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https://en.wikipedia.org/wiki/Continuous%20dual%20q-Hahn%20polynomials | In mathematics, the continuous dual q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.
Definition
The polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by
In which
Gallery
References
Orthogonal polynomials
Q-analogs
Special hypergeometric functions |
https://en.wikipedia.org/wiki/Q-Hahn%20polynomials | In mathematics, the q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.
Definition
The polynomials are given in terms of basic hypergeometric functions by
Relation to other polynomials
q-Hahn polynomials→ Quantum q-Krawtchouk polynomials：
q-Hahn polynomials→ Hahn polynomials
make the substitution, into definition of q-Hahn polynomials, and find the limit q→1, we obtain
，which is exactly Hahn polynomials.
References
Orthogonal polynomials
Q-analogs
Special hypergeometric functions |
https://en.wikipedia.org/wiki/Dual%20q-Hahn%20polynomials | In mathematics, the dual q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.
Definition
The polynomials are given in terms of basic hypergeometric functions.
References
Orthogonal polynomials
Q-analogs
Special hypergeometric functions |
https://en.wikipedia.org/wiki/Panos%20Ipeirotis | Panagiotis G. Ipeirotis (Born on May 3rd, 1976 in Serres, Greece) is a professor and George A. Kellner Faculty Fellow at the Department of Technology, Operations, and Statistics at Leonard N. Stern School of Business of New York University.
He is known for his work on crowdsourcing (especially Amazon Mechanical Turk) and on integrating human and machine intelligence.
He also worked on the intersection of data mining with economics, through the EconoMining project. The finding that good spelling and grammar can lead to improved product sales was discussed in the media.
He is the author of the blog "A Computer Scientist in a Business School", where he often writes about crowdsourcing and other topics. Many of his blog posts are frequently cited in the press and in academic papers.
Career
In 2004, Panos Ipeirotis was awarded a Ph.D. in Computer Science from Columbia University. In the same year, he joined New York University Stern School of Business where he is currently a professor and George A. Kellner Faculty Fellow at the Department of Information, Operations, and Management Sciences. He also worked for oDesk (now UpWork) as Academic-in-Residence, and at Google as a visiting scientist. He is also the greatest father ever.
Awards
Ipeirotis is the recipient of the 2015 Lagrange Prize in Complex systems for his contributions in the field of Social media, User-generated content, and Crowdsourcing. Additionally, he has received nine "Best Paper" awards and nominations and a CAREER award from the National Science Foundation.
References
1976 births
Living people
Columbia University alumni
Greek emigrants to the United States
New York University faculty
American computer scientists |
https://en.wikipedia.org/wiki/Al-Salam%E2%80%93Chihara%20polynomials | In mathematics, the Al-Salam–Chihara polynomials Qn(x;a,b;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of the properties of Al-Salam–Chihara polynomials.
Definition
The Al-Salam–Chihara polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by
where x = cos(θ).
References
Further reading
Bryc, W., Matysiak, W., & Szabłowski, P. (2005). Probabilistic aspects of Al-Salam–Chihara polynomials. Proceedings of the American Mathematical Society, 133(4), 1127-1134.
Floreanini, R., LeTourneux, J., & Vinet, L. (1997). Symmetry techniques for the Al-Salam-Chihara polynomials. Journal of Physics A: Mathematical and General, 30(9), 3107.
Christiansen, J. S., & Koelink, E. (2008). Self-adjoint difference operators and symmetric Al-Salam–Chihara polynomials. Constructive Approximation, 28(2), 199-218.
Ishikawa, M., & Zeng, J. (2009). The Andrews–Stanley partition function and Al-Salam–Chihara polynomials. Discrete Mathematics, 309(1), 151-175.
Atakishiyeva, M. K., & Atakishiyev, N. M. (1997). Fourier-Gauss transforms of the Al-Salam-Chihara polynomials. Journal of Physics A: Mathematical and General, 30(19), L655.
Orthogonal polynomials
Q-analogs
Special hypergeometric functions |
https://en.wikipedia.org/wiki/Al-Salam%E2%80%93Carlitz%20polynomials | In mathematics, Al-Salam–Carlitz polynomials U(x;q) and V(x;q) are two families of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of their properties.
Definition
The Al-Salam–Carlitz polynomials are given in terms of basic hypergeometric functions by
References
Further reading
Wang, M. (2009). -integral representation of the Al-Salam–Carlitz polynomials. Applied Mathematics Letters, 22(6), 943-945.
Askey, R., & Suslov, S. K. (1993). The -harmonic oscillator and the Al-Salam and Carlitz polynomials. Letters in Mathematical Physics, 29(2), 123-132.
Chen, W. Y., Saad, H. L., & Sun, L. H. (2010). An operator approach to the Al-Salam–Carlitz polynomials. Journal of Mathematical Physics, 51(4).
Kim, D. (1997). On combinatorics of Al-Salam Carlitz polynomials. European Journal of Combinatorics, 18(3), 295-302.
Andrews, G. E. (2000). Schur's theorem, partitions with odd parts and the Al-Salam-Carlitz polynomials. Contemporary Mathematics, 254, 45-56.
Baker, T. H., & Forrester, P. J. (2000). Multivariable Al–Salam & Carlitz Polynomials Associated with the Type A –Dunkl Kernel. Mathematische Nachrichten, 212(1), 5-35.
Orthogonal polynomials
Special functions
Q-analogs |
https://en.wikipedia.org/wiki/Al-Salam%20polynomial | In mathematics, Al-Salam polynomials, named for Waleed Al Salam, may refer to:
Al-Salam–Carlitz polynomials
Al-Salam–Chihara polynomials
Al-Salam–Ismail polynomials |
https://en.wikipedia.org/wiki/Bal%C3%A1zs%20Zamostny | Balázs Zamostny (born 31 January 1992) is a Hungarian forward who plays for Tiszakécske.
Club career
On 28 June 2022, Zamostny moved to Tiszakécske.
Career statistics
.
Honours
Újpest
Hungarian Cup (1): 2013–14
References
External links
Player profile at HLSZ
1992 births
Living people
Footballers from Pécs
Hungarian men's footballers
Hungary men's youth international footballers
Hungary men's under-21 international footballers
Men's association football forwards
Pécsi MFC players
Újpest FC players
BFC Siófok players
Vasas SC players
Szombathelyi Haladás footballers
Soproni VSE players
Győri ETO FC players
FC Ajka players
Nyíregyháza Spartacus FC players
Tiszakécske FC footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players |
https://en.wikipedia.org/wiki/Sweedler%27s%20Hopf%20algebra | In mathematics, introduced an example of an infinite-dimensional Hopf algebra, and Sweedler's Hopf algebra H4 is a certain 4-dimensional quotient of it that is neither commutative nor cocommutative.
Definition
The following infinite dimensional Hopf algebra was introduced by . The Hopf algebra is generated as an algebra by three elements x, g and g-1.
The coproduct Δ is given by
Δ(g) = g ⊗g, Δ(x) = 1⊗x + x ⊗g
The antipode S is given by
S(x) = –x g−1, S(g) = g−1
The counit ε is given by
ε(x)=0, ε(g) = 1
Sweedler's 4-dimensional Hopf algebra H4 is the quotient of this by the relations
x2 = 0, g2 = 1, gx = –xg
so it has a basis 1, x, g, xg . Note that Montgomery describes a slight variant of this Hopf algebra using the opposite coproduct, i.e. the coproduct described above composed with the tensor flip on H4⊗H4.
Sweedler's 4-dimensional Hopf algebra is a quotient of the Pareigis Hopf algebra, which is in turn a quotient of the infinite dimensional Hopf algebra.
References
Hopf algebras |
https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Zurbenko%20filter | Within statistics, the Kolmogorov–Zurbenko (KZ) filter was first proposed by A. N. Kolmogorov and formally defined by Zurbenko. It is a series of iterations of a moving average filter of length m, where m is a positive, odd integer. The KZ filter belongs to the class of low-pass filters. The KZ filter has two parameters, the length m of the moving average window and the number of iterations k of the moving average itself. It also can be considered as a special window function designed to eliminate spectral leakage.
