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https://en.wikipedia.org/wiki/274%20%28number%29 | 274 is the natural number following 273 and preceding 275.
In mathematics
274 is an even composite number.
274's sum of its proper divisors is 140.
The number 274 is the 13th tribonacci number. This is defined by the equations P(0)=P(1)=0 P(2)=1 and P(n)=P(n-1)+P(n-2)+P(n-3).
274 is the sum of 5 perfect cubes. It is the sum of 2³+2³+2³+5³+5³.
274 is a Stirling number of the first kind which counts the number of permutations and their number of cycles.
In technology
The number 274 is an area code in Northeast Wisconsin. The previous code in this area was 920 and the switch was made in May 2023.
MiR-274 is a type of microRNA which can mediate the communication between certain organs and tissues. This one in particular is used between neurons and tracheal cells.
The Heinkel He 274 was a German Bomber that was created during World War II. It was designed for long-distance trips to disrupt American involvement in Europe.
World Records
On June 2, 2021, Namco received a Guinness World Record by releasing its 274th intellectual property license in a single role-playing game series.
On July 13, 2019, 274 participants participated in a perfume appreciation relay race. Bath & Body Works (USA) received the record for organizing this event.
On April 15, 2023, Ronald Sarchain received the Guinness World Record for snapping 274 chopsticks using karate chops in one minute.
On September 14, 2013, Kabushikigaisha Shiseido Kamakurakoujou received the Guinness World Record for organizing an event in Fujisawa, Japan, where 274 people were painting their toenails simultaneously.
On April 19, 2022, Fayis Nazer achieved 274 hula hoop rotations on one arm in one minute in Abu Dhabi.
Other fields
The calendar years 274 AD and 274 BC.
In the French Republican calendar, The year 274 would be a year 9 cycle and be in 2064.
274 is the number of several highways in Canada, Japan, and the United States
References
Integers |
https://en.wikipedia.org/wiki/Bettina%20Richmond | Martha Bettina Richmond (née Zoeller, January 30, 1958 – November 22, 2009) was a German-American mathematician, mathematics textbook author, professor at Western Kentucky University, and murder victim.
Life
Richmond was born in Dresden on January 30, 1958, earned a vordiplom (the German equivalent of a bachelor's degree) from the University of Würzburg, and completed her Ph.D. at Florida State University in 1985. Her doctoral dissertation, Freeness of Hopf algebras over grouplike subalgebras, was supervised by Warren Nichols, a student of Irving Kaplansky.
She became a professor at Western Kentucky University, teaching there for 23 years. Topics in her mathematical research included abstract algebra, transformation semigroups, ring theory, and Hopf algebra, including the proof of the Nichols–Zoeller freeness theorem in Hopf algebra. With her husband, Thomas Richmond, she was the author of a mathematics textbook, A Discrete Transition to Advanced Mathematics. She also published works in recreational mathematics.
Murder
Richmond was stabbed to death on November 22, 2009, in the parking lot of a racquetball facility in downtown Bowling Green, Kentucky. According to the FBI, her murder was likely an opportunistic crime motivated by armed robbery. At the time of her death, she had been on leave from her faculty position to assist her father in Germany. The murder is still unsolved.
Selected publications
References
1958 births
2009 deaths
2009 murders in the United States
German emigrants to the United States
Scientists from Dresden
American mathematicians
American women mathematicians
Algebraists
University of Würzburg alumni
Florida State University alumni
Western Kentucky University faculty
People murdered in Kentucky
Female murder victims
Unsolved murders in the United States |
https://en.wikipedia.org/wiki/Leylah%20Fernandez%20career%20statistics | This is a list of career statistics of Canadian tennis professional Leylah Fernandez.
Performance timelines
Only main-draw results in WTA Tour, Grand Slam tournaments, Billie Jean King Cup and Olympic Games are included in win–loss records.
Singles
Current through the 2023 Hong Kong Open.
Doubles
Current through the 2023 French Open.
Significant finals
Grand Slam tournaments
Singles: 1 (1 runner-up)
Doubles: 1 (1 runner-up)
WTA 1000 tournaments
Doubles: 1 (runner-up)
WTA Tour finals
Singles: 5 (3 titles, 2 runner-ups)
Doubles: 3 (3 runner-ups)
ITF Circuit finals
Singles: 3 (1 title, 2 runner–ups)
Doubles: 4 (2 titles, 2 runner-ups)
Junior Grand Slam tournament finals
Girls' singles: 2 (1 title, 1 runner–up)
WTA Tour career earnings
As of 9 October 2023
Career Grand Slam statistics
Seedings
The tournaments won by Fernandez are in boldface, and advanced into finals by Fernandez are in italics.
Best Grand Slam singles results details
Head-to-head records
Record vs. top-10 ranked players
Active players are in boldface.
Record against No. 11–20 players
Fernandez's record against players who have been ranked world No. 11–20. Active players are in boldface:
Varvara Lepchenko
Anastasia Pavlyuchenkova
Alizé Cornet
Kaia Kanepi
Wang Qiang
Ana Konjuh
Elise Mertens
Alison Riske
Mihaela Buzărnescu
Top 10 wins
She has a record against players who were, at the time the match was played, ranked in the top 10.
Notes
References
Fernandez, Leylah |
https://en.wikipedia.org/wiki/289%20%28number%29 | 289 is the natural number following 288 and preceding 290.
In mathematics
289 is an odd composite number with only one prime factor.
289 is the 9th Friedman number. Friedman numbers are numbers that can be written by using its own digits the exact number of times they show up in the number. This one can be expressed as (8+9)².
289 is a perfect square being equal to 17². It is also the 7th number to only have 3 factors because it is a square of a prime number.
289 is the sum of perfect cubes. It is the sum of 1³+2³+4³+6³.
289 is equivalent to the sum of the first 5 whole numbers to their respective powers. It is equal to 0⁰+1¹+2²+3³+4⁴.
In technology
The area code 289 is shared with 905 in the area in southern Ontario surrounding the greater Toronto metropolitan area.
The Fluke 289 True-RMS Industrial Logging Digital Multimeter with Trendcapture is a tool designed to find signal in certain areas. It has a graphic visualization of this information that it displays on its screen. It can store up to 15,000 data points.
The 289 series is a style of train that is operated by the West Japan Railway Company. It uses a DC EMU as opposed to the dual-voltage 683 series that preceded it.
World Records
On February 2, 2021, Ariel Chahi grew a strawberry that weighed 289 g in Israel. This was confirmed to be the world's heaviest strawberry.
On December 2, 2022, the longest flight by a paper aircraft was achieved by Dillon Ruble, Nathaniel Erickson, and Garret Jensen at a distance of 289 ft.
On March 21, 2017, e.motion21 Inc received the Guinness World Record for 289 people drumming on Swiss Balls simultaneously.
In the 2001-2002 horse racing season, Tom McCoy received the world record for achieving 289 steeplechase wins.
In the US Masters golf tournament, Sam Snead, Jack Burke Jr., and Zach Johnson all won the competition in 1954, 1956, and 2007 respectively. They all won with a score of 289 which is the highest winning score for this event.
Other fields
The calendar years 289 AD and 289 BC.
In the French Republican calendar, The year 289 would be a year 4 cycle and be in 2071.
289 is the number for several highways across the countries of Canada, Japan, the United Kingdom and the United States.
289 Nenetta is an asteroid in the Asteroid belt that was discovered by Auguste Charlois in Nice, France. It is considered an A-Type Asteroid being only 38 km across.
The 289th Engineer Combat Battalion was a combat engineer battalion during World War II in the United States. They participated in France and Germany, in particular the Battle of the Bulge, Rhineland Campaign, and the Invasion of Germany.
289P/Blanpain is a comet that orbits the sun every 5.31 years. It was discovered by Jean-Jacques Blanpain.
The 289 Commando Troop began as a section of the Territorial Army in London in the mid 1900's. It changed a few times before eventually becoming Plymouth-based and part of the Gloucestershire Volunteer Artillery.
References |
https://en.wikipedia.org/wiki/293%20%28number%29 | 293 is the natural number following 292 and preceding 294.
In mathematics
293 is an odd prime number with one prime factor, being itself.
293's sum of its proper divisors is 1.
293 is equivalent to the sum of the first three tetradic primes. Tetradic numbers are numbers that are the same if written backwards, flipped upside-down, or mirrored upside-down and tetradic primes are tetradic numbers that are also prime. It is the sum of 11 + 101 + 181.
293 can be written as the sum of perfect cubes. It is 2³+2³+3³+5³+5³
The total number of ways to make change for a dollar in US minted coins including the half-dollar and the dollar coin.
In technology
There is no area code in the United States with 293. There is an area code for +46 293 in Tierp, Sweden. The +46 part locates the code in Sweden and the 293 part locates it as Tierp.
293T is a cell line in humans that can assist in gene expression or DNA replication. It is has been tested on its ability to use certain viruses such as adenoviruses and other mammalian viruses.
The Omni-293 is a type of omni-directional antenna that includes 3.5 GHz of 5G bands. It is designed to be used in both rural and urban locations.<ref</ref>
World Records
In 1995, the Bagger 293 was developed by Takraf and was the heaviest machine that could move by itself. It weighs in a 31.3 million lbs.
On December 31, 2012, it was reported that in 2011, the United States was told to have the highest average watch time of TV per day at a 293 minutes.
On April 23, 2019, Jules and You in the Netherlands organized an event where 293 participants were fire breathing simultaneously.
On August 23, 2015, the Purple House Cancer Support in Ireland received the Guinness World Record for the greatest number of people dressed as sumo wrestlers at 293 people.
On May 15, 2018, Alberto Pires in Portugal, at the Lisbon Bar Show, hosted the largest port tasting event with a total of 293 people.
Other fields
The calendar years 293 AD and 293 BC.
In the French Republican calendar, The year 293 would be a year 5 cycle and be in 2085.
265 is the number of several different highways in Brazil, Canada, Israel, Japan, and the United States.
293 Brasilia is an asteroid in the Asteroid belt. It was discovered by Auguste Charlois in Nice, France. It is an X-type asteroid in the outer parts of the Asteroid belt.
The 293rd Rifle Division was a Red Army rifle division that was formed by Soviet Russia after the German invasion. It often fought with the German 2nd Panzer Group.
References
Integers |
https://en.wikipedia.org/wiki/Izbat%20Beit%20Hanoun | Izbat Beit Hanoun () is a Palestinian village in the North Gaza Governorate of the State of Palestine, in the Gaza Strip. According to the Palestinian Central Bureau of Statistics, Izbat Beit Hanoun had a population of 7,383 in mid-2006.
References
Villages in the Gaza Strip
Municipalities of the State of Palestine
North Gaza Governorate |
https://en.wikipedia.org/wiki/282%20%28number%29 | 282 is the natural number following 281 and preceding 283.
In mathematics
282 is an even composite number with three prime factors.
282 is a palindromic number. This is a number that is the same backwards as it is forwards. 282 is the smallest multi-digit palindromic number that is between twin primes, numbers that are prime and are 2 away from another prime number.
282 is equal to the sum of its divisors containing the number 4. It is the sum of 47 + 94 + 141.
282 is the number of planar partitions of 9. This means that 282 is the number of ways to separate 9 units.
In technology
The area code 282 is not in use in the North American numbering plan, but the area code is in use in Libya as +218 282. The +218 portion is the country code that places the phone number in Libya and the 282 piece labels it as Agelat.
The Haynes 282 alloy is a superalloy that is nickel-based and can withstand high temperatures. It was designed for use in industrial gas turbine engines.
A method for Dynamic Nuclear Polarization which uses microwaves to irradiate substances utilizes an electron frequency of 282 GHz.
World Records
On April 20, 2010, Dr.M completed, with a time of 2.82 seconds, the "Break the Targets" challenge in Super Smash Bros. Melee with the character Mr. Game and Watch.
On July 26, 2012, Curt Markwardt received the Guinness World Record for the highest backflip on a pogo stick. It totaled a height of 2.82 meters.
In 1998, several surfers managed to stack 282 surfboards on top of a Humvee and managed to drive 100 ft without the surfboards falling.
In the 2018 season, basketball player Sylvia Fowles received the Guinness World Record for grabbing 282 defensive rebounds for the Minnesota Lynx.
On June 18, 2005, Cristian Sterling achieved the longest golf carry with a total distance of 282 yards. This was achieved as St. Andrews Bay Golf Resort and Spa in Scotland.
On March 21, 2019, the fastest autonomous car was declared as the Robocar at Roborace with a speed of 282.42 km/h according to the UK Timing Association in Yorkshire.
Other fields
The calendar years 282 AD and 282 BC.
In the French Republican calendar, The year 282 would be a year 8 cycle and be in 2074.
282 is the number for several highways across the countries of Brazil, Canada, Japan, and the United States.
282 Clorinde is an asteroid in the Asteroid belt. This one was discovered by Auguste Charlois in Nice, France. This asteroid was named after Clorinda, who was a heroine of a poem by Torquato Tasso.
Hammurabi's Code was a series of laws in Ancient Mesopotamia, by the leader, Hammurabi. It has been an important insight into how Ancient Mesopotamia was run. A famous concept from this code is "an eye for an eye," or punishments equal to the crime. This code has 282 points.
References
Integers |
https://en.wikipedia.org/wiki/Einstein%E2%80%93Weyl%20geometry | An Einstein–Weyl geometry is a smooth conformal manifold, together with a compatible Weyl connection that satisfies an appropriate version of the Einstein vacuum equations, first considered by and named after Albert Einstein and Hermann Weyl. Specifically, if is a manifold with a conformal metric , then a Weyl connection is by definition a torsion-free affine connection such that
where is a one-form.
The curvature tensor is defined in the usual manner by
and the Ricci curvature is
The Ricci curvature for a Weyl connection may fail to be symmetric (its skew part is essentially the exterior derivative of .)
An Einstein–Weyl geometry is then one for which the symmetric part of the Ricci curvature is a multiple of the metric, by an arbitrary smooth function:
The global analysis of Einstein–Weyl geometries is generally more subtle than that of conformal geometry. For example, the Einstein cylinder is a global static conformal structure, but only one period of the cylinder (with the conformal structure of the de Sitter metric) is Einstein–Weyl.
Citations
References
.
.
Conformal geometry |
https://en.wikipedia.org/wiki/Margaret%20Gamalo | Margaret (Meg) Gamalo-Siebers is a Filipino-American biostatistician and drug development executive specializing in inflammation and immunology. She works for Pfizer as senior director – biostatistics, global product development – inflammation and immunology, and is editor-in-chief of the Journal of Biopharmaceutical Statistics.
Education and career
Gamalo earned a master's degree in applied mathematics and operations research from the University of the Philippines. She completed a Ph.D. in statistics at the University of Pittsburgh, in 2006.
She worked in the Center for Drug Evaluation and Research at the US Food and Drug Administration, and as a principal research scientist and research advisor for global statistical sciences at Eli Lilly, before taking her present position at Pfizer.
Recognition
Gamalo was named as a Fellow of the American Statistical Association, in the 2023 class of fellows.
References
Year of birth missing (living people)
Living people
Filipino emigrants to the United States
American statisticians
American women statisticians
Biostatisticians
University of Pittsburgh alumni
Fellows of the American Statistical Association |
https://en.wikipedia.org/wiki/267%20%28number%29 | 267 is the natural number following 266 and preceding 268.
In mathematics
267 is an odd composite number with two prime factors.
267 is the number of planar partitions of the number 12. Planar partitions are the number of ways in which the given number can be organized as split in an array.
267 is the sum of perfect cubes in two different ways. It is the sum of 1³+2³+2³+5³+5³ and 2³+2³+2³+3³+6³
In technology
The area code 267 is used for Philadelphia and the greater Philadelphia metropolitan area in Pennsylvania. It began use in 1999.
+267 is also a country code. This calling code comes from Botswana.
A scientific discovery that in 267 GHz observation of Venus, it presents phosphine was shown by ALMA. It has been declared that it is not statistically significant enough to consider.
The Setra's Model is a low differential pressure transducer, which measures the pressure in an area. It is primarily composed of its stainless steel capacitive sensing element that is good for long-term usage.
The Masters 267 is a Fishing Boat that is designed to comfortably fit 4 people and fishing gear. Its entire designs based on the comfort of the user.
World records
In 1985, E. Stone grew the heaviest rhubarb in Wiltshire, in the UK. It weighed in at 2.67 kg.
On November 13, 2021, the largest number of birthday wishes uploaded to a bespoke platform in an hour was 267 by Mecca Bingo (UK).
On On May 10, 2017, Leon Walraven received the Guinness World Record for the greatest number of football (soccer) touches with the shin with 267 touches in one minute.
On March 16, 2022, Current Food, Inc. produced the largest serving of ceviche with a total mass of 267 kg.
The highest concentration of male centenarians is in Sardinia, Italy with a percentage of 1.267%. Most areas on the other hand have a rate closer to .5%.
Other fields
The calendar years 267 AD and 267 BC.
In the French Republican calendar, The year 267 would be a year 3 cycle and be in 2059.
267 is the number for several highways across the countries of Canada, Ireland, Japan, and the United States.
267 Tirza is an asteroid in the asteroid belt. It was named after Tirzah from the bible.
The NGC 267 is an open cluster that is in the constellation Tucana. This constellation was discovered by John Herschel and written about by John Louis Emil Dreyer.
The Trigana Air Flight 267 was a flight that crashed an ATR 42 in Papua, Indonesia. The flight crashed into the Maoke Mountains and all 54 people on the plane died.
References |
https://en.wikipedia.org/wiki/Jorge%20M.%20L%C3%B3pez | Jorge Marcial López Fernández (1943-2021) (see Naming customs of Hispanic America) was a mathematician and mathematics educator. He directed several master theses as faculty at the Department of Mathematics, University of Puerto Rico, Río Piedras Campus which he chaired for eight years. Later he became an advocate for the Realistic Mathematics Education movement in Puerto Rico organizing influential projects in school mathematics education and publishing in academic journals.
Biography
Early life
The 1950 Census registers Jorge M. López, then six years old, living in Manuel Corchado street, Santurce, San Juan, Puerto Rico with his mother Isabel Hernández Sánchez who was a psichologist at the Hospital Maternal Infantil.
Education
Jorge López attended the then known as Escuela Modelo of the University of Puerto Rico (UPR), now University High School, in Río Piedras. In 1967 he earned a Bachelor in Arts in Mathematics at Reed College in Oregon, U.S.A.. He wrote the thesis Integration over Locally Compact Spaces and Haar Measure under the supervision of Prof. Larry Edison. At Reed he was active in student opposition to the Vietnam War.
In 1975 he earned his PhD. in Mathematics from the University of Oregon with a thesis titled Fatou-Zygmund Properties on Groups as a student of Kenneth A. Ross.
Death
Dr. López died on December 30, 2021 in the town of Aguas Buenas, Puerto Rico. A tribute was held at the La Torre building of the UPR - Río Piedras, as special honor for notable faculty.
Professional career
Jorge Lopez spent his entire career as a faculty in the Mathematics Department of the UPR-Río Piedras from 1975 until his retirement in 2014. Starting from 1978 and for ten years, López, Frank Anger and Víctor M. García Muñiz supervised most of the master's theses of that institution, which was the highest degree offered at the island during that period.
He was also an influential figure in mathematics education in Puerto Rico.
Harmonic analysis
During his studies at the University of Oregon, Jorge M. López co-wrote a book on Sidon sequences in harmonic analysis with Kenneth Ross.
These sequences were first introduced by Simon Sidon during his Fourier series research. His collaboration with Ross continued and in 2013 Ross acknowledged this partnership in the second edition of his renowned textbook, Elementary Analysis: The Theory of Calculus, by adding the annotation In collaboration with Jorge M. López.
After obtaining his PhD, he joined the Mathematics Department at the University of Puerto Rico, Río Piedras Campus where he continued to pursue his interest in this field through the late 1980s. During this time, he directed thirteen master's theses on subjects related to Wiener's Tauberian theorem and its generalization the Gelfand representation to commutative Banach algebras, C*-algebras and locally compact Hausdorff spaces.
Mathematics Education
Jorge López had a significant impact on primary education in mathematics in Pu |
https://en.wikipedia.org/wiki/Slaven%20Bari%C5%A1i%C4%87 | Slaven Barišić (January 26, 1942 – April 5, 2015) was a Croatian scientist, physicist, academician, full professor of the Faculty of Science and Mathematics and member of the Croatian Academy of Sciences and Arts.
Education
Barišić was born in Pleternica and attended elementary school there, and later in Zagreb. He graduated in theoretical physics in 1964 at the Faculty of Science in Zagreb, and obtained his master's degree (1968) and doctorate (1971) at the Faculty of Sciences, at the Université de Paris-Sud in Orsay under the guidance of Jacques Friedel.
Scientific career
From 1965 to 1972, he was a research assistant at the Institute of Physics in Zagreb. From 1967 to 1971, he was a researcher at the National Center for Scientific Research in France , where he received his doctorate. In 1972, he returned to his homeland and became an assistant professor. From 1976 he was an associate professor, and from 1979 he was a full professor in the Physics Department of the Faculty of Medicine.
Since his doctorate, he has been working in the field of theoretical physics of the solid state, namely on the theoretical physics of metals, especially high-temperature superconductors and phase transitions in low-dimensional conductive materials.
He has published over 100 scientific articles in international journals and anthologies in the field of the general theory of electron-photon binding in a strong (covalent) bond, the theory of chain (anisotropic) conductors, the analysis of low-dimensional phenomenological models of the Landau type and the theory of high-temperature superconductivity, mostly in renowned scientific journals, which have been cited about 1,800 times.
Social engagement
As vice-chancellor of the University of Zagreb from 1984 to 1986, and later, he publicly opposed the reform of secondary education.
He was one of the founders and vice president of the independent civic association Croatian Cultural Council.
Barišić died in Zagreb.
References
Croatian physicists
1942 births
2015 deaths |
https://en.wikipedia.org/wiki/294%20%28number%29 | 294 is the natural number following 293 and preceding 295.
