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https://en.wikipedia.org/wiki/Paul%20Evans%20%28ice%20hockey%2C%20born%201954%29
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John Paul Evans (born May 2, 1954) is a Canadian former professional ice hockey centre who played three seasons in the National Hockey League (NHL) for the Philadelphia Flyers.
Career statistics
References
External links
1954 births
Canadian ice hockey centres
Chicago Cougars draft picks
Ice hockey people from Toronto
Kitchener Rangers players
Living people
Los Angeles Kings draft picks
Maine Mariners (AHL) players
Philadelphia Flyers players
Saginaw Gears players
Springfield Indians players
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https://en.wikipedia.org/wiki/Daniel%20Kane%20%28mathematician%29
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Daniel Mertz Kane (born 1986) is an American mathematician. He is an associate professor with a joint position in the Mathematics Department and the Computer Science and Engineering Department at the University of California, San Diego.
Early life and education
Kane was born in Madison, Wisconsin, to Janet E. Mertz and Jonathan M. Kane, professors of oncology and of mathematics and computer science, respectively.
He attended Wingra School, a small alternative K-8 school in Madison that focuses on self-guided education. By 3rd grade, he had mastered K through 9th-grade mathematics. Starting at age 13, he took honors math courses at the University of Wisconsin–Madison and did research under the mentorship of Ken Ono while dual enrolled at Madison West High School. He earned gold medals in the 2002 and 2003 International Mathematical Olympiads. Prior to his 17th birthday, he resolved an open conjecture proposed years earlier by Andrews and Lewis; for this research, he was named Fellow Laureate of the Davidson Institute for Talent Development.
He graduated Phi Beta Kappa from the Massachusetts Institute of Technology in 2007 with two bachelor's degrees, one in mathematics with computer science and the other in physics.
While at MIT, Kane was one of four people since 2003 (and one of eight in the history of the competition) to be named a four-time Putnam Fellow in the William Lowell Putnam Mathematical Competition. He also won the 2007 Morgan Prize and competed as part of the MIT team in the Mathematical Contest in Modeling four times, earning the highest score three times and winning the Ben Fusaro Award in 2004, INFORMS Award in 2006, and SIAM Award in 2007. He also won the Machtey Award as an undergraduate in 2005, with Tim Abbott and Paul Valiant, for the best student-authored
paper at the Symposium on Foundations of Computer Science that year, on the complexity of two-player win-loss games.
Kane received his doctorate in mathematics from Harvard University in 2011; his dissertation, on number theory, was supervised by Barry Mazur. In his curriculum vitae, Kane lists as mentors Ken Ono while in high school; Erik Demaine, Joseph Gallian, and Cesar Silva while an undergraduate student at MIT; and Barry Mazur, Benedict Gross, and Henry Cohn while a graduate student at Harvard.
Research contributions
In 2010, joint work with Jelani Nelson and David Woodruff won both the IBM Pat Goldberg Memorial and Symposium on Principles of Database Systems (PODS) best paper awards for work on an optimal algorithm for the count-distinct problem.
References
External links
Daniel Kane's homepage at UCSD
1986 births
Living people
Scientists from Madison, Wisconsin
Massachusetts Institute of Technology School of Science alumni
Harvard Graduate School of Arts and Sciences alumni
20th-century American mathematicians
21st-century American mathematicians
Putnam Fellows
International Mathematical Olympiad participants
Mathematicians from Wisconsin
Madison West High
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https://en.wikipedia.org/wiki/Advanced%20level%20mathematics
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Advanced Level (A-Level) Mathematics is a qualification of further education taken in the United Kingdom (and occasionally other countries as well). In the UK, A-Level exams are traditionally taken by 17-18 year-olds after a two-year course at a sixth form or college. Advanced Level Further Mathematics is often taken by students who wish to study a mathematics-based degree at university, or related degree courses such as physics or computer science.
Like other A-level subjects, mathematics has been assessed in a modular system since the introduction of Curriculum 2000, whereby each candidate must take six modules, with the best achieved score in each of these modules (after any retake) contributing to the final grade. Most students will complete three modules in one year, which will create an AS-level qualification in their own right and will complete the A-level course the following year—with three more modules.
The system in which mathematics is assessed is changing for students starting courses in 2017 (as part of the A-level reforms first introduced in 2015), where the reformed specifications have reverted to a linear structure with exams taken only at the end of the course in a single sitting.
In addition, while schools could choose freely between taking Statistics, Mechanics or Discrete Mathematics (also known as Decision Mathematics) modules with the ability to specialise in one branch of applied Mathematics in the older modular specification, in the new specifications, both Mechanics and Statistics were made compulsory, with Discrete Mathematics being made exclusive as an option to students pursuing a Further Mathematics course. The first assessment opportunity for the new specification is 2018 and 2019 for A-levels in Mathematics and Further Mathematics, respectively.
2000s specification
Prior to the 2017 reform, the basic A-Level course consisted of six modules, four pure modules (C1, C2, C3, and C4) and two applied modules in Statistics, Mechanics and/or Decision Mathematics. The C1 through C4 modules are referred to by A-level textbooks as "Core" modules, encompassing the major topics of mathematics such as logarithms, differentiation/integration and geometric/arithmetic progressions.
The two chosen modules for the final two parts of the A-Level are determined either by a student's personal choices, or the course choice of their school/college, though it commonly took the form of S1 (Statistics) and M1 (Mechanics).
Further mathematics
Students that were studying for (or had completed) an A-level in Mathematics had the opportunity to study an A-level in Further Mathematics, which required taking a further 6 modules to give a second qualification. The grades of the two A-levels will be independent of each other, with Further Mathematics requiring students to take a minimum of two Further Pure modules, one of which must be FP1, and the other either FP2 or FP3, which are simply extensions of the four Core modules from the norm
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https://en.wikipedia.org/wiki/Planar%20straight-line%20graph
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In computational geometry and geometric graph theory, a planar straight-line graph (or straight-line plane graph, or plane straight-line graph), in short PSLG, is an embedding of a planar graph in the plane such that its edges are mapped into straight-line segments. Fáry's theorem (1948) states that every planar graph has this kind of embedding.
In computational geometry, PSLGs have often been called planar subdivisions, with an assumption or assertion that subdivisions are polygonal rather than having curved boundaries.
PSLGs may serve as representations of various maps, e.g., geographical maps in geographical information systems.
Special cases of PSLGs are triangulations (polygon triangulation, point-set triangulation). Point-set triangulations are maximal PSLGs in the sense that it is impossible to add straight edges to them while keeping the graph planar. Triangulations have numerous applications in various areas.
PSLGs may be seen as a special kind of Euclidean graphs. However, in discussions involving Euclidean graphs, the primary interest is their metric properties, i.e., distances between vertices, while for PSLGs the primary interest is the topological properties. For some graphs, such as Delaunay triangulations, both metric and topological properties are of importance.
Representations
There exist three well-known data structures for representing PSLGs, these are the Winged-edge data structure, Halfedge, and Quadedge. The winged-edge data structure is the oldest of the three, but manipulating it often requires complicated case distinctions. This is because edge references do not store the edge direction, and the directions of edges around a face need not be consistent. The halfedge data structure stores both orientations of an edge and links them properly, simplifying operations and the storage scheme. The Quadedge data structure stores both the planar subdivision and its dual simultaneously. Its records consist explicitly only of edge records, four for each edge, and in a simplified form it is suitable for storing PSLGs.
Problems in terms of PSLG
Point location. For a query point, find which face of the PSLG it belongs to.
Map overlay. Find the overlay of two PSLGs (maps), which is the subdivision of the plane by the two simultaneously embedded PSLGs. In GIS this problem is known as "thematic map overlay".
See also
Doubly connected edge list, a data structure to represent a PSLG
Local feature size
References
Geometric algorithms
Geometric graphs
Planar graphs
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https://en.wikipedia.org/wiki/41st%20British%20Academy%20Film%20Awards
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The 41st British Academy Film Awards, given by the British Academy of Film and Television Arts in 1988, honoured the best in film for 1987.
Winners and nominees
Statistics
See also
60th Academy Awards
13th César Awards
40th Directors Guild of America Awards
1st European Film Awards
45th Golden Globe Awards
8th Golden Raspberry Awards
2nd Goya Awards
3rd Independent Spirit Awards
14th Saturn Awards
40th Writers Guild of America Awards
References
041
BAFTA
1988 in British cinema
1987 awards in the United Kingdom
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https://en.wikipedia.org/wiki/Jung%20Kyung-ho%20%28footballer%2C%20born%201987%29
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Jung Kyung-ho () (born 12 January 1987) is a South Korean footballer who plays for Ansan Greeners.
He was a member of South Korea U20 team at the 2007 FIFA U-20 World Cup.
Club career statistics
References
External links
FIFA Player Statistics
1987 births
Living people
Men's association football midfielders
South Korean men's footballers
South Korea men's under-20 international footballers
Gyeongnam FC players
Jeonnam Dragons players
Jeju United FC players
Gimcheon Sangmu FC players
Gwangju FC players
Ulsan Hyundai Mipo Dockyard FC players
Ansan Greeners FC players
K League 1 players
K League 2 players
Korea National League players
Footballers from North Gyeongsang Province
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https://en.wikipedia.org/wiki/Kim%20Poong-joo
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Kim Poong-Joo (; born 1 October 1964) is a South Korean football coach and former player. He played as a goalkeeper for Pusan Daewoo Royals.
Career statistics
External links
1964 births
Living people
Men's association football goalkeepers
South Korean men's footballers
South Korea men's international footballers
Busan IPark players
K League 1 players
1990 FIFA World Cup players
Footballers at the 1988 Summer Olympics
Olympic footballers for South Korea
Asian Games medalists in football
Footballers at the 1990 Asian Games
Asian Games bronze medalists for South Korea
Medalists at the 1990 Asian Games
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https://en.wikipedia.org/wiki/Hwangbo%20Kwan
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Hwangbo Kwan (; born 1 March 1965) is a South Korean football manager and former player. He is famous for his long-range goal against Spain in the 1990 FIFA World Cup.
Career statistics
International
Results list South Korea's goal tally first.
Managerial statistics
Honours
Yukong Elephants
K League 1: 1989
Korean League Cup: 1994
South Korea
AFC Asian Cup runner-up: 1988
Asian Games bronze medal: 1990
Dynasty Cup: 1990
Individual
K League Rookie of the Year: 1988
K League 1 Best XI: 1988, 1994
K League '90s All-Star Team: 2003
References
External links
1965 births
Living people
Men's association football midfielders
South Korean men's footballers
South Korean expatriate men's footballers
South Korea men's international footballers
South Korean football managers
Jeju United FC players
Oita Trinita players
K League 1 players
Japan Football League (1992–1998) players
Expatriate men's footballers in Japan
J1 League managers
J2 League managers
Oita Trinita managers
FC Seoul managers
K League 1 managers
Expatriate football managers in Japan
1988 AFC Asian Cup players
1990 FIFA World Cup players
Footballers from Daegu
South Korean expatriate sportspeople in Japan
Seoul National University alumni
Asian Games medalists in football
Footballers at the 1990 Asian Games
Yeongcheon Hwangbo clan
Asian Games bronze medalists for South Korea
Medalists at the 1990 Asian Games
South Korean expatriate football managers
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https://en.wikipedia.org/wiki/Institute%20for%20Pure%20and%20Applied%20Mathematics
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The Institute for Pure and Applied Mathematics (IPAM) is an American mathematics institute funded by the National Science Foundation. The initial funding for the institute was approved in May 1999 and it was inaugurated in August, 2000.
IPAM is located on the UCLA campus, in close proximity to UCLA's Department of Mathematics. The building currently housing the institute was designed in 1973 by world-renowned Pritzker Prize-winning architect Frank Gehry.
Mission
The mission of the institute is to make connections between a broad spectrum of mathematicians and scientists, to launch new collaborations, to better inform mathematicians and scientists about interdisciplinary problems, and to broaden the range of applications in which mathematics is used.
IPAM seeks to bring the full range of mathematical techniques to bear on the great scientific challenges of our time, to stimulate exciting new mathematics via new problems motivated by other sciences, and to train the people who will do this.
Background
IPAM is currently one of seven NSF Mathematical Sciences Institutes in the United States. The initial five year grant was renewed in 2005, and this grant was once again renewed in 2010, 2015 and 2020, for an additional five years.
The institute was co-founded by Tony F. Chan, Mark Green, and Eitan Tadmor; Dima Shlyakhtenko is its current director. Christian Ratsch is a deputy director, and Selenne Bañuelos is its current associate director. Stanley Osher is its current special projects director.
Programs
Every year IPAM offers two three-month scientific programs, or long programs. These programs bring together senior and junior mathematicians and scientists and engineers from the scientific disciplines related to the program. In addition, IPAM supports graduate students, post-doctoral scholars and young academics to encourage their participation in long programs.
The programs consist of three phases: Tutorials from both streams are offered at the beginning. These are followed by four five-day workshops focusing on particular topics related to the overall theme of the program. The programs culminate with a 1-week Oberwolfach-like workshop at the UCLA conference center at Lake Arrowhead, California.
Between the long programs, IPAM sponsors independent five-day workshops on a broad range of scientific themes. During the summer IPAM holds a research program for undergraduates (RIPS) focusing on industrial problems as well as a summer school for graduate students. The graduate student summer school is dedicated to an important scientific theme involving problems of mathematical interest.
References
External links
IPAM home page
NSF Mathematical Sciences Institutes
Mathematical institutes
Research institutes in California
National Science Foundation mathematical sciences institutes
Educational institutions established in 2000
Frank Gehry buildings
2000 establishments in California
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https://en.wikipedia.org/wiki/Small-bias%20sample%20space
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In theoretical computer science, a small-bias sample space (also known as -biased sample space, -biased generator, or small-bias probability space) is a probability distribution that fools parity functions.
In other words, no parity function can distinguish between a small-bias sample space and the uniform distribution with high probability, and hence, small-bias sample spaces naturally give rise to pseudorandom generators for parity functions.
The main useful property of small-bias sample spaces is that they need far fewer truly random bits than the uniform distribution to fool parities. Efficient constructions of small-bias sample spaces have found many applications in computer science, some of which are derandomization, error-correcting codes, and probabilistically checkable proofs.
The connection with error-correcting codes is in fact very strong since -biased sample spaces are equivalent to -balanced error-correcting codes.
Definition
Bias
Let be a probability distribution over .
The bias of with respect to a set of indices is defined as
where the sum is taken over , the finite field with two elements. In other words, the sum equals if the number of ones in the sample at the positions defined by is even, and otherwise, the sum equals .
For , the empty sum is defined to be zero, and hence .
ϵ-biased sample space
A probability distribution over is called an -biased sample space if
holds for all non-empty subsets .
ϵ-biased set
An -biased sample space that is generated by picking a uniform element from a multiset is called -biased set.
The size of an -biased set is the size of the multiset that generates the sample space.
ϵ-biased generator
An -biased generator is a function that maps strings of length to strings of length such that the multiset is an -biased set. The seed length of the generator is the number and is related to the size of the -biased set via the equation .
Connection with epsilon-balanced error-correcting codes
There is a close connection between -biased sets and -balanced linear error-correcting codes.
A linear code of message length and block length is
-balanced if the Hamming weight of every nonzero codeword is between and .
Since is a linear code, its generator matrix is an -matrix over with .
Then it holds that a multiset is -biased if and only if the linear code , whose columns are exactly elements of , is -balanced.
Constructions of small epsilon-biased sets
Usually the goal is to find -biased sets that have a small size relative to the parameters and .
This is because a smaller size means that the amount of randomness needed to pick a random element from the set is smaller, and so the set can be used to fool parities using few random bits.
Theoretical bounds
The probabilistic method gives a non-explicit construction that achieves size .
The construction is non-explicit in the sense that finding the -biased set requires a lot of true randomness, which does not help towa
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https://en.wikipedia.org/wiki/X-bar%20chart
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In industrial statistics, the X-bar chart is a type of Shewhart control chart that is used to monitor the arithmetic means of successive samples of constant size, n. This type of control chart is used for characteristics that can be measured on a continuous scale, such as weight, temperature, thickness etc. For example, one might take a sample of 5 shafts from production every hour, measure the diameter of each, and then plot, for each sample, the average of the five diameter values on the chart.
For the purposes of control limit calculation, the sample means are assumed to be normally distributed, an assumption justified by the Central Limit Theorem.
The X-bar chart is always used in conjunction with a variation chart such as the and R chart or and s chart. The R-chart shows sample ranges (difference between the largest and the smallest values in the sample), while the s-chart shows the samples' standard deviation. The R-chart was preferred in times when calculations were performed manually, as the range is far easier to calculate than the standard deviation; with the advent of computers, ease of calculation ceased to be an issue, and the s-chart is preferred these days, as it is statistically more meaningful and efficient. Depending on the type of variation chart used, the average sample range or the average sample standard deviation is used to derive the X-bar chart's control limits.
Statistical charts and diagrams
Quality control tools
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https://en.wikipedia.org/wiki/David%20Gibson%20%28Scrabble%20player%29
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David Lawrence Gibson (February 8, 1951 – November 22, 2019) was an American professional Scrabble player and mathematics professor. Ranked the top player in North America and widely regarded as one of the greatest Scrabble players, Gibson won the North American Scrabble Championship twice.
Early life
The eldest of three children, David Lawrence Gibson was born in 1951 in Raleigh, North Carolina and raised in Charlotte, North Carolina, where his parents owned an auto parts locating company. A graduate of North Mecklenburg High School and Furman University, he excelled in mathematics but fared considerably poorly in English. In 1975, Gibson and his family moved to Spartanburg, South Carolina.
Career
Gibson was introduced to Scrabble at age 6. He began playing tournament Scrabble in 1983 and won the North American Scrabble Championship (NASC) in August 1994; although he had kept a relatively low profile in the Scrabble community, the win propelled him to No. 1 in the North American rankings. In 1995, Gibson won the "Scrabble Superstars" tournament and its top prize of $50,000. Gibson finished second in the 2012 NASC; only needing to lose by fewer than 169 points to win the title, he ended up losing 298-475 in the final round against Nigel Richards. In 2016, Gibson won his second NASC title in Fort Wayne, Indiana. The term "Gibsonization" for a process of pairings in a single-elimination final was named after Gibson due to his tendency to clinch tournaments before their final rounds. Gibson also taught mathematics for some four decades at the Spartanburg Methodist College, and was awarded with an honorary distinction of Professor Emeritus by the college in early 2019.
Death
Gibson died aged 68 on November 22, 2019, due to complications from stage four pancreatic cancer.
Major championship wins
References
External links
1951 births
2019 deaths
American Scrabble players
People from Charlotte, North Carolina
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https://en.wikipedia.org/wiki/Kiiti%20Morita
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was a Japanese mathematician working in algebra and topology.
Morita was born in 1915 in Hamamatsu, Shizuoka Prefecture and graduated from the Tokyo Higher Normal School in 1936. Three years later he was appointed assistant at the Tokyo University of Science. He received his Ph.D. from Osaka University in 1950, with a thesis in topology. After teaching at the Tokyo Higher Normal School, he became professor at the University of Tsukuba in 1951. He held this position until 1978, after which he taught at Sophia University. Morita died of heart failure in 1995 at the Sakakibara Heart Institute in Tokyo; he was survived by his wife, Tomiko, his son, Yasuhiro, and a grandson.
He introduced the concepts now known as Morita equivalence and Morita duality which were given wide circulation in the 1960s by Hyman Bass in a series of lectures. The Morita conjectures on normal topological spaces are also named after him.
Publications
References
1915 births
1995 deaths
People from Hamamatsu
University of Tsukuba alumni
Osaka University alumni
20th-century Japanese mathematicians
Algebraists
Topologists
Academic staff of the University of Tsukuba
Academic staff of Sophia University
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https://en.wikipedia.org/wiki/Morita%20conjectures
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The Morita conjectures in general topology are certain problems about normal spaces, now solved in the affirmative. The conjectures, formulated by Kiiti Morita in 1976, asked
If is normal for every normal space Y, is X a discrete space?
If is normal for every normal P-space Y, is X metrizable?
If is normal for every normal countably paracompact space Y, is X metrizable and sigma-locally compact?
The answers were believed to be affirmative. Here a normal P-space Y is characterised by the property that the product with every metrizable X is normal; thus the conjecture was that the converse holds.
Keiko Chiba, Teodor C. Przymusiński, and Mary Ellen Rudin proved conjecture (1) and showed that conjectures (2) and (3) cannot be proven false under the standard ZFC axioms for mathematics (specifically, that the conjectures hold under the axiom of constructibility V=L).
Fifteen years later, Zoltán Tibor Balogh succeeded in showing that conjectures (2) and (3) are true.
