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https://en.wikipedia.org/wiki/Clery%20Act
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The Jeanne Clery Disclosure of Campus Security Policy and Campus Crime Statistics Act or Clery Act, signed in 1990, is a federal statute codified at , with implementing regulations in the U.S. Code of Federal Regulations at .
The Clery Act requires all colleges and universities that participate in federal financial aid programs to keep and disclose information about crime on and near their respective campuses. Compliance is monitored by the United States Department of Education, which can impose civil penalties, up to $67,544 per violation, against institutions for each infraction and can suspend institutions from participating in federal student financial aid programs.
The law is named after Jeanne Clery, a 19-year-old Lehigh University student who was raped and murdered in her campus residence hall in 1986. Her murder triggered a backlash against unreported crime on campuses across the country.
Jeanne Clery
Josoph M. Henry, another student, raped and murdered Jeanne Clery in April 1986 in Stoughton Hall at Lehigh University in Bethlehem, Pennsylvania. Henry was given a death sentence via the electric chair by a trial court, a decision which was upheld by the Pennsylvania Supreme Court when appealed. The attack on Clery was one of 38 violent crimes recorded at the university in three years. Her parents argued that, had the university's crime record been known, Clery would not have attended. They sued, were awarded $2 million, and founded Security on Campus, a non-profit group.
Requirements of act
Annual security report
By October 1 of each year, institutions must publish and distribute their Annual Campus Security Report to current and prospective students and employees. Institutions are also allowed to provide notice of the report, a URL if available, and how to obtain a paper copy if desired. This report is required to provide crime statistics for the prior three years, policy statements regarding various safety and security measures, campus crime prevention program descriptions, and procedures to be followed in the investigation and prosecution of alleged sex offenses.
Crime log
The institution's police department or security departments are required to maintain a public log of all crimes reported to them, or those of which they are made aware. The log is required to have the most recent 60 days' worth of information. Each entry in the log must contain the nature, date, time and general location of each crime and disposition of the complaint, if known. Information in the log older than 60 days must be made available within two business days. Crime logs must be kept for seven years, three years following the publication of the last annual security report.
Timely warnings
The Clery Act requires institutions to give timely warnings of crimes that represent a threat to the safety of students or employees. Institutions are required to publish their policies regarding timely warnings in their Annual Campus Security Report. The instituti
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https://en.wikipedia.org/wiki/List%20of%20bird%20species%20described%20in%20the%202000s
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This page details the bird species described as new to science in the years 2000 to 2010:
Summary statistics
Number of species described per year
Countries with high numbers of newly described species
Brazil
Colombia
Peru
Indonesia
The birds, year-by-year
2000
Foothill elaenia, Myiopagis olallai
Coopmans, P. & Krabbe, N. (2000) A new species of flycatcher (Tyrannidae: Myiopagis) from eastern Ecuador and eastern Peru Wilson Bulletin 112: 305–312
Caatinga antwren, Herpsilochmus sellowi
Whitney, B.M.; Pacheco, J.F.; Buzzetti, D.R.C. & Parrini, R. (2000) Systematic revision and biogeography of the Herpsilochmus pileatus complex, with description of a new species from northeastern Brazil Auk 117: 869–891
Taiwan bush-warbler, Bradypterus alishanensis
Rasmussen, P.C.; Round, P.D.; Dickinson, E.C. & Rozendaal, F.G. (2000) A new bush-warbler (Sylviidae, Bradypterus) from Taiwan The Auk 117: 279–289
Scarlet-banded barbet or Wallace's scarlet-banded barbet, Capito wallacei
O'Neill, Lane, Kratter, Capparella & Fox Joo, 2000.
Gunnison sage-grouse, Centrocercus minimus
Young, Braun, Oyler-McCance, Hupp & Quinn, 2000.
Newly split species:
Gray-crested cacholote, Pseudoseisura unirufa, formerly included in the Caatinga cacholote
Zimmer, Kevin J. & Whittaker, Andrew (2000): The Rufous Cacholote (Furnariidae: Pseudoseisura) is two species. Condor 102(2): 409–422. PDF fulltext
2001
Bukidnon woodcock, Scolopax bukidnonensis, from Mindanao and Luzon, Philippines.
Kennedy, Robert S.; Fisher, Timothy H.; Harrap, Simon C.B.; Diesmos, Arvin C: & Manamtam, Arturo S. (2001): A new species of woodcock from the Philippines and a re-evaluation of other Asian/Papuasian woodcock Forktail 17(1): 1–12. PDF fulltext
Mekong wagtail, Motacilla samveasnae.
Duckworth, J.W.; Alström, P.; Davidson, P.; Evans, T.D.; Poole, C.M.; Tan, S. & Timmins, R.J. (2001) A new species of wagtail from the lower Mekong basin Bulletin of the British Ornithologists' Club 121: 152–182
Chestnut-eared laughingthrush, Garrulax konkakinhensis.
Eames, JC & Eames, C, 2001.
Chestnut-capped piha, Lipaugus weberi.
Cuervo, Andres, Salaman, P., Donegan, T.M. & Ochoa, J.M. 2001. A new species of piha (Cotingidae: Lipaugus) from the Cordillera Central of Colombia. Ibis 143: 353–368.
Chapada flycatcher, Suiriri islerorum, from the cerrado region of Brazil and adjacent eastern Bolivia.
Zimmer, K.J.; Whittaker, A. & Oren, D.C. (2001): A cryptic new species of flycatcher (Tyrannidae: Suiriri) from the cerrado region of central South America Auk 118: 56–78
Mishana tyrannulet, Zimmerius villarejoi, from Amazonian 'white sand forests' in northern Peru.
Alonso, J.A. & Whitney, B.M. (2001) A new Zimmerius tyrannulet (Aves: Tyrannidae) from white sand forests of northern Amazonian Peru Wilson Bulletin 113: 1–9
Lulu's tody-tyrant. Poecilotriccus luluae, from the north-eastern Andes in Peru.
Johnson, N.K. & Jones, R.E. (2001) A new species of tody-tyrant (Tyrannidae: Poecilotriccus) from north
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https://en.wikipedia.org/wiki/Marshall%20Moore
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Marshall Moore (born June 29, 1970), in Havelock, North Carolina, is an American author and academic living in Cornwall, England. He attended the North Carolina School of Science and Mathematics (NCSSM) and went on to obtain a BA in psychology from East Carolina University, an MA in applied linguistics from the University of New England, and a PhD in creative writing from Aberystwyth University in Wales. He has also studied at Gallaudet University. He has lived in New Bern, Winston-Salem, Greensboro, Washington DC, Maryland, Oakland, Portland, Seattle, the suburbs of Seoul, and Hong Kong. Fluent in American Sign Language, he worked for many years as an interpreter before moving abroad.
Bibliography
Novels
The Concrete Sky, Binghamton, NY: Haworth Press, 2003
An Ideal for Living, Maple Shade, NJ: Lethe Press, 2010
Bitter Orange, Hong Kong: Signal 8 Press, 2013
Inhospitable, Manchester: Camphor Press, 2018
Nonfiction
I Wouldn't Normally Do This Kind of Thing: A Memoir, New Orleans: Rebel Satori Press, 2022
Blood and Black T-Shirts: Dispatches from Hong Kong's Descent into Hell 2019-2020, Manchester: Camphor Press, 2023
Sunset House: Essays, New Orleans: Rebel Satori Press, 2024
Short story collections
Black Shapes in a Darkened Room, San Francisco: Suspect Thoughts Press, 2004
The Infernal Republic, Hong Kong: Signal 8 Press, 2012
A Garden Fed by Lightning, Hong Kong: Signal 8 Press, 2016
Love Is a Poisonous Color, New Orleans: Rebel Satori Press, 2023
In translation
Sagome nere, Turin: 96, Rue de-la-Fontaine Edizioni, 2017 (Translator: Rossella Cirigliano)
Edited anthologies (short fiction)
The Queen of Statue Square: New Short Fiction from Hong Kong (Co-editor: Xu Xi), Nottingham: Critical, Cultural & Communications Press, 2014
Edited academic nonfiction
The Place and the Writer: International Intersections of Teacher Lore and Creative Writing Pedagogy (Co-editor: Sam Meekings) London: Bloomsbury - Continuum, 2021
Creative Writing Scholars on the Publishing Trade: Practice, Praxis, Print (Co-editor: Sam Meekings) London: Routledge, 2022
The Scholarship of Creative Writing Practice: Beyond Craft, Pedagogy, and the Academy (Co-editor: Sam Meekings) London: Bloomsbury - Continuum, 2024
Chapbooks
Il look del diavolo, Hong Kong: Signal 8 Press, 2011
Never Turn Away, Hong Kong: Signal 8 Press, 2013
In addition to these books, Moore has published dozens of short stories, book reviews, and essays.
His work has been translated into Greek, Polish, and Italian.
External links
Marshall Moore website
1970 births
East Carolina University alumni
21st-century American novelists
American male novelists
Living people
People from Havelock, North Carolina
American gay writers
Novelists from North Carolina
LGBT people from North Carolina
American LGBT novelists
American male short story writers
21st-century American short story writers
American expatriates in Hong Kong
21st-century American male writers
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https://en.wikipedia.org/wiki/No-three-in-line%20problem
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The no-three-in-line problem in discrete geometry asks how many points can be placed in the grid so that no three points lie on the same line. The problem concerns lines of all slopes, not only those aligned with the grid. It was introduced by Henry Dudeney in 1900. Brass, Moser, and Pach call it "one of the oldest and most extensively studied geometric questions concerning lattice points".
At most points can be placed, because points in a grid would include a row of three or more points, by the pigeonhole principle. Although the problem can be solved with points for every up it is conjectured that fewer than points can be placed in grids of large size. Known methods can place linearly many points in grids of arbitrary size, but the best of these methods place slightly fewer than points,
Several related problems of finding points with no three in line, among other sets of points than grids, have also been studied. Although originating in recreational mathematics, the no-three-in-line problem has applications in graph drawing and to the Heilbronn triangle problem.
Small instances
The problem was first posed by Henry Dudeney in 1900, as a puzzle in recreational mathematics, phrased in terms of placing the 16 pawns of a chessboard onto the board so that no three are in a line. This is exactly the no-three-in-line problem, for the In a later version of the puzzle, Dudeney modified the problem, making its solution unique, by asking for a solution in which two of the pawns are on squares d4 and e5, attacking each other in the center of the board.
Many authors have published solutions to this problem for small values and by 1998 it was known that points could be placed on an grid with no three in a line
for all up and some larger values. The numbers of solutions with points for small values of , starting are
The numbers of equivalence classes of solutions with points under reflections and rotations are
Upper and lower bounds
The exact number of points that can be placed, as a function is not known. However, both proven and conjectured bounds limit this number to within a range proportional
General placement methods
A solution of Paul Erdős, published by , is based on the observation that when is a prime number, the set of grid points for contains no three collinear points. When is not prime, one can perform this construction for a grid contained in the grid, where is the largest prime that is at Because the gap between consecutive primes is much smaller than the primes themselves, will always be close so this method can be used to place points in the grid with no three points collinear.
Erdős' bound has been improved subsequently: show that, when is prime, one can obtain a solution with points, by placing points in multiple copies of the hyperbola where may be chosen arbitrarily as long as it is nonzero Again, for arbitrary one can perform this construction for a prime near to obtain a solution with
Up
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https://en.wikipedia.org/wiki/1960%20in%20Singapore
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The following lists events that happened during 1960 in Singapore.
Statistics
Births
There were 61770 recorded births
Deaths
There were 10210 recorded deaths.
Incumbents
Yang di-Pertuan Negara – Yusof Ishak
Prime Minister – Lee Kuan Yew
Events
February
1 February – The Housing and Development Board is set up by Lim Kim San, replacing the Singapore Improvement Trust. The Planning Authority takes over SIT's functions of land use in Singapore.
July
1 July – The People's Association is formed.
September
6 September – The National Library launches a new mobile library service.
8 September – Tan Howe Liang wins a silver medal during the 1960 Summer Olympics.
November
12 November – The Old National Library Building (demolished in 2004) was opened.
Date unknown
– Far East Organization, a property developer was founded.
Births
7 January – Hong Huifang, actress.
24 January – Jack Neo, film director.
17 March – Ruth Langsford, presenter in the United Kingdom.
29 March – Paddy Chew, actor (d. 1999).
20 June – Jeremy Monteiro, singer.
19 August – Wee Siew Kim, former politician.
24 November – Shirley Ng, sports shooter.
Isa Kamari, author.
See also
List of years in Singapore
References
Singapore
Years in Singapore
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https://en.wikipedia.org/wiki/Gateaux%20derivative
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In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, a French mathematician who died at age 25 in World War I, it is defined for functions between locally convex topological vector spaces such as Banach spaces. Like the Fréchet derivative on a Banach space, the Gateaux differential is often used to formalize the functional derivative commonly used in the calculus of variations and physics.
Unlike other forms of derivatives, the Gateaux differential of a function may be nonlinear. However, often the definition of the Gateaux differential also requires that it be a continuous linear transformation. Some authors, such as , draw a further distinction between the Gateaux differential (which may be nonlinear) and the Gateaux derivative (which they take to be linear). In most applications, continuous linearity follows from some more primitive condition which is natural to the particular setting, such as imposing complex differentiability in the context of infinite dimensional holomorphy or continuous differentiability in nonlinear analysis.
Definition
Suppose and are locally convex topological vector spaces (for example, Banach spaces), is open, and The Gateaux differential of at in the direction is defined as
If the limit exists for all then one says that is Gateaux differentiable at
The limit appearing in () is taken relative to the topology of If and are real topological vector spaces, then the limit is taken for real On the other hand, if and are complex topological vector spaces, then the limit above is usually taken as in the complex plane as in the definition of complex differentiability. In some cases, a weak limit is taken instead of a strong limit, which leads to the notion of a weak Gateaux derivative.
Linearity and continuity
At each point the Gateaux differential defines a function
This function is homogeneous in the sense that for all scalars
However, this function need not be additive, so that the Gateaux differential may fail to be linear, unlike the Fréchet derivative. Even if linear, it may fail to depend continuously on if and are infinite dimensional. Furthermore, for Gateaux differentials that linear and continuous in there are several inequivalent ways to formulate their continuous differentiability.
For example, consider the real-valued function of two real variables defined by
This is Gateaux differentiable at with its differential there being
However this is continuous but not linear in the arguments In infinite dimensions, any discontinuous linear functional on is Gateaux differentiable, but its Gateaux differential at is linear but not continuous.
Relation with the Fréchet derivative
If is Fréchet differentiable, then it is also Gateaux differentiable, and its Fréchet and Gateaux derivatives agree. The converse is clearly not true, since the Gateaux d
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https://en.wikipedia.org/wiki/Monomorphic
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Monomorphic or Monomorphism may refer to:
Monomorphism, an injective homomorphism in mathematics
Monomorphic QRS complex, a wave pattern seen on an electrocardiogram
Monomorphic, a linguistic term meaning "consisting of only one morpheme"
Monomorphic phenotype, when only one phenotype exists in a population of a species
Sexual monomorphism, when both biological sexes are phenotypically indistinguishable from each other.
Monomorphism (computer science), a programming concept
See also
Dimorphism (disambiguation)
Polymorphism (disambiguation)
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https://en.wikipedia.org/wiki/Situation%20calculus
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The situation calculus is a logic formalism designed for representing and reasoning about dynamical domains. It was first introduced by John McCarthy in 1963. The main version of the situational calculus that is presented in this article is based on that introduced by Ray Reiter in 1991. It is followed by sections about McCarthy's 1986 version and a logic programming formulation.
Overview
The situation calculus represents changing scenarios as a set of first-order logic formulae. The basic elements of the calculus are:
The actions that can be performed in the world
The fluents that describe the state of the world
The situations
A domain is formalized by a number of formulae, namely:
Action precondition axioms, one for each action
Successor state axioms, one for each fluent
Axioms describing the world in various situations
The foundational axioms of the situation calculus
A simple robot world will be modeled as a running example. In this world there is a single robot and several inanimate objects. The world is laid out according to a grid so that locations can be specified in terms of coordinate points. It is possible for the robot to move around the world, and to pick up and drop items. Some items may be too heavy for the robot to pick up, or fragile so that they break when they are dropped. The robot also has the ability to repair any broken items that it is holding.
Elements
The main elements of the situation calculus are the actions, fluents and the situations. A number of objects are also typically involved in the description of the world. The situation calculus is based on a sorted domain with three sorts: actions, situations, and objects, where the objects include everything that is not an action or a situation. Variables of each sort can be used. While actions, situations, and objects are elements of the domain, the fluents are modeled as either predicates or functions.
Actions
The actions form a sort of the domain. Variables of sort action can be used and also functions whose result is of sort action. Actions can be quantified. In the example robot world, possible action terms would be to model the robot moving to a new location , and to model the robot picking up an object . A special predicate is used to indicate when an action is executable.
Situations
In the situation calculus, a dynamic world is modeled as progressing through a series of situations as a result of various actions being performed within the world. A situation represents a history of action occurrences. In the Reiter version of the situation calculus described here, a situation does not represent a state, contrarily to the literal meaning of the term and contrarily to the original definition by McCarthy and Hayes. This point has been summarized by Reiter as follows:
A situation is a finite sequence of actions. Period. It's not a state, it's not a snapshot, it's a history.
The situation before any actions have been performed is typically denoted an
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https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s%20theorem%20%28conformal%20mapping%29
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In mathematics, Carathéodory's theorem is a theorem in complex analysis, named after Constantin Carathéodory, which extends the Riemann mapping theorem. The theorem, first proved in 1913, states that any conformal mapping sending the unit disk to some region in the complex plane bounded by a Jordan curve extends continuously to a homeomorphism from the unit circle onto the Jordan curve. The result is one of Carathéodory's results on prime ends and the boundary behaviour of univalent holomorphic functions.
Proofs of Carathéodory's theorem
The first proof of Carathéodory's theorem presented here is a summary of the short self-contained account in ; there are related proofs in and .
Clearly if f admits an extension to a homeomorphism, then ∂U must be a Jordan curve.
Conversely if ∂U is a Jordan curve, the first step is to prove f extends continuously to the closure of D. In fact this will hold if and only if f is uniformly continuous on D: for this is true if it has a continuous extension to the closure of D; and, if f is uniformly continuous, it is easy to check f has limits on the unit circle and the same inequalities for uniform continuity hold on the closure of D.
Suppose that f is not uniformly continuous. In this case there must be an ε > 0 and a point ζ on the unit circle and sequences zn, wn tending to ζ with |f(zn) − f(wn)| ≥ 2ε. This is shown below to lead to a contradiction, so that f must be uniformly continuous and hence has a continuous extension to the closure of D.
For 0 < r < 1, let γr be the curve given by the arc of the circle | z − ζ | = r lying within D. Then f ∘ γr is a Jordan curve. Its length can be estimated using the Cauchy–Schwarz inequality:
Hence there is a "length-area estimate":
The finiteness of the integral on the left hand side implies that there is a sequence rn decreasing to 0 with tending to 0. But the length of a curve g(t) for t in (a, b) is given by
The finiteness of therefore implies that the curve has limiting points an, bn at its two ends with |an – bn| ≤ , so this distance, as well as diameter of the curve, tends to 0. These two limit points must lie on ∂U, because f is a homeomorphism between D and U and thus a sequence converging in U has to be the image under f of a sequence converging in D. By assumption there exist a homeomorphism β between the circle ∂D and ∂U. Since β−1 is uniformly continuous, the distance between the two points ξn and ηn corresponding to an and bn in ∂U must tend to 0. So eventually the smallest circular arc in ∂D joining ξn and ηn is defined. Denote τn image of this arc under β. By uniform continuity of β, diameter of τn in ∂U tends to 0. Together τn and f ∘ γrn form a simple Jordan curve. Its interior Un is contained in U by the Jordan curve theorem for ∂U and ∂Un: to see this, notice that U is the interior of ∂U, as it is bounded, connected and it is both open and closed in the complement of ∂U; so the exterior region of ∂U is unbounded, connected and does not in
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https://en.wikipedia.org/wiki/Job%20security
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Job security is the probability that an individual will keep their job; a job with a high level of security is such that a person with the job would have a small chance of losing it. Many factors threaten job security: globalization, outsourcing, downsizing, recession, and new technology, to name a few.
Basic economic theory holds that during periods of economic expansion businesses experience increased demand, which in turn necessitates investment in more capital or labor. When businesses are experiencing growth, job confidence and security typically increase. The opposite often holds true during a recession: businesses experience reduced demand and look to downsize their workforces in the short term.
Governments and individuals are both motivated to achieve higher levels of job security. Governments attempt to do this by passing laws (such as the U.S. Civil Rights Act of 1964) which make it illegal to fire employees for certain reasons. Individuals can influence their degree of job security by increasing their skills through education and experience, or by moving to a more favorable location. The official unemployment rate and employee confidence indexes are good indicators of job security in particular fields. These statistics are closely watched by economists, government officials, and banks.
Unions also strongly influence job security. Jobs that traditionally have a strong union presence such as many government jobs and jobs in education, healthcare and law enforcement are considered very secure while many non-unionized private sector jobs are generally believed to offer lower job security, although this varies by industry and country.
Measuring job security
This is a list of countries by job security, an important component in measuring quality of life and the well-being of its citizens. It lists OECD countries' workers' chance of losing their job in 2012, with some non-OECD countries also included. Workers facing a high risk of job loss are more vulnerable, especially in countries with smaller social safety nets . This indicator presents the probability to become unemployed, calculated as the number of people who were unemployed in 2012, but were employed in 2011 over the total number of employed in 2011.
In the United States
While all economies are impacted by market forces (which change the supply and demand of labor) the United States is particularly susceptible to these forces due to a long history of fiscal conservatism and minimal government intervention.
Minimal government intervention has helped the United States create an at-will employment system that applies across many industries. Consequently, with limited exceptions, an employee's job security closely follows an employer's demand for their skills. For example, in the aftermath of the dot com boom of 1997–2000, employees in the technology industry experienced a massive drop in job security and confidence. More recently, in 2009 many manufacturing workers experienced a si
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https://en.wikipedia.org/wiki/Lambda%20expression
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Lambda expression may refer to:
Lambda expression in computer programming, also called an anonymous function, is a defined function not bound to an identifier.
Lambda expression in lambda calculus, a formal system in mathematical logic and computer science for expressing computation by way of variable binding and substitution.
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https://en.wikipedia.org/wiki/Donald%20A.%20Martin
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Donald Anthony Martin (born December 24, 1940), also known as Tony Martin, is an American set theorist and philosopher of mathematics at UCLA, where he is an emeritus professor of mathematics and philosophy.
Education and career
Martin received his B.S. from the Massachusetts Institute of Technology in 1962 and was a Junior Fellow of the Harvard Society of Fellows in 1965–67. In 2014, he became a Fellow of the American Mathematical Society.
