic modelling tool. Furthermore, its lack of Human Error Probability (HEP) as an output is a marked difference of the method compared to first generation HRA methodologies. The outcome provided by ATHEANA identifies various human actions within a system while also eliciting many contextual situations within this system, which influence whether the action will be carried out successfully or will lead to failure.
Background
ATHEANA is a post-incident Human Reliability Assessment (HRA) methodology developed by the US Nuclear Regulatory Commission in 2000. It was developed in the hope that certain types of human behaviour in nuclear plants and industries, which use similar processes, could be represented in a way in which they could be more easily understood. It seeks to provide a robust psychological framework to evaluate and identify Performance Shaping Factors (PSFs) - including organisational/environmental factors - which have driven incidents involving human factors, primarily with the intention of suggesting process improvement. Essentially it is a method of representing complex accident |
https://en.wikipedia.org/wiki/Elshan%20Mamedov | Elshan Mamedov (; born 4 May 1980 in Baku) is an Azerbaijani football player who currently plays for Sharurspor PFK as a forward.
Career statistics
References
External links
Player`s interview
Player`s profile
1980 births
Living people
Azerbaijani men's footballers
Qarabağ FK players
Simurq PIK players
FK Standard Sumgayit players
Footballers from Baku
Men's association football forwards
Azerbaijan men's international footballers |
https://en.wikipedia.org/wiki/Topological%20pair | In mathematics, more specifically algebraic topology, a pair is shorthand for an inclusion of topological spaces . Sometimes is assumed to be a cofibration. A morphism from to is given by two maps and
such that .
A pair of spaces is an ordered pair where is a topological space and a subspace (with the subspace topology). The use of pairs of spaces is sometimes more convenient and technically superior to taking a quotient space of by . Pairs of spaces occur centrally in relative homology, homology theory and cohomology theory, where chains in are made equivalent to 0, when considered as chains in .
Heuristically, one often thinks of a pair as being akin to the quotient space .
There is a functor from the category of topological spaces to the category of pairs of spaces, which sends a space to the pair .
A related concept is that of a triple , with . Triples are used in homotopy theory. Often, for a pointed space with basepoint at , one writes the triple as , where .
References
.
Algebraic topology |
https://en.wikipedia.org/wiki/2%2031%20polytope | {{DISPLAYTITLE:2 31 polytope}}
In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.
Its Coxeter symbol is 231, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node branch.
The rectified 231 is constructed by points at the mid-edges of the 231.
These polytopes are part of a family of 127 (or 27−1) convex uniform polytopes in 7-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .
2_31 polytope
The 231 is composed of 126 vertices, 2016 edges, 10080 faces (Triangles), 20160 cells (tetrahedra), 16128 4-faces (3-simplexes), 4788 5-faces (756 pentacrosses, and 4032 5-simplexes), 632 6-faces (576 6-simplexes and 56 221). Its vertex figure is a 6-demicube.
Its 126 vertices represent the root vectors of the simple Lie group E7.
This polytope is the vertex figure for a uniform tessellation of 7-dimensional space, 331.
Alternate names
E. L. Elte named it V126 (for its 126 vertices) in his 1912 listing of semiregular polytopes.
It was called 231 by Coxeter for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence.
Pentacontihexa-pentacosiheptacontihexa-exon (Acronym laq) - 56-576 facetted polyexon (Jonathan Bowers)
Construction
It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram, .
Removing the node on the short branch leaves the 6-simplex. There are 576 of these facets. These facets are centered on the locations of the vertices of the 321 polytope, .
Removing the node on the end of the 3-length branch leaves the 221. There are 56 of these facets. These facets are centered on the locations of the vertices of the 132 polytope, .
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 6-demicube, 131, .
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.
Images
Related polytopes and honeycombs
Rectified 2_31 polytope
The rectified 231 is a rectification of the 231 polytope, creating new vertices on the center of edge of the 231.
Alternate names
Rectified pentacontihexa-pentacosiheptacontihexa-exon - as a rectified 56-576 facetted polyexon (acronym ) (Jonathan Bowers)
Construction
It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram, .
Removing the node on the short branch leaves the rectified 6-simplex, .
Removing the node on the end of the 2-length branch leaves the, 6-demicube,
.
Removing the node on the end of the 3-length branch leaves the rectified 221, .
The vertex figure is determined by removing the ringed node and ringing the neighboring node.
Images
See also
List of E7 polytopes |
https://en.wikipedia.org/wiki/1%2022%20polytope | {{DISPLAYTITLE:1 22 polytope}}
In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 (for its 72 vertices).
Its Coxeter symbol is 122, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. There are two rectifications of the 122, constructed by positions points on the elements of 122. The rectified 122 is constructed by points at the mid-edges of the 122. The birectified 122 is constructed by points at the triangle face centers of the 122.
These polytopes are from a family of 39 convex uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .
122 polytope
The 122 polytope contains 72 vertices, and 54 5-demicubic facets. It has a birectified 5-simplex vertex figure. Its 72 vertices represent the root vectors of the simple Lie group E6.
Alternate names
Pentacontatetra-peton (Acronym Mo) - 54-facetted polypeton (Jonathan Bowers)
Images
Construction
It is created by a Wythoff construction upon a set of 6 hyperplane mirrors in 6-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram, .
Removing the node on either of 2-length branches leaves the 5-demicube, 131, .
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 5-simplex, 022, .
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.
Related complex polyhedron
The regular complex polyhedron 3{3}3{4}2, , in has a real representation as the 122 polytope in 4-dimensional space. It has 72 vertices, 216 3-edges, and 54 3{3}3 faces. Its complex reflection group is 3[3]3[4]2, order 1296. It has a half-symmetry quasiregular construction as , as a rectification of the Hessian polyhedron, .
Related polytopes and honeycomb
Along with the semiregular polytope, 221, it is also one of a family of 39 convex uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .
Geometric folding
The 122 is related to the 24-cell by a geometric folding E6 → F4 of Coxeter-Dynkin diagrams, E6 corresponding to 122 in 6 dimensions, F4 to the 24-cell in 4 dimensions. This can be seen in the Coxeter plane projections. The 24 vertices of the 24-cell are projected in the same two rings as seen in the 122.
Tessellations
This polytope is the vertex figure for a uniform tessellation of 6-dimensional space, 222, .