Notes
References
A.V. Arhangelskii, K.R. Goodearl, B. Huisgen-Zimmerman, Kiiti Morita 1915-1995, Notices of the AMS, June 1997
Topology
Conjectures that have been proved
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https://en.wikipedia.org/wiki/Enhanced%20heat%20transfer
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Heat exchangers were initially developed to use plain (or smooth) heat transfer surfaces. An Enhanced heat transfer surface has a special surface geometry that provides a higher thermal performance, per unit base surface area than a plain surface.
Objectives
This higher thermal performance, per unit base surface area A may be used to achieve one of the three objectives below:
Size Reduction: If the heat exchange rate (Q) is held constant the heat exchanger length may be reduced. This will provide a smaller heat exchanger.
Increased heat exchange rate: This may be exploited either of two ways:
Reduced driving temperature difference (ΔTm): If Q and the total tube length (L) are held constant, the ΔTm may be reduced. This provides increased thermodynamic process efficiency, and yields a savings of operating costs.
Increased Heat Exchange Rate: Keeping L constant, the increased UA/L will result in increased heat exchange rate for fixed fluid inlet temperatures.
This increase in heat transfer capacity per unit volume comes at the expense of an increase in pressure drop across the heat exchanger. This pressure drop may be significant, so the heat transfer enhancement is often limited by the pumping power available.
Usage
The subject of “enhanced” heat transfer has become much more important to industry with progressing time. Use of relatively complex geometries were initially limited by manufacturing process. However, new manufacturing methods now allow manufacture of many complex surface geometries. Some enhanced surfaces (e.g., boiling and condensing tubes) are now in their 4th generation. Nearly all heat exchangers used in the air-conditioning and automotive industries are “enhanced” geometries. Further inroads are being seen in the electronic cooling, process and power industries.
References
Webb, R. L., and Kim, N-H. 2005. Principles of Enhanced Heat Transfer, 2nd ed., Taylor & Francis, New York, 785 pages plus reference CD, .
The Journal of Enhanced Heat Transfer. This journal published by Begell House publishes scholarly articles on the subject area.
External links
Journal of Enhanced Heat Transfer
Heating, ventilation, and air conditioning
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https://en.wikipedia.org/wiki/Vusal%20Garaev
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Vusal Garaev (born 1986) is an Azerbaijani football forward playing for Turan in the Azerbaijan First Division.
Career statistics
References
External links
http://www.futbol-agent.com/profi/nap_garayevvusal.htm
1986 births
Living people
Azerbaijani men's footballers
Gabala SC players
Ravan Baku FK players
Men's association football forwards
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https://en.wikipedia.org/wiki/Gromov%27s%20inequality%20for%20complex%20projective%20space
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In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality
,
valid for an arbitrary Riemannian metric on the complex projective space, where the optimal bound is attained
by the symmetric Fubini–Study metric, providing a natural geometrisation of quantum mechanics. Here is the stable 2-systole, which in this case can be defined as the infimum of the areas of rational 2-cycles representing the class of the complex projective line in 2-dimensional homology.
The inequality first appeared in as Theorem 4.36.
The proof of Gromov's inequality relies on the Wirtinger inequality for exterior 2-forms.
Projective planes over division algebras
In the special case n=2, Gromov's inequality becomes . This inequality can be thought of as an analog of Pu's inequality for the real projective plane . In both cases, the boundary case of equality is attained by the symmetric metric of the projective plane. Meanwhile, in the quaternionic case, the symmetric metric on is not its systolically optimal metric. In other words, the manifold admits Riemannian metrics with higher systolic ratio than for its symmetric metric .
See also
Loewner's torus inequality
Pu's inequality
Gromov's inequality (disambiguation)
Gromov's systolic inequality for essential manifolds
Systolic geometry
References
Geometric inequalities
Differential geometry
Riemannian geometry
Systolic geometry
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https://en.wikipedia.org/wiki/Bill%20Loughery
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William Gordon Ridley Loughery (1 November 1907 – 1 August 1977) was an Irish academic and cricketer. He was a scholar in mathematics at Trinity College Dublin, earning his BA there in 1930. From 1930 to 1934 he taught at Campbell College in Belfast.
Cricket career
A right-handed batsman, he played six times for the Ireland cricket team between 1929 and 1933 including two first-class matches against Scotland.
Loughery made his debut for Ireland in July 1929, playing a first-class match against Scotland. He played a match against "The Cataramans" later that month, twice against Sir Julien Cahn's XI in July 1930 and against the MCC in August 1930, before spending three years out of the Irish side. He returned in June 1933, bookending his career with another first-class match against Scotland.
In all matches for Ireland, Loughery scored 134 runs at an average of 12.18, with a top score of 29 against the MCC in August 1930. His top score in his two first-class games was 18 not out. He bowled just nine balls, not taking a wicket, and his bowling style is not known.
References
Cricketers from Belfast
1907 births
1977 deaths
Irish cricketers
Cricketers from Northern Ireland
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https://en.wikipedia.org/wiki/Dowker%20space
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In the mathematical field of general topology, a Dowker space is a topological space that is T4 but not countably paracompact. They are named after Clifford Hugh Dowker.
The non-trivial task of providing an example of a Dowker space (and therefore also proving their existence as mathematical objects) helped mathematicians better understand the nature and variety of topological spaces.
Equivalences
Dowker showed, in 1951, the following:
If X is a normal T1 space (that is, a T4 space), then the following are equivalent:
X is a Dowker space
The product of X with the unit interval is not normal.
X is not countably metacompact.
Dowker conjectured that there were no Dowker spaces, and the conjecture was not resolved until Mary Ellen Rudin constructed one in 1971. Rudin's counterexample is a very large space (of cardinality ). Zoltán Balogh gave the first ZFC construction of a small (cardinality continuum) example, which was more well-behaved than Rudin's. Using PCF theory, M. Kojman and S. Shelah constructed a subspace of Rudin's Dowker space of cardinality that is also Dowker.
References
Properties of topological spaces
Separation axioms
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https://en.wikipedia.org/wiki/Kappa%20Mu%20Epsilon
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Kappa Mu Epsilon () is a mathematics honor society founded by Emily Kathryn Wyant in 1931 at Northeastern Oklahoma State Teachers College to focus on the needs of undergraduate mathematics students. There are now over 80,000 members in about 150 chapters at various American universities and colleges in 35 states, primarily at mid-sized public universities or smaller private institutions. The five goals of Kappa Mu Epsilon are to further interest in mathematics, emphasize the role of mathematics in the development of civilization, develop an appreciation of the power and the beauty of mathematics, recognize the outstanding mathematical achievement of its members, and familiarize members with the advancements being made in mathematics.
The society sponsors a biennial national conference as well as regional conventions on alternate years which allow both for the sharing of ideas and the opportunity for students to present papers, preparing them for the graduate research experience.
KME also produces a yearly journal called The Pentagon which publishes student articles on a variety of mathematical topics.
The current president of KME is Dr. Brian Hollenbeck, a mathematics professor at Emporia State University in Emporia, Kansas. The president-elect is Dr. Don Tosh.
The organization had become an official member of the Association of College Honor Societies back in 1968.
See also
Mu Alpha Theta, (mathematics, high school)
Mu Sigma Rho, (statistics)
Pi Mu Epsilon, (mathematics)
Association of College Honor Societies
References
External links
KME Website (including history link)
Eastern Connecticut State University | Kappa Mu Epsilon
ACHS Kappa Mu Epsilon entry
Kappa Mu Epsilon chapter list at ACHS
Association of College Honor Societies
Honor societies
Mathematical societies
1931 establishments in Oklahoma
Student organizations established in 1931
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https://en.wikipedia.org/wiki/Bombieri%20norm
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In mathematics, the Bombieri norm, named after Enrico Bombieri, is a norm on homogeneous polynomials with coefficient in or (there is also a version for non homogeneous univariate polynomials). This norm has many remarkable properties, the most important being listed in this article.
Bombieri scalar product for homogeneous polynomials
To start with the geometry, the Bombieri scalar product for homogeneous polynomials with N variables can be defined as follows using multi-index notation:
by definition different monomials are orthogonal, so that
if
while by definition
In the above definition and in the rest of this article the following notation applies:
if write and and
Bombieri inequality
The fundamental property of this norm is the Bombieri inequality:
let be two homogeneous polynomials respectively of degree and with variables, then, the following inequality holds:
Here the Bombieri inequality is the left hand side of the above statement, while the right side means that the Bombieri norm is an algebra norm. Giving the left hand side is meaningless without that constraint, because in this case, we can achieve the same result with any norm by multiplying the norm by a well chosen factor.
This multiplicative inequality implies that the product of two polynomials is bounded from below by a quantity that depends on the multiplicand polynomials. Thus, this product can not be arbitrarily small. This multiplicative inequality is useful in metric algebraic geometry and number theory.
Invariance by isometry
Another important property is that the Bombieri norm is invariant by composition with an
isometry:
let be two homogeneous polynomials of degree with variables and let be an isometry
of (or ). Then we have . When this implies .
This result follows from a nice integral formulation of the scalar product:
where is the unit sphere of with its canonical measure .
Other inequalities
Let be a homogeneous polynomial of degree with variables and let . We have:
where denotes the Euclidean norm.
The Bombieri norm is useful in polynomial factorization, where it has some advantages over the Mahler measure, according to Knuth (Exercises 20-21, pages 457-458 and 682-684).
See also
Grassmann manifold
Hardy space
Homogeneous polynomial
Plücker embedding
References
Norms (mathematics)
Analytic number theory
Polynomials
Homogeneous polynomials
Complex analysis
Several complex variables
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https://en.wikipedia.org/wiki/Institute%20of%20Mathematics%20%28National%20Academy%20of%20Sciences%20of%20Belarus%29
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The Institute of Mathematics of the National Academy of Sciences of Belarus was founded in 1959. It is headquartered in Minsk, with a division in Gomel.
Departments
Algebra
Control Process Theory
Computational Mathematics and Mathematical Modelling
Differential Equations
Finite Croup Theory and Applications (in Gomel)
Combinatorial Models and Algorithms
Mathematical Theory of Systems
Mathematical Physics
Nonlinear and Stochastic Analysis
Parallel Computing Processes
External links
Official website
Research institutes in Belarus
Mathematical institutes
Universities and institutes established in the Soviet Union
Research institutes in the Soviet Union
1959 establishments in the Soviet Union
Research institutes established in 1959
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https://en.wikipedia.org/wiki/Polygonal%20chain
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In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points called its vertices. The curve itself consists of the line segments connecting the consecutive vertices.
Variations
Simple
A simple polygonal chain is one in which only consecutive segments intersect and only at their endpoints.
Closed
A closed polygonal chain is one in which the first vertex coincides with the last one, or, alternatively, the first and the last vertices are also connected by a line segment.
A simple closed polygonal chain in the plane is the boundary of a simple polygon. Often the term "polygon" is used in the meaning of "closed polygonal chain", but in some cases it is important to draw a distinction between a polygonal area and a polygonal chain.
Monotone
A polygonal chain is called monotone if there is a straight line L such that every line perpendicular to L intersects the chain at most once. Every nontrivial monotone polygonal chain is open. In comparison, a monotone polygon is a polygon (a closed chain) that can be partitioned into exactly two monotone chains. The graphs of piecewise linear functions form monotone chains with respect to a horizontal line.
Properties
Every set of at least points contains a polygonal path of at least edges in which all slopes have the same sign. This is a corollary of the Erdős–Szekeres theorem.
Applications
Polygonal chains can often be used to approximate more complex curves. In this context, the Ramer–Douglas–Peucker algorithm can be used to find a polygonal chain with few segments that serves as an accurate approximation.
In graph drawing, polygonal chains are often used to represent the edges of graphs, in drawing styles where drawing the edges as straight line segments would cause crossings, edge-vertex collisions, or other undesired features. In this context, it is often desired to draw edges with as few segments and bends as possible, to reduce the visual clutter in the drawing; the problem of minimizing the number of bends is called bend minimization.
Polygonal chains are also a fundamental data type in computational geometry. For instance, a point location algorithm of Lee and Preparata operates by decomposing arbitrary planar subdivisions into an ordered sequence of monotone chains, in which a point location query problem may be solved by binary search; this method was later refined to give optimal time bounds for the point location problem.
With geographic information system, linestrings may represent any linear geometry, and can be described using the well-known text markup as a LineString or MultiLineString. Linear rings (or LinearRing) are closed and simple polygonal chains used to build polygon geometries.
See also
Chain (algebraic topology), a formal combination of simplices that in the 1-dimensional case includes polygonal chains
Composite Bézier curve, a generalization that replaces each straight line of a polyg
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https://en.wikipedia.org/wiki/Institute%20of%20Mathematics
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This is a list of Institutes of Mathematics or Mathematical Institutes.
Americas
American Institute of Mathematics
Clay Mathematics Institute, Cambridge, Massachusetts
Centre de Recherches Mathématiques, at the Université de Montréal
Center for Mathematical Modeling, at the University of Chile
Centro de Investigación en Matemáticas, Guanajuato, Guanajuato in Mexico
Courant Institute of Mathematical Sciences, at New York University
Fields Institute, at the University of Toronto
Institute for Advanced Study, in Princeton, New Jersey
Institute for Mathematics and its Applications, at the University of Minnesota
Institute for Pure and Applied Mathematics, at the University of California, Los Angeles
Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, Brazil
Mathematical Sciences Research Institute, at the University of California, Berkeley
PPGMAp, at the Universidade Federal do Rio Grande do Sul in Brazil
Europe
Brunel Institute of Computational Mathematics, in Uxbridge, UK
Central Economic Mathematical Institute, at the Russian Academy of Sciences
Centre de Recerca Matemàtica, at the Autonomous University of Barcelona
Centrum Wiskunde & Informatica, at Science Park, Amsterdam
CoMPLEX, at University College London
Fachinformationszentrum Karlsruhe, Germany
Hamilton Mathematics Institute, at Trinity College, Dublin, Ireland
Hausdorff Center for Mathematics, Bonn, Germany
Institut de Mathématiques de Toulouse, France
Institute for Experimental Mathematics, at the University of Duisburg-Essen in Germany
Institute of Mathematics (National Academy of Sciences of Belarus)
Institute of Mathematics of the National Academy of Sciences of Ukraine
Institute of Mathematics and its Applications, a UK society
The Institute of Mathematics and Computer Science, University of Latvia
Institute of Mathematics and Informatics (Bulgarian Academy of Sciences)
Institute of Mathematics of National Academy of Sciences of Armenia
Institute of Mathematics of the Romanian Academy at Bucharest
Institute of Mathematics, Physics, and Mechanics in Slovenia
Institut des Hautes Études Scientifiques, near Paris, France
Institut Henri Poincaré, Paris, France
International Centre for Mathematical Sciences, at Edinburgh
Isaac Newton Institute, at the University of Cambridge
János Bolyai Mathematical Institute, at the University of Szeged in Hungary
Keldysh Institute of Applied Mathematics, at the Russian Academy of Sciences
The Mathematical Institute, University of Oxford
Max Planck Institute for Mathematics, Bonn, Germany
Max Planck Institute for Mathematics in the Sciences at Leipzig
Mittag-Leffler Institute, Stockholm. Sweden
Moscow State Institute of Electronics and Mathematics
Oberwolfach Research Institute for Mathematics, Oberwolfach, Germany
Steklov Institute of Mathematics, Moscow, Russia
University of Copenhagen Institute for Mathematical Sciences
Asia
Chennai Mathematical Institute, India
CR Rao Advanced Institute
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https://en.wikipedia.org/wiki/SAT%20Subject%20Test%20in%20Mathematics%20Level%202
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In the U.S., the SAT Subject Test in Mathematics Level 2 (formerly known as Math II or Math IIChest, the "C" representing chest) was a one-hour multiple choice test. The questions covered a broad range of topics. Approximately 10-14% of questions focused on numbers and operations, 48-52% focused on algebra and functions, 28-32% focused on geometry (coordinate, three-dimensional, and trigonometric geometry were covered; plane geometry was not directly tested), and 8-12% focused on data analysis, statistics and probability. Compared to Mathematics 1, Mathematics 2 was more advanced. Whereas the Mathematics 1 test covered Algebra II and basic trigonometry, a pre-calculus class was good preparation for Mathematics 2. On January 19, 2021, the College Board discontinued all SAT Subject tests, including the SAT Subject Test in Mathematics Level 2. This was effective immediately in the United States, and the tests were to be phased out by the following summer for international students. This was done as a response to changes in college admissions due to the impact of the COVID-19 pandemic on education.
Format
The test had 50 multiple choice questions that were to be answered in one hour. All questions had five answer choices. Students received 1 point for every correct answer, lost ¼ of a point for each incorrect answer, and received 0 points for questions left blank.
Calculator use
The College Board stated that a calculator "may be useful or necessary" for about 55-60% of the questions on the test. The College Board also encouraged the use of a graphing calculator over a scientific calculator, saying that the test was "developed with the expectation that most students are using graphing calculators."
For the Mathematics Level Two test, students were not permitted to use calculators that have a QWERTY format keyboard, require an electrical outlet, make noise, use paper tape, have non-traditional methods of input (such as a stylus), or are part of a communication device (such as PDAs, laptops, or cell phones).
Preparation
The College Board suggested as preparation for the test four years of mathematics, including two years of algebra, one year of geometry, and one year of either precalculus or trigonometry.
While the precalculus or trigonometry course may have been good preparation for this test, students could have needed to buy extra resource materials if they want to score beyond a 700. The exam covered several years of mathematics, and students were expected to work quickly and efficiently.
Scoring
For each of the 50 multiple choice questions, students received 1 point for every correct answer, lost ¼ of a point for each incorrect answer, and received 0 points for questions left blank. This created a raw score, which was then converted into a scaled score. The conversion between these numbers varied depending on the difficulty of a particular test administration. The scaled score was the only score reported to either students or colleges, a
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https://en.wikipedia.org/wiki/Demography%20of%20Greater%20Manchester
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The demography of Greater Manchester is analysed by the Office for National Statistics and data is produced for each of its ten metropolitan boroughs, each of the Greater Manchester electoral wards, the NUTS3 statistical sub-regions, each of the Parliamentary constituencies in Greater Manchester, the 15 civil parishes in Greater Manchester, and for all of Greater Manchester as a whole; the latter of which had a population of 2,682,500 at the 2011 UK census. Additionally, data is produced for the Greater Manchester Urban Area. Statistical information is produced about the size and geographical breakdown of the population, the number of people entering and leaving country and the number of people in each demographic subgroup.
Key statistics
Compared against the demography of England, Greater Manchester's demographics are broadly inline with national averages on many topics. In terms of ethnicity, its Asian and British Asian population is considerably above the regional and national averages, as is the portion of residents who identify as Muslim. Compared against the demography of the United Kingdom, Greater Manchester's ethnic minority population consists of 11.09% of the total population. NB. Information in the table on the right is from the 2001 census and not the most recent 2011 census.
As of 2020, the ONS estimates the population of the Greater Manchester Metropolitan County to be 2,835,686.
Population change
The following is a table outlining population totals of the area for every ten years since 1801, using material from the census in the United Kingdom via the Great Britain Historical GIS; pre-1974 statistics were gathered from local government areas that now comprise Greater Manchester. The total population of Greater Manchester is predicted to grow to around 2,950,000 by 2031, with the City of Manchester alone accounting for 36% of the growth.
Boroughs
According to the 2011 census, of Greater Manchester's ten metropolitan boroughs, the City of Manchester is the most populous with a population of 503,127, whilst the Metropolitan Borough of Bury is the least populous with 185,100. The City of Manchester's population in 2021 is predicted at 532,200. The city experienced the greatest percentage population growth outside London, with an increase of 19% to over 500,000. Manchester's population is projected to reach 532,200 by 2021, an increase of 5.8% from 2011. This represents a slower rate of growth than the previous decade.
In terms of ethnic composition, the City of Manchester has the highest non-white population in Greater Manchester (34th in England), followed by the metropolitan boroughs of Oldham (45th in England), Rochdale (53rd in England), and Trafford (68th in England). Wigan is the least ethnically diverse borough in the county, and 274th in England.
The Metropolitan Borough of Oldham is the borough with the highest proportion of people under fifteen years of age. Almost 12% of people in the Metropolitan Borough of Oldham
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https://en.wikipedia.org/wiki/Gromov%27s%20systolic%20inequality%20for%20essential%20manifolds
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In the mathematical field of Riemannian geometry, M. Gromov's systolic inequality bounds the length of the shortest non-contractible loop on a Riemannian manifold in terms of the volume of the manifold. Gromov's systolic inequality was proved in 1983; it can be viewed as a generalisation, albeit non-optimal, of Loewner's torus inequality and Pu's inequality for the real projective plane.