Background
A. N. Kolmogorov had the original idea for the KZ filter during a study of turbulence in the Pacific Ocean. Kolmogorov had just received the International Balzan Prize for his law of 5/3 in the energy spectra of turbulence. Surprisingly the 5/3 law was not obeyed in the Pacific Ocean, causing great concern. Standard fast Fourier transform (FFT) was completely fooled by the noisy and non-stationary ocean environment. KZ filtration resolved the problem and enabled proof of Kolmogorov's law in that domain. Filter construction relied on the main concepts of the continuous Fourier transform and their discrete analogues. The algorithm of the KZ filter came from the definition of higher-order derivatives for discrete functions as higher-order differences. Believing that infinite smoothness in the Gaussian window was a beautiful but unrealistic approximation of a truly discrete world, Kolmogorov chose a finitely differentiable tapering window with finite support, and created this mathematical construction for the discrete case. The KZ filter is robust and nearly optimal. Because its operation is a simple Moving Average (MA), the KZ filter performs well in a missing data environment, especially in multidimensional time series where missing data problem arises from spatial sparseness. Another nice feature of the KZ filter is that the two parameters have clear interpretation so that it can be easily adopted by specialists in different areas. A few software packages for time series, longitudinal and spatial data have been developed in the popular statistical software R, which facilitate the use of the KZ filter and its extensions in different areas. I.Zurbenko Postdoctoral position at UC Berkeley with Jerzy Neyman and Elizabeth Scott provided a lot of ideas of applications supported in contacts with Murray Rosenblatt, Robert Shumway, Harald Cramér, David Brillinger, Herbert Robbins, Wilfrid Dixon, Emanuel Parzen.
Definition
KZ Filter
Let be a real-valued time series, the KZ filter with parameters and is defined as
where coefficients
are given by the polynomial coefficients obtained from equation
From another point of view, the KZ filter with parameters and can be defined as time iterations of a moving average (MA) filter of points. It can be obtained through iterations.
First iteration is to apply a MA filter over process
The second iteration is to apply the MA operation to the result of the first iteration,
|
https://en.wikipedia.org/wiki/Graph%20energy | In mathematics, the energy of a graph is the sum of the absolute values of the eigenvalues of the adjacency matrix of the graph. This quantity is studied in the context of spectral graph theory.
More precisely, let G be a graph with n vertices. It is assumed that G is simple, that is, it does not contain loops or parallel edges. Let A be the adjacency matrix of G and let , , be the eigenvalues of A. Then the energy of the graph is defined as:
References
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.
.
.
Algebraic graph theory |
https://en.wikipedia.org/wiki/Unemployment%20in%20the%20United%20Kingdom | Unemployment in the United Kingdom is measured by the Office for National Statistics.
In the most recent three-month figures (July to September 2022) the unemployment rate was estimated at 3.6%, which is 0.2 percentage points lower than the previous three-month period. The ONS said the employment rate, or percentage of people in work for those aged between 16 and 64, was estimated to be 75.5%. This was largely unchanged compared with the previous three-month period and 1.1 percentage points lower than before the pandemic (December 2019 to February 2020). The economic inactivity rate (is the proportion of people aged between 16 and 64 years who are not in the labour force) is 21.6%, an increase of 0.2 percentage points on the quarter
The figures are compiled through the Labour Force Survey, which asks a sample of 53,000 households and is conducted every 3 months.
Unemployment levels and rates are published each month by the Office for National Statistics in the Labour Market Statistical Bulletin. Estimates are available by sex, age, duration of unemployment and by area of the UK.
The definition and measurement of UK unemployment
The definition of unemployment used by the Office for National Statistics is based on the internationally agreed and recommended definition from the International Labour Organization (ILO)—an agency of the United Nations. Use of this definition allows international comparisons of unemployment rates.
Unemployed people are defined as those aged 16 or over who are without work, available to start work in the next two weeks and who have either:
a) been actively seeking work in the past four weeks, or
b) are waiting to start a new job they have already obtained.
Those who are without work who do not meet the criteria of unemployment are classed as "out of the labour force", otherwise known as "economically inactive". For example, a person who wants a job but is not available for work due to sickness or disability would be classed as economically inactive, not unemployed.
A short video explaining the basic labour market concepts of employment, unemployment and economic inactivity is available on the ONS YouTube channel.
The UK unemployment rate
In the UK the official unemployment rate is defined as the percentage of the labour force that is classed as unemployed.
The denominator here is also known as the "Labour Force" or the "Economically Active Population".
In the three months to February 2017 there were 33.4 million people in the UK labour force and 1.56 million people classed as unemployed. These figures gave an official UK unemployment rate of 4.7%.
UK unemployment rates consistent with this definition are available from 1971. Considering this consistent time series, the highest unemployment rate recorded since 1971 was 11.9% in 1984 and the lowest was 3.4% in late 1973/early 1974.
Data consistent with current international definitions is not available for years before 1971 due to there not being a Labour Force |
https://en.wikipedia.org/wiki/Sarah%20West | Sarah West (born 1972) is a retired Royal Navy officer, the first woman to be appointed to command a major warship in the Royal Navy.
West was born in Lincolnshire and studied mathematics at the University of Hertfordshire before entering Britannia Royal Naval College in September 1995. She joined the Royal Navy as a warfare officer. She also took a law degree whilst on active service in the Middle East and is an expert in large-scale naval planning, mine clearance, weapons systems and underwater warfare. She was trained on HMS Battleaxe, and , deployed in the Persian Gulf. West went on to join the minesweeper as a navigator in 1997, and later served as officer of the watch of HMS Sheffield and navigating officer of .
West went on to complete the Principal Warfare Officers' course, specialising in underwater warfare, and joined in 2003. She became Operations Officer in in 2004, and spent some time as the ship's executive officer. She was with Commander Amphibious Task Group from 2005 as the underwater warfare specialist, then at Permanent Joint Headquarters at Northwood from 2007, followed by work with the Middle East Operations Team. West went on to command several of the ships of the First Mine Counter Measures Squadron between April 2009 and December 2011, , , and . She has also worked on the co-ordination and logistics of the evacuation of UK citizens during the 2006 Lebanon War, naval operations at the time of the 2008 Kosovo declaration of independence and the British naval contribution to the Iraq War. She was promoted to commander in January 2012.
In May 2012 West assumed command of HMS Portland, a Type 23 frigate. West left the ship during a seven-month deployment to the Caribbean in mid-2014, after allegations that she had had an affair with one of her officers. The Royal Navy was reported to be investigating the allegations. On 8 August 2014 the Navy announced that West had been removed from command, citing an "internal matter", and that she would be reappointed to another post.
References
External links
1972 births
Alumni of the University of Hertfordshire
Living people
People from Grimsby
Royal Navy officers
Royal Navy personnel of the Iraq War
Women in the Royal Navy
Women in the Iraq War |
https://en.wikipedia.org/wiki/Thomas%20Graham%20Balfour | Thomas Graham Balfour (18 March 1813 – 17 January 1891) was a Scottish physician noted for his work with medical statistics, and a member of Florence Nightingale's inner circle.
Biography
Balfour was born in Edinburgh on 18 March 1813. He was son of John Balfour, a merchant of Leith, and his wife Helen, daughter of Thomas Buchanan of Ardoch. He was the great-grandson of James Balfour, professor of moral philosophy at Edinburgh in 1754, and of Robert Whytt, a celebrated medical writer and professor of physiology at Edinburgh.
He graduated MD at Edinburgh in 1834; FRS, 1859; Surgeon-General, Honorary Physician to Queen Victoria and compiler of the first four volumes of Statistics of the British Army. According to Francis de Chaumont, the publication of this statistical work "marked an epoch in hygiene".
The Statistics of the British Army were reputed to be the most accurate and complete of their sort throughout Europe.
From 1840 to 1848, Balfour served as assistant surgeon in the Grenadier Guards. In 1857 he was appointed secretary to Sidney Herbert's committee on the sanitary state of the army, and in 1859 he became deputy inspector-general in charge of the new statistical branch of the army medical department, a post which he held for fourteen years. He was elected a Fellow of the Royal Society on 3 June 1858, and in 1860 a fellow of the Royal College of Physicians of London.