Philosophical and mathematical work
Among Martin's most notable works are the proofs of analytic determinacy (from the existence of a measurable cardinal), Borel determinacy (from ZFC alone), the proof (with John R. Steel) of projective determinacy (from suitable large cardinal axioms), and his work on Martin's axiom. The Martin measure on Turing degrees is also named after Martin.
See also
American philosophy
List of American philosophers
References
External links
List of publications
UCLA Logic center
Personal Website at UCLA
1940 births
Living people
20th-century American mathematicians
21st-century American mathematicians
American logicians
UCLA Philosophy
Philosophers of mathematics
Set theorists
Fellows of the American Mathematical Society
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https://en.wikipedia.org/wiki/Giuseppe%20Melfi
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Giuseppe Melfi (June 11, 1967) is an Italo-Swiss mathematician who works on practical numbers and modular forms.
Career
He gained his PhD in mathematics in 1997 at the University of Pisa. After some time spent at the University of Lausanne during 1997-2000, Melfi was appointed at the University of Neuchâtel, as well as at the University of Applied Sciences Western Switzerland and at the local University of Teacher Education.
Work
His major contributions are in the field of practical numbers. This prime-like sequence of numbers is known for having an asymptotic behavior and other distribution properties similar to the sequence of primes. Melfi proved two conjectures both raised in 1984 one of which is the corresponding of the Goldbach conjecture for practical numbers: every even number is a sum of two practical numbers. He also proved that there exist infinitely many triples of practical numbers of the form .
Another notable contribution has been in an application of the theory of modular forms, where he found new Ramanujan-type identities for the sum-of-divisor functions. His seven new identities extended the ten other identities found by Ramanujan in 1913. In particular he found the remarkable identity
where is the sum of the divisors of and is the sum of the third powers of the divisors of .
Among other problems in elementary number theory, he is the author of a theorem that allowed him to get a 5328-digit number that has been for a while the largest known primitive weird number.
In applied mathematics his research interests include probability and simulation.
Selected research publications
.
See also
Applications of randomness
References
External links
Giuseppe Melfi's home page
The proof of conjectures on practical numbers and the joint work with Paul Erdős on Zentralblatt.
Tables of practical numbers compiled by Giuseppe Melfi
Academic research query for "Giuseppe Melfi"
1967 births
20th-century Italian mathematicians
21st-century Italian mathematicians
Living people
Number theorists
Mathematicians from Sicily
Academic staff of the University of Neuchâtel
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https://en.wikipedia.org/wiki/Daina%20Taimi%C5%86a
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Daina Taimiņa (born August 19, 1954) is a Latvian mathematician, retired adjunct associate professor of mathematics at Cornell University, known for developing a way of modeling hyperbolic geometry with crocheted objects.
Education and career
Taimiņa received all of her formal education in Riga, Latvia, where in 1977 she graduated summa cum laude from the University of Latvia and completed her graduate work in Theoretical Computer Science (with thesis advisor Prof. Rūsiņš Mārtiņš Freivalds) in 1990. As one of the restrictions of the Soviet system at that time, a doctoral thesis was not allowed to be defended in Latvia, so she defended hers in Minsk, receiving the title of Candidate of Sciences. This explains the fact that Taimiņa's doctorate was formally issued by the Institute of Mathematics of the National Academy of Sciences of Belarus. After Latvia regained independence in 1991, Taimiņa received her higher doctoral degree (doktor nauk) in mathematics from the University of Latvia, where she taught for 20 years.
Daina Taimiņa joined the Cornell Math Department in December 1996.
Combining her interests in mathematics and crocheting, she is one of 24 mathematicians and artists who make up the Mathemalchemy Team.
Hyperbolic crochet
While attending a geometry workshop at Cornell University about teaching geometry for university professors in 1997, Taimiņa was presented with a fragile paper model of a hyperbolic plane, made by the professor in charge of the workshop, David Henderson (designed by geometer William Thurston.) It was made «out of thin, circular strips of paper taped together». She decided to make more durable models, and did so by crocheting them. The first night after first seeing the paper model at the workshop she began experimenting with algorithms for a crocheting pattern, after visualising hyperbolic planes as exponential growth.
The following fall, Taimiņa was scheduled to teach a geometry class at Cornell. She was determined to find what she thought was the best possible way to teach her class. So while she, together with her family, spent the preceding summer at a tree farm in Pennsylvania, she also spent her days by the pool watching her two daughters learning how to swim whilst simultaneously making a classroom set of models of the hyperbolic plane. This was the first ever made from yarn and crocheting.
The models made a significant difference to her students, according to themselves. They said they "liked the tactile way of exploring hyperbolic geometry" and that it helped them acquire experiences that helped them move on in said geometry. This was what Taimina herself had been missing when first learning about hyperbolic planes and is also what has made her models so effective, as these models have later become the preferred way of explaining hyperbolic space within geometry.
In a TedxRiga by Taimiņa she tells the story of how the need for a visual, intuitive way of understanding hyperbolic planes spurred her toward
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https://en.wikipedia.org/wiki/Posner
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Posner or Pozner may refer to:
Posner (surname)
Posner Park, in Florida, US
Posner's theorem in algebra
Posner cueing task, a neuropsychological test
See also
Posener, a surname
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https://en.wikipedia.org/wiki/Coordination%20geometry
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The coordination geometry of an atom is the geometrical pattern defined by the atoms around the central atom. The term is commonly applied in the field of inorganic chemistry, where diverse structures are observed. The coodination geometry depends on the number, not the type, of ligands bonded to the metal centre as well as their locations. The number of atoms bonded is the coordination number.
The geometrical pattern can be described as a polyhedron where the vertices of the polyhedron are the centres of the coordinating atoms in the ligands.
The coordination preference of a metal often varies with its oxidation state. The number of coordination bonds (coordination number) can vary from two in as high as 20 in .
One of the most common coordination geometries is octahedral, where six ligands are coordinated to the metal in a symmetrical distribution, leading to the formation of an octahedron if lines were drawn between the ligands. Other common coordination geometries are tetrahedral and square planar.
Crystal field theory may be used to explain the relative stabilities of transition metal compounds of different coordination geometry, as well as the presence or absence of paramagnetism, whereas VSEPR may be used for complexes of main group element to predict geometry.
Crystallography usage
In a crystal structure the coordination geometry of an atom is the geometrical pattern of coordinating atoms where the definition of coordinating atoms depends on the bonding model used. For example, in the rock salt ionic structure each sodium atom has six near neighbour chloride ions in an octahedral geometry and each chloride has similarly six near neighbour sodium ions in an octahedral geometry. In metals with the body centred cubic (bcc) structure each atom has eight nearest neighbours in a cubic geometry. In metals with the face centred cubic (fcc) structure each atom has twelve nearest neighbours in a cuboctahedral geometry.
Table of coordination geometries
A table of the coordination geometries encountered is shown below with examples of their occurrence in complexes found as discrete units in compounds and coordination spheres around atoms in crystals (where there is no discrete complex).
Naming of inorganic compounds
IUPAC have introduced the polyhedral symbol as part of their IUPAC nomenclature of inorganic chemistry 2005 recommendations to describe the geometry around an atom in a compound.
IUCr have proposed a symbol which is shown as a superscript in square brackets in the chemical formula. For example, would be Ca[8cb]F2[4t], where [8cb] means cubic coordination and [4t] means tetrahedral. The equivalent symbols in IUPAC are CU−8 and T−4 respectively.
The IUPAC symbol is applicable to complexes and molecules whereas the IUCr proposal applies to crystalline solids.
See also
Molecular geometry
VSEPR
Ligand field theory
Cis effect
Addition to pi ligands
References
Molecular physics
Chemical bonding
Coordination chemistry
Inorganic chem
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https://en.wikipedia.org/wiki/Cabal%20%28set%20theory%29
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The Cabal was, or perhaps is, a set of set theorists in Southern California, particularly at UCLA and Caltech, but also at UC Irvine. Organization and procedures range from informal to nonexistent, so it is difficult to say whether it still exists or exactly who has been a member, but it has included such notable figures as Donald A. Martin, Yiannis N. Moschovakis, John R. Steel, and Alexander S. Kechris. Others who have published in the proceedings of the Cabal seminar include Robert M. Solovay, W. Hugh Woodin, Matthew Foreman, and Steve Jackson.
The work of the group is characterized by free use of large cardinal axioms, and research into the descriptive set theoretic behavior of sets of reals if such assumptions hold.
Some of the philosophical views of the Cabal seminar were described in and .
Publications
References
Descriptive set theory
Set theory
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https://en.wikipedia.org/wiki/Szeged%20Faculty%20of%20Sciences
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The Faculty of Sciences of the University of Szeged.
Notable persons
István Apáthy, zoology
Zoltán Bay, physicist
Jenő Cholnoky, geography
Lipót Fejér, mathematics
István Györffy, botany
Alfréd Haar, mathematics
László Kalmár, computer science
Béla Kerékjártó, geometry
László Lovász, mathematics; Wolf Prize 1999, Knuth Prize 199, Kyoto prize 2010
Tibor Radó, mathematics
László Rédei, mathematics
Frigyes Riesz, mathematics
Brúnó F. Straub, biology
Béla Szőkefalvi-Nagy, mathematics
Faculties of the University of Szeged
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https://en.wikipedia.org/wiki/Jacques%20Touchard
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Jacques Touchard (1885–1968) was a French mathematician. In 1953, he proved that an odd perfect number must be of the form 12k + 1 or 36k + 9. In combinatorics and probability theory, he introduced the Touchard polynomials. He is also known for his solution to the ménage problem of counting seating arrangements in which men and women alternate and are not seated next to their spouses.
Touchard's Catalan identity
The following algebraic identity involving the Catalan numbers
is apparently due to Touchard (according to Richard P. Stanley, who mentions it in his panorama article "Exercises on Catalan and Related Numbers" giving an overwhelming plenitude of different definitions for the Catalan numbers).
For n ≥ 0 one has
Using the generating function
it can be proved by algebraic manipulations of generating series that Touchard's identity is
equivalent to the functional equation
satisfied by the Catalan generating series C(t).
Further reading
Canadian Journal of Mathematics 1956, Vol 8, No 3.; Journal in French
French mathematicians
1885 births
1968 deaths
Place of birth missing
People from Basel-Landschaft
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https://en.wikipedia.org/wiki/Andreas%20Floer
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Andreas Floer (; 23 August 1956 – 15 May 1991) was a German mathematician who made seminal contributions to symplectic topology, and mathematical physics, in particular the invention of Floer homology. Floer's first pivotal contribution was a solution of a special case of Arnold's conjecture on fixed points of a symplectomorphism. Because of his work on Arnold's conjecture and his development of instanton homology, he achieved wide recognition and was invited as a plenary speaker for the International Congress of Mathematicians held in Kyoto in August 1990. He received a Sloan Fellowship in 1989.
Life
He was an undergraduate student at the Ruhr-Universität Bochum and received a Diplom in mathematics in 1982. He then went to the University of California, Berkeley, living at Barrington Hall of the Berkeley Student Cooperative, and undertook Ph.D. work on monopoles on 3-manifolds, under the supervision of Clifford Taubes; but he did not complete it when interrupted by his obligatory alternative service in Germany. He received his Dr. rer. nat. at Bochum in 1984, under the supervision of Eduard Zehnder.
In 1988 he became an Assistant Professor at the University of California, Berkeley and was promoted to Full Professor of Mathematics in 1990. From 1990 he was Professor of Mathematics at the Ruhr-Universität Bochum, until his suicide in 1991 as a result of depression.
Quotes
"Andreas Floer's life was tragically interrupted, but his mathematical visions and striking contributions have provided powerful methods which are being applied to problems which seemed to be intractable only a few years ago."
Simon Donaldson wrote: "The concept of Floer homology is one of the most striking developments in differential geometry over the past 20 years. ... The ideas have led to great advances in the areas of low-dimensional topology and symplectic geometry and are intimately related to developments in Quantum Field Theory" and "the full richness of Floer's theory is only beginning to be explored".
"Since its introduction by Andreas Floer in the late nineteen eighties, Floer theory has had a tremendous influence on many branches of mathematics including geometry, topology and dynamical systems. The development of new Floer theoretic tools continues at a remarkable pace and underlies many of the recent breakthroughs in these diverse fields."
Selected publications
Floer, Andreas. An instanton-invariant for 3-manifolds. Comm. Math. Phys. 118 (1988), no. 2, 215–240. Project Euclid
Floer, Andreas. Morse theory for Lagrangian intersections. J. Differential Geom. 28 (1988), no. 3, 513–547.
Floer, Andreas. Cuplength estimates on Lagrangian intersections. Comm. Pure Appl. Math. 42 (1989), no. 4, 335–356.
Posthumous publications
Hofer, Helmut. Coherent orientation for periodic orbit problems in symplectic geometry (jointly with A. Floer) Math. Zeit. 212, 13–38, 1993.
Hofer, Helmut. Symplectic homology I: Open sets in C^n (jointly with A. Floer) Math. Zeit. 215
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https://en.wikipedia.org/wiki/Proj%20construction
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In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not functorial, is a fundamental tool in scheme theory.
In this article, all rings will be assumed to be commutative and with identity.
Proj of a graded ring
Proj as a set
Let be a graded ring, whereis the direct sum decomposition associated with the gradation. The irrelevant ideal of is the ideal of elements of positive degreeWe say an ideal is homogeneous if it is generated by homogeneous elements. Then, as a set, For brevity we will sometimes write for .
Proj as a topological space
We may define a topology, called the Zariski topology, on by defining the closed sets to be those of the form
where is a homogeneous ideal of . As in the case of affine schemes it is quickly verified that the form the closed sets of a topology on .
Indeed, if are a family of ideals, then we have and if the indexing set I is finite, then .
Equivalently, we may take the open sets as a starting point and define
A common shorthand is to denote by , where is the ideal generated by . For any ideal , the sets and are complementary, and hence the same proof as before shows that the sets form a topology on . The advantage of this approach is that the sets , where ranges over all homogeneous elements of the ring , form a base for this topology, which is an indispensable tool for the analysis of , just as the analogous fact for the spectrum of a ring is likewise indispensable.
Proj as a scheme
We also construct a sheaf on , called the “structure sheaf” as in the affine case, which makes it into a scheme. As in the case of the Spec construction there are many ways to proceed: the most direct one, which is also highly suggestive of the construction of regular functions on a projective variety in classical algebraic geometry, is the following. For any open set of (which is by definition a set of homogeneous prime ideals of not containing ) we define the ring to be the set of all functions
(where denotes the subring of the ring of fractions consisting of fractions of homogeneous elements of the same degree) such that for each prime ideal of :
is an element of ;
There exists an open subset containing and homogeneous elements of of the same degree such that for each prime ideal of :
is not in ;
It follows immediately from the definition that the form a sheaf of rings on , and it may be shown that the pair (, ) is in fact a scheme (this is accomplished by showing that each of the open subsets is in fact an affine scheme).
The sheaf associated to a graded module
The essential property of for the above construction was the ability to form localizations for each prime ideal of . This property is also possessed by any graded module over , and therefore with the appropriate minor modifications
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https://en.wikipedia.org/wiki/John%20Colson
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John Colson (1680 – 20 January 1760) was an English clergyman, mathematician, and the Lucasian Professor of Mathematics at Cambridge University.
Life
John Colson was educated at Lichfield School before becoming an undergraduate at Christ Church, Oxford, though he did not take a degree there.
He became a schoolmaster at Sir Joseph Williamson's Mathematical School in Rochester, and was elected Fellow of the Royal Society in 1713.
He was Vicar of Chalk, Kent from 1724 to 1740.
He relocated to Cambridge and lectured at Sidney Sussex College, Cambridge.
From 1739 to 1760, he was Lucasian Professor of Mathematics. He was also Rector of Lockington, Yorkshire.
Works
In 1726 he published his Negativo-Affirmativo Arithmetik advocating a modified decimal system of numeration. It involved "reduction [to] small figures" by "throwing all the large figures out of a given number, and introducing in their room the equivalent small figures respectively".
John Colson translated several of Isaac Newton's works into English, including De Methodis Serierum et Fluxionum in 1736.
See also
Method of Fluxions
Witch of Agnesi
Notes
References
"A Brief History of The Lucasian Professorship of Mathematics at Cambridge University" – Robert Bruen, Boston College, May 1995
External links
1680 births
1760 deaths
18th-century English mathematicians
Academics of the University of Cambridge
English Anglicans
Lucasian Professors of Mathematics
Fellows of the Royal Society
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https://en.wikipedia.org/wiki/Mathematical%20joke
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A mathematical joke is a form of humor which relies on aspects of mathematics or a stereotype of mathematicians. The humor may come from a pun, or from a double meaning of a mathematical term, or from a lay person's misunderstanding of a mathematical concept. Mathematician and author John Allen Paulos in his book Mathematics and Humor described several ways that mathematics, generally considered a dry, formal activity, overlaps with humor, a loose, irreverent activity: both are forms of "intellectual play"; both have "logic, pattern, rules, structure"; and both are "economical and explicit".
Some performers combine mathematics and jokes to entertain and/or teach math.
Humor of mathematicians may be classified into the esoteric and exoteric categories. Esoteric jokes rely on the intrinsic knowledge of mathematics and its terminology. Exoteric jokes are intelligible to the outsiders, and most of them compare mathematicians with representatives of other disciplines or with common folk.
Pun-based jokes
Some jokes use a mathematical term with a second non-technical meaning as the punchline of a joke.
Occasionally, multiple mathematical puns appear in the same jest:
This invokes four double meanings: adder (snake) vs. addition (algebraic operation); multiplication (biological reproduction) vs. multiplication (algebraic operation); log (a cut tree trunk) vs. log (logarithm); and table (set of facts) vs. table (piece of furniture).
Other jokes create a double meaning from a direct calculation involving facetious variable names, such as this retold from Gravity's Rainbow:
The first part of this joke relies on the fact that the primitive (formed when finding the antiderivative) of the function 1/x is log(x). The second part is then based on the fact that the antiderivative is actually a class of functions, requiring the inclusion of a constant of integration, usually denoted as C—something which calculus students may forget. Thus, the indefinite integral of 1/cabin is "log(cabin) + C", or "A log cabin plus the sea", i.e., "A houseboat".
Jokes with numeral bases
Some jokes depend on ambiguity of numeral bases.
This joke subverts the trope of phrases that begin with "there are two types of people in the world..." and relies on an ambiguous meaning of the expression 10, which in the binary numeral system is equal to the decimal number 2. There are many alternative versions of the joke, such as "There are two types of people in this world. Those who can extrapolate from incomplete information."
Another pun using different radices, asks:
The play on words lies in the similarity of the abbreviation for October/Octal and December/Decimal, and the coincidence that both equal the same amount ().
Imaginary numbers
Some jokes are based on imaginary number , treating it as if it is a real number. A telephone intercept message of "you have dialed an imaginary number, please rotate your handset ninety degrees and try again" is a typical example. Another
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https://en.wikipedia.org/wiki/Joseph%20H.%20Silverman
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Joseph Hillel Silverman (born March 27, 1955, New York City) is a professor of mathematics at Brown University working in arithmetic geometry, arithmetic dynamics, and cryptography.
Biography
Joseph Silverman received an Sc.B. from Brown University in 1977 and a Ph.D. from Harvard University in 1982 under the direction of John Tate. He taught at M.I.T. (1982–1986) and at Boston University (1986–1988) before taking a position at Brown in 1988.
Silverman has published more than 100 research articles, written or coauthored six books, and edited three conference proceedings; his work has been cited more than 5000 times, by over 2000 distinct authors. He currently serves on the editorial boards of Algebra and Number Theory and New York Journal of Mathematics.
Industry
In 1996, Silverman, along with Jeffrey Hoffstein, Jill Pipher and Daniel Lieman, founded NTRU Cryptosystems, Inc. to market their cryptographic algorithms, NTRUEncrypt and NTRUSign.
Awards
In 2012 he became a fellow of the American Mathematical Society.
Books
Silverman has written two graduate texts on elliptic curves, The Arithmetic of Elliptic Curves (1986) and Advanced Topics in the Arithmetic of Elliptic Curves (1994). For these two books he received a Steele Prize for Mathematical Exposition from the American Mathematical Society, which cited them by saying that “Silverman's volumes have become standard references on one of the most exciting areas of algebraic geometry and number theory.” Silverman has also written three undergraduate texts: Rational Points on Elliptic Curves (1992, co-authored with John Tate), A Friendly Introduction to Number Theory (3rd ed. 2005), and An Introduction to Mathematical Cryptography (2008, co-authored with Jeffrey Hoffstein and Jill Pipher). Additional graduate-level texts authored by Silverman are Diophantine Geometry: An Introduction (2000, co-authored with Marc Hindry) and The Arithmetic of Dynamical Systems (2007).
Publications
.
.
.
.
.
.
.
.
Notes
External links
Joseph Silverman's home page
1998 Steele Prizes awarded by the American Mathematical Society
Living people
1955 births
20th-century American mathematicians
21st-century American mathematicians
Arithmetic geometers
Brown University alumni
Harvard University alumni
Brown University faculty
Fellows of the American Mathematical Society
Massachusetts Institute of Technology School of Science faculty
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https://en.wikipedia.org/wiki/179%20%28number%29
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179 (one hundred [and] seventy-nine) is the natural number following 178 and preceding 180.
In mathematics
179 is part of the Cunningham chain of prime numbers 89, 179, 359, 719, 1439, 2879, in which each successive number is two times the previous number, plus one. Among Cunningham chains of this length, this one has the smallest numbers. Because 179 is neither the start nor the end of this chain, it is both a safe prime and a Sophie Germain prime. It is also a super-prime number, because it is the 41st smallest prime and 41 is also prime. Since 971 (the digits of 179 reversed) is prime, 179 is an emirp.
In other fields
Astronomers have suggested that sunspot frequency undergoes a cycle of approximately 179 years in length.
See also
AD 179 and 179 BC
List of highways numbered 179
External links
References
Integers
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https://en.wikipedia.org/wiki/181%20%28number%29
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181 (one hundred [and] eighty-one) is the natural number following 180 and preceding 182.