Technically, let M be an essential Riemannian manifold of dimension n; denote by sysπ1(M) the homotopy 1-systole of M, that is, the least length of a non-contractible loop on M. Then Gromov's inequality takes the form
where Cn is a universal constant only depending on the dimension of M.
Essential manifolds
A closed manifold is called essential if its fundamental class defines a nonzero element in the homology of its fundamental group, or more precisely in the homology of the corresponding Eilenberg–MacLane space. Here the fundamental class is taken in homology with integer coefficients if the manifold is orientable, and in coefficients modulo 2, otherwise.
Examples of essential manifolds include aspherical manifolds, real projective spaces, and lens spaces.
Proofs of Gromov's inequality
Gromov's original 1983 proof is about 35 pages long. It relies on a number of techniques and inequalities of global Riemannian geometry. The starting point of the proof is the imbedding of X into the Banach space of Borel functions on X, equipped with the sup norm. The imbedding is defined by mapping a point p of X, to the real function on X given by the distance from the point p. The proof utilizes the coarea inequality, the isoperimetric inequality, the cone inequality, and the deformation theorem of Herbert Federer.
Filling invariants and recent work
One of the key ideas of the proof is the introduction of filling invariants, namely the filling radius and the filling volume of X. Namely, Gromov proved a sharp inequality relating the systole and the filling radius,
valid for all essential manifolds X; as well as an inequality
valid for all closed manifolds X.
It was shown by that the filling invariants, unlike the systolic invariants, are independent of the topology of the manifold in a suitable sense.
and developed approaches to the proof of Gromov's systolic inequality for essential manifolds.
Inequalities for surfaces and polyhedra
Stronger results are available for surfaces, where the asymptotics when the genus tends to infinity are by now well understood, see systoles of surfaces. A uniform inequality for arbitrary 2-complexes with non-free fundamental groups is available, whose proof relies on the Grushko decomposition theorem.
Notes
See also
Filling area conjecture
Gromov's inequality (disambiguation)
Gromov's inequality for complex projective space
Loewner's torus inequality
Pu's inequality
Systolic geometry
References
.
Geometric inequalities
Riemannian geometry
Systolic geometry
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https://en.wikipedia.org/wiki/K-medians%20clustering
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In statistics, k-medians clustering is a cluster analysis algorithm. It is a variation of k-means clustering where instead of calculating the mean for each cluster to determine its centroid, one instead calculates the median. This has the effect of minimizing error over all clusters with respect to the 1-norm distance metric, as opposed to the squared 2-norm distance metric (which k-means does.)
This relates directly to the k-median problem with respect to the 1-norm, which is the problem of finding k centers such that the clusters formed by them are the most compact. Formally, given a set of data points x, the k centers ci are to be chosen so as to minimize the sum of the distances from each x to the nearest ci.
The criterion function formulated in this way is sometimes a better criterion than that used in the k-means clustering algorithm, in which the sum of the squared distances is used. The sum of distances is widely used in applications such as the facility location problem.
The proposed algorithm uses Lloyd-style iteration which alternates between an expectation (E) and maximization (M) step, making this an expectation–maximization algorithm. In the E step, all objects are assigned to their nearest median. In the M step, the medians are recomputed by using the median in each single dimension.
Medians and medoids
The median is computed in each single dimension in the Manhattan-distance formulation of the k-medians problem, so the individual attributes will come from the dataset (or be an average of two values from the dataset). This makes the algorithm more reliable for discrete or even binary data sets. In contrast, the use of means or Euclidean-distance medians will not necessarily yield individual attributes from the dataset. Even with the Manhattan-distance formulation, the individual attributes may come from different instances in the dataset; thus, the resulting median may not be a member of the input dataset.
This algorithm is often confused with the k-medoids algorithm. However, a medoid has to be an actual instance from the dataset, while for the multivariate Manhattan-distance median this only holds for single attribute values. The actual median can thus be a combination of multiple instances. For example, given the vectors (0,1), (1,0) and (2,2), the Manhattan-distance median is (1,1), which does not exist in the original data, and thus cannot be a medoid.
Software
ELKI includes various k-means variants, including k-medians.
FORTRAN kmedians
GNU R includes k-medians in the "flexclust" package.
Stata kmedians
See also
cluster analysis
k-means
medoid
silhouette
References
Cluster analysis algorithms
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https://en.wikipedia.org/wiki/Imaginary%20curve
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In algebraic geometry an imaginary curve is an algebraic curve which does not contain any real points.
For example, the set of pairs of complex numbers satisfying the equation forms an imaginary circle, containing points such as and but not containing any points both of whose coordinates are real.
In some cases, more generally, an algebraic curve with only finitely many real points is considered to be an imaginary curve. For instance, an imaginary line is a line (in a complex projective space) that contains only one real point.
See also
Imaginary point
Real line
Real curve
References
Projective geometry
Curves
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https://en.wikipedia.org/wiki/Ivars%20Peterson
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Ivars Peterson (born 4 December 1948) is an American mathematics writer.
Early life
Peterson received a B.Sc. in Physics and Chemistry and a B.Ed. in Education from the University of Toronto. Peterson received an M.A. in Journalism from the University of Missouri-Columbia.
Career
Peterson worked as a high school science and mathematics teacher.
Peterson has been a columnist and online editor at Science News and Science News for Kids, and has been columnist for the children's magazine Muse. He wrote the weekly online column Ivars Peterson's MathTrek. Peterson is the author of a number of popular mathematics and related books. Peterson has been a weekly mathematics columnist for MAA Online.
Peterson received the Joint Policy Board for Mathematics Communications Award in 1991 for "exceptional skill in communicating mathematics to the general public over the last decade".
For the spring 2008 semester, he accepted the Wayne G. Basler Chair of Excellence for the Integration of the Arts, Rhetoric and Science at East Tennessee State University. He gave a four lectures on how math is integral in our society and our universe. He also taught a course entitled "Communicating Mathematics".
In 2007, Peterson was named Director of Publications for Journals and Communications at the Mathematical Association of America.
Bibliography
Mathematical Treks: From Surreal Numbers to Magic Circles (2002) Mathematical Association of America
Fragments of Infinity: A Kaleidoscope of Math and Art (2000) John Wiley & Sons
The Jungles of Randomness: A Mathematical Safari (1997) John Wiley & Sons
Fatal Defect: Chasing Killer Computer Bugs (1995) Times Books
Newton's Clock: Chaos in the Solar System (1993) W.H. Freeman
Islands of Truth: Mathematical Mystery Cruise (1990) W.H.Freeman
The Mathematical Tourist: Snapshots of Modern Mathematics (1988) W.H.Freeman
References
External links
Ivars Peterson homepage - googlepages
Ivars Peterson The Mathematical Tourist - blogspot
Ivars Peterson The Mathematical Tourist - Mathematical Association of America via: archive.org
Ivars Peterson MathTrek archives Mathematical Association of America
JPBM Communications Award.
Living people
University of Toronto alumni
University of Missouri alumni
Canadian science writers
Mathematics writers
Mathematics popularizers
East Tennessee State University faculty
1948 births
Canadian mathematicians
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https://en.wikipedia.org/wiki/Numerical%20error
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In software engineering and mathematics, numerical error is the error in the numerical computations.
Types
It can be the combined effect of two kinds of error in a calculation.
the first is caused by the finite precision of computations involving floating-point or integer values
the second usually called truncation error is the difference between the exact mathematical solution and the approximate solution obtained when simplifications are made to the mathematical equations to make them more amenable to calculation. The term truncation comes from the fact that either these simplifications usually involve the truncation of an infinite series expansion so as to make the computation possible and practical, or because the least significant bits of an arithmetic operation are thrown away.
Measure
Floating-point numerical error is often measured in ULP (unit in the last place).
See also
Loss of significance
Numerical analysis
Error analysis (mathematics)
Round-off error
Kahan summation algorithm
Numerical sign problem
References
Accuracy and Stability of Numerical Algorithms, Nicholas J. Higham,
"Computational Error And Complexity In Science And Engineering", V. Lakshmikantham, S.K. Sen,
Computer arithmetic
Numerical analysis
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https://en.wikipedia.org/wiki/Filling%20area%20conjecture
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In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introducing shortcuts between its points.
Definitions and statement of the conjecture
Every smooth surface or curve in Euclidean space is a metric space, in which the (intrinsic) distance between two points of is defined as the infimum of the lengths of the curves that go from to along . For example, on a closed curve of length , for each point of the curve there is a unique other point of the curve (called the antipodal of ) at distance from .
A compact surface fills a closed curve if its border (also called boundary, denoted ) is the curve . The filling is said to be isometric if for any two points of the boundary curve , the distance between them along is the same (not less) than the distance along the boundary. In other words, to fill a curve isometrically is to fill it without introducing shortcuts.
Question: How small can be the area of a surface that isometrically fills its boundary curve, of given length?
For example, in three-dimensional Euclidean space, the circle
(of length 2) is filled by the flat disk
which is not an isometric filling, because any straight chord along it is a shortcut. In contrast, the hemisphere
is an isometric filling of the same circle , which has twice the area of the flat disk. Is this the minimum possible area?
The surface can be imagined as made of a flexible but non-stretchable material, that allows it to be moved around and bended in Euclidean space. None of these transformations modifies the area of the surface nor the length of the curves drawn on it, which are the magnitudes relevant to the problem. The surface can be removed from Euclidean space altogether, obtaining a Riemannian surface, which is an abstract smooth surface with a Riemannian metric that encodes the lengths and area. Reciprocally, according to the Nash-Kuiper theorem, any Riemannian surface with boundary can be embedded in Euclidean space preserving the lengths and area specified by the Riemannian metric. Thus the filling problem can be stated equivalently as a question about Riemannian surfaces, that are not placed in Euclidean space in any particular way.
Conjecture (Gromov's filling area conjecture, 1983): The hemisphere has minimum area among the orientable compact Riemannian surfaces that fill isometrically their boundary curve, of given length.
Gromov's proof for the case of Riemannian disks
In the same paper where Gromov stated the conjecture, he proved that
the hemisphere has least area among the Riemannian surfaces that isometrically fill a circle of given length, and are homeomorphic to a disk.
Proof: Let be a Riemannian disk that isometrically fills its boundary of length . Glue each point with its antipodal point , defined as the unique point of that is at the maximum possible distance from . Gluing in t
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https://en.wikipedia.org/wiki/Slepian%27s%20lemma
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In probability theory, Slepian's lemma (1962), named after David Slepian, is a Gaussian comparison inequality. It states that for Gaussian random variables and in satisfying ,
the following inequality holds for all real numbers :
or equivalently,
While this intuitive-seeming result is true for Gaussian processes, it is not in general true for other random variables—not even those with expectation 0.
As a corollary, if is a centered stationary Gaussian process such that for all , it holds for any real number that
History
Slepian's lemma was first proven by Slepian in 1962, and has since been used in reliability theory, extreme value theory and areas of pure probability. It has also been re-proven in several different forms.
References
Slepian, D. "The One-Sided Barrier Problem for Gaussian Noise", Bell System Technical Journal (1962), pp 463–501.
Huffer, F. "Slepian's inequality via the central limit theorem", Canadian Journal of Statistics (1986), pp 367–370.
Ledoux, M., Talagrand, M. "Probability in Banach Spaces", Springer Verlag, Berlin 1991, pp 75.
Lemmas
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https://en.wikipedia.org/wiki/Borel%20right%20process
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In the mathematical theory of probability, a Borel right process, named after Émile Borel, is a particular kind of continuous-time random process.
Let be a locally compact, separable, metric space.
We denote by the Borel subsets of .
Let be the space of right continuous maps from to that have left limits in ,
and for each , denote by the coordinate map at ; for
each , is the value of at .
We denote the universal completion of by .
For each , let
and then, let
For each Borel measurable function on , define, for each ,
Since and the mapping given by is right continuous, we see that
for any uniformly continuous function , we have the mapping given by is right continuous.
Therefore, together with the monotone class theorem, for any universally measurable function , the mapping given by , is jointly measurable, that is, measurable, and subsequently, the mapping is also -measurable for all finite measures on and on .
Here,
is the completion of
with respect
to the product measure .
Thus, for any bounded universally measurable function on ,
the mapping is Lebeague measurable, and hence,
for each , one can define
There is enough joint measurability to check that is a Markov resolvent on ,
which uniquely associated with the Markovian semigroup .
Consequently, one may apply Fubini's theorem to see that
The following are the defining properties of Borel right processes:
Hypothesis Droite 1:
For each probability measure on , there exists a probability measure on such that is a Markov process with initial measure and transition semigroup .
Hypothesis Droite 2:
Let be -excessive for the resolvent on . Then, for each probability measure on , a mapping given by is almost surely right continuous on .
Notes
References
Stochastic processes
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https://en.wikipedia.org/wiki/Minimum%20bounding%20box
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In geometry, the minimum bounding box or smallest bounding box (also known as the minimum enclosing box or smallest enclosing box) for a point set in dimensions is the box with the smallest measure (area, volume, or hypervolume in higher dimensions) within which all the points lie. When other kinds of measure are used, the minimum box is usually called accordingly, e.g., "minimum-perimeter bounding box".
The minimum bounding box of a point set is the same as the minimum bounding box of its convex hull, a fact which may be used heuristically to speed up computation.
In the two-dimensional case it is called the minimum bounding rectangle.
Axis-aligned minimum bounding box
The axis-aligned minimum bounding box (or AABB) for a given point set is its minimum bounding box subject to the constraint that the edges of the box are parallel to the (Cartesian) coordinate axes. It is the Cartesian product of N intervals each of which is defined by the minimal and maximal value of the corresponding coordinate for the points in S.
Axis-aligned minimal bounding boxes are used to an approximate location of an object in question and as a very simple descriptor of its shape. For example, in computational geometry and its applications when it is required to find intersections in the set of objects, the initial check is the intersections between their MBBs. Since it is usually a much less expensive operation than the check of the actual intersection (because it only requires comparisons of coordinates), it allows quickly excluding checks of the pairs that are far apart.
Arbitrarily oriented minimum bounding box
The arbitrarily oriented minimum bounding box is the minimum bounding box, calculated subject to no constraints as to the orientation of the result. Minimum bounding box algorithms based on the rotating calipers method can be used to find the minimum-area or minimum-perimeter bounding box of a two-dimensional convex polygon in linear time, and of a three-dimensional point set in the time it takes to construct its convex hull followed by a linear-time computation. A three-dimensional rotating calipers algorithm can find the minimum-volume arbitrarily-oriented bounding box of a three-dimensional point set in cubic time. Matlab implementations of the latter as well as the optimal compromise between accuracy and CPU time are available.
Object-oriented minimum bounding box
In the case where an object has its own local coordinate system, it can be useful to store a bounding box relative to these axes, which requires no transformation as the object's own transformation changes.
Digital image processing
In digital image processing, the bounding box is merely the coordinates of the rectangular border that fully encloses a digital image when it is placed over a page, a canvas, a screen or other similar bidimensional background.
See also
Bounding sphere
Bounding volume
Minimum bounding rectangle
Darboux integral
References
Geometry
Geometric algorithms
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https://en.wikipedia.org/wiki/Johnson%20circles
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In geometry, a set of Johnson circles comprises three circles of equal radius sharing one common point of intersection . In such a configuration the circles usually have a total of four intersections (points where at least two of them meet): the common point that they all share, and for each of the three pairs of circles one more intersection point (referred here as their 2-wise intersection). If any two of the circles happen to osculate, they only have as a common point, and it will then be considered that be their 2-wise intersection as well; if they should coincide we declare their 2-wise intersection be the point diametrically opposite . The three 2-wise intersection points define the reference triangle of the figure. The concept is named after Roger Arthur Johnson.
Properties
The centers of the Johnson circles lie on a circle of the same radius as the Johnson circles centered at . These centers form the Johnson triangle.
The circle centered at with radius , known as the anticomplementary circle is tangent to each of the Johnson circles. The three tangent points are reflections of point about the vertices of the Johnson triangle.
The points of tangency between the Johnson circles and the anticomplementary circle form another triangle, called the anticomplementary triangle of the reference triangle. It is similar to the Johnson triangle, and is homothetic by a factor 2 centered at , their common circumcenter.
Johnson's theorem: The 2-wise intersection points of the Johnson circles (vertices of the reference triangle ) lie on a circle of the same radius as the Johnson circles. This property is also well known in Romania as The 5 lei coin problem of Gheorghe Țițeica.
The reference triangle is in fact congruent to the Johnson triangle, and is homothetic to it by a factor −1.
The point is the orthocenter of the reference triangle and the circumcenter of the Johnson triangle.
The homothetic center of the Johnson triangle and the reference triangle is their common nine-point center.
Proofs
Property 1 is obvious from the definition.
Property 2 is also clear: for any circle of radius , and any point on it, the circle of radius centered at is tangent to the circle in its point opposite to ; this applies in particular to , giving the anticomplementary circle .
Property 3 in the formulation of the homothety immediately follows; the triangle of points of tangency is known as the anticomplementary triangle.
For properties 4 and 5, first observe that any two of the three Johnson circles are interchanged by the reflection in the line connecting and their 2-wise intersection (or in their common tangent at if these points should coincide), and this reflection also interchanges the two vertices of the anticomplementary triangle lying on these circles. The 2-wise intersection point therefore is the midpoint of a side of the anticomplementary triangle, and lies on the perpendicular bisector of this side. Now the midpoints of the side
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https://en.wikipedia.org/wiki/Minimum%20bounding%20box%20algorithms
|
In computational geometry, the smallest enclosing box problem is that of finding the oriented minimum bounding box enclosing a set of points. It is a type of bounding volume. "Smallest" may refer to volume, area, perimeter, etc. of the box.
It is sufficient to find the smallest enclosing box for the convex hull of the objects in question. It is straightforward to find the smallest enclosing box that has sides parallel to the coordinate axes; the difficult part of the problem is to determine the orientation of the box.
Two dimensions
For the convex polygon, a linear time algorithm for the minimum-area enclosing rectangle is known. It is based on the observation that a side of a minimum-area enclosing box must be collinear with a side of the convex polygon. It is possible to enumerate boxes of this kind in linear time with the approach called rotating calipers by
Godfried Toussaint in 1983. The same approach is applicable for finding the minimum-perimeter enclosing rectangle. A C++ implementation of the algorithm that is robust against floating point errors is available.
Three dimensions
In 1985, Joseph O'Rourke published a cubic-time algorithm to find the minimum-volume enclosing box of a 3-dimensional point set. O'Rourke's approach uses a 3-dimensional rotating calipers technique, and is based on lemmas characterizing the minimum enclosing box:
There must exist two neighbouring faces of the smallest-volume enclosing box which both contain an edge of the convex hull of the point set. This criterion is satisfied by a single convex hull edge collinear with an edge of the box, or by two distinct hull edges lying in adjacent box faces.
The other four faces need only contain a point of the convex hull. Again, the points which they contain need not be distinct: a single hull point lying in the corner of the box already satisfies three of these four criteria.
It follows in the most general case where no convex hull vertices lie in edges of the minimal enclosing box, that at least 8 convex hull points must lie within faces of the box: two endpoints of each of the two edges, and four more points, one for each of the remaining four box faces. Conversely, if the convex hull consists of 7 or fewer vertices, at least one of them must lie within an edge of the hull's minimal enclosing box.
It is also possible to approximate the minimum bounding box volume, to within any constant factor greater than one, in linear time. The algorithm for doing this involves finding an approximation to the diameter of the point set, and using a box oriented towards this diameter as an initial approximation to the minimum volume bounding box. Then, this initial bounding box is partitioned into a grid of smaller cubes, and grid points near the boundary of the convex hull of the input are used as a coreset, a small set of points whose optimum bounding box approximates the optimum bounding box of the original input. Finally, O'Rourke's algorithm is applied to find the exact
|
https://en.wikipedia.org/wiki/Continuous%20function%20%28set%20theory%29
|
In set theory, a continuous function is a sequence of ordinals such that the values assumed at limit stages are the limits (limit suprema and limit infima) of all values at previous stages. More formally, let γ be an ordinal, and be a γ-sequence of ordinals. Then s is continuous if at every limit ordinal β < γ,
and
Alternatively, if s is an increasing function then s is continuous if s: γ → range(s) is a continuous function when the sets are each equipped with the order topology. These continuous functions are often used in cofinalities and cardinal numbers.
A normal function is a function that is both continuous and increasing.
References
Thomas Jech. Set Theory, 3rd millennium ed., 2002, Springer Monographs in Mathematics,Springer,
Set theory
Ordinal numbers
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https://en.wikipedia.org/wiki/Wirtinger%20inequality%20%282-forms%29
|
For other inequalities named after Wirtinger, see Wirtinger's inequality.