He along with Dr John Sutherland were secretaries to the Sanitary Section of the International Statistical Congress in 1860, which was attended by Adolphe Quetelet, and to which Florence Nightingale submitted a paper.
In 1887 he was appointed honorary physician to the queen. He was placed on half-pay as surgeon-general in 1876, and in his forty years of service had done much to improve the sanitary condition of the forces. He was President of the Royal Statistical Society from 1888 to 1890 shortly after it had changed from the Statistical Society of London. He married in 1856 Georgina, daughter of George Prentice of Armagh, and had one son, Sir Graham Balfour. He died at Coombe Lodge, Wimbledon, on 17 January 1891.
References
Attribution
1813 births
1891 deaths
British statisticians
19th-century Scottish medical doctors
British Army generals
Medical doctors from Edinburgh
Alumni of the University of Edinburgh
Grenadier Guards officers
Fellows of the Royal Society
Fellows of the Royal College of Physicians
British Army regimental surgeons
Florence Nightingale
19th-century British Army personnel
Thomas Graham |
https://en.wikipedia.org/wiki/Greville%20Ewing | Greville Ewing (1767–1841), was a Scottish congregational minister of the Church of Scotland.
Career
Ewing, the son of Alexander Ewing, a teacher of mathematics, was born in 1767 at Edinburgh, and lived on the Cowgate, south of Canongate, the east part of the Old Town.
He studied with considerable distinction at the high school and university there. Of a deeply religious temperament, he decided to prepare for the ministry, much against his father's wishes. On being licensed as a probationer he was chosen, first as assistant and afterwards as colleague to the Rev. Dr. Thomas Snell Jones, minister of Lady Glenorchy's Church in Edinburgh.
Here he soon acquired wide popularity as a preacher, and exercised his ministry with great success. Missions attracted much of his attention, and in 1796 he took an active part in the formation of the Edinburgh Missionary Society, becoming its first secretary. He was also editor of the 'Missionary Magazine' from 1796 to 1799. When Robert Haldane of Airthrey projected a mission to India, Ewing was appointed to go out, but the directors of the East India Company refused to sanction the undertaking, and it was abandoned. He then joined with the brothers Haldane in an important missionary movement at home. Among its supporters were many who had not received Presbyterian ordination. It was condemned in a pastoral admonition from the general assembly of the established church.
Ewing, who regarded the congregational system as more scriptural and more elastic than the presbyterian, had in 1798 resigned his charge as minister of Lady Glenorchy's Chapel, as well as his connection with the Church of Scotland. In 1799 he became minister of a congregational church in Glasgow, and retained the charge till 1836. As a result of his labours with the Haldanes and afterwards with Dr. Ralph Wardlaw, congregationalism was introduced into Scotland; he guided the formation of several congregations, including St. James' Congregational Church. He was tutor of the Glasgow Theological Academy – a congregationalist foundation – from its foundation in 1809 till 1836, and did much to promote the study of the Bible in the original languages. In 1812 he helped to form the Congregational Union of Scotland.
In 1801 he published a Greek grammar and lexicon (The Rudiments of the Greek Language Shortly Illustrated ...) for students of the New Testament. He also published several pamphlets and sermons, and two larger works—'Essays to the Jews, on the Law and the Prophets,’ 2 vols. (1809–1810), and an 'Essay on Baptism' (1823).
Family and death
Ewing married three times: in 1794 to Anne Innes, who died in 1795; in 1799 to Janet Jamieson, who died in 1801; and in 1802, to Barbara, daughter of Sir James Maxwell, bart., of Pollok, and stepdaughter of Sir John Shaw-Stewart, bart., of Ardgowan. Ewing's third wife died 14 September 1828, in consequence of an accident at the Falls of Clyde, and her husband published a memoir, of which a second edition ap |
https://en.wikipedia.org/wiki/Thomas%20Exley | Thomas Exley (9 December 1774 – 17 February 1855) was an English schoolmaster and schoolkeeper, who taught and occasionally published on mathematics, but was better known for advancing controversial scientific theories and for theological discussions, with special reference to Methodism.
Exley was born in Gowdall, a village one mile west of Snaith, Yorkshire. He settled at Bristol in the last week of 1799, quickly resuming work teaching mathematics. In 1812 he brought out with the Rev. William Moore Johnson, then curate of Henbury, Gloucestershire, a compilation entitled The Imperial Encyclopædia; or, Dictionary of the Sciences and Arts; comprehending also the whole circle of Miscellaneous Literature, &c., 4 vols. 4to, London [1812]; the following year, on 6 January 1813, he was awarded an honorary MA degree from King's College, Aberdeen, on the nomination of Johnson and their mutual brother-in-law Dr. Adam Clarke, both Johnson and Clarke already holding honorary degrees from Aberdeen. By 1848 Exley had given up keeping school and retired to Cotham Park Road, Bristol. He was an early member of the British Association for the Advancement of Science, and read several papers at its meetings.
The Imperial Magazine; or Compendium of Religious, Moral & Philosophical Knowledge carried a profile of Exley in the sixth issue of its first volume in 1819. Exley was then only forty-four, but the final paragraph observes that, the life of a retired mathematician can hardly be expected to furnish any extensive variety. He died on 17 February 1855, aged 80.
Mathematical activities
Exley showed an early taste for mathematics. He was a pupil at a classical and mathematical school near Barnard Castle, North Yorkshire and then moved to Manchester in search of further study and employment, finding both in a recently opened classical school. This was also his entry to the Clarke family, as, in 1796, he married Hannah Clarke, a daughter of the schoolkeeper and sister of Dr. Clarke. Exley opened his own school in Huddersfield, but, despite its success, ill health obliged him to move again, which is how he came to settle in Bristol. His reputation was assured there when, in 1802, he solved some challenging questions that appeared in a local paper.
Exley has not left much in the way of mathematical legacy, but work on the extraction of cube roots did attract favourable attention. He published his method in the entry on arithmetic in The Imperial Encyclopædia and returned to the subject in a note in The Imperial Magazine in 1819. His friend and associate W. G. Horner remarks on this contribution in his more celebrated paper on root extraction presented to the Royal Society of London that year. Exley and Horner were examiners at Kingswood School, then in Bristol: Exley for mathematics; Horner for classics.
Selected works
A Vindication of Dr. Adam Clarke, in answer to Mr. Moore's Thoughts on the Eternal Sonship of the Second Person of the Holy Trinity, addressed to the |
https://en.wikipedia.org/wiki/Availability%20%28system%29 | Availability is the probability that a system will work as required when required during the period of a mission. The mission could be the 18-hour span of an aircraft flight. The mission period could also be the 3 to 15-month span of a military deployment. Availability includes non-operational periods associated with reliability, maintenance, and logistics.
This is measured in terms of nines. Five-9's (99.999%) means less than 5 minutes when the system is not operating correctly over the span of one year.
Availability is only meaningful for supportable systems. As an example, availability of 99.9% means nothing after the only known source stops manufacturing a critical replacement part.
Definition
There are two kinds of availability.
Operational
Predicted
Operational availability is presumed to be the same as predicted availability until after operational metrics become available.
Availability
Operational availability is based on observations after at least one system has been built. This usually begins with the brassboard system that is used to complete system development, and continues with the first of kind used for live fire test and evaluation (LFTE). Organizations responsible for maintenance use this to evaluate the effectiveness of the maintenance philosophy.
Predicted availability is based on a model of the system before it is built.
Downtime is the total of all of the different contributions that compromise operation. For modeling, these are different aspects of the model, such as human-system interface for MTTR and reliability modeling for MTBF. For observation, these reflect the different areas of the organization, such as maintenance personnel and documentation for MTTR, and manufacturers and shippers for MLDT.
MTBF
Mean Time Between Failure (MTBF) depends upon the maintenance philosophy.
If a system is designed with both redundancy and automatic fault bypass, then MTBF is the anticipated lifespan of the system if these features cover all possible failure modes (infinity for all practical purposes). Such systems will continue without noticeable interruption when these conditions are satisfied unless there are secondary failures. This is called active redundancy, which requires no maintenance to prevent mission failure. Active redundancy is required for systems that cannot be maintained, such as satellites.
If a system has no redundancy, then MTBF is the inverse of failure rate, .