In mathematics
181 is an odd number
181 is a centered number
181 is a centered pentagonal number
181 is a centered 12-gonal number
181 is a centered 18-gonal number
181 is a centered 30-gonal number
181 is a centered square number
181 is a star number that represents a centered hexagram (as in the game of Chinese checkers)
181 is a deficient number, as 1 is less than 181
181 is an odious number
181 is a prime number
181 is a Chen prime
181 is a dihedral prime
181 is a full reptend prime
181 is a palindromic prime
181 is a strobogrammatic prime, the same when viewed upside down
181 is a twin prime with 179
181 is a square-free number
181 is an undulating number, if written in the ternary, the negaternary, or the nonary numeral systems
181 is the difference of 2 square numbers: 912 – 902
181 is the sum of 2 consecutive square numbers: 92 + 102
181 is the sum of 5 consecutive prime numbers: 29 + 31 + 37 + 41 + 43
In geography
Langenburg No. 181, Saskatchewan rural municipality in Saskatchewan, Canada
181 Fremont Street proposed skyscraper in San Francisco, California
181 West Madison Street, Chicago
In the military
181st (Brandon) Battalion, CEF was a unit in the Canadian Expeditionary Force during World War I
181st Airlift Squadron is a unit of the Texas Air National Guard
181st Infantry Brigade of the United States Army based at Fort McCoy, Wisconsin
181st Intelligence Wing is a unit of the United States Air Force located at Hulman Field, Terre Haute, Indiana
AN/APQ-181 an all-weather, low probability of intercept (LPI) radar system for the U.S. Air Force B-2A Spirit bomber aircraft
Bücker Bü 181 Bestmann single-engine trainer aircraft during World War II
was a ship scheduled to be acquired by the United States Navy, however, the program was cancelled
was a United States Navy troop transport during World War II
was a United States Navy oiler following the Vietnam War
was a United States Navy ATA-174-class auxiliary ocean tugboat during World War II
was a United States Navy Alamosa-class cargo ship following World War II
was a United States Navy following World War I
was a United States Navy Porpoise-class submarine during World War II
was a United States Navy during World War II
In movies
“181: Fragments of a Journey in Palestine-Israel", winner of the 2005 Yamagata International Documentary Film Festival
The war film “The Enemy Below” revolves around the fictitious , USS Hayes (DE-181) and a German U-boat
In transportation
The Volkswagen 181
Lufthansa Flight 181, which was hijacked on October 13, 1977
London Buses route 181
181 Union City-New York, a New Jersey Transit bus route from New Jersey to New York
The Córas Iompair Éireann CIE 181 Class diesel locomotives, and numbered B181 to B192, built by General Motors Electro-Motive Division (EMD) in 1966
East 181st Avenue, a light rail station
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https://en.wikipedia.org/wiki/Theodore%20Streleski
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Theodore Landon "Ted" Streleski (b. 1936) is an American former graduate student in mathematics at Stanford University who murdered his former faculty advisor, Professor Karel de Leeuw, with a ball-peen hammer on August 18, 1978. Shortly after the murder, Streleski turned himself in to the authorities, claiming he felt the murder was justifiable homicide because de Leeuw had withheld departmental awards from him, demeaned Streleski in front of his peers, and refused his requests for financial support. Streleski was in his 19th year pursuing his doctorate in the mathematics department, alternating with low-paying jobs to support himself.
During his trial Streleski told the court he felt the murder was "logically and morally correct" and "a political statement" about the department's treatment of its graduate students, and he forced his court-appointed lawyer to enter a plea of "not guilty" rather than "not guilty by reason of insanity" as the lawyer had urged. Streleski was convicted of second degree murder with a sentence of eight years.
He served seven years in prison at California Medical Facility.
Streleski was eligible for parole on three occasions, but turned it down as the conditions of his parole required him to get psychiatric treatment. Upon his release in 1985, he said, "I have no intention of killing again. On the other hand, I cannot predict the future."
In 1993 Streleski was turned down for a fare box repair position with the San Francisco Municipal Railway after his crime came to light.
References
External links
A commemorative article by Stanford's alumni association
When Student-Adviser tensions erupt
1936 births
American people convicted of murder
People convicted of murder by California
Place of birth missing (living people)
Stanford University alumni
Living people
Criminals of the San Francisco Bay Area
Violence at universities and colleges
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https://en.wikipedia.org/wiki/191%20%28number%29
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191 (one hundred [and] ninety-one) is the natural number following 190 and preceding 192.
In mathematics
191 is a prime number, part of a prime quadruplet of four primes: 191, 193, 197, and 199. Because doubling and adding one produces another prime number (383), 191 is a Sophie Germain prime. It is the smallest prime that is not a full reptend prime in any base from 2 to 10; in fact, the smallest base for which 191 is a full period prime is base 19.
See also
191 (disambiguation)
References
Integers
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https://en.wikipedia.org/wiki/193%20%28number%29
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193 (one hundred [and] ninety-three) is the natural number following 192 and preceding 194.
In mathematics
193 is the number of compositions of 14 into distinct parts. In decimal, it is the seventeenth full repetend prime, or long prime.
It is the only odd prime known for which 2 is not a primitive root of .
It is the thirteenth Pierpont prime, which implies that a regular 193-gon can be constructed using a compass, straightedge, and angle trisector.
It is part of the fourteenth pair of twin primes , the seventh trio of prime triplets , and the fourth set of prime quadruplets .
Aside from itself, the friendly giant (the largest sporadic group) holds a total of 193 conjugacy classes. It also holds at least 44 maximal subgroups aside from the double cover of (the forty-fourth prime number is 193).
193 is also the eighth numerator of convergents to Euler's number; correct to three decimal places: The denominator is 71, which is the largest supersingular prime that uniquely divides the order of the friendly giant.
In other fields
193 is the telephonic number of the 27 Brazilian Military Firefighters Corpses.
193 is the number of internationally recognized nations by the United Nations Organization (UNO).
See also
193 (disambiguation)
References
Integers
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https://en.wikipedia.org/wiki/197%20%28number%29
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197 (one hundred [and] ninety-seven) is the natural number following 196 and preceding 198.
In mathematics
197 is a prime number, the third of a prime quadruplet: 191, 193, 197, 199
197 is the smallest prime number that is the sum of 7 consecutive primes: 17 + 19 + 23 + 29 + 31 + 37 + 41, and is the sum of the first twelve prime numbers: 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37
197 is a centered heptagonal number, a centered figurate number that represents a heptagon with a dot in the center and all other dots surrounding the center dot in successive heptagonal layers
197 is a Schröder–Hipparchus number, counting for instance the number of ways of subdividing a heptagon by a non-crossing set of its diagonals.
In other fields
197 is also:
A police emergency telephone number in Tunisia
Number enquiry telephone number in Nepal
a song by Norwegian alternative rock group Major Parkinson from their self-titled debut album
See also
The year AD 197 or 197 BC
List of highways numbered 197
References
Integers
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https://en.wikipedia.org/wiki/199%20%28number%29
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199 (one hundred [and] ninety-nine) is the natural number following 198 and preceding 200.
In mathematics
199 is a centered triangular number.
It is a prime number and the fourth part of a prime quadruplet: 191, 193, 197, 199.
199 is the smallest natural number that takes more than two iterations to compute its digital root as a repeated digit sum:
Thus, its additive persistence is three, and it is the smallest number of persistence three.
See also
The year AD 199 or 199 BC
List of highways numbered 199
References
Integers
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https://en.wikipedia.org/wiki/Yiannis%20N.%20Moschovakis
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Yiannis Nicholas Moschovakis (; born January 18, 1938) is a set theorist, descriptive set theorist, and recursion (computability) theorist, at UCLA.
His book Descriptive Set Theory (North-Holland) is the primary reference for the subject. He is especially associated with the development of the effective, or lightface, version of descriptive set theory, and he is known for the Moschovakis coding lemma that is named after him.
Biography
Moschovakis earned his Ph.D. from the University of Wisconsin–Madison in 1963 under the direction of Stephen Kleene, with a dissertation entitled Recursive Analysis. In 2015, he was elected as a fellow of the American Mathematical Society "for contributions to mathematical logic, especially set theory and computability theory, and for exposition".
For many years, he has split his time between UCLA and the University of Athens (he retired from the latter in July 2005).
Moschovakis is married to Joan Moschovakis, with whom he gave the 2014 Lindström Lectures at the University of Gothenburg.
Publications
Second edition available online
References
External links
Home page
Living people
20th-century American mathematicians
American logicians
Greek logicians
Academic staff of the National and Kapodistrian University of Athens
Set theorists
University of California, Los Angeles faculty
1938 births
Fellows of the American Mathematical Society
People from Athens
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https://en.wikipedia.org/wiki/Global%20dimension
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In ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring A denoted gl dim A, is a non-negative integer or infinity which is a homological invariant of the ring. It is defined to be the supremum of the set of projective dimensions of all A-modules. Global dimension is an important technical notion in the dimension theory of Noetherian rings. By a theorem of Jean-Pierre Serre, global dimension can be used to characterize within the class of commutative Noetherian local rings those rings which are regular. Their global dimension coincides with the Krull dimension, whose definition is module-theoretic.
When the ring A is noncommutative, one initially has to consider two versions of this notion, right global dimension that arises from consideration of the right , and left global dimension that arises from consideration of the left . For an arbitrary ring A the right and left global dimensions may differ. However, if A is a Noetherian ring, both of these dimensions turn out to be equal to weak global dimension, whose definition is left-right symmetric. Therefore, for noncommutative Noetherian rings, these two versions coincide and one is justified in talking about the global dimension.
Examples
Let A = K[x1,...,xn] be the ring of polynomials in n variables over a field K. Then the global dimension of A is equal to n. This statement goes back to David Hilbert's foundational work on homological properties of polynomial rings; see Hilbert's syzygy theorem. More generally, if R is a Noetherian ring of finite global dimension k and A = R[x] is a ring of polynomials in one variable over R then the global dimension of A is equal to k + 1.
A ring has global dimension zero if and only if it is semisimple.
The global dimension of a ring A is less than or equal to one if and only if A is hereditary. In particular, a commutative principal ideal domain which is not a field has global dimension one. For example has global dimension one.
The first Weyl algebra A1 is a noncommutative Noetherian domain of global dimension one.
If a ring is right Noetherian, then the right global dimension is the same as the weak global dimension, and is at most the left global dimension. In particular if a ring is right and left Noetherian then the left and right global dimensions and the weak global dimension are all the same.
The triangular matrix ring has right global dimension 1, weak global dimension 1, but left global dimension 2. It is right Noetherian but not left Noetherian.
Alternative characterizations
The right global dimension of a ring A can be alternatively defined as:
the supremum of the set of projective dimensions of all cyclic right A-modules;
the supremum of the set of projective dimensions of all finite right A-modules;
the supremum of the injective dimensions of all right A-modules;
when A is a commutative Noetherian local ring with maximal ideal m,
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https://en.wikipedia.org/wiki/Square%20principle
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In mathematical set theory, a square principle is a combinatorial principle asserting the existence of a cohering sequence of
short closed unbounded (club) sets so that no one (long) club set coheres with them all. As such they may be viewed as a kind of
incompactness phenomenon. They were introduced by Ronald Jensen in his analysis of the fine structure of the constructible universe L.
Definition
Define Sing to be the class of all limit ordinals which are not regular. Global square states that there is a system satisfying:
is a club set of .
ot
If is a limit point of then and
Variant relative to a cardinal
Jensen introduced also a local version of the principle. If
is an uncountable cardinal,
then asserts that there is a sequence satisfying:
is a club set of .
If , then
If is a limit point of then
Jensen proved that this principle holds in the constructible universe for any uncountable cardinal κ.
Notes
Set theory
Constructible universe
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https://en.wikipedia.org/wiki/List%20of%20IFK%20G%C3%B6teborg%20records%20and%20statistics
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This article is about the records and statistics of the football section of IFK Göteborg. For the statistics of other sections, see IFK Göteborg (sports club).
Honours
Domestic
Swedish Champions
Winners (18): 1908, 1910, 1918, 1934–35, 1941–42, 1957–58, 1969, 1982, 1983, 1984, 1987, 1990, 1991, 1993, 1994, 1995, 1996, 2007
League
Allsvenskan:
Winners (13): 1934–35, 1941–42, 1957–58, 1969, 1982, 1984, 1990, 1991, 1993, 1994, 1995, 1996, 2007
Runners-up (13): 1924–25, 1926–27, 1929–30, 1939–49, 1979, 1981, 1986, 1988, 1997, 2005, 2009, 2014, 2015
Svenska Serien:
Winners (5): 1912–13, 1913–14, 1914–15, 1915–16, 1916–17
Fyrkantserien:
Winners (2): 1918, 1919
Mästerskapsserien:
Winners (1): 1991
Division 2
Winners (3): 1938–39, 1950–51, 1976
Runners-up (2): 1972, 1975
Cups
Svenska Cupen:
Winners (8): 1978–79, 1981–82, 1982–83, 1991, 2008, 2012–13, 2014–15, 2019–20
Runners-up (5): 1985–86, 1998–99, 2004, 2007, 2009
Allsvenskan play-offs:
Winners (5): 1982, 1983, 1984, 1987, 1990
Runners-up (1): 1985
Svenska Mästerskapet:
Winners (3): 1908, 1910, 1918
Svenska Supercupen:
Winners (1): 2008
Runners-up (4): 2009, 2010, 2013, 2015
Kamratmästerskapen:
Winners (11): 1909, 1910, 1912, 1913, 1914, 1915, 1920, 1921, 1922, 1924, 1940
Runners-up (2): 1906, 1908
European
UEFA Cup:
Winners (2): 1981–82, 1986–87
European Cup/UEFA Champions League:
Semi-finals (2): 1985–86, 1992–93
Quarter-finals (3): 1984–85, 1988–89, 1994–95
UEFA Cup Winners' Cup:
Quarter-finals (1): 1979–80
Royal League:
Runners-up (1): 2004–05
Doubles, trebles and quadruples
Doubles
Fyrkantserien and Svenska Mästerskapet (Swedish Champions):
Winners (1): 1918
Allsvenskan play-offs (Swedish Champions) and Svenska Cupen:
Winners (1): 1983
Allsvenskan and Allsvenskan play-offs (Swedish Champions):
Winners (2): 1984, 1990
Svenska Cupen and Svenska Supercupen:
Winners (1): 2008
Trebles
Allsvenskan, Allsvenskan play-offs (Swedish Champions) and the UEFA Cup:
Winners (1): 1987
Allsvenskan, Mästerskapsserien (Swedish Champions) and Svenska Cupen:
Winners (1): 1991
Quadruples
Allsvenskan, Allsvenskan play-offs (Swedish Champions), Svenska Cupen and the UEFA Cup:
Winners (1): 1982
Records
Home victory, Allsvenskan: 9-1 vs. IK Sleipner, 10 May 1925; 8-0 vs. Hammarby IF, 2 June 1925; 8-0 vs. Stattena IF, 21 April 1930
Away victory, Allsvenskan: 9-2 vs. IFK Eskilstuna, 8 October 1933; 7-0 vs. IK Sleipner, 20 April 1941
Home loss, Allsvenskan: 2-9 vs. Malmö FF, 10 September 1949
Away loss, Allsvenskan: 0-7 vs. IFK Norrköping, 1 May 1960
Highest attendance, Nya Ullevi: 52,194 vs. Örgryte IS, 3 June 1959
Highest attendance, Gamla Ullevi: 31,064 vs. GAIS, 27 May 1955
Highest attendance, Slottsskogsvallen: 21,580 vs. AIK, 25 October 1931
Highest attendance, The New Gamla Ullevi: 18,276 vs Djurgårdens IF
Highest average attendance, season: 23,796, 1977
Most appearances, total: 609, Mikael Nilsson 1987–2001
Most appearances, Allsvenskan:
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https://en.wikipedia.org/wiki/Sequence%20transformation
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In mathematics, a sequence transformation is an operator acting on a given space of sequences (a sequence space). Sequence transformations include linear mappings such as convolution with another sequence, and resummation of a sequence and, more generally, are commonly used for series acceleration, that is, for improving the rate of convergence of a slowly convergent sequence or series. Sequence transformations are also commonly used to compute the antilimit of a divergent series numerically, and are used in conjunction with extrapolation methods.
Overview
Classical examples for sequence transformations include the binomial transform, Möbius transform, Stirling transform and others.
Definitions
For a given sequence
the transformed sequence is
where the members of the transformed sequence are usually computed from some finite number of members of the original sequence, i.e.
for some which often depends on (cf. e.g. Binomial transform). In the simplest case, the and the are real or complex numbers. More generally, they may be elements of some vector space or algebra.
In the context of acceleration of convergence, the transformed sequence is said to converge faster than the original sequence if
where is the limit of , assumed to be convergent. In this case, convergence acceleration is obtained. If the original sequence is divergent, the sequence transformation acts as extrapolation method to the antilimit .
If the mapping is linear in each of its arguments, i.e., for
for some constants (which may depend on n), the sequence transformation is called a linear sequence transformation. Sequence transformations that are not linear are called nonlinear sequence transformations.
Examples
Simplest examples of (linear) sequence transformations include shifting all elements, (resp. = 0 if n + k < 0) for a fixed k, and scalar multiplication of the sequence.
A less trivial example would be the discrete convolution with a fixed sequence. A particularly basic form is the difference operator, which is convolution with the sequence and is a discrete analog of the derivative. The binomial transform is another linear transformation of a still more general type.
An example of a nonlinear sequence transformation is Aitken's delta-squared process, used to improve the rate of convergence of a slowly convergent sequence. An extended form of this is the Shanks transformation. The Möbius transform is also a nonlinear transformation, only possible for integer sequences.
See also
Aitken's delta-squared process
Minimum polynomial extrapolation
Richardson extrapolation
Series acceleration
Steffensen's method
References
Hugh J. Hamilton, "Mertens' Theorem and Sequence Transformations", AMS (1947)
External links
Transformations of Integer Sequences, a subpage of the On-Line Encyclopedia of Integer Sequences
Mathematical series
Asymptotic analysis
Perturbation theory
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https://en.wikipedia.org/wiki/211%20%28number%29
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211 (two hundred [and] eleven) is the natural number following 210 and preceding 212. It is also a prime number.
In mathematics
211 is an odd number.
211 is a primorial prime, the sum of three consecutive primes (), a Chen prime, a centered decagonal prime, and a self prime.
211 is the smallest prime separated by eight or more from the nearest primes (199 and 223). It is thus a balanced prime and an isolated prime.
211 is a repdigit in tetradecimal (111).
In decimal, multiplying 211's digits results in a prime (); adding its digits results in a square (). Rearranging its digits, 211 becomes 121, which also is a square (). Adding any two of 211's digits will result in a prime (2 or 3).
211 is a super-prime.
In science and technology
2-1-1 is special abbreviated telephone number reserved in Canada and the United States as an easy-to-remember three-digit telephone number. It is meant to provide quick information and referrals to health and human service organizations for both services from charities and from governmental agencies.
In chemistry, 211 is also associated with E211, the preservative sodium benzoate.
In religions
In Islam, Sermon 211 is about the strength and greatness of Allah.
In other fields
211 is also the California Penal Code section defining robbery. It is sometimes paired with 187, California PC section for murder.
211 is also an EDI (Electronic data interchange) document known as an Electronic Bill of Lading.
211 is also a nickname for Steel Reserve, a malt liquor alcoholic beverage.
211 is also SMTP status code for system status.
+211 is the code for international direct-dial phone calls to South Sudan.
See also
211 Crew
References
Integers
ca:Nombre 210#Nombres del 211 al 219
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https://en.wikipedia.org/wiki/Pseudoholomorphic%20curve
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In mathematics, specifically in topology and geometry, a pseudoholomorphic curve (or J-holomorphic curve) is a smooth map from a Riemann surface into an almost complex manifold that satisfies the Cauchy–Riemann equation. Introduced in 1985 by Mikhail Gromov, pseudoholomorphic curves have since revolutionized the study of symplectic manifolds. In particular, they lead to the Gromov–Witten invariants and Floer homology, and play a prominent role in string theory.
Definition
Let be an almost complex manifold with almost complex structure . Let be a smooth Riemann surface (also called a complex curve) with complex structure . A pseudoholomorphic curve in is a map that satisfies the Cauchy–Riemann equation
Since , this condition is equivalent to
which simply means that the differential is complex-linear, that is, maps each tangent space
to itself. For technical reasons, it is often preferable to introduce some sort of inhomogeneous term and to study maps satisfying the perturbed Cauchy–Riemann equation
A pseudoholomorphic curve satisfying this equation can be called, more specifically, a -holomorphic curve. The perturbation is sometimes assumed to be generated by a Hamiltonian (particularly in Floer theory), but in general it need not be.
A pseudoholomorphic curve is, by its definition, always parametrized. In applications one is often truly interested in unparametrized curves, meaning embedded (or immersed) two-submanifolds of , so one mods out by reparametrizations of the domain that preserve the relevant structure. In the case of Gromov–Witten invariants, for example, we consider only closed domains of fixed genus and we introduce marked points (or punctures) on . As soon as the punctured Euler characteristic is negative, there are only finitely many holomorphic reparametrizations of that preserve the marked points. The domain curve is an element of the Deligne–Mumford moduli space of curves.
Analogy with the classical Cauchy–Riemann equations
The classical case occurs when and are both simply the complex number plane. In real coordinates
and
where . After multiplying these matrices in two different orders, one sees immediately that the equation
written above is equivalent to the classical Cauchy–Riemann equations
Applications in symplectic topology
Although they can be defined for any almost complex manifold, pseudoholomorphic curves are especially interesting when interacts with a symplectic form . An almost complex structure is said to be -tame if and only if
for all nonzero tangent vectors . Tameness implies that the formula
defines a Riemannian metric on . Gromov showed that, for a given , the space of -tame is nonempty and contractible. He used this theory to prove a non-squeezing theorem concerning symplectic embeddings of spheres into cylinders.
Gromov showed that certain moduli spaces of pseudoholomorphic curves (satisfying additional specified conditions) are compact, and described the way in which pseudoh
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https://en.wikipedia.org/wiki/Karel%20deLeeuw
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Karel deLeeuw, or de Leeuw ( – ), was a mathematics professor at Stanford University, specializing in harmonic analysis and functional analysis.
Life and career
Born in Chicago, Illinois, he attended the Illinois Institute of Technology and the University of Chicago, earning a B.S. degree in 1950. He stayed at Chicago to earn an M.S. degree in mathematics in 1951, then went to Princeton University, where he obtained a Ph.D. degree in 1954. His thesis, titled "The relative cohomology structure of formations", was written under the direction of Emil Artin.
After first teaching mathematics at Dartmouth College and the University of Wisconsin–Madison, he joined the Stanford University faculty in 1957, becoming a full professor in 1966. During sabbaticals and leaves he also spent time at the Institute for Advanced Study and at Churchill College, Cambridge (where he was a Fulbright Fellow). He was also a Member-at-Large of the Council of the American Mathematical Society.
Death and legacy
DeLeeuw was murdered by Theodore Streleski, a Stanford doctoral student for 19 years, whom he briefly advised. DeLeeuw's widow Sita deLeeuw was critical of media coverage of the crime, saying, "The media, in their eagerness to give Streleski a forum, become themselves accomplices in the murder—giving Streleski what he wanted in the first place."
A memorial lecture series was established in 1978 by the Stanford Department of Mathematics to honor deLeeuw's memory.
Selected publications
References
External links
20th-century American mathematicians
Mathematical analysts
Stanford University Department of Mathematics faculty
Princeton University alumni
Dartmouth College faculty
University of Wisconsin–Madison faculty
People from Chicago
1930 births
1978 deaths
People murdered in California
Deaths by beating in the United States
Illinois Institute of Technology alumni
Mathematicians from Illinois
American textbook writers
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https://en.wikipedia.org/wiki/223%20%28number%29
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223 (two hundred [and] twenty-three) is the natural number following 222 and preceding 224.