In mathematics, the Wirtinger inequality, named after Wilhelm Wirtinger, is a fundamental result in complex linear algebra which relates the symplectic and volume forms of a hermitian inner product. It has important consequences in complex geometry, such as showing that the normalized exterior powers of the Kähler form of a Kähler manifold are calibrations.
Statement
Consider a real vector space with positive-definite inner product , symplectic form , and almost-complex structure , linked by for any vectors and . Then for any orthonormal vectors there is
There is equality if and only if the span of is closed under the operation of .
In the language of the comass of a form, the Wirtinger theorem (although without precision about when equality is achieved) can also be phrased as saying that the comass of the form is equal to .
Proof
In the special case , the Wirtinger inequality is a special case of the Cauchy–Schwarz inequality:
According to the equality case of the Cauchy–Schwarz inequality, equality occurs if and only if and are collinear, which is equivalent to the span of being closed under .
Let be fixed, and let denote their span. Then there is an orthonormal basis of with dual basis such that
where denotes the inclusion map from into . This implies
which in turn implies
where the inequality follows from the previously-established case. If equality holds, then according to the equality case, it must be the case that for each . This is equivalent to either or , which in either case (from the case) implies that the span of is closed under , and hence that the span of is closed under .
Finally, the dependence of the quantity
on is only on the quantity , and from the orthonormality condition on , this wedge product is well-determined up to a sign. This relates the above work with to the desired statement in terms of .
Consequences
Given a complex manifold with hermitian metric, the Wirtinger theorem immediately implies that for any -dimensional embedded submanifold , there is
where is the Kähler form of the metric. Furthermore, equality is achieved if and only if is a complex submanifold. In the special case that the hermitian metric satisfies the Kähler condition, this says that is a calibration for the underlying Riemannian metric, and that the corresponding calibrated submanifolds are the complex submanifolds of complex dimension . This says in particular that every complex submanifold of a Kähler manifold is a minimal submanifold, and is even volume-minimizing among all submanifolds in its homology class.
Using the Wirtinger inequality, these facts even extend to the more sophisticated context of currents in Kähler manifolds.
See also
Gromov's inequality for complex projective space
Systolic geometry
Notes
References
Inequalities
Differential geometry
Systolic geometry
|
https://en.wikipedia.org/wiki/Joins%20%28concurrency%20library%29
|
Joins is an asynchronous concurrent computing API (Join-pattern) from Microsoft Research for the .NET Framework. It is based on join calculus and makes the concurrency constructs of the Cω language available as a CLI assembly that any CLI compliant language can use.
Overview
Joins can be used to express concurrency in an application using the joins pattern, usable both for multi-threaded applications as well as for event based distributed applications. The Joins API emulates declarative type-safe expression of synchronization patterns.
The Joins library emulates asynchronous and synchronous methods. An asynchronous method, in Cω and Joins parlance, is one which does not block the caller method, nor does it return any result, whereas a synchronous method blocks the caller method. In the Joins API, synchronous as well as asynchronous methods are implemented as generic delegates. Usage of generics provide type safety. For example, a set of synchronous and asynchronous method can be created and using them to create an object that implements the pattern, as:
public class JoinDemo
{
public readonly Asynchronous.Channel<int> Queue;
public readonly Asynchronous.Channel<string> Send;
public readonly Synchronous<int>.Channel Retrieve;
private Join joinPattern = Join.Create();
public JoinDemo()
{
joinPattern.Initialize(out Queue);
joinPattern.Initialize(out Send);
joinPattern.Initialize(out Retrieve);
}
}
When asynchronous methods are called, the parameters are put in a channel, which is a queue managed by the Joins runtime. The method can optionally start a new thread to process the parameters in the background, and return the results. When the corresponding synchronous method is called the parameter is returned for further processing. If no parameter is present in the queue when the synchronous method is called, the caller stalls. The Joins runtime schedules which parameter is returned based on whether it is ready.
The synchronization pattern of the methods are defined by joins patterns, which describes what happens when a set of channels are invoked. For example, what happens when Send and Retrieve are called together can be different than Send and Queue.
public void SetPatterns()
{
join.When(Send).And(Retrieve).Do(s => s);
join.When(Queue).And(Retrieve).Do(n => n.ToString());
join.When(Send).And(Queue).And(Retrieve).Do(s =>
{
Send(s);
return Retrieve();
});
}
References
The Joins Concurrency Library
Joins — A Concurrency Library
External links
Joins at Microsoft Research
Concurrent programming libraries
Microsoft Research
Articles with example Java code
|
https://en.wikipedia.org/wiki/Convolution%20power
|
In mathematics, the convolution power is the n-fold iteration of the convolution with itself. Thus if is a function on Euclidean space Rd and is a natural number, then the convolution power is defined by
where ∗ denotes the convolution operation of functions on Rd and δ0 is the Dirac delta distribution. This definition makes sense if x is an integrable function (in L1), a rapidly decreasing distribution (in particular, a compactly supported distribution) or is a finite Borel measure.
If x is the distribution function of a random variable on the real line, then the nth convolution power of x gives the distribution function of the sum of n independent random variables with identical distribution x. The central limit theorem states that if x is in L1 and L2 with mean zero and variance σ2, then
where Φ is the cumulative standard normal distribution on the real line. Equivalently, tends weakly to the standard normal distribution.
In some cases, it is possible to define powers x*t for arbitrary real t > 0. If μ is a probability measure, then μ is infinitely divisible provided there exists, for each positive integer n, a probability measure μ1/n such that
That is, a measure is infinitely divisible if it is possible to define all nth roots. Not every probability measure is infinitely divisible, and a characterization of infinitely divisible measures is of central importance in the abstract theory of stochastic processes. Intuitively, a measure should be infinitely divisible provided it has a well-defined "convolution logarithm." The natural candidate for measures having such a logarithm are those of (generalized) Poisson type, given in the form
In fact, the Lévy–Khinchin theorem states that a necessary and sufficient condition for a measure to be infinitely divisible is that it must lie in the closure, with respect to the vague topology, of the class of Poisson measures .
Many applications of the convolution power rely on being able to define the analog of analytic functions as formal power series with powers replaced instead by the convolution power. Thus if is an analytic function, then one would like to be able to define
If x ∈ L1(Rd) or more generally is a finite Borel measure on Rd, then the latter series converges absolutely in norm provided that the norm of x is less than the radius of convergence of the original series defining F(z). In particular, it is possible for such measures to define the convolutional exponential
It is not generally possible to extend this definition to arbitrary distributions, although a class of distributions on which this series still converges in an appropriate weak sense is identified by .
Properties
If x is itself suitably differentiable, then from the properties of convolution, one has
where denotes the derivative operator. Specifically, this holds if x is a compactly supported distribution or lies in the Sobolev space W1,1 to ensure that the derivative is sufficiently regular for the convo
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https://en.wikipedia.org/wiki/Mpstat
|
mpstat is a computer command-line software used in Unix-type operating systems to report (on the screen) processor-related statistics. It is used in computer monitoring in order to diagnose problems or to build statistics about a computer's CPU usage.
Description
The mpstat command writes to standard output activities for each available processor.
The mpstat command can be used both on SMP and UP machines, but in the latter, only global average activities will be printed.
Usage
$ mpstat <interval> <count>
Interval is the time in seconds between printing out a line of statistics. Count is the number of lines of output you want.
Note that the first line of output from mpstat (like iostat, vmstat, etc.) contains averages since system boot. The subsequent lines will show current values.
Examples
Different examples of output under different operating systems:
under Linux kernel 4.14 on a two CPU machine:
Linux 4.14.24.mptcp (hostname) 05/23/2018 _x86_64_ (2 CPU)
03:51:19 PM CPU %usr %nice %sys %iowait %irq %soft %steal %guest %gnice %idle
03:51:20 PM all 2.51 0.00 2.01 0.00 0.00 0.00 0.00 0.00 0.00 95.48
03:51:21 PM all 2.53 0.00 2.02 0.00 0.00 0.00 0.00 0.00 0.00 95.45
under Linux kernel 2.4:
$ mpstat
Linux 2.4.21-32.ELsmp (linux00) 07/04/07
10:26:54 CPU %user %nice %system %iowait %irq %soft %idle intr/s
10:26:54 all 0.07 0.00 0.16 8.48 0.00 0.09 91.18 165.49
under Solaris 11:
$ mpstat
CPU minf mjf xcal intr ithr csw icsw migr smtx srw syscl usr sys wt idl
0 0 0 0 329 121 169 6 0 0 0 406 0 1 0 98
under AIX 6:
$ mpstat 1 1
System configuration: lcpu=8 ent=1.0 mode=Uncapped
cpu min maj mpc int cs ics rq mig lpa sysc us sy wa id pc %ec lcs
0 8 0 0 182 336 102 0 0 100 1434 38 51 0 12 0.02 1.8 185
1 0 0 0 11 5 5 0 0 - 0 0 19 0 81 0.00 0.1 12
2 0 0 0 1 0 0 0 0 - 0 0 42 0 58 0.00 0.0 0
3 0 0 0 1 0 0 0 0 - 0 0 43 0 57 0.00 0.0 0
4 0 0 0 1 0 0 0 0 - 0 0 45 0 55 0.00 0.0 0
5 0 0 0 1 0 0 0 0 - 0 0 44 0 56 0.00 0.0 0
6 0 0 0 1 0 0 0 0 - 0 0 2 0 98 0.00 0.0 0
7 0 0 0 53 5 5 0 0 - 0 0 66 0 34 0.00 0.2 54
U - - - - - - - - - - - - 0 99 0.99 99.0 -
ALL 8 0 0 251 346 112 0 0 100 1434 0 0 0 99 0.02 2.0 251
See also
nmon
top
References
Linux mpstat man page
AIX mpstat manual page
External links
sysstat - includes mpstat
Unix software
|
https://en.wikipedia.org/wiki/Laurens%20de%20Haan
|
Laurens de Haan (born 15 January 1937) is a Dutch economist and Emeritus Professor of Probability and Mathematical Statistics at the Erasmus University Rotterdam, specializing in extreme value theory.
Biography
Born in Rotterdam, De Haan received his MA in mathematics from the University of Amsterdam in 1966, and his PhD in mathematics in 1970 under supervision of Johannes Runnenburg for the thesis "On regular variation and sample extremes".
De Haan started his academic career in 1966 as researcher in probability and statistics at the Mathematisch Centrum, Amsterdam. In the year 1971–72 he was visiting assistant professor at Stanford University. In 1977 he was appointed professor of probability and mathematical statistics at the Erasmus Universiteit, where he stayed until his retirement in 1998. From 1990 to 1992 he was associate dean of the school of economics. From 2008 to 2011 he was part-time professor of statistics at the University of Tilburg.
In 1977 he was elected Fellow of the Institute of Mathematical Statistics (I.M.S.), and he was guest professor at Peking University in 1994. He was awarded a Doctor honoris causa from the Universidade de Lisboa in 2000 and the Medallion lecture at the I.M.S. annual meeting in Gothenburg in 2000.
Work
Overschrijdingslijnen project
The "Overschrijdingslijnen" was a research project based on extreme-value analysis, meant to provide new standards for the Dutch sea defenses. It was commissioned by the Ministry of Public Works, starting in 1984 and finished in 1992.
In this joint project people participated from "Rijkswaterstaat", the Dutch government agency overseeing the sea defenses; the Royal Netherlands Meteorological service; and the CWI Centre for Mathematics and Computer Science in Amsterdam.
Neptune
Neptune was a larger scale but similar project, sponsored by the European Union via the MAST program and in cooperation with BMT Port & Coastal Limited, Delft Hydraulics, Rijkswaterstaat, GKSS-Forschungszentrum Geest-hacht GmbH, University of Lancaster and University of East Anglia. Novel aspects are: firstly the wide-ranging set-up starting from climatological data going down to the water levels and movements near the British and Dutch coasts and secondly the higher-dimensional statistical set-up in the extreme-value analysis (1995-1997). Finished March 1997.
Other projects
Other projects were initiated, such as:
NATO collaborative research grant (1991-1995) with Sidney Resnick, Cornell University.
European Union grant "Training through research" (cat. 40) in the "Training and Mobility of Researchers" program. University of Lisbon, January through June 1997.
Extreme interest rates, a project for ING insurance company; jointly with H. Drees, Heidelberg (1999-2000).
Publications
De Haan published multiple books. A selection:
1970. On regular variation and its application to the weak convergence of sample extremes. Mathematical Centre tracts
1979. A simple asymptotic estimate for the
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https://en.wikipedia.org/wiki/Husan
|
Husan () is a Palestinian town located west of Bethlehem, in the Bethlehem Governorate. According to the Palestinian Central Bureau of Statistics, Husan had a population of 7,048 in 2017.
Husan is located in the Seam Zone of the Israeli-occupied West Bank, such that it is separated from the rest of the West Bank by the Israeli West Bank barrier.
Etymology
In Arabic, husan means "goodness and beauty". The name may also be derived from Hassan monastery, which later became Husan. In 1881, Edward Henry Palmer wrote that it came from "hovering round".
History
The oldest remains found in the village date back to the Iron Age. Other remains date from the post-Babylonian captivity period and the Middle Ages. The original inhabitants came from the Arabian Peninsula and Yemen in the 3rd century. Ceramics from the Byzantine era have been found here.
Ottoman era
Husan, like the rest of Palestine, was incorporated into the Ottoman Empire in 1517, and in the census of 1596, the village appeared in the tax registers as being in the Nahiya of Quds of the Liwa of Quds. It had an all-Muslim population of 12 households and paid taxes on wheat, barley, summer crops, vineyards and fruit trees, vegetable and fruit garden, orchard, occasional revenues, goats and/or beehives.
In 1838 it was noted as a Muslim village in the District of el-'Arkub; Southwest of Jerusalem.
When Victor Guérin first passed by Husan in June 1863, he noted that it was located on a mountain. Later, in August the same year, he found that it contained only a small number of people, with houses grouped around a tower.
An Ottoman village list from about 1870 found that Husan had a population of 115, in 28 houses, though the population count included men only. Husan was listed as a plain village in the Hebron District to the West of Solomon's Pools.
In 1883, the PEF's Survey of Western Palestine described Hausan as a small stone village on a flat ridge, with a steep valley to the north and a well to the south.
In 1896 the population of Husan was estimated to be about 258 persons.
British Mandate era
In the 1922 census of Palestine, conducted by the British Mandate authorities, Husan had an all-Muslim population of 396, 219 males and 177 females. This had increased in the 1931 census to a total of 540, still all-Muslim, in 122 houses.
In the 1945 statistics the population of Husan was 770, all Muslims, who owned of land according to an official land and population survey. Of this, were for cereals, while were built-up (urban) land.
Jordanian era
In the wake of the 1948 Arab–Israeli War, and after the 1949 Armistice Agreements, Husan came under Jordanian rule.
In 1961, the population of Husan was 1,073.
Post-1967
Since the Six-Day War in 1967, Husan has been under Israeli occupation. As a result of the restrictions following the occupation, many people in Husan have become unemployed, and the economic situation has severely declined, forcing many to work in the service sector
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https://en.wikipedia.org/wiki/Statistical%20benchmarking
|
In statistics, benchmarking is a method of using auxiliary information to adjust the sampling weights used in an estimation process, in order to yield more accurate estimates of totals.
Suppose we have a population where each unit has a "value" associated with it. For example, could be a wage of an employee , or the cost of an item . Suppose we want to estimate the sum of all the . So we take a sample of the , get a sampling weight W(k) for all sampled , and then sum up for all sampled .
One property usually common to the weights described here is that if we sum them over all sampled , then this sum is an estimate of the total number of units in the population (for example, the total employment, or the total number of items). Because we have a sample, this estimate of the total number of units in the population will differ from the true population total. Similarly, the estimate of total (where we sum for all sampled ) will also differ from true population total.
We do not know what the true population total value is (if we did, there would be no point in sampling!). Yet often we do know what the sum of the are over all units in the population. For example, we may not know the total earnings of the population or the total cost of the population, but often we know the total employment or total volume of sales. And even if we don't know these exactly, there often are surveys done by other organizations or at earlier times, with very accurate estimates of these auxiliary quantities. One important function of a population census is to provide data that can be used for benchmarking smaller surveys.
The benchmarking procedure begins by first breaking the population into benchmarking cells. Cells are formed by grouping units together that share common characteristics, for example, similar , yet anything can be used that enhances the accuracy of the final estimates. For each cell , we let be the sum of all , where the sum is taken over all sampled in the cell . For each cell , we let be the auxiliary value for cell , which is commonly called the "benchmark target" for cell . Next, we compute a benchmark factor . Then, we adjust all weights by multiplying it by its benchmark factor , for its cell . The net result is that the estimated [formed by summing ] will now equal the benchmark target total . But the more important benefit is that the estimate of the total of [formed by summing ] will tend to be more accurate.
Relationship to stratified sampling
Benchmarking is sometimes referred to as 'post-stratification' because of its similarities to stratified sampling. The difference between the two is that in stratified sampling, we decide in advance how many units will be sampled from each stratum (equivalent to benchmarking cells); in benchmarking, we select units from the broader population, and the number chosen from each cell is a matter of chance.
The advantage of stratified sampling is that the sample numbers in each stratum can b
|
https://en.wikipedia.org/wiki/Herbert%20Ginsburg
|
Herbert P. Ginsburg is Jacob H. Schiff Foundation Professor of Psychology & Education at Teachers College, Columbia University.
He is a leading interpreter of children's understanding of mathematics, with research and teaching interests in intellectual development, mathematics education, and testing and assessment. He is a co-author of the Big Math for Little Kids curriculum for prekindergarten and kindergarten.
He received a Ph.D. and an M.S. in Developmental Psychology from the University of North Carolina, and a B.A. from Harvard University.
External links
Faculty Emeritus Bio
Big Math for Little Kids
Teachers College, Columbia University faculty
Living people
Harvard University alumni
University of North Carolina at Chapel Hill alumni
Year of birth missing (living people)
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https://en.wikipedia.org/wiki/KKM
|
KKM may refer to:
Knaster–Kuratowski–Mazurkiewicz lemma, in mathematics
Kyo Kara Maoh!, a series of Japanese light novels
Kola Kolaya Mundhirika, a 2010 Tamil comedy film
KKM, a song by the music group Miracle Legion from their 1996 album Portrait of a Damaged Family
|
https://en.wikipedia.org/wiki/LOP
|
LOP may refer to:
Landscape of practice in social science
Language oriented programming
Legion of Pain, a professional wrestling tag team
C2.LOP, Windows malware
Line of position in geometry and navigation
Local operational picture
Lombok International Airport (IATA code)
Long Ping station (Hong Kong MTR station code)
See also
Lop (disambiguation)
|
https://en.wikipedia.org/wiki/SAT%20Subject%20Test%20in%20Mathematics%20Level%201
|
The SAT Subject Test in Mathematics Level 1 (formerly known as Math I or MathIC (the "C" representing the use of a calculator)) was the name of a one-hour multiple choice test given on algebra, geometry, basic trigonometry, algebraic functions, elementary statistics and basic foundations of calculus by The College Board. A student chose whether to take the test depending upon college entrance requirements for the schools in which the student is planning to apply. Until 1994, the SAT Subject Tests were known as Achievement Tests; and from 1995 until January 2005, they were known as SAT IIs. Mathematics Level 1 was taken 109,048 times in 2006. The SAT Subject Test in Mathematics Level 2 covered more advanced content.
On January 19, 2021, the College Board discontinued all SAT Subject tests, including the SAT Subject Test in Mathematics Level 1. This was effective immediately in the United States, and the tests were to be phased out by the following summer for international students. This was done as a response to changes in college admissions due to the impact of the COVID-19 pandemic on education.
Format
The test had 50 multiple choice questions that were to be answered in one hour. All questions had five answer choices. Students received 1 point for every correct answer, lost ¼ of a point for each incorrect answer, and received 0 points for questions left blank.
The questions covered a broad range of topics. Approximately 10-14% of questions focused on Numbers and Operations, 38-42% focused on Algebra and functions, 38-42% focused on Geometry (including Euclidean, coordinate, three-dimensional, and trigonometry), and 6-10% focused on Data analysis, Statistics, and probability.
Calculator Use
The College Board stated that a calculator "may be useful or necessary" for about 40-50% of the questions on the test. The College Board also encouraged the use of a graphing calculator over a scientific calculator. It also said that this test was "developed with the expectation that most students are using graphing calculators."
Students were not permitted to use calculators on the Mathematics Level 1 test that have a QWERTY format keyboard, that require an electrical outlet, that make noise, use paper tape, that have non-traditional methods of input (such as a stylus), or those that are part of a communication device (such as PDA's, laptops, or cell phones). Calculators that had slanted screens or large displays (numbers 1 in. or more) are seated at the discretion of the test supervisor.