Systems with spare parts that are energized but that lack automatic fault bypass are not actually redundant because human action is required to restore operation after every failure. This depends upon Condition-based maintenance and Planned Maintenance System support.
MTTR
Mean Time To Recover (MTTR) is the length of time required to restore operation to specification.
This includes three values.
Mean Time To Discover
Mean Time To Isolate
Mean Time To Repair
Mean Time To Discover is the length of time that transpires between when a f |
https://en.wikipedia.org/wiki/Affine%20root%20system | In mathematics, an affine root system is a root system of affine-linear functions on a Euclidean space. They are used in the classification of affine Lie algebras and superalgebras, and semisimple p-adic algebraic groups, and correspond to families of Macdonald polynomials. The reduced affine root systems were used by Kac and Moody in their work on Kac–Moody algebras. Possibly non-reduced affine root systems were introduced and classified by and (except that both these papers accidentally omitted the Dynkin diagram ).
Definition
Let E be an affine space and V the vector space of its translations.
Recall that V acts faithfully and transitively on E.
In particular, if , then it is well defined an element in V denoted as which is the only element w such that .
Now suppose we have a scalar product on V.
This defines a metric on E as .
Consider the vector space F of affine-linear functions .
Having fixed a , every element in F can be written as with a linear function on V that doesn't depend on the choice of .
Now the dual of V can be identified with V thanks to the chosen scalar product and we can define a product on F as .
Set and for any and respectively.
The identification let us define a reflection over E in the following way:
By transposition acts also on F as
An affine root system is a subset such that:
The elements of S are called affine roots.
Denote with the group generated by the with .
We also ask
This means that for any two compacts the elements of such that are a finite number.
Classification
The affine roots systems A1 = B1 = B = C1 = C are the same, as are the pairs B2 = C2, B = C, and A3 = D3
The number of orbits given in the table is the number of orbits of simple roots under the Weyl group.
In the Dynkin diagrams, the non-reduced simple roots α (with 2α a root) are colored green. The first Dynkin diagram in a series sometimes does not follow the same rule as the others.
Irreducible affine root systems by rank
Rank 1: A1, BC1, (BC1, C1), (C, BC1), (C, C1).
Rank 2: A2, C2, C, BC2, (BC2, C2), (C, BC2), (B2, B), (C, C2), G2, G.
Rank 3: A3, B3, B, C3, C, BC3, (BC3, C3), (C, BC3), (B3, B), (C, C3).
Rank 4: A4, B4, B, C4, C, BC4, (BC4, C4), (C, BC4), (B4, B), (C, C4), D4, F4, F.
Rank 5: A5, B5, B, C5, C, BC5, (BC5, C5), (C, BC5), (B5, B), (C, C5), D5.
Rank 6: A6, B6, B, C6, C, BC6, (BC6, C6), (C, BC6), (B6, B), (C, C6), D6, E6,
Rank 7: A7, B7, B, C7, C, BC7, (BC7, C7), (C, BC7), (B7, B), (C, C7), D7, E7,
Rank 8: A8, B8, B, C8, C, BC8, (BC8, C8), (C, BC8), (B8, B), (C, C8), D8, E8,
Rank n (n>8): An, Bn, B, Cn, C, BCn, (BCn, Cn), (C, BCn), (Bn, B), (C, Cn), Dn.
Applications
showed that the affine root systems index Macdonald identities
used affine root systems to study p-adic algebraic groups.
Reduced affine root systems classify affine Kac–Moody algebras, while the non-reduced affine root systems correspond to affine Lie superalgebras.
showed that affine roots systems index families of Macdonald po |
https://en.wikipedia.org/wiki/Orthogonal%20polynomials%20on%20the%20unit%20circle | In mathematics, orthogonal polynomials on the unit circle are families of polynomials that are orthogonal with respect to integration over the unit circle in the complex plane, for some probability measure on the unit circle. They were introduced by .
Definition
Suppose that is a probability measure on the unit circle in the complex plane, whose support is not finite. The orthogonal polynomials associated to are the polynomials with leading term that are orthogonal with respect to the measure .
The Szegő recurrence
The monic orthogonal Szegő polynomials satisfy a recurrence relation of the form
for and initial condition , with
and constants given by
called the Verblunsky coefficients. Subsequently, Geronimus' theorem states that the Verblunsky coefficients associated with are the Schur parameters:
Verblunsky's theorem
Verblunsky's theorem states that any sequence of complex numbers in the open unit disk is the sequence of Verblunsky coefficients for a unique probability measure on the unit circle with infinite support.
Baxter's theorem
Baxter's theorem states that the Verblunsky coefficients form an absolutely convergent series if and only if the moments of form an absolutely convergent series and the weight function is strictly positive everywhere.
Szegő's theorem
Verblunsky's form of Szegő's theorem states that
where is the absolutely continuous part of the measure . Verblunsky's form also allows for a non-zero singular part while in Szegő's original version.
Rakhmanov's theorem
Rakhmanov's theorem states that if the absolutely continuous part of the measure is positive almost everywhere then the Verblunsky coefficients tend to 0.
Examples
The Rogers–Szegő polynomials are an example of orthogonal polynomials on the unit circle.
See also
Schur class
References
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Christophe%20Soul%C3%A9 | Christophe Soulé (born 1951) is a French mathematician working in arithmetic geometry.
Education
Soulé started his studies in 1970 at École Normale Supérieure in Paris.
He completed his Ph.D. at the University of Paris in 1979 under the supervision of Max Karoubi and Roger Godement, with a dissertation titled K-Théorie des anneaux d'entiers de corps de nombres et cohomologie étale.
Awards and recognition
In 1979, he was awarded a CNRS Bronze Medal. He received the Prix J. Ponti in 1985 and the Prize Ampère in 1993.
Since 2001, he is member of the French Academy of Sciences. In 1983, he was invited speaker at the International Congress of Mathematicians (ICM) in Warsaw.
Publications
Christophe Soulé, with the collaboration of Dan Abramovich, Jean-François Burnol, and Jürg Kramer: Lectures on Arakelov Geometry. Cambridge Studies in Advanced Mathematics 33. Cambridge University Press, 1992. ,
Henri Gillet, Christophe Soulé: An arithmetic Riemann–Roch Theorem, Inventiones Mathematicae 110 (1992), no. 3, 473–543. ,
References
External links
personal page
Living people
1951 births
École Normale Supérieure alumni
University of Paris alumni
20th-century French mathematicians
21st-century French mathematicians
Members of the French Academy of Sciences
Arithmetic geometers |
https://en.wikipedia.org/wiki/Hovhannes%20Imastaser | Hovhannes Imastaser (, c. 1047–1129), also known as Hovhannes Sarkavag (), was a medieval Armenian multi-disciplinary scholar known for his works on philosophy, theology, mathematics, cosmology, and literature. Imastaser was also a gifted hymnologist and pedagogue.
Biography
Hovhannes Imastaser was born around 1047 in the district of Gardman (village of Pib) of historical Armenia’s eastern province of Utik, which today is located in Azerbaijan, north of Nagorno Karabakh.
The most extensive historical account of Hovhannes Imastaser’s life and work is in the 12-13th century Armenian historian Kirakos Gandzaketsi’s “History of Armenia.” There also exists a 13th-century anonymous biography of Hovhannes Imastaser, which is attributed sometimes to Kirakos Gandzaketsi.
Hovhannes received his education in theology and science in Haghbat and Sanahin, two important monastic centers of Armenian medieval scholarship. Upon the completion of his studies, Hovhannes moved to medieval Armenia's capital city of Ani, where he taught philosophy, mathematics, music, cosmography and grammar. In Ani, Hovhannes received the ecclesiastical rank of sarkavag (deacon), and eventually rose to become a vardapet (archimandrite, Doctor of Theology) of the Armenian Apostolic Church. But it was the title sarkavag, however, that became attached to his name.
While Hovhannes Imastaser was recognized as a master of Armenian literature, his works acquired wider publicity only in the 19th century when they were published by Abbot Ghevont Alishan, a member of the Mekhitarist Congregation in Venice that is associated with Armenian Catholics. Imastaser's innovative approach to literature, for which he is often referred to as a key representative of Armenian literary renaissance, is fully demonstrated in his poem Ban Imastutian (Discourse on Wisdom). In the poem, written as a dialogue between the author and a blackbird, the bird symbolizes nature, which, per author, is the main inspiration behind art. In Imastaser's time, artistic inspiration was usually attributed to divine reasons.