Rectified 122 polytope
The rectified 122 polytope (also called 0221) can tessellate 6-dimensional space as the Voronoi cell of the E6* honeycomb lattice (dual of E6 lattice).
Alternate names
Birectified 221 polytope
Rectified pentacontatetrapeton (acro |
https://en.wikipedia.org/wiki/1%2032%20polytope | {{DISPLAYTITLE:1 32 polytope}}
In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.
Its Coxeter symbol is 132, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences.
The rectified 132 is constructed by points at the mid-edges of the 132.
These polytopes are part of a family of 127 (27-1) convex uniform polytopes in 7-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .
1_32 polytope
This polytope can tessellate 7-dimensional space, with symbol 133, and Coxeter-Dynkin diagram, . It is the Voronoi cell of the dual E7* lattice.
Alternate names
Emanuel Lodewijk Elte named it V576 (for its 576 vertices) in his 1912 listing of semiregular polytopes.
Coxeter called it 132 for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch.
Pentacontihexa-hecatonicosihexa-exon (Acronym lin) - 56-126 facetted polyexon (Jonathan Bowers)
Images
Construction
It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram,
Removing the node on the end of the 2-length branch leaves the 6-demicube, 131,
Removing the node on the end of the 3-length branch leaves the 122,
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 6-simplex, 032,
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.
Related polytopes and honeycombs
The 132 is third in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. The next figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb, 134.
Rectified 1_32 polytope
The rectified 132 (also called 0321) is a rectification of the 132 polytope, creating new vertices on the center of edge of the 132. Its vertex figure is a duoprism prism, the product of a regular tetrahedra and triangle, doubled into a prism: {3,3}×{3}×{}.
Alternate names
Rectified pentacontihexa-hecatonicosihexa-exon for rectified 56-126 facetted polyexon (acronym rolin) (Jonathan Bowers)
Construction
It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space. These mirrors are represented by its Coxeter-Dynkin diagram, , and the ring represents the position of the active mirror(s).
Removing the node on the end of the 3-length branch leaves the rectified 122 polytope,
Removing the node on the end of the 2-length branch leaves the demihexeract, 131,
Removing the node on the end of the 1-length branch leaves the birectified 6-simplex,
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the tetrahedron-triangle duoprism prism, {3,3}×{3}×{},
Seen in |
https://en.wikipedia.org/wiki/KSEG | KSEG may refer to:
KSEG (FM), a radio station (96.9 FM) licensed to Sacramento, California, United States
KSEG (software), a geometry software
Penn Valley Airport's ICAO code |
https://en.wikipedia.org/wiki/Microbundle | In mathematics, a microbundle is a generalization of the concept of vector bundle, introduced by the American mathematician John Milnor in 1964. It allows the creation of bundle-like objects in situations where they would not ordinarily be thought to exist. For example, the tangent bundle is defined for a smooth manifold but not a topological manifold; use of microbundles allows the definition of a topological tangent bundle.
Definition
A (topological) -microbundle over a topological space (the "base space") consists of a triple , where is a topological space (the "total space"), and are continuous maps (respectively, the "zero section" and the "projection map") such that:
the composition is the identity of ;
for every , there are a neighborhood of and a neighbourhood of such that , , is homeomorphic to and the maps and commute with and .
In analogy with vector bundles, the integer is also called the rank or the fibre dimension of the microbundle. Similarly, note that the first condition suggests should be thought of as the zero section of a vector bundle, while the second mimics the local triviality condition on a bundle. An important distinction here is that "local triviality" for microbundles only holds near a neighborhood of the zero section. The space could look very wild away from that neighborhood. Also, the maps gluing together locally trivial patches of the microbundle may only overlap the fibers.
The definition of microbundle can be adapted to other categories more general than the smooth one, such as that of piecewise linear manifolds, by replacing topological spaces and continuous maps by suitable objects and morphisms.
Examples
Any vector bundle of rank has an obvious underlying -microbundle, where is the zero section.
Given any topological space , the cartesian product (together with the projection on and the map ) defines an -microbundle, called the standard trivial microbundle of rank . Equivalently, it is the underlying microbundle of the trivial vector bundle of rank .
Given a topological manifold of dimension , the cartesian product together with the projection on the first component and the diagonal map defines an -microbundle, called the tangent microbundle of .
Given an -microbundle over and a continuous map , the space defines an -microbundle over , called the pullback (or induced) microbundle by , together with the projection and the zero section . If is a vector bundle, the pullback microbundle of its underlying microbundle is precisely the underlying microbundle of the standard pullback bundle.
Given an -microbundle over and a subspace , the restricted microbundle, also denoted by , is the pullback microbundle with respect to the inclusion .
Morphisms
Two -microbundles and over the same space are isomorphic (or equivalent) if there exist a neighborhood of and a neighborhood of , together with a homeomorphism commuting with the projections and the zero sections.