In mathematics
294 is an even composite number with three prime factors.
294 is the number of planar biconnected graphs with 7 vertices. Biconnected graphs are two dimensional graphs with a given number of points and 294 is the number of ways to organize 7 vertices in different ways.
11115² - 294² = 123456789
The Magic Inscribed Lotus was created by Nārāyaṇa, and Indian Mathematician in the 14th century. In this inscription, each group of 12 numbers has a sum of 294. It was constructed with a 12 x 4 magic rectangle.
In 1930, George A. Miller determined that there are 294 isomorphic groups in the order of 64. Isomorphism is making a map that preserves relationships. This was later disproven as there are 267 isomorphic groups in the order of 64. See List of incomplete proofs.
In technology
The area code +52 294 is in use in Mexico, specifically in Veracruz.The +52 locates the phone number in Mexico and the 294 locates it as Veracruz.
A CMOS THz transmitter is used for short range links. It produces .47 mW with an output power of 294 GHz.
The UL Standard 294 is a number of guidelines that many facilities follow regarding the electronic access of control devices.
World Records
On October 20, 2019, Doshisha Kori Alumni Association in Japan received the Guinness World Record for writing 294 signatures on a t-shirt in one hour.
On November 29, 2015, in Mexico City, 294 participants were Beatles impersonators. A Beatles cover band lead them in a sing-along.
2008 was declared as the worst year regarding Afghan insurrection. By NATO's calculations, there were a total of 294 military deaths.
On December 3, 2011, in Saint-Genis-les-Ollières, the Guinness World Record for the greatest number of people in an indoor volleyball exhibition match was achieved with 294 people.
On December 21, 2000, Jessie Frankson kicked the highest martial arts kick at a height of 2.94m.
Other fields
The calendar years 294 AD and 294 BC.
In the French Republican calendar, The year 294 would be a year 6 cycle and be in 2086.
294 is the number for several highways across the countries of Japan and the United States.
294 Felicia is an asteroid in the Asteroid Belt that was discovered by Auguste Charlois in Nice, France.
Ogsnesson-294 is a radioactive isotope of the largest element on the periodic table. It is so radioactive that scientists reported a half-life of .69ms. It is extremely difficult to measure anything about this element before it decays.
References
Integers |
https://en.wikipedia.org/wiki/279%20%28number%29 | 279 is the natural number following 278 and preceding 280.
In mathematics
279 is an odd composite number with two prime factors.
Waring’s Conjecture is g(n)=2n+⌊(3/2)n⌋-2. When 8 is plugged in for n, the result is 279. That means that any positive integer can be formed with at most 279 numbers to the 8th power.
279 is the smallest number whose product of digits is 7 times the sum of its digits.
279 can be written as the sum of 4 nonzero perfect squares.
In technology
The area code of 279 was added to the Sacramento metropolitan area in California. It was added to the code of 916 in that area.
Tyrosine Kinase 2 Inhibitor or TAK-279 Inhibitor is a mediator of IL12 and IL23. TAK-279 is involved with certain inflammatory diseases like lupus and arthritis. TAK-279 inhibition is a possible way to treat these diseases.
World Records
On March 1, 2019, Benjamin Comparot and Carnival du Cor received the Guinness World Record for the largest French horn ensemble. It contained 279 people.
On October 6, 2023, EO Discover Okinawa had 279 people we simultaneously breaking roof tiles as a multi-day event.
On February 21, 2017? Dude Perfect achieved the Guinness World Record for the fastest time to make 5 3 point basketball shots in a time of 2.79 seconds.
On November 11, 2016, the tallest cake pyramid was built by Stratford University. It reached a total height of 2.79m.
In 2011, according to Twinning across the Developing World, Benin has the greatest rate of twins. It has 279 twins per 10000 births. Although some countries have a greater rate, this study is done without modern medicine.
Other fields
The calendar years 279 AD and 279 BC.
In the French Republican calendar, The year 279 would be a year 2 cycle and be in 2069.
279 is the number for several highways across the countries of Canada, Japan, and the United States.
279 Thule is a D-type asteroid in the asteroid belt. It was discovered by Johann Palisa in Vienna and was named after the land of Thule.
In Greenland, for the past 30 years, it has been losing an average of 279 billion tons of ice every year due to global warming.
References |
https://en.wikipedia.org/wiki/266%20%28number%29 | 266 is the natural number following 265 and preceding 267.
In mathematics
266 is an even composite number with three prime factors.
266 is a repdigit in base 11. In base 11, 266 is 222.
266 is a sphenic number being the product of 3 prime numbers.
266 is a nontotient number which is an even number, not in Euler’s totient function.
266 is an inconsummate number.
In technology
+266 is the calling code in Lesotho.
H.266 or Versatile Video Coding (VVC) is a video compression service. It was created by the Joint Video Experts Team as ITU and was created in 2017. It is designed to handle anywhere between 4k and 16k streaming.
World Records
On September 10, 2022, Delgadill’s Snow Cap offered 266 milkshake varieties. In support of Route 66, they created 266 flavors of milkshake.
On November 2, 2019, the Guinness World Record for the greatest number of people in the shape of a gemstone ring was completed in Kunming, China. There were 266 participants
On October 27, 2017, Casio Computer Co., LTD, and Hamura R&D Center created the largest human image of a watch with 266 people.
On February 13, 2022, Team Hantare Nacs received the Guinnes World Record for setting up 266 dominoes and knocking them down in one minute by a team of 8.
On October 17, 2013, 266 people in Quitman, Georgia were tossing skillets simultaneously.
Other fields
The calendar years 266 AD and 266 BC.
In the French Republican calendar, The year 266 would be a year 2 cycle and be in 2058.
266 is the number for several highways across the countries of Canada, Japan, and the United States.
266 Aline is an asteroid in the asteroid belt. It was discovered by Johann Palisa in Vienna. He named it after fellow astronomer Edmund Weiss' daughter.
Jorge Mario Bergoglio was elected the 266th pope on March 13, 2013 as Pope Francis.
References |
https://en.wikipedia.org/wiki/Fernando%20Zalamea | Fernando Zalamea Traba (Bogota, 28 February 1959) is a Colombian mathematician, essayist, critic, philosopher and popularizer, known by his contributions to the philosophy of mathematics, being the creator of the synthetic philosophy of mathematics. He is the author of around twenty books and is one of the world's leading experts on the mathematical and philosophical work of Alexander Grothendieck, as well as in the logical work of Charles S. Peirce.
Currently, he is a full professor in the Department of Mathematics of the National University of Colombia, where he has established a mathematical school, primarily through his ongoing seminar of epistemology, history and philosophy of mathematics, which he conducted for eleven years at the university. He is also known for his creative, critical, and constructive teaching of mathematics. Zalamea has supervised approximately 50 thesis projects at the undergraduate, master's and doctoral levels in various fields, including mathematics, philosophy, logic, category theory, semiology, medicine, culture, among others. Since 2018, he has been an honorary member of the Colombian Academy of Physical Exact Sciences and Natural. In 2016, he was recognized as one of the 100 most outstanding contemporary interdisciplinary global minds by "100 Global Minds, the most daring cross-disciplinary thinkers in the world," being the only Latin American included in this recognition.
References
National University of Colombia
Living people
Mathematics and culture
Academic staff of the National University of Colombia
1959 births
Colombian mathematicians
es:Fernando Zalamea |
https://en.wikipedia.org/wiki/268%20%28number%29 | 268 is the natural number following 267 and preceding 269.
In mathematics
268 is an even composite number with two prime factors, but one of the prime factors is repeated: 268 = 67*2*2.
268 is the smallest number whose product of digits is 6 times the sum of its digits.
268 is untouchable which means that it is not the sum of the proper divisors of any number
268 is the sum of the consecutive primes 131 and 137.
In technology
The area code for 268 is in Antigua and Barbuda.
The country code +268 is the calling code for Eswatini, formerly Swaziland.
MF 268 is a chemical compound with the formula of C28H46N4O3. It reacts at the abiotic site of the enzyme.
World records
Youtuber Hassan Suliman (AboFlah) received 2 world records during a live stream. The first was the longest video live stream with a total length of 268 hours. The second was the most viewers on a charity live stream on youtube with 698000 viewers.
On January 17, 2013, The Procter & Gamble Company received the Guinness World Record for the most people using teeth-whitening strips simultaneously. There were 268 participants.
On December 9, 2012, 268 people were brushing dog’s teeth. It was an event to promote dog’s dental health awareness.
On April 17, 2016, there were 268 people in Hikone, Japan dressed as ninjas. https://www.guinnessworldrecords.com/world-records/95237-largest-gathering-of-people-dressed-as-ninjas
On June 19, 2011, the lowest score in the golf US Open was achieved by Rory McIlroy. The score was 268 in 72 holes.
Other fields
The calendar years 268 AD and 268 BC.
In the French Republican calendar, The year 268 would be a year 4 cycle and be in 2060.
267 is the number for several highways across the countries of Canada, Japan, and the United States.
268 Adorea is an asteroid in the asteroid belt. It was discovered by Alphonse Borrelly in Marseilles. It was named after adorea liba which were split cakes made by the Romans as sacrificial offerings. This was controversial because all asteroids before this point were named after people or mythological people.
References |
https://en.wikipedia.org/wiki/Cornelia%20Fabri | Cornelia Fabri (Ravenna, 9 September 1869 – Florence, 24 May 1915) was an Italian mathematician and the first woman to graduate in mathematics from University of Pisa (1891).
Life and work
Cornelia Fabri was born in Ravenna, Italy, into a noble family headed by Ruggero Fabri and Lucrezia Satanassi de Sordi. Her immediate family was well-schooled in math and science. Her grandfather, Santi Fabri, had been a mathematics graduate from the University of Bologna and taught at the College of Ravenna. Her father Ruggero Fabri, focused on scientific studies and graduated from the University of Rome in Physical and Mathematical Sciences. As a child, Cornelia demonstrated such an "uncommon ability" for scientific subjects that, with her father's approval, she enrolled in the city's technical institute becoming the only female in a class of males. She earned top marks, easily passed the entrance exam and was allowed to enroll in the Faculty of Physical, Mathematical and Natural Sciences at the University of Pisa. Again, she was the only woman and attended her lessons accompanied by her mother. She graduated in 1891.
Her university teacher Vito Volterra, mathematical physicist and president of the Accademia dei Lincei, supervised her dissertation and followed Fabri's progress throughout her university years and remembered her as follows:"I have a very vivid memory of Signorina Cornelia Fabri, my student at the University of Pisa around 1880, the first, and perhaps the best, among the many students I subsequently had in Turin and Rome. I remember that her degree exam was an event for the University of Pisa, not only because it was the first time a woman had come there to get her doctorate, but also because the test was supported admirably by the candidate, who achieved full marks, absolutes and praise. On that occasion the Illustrious Dean of the Faculty of Science, Professor Antonio Pacinotti, uttered lofty and timely words, noting all the importance of the event, and foreseeing the opening of a new era with the entry into the field of science, of eminent female personalities."Fabri's scientific work focused primarily on hydraulics and was intense but brief. Her last academic work was published in 1895. In 1902, after the death of her mother, she left Pisa and returned to Ravenna to look after the family properties and her father. He died in 1904. She continued to keep in contact with Professor Volterra through detailed correspondence and the two mathematicians exchanged letters until 1902. They met for the last time in 1905.
In Ravenna, Fabri dedicated herself to charities and charitable activities.
Just three months before she died, Fabri used the language of science to describe to her confessor the reasons she was considering becoming a nun. "My heart has always been suspended between two equal and opposing forces, which balance each other and keep me in perfect blindness as to what my future will be."
Fabri died at age 46 in Florence from pneumo |
https://en.wikipedia.org/wiki/Shavkat%20Ayupov | Shavkat Abdullayevich Ayupov (born September 14, 1952, in Tashkent) is a Soviet Uzbek scientist in the field of mathematics. He is an Academician of the Uzbekistan Academy of Sciences (1995). He is also a Senator in the Senate of the Oliy Majlis of the Republic of Uzbekistan (2020). He was awarded the title of Hero of Uzbekistan in 2021, and he holds the title of Distinguished Scientist of the Republic of Uzbekistan (2011).
Biography
He was born into an intellectual family. His father, Abdulla Talipovich Ayupov, was a participant in the Great Patriotic War and headed the Department of Philosophy at Tashkent University. His mother, Marguba Khamidova, was a doctor who worked as a therapist at the 4th City Clinical Hospital.
He graduated from Tashkent University in 1974 and was a student of Academician T.A. Sarymsakov. He earned a Candidate of Physical and Mathematical Sciences degree (1977) and a Doctor of Physical and Mathematical Sciences degree (1983).
He was a laureate of the Lenin Komsomol Prize as part of the authorial team consisting of Berdikulov, Musulmonkul Abdullaevich, Usmanov, Shukhrat Muttalibovich, and a research fellow at the Institute of Mathematics named after V.I. Romanovsky at the Academy of Sciences of the Uzbek SSR; Abdullaev, Rustambai Zairovich, an assistant at the Tashkent State Pedagogical Institute named after V.I. Lenin; Tikhonov, Oleg Yevgenyevich, an assistant, and Trunov, Nikolai Vasilievich, an associate professor at the V.I. Ulyanov Kazan State University, for their work on "Research on Operator Algebras and Non-Commutative Integration" (1989).
He became an Academician of the World Academy of Sciences in 2003. From 2008 to 2013, he was an associated member of the Abdus Salam International Centre for Theoretical Physics (ICTP) in Trieste, Italy. He became a member of the Senate of the Oliy Majlis of the Republic of Uzbekistan in 2020.
In 1992, he became the head of the Institute of Mathematics named after V.I. Romanovsky at the Academy of Sciences of the Republic of Uzbekistan.
In 1994, he embarked on a lengthy assignment at the University of Louis Pasteur in Strasbourg, France, where he conducted joint research with Professor J.-L. Lode.
He teaches at the National University of Uzbekistan and is a professor of the Department of Algebra and Functional Analysis.
Awards
"Oʻzbekiston Qahramon" (Hero of Uzbekistan, August 24, 2021)
"Mehnat shuhrati ordeni" (Order of "Mekhnat Shukhrati") (2003)
"Shuhrat" medali (Medal "Shukhrat" 1996)
"Oʻzbekiston fan arbobi" (Distinguished Scientist of the Republic of Uzbekistan, 2011)
Recipient of the State Prize of the 1st Degree in the field of science and technology (2017)
Literature
National Encyclopedia of Uzbekistan. The first volume. Tashkent, 2000
References
Living people
1952 births
National University of Uzbekistan alumni
Soviet mathematicians
Uzbekistani mathematicians |
https://en.wikipedia.org/wiki/275%20%28number%29 | 275 is the natural number following 274 and preceding 276.
In mathematics
275 is an odd composite number with 2 prime factors.
275 is equivalent to the number of partitions of 28 when no partition occurs only once. Partitions are the number of ways of writing a number as a sum of other positive integers.
275 is the sum of fifth powers of the first two primes (2^5 + 3^5 = 275).
275 is the maximum number of pieces made by cutting an annulus with 22 cuts.
275 is the smallest non semiprime that follows the equations n>1 and the greatest common denominator of n and b^n-b is 1 for some value of b.
World Records
In May 2011, Angry Birds ended its 275 day streak of the best-selling app in the Apple App Store.
On November 14, 2007, the Guinness World Record for the greatest number of people dressed as mobile phones at 275 people. It was completed in Puerto Rico.
On April 11, 2015, the largest welding lesson was completed in Willowbrook, Illinois. There were 275 attendees.
The greatest number of certificates that a show dog earned as 275 certificates. It was earned by German Shepherd Mystique.
Other fields
The calendar years 275 AD and 275 BC.
In the French Republican calendar, The year 275 would be a year 10 cycle and be in 2065.
275 is the number of several highways in Canada, India, Japan, and the United States
275 Sapientia is an asteroid in the asteroid belt. It is a C-type asteroid that was discovered by Johann Palisa.
References |
https://en.wikipedia.org/wiki/278%20%28number%29 | 278 is the natural number following 277 and preceding 279.
In mathematics
278 is an even composite number with 2 prime factors.
278 is equal to Φ(30). It is the sum of the totient function.
278 is a nontotient number which means that it is an even number that doesn't follow Euler's totient function.
278 is the smallest semiprime number that has an anagram that is also semiprime. The other number is 287.
World Records
On October 29, 2011, April Mathis lifted the heaviest squat weight by a female at 278 kg in Orlando, Florida.
On November, 10, 1993, Tom Rodden received the Guinness World Record for shaving 278 people in one hour.
On March 25, 2000, in Meadowhall Center, there were 278 human mannequins in the shopping center. They were in a total of 114 stores.
On December 5, 2006, James Cripps received the Guinness World Record for the highest score in backwards bowling. The score was 278.
Other fields
The calendar years 278 AD and 278 BC.
278 is the number of several highways in Canada, Japan, and the United States.
278 Paulina is an asteroid in the asteroid belt. It was discovered by Johann Palisa in Vienna.
References |
https://en.wikipedia.org/wiki/283%20%28number%29 | 283 is the natural number following 282 and preceding 284.
In mathematics
283 is an odd prime number with 1 prime factors.
283 is a twin prime number and a super prime. The former are two prime numbers that are only separated by a single number with 281. The latter is a prime number that is the nth prime where n is a prime number as well.
283 is a strictly non palindromic number. That means that between base 2 and base n-2, that number is never palindromic.
283 is such a number where 4283-3238 is prime.
283 is equivalent to 25+8+35.
World Records
In 2015, Kris Russell set the Guinness World Record for the greatest number of shots blocked in an NHL season. The record was 283 shots blocked.
According to Professor Olu Tomori, the English language has the greatest number of irregular verbs as 283.
On November 21, 2012, the fastest time to climb a vertical corridor was achieved in Beijing by Zhisheng Fang with a time of 28.3 seconds.
In the Marshall Islands, the largest atoll is found with a length of 283 km.
On May 13, 2023, the longest standing jump was completed by Lorenz Wetsher with a grand total of 2.83 m distance.
Other fields
The calendar years 283 AD and 283 BC.
283 is the number of several highways in Canada, Japan, and the United States.
283 Emma is an asteroid in the asteroid belt. It was discovered by Auguste Charlois in Nice, France.
References |
https://en.wikipedia.org/wiki/Jankov%E2%80%93von%20Neumann%20uniformization%20theorem | In descriptive set theory the Jankov–von Neumann uniformization theorem is a result saying that every measurable relation on a pair of standard Borel spaces (with respect to the sigma algebra of analytic sets) admits a measurable section. It is named after V. A. Jankov and John von Neumann. While the axiom of choice guarantees that every relation has a section, this is a stronger conclusion in that it asserts that the section is measurable, and thus "definable" in some sense without using the axiom of choice.
Statement
Let be standard Borel spaces and a subset that is measurable with respect to the analytic sets. Then there exists a measurable function such that, for all , if and only if .
An application of the theorem is that, given any measurable function , there exists a universally measurable function such that for all .
References
.
.
Descriptive set theory
Inverse functions
Measure theory |
https://en.wikipedia.org/wiki/284%20%28number%29 | 284 is the natural number following 283 and preceding 285.
In mathematics
284 is an even composite number with 2 prime factors.
284 is in the first pair of amicable numbers with 220. That means that the sum of the proper divisors are the same between the two numbers.
284 can be written as a sum of exactly 4 nonzero perfect squares.
284 is a nontotient number which are numbers where phi(x) equaling that number has no solution.
284 is a number that is the nth prime plus n. It is the 51st prime number (233) plus 51.
World Records
On July 3, 2010, Eduaedo Nakagawa achieved the Guinness World Record as the most prolific holder of records on Picross. He has the record for 284 of the 345 levels.
On June 21, 2019, the largest rhythmic yoga lesson was completed in Hefei, China. It had 284 attendees.
On October 25, 2005, Soulcalibur III was released on the PlayStation 2 with the most weapons in any video game. It had 284 weapons.
On October 17, 2022, the students of Tabla Talim Sanstha in Ahmedabad, India completed the longest hand-drumming relay with 284 people.
Other fields
The calendar years 284 AD and 284 BC.
283 is the number of several highways in Japan and the United States.
284 Amalia is an asteroid in the asteroid belt. It was discovered by Auguste Charlois in Nice, France and is a Ch-type asteroid.
References |
https://en.wikipedia.org/wiki/286%20%28number%29 | 286 is the natural number following 285 and preceding 287.
In mathematics
286 is an even composite number with 3 prime factors.
286 is in the smallest pair of nontotient anagrams with 268.
286 is a tetrahedral number which means that represents a tetrahedron.
286 is a sphenic number which means that it has exactly 3 prime factors.
286 the first even pseudoprime to base 3.
World Records
On November 7, 2013, Alastair Galpin received the Guinness World Record for the fastest time to slice 10 matches in half. He completed it in 2.86 seconds.
On April 27, 2013, the largest tiramisu-making lesson ensued in Japan with 286 participants.
On September 21, 2007, the largest steel drum ensemble was achieved in Poznan, Poland. There were 286 members.
On November 12, 2017, Toyota achieved the Guinness World Record for the greatest number of toy cars launched simultaneously. There were 286 toy cars.
Other fields
The calendar years 286 AD and 286 BC.
286 is the number of several highways in Japan and the United States.
286 Iclea is an asteroid in the asteroid belt. It was discovered by Johann Palisa in Vienna, Austria.
References |
https://en.wikipedia.org/wiki/Derek%20J.%20S.%20Robinson | Derek John Scott Robinson (born 25 September 1938 in Montrose, Scotland) is a British mathematician, specialising in group theory and homological algebra.
Education and career
Robinson graduated in 1960 with a bachelor's degree from the University of Edinburgh and in 1963 with a Ph.D. from the University of Cambridge. His Ph.D. thesis Theory of Subnormal Subgroups was supervised by Philip Hall. As a postdoc, Robinson was from 1963 to 1965 an instructor at the University of Illinois Urbana-Champaign. From 1965 to 1968 he was a lecturer at Queen Mary College (now named Queen Mary University of London). At the University of Illinois Urbana-Champaign he was an assistant professor from 1968 to 1969, an associate professor from 1969 to 1974, and a full professor from 1974 to 2007, when he retired as professor emeritus. He held visiting appointments in Switzerland, Italy, Germany, and Singapore.