As a hymnologist, Imastaser wrote several important sharakans (hymns): Tagh Harutean (Ode to the Resurrection), Paitsaratsan Aisor (Brightened on This Day), Anskizbn Bann Astvatz (God, The Infinite Word), Anchareli Bann Astavatz (God, The Inexpressible Word). The latter two are acrostic compositions, each encompassing within their ten stanzas thirty six letters of the Armenian Alphabet. In them, Imastaser glorifies heroes and martyrs who sacrificed their lives defending Armenian homeland and their Christian faith. Imastaser also introduced another patriotic theme to Armenian literature and music: emigration. In his hymns Imastaser prays to God so that Armenians who left their country could find strength to return home.
Hovhannes Imastaser also contributed to the standardization of the Armenian prayer book and Psalter.
Hovhannes Imastaser's work in mathematics is represented by the volume Haghaks Ankiu |
https://en.wikipedia.org/wiki/Alexiewicz%20norm | In mathematics — specifically, in integration theory — the Alexiewicz norm is an integral norm associated to the Henstock–Kurzweil integral. The Alexiewicz norm turns the space of Henstock–Kurzweil integrable functions into a topological vector space that is barrelled but not complete. The Alexiewicz norm is named after the Polish mathematician Andrzej Alexiewicz, who introduced it in 1948.
Definition
Let HK(R) denote the space of all functions f: R → R that have finite Henstock–Kurzweil integral. Define the Alexiewicz semi-norm of f ∈ HK(R) by
This defines a semi-norm on HK(R); if functions that are equal Lebesgue-almost everywhere are identified, then this procedure defines a bona fide norm on the quotient of HK(R) by the equivalence relation of equality almost everywhere. (Note that the only constant function f: R → R that is integrable is the one with constant value zero.)
Properties
The Alexiewicz norm endows HK(R) with a topology that is barrelled but incomplete.
The Alexiewicz norm as defined above is equivalent to the norm defined by
The completion of HK(R) with respect to the Alexiewicz norm is often denoted A(R) and is a subspace of the space of tempered distributions, the dual of Schwartz space. More precisely, A(R) consists of those tempered distributions that are distributional derivatives of functions in the collection
Therefore, if f ∈ A(R), then f is a tempered distribution and there exists a continuous function F in the above collection such that
for every compactly supported C∞ test function φ: R → R. In this case, it holds that
The translation operator is continuous with respect to the Alexiewicz norm. That is, if for f ∈ HK(R) and x ∈ R the translation Txf of f by x is defined by
then
References
Norms (mathematics) |
https://en.wikipedia.org/wiki/Biorthogonal%20polynomial | In mathematics, a biorthogonal polynomial is a polynomial that is orthogonal to several different measures. Biorthogonal polynomials are a generalization of orthogonal polynomials and share many of their properties. There are two different concepts of biorthogonal polynomials in the literature: introduced the concept of polynomials biorthogonal with respect to a sequence of measures, while Szegő introduced the concept of two sequences of polynomials that are biorthogonal with respect to each other.
Polynomials biorthogonal with respect to a sequence of measures
A polynomial p is called biorthogonal with respect to a sequence of measures μ1, μ2, ... if
whenever i ≤ deg(p).
Biorthogonal pairs of sequences
Two sequences ψ0, ψ1, ... and φ0, φ1, ... of polynomials are called biorthogonal (for some measure μ) if
whenever m ≠ n.
The definition of biorthogonal pairs of sequences is in some sense a special case of the definition of biorthogonality with respect to a sequence of measures. More precisely two sequences ψ0, ψ1, ... and φ0, φ1, ... of polynomials are biorthogonal for the measure μ if and only if the sequence ψ0, ψ1, ... is biorthogonal for the sequence of measures φ0μ, φ1μ, ..., and the sequence φ0, φ1, ... is biorthogonal for the sequence of measures ψ0μ, ψ1μ,....
References
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Historical%20table%20of%20the%20Copa%20Sudamericana | The Historical Table of the Copa Sudamericana is a record of statistics of every team that has played in the Copa Sudamericana since its inception in 2002, up to the 2022 season. The list is ordered according to the most points each team has accumulated.
Playing at 2022 edition.
References
Copa Sudamericana
All-time football league tables |
https://en.wikipedia.org/wiki/Leo%20T%C3%B6rnqvist | Leo Waldemar Törnqvist (14 February 1911 – 18 April 1983) was one of the first professors of statistics in Finland, and the first to achieve international recognition. He taught at the University of Helsinki from 1943 to 1974, and developed techniques that are used in official price and productivity statistics.
Life, education, and career
Törnqvist was born on 14 February 1911 in Jeppo, a Swedish-speaking village in Finland. He studied mathematics, physics, and chemistry at Åbo Akademi University in Turku, where his interests shifted to economics and statistics under the influence of Swedish economist Arthur Montgomery. He finished his studies in Turku in 1933 and continued with graduate work in mathematics at Stockholm University, earning a doctorate in 1937 under the supervision of Harald Cramér and Gunnar Myrdal.
After a short-term teaching position at Åbo Akademi University from 1937 to 1938,
he began his career working for the Finnish railway service from 1938 until 1943. He was appointed as an associate professor of statistics at the University of Helsinki in 1943 and promoted to full professor in 1950.
In the early 1950s he visited researchers in the US and, in the early 1960s, worked as a consultant for the United Nations in Indonesia.
He died on 18 April 1983.
Contributions
Törnqvist developed an approach to creating weighted price indexes across discrete time periods using weighted averages of growth rates in prices where the weights were quantity averages across the two periods, in work he did with the Bank of Finland published in 1936. These Törnqvist indexes are used in official price and productivity statistics in many countries.
In a 1949 work, he also made "the first serious attempt to describe population forecasting from a stochastic point of view", providing "seminal works" in Bayesian inference in demography.
As a professor at the University of Helsinki, his students included economist Timo Teräsvirta. His student Vieno Rajaoja was the first Finnish woman to earn a doctorate in statistics, in 1958.
Recognition
Törnqvist was elected member of the Finnish Society of Sciences and Letters in 1956, fellow of the Econometric Society in 1951, and member of the International Statistical Institute in 1956. He was decorated Commander of the Order of the Lion of Finland in 1961, and given honorary doctorates by the University of Helsinki in 1971 and by Åbo Akademi University in 1978.
Family
Törnqvist bought a VIC-20 about 1981 and asked his daughter Anna's son, Linus Torvalds, to help him program it. Törnqvist wrote out BASIC language programs, and grandson Linus, aged about eleven, typed them in. "He wanted me to share in the experience [and] get me interested in math," wrote Torvalds later. These were Linus's first programming experiences. Ten years later, Torvalds began to write the Linux kernel.
Leo Törnqvist’s brother was diplomat Erik Törnqvist. His son was the nuclear physicist (1938–2018).
References
Further reading |
https://en.wikipedia.org/wiki/Teemu%20Normio | Teemu Normio (born May 9, 1980) is a Finnish professional ice hockey player who is currently playing for Frederikshavn White Hawks in the AL-Bank Ligaen.
Career statistics
External links
1980 births
Porin Ässät (men's ice hockey) players
Finnish ice hockey left wingers
Frederikshavn White Hawks players
KalPa players
Kokkolan Hermes players
Living people
Lukko players
Oulun Kärpät players
Tappara players
Ice hockey people from Tampere |
https://en.wikipedia.org/wiki/Ben%20Binyamin | Ben Binyamin is an Israeli footballer currently playing for Maccabi Sha'arayim in the Liga Alef.