More g |
https://en.wikipedia.org/wiki/Order%20%28mathematics%29 | Order in mathematics may refer to:
Set theory
Total order and partial order, a binary relation generalizing the usual ordering of numbers and of words in a dictionary
Ordered set
Order in Ramsey theory, uniform structures in consequence to critical set cardinality
Algebra
Order (group theory), the cardinality of a group or period of an element
Order of a polynomial (disambiguation)
Order of a square matrix, its dimension
Order (ring theory), an algebraic structure
Ordered group
Ordered field
Analysis
Order (differential equation) or order of highest derivative, of a differential equation
Leading-order terms
NURBS order, a number one greater than the degree of the polynomial representation of a non-uniform rational B-spline
Order of convergence, a measurement of convergence
Order of derivation
Order of an entire function
Order of a power series, the lowest degree of its terms
Ordered list, a sequence or tuple
Orders of approximation in Big O notation
Z-order (curve), a space-filling curve
Arithmetic
Multiplicative order in modular arithmetic
Order of operations
Orders of magnitude, a class of scale or magnitude of any amount
Combinatorics
Order in the Josephus permutation
Ordered selections and partitions of the twelvefold way in combinatorics
Ordered set, a bijection, cyclic order, or permutation
Unordered subset or combination
Weak order of permutations
Fractals
Complexor, or complex order in fractals
Order of extension in Lakes of Wada
Order of fractal dimension (Rényi dimensions)
Orders of construction in the Pythagoras tree
Geometry
Long-range aperiodic order, in pinwheel tiling, for instance
Graphs
Graph order, the number of nodes in a graph
First order and second order logic of graphs
Topological ordering of directed acyclic graphs
Degeneracy ordering of undirected graphs
Elimination ordering of chordal graphs
Order, the complexity of a structure within a graph: see haven (graph theory) and bramble (graph theory)
Logic
In logic, model theory and type theory:
Zeroth-order logic
First-order logic
Second-order logic
Higher-order logic
Order theory
Order (journal), an academic journal on order theory
Dense order, a total order wherein between any unequal pair of elements there is always an intervening element in the order
Glossary of order theory
Lexicographical order, an ordering method on sequences analogous to alphabetical order on words
List of order topics, list of order theory topics
Order theory, study of various binary relations known as orders
Order topology, a topology of total order for totally ordered sets
Ordinal numbers, numbers assigned to sets based on their set-theoretic order
Partial order, often called just "order" in order theory texts, a transitive antisymmetric relation
Total order, a partial order that is also total, in that either the relation or its inverse holds between any unequal elements
Statistics
Order statistics
First-order statistics, e.g., arithmetic mean, median, quantiles
Second-order |
https://en.wikipedia.org/wiki/PGF/TikZ | PGF/TikZ is a pair of languages for producing vector graphics (e.g., technical illustrations and drawings) from a geometric/algebraic description, with standard features including the drawing of points, lines, arrows, paths, circles, ellipses and polygons. PGF is a lower-level language, while TikZ is a set of higher-level macros that use PGF. The top-level PGF and TikZ commands are invoked as TeX macros, but in contrast with PSTricks, the PGF/TikZ graphics themselves are described in a language that resembles MetaPost. Till Tantau is the designer of the PGF and TikZ languages. He is also the main developer of the only known interpreter for PGF and TikZ, which is written in TeX. PGF is an acronym for "Portable Graphics Format". TikZ was introduced in version 0.95 of PGF, and it is a recursive acronym for "TikZ ist kein Zeichenprogramm" (German for "TikZ is not a drawing program").
Overview
The PGF/TikZ interpreter can be used from the popular LaTeX and ConTeXt macro packages, and also directly from the original TeX. Since TeX itself is not concerned with graphics, the interpreter supports multiple TeX output backends: dvips, dvipdfm/dvipdfmx/xdvipdfmx, TeX4ht, and pdftex's internal PDF output driver. Unlike PSTricks, PGF can thus directly produce either PostScript or PDF output, but it cannot use some of the more advanced PostScript programming features that PSTricks can use due to the "least common denominator" effect. PGF/TikZ comes with an extensive documentation; the version 3.1.4a of the manual has over 1300 pages.
The standard LaTeX picture environment can also be used as a front end for PGF — by merely using the pgfpict2e package.
The project has been under constant development since 2005. Most of the development until 2018 was done by Till Tantau and since then Henri Menke has been the main contributor. Version 3.0.0 was released on 20 December 2013. One of the major new features of this version was graph drawing using the graphdrawing package, which however requires LuaTeX. This version also added a new data visualization method and support for direct SVG output via the new dvisvgm driver.
Export
Several graphical editors can produce output for PGF/TikZ, such as the KDE program Cirkuit and the math drawing program GeoGebra. Export to TikZ is also available as extensions for Inkscape, Blender, MATLAB, matplotlib, Gnuplot, and R. The circuit-macros package of m4 macros exports circuit diagrams to TikZ using the dpic -g command line option. The dot2tex program can convert files in the DOT graph description language to PGF/TikZ.
Libraries
TikZ features libraries for easy drawing of many kinds of diagrams, such as the following (alphabetized by library name):
3D drawing3d
Finite automata and Turing machinesautomata
Coordinate system calculationscalc
Calendarscalendar
Chains: nodes typically connected by edges and arranged in rows and columnschain
Logic circuit and electrical circuit diagramscircuits.logic and circuits.ee
Entity–re |
https://en.wikipedia.org/wiki/Integration%20using%20parametric%20derivatives | In calculus, integration by parametric derivatives, also called parametric integration, is a method which uses known Integrals to integrate derived functions. It is often used in Physics, and is similar to integration by substitution.
Statement of the theorem
By using The Leibniz integral rule with the upper and lower bounds fixed we get that
It is also true for non-finite bounds.
Examples
Example One: Exponential Integral
For example, suppose we want to find the integral
Since this is a product of two functions that are simple to integrate separately, repeated integration by parts is certainly one way to evaluate it. However, we may also evaluate this by starting with a simpler integral and an added parameter, which in this case is t = 3:
This converges only for t > 0, which is true of the desired integral. Now that we know
we can differentiate both sides twice with respect to t (not x) in order to add the factor of x2 in the original integral.
This is the same form as the desired integral, where t = 3. Substituting that into the above equation gives the value:
Example Two: Gaussian Integral
Starting with the integral ,
taking the derivative with respect to t on both sides yields
.
In general, taking the n-th derivative with respect to t gives us
.
Example Three: A Polynomial
Using the classical and taking the derivative with respect to t we get
.
Example Four: Sums
The method can also be applied to sums, as exemplified below.
Use the Weierstrass factorization of the sinh function:
.
Take the logarithm:
.
Derive with respect to z:
.
Let :
.
References
External links
WikiBooks: Parametric_Integration
Integral calculus |
https://en.wikipedia.org/wiki/Integration%20using%20Euler%27s%20formula | In integral calculus, Euler's formula for complex numbers may be used to evaluate integrals involving trigonometric functions. Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely and and then integrated. This technique is often simpler and faster than using trigonometric identities or integration by parts, and is sufficiently powerful to integrate any rational expression involving trigonometric functions.
Euler's formula
Euler's formula states that
Substituting for gives the equation
because cosine is an even function and sine is odd. These two equations can be solved for the sine and cosine to give
Examples
First example
Consider the integral
The standard approach to this integral is to use a half-angle formula to simplify the integrand. We can use Euler's identity instead:
At this point, it would be possible to change back to real numbers using the formula . Alternatively, we can integrate the complex exponentials and not change back to trigonometric functions until the end:
Second example
Consider the integral
This integral would be extremely tedious to solve using trigonometric identities, but using Euler's identity makes it relatively painless:
At this point we can either integrate directly, or we can first change the integrand to and continue from there.