Robinson's 1964 paper on T-groups has over 250 citations. He was awarded in 1970 the Sir Edmund Whittaker Memorial Prize and received in 1979 a Humboldt Prize.
Books
with S. Cruz Rambaud, J. García Pérez, and R.A. Nehmer: Algebraic models for accounting systems, World Scientific, Singapore, 2010.
A course in linear algebra with applications, World Scientific, 1991 ; 2nd edition 2006
An introduction to abstract algebra, De Gruyter 2003 ; Abstract algebra: an introduction with applications, 3rd edition 2022
with John C. Lennox: Theory of infinite soluble groups, Clarendon Press, Oxford 2004
A course in the theory of groups, Springer Verlag, 1982, 2nd edition 1996
Finiteness conditions and generalized soluble groups, Parts I & II, Springer Verlag 1972
Infinite soluble and nilpotent groups, London 1968
as editor with Phillip Griffith: The mathematical legacy of Reinhold Baer: a collection of articles in honor of the centenary of the birth of Reinhold Baer, University of Illinois, Urbana-Champaign, 2004
References
20th-century British mathematicians
21st-century British mathematicians
Alumni of the University of Edinburgh
Alumni of the University of Cambridge
University of Illinois Urbana-Champaign faculty
1938 births
Living people
Group theorists |
https://en.wikipedia.org/wiki/International%20Conference%20on%20Formal%20Power%20Series%20and%20Algebraic%20Combinatorics | The International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC) is an annual academic conference in the areas of algebraic and enumerative combinatorics and their applications and relations with other areas of mathematics, physics, biology and computer science.
History
FPSAC was first held in 1988 and has been held annually since 1990, typically in June, July or August.
The most recent conference in the series, FPSAC'23, was held in July 2023 at the University of California, Davis. The 2024 meeting is slated to take place at Ruhr-Universität Bochum in Bochum, Germany on July 22-26, 2024.
The proceedings of conferences in the series have appeared as a Springer volume,
and in the journals Discrete Mathematics, Discrete Mathematics and Theoretical Computer Science, and Séminaire Lotharingien de Combinatoire.
References
External Links
Mathematics conferences |
https://en.wikipedia.org/wiki/287%20%28number%29 | 287 is the natural number following 286 and preceding 288.
In mathematics
287 is an odd composite number with 2 prime factors.
287 is the sum of consecutive primes in three different ways, 89+97+101, 43+53+59+61+67, and 17+19+23+29+31+37+41+43+47
287 is a pentagonal number which follows the concept of triangular numbers.
287 is an odd semiprime number.
287 is the sum of exactly 4 nonzero squares.
287 is a number where 2(287)-1 and 2(287)+1 are both prime.
World Records
On November 15, 2012, in the United Kingdom, the largest toy construction lesson was achieved with 287 participants.
On August 8, 2018, in Japan, the largest blind soccer (football) lesson was completed with 287 members.
On February 28, 2020, the largest Welsh folk dance was completed in Swansea in the United Kingdom. There were 287 participants.
On June 20, 2009, Bora Milutinović completed the Guinness World Record for the greatest number of international soccer matches a coach has been in charge of. He was at 287 games.
On December 20, 2020, the largest group of people dressed as the sun was completed in Guangzhou, China. There were 287 participants.
Other fields
The calendar years 287 AD and 287 BC.
287 is the number of several highways in Brazil, Canada, Japan, and the United States.
287 Nephthys is an S-type asteroid in the asteroid belt. It was discovered by Christian Heinrich Friedrich Peters in Clinton, Oneida County, New York. It was named after Nephthys in Egyptian mythology.
References |
https://en.wikipedia.org/wiki/A.E.K.%20Athens%20H.C.%20in%20international%20handball%20competitions | A.E.K. Athens H.C. in international handball competitions is the history and statistics of AEK H.C. in EHF competitions.
AEK Athens has won one EHF European Cup.
Honours
EHF European Cup: 2017–18, 2018–19, 2020–21
EHF European competitions record
Statistics record by competition
References
External links
AEK Athens H.C. at EHF
Handball
Handball clubs in Greece
2005 establishments in Greece
Handball clubs established in 2005 |
https://en.wikipedia.org/wiki/310%20%28number%29 | 310 is the natural number following 309 and preceding 311.
In mathematics
310 is an even composite number with 3 prime factors.
310 is a sphenic number meaning that it has 3 prime factors.
310 is a noncototient number which means that m − φ(m) = n has no solution for n=310.
310 is the number of Dyks 11 paths with strictly intersecting peaks.
310 base 6 it is 1234
The sum of the divisors of 310 is a perfect square.
World Records
On September 14, 2016, Siamand Rahman achieved the heaviest power lift for a paralympic male at 310 kg lifted.
On July 21, 2023, the Guinness World Record for the longest microphone passing relay was completed in Zhangjiakou, China. There were 310 participants.
On October 21, 2018, the largest football tournament was achieved in Colorado. There were 310 participants.
On September 29, 2012, the greatest number of consecutive football appearances was 310 completed by Brad Friedel.
Other fields
The calendar years 310 AD and 310 BC.
310 is the number of several highways in Canada, China, Costa Rica, Japan, and the United States.
310 Margarita is an asteroid in the asteroid belt. It was discovered by Auguste Charlois in Nice, France.
References |
https://en.wikipedia.org/wiki/Rouse%20Ball%20Professor | Rouse Ball Professor may refer to:
Rouse Ball Professor of English Law
Rouse Ball Professor of Mathematics |
https://en.wikipedia.org/wiki/Tverberg | Tverberg may refer to:
People
Helge Tverberg (1935–2020),Norwegian mathematician
Ryan Tverberg (born 2002), Canadian ice hockey player
Other uses
Tverberg's theorem, mathematics theorem |
https://en.wikipedia.org/wiki/Finite%20subgroups%20of%20SU%282%29 | In applied mathematics, finite subgroups of are groups composed of rotations and related transformations, employed particularly in the field of physical chemistry. The symmetry group of a physical body generally contains a subgroup (typically finite) of the 3D rotation group. It may occur that the group with two elements acts also on the body; this is typically the case in magnetism for the exchange of north and south poles, or in quantum mechanics for the change of spin sign. In this case, the symmetry group of a body may be a central extension of the group of spatial symmetries by the group with two elements. Hans Bethe introduced the term "double group" (Doppelgruppe) for such a group, in which two different elements induce the spatial identity, and a rotation of may correspond to an element of the double group that is not the identity.
The classification of the finite double groups and their character tables is therefore physically meaningful and is thus the main part of the theory of double groups. Finite double groups include the binary polyhedral groups.
In physical chemistry, double groups are used in the treatment of the magnetochemistry of complexes of metal ions that have a single unpaired electron in the d-shell or f-shell. Instances when a double group is commonly used include 6-coordinate complexes of copper(II), titanium(III) and cerium(III). In these double groups rotation by 360° is treated as a symmetry operation separate from the identity operation; the double group is formed by combining these two symmetry operations with a point group such as a dihedral group or the full octahedral group.
Definition and theory
Let be a finite subgroup of SO(3), the three-dimensional rotation group. There is a natural homomorphism of SU(2) onto SO(3) which has kernel {±I}. This double cover can be realised using the adjoint action of SU(2) on the Lie algebra of traceless 2-by-2 skew-adjoint matrices or using the action by conjugation of unit quaternions. The double group is defined as –1 (). By construction {±I} is a central subgroup of and the quotient is isomorphic to . Thus is a central extension of the group by {±1}, the cyclic group of order 2. Ordinary representations of are just mappings of into the general linear group that are homomorphisms up to a sign; equivalently, they are projective representations of with a factor system or Schur multiplier in {±1}. Two projective representations of are closed under the tensor product operation, with their corresponding factor systems in {±1} multiplying. The central extensions of by {±1} also have a natural product.
The finite subgroups of SU(2) and SO(3) were determined in 1876 by Felix Klein in an article in Mathematische Annalen, later incorporated in his celebrated 1884 "Lectures on the Icosahedron": for SU(2), the subgroups correspond to the cyclic groups, the binary dihedral groups, the binary tetrahedral group, the binary octahedral group, and the binary icosahedral g |
https://en.wikipedia.org/wiki/Lucien%20Hibbert | Lucien Hibbert (18 August 1899-5 February 1964) was a Haitian public servant and mathematician. He was the first Haitian to receive a doctoral degree in mathematics and is remembered for his roles in government and higher education administration. In the administration of President Sténio Vincent, Hibbert served as the minister of finance for Haiti from 1932 to 1934 and was subsequently foreign minister until 1935.
Hibbert earned his doctoral degree from the Université de Paris in 1937. His dissertation consisted of two theses, Univalence et automorphie pour les polynômes et les fonctions entières and Sur les équations du problème de l'Interdépendance des Marchés.
References
Finance ministers of Haiti |
https://en.wikipedia.org/wiki/Wolfgang%20Gasch%C3%BCtz | Wolfgang Gaschütz (11 June 1920 – 7 November 2016) was a German mathematician, known for his research in group theory, especially the theory of finite groups.
Biography
Gaschütz was born on 11 June 1920 in Karlshof, Oderbruch. He moved with his family in 1931 to Berlin, where he completed his Abitur in 1938. He served as an artillery officer in WW II, which ended for him in 1945 near Kiel. There in autumn 1945 he matriculated at the University of Kiel. He was inspired by Andreas Speiser's book Die Theorie der Gruppen von endlicher Ordnung. Gaschütz received his Ph.D. (Promotion) in 1949 under the supervision of Karl-Heinrich Weise with doctoral dissertation entitled (Zur -Untergruppe endlicher Gruppen). In 1953 Gaschütz completed his habilitation in Kiel. At the University of Kiel he held the junior academic appointments Wissenschaftliche Hilfskraft from 1949 to 1956 and Diätendozent from 1956 to 1959. He was Außerplanmäßiger Professor from 1959 to 1962, professor extraordinarius from 1962 to 1964, and professor ordinarius (full professor) from 1964 to 1988. He taught at Kiel until his retirement as professor emeritus in 1988. He rejected calls to Karlsruhe and Mainz. He was a visiting professor at various universities in Europe (Queen Mary College London 1965 and 1970, University of Padua 1966, University of Florence 1971, University of Naples Federico II 1974, University of Warwick 1967, 1973 & 1977); in the USA (Michigan State University 1963, University of Chicago 1968); and in Australia (Australian National University in Canberra).
Gaschütz created a school of group theorists in Kiel, where there had been a gap in mathematical expertise in algebra since the death of Ernst Steinitz in 1928. Gaschütz, influenced by Helmut Wielandt in the 1950s, is best known for his research on Frattini subgroups, on questions of complementability, on group cohomology, and on the theory of finite solvable groups. In 1959 he gave a formula for the Eulerian function introduced in 1936 by Philip Hall and determined the number of generators of a finite solvable group in terms of structure and embedding of the chief factors of the Eulerian function.
In 1962, Gaschütz published his theory of formations, giving a unified theory of Hall subgroups and Carter subgroups. Gaschütz's theory is important for understanding finite solvable groups. He characterized solvable T-groups. He is one of the pioneers of the theory of Fitting classes begun by Bernd Fischer in 1966 and the theory of Schunk classes.
Gaschütz organized the Oberwolfach conferences on group theory for many years with Bertram Huppert and Karl W. Gruenberg. In 2000 Gaschütz received an honorary doctorate from Francisk Skorina Gomel State University in Belarus. His doctoral students include Joachim Neubüser.
Gaschütz and his wife Gudrun were married in 1943 and became the parents of a son and two daughters.
Gaschütz died in Kiel on 7 November 2016, at the age of 96.
Selected publications
Zur Erweiter |
https://en.wikipedia.org/wiki/Susan%20Murabana | Susan Murabana Owen is a Kenyan astronomer. The co-founder of Travelling Telescope, she is known for her efforts to promote science, technology, engineering and mathematics in Africa, particularly among girls.
Early life and education
Murabana grew up in Nairobi, Kenya and studied sociology and economics at the city's Catholic University of Eastern Africa. In 2011, she graduated with a master's degree in astronomy from James Cook University in Australia, having studied online. Murabana has cited African American astronaut Mae Jemison and Kenyan environmentalist Wangari Maathai as her heroes.
Astronomy career
Voluntary work
Murabana first became interested in space when she was in her twenties, when her uncle invited her to attend a science outreach programme in Mumias, Kakamega County, facilitated by Cosmos Education. She subsequently became a volunteer for the organisation, and later went on to work with the International Astronomical Union's Global Hands-on Universe programme.
Travelling Telescope
In 2013, Murabana met her future husband, Daniel "Chu" Owen, at a solar eclipse at Lake Turkana in the Kenyan Rift Valley. Owen had previously established Travelling Telescope, in which he had travelled around his home country, the United Kingdom, allowing the public to look at space through his telescope. In 2014 and 2015, Travelling Telescope was relaunched in Kenya as a social enterprise aiming to educate poor and remote communities about science and astronomy. The organisation raises money by holding private events, such as the annual Shooting Star Safari in Samburu County during the Perseids meteor shower, as well as providing astronomy services to private schools and safari lodges. The money is subsequently used to provide free outreach work to state schools and remote communities throughout Kenya.
Travelling Telescope's outreach work includes utilising Sky-Watcher Flextubes to allow people to see planets including Jupiter, Mars, Saturn and Venus; the Orion and Trifid nebulae; and the Pinwheel and Andromeda galaxies. The organisation has an inflatable planetarium, and in 2020 established the Nairobi Planetarium, Africa's first permanent planetarium, constructed out of bamboo. Murabana has also run space camps in Nairobi.
During the COVID-19 pandemic, Travelling Telescope received funding from the Airbus Foundation to establish online classes in astronomy, rocketry and robotics for African schoolchildren.
Other work
Murabana has served as a national advisor for Kenya at Universe Awareness; the African representative on the International Planetarium Association; and as a board member for World Space Week. She is also the president of the African Planetarium Association, which aims to establish more permanent planetariums around the continent.
In 2021, Murabana was named as a Space4Women mentor, as part of the United Nations' women in STEM programming.
Recognition
In 2020, Murabana and Travelling Telescope was awarded the Europlanet A |
https://en.wikipedia.org/wiki/Khun%20Kyaw%20Zin%20Hein | Khun Kyaw Zin Hein (born 15 July 2002) is a Burmese professional footballer currently playing as a Left midfielder for Myanmar National League side Hanthawaddy United.
Club statistics
Myanmar National Team
Khun Kyaw Zin Hein played his first international World Cup Qualifier match against Macau national football team and assist Lwin Moe Aung goal.
International
References
2002 births
Living people
Burmese men's footballers
Myanmar men's international footballers
Myanmar National League players
Hantharwady United F.C. players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Beat-Sofi%20Granqvist | Beat-Sofi Granqvist (1869 – 1960) was a Finnish actress and florist.
Beat-Sofi Granqvist's parents were Karl Emil Granqvist (1830–1889), a mathematics and natural history teacher at Pori upper elementary school, and his first wife, Maria Sofia Nyström. Beat-Sofi Granqvist was a student of Kaarlo Bergbom. She acted in the Swedish Theatre in Helsinki. She also went on tour in Sweden.
In the 1910s, Granqvist studied the manufacture of artificial flowers in Germany. After returning to Finland, she founded the country's first artificial flower factory, next to her apartment at Pieni Roobertinkatu 4-6.
References
1869 births
1960 deaths
Finnish actresses
Finnish stage actresses
Florists |
https://en.wikipedia.org/wiki/Alena%20Varmu%C5%BEov%C3%A1 | Alena Varmužová (24 April 1939 – 7 August 1997) was a Czech mathematician. She was specialized in creating teaching systems for mathematics education of young students.
Life and work
Alena Varmužová was born on 24 April 1939 in Rožnov pod Radhoštěm. She graduated from the Faculty of Science in Olomouc, having completed the coursework for mathematics and descriptive geometry. Then she enrolled in the Pedagogical Institute in Ostrava where she earned the title Doctor of Natural Sciences (1975), in 1988 she defended her candidate's thesis Content definition of the didactic system of mathematical preparation of preschool children, which dealt with designing systems to effectively teach mathematics to young students.
In 1990, she was appointed associate professor for mathematics didactics at the Faculty of Education of the University of Ostrava, and she became head of the Department of Mathematics Didactics. She worked at the Union of Czech mathematicians and physicists.
Varmužová authored the section of a textbook for secondary schools that was titled Reasoning Education - Mathematical Concepts. In 1991 she published her book, Mathematics for Preschool Children.
Varmužová died on 7 August 1997 at the age of 58. Her husband was the sculptor Vratislav Varmuža.
References
1939 births
1997 deaths
Czech mathematicians
Women mathematicians
20th-century Czech mathematicians
20th-century women mathematicians
Czech women educators
People from Rožnov pod Radhoštěm |
https://en.wikipedia.org/wiki/Guy%20Brousseau | Guy Brousseau is a French mathematics educationalist, born on February 4, 1933 in Taza, Morocco.
Early life and education
Guy Brousseau was born on February 4, 1933 in Taza, Morocco. From an early age, he wanted to become a primary school teacher, which he did for several years until he was recruited as an assistant at Bordeaux University. From 1967 to 1969, he was Director of the Centre de recherches pour l'enseignement des mathématiques at the CRDP de Bordeaux, and in 1969 became Assistant of Mathematics at the Faculté des Sciences de Bordeaux. In 1968, he obtained a bachelor's degree in mathematics and a bachelor's degree in educational science.
Career and research
He began his career as a teacher in 1953. He began publishing in 1961, followed by a textbook for the first year of elementary school (1965), and continued to publish in the scientific field. In the late 1960s, after obtaining a degree in mathematics, he joined the University of Bordeaux.
He founded COREM (Centre pour l'Observation et la Recherche sur l'Enseignement des Mathématiques), which he ran from 1973 to 1998 at the Jules Michelet elementary school in Talence (Gironde). The school went on to achieve international renown. He subsequently founded the LADIST (Laboratoire Aquitain de Didactique des Sciences et Techniques), which supported COREM.
In 1986, he obtained a doctorate in science and, in 1991, became a university professor at the newly-created IUFM d'Aquitaine, where he worked until 1998. He decided to create the CREM (Centre de Recherche pour l'Enseignement des Mathématiques) in Bordeaux following his meeting with André Lichnerowicz. He is currently Professor Emeritus at the IUFM d'Aquitaine. He also holds honorary doctorates from the University of Montreal (1997) and the University of Geneva (2004).
His main theoretical contribution is the theory of didactic situations, a theory initiated in the early 1970s. Together with Gérard Vergnaud's conceptual field theory and Yves Chevallard's anthropological theory of didactics, it forms one of the three main theoretical frameworks of the French school of mathematics didactics.
He published numerous works on mathematics from 1965 to 2001, with the help of numerous collaborators.
He has carried out a large number of research and training missions in Europe, Latin America and North America, as well as in North Africa and South-East Asia. His research focuses on the teaching of natural and decimal numbers, probability, statistics, geometry, elementary algebra, logic and reasoning.
Awards and honours
In 2003, he was awarded the first Felix Klein Medal by the International Commission on Mathematical Instruction.
Publications
Brousseau, Guy (1965). Les mathématiques du cours préparatoire collaboration de G. Ratier.
Brousseau, Guy; Felix, Lucienne (1972). Mathématique et thèmes d'activité à l'école maternelle (in French).
Brousseau, Guy (1986). Théorisation des phénomènes d'enseignement des Mathématiques (in Frenc |
https://en.wikipedia.org/wiki/291%20%28number%29 | 291 is the natural number following 290 and preceding 292.
In mathematics
291 is an odd composite number with two prime factor.
291 is a semiprime number meaning that it has 2 prime factors.
291 can be written as the sum of the nth prime plus n. It is the 52nd prime (239) plus 52.
291 is one of the positions of “c” in the tribonacci word abacabaab… defined by a->ab, b->ac, c->a.
291 is the sum of 6 different 4th powers. It is the sum of 44+24+24+14+14+14.
World Records
On March 6, 2022, Tamara Walcott received the Guinness World Record for the elephant bar deadlift achieved by a woman. She lifted a weight of 291 kg.
On December 19, 2021, the greatest number of cakes eaten online simultaneously was achieved in Tanba, Japan. There were 291 cakes.
On October 25, 2014, the longest string instrument was created in Singapore. It was the ‘Earth Harp’ at a length of 291 m.
On August 7, 2011, Fred Grsybowski achieved the Guinness World Record for the tallest usable pogo stick in Toronto. It was 2.91 m tall.
Other fields
The calendar years 291 AD and 291 BC.
291 is the number for several highways across the countries of Canada, Japan, and the United States.
291 Alice is an asteroid in the asteroid belt. It was discovered by Johann Palisa in Vienna, Austria.
References |
https://en.wikipedia.org/wiki/304%20%28number%29 | 304 is the natural number following 303 and preceding 305.
In mathematics
304 is an even composite number with two prime factor.
304 is the sum of consecutive primes in two different ways. It is the sum of 41+43+47+53+59+61 and of 23+29+31+37+41+43+47+53.3
304 is a primitive semiperfect number meaning that it is a semiperfect number that is not divisible by any other semiperfect number.
304 is an untouchable number meaning that it is not equal to the sum of any number’s proper divisors.
304 is a nontotient number meaning that it is an even number where phi(x) cannot result in that number.
World Records
In 2021, Tom Brady completed his 304th touchdown during his football career in Gillette Stadium. This is the most by one quarterback in one stadium.
On October 20, 2018, the Guinness World Record for the greatest number of participants in a Beetle drive game was achieved. There were 304 participants.
On August 25, 2014, David Chapple attended the most performances in the Edinburgh Fringe Festival. There were 304 shows with David attending around 11 shows each day.