Preparation
The College Board suggested as preparation for the test three years of mathematics, including two years of algebra, and one year of geometry.
Scoring
For each of the 50 multiple choice questions, students received 1 point for every correct answer, lost ¼ of a point for each incorrect answer, and received 0 points for questions left blank. This created a raw score, which was then converted into a scaled score. The conversion between these numbers
|
https://en.wikipedia.org/wiki/Uniform%20tiling
|
In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.
Uniform tilings can exist in both the Euclidean plane and hyperbolic plane. Uniform tilings are related to the finite uniform polyhedra which can be considered uniform tilings of the sphere.
Most uniform tilings can be made from a Wythoff construction starting with a symmetry group and a singular generator point inside of the fundamental domain. A planar symmetry group has a polygonal fundamental domain and can be represented by the group name represented by the order of the mirrors in sequential vertices.
A fundamental domain triangle is (p q r), and a right triangle (p q 2), where p, q, r are whole numbers greater than 1. The triangle may exist as a spherical triangle, a Euclidean plane triangle, or a hyperbolic plane triangle, depending on the values of p, q and r.
There are a number of symbolic schemes for naming these figures, from a modified Schläfli symbol for right triangle domains: (p q 2) → {p, q}. The Coxeter-Dynkin diagram is a triangular graph with p, q, r labeled on the edges. If r = 2, the graph is linear since order-2 domain nodes generate no reflections. The Wythoff symbol takes the 3 integers and separates them by a vertical bar (|). If the generator point is off the mirror opposite a domain node, it is given before the bar.
Finally tilings can be described by their vertex configuration, the sequence of polygons around each vertex.
All uniform tilings can be constructed from various operations applied to regular tilings. These operations as named by Norman Johnson are called truncation (cutting vertices), rectification (cutting vertices until edges disappear), and cantellation (cutting edges). Omnitruncation is an operation that combines truncation and cantellation. Snubbing is an operation of alternate truncation of the omnitruncated form. (See Uniform polyhedron#Wythoff construction operators for more details.)
Coxeter groups
Coxeter groups for the plane define the Wythoff construction and can be represented by Coxeter-Dynkin diagrams:
For groups with whole number orders, including:
Uniform tilings of the Euclidean plane
There are symmetry groups on the Euclidean plane constructed from fundamental triangles: (4 4 2), (6 3 2), and (3 3 3). Each is represented by a set of lines of reflection that divide the plane into fundamental triangles.
These symmetry groups create 3 regular tilings, and 7 semiregular ones. A number of the semiregular tilings are repeated from different symmetry constructors.
A prismatic symmetry group represented by (2 2 2 2) represents by two sets of parallel mirrors, which in general can have a rectangular fundamental domain. It generates no new tilings.
A further prismatic symmetry group represented by (∞ 2 2) which has an infinite fundamental domain. It constructs two uniform tilings, the apeirogonal prism and apeirogonal antiprism.
The stacking of the f
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https://en.wikipedia.org/wiki/Dynkin%27s%20formula
|
In mathematics — specifically, in stochastic analysis — Dynkin's formula is a theorem giving the expected value of any suitably smooth statistic of an Itō diffusion at a stopping time. It may be seen as a stochastic generalization of the (second) fundamental theorem of calculus. It is named after the Russian mathematician Eugene Dynkin.
Statement of the theorem
Let X be the Rn-valued Itō diffusion solving the stochastic differential equation
For a point x ∈ Rn, let Px denote the law of X given initial datum X0 = x, and let Ex denote expectation with respect to Px.
Let A be the infinitesimal generator of X, defined by its action on compactly-supported C2 (twice differentiable with continuous second derivative) functions f : Rn → R as
or, equivalently,
Let τ be a stopping time with Ex[τ] < +∞, and let f be C2 with compact support. Then Dynkin's formula holds:
In fact, if τ is the first exit time for a bounded set B ⊂ Rn with Ex[τ] < +∞, then Dynkin's formula holds for all C2 functions f, without the assumption of compact support.
Example
Dynkin's formula can be used to find the expected first exit time τK of Brownian motion B from the closed ball
which, when B starts at a point a in the interior of K, is given by
Choose an integer j. The strategy is to apply Dynkin's formula with X = B, τ = σj = min(j, τK), and a compactly-supported C2 f with f(x) = |x|2 on K. The generator of Brownian motion is Δ/2, where Δ denotes the Laplacian operator. Therefore, by Dynkin's formula,
Hence, for any j,
Now let j → +∞ to conclude that τK = limj→+∞σj < +∞ almost surely and
as claimed.
References
(See Vol. I, p. 133)
(See Section 7.4)
Stochastic differential equations
Probability theorems
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https://en.wikipedia.org/wiki/Tanaka%20equation
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In mathematics, Tanaka's equation is an example of a stochastic differential equation which admits a weak solution but has no strong solution. It is named after the Japanese mathematician Hiroshi Tanaka (Tanaka Hiroshi).
Tanaka's equation is the one-dimensional stochastic differential equation
driven by canonical Brownian motion B, with initial condition X0 = 0, where sgn denotes the sign function
(Note the unconventional value for sgn(0).) The signum function does not satisfy the Lipschitz continuity condition required for the usual theorems guaranteeing existence and uniqueness of strong solutions. The Tanaka equation has no strong solution, i.e. one for which the version B of Brownian motion is given in advance and the solution X is adapted to the filtration generated by B and the initial conditions. However, the Tanaka equation does have a weak solution, one for which the process X and version of Brownian motion are both specified as part of the solution, rather than the Brownian motion being given a priori. In this case, simply choose X to be any Brownian motion and define by
i.e.
Hence,
and so X is a weak solution of the Tanaka equation. Furthermore, this solution is weakly unique, i.e. any other weak solution must have the same law.
Another counterexample of this type is Tsirelson's stochastic differential equation.
References
(Example 5.3.2)
Stochastic differential equations
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https://en.wikipedia.org/wiki/It%C3%B4%20diffusion
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In mathematics – specifically, in stochastic analysis – an Itô diffusion is a solution to a specific type of stochastic differential equation. That equation is similar to the Langevin equation used in physics to describe the Brownian motion of a particle subjected to a potential in a viscous fluid. Itô diffusions are named after the Japanese mathematician Kiyosi Itô.
Overview
A (time-homogeneous) Itô diffusion in n-dimensional Euclidean space Rn is a process X : [0, +∞) × Ω → Rn defined on a probability space (Ω, Σ, P) and satisfying a stochastic differential equation of the form
where B is an m-dimensional Brownian motion and b : Rn → Rn and σ : Rn → Rn×m satisfy the usual Lipschitz continuity condition
for some constant C and all x, y ∈ Rn; this condition ensures the existence of a unique strong solution X to the stochastic differential equation given above. The vector field b is known as the drift coefficient of X; the matrix field σ is known as the diffusion coefficient of X. It is important to note that b and σ do not depend upon time; if they were to depend upon time, X would be referred to only as an Itô process, not a diffusion. Itô diffusions have a number of nice properties, which include
sample and Feller continuity;
the Markov property;
the strong Markov property;
the existence of an infinitesimal generator;
the existence of a characteristic operator;
Dynkin's formula.
In particular, an Itô diffusion is a continuous, strongly Markovian process such that the domain of its characteristic operator includes all twice-continuously differentiable functions, so it is a diffusion in the sense defined by Dynkin (1965).
Continuity
Sample continuity
An Itô diffusion X is a sample continuous process, i.e., for almost all realisations Bt(ω) of the noise, Xt(ω) is a continuous function of the time parameter, t. More accurately, there is a "continuous version" of X, a continuous process Y so that
This follows from the standard existence and uniqueness theory for strong solutions of stochastic differential equations.
Feller continuity
In addition to being (sample) continuous, an Itô diffusion X satisfies the stronger requirement to be a Feller-continuous process.
For a point x ∈ Rn, let Px denote the law of X given initial datum X0 = x, and let Ex denote expectation with respect to Px.
Let f : Rn → R be a Borel-measurable function that is bounded below and define, for fixed t ≥ 0, u : Rn → R by
Lower semi-continuity: if f is lower semi-continuous, then u is lower semi-continuous.
Feller continuity: if f is bounded and continuous, then u is continuous.
The behaviour of the function u above when the time t is varied is addressed by the Kolmogorov backward equation, the Fokker–Planck equation, etc. (See below.)
The Markov property
The Markov property
An Itô diffusion X has the important property of being Markovian: the future behaviour of X, given what has happened up to some time t, is the same as if the process had been starte
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https://en.wikipedia.org/wiki/Maths%20and%20Social%20Sciences%20Building
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The Maths and Social Sciences Building is a high-rise tower in Manchester, England. It was part of the University of Manchester Institute of Science and Technology (UMIST) until that university merged with the Victoria University of Manchester, to form the University of Manchester, in 2004. It was vacated by the university in 2010 but is currently in use by the School of Materials while waiting for a new building to be constructed.
The MSS Building was built in 1969, as part of the UMIST campus. Constructed from reinforced concrete and designed by architects Cruikshank and Seward, it has fifteen stories and an overall height of , making it the tallest building on the former UMIST campus. Unlike many examples of Brutalist architecture on university campuses of that period, the building deviates from a purely cuboid outline with decorative towers at either end (now used as convenient locations for mobile phone antennae) and the floors up to the 10th being larger, which also breaks up the outline. The building was used largely for staff offices, with some teaching rooms. The 10th to 14th floors (called floors M–Q) accommodated the Department of Mathematics. The University of Manchester Regional Computer Centre (UMRCC) was based on J floor. The "Social Sciences" in the building's name indicates that the building once housed the Management Department, but in recent years the Department of Computation occupied the lower floors of the building. They were to become the School of Informatics in the new university and have since been split between the Schools of Computer Science and Manchester Business School. A two-floor annex to the MSS building connected to the ground floor houses tiered lecture theatres.
It was built on the site of cramped terraced housing that accommodated factory workers that was studied by Friedrich Engels in his book The Condition of the Working Class in England in 1844.
The new, merged University of Manchester announced in June 2007 that it plans to sell the Mathematics and Social Sciences Building. In July 2007, School of Mathematics relocated from MSS as well from the Ferranti building and the temporary buildings Newman and Lamb, to the new purpose-designed Alan Turing Building. Later in 2007, the staff of the former School of Informatics relocated, some of them to the Lamb building vacated by the mathematicians.
As of 2015, the building houses the Materials Science department, recently relocated from the old Materials Science Building, awaiting demolition.
External links
Buildings at the University of Manchester
Buildings and structures completed in 1968
Brutalist architecture in England
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https://en.wikipedia.org/wiki/Harmonic%20measure
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In mathematics, especially potential theory, harmonic measure is a concept related to the theory of harmonic functions that arises from the solution of the classical Dirichlet problem. In probability theory, the harmonic measure of a subset of the boundary of a bounded domain in Euclidean space , is the probability that a Brownian motion started inside a domain hits that subset of the boundary. More generally, harmonic measure of an Itō diffusion X describes the distribution of X as it hits the boundary of D. In the complex plane, harmonic measure can be used to estimate the modulus of an analytic function inside a domain D given bounds on the modulus on the boundary of the domain; a special case of this principle is Hadamard's three-circle theorem. On simply connected planar domains, there is a close connection between harmonic measure and the theory of conformal maps.
The term harmonic measure was introduced by Rolf Nevanlinna in 1928 for planar domains, although Nevanlinna notes the idea appeared implicitly in earlier work by Johansson, F. Riesz, M. Riesz, Carleman, Ostrowski and Julia (original order cited). The connection between harmonic measure and Brownian motion was first identified by Kakutani ten years later in 1944.
Definition
Let D be a bounded, open domain in n-dimensional Euclidean space Rn, n ≥ 2, and let ∂D denote the boundary of D. Any continuous function f : ∂D → R determines a unique harmonic function Hf that solves the Dirichlet problem
If a point x ∈ D is fixed, by the Riesz–Markov–Kakutani representation theorem and the maximum principle Hf(x) determines a probability measure ω(x, D) on ∂D by
The measure ω(x, D) is called the harmonic measure (of the domain D with pole at x).
Properties
For any Borel subset E of ∂D, the harmonic measure ω(x, D)(E) is equal to the value at x of the solution to the Dirichlet problem with boundary data equal to the indicator function of E.
For fixed D and E ⊆ ∂D, ω(x, D)(E) is a harmonic function of x ∈ D and
Hence, for each x and D, ω(x, D) is a probability measure on ∂D.
If ω(x, D)(E) = 0 at even a single point x of D, then is identically zero, in which case E is said to be a set of harmonic measure zero. This is a consequence of Harnack's inequality.
Since explicit formulas for harmonic measure are not typically available, we are interested in determining conditions which guarantee a set has harmonic measure zero.
F. and M. Riesz Theorem: If is a simply connected planar domain bounded by a rectifiable curve (i.e. if ), then harmonic measure is mutually absolutely continuous with respect to arc length: for all , if and only if .
Makarov's theorem: Let be a simply connected planar domain. If and for some , then . Moreover, harmonic measure on D is mutually singular with respect to t-dimensional Hausdorff measure for all t > 1.
Dahlberg's theorem: If is a bounded Lipschitz domain, then harmonic measure and (n − 1)-dimensional Hausdorff measure are mutually absolute
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https://en.wikipedia.org/wiki/Increment%20theorem
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In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function is differentiable at and that is infinitesimal. Then
for some infinitesimal , where
If then we may write
which implies that , or in other words that is infinitely close to , or is the standard part of .
A similar theorem exists in standard Calculus. Again assume that is differentiable, but now let be a nonzero standard real number. Then the same equation
holds with the same definition of , but instead of being infinitesimal, we have
(treating and as given so that is a function of alone).
See also
Nonstandard calculus
Elementary Calculus: An Infinitesimal Approach
Abraham Robinson
Taylor's theorem
References
Howard Jerome Keisler: Elementary Calculus: An Infinitesimal Approach. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html
Theorems in calculus
Nonstandard analysis
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https://en.wikipedia.org/wiki/Elvis%20Kokalovi%C4%87
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Elvis Kokalović (born 17 July 1988) is a Croatian football defender who currently plays for NK Novigrad.
Club career
In February 2018, Kokalović joined NK Novigrad.
Club statistics
1 Including Polish Super Cup.
References
External links
1988 births
Living people
Sportspeople from Novo Mesto
Men's association football defenders
Croatian men's footballers
Croatia men's youth international footballers
Croatia men's under-21 international footballers
NK Slaven Belupo players
NK Koprivnica players
Konyaspor footballers
Kardemir Karabükspor footballers
Lech Poznań players
NK Novigrad players
Croatian Football League players
Süper Lig players
TFF First League players
Croatian expatriate men's footballers
Expatriate men's footballers in Turkey
Croatian expatriate sportspeople in Turkey
Expatriate men's footballers in Poland
Croatian expatriate sportspeople in Poland
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https://en.wikipedia.org/wiki/The%20Contortionist%27s%20Handbook
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The Contortionist's Handbook is the debut novel by American novelist Craig Clevenger.
Plot summary
John Dolan Vincent is a talented young forger with a proclivity for mathematics and drug addiction. In the face of his impending institutionalization, he continually reinvents himself to escape the legal and mental health authorities and to save himself from a life of incarceration. But running turns out to be costly. Vincent's clients in the L.A. underworld lose patience, the hospital evaluator may not be fooled by his story, and the only person in as much danger as himself is the woman who knows his real name.
Characters
John Dolan Vincent
Daniel John Fletcher:
Brian Delvine: The alias Vincent used while living in Los Angeles. He then became Martin Kelly to cover up traffic warrants, an eviction, and a drug screening.
Martin Kelly: "Born to" Liam and Fiona Kelly, a deceased couple that Vincent had picked out of The Boston Globe.
Paul MacIntyre: The identity Vincent created for the son of a stripper that had left and taken her son to live in Virginia before all of the paperwork could get pushed through.
Keara/Molly Wheeler
"The Evaluator"
Jeremy
Shelly
Mom
Dad
Officer Durrell
Brett A childhood neighbor to Vincent, first seen meticulously cutting the front lawn while being berated by his mother. Eventually he is found to be visiting the same psychiatric doctor's office that the young John Dolan Vincent went to for treatment of his headaches. Brett is later pacing his front yard sans a lawnmower, being used as a foil to highlight the direction that John could have gone.
Brett's Mom
Dr. Gaines
Dr. Fred Smith
"The Prosecutor"
"The Defense Attorney"
Erica
Dr. Carlisle
Andrea The sister of Keara's that moved to San Diego from the East Coast for graduate school. Keara visits her while John makes a new home and identities for he and Keara. Andrea also serves as the source of an argument between John and Keara, Keara clearly jealous of her sister (possibly alluding to a prior tryst between Andrea and one of Keara's lovers).
Sudden The girl that turned John Dolan Vincent onto cocaine. She was a stripper that choked Vincent until he was unconscious while giving him a lap dance and she then stole his money. Three nights later they slept together.
Jimmy The doorman of the nightclub where Sudden danced. He was Sudden's cocaine hookup, who then became Vincent's hookup, who then introduced him to Ray, Jimmy's hookup. The employers of the club became the source of Vincent's mob connections, as Vincent made social security numbers, passports, etc. for them. Jimmy insisted that Vincent meet with more people to work for, but when Vincent became Paul MacIntyre, Vincent decided to disappear instead.
Sharon Worked with Sudden at the nightclub Vincent frequented. She needed her life to get cleaned up in order to keep her son that the State was threatening to take, so Vincent gave her and her son Paul new identities. Sharon t
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https://en.wikipedia.org/wiki/Marcel%20Berger
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Marcel Berger (14 April 1927 – 15 October 2016) was a French mathematician, doyen of French differential geometry, and a former director of the Institut des Hautes Études Scientifiques (IHÉS), France. Formerly residing in Le Castera in Lasseube, Berger was instrumental in Mikhail Gromov's accepting positions both at the University of Paris and at the IHÉS.
Awards and honors
1956 Prix Peccot, Collège de France
1962 Prix Maurice Audin
1969 Prix Carrière, Académie des Sciences
1978 Prix Leconte, Académie des Sciences
1979 Prix Gaston Julia
1979–1980 President of the French Mathematical Society.
1991 Lester R. Ford Award
Selected publications
Berger, M.: Geometry revealed. Springer, 2010.
Berger, M.: What is... a Systole? Notices of the AMS 55 (2008), no. 3, 374–376. online text
Berger, Marcel; Gauduchon, Paul; Mazet, Edmond: Le spectre d'une variété riemannienne. (French) Lecture Notes in Mathematics, Vol. 194 Springer-Verlag, Berlin-New York 1971.
Berger, Marcel: Sur les groupes d'holonomie homogène des variétés à connexion affine et des variétés riemanniennes. (French) Bull. Soc. Math. France 83 (1955), 279–330.
Berger, Marcel: Les espaces symétriques noncompacts. (French) Ann. Sci. École Norm. Sup. (3) 74 1957 85–177.
Berger, Marcel; Gostiaux, Bernard: Differential geometry: manifolds, curves, and surfaces. Translated from the French by Silvio Levy. Graduate Texts in Mathematics, 115. Springer-Verlag, New York, 1988. xii+474 pp.
Berger, Marcel: Geometry. II. Translated from the French by M. Cole and S. Levy. Universitext. Springer-Verlag, Berlin, 1987.
Berger, M.: Les variétés riemanniennes homogènes normales simplement connexes à courbure strictement positive. (French) Ann. Scuola Norm. Sup. Pisa (3) 15 1961 179–246.
Berger, Marcel: Geometry. I. Translated from the French by M. Cole and S. Levy. Universitext. Springer-Verlag, Berlin, 1987. xiv+428 pp.
Berger, Marcel: Systoles et applications selon Gromov. (French) [Systoles and their applications according to Gromov] Séminaire Bourbaki, Vol. 1992/93. Astérisque No. 216 (1993), Exp. No. 771, 5, 279–310.
Berger, Marcel: Geometry. I. Translated from the 1977 French original by M. Cole and S. Levy. Corrected reprint of the 1987 translation. Universitext. Springer-Verlag, Berlin, 1994. xiv+427 pp.
Berger, Marcel: Riemannian geometry during the second half of the twentieth century. Reprint of the 1998 original. University Lecture Series, 17. American Mathematical Society, Providence, Rhode Island, 2000. x+182 pp.
Besse, A.L.: Einstein Manifolds. Springer-Verlag, Berlin, 1987.
See also
Arthur Besse
Berger's inequality for Einstein manifolds
Berger–Kazdan comparison theorem
Musical isomorphism
Parametrix
Quaternion-Kähler manifold
Spin(7)-manifold
Symmetric space
Systolic geometry
References
Further reading
Claude LeBrun (Editor and Translator). "Marcel Berger Remembered", Notices of the American Mathematical Society, December 2017, Volume 64, Number 11, pp. 1285–1295.