Either method gives
Using real parts
In addition to Euler's identity, it can be helpful to make judicious use of the real parts of complex expressions. For example, consider the integral
Since is the real part of , we know that
The integral on the right is easy to evaluate:
Thus:
Fractions
In general, this technique may be used to evaluate any fractions involving trigonometric functions. For example, consider the integral
Using Euler's identity, this integral becomes
If we now make the substitution , the result is the integral of a rational function:
One may proceed using partial fraction decomposition.
See also
Trigonometric substitution
Weierstrass substitution
Euler substitution
References
Integral calculus
Theorems in analysis
Theorems in calculus |
https://en.wikipedia.org/wiki/Lambert%20summation | In mathematical analysis and analytic number theory, Lambert summation is a summability method for summing infinite series related to Lambert series specially relevant in analytic number theory.
Definition
Define the Lambert kernel by with . Note that is decreasing as a function of when . A sum is Lambert summable to if , written .
Abelian and Tauberian theorem
Abelian theorem: If a series is convergent to then it is Lambert summable to .
Tauberian theorem: Suppose that is Lambert summable to . Then it is Abel summable to . In particular, if is Lambert summable to and then converges to .
The Tauberian theorem was first proven by G. H. Hardy and John Edensor Littlewood but it was not independent of number theory, in fact they used a number-theoretic estimate which is somewhat stronger than the prime number theorem itself. The unsatisfactory situation aronund the Lambert Tauberian was resolved by Norbert Wiener.
Examples
, where μ is the Möbius function. Hence if this series converges at all, it converges to zero. Note that the sequence satisfies the Tauberian condition, therefore the Tauberian theorem implies in the oridnary sense. This is equivalent to the prime number theorem.
where is von Mangoldt function and is Euler's constant. By the Tauberian theorem, the ordinary sum converges and in particular converges to . This is equivalent to where is the second Chebyshev function.
See also
Lambert series
Abel–Plana formula
Abelian and tauberian theorems
References
Mathematical series
Summability methods |
https://en.wikipedia.org/wiki/Girth | Girth may refer to:
Mathematics
Girth (functional analysis), the length of the shortest centrally symmetric simple closed curve on the unit sphere of a Banach space
Girth (geometry), the perimeter of a parallel projection of a shape
Girth (graph theory), the length of a shortest cycle contained in a graph
Matroid girth, the size of the smallest circuit in a matroid
Music and entertainment
Girth (album), 1997 album by heavy metal band Winters Bane
Girth (Pushing Daisies), an episode of the TV show Pushing Daisies
Girth (song), the former name of the Guns N' Roses song "Coma"
Other
Girth (tack), a piece of equipment used to keep a saddle in place on a horse
Girth (tree), measurement of the circumference of a tree trunk above its base |
https://en.wikipedia.org/wiki/Science.ie | The Science.ie portal provides all sorts of information about careers in science, technology, engineering and mathematics (STEM).
Overview
Science.ie is an initiative of the Irish Government’s Discover Science & Engineering (DSE) awareness programme in Ireland. DSE is managed by Forfás on behalf of the Office of Science and Technology at the Department of Enterprise, Trade and Employment.
The careers-related information on Science.ie has been moved to a new DSE website, which was launched in early October 2009. On MyScienceCareer.ie is:
Profiles of people currently working in STEM in Ireland
Information on famous Irish scientists
Links to many other career resources
A redeveloped Science.ie was also launched in October 2009. The site has been redesigned and includes social media bookmarking and RSS feeds.
Science.ie provides more general information on science in Ireland. This includes listings of science links, news and events. Its "Resources" section gives information on activities and visitor centres where you can learn about science.
The site also provides a free newsletter relating to Irish science, technology and innovation news, events, research and facts which is issued monthly by email.
DSE runs numerous other initiatives, including My Science Career, Project Blogger, Science Week Ireland and Discover Primary Science.
References
External links
Science.ie website
My Science Career website from DSE
Information about Science.ie on Discover-Science.ie
Science education
Science and technology in the Republic of Ireland |
https://en.wikipedia.org/wiki/Laver%20function | In set theory, a Laver function (or Laver diamond, named after its inventor, Richard Laver) is a function connected with supercompact cardinals.
Definition
If κ is a supercompact cardinal, a Laver function is a function ƒ:κ → Vκ such that for every set x and every cardinal λ ≥ |TC(x)| + κ there is a supercompact measure U on [λ]<κ such that if j U is the associated elementary embedding then j U(ƒ)(κ) = x. (Here Vκ denotes the κ-th level of the cumulative hierarchy, TC(x) is the transitive closure of x)
Applications
The original application of Laver functions was the following theorem of Laver.
If κ is supercompact, there is a κ-c.c. forcing notion (P, ≤) such after forcing with (P, ≤) the following holds: κ is supercompact and remains supercompact after forcing with any κ-directed closed forcing.
There are many other applications, for example the proof of the consistency of the proper forcing axiom.
References
Set theory
Large cardinals
Functions and mappings |
https://en.wikipedia.org/wiki/1%2042%20polytope | {{DISPLAYTITLE:142 polytope}}
In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.
Its Coxeter symbol is 142, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequences.
The rectified 142 is constructed by points at the mid-edges of the 142 and is the same as the birectified 241, and the quadrirectified 421.
These polytopes are part of a family of 255 (28 − 1) convex uniform polytopes in 8 dimensions, made of uniform polytope facets and vertex figures, defined by all non-empty combinations of rings in this Coxeter-Dynkin diagram: .
142 polytope
The 142 is composed of 2400 facets: 240 132 polytopes, and 2160 7-demicubes (141). Its vertex figure is a birectified 7-simplex.
This polytope, along with the demiocteract, can tessellate 8-dimensional space, represented by the symbol 152, and Coxeter-Dynkin diagram: .
Alternate names
E. L. Elte (1912) excluded this polytope from his listing of semiregular polytopes, because it has more than two types of 6-faces, but under his naming scheme it would be called V17280 for its 17280 vertices.
Coxeter named it 142 for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch.
Diacositetracont-dischiliahectohexaconta-zetton (acronym bif) - 240-2160 facetted polyzetton (Jonathan Bowers)
Coordinates
The 17280 vertices can be defined as sign and location permutations of:
All sign combinations (32): (280×32=8960 vertices)
(4, 2, 2, 2, 2, 0, 0, 0)
Half of the sign combinations (128): ((1+8+56)×128=8320 vertices)
(2, 2, 2, 2, 2, 2, 2, 2)
(5, 1, 1, 1, 1, 1, 1, 1)
(3, 3, 3, 1, 1, 1, 1, 1)
The edge length is 2 in this coordinate set, and the polytope radius is 4.