On December 31, 2021, the greatest number of people passing a squash ball over a video chain was achieved by SquashSmarts.
Other fields
The calendar years 304 AD and 304 BC.
289 is the number for several highways across the countries of Brazil, Canada, China, Costa Rica, Hungary, Japan, the Thailand and the United States.
304 Olga is an asteroid in the asteroid belt. It was discovered by Johann Palisa in Vienna.
References |
https://en.wikipedia.org/wiki/2023%20FC%20Neftchi%20Fergana%20season |
Season events
Squad
Transfers
In
Loans in
Loans out
Out
Friendlies
Competitions
Overview
League table
Results summary
Results by round
Results
Uzbek Cup
Group stage
Squad statistics
Appearances and goals
|-
|colspan="14"|Players away on loan:
|-
|colspan="14"|Players who left Neftchi Fergana during the season:
|}
Goal scorers
Clean sheets
Disciplinary Record
References
Sport in Fergana
FC Neftchi Fergana seasons |
https://en.wikipedia.org/wiki/292%20%28number%29 | 292 is the natural number following 291 and preceding 293.
In mathematics
292 is an even composite number with two prime factor.
292 is a noncototient number meaning that phi(x) cannot result in 292.
292 is an untouchable number meaning that the proper divisors of any number do not add up to 292.
292 is a repdigit in base 8 with it being 444.
In the simplified continued fraction for pi, 292 is the 5th number.
World Records
Between December 16, 2021, and March 15, 2022, Arnaud Clein watched Spider-Man: No Way Home a grand total of 292 times with a total run time of 720 hours.
On May 30, 2016, in Tokyo, Japan, the greatest number of people using vacuum cleaners was achieved with 292 participants.
On April 23, 2019, the largest Shakespeare recital was completed in London. There were 292 participants.
On December 19, 2010, the Guinness World Record for the longest paper airplane flight time was 29.2 seconds achieved in Fukuyama City, Japan.
Other fields
The calendar years 292 AD and 292 BC.
292 is the number for several highways across the countries of Japan and the United States.
292 Ludovica is an asteroid in the asteroid belt. It was discovered by Johann Palisa in Vienna.
References |
https://en.wikipedia.org/wiki/295%20%28number%29 | 295 is the natural number following 294 and preceding 296.
In mathematics
295 is an odd composite number with two prime factor.
295 is a centered tetrahedral number meaning that it can be represented as a tetrahedron.
295 Is a structured deltoidal hexecontahedral number which can be represented as a deltoidal hexecontahedron.
295 can be written as the sum of 4 nonzero perfect squares.
In binary, 295 would be a decimal prime number.
World Records
On September 28, 2018, the greatest number of people filling in the eyebrows was achieved in San Francisco. There were 295 participants.
On December 18, 2014, in Chengdu, China, the greatest number of handheld lasers were lit simultaneously. There were 295 lasers.
The largest lunar crater is Bailly, near the South Pole of the moon. It is 295 km in diameter.
On September 16, 2006, the Guinness World Record for the longest jump on a unicycle was achieved. It was a distance of 2.95 m achieved by David Weichenberger in Vienna.
Other fields
The calendar years 295 AD and 295 BC.
295 is the number for several highways across the countries of Canada, Japan, and the United States.
295 Theresia is an asteroid in the asteroid belt. It was discovered by Johann Palisa in Vienna.
295 is a song recorded by Punjabi singer Sidhu Moose Wala.
References |
https://en.wikipedia.org/wiki/Math%20and%20Science%20Academy | Math and Science Academy may refer to:
Math and Science Academy (Woodbury, Minnesota)
Chicago Math and Science Academy
Hawthorne Math and Science Academy
Illinois Mathematics and Science Academy
Robert Lindblom Math & Science Academy
See also
California Academy of Mathematics and Science
Massachusetts Academy of Math and Science at WPI |
https://en.wikipedia.org/wiki/Lorougnon%20Doukouo | Lorougnon Doukouo (born 16 November 2002) is an Ivorian professional footballer who plays as a forward for Albanian club Egnatia in the Kategoria Superiore.
Career statistics
Club
Honours
KF Egnatia
Albanian Cup: 2022–23
References
External links
2002 births
Living people
Ivorian men's footballers
Men's association football forwards
Al-Yarmouk SC (Amman) players
KF Egnatia players
Kategoria Superiore players
Ivorian expatriate men's footballers
Ivorian expatriate sportspeople in Jordan
Ivorian expatriate sportspeople in Albania
Expatriate men's footballers in Jordan
Expatriate men's footballers in Albania |
https://en.wikipedia.org/wiki/Stirling%27s%20approximation | In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of . It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre.
One way of stating the approximation involves the logarithm of the factorial:
where the big O notation means that, for all sufficiently large values of , the difference between and will be at most proportional to the logarithm. In computer science applications such as the worst-case lower bound for comparison sorting, it is convenient to instead use the binary logarithm, giving the equivalent form
The error term in either base can be expressed more precisely as , corresponding to an approximate formula for the factorial itself,
Here the sign means that the two quantities are asymptotic, that is, that their ratio tends to 1 as tends to infinity. The following version of the bound holds for all , rather than only asymptotically:
Derivation
Roughly speaking, the simplest version of Stirling's formula can be quickly obtained by approximating the sum
with an integral:
The full formula, together with precise estimates of its error, can be derived as follows. Instead of approximating , one considers its natural logarithm, as this is a slowly varying function:
The right-hand side of this equation minus
is the approximation by the trapezoid rule of the integral
and the error in this approximation is given by the Euler–Maclaurin formula:
where is a Bernoulli number, and is the remainder term in the Euler–Maclaurin formula. Take limits to find that
Denote this limit as . Because the remainder in the Euler–Maclaurin formula satisfies
where big-O notation is used, combining the equations above yields the approximation formula in its logarithmic form:
Taking the exponential of both sides and choosing any positive integer , one obtains a formula involving an unknown quantity . For , the formula is
The quantity can be found by taking the limit on both sides as tends to infinity and using Wallis' product, which shows that . Therefore, one obtains Stirling's formula:
Alternative derivations
An alternative formula for using the gamma function is
(as can be seen by repeated integration by parts). Rewriting and changing variables , one obtains
Applying Laplace's method one has
which recovers Stirling's formula:
Higher orders
In fact, further corrections can also be obtained using Laplace's method. From previous result, we know that , so we "peel off" this dominant term, then perform a change of variables, to obtain:Now the function is unimodal, with maximum value zero. Locally around zero, it looks like , which is why we are able to perform Laplace's method. In order to extend Laplace's method to higher orders, we perform another change of variables by . This equation cannot be solved in closed form, but it can be solved by serial expansion |
https://en.wikipedia.org/wiki/Divergence%20theorem | In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem which relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.
More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the "flux" through the surface, is equal to the volume integral of the divergence over the region inside the surface. Intuitively, it states that "the sum of all sources of the field in a region (with sinks regarded as negative sources) gives the net flux out of the region".
The divergence theorem is an important result for the mathematics of physics and engineering, particularly in electrostatics and fluid dynamics. In these fields, it is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to integration by parts. In two dimensions, it is equivalent to Green's theorem.
Explanation using liquid flow
Vector fields are often illustrated using the example of the velocity field of a fluid, such as a gas or liquid. A moving liquid has a velocity—a speed and a direction—at each point, which can be represented by a vector, so that the velocity of the liquid at any moment forms a vector field. Consider an imaginary closed surface S inside a body of liquid, enclosing a volume of liquid. The flux of liquid out of the volume at any time is equal to the volume rate of fluid crossing this surface, i.e., the surface integral of the velocity over the surface.
Since liquids are incompressible, the amount of liquid inside a closed volume is constant; if there are no sources or sinks inside the volume then the flux of liquid out of S is zero. If the liquid is moving, it may flow into the volume at some points on the surface S and out of the volume at other points, but the amounts flowing in and out at any moment are equal, so the net flux of liquid out of the volume is zero.
However if a source of liquid is inside the closed surface, such as a pipe through which liquid is introduced, the additional liquid will exert pressure on the surrounding liquid, causing an outward flow in all directions. This will cause a net outward flow through the surface S. The flux outward through S equals the volume rate of flow of fluid into S from the pipe. Similarly if there is a sink or drain inside S, such as a pipe which drains the liquid off, the external pressure of the liquid will cause a velocity throughout the liquid directed inward toward the location of the drain. The volume rate of flow of liquid inward through the surface S equals the rate of liquid removed by the sink.
If there are multiple sources and sinks of liquid inside S, the flux through the surface can be calculated by adding up the volume rate of liquid added by the sources and subtracting the rate of liquid drained off by the sinks. The volume rate of flow of liquid |
https://en.wikipedia.org/wiki/Del | Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes the standard derivative of the function as defined in calculus. When applied to a field (a function defined on a multi-dimensional domain), it may denote any one of three operations depending on the way it is applied: the gradient or (locally) steepest slope of a scalar field (or sometimes of a vector field, as in the Navier–Stokes equations); the divergence of a vector field; or the curl (rotation) of a vector field.
Del is a very convenient mathematical notation for those three operations (gradient, divergence, and curl) that makes many equations easier to write and remember. The del symbol (or nabla) can be formally defined as a three-dimensional vector operator whose three components are the corresponding partial derivative operators. As a vector operator, it can act on scalar and vector fields in three different ways, giving rise to three different differential operations: first, it can act on scalar fields by a "formal" scalar multiplication—to give a vector field called the gradient; second, it can act on vector fields by a "formal" dot product—to give a scalar field called the divergence; and lastly, it can act on vector fields by a "formal" cross product—to give a vector field called the curl. These "formal" products do not necessarily commute with other operators or products. These three uses, detailed below, are summarized as:
Gradient:
Divergence:
Curl:
Definition
In the Cartesian coordinate system with coordinates and standard basis , del is a vector operator whose components are the partial derivative operators ; that is,
Where the expression in parentheses is a row vector. In three-dimensional Cartesian coordinate system with coordinates and standard basis or unit vectors of axes , del is written as
As a vector operator, del naturally acts on scalar fields via scalar multiplication, and naturally acts on vector fields via dot products and cross products.
More specifically, for any scalar field and any vector field , if one defines
then using the above definition of , one may write
and
and
Example:
Del can also be expressed in other coordinate systems, see for example del in cylindrical and spherical coordinates.
Notational uses
Del is used as a shorthand form to simplify many long mathematical expressions. It is most commonly used to simplify expressions for the gradient, divergence, curl, directional derivative, and Laplacian.
Gradient
The vector derivative of a scalar field is called the gradient, and it can be represented as:
It always points in the direction of greatest increase of , and it has a magnitude equal to the maximum rate of increase at the point—just like a standard derivative. In particular, if a hill is defined as a height function over a plane , the gr |
https://en.wikipedia.org/wiki/Linear%20function | In mathematics, the term linear function refers to two distinct but related notions:
In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For distinguishing such a linear function from the other concept, the term affine function is often used.
In linear algebra, mathematical analysis, and functional analysis, a linear function is a linear map.
As a polynomial function
In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or less, including the zero polynomial (the latter not being considered to have degree zero).
When the function is of only one variable, it is of the form
where and are constants, often real numbers. The graph of such a function of one variable is a nonvertical line. is frequently referred to as the slope of the line, and as the intercept.
If a > 0 then the gradient is positive and the graph slopes upwards.
If a < 0 then the gradient is negative and the graph slopes downwards.
For a function of any finite number of variables, the general formula is
and the graph is a hyperplane of dimension .
A constant function is also considered linear in this context, as it is a polynomial of degree zero or is the zero polynomial. Its graph, when there is only one variable, is a horizontal line.
In this context, a function that is also a linear map (the other meaning) may be referred to as a homogeneous linear function or a linear form. In the context of linear algebra, the polynomial functions of degree 0 or 1 are the scalar-valued affine maps.
As a linear map
In linear algebra, a linear function is a map f between two vector spaces s.t.
Here denotes a constant belonging to some field of scalars (for example, the real numbers) and and are elements of a vector space, which might be itself.
In other terms the linear function preserves vector addition and scalar multiplication.
Some authors use "linear function" only for linear maps that take values in the scalar field; these are more commonly called linear forms.
The "linear functions" of calculus qualify as "linear maps" when (and only when) , or, equivalently, when the constant equals zero in the one-degree polynomial above. Geometrically, the graph of the function must pass through the origin.
See also
Homogeneous function
Nonlinear system
Piecewise linear function
Linear approximation
Linear interpolation
Discontinuous linear map
Linear least squares
Notes
References
Izrail Moiseevich Gelfand (1961), Lectures on Linear Algebra, Interscience Publishers, Inc., New York. Reprinted by Dover, 1989.
Thomas S. Shores (2007), Applied Linear Algebra and Matrix Analysis, Undergraduate Texts in Mathematics, Springer.
James Stewart (2012), Calculus: Early Transcendentals, edition 7E, Brooks/Cole.
Leonid N. Vaserstein (2006), "Linear Programming", in Leslie Hogben, ed., Handbook of Linear Algebra, Discrete Mathematics |
https://en.wikipedia.org/wiki/Forcing%20%28mathematics%29 | In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. Intuitively, forcing can be thought of as a technique to expand the set theoretical universe to a larger universe by introducing a new "generic" object .
Forcing was first used by Paul Cohen in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory. It has been considerably reworked and simplified in the following years, and has since served as a powerful technique, both in set theory and in areas of mathematical logic such as recursion theory. Descriptive set theory uses the notions of forcing from both recursion theory and set theory. Forcing has also been used in model theory, but it is common in model theory to define genericity directly without mention of forcing.
Intuition
Forcing is usually used to construct an expanded universe that satisfies some desired property. For example, the expanded universe might contain many new real numbers (at least of them), identified with subsets of the set of natural numbers, that were not there in the old universe, and thereby violate the continuum hypothesis.
In order to intuitively justify such an expansion, it is best to think of the "old universe" as a model of the set theory, which is itself a set in the "real universe" . By the Löwenheim–Skolem theorem, can be chosen to be a "bare bones" model that is externally countable, which guarantees that there will be many subsets (in ) of that are not in . Specifically, there is an ordinal that "plays the role of the cardinal " in , but is actually countable in . Working in , it should be easy to find one distinct subset of per each element of . (For simplicity, this family of subsets can be characterized with a single subset .)
However, in some sense, it may be desirable to "construct the expanded model within ". This would help ensure that "resembles" in certain aspects, such as being the same as (more generally, that cardinal collapse does not occur), and allow fine control over the properties of . More precisely, every member of should be given a (non-unique) name in . The name can be thought as an expression in terms of , just like in a simple field extension every element of can be expressed in terms of . A major component of forcing is manipulating those names within , so sometimes it may help to directly think of as "the universe", knowing that the theory of forcing guarantees that will correspond to an actual model.
A subtle point of forcing is that, if is taken to be an arbitrary "missing subset" of some set in , then the constructed "within " may not even be a model. This is because may encode "special" information about that is invisible within (e.g. the countability of ), and thus prove the existence of sets that are "too complex for to describe".
Forcing avoids such problems by requiring the newly introduced set to be a generic set r |
https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel%20set%20theory | In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.
Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models from containing urelements (elements of sets that are not themselves sets). Furthermore, proper classes (collections of mathematical objects defined by a property shared by their members where the collections are too big to be sets) can only be treated indirectly. Specifically, Zermelo–Fraenkel set theory does not allow for the existence of a universal set (a set containing all sets) nor for unrestricted comprehension, thereby avoiding Russell's paradox. Von Neumann–Bernays–Gödel set theory (NBG) is a commonly used conservative extension of Zermelo–Fraenkel set theory that does allow explicit treatment of proper classes.
There are many equivalent formulations of the axioms of Zermelo–Fraenkel set theory. Most of the axioms state the existence of particular sets defined from other sets. For example, the axiom of pairing says that given any two sets and there is a new set containing exactly and . Other axioms describe properties of set membership. A goal of the axioms is that each axiom should be true if interpreted as a statement about the collection of all sets in the von Neumann universe (also known as the cumulative hierarchy). Formally, ZFC is a one-sorted theory in first-order logic. The signature has equality and a single primitive binary relation, intended to formalize set membership, which is usually denoted . The formula means that the set is a member of the set (which is also read, " is an element of " or " is in ").
The metamathematics of Zermelo–Fraenkel set theory has been extensively studied. Landmark results in this area established the logical independence of the axiom of choice from the remaining Zermelo-Fraenkel axioms (see ) and of the continuum hypothesis from ZFC. The consistency of a theory such as ZFC cannot be proved within the theory itself, as shown by Gödel's second incompleteness theorem.
History
The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. However, the discovery of paradoxes in naive set theory, such as Russe |
https://en.wikipedia.org/wiki/Giovanni%20Ceva | Giovanni Ceva (September 1, 1647 – May 13, 1734) was an Italian mathematician widely known for proving Ceva's theorem in elementary geometry. His brother, Tommaso Ceva, was also a well-known poet and mathematician.
Life
Ceva received his education at a Jesuit college in Milan. Later in his life, he studied at the University of Pisa, where he subsequently became a professor. In 1686, however, he was designated as the Professor of Mathematics at the University of Mantua and worked there for the rest of his life.
Work
Ceva studied geometry for most of his long life. In 1678, he published a now famous theorem on synthetic geometry in a triangle called Ceva's Theorem. The theorem,
already known to Yusuf Al-Mu'taman ibn Hűd in 11th century, states that if three line segments are drawn from the vertices of a triangle to the opposite sides, then the three line segments are concurrent if, and only if, the product of the ratios of the newly created line segments on each side of the triangle is equal to one. He published this theorem in De lineis rectis.
Ceva also rediscovered and published Menelaus's theorem. He published Opuscula mathematica in 1682 and Geometria Motus in 1692, as well. In Geometria Motus, he anticipated the infinitesimal calculus. Finally, Ceva wrote De Re Nummaria in 1711, which was one of the first books in mathematical economics.
Giovanni Ceva also studied applications of mechanics and statics to geometric systems. At one point, however, he incorrectly resolved that the periods of oscillation of two pendulums were in the same ratio as their lengths, but he later realized and corrected the error. Ceva also worked on hydraulics. In 1728, he published Opus hydrostaticum which discusses his work in hydraulics. In fact, he used his knowledge of hydraulics to stop a project from diverting the river Reno into the river Po.
List of works
Geometria Motus, 1692
See also
Cevian
References
1647 births
1734 deaths
Scientists from Milan
17th-century Italian mathematicians
18th-century Italian mathematicians
Academic staff of the University of Pisa
University of Pisa alumni |
https://en.wikipedia.org/wiki/Ceva%27s%20theorem | In Euclidean geometry, Ceva's theorem is a theorem about triangles. Given a triangle , let the lines be drawn from the vertices to a common point (not on one of the sides of ), to meet opposite sides at respectively. (The segments are known as cevians.) Then, using signed lengths of segments,
In other words, the length is taken to be positive or negative according to whether is to the left or right of in some fixed orientation of the line. For example, is defined as having positive value when is between and and negative otherwise.
Ceva's theorem is a theorem of affine geometry, in the sense that it may be stated and proved without using the concepts of angles, areas, and lengths (except for the ratio of the lengths of two line segments that are collinear). It is therefore true for triangles in any affine plane over any field.
A slightly adapted converse is also true: If points are chosen on respectively so that
then are concurrent, or all three parallel. The converse is often included as part of the theorem.
The theorem is often attributed to Giovanni Ceva, who published it in his 1678 work De lineis rectis. But it was proven much earlier by Yusuf Al-Mu'taman ibn Hűd, an eleventh-century king of Zaragoza.
Associated with the figures are several terms derived from Ceva's name: cevian (the lines are the cevians of ), cevian triangle (the triangle is the cevian triangle of ); cevian nest, anticevian triangle, Ceva conjugate. (Ceva is pronounced Chay'va; cevian is pronounced chev'ian.)
The theorem is very similar to Menelaus' theorem in that their equations differ only in sign. By re-writing each in terms of cross-ratios, the two theorems may be seen as projective duals.
Proofs
Several proofs of the theorem have been given.
Two proofs are given in the following.
The first one is very elementary, using only basic properties of triangle areas. However, several cases have to be considered, depending on the position of the point .
The second proof uses barycentric coordinates and vectors, but is somehow more natural and not case dependent. Moreover, it works in any affine plane over any field.
Using triangle areas
First, the sign of the left-hand side is positive since either all three of the ratios are positive, the case where is inside the triangle (upper diagram), or one is positive and the other two are negative, the case is outside the triangle (lower diagram shows one case).
To check the magnitude, note that the area of a triangle of a given height is proportional to its base. So
Therefore,
(Replace the minus with a plus if and are on opposite sides of .)
Similarly,
and
Multiplying these three equations gives
as required.
The theorem can also be proven easily using Menelaus' theorem. From the transversal of triangle ,
and from the transversal of triangle ,
The theorem follows by dividing these two equations.
The converse follows as a corollary. Let be given on the lines so that the equatio |
https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini%20theorem | In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, general means that the coefficients of the equation are viewed and manipulated as indeterminates.
The theorem is named after Paolo Ruffini, who made an incomplete proof in 1799 (which was refined and completed in 1813 and accepted by Cauchy) and Niels Henrik Abel, who provided a proof in 1824.
Abel–Ruffini theorem refers also to the slightly stronger result that there are equations of degree five and higher that cannot be solved by radicals. This does not follow from Abel's statement of the theorem, but is a corollary of his proof, as his proof is based on the fact that some polynomials in the coefficients of the equation are not the zero polynomial. This improved statement follows directly from . Galois theory implies also that
is the simplest equation that cannot be solved in radicals, and that almost all polynomials of degree five or higher cannot be solved in radicals.
The impossibility of solving in degree five or higher contrasts with the case of lower degree: one has the quadratic formula, the cubic formula, and the quartic formula for degrees two, three, and four, respectively.
Context
Polynomial equations of degree two can be solved with the quadratic formula, which has been known since antiquity. Similarly the cubic formula for degree three, and the quartic formula for degree four, were found during the 16th century. At that time a fundamental problem was whether equations of higher degree could be solved in a similar way.