Externa
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https://en.wikipedia.org/wiki/Ivan%20Zhegalkin
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Ivan Ivanovich Zhegalkin (; 3 August 1869, Mtsensk – 28 March 1947, Moscow) was a Russian and Soviet mathematician. He is best known for his formulation of Boolean algebra as the theory of the ring of integers mod 2, via what are now called Zhegalkin polynomials.
Zhegalkin was professor of mathematics at Moscow State University. He helped found the thriving mathematical logic group there, which became the Department of Mathematical Logic established by Sofya Yanovskaya in 1959. Reminiscing on his student days, Nikolai Luzin recalls Zhegalkin as the only professor he was not afraid of.
References
(NB. German translation of булевы алгебры, 1969.)
(NB. Circulation: 1000.)
External links
http://letopis.msu.ru/peoples/923
1869 births
1947 deaths
Mathematicians from the Russian Empire
People from Mtsensk
Recipients of the Order of the Red Banner of Labour
Soviet logicians
Soviet mathematicians
Burials at Vagankovo Cemetery
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https://en.wikipedia.org/wiki/Zhegalkin%20polynomial
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Zhegalkin (also Žegalkin, Gégalkine or Shegalkin) polynomials (), also known as algebraic normal form, are a representation of functions in Boolean algebra. Introduced by the Russian mathematician Ivan Ivanovich Zhegalkin in 1927, they are the polynomial ring over the integers modulo 2. The resulting degeneracies of modular arithmetic result in Zhegalkin polynomials being simpler than ordinary polynomials, requiring neither coefficients nor exponents. Coefficients are redundant because 1 is the only nonzero coefficient. Exponents are redundant because in arithmetic mod 2, x2 = x. Hence a polynomial such as 3x2y5z is congruent to, and can therefore be rewritten as, xyz.
Boolean equivalent
Prior to 1927, Boolean algebra had been considered a calculus of logical values with logical operations of conjunction, disjunction, negation, and so on. Zhegalkin showed that all Boolean operations could be written as ordinary numeric polynomials, representing the false and true values as 0 and 1, the integers mod 2. Logical conjunction is written as xy, and logical exclusive-or as arithmetic addition mod 2, (written here as x⊕y to avoid confusion with the common use of + as a synonym for inclusive-or ∨). The logical complement ¬x is then x⊕1. Since ∧ and ¬ form a basis for Boolean algebra, all other logical operations are compositions of these basic operations, and so the polynomials of ordinary algebra can represent all Boolean operations, allowing Boolean reasoning to be performed using elementary algebra.
For example, the Boolean 2-out-of-3 threshold or median operation is written as the Zhegalkin polynomial xy⊕yz⊕zx.
Formal properties
Formally a Zhegalkin monomial is the product of a finite set of distinct variables (hence square-free), including the empty set whose product is denoted 1. There are 2n possible Zhegalkin monomials in n variables, since each monomial is fully specified by the presence or absence of each variable. A Zhegalkin polynomial is the sum (exclusive-or) of a set of Zhegalkin monomials, with the empty set denoted by 0. A given monomial's presence or absence in a polynomial corresponds to that monomial's coefficient being 1 or 0 respectively. The Zhegalkin monomials, being linearly independent, span a 2n-dimensional vector space over the Galois field GF(2) (NB: not GF(2n), whose multiplication is quite different). The 22n vectors of this space, i.e. the linear combinations of those monomials as unit vectors, constitute the Zhegalkin polynomials. The exact agreement with the number of Boolean operations on n variables, which exhaust the n-ary operations on {0,1}, furnishes a direct counting argument for completeness of the Zhegalkin polynomials as a Boolean basis.
This vector space is not equivalent to the free Boolean algebra on n generators because it lacks complementation (bitwise logical negation) as an operation (equivalently, because it lacks the top element as a constant). This is not to say that the space is not
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https://en.wikipedia.org/wiki/List%20of%20problems%20in%20loop%20theory%20and%20quasigroup%20theory
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In mathematics, especially abstract algebra, loop theory and quasigroup theory are active research areas with many open problems. As in other areas of mathematics, such problems are often made public at professional conferences and meetings. Many of the problems posed here first appeared in the Loops (Prague) conferences and the Mile High (Denver) conferences.
Open problems (Moufang loops)
Abelian by cyclic groups resulting in Moufang loops
Let L be a Moufang loop with normal abelian subgroup (associative subloop) M of odd order such that L/M is a cyclic group of order bigger than 3. (i) Is L a group? (ii) If the orders of M and L/M are relatively prime, is L a group?
Proposed: by Michael Kinyon, based on (Chein and Rajah, 2000)
Comments: The assumption that L/M has order bigger than 3 is important, as there is a (commutative) Moufang loop L of order 81 with normal commutative subgroup of order 27.
Embedding CMLs of period 3 into alternative algebras
Conjecture: Any finite commutative Moufang loop of period 3 can be embedded into a commutative alternative algebra.
Proposed: by Alexander Grishkov at Loops '03, Prague 2003
Frattini subloop for Moufang loops
Conjecture: Let L be a finite Moufang loop and Φ(L) the intersection of all maximal subloops of L. Then Φ(L) is a normal nilpotent subloop of L.
Proposed: by Alexander Grishkov at Loops '11, Třešť 2011
Minimal presentations for loops M(G,2)
For a group , define on x by
, , , . Find a minimal presentation for the Moufang loop with respect to a presentation for .
Proposed: by Petr Vojtěchovský at Loops '03, Prague 2003
Comments: Chein showed in (Chein, 1974) that is a Moufang loop that is nonassociative if and only if is nonabelian. Vojtěchovský (Vojtěchovský, 2003) found a minimal presentation for when is a 2-generated group.
Moufang loops of order p2q3 and pq4
Let p and q be distinct odd primes. If q is not congruent to 1 modulo p, are all Moufang loops of order p2q3 groups? What about pq4?
Proposed: by Andrew Rajah at Loops '99, Prague 1999
Comments: The former has been solved by Rajah and Chee (2011) where they showed that for distinct odd primes p1 < ··· < pm < q < r1 < ··· < rn, all Moufang loops of order p12···pm2q3r12···rn2 are groups if and only if q is not congruent to 1 modulo pi for each i.
(Phillips' problem) Odd order Moufang loop with trivial nucleus
Is there a Moufang loop of odd order with trivial nucleus?
Proposed: by Andrew Rajah at Loops '03, Prague 2003
Presentations for finite simple Moufang loops
Find presentations for all nonassociative finite simple Moufang loops in the variety of Moufang loops.
Proposed: by Petr Vojtěchovský at Loops '03, Prague 2003
Comments: It is shown in (Vojtěchovský, 2003) that every nonassociative finite simple Moufang loop is generated by 3 elements, with explicit formulas for the generators.
The restricted Burnside problem for Moufang loops
Conjecture: Let M be a finite Moufang loop of exponent n with m generators. Then
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https://en.wikipedia.org/wiki/Green%20measure
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In mathematics — specifically, in stochastic analysis — the Green measure is a measure associated to an Itō diffusion. There is an associated Green formula representing suitably smooth functions in terms of the Green measure and first exit times of the diffusion. The concepts are named after the British mathematician George Green and are generalizations of the classical Green's function and Green formula to the stochastic case using Dynkin's formula.
Notation
Let X be an Rn-valued Itō diffusion satisfying an Itō stochastic differential equation of the form
Let Px denote the law of X given the initial condition X0 = x, and let Ex denote expectation with respect to Px. Let LX be the infinitesimal generator of X, i.e.
Let D ⊆ Rn be an open, bounded domain; let τD be the first exit time of X from D:
The Green measure
Intuitively, the Green measure of a Borel set H (with respect to a point x and domain D) is the expected length of time that X, having started at x, stays in H before it leaves the domain D. That is, the Green measure of X with respect to D at x, denoted G(x, ·), is defined for Borel sets H ⊆ Rn by
or for bounded, continuous functions f : D → R by
The name "Green measure" comes from the fact that if X is Brownian motion, then
where G(x, y) is Green's function for the operator LX (which, in the case of Brownian motion, is ½Δ, where Δ is the Laplace operator) on the domain D.
The Green formula
Suppose that Ex[τD] < +∞ for all x ∈ D, and let f : Rn → R be of smoothness class C2 with compact support. Then
In particular, for C2 functions f with support compactly embedded in D,
The proof of Green's formula is an easy application of Dynkin's formula and the definition of the Green measure:
References
(See Section 9)
Measures (measure theory)
Stochastic differential equations
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https://en.wikipedia.org/wiki/Edge%20%28geometry%29
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In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a line segment on the boundary, and is often called a polygon side. In a polyhedron or more generally a polytope, an edge is a line segment where two faces (or polyhedron sides) meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal.
Relation to edges in graphs
In graph theory, an edge is an abstract object connecting two graph vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment.
However, any polyhedron can be represented by its skeleton or edge-skeleton, a graph whose vertices are the geometric vertices of the polyhedron and whose edges correspond to the geometric edges. Conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by Steinitz's theorem as being exactly the 3-vertex-connected planar graphs.
Number of edges in a polyhedron
Any convex polyhedron's surface has Euler characteristic
where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Euler's polyhedron formula. Thus the number of edges is 2 less than the sum of the numbers of vertices and faces. For example, a cube has 8 vertices and 6 faces, and hence 12 edges.
Incidences with other faces
In a polygon, two edges meet at each vertex; more generally, by Balinski's theorem, at least d edges meet at every vertex of a d-dimensional convex polytope.
Similarly, in a polyhedron, exactly two two-dimensional faces meet at every edge, while in higher dimensional polytopes three or more two-dimensional faces meet at every edge.
Alternative terminology
In the theory of high-dimensional convex polytopes, a facet or side of a d-dimensional polytope is one of its (d − 1)-dimensional features, a ridge is a (d − 2)-dimensional feature, and a peak is a (d − 3)-dimensional feature. Thus, the edges of a polygon are its facets, the edges of a 3-dimensional convex polyhedron are its ridges, and the edges of a 4-dimensional polytope are its peaks.
See also
Extended side
References
External links
Elementary geometry
Multi-dimensional geometry
1
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https://en.wikipedia.org/wiki/EcosimPro
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EcosimPro is a simulation tool developed by Empresarios Agrupados A.I.E for modelling simple and complex physical processes that can be expressed in terms of Differential algebraic equations or Ordinary differential equations and Discrete event simulation.
The application runs on the various Microsoft Windows platforms and uses its own graphic environment for model design.
The modelling of physical components is based on the EcosimPro language (EL) which is very similar to other conventional Object-oriented programming languages but is powerful enough to model continuous and discrete processes.
This tool employs a set of libraries containing various types of components (mechanical, electrical, pneumatic, hydraulic, etc.) that can be reused to model any type of system.
It is used within ESA for propulsion systems analysis and is the recommended ESA analysis tool for ECLS systems.
Origins
The EcosimPro Tool Project began in 1989 with funds from the European Space Agency (ESA) and with the goal of simulating environmental control and life support systems for crewed spacecraft, such as the Hermes shuttle.
The multidisciplinary nature of this modelling tool led to its use in many other disciplines, including fluid mechanics, chemical processing, control, energy, propulsion and flight dynamics. These complex applications have demonstrated that EcosimPro is very robust and ready for use in many other fields.
The modelling language
Code examples
Differential equation
To familiarize yourself with the use of EcosimPro, first create a simple component to solve a differential equation. Although EcosimPro is designed to simulate complex systems, it can also be used independently of a physical system as if it were a pure equation solver. The example in this section illustrates this type of use. It solves the following differential equation to introduce a delay to variable x:
which is equivalent to
where x and y have a time dependence that will be defined in the experiment. Tau is datum provided given by the user; we will use a value of 0.6 seconds. This equation introduces a delay in the x variable with respect to y with value tau. To simulate this equation we will create an EcosimPro component with the equation in it.
The component to be simulated in EL is like thus:
COMPONENT equation_test
DATA
REAL tau = 0.6 "delay time (seconds)"
DECLS
REAL x, y
CONTINUOUS
y' = (x - y) / tau
END COMPONENT
Pendulum
One example of applied calculus could be the movement of a perfect pendulum (no friction taken into account). We would have the following data: the force of gravity ‘g’; the length of the pendulum ‘L’; and the pendulum's mass ‘M’. As variables to be calculated we would have: the Cartesian position at each moment in time of the pendulum ‘x’ and ‘y’ and the tension on the wire of the pendulum ‘T’. The equations that define the model would be:
- Projecting the length of the cable on the Cartesian axes and applyin
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https://en.wikipedia.org/wiki/PCBS
|
PCBS may refer to:
Palestinian Central Bureau of Statistics
PCBs, polychlorinated biphenyls
Wellington Street bus station (originally Perth Central Bus Station)
PC Building Simulator
See also
PCB (disambiguation)
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https://en.wikipedia.org/wiki/SymPy
|
SymPy is an open-source Python library for symbolic computation. It provides computer algebra capabilities either as a standalone application, as a library to other applications, or live on the web as SymPy Live or SymPy Gamma. SymPy is simple to install and to inspect because it is written entirely in Python with few dependencies. This ease of access combined with a simple and extensible code base in a well known language make SymPy a computer algebra system with a relatively low barrier to entry.
SymPy includes features ranging from basic symbolic arithmetic to calculus, algebra, discrete mathematics, and quantum physics. It is capable of formatting the result of the computations as LaTeX code.
SymPy is free software and is licensed under the New BSD license. The lead developers are Ondřej Čertík and Aaron Meurer. It was started in 2005 by Ondřej Čertík.
Features
The SymPy library is split into a core with many optional modules.
Currently, the core of SymPy has around 260,000 lines of code (it also includes a comprehensive set of self-tests: over 100,000 lines in 350 files as of version 0.7.5), and its capabilities include:
Core capabilities
Basic arithmetic: *, /, +, -, **
Simplification
Expansion
Functions: trigonometric, hyperbolic, exponential, roots, logarithms, absolute value, spherical harmonics, factorials and gamma functions, zeta functions, polynomials, hypergeometric, special functions, etc.
Substitution
Arbitrary precision integers, rationals and floats
Noncommutative symbols
Pattern matching
Polynomials
Basic arithmetic: division, gcd, etc.
Factorization
Square-free factorization
Gröbner bases
Partial fraction decomposition
Resultants
Calculus
Limits
Differentiation
Integration: Implemented Risch–Norman heuristic
Taylor series (Laurent series)
Solving equations
Systems of linear equations
Systems of algebraic equations that are solvable by radicals
Differential equations
Difference equations
Discrete math
Binomial coefficients
Summations
Products
Number theory: generating Prime numbers, primality testing, integer factorization, etc.
Logic expressions
Matrices
Basic arithmetic
Eigenvalues and their eigenvectors when the characteristic polynomial is solvable by radicals
Determinants
Inversion
Solving
Geometry
Points, lines, rays, ellipses, circles, polygons, etc.
Intersections
Tangency
Similarity
Plotting
Note, plotting requires the external Matplotlib or Pyglet module.
Coordinate models
Plotting Geometric Entities
2D and 3D
Interactive interface
Colors
Animations
Physics
Units
Classical mechanics
Continuum mechanics
Quantum mechanics
Gaussian optics
Pauli algebra
Linear control
Statistics
Normal distributions
Uniform distributions
Probability
Combinatorics
Permutations
Combinations
Partitions
Subsets
Permutation group: Polyhedral, Rubik, Symmetric, etc.
Prufer sequence and Gray codes
Printing
Pretty-printing: ASCII/Unicode pretty-printing, LaTeX
Code generati
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https://en.wikipedia.org/wiki/Mitrofan%20Cioban
|
Mitrofan Cioban (5 January 1942 – 2 February 2021) was a Moldovan mathematician specializing in topology, a member of the Academy of Sciences of Moldova (2000).
He was born in Copceac (then in Tighina County, Romania, now in Ștefan Vodă District, Moldova), the son of Mihail and Tecla Cioban. At age 17 he enrolled in the Faculty of Mathematics and Physics of Tiraspol State University. After one year Cioban transferred to Moscow State University, where he started attending the Topology seminar of Pavel Alexandrov. He obtained his Ph.D. in 1969 with thesis Properties of Quotient Mappings and Classification of Spaces written under the direction of Alexander Arhangelskii. Upon graduation, he returned in 1970 to Tiraspol State University as a faculty member, where he directed 17 Ph.D. theses and served as prorector and then rector.
He published over 200 papers in academic journals from 1966 to 2020, mostly under the names of Choban or Čoban, and occasionally Cioban, Ciobanu, or Coban. Starting in 1999 he served as president of the Mathematical Society of the Republic of Moldova, with headquarters in Chișinău.
He died from COVID-19 on 2 February 2021 in Chișinău during the COVID-19 pandemic in Moldova.
Publications
References
External links
1942 births
2021 deaths
20th-century Moldovan mathematicians
21st-century Moldovan mathematicians
People from Ștefan Vodă District
Moscow State University alumni
Academic staff of Shevchenko Transnistria State University
Algebraists
Topologists
Titular members of the Academy of Sciences of Moldova
Recipients of the Order of Honour (Moldova)
Deaths from the COVID-19 pandemic in Moldova
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https://en.wikipedia.org/wiki/Geometric%20measure%20theory
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In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfaces that are not necessarily smooth.
History
Geometric measure theory was born out of the desire to solve Plateau's problem (named after Joseph Plateau) which asks if for every smooth closed curve in there exists a surface of least area among all surfaces whose boundary equals the given curve. Such surfaces mimic soap films.
The problem had remained open since it was posed in 1760 by Lagrange. It was solved independently in the 1930s by Jesse Douglas and Tibor Radó under certain topological restrictions. In 1960 Herbert Federer and Wendell Fleming used the theory of currents with which they were able to solve the orientable Plateau's problem analytically without topological restrictions, thus sparking geometric measure theory. Later Jean Taylor after Fred Almgren proved Plateau's laws for the kind of singularities that can occur in these more general soap films and soap bubbles clusters.
Important notions
The following objects are central in geometric measure theory:
Hausdorff measure and Hausdorff dimension
Rectifiable sets (or Radon measures), which are sets with the least possible regularity required to admit approximate tangent spaces.
Characterization of rectifiability through existence of approximate tangents, densities, projections, etc.
Orthogonal projections, Kakeya sets, Besicovitch sets
Uniform rectifiability
Rectifiability and uniform rectifiability of (subsets of) metric spaces, e.g. SubRiemannian manifolds, Carnot groups, Heisenberg groups, etc.
Connections to singular integrals, Fourier transform, Frostman measures, harmonic measures, etc
Currents, a generalization of the concept of oriented manifolds, possibly with boundary.
Flat chains, an alternative generalization of the concept of manifolds, possibly with boundary.
Caccioppoli sets (also known as sets of locally finite perimeter), a generalization of the concept of manifolds on which the divergence theorem applies.
Plateau type minimization problems from calculus of variations
The following theorems and concepts are also central:
The area formula, which generalizes the concept of change of variables in integration.
The coarea formula, which generalizes and adapts Fubini's theorem to geometric measure theory.
The isoperimetric inequality, which states that the smallest possible circumference for a given area is that of a round circle.
Flat convergence, which generalizes the concept of manifold convergence.
Examples
The Brunn–Minkowski inequality for the n-dimensional volumes of convex bodies K and L,
can be proved on a single page and quickly yields the classical isoperimetric inequality. The Brunn–Minkowski inequality also leads to Anderson's theorem in statistics. The proof of the Brunn–Minkowski inequali
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https://en.wikipedia.org/wiki/Calthorpe%20Park%20School
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Calthorpe Park School is one of two secondary schools in Fleet, Hampshire, England. for pupils aged 11–16. The school was awarded specialist status as a Maths and Computing College in 2005. In the 2013 OFSTED report the school received an overall effectiveness of 'good'.
A major capital scheme by Hampshire County Council to extend the school to accommodate more pupils was completed in 2004. It then underwent another scheme in 2008–9 to extend the school further. The school underwent another expansion project in 2014–2015.
History
Calthorpe Park School opened on Wednesday 3 September 1969 as North Fleet Bilateral School with J. Ormerod as Headteacher. A few months later, the new Governing Body changed the name. 153 pupils and 11 staff were the first occupants before all year groups were complete five years later.
The official opening took place on Friday 18 June 1971 by W. van Straubenzee, MBE, MP from the Department of Education and Science. After the National Anthem and a Dedication from the Vicar of Fleet, the Rev A.C.B. Deedes, speeches were given by a variety of people connected to the school including the Chairman of the School Governors, Major B.C. Debenham, MBE, the Chairman of Hampshire County Council, Brig Sir Richard A-G Calthorpe, BT, CBE and Ormerod.