Construction
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram: .
Removing the node on the end of the 2-length branch leaves the 7-demicube, 141, .
Removing the node on the end of the 4-length branch leaves the 132, .
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 7-simplex, 042, .
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.
Projections
Orthographic projections are shown for the sub-symmetries of E8: E7, E6, B8, B7, B6, B5, B4, B3, B2, A7, and A5 Coxeter planes, as well as two more symmetry planes of order 20 and 24. Vertices are shown as circles, colored by their order of overlap in each projective plane.
Related polytopes and honeycombs
Rectified 142 polytope
The rectified 142 is named from being a rectification of the 142 polytope, with vertices positioned at the mid-edges of the 142. It can also be called a 0421 polytope with the ring at the center of 3 branches of length 4, 2, and 1.
Alternate names
0421 polytope
Birectif |
https://en.wikipedia.org/wiki/2%2041%20polytope | {{DISPLAYTITLE:2 41 polytope}}
In 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group.
Its Coxeter symbol is 241, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequences.
The rectified 241 is constructed by points at the mid-edges of the 241. The birectified 241 is constructed by points at the triangle face centers of the 241, and is the same as the rectified 142.
These polytopes are part of a family of 255 (28 − 1) convex uniform polytopes in 8-dimensions, made of uniform polytope facets, defined by all permutations of rings in this Coxeter-Dynkin diagram: .
241 polytope
The 241 is composed of 17,520 facets (240 231 polytopes and 17,280 7-simplices), 144,960 6-faces (6,720 221 polytopes and 138,240 6-simplices), 544,320 5-faces (60,480 211 and 483,840 5-simplices), 1,209,600 4-faces (4-simplices), 1,209,600 cells (tetrahedra), 483,840 faces (triangles), 69,120 edges, and 2160 vertices. Its vertex figure is a 7-demicube.
This polytope is a facet in the uniform tessellation, 251 with Coxeter-Dynkin diagram:
Alternate names
E. L. Elte named it V2160 (for its 2160 vertices) in his 1912 listing of semiregular polytopes.
It is named 241 by Coxeter for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence.
Diacositetracont-myriaheptachiliadiacosioctaconta-zetton (Acronym Bay) - 240-17280 facetted polyzetton (Jonathan Bowers)
Coordinates
The 2160 vertices can be defined as follows:
16 permutations of (±4,0,0,0,0,0,0,0) of (8-orthoplex)
1120 permutations of (±2,±2,±2,±2,0,0,0,0) of (trirectified 8-orthoplex)
1024 permutations of (±3,±1,±1,±1,±1,±1,±1,±1) with an odd number of minus-signs
Construction
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram: .
Removing the node on the short branch leaves the 7-simplex: . There are 17280 of these facets
Removing the node on the end of the 4-length branch leaves the 231, . There are 240 of these facets. They are centered at the positions of the 240 vertices in the 421 polytope.
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 7-demicube, 141, .
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.
Visualizations
Petrie polygon projections are 12, 18, or 30-sided based on the E6, E7, and E8 symmetries (respectively). The 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown.
Related polytopes and honeycombs
Rectified 2_41 polytope
The rectified 241 is a rectification of the 241 polytope, with vertices positioned at the mid-edges of the 241.
Alternate names
Rectified Diacositet |
https://en.wikipedia.org/wiki/Founders%20of%20statistics | Statistics is the theory and application of mathematics to the scientific method including hypothesis generation, experimental design, sampling, data collection, data summarization, estimation, prediction and inference from those results to the population from which the experimental sample was drawn. This article lists statisticians who have been instrumental in the development of theoretical and applied statistics.
Founders of departments of statistics
The role of a department of statistics is discussed in a 1949 article by Harold Hotelling, which helped to spur the creation of many departments of statistics.
See also
List of statisticians
History of statistics
Timeline of probability and statistics
List of people considered father or mother of a scientific field
References
External links
StatProb – peer-reviewed encyclopedia sponsored by statistics and probability societies
History of probability and statistics
Statistics
Statistics-related lists |
https://en.wikipedia.org/wiki/Data%20Desk | Data Desk is a software program for visual data analysis, visual data exploration, and statistics. It carries out Exploratory Data Analysis (EDA) and standard statistical analyses by means of dynamically linked graphic data displays that update any change simultaneously.
History
Data Desk was developed in 1985 by Paul F. Velleman, a statistics professor at Cornell University who had studied exploratory data analysis with John Tukey. Data Desk was released in 1986 for the Macintosh. It provided most standard statistical methods accessed through its own desktop interface.
In 1997, Data Desk was released for Windows, and included a General Linear Model (GLM), multivariate statistics, and nonlinear curve fitting. DD/XL is an add-in for Microsoft Excel that adds Data Desk Functionality directly to the Spreadsheet
Data Desk's developer, Data Description, pioneered linked graphic displays including a 3-D rotating plot and graphical slider control of parameters. It has also developed proprietary technology for computer-based multimedia instruction and currently provides contract data analysis services.
Reviews
Macworld reviewed DD/XL on December 1, 2000 with a 4.5 out of 5.
InfoWorld reviewed Data Desk 6.0 and said "DataDesk Plus is by far the best Windows package for in-depth data exploration". Also, DataDesk Plus is easily the best Windows statistics package for teaching statistics"
Macworld reviewed Data Desk in October 1997, and gave it 9.1 out of 10, and a 5 star rating.
See also
Data visualization
References
Further reading
External links
Data Description's Website
Data Desk's History page
Plotting software |
https://en.wikipedia.org/wiki/Oil%20reserves%20in%20Iraq | Oil reserves in Iraq are considered the world's fifth-largest proven oil reserves, with 140 billion barrels.
As a result of military occupation and civil unrest, the official statistics have not been revised since 2001 and are largely based on 2-D seismic data from three decades ago. International geologists and consultants have estimated that unexplored territory may contain vastly larger reserves. The majority of Iraq's proven reserves of oil comes from the following cities: Basra (Being #1), Baghdad (Being #2), Ramadi (Being #3), and finally, Ba'aj (Being the last oil rich city).