The fact that every polynomial equation of positive degree has solutions, possibly non-real, was asserted during the 17th century, but completely proved only at the beginning of the 19th century. This is the fundamental theorem of algebra, which does not provide any tool for computing exactly the solutions, although Newton's method allows approximating the solutions to any desired accuracy.
From the 16th century to beginning of the 19th century, the main problem of algebra was to search for a formula for the solutions of polynomial equations of degree five and higher, hence the name the "fundamental theorem of algebra". This meant a solution in radicals, that is, an expression involving only the coefficients of the equation, and the operations of addition, subtraction, multiplication, division, and th root extraction.
The Abel–Ruffini theorem proves that this is impossible. However, this impossibility does not imply that a specific equation of any degree cannot be solved in radicals. On the contrary, there are equations of any degree that can be solved in radicals. This is the case of the equation for any , and the equations defined by cyclotomic polynomials, all of whose solutions can be expressed in radicals.
Abel's proof of the theorem does not explicitly contain the assertion that the |
https://en.wikipedia.org/wiki/Bisection | In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a bisector. The most often considered types of bisectors are the segment bisector, a line that passes through the midpoint of a given segment, and the angle bisector, a line that passes through the apex of an angle (that divides it into two equal angles).
In three-dimensional space, bisection is usually done by a bisecting plane, also called the bisector.
Perpendicular line segment bisector
Definition
The perpendicular bisector of a line segment is a line which meets the segment at its midpoint perpendicularly.
The perpendicular bisector of a line segment also has the property that each of its points is equidistant from segment AB's endpoints:
(D).
The proof follows from and Pythagoras' theorem:
Property (D) is usually used for the construction of a perpendicular bisector:
Construction by straight edge and compass
In classical geometry, the bisection is a simple compass and straightedge construction, whose possibility depends on the ability to draw arcs of equal radii and different centers:
The segment is bisected by drawing intersecting circles of equal radius , whose centers are the endpoints of the segment. The line determined by the points of intersection of the two circles is the perpendicular bisector of the segment.
Because the construction of the bisector is done without the knowledge of the segment's midpoint , the construction is used for determining as the intersection of the bisector and the line segment.
This construction is in fact used when constructing a line perpendicular to a given line at a given point : drawing a circle whose center is such that it intersects the line in two points , and the perpendicular to be constructed is the one bisecting segment .
Equations
If are the position vectors of two points , then its midpoint is and vector is a normal vector of the perpendicular line segment bisector. Hence its vector equation is . Inserting and expanding the equation leads to the vector equation
(V)
With one gets the equation in coordinate form:
(C)
Or explicitly:
(E),
where , , and .
Applications
Perpendicular line segment bisectors were used solving various geometric problems:
Construction of the center of a Thales' circle,
Construction of the center of the Excircle of a triangle,
Voronoi diagram boundaries consist of segments of such lines or planes.
Perpendicular line segment bisectors in space
The perpendicular bisector of a line segment is a plane, which meets the segment at its midpoint perpendicularly.
Its vector equation is literally the same as in the plane case:
(V)
With one gets the equation in coordinate form:
(C3)
Property (D) (see above) is literally true in space, too:
(D) The perpendicular bisector plane of a segment has for any point the property: .
Angle bisector
An angle bisector divides th |
https://en.wikipedia.org/wiki/Generalized%20Riemann%20hypothesis | The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function. One can then ask the same question about the zeros of these L-functions, yielding various generalizations of the Riemann hypothesis. Many mathematicians believe these generalizations of the Riemann hypothesis to be true. The only cases of these conjectures which have been proven occur in the algebraic function field case (not the number field case).
Global L-functions can be associated to elliptic curves, number fields (in which case they are called Dedekind zeta-functions), Maass forms, and Dirichlet characters (in which case they are called Dirichlet L-functions). When the Riemann hypothesis is formulated for Dedekind zeta-functions, it is known as the extended Riemann hypothesis (ERH) and when it is formulated for Dirichlet L-functions, it is known as the generalized Riemann hypothesis or generalised Riemann hypothesis (see spelling differences) (GRH). These two statements will be discussed in more detail below. (Many mathematicians use the label generalized Riemann hypothesis to cover the extension of the Riemann hypothesis to all global L-functions,
not just the special case of Dirichlet L-functions.)
Generalized Riemann hypothesis (GRH)
The generalized Riemann hypothesis (for Dirichlet L-functions) was probably formulated for the first time by Adolf Piltz in 1884. Like the original Riemann hypothesis, it has far reaching consequences about the distribution of prime numbers.
The formal statement of the hypothesis follows. A Dirichlet character is a completely multiplicative arithmetic function χ such that there exists a positive integer k with for all n and whenever . If such a character is given, we define the corresponding Dirichlet L-function by
for every complex number s such that . By analytic continuation, this function can be extended to a meromorphic function (only when is primitive) defined on the whole complex plane. The generalized Riemann hypothesis asserts that, for every Dirichlet character χ and every complex number s with , if s is not a negative real number, then the real part of s is 1/2.
The case for all n yields the ordinary Riemann hypothesis.
Consequences of GRH
Dirichlet's theorem states that if a and d are coprime natural numbers, then the arithmetic progression a, , , , ... contains infinitely many prime numbers. Let denote the number of prime numbers in this progression which are less than or equal to x. If the generalized Riemann hypothesis is true, then for every coprime a and d and for every ,
where is Euler's totient function and is the Big O notation. This is a considerable strengthening of the prime number theorem.
If GRH is true, then every proper subgroup of the multiplicative group omits a number l |
https://en.wikipedia.org/wiki/Connector%20%28mathematics%29 | In mathematics, a connector is a map which can be defined for a linear connection and used to define the covariant derivative on a vector bundle from the linear connection.
Definition
Let ∇ be a connection on the tangent space TN of a smooth manifold N. For smooth mappings h:M→TN from any smooth manifold M, the connector K:TTN→TN satisfies : ∇ h = K○Th:TM→TN where Th:TM→TTN is the differential of h.
Connection (mathematics) |
https://en.wikipedia.org/wiki/General%20number%20field%20sieve | In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than . Heuristically, its complexity for factoring an integer (consisting of bits) is of the form
in O and L-notations. It is a generalization of the special number field sieve: while the latter can only factor numbers of a certain special form, the general number field sieve can factor any number apart from prime powers (which are trivial to factor by taking roots).
The principle of the number field sieve (both special and general) can be understood as an improvement to the simpler rational sieve or quadratic sieve. When using such algorithms to factor a large number , it is necessary to search for smooth numbers (i.e. numbers with small prime factors) of order . The size of these values is exponential in the size of (see below). The general number field sieve, on the other hand, manages to search for smooth numbers that are subexponential in the size of . Since these numbers are smaller, they are more likely to be smooth than the numbers inspected in previous algorithms. This is the key to the efficiency of the number field sieve. In order to achieve this speed-up, the number field sieve has to perform computations and factorizations in number fields. This results in many rather complicated aspects of the algorithm, as compared to the simpler rational sieve.
The size of the input to the algorithm is or the number of bits in the binary representation of . Any element of the order for a constant is exponential in . The running time of the number field sieve is super-polynomial but sub-exponential in the size of the input.
Number fields
Suppose is a -degree polynomial over (the rational numbers), and is a complex root of . Then, , which can be rearranged to express as a linear combination of powers of less than . This equation can be used to reduce away any powers of with exponent . For example, if and is the imaginary unit , then , or . This allows us to define the complex product:
In general, this leads directly to the algebraic number field , which can be defined as the set of complex numbers given by:
The product of any two such values can be computed by taking the product as polynomials, then reducing any powers of with exponent as described above, yielding a value in the same form. To ensure that this field is actually -dimensional and does not collapse to an even smaller field, it is sufficient that is an irreducible polynomial over the rationals. Similarly, one may define the ring of integers as the subset of which are roots of monic polynomials with integer coefficients. In some cases, this ring of integers is equivalent to the ring . However, there are many exceptions, such as for when is equal to 1 modulo 4.
Method
Two polynomials f(x) and g(x) of small degrees d and e are chosen, which have integer coefficients, which are irreducible over the rationals, and which, when i |
https://en.wikipedia.org/wiki/Hilbert%27s%20second%20problem | In mathematics, Hilbert's second problem was posed by David Hilbert in 1900 as one of his 23 problems. It asks for a proof that the arithmetic is consistent – free of any internal contradictions. Hilbert stated that the axioms he considered for arithmetic were the ones given in , which include a second order completeness axiom.
In the 1930s, Kurt Gödel and Gerhard Gentzen proved results that cast new light on the problem. Some feel that Gödel's theorems give a negative solution to the problem, while others consider Gentzen's proof as a partial positive solution.
Hilbert's problem and its interpretation
In one English translation, Hilbert asks:
"When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. ... But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a definite number of logical steps based upon them can never lead to contradictory results. In geometry, the proof of the compatibility of the axioms can be effected by constructing a suitable field of numbers, such that analogous relations between the numbers of this field correspond to the geometrical axioms. ... On the other hand a direct method is needed for the proof of the compatibility of the arithmetical axioms."
Hilbert's statement is sometimes misunderstood, because by the "arithmetical axioms" he did not mean a system equivalent to Peano arithmetic, but a stronger system with a second-order completeness axiom. The system Hilbert asked for a completeness proof of is more like second-order arithmetic than first-order Peano arithmetic.
As a nowadays common interpretation, a positive solution to Hilbert's second question would in particular provide a proof that Peano arithmetic is consistent.
There are many known proofs that Peano arithmetic is consistent that can be carried out in strong systems such as Zermelo–Fraenkel set theory. These do not provide a resolution to Hilbert's second question, however, because someone who doubts the consistency of Peano arithmetic is unlikely to accept the axioms of set theory (which is much stronger) to prove its consistency. Thus a satisfactory answer to Hilbert's problem must be carried out using principles that would be acceptable to someone who does not already believe PA is consistent. Such principles are often called finitistic because they are completely constructive and do not presuppose a completed infinity of natural numbers. Gödel's second incompleteness theorem (see Gödel's incompleteness theorems) places a severe limit on how weak a finitistic system can be while still proving the consistency of Peano arithmetic.
Gödel's incompleteness theorem
Gödel's second incompleteness theorem shows that it is not possible |
https://en.wikipedia.org/wiki/Subclass | Subclass may refer to:
Subclass (taxonomy), a taxonomic rank below "class"
Subclass (computer science)
Subclass (set theory)
See also
Superclass |
https://en.wikipedia.org/wiki/Finitism | Finitism is a philosophy of mathematics that accepts the existence only of finite mathematical objects. It is best understood in comparison to the mainstream philosophy of mathematics where infinite mathematical objects (e.g., infinite sets) are accepted as legitimate.
Main idea
The main idea of finitistic mathematics is not accepting the existence of infinite objects such as infinite sets. While all natural numbers are accepted as existing, the set of all natural numbers is not considered to exist as a mathematical object. Therefore quantification over infinite domains is not considered meaningful. The mathematical theory often associated with finitism is Thoralf Skolem's primitive recursive arithmetic.
History
The introduction of infinite mathematical objects occurred a few centuries ago when the use of infinite objects was already a controversial topic among mathematicians. The issue entered a new phase when Georg Cantor in 1874 introduced what is now called naive set theory and used it as a base for his work on transfinite numbers. When paradoxes such as Russell's paradox, Berry's paradox and the Burali-Forti paradox were discovered in Cantor's naive set theory, the issue became a heated topic among mathematicians.
There were various positions taken by mathematicians. All agreed about finite mathematical objects such as natural numbers. However there were disagreements regarding infinite mathematical objects.
One position was the intuitionistic mathematics that was advocated by L. E. J. Brouwer, which rejected the existence of infinite objects until they are constructed.
Another position was endorsed by David Hilbert: finite mathematical objects are concrete objects, infinite mathematical objects are ideal objects, and accepting ideal mathematical objects does not cause a problem regarding finite mathematical objects. More formally, Hilbert believed that it is possible to show that any theorem about finite mathematical objects that can be obtained using ideal infinite objects can be also obtained without them. Therefore allowing infinite mathematical objects would not cause a problem regarding finite objects. This led to Hilbert's program of proving both consistency and completeness of set theory using finitistic means as this would imply that adding ideal mathematical objects is conservative over the finitistic part. Hilbert's views are also associated with the formalist philosophy of mathematics. Hilbert's goal of proving the consistency and completeness of set theory or even arithmetic through finitistic means turned out to be an impossible task due to Kurt Gödel's incompleteness theorems. However, Harvey Friedman's grand conjecture would imply that most mathematical results are provable using finitistic means.
Hilbert did not give a rigorous explanation of what he considered finitistic and referred to as elementary. However, based on his work with Paul Bernays some experts such as have argued that primitive recursive arithmetic c |
https://en.wikipedia.org/wiki/Pedal%20triangle | In plane geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle.
More specifically, consider a triangle , and a point that is not one of the vertices . Drop perpendiculars from to the three sides of the triangle (these may need to be produced, i.e., extended). Label the intersections of the lines from with the sides . The pedal triangle is then .
If is not an obtuse triangle, is the orthocenter then the angles of are , and .
The location of the chosen point relative to the chosen triangle gives rise to some special cases:
If is the orthocenter, then is the orthic triangle.
If is the incenter, then is the intouch triangle.
If is the circumcenter, then is the medial triangle.
If is on the circumcircle of the triangle, collapses to a line (the pedal line or Simson line).
The vertices of the pedal triangle of an interior point , as shown in the top diagram, divide the sides of the original triangle in such a way as to satisfy Carnot's theorem:
Trilinear coordinates
If has trilinear coordinates , then the vertices of the pedal triangle of are given by
Antipedal triangle
One vertex, , of the antipedal triangle of is the point of intersection of the perpendicular to through and the perpendicular to through . Its other vertices, and , are constructed analogously. Trilinear coordinates are given by
For example, the excentral triangle is the antipedal triangle of the incenter.
Suppose that does not lie on any of the extended sides , and let denote the isogonal conjugate of . The pedal triangle of is homothetic to the antipedal triangle of . The homothetic center (which is a triangle center if and only if is a triangle center) is the point given in trilinear coordinates by
The product of the areas of the pedal triangle of and the antipedal triangle of equals the square of the area of .
Pedal circle
The pedal circle is defined as the circumcircle of the pedal triangle. Note that the pedal circle is not defined for points lying on the circumcircle of the triangle.
Pedal circle of isogonal conjugates
For any point not lying on the circumcircle of the triangle, it is known that and its isogonal conjugate have a common pedal circle, whose center is the midpoint of these two points.
References
External links
Mathworld: Pedal Triangle
Simson Line
Pedal Triangle and Isogonal Conjugacy
Objects defined for a triangle |
https://en.wikipedia.org/wiki/Robert%20Simson | Robert Simson (14 October 1687 – 1 October 1768) was a Scottish mathematician and professor of mathematics at the University of Glasgow. The Simson line is named after him.
Biography
Robert Simson was born on 14 October 1687, probably the eldest of the seventeen children, all male, of John Simson, a Glasgow merchant, and Agnes, daughter of Patrick Simson, minister of Renfrew; only six of them reached adulthood.
Simson matriculated at the University of Glasgow in 1701, intending to enter the Church. He followed the course in the faculty of arts (Latin, Greek, logic, natural philosophy) and then concentrated on studying theology and Semitic languages. Mathematics was not taught at the university, but by reading Sinclair's Tuyrocinia Mathematica in Novem Tractatus and then Euclid’s Elements Simson soon became deeply interested in mathematics and especially geometry. His efforts impressed the university Senate to such an extent that they offered him the chair of mathematics, to replace the recently-dismissed Sinclair. As he had had no formal training in the subject, Simson turned down the offer but agreed to take up the post a year later, during which time he would increase his knowledge of mathematics.
After a failed attempt to go to Oxford, Simson spent his year in London at Christ's Hospital. During this time he made valuable contacts with several prominent mathematicians, including John Caswell, James Jurin (secretary of the Royal Society), Humphrey Ditton and, most importantly, Edmond Halley.
Simson was admitted professor of mathematics at Glasgow, aged 23, on 20 November 1711, where his first task was to design a two-year course in mathematics, some of which he taught himself; his lectures included geometry, of course, and algebra, logarithms and optics. Among his students were Maclaurin, Matthew Stewart, and William Trail. He resigned the post in 1761, and was succeeded by another of his pupils Rev Prof James Williamson FRSE (1725-1795).
During his time at Glasgow Simson noted in 1753 that, as the Fibonacci numbers increased in magnitude, the ratio between adjacent numbers approached the golden ratio, whose value is
As for the man himself, “Simson appears to have been tall and of good stature. In spite of his great scholarship he was a modest, unassuming man who was very cautious in promoting his own work. He enjoyed good company and presided over the weekly meetings of a dining club that he had instituted … He had a special interest in botany, in which he was an acknowledged expert”.
Robert Simson did not marry. He died, aged 80, in his college residence at Glasgow on 1 October 1768, and was interred in the Blackfriars Burying Ground (now known as Ramshorn Cemetery), where, in the south wall, is placed to his memory a plain marble tablet, with a highly and justly complimentary inscription”. Simson's library, including some of his own works, was bequeathed to the university on his death. It consists of about 850 printed books, mainly e |
https://en.wikipedia.org/wiki/Knot%20theory | In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, the simplest knot being a ring (or "unknot"). In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, . Two mathematical knots are equivalent if one can be transformed into the other via a deformation of upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself.
Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram, in which any knot can be drawn in many different ways. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot.
A complete algorithmic solution to this problem exists, which has unknown complexity. In practice, knots are often distinguished using a knot invariant, a "quantity" which is the same when computed from different descriptions of a knot. Important invariants include knot polynomials, knot groups, and hyperbolic invariants.
The original motivation for the founders of knot theory was to create a table of knots and links, which are knots of several components entangled with each other. More than six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century.
To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other three-dimensional spaces and objects other than circles can be used; see knot (mathematics). For example, a higher-dimensional knot is an n-dimensional sphere embedded in (n+2)-dimensional Euclidean space.
History
Archaeologists have discovered that knot tying dates back to prehistoric times. Besides their uses such as recording information and tying objects together, knots have interested humans for their aesthetics and spiritual symbolism. Knots appear in various forms of Chinese artwork dating from several centuries BC (see Chinese knotting). The endless knot appears in Tibetan Buddhism, while the Borromean rings have made repeated appearances in different cultures, often representing strength in unity. The Celtic monks who created the Book of Kells lavished entire pages with intricate Celtic knotwork.
A mathematical theory of knots was first developed in 1771 by Alexandre-Théophile Vandermonde who explicitly noted the importance of topological features when discussing the properties of knots related to the geometry of position. Mathematical studies of knots began in the 19th century with Carl Friedrich Gauss, who defined the linking integral . In the 1860s, Lord Kelvin's theory that atoms were knots |
https://en.wikipedia.org/wiki/Normal%20closure%20%28group%20theory%29 | In group theory, the normal closure of a subset of a group is the smallest normal subgroup of containing
Properties and description
Formally, if is a group and is a subset of the normal closure of is the intersection of all normal subgroups of containing :
The normal closure is the smallest normal subgroup of containing in the sense that is a subset of every normal subgroup of that contains
The subgroup is generated by the set of all conjugates of elements of in
Therefore one can also write
Any normal subgroup is equal to its normal closure. The conjugate closure of the empty set is the trivial subgroup.
A variety of other notations are used for the normal closure in the literature, including and
Dual to the concept of normal closure is that of or , defined as the join of all normal subgroups contained in
Group presentations
For a group given by a presentation with generators and defining relators the presentation notation means that is the quotient group where is a free group on
References
Group theory
Closure operators |
https://en.wikipedia.org/wiki/Dedekind%20group | In group theory, a Dedekind group is a group G such that every subgroup of G is normal.
All abelian groups are Dedekind groups.
A non-abelian Dedekind group is called a Hamiltonian group.
The most familiar (and smallest) example of a Hamiltonian group is the quaternion group of order 8, denoted by Q8.
Dedekind and Baer have shown (in the finite and respectively infinite order case) that every Hamiltonian group is a direct product of the form , where B is an elementary abelian 2-group, and D is a torsion abelian group with all elements of odd order.
Dedekind groups are named after Richard Dedekind, who investigated them in , proving a form of the above structure theorem (for finite groups). He named the non-abelian ones after William Rowan Hamilton, the discoverer of quaternions.
In 1898 George Miller delineated the structure of a Hamiltonian group in terms of its order and that of its subgroups. For instance, he shows "a Hamilton group of order 2a has quaternion groups as subgroups". In 2005 Horvat et al used this structure to count the number of Hamiltonian groups of any order where o is an odd integer. When then there are no Hamiltonian groups of order n, otherwise there are the same number as there are Abelian groups of order o.
Notes
References
.
Baer, R. Situation der Untergruppen und Struktur der Gruppe, Sitz.-Ber. Heidelberg. Akad. Wiss.2, 12–17, 1933.
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Group theory
Properties of groups |
https://en.wikipedia.org/wiki/Quaternion%20group | In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset
of the quaternions under multiplication. It is given by the group presentation
where e is the identity element and commutes with the other elements of the group.
Another presentation of Q8 is
The quaternion group sprang full-blown from the mind of W. R. Hamilton, and there has been an effort to connect it with the wellspring of discrete groups in field extensions and the study of algebraic numbers. Richard Dedekind considered the field ℚ[√2, √3] in this effort. In field theory, extensions are generated by roots of polynomial irreducible over a ground field. Isomorphic fields are associated with permutation groups that move around the roots of the polynomial. This notion, pioneered by Évariste Galois (1830), now forms a standard study Galois theory in mathematics education. In 1936 Ernst Witt published his approach to the quaternion group through Galois theory. A definitive connection was published in 1981, see § Galois group.