Club career statistics
(correct as of Feb 2013)
Honours
Liga Leumit
Runner-up (1): 2008–09
References
1985 births
Living people
Israeli Jews
Israeli men's footballers
Hapoel Acre F.C. players
Maccabi Ironi Shlomi F.C. players
Hapoel Afula F.C. players
Hapoel Herzliya F.C. players
Maccabi Netanya F.C. players
Hapoel Ra'anana A.F.C. players
Hapoel Nir Ramat HaSharon F.C. players
Maccabi Sha'arayim F.C. players
Liga Leumit players
Israeli Premier League players
Israeli people of Iranian-Jewish descent
Footballers from Nahariya
Men's association football midfielders |
https://en.wikipedia.org/wiki/Riku%20Toivo | Riku Toivo (born July 25, 1989) is a Finnish professional ice hockey player who is currently playing for Kärpät in the SM-liiga.
Career statistics
References
1989 births
Finnish ice hockey right wingers
HC Temirtau players
Hokki players
Kiekko-Laser players
Oulun Kärpät players
Living people
Ice hockey people from North Ostrobothnia |
https://en.wikipedia.org/wiki/Discrete%20orthogonal%20polynomials | In mathematics, a sequence of discrete orthogonal polynomials is a sequence of polynomials that are pairwise orthogonal with respect to a discrete measure.
Examples include the discrete Chebyshev polynomials, Charlier polynomials, Krawtchouk polynomials, Meixner polynomials, dual Hahn polynomials, Hahn polynomials, and Racah polynomials.
If the measure has finite support, then the corresponding sequence of discrete orthogonal polynomials has only a finite number of elements. The Racah polynomials give an example of this.
Definition
Consider a discrete measure on some set with weight function .
A family of orthogonal polynomials is called discrete, if they are orthogonal with respect to (resp. ), i.e.
where is the Kronecker delta.
Remark
Any discrete measure is of the form
,
so one can define a weight function by .
Listeratur
References
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Fernando%20Canesin | Fernando Canesin Matos (born 27 February 1992) is a Brazilian footballer who plays for Cruzeiro as an attacking midfielder.
Career statistics
Honours
R.S.C. Anderlecht
Belgian First Division: 2011–12
Belgian Supercup: 2012
Athletico Paranaense
Campeonato Paranaense: 2020
References
External links
1992 births
Living people
Footballers from Ribeirão Preto
Brazilian men's footballers
Men's association football midfielders
Belgian Pro League players
R.S.C. Anderlecht players
K.V. Oostende players
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
Club Athletico Paranaense players
Cruzeiro Esporte Clube players
Brazilian expatriate men's footballers
Brazilian expatriate sportspeople in Belgium
Expatriate men's footballers in Belgium |
https://en.wikipedia.org/wiki/Mateusz%20Mak | Mateusz Mak (born 14 November 1991) is a Polish professional footballer who plays as a midfielder for GKS Katowice.
Career statistics
Club
Honours
GKS Bełchatów
I liga: 2013–14
Piast Gliwice
Ekstraklasa: 2018–19
Stal Mielec
I liga: 2019–20
References
External links
Living people
1991 births
People from Sucha Beskidzka
Footballers from Lesser Poland Voivodeship
Men's association football midfielders
Polish men's footballers
Ruch Radzionków players
GKS Bełchatów players
Piast Gliwice players
Stal Mielec players
GKS Katowice players
Ekstraklasa players
I liga players
II liga players
Poland men's under-21 international footballers |
https://en.wikipedia.org/wiki/Pseudo%20Jacobi%20polynomials | In mathematics, the term Pseudo Jacobi polynomials was introduced by Lesky for one of three finite sequences of orthogonal polynomials y. Since they form an orthogonal subset of Routh polynomials it seems consistent to refer to them as Romanovski-Routh polynomials, by analogy with the terms Romanovski-Bessel and Romanovski-Jacobi used by Lesky. As shown by Askey for two other sequencesth is finite sequence orthogonal polynomials of can be expressed in terms of Jacobi polynomials of imaginary argument. In following Raposo et al. they are often referred to simply as Romanovski polynomials.
References
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Ordinary%20differential%20equation | In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equations which may be with respect to one independent variable.
Differential equations
A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable .
Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are solutions of linear differential equations (see Holonomic function). When physical phenomena are modeled with non-linear equations, they are generally approximated by linear differential equations for an easier solution. The few non-linear ODEs that can be solved explicitly are generally solved by transforming the equation into an equivalent linear ODE (see, for example Riccati equation).
Some ODEs can be solved explicitly in terms of known functions and integrals. When that is not possible, the equation for computing the Taylor series of the solutions may be useful. For applied problems, numerical methods for ordinary differential equations can supply an approximation of the solution.
Background
Ordinary differential equations (ODEs) arise in many contexts of mathematics and social and natural sciences. Mathematical descriptions of change use differentials and derivatives. Various differentials, derivatives, and functions become related via equations, such that a differential equation is a result that describes dynamically changing phenomena, evolution, and variation. Often, quantities are defined as the rate of change of other quantities (for example, derivatives of displacement with respect to time), or gradients of quantities, which is how they enter differential equations.
Specific mathematical fields include geometry and analytical mechanics. Scientific fields include much of physics and astronomy (celestial mechanics), meteorology (weather modeling), chemistry (reaction rates), biology (infectious diseases, genetic variation), ecology and population modeling (population competition), economics (stock trends, interest rates and the market equilibrium price changes).
Many mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d'Alembert, and Euler.
A simple example is Newton's second law of motion—the relationship between the displacement x and the time t of an object under the force F, is |
https://en.wikipedia.org/wiki/Rogers%20polynomials | In mathematics, the Rogers polynomials, also called Rogers–Askey–Ismail polynomials and continuous q-ultraspherical polynomials, are a family of orthogonal polynomials introduced by in the course of his work on the Rogers–Ramanujan identities. They are q-analogs of ultraspherical polynomials, and are the Macdonald polynomials for the special case of the A1 affine root system .
and discuss the properties of Rogers polynomials in detail.
Definition
The Rogers polynomials can be defined in terms of the q-Pochhammer symbol and the basic hypergeometric series by
where x = cos(θ).
References
Orthogonal polynomials
Q-analogs |
https://en.wikipedia.org/wiki/Matheass | MatheAss (former Math-Assist) is a computer program for numerical solutions in school mathematics and functions in some points similar to Microsoft Mathematics. "MatheAss is widely spread in math classes" in Germany. For schools in the federal state of Hessen (Germany) exists a state license, which allows all secondary schools to use MatheAss
Its functionality is limited compared to other numerical programs, for example, MatheAss has no script language and does no symbolic computation. On the other side it is easy to use and offers the user fully worked out solutions, in which only the necessary quantities need to be entered. MatheAss covers the topics algebra, geometry, analysis, stochastics, and linear algebra.
After a precursor for the home computers, usual around 1980, MatheAss appeared in 1983 as a shareware version for the PC, so it was one of the first shareware programs on the German market. MatheAss is available on the manufacturer's website for download for various versions of the Windows operating system.
Since version 8.2 (released in February 2011) MatheAss again offers a context-sensitive help, which was supplemented in many places by showing mathematical examples and background information. The MatheAss help file can also be viewed online.
References
Educational math software
Mathematical tools |
https://en.wikipedia.org/wiki/Femmes%20et%20Math%C3%A9matiques | L'association femmes et mathématiques (in English: Association of Women and Mathematics), created in 1987, is a voluntary association promoting women in scientific studies and research in general, and mathematics in particular. This organization currently has about 200 members, including university professors of math, math teachers, sociologists, philosophers and historians that are interested in the "woman question" in scientific domains.
According to its mandate, its principal objectives are:
To act for the promotion of women in the scientific sphere and more specifically mathematics.
To encourage the presence of women in mathematical studies and more generally scientific and technical studies.
To be a place for meetings between mathematicians and teachers of mathematics.
Make the scientific and pedagogic community aware of the question of equality between men and women.
To rally for equal representation, or parity, of women within mathematical jobs and the increase of recruitment of female math students at university.
It specifically organizes a forum of young women mathematicians, as well as conferences on different topics related to its objectives. They regularly hold a general assembly, either in Paris or outside of the city, based on various themes. It also publishes an academic journal.