A measure of the uncertainty about Iraq's oil reserves is indicated by widely differing estimates. The U.S. Department of Energy (DOE) estimated in 2003 that Iraq had . The United States Geological Survey (USGS) in 1995 estimated proven reserves were . Iraq's prewar deputy oil minister said that potential reserves might be . The source of the uncertainty is that due to decades of war and unrest, many of Iraq's oil wells are run down and unkept. Repairs to the wells and oil facilities should make far more oil available economically from the same deposits. Iraq may prove to contain the largest extractable deposits of oil in the entire Middle East once these upgrading and facility improvements have advanced.
After more than a decade of sanctions and two Gulf Wars, Iraq's oil infrastructure needs modernization and investment. Despite a large reconstruction effort, the Iraqi oil industry has not been able to meet hydrocarbon production and export targets. The World Bank estimates that an additional $1 billion per year would need to be invested just to maintain current production. Long-term Iraq reconstruction costs could reach $100 billion or higher, of which more than a third will go to the oil, gas and electricity sectors. Another challenge to Iraq's development of the oil sector is that resources are not evenly divided across sectarian lines. Most known resources are in the Shiite areas of the south and the Kurdish areas of the north, with few resources in control of the Sunni population in the center.
In 2006, Iraq's oil production averaged , down from around of production prior to the coalition invasion in 2003. Iraq's reserve to production ratio is 158 years. After the end of the invasion the production increased on a high level, even though there is an invasion from the self-proclaimed Islamic State of Iraq and the Levant the production in March 2016 stood at 4.55 million barrels a day. Which seems to well become a new all-time peak year for Iraq if OPEC talks about freezing or reduce production held in April 2016 will not lead to a reduction. The old peak was 1979 with 171.6 million tons of oil compared to 136.9 million tons produced in 2011 and 152.4 million tons in 2012.
Oil extraction contracts awarded
2009
On June 30 and December 11, 2009, the Iraqi Ministry of Oil awarded contracts to international oil companies for some of Iraq's many oil fields. The |
https://en.wikipedia.org/wiki/Carmichael%27s%20totient%20function%20conjecture | In mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function φ(n), which counts the number of integers less than and coprime to n. It states that, for every n there is at least one other integer m ≠ n such that φ(m) = φ(n).
Robert Carmichael first stated this conjecture in 1907, but as a theorem rather than as a conjecture. However, his proof was faulty, and in 1922, he retracted his claim and stated the conjecture as an open problem.
Examples
The totient function φ(n) is equal to 2 when n is one of the three values 3, 4, and 6. Thus, if we take any one of these three values as n, then either of the other two values can be used as the m for which φ(m) = φ(n).
Similarly, the totient is equal to 4 when n is one of the four values 5, 8, 10, and 12, and it is equal to 6 when n is one of the four values 7, 9, 14, and 18. In each case, there is more than one value of n having the same value of φ(n).
The conjecture states that this phenomenon of repeated values holds for every n.
Lower bounds
There are very high lower bounds for Carmichael's conjecture that are relatively easy to determine. Carmichael himself proved that any counterexample to his conjecture (that is, a value n such that φ(n) is different from the totients of all other numbers) must be at least 1037, and Victor Klee extended this result to 10400. A lower bound of was given by Schlafly and Wagon, and a lower bound of was determined by Kevin Ford in 1998.
The computational technique underlying these lower bounds depends on some key results of Klee that make it possible to show that the smallest counterexample must be divisible by squares of the primes dividing its totient value. Klee's results imply that 8 and Fermat primes (primes of the form 2k + 1) excluding 3 do not divide the smallest counterexample. Consequently, proving the conjecture is equivalent to proving that the conjecture holds for all integers congruent to 4 (mod 8).
Other results
Ford also proved that if there exists a counterexample to the conjecture, then a positive proportion (in the sense of asymptotic density) of the integers are likewise counterexamples.
Although the conjecture is widely believed, Carl Pomerance gave a sufficient condition for an integer n to be a counterexample to the conjecture . According to this condition, n is a counterexample if for every prime p such that p − 1 divides φ(n), p2 divides n. However Pomerance showed that the existence of such an integer is highly improbable. Essentially, one can show that if the first k primes p congruent to 1 (mod q) (where q is a prime) are all less than qk+1, then such an integer will be divisible by every prime and thus cannot exist. In any case, proving that Pomerance's counterexample does not exist is far from proving Carmichael's conjecture. However if it exists then infinitely many counterexamples exist as asserted by Ford.
Another way of stating Carmichael's conjecture is that, if
A(f) |
https://en.wikipedia.org/wiki/Circuit%20topology%20%28electrical%29 | The circuit topology of an electronic circuit is the form taken by the network of interconnections of the circuit components. Different specific values or ratings of the components are regarded as being the same topology. Topology is not concerned with the physical layout of components in a circuit, nor with their positions on a circuit diagram; similarly to the mathematical concept of topology, it is only concerned with what connections exist between the components. There may be numerous physical layouts and circuit diagrams that all amount to the same topology.
Strictly speaking, replacing a component with one of an entirely different type is still the same topology. In some contexts, however, these can loosely be described as different topologies. For instance, interchanging inductors and capacitors in a low-pass filter results in a high-pass filter. These might be described as high-pass and low-pass topologies even though the network topology is identical. A more correct term for these classes of object (that is, a network where the type of component is specified but not the absolute value) is prototype network.
Electronic network topology is related to mathematical topology. In particular, for networks which contain only two-terminal devices, circuit topology can be viewed as an application of graph theory. In a network analysis of such a circuit from a topological point of view, the network nodes are the vertices of graph theory, and the network branches are the edges of graph theory.
Standard graph theory can be extended to deal with active components and multi-terminal devices such as integrated circuits. Graphs can also be used in the analysis of infinite networks.
Circuit diagrams
The circuit diagrams in this article follow the usual conventions in electronics; lines represent conductors, filled small circles represent junctions of conductors, and open small circles represent terminals for connection to the outside world. In most cases, impedances are represented by rectangles. A practical circuit diagram would use the specific symbols for resistors, inductors, capacitors etc., but topology is not concerned with the type of component in the network, so the symbol for a general impedance has been used instead.
The Graph theory section of this article gives an alternative method of representing networks.
Topology names
Many topology names relate to their appearance when drawn diagrammatically. Most circuits can be drawn in a variety of ways and consequently have a variety of names. For instance, the three circuits shown in Figure 1.1 all look different but have identical topologies.