In mathematics
223 is a prime number. Among the 720 permutations of the numbers from 1 to 6, exactly 223 of them have the property that at least one of the numbers is fixed in place by the permutation and the numbers less than it and greater than it are separately permuted among themselves.
In connection with Waring's problem, 223 requires the maximum number of terms (37 terms) when expressed as a sum of positive fifth powers, and is the only number that requires that many terms.
In other fields
.223 (disambiguation), the caliber of several firearm cartridges
The years 223 and 223 BC
The number of synodic months of a Saros
References
Integers
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https://en.wikipedia.org/wiki/227%20%28number%29
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227 (two hundred [and] twenty-seven) is the natural number between 226 and 228. It is also a prime number.
In mathematics
227 is a twin prime and the start of a prime triplet (with 229 and 233). It is a safe prime, as dividing it by two and rounding down produces the Sophie Germain prime 113. It is also a regular prime, a Pillai prime, a Stern prime, and a Ramanujan prime.
227 and 229 form the first twin prime pair for which neither is a cluster prime.
The 227th harmonic number is the first to exceed six. There are 227 different connected graphs with eight edges, and 227 independent sets in a 3 × 4 grid graph.
References
Integers
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https://en.wikipedia.org/wiki/Nitin%20Saxena
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Nitin Saxena (born 3 May 1981) is an Indian scientist in mathematics and theoretical computer science. His research focuses on computational complexity.
He attracted international attention for proposing the AKS Primality Test in 2002 in a joint work with Manindra Agrawal and Neeraj Kayal, for which the trio won the 2006 Fulkerson Prize, and the 2006 Gödel Prize. They provided the first unconditional deterministic algorithm to test an n-digit number for primality in a time that has been proven to be polynomial in n. This research work came out as a part of his undergraduate study.
Early life and education
He is an alumnus of Boys' High School And College, Allahabad. He graduated with his B.Tech in Computer Science and Engineering from Indian Institute of Technology Kanpur in 2002. He received his PhD from the Department of Computer Science and Engineering of the same institute in 2006 with the Dissertation titled "Morphisms of Rings and Applications to Complexity".
Career
He was awarded the Distinguished Alumnus Award of the Indian Institute of Technology Kanpur in 2003 for his work in computational complexity theory. He was appointed at the Centrum Wiskunde & Informatica (CWI) starting as a postdoc researcher from September 2006 onwards. He was a Bonn Junior Fellow at the University of Bonn from Summer 2008 onwards. He joined the Department of Computer Science and Engineering at IIT Kanpur as faculty in April 2013.
Saxena was awarded the 2018 Shanti Swarup Bhatnagar Prize for his work in Algebraic Complexity Theory. One of the youngest awardees, Saxena’s research interests include Computational Complexity and Algebraic Geometry.
References
External links
Nitin Saxena's Homepage
Profile of Nitin Saxena at the IIT Kanpur Alumni Association
.
1981 births
Living people
Indian computer scientists
Gödel Prize laureates
IIT Kanpur alumni
Academic staff of the University of Bonn
Recipients of the Shanti Swarup Bhatnagar Award in Mathematical Science
Theoretical computer scientists
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https://en.wikipedia.org/wiki/Covering%20system
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In mathematics, a covering system (also called a complete residue system) is a collection
of finitely many residue classes
whose union contains every integer.
Examples and definitions
The notion of covering system was introduced by Paul Erdős in the early 1930s.
The following are examples of covering systems:
A covering system is called disjoint (or exact) if no two members overlap.
A covering system is called distinct (or incongruent) if all the moduli are different (and bigger than 1). Hough and Nielsen (2019) proved that any distinct covering system has a modulus that is divisible by either 2 or 3.
A covering system is called irredundant (or minimal) if all the residue classes are required to cover the integers.
The first two examples are disjoint.
The third example is distinct.
A system (i.e., an unordered multi-set)
of finitely many residue classes is called an -cover if it covers every integer at least
times, and an exact -cover if it covers each integer exactly times. It is known that for each
there are exact -covers which cannot be written as a union of two covers. For example,
is an exact 2-cover which is not a union of two covers.
The first example above is an exact 1-cover (also called an exact cover). Another exact cover in common use is that of odd and even numbers, or
This is just one case of the following fact: For every positive integer modulus , there is an exact cover:
Mirsky–Newman theorem
The Mirsky–Newman theorem, a special case of the Herzog–Schönheim conjecture, states that there is no disjoint distinct covering system. This result was conjectured in 1950 by Paul Erdős and proved soon thereafter by Leon Mirsky and Donald J. Newman. However, Mirsky and Newman never published their proof. The same proof was also found independently by Harold Davenport and Richard Rado.
Primefree sequences
Covering systems can be used to find primefree sequences, sequences of integers satisfying the same recurrence relation as the Fibonacci numbers, such that consecutive numbers in the sequence are relatively prime but all numbers in the sequence are composite numbers. For instance, a sequence of this type found by Herbert Wilf has initial terms
a1 = 20615674205555510, a2 = 3794765361567513 .
In this sequence, the positions at which the numbers in the sequence are divisible by a prime p form an arithmetic progression; for instance, the even numbers in the sequence are the numbers ai where i is congruent to 1 mod 3. The progressions divisible by different primes form a covering system, showing that every number in the sequence is divisible by at least one prime.
Boundedness of the smallest modulus
Paul Erdős asked whether for any arbitrarily large N there exists an incongruent covering system the minimum of whose moduli is at least N. It is easy to construct examples where the minimum of the moduli in such a system is 2, or 3 (Erdős gave an example where the moduli are in the set of the divisors of 120; a suit
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https://en.wikipedia.org/wiki/Mueller%20calculus
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Mueller calculus is a matrix method for manipulating Stokes vectors, which represent the polarization of light. It was developed in 1943 by Hans Mueller. In this technique, the effect of a particular optical element is represented by a Mueller matrix—a 4×4 matrix that is an overlapping generalization of the Jones matrix.
Introduction
Disregarding coherent wave superposition, any fully polarized, partially polarized, or unpolarized state of light can be represented by a Stokes vector ; and any optical element can be represented by a Mueller matrix (M).
If a beam of light is initially in the state and then passes through an optical element M and comes out in a state , then it is written
If a beam of light passes through optical element M1 followed by M2 then M3 it is written
given that matrix multiplication is associative it can be written
Matrix multiplication is not commutative, so in general
Mueller vs. Jones calculi
With disregard for coherence, light which is unpolarized or partially polarized must be treated using the Mueller calculus, while fully polarized light can be treated with either the Mueller calculus or the simpler Jones calculus. Many problems involving coherent light (such as from a laser) must be treated with Jones calculus, however, because it works directly with the electric field of the light rather than with its intensity or power, and thereby retains information about the phase of the waves.
More specifically, the following can be said about Mueller matrices and Jones matrices:
Stokes vectors and Mueller matrices operate on intensities and their differences, i.e. incoherent superpositions of light; they are not adequate to describe either interference or diffraction effects.
(...)
Any Jones matrix [J] can be transformed into the corresponding Mueller–Jones matrix, M, using the following relation:
,
where * indicates the complex conjugate [sic], [A is:]
and ⊗ is the tensor (Kronecker) product.
(...)
While the Jones matrix has eight independent parameters [two Cartesian or polar components for each of the four complex values in the 2-by-2 matrix], the absolute phase information is lost in the [equation above], leading to only seven independent matrix elements for a Mueller matrix derived from a Jones matrix.
Mueller matrices
Below are listed the Mueller matrices for some ideal common optical elements:
General expression for reference frame rotation from the local frame to the laboratory frame:
where is the angle of rotation. For rotation from the laboratory frame to the local frame, the sign of the sine terms inverts.
Linear polarizer (horizontal transmission)
The Mueller matrices for other polarizer rotation angles can be generated by reference frame rotation.
Linear polarizer (vertical transmission)
Linear polarizer (+45° transmission)
Linear polarizer (−45° transmission)
General linear polarizer matrix
where is the angle of rotation of the polarizer.
General linear retarder (wave plate calc
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https://en.wikipedia.org/wiki/Closure%20with%20a%20twist
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Closure with a twist is a property of subsets of an algebraic structure. A subset of an algebraic structure is said to exhibit closure with a twist if for every two elements
there exists an automorphism of and an element such that
where "" is notation for an operation on preserved by .
Two examples of algebraic structures which exhibit closure with a twist are the cwatset and the generalized cwatset, or GC-set.
Cwatset
In mathematics, a cwatset is a set of bitstrings, all of the same length, which is closed with a twist.
If each string in a cwatset, C, say, is of length n, then C will be a subset of . Thus, two strings in C are added by adding the bits in the strings modulo 2 (that is, addition without carry, or exclusive disjunction). The symmetric group on n letters, , acts on by bit permutation:
where is an element of and p is an element of . Closure with a twist now means that for each element c in C, there exists some permutation such that, when you add c to an arbitrary element e in the cwatset and then apply the permutation, the result will also be an element of C. That is, denoting addition without carry by , C will be a cwatset if and only if
This condition can also be written as
Examples
All subgroups of — that is, nonempty subsets of which are closed under addition-without-carry — are trivially cwatsets, since we can choose each permutation pc to be the identity permutation.
An example of a cwatset which is not a group is
F = {000,110,101}.
To demonstrate that F is a cwatset, observe that
F + 000 = F.
F + 110 = {110,000,011}, which is F with the first two bits of each string transposed.
F + 101 = {101,011,000}, which is the same as F after exchanging the first and third bits in each string.
A matrix representation of a cwatset is formed by writing its words as the rows of a 0-1 matrix. For instance a matrix representation of F is given by
To see that F is a cwatset using this notation, note that
where and denote permutations of the rows and columns of the matrix, respectively, expressed in cycle notation.
For any another example of a cwatset is , which has -by- matrix representation
Note that for , .
An example of a nongroup cwatset with a rectangular matrix representation is
Properties
Let be a cwatset.
The degree of C is equal to the exponent n.
The order of C, denoted by |C|, is the set cardinality of C.
There is a necessary condition on the order of a cwatset in terms of its degree, which is
analogous to Lagrange's Theorem in group theory. To wit,
Theorem. If C is a cwatset of degree n and order m, then m divides .
The divisibility condition is necessary but not sufficient. For example, there does not exist a cwatset of degree 5 and order 15.
Generalized cwatset
In mathematics, a generalized cwatset (GC-set) is an algebraic structure generalizing the notion of closure with a twist, the defining characteristic of the cwatset.
Definitions
A subset H of a group G is a GC-set if fo
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https://en.wikipedia.org/wiki/Anti-diagonal%20matrix
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In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗), known as the anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal or bad diagonal).
Formal definition
An n-by-n matrix A is an anti-diagonal matrix if the (i, j) element is zero
Example
An example of an anti-diagonal matrix is
Properties
All anti-diagonal matrices are also persymmetric.
The product of two anti-diagonal matrices is a diagonal matrix. Furthermore, the product of an anti-diagonal matrix with a diagonal matrix is anti-diagonal, as is the product of a diagonal matrix with an anti-diagonal matrix.
An anti-diagonal matrix is invertible if and only if the entries on the diagonal from the lower left corner to the upper right corner are nonzero. The inverse of any invertible anti-diagonal matrix is also anti-diagonal, as can be seen from the paragraph above. The determinant of an anti-diagonal matrix has absolute value given by the product of the entries on the diagonal from the lower left corner to the upper right corner. However, the sign of this determinant will vary because the one nonzero signed elementary product from an anti-diagonal matrix will have a different sign depending on whether the permutation related to it is odd or even:
More precisely, the sign of the elementary product needed to calculate the determinant of an anti-diagonal matrix is related to whether the corresponding triangular number is even or odd. This is because the number of inversions in the permutation for the only nonzero signed elementary product of any n × n anti-diagonal matrix is always equal to the nth such number.
See also
Main diagonal, all off-diagonal elements are zero in a diagonal matrix.
Exchange matrix, an anti-diagonal matrix with 1s along the counter-diagonal.
External links
Matrix calculator
Sparse matrices
Matrices
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https://en.wikipedia.org/wiki/Smith%27s%20Prize
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Smith's Prize was the name of each of two prizes awarded annually to two research students in mathematics and theoretical physics at the University of Cambridge from 1769. Following the reorganization in 1998, they are now awarded under the names Smith-Knight Prize and Rayleigh-Knight Prize.
History
The Smith Prize fund was founded by bequest of Robert Smith upon his death in 1768, having by his will left £3,500 of South Sea Company stock to the University. Every year two or more junior Bachelor of Arts students who had made the greatest progress in mathematics and natural philosophy were to be awarded a prize from the fund. The prize was awarded every year from 1769 to 1998 except 1917.
From 1769 to 1885, the prize was awarded for the best performance in a series of examinations. In 1854 George Stokes included an examination question on a particular theorem that William Thomson had written to him about, which is now known as Stokes' theorem. T. W. Körner notes
Only a small number of students took the Smith's prize examination in the nineteenth century. When Karl Pearson took the examination in 1879, the examiners were Stokes, Maxwell, Cayley, and Todhunter and the examinees went on each occasion to an examiner's dwelling, did a morning paper, had lunch there and continued their work on the paper in the afternoon.
In 1885, the examination was renamed Part III, (now known as the Master of Advanced Study in Mathematics for students who studied outside of Cambridge before taking it) and the prize was awarded for the best submitted essay rather than examination performance. According to Barrow-Green
By fostering an interest in the study of applied mathematics, the competition contributed towards the success in mathematical physics that was to become the hallmark of Cambridge mathematics during the second half of the nineteenth century.
In the twentieth century, the competition stimulated postgraduate research in mathematics in Cambridge and the competition has played a significant role by providing a springboard for graduates considering an academic career. The majority of prize-winners have gone on to become professional mathematicians or physicists.
The Rayleigh Prize was an additional prize, which was awarded for the first time in 1911.
The Smith's and Rayleigh prizes were only available to Cambridge graduate students who had been undergraduates at Cambridge. The J.T. Knight Prize was established in 1974 for Cambridge graduates who had been undergraduates at other universities. The prize commemorates J.T. Knight (1942–1970), who had been an undergraduate student at Glasgow and a graduate student at Cambridge. He was killed in a motor car accident in Ireland in April 1970.
Value of the prizes
Originally, in 1769, the prizes were worth £25 each and remained at that level for 100 years. In 1867, they fell to £23 and in 1915 were still reported to be worth that amount. By 1930, the value had risen to about £30, and by 1940, the value had risen
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https://en.wikipedia.org/wiki/Viscosity%20solution
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In mathematics, the viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael G. Crandall as a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation (PDE). It has been found that the viscosity solution is the natural solution concept to use in many applications of PDE's, including for example first order equations arising in dynamic programming (the Hamilton–Jacobi–Bellman equation), differential games (the Hamilton–Jacobi–Isaacs equation) or front evolution problems, as well as second-order equations such as the ones arising in stochastic optimal control or stochastic differential games.
The classical concept was that a PDE
over a domain has a solution if we can find a function u(x) continuous and differentiable over the entire domain such that , , , satisfy the above equation at every point.
If a scalar equation is degenerate elliptic (defined below), one can define a type of weak solution called viscosity solution.
Under the viscosity solution concept, u does not need to be everywhere differentiable. There may be points where either or does not exist and yet u satisfies the equation in an appropriate generalized sense. The definition allows only for certain kind of singularities, so that existence, uniqueness, and stability under uniform limits, hold for a large class of equations.
Definition
There are several equivalent ways to phrase the definition of viscosity solutions. See for example the section II.4 of Fleming and Soner's book or the definition using semi-jets in the Users Guide.
Degenerate elliptic An equation in a domain is defined to be degenerate elliptic if for any two symmetric matrices and such that is positive definite, and any values of , and , we have the inequality . For example, (where denotes the Laplacian) is degenerate elliptic since in this case, , and the trace of is the sum of its eigenvalues. Any real first- order equation is degenerate elliptic.
Viscosity subsolution An upper semicontinuous function in is defined to be a subsolution of the above degenerate elliptic equation in the viscosity sense if for any point and any function such that and in a neighborhood of , we have .
Viscosity supersolution A lower semicontinuous function in is defined to be a supersolution of the above degenerate elliptic equation in the viscosity sense if for any point and any function such that and in a neighborhood of , we have .
Viscosity solution A continuous function u is a viscosity solution of the PDE in if it is both a supersolution and a subsolution. Note that the boundary condition in the viscosity sense has not been discussed here.
Example
Consider the boundary value problem , or , on with boundary conditions . Then, the function is a viscosity solution.
Indeed, note that the boundary conditions are satisfied classically, and is well-defined in the interior except at . Thus, it remains to
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https://en.wikipedia.org/wiki/Paravector
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The name paravector is used for the combination of a scalar and a vector in any Clifford algebra, known as geometric algebra among physicists.
This name was given by J. G. Maks in a doctoral dissertation at Technische Universiteit Delft, Netherlands, in 1989.
The complete algebra of paravectors along with corresponding higher grade generalizations, all in the context of the Euclidean space of three dimensions, is an alternative approach to the spacetime algebra (STA) introduced by David Hestenes. This alternative algebra is called algebra of physical space (APS).
Fundamental axiom
For Euclidean spaces, the fundamental axiom indicates that the product of a vector with itself is the scalar value of the length squared (positive)
Writing
and introducing this into the expression of the fundamental axiom
we get the following expression after appealing to the fundamental axiom again
which allows to
identify the scalar product of two vectors as
As an important consequence we conclude that two orthogonal vectors (with zero scalar product) anticommute
The three-dimensional Euclidean space
The following list represents an instance of a complete basis for the space,
which forms an eight-dimensional space, where the multiple indices indicate the product of the respective basis vectors, for example
The grade of a basis element is defined in terms of the vector multiplicity, such that
According to the fundamental axiom, two different basis vectors anticommute,
or in other words,
This means that the volume element squares to
Moreover, the volume element commutes with any other element of the algebra, so that it can be identified with the complex number , whenever there is no danger of confusion. In fact, the volume element along with the real scalar forms an algebra isomorphic to the standard complex algebra. The volume element can be used to rewrite an equivalent form of the
basis as
Paravectors
The corresponding paravector basis that combines a real scalar and vectors is
,
which forms a four-dimensional linear space. The paravector space in the three-dimensional Euclidean space can be used to represent the space-time of special relativity as expressed in the algebra of physical space (APS).
It is convenient to write the unit scalar as , so that
the complete basis can be written in a compact form as
where the Greek indices such as run from to .
Antiautomorphism
Reversion conjugation
The Reversion antiautomorphism is denoted by . The action of this conjugation is to reverse the order of the geometric product (product between Clifford numbers in general).
,
where vectors and real scalar numbers are invariant under
reversion conjugation and are said to be real, for example:
On the other hand, the trivector and bivectors change sign under reversion
conjugation and are said to be purely imaginary. The reversion conjugation applied to each basis element is given
below
Clifford conjugation
The Clifford Conjugation is denoted by
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https://en.wikipedia.org/wiki/Stable%20map
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In mathematics, specifically in symplectic topology and algebraic geometry, one can construct the moduli space of stable maps, satisfying specified conditions, from Riemann surfaces into a given symplectic manifold. This moduli space is the essence of the Gromov–Witten invariants, which find application in enumerative geometry and type IIA string theory. The idea of stable map was proposed by Maxim Kontsevich around 1992 and published in .
Because the construction is lengthy and difficult, it is carried out here rather than in the Gromov–Witten invariants article itself.
Smooth pseudoholomorphic curves
Fix a closed symplectic manifold with symplectic form . Let and be natural numbers (including zero) and a two-dimensional homology class in . Then one may consider the set of pseudoholomorphic curves
where is a smooth, closed Riemann surface of genus with marked points , and
is a function satisfying, for some choice of -tame almost complex structure and inhomogeneous term , the perturbed Cauchy–Riemann equation
Typically one admits only those and that make the punctured Euler characteristic of negative. Then the domain is stable, meaning that there are only finitely many holomorphic automorphisms of that preserve the marked points.
The operator is elliptic and thus Fredholm. After significant analytical argument (completing in a suitable Sobolev norm, applying the implicit function theorem and Sard's theorem for Banach manifolds, and using elliptic regularity to recover smoothness) one can show that, for a generic choice of -tame and perturbation , the set of -holomorphic curves of genus with marked points that represent the class forms a smooth, oriented orbifold
of dimension given by the Atiyah-Singer index theorem,
Motivation
The moduli space is not compact, because a sequence of curves can degenerate to a singular curve, which is not in the moduli space as defined above. This happens, for example, when the energy of (meaning the L2 norm of the derivative) concentrates at some point on the domain.
One can capture the energy by rescaling the map around the concentration point. The effect is to attach a sphere, called a bubble, to the original domain at the concentration point and to extend the map across the sphere. The rescaled map may still have energy concentrating at one or more points, so one must rescale iteratively, eventually attaching an entire bubble tree onto the original domain, with the map well-behaved on each smooth component of the new domain.
Definition
Define a stable map to be a pseudoholomorphic map from a Riemann surface with at worst nodal singularities, such that there are only finitely many automorphisms of the map.
Concretely, this means the following. A smooth component of a nodal Riemann surface is said to be stable if there are at most finitely many automorphisms preserving its marked and nodal points. Then a stable map is a pseudoholomorphic map with at least one stable domain compone
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https://en.wikipedia.org/wiki/Murderous%20Maths
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Murderous Maths is a series of British educational books by author Kjartan Poskitt. Most of the books in the series are illustrated by illustrator Philip Reeve, with the exception of "The Secret Life of Codes", which is illustrated by Ian Baker, "Awesome Arithmetricks" illustrated by Daniel Postgate and Rob Davis, and "The Murderous Maths of Everything", also illustrated by Rob Davis.
The Murderous Maths books have been published in over 25 countries. The books, which are aimed at children aged 8 and above, teach maths, spanning from basic arithmetic to relatively complex concepts such as the quadratic formula and trigonometry. The books are written in an informal similar style to the Horrible Histories, Horrible Science and Horrible Geography series, involving evil geniuses, gangsters, and a generally comedic tone.
Development
The first two books of the series were originally part of "The Knowledge" (now "Totally") series, itself a spin-off of Horrible Histories. However, these books were eventually redesigned and they, as well as the rest of the titles in the series, now use the Murderous Maths banner. According to Poskitt, "these books have even found their way into schools and proved to be a boost to GCSE studies". The books are also available in foreign editions, including: German, Spanish, Polish, Czech, Greek, Dutch, Norwegian, Turkish, Croatian, Italian, Lithuanian, Korean, Danish, Hungarian, Finnish, Thai and Portuguese (Latin America). In 2009, the books were redesigned again, changing the cover art style and the titles of most of the books in the series.
Poskitt's goal, according to the Murderous Maths website, is to write books that are "something funny to read", have "good amusing illustrations", include "tricks", and "explaining the maths involved as clearly as possible". He adds that although he doesn't "work to any government imposed curriculum or any stage achievement levels", he has "been delighted to receive many messages of support and thanks from parents and teachers in the UK, the United States and elsewhere".
Titles
The following are the thirteen books that are available in the series.