The association has its headquarters at the Maison des mathématiciens at l'Institut Henri Poincaré in Paris. It participates in different initiatives with other scholarly and professional societies, in particular the Société Mathématique de France(Mathematical Society of France), la Société de mathématiques appliquées et industrielles (Society of Applied and Industrial Maths), l'Association des professeurs de mathématiques de l'enseignement public (Association of Math Professors and Public Teachers) and l'Union des professeurs de spéciales, as well as La Commission française pour l'enseignement des mathématiques (the French commission for the teaching of mathematics).
Board of the Association for 2013:
President: Laurence Broze
Vice president: Véronique Slovacek-Chauveau
Treasurer: Florence Lecomte
Co-treasurer: Christine Charretton
Secretary: Annick Boisseau
Co-secretary: Camille Ternynck
Notes
Bibliography
Du côté des mathématiciennes, Edited by Annick Boisseau, Véronique Chauveau, Françoise Delon, Gwenola Madec, avec la participation de Marie-Françoise Roy, through l'Association femmes et mathématiques, Aléas, 2002.
Femmes Et Mathématiques. "Nous Sommes... (We Are...)." Femmes Et Maths. Femmes Et Maths, 30 Jan. 2013. Web. 17 Nov. 2013.
Rencontres entre artistes et mathématiciennes : Toutes un peu les autres, by T.Chotteau, F.Delmer, P. Jakubowski, S. Paycha, J.Peiffer, Y.Perrin, V. Roca, B. Taquet, through l'Initiative de femmes et mathematiques, Paris : L'Harmattan, 2001.
External links
Official Website of the Association
Official About Page of the Association
Association femmes et mathematiques
Women's organizations based in Franc |
https://en.wikipedia.org/wiki/Rogers%E2%80%93Szeg%C5%91%20polynomials | In mathematics, the Rogers–Szegő polynomials are a family of polynomials orthogonal on the unit circle introduced by , who was inspired by the continuous q-Hermite polynomials studied by Leonard James Rogers. They are given by
where (q;q)n is the descending q-Pochhammer symbol.
Furthermore, the satisfy (for ) the recurrence relation
with and .
References
Orthogonal polynomials
Q-analogs |
https://en.wikipedia.org/wiki/Michael%20A.%20Newton | Michael Abbott Newton (born July 19, 1964, Baddeck, Nova Scotia) is a Canadian statistician. He is a Professor in the Department of Statistics and the Department of Biostatistics and Medical Informatics at the University of Wisconsin–Madison, and he received the COPSS Presidents' Award in 2004. He has written many research papers about the statistical analysis of cancer biology, including linkage analysis and signal identification.
Newton received his B.Sc. in mathematics and statistics from Dalhousie University in 1986, and his PhD in statistics from the University of Washington, Seattle in 1991 (under the supervision of Adrian E. Raftery).
In 2003 Newton won the Spiegelman Award (presented annually by the American Public Health Association to an outstanding public health statistician under age 40). He was elected a fellow of the American Statistical Association in 2007. Newton gave a Presidential Invited Address at the International Biometric Society WNAR (Western North American Region) conference in 2002 and was an IMS Medallion Lecturer in 2011.
References
External links
Michael A. Newton's home page
1964 births
Living people
Canadian statisticians
Fellows of the American Statistical Association
University of Wisconsin–Madison faculty
People from Baddeck, Nova Scotia |
https://en.wikipedia.org/wiki/Quintuple%20product%20identity | In mathematics the Watson quintuple product identity is an infinite product identity introduced by and rediscovered by and . It is analogous to the Jacobi triple product identity, and is the Macdonald identity for a certain non-reduced affine root system. It is related to Euler's pentagonal number theorem.
Statement
References
Foata, D., & Han, G. N. (2001). The triple, quintuple and septuple product identities revisited. In The Andrews Festschrift (pp. 323–334). Springer, Berlin, Heidelberg.
Cooper, S. (2006). The quintuple product identity. International Journal of Number Theory, 2(01), 115-161.
Further reading
Subbarao, M. V., & Vidyasagar, M. (1970). On Watson’s quintuple product identity. Proceedings of the American Mathematical Society, 26(1), 23-27.
Hirschhorn, M. D. (1988). A generalisation of the quintuple product identity. Journal of the Australian Mathematical Society, 44(1), 42-45.
Alladi, K. (1996). The quintuple product identity and shifted partition functions. Journal of Computational and Applied Mathematics, 68(1-2), 3-13.
Farkas, H., & Kra, I. (1999). On the quintuple product identity. Proceedings of the American Mathematical Society, 127(3), 771-778.
Chen, W. Y., Chu, W., & Gu, N. S. (2005). Finite form of the quintuple product identity. arXiv preprint math/0504277.
Elliptic functions
Theta functions
Mathematical identities
Theorems in number theory
Infinite products |
https://en.wikipedia.org/wiki/Larry%20A.%20Wasserman | Larry Alan Wasserman (born 1959) is a Canadian-American statistician and a professor in the Department of Statistics & Data Science and the Machine Learning Department at Carnegie Mellon University.
Biography
Wasserman received his Ph.D. from the University of Toronto in 1988 under the supervision of Robert Tibshirani.
He received the COPSS Presidents' Award in 1999 and the CRM-SSC Prize in 2002.
He was elected a fellow of the American Statistical Association in 1996, of the Institute of Mathematical Statistics in 2004, and of the American Association for the Advancement of Science in 2011. He was elected to National Academy of Sciences in May, 2016.
Selected works
Wasserman has written many research papers about nonparametric inference, asymptotic theory, causality, and applications of statistics to astrophysics, bioinformatics, and genetics. He has also written two advanced statistics textbooks, All of Statistics and All of Nonparametric Statistics.
2004. All of Statistics: A Concise Course in Statistical Inference. Springer-Verlag, New York.
won DeGroot Prize 2005.
2006. All of Nonparametric Statistics. Springer.
2013. Topological Inference. Reitz Lecture 2013.
Honors and awards
2016, Member of National Academy of Sciences
Wasserman was elected to National Academy of Sciences in recognition of his distinguished and continuing achievement in original research.
See also
Peer review
Computational thinking
References
External links
Wasserman's home page
Wasserman's Blog on Statistics and Machine Learning
Canadian statisticians
Fellows of the Institute of Mathematical Statistics
Fellows of the American Statistical Association
Members of the United States National Academy of Sciences
Carnegie Mellon University faculty
Living people
University of Toronto alumni
Scientists from Ontario
21st-century Canadian mathematicians
1959 births
Mathematical statisticians |
https://en.wikipedia.org/wiki/Mott%20polynomials | In mathematics the Mott polynomials sn(x) are polynomials introduced by who applied them to a problem in the theory of electrons.
They are given by the exponential generating function
Because the factor in the exponential has the power series
in terms of Catalan numbers , the coefficient in front of of the polynomial can be written as
,
according to the general formula for generalized Appell polynomials,
where the sum is over all compositions of into positive odd integers. The empty product appearing for equals 1. Special values, where all contributing Catalan numbers equal 1, are
By differentiation the recurrence for the first derivative becomes
The first few of them are
The polynomials sn(x) form the associated Sheffer sequence for –2t/(1–t2) .
give an explicit expression for them in terms of the generalized hypergeometric function 3F0:
References
Polynomials |
https://en.wikipedia.org/wiki/Gould%20polynomials | In mathematics the Gould polynomials Gn(x; a,b) are polynomials introduced by H. W. Gould and named by Roman in 1984.
They are given by
where
so
References
Polynomials |
https://en.wikipedia.org/wiki/Sister%20Celine%27s%20polynomials | In mathematics, Sister Celine's polynomials are a family of hypergeometric polynomials introduced by . They include Legendre polynomials, Jacobi polynomials, and Bateman polynomials as special cases.
References
Polynomials |
https://en.wikipedia.org/wiki/Mittag-Leffler%20polynomials | In mathematics, the Mittag-Leffler polynomials are the polynomials gn(x) or Mn(x) studied by .
Mn(x) is a special case of the Meixner polynomial Mn(x;b,c) at b = 0, c = -1.
Definition and examples
Generating functions
The Mittag-Leffler polynomials are defined respectively by the generating functions
and
They also have the bivariate generating function
Examples
The first few polynomials are given in the following table. The coefficients of the numerators of the can be found in the OEIS, though without any references, and the coefficients of the are in the OEIS as well.