Compared to dihedral group
The quaternion group Q8 has the same order as the dihedral group D4, but a different structure, as shown by their Cayley and cycle graphs:
In the diagrams for D4, the group elements are marked with their action on a letter F in the defining representation R2. The same cannot be done for Q8, since it has no faithful representation in R2 or R3. D4 can be realized as a subset of the split-quaternions in the same way that Q8 can be viewed as a subset of the quaternions.
Cayley table
The Cayley table (multiplication table) for Q8 is given by:
Properties
The elements i, j, and k all have order four in Q8 and any two of them generate the entire group. Another presentation of Q8 based in only two elements to skip this redundancy is:
One may take, for instance, and .
The quaternion group has the unusual property of being Hamiltonian: Q8 is non-abelian, but every subgroup is normal. Every Hamiltonian group contains a copy of Q8.
The quaternion group Q8 and the dihedral group D4 are the two smallest examples of a nilpotent non-abelian group.
The center and the commutator subgroup of Q8 is the subgroup . The inner automorphism group of Q8 is given by the group modulo its center, i.e. the factor group which is isomorphic to the Klein four-group V. The full automorphism group of Q8 is isomorphic to S4, the symmetric group on four letters (see Matrix representations below), and the outer automorphism group of Q8 is thus S4/V, which is isomorphic to S3.
The quaternion group Q8 has five conjugacy classes, and so five irreducible representations over the complex numbers, with dimensions 1, 1, 1, 1, 2:
Trivial representation.
Sign representations with i, j, k-kernel: Q8 has three maximal normal subgroups: the cyclic subgroups generated by i, j, and k respectively. For each maximal normal subgroup N, we obtain a one-dimensional representation fac |
https://en.wikipedia.org/wiki/Root-finding%20algorithms | In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex numbers, is a number such that . As, generally, the zeros of a function cannot be computed exactly nor expressed in closed form, root-finding algorithms provide approximations to zeros, expressed either as floating-point numbers or as small isolating intervals, or disks for complex roots (an interval or disk output being equivalent to an approximate output together with an error bound).
Solving an equation is the same as finding the roots of the function . Thus root-finding algorithms allow solving any equation defined by continuous functions. However, most root-finding algorithms do not guarantee that they will find all the roots; in particular, if such an algorithm does not find any root, that does not mean that no root exists.
Most numerical root-finding methods use iteration, producing a sequence of numbers that hopefully converges towards the root as its limit. They require one or more initial guesses of the root as starting values, then each iteration of the algorithm produces a successively more accurate approximation to the root. Since the iteration must be stopped at some point, these methods produce an approximation to the root, not an exact solution. Many methods compute subsequent values by evaluating an auxiliary function on the preceding values. The limit is thus a fixed point of the auxiliary function, which is chosen for having the roots of the original equation as fixed points, and for converging rapidly to these fixed points.
The behavior of general root-finding algorithms is studied in numerical analysis. However, for polynomials, root-finding study belongs generally to computer algebra, since algebraic properties of polynomials are fundamental for the most efficient algorithms. The efficiency of an algorithm may depend dramatically on the characteristics of the given functions. For example, many algorithms use the derivative of the input function, while others work on every continuous function. In general, numerical algorithms are not guaranteed to find all the roots of a function, so failing to find a root does not prove that there is no root. However, for polynomials, there are specific algorithms that use algebraic properties for certifying that no root is missed, and locating the roots in separate intervals (or disks for complex roots) that are small enough to ensure the convergence of numerical methods (typically Newton's method) to the unique root so located.
Bracketing methods
Bracketing methods determine successively smaller intervals (brackets) that contain a root. When the interval is small enough, then a root has been found. They generally use the intermediate value theorem, which asserts that if a continuous function has values of opposite signs at the end points of an in |
https://en.wikipedia.org/wiki/Characteristic%20function | In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
The indicator function of a subset, that is the function
which for a given subset A of X, has value 1 at points of A and 0 at points of X − A.
The characteristic function in convex analysis, closely related to the indicator function of a set:
In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question:
where denotes expected value. For multivariate distributions, the product tX is replaced by a scalar product of vectors.
The characteristic function of a cooperative game in game theory.
The characteristic polynomial in linear algebra.
The characteristic state function in statistical mechanics.
The Euler characteristic, a topological invariant.
The receiver operating characteristic in statistical decision theory.
The point characteristic function in statistics.
References |
https://en.wikipedia.org/wiki/Local%20ring | In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules.
In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal.
The concept of local rings was introduced by Wolfgang Krull in 1938 under the name Stellenringe. The English term local ring is due to Zariski.
Definition and first consequences
A ring R is a local ring if it has any one of the following equivalent properties:
R has a unique maximal left ideal.
R has a unique maximal right ideal.
1 ≠ 0 and the sum of any two non-units in R is a non-unit.
1 ≠ 0 and if x is any element of R, then x or is a unit.
If a finite sum is a unit, then it has a term that is a unit (this says in particular that the empty sum cannot be a unit, so it implies 1 ≠ 0).
If these properties hold, then the unique maximal left ideal coincides with the unique maximal right ideal and with the ring's Jacobson radical. The third of the properties listed above says that the set of non-units in a local ring forms a (proper) ideal, necessarily contained in the Jacobson radical. The fourth property can be paraphrased as follows: a ring R is local if and only if there do not exist two coprime proper (principal) (left) ideals, where two ideals I1, I2 are called coprime if .
In the case of commutative rings, one does not have to distinguish between left, right and two-sided ideals: a commutative ring is local if and only if it has a unique maximal ideal.
Before about 1960 many authors required that a local ring be (left and right) Noetherian, and (possibly non-Noetherian) local rings were called quasi-local rings. In this article this requirement is not imposed.
A local ring that is an integral domain is called a local domain.
Examples
All fields (and skew fields) are local rings, since {0} is the only maximal ideal in these rings.
The ring is a local ring ( prime, ). The unique maximal ideal consists of all multiples of .
More generally, a nonzero ring in which every element is either a unit or nilpotent is a local ring.
An important class of local rings are discrete valuation rings, which are local principal ideal domains that are not fields.
The ring , whose elements are infinite series where multiplications are given by such that , is local. Its unique maximal ideal consists of all elements which are not invertible. In other words, it consists of all elements with constant term zero.
More generally, every ring of formal power series over a local ring is local; the maximal ideal consists of those power series with constant term in the maximal ideal of the base ring.
Similarly, the algebra of dual numbers over a |
https://en.wikipedia.org/wiki/Hilbert%27s%20problems | Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 at the Sorbonne. The complete list of 23 problems was published later, in English translation in 1902 by Mary Frances Winston Newson in the Bulletin of the American Mathematical Society.
Nature and influence of the problems
Hilbert's problems ranged greatly in topic and precision. Some of them, like the 3rd problem, which was the first to be solved, or the 8th problem (the Riemann hypothesis), which still remains unresolved, were presented precisely enough to enable a clear affirmative or negative answer. For other problems, such as the 5th, experts have traditionally agreed on a single interpretation, and a solution to the accepted interpretation has been given, but closely related unsolved problems exist. Some of Hilbert's statements were not precise enough to specify a particular problem, but were suggestive enough that certain problems of contemporary nature seem to apply; for example, most modern number theorists would probably see the 9th problem as referring to the conjectural Langlands correspondence on representations of the absolute Galois group of a number field. Still other problems, such as the 11th and the 16th, concern what are now flourishing mathematical subdisciplines, like the theories of quadratic forms and real algebraic curves.
There are two problems that are not only unresolved but may in fact be unresolvable by modern standards. The 6th problem concerns the axiomatization of physics, a goal that 20th-century developments seem to render both more remote and less important than in Hilbert's time. Also, the 4th problem concerns the foundations of geometry, in a manner that is now generally judged to be too vague to enable a definitive answer.
The other 21 problems have all received significant attention, and late into the 20th century work on these problems was still considered to be of the greatest importance. Paul Cohen received the Fields Medal in 1966 for his work on the first problem, and the negative solution of the tenth problem in 1970 by Yuri Matiyasevich (completing work by Julia Robinson, Hilary Putnam, and Martin Davis) generated similar acclaim. Aspects of these problems are still of great interest today.
Ignorabimus
Following Gottlob Frege and Bertrand Russell, Hilbert sought to define mathematics logically using the method of formal systems, i.e., finitistic proofs from an agreed-upon set of axioms. One of the main goals of Hilbert's program was a finitistic proof of the consistency of the axioms of arithmetic: that is his second problem.
However, Gödel's second incompleteness theorem gives a precise sense in which such a |
https://en.wikipedia.org/wiki/Negative%20number | In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset. If a quantity, such as the charge on an electron, may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common-sense idea of an opposite is reflected in arithmetic. For example, −(−3) = 3 because the opposite of an opposite is the original value.
Negative numbers are usually written with a minus sign in front. For example, −3 represents a negative quantity with a magnitude of three, and is pronounced "minus three" or "negative three". To help tell the difference between a subtraction operation and a negative number, occasionally the negative sign is placed slightly higher than the minus sign (as a superscript). Conversely, a number that is greater than zero is called positive; zero is usually (but not always) thought of as neither positive nor negative. The positivity of a number may be emphasized by placing a plus sign before it, e.g. +3. In general, the negativity or positivity of a number is referred to as its sign.
Every real number other than zero is either positive or negative. The non-negative whole numbers are referred to as natural numbers (i.e., 0, 1, 2, 3...), while the positive and negative whole numbers (together with zero) are referred to as integers. (Some definitions of the natural numbers exclude zero.)
In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, as an alternative notation to represent negative numbers.
Negative numbers were used in the Nine Chapters on the Mathematical Art, which in its present form dates from the period of the Chinese Han Dynasty (202 BC – AD 220), but may well contain much older material. Liu Hui (c. 3rd century) established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers. Islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Prior to the concept of negative numbers, mathematicians such as Diophantus considered negative solutions to problems "false" and equations requiring negative solutions were described as absurd. Western mathematicians like Leibniz held that negative numbers were invalid, but still used them in calculations.
Introduction
The number line
The relationship between negative numbers, positive numbers, and zero is often expressed in the form of a number line:
Numbers appearing farthe |
https://en.wikipedia.org/wiki/Probability%20mass%20function | In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete probability density function. The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose domain is discrete.
A probability mass function differs from a probability density function (PDF) in that the latter is associated with continuous rather than discrete random variables. A PDF must be integrated over an interval to yield a probability.
The value of the random variable having the largest probability mass is called the mode.
Formal definition
Probability mass function is the probability distribution of a discrete random variable, and provides the possible values and their associated probabilities. It is the function defined by
for , where is a probability measure. can also be simplified as .
The probabilities associated with all (hypothetical) values must be non-negative and sum up to 1,
and
Thinking of probability as mass helps to avoid mistakes since the physical mass is conserved as is the total probability for all hypothetical outcomes .
Measure theoretic formulation
A probability mass function of a discrete random variable can be seen as a special case of two more general measure theoretic constructions:
the distribution of and the probability density function of with respect to the counting measure. We make this more precise below.
Suppose that is a probability space
and that is a measurable space whose underlying σ-algebra is discrete, so in particular contains singleton sets of . In this setting, a random variable is discrete provided its image is countable.
The pushforward measure —called the distribution of in this context—is a probability measure on whose restriction to singleton sets induces the probability mass function (as mentioned in the previous section) since for each .
Now suppose that is a measure space equipped with the counting measure μ. The probability density function of with respect to the counting measure, if it exists, is the Radon–Nikodym derivative of the pushforward measure of (with respect to the counting measure), so and is a function from to the non-negative reals. As a consequence, for any we have
demonstrating that is in fact a probability mass function.
When there is a natural order among the potential outcomes , it may be convenient to assign numerical values to them (or n-tuples in case of a discrete multivariate random variable) and to consider also values not in the image of . That is, may be defined for all real numbers and for all as shown in the figure.
The image of has a countable subset on which the probability mass function is one. Consequently, the probability mass function is zero for all but a countable number of values |
https://en.wikipedia.org/wiki/Dicyclic%20group | In group theory, a dicyclic group (notation Dicn or Q4n, ) is a particular kind of non-abelian group of order 4n (n > 1). It is an extension of the cyclic group of order 2 by a cyclic group of order 2n, giving the name di-cyclic. In the notation of exact sequences of groups, this extension can be expressed as:
More generally, given any finite abelian group with an order-2 element, one can define a dicyclic group.
Definition
For each integer n > 1, the dicyclic group Dicn can be defined as the subgroup of the unit quaternions generated by
More abstractly, one can define the dicyclic group Dicn as the group with the following presentation
Some things to note which follow from this definition:
if , then
Thus, every element of Dicn can be uniquely written as amxl, where 0 ≤ m < 2n and l = 0 or 1. The multiplication rules are given by
It follows that Dicn has order 4n.
When n = 2, the dicyclic group is isomorphic to the quaternion group Q. More generally, when n is a power of 2, the dicyclic group is isomorphic to the generalized quaternion group.
Properties
For each n > 1, the dicyclic group Dicn is a non-abelian group of order 4n. (For the degenerate case n = 1, the group Dic1 is the cyclic group C4, which is not considered dicyclic.)
Let A = be the subgroup of Dicn generated by a. Then A is a cyclic group of order 2n, so [Dicn:A] = 2. As a subgroup of index 2 it is automatically a normal subgroup. The quotient group Dicn/A is a cyclic group of order 2.
Dicn is solvable; note that A is normal, and being abelian, is itself solvable.
Binary dihedral group
The dicyclic group is a binary polyhedral group — it is one of the classes of subgroups of the Pin group Pin−(2), which is a subgroup of the Spin group Spin(3) — and in this context is known as the binary dihedral group.
The connection with the binary cyclic group C2n, the cyclic group Cn, and the dihedral group Dihn of order 2n is illustrated in the diagram at right, and parallels the corresponding diagram for the Pin group. Coxeter writes the binary dihedral group as ⟨2,2,n⟩ and binary cyclic group with angle-brackets, ⟨n⟩.
There is a superficial resemblance between the dicyclic groups and dihedral groups; both are a sort of "mirroring" of an underlying cyclic group. But the presentation of a dihedral group would have x2 = 1, instead of x2 = an; and this yields a different structure. In particular, Dicn is not a semidirect product of A and , since A ∩ is not trivial.
The dicyclic group has a unique involution (i.e. an element of order 2), namely x2 = an. Note that this element lies in the center of Dicn. Indeed, the center consists solely of the identity element and x2. If we add the relation x2 = 1 to the presentation of Dicn one obtains a presentation of the dihedral group Dihn, so the quotient group Dicn/<x2> is isomorphic to Dihn.
There is a natural 2-to-1 homomorphism from the group of unit quaternions to the 3-dimensional rotation group described at quaternions and spa |
https://en.wikipedia.org/wiki/Kleene%20algebra | In mathematics, a Kleene algebra ( ; named after Stephen Cole Kleene) is an idempotent (and thus partially ordered) semiring endowed with a closure operator. It generalizes the operations known from regular expressions.
Definition
Various inequivalent definitions of Kleene algebras and related structures have been given in the literature. Here we will give the definition that seems to be the most common nowadays.
A Kleene algebra is a set A together with two binary operations + : A × A → A and · : A × A → A and one function * : A → A, written as a + b, ab and a* respectively, so that the following axioms are satisfied.
Associativity of + and ·: a + (b + c) = (a + b) + c and a(bc) = (ab)c for all a, b, c in A.
Commutativity of +: a + b = b + a for all a, b in A
Distributivity: a(b + c) = (ab) + (ac) and (b + c)a = (ba) + (ca) for all a, b, c in A
Identity elements for + and ·: There exists an element 0 in A such that for all a in A: a + 0 = 0 + a = a. There exists an element 1 in A such that for all a in A: a1 = 1a = a.
Annihilation by 0: a0 = 0a = 0 for all a in A.
The above axioms define a semiring. We further require:
+ is idempotent: a + a = a for all a in A.
It is now possible to define a partial order ≤ on A by setting a ≤ b if and only if a + b = b (or equivalently: a ≤ b if and only if there exists an x in A such that a + x = b; with any definition, a ≤ b ≤ a implies a = b). With this order we can formulate the last four axioms about the operation *:
1 + a(a*) ≤ a* for all a in A.
1 + (a*)a ≤ a* for all a in A.
if a and x are in A such that ax ≤ x, then a*x ≤ x
if a and x are in A such that xa ≤ x, then x(a*) ≤ x
Intuitively, one should think of a + b as the "union" or the "least upper bound" of a and b and of ab as some multiplication which is monotonic, in the sense that a ≤ b implies ax ≤ bx. The idea behind the star operator is a* = 1 + a + aa + aaa + ... From the standpoint of programming language theory, one may also interpret + as "choice", · as "sequencing" and * as "iteration".
Examples
Let Σ be a finite set (an "alphabet") and let A be the set of all regular expressions over Σ. We consider two such regular expressions equal if they describe the same language. Then A forms a Kleene algebra. In fact, this is a free Kleene algebra in the sense that any equation among regular expressions follows from the Kleene algebra axioms and is therefore valid in every Kleene algebra.
Again let Σ be an alphabet. Let A be the set of all regular languages over Σ (or the set of all context-free languages over Σ; or the set of all recursive languages over Σ; or the set of all languages over Σ). Then the union (written as +) and the concatenation (written as ·) of two elements of A again belong to A, and so does the Kleene star operation applied to any element of A. We obtain a Kleene algebra A with 0 being the empty set and 1 being the set that only contains the empty string.
Let M be a monoid with identity element e and let A |
https://en.wikipedia.org/wiki/Sievert | The sievert (symbol: Sv) is a unit in the International System of Units (SI) intended to represent the stochastic health risk of ionizing radiation, which is defined as the probability of causing radiation-induced cancer and genetic damage. The sievert is important in dosimetry and radiation protection. It is named after Rolf Maximilian Sievert, a Swedish medical physicist renowned for work on radiation dose measurement and research into the biological effects of radiation.
The sievert is used for radiation dose quantities such as equivalent dose and effective dose, which represent the risk of external radiation from sources outside the body, and committed dose, which represents the risk of internal irradiation due to inhaled or ingested radioactive substances. According to the International Commission on Radiological Protection (ICRP), one sievert results in a 5.5% probability of eventually developing fatal cancer based on the disputed linear no-threshold model of ionizing radiation exposure.
To calculate the value of stochastic health risk in sieverts, the physical quantity absorbed dose is converted into equivalent dose and effective dose by applying factors for radiation type and biological context, published by the ICRP and the International Commission on Radiation Units and Measurements (ICRU). One sievert equals 100 rem, which is an older, CGS radiation unit.
Conventionally, deterministic health effects due to acute tissue damage that is certain to happen, produced by high dose rates of radiation, are compared to the physical quantity absorbed dose measured by the unit gray (Gy).
Definition
CIPM definition of the sievert
The SI definition given by the International Committee for Weights and Measures (CIPM) says:
"The quantity dose equivalent H is the product of the absorbed dose D of ionizing radiation and the dimensionless factor Q (quality factor) defined as a function of linear energy transfer by the ICRU"
H = Q × D
The value of Q is not defined further by CIPM, but it requires the use of the relevant ICRU recommendations to provide this value.
The CIPM also says that "in order to avoid any risk of confusion between the absorbed dose D and the dose equivalent H, the special names for the respective units should be used, that is, the name gray should be used instead of joules per kilogram for the unit of absorbed dose D and the name sievert instead of joules per kilogram for the unit of dose equivalent H".
In summary:
gray: quantity D – absorbed dose
1 Gy = 1 joule/kilogram – a physical quantity. 1 Gy is the deposit of a joule of radiation energy per kilogram of matter or tissue.
sievert: quantity H – equivalent dose
1 Sv = 1 joule/kilogram – a biological effect. The sievert represents the equivalent biological effect of the deposit of a joule of radiation energy in a kilogram of human tissue. The ratio to absorbed dose is denoted by Q.
ICRP definition of the sievert
The ICRP definition of the sievert is:
"The sievert |
https://en.wikipedia.org/wiki/Converse%20%28logic%29 | In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication P → Q, the converse is Q → P. For the categorical proposition All S are P, the converse is All P are S. Either way, the truth of the converse is generally independent from that of the original statement.
Implicational converse
Let S be a statement of the form P implies Q (P → Q). Then the converse of S is the statement Q implies P (Q → P). In general, the truth of S says nothing about the truth of its converse, unless the antecedent P and the consequent Q are logically equivalent.
For example, consider the true statement "If I am a human, then I am mortal." The converse of that statement is "If I am mortal, then I am a human," which is not necessarily true.
On the other hand, the converse of a statement with mutually inclusive terms remains true, given the truth of the original proposition. This is equivalent to saying that the converse of a definition is true. Thus, the statement "If I am a triangle, then I am a three-sided polygon" is logically equivalent to "If I am a three-sided polygon, then I am a triangle", because the definition of "triangle" is "three-sided polygon".
A truth table makes it clear that S and the converse of S are not logically equivalent, unless both terms imply each other:
Going from a statement to its converse is the fallacy of affirming the consequent. However, if the statement S and its converse are equivalent (i.e., P is true if and only if Q is also true), then affirming the consequent will be valid.
Converse implication is logically equivalent to the disjunction of and
In natural language, this could be rendered "not Q without P".
Converse of a theorem
In mathematics, the converse of a theorem of the form P → Q will be Q → P. The converse may or may not be true, and even if true, the proof may be difficult. For example, the Four-vertex theorem was proved in 1912, but its converse was proved only in 1997.
In practice, when determining the converse of a mathematical theorem, aspects of the antecedent may be taken as establishing context. That is, the converse of "Given P, if Q then R" will be "Given P, if R then Q". For example, the Pythagorean theorem can be stated as:
Given a triangle with sides of length , , and , if the angle opposite the side of length is a right angle, then .
The converse, which also appears in Euclid's Elements (Book I, Proposition 48), can be stated as:
Given a triangle with sides of length , , and , if , then the angle opposite the side of length is a right angle.
Converse of a relation
If is a binary relation with then the converse relation is also called the transpose.
Notation
The converse of the implication P → Q may be written Q → P, , but may also be notated , or "Bpq" (in Bocheński notation).