Guaranteed to Bend Your Brain (previously Murderous Maths), - (addition, subtraction, multiplication, division, percentages, powers, tessellation, Roman numerals, the development of the "10" and the place system, shortcomings of calculators, prime numbers, time - how the year and day got divided, digital/analogue clocks, angles, introduction to real Mathematicians, magic squares, mental arithmetic, card trick with algebra explanation, rounding and symmetry.) Guaranteed to Mash your Mind (previously More Murderous Maths), (the monomino, domino, tromino, tetromino, pentomino, hexomino and heptomino, length area and volume, dimensions, measuring areas and volumes, basic rectangle and triangle formulas, speed, conversion of units, Möbius strip, Pythagoras, right-angled triangles, irrational numbers, pi, area and perimeter, bisecting angles
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https://en.wikipedia.org/wiki/Gromov%E2%80%93Witten%20invariant
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In mathematics, specifically in symplectic topology and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations, count pseudoholomorphic curves meeting prescribed conditions in a given symplectic manifold. The GW invariants may be packaged as a homology or cohomology class in an appropriate space, or as the deformed cup product of quantum cohomology. These invariants have been used to distinguish symplectic manifolds that were previously indistinguishable. They also play a crucial role in closed type IIA string theory. They are named after Mikhail Gromov and Edward Witten.
The rigorous mathematical definition of Gromov–Witten invariants is lengthy and difficult, so it is treated separately in the stable map article. This article attempts a more intuitive explanation of what the invariants mean, how they are computed, and why they are important.
Definition
Consider the following:
X: a closed symplectic manifold of dimension 2k,
A: a 2-dimensional homology class in X,
g: a non-negative integer,
n: a non-negative integer.
Now we define the Gromov–Witten invariants associated to the 4-tuple: (X, A, g, n). Let be the Deligne–Mumford moduli space of curves of genus g with n marked points and denote the moduli space of stable maps into X of class A, for some chosen almost complex structure J on X compatible with its symplectic form. The elements of are of the form:
,
where C is a (not necessarily stable) curve with n marked points x1, ..., xn and f : C → X is pseudoholomorphic. The moduli space has real dimension
Let
denote the stabilization of the curve. Let
which has real dimension . There is an evaluation map
The evaluation map sends the fundamental class of to a d-dimensional rational homology class in Y, denoted
In a sense, this homology class is the Gromov–Witten invariant of X for the data g, n, and A. It is an invariant of the symplectic isotopy class of the symplectic manifold X.
To interpret the Gromov–Witten invariant geometrically, let β be a homology class in and homology classes in X, such that the sum of the codimensions of equals d. These induce homology classes in Y by the Künneth formula. Let
where denotes the intersection product in the rational homology of Y. This is a rational number, the Gromov–Witten invariant for the given classes. This number gives a "virtual" count of the number of pseudoholomorphic curves (in the class A, of genus g, with domain in the β-part of the Deligne–Mumford space) whose n marked points are mapped to cycles representing the .
Put simply, a GW invariant counts how many curves there are that intersect n chosen submanifolds of X. However, due to the "virtual" nature of the count, it need not be a natural number, as one might expect a count to be. For the space of stable maps is an orbifold, whose points of isotropy can contribute noninteger values to the invariant.
There are numerous variations on this construction, in which cohomology is u
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https://en.wikipedia.org/wiki/Lehmer%20sequence
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In mathematics, a Lehmer sequence is a generalization of a Lucas sequence.
Algebraic relations
If a and b are complex numbers with
under the following conditions:
Q and R are relatively prime nonzero integers
is not a root of unity.
Then, the corresponding Lehmer numbers are:
for n odd, and
for n even.
Their companion numbers are:
for n odd and
for n even.
Recurrence
Lehmer numbers form a linear recurrence relation with
with initial values . Similarly the companion sequence satisfies
with initial values
References
Integer sequences
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https://en.wikipedia.org/wiki/%CE%95-net
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An -net or epsilon net in mathematics may refer to:
ε-net (computational geometry) in computational geometry and in geometric probability theory
ε-net (metric spaces) in metric spaces
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https://en.wikipedia.org/wiki/Peter%20Stoner
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Peter Stoner (June 16, 1888 – March 21, 1980) was a Christian writer and Chairman of the departments of mathematics and astronomy at Pasadena City College until 1953; Chairman of the science division, Westmont College, 1953–57; Professor Emeritus of Science, Westmont College; and Professor Emeritus of Mathematics and Astronomy, Pasadena City College.
Career
Stoner is probably best known for his book Science Speaks, which discusses, among other things, Bible prophecies vis a vis probability estimates and calculations. The work is often cited in the field of Christian apologetics in regard to Bible prophecy. Stoner's book became widely known when it was mentioned by Josh McDowell in his 1972 book Evidence that Demands a Verdict (revised as New Evidence that Demands a Verdict).
American Scientific Affiliation
Peter Stoner was a co-founder of the American Scientific Affiliation, a Christian organization that describes itself as "a fellowship of men and women in science and disciplines that relate to science who share a common fidelity to the Word of God and a commitment to integrity in the practice of science." The foreword to Stoner's Science Speaks includes a partial endorsement from this body (covering the book's scientific content and prophecy probability calculations, but not addressing issues of Biblical exegesis or historical accuracy): They considered it "...in general, to be dependable and accurate in regard to the scientific material presented" and the probability material presented in regard to prophecy. While the ASA includes members with a diverse range of attitudes towards science (theistic evolutionists, Intelligent Design advocates, old-Earth creationists and young-Earth creationists), Stoner himself was apparently an old-Earth creationist.
Criticism
C. P. Swanson, reviewing Science Speaks in The Quarterly Review of Biology, wrote: "... the author has fallen into the commonest error of using only these facts which bolster his hypothesis, and of discarding or controverting those which do not. For example, his discussion of the theory of evolution is not only misleading; it displays an abysmal ignorance of recent evolutionary studies."
Various critics have taken issue with Stoner's interpretation of prophecy. Stoner's apologetic work did not receive critical attention until its inclusion in Josh McDowell's Evidence that Demands a Verdict and criticism of these claims tends to be addressed to McDowell rather than Stoner, with Stoner's name mentioned in passing. These criticisms against McDowell, Stoner and others include historical errors, claims regarding after-the-event authorship or tampering with biblical prophecies, and disputed meanings of certain biblical phrases.
Others who disagree with specific claims made by Stoner include fellow Christians and secular historians. For instance, while Stoner says of Ezekiel's prophecy of the permanent destruction of Tyre "If Ezekiel had looked at Tyre in his day and had made these seve
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https://en.wikipedia.org/wiki/Oliver%20Schr%C3%B6der
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Oliver Schröder (born 11 June 1980 in West Berlin, West Germany) is a retired German footballer, currently serving as assistant coach for the under-16 team of Hertha BSC.
Career statistics
1.Includes German Cup.
2.Includes UEFA Cup.
3.Includes German League Cup.
References
External links
Player profile at fc-hansa.de
1980 births
Living people
German men's footballers
Hertha BSC II players
1. FC Köln II players
1. FC Köln players
Hertha BSC players
VfL Bochum players
FC Hansa Rostock players
FC Erzgebirge Aue players
SC Fortuna Köln players
Bundesliga players
2. Bundesliga players
3. Liga players
Regionalliga players
Footballers from Berlin
Men's association football defenders
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https://en.wikipedia.org/wiki/Virtual%20displacement
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In analytical mechanics, a branch of applied mathematics and physics, a virtual displacement (or infinitesimal variation) shows how the mechanical system's trajectory can hypothetically (hence the term virtual) deviate very slightly from the actual trajectory of the system without violating the system's constraints. For every time instant is a vector tangential to the configuration space at the point The vectors show the directions in which can "go" without breaking the constraints.
For example, the virtual displacements of the system consisting of a single particle on a two-dimensional surface fill up the entire tangent plane, assuming there are no additional constraints.
If, however, the constraints require that all the trajectories pass through the given point at the given time i.e. then
Notations
Let be the configuration space of the mechanical system, be time instants, consists of smooth functions on , and
The constraints are here for illustration only. In practice, for each individual system, an individual set of constraints is required.
Definition
For each path and a variation of is a function such that, for every and The virtual displacement being the tangent bundle of corresponding to the variation assigns to every the tangent vector
In terms of the tangent map,
Here is the tangent map of where and
Properties
Coordinate representation. If are the coordinates in an arbitrary chart on and then
If, for some time instant and every then, for every
If then
Examples
Free particle in R3
A single particle freely moving in has 3 degrees of freedom. The configuration space is and For every path and a variation of there exists a unique such that as
By the definition,
which leads to
Free particles on a surface
particles moving freely on a two-dimensional surface have degree of freedom. The configuration space here is
where is the radius vector of the particle. It follows that
and every path may be described using the radius vectors of each individual particle, i.e.
This implies that, for every
where Some authors express this as
Rigid body rotating around fixed point
A rigid body rotating around a fixed point with no additional constraints has 3 degrees of freedom. The configuration space here is the special orthogonal group of dimension 3 (otherwise known as 3D rotation group), and We use the standard notation to refer to the three-dimensional linear space of all skew-symmetric three-dimensional matrices. The exponential map guarantees the existence of such that, for every path its variation and there is a unique path such that and, for every By the definition,
Since, for some function , as ,
See also
D'Alembert principle
Virtual work
References
Dynamical systems
Mechanics
Classical mechanics
Lagrangian mechanics
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https://en.wikipedia.org/wiki/Rotations%20in%204-dimensional%20Euclidean%20space
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In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4.
In this article rotation means rotational displacement. For the sake of uniqueness, rotation angles are assumed to be in the segment except where mentioned or clearly implied by the context otherwise.
A "fixed plane" is a plane for which every vector in the plane is unchanged after the rotation. An "invariant plane" is a plane for which every vector in the plane, although it may be affected by the rotation, remains in the plane after the rotation.
Geometry of 4D rotations
Four-dimensional rotations are of two types: simple rotations and double rotations.
Simple rotations
A simple rotation about a rotation centre leaves an entire plane through (axis-plane) fixed. Every plane that is completely orthogonal to intersects in a certain point . For each such point is the centre of the 2D rotation induced by in . All these 2D rotations have the same rotation angle .
Half-lines from in the axis-plane are not displaced; half-lines from orthogonal to are displaced through ; all other half-lines are displaced through an angle less than .
Double rotations
For each rotation of 4-space (fixing the origin), there is at least one pair of orthogonal 2-planes and each of which is invariant and whose direct sum is all of 4-space. Hence operating on either of these planes produces an ordinary rotation of that plane. For almost all (all of the 6-dimensional set of rotations except for a 3-dimensional subset), the rotation angles in plane and in plane – both assumed to be nonzero – are different. The unequal rotation angles and satisfying , are almost uniquely determined by . Assuming that 4-space is oriented, then the orientations of the 2-planes and can be chosen consistent with this orientation in two ways. If the rotation angles are unequal (), is sometimes termed a "double rotation".
In that case of a double rotation, and are the only pair of invariant planes, and half-lines from the origin in , are displaced through and respectively, and half-lines from the origin not in or are displaced through angles strictly between and .
Isoclinic rotations
If the rotation angles of a double rotation are equal then there are infinitely many invariant planes instead of just two, and all half-lines from are displaced through the same angle. Such rotations are called isoclinic or equiangular rotations, or Clifford displacements. Beware: not all planes through are invariant under isoclinic rotations; only planes that are spanned by a half-line and the corresponding displaced half-line are invariant.
Assuming that a fixed orientation has been chosen for 4-dimensional space, isoclinic 4D rotations may be put into two categories. To see this, consider an isoclinic rotation , and take an orientation-consistent ordered set of mutually perpendic
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https://en.wikipedia.org/wiki/Calabi%20flow
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In the mathematical fields of differential geometry and geometric analysis, the Calabi flow is a geometric flow which deforms a Kähler metric on a complex manifold. Precisely, given a Kähler manifold , the Calabi flow is given by:
,
where is a mapping from an open interval into the collection of all Kähler metrics on , is the scalar curvature of the individual Kähler metrics, and the indices correspond to arbitrary holomorphic coordinates . This is a fourth-order geometric flow, as the right-hand side of the equation involves fourth derivatives of .
The Calabi flow was introduced by Eugenio Calabi in 1982 as a suggestion for the construction of extremal Kähler metrics, which were also introduced in the same paper. It is the gradient flow of the ; extremal Kähler metrics are the critical points of the Calabi functional.
A convergence theorem for the Calabi flow was found by Piotr Chruściel in the case that has complex dimension equal to one. Xiuxiong Chen and others have made a number of further studies of the flow, although as of 2020 the flow is still not well understood.
References
Eugenio Calabi. Extremal Kähler metrics. Ann. of Math. Stud. 102 (1982), pp. 259–290. Seminar on Differential Geometry. Princeton University Press (PUP), Princeton, N.J.
E. Calabi and X.X. Chen. The space of Kähler metrics. II. J. Differential Geom. 61 (2002), no. 2, 173–193.
X.X. Chen and W.Y. He. On the Calabi flow. Amer. J. Math. 130 (2008), no. 2, 539–570.
Piotr T. Chruściel. Semi-global existence and convergence of solutions of the Robinson-Trautman (2-dimensional Calabi) equation. Comm. Math. Phys. 137 (1991), no. 2, 289–313.
Geometric flow
Partial differential equations
String theory
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https://en.wikipedia.org/wiki/Stigler%20Commission
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Formally known as the Price Statistics Review Committee, the Stigler Commission was convened in 1961 to study the measurement of inflation in the United States. Headed by economist George Stigler, its mandate was to conduct research into all types of price indices, including the Consumer Price Index (CPI). Based on its recommendations, the Bureau of Labor Statistics established a price research division.
The next major commission like it was the Boskin Commission in 1996, which was solely focused on evaluating the CPI.
Economy of the United States
1961 establishments in the United States
1961 in economics
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https://en.wikipedia.org/wiki/Chain%20ganging
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Chain ganging is a term in the field of international relations describing the elevated probability for interstate conflict or conflagration due to several states having joined in alliances or coalitions.
The agreed principles of such alliances typically include mutual defence clauses requiring that, in the case of one member state suffering military attack from another power, all members must declare hostilities against that offending power. The result of such an arrangement is an elevated probability for an international conflagration, since the case of an actor attacking another power would almost certainly trigger, whether intentionally or not, a multinational conflict potentially involving many more actors than the original two states which had attacked and been attacked, respectively.
According to sworn agreements or treaties no member state has the option to refuse to participate in this involvement: once the states have agreed to the alliance, they are bound by obligation to join in the hostilities or conflagration as soon as they have begun in one state (though this obligation is not always honoured).
Chain ganging is believed to be most effective in a scenario with a multipolar balance of power where the allied states have a military advantage over potential enemies. Chain ganging is also often discussed in relation and comparison to its counterpart Buck passing. Both are derived from Neorealism and are existent in a multipolar system but chain ganging requires aggressive state behaviour to occur.
Examples
An empirical example of the chain-ganging dilemma is World War I. When Italy decided to part from the Triple Alliance, both Austria-Hungary and Germany were alone for the most part in Europe (though Bulgaria and the Ottoman Empire joined Austria-Hungary and Germany to form the Central Powers), and surrounded by the Allied Powers. The defeat of either of the two would severely weaken the remaining member. According to Kenneth Waltz, "The defeat or defection of a major ally would have shaken the balance, each state was constrained to adjust its strategy and the use of its forces to the aims and fears of its partners."
Due to conflict between China and the United States, both sides have been engaging in chaining themselves to far eastern states prompting some to declare the region likely play host to the chain gang dilemma should conflict escalate in that region with the Philippines to claim an attack on America is an attack on the Philippines and vice versa during the Spratly Islands dispute.
Etymology
The term is a metaphor deriving from chain gangs, groups of people, usually prisoners or slaves, bound together with chains or other devices as they work or march. Like a real-life chain gang, the states joined in a chain gang, according to bound obligation, have no option to refuse to follow along with the intent of the others. However, in reality, the members of a chain gang coalition can and sometimes do choose to refuse to acqui
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https://en.wikipedia.org/wiki/Radial%20basis%20function
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In mathematics a radial basis function (RBF) is a real-valued function whose value depends only on the distance between the input and some fixed point, either the origin, so that , or some other fixed point , called a center, so that . Any function that satisfies the property is a radial function. The distance is usually Euclidean distance, although other metrics are sometimes used. They are often used as a collection which forms a basis for some function space of interest, hence the name.
Sums of radial basis functions are typically used to approximate given functions. This approximation process can also be interpreted as a simple kind of neural network; this was the context in which they were originally applied to machine learning, in work by David Broomhead and David Lowe in 1988, which stemmed from Michael J. D. Powell's seminal research from 1977.
RBFs are also used as a kernel in support vector classification. The technique has proven effective and flexible enough that radial basis functions are now applied in a variety of engineering applications.
Definition
A radial function is a function . When paired with a metric on a vector space a function is said to be a radial kernel centered at . A Radial function and the associated radial kernels are said to be radial basis functions if, for any set of nodes
Examples
Commonly used types of radial basis functions include (writing and using to indicate a shape parameter that can be used to scale the input of the radial kernel):
Approximation
Radial basis functions are typically used to build up function approximations of the form
where the approximating function is represented as a sum of radial basis functions, each associated with a different center , and weighted by an appropriate coefficient The weights can be estimated using the matrix methods of linear least squares, because the approximating function is linear in the weights .
Approximation schemes of this kind have been particularly used in time series prediction and control of nonlinear systems exhibiting sufficiently simple chaotic behaviour and 3D reconstruction in computer graphics (for example, hierarchical RBF and Pose Space Deformation).
RBF Network
The sum
can also be interpreted as a rather simple single-layer type of artificial neural network called a radial basis function network, with the radial basis functions taking on the role of the activation functions of the network. It can be shown that any continuous function on a compact interval can in principle be interpolated with arbitrary accuracy by a sum of this form, if a sufficiently large number of radial basis functions is used.
The approximant is differentiable with respect to the weights . The weights could thus be learned using any of the standard iterative methods for neural networks.
Using radial basis functions in this manner yields a reasonable interpolation approach provided that the fitting set has been chosen such that it covers the ent
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https://en.wikipedia.org/wiki/Gauss%27s%20lemma%20%28polynomials%29
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In algebra, Gauss's lemma, named after Carl Friedrich Gauss, is a statement about polynomials over the integers, or, more generally, over a unique factorization domain (that is, a ring that has a unique factorization property similar to the fundamental theorem of arithmetic). Gauss's lemma underlies all the theory of factorization and greatest common divisors of such polynomials.
Gauss's lemma asserts that the product of two primitive polynomials is primitive (a polynomial with integer coefficients is primitive if it has 1 as a greatest common divisor of its coefficients).
A corollary of Gauss's lemma, sometimes also called Gauss's lemma, is that a primitive polynomial is irreducible over the integers if and only if it is irreducible over the rational numbers. More generally, a primitive polynomial has the same complete factorization over the integers and over the rational numbers. In the case of coefficients in a unique factorization domain , "rational numbers" must be replaced by "field of fractions of ". This implies that, if is either a field, the ring of integers, or a unique factorization domain, then every polynomial ring (in one or several indeterminates) over is a unique factorization domain. Another consequence is that factorization and greatest common divisor computation of polynomials with integers or rational coefficients may be reduced to similar computations on integers and primitive polynomials. This is systematically used (explicitly or implicitly) in all implemented algorithms (see Polynomial greatest common divisor and Factorization of polynomials).
Gauss's lemma, and all its consequences that do not involve the existence of a complete factorization remain true over any GCD domain (an integral domain over which greatest common divisors exist). In particular, a polynomial ring over a GCD domain is also a GCD domain. If one calls primitive a polynomial such that the coefficients generate the unit ideal, Gauss's lemma is true over every commutative ring. However, some care must be taken when using this definition of primitive, as, over a unique factorization domain that is not a principal ideal domain, there are polynomials that are primitive in the above sense and not primitive in this new sense.
The lemma over the integers
If is a polynomial with integer coefficients, then is called primitive if the greatest common divisor of all the coefficients is 1; in other words, no prime number divides all the coefficients.
Proof: Clearly the product f(x)g(x) of two primitive polynomials has integer coefficients. Therefore, if it is not primitive, there must be a prime p which is a common divisor of all its coefficients. But p can not divide all the coefficients of either f(x) or g(x) (otherwise they would not be primitive). Let arxr be the first term of f(x) not divisible by p and let bsxs be the first term of g(x) not divisible by p. Now consider the term xr+s in the product, whose coefficient is
The term arbs is not di
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https://en.wikipedia.org/wiki/Gauss%27s%20lemma%20%28number%20theory%29
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Gauss's lemma in number theory gives a condition for an integer to be a quadratic residue. Although it is not useful computationally, it has theoretical significance, being involved in some proofs of quadratic reciprocity.
It made its first appearance in Carl Friedrich Gauss's third proof (1808) of quadratic reciprocity and he proved it again in his fifth proof (1818).
Statement of the lemma
For any odd prime let be an integer that is coprime to .
Consider the integers
and their least positive residues modulo . These residues are all distinct, so there are ( of them.
Let be the number of these residues that are greater than . Then
where is the Legendre symbol.
Example
Taking = 11 and = 7, the relevant sequence of integers is
7, 14, 21, 28, 35.
After reduction modulo 11, this sequence becomes
7, 3, 10, 6, 2.
Three of these integers are larger than 11/2 (namely 6, 7 and 10), so = 3. Correspondingly Gauss's lemma predicts that
This is indeed correct, because 7 is not a quadratic residue modulo 11.
The above sequence of residues
7, 3, 10, 6, 2
may also be written
−4, 3, −1, −5, 2.
In this form, the integers larger than 11/2 appear as negative numbers. It is also apparent that the absolute values of the residues are a permutation of the residues
1, 2, 3, 4, 5.
Proof
A fairly simple proof, reminiscent of one of the simplest proofs of Fermat's little theorem, can be obtained by evaluating the product
modulo p in two different ways. On one hand it is equal to
The second evaluation takes more work. If is a nonzero residue modulo , let us define the "absolute value" of to be
Since counts those multiples which are in the latter range, and since for those multiples, is in the first range, we have
Now observe that the values are distinct for . Indeed, we have
because is coprime to .
This gives = , since and are positive least residues. But there are exactly of them, so their values are a rearrangement of the integers . Therefore,
Comparing with our first evaluation, we may cancel out the nonzero factor
and we are left with
This is the desired result, because by Euler's criterion the left hand side is just an alternative expression for the Legendre symbol .
Generalization
For any odd prime let be an integer that is coprime to .
Let be a set such that is the disjoint union of and .
Then , where .
In the original statement, .
The proof is almost the same.
Applications
Gauss's lemma is used in many, but by no means all, of the known proofs of quadratic reciprocity.
For example, Gotthold Eisenstein used Gauss's lemma to prove that if is an odd prime then
and used this formula to prove quadratic reciprocity. By using elliptic rather than circular functions, he proved the cubic and quartic reciprocity laws.