This example also demonstrates a common convention of naming topologies after a letter of the alphabet to which they have a resemblance. Greek alphabet letters can also be used in this way, for example Π (pi) topology and Δ (delta) topology.
Series and parallel topologies
For a network with two branches, there are only two possible topolo |
https://en.wikipedia.org/wiki/Steiner%20ellipse | In geometry, the Steiner ellipse of a triangle, also called the Steiner circumellipse to distinguish it from the Steiner inellipse, is the unique circumellipse (ellipse that touches the triangle at its vertices) whose center is the triangle's centroid. Named after Jakob Steiner, it is an example of a circumconic. By comparison the circumcircle of a triangle is another circumconic that touches the triangle at its vertices, but is not centered at the triangle's centroid unless the triangle is equilateral.
The area of the Steiner ellipse equals the area of the triangle times and hence is 4 times the area of the Steiner inellipse. The Steiner ellipse has the least area of any ellipse circumscribed about the triangle.
The Steiner ellipse is the scaled Steiner inellipse (factor 2, center is the centroid). Hence both ellipses are similar (have the same eccentricity).
Properties
A Steiner ellipse is the only ellipse, whose center is the centroid of a triangle and contains the points . The area of the Steiner ellipse is -fold of the triangle's area.
Proof
A) For an equilateral triangle the Steiner ellipse is the circumcircle, which is the only ellipse, that fulfills the preconditions. The desired ellipse has to contain the triangle reflected at the center of the ellipse. This is true for the circumcircle. A conic is uniquely determined by 5 points. Hence the circumcircle is the only Steiner ellipse.
B) Because an arbitrary triangle is the affine image of an equilateral triangle, an ellipse is the affine image of the unit circle and the centroid of a triangle is mapped onto the centroid of the image triangle, the property (a unique circumellipse with the centroid as center) is true for any triangle.
The area of the circumcircle of an equilateral triangle is -fold of the area of the triangle. An affine map preserves the ratio of areas. Hence the statement on the ratio is true for any triangle and its Steiner ellipse.
Determination of conjugate points
An ellipse can be drawn (by computer or by hand), if besides the center at least two conjugate points on conjugate diameters are known. In this case
either one determines by Rytz's construction the vertices of the ellipse and draws the ellipse with a suitable ellipse compass
or uses an parametric representation for drawing the ellipse.
Let be a triangle and its centroid . The shear mapping with axis through and parallel to transforms the triangle onto the isosceles triangle (see diagram). Point is a vertex of the Steiner ellipse of triangle . A second vertex of this ellipse lies on , because is perpendicular to (symmetry reasons). This vertex can be determined from the data (ellipse with center through and , ) by calculation. It turns out that
Or by drawing: Using de la Hire's method (see center diagram) vertex of the Steiner ellipse of the isosceles triangle is determined.
The inverse shear mapping maps back to and point is fixed, because it is a point on the shear axis. |
https://en.wikipedia.org/wiki/Schoch%20circles | In geometry, the Schoch circles are twelve Archimedean circles constructed by Thomas Schoch.
History
In 1979, Thomas Schoch discovered a dozen new Archimedean circles; he sent his discoveries to Scientific American's "Mathematical Games" editor Martin Gardner. The manuscript was forwarded to Leon Bankoff. Bankoff gave a copy of the manuscript to Professor Clayton Dodge of the University of Maine in 1996. The two were planning to write an article about the Arbelos, in which the Schoch circles would be included; however, Bankoff died the year after.
In 1998, Peter Y. Woo of Biola University published Schoch's findings on his website. By generalizing two of Schoch's circles, Woo discovered an infinite family of Archimedean circles named the Woo circles in 1999.
Circles
See also
Schoch line
References
External links
Online catalogue of Archimedean circles
Hiroshi Okumura and Masayuki Watanabe (2004). "The Archimedean Circles of Schoch and Woo". Forum Geometricorum Volume 4.
Arbelos
de:Archimedischer Kreis#Schoch-Kreise und Schoch-Gerade |
https://en.wikipedia.org/wiki/Bill%20Eddy | William Eddy is an American author. Currently the John C. Warner Professor of Statistics, Emeritus at Carnegie Mellon University and is an Elected Fellow of the American Association for the Advancement of Science, American Statistical Association and Institute of Mathematical Statistics.
Eddy received a A.B. degree in statistics from Princeton University, followed by M.A., M.Phil., and Ph.D. degrees in statistics from Yale University under the supervision of John A. Hartigan. He began teaching at Carnegie Mellon University in 1976.
References
Year of birth missing (living people)
Living people
Carnegie Mellon University faculty
Princeton University alumni
Yale Graduate School of Arts and Sciences alumni
American statisticians |
https://en.wikipedia.org/wiki/K-equivalence | In mathematics, -equivalence, or contact equivalence, is an equivalence relation between map germs. It was introduced by John Mather in his seminal work in Singularity theory in the 1960s as a technical tool for studying stable maps. Since then it has proved important in its own right. Roughly speaking, two map germs ƒ, g are -equivalent if ƒ−1(0) and g−1(0) are diffeomorphic.
Definition
Two map germs are -equivalent if there is a diffeomorphism
of the form Ψ(x,y) = (φ(x),ψ(x,y)), satisfying,
, and
.
In other words, Ψ maps the graph of f to the graph of g, as well as the graph of the zero map to itself. In particular, the diffeomorphism φ maps f−1(0) to g−1(0). The name contact is explained by the fact that this equivalence is measuring the contact between the graph of f and the graph of the zero map.
Contact equivalence is the appropriate equivalence relation for studying the sets of solution of equations, and finds many applications in dynamical systems and bifurcation theory, for example.
It is easy to see that this equivalence relation is weaker than A-equivalence, in that any pair of -equivalent map germs are necessarily -equivalent.
KV-equivalence
This modification of -equivalence was introduced by James Damon in the 1980s. Here V is a subset (or subvariety) of Y, and the diffeomorphism Ψ above is required to preserve not but (that is, ). In particular, Ψ maps f−1(V) to g−1(V).
See also
A-equivalence
References
J. Martinet, Singularities of Smooth Functions and Maps, Volume 58 of LMS Lecture Note Series. Cambridge University Press, 1982.