Categorical converse
In traditional logic, the process of switching the subject term with the predicate term is |
https://en.wikipedia.org/wiki/Galois%20connection | In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fundamental theorem of Galois theory about the correspondence between subgroups and subfields, discovered by the French mathematician Évariste Galois.
A Galois connection can also be defined on preordered sets or classes; this article presents the common case of posets.
The literature contains two closely related notions of "Galois connection". In this article, we will refer to them as (monotone) Galois connections and antitone Galois connections.
A Galois connection is rather weak compared to an order isomorphism between the involved posets, but every Galois connection gives rise to an isomorphism of certain sub-posets, as will be explained below.
The term Galois correspondence is sometimes used to mean a bijective Galois connection; this is simply an order isomorphism (or dual order isomorphism, depending on whether we take monotone or antitone Galois connections).
Definitions
(Monotone) Galois connection
Let and be two partially ordered sets. A monotone Galois connection between these posets consists of two monotone functions: and , such that for all in and in , we have
if and only if .
In this situation, is called the lower adjoint of and is called the upper adjoint of F. Mnemonically, the upper/lower terminology refers to where the function application appears relative to ≤. The term "adjoint" refers to the fact that monotone Galois connections are special cases of pairs of adjoint functors in category theory as discussed further below. Other terminology encountered here is left adjoint (resp. right adjoint) for the lower (resp. upper) adjoint.
An essential property of a Galois connection is that an upper/lower adjoint of a Galois connection uniquely determines the other:
is the least element with , and
is the largest element with .
A consequence of this is that if or is invertible, then each is the inverse of the other, i.e. .
Given a Galois connection with lower adjoint and upper adjoint , we can consider the compositions , known as the associated closure operator, and , known as the associated kernel operator. Both are monotone and idempotent, and we have for all in and for all in .
A Galois insertion of into is a Galois connection in which the kernel operator is the identity on , and hence is an order isomorphism of onto the set of closed elements  [] of .
Antitone Galois connection
The above definition is common in many applications today, and prominent in lattice and domain theory. However the original notion in Galois theory is slightly different. In this alternative definition, a Galois connection is a pair of antitone, i.e. order-reversing, functions and between two posets and , such that
if and only if .
The symmetry of and in this vers |
https://en.wikipedia.org/wiki/Chebyshev%27s%20inequality | In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from the mean. Specifically, no more than 1/k2 of the distribution's values can be k or more standard deviations away from the mean (or equivalently, at least 1 − 1/k2 of the distribution's values are less than k standard deviations away from the mean). The rule is often called Chebyshev's theorem, about the range of standard deviations around the mean, in statistics. The inequality has great utility because it can be applied to any probability distribution in which the mean and variance are defined. For example, it can be used to prove the weak law of large numbers.
Its practical usage is similar to the 68–95–99.7 rule, which applies only to normal distributions. Chebyshev's inequality is more general, stating that a minimum of just 75% of values must lie within two standard deviations of the mean and 88.89% within three standard deviations for a broad range of different probability distributions.
The term Chebyshev's inequality may also refer to Markov's inequality, especially in the context of analysis. They are closely related, and some authors refer to Markov's inequality as "Chebyshev's First Inequality," and the similar one referred to on this page as "Chebyshev's Second Inequality."
History
The theorem is named after Russian mathematician Pafnuty Chebyshev, although it was first formulated by his friend and colleague Irénée-Jules Bienaymé. The theorem was first stated without proof by Bienaymé in 1853 and later proved by Chebyshev in 1867. His student Andrey Markov provided another proof in his 1884 Ph.D. thesis.
Statement
Chebyshev's inequality is usually stated for random variables, but can be generalized to a statement about measure spaces.
Probabilistic statement
Let X (integrable) be a random variable with finite non-zero variance σ2 (and thus finite expected value μ). Then for any real number ,
Only the case is useful. When the right-hand side and the inequality is trivial as all probabilities are ≤ 1.
As an example, using shows that the probability that values lie outside the interval does not exceed . Equivalently, it implies that the probability of values lying within the interval (i.e. its "coverage") is at least .
Because it can be applied to completely arbitrary distributions provided they have a known finite mean and variance, the inequality generally gives a poor bound compared to what might be deduced if more aspects are known about the distribution involved.
Measure-theoretic statement
Let (X, Σ, μ) be a measure space, and let f be an extended real-valued measurable function defined on X. Then for any real number t > 0 and 0 < p < ∞,
More generally, if g is an extended real-valued measurable function, nonnegative and nondecreasing, with then:
The previous statement then f |
https://en.wikipedia.org/wiki/Cuboid | In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the lengths of the edges or the angles between faces, a cuboid can be transformed into a cube. In mathematical language a cuboid is a convex polyhedron whose polyhedral graph is the same as that of a cube.
A special case of a cuboid is a rectangular cuboid, with six rectangles as faces and adjacent faces meeting at right angles. A cube is a special case of a rectangular cuboid, with six square faces meeting at right angles.
By Euler's formula the numbers of faces F, of vertices V, and of edges E of any convex polyhedron are related by the formula F + V = E + 2. In the case of a cuboid this gives 6 + 8 = 12 + 2; that is, like a cube, a cuboid has six faces, eight vertices, and twelve edges.
Along with the rectangular cuboids, any parallelepiped is a cuboid of this type, as is a square frustum (the shape formed by truncation of the apex of a square pyramid).
See also
Hyperrectangle
Trapezohedron
Lists of shapes
References
External links
Rectangular prism and cuboid Paper models and pictures
Elementary shapes
Polyhedra
Prismatoid polyhedra
Space-filling polyhedra
Zonohedra |
https://en.wikipedia.org/wiki/Peter%20Barlow%20%28mathematician%29 | Peter Barlow (13 October 1776 – 1 March 1862) was an English mathematician and physicist.
Work in mathematics
In 1801, Barlow was appointed assistant mathematics master at the Royal Military Academy, Woolwich, and retained this post until 1847. He contributed articles on mathematics to The Ladies' Diary as well as publishing books such as:
An Elementary Investigation of the Theory of Numbers (1811);
A New Mathematical and Philosophical Dictionary (1814); and
New Mathematical Tables (1814).
The latter became known as Barlow's Tables and gives squares, cubes, square roots, cube roots, and reciprocals of all integer numbers from 1 to 10,000. These tables were regularly reprinted until 1965, when computers rendered them obsolete. He contributed to Rees's Cyclopædia articles on Algebra, Analysis, Geometry and Strength of Materials. Barlow also contributed largely to the Encyclopædia Metropolitana.
Work in physics and engineering
In collaboration (1827–1832) with optician George Dollond, Barlow built an achromatic lens that utilized liquid carbon disulfide. (Achromatic lenses were important optical elements of improved telescopes.) In 1833, Barlow built an achromatic doublet lens of joined flint glass and crown glass. A derivative of this design, named a Barlow lens, is widely used in modern astronomy and photography as an optical element to increase both achromatism and magnification.
In 1823, he was made a fellow of the Royal Society. Two years later, he received its Copley Medal for his work on correcting the deviation in ship compasses caused by the presence of iron in the hull. Some of his magnetic research was done in collaboration with Samuel Hunter Christie. He conducted early experimental and observational studies on the origins of terrestrial magnetism. He is credited with the eponymous Barlow's wheel (an early homopolar electric motor) and with Barlow's law (an incorrect formula of electrical conductance).
Barlow investigated a suggestion made by André-Marie Ampère in 1820 that an electromagnetic telegraph could be made by deflecting a compass needle with an electric current. In 1824 Barlow proclaimed the idea impractical after he found that the effect on the compass seriously diminished "with only 200 feet of wire". Barlow, and other eminent scientists of the time who agreed with him, are criticised for retarding the development of the telegraph. A decade passed between Ampère's paper being read at the Paris Academy of Sciences and William Ritchie building the first demonstration electromagnetic telegraph. In Barlow's defence, Ampère's design did not enclose the compass in a multiplying coil, as Ritchie's demonstrator did, so the effect would have been very weak at a distance.
Steam locomotion received much attention at Barlow's hands and he sat on the railway commissions of 1836, 1839, 1842 and 1845. He also conducted several investigations for the newly formed Railway Inspectorate in the early 1840s.
Barlow made several c |
https://en.wikipedia.org/wiki/Law%20of%20large%20numbers | In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value and tends to become closer to the expected value as more trials are performed.
The LLN is important because it guarantees stable long-term results for the averages of some random events. For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. Importantly, the law applies (as the name indicates) only when a large number of observations are considered. There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be "balanced" by the others (see the gambler's fallacy).
The LLN only applies to the average. Therefore, while
other formulas that look similar are not verified, such as the raw deviation from "theoretical results":
not only does it not converge toward zero as n increases, but it tends to increase in absolute value as n increases.
Examples
For example, a single roll of a fair, six-sided die produces one of the numbers 1, 2, 3, 4, 5, or 6, each with equal probability. Therefore, the expected value of the average of the rolls is:
According to the law of large numbers, if a large number of six-sided dice are rolled, the average of their values (sometimes called the sample mean) will approach 3.5, with the precision increasing as more dice are rolled.
It follows from the law of large numbers that the empirical probability of success in a series of Bernoulli trials will converge to the theoretical probability. For a Bernoulli random variable, the expected value is the theoretical probability of success, and the average of n such variables (assuming they are independent and identically distributed (i.i.d.)) is precisely the relative frequency.
For example, a fair coin toss is a Bernoulli trial. When a fair coin is flipped once, the theoretical probability that the outcome will be heads is equal to . Therefore, according to the law of large numbers, the proportion of heads in a "large" number of coin flips "should be" roughly . In particular, the proportion of heads after n flips will almost surely converge to as n approaches infinity.
Although the proportion of heads (and tails) approaches , almost surely the absolute difference in the number of heads and tails will become large as the number of flips becomes large. That is, the probability that the absolute difference is a small number approaches zero as the number of flips becomes large. Also, almost surely the ratio of the absolute difference to the number of flips will approach zero. Intuitively, the expected difference grows, bu |
https://en.wikipedia.org/wiki/Correlation | In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it usually refers to the degree to which a pair of variables are linearly related.
Familiar examples of dependent phenomena include the correlation between the height of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the so-called demand curve.
Correlations are useful because they can indicate a predictive relationship that can be exploited in practice. For example, an electrical utility may produce less power on a mild day based on the correlation between electricity demand and weather. In this example, there is a causal relationship, because extreme weather causes people to use more electricity for heating or cooling. However, in general, the presence of a correlation is not sufficient to infer the presence of a causal relationship (i.e., correlation does not imply causation).
Formally, random variables are dependent if they do not satisfy a mathematical property of probabilistic independence. In informal parlance, correlation is synonymous with dependence. However, when used in a technical sense, correlation refers to any of several specific types of mathematical operations between the tested variables and their respective expected values. Essentially, correlation is the measure of how two or more variables are related to one another. There are several correlation coefficients, often denoted or , measuring the degree of correlation. The most common of these is the Pearson correlation coefficient, which is sensitive only to a linear relationship between two variables (which may be present even when one variable is a nonlinear function of the other). Other correlation coefficients – such as Spearman's rank correlation – have been developed to be more robust than Pearson's, that is, more sensitive to nonlinear relationships. Mutual information can also be applied to measure dependence between two variables.
Pearson's product-moment coefficient
The most familiar measure of dependence between two quantities is the Pearson product-moment correlation coefficient (PPMCC), or "Pearson's correlation coefficient", commonly called simply "the correlation coefficient". It is obtained by taking the ratio of the covariance of the two variables in question of our numerical dataset, normalized to the square root of their variances. Mathematically, one simply divides the covariance of the two variables by the product of their standard deviations. Karl Pearson developed the coefficient from a similar but slightly different idea by Francis Galton.
A Pearson product-moment correlation coefficient attempts to establish a line of best fit through a dataset of two variables by essentially laying out the exp |
https://en.wikipedia.org/wiki/Covariance | Covariance in probability theory and statistics is a measure of the joint variability of two random variables.
Definition
If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the lesser values (that is, the variables tend to show similar behavior), the covariance is positive. In the opposite case, when the greater values of one variable mainly correspond to the lesser values of the other, (that is, the variables tend to show opposite behavior), the covariance is negative. The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables. The magnitude of the covariance is the geometric mean of the variances that are in-common for the two random variables. The correlation coefficient normalizes the covariance by dividing by the geometric mean of the total variances for the two random variables.
A distinction must be made between (1) the covariance of two random variables, which is a population parameter that can be seen as a property of the joint probability distribution, and (2) the sample covariance, which in addition to serving as a descriptor of the sample, also serves as an estimated value of the population parameter.
Mathematics
For two jointly distributed real-valued random variables and with finite second moments, the covariance is defined as the expected value (or mean) of the product of their deviations from their individual expected values:
where is the expected value of , also known as the mean of . The covariance is also sometimes denoted or , in analogy to variance. By using the linearity property of expectations, this can be simplified to the expected value of their product minus the product of their expected values:
but this equation is susceptible to catastrophic cancellation (see the section on numerical computation below).
The units of measurement of the covariance are those of times those of . By contrast, correlation coefficients, which depend on the covariance, are a dimensionless measure of linear dependence. (In fact, correlation coefficients can simply be understood as a normalized version of covariance.)
Complex random variables
The covariance between two complex random variables is defined as
Notice the complex conjugation of the second factor in the definition.
A related pseudo-covariance can also be defined.
Discrete random variables
If the (real) random variable pair can take on the values for , with equal probabilities , then the covariance can be equivalently written in terms of the means and as
It can also be equivalently expressed, without directly referring to the means, as
More generally, if there are possible realizations of , namely but with possibly unequal probabilities for , then the covariance is
Examples
Consider 3 independent random variables and two constants .
In the special case, and , the covariance between and , is just the variance of and the name cova |
https://en.wikipedia.org/wiki/Cross%20product | In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol . Given two linearly independent vectors and , the cross product, (read "a cross b"), is a vector that is perpendicular to both and , and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product).
The magnitude of the cross product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The units of the cross-product are the product of the units of each vector. If two vectors are parallel or are anti-parallel (that is, they are linearly dependent), or if either one has zero length, then their cross product is zero.
The cross product is anticommutative (that is, ) and is distributive over addition (that is, ). The space together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.
Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on a choice of orientation (or "handedness") of the space (it is why an oriented space is needed). In connection with the cross product, the exterior product of vectors can be used in arbitrary dimensions (with a bivector or 2-form result) and is independent of the orientation of the space.
The product can be generalized in various ways, using the orientation and metric structure just as for the traditional 3-dimensional cross product, one can, in dimensions, take the product of vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions. The cross-product in seven dimensions has undesirable properties, however (e.g. it fails to satisfy the Jacobi identity), so it is not used in mathematical physics to represent quantities such as multi-dimensional space-time. (See § Generalizations, below, for other dimensions.)
Definition
The cross product of two vectors a and b is defined only in three-dimensional space and is denoted by . In physics and applied mathematics, the wedge notation is often used (in conjunction with the name vector product), although in pure mathematics such notation is usually reserved for just the exterior product, an abstraction of the vector product to dimensions.
The cross product is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors s |
https://en.wikipedia.org/wiki/Dot%20product | In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more).
Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In modern geometry, Euclidean spaces are often defined by using vector spaces. In this case, the dot product is used for defining lengths (the length of a vector is the square root of the dot product of the vector by itself) and angles (the cosine of the angle between two vectors is the quotient of their dot product by the product of their lengths).
The name "dot product" is derived from the centered dot " · " that is often used to designate this operation; the alternative name "scalar product" emphasizes that the result is a scalar, rather than a vector (as with the vector product in three-dimensional space).
Definition
The dot product may be defined algebraically or geometrically. The geometric definition is based on the notions of angle and distance (magnitude) of vectors. The equivalence of these two definitions relies on having a Cartesian coordinate system for Euclidean space.
In modern presentations of Euclidean geometry, the points of space are defined in terms of their Cartesian coordinates, and Euclidean space itself is commonly identified with the real coordinate space . In such a presentation, the notions of length and angle are defined by means of the dot product. The length of a vector is defined as the square root of the dot product of the vector by itself, and the cosine of the (non oriented) angle between two vectors of length one is defined as their dot product. So the equivalence of the two definitions of the dot product is a part of the equivalence of the classical and the modern formulations of Euclidean geometry.
Coordinate definition
The dot product of two vectors and specified with respect to an orthonormal basis, is defined as:
where denotes summation and is the dimension of the vector space. For instance, in three-dimensional space, the dot product of vectors and is:
Likewise, the dot product of the vector with itself is:
If vectors are identified with column vectors, the dot product can also be written as a matrix product
where denotes the transpose of .
Expressing the above example in this way, a 1 × 3 matrix (row vector) is multiplied by a 3 × 1 matrix (column vector) to get a 1 × 1 matrix that is identified with |
https://en.wikipedia.org/wiki/Van%20der%20Waerden%27s%20theorem | Van der Waerden's theorem is a theorem in the branch of mathematics called Ramsey theory. Van der Waerden's theorem states that for any given positive integers r and k, there is some number N such that if the integers {1, 2, ..., N} are colored, each with one of r different colors, then there are at least k integers in arithmetic progression whose elements are of the same color. The least such N is the Van der Waerden number W(r, k), named after the Dutch mathematician B. L. van der Waerden.
Example
For example, when r = 2, you have two colors, say red and blue. W(2, 3) is bigger than 8, because you can color the integers from {1, ..., 8} like this:
and no three integers of the same color form an arithmetic progression. But you can't add a ninth integer to the end without creating such a progression. If you add a red 9, then the red 3, 6, and 9 are in arithmetic progression. Alternatively, if you add a blue 9, then the blue 1, 5, and 9 are in arithmetic progression.
In fact, there is no way of coloring 1 through 9 without creating such a progression (it can be proved by considering examples). Therefore, W(2, 3) is 9.
Open problem
It is an open problem to determine the values of W(r, k) for most values of r and k. The proof of the theorem provides only an upper bound. For the case of r = 2 and k = 3, for example, the argument given below shows that it is sufficient to color the integers {1, ..., 325} with two colors to guarantee there will be a single-colored arithmetic progression of length 3. But in fact, the bound of 325 is very loose; the minimum required number of integers is only 9. Any coloring of the integers {1, ..., 9} will have three evenly spaced integers of one color.
For r = 3 and k = 3, the bound given by the theorem is 7(2·37 + 1)(2·37·(2·37 + 1) + 1), or approximately 4.22·1014616. But actually, you don't need that many integers to guarantee a single-colored progression of length 3; you only need 27. (And it is possible to color {1, ..., 26} with three colors so that there is no single-colored arithmetic progression of length 3; for example:
An open problem is the attempt to reduce the general upper bound to any 'reasonable' function. Ronald Graham offered a prize of US$1000 for showing W(2, k) < 2k2. In addition, he offered a US$250 prize for a proof of his conjecture involving more general off-diagonal van der Waerden numbers, stating W(2; 3, k) ≤ kO(1), while mentioning numerical evidence suggests W(2; 3, k) = k2 + o(1). Ben Green disproved this latter conjecture and proved super-polynomial counterexamples to W(2; 3, k) < kr for any r. The best upper bound currently known is due to Timothy Gowers, who establishes
by first establishing a similar result for Szemerédi's theorem, which is a stronger version of Van der Waerden's theorem. The previously best-known bound was due to Saharon Shelah and proceeded via first proving a result for the Hales–Jewett theorem, which is another strengthening of Van der Waerden's |
https://en.wikipedia.org/wiki/Elmer%20Rees | Elmer Gethin Rees, (19 November 1941 – 4 October 2019) was a Welsh mathematician with publications in areas ranging from topology, differential geometry, algebraic geometry, linear algebra and Morse theory to robotics. He held the post of Director of the Heilbronn Institute for Mathematical Research, a partnership between the University of Bristol and the British signals intelligence agency GCHQ, from its creation in 2005 until 2009.
Biography
Rees was born in Llandybie and grew up in Wales. He studied at St Catharine's College, Cambridge gaining a BA before moving on to the University of Warwick, where he completed his PhD in 1967. His thesis on Projective Spaces and Associated Maps, was written under the supervision of David B. A. Epstein.
Rees's career had taken him to University of Hull, the Institute for Advanced Study in Princeton, New Jersey, Swansea University and St Catherine's College, Oxford, before becoming a professor at the University of Edinburgh in 1979, where he remained until retiring from the post in 2005.
He was elected as a fellow of the Royal Society of Edinburgh in 1982. One of his most notable legacies was the establishment of the International Centre for Mathematical Sciences.
Rees was appointed Commander of the Order of the British Empire (CBE) in the 2009 Birthday Honours.
While at the universities of Oxford and Edinburgh, he supervised at least 15 PhD students, including Anthony Bahri, John D. S. Jones, Gregory Lupton, Jacob Mostovoy, Simon Willerton and Richard Hepworth.
Footnotes
External links and references
Senatus Academicus, University of Edinburgh. "Special Minute - Professor Elmer Gethin Rees MA PhD FRSE". Retrieved 2006-10-21.
University of Edinburgh Honorary Degree (24 June 2008)
Heilbronn Institute for Mathematical Research
Elmer Rees 70th birthday conference
70th birthday conference poster
Welsh mathematicians
20th-century British mathematicians
21st-century British mathematicians
Algebraic geometers
Differential geometers
Topologists
Alumni of St Catharine's College, Cambridge
Alumni of the University of Warwick
Fellows of the Royal Society of Edinburgh
Welsh scientists
Commanders of the Order of the British Empire
1941 births
2019 deaths
Academics of the University of Edinburgh
20th-century Welsh writers
21st-century Welsh writers
20th-century Welsh educators
21st-century Welsh educators
21st-century Welsh scientists
20th-century Welsh scientists |
https://en.wikipedia.org/wiki/John%20Edensor%20Littlewood | John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to analysis, number theory, and differential equations and had lengthy collaborations with G. H. Hardy, Srinivasa Ramanujan and Mary Cartwright.