Leopold Kronecker used the lemma to show that
Switching and immediately gives quadratic reciprocity.
It is also used in what are probably the simplest proofs of the "second supplementary
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https://en.wikipedia.org/wiki/Richard%20Turner%20%28computer%20scientist%29
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Richard Turner (born 1954) is a distinguished service professor in the School of Systems and Enterprises of Stevens Institute of Technology in Hoboken, New Jersey.
Turner has a BA in mathematics from Huntingdon College, an MS in computer science from the University of Louisiana-Lafayette, and a DSc in engineering management from the George Washington University.
Before joining Stevens, he was a Fellow of the Systems and Software Consortium Inc., a Research Professor at The George Washington University, a computer scientist at the Federal Aviation Administration, and technical manager and practitioner with various DC area businesses working with defense, intelligence, and commercial clients. He has also served as a visiting scientist at the Software Engineering Institute of Carnegie Mellon University, and consulted independently.
Much of his research at Stevens has been through the Systems Engineering Research Center (SERC) supporting the U.S. Department of Defense, particularly on the integration of systems and software engineering and the acquisition of complex defense systems. He was on the original author team of the CMMI and a core author of the Software Extension to the Guide to the Project Management Body of Knowledge PMI and IEEE Computer Society.
He is a Senior Member of the IEEE, a Golden Core Awardee of the IEEE Computer Society, and a Fellow of the Lean Systems Society.
He has authored / co authored several books:-
The Incremental Commitment Spiral Model: Principles and Practices for Successful Systems and Software, by Barry Boehm, Jo Ann Lane, Supannika Koolmanojwong, and Richard Turner: Addison-Wesley, (2014).
CMMI (Capability Maturity Model Integration) Distilled: A Practical Introduction to Integrated Process Improvement, by Dennis M. Ahern, Aaron Clouse, Richard Turner: Addison-Wesley, (Third Edition 2008).
CMMI Survival Guide: Just enough process improvement, by Suzanne Garcia, Richard Turner: Addison-Wesley, (2007).
Balancing Agility and Discipline: A Guide for the Perplexed by Barry Boehm, Richard Turner: Addison-Wesley, (Paperback - September 26, 2003)
An interactive simulator for MATHILDA-RIKKE on multics: Concept, design and implementation by Richard Turner, Publisher: Computer Science Dept., University of Southwestern, Louisiana (1977)
Turner lives in the District of Columbia with his wife, Johanna - they have three grown children and two grandchildren.
Notes
References
External links
http://www.sercuarc.org
http://leansystemssociety.org
1954 births
Living people
Huntingdon College alumni
Software engineering researchers
Stevens Institute of Technology faculty
American male writers
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https://en.wikipedia.org/wiki/Religion%20in%20Albania
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The most common religion in Albania is Islam, with the second-most-common religion being Christianity. There are also a number of irreligious Albanians. There are no official statistics regarding the number of practicing religious people per each religious group.
Albania has been a secular state since 1912, and as such, is "neutral in questions of belief and conscience": The former socialist government declared Albania the world's first "atheist state", even though the Soviet Union had already done so. Believers faced harsh punishments, and many clergymen were killed. Religious observance and practice is generally lax today, and polls have shown that, compared to the populations of other countries, few Albanians consider religion to be a dominant factor in their lives. When asked about religion, people generally refer to their family's historical religious legacy and not to their own choice of faith.
The 2011 census on religion and ethnicity has been deemed unreliable by the Council of Europe, as well as other internal and external organisations and groups.
History
Antiquity
Christianity spread to urban centers in the region of Albania, at the time composed mostly Epirus Nova and part of south Illyricum, during the later period of Roman era and reached the region relatively early. St. Paul preached the Gospel 'even unto Illyricum' (Romans 15:19). Schnabel asserts that Paul probably preached in Shkodra and Durrës. The steady growth of the Christian community in Dyrrhachium (the Roman name for Epidamnus) led to the creation of a local bishopric in 58 AD. Later, episcopal seats were established in Apollonia, Buthrotum (modern Butrint), and Scodra (modern Shkodra).
One notable Martyr was Saint Astius, who was Bishop of Dyrrachium, who was crucified during the persecution of Christians by the Roman Emperor Trajan. Saint Eleutherius (not to be confused with the later Saint-Pope) was bishop of Messina and Illyria. He was martyred along with his mother Anthia during the anti-Christian campaign of Hadrian.
From the 2nd to the 4th centuries, the main language used to spread the Christian religion was Latin, whereas in the 4th to the 5th centuries it was Greek in Epirus and Macedonia and Latin in Praevalitana and Dardania. Christianity spread to the region during the 4th century, however the Bible cites in Romans that Christianity was spread in the first century. The following centuries saw the erection of characteristic examples of Byzantine architecture such as the churches in Kosine, Mborje and Apollonia.
Christian bishops from what would later become eastern Albania took part in the First Council of Nicaea. Arianism had at that point extended to Illyria, where Arius himself had been exiled to by Constantine.
Middle Ages
Since the early 4th century AD, Christianity had become the established religion in the Roman Empire, supplanting pagan polytheism and eclipsing for the most part the humanistic world outlook and institutions inherited from
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https://en.wikipedia.org/wiki/LF-space
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In mathematics, an LF-space, also written (LF)-space, is a topological vector space (TVS) X that is a locally convex inductive limit of a countable inductive system of Fréchet spaces.
This means that X is a direct limit of a direct system in the category of locally convex topological vector spaces and each is a Fréchet space. The name LF stands for Limit of Fréchet spaces.
If each of the bonding maps is an embedding of TVSs then the LF-space is called a strict LF-space. This means that the subspace topology induced on by is identical to the original topology on .
Some authors (e.g. Schaefer) define the term "LF-space" to mean "strict LF-space," so when reading mathematical literature, it is recommended to always check how LF-space is defined.
Definition
Inductive/final/direct limit topology
Throughout, it is assumed that
is either the category of topological spaces or some subcategory of the category of topological vector spaces (TVSs);
If all objects in the category have an algebraic structure, then all morphisms are assumed to be homomorphisms for that algebraic structure.
is a non-empty directed set;
is a family of objects in where is a topological space for every index ;
To avoid potential confusion, should not be called 's "initial topology" since the term "initial topology" already has a well-known definition. The topology is called the original topology on or 's given topology.
is a set (and if objects in also have algebraic structures, then is automatically assumed to have has whatever algebraic structure is needed);
is a family of maps where for each index , the map has prototype . If all objects in the category have an algebraic structure, then these maps are also assumed to be homomorphisms for that algebraic structure.
If it exists, then the final topology on in , also called the colimit or inductive topology in , and denoted by or , is the finest topology on such that
is an object in , and
for every index , the map is a continuous morphism in .
In the category of topological spaces, the final topology always exists and moreover, a subset is open (resp. closed) in if and only if is open (resp. closed) in for every index .
However, the final topology may not exist in the category of Hausdorff topological spaces due to the requirement that belong to the original category (i.e. belong to the category of Hausdorff topological spaces).
Direct systems
Suppose that is a directed set and that for all indices there are (continuous) morphisms in
such that if then is the identity map on and if then the following compatibility condition is satisfied:
where this means that the composition
If the above conditions are satisfied then the triple formed by the collections of these objects, morphisms, and the indexing set
is known as a direct system in the category that is directed (or indexed) by . Since the indexing set is a directed set, the direct system is said to be dir
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https://en.wikipedia.org/wiki/Hosohedron
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In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices.
A regular -gonal hosohedron has Schläfli symbol with each spherical lune having internal angle radians ( degrees).
Hosohedra as regular polyhedra
For a regular polyhedron whose Schläfli symbol is {m, n}, the number of polygonal faces is :
The Platonic solids known to antiquity are the only integer solutions for m ≥ 3 and n ≥ 3. The restriction m ≥ 3 enforces that the polygonal faces must have at least three sides.
When considering polyhedra as a spherical tiling, this restriction may be relaxed, since digons (2-gons) can be represented as spherical lunes, having non-zero area.
Allowing m = 2 makes
and admits a new infinite class of regular polyhedra, which are the hosohedra. On a spherical surface, the polyhedron {2, n} is represented as n abutting lunes, with interior angles of . All these spherical lunes share two common vertices.
Kaleidoscopic symmetry
The digonal spherical lune faces of a -hosohedron, , represent the fundamental domains of dihedral symmetry in three dimensions: the cyclic symmetry , , , order . The reflection domains can be shown by alternately colored lunes as mirror images.
Bisecting each lune into two spherical triangles creates an -gonal bipyramid, which represents the dihedral symmetry , order .
Relationship with the Steinmetz solid
The tetragonal hosohedron is topologically equivalent to the bicylinder Steinmetz solid, the intersection of two cylinders at right-angles.
Derivative polyhedra
The dual of the n-gonal hosohedron {2, n} is the n-gonal dihedron, {n, 2}. The polyhedron {2,2} is self-dual, and is both a hosohedron and a dihedron.
A hosohedron may be modified in the same manner as the other polyhedra to produce a truncated variation. The truncated n-gonal hosohedron is the n-gonal prism.
Apeirogonal hosohedron
In the limit, the hosohedron becomes an apeirogonal hosohedron as a 2-dimensional tessellation:
Hosotopes
Multidimensional analogues in general are called hosotopes. A regular hosotope with Schläfli symbol {2,p,...,q} has two vertices, each with a vertex figure {p,...,q}.
The two-dimensional hosotope, {2}, is a digon.
Etymology
The term “hosohedron” appears to derive from the Greek ὅσος (hosos) “as many”, the idea being that a hosohedron can have “as many faces as desired”. It was introduced by Vito Caravelli in the eighteenth century.
See also
Polyhedron
Polytope
References
Coxeter, H.S.M, Regular Polytopes (third edition), Dover Publications Inc.,
External links
Polyhedra
Tessellation
Regular polyhedra
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https://en.wikipedia.org/wiki/Simplex%20%28disambiguation%29
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Simplex may refer to:
Mathematics
Simplex, a term in geometry meaning an n-dimensional analogue of a triangle
Pascal's simplex, a version of Pascal's triangle of more than three dimensions
Simplex algorithm, a popular algorithm for numerical solution of linear programming problems
Simplex graph, derived from the cliques of another graph
Simplex noise, a method for constructing an n-dimensional noise function
Simplex plot, a ternary plot used in game theory
Companies and trade names
Simplex (bicycle), a French bicycle derailleur brand
Simplex Manufacturing Corporation, an American manufacturer of motorcycles in Louisiana from 1935 to 1975
American Simplex, an American automobile made in Mishawaka, Indiana, US
Simplex Automobile Company was a defunct luxury car manufacturer from 1907 to 1921.
Crane-Simplex, a defunct luxury car manufacturer in New York, US at the start of the 20th century
Sheffield-Simplex, a British vehicle manufacturer operating 1907–1920
Simplex Typewriter Company, an index typewriter manufacturer
A trade name used by British railway locomotive manufacturers The Motor Rail & Tramcar Co Ltd
The trade name of the swinging-tray record changer used in Wurlitzer jukeboxes from the 1930s to the 1950s
A brand of fire alarm systems made by SimplexGrinnell
A manufacturer of jacks used in railroads and mining industries, now owned by Actuant Corporation
Technology
Bergmann Simplex, an early 20th-century, German-made handgun
Simplex communication, a one-way communications channel
Simplex signaling, signalling in which two conductors are used for a single channel
Simplex, using a single frequency for transmit and receive instead of an amateur radio repeater
SimpleX Messaging Protocol, a privacy focused messaging protocol
Small Innovative Missions for Planetary Exploration (SIMPLEx) program at NASA
Other uses
List of species named simplex, a common species name
Mercedes Simplex, an automobile model produced between 1902 and 1909
Herpes simplex, a viral disease caused by Herpes simplex viruses
Simplex printing, printing one sided pages, a technique that is contrast to duplex printing
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https://en.wikipedia.org/wiki/Classification%20of%20electromagnetic%20fields
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In differential geometry and theoretical physics, the classification of electromagnetic fields is a pointwise classification of bivectors at each point of a Lorentzian manifold. It is used in the study of solutions of Maxwell's equations and has applications in Einstein's theory of relativity.
The classification theorem
The electromagnetic field at a point p (i.e. an event) of a Lorentzian spacetime is represented by a real bivector defined over the tangent space at p.
The tangent space at p is isometric as a real inner product space to E1,3. That is, it has the same notion of vector magnitude and angle as Minkowski spacetime. To simplify the notation, we will assume the spacetime is Minkowski spacetime. This tends to blur the distinction between the tangent space at p and the underlying manifold; fortunately, nothing is lost by this specialization, for reasons we discuss as the end of the article.
The classification theorem for electromagnetic fields characterizes the bivector F in relation to the Lorentzian metric by defining and examining the so-called "principal null directions". Let us explain this.
The bivector Fab yields a skew-symmetric linear operator defined by lowering one index with the metric. It acts on the tangent space at p by . We will use the symbol F to denote either the bivector or the operator, according to context.
We mention a dichotomy drawn from exterior algebra. A bivector that can be written as , where v, w are linearly independent, is called simple. Any nonzero bivector over a 4-dimensional vector space either is simple, or can be written as , where v, w, x, and y are linearly independent; the two cases are mutually exclusive. Stated like this, the dichotomy makes no reference to the metric η, only to exterior algebra. But it is easily seen that the associated skew-symmetric linear operator Fab has rank 2 in the former case and rank 4 in the latter case.
To state the classification theorem, we consider the eigenvalue problem for F, that is, the problem of finding eigenvalues λ and eigenvectors r which satisfy the eigenvalue equation
The skew-symmetry of F implies that:
either the eigenvector r is a null vector (i.e. ), or the eigenvalue λ is zero, or both.
A 1-dimensional subspace generated by a null eigenvector is called a principal null direction of the bivector.
The classification theorem characterizes the possible principal null directions of a bivector. It states that one of the following must hold for any nonzero bivector:
the bivector has one "repeated" principal null direction; in this case, the bivector itself is said to be null,
the bivector has two distinct principal null directions; in this case, the bivector is called non-null.
Furthermore, for any non-null bivector, the two eigenvalues associated with the two distinct principal null directions have the same magnitude but opposite sign, , so we have three subclasses of non-null bivectors:
spacelike: ν = 0
timelike : ν ≠ 0 and
non-simp
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https://en.wikipedia.org/wiki/Euler%20function
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In mathematics, the Euler function is given by
Named after Leonhard Euler, it is a model example of a q-series and provides the prototypical example of a relation between combinatorics and complex analysis.
Properties
The coefficient in the formal power series expansion for gives the number of partitions of k. That is,
where is the partition function.
The Euler identity, also known as the Pentagonal number theorem, is
is a pentagonal number.
The Euler function is related to the Dedekind eta function as
The Euler function may be expressed as a q-Pochhammer symbol:
The logarithm of the Euler function is the sum of the logarithms in the product expression, each of which may be expanded about q = 0, yielding
which is a Lambert series with coefficients -1/n. The logarithm of the Euler function may therefore be expressed as
where -[1/1, 3/2, 4/3, 7/4, 6/5, 12/6, 8/7, 15/8, 13/9, 18/10, ...] (see OEIS A000203)
On account of the identity , where is the sum-of-divisors function, this may also be written as
.
Also if and , then
Special values
The next identities come from Ramanujan's Notebooks:
Using the Pentagonal number theorem, exchanging sum and integral, and then invoking complex-analytic methods, one derives
References
Number theory
Q-analogs
Leonhard Euler
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https://en.wikipedia.org/wiki/Anosov%20diffeomorphism
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In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold M is a certain type of mapping, from M to itself, with rather clearly marked local directions of "expansion" and "contraction". Anosov systems are a special case of Axiom A systems.
Anosov diffeomorphisms were introduced by Dmitri Victorovich Anosov, who proved that their behaviour was in an appropriate sense generic (when they exist at all).
Overview
Three closely related definitions must be distinguished:
If a differentiable map f on M has a hyperbolic structure on the tangent bundle, then it is called an Anosov map. Examples include the Bernoulli map, and Arnold's cat map.
If the map is a diffeomorphism, then it is called an Anosov diffeomorphism.
If a flow on a manifold splits the tangent bundle into three invariant subbundles, with one subbundle that is exponentially contracting, and one that is exponentially expanding, and a third, non-expanding, non-contracting one-dimensional sub-bundle (spanned by the flow direction), then the flow is called an Anosov flow.
A classical example of Anosov diffeomorphism is the Arnold's cat map.
Anosov proved that Anosov diffeomorphisms are structurally stable and form an open subset of mappings (flows) with the C1 topology.
Not every manifold admits an Anosov diffeomorphism; for example, there are no such diffeomorphisms on the sphere . The simplest examples of compact manifolds admitting them are the tori: they admit the so-called linear Anosov diffeomorphisms, which are isomorphisms having no eigenvalue of modulus 1. It was proved that any other Anosov diffeomorphism on a torus is topologically conjugate to one of this kind.
The problem of classifying manifolds that admit Anosov diffeomorphisms turned out to be very difficult, and still has no answer for dimension over 3. The only known examples are infranilmanifolds, and it is conjectured that they are the only ones.
A sufficient condition for transitivity is that all points are nonwandering: .
Also, it is unknown if every volume-preserving Anosov diffeomorphism is ergodic. Anosov proved it under a assumption. It is also true for volume-preserving Anosov diffeomorphisms.
For transitive Anosov diffeomorphism there exists a unique SRB measure (the acronym stands for Sinai, Ruelle and Bowen) supported on such that its basin is of full volume, where
Anosov flow on (tangent bundles of) Riemann surfaces
As an example, this section develops the case of the Anosov flow on the tangent bundle of a Riemann surface of negative curvature. This flow can be understood in terms of the flow on the tangent bundle of the Poincaré half-plane model of hyperbolic geometry. Riemann surfaces of negative curvature may be defined as Fuchsian models, that is, as the quotients of the upper half-plane and a Fuchsian group. For the following, let H be the upper half-plane; let Γ be a Fuchsian group; let M = H/Γ be a Riemann surface of negat
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https://en.wikipedia.org/wiki/Degrees%20of%20freedom%20%28statistics%29
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In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.
Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the estimate of a parameter is called the degrees of freedom. In general, the degrees of freedom of an estimate of a parameter are equal to the number of independent scores that go into the estimate minus the number of parameters used as intermediate steps in the estimation of the parameter itself. For example, if the variance is to be estimated from a random sample of independent scores, then the degrees of freedom is equal to the number of independent scores (N) minus the number of parameters estimated as intermediate steps (one, namely, the sample mean) and is therefore equal to .
Mathematically, degrees of freedom is the number of dimensions of the domain of a random vector, or essentially the number of "free" components (how many components need to be known before the vector is fully determined).
The term is most often used in the context of linear models (linear regression, analysis of variance), where certain random vectors are constrained to lie in linear subspaces, and the number of degrees of freedom is the dimension of the subspace. The degrees of freedom are also commonly associated with the squared lengths (or "sum of squares" of the coordinates) of such vectors, and the parameters of chi-squared and other distributions that arise in associated statistical testing problems.
While introductory textbooks may introduce degrees of freedom as distribution parameters or through hypothesis testing, it is the underlying geometry that defines degrees of freedom, and is critical to a proper understanding of the concept.
History
Although the basic concept of degrees of freedom was recognized as early as 1821 in the work of German astronomer and mathematician Carl Friedrich Gauss, its modern definition and usage was first elaborated by English statistician William Sealy Gosset in his 1908 Biometrika article "The Probable Error of a Mean", published under the pen name "Student". While Gosset did not actually use the term 'degrees of freedom', he explained the concept in the course of developing what became known as Student's t-distribution. The term itself was popularized by English statistician and biologist Ronald Fisher, beginning with his 1922 work on chi squares.
Notation
In equations, the typical symbol for degrees of freedom is ν (lowercase Greek letter nu). In text and tables, the abbreviation "d.f." is commonly used. R. A. Fisher used n to symbolize degrees of freedom but modern usage typically reserves n for sample size.
Of random vectors
Geometrically, the degrees of freedom can be interpreted as the dimension of certain vector subspaces. As a starting point, suppose that we have a sample of independent normally distributed observations,
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https://en.wikipedia.org/wiki/Sequent%20%28disambiguation%29
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A sequent is a formalized statement of provability used within sequent calculus.
Sequent may also refer to:
Sequent (MUD), text-based online game software
Sequent Computer Systems, a defunct computer hardware company
Sequent calculus
See also
Sequence (disambiguation)
Sequential
Sequentional
Sequention
Sequentor
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https://en.wikipedia.org/wiki/St%20Columb%27s%20College
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St Columb's College () is a Roman Catholic boys' grammar school in Derry, Northern Ireland. Since 2008, it has been a specialist school in mathematics. It is named after Saint Columba, the missionary monk from County Donegal who founded a monastery in the area. The college was originally built to educate young men into the priesthood, but now educates boys in a variety of disciplines.
St Columb's College was established in 1879 on Bishop Street (now the site of Lumen Christi College), but later moved to Buncrana Road in the suburbs of the city.
Early history
St Columb's College was preceded by several failed attempts to create such an institution in Derry. Repeated but sporadic efforts were made to maintain a seminary for almost a century; at Clady, near Strabane, in the late eighteenth century, at Ferguson's Lane in Derry in the early nineteenth century and at Pump Street (first reference to St Columb's College as such) in the city from 1841 to 1864.
St Columb's finally opened its doors on 3 November 1879 with two priest teachers, Edward O'Brien and John Hassan. The school was considered to be quite large at the time and was expected to accommodate 20–30 boarders. The school quickly gained a reputation for academic achievement. On 18 September 1931 the Derry Journal listed St Columb's College's academic results. They were as follows:
2 university scholarships
3 exhibitions and prizes
6 calls in King's Scholarship exam (calls to teacher training)
2 pupil teacherships
8 regional committee scholarships
31 passed matriculation
26 passed Senior Leaving Cert. exam
52 passed Junior Leaving Cert. exam
The Education Act 1947 and expansion
One of the most notable alumni of St Columb's College, John Hume, noted, "When the history of St. Columb's College in this century is written, it will be clear that one of its major transformations, if not its major transformation, took place as a result of the Eleven Plus examination."
The Education Act 1947 provided for free secondary education to all throughout the United Kingdom. Entry to St. Columb's College, a grammar school, would be determined by one's performance in the 11-plus or Transfer Test. The immediate result was an explosion in pupil numbers, a shortfall in teaching staff and greater pressure on existing resources. In 1941 the student body numbered 263. By 1960 the number stood at 770 with a teaching staff of 35. In under twenty years the school's size had tripled. It was now clear that additional facilities would be needed.
In September 1973 St. Columb's College opened a new campus on the Buncrana Road in the city. The new site would cater for the senior years; its initial enrolment was of 900. The new building was designed by Frank Corr of Corr & McCormick and constructed by J Kennedy & Co. The total cost was £762,000. This figure does not include the £56,000 spent employing W & J McMonagle Ltd to construct the playing fields.