J. Damon, The Unfolding and Determinacy Theorems for Subgroups of and . Memoirs Amer. Math. Soc. 50, no. 306 (1984).
Functions and mappings
Singularity theory
Equivalence (mathematics) |
https://en.wikipedia.org/wiki/List%20of%20books%20about%20nuclear%20issues | This is a list of books about nuclear issues. They are non-fiction books which relate to uranium mining, nuclear weapons and/or nuclear power.
The Algebra of Infinite Justice (2001)
American Prometheus: The Triumph and Tragedy of J. Robert Oppenheimer (2005)
The Angry Genie: One Man's Walk Through the Nuclear Age (1999)
The Atom Besieged: Extraparliamentary Dissent in France and Germany (1981)
Atomic Obsession: Nuclear Alarmism From Hiroshima to Al-Qaeda (2010)
The Bells of Nagasaki (1949)
Brighter than a Thousand Suns: A Personal History of the Atomic Scientists (1958)
Britain, Australia and the Bomb (2006)
Brittle Power: Energy Strategy for National Security (1982)
Canada’s Deadly Secret: Saskatchewan Uranium and the Global Nuclear System (2007)
Carbon-Free and Nuclear-Free (2007)
Chernobyl: Consequences of the Catastrophe for People and the Environment (2009)
Chernobyl. Vengeance of peaceful atom. (2006)
The Cold and the Dark: The World after Nuclear War (1984)
Command and Control (book) (2013)
Confronting the Bomb: A Short History of the World Nuclear Disarmament Movement (2009)
Conservation Fallout: Nuclear Protest at Diablo Canyon (2006)
Contesting the Future of Nuclear Power (2011)
Critical Masses: Opposition to Nuclear Power in California, 1958–1978 (1998)
The Cult of the Atom: The Secret Papers of the Atomic Energy Commission (1982)
The Day of the Bomb (1961)
The Doomsday Machine: Confessions of a Nuclear War Planner (2017)
The Doomsday Machine: The High Price of Nuclear Energy, The World's Most Dangerous Fuel (2012)
Essence of Decision: Explaining the Cuban Missile Crisis (1971)
Explaining the Atom (1947)
Fallout: An American Nuclear Tragedy (2004)
Fallout Protection (1961)
The Fate of the Earth (1982)
The Four Faces of Nuclear Terrorism (2004)
The Fourth Protocol (1984)
Fukushima: Japan's Tsunami and the Inside Story of the Nuclear Meltdowns (2013)
Full Body Burden: Growing Up in the Nuclear Shadow of Rocky Flats (2012)
The Gift of Time: The Case for Abolishing Nuclear Weapons Now (1998)
Hiroshima (1946)
The Hundredth Monkey (1982)
In Mortal Hands: A Cautionary History of the Nuclear Age (2009)
The International Politics of Nuclear Waste (1991)
Joseph Rotblat: A Man of Conscience in the Nuclear Age (2009)
Killing Our Own: The Disaster of America’s Experience with Atomic Radiation (1982)
The Last Train From Hiroshima (2010)
The Lean Guide to Nuclear Energy: A Life-Cycle in Trouble (2007)
Licensed to Kill? The Nuclear Regulatory Commission and the Shoreham Power Plant (1997)
Life After Doomsday (1980)
Los Alamos Primer (1992)
The Making of the Atomic Bomb (1988)
Making a Real Killing: Rocky Flats and the Nuclear West (1999)
Maralinga: Australia’s Nuclear Waste Cover-up (2007)
Megawatts and Megatons (2001)
My Australian Story: Atomic Testing (2009)
The Navajo People and Uranium Mining (2006)
Non-Nuclear Futures: The Case for an Ethical Energy Strategy (1975)
Normal Accidents: Living with High-Risk Technologies (1984)
Nuclear Implosions |
https://en.wikipedia.org/wiki/Antti%20Sakari%20Saario | Antti Saario is a contemporary electroacoustic composer and academic.
Biography
Born in 1974 in Lahti, Finland, Antti Sakari Saario graduated in mathematics and electronic music at Keele University in 1997.
He continued his studies in composition under Jonty Harrison at the University of Birmingham - working with Birmingham ElectroAcoustic Sound Theatre ( BEAST ) - receiving his PhD from Birmingham University in 2002 (PhD, University of Birmingham, School of Humanities, Department of Music, 2002). His numerous postgraduate compositions have been performed in a variety of venues and through a wide range of mediums - Most notably B-Side (1998) — A Fixed media composition. This was Released on Spike Works from BEAST — vol. 1. London: Sargasso Records. It was further featured in the Canadian Electroacoustic Community’s release Presence II . B-Side (1998) was positively reviewed by Rajmil Fischman in Sonic Arts Network's publication Diffusion, reprinted in .
After a period of experimental work in Norway, Antti Sakari Saario undertook his current teaching post at Falmouth University. His second wave of non-commercial compositional output began to emerge in the 2006 and has continued to date. Notable compositions have included :
Making Space: When The Curtain Falls (2007) - an acousmatic play for multichannel playback system and blacked-out space, performed as part of Nuffield Theatre's Making Space 2007 bursary/mini-residency scheme. This work was featured at ACMC 2008, Sydney Conservatorium of Music - The University of Sydney, Australia in July 2008.
Flocking: an improvatorio (2007-) for voices, string quartet and live-electronics. A free improvisation led by Steve Lewis in collaboration with David Prior. This was a site specific performance at Ashton Memorial as part of Lancaster Jazz Festival. . It subsequently featured at View Two Gallery, Liverpool, UK on Sept 6th, 2007. . A DVD of the Jazz Festival performance was produced with a 5.1 surround audio recording by David Prior, Saario and video artist Jenny McCabe.
Influences
Saario's work reflect his evolution as a composer - he readily acknowledges the influence of his tutors and collaborators on his work. Jonty Harrison, David Prior, Iain Armstrong and Robert Penman have left a telling influence on his sonic production. One characteristic feature of Dr Saarios work is his willingness to counterbalance within his work a wide range of aesthetic and academic approaches. The literary output of William S. Burroughs has been evident since his earliest compositions. The philosophy of Deleuze and Guattari and the filmwork of David Lynch and the Dogma 95 collaboration in particular underpin his composition as well as his academic approach. Since 2001 majority of his output has been collaborative, and in many instances interdisciplinary (e.g. 'The Hollywoods' and 'Aboriginal Terraformations' with visual artist Amanda Newall, and the [zygote] concept with composer and musicologist Martin Iddon). |
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