The school formed part of the 1967/68 Major Building Programme, designed by the County Architect in the SCOLA Mark IA form of construction using standard components with some brick cladding. The total building cost of the project, including fees, furniture and equipment was approximately £410,000.
The original buildings were extended over the years with Phase 2 (English, Tech, Library) being built in 1974, followed by the Maths block, the old and new Drama blocks and the Leisure Centre next door to the school.
The Hart Leisure centre was previously located on site, before having its extended car park being demolished in Early 2018 which has since been replaced with a sports hall. The total cost of the project was approximately £3.4 Million. Followed by the demolition of the main leisure centre itself in late 2019 to make way for an expansion of the school onto that site.
C. Heasman took over as Headteacher in 1988 before retiring in 2005 when C. Anwar became Headteacher. M. Amos and M. Hooper were appointed Joint Headteachers in 2014. Mr Kevan John became the Headteacher in 2021.
In 2020, Morgan Sindall was awarded a contract for the expansion of the school. As part of the works, Morgan Sindall Construction will also carry out re-modelling to the existing school to provide two new science labs and additional library space in the heart of the existing school. The project is set to complete in September 2021.
References
External links
Secondary schools in Hampshire
Community schools in Hampshire
Fleet, Hart
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https://en.wikipedia.org/wiki/Killed%20process
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In probability theory — specifically, in stochastic analysis — a killed process is a stochastic process that is forced to assume an undefined or "killed" state at some (possibly random) time.
Definition
Let X : T × Ω → S be a stochastic process defined for "times" t in some ordered index set T, on a probability space (Ω, Σ, P), and taking values in a measurable space S. Let ζ : Ω → T be a random time, referred to as the killing time. Then the killed process Y associated to X is defined by
and Yt is left undefined for t ≥ ζ. Alternatively, one may set Yt = c for t ≥ ζ, where c is a "coffin state" not in S.
See also
Stopped process
References
(See Section 8.2)
Stochastic processes
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https://en.wikipedia.org/wiki/Toninho%20%28footballer%2C%20born%201977%29
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Antonio Bezerra Brandão or simply Toninho (born December 21, 1977), is a former Brazilian central defender.
Club statistics
External links
sambafoot
CBF
placar
1977 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Associação Ferroviária de Esportes players
J1 League players
J2 League players
Centro Sportivo Alagoano players
Sport Club Corinthians Paulista players
Clube Atlético Juventus players
Expatriate men's footballers in Japan
Omiya Ardija players
Paraná Clube players
Agremiação Sportiva Arapiraquense players
Sociedade Esportiva e Recreativa Caxias do Sul players
Rio Claro Futebol Clube players
Grêmio Barueri Futebol players
Esporte Clube XV de Novembro (Piracicaba) players
Men's association football defenders
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https://en.wikipedia.org/wiki/Rethinking%20Mathematics
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Rethinking Mathematics: Teaching Social Justice by the Numbers is a 2005 book (2nd edition 2013) edited by Eric Gutstein and Bob Peterson, advocating a mathematics education curriculum that intertwines mathematics with social justice. The various essays in the book, including "Home Buying While Brown or Black" and "Sweatshop Accounting", advocate using social-justice issues to motivate the teaching of rigorous mathematical concepts, and the use of mathematics education as a way of promoting ideas of social justice.
Critics derided the work as an attempt to subvert mathematics education for partisan political purposes, while the authors defended it as a useful way to motivate a wide range of students in mathematics.
See also
Critical mathematics pedagogy
Mathematics for social justice
Mathematics education
References
External links
Book website
2005 books
Mathematics education reform
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https://en.wikipedia.org/wiki/List%20of%20Major%20League%20Baseball%20career%20stolen%20bases%20leaders
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In baseball statistics, a stolen base is credited to a baserunner when he successfully advances to the next base while the pitcher is throwing the ball to home plate. Under Rule 7.01 of Major League Baseball's (MLB) Official Rules, a runner acquires the right to an unoccupied base when he touches it before he is out. Stolen bases were more common in baseball's dead-ball era, when teams relied more on stolen bases and hit and run plays than on home runs.
Rickey Henderson holds the MLB career stolen base record with 1,406. He is the only MLB player to have reached the 1,000 stolen bases milestone in his career. Following Henderson is Lou Brock with 938 stolen bases; Billy Hamilton is third on the all-time steals listing. The number of career steals attributed to Hamilton varies by source, but all sources hold his career steals placing him in third on the list before Ty Cobb (897), Tim Raines (808), Vince Coleman (752), Arlie Latham (742), Eddie Collins (741), Max Carey (738), and Honus Wagner (723), who are the only other players to have stolen at least 700 bases. Coleman is the leader for retired players that are not members of the Hall of Fame.Brock held the all-time career stolen bases before being surpassed by Henderson in 1991. Brock had held the record from 1977 to 1991. Before Brock, Hamilton held the record for eighty-one years, from 1897 to 1977. Before that, Latham held the record from 1887 to 1896. Latham was also the first player to collect 300 career stolen bases. With Kenny Lofton's retirement in 2007, 2008 was the first season since 1967 in which no active player had more than 500 career stolen bases. Between 2008 and 2010, no active player had more than 500 stolen bases until Juan Pierre collected his 500th stolen base on August 5, 2010. He was the leader in stolen bases for active players until his retirement at the end of the 2013 season.
Key
List
References
External links
All-time stolen base leaders at Baseball-Reference.com
Major League Baseball lists
Major League Baseball statistics
Career
Major League Baseball records
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https://en.wikipedia.org/wiki/Remez%20inequality
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In mathematics, the Remez inequality, discovered by the Soviet mathematician Evgeny Yakovlevich Remez , gives a bound on the sup norms of certain polynomials, the bound being attained by the Chebyshev polynomials.
The inequality
Let σ be an arbitrary fixed positive number. Define the class of polynomials πn(σ) to be those polynomials p of the nth degree for which
on some set of measure ≥ 2 contained in the closed interval [−1, 1+σ]. Then the Remez inequality states that
where Tn(x) is the Chebyshev polynomial of degree n, and the supremum norm is taken over the interval [−1, 1+σ].
Observe that Tn is increasing on , hence
The R.i., combined with an estimate on Chebyshev polynomials, implies the following corollary: If J ⊂ R is a finite interval, and E ⊂ J is an arbitrary measurable set, then
for any polynomial p of degree n.
Extensions: Nazarov–Turán lemma
Inequalities similar to () have been proved for different classes of functions, and are known as Remez-type inequalities. One important example is Nazarov's inequality for exponential sums :
Nazarov's inequality. Let
be an exponential sum (with arbitrary λk ∈C), and let J ⊂ R be a finite interval, E ⊂ J—an arbitrary measurable set. Then
where C > 0 is a numerical constant.
In the special case when λk are pure imaginary and integer, and the subset E is itself an interval, the inequality was proved by Pál Turán and is known as Turán's lemma.
This inequality also extends to in the following way
for some A>0 independent of p, E, and n. When
a similar inequality holds for p > 2. For p=∞ there is an extension to multidimensional polynomials.
Proof: Applying Nazarov's lemma to leads to
thus
Now fix a set and choose such that , that is
Note that this implies:
Now
which completes the proof.
Pólya inequality
One of the corollaries of the R.i. is the Pólya inequality, which was proved by George Pólya , and states that the Lebesgue measure of a sub-level set of a polynomial p of degree n is bounded in terms of the leading coefficient LC(p) as follows:
References
Theorems in analysis
Inequalities
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https://en.wikipedia.org/wiki/Solid%20harmonics
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In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions . There are two kinds: the regular solid harmonics , which are well-defined at the origin and the irregular solid harmonics , which are singular at the origin. Both sets of functions play an important role in potential theory, and are obtained by rescaling spherical harmonics appropriately:
Derivation, relation to spherical harmonics
Introducing , , and for the spherical polar coordinates of the 3-vector , and assuming that is a (smooth) function , we can write the Laplace equation in the following form
where is the square of the nondimensional angular momentum operator,
It is known that spherical harmonics are eigenfunctions of :
Substitution of into the Laplace equation gives, after dividing out the spherical harmonic function, the following radial equation and its general solution,
The particular solutions of the total Laplace equation are regular solid harmonics:
and irregular solid harmonics:
The regular solid harmonics correspond to harmonic homogeneous polynomials, i.e. homogeneous polynomials which are solutions to Laplace's equation.
Racah's normalization
Racah's normalization (also known as Schmidt's semi-normalization) is applied to both functions
(and analogously for the irregular solid harmonic) instead of normalization to unity. This is convenient because in many applications the Racah normalization factor appears unchanged throughout the derivations.
Addition theorems
The translation of the regular solid harmonic gives a finite expansion,
where the Clebsch–Gordan coefficient is given by
The similar expansion for irregular solid harmonics gives an infinite series,
with . The quantity between pointed brackets is again a Clebsch-Gordan coefficient,
The addition theorems were proved in different manners by several authors.
Complex form
The regular solid harmonics are homogeneous, polynomial solutions to the Laplace equation . Separating the indeterminate and writing , the Laplace equation is easily seen to be equivalent to the recursion formula
so that any choice of polynomials of degree and of degree gives a solution to the equation. One particular basis of the space of homogeneous polynomials (in two variables) of degree is . Note that it is the (unique up to normalization) basis of eigenvectors of the rotation group : The rotation of the plane by acts as multiplication by on the basis vector .
If we combine the degree basis and the degree basis with the recursion formula, we obtain a basis of the space of harmonic, homogeneous polynomials (in three variables this time) of degree consisting of eigenvectors for (note that the recursion formula is compatible with the -action because the Laplace operator is rotationally invariant). These are the complex solid harmonics:
and in general
for .
Plugging in spherical coordinates , , and using one fin
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https://en.wikipedia.org/wiki/Herbert%20Federer
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Herbert Federer (July 23, 1920 – April 21, 2010) was an American mathematician. He is one of the creators of geometric measure theory, at the meeting point of differential geometry and mathematical analysis.
Career
Federer was born July 23, 1920, in Vienna, Austria. After emigrating to the US in 1938, he studied mathematics and physics at the University of California, Berkeley, earning the Ph.D. as a student of Anthony Morse in 1944. He then spent virtually his entire career as a member of the Brown University Mathematics Department, where he eventually retired with the title of Professor Emeritus.
Federer wrote more than thirty research papers in addition to his book Geometric measure theory. The Mathematics Genealogy Project assigns him nine Ph.D. students and well over a hundred subsequent descendants. His most productive students include the late Frederick J. Almgren, Jr. (1933–1997), a professor at Princeton for 35 years, and his last student, Robert Hardt, now at Rice University.
Federer was a member of the National Academy of Sciences. In 1987, he and his Brown colleague Wendell Fleming won the American Mathematical Society's Steele Prize "for their pioneering work in Normal and Integral currents."
Mathematical work
In the 1940s and 1950s, Federer made many contributions at the technical interface of geometry and measure theory. Particular themes included surface area, rectifiability of sets, and the extent to which one could substitute rectifiability for smoothness in the classical analysis of surfaces. A particularly noteworthy early accomplishment (improving earlier work of Abram Besicovitch) was the characterization of purely unrectifiable sets as those which "vanish" under almost all projections. Federer also made noteworthy contributions to the study of Green's theorem in low regularity. The theory of capacity with modified exponents was developed by Federer and William Ziemer. In his first published paper, written with his Ph.D. advisor Anthony Morse, Federer proved the Federer–Morse theorem which states that any continuous surjection between compact metric spaces can be restricted to a Borel subset so as to become an injection, without changing the image.
One of Federer's best-known papers, Curvature Measures, was published in 1959. The intention is to establish measure-theoretic formulations of second-order analysis in differential geometry, particularly curvature. The Steiner formula formed a fundamental precedent for Federer's work; it established that the volume of a neighborhood of a convex set in Euclidean space is given by a polynomial. If the boundary of the convex set is a smooth submanifold, then the coefficients of the Steiner formula are defined by its curvature. Federer's work was aimed towards developing a general formulation of this result. The class of subsets that he identified are those of positive reach, subsuming both the class of convex sets and the class of smooth submanifolds. He proved the Steiner formu
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https://en.wikipedia.org/wiki/Math%20%28disambiguation%29
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Math (or maths in non-American English-speaking countries) is an abbreviation of mathematics.
Math or Maths may also refer to:
Arts, entertainment and media
Fictional characters
Math, a character on the TV series Life Unexpected
Math fab Mathonwy, a king in Welsh mythology
Mathematical Anti Telharsic Harfatum Septomin, a topic on the TV series Look Around You
Music
Groups
MATH (band), an American band formed in the 1990s
Math, the original name of American rock band Mutemath
Songs
"Maths" (song), a 2012 single by Canadian producer deadmau5
"The Math", a song on Hilary Duff's 2003 album Metamorphosis
Television
"Math!", an episode of Blue's Clues
Computing and technology
, a tag used in MathML
, a module in the Python programming language
Places
Math, Khyber Pakhtunkhwa, a village of the Khyber Pakhtunkhwa Province of Pakistan
Matha, Hindu monastic establishments, also known as "math" or "mutt"
Other uses
MATH, or Make America Think Harder, a slogan for Andrew Yang's 2020 presidential campaign
Maths O. Sundqvist (1950–2012), a Swedish businessman
See also
Matha (disambiguation)
Cmath (disambiguation)
Mathematica (disambiguation)
Mathematics (disambiguation)
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https://en.wikipedia.org/wiki/Infinitesimal%20generator%20%28stochastic%20processes%29
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In mathematics — specifically, in stochastic analysis — the infinitesimal generator of a Feller process (i.e. a continuous-time Markov process satisfying certain regularity conditions) is a Fourier multiplier operator that encodes a great deal of information about the process.
The generator is used in evolution equations such as the Kolmogorov backward equation, which describes the evolution of statistics of the process; its L2 Hermitian adjoint is used in evolution equations such as the Fokker–Planck equation, also known as Kolmogorov forward equation, which describes the evolution of the probability density functions of the process.
The Kolmogorov forward equation in the notation is just , where is the probability density function, and is the adjoint of the infinitesimal generator of the underlying stochastic process. The Klein–Kramers equation is a special case of that.
Definition
General case
For a Feller process with Feller semigroup and state space we define the generator by
,
, for any .
Here denotes the Banach space of continuous functions on vanishing at infinity, equipped with the supremum norm, and . In general, it is not easy to describe the domain of the Feller generator. However, the Feller generator is always closed and densely defined. If is -valued and contains the test functions (compactly supported smooth functions) then
,
where , and is a Lévy triplet for fixed .
Lévy processes
The generator of Lévy semigroup is of the form
where is positive semidefinite and is a Lévy measure satisfying
and for some with is bounded. If we define
for then the generator can be written as
where denotes the Fourier transform. So the generator of a Lévy process (or semigroup) is a Fourier multiplier operator with symbol .
Stochastic differential equations driven by Lévy processes
Let be a Lévy process with symbol (see above). Let be locally Lipschitz and bounded. The solution of the SDE exists for each deterministic initial condition and yields a Feller process with symbol
Note that in general, the solution of an SDE driven by a Feller process which is not Lévy might fail to be Feller or even Markovian.
As a simple example consider with a Brownian motion driving noise. If we assume are Lipschitz and of linear growth, then for each deterministic initial condition there exists a unique solution, which is Feller with symbol
Mean first passage time
The mean first passage time satisfies . This can be used to calculate, for example, the time it takes for a Brownian motion particle in a box to hit the boundary of the box, or the time it takes for a Brownian motion particle in a potential well to escape the well. Under certain assumptions, the escape time satisfies the Arrhenius equation.
Generators of some common processes
For finite-state continuous time Markov chains the generator may be expressed as a transition rate matrix.
The general n-dimensional diffusion process has generatorwhere is the diffusi
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https://en.wikipedia.org/wiki/Stochastic%20processes%20and%20boundary%20value%20problems
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In mathematics, some boundary value problems can be solved using the methods of stochastic analysis. Perhaps the most celebrated example is Shizuo Kakutani's 1944 solution of the Dirichlet problem for the Laplace operator using Brownian motion. However, it turns out that for a large class of semi-elliptic second-order partial differential equations the associated Dirichlet boundary value problem can be solved using an Itō process that solves an associated stochastic differential equation.
Introduction: Kakutani's solution to the classical Dirichlet problem
Let be a domain (an open and connected set) in . Let be the Laplace operator, let be a bounded function on the boundary , and consider the problem:
It can be shown that if a solution exists, then is the expected value of at the (random) first exit point from for a canonical Brownian motion starting at . See theorem 3 in Kakutani 1944, p. 710.
The Dirichlet–Poisson problem
Let be a domain in and let be a semi-elliptic differential operator on of the form:
where the coefficients and are continuous functions and all the eigenvalues of the matrix are non-negative. Let and . Consider the Poisson problem:
The idea of the stochastic method for solving this problem is as follows. First, one finds an Itō diffusion whose infinitesimal generator coincides with on compactly-supported functions . For example, can be taken to be the solution to the stochastic differential equation:
where is n-dimensional Brownian motion, has components as above, and the matrix field is chosen so that:
For a point , let denote the law of given initial datum , and let denote expectation with respect to . Let denote the first exit time of from .
In this notation, the candidate solution for (P1) is:
provided that is a bounded function and that:
It turns out that one further condition is required:
For all , the process starting at almost surely leaves in finite time. Under this assumption, the candidate solution above reduces to:
and solves (P1) in the sense that if denotes the characteristic operator for (which agrees with on functions), then:
Moreover, if satisfies (P2) and there exists a constant such that, for all :
then .
References
(See Section 9)
Boundary value problems
Partial differential equations
Stochastic differential equations
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https://en.wikipedia.org/wiki/Haj%C3%B3s%27s%20theorem
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In group theory, Hajós's theorem states that if a finite abelian group is expressed as the Cartesian product of simplexes, that is, sets of the form where is the identity element, then at least one of the factors is a subgroup. The theorem was proved by the Hungarian mathematician György Hajós in 1941 using group rings. Rédei later proved the statement when the factors are only required to contain the identity element and be of prime cardinality. Rédei's proof of Hajós's theorem was simplified by Tibor Szele.
An equivalent statement on homogeneous linear forms was originally conjectured by Hermann Minkowski. A consequence is Minkowski's conjecture on lattice tilings, which says that in any lattice tiling of space by cubes, there are two cubes that meet face to face. Keller's conjecture is the same conjecture for non-lattice tilings, which turns out to be false in high dimensions.
References
Theorems in group theory
Conjectures that have been proved
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https://en.wikipedia.org/wiki/McShane%27s%20identity
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In geometric topology, McShane's identity for a once punctured torus with a complete, finite-volume hyperbolic structure is given by
where
the sum is over all simple closed geodesics γ on the torus; and
ℓ(γ) denotes the hyperbolic length of γ.
This identity was generalized by Maryam Mirzakhani on her PhD thesis
References
Further reading
Geometric topology
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https://en.wikipedia.org/wiki/Bracket%20algebra
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In mathematics, a bracket algebra is an algebraic system that connects the notion of a supersymmetry algebra with a symbolic representation of projective invariants.
Given that L is a proper signed alphabet and Super[L] is the supersymmetric algebra, the bracket algebra Bracket[L] of dimension n over the field K is the quotient of the algebra Brace{L} obtained by imposing the congruence relations below, where w, w, ..., w" are any monomials in Super[L]:
{w} = 0 if length(w) ≠ n
{w}{w'}...{w"} = 0 whenever any positive letter a of L occurs more than n times in the monomial {w}{w'''}...{w"}.
Let {w}{w}...{w"} be a monomial in Brace{L} in which some positive letter a occurs more than n times, and let b, c, d, e, ..., f, g be any letters in L''.
See also
Bracket ring
References
.
.
Invariant theory
Algebras
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https://en.wikipedia.org/wiki/Hutchinson%20metric
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In mathematics, the Hutchinson metric otherwise known as Kantorovich metric is a function which measures "the discrepancy between two images for use in fractal image processing" and "can also be applied to describe the similarity between DNA sequences expressed as real or complex genomic signals".
Formal definition
Consider only nonempty, compact, and finite metric spaces. For such a space , let denote the space of Borel probability measures on , with
the embedding associating to the point measure . The support of a measure in is the smallest closed subset of measure 1.
If is Borel measurable then the induced map
associates to the measure defined by
for all Borel in .
Then the Hutchinson metric is given by
where the is taken over all real-valued functions with Lipschitz constant
Then is an isometric embedding of into , and if is Lipschitz then is Lipschitz with the same Lipschitz constant.