Sport
The school has a long and successful sporting his
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https://en.wikipedia.org/wiki/Ramanujan%20theta%20function
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In mathematics, particularly -analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the Jacobi triple product takes on a particularly elegant form when written in terms of the Ramanujan theta. The function is named after mathematician Srinivasa Ramanujan.
Definition
The Ramanujan theta function is defined as
for . The Jacobi triple product identity then takes the form
Here, the expression denotes the -Pochhammer symbol. Identities that follow from this include
and
and
This last being the Euler function, which is closely related to the Dedekind eta function. The Jacobi theta function may be written in terms of the Ramanujan theta function as:
Integral representations
We have the following integral representation for the full two-parameter form of Ramanujan's theta function:
The special cases of Ramanujan's theta functions given by and also have the following integral representations:
This leads to several special case integrals for constants defined by these functions when (cf. theta function explicit values). In particular, we have that
and that
Application in string theory
The Ramanujan theta function is used to determine the critical dimensions in Bosonic string theory, superstring theory and M-theory.
References
Q-analogs
Elliptic functions
Theta functions
Srinivasa Ramanujan
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https://en.wikipedia.org/wiki/Rogers%E2%80%93Ramanujan%20identities
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In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered and proved by , and were subsequently rediscovered (without a proof) by Srinivasa Ramanujan some time before 1913. Ramanujan had no proof, but rediscovered Rogers's paper in 1917, and they then published a joint new proof . independently rediscovered and proved the identities.
Definition
The Rogers–Ramanujan identities are
and
.
Here, denotes the q-Pochhammer symbol.
Combinatorial interpretation
Consider the following:
is the generating function for partitions with exactly parts such that adjacent parts have difference at least 2.
is the generating function for partitions such that each part is congruent to either 1 or 4 modulo 5.
is the generating function for partitions with exactly parts such that adjacent parts have difference at least 2 and such that the smallest part is at least 2.
is the generating function for partitions such that each part is congruent to either 2 or 3 modulo 5.
The Rogers–Ramanujan identities could be now interpreted in the following way. Let be a non-negative integer.
The number of partitions of such that the adjacent parts differ by at least 2 is the same as the number of partitions of such that each part is congruent to either 1 or 4 modulo 5.
The number of partitions of such that the adjacent parts differ by at least 2 and such that the smallest part is at least 2 is the same as the number of partitions of such that each part is congruent to either 2 or 3 modulo 5.
Alternatively,
The number of partitions of such that with parts the smallest part is at least is the same as the number of partitions of such that each part is congruent to either 1 or 4 modulo 5.
The number of partitions of such that with parts the smallest part is at least is the same as the number of partitions of such that each part is congruent to either 2 or 3 modulo 5.
Application to partitions
Since the terms occurring in the identity are generating functions of certain partitions, the identities make statements about partitions (decompositions) of natural numbers. The number sequences resulting from the coefficients of the Maclaurin series of the Rogers-Ramanujan functions G and H are special partition number sequences of level 5:
The number sequence (OEIS code: A003114) represents the number of possibilities for the affected natural number n to decompose this number into summands of the patterns 4a + 1 or 4a + 4 with a ∈ ℕ₀. Thus gives the number of decays of an integer n in which adjacent parts of the partition differ by at least 2, equal to the number of decays in which each part is equal to 1 or 4 mod 5 is.
And the number sequence (OEIS code: A003106) analogously represents the number of possibilities for the affected natural number n to decompose this number into summands of the patterns 4a + 2 or 4a + 3 with a ∈ ℕ₀. Thus gives t
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https://en.wikipedia.org/wiki/Pseudo-Anosov%20map
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In mathematics, specifically in topology, a pseudo-Anosov map is a type of a diffeomorphism or homeomorphism of a surface. It is a generalization of a linear Anosov diffeomorphism of the torus. Its definition relies on the notion of a measured foliation introduced by William Thurston, who also coined the term "pseudo-Anosov diffeomorphism" when he proved his classification of diffeomorphisms of a surface.
Definition of a measured foliation
A measured foliation F on a closed surface S is a geometric structure on S which consists of a singular foliation and a measure in the transverse direction. In some neighborhood of a regular point of F, there is a "flow box" φ: U → R2 which sends the leaves of F to the horizontal lines in R2. If two such neighborhoods Ui and Uj overlap then there is a transition function φij defined on φj(Uj), with the standard property
which must have the form
for some constant c. This assures that along a simple curve, the variation in y-coordinate, measured locally in every chart, is a geometric quantity (i.e. independent of the chart) and permits the definition of a total variation along a simple closed curve on S. A finite number of singularities of F of the type of "p-pronged saddle", p≥3, are allowed. At such a singular point, the differentiable structure of the surface is modified to make the point into a conical point with the total angle πp. The notion of a diffeomorphism of S is redefined with respect to this modified differentiable structure. With some technical modifications, these definitions extend to the case of a surface with boundary.
Definition of a pseudo-Anosov map
A homeomorphism
of a closed surface S is called pseudo-Anosov if there exists a transverse pair of measured foliations on S, Fs (stable) and Fu (unstable), and a real number λ > 1 such that the foliations are preserved by f and their transverse measures are multiplied by 1/λ and λ. The number λ is called the stretch factor or dilatation of f.
Significance
Thurston constructed a compactification of the Teichmüller space T(S) of a surface S such that the action induced on T(S) by any diffeomorphism f of S extends to a homeomorphism of the Thurston compactification. The dynamics of this homeomorphism is the simplest when f is a pseudo-Anosov map: in this case, there are two fixed points on the Thurston boundary, one attracting and one repelling, and the homeomorphism behaves similarly to a hyperbolic automorphism of the Poincaré half-plane. A "generic" diffeomorphism of a surface of genus at least two is isotopic to a pseudo-Anosov diffeomorphism.
Generalization
Using the theory of train tracks, the notion of a pseudo-Anosov map has been extended to self-maps of graphs (on the topological side) and outer automorphisms of free groups (on the algebraic side). This leads to an analogue of Thurston classification for the case of automorphisms of free groups, developed by Bestvina and Handel.
References
A. Casson, S. Bleiler, "Automorph
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https://en.wikipedia.org/wiki/Dill%20Faulkes
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Martin C. "Dill" Faulkes (born 1944) is a British businessman.
Faulkes has a Special Mathematics degree from Hull University, a PhD in mathematics from Queen Elizabeth College, London and did postdoctoral work in general relativity.
He then left academia and went into software. He worked for the company Logica, then SPL, which was bought by Systems Designers. He then invested money in a variety of software companies and made a lot of money on the flotation of Triad and the private sale of SmartGroups.com.
Philantropy
He is chair the Dill Faulkes Educational Trust which has made donations to a number of scientific causes including the Faulkes Telescope Project.
Faulkes contributed towards the building of Cambridge University's Centre for Mathematical Sciences and has the Faulkes Gatehouse and Faulkes Institute for Geometry named after him.
Faulkes is a gliding enthusiast having been part of Hull University's Flying Squadron as a student and for a time subsidised "mini-lessons" for children in gliding via the Faulkes Flying Foundation.
His trust has also funded "Bell projects" including replacing the bells at Trinity Church (Manhattan).
Honours and awards
Faulkes has been made an honorary fellow by some UK academic institutions such as the E.A. Milne Centre for Astrophysics at Hull University. and in 2017 at Cardiff University. He subsequently returned a honorary doctorate awarded by the University of South Wales in protest at the university's decision to close its observational astronomy course.
He has had an asteroid (47144 Faulkes) named after him.
References
External links
Interview & biography
Faulkes Educational Trust
Faulkes Institute of Geometry
Faulkes Telescope
Faulkes Flying Foundation
1944 births
Living people
British philanthropists
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https://en.wikipedia.org/wiki/Chen%20prime
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In mathematics, a prime number p is called a Chen prime if p + 2 is either a prime or a product of two primes (also called a semiprime). The even number 2p + 2 therefore satisfies Chen's theorem.
The Chen primes are named after Chen Jingrun, who proved in 1966 that there are infinitely many such primes. This result would also follow from the truth of the twin prime conjecture as the lower member of a pair of twin primes is by definition a Chen prime.
The first few Chen primes are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 67, 71, 83, 89, 101, … .
The first few Chen primes that are not the lower member of a pair of twin primes are
2, 7, 13, 19, 23, 31, 37, 47, 53, 67, 83, 89, 109, 113, 127, ... .
The first few non-Chen primes are
43, 61, 73, 79, 97, 103, 151, 163, 173, 193, 223, 229, 241, … .
All of the supersingular primes are Chen primes.
Rudolf Ondrejka discovered the following 3 × 3 magic square of nine Chen primes:
, the largest known Chen prime is 2996863034895 × 21290000 − 1, with 388342 decimal digits.
The sum of the reciprocals of Chen primes converges.
Further results
Chen also proved the following generalization: For any even integer h, there exist infinitely many primes p such that p + h is either a prime or a semiprime.
Green and Tao showed that the Chen primes contain infinitely many arithmetic progressions of length 3. Binbin Zhou generalized this result by showing that the Chen primes contain arbitrarily long arithmetic progressions.
Notes
1.Chen primes were first described by Yuan, W. On the Representation of Large Even Integers as a Sum of a Product of at Most 3 Primes and a Product of at Most 4 Primes, Scienca Sinica 16, 157-176, 1973.
References
External links
The Prime Pages
Classes of prime numbers
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https://en.wikipedia.org/wiki/Hilbert%20modular%20form
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In mathematics, a Hilbert modular form is a generalization of modular forms to functions of two or more variables. It is a (complex) analytic function on the m-fold product of upper half-planes satisfying a certain kind of functional equation.
Definition
Let F be a totally real number field of degree m over the rational field. Let be the real embeddings of F. Through them we have a map
Let be the ring of integers of F. The group is called the full Hilbert modular group.
For every element , there is a group action of defined by
For
define:
A Hilbert modular form of weight is an analytic function on such that for every
Unlike the modular form case, no extra condition is needed for the cusps because of Koecher's principle.
History
These modular forms, for real quadratic fields, were first treated in the 1901 Göttingen University Habilitationssschrift of Otto Blumenthal. There he mentions that David Hilbert had considered them initially in work from 1893-4, which remained unpublished. Blumenthal's work was published in 1903. For this reason Hilbert modular forms are now often called Hilbert-Blumenthal modular forms.
The theory remained dormant for some decades; Erich Hecke appealed to it in his early work, but major interest in Hilbert modular forms awaited the development of complex manifold theory.
See also
Siegel modular form
Hilbert modular surface
References
Jan H. Bruinier: Hilbert modular forms and their applications.
Paul B. Garrett: Holomorphic Hilbert Modular Forms. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990.
Eberhard Freitag: Hilbert Modular Forms. Springer-Verlag.
Automorphic forms
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https://en.wikipedia.org/wiki/Particular%20values%20of%20the%20Riemann%20zeta%20function
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In mathematics, the Riemann zeta function is a function in complex analysis, which is also important in number theory. It is often denoted and is named after the mathematician Bernhard Riemann. When the argument is a real number greater than one, the zeta function satisfies the equation
It can therefore provide the sum of various convergent infinite series, such as Explicit or numerically efficient formulae exist for at integer arguments, all of which have real values, including this example. This article lists these formulae, together with tables of values. It also includes derivatives and some series composed of the zeta function at integer arguments.
The same equation in above also holds when is a complex number whose real part is greater than one, ensuring that the infinite sum still converges. The zeta function can then be extended to the whole of the complex plane by analytic continuation, except for a simple pole at . The complex derivative exists in this more general region, making the zeta function a meromorphic function. The above equation no longer applies for these extended values of , for which the corresponding summation would diverge. For example, the full zeta function exists at (and is therefore finite there), but the corresponding series would be whose partial sums would grow indefinitely large.
The zeta function values listed below include function values at the negative even numbers (, ), for which and which make up the so-called trivial zeros. The Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane. The successful characterisation of its non-trivial zeros in the wider plane is important in number theory, because of the Riemann hypothesis.
The Riemann zeta function at 0 and 1
At zero, one has
At 1 there is a pole, so ζ(1) is not finite but the left and right limits are:
Since it is a pole of first order, it has a complex residue
Positive integers
Even positive integers
For the even positive integers , one has the relationship to the Bernoulli numbers:
The computation of is known as the Basel problem. The value of is related to the Stefan–Boltzmann law and Wien approximation in physics. The first few values are given by:
Taking the limit , one obtains .
The relationship between zeta at the positive even integers and the Bernoulli numbers may be written as
where and are integers for all even . These are given by the integer sequences and , respectively, in OEIS. Some of these values are reproduced below:
If we let be the coefficient of as above,
then we find recursively,
This recurrence relation may be derived from that for the Bernoulli numbers.
Also, there is another recurrence:
which can be proved, using that
The values of the zeta function at non-negative even integers have the generating function:
Since
The formula also shows that for ,
Odd positive integers
The sum of the harmonic series
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https://en.wikipedia.org/wiki/Instituto%20Nacional%20de%20Matem%C3%A1tica%20Pura%20e%20Aplicada
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The Instituto Nacional de Matemática Pura e Aplicada (National Institute for Pure and Applied Mathematics) is widely considered to be the foremost research and educational institution of Brazil in the area of mathematics. It is located in the city of Rio de Janeiro, and was formerly known simply as Instituto de Matemática Pura e Aplicada (IMPA), whose abbreviation remains in use.
It is a research and education institution qualified as a Social Organization (SO) under the auspices of the Ministry of Science, Technology, Innovations and Communications (MCTIC) and the Ministry of Education (MEC) of Brazil. Currently located in the Jardim Botânico neighborhood (South Zone) of Rio de Janeiro, Brazil, IMPA was founded on October 15, 1952. It was the first research unit of the National Research Council (CNPq), a federal funding agency created a year earlier. Its logo is a stylized Möbius strip, reproducing a large sculpture of a Möbius strip on display within the IMPA headquarters.
Founded by Lélio Gama, Leopoldo Nachbin and Maurício Peixoto, IMPA's primary mission is to stimulate scientific research, the training of new researchers and the dissemination and improvement of mathematical culture in Brazil. Mathematical knowledge is fundamental for scientific and technological development, which are indispensable components for economic, social and human progress. Since 2015, IMPA is directed by Marcelo Viana.
History
At the time of creation, IMPA did not have its own headquarters: it was temporarily housed in a room in the headquarters of the Brazilian Center for Research in Physics (created in 1949), in Praia Vermelha, south zone of Rio de Janeiro. The scientific body was also diminutive, though illustrious: in addition to the director, astronomer Lélio Gama, who also headed the National Observatory, the institute counted only the young mathematicians Leopoldo Nachbin and Maurício Peixoto.
Gama's performance at the helm of IMPA, with his experience and wisdom, played a crucial role in the creation and consolidation of the young institute. And Nachbin and Peixoto would later be the first Brazilians invited to lecture at the International Congress of Mathematicians, one of the greatest distinctions in a mathematician's career. The academic prestige of IMPA grew from 1957, with the organization of the first Brazilian Colloquium of Mathematics, with about 50 participants. The Colloquium has been taking place every two years since then, in an uninterrupted fashion. Much of Brazilian mathematics was built around it. Also in 1957, IMPA moved to Rua São Clemente, in Botafogo, also in the South Zone of Rio.
In 1962, the master's and doctoral programs began in Mathematics, through an agreement signed with the Federal University of Rio de Janeiro (UFRJ), which officially awarded the titles of master and doctor. In 1967, IMPA moved again to a historic building on Rua Luiz de Camões, in the center of Rio de Janeiro, which currently houses the Hélio Oiticica Cultu
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https://en.wikipedia.org/wiki/Congruent%20number
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In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property.
The sequence of (integer) congruent numbers starts with
5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 69, 70, 71, 77, 78, 79, 80, 84, 85, 86, 87, 88, 92, 93, 94, 95, 96, 101, 102, 103, 109, 110, 111, 112, 116, 117, 118, 119, 120, ...
For example, 5 is a congruent number because it is the area of a (20/3, 3/2, 41/6) triangle. Similarly, 6 is a congruent number because it is the area of a (3,4,5) triangle. 3 and 4 are not congruent numbers.
If is a congruent number then is also a congruent number for any natural number (just by multiplying each side of the triangle by ), and vice versa. This leads to the observation that whether a nonzero rational number is a congruent number depends only on its residue in the group
,
where is the set of nonzero rational numbers.
Every residue class in this group contains exactly one square-free integer, and it is common, therefore, only to consider square-free positive integers, when speaking about congruent numbers.
Congruent number problem
The question of determining whether a given rational number is a congruent number is called the congruent number problem. This problem has not (as of 2019) been brought to a successful resolution. Tunnell's theorem provides an easily testable criterion for determining whether a number is congruent; but his result relies on the Birch and Swinnerton-Dyer conjecture, which is still unproven.
Fermat's right triangle theorem, named after Pierre de Fermat, states that no square number can be a congruent number. However, in the form that every congruum (the difference between consecutive elements in an arithmetic progression of three squares) is non-square, it was already known (without proof) to Fibonacci. Every congruum is a congruent number, and every congruent number is a product of a congruum and the square of a rational number. However, determining whether a number is a congruum is much easier than determining whether it is congruent, because there is a parameterized formula for congrua for which only finitely many parameter values need to be tested.
Solutions
n is a congruent number if and only if the system
,
has a solution where , and are integers.
Given a solution, the three numbers , , and will be in an arithmetic progression with common difference .
Furthermore, if there is one solution (where the right-hand sides are squares), then there are infinitely many: given any solution ,
another solution can be computed from
,
.
For example, with , the equations are:
,
.
One solution is (so that ). Another solution is
,
.
With this new and , the right-hand sides are still both squares:
.
Given , and , one can obtain , and such that
, and
from
, , .
The
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https://en.wikipedia.org/wiki/Highly%20optimized%20tolerance
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In applied mathematics, highly optimized tolerance (HOT) is a method of generating power law behavior in systems by including a global optimization principle. It was developed by Jean M. Carlson and John Doyle in the early 2000s. For some systems that display a characteristic scale, a global optimization term could potentially be added that would then yield power law behavior. It has been used to generate and describe internet-like graphs, forest fire models and may also apply to biological systems.
Example
The following is taken from Sornette's book.
Consider a random variable, , that takes on values with probability . Furthermore, let’s assume for another parameter
for some fixed . We then want to minimize
subject to the constraint
Using Lagrange multipliers, this gives
giving us a power law. The global optimization of minimizing the energy along with the power law dependence between and gives us a power law distribution in probability.
See also
self-organized criticality
References
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Mathematical optimization
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https://en.wikipedia.org/wiki/Hilbert%27s%20fifteenth%20problem
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Hilbert's fifteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. The problem is to put Schubert's enumerative calculus on a rigorous foundation.
Introduction
Schubert calculus is the intersection theory of the 19th century, together with applications to enumerative geometry. Justifying this calculus was the content of Hilbert's 15th problem, and was also the major topic of the 20 century algebraic geometry. In the course of securing the foundations of intersection theory, Van der Waerden and André Weil related the problem to the determination of the cohomology ring H*(G/P) of a flag manifold G/P, where G is a Lie group and P a
parabolic subgroup of G.
The additive structure of the ring H*(G/P) is given by the basis theorem of Schubert calculus due to Ehresmann, Chevalley, and Bernstein-Gel'fand-Gel'fand, stating that the classical Schubert classes on G/P form a free basis of
the cohomology ring H*(G/P). The remaining problem of expanding products of Schubert classes as linear combination of
basis elements was called the characteristic problem, by Schubert and regarded by him as "the main theoretic problem of enumerative geometry".
While enumerative geometry made no connection with physics during the first century of its development, it has since emerged as a central element of string theory.
Problem statement
The entirety of the original problem statement is as follows:
The problem consists in this: To establish rigorously and with an exact determination of the limits of their validity those geometrical numbers which Schubert especially has determined on the basis of the so-called principle of special position, or conservation of number, by means of the enumerative calculus developed by him.
Although the algebra of today guarantees, in principle, the possibility of carrying out the processes of elimination, yet for the proof of the theorems of enumerative geometry decidedly more is requisite, namely, the actual carrying out of the process of elimination in the case of equations of special form in such a way that the degree of the final equations and the multiplicity of their solutions may be foreseen.
Schubert calculus
Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). It was a precursor of several more modern theories, for example characteristic classes, and in particular its algorithmic aspects are still of current interest.
The objects introduced by Schubert are the Schubert cells, which are locally closed sets in a Grassmannian defined by conditions of incidence of a linear subspace in projective space with a given flag. For details see Schubert variety.
According to Van der Waerden and André Weil Hilbert problem fifteen has been solved. In particular,
a) Schubert’s characteristic problem has been solved by Haibao
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https://en.wikipedia.org/wiki/Hilbert%27s%20nineteenth%20problem
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Hilbert's nineteenth problem is one of the 23 Hilbert problems, set out in a list compiled in 1900 by David Hilbert. It asks whether the solutions of regular problems in the calculus of variations are always analytic. Informally, and perhaps less directly, since Hilbert's concept of a "regular variational problem" identifies precisely a variational problem whose Euler–Lagrange equation is an elliptic partial differential equation with analytic coefficients, Hilbert's nineteenth problem, despite its seemingly technical statement, simply asks whether, in this class of partial differential equations, any solution function inherits the relatively simple and well understood structure from the solved equation. Hilbert's nineteenth problem was solved independently in the late 1950s by Ennio De Giorgi and John Forbes Nash, Jr.
History
The origins of the problem
David Hilbert presented the now called Hilbert's nineteenth problem in his speech at the second International Congress of Mathematicians. In he states that, in his opinion, one of the most remarkable facts of the theory of analytic functions is that there exist classes of partial differential equations which admit only such kind of functions as solutions, adducing Laplace's equation, Liouville's equation, the minimal surface equation and a class of linear partial differential equations studied by Émile Picard as examples. He then notes the fact that most of the partial differential equations sharing this property are the Euler–Lagrange equation of a well defined kind of variational problem, featuring the following three properties:
,
,
is an analytic function of all its arguments and .
Hilbert calls this kind of variational problem a "regular variational problem": property means that such kind of variational problems are minimum problems, property is the ellipticity condition on the Euler–Lagrange equations associated to the given functional, while property is a simple regularity assumption the function . Having identified the class of problems to deal with, he then poses the following question:-"... does every Lagrangian partial differential equation of a regular variation problem have the property of admitting analytic integrals exclusively?" and asks further if this is the case even when the function is required to assume, as it happens for Dirichlet's problem on the potential function, boundary values which are continuous, but not analytic.
The path to the complete solution
Hilbert stated his nineteenth problem as a regularity problem for a class of elliptic partial differential equation with analytic coefficients, therefore the first efforts of the researchers who sought to solve it were directed to study the regularity of classical solutions for equations belonging to this class. For solutions Hilbert's problem was answered positively by in his thesis: he showed that solutions of nonlinear elliptic analytic equations in 2 variables are analytic. Bernstein's result was improved
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https://en.wikipedia.org/wiki/Eduard%20Jan%20Dijksterhuis
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Eduard Jan Dijksterhuis (28 October 1892, in Tilburg – 18 May 1965, in De Bilt) was a Dutch historian of science.