Biography
Littlewood was born on 9 June 1885 in Rochester, Kent, the eldest son of Edward Thornton Littlewood and Sylvia Maud (née Ackland). In 1892, his father accepted the headmastership of a school in Wynberg, Cape Town, in South Africa, taking his family there. Littlewood returned to Britain in 1900 to attend St Paul's School in London, studying under Francis Sowerby Macaulay, an influential algebraic geometer.
In 1903, Littlewood entered the University of Cambridge, studying in Trinity College. He spent his first two years preparing for the Tripos examinations which qualify undergraduates for a bachelor's degree where he emerged in 1905 as Senior Wrangler bracketed with James Mercer (Mercer had already graduated from the University of Manchester before attending Cambridge). In 1906, after completing the second part of the Tripos, he started his research under Ernest Barnes. One of the problems that Barnes suggested to Littlewood was to prove the Riemann hypothesis, an assignment at which he did not succeed. He was elected a Fellow of Trinity College in 1908. From October 1907 to June 1910, he worked as a Richardson Lecturer in the School of Mathematics at the University of Manchester before returning to Cambridge in October 1910, where he remained for the rest of his career. He was appointed Rouse Ball Professor of Mathematics in 1928, retiring in 1950. He was elected a Fellow of the Royal Society in 1916, awarded the Royal Medal in 1929, the Sylvester Medal in 1943, and the Copley Medal in 1958. He was president of the London Mathematical Society from 1941 to 1943 and was awarded the De Morgan Medal in 1938 and the Senior Berwick Prize in 1960.
Littlewood died on 6 September 1977.
Work
Most of Littlewood's work was in the field of mathematical analysis. He began research under the supervision of Ernest William Barnes, who suggested that he attempt to prove the Riemann hypothesis: Littlewood showed that if the Riemann hypothesis is true, then the prime number theorem follows and obtained the error term. This work won him his Trinity fellowship. However, the link between the Riemann hypothesis and the prime number theorem had been known before in Continental Europe, and Littlewood wrote later in his book, A Mathematician's Miscellany that his rediscovery of the result did not shed a positive light on the isolated nature of British mathematics at the time.
Theory of the distribution of prime numbers
In 1914, Littlewood published his first result in the field of analytic number theory concerning the error term of the prime-counting function. If denotes the number of primes up , then the prime number theorem implies that ,
where is known as the Eulerian logarithmic integral.
Num |
https://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac%20statistics | Fermi–Dirac statistics is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the Fermi–Dirac distribution of particles over energy states. It is named after Enrico Fermi and Paul Dirac, each of whom derived the distribution independently in 1926. Fermi–Dirac statistics is a part of the field of statistical mechanics and uses the principles of quantum mechanics.
Fermi–Dirac statistics applies to identical and indistinguishable particles with half-integer spin (1/2, 3/2, etc.), called fermions, in thermodynamic equilibrium. For the case of negligible interaction between particles, the system can be described in terms of single-particle energy states. A result is the Fermi–Dirac distribution of particles over these states where no two particles can occupy the same state, which has a considerable effect on the properties of the system. Fermi–Dirac statistics is most commonly applied to electrons, a type of fermion with spin 1/2.
A counterpart to Fermi–Dirac statistics is Bose–Einstein statistics, which applies to identical and indistinguishable particles with integer spin (0, 1, 2, etc.) called bosons. In classical physics, Maxwell–Boltzmann statistics is used to describe particles that are identical and treated as distinguishable. For both Bose–Einstein and Maxwell–Boltzmann statistics, more than one particle can occupy the same state, unlike Fermi–Dirac statistics.
History
Before the introduction of Fermi–Dirac statistics in 1926, understanding some aspects of electron behavior was difficult due to seemingly contradictory phenomena. For example, the electronic heat capacity of a metal at room temperature seemed to come from 100 times fewer electrons than were in the electric current. It was also difficult to understand why the emission currents generated by applying high electric fields to metals at room temperature were almost independent of temperature.
The difficulty encountered by the Drude model, the electronic theory of metals at that time, was due to considering that electrons were (according to classical statistics theory) all equivalent. In other words, it was believed that each electron contributed to the specific heat an amount on the order of the Boltzmann constant kB.
This problem remained unsolved until the development of Fermi–Dirac statistics.
Fermi–Dirac statistics was first published in 1926 by Enrico Fermi and Paul Dirac. According to Max Born, Pascual Jordan developed in 1925 the same statistics, which he called Pauli statistics, but it was not published in a timely manner. According to Dirac, it was first studied by Fermi, and Dirac called it "Fermi statistics" and the corresponding particles "fermions".
Fermi–Dirac statistics was applied in 1926 by Ralph Fowler to describe the collapse of a star to a white dwarf. In 1927 Arnold Sommerfeld applied it to electrons in metals and developed the free e |
https://en.wikipedia.org/wiki/Diffeology | In mathematics, a diffeology on a set generalizes the concept of smooth charts in a differentiable manifold, declaring what the "smooth parametrizations" in the set are.
The concept was first introduced by Jean-Marie Souriau in the 1980s under the name Espace différentiel and later developed by his students Paul Donato and Patrick Iglesias. A related idea was introduced by Kuo-Tsaï Chen (陳國才, Chen Guocai) in the 1970s, using convex sets instead of open sets for the domains of the plots.
Intuitive definition
Recall that a topological manifold is a topological space which is locally homeomorphic to . Differentiable manifolds generalize the notion of smoothness on in the following sense: a differentiable manifold is a topological manifold with a differentiable atlas, i.e. a collection of maps from open subsets of to the manifold which are used to "pull back" the differential structure from to the manifold.
A diffeological space consists of a set together with a collection of maps (called a diffeology) satisfying suitable axioms, which generalise the notion of an atlas on a manifold. In this way, the relationship between smooth manifolds and diffeological spaces is analogous to the relationship between topological manifolds and topological spaces.
More precisely, a smooth manifold can be equivalently defined as a diffeological space which is locally diffeomorphic to . Indeed, every smooth manifold has a natural diffeology, consisting of its maximal atlas (all the smooth maps from open subsets of to the manifold). This abstract point of view makes no reference to a specific atlas (and therefore to a fixed dimension ) nor to the underlying topological space, and is therefore suitable to treat examples of objects more general than manifolds.
Formal definition
A diffeology on a set consists of a collection of maps, called plots or parametrizations, from open subsets of () to such that the following axioms hold:
Covering axiom: every constant map is a plot.
Locality axiom: for a given map , if every point in has a neighborhood such that is a plot, then itself is a plot.
Smooth compatibility axiom: if is a plot, and is a smooth function from an open subset of some into the domain of , then the composite is a plot.
Note that the domains of different plots can be subsets of for different values of ; in particular, any diffeology contains the elements of its underlying set as the plots with . A set together with a diffeology is called a diffeological space.
More abstractly, a diffeological space is a concrete sheaf on the site of open subsets of , for all , and open covers.
Morphisms
A map between diffeological spaces is called smooth if and only if its composite with any plot of the first space is a plot of the second space. It is called a diffeomorphism if it is smooth, bijective, and its inverse is also smooth. By construction, given a diffeological space , its plots defined on are precisely all the smooth maps from to .
Di |
https://en.wikipedia.org/wiki/Quasi-empiricism%20in%20mathematics | Quasi-empiricism in mathematics is the attempt in the philosophy of mathematics to direct philosophers' attention to mathematical practice, in particular, relations with physics, social sciences, and computational mathematics, rather than solely to issues in the foundations of mathematics. Of concern to this discussion are several topics: the relationship of empiricism (see Penelope Maddy) with mathematics, issues related to realism, the importance of culture, necessity of application, etc.
Primary arguments
A primary argument with respect to quasi-empiricism is that whilst mathematics and physics are frequently considered to be closely linked fields of study, this may reflect human cognitive bias. It is claimed that, despite rigorous application of appropriate empirical methods or mathematical practice in either field, this would nonetheless be insufficient to disprove alternate approaches.
Eugene Wigner (1960) noted that this culture need not be restricted to mathematics, physics, or even humans. He stated further that "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning." Wigner used several examples to demonstrate why 'bafflement' is an appropriate description, such as showing how mathematics adds to situational knowledge in ways that are either not possible otherwise or are so outside normal thought to be of little notice. The predictive ability, in the sense of describing potential phenomena prior to observation of such, which can be supported by a mathematical system would be another example.
Following up on Wigner, Richard Hamming (1980) wrote about applications of mathematics as a central theme to this topic and suggested that successful use can sometimes trump proof, in the following sense: where a theorem has evident veracity through applicability, later evidence that shows the theorem's proof to be problematic would result more in trying to firm up the theorem rather than in trying to redo the applications or to deny results obtained to date. Hamming had four explanations for the 'effectiveness' that we see with mathematics and definitely saw this topic as worthy of discussion and study.
"We see what we look for." Why 'quasi' is apropos in reference to this discussion.
"We select the kind of mathematics to use." Our use and modification of mathematics are essentially situational and goal-driven.
"Science in fact answers comparatively few problems." What still needs to be looked at is a larger set.
"The evolution of man provided the model." There may be limits attributable to the human element.
For Willard Van Orman Quine (1960), existence is only existence in a structure. This position is re |
https://en.wikipedia.org/wiki/Quasi-empirical%20method | Quasi-empirical methods are methods applied in science and mathematics to achieve epistemology similar to that of empiricism (thus quasi- + empirical) when experience cannot falsify the ideas involved. Empirical research relies on empirical evidence, and its empirical methods involve experimentation and disclosure of apparatus for reproducibility, by which scientific findings are validated by other scientists. Empirical methods are studied extensively in the philosophy of science, but they cannot be used directly in fields whose hypotheses cannot be falsified by real experiment (for example, mathematics, philosophy, theology, and ideology). Because of such empirical limits in science, the scientific method must rely not only on empirical methods but sometimes also on quasi-empirical ones. The prefix quasi- came to denote methods that are "almost" or "socially approximate" an ideal of truly empirical methods.
It is unnecessary to find all counterexamples to a theory; all that is required to disprove a theory logically is one counterexample. The converse does not prove a theory; Bayesian inference simply makes a theory more likely, by weight of evidence.
One can argue that no science is capable of finding all counter-examples to a theory, therefore, no science is strictly empirical, it's all quasi-empirical. But usually, the term "quasi-empirical" refers to the means of choosing problems to focus on (or ignore), selecting prior work on which to build an argument or proof, notations for informal claims, peer review and acceptance, and incentives to discover, ignore, or correct errors. These are common to both science and mathematics, and do not include experimental method.
Albert Einstein's discovery of the general relativity theory relied upon thought experiments and mathematics. Empirical methods only became relevant when confirmation was sought. Furthermore, some empirical confirmation was found only some time after the general acceptance of the theory.
Thought experiments are almost standard procedure in philosophy, where a conjecture is tested out in the imagination for possible effects on experience; when these are thought to be implausible, unlikely to occur, or not actually occurring, then the conjecture may be either rejected or amended. Logical positivism was a perhaps extreme version of this practice, though this claim is open to debate.
Post-20th-century philosophy of mathematics is mostly concerned with quasi-empirical mathematical methods, especially as reflected in the actual mathematical practice of working mathematicians.
See also
Philosophy of science
Scientific method
Philosophy of science
Philosophy of mathematics
Thought experiments |
https://en.wikipedia.org/wiki/Linear%20interpolation | In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.
Linear interpolation between two known points
If the two known points are given by the coordinates and , the linear interpolant is the straight line between these points. For a value in the interval , the value along the straight line is given from the equation of slopes
which can be derived geometrically from the figure on the right. It is a special case of polynomial interpolation with .
Solving this equation for , which is the unknown value at , gives
which is the formula for linear interpolation in the interval . Outside this interval, the formula is identical to linear extrapolation.
This formula can also be understood as a weighted average. The weights are inversely related to the distance from the end points to the unknown point; the closer point has more influence than the farther point. Thus, the weights are and , which are normalized distances between the unknown point and each of the end points. Because these sum to 1,
yielding the formula for linear interpolation given above.
Interpolation of a data set
Linear interpolation on a set of data points is defined as the concatenation of linear interpolants between each pair of data points. This results in a continuous curve, with a discontinuous derivative (in general), thus of differentiability class
Linear interpolation as approximation
Linear interpolation is often used to approximate a value of some function using two known values of that function at other points. The error of this approximation is defined as
where denotes the linear interpolation polynomial defined above:
It can be proven using Rolle's theorem that if has a continuous second derivative, then the error is bounded by
That is, the approximation between two points on a given function gets worse with the second derivative of the function that is approximated. This is intuitively correct as well: the "curvier" the function is, the worse the approximations made with simple linear interpolation become.
History and applications
Linear interpolation has been used since antiquity for filling the gaps in tables. Suppose that one has a table listing the population of some country in 1970, 1980, 1990 and 2000, and that one wanted to estimate the population in 1994. Linear interpolation is an easy way to do this. It is believed that it was used in the Seleucid Empire (last three centuries BC) and by the Greek astronomer and mathematician Hipparchus (second century BC). A description of linear interpolation can be found in the ancient Chinese mathematical text called The Nine Chapters on the Mathematical Art (九章算術), dated from 200 BC to AD 100 and the Almagest (2nd century AD) by Ptolemy.
The basic operation of linear interpolation between two values is commonly used in computer graphics. In that field's jargon it is sometimes call |
https://en.wikipedia.org/wiki/Combinatorial%20species | In combinatorial mathematics, the theory of combinatorial species is an abstract, systematic method for deriving the generating functions of discrete structures, which allows one to not merely count these structures but give bijective proofs involving them. Examples of combinatorial species are (finite) graphs, permutations, trees, and so on; each of these has an associated generating function which counts how many structures there are of a certain size. One goal of species theory is to be able to analyse complicated structures by describing them in terms of transformations and combinations of simpler structures. These operations correspond to equivalent manipulations of generating functions, so producing such functions for complicated structures is much easier than with other methods. The theory was introduced, carefully elaborated and applied by Canadian researchers around André Joyal.
The power of the theory comes from its level of abstraction. The "description format" of a structure (such as adjacency list versus adjacency matrix for graphs) is irrelevant, because species are purely algebraic. Category theory provides a useful language for the concepts that arise here, but it is not necessary to understand categories before being able to work with species.
The category of species is equivalent to the category of symmetric sequences in finite sets.
Definition of species
Any species consists of individual combinatorial structures built on the elements of some finite set: for example, a combinatorial graph is a structure of edges among a given set of vertices, and the species of graphs includes all graphs on all finite sets. Furthermore, a member of a species can have its underying set relabeled by the elements of any other equinumerous set, for example relabeling the vertices of a graph gives "the same graph structure" on the new vertices, i.e. an isomorphic graph.
This leads to the formal definition of a combinatorial species. Let be the category of finite sets, with the morphisms of the category being the bijections between these sets. A species is a functor
For each finite set A in , the finite set F[A] is called the set of F-structures on A, or the set of structures of species F on A. Further, by the definition of a functor, if φ is a bijection between sets A and B, then F[φ] is a bijection between the sets of F-structures F[A] and F[B], called transport of F-structures along φ.
For example, the "species of permutations" maps each finite set A to the set S[A] of all permutations of A (all ways of ordering A into a list), and each bijection f from A to another set B naturally induces a bijection (a relabeling) taking each permutation of A to a corresponding permutation of B, namely a bijection . Similarly, the "species of partitions" can be defined by assigning to each finite set the set of all its partitions, and the "power set species" assigns to each finite set its power set. The adjacent diagram shows a structure (represented by |
https://en.wikipedia.org/wiki/Sampling%20%28statistics%29 | In statistics, quality assurance, and survey methodology, sampling is the selection of a subset or a statistical sample (termed sample for short) of individuals from within a statistical population to estimate characteristics of the whole population. Statisticians attempt to collect samples that are representative of the population. Sampling has lower costs and faster data collection compared to recording data from the entire population, and thus, it can provide insights in cases where it is infeasible to measure an entire population.
Each observation measures one or more properties (such as weight, location, colour or mass) of independent objects or individuals. In survey sampling, weights can be applied to the data to adjust for the sample design, particularly in stratified sampling. Results from probability theory and statistical theory are employed to guide the practice. In business and medical research, sampling is widely used for gathering information about a population. Acceptance sampling is used to determine if a production lot of material meets the governing specifications.
Population definition
Successful statistical practice is based on focused problem definition. In sampling, this includes defining the "population" from which our sample is drawn. A population can be defined as including all people or items with the characteristics one wishes to understand. Because there is very rarely enough time or money to gather information from everyone or everything in a population, the goal becomes finding a representative sample (or subset) of that population.
Sometimes what defines a population is obvious. For example, a manufacturer needs to decide whether a batch of material from production is of high enough quality to be released to the customer or should be scrapped or reworked due to poor quality. In this case, the batch is the population.
Although the population of interest often consists of physical objects, sometimes it is necessary to sample over time, space, or some combination of these dimensions. For instance, an investigation of supermarket staffing could examine checkout line length at various times, or a study on endangered penguins might aim to understand their usage of various hunting grounds over time. For the time dimension, the focus may be on periods or discrete occasions.
In other cases, the examined 'population' may be even less tangible. For example, Joseph Jagger studied the behaviour of roulette wheels at a casino in Monte Carlo, and used this to identify a biased wheel. In this case, the 'population' Jagger wanted to investigate was the overall behaviour of the wheel (i.e. the probability distribution of its results over infinitely many trials), while his 'sample' was formed from observed results from that wheel. Similar considerations arise when taking repeated measurements of some physical characteristic such as the electrical conductivity of copper.
This situation often arises when seeking knowledge about |
https://en.wikipedia.org/wiki/Point%20estimation | In statistics, point estimation involves the use of sample data to calculate a single value (known as a point estimate since it identifies a point in some parameter space) which is to serve as a "best guess" or "best estimate" of an unknown population parameter (for example, the population mean). More formally, it is the application of a point estimator to the data to obtain a point estimate.
Point estimation can be contrasted with interval estimation: such interval estimates are typically either confidence intervals, in the case of frequentist inference, or credible intervals, in the case of Bayesian inference. More generally, a point estimator can be contrasted with a set estimator. Examples are given by confidence sets or credible sets. A point estimator can also be contrasted with a distribution estimator. Examples are given by confidence distributions, randomized estimators, and Bayesian posteriors.
Properties of point estimates
Biasedness
“Bias” is defined as the difference between the expected value of the estimator and the true value of the population parameter being estimated. It can also be described that the closer the expected value of a parameter is to the measured parameter, the lesser the bias. When the estimated number and the true value is equal, the estimator is considered unbiased. This is called an unbiased estimator. The estimator will become a best unbiased estimator if it has minimum variance. However, a biased estimator with a small variance may be more useful than an unbiased estimator with a large variance. Most importantly, we prefer point estimators that have the smallest mean square errors.
If we let T = h(X1,X2, . . . , Xn) be an estimator based on a random sample X1,X2, . . . , Xn, the estimator T is called an unbiased estimator for the parameter θ if E[T] = θ, irrespective of the value of θ. For example, from the same random sample we have E(x̄) = µ (mean) and E(s2) = σ2 (variance), then x̄ and s2 would be unbiased estimators for µ and σ2. The difference E[T ] − θ is called the bias of T ; if this difference is nonzero, then T is called biased.
Consistency
Consistency is about whether the point estimate stays close to the value when the parameter increases its size. The larger the sample size, the more accurate the estimate is. If a point estimator is consistent, its expected value and variance should be close to the true value of the parameter. An unbiased estimator is consistent if the limit of the variance of estimator T equals zero.
Efficiency
Let T1 and T2 be two unbiased estimators for the same parameter θ. The estimator T2 would be called more efficient than estimator T1 if Var(T2) < Var(T1), irrespective of the value of θ. We can also say that the most efficient estimators are the ones with the least variability of outcomes. Therefore, if the estimator has smallest variance among sample to sample, it is both most efficient and unbiased. We extend the notion of efficiency by saying that estimator T2 |
https://en.wikipedia.org/wiki/Interval%20estimation | In statistics, interval estimation is the use of sample data to estimate an interval of possible values of a parameter of interest. This is in contrast to point estimation, which gives a single value.
The most prevalent forms of interval estimation are confidence intervals (a frequentist method) and credible intervals (a Bayesian method);
less common forms include likelihood intervals and fiducial intervals.
Other forms of statistical intervals include tolerance intervals (covering a proportion of a sampled population) and prediction intervals (an estimate of a future observation, used mainly in regression analysis).
Non-statistical methods that can lead to interval estimates include fuzzy logic.
Discussion
The scientific problems associated with interval estimation may be summarised as follows:
When interval estimates are reported, they should have a commonly held interpretation in the scientific community and more widely. In this regard, credible intervals are held to be most readily understood by the general public. Interval estimates derived from fuzzy logic have much more application-specific meanings.
For commonly occurring situations there should be sets of standard procedures that can be used, subject to the checking and validity of any required assumptions. This applies for both confidence intervals and credible intervals.
For more novel situations there should be guidance on how interval estimates can be formulated. In this regard confidence intervals and credible intervals have a similar standing but there are differences:
credible intervals can readily deal with prior information, while confidence intervals cannot.
confidence intervals are more flexible and can be used practically in more situations than credible intervals: one area where credible intervals suffer in comparison is in dealing with non-parametric models (see non-parametric statistics).
There should be ways of testing the performance of interval estimation procedures. This arises because many such procedures involve approximations of various kinds and there is a need to check that the actual performance of a procedure is close to what is claimed. The use of stochastic simulations makes this is straightforward in the case of confidence intervals, but it is somewhat more problematic for credible intervals where prior information needs to be taken properly into account. Checking of credible intervals can be done for situations representing no-prior-information but the check involves checking the long-run frequency properties of the procedures.
Severini (1991) discusses conditions under which credible intervals and confidence intervals will produce similar results, and also discusses both the coverage probabilities of credible intervals and the posterior probabilities associated with confidence intervals.
In decision theory, which is a common approach to and justification for Bayesian statistics, interval estimation is not of direct interest. The outcome is a decisi |
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