See also
Wasserstein metric
Acoustic metric
Apophysis (software)
Complete metric
Fractal image compression
Image differencing
Metric tensor
Multifractal system
Sources and notes
Metric geometry
Topology
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https://en.wikipedia.org/wiki/QPD
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QPD may refer to:
Quantile-parameterized distribution, probability distributions that are directly parameterized by data
Quasiprobability distribution, a mathematical object similar to a probability distribution
Quebec platelet disorder, a rare autosomal dominant bleeding disorder
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https://en.wikipedia.org/wiki/Institute%20of%20Mathematics%20and%20Informatics
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The Institute of Mathematics and Informatics was established in 1947 as Institute of Mathematics at the Bulgarian Academy of Sciences.
Its name changed to Institute of Mathematics with Computing Centre in 1961, while from 1970 to 1988 the Institute together with the Faculty of Mathematics at Sofia University functioned in a unified structure, the Joint Centre of Mathematics and Mechanics. The present name was adopted in 1995.
The Institute carries out scientific research in the fields of mathematics and informatics, as well as their applications to education, science, business and industry, government, etc.
Members of the Institute hold teaching positions at a number of Bulgarian universities and colleges, providing also direction to Ph.D. and M.S. students. The Institute plays an important role in training gifted pupils, graduates, etc. for participating in major international competitions in mathematics and informatics. In particular, the Bulgarian team in mathematics has been ranking among the top five at International Mathematical Olympiad events in the recent years.
Departments
Algebra and Logic (former Algebra and Logic)
Analysis, Geometry and Topology
Biomathematics
Computational Mathematics
Differential Equations and Mathematical Physics
Education in Mathematics and Informatics
Information Systems
Information Modelling
Laboratory of Digitization of Scientific and Cultural Heritage
Mathematical Foundations of Informatics
Mathematical Linguistics
Mathematical Modelling
Operation Research
Probability and Statistics
Software Engineering
Library
Directors
Academician Lyubomir Chakalov, 1947-50
Academician Nikola Obreshkov, 1951-63
Academician Lyubomir Iliev, 1964-88
Academician Petar Kenderov, 1989-93
Sen.Res. Assoc. Nikolay Yanev, 1994-99
Academician Stefan Dodunekov, 1999 –
Publications
Serdica Mathematical Journal
Serdica Journal of Computing
Mathematica Plus
Fractional Calculus and Applied Analysis
Pliska Studia Mathematica Bulgarica
Mathematica Balkanica (founded: New Series, 1987, published quarterly)
Physico-Mathematical Journal (founded: 1904, published quarterly)
Applications
The first computing centre in Bulgaria was created at the Institute in 1961.
The first Bulgarian electronic digital computer Vitosha, and the first Bulgarian electronic calculator Elka were constructed at the Institute in 1962-64.
The official Bulgarian system for the Romanization of Cyrillic alphabet was developed at the Institute in 1995, and codified by the Law of Transliteration in 2009.
Sources
Yanev, Nikolay. (1998). Fifty years of Institute of Mathematics and Informatics at BAS. Journal of BAS, No 3-4, 96-100. (in Bulgarian)
Andreev A., I. Derzhanski eds. Bulgarian Academy of Sciences: Institute of Mathematics and Informatics, founded 1947. Sofia: Multiprint Ltd., 2007. 64 pp. (Bilingual publication in Bulgarian and English)
External links
Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
Bulgarian Acad
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https://en.wikipedia.org/wiki/Mohsen%20Bayatinia
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Mohsen Bayatinia (, born April 9, 1980, in Abadan) is a former Iranian football player and coach.
Club career
Club career statistics
Assist Goals
International career
He was a member of Iran national football team at the West Asian Football Federation Championship 2002. He was also a member of Iran Under-23 team that won the Gold Medal at the 2002 Asian Games in Busan. He scored Iran's winning goal at the final against Japan.
References
Iranian men's footballers
Iran men's international footballers
Men's association football forwards
Iranian expatriate men's footballers
PAS Tehran F.C. players
Paykan F.C. players
Esteghlal F.C. players
Saba Qom F.C. players
Sanat Mes Kerman F.C. players
Persian Gulf Pro League players
Footballers from Abadan, Iran
1980 births
Living people
Asian Games gold medalists for Iran
Asian Games medalists in football
Footballers at the 2002 Asian Games
Medalists at the 2002 Asian Games
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https://en.wikipedia.org/wiki/Either%E2%80%93or%20topology
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In mathematics, the either–or topology is a topological structure defined on the closed interval [−1, 1] by declaring a set open if it either does not contain {0} or does contain (−1, 1).
References
General topology
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https://en.wikipedia.org/wiki/Compact%20complement%20topology
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In mathematics, the compact complement topology is a topology defined on the set of real numbers, defined by declaring a subset open if and only if it is either empty or its complement is compact in the standard Euclidean topology on .
References
Topology
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https://en.wikipedia.org/wiki/Uniform%20k%2021%20polytope
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{{DISPLAYTITLE:Uniform k 21 polytope}}
In geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the En Coxeter group, and having only regular polytope facets. The family was named by their Coxeter symbol k21 by its bifurcating Coxeter–Dynkin diagram, with a single ring on the end of the k-node sequence.
Thorold Gosset discovered this family as a part of his 1900 enumeration of the regular and semiregular polytopes, and so they are sometimes called Gosset's semiregular figures. Gosset named them by their dimension from 5 to 9, for example the 5-ic semiregular figure.
Family members
The sequence as identified by Gosset ends as an infinite tessellation (space-filling honeycomb) in 8-space, called the E8 lattice. (A final form was not discovered by Gosset and is called the E9 lattice: 621. It is a tessellation of hyperbolic 9-space constructed of ∞ 9-simplex and ∞ 9-orthoplex facets with all vertices at infinity.)
The family starts uniquely as 6-polytopes. The triangular prism and rectified 5-cell are included at the beginning for completeness. The demipenteract also exists in the demihypercube family.
They are also sometimes named by their symmetry group, like E6 polytope, although there are many uniform polytopes within the E6 symmetry.
The complete family of Gosset semiregular polytopes are:
triangular prism: −121 (2 triangles and 3 square faces)
rectified 5-cell: 021, Tetroctahedric (5 tetrahedra and 5 octahedra cells)
demipenteract: 121, 5-ic semiregular figure (16 5-cell and 10 16-cell facets)
2 21 polytope: 221, 6-ic semiregular figure (72 5-simplex and 27 5-orthoplex facets)
3 21 polytope: 321, 7-ic semiregular figure (576 6-simplex and 126 6-orthoplex facets)
4 21 polytope: 421, 8-ic semiregular figure (17280 7-simplex and 2160 7-orthoplex facets)
5 21 honeycomb: 521, 9-ic semiregular check tessellates Euclidean 8-space (∞ 8-simplex and ∞ 8-orthoplex facets)
6 21 honeycomb: 621, tessellates hyperbolic 9-space (∞ 9-simplex and ∞ 9-orthoplex facets)
Each polytope is constructed from (n − 1)-simplex and (n − 1)-orthoplex facets.
The orthoplex faces are constructed from the Coxeter group Dn−1 and have a Schläfli symbol of {31,n−1,1} rather than the regular {3n−2,4}. This construction is an implication of two "facet types". Half the facets around each orthoplex ridge are attached to another orthoplex, and the others are attached to a simplex. In contrast, every simplex ridge is attached to an orthoplex.
Each has a vertex figure as the previous form. For example, the rectified 5-cell has a vertex figure as a triangular prism.
Elements
See also
Uniform 2k1 polytope family
Uniform 1k2 polytope family
References
T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
Alicia Boole Stott Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit
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https://en.wikipedia.org/wiki/Abel%27s%20inequality
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In mathematics, Abel's inequality, named after Niels Henrik Abel, supplies a simple bound on the absolute value of the inner product of two vectors in an important special case.
Mathematical description
Let {a1, a2,...} be a sequence of real numbers that is either nonincreasing or nondecreasing, and let {b1, b2,...} be a sequence of real or complex numbers.
If {an} is nondecreasing, it holds that
and if {an} is nonincreasing, it holds that
where
In particular, if the sequence is nonincreasing and nonnegative, it follows that
Relation to Abel's transformation
Abel's inequality follows easily from Abel's transformation, which is the discrete version of integration by parts: If
and are sequences of real or complex numbers, it holds that
References
Abel's inequality in Encyclopedia of Mathematics.
Inequalities
Niels Henrik Abel
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https://en.wikipedia.org/wiki/2007%E2%80%9308%20Real%20Madrid%20CF%20season
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The 2007–08 season was Real Madrid Club de Fútbol's 77th season in La Liga. This article lists all matches that the club played in the 2007–08 season, and also shows statistics of the club's players. Bwin.com became the new kit sponsor.
This season was played since 1995–96 without featuring former legend Brazilian defender and World Cup winner Roberto Carlos who had signed to join Turkish club Fenerbahçe.
Players
Squad information
Transfers
In
Total spending: €119 million
Out
Total income: €37.8 million
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Club
Technical staff
Kits
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† Only used against Alicante CF during Copa del Rey round of 32 first leg.
Other information
Competitions
La Liga
League table
Results by round
Matches
Champions League
Group C
Round of 16
Copa del Rey
Round of 32
Round of 16
Supercopa de España
Friendlies
Russian Railways Cup
Teresa Herrera Trophy
Trofeo Ramón de Carranza
Trofeo Santiago Bernabéu
Majed Abdullah retiring festival
Statistics
Squad stats
Disciplinary record
See also
2007–08 La Liga
2007–08 Copa del Rey
2007–08 UEFA Champions League
References
Real Madrid
Real Madrid CF seasons
Spanish football championship-winning seasons
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https://en.wikipedia.org/wiki/Bounded%20deformation
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In mathematics, a function of bounded deformation is a function whose distributional derivatives are not quite well-behaved-enough to qualify as functions of bounded variation, although the symmetric part of the derivative matrix does meet that condition. Thought of as deformations of elasto-plastic bodies, functions of bounded deformation play a major role in the mathematical study of materials, e.g. the Francfort-Marigo model of brittle crack evolution.
More precisely, given an open subset Ω of Rn, a function u : Ω → Rn is said to be of bounded deformation if the symmetrized gradient ε(u) of u,
is a bounded, symmetric n × n matrix-valued Radon measure. The collection of all functions of bounded deformation is denoted BD(Ω; Rn), or simply BD, introduced essentially by P.-M. Suquet in 1978. BD is a strictly larger space than the space BV of functions of bounded variation.
One can show that if u is of bounded deformation then the measure ε(u) can be decomposed into three parts: one absolutely continuous with respect to Lebesgue measure, denoted e(u) dx; a jump part, supported on a rectifiable (n − 1)-dimensional set Ju of points where u has two different approximate limits u+ and u−, together with a normal vector νu; and a "Cantor part", which vanishes on Borel sets of finite Hn−1-measure (where Hk denotes k-dimensional Hausdorff measure).
A function u is said to be of special bounded deformation if the Cantor part of ε(u) vanishes, so that the measure can be written as
where H n−1 | Ju denotes H n−1 on the jump set Ju and denotes the symmetrized dyadic product:
The collection of all functions of special bounded deformation is denoted SBD(Ω; Rn), or simply SBD.
References
Functional analysis
Materials science
Solid mechanics
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https://en.wikipedia.org/wiki/Cylindrical%20algebraic%20decomposition
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In mathematics, cylindrical algebraic decomposition (CAD) is a notion, and an algorithm to compute it, that are fundamental for computer algebra and real algebraic geometry. Given a set S of polynomials in Rn, a cylindrical algebraic decomposition is a decomposition of Rn into connected semialgebraic sets called cells, on which each polynomial has constant sign, either +, − or 0. To be cylindrical, this decomposition must satisfy the following condition: If 1 ≤ k < n and π is the projection from Rn onto Rn−k consisting in removing the last k coordinates, then for every pair of cells c and d, one has either π(c) = π(d) or π(c) ∩ π(d) = ∅. This implies that the images by π of the cells define a cylindrical decomposition of Rn−k.
The notion was introduced by George E. Collins in 1975, together with an algorithm for computing it.
Collins' algorithm has a computational complexity that is double exponential in n. This is an upper bound, which is reached on most entries. There are also examples for which the minimal number of cells is doubly exponential, showing that every general algorithm for cylindrical algebraic decomposition has a double exponential complexity.
CAD provides an effective version of quantifier elimination over the reals that has a much better computational complexity than that resulting from the original proof of Tarski–Seidenberg theorem. It is efficient enough to be implemented on a computer. It is one of the most important algorithms of computational real algebraic geometry. Searching to improve Collins' algorithm, or to provide algorithms that have a better complexity for subproblems of general interest, is an active field of research.
Implementations
Mathematica: CylindricalDecomposition
QEPCAD -- Quantifier Elimination by Partial Cylindrical Algebraic Decomposition
redlog
Maple: The RegularChains Library and ProjectionCAD
References
Basu, Saugata; Pollack, Richard; Roy, Marie-Françoise Algorithms in real algebraic geometry. Second edition. Algorithms and Computation in Mathematics, 10. Springer-Verlag, Berlin, 2006. x+662 pp. ; 3-540-33098-4
Strzebonski, Adam. Cylindrical Algebraic Decomposition from MathWorld.
Cylindrical Algebraic Decomposition in Chapter 6 ("Combinatorial Motion Planning") of Planning algorithms by Steven M. LaValle. Accessed 8 February 2023
Caviness, Bob; Johnson, Jeremy; Quantifier Elimination and Cylindrical Algebraic Decomposition. Texts and Monographs in Symbolic Computation. Springer-Verlag, Berlin, 1998.
Collins, George E.: Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition, Second GI Conf. Automata Theory and Formal Languages, Springer LNCS 33, 1975.
Davenport, James H.; Heintz, Joos: Real quantifier elimination is doubly exponential, Journal of Symbolic Computation, 1988. Volume 5, Issues 1–2, ISSN 0747-7171,
Algebra
Real algebraic geometry
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https://en.wikipedia.org/wiki/David%20Karoly
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David John Karoly (born 1955) is an Australian atmospheric scientist, currently based at CSIRO.
Education and academic career
In the early 1970s David Karoly enrolled in applied mathematics at Monash University, Melbourne, but later became interested in meteorology. In 1980 he was awarded a doctorate in meteorology from the University of Reading in Reading, England.
After returning to Australia, from 1995 to 2000 Karoly became Director of the Cooperative Research Centre for Southern Hemisphere Meteorology at Monash University. Between 2003 and 2007 he was Professor of Meteorology in the School of Meteorology at the University of Oklahoma (OU). In May 2007 he joined the School of Earth Sciences at the University of Melbourne as a Federation Professor. In 2017 he became Leader of the Earth Systems and Climate Change Hub in the Australian Government's National Environmental Science Program.
Contributions
He is an expert in climate change, stratospheric ozone depletion, and climate variations due to the El Niño-Southern Oscillation (ENSO).
Karoly has served as a lead author for the Intergovernmental Panel on Climate Change (IPCC) Working Group 2 (on societal impacts) and he is a member of the faculty of the School of Earth Sciences at the University of Melbourne. His work, along with that of the many other lead authors and review editors, contributed to the award of the 2007 Nobel Peace Prize, which was won jointly by the IPCC and Al Gore. He is member of the board of the Climate Change Authority.
David Karoly as communicator
In the Australian scene, Karoly is credited with standing up to Alan Jones, a conservative and climate change denying Sydney radio commentator with the largest daily audience in that city of ca.150,000.
In doing so Karoly, originally a sceptic (1980), has earned the reputation of being a climate scientist communicator with the ability to explain the complexities of his research to the general public. On The Australian Broadcasting Corporation’s high ranking television program QndA wherein audience members can ask direct questions of experts, Karoly claimed his authority by stating in regard to his nemesis, “I am a climate scientist and Alan Jones is wrong.”
Karoly pointed out that one hundred years ago carbon dioxide in the atmosphere was 280 parts per million whereas now it is 400 parts per million, an increase of 40% which he asserted was unquestionably caused by human activity. He also admonished Australians for producing 1.5% of the world's greenhouse gases when they were only 0.3% of the world's population.
On the population issue Karoly sounded a warning, saying, "climate scientists in Europe have said that the long-term sustainable population of people on the Earth is about 1 billion people in 2100 – not the foreshadowed United Nations population estimates of about 10 to 12 billion people. That’s not good news."
References
External links
David Karoly on SlowTV
Australian meteorologists
Australian climatol
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https://en.wikipedia.org/wiki/Magnus%20Hestenes
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Magnus Rudolph Hestenes (February 13, 1906 – May 31, 1991) was an American mathematician best known for his contributions to calculus of variations and optimal control. As a pioneer in computer science, he devised the conjugate gradient method, published jointly with Eduard Stiefel.
Biography
Born in Bricelyn, Minnesota, Hestenes graduated with a B.S. in 1927 from St. Olaf College and with an M.A. in 1928 from the University of Wisconsin–Madison. He earned his Ph.D. at the University of Chicago in 1932 under Gilbert Bliss. His dissertation was titled "Sufficient Conditions for the General Problem of Mayer with Variable End-Points." After teaching as an associate professor at Chicago, in 1947 he moved to a professorship at UCLA. He continued there until his retirement in 1973, and during that time he served as department chair from 1950 to 1958. While a professor, Hestenes supervised the thesis research of 34 students, among them Glen Culler, Richard Tapia and Jesse Wilkins, Jr.
Hestenes received the Guggenheim (1954) and Fulbright awards, was a vice president of the American Mathematical Society, and was an invited speaker at the 1954 International Congress of Mathematicians in Amsterdam.
He is the father of mathematician and physicist David Hestenes.
He died on May 31, 1991, in Los Angeles, California.
Selected publications
(M.A. thesis)
References
External links
Obituary in the Numerical Analysis Digest
1906 births
1991 deaths
St. Olaf College alumni
University of Wisconsin–Madison alumni
University of Chicago alumni
University of California, Los Angeles faculty
20th-century American mathematicians
Burials at Pacific View Memorial Park
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https://en.wikipedia.org/wiki/Devil%27s%20curve
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In geometry, a Devil's curve, also known as the Devil on Two Sticks, is a curve defined in the Cartesian plane by an equation of the form
The polar equation of this curve is of the form
.
Devil's curves were discovered in 1750 by Gabriel Cramer, who studied them extensively.
The name comes from the shape its central lemniscate takes when graphed. The shape is named after the juggling game diabolo, which was named after the Devil and which involves two sticks, a string, and a spinning prop in the likeness of the lemniscate.
For , the central lemniscate, often called hourglass, is horizontal. For it is vertical. If , the shape becomes a circle.
The vertical hourglass intersects the y-axis at . The horizontal hourglass intersects the x-axis at .
Electric Motor Curve
A special case of the Devil's curve occurs at , where the curve is called the electric motor curve. It is defined by an equation of the form
.
The name of the special case comes from the middle shape's resemblance to the coils of wire, which rotate from forces exerted by magnets surrounding it.
References
External links
The MacTutor History of Mathematics (University of St. Andrews) – Devil's curve
Plane curves
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https://en.wikipedia.org/wiki/Isomorphism%20extension%20theorem
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In field theory, a branch of mathematics, the isomorphism extension theorem is an important theorem regarding the extension of a field isomorphism to a larger field.
Isomorphism extension theorem
The theorem states that given any field , an algebraic extension field of and an isomorphism mapping onto a field then can be extended to an isomorphism mapping onto an algebraic extension of (a subfield of the algebraic closure of ).
The proof of the isomorphism extension theorem depends on Zorn's lemma.
References
D.J. Lewis, Introduction to algebra, Harper & Row, 1965, Chap.IV.12, p.193.
Field (mathematics)
Theorems in abstract algebra
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https://en.wikipedia.org/wiki/The%20Ouse
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The Ouse is a tidal estuary in northern Shapinsay, Orkney Islands. This water body has been shown on early maps of the island in a very similar shape to its current geometry. The Ouse is fed by small rivulets and upland springs that rise on the western part of the island's northeast spur. pH levels of these feed waters are moderately alkaline, in the range of 9.1.
See also
Burroughston Broch
Mor Stein
Vasa Loch
Footnotes
Shapinsay
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https://en.wikipedia.org/wiki/Anthropomorphic%20polygon
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In geometry, an anthropomorphic polygon is a simple polygon with precisely two ears and one mouth. That is, for exactly three polygon vertices, the line segment connecting the two neighbors of the vertex does not cross the polygon. For two of these vertices (the ears) the line segment connecting the neighbors forms a diagonal of the polygon, contained within the polygon. For the third vertex (the mouth) the line segment connecting the neighbors lies outside the polygon, forming the entrance to a concavity of the polygon.
Every simple polygon has at least two ears (this is the two ears theorem) and every non-convex simple polygon has at least one mouth, so in some sense the anthropomorphic polygons are the simplest possible non-convex simple polygons.
It is possible to recognize anthropomorphic polygons in linear time.
References
External links
Euclidean geometry
Types of polygons
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