Career
Dijksterhuis studied mathematics at the University of Groningen from 1911 to 1918. His Ph.d. thesis was entitled "A Contribution to the Knowledge of the Flat Helicoid."
From 1916 to 1953 he was a professor and taught mathematics, physics and cosmography. He advocated changes in the way mathematics was taught to reinforce its formal characteristics. In 1950, he was appointed as a German member of the Royal Netherlands Academy of Arts and Sciences. In 1953, he was appointed to teach mathematics history and the nature of science at Utrecht University and in 1955 at Leiden University.
His first biography was on the life and work of Archimedes, published in Dutch in 1938. It was translated into English by C. Dikshoorn in 1956, published in Copenhagen by Munksgard. Princeton University Press republished it, with additional commentary, in 1987.
In 1943 he wrote on the life and times of Simon Stevin, again first in Dutch, which Dikshoorn translated for English publication in 1970.
Upon the completion of Huygens Collected Works in 1950, at the annual meeting of the Dutch Society of Sciences at Haarlem, Dijksterhuis spoke on the 60-year project. The text of his speech was published in Centaurus in March 1953, when he gave a "sketch of the position occupied by Huygens in the scientific life of the 17th century." To do so, he explained "the dual nature of science which is both cumulative and collective. The first characteristic entails that intellectual heights once reserved to the privileged few become in due time accessible to beginners; the second, that in the strict sense of the word no scientist works alone." He noted that Huygens "often preferred leaving his findings unpublished and restricted himself to communicating the results in his letters or in a work of much later date." On Huygens' contribution to timekeeping, Dijksterhuis wrote, "he contrived to make the wheelwork of the clock keep up the motion of the pendulum, which on the other hand communicates to it its own periodicity, not materially changed by the connection." He named the editors of the Omnia Opera: Bierens de Haan, Johannes Bosscha, Diederik Korteweg, Vollgraff, and A. A. Nijland.
In 1950 Dijkserhuis published De mechanisering van het wereldbeeld (Dutch language) which was positively reviewed in a French journal. In 1956 Springer books produced a German translation, and in 1961 The Mechanization of the World Picture appeared in English. "Dijksterhuis’s primary interest is in the conflict and evolution of ideas as Aristotelian conceptions rose to dominance and then were overthrown ... Within the limited framework that Dijksterhuis chose for his work he has produced a masterpiece of lasting value ... Dijksterhuis knows the contents of a tremendous number of philosophical and scientific works, understands the physical concepts involved, and is aware of their inte
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https://en.wikipedia.org/wiki/Hua%20Luogeng
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Hua Luogeng or Hua Loo-Keng (; 12 November 1910 – 12 June 1985) was a Chinese mathematician and politician famous for his important contributions to number theory and for his role as the leader of mathematics research and education in the People's Republic of China. He was largely responsible for identifying and nurturing the renowned mathematician Chen Jingrun who proved Chen's theorem, the best known result on the Goldbach conjecture. In addition, Hua's later work on mathematical optimization and operations research made an enormous impact on China's economy. He was elected a foreign associate of the US National Academy of Sciences in 1982. He was elected a member of the standing Committee of the first to sixth National people's Congress, Vice-Chairman of the sixth National Committee of the Chinese People's Political Consultative Conference (April 1985) and Vice-Chairman of the China Democratic League (1979). He joined the Communist Party of China in 1979.
Hua did not receive a formal university education. Although awarded several honorary PhDs, he never got a formal degree from any university. In fact, his formal education only consisted of six years of primary school and three years of secondary school. For that reason, Xiong Qinglai, after reading one of Hua's early papers, was amazed by Hua's mathematical talent, and in 1931 Xiong invited him to study mathematics at Tsinghua University.
Biography
Early years (1910–1936)
Hua Luogeng was born in Jintan, Jiangsu on 12 November, 1910. Hua's father was a small businessman. Hua met a capable math teacher in middle school who recognized his talent early and encouraged him to read advanced texts. After middle school, Hua enrolled in Chinese Vocational College in Shanghai, and there he distinguished himself by winning a national abacus competition. Although tuition fees at the college were low, living costs proved too high for his means, and Hua was forced to leave a term before graduating. After failing to find a job in Shanghai, Hua returned home in 1927 to help in his father's store. In 1929, Hua contracted typhoid fever and was in bed for half a year. The culmination of Hua's illness resulted in the partial paralysis of his left leg, which impeded his movement quite severely for the rest of his life.
After middle school, Hua continued to study mathematics independently with the few books he had, and studied the entire high school and early undergraduate math curriculum. By the time Hua returned to Jintan, he was already engaged in independent mathematics research, and his first publication Some Researches on the Theorem of Sturm, appeared in the December 1929 issue of the Shanghai periodical Science. In the following year Hua showed in a short note in the same journal that a certain 1926 paper claiming to have solved the quintic was fundamentally flawed. Hua's lucid analysis caught the eye of Prof. Xiong Qinglai at Tsinghua University in Beijing, and in 1931 Hua was invited, despite his lac
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https://en.wikipedia.org/wiki/Xiong%20Qinglai
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Xiong Qinglai, or Hiong King-Lai (, October 20, 1893 – February 3, 1969), courtesy name Dizhi (), was a Chinese mathematician from Yunnan. He was the first person to introduce modern mathematics into China, and served as an influential president of Yunnan University from 1937 through 1947. A Chinese stamp was issued in his honour.
Biography
Xiong was born in Xizhai village (nowadays named Qinglai village to honour him) of Mile County, Yunnan province. He was the son of Xiong Guodong (熊国栋), a government official in Zhaozhou.
In 1907, Xiong accompanied his father to Kunming and enrolled in the Yunnan Higher School for preparatory studies. After two years, he began studying English and French. In 1911, he entered the Yunnan Provincial Institute of Higher Learning.
At the age of sixteen, Xiong Qinglai followed his parents' instructions and married Jiang Juyuan (b. 1893). The couple had 5 children.
In 1913, Xiong was successfully selected and funded by Yunnan provincial government to study mining in Belgium. However, following German invasion of Belgium in 1914, Xiong embarked on a journey to Paris, France where he enrolled at Lycée St Louis, focusing on mathematics. After completing his studies at Lycée St Louis in 1915, Xiong began his undergraduate studies at the University of Grenoble later that year. He then moved to Paris, where he continued his studies in mathematics, analytical mechanics, physics, and astronomy at the Faculty of Science in Paris (Faculté des sciences de Paris). He further pursued his education at the University of Montpellier and the University of Marseille. In 1920, Xiong was awarded a Master of Science degree by the University of Montpellier.
In the spring of 1921, Xiong first returned to Kunming, China, where he took up teaching positions at the Kunming Yunnan Industrial School and the Yunnan Road School. Xiong then relocated to Nanjing in September 1921, upon acceptance of Kuo Ping-Wen' s invitation to establish a Department of Mathematics at the National Southeastern University (Later renamed National Central University and Nanjing University). During his tenure as professor of mathematics at the National Southeastern University, Xiong wrote more than ten textbooks on geometry, calculus, differential equations, mechanics, etc. It was the first endeavor in history to introduce modern mathematics in Chinese textbooks.
In the autumn of 1925, Xiong Qinglai briefly taught at Northwestern University for one semester before returning to Southeast University in the following spring. In the autumn of 1926, he received an invitation from Cao Yunxiang and Ye Qisun to join the mathematics department at Tsinghua University as a professor, teaching advanced courses in calculus, differential equations, and analytical functions. Xiong became the head of the Department of Mathematics in 1928 and later the Dean of Science in 1930, replacing Ye Qisun.
During this time, after reading Hua Luogeng's paper in the Shanghai Journal of S
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https://en.wikipedia.org/wiki/Matthew%20J.%20Holman
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Matthew J. Holman (born 1967) is a Smithsonian astrophysicist and lecturer at Harvard University. Holman studied at MIT, where he received his bachelor's degree in mathematics in 1989 and his PhD in planetary science in 1994. He was awarded the Newcomb Cleveland Prize in 1998.
From 25 January 2015 to 9 February 2021, he held the position of an interim director of IAU's Minor Planet Center (MPC), after former director Timothy B. Spahr had stepped down. Holman was followed by Matthew Payne as new director of the MPC.
He was a Salina Central High School (Kansas) classmate and fellow debate team member of Joe Miller, Alaskan Senate candidate. The main-belt asteroid 3666 Holman was named in his honour in 1999 ().
Discoveries
For the period between 1999 and 2000, Holman is credited by the MPC with the discovery and co-discovery of several trans-Neptunian objects such as and (see table) and has been an active observer of centaurs.
He was also part of a team that discovered numerous irregular moons:
Discovered moons of Neptune (full list):
Halimede – in 2002 with J.J. Kavelaars, T. Grav, W. Fraser and D. Milisavljevic
Sao – in 2002 with J.J. Kavelaars, T. Grav, W. Fraser, D. Milisavljevic
Laomedeia – in 2002, with J.J. Kavelaars, T. Grav, W. Fraser, D. Milisavljevic
Neso – in 2002, with B. Gladman et al.
Discovered moons of Uranus (full list):
Prospero – in 1999, with J.J. Kavelaars, B. Gladman, J.-M. Petit, H. Scholl
Setebos – in 1999, with J.J. Kavelaars, B. Gladman, J.-M. Petit, H. Scholl
Stephano – in 1999, with B. Gladman, J.J. Kavelaars, J.-M. Petit, H. Scholl
Trinculo – in 2001, with J.J. Kavelaars, D. Milisavljevic
Francisco – in 2001, with J.J. Kavelaars, D. Milisavljevic, T. Grav
Ferdinand – in 2001, with D. Milisavljevic, J.J. Kavelaars, T. Grav
Discovered moons of Saturn (full list):
Albiorix – in 2000, with T.B. Spahr
See also
List of minor planet discoverers
References
External links
Matthew J. Holman, homepage at Center for Astrophysics Harvard & Smithsonian
The Minor Planet Center Status Report Matthew Holman, 8 November 2015
1967 births
American astronomers
Discoverers of moons
Discoverers of minor planets
Harvard University faculty
Living people
Massachusetts Institute of Technology School of Science alumni
Planetary scientists
Harvard–Smithsonian Center for Astrophysics people
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https://en.wikipedia.org/wiki/XLA
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XLA may refer to:
XLA (singer) (born 1981), Canadian indie singer
XLA (Accelerated Linear Algebra), a domain-specific compiler for linear algebra that can accelerate TensorFlow models
.xla, a file format for Microsoft Excel add-ins
X-linked agammaglobulinemia, an immune deficiency
Xbox Live Avatar, a character representing a user of the Xbox video game consoles
Xin Los Angeles, a 2006 container ship registered in Hong Kong
Dow XLA elastic fiber, a marketing name for Lastol
X-stem Logic Alphabet
XLA, the ICAO three letter callsign of former airline XL Airways UK
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https://en.wikipedia.org/wiki/List%20of%20Arsenal%20F.C.%20records%20and%20statistics
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Arsenal Football Club is an English professional association football club based in Islington, London. The club was formed in Woolwich in 1886 as Dial Square before being renamed as Royal Arsenal, and then Woolwich Arsenal in 1893. In 1914, the club's name was shortened to Arsenal F.C. after moving to Highbury a year earlier. After spending their first four seasons solely participating in cup tournaments and friendlies, Arsenal became the first southern member admitted into the Football League in 1893. In spite of finishing fifth in the Second Division in 1919, the club was voted to rejoin the First Division at the expense of local rivals Tottenham Hotspur. Since that time, they have not fallen below the first tier of the English football league system and hold the record for the longest uninterrupted period in the top flight. The club remained in the Football League until 1992, when its First Division was superseded as English football's top level by the newly formed Premier League, of which they were an inaugural member.
The list encompasses the honours won by Arsenal at national, regional, county and friendly level, records set by the club, their managers and their players. The player records section itemises the club's leading goalscorers and those who have made most appearances in first-team competitions. It also records notable achievements by Arsenal players on the international stage, and the highest transfer fees paid and received by the club. Attendance records at Highbury, the Emirates Stadium, the club's home ground since 2006, and Wembley Stadium, their temporary home for UEFA Champions League games between 1998 and 1999, are also included.
Arsenal have won 13 top-flight titles, and hold the record for the most FA Cup wins, with 14. The club's record appearance maker is David O'Leary, who made 722 appearances between 1975 and 1993. Thierry Henry is Arsenal's record goalscorer, scoring 228 goals in total.
All figures are correct as of 26 December 2020.
Honours and achievements
Arsenal's first ever silverware was won as the Royal Arsenal in 1890. The Kent Junior Cup, won by Royal Arsenal's reserves, was the club's first trophy, while the first team's first trophy came three weeks later when they won the Kent Senior Cup. Their first national major honour came in 1930, when they won the FA Cup. The club enjoyed further success in the 1930s, winning another FA Cup and five Football League First Division titles. Arsenal won their first league and cup double in the 1970–71 season and twice repeated the feat, in 1997–98 and 2001–02, as well as winning a cup double of the FA Cup and League Cup in 1992–93. In 2003–04, Arsenal recorded an unbeaten top-flight league season, something achieved only once before by Preston North End in 1888–89, who only had to play 22 games. To mark the achievement, a special gold version of the Premier League trophy was commissioned and presented to the club the following season. Their most recent success cam
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https://en.wikipedia.org/wiki/Fernand%20Boden
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Fernand Boden (born 13 September 1943) is a politician from Luxembourg. He was a minister in the government of Luxembourg from 1979 to 2009.
Boden was born in Echternach. He studied Mathematics and Physics at the University of Liège, and between 1966 and 1978 he taught at Echternach grammar school. He served as deputy mayor of Echternach from 1970 to 1976 and was a member of the local council.
He was first elected to the Chamber of Deputies of Luxembourg from the Eastern Constituency as a member of the Christian Social People's Party in 1978; he was re-elected in 1979. He joined the government in 1979 as Minister of National Education and Youth and Minister of Tourism, holding those portfolios until 1989. In the latter year he was moved to the posts of Minister for Family and Solidarity and Minister of the Middle Classes and Tourism, and in 1994 he became Minister for the Civil Service. He served in the latter position until 26 January 1995, when he became Minister of Agriculture, Viticulture and Rural Development and Minister for the Middle Classes, Tourism, and Housing. He retained those portfolios for over 14 years, until being replaced in July 2009.
References
Government ministers of Luxembourg
Members of the Chamber of Deputies (Luxembourg) from Est
Christian Social People's Party politicians
Luxembourgian educators
University of Liège alumni
1943 births
Living people
People from Echternach
Ministers for Agriculture of Luxembourg
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https://en.wikipedia.org/wiki/Peter%20Wadhams
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Peter Wadhams ScD (born 14 May 1948), is emeritus professor of Ocean Physics, and Head of the Polar Ocean Physics Group
in the Department of Applied Mathematics and Theoretical Physics, University of Cambridge. He is best known for his work on sea ice.
Career
Wadhams has been the leader of 40 polar field expeditions.
Wadhams advocates for the use of climate engineering to mitigate climate change.
Attempting to estimate when the Arctic Ocean will be "ice-free", Wadhams in 2014 predicted that by 2020 "summer sea ice to disappear," Wadhams and several others have noted that climate model predictions have been overly conservative regarding sea ice decline.
In 2021, Wadhams is the Chairman of Science Committee for Extreme E.
Honours and awards
1977 W. S. Bruce Medal for his oceanographic investigations, especially in studying the behaviour of pack ice near Spitsbergen, the North Pole and off east Greenland.
1987 Polar Medal.
1990 Italgas Prize for Environmental Sciences
See also
Global warming controversy
List of climate scientists
Shutdown of thermohaline circulation
References
External links
Homepage, including bio
1948 births
Alumni of Churchill College, Cambridge
British physicists
Living people
Cambridge mathematicians
Fellows of Clare Hall, Cambridge
People of the Scott Polar Research Institute
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https://en.wikipedia.org/wiki/Jacobi%27s%20formula
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In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A.
If is a differentiable map from the real numbers to matrices, then
where is the trace of the matrix . (The latter equality only holds if A(t) is invertible.)
As a special case,
Equivalently, if stands for the differential of , the general formula is
The formula is named after the mathematician Carl Gustav Jacob Jacobi.
Derivation
Via Matrix Computation
We first prove a preliminary lemma:
Lemma. Let A and B be a pair of square matrices of the same dimension n. Then
Proof. The product AB of the pair of matrices has components
Replacing the matrix A by its transpose AT is equivalent to permuting the indices of its components:
The result follows by taking the trace of both sides:
Theorem. (Jacobi's formula) For any differentiable map A from the real numbers to n × n matrices,
Proof. Laplace's formula for the determinant of a matrix A can be stated as
Notice that the summation is performed over some arbitrary row i of the matrix.
The determinant of A can be considered to be a function of the elements of A:
so that, by the chain rule, its differential is
This summation is performed over all n×n elements of the matrix.
To find ∂F/∂Aij consider that on the right hand side of Laplace's formula, the index i can be chosen at will. (In order to optimize calculations: Any other choice would eventually yield the same result, but it could be much harder). In particular, it can be chosen to match the first index of ∂ / ∂Aij:
Thus, by the product rule,
Now, if an element of a matrix Aij and a cofactor adjT(A)ik of element Aik lie on the same row (or column), then the cofactor will not be a function of Aij, because the cofactor of Aik is expressed in terms of elements not in its own row (nor column). Thus,
so
All the elements of A are independent of each other, i.e.
where δ is the Kronecker delta, so
Therefore,
and applying the Lemma yields
Via Chain Rule
Lemma 1. , where is the differential of .
This equation means that the differential of , evaluated at the identity matrix, is equal to the trace. The differential is a linear operator that maps an n × n matrix to a real number.
Proof. Using the definition of a directional derivative together with one of its basic properties for differentiable functions, we have
is a polynomial in of order n. It is closely related to the characteristic polynomial of . The constant term () is 1, while the linear term in is .
Lemma 2. For an invertible matrix A, we have: .
Proof. Consider the following function of X:
We calculate the differential of and evaluate it at using Lemma 1, the equation above, and the chain rule:
Theorem. (Jacobi's formula)
Proof. If is invertible, by Lemma 2, with
using the equation relating the adjugate of to . Now, the formula holds for all matrices, since the set of invertible linear matrices is d
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https://en.wikipedia.org/wiki/Jeffrey%20H.%20Smith
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Jeffrey Henderson Smith is a former professor of mathematics at Purdue University in Lafayette, Indiana. He received his Ph.D. from the Massachusetts Institute of Technology in 1981, under the supervision of Daniel Kan, and was promoted to full professor at Purdue in 1999. His primary research interest is algebraic topology; his best-cited work consists of two papers in the Annals of Mathematics on "nilpotence and stable homotopy".
Publications
References
Year of birth missing (living people)
Living people
Topologists
Massachusetts Institute of Technology alumni
Purdue University faculty
20th-century American mathematicians
21st-century American mathematicians
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https://en.wikipedia.org/wiki/Adam%20Hope
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Adam Hope (8 January 1813 – 7 August 1882) was a Canadian businessman and senator. "Adam Hope was trained in mathematics, bookkeeping, and German, studies that were all useful for what his father anticipated would be his pursuits as a merchant." Born to a prosperous Scottish tenant farming family in Dirleton parish, Adam Hope worked as a clerk in a sawmill in Leith, the port of Edinburgh, from 1828 until emigrating to North America. He migrated from Scotland in 1834, settling in Upper Canada. Adam Hope sent his father sixty-six letters, vivid and detailed, which trace Hope's passage across the Atlantic and efforts to settle in Upper Canada. A Liberal, he was appointed to the Senate of Canada on 3 January 1877 on the recommendation of Alexander Mackenzie. He represented the senatorial division of Hamilton, Ontario until his death. In 2007, Adam Crerar edited the selected letters of Adam Hope, written to his father in Scotland between 1834 and 1845. This collection of letters totals approximately 200,000 words, and represents one of the single richest accounts of Upper Canada in the 1830s and 1840s, touching on telling aspects of colonial politics, religion, society, economics, and communications. This edited compilation was published as part of the Champlain Society's General Series.
References
1813 births
1882 deaths
Canadian senators from Ontario
Liberal Party of Canada senators
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https://en.wikipedia.org/wiki/Unit%20function
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In number theory, the unit function is a completely multiplicative function on the positive integers defined as:
It is called the unit function because it is the identity element for Dirichlet convolution.
It may be described as the "indicator function of 1" within the set of positive integers. It is also written as u(n) (not to be confused with μ(n), which generally denotes the Möbius function).
See also
Möbius inversion formula
Heaviside step function
Kronecker delta
References
Multiplicative functions
1 (number)
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https://en.wikipedia.org/wiki/Eisenstein%20integer
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In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form
where and are integers and
is a primitive (hence non-real) cube root of unity. The Eisenstein integers form a triangular lattice in the complex plane, in contrast with the Gaussian integers, which form a square lattice in the complex plane. The Eisenstein integers are a countably infinite set.
Properties
The Eisenstein integers form a commutative ring of algebraic integers in the algebraic number field – the third cyclotomic field. To see that the Eisenstein integers are algebraic integers note that each is a root of the monic polynomial
In particular, satisfies the equation
The product of two Eisenstein integers and is given explicitly by
The 2-norm of an Eisenstein integer is just its squared modulus, and is given by
which is clearly a positive ordinary (rational) integer.
Also, the complex conjugate of satisfies
The group of units in this ring is the cyclic group formed by the sixth roots of unity in the complex plane: , the Eisenstein integers of norm .
Euclidean domain
The ring of Eisenstein integers forms a Euclidean domain whose norm is given by the square modulus, as above:
A division algorithm, applied to any dividend and divisor , gives a quotient and a remainder smaller than the divisor, satisfying:
Here, , , , are all Eisenstein integers. This algorithm implies the Euclidean algorithm, which proves Euclid's lemma and the unique factorization of Eisenstein integers into Eisenstein primes.
One division algorithm is as follows. First perform the division in the field of complex numbers, and write the quotient in terms of :
for rational . Then obtain the Eisenstein integer quotient by rounding the rational coefficients to the nearest integer:
Here may denote any of the standard rounding-to-integer functions.
The reason this satisfies , while the analogous procedure fails for most other quadratic integer rings, is as follows. A fundamental domain for the ideal , acting by translations on the complex plane, is the 60°–120° rhombus with vertices , , , . Any Eisenstein integer lies inside one of the translates of this parallelogram, and the quotient is one of its vertices. The remainder is the square distance from to this vertex, but the maximum possible distance in our algorithm is only , so . (The size of could be slightly decreased by taking to be the closest corner.)
Eisenstein primes
If and are Eisenstein integers, we say that divides if there is some Eisenstein integer such that . A non-unit Eisenstein integer is said to be an Eisenstein prime if its only non-unit divisors are of the form , where is any of the six units. They are the corresponding concept to the Gaussian primes in the Gaussian integers.
There are two types of Eisenstein prime. First, an ordinary prime number (or rational prime)
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