{| class="wikitable"
!n !! gn(x) !! Mn(x)
|-
| 0 || ||
|-
| 1 || ||
|-
| 2 || ||
|-
| 3 || ||
|-
| 4 || ||
|-
| 5 || ||
|-
| 6 || ||
|-
| 7 || ||
|-
| 8 || ||
|-
| 9 || ||
|-
| 10 || ||
|}
Properties
The polynomials are related by and we have for . Also .
Explicit formulas
Explicit formulas are
(the last one immediately shows , a kind of reflection formula), and
, which can be also written as
, where denotes the falling factorial.
In terms of the Gaussian hypergeometric function, we have
Reflection formula
As stated above, for , we have the reflection formula .
Recursion formulas
The polynomials can be defined recursively by
, starting with and .
Another recursion formula, which produces an odd one from the preceding even ones and vice versa, is
, again starting with .
As for the , we have several different recursion formulas:
Concerning recursion formula (3), the polynomial is the unique polynomial solution of the difference equation , normalized so that . Further note that (2) and (3) are dual to each other in the sense that for , we can apply the reflection formula to one of the identities and then swap and to obtain the other one. (As the are polynomials, the validity extends from natural to all real values of .)
Initial values
The table of the initial values of (these values are also called the "figurate numbers for the n-dimensional cross polytopes" in the OEIS) may illustrate the recursion formula (1), which can be taken to mean that each entry is the sum of the three neighboring entries: to its left, above and above left, e.g. . It also illustrates the reflection formula with respect to the main diagonal, e.g. .
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|-
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| style="text-align: right;" |2 || style="text-align: right;" |4 || style="text-align: right;" |6 || style="text-align: right;" |8 || style="text-align: right;" |10 || style="text-align: right;" |12 || style="text-align: right;" |14 || style="text-align: right;" |16 || style="text-align |
https://en.wikipedia.org/wiki/Tricomi%E2%80%93Carlitz%20polynomials | In mathematics, the Tricomi–Carlitz polynomials or (Carlitz–)Karlin–McGregor polynomials are polynomials studied by and and , related to random walks on the positive integers.
They are given in terms of Laguerre polynomials by
They are special cases of the Chihara–Ismail polynomials.
References
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Carlitz%20polynomial | In mathematics, Carlitz polynomial, named for Leonard Carlitz, may refer to:
Al-Salam–Carlitz polynomials
Tricomi–Carlitz polynomials |
https://en.wikipedia.org/wiki/Wall%20polynomial | In mathematics, a Wall polynomial is a polynomial studied by in his work on conjugacy classes in classical groups, and named by .
References
Polynomials |
https://en.wikipedia.org/wiki/Bender%E2%80%93Dunne%20polynomials | In mathematics, Bender–Dunne polynomials are a two-parameter family of sequences of orthogonal polynomials studied by . They may be defined by the recursion:
,
,
and for :
where and are arbitrary parameters.
References
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Gottlieb%20polynomials | In mathematics, Gottlieb polynomials are a family of discrete orthogonal polynomials introduced by . They are given by
References
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Harish-Chandra%27s%20c-function | In mathematics, Harish-Chandra's c-function is a function related to the intertwining operator between two principal series representations, that appears in the Plancherel measure for semisimple Lie groups. introduced a special case of it defined in terms of the asymptotic behavior of a zonal spherical function of a Lie group, and introduced a more general c-function called Harish-Chandra's (generalized) C-function. introduced the Gindikin–Karpelevich formula, a product formula for Harish-Chandra's c-function.
Gindikin–Karpelevich formula
The c-function has a generalization cw(λ) depending on an element w of the Weyl group.
The unique element of greatest length
s0, is the unique element that carries the Weyl chamber onto . By Harish-Chandra's integral formula, cs0 is Harish-Chandra's c-function:
The c-functions are in general defined by the equation
where ξ0 is the constant function 1 in L2(K/M). The cocycle property of the intertwining operators implies a similar multiplicative property for the c-functions:
provided
This reduces the computation of cs to the case when s = sα, the reflection in a (simple) root α, the so-called
"rank-one reduction" of . In fact the integral involves only the closed connected subgroup Gα corresponding to the Lie subalgebra generated by where α lies in Σ0+. Then Gα is a real semisimple Lie group with real rank one, i.e. dim Aα = 1,
and cs is just the Harish-Chandra c-function of Gα. In this case the c-function can be computed directly and is given by
where
and α0=α/〈α,α〉.
The general Gindikin–Karpelevich formula for c(λ) is an immediate consequence of this formula and the multiplicative properties of cs(λ), as follows:
where the constant c0 is chosen so that c(–iρ)=1 .
Plancherel measure
The c-function appears in the Plancherel theorem for spherical functions, and the Plancherel measure is 1/c2 times Lebesgue measure.
p-adic Lie groups
There is a similar c-function for p-adic Lie groups.
and found an analogous product formula for the c-function of a p-adic Lie group.
References
Lie groups |
https://en.wikipedia.org/wiki/Fridrikh%20Karpelevich | Fridrikh Israilevich Karpelevich (; 2 October 1927 – 5 July 2000) was a Russian mathematician known for his work on semisimple Lie algebras, geometry, and probability theory. Together with Simon Gindikin, he discovered the Gindikin–Karpelevich formula.
Notes
References
Russian mathematicians
Algebraists
Probability theorists
Queueing theorists |
https://en.wikipedia.org/wiki/Daniel%20Drescher | Daniel Drescher (born 7 October 1989) is an Austrian professional footballer who plays for TWL Elektra and is noted for his tackling abilities and aerial prowess.
Club statistics
Updated to games played as of 16 June 2014.
References
1989 births
Living people
Footballers from Vienna
Austrian men's footballers
Men's association football defenders
Austrian Football Bundesliga players
FC Admira Wacker Mödling players
Wolfsberger AC players
SKN St. Pölten players
Austrian Regionalliga players
Austria men's youth international footballers |
https://en.wikipedia.org/wiki/Nicola%20Fusco | Nicola Fusco (born August 14, 1956 in Napoli) is an Italian mathematician
mainly known for his contributions to the fields of calculus of variations, regularity theory of partial differential equations, and the theory of symmetrization. He is currently professor at the Università di Napoli "Federico II". Fusco also taught and conducted research at the Australian National University at Canberra, the Carnegie Mellon University at Pittsburgh and at the University of Florence.
He is the Managing Editor of the scientific journal Advances in Calculus of Variations, and member of the editorial boards of various scientific journals.
Awards
Fusco won the 1994 edition of the Caccioppoli Prize of the Italian Mathematical Union, and, in 2010, the Tartufari Prize from the Accademia Nazionale dei Lincei. In 2008 he was an invited speaker at European Congress of Mathematics and in 2010 he was invited speaker at the International Congress of Mathematicians on the topic of "Partial Differential Equations."
From 2010 he is a corresponding member of Accademia Nazionale dei Lincei.
Selected publications
Acerbi, E.; Fusco, N. "Semicontinuity problems in the Calculus of Variations" Archive for Rational Mechanics and Analysis 86 (1984)
Haïm Brezis; Fusco, N.; Sbordone, C. "Integrability for the Jacobian of orientation preserving mappings" Journal of Functional Analysis 115 (1993)
Fusco, N.; Pierre-Louis Lions; Sbordone, C. "Sobolev imbedding theorems in borderline cases" Proceedings of the American Mathematical Society 124 (1996)
Luigi Ambrosio, L.; Fusco, N.; Pallara, D. "Partial regularity of free discontinuity sets" Annali della Scuola Normale Superiore di Pisa Classe di Scienze (2) 24 (1997)
Ambrosio, L.; Fusco, N.; Pallara, D. Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)
Irene Fonseca; Fusco, N.; Paolo Marcellini; "On the total variation of the Jacobian" Journal of Functional Analysis 207 (2004)
Chlebik, M.; Cianchi, A.; Fusco, N. "The perimeter inequality under Steiner symmetrization: cases of equality" Annals of Mathematics (2) 162 (2005)
Fusco, N.; Maggi, F.; Pratelli, A. "The sharp quantitative isoperimetric inequality" Annals of Mathematics (2) 168 (2008)
References
External links
Site of Caccioppoli Prize
21st-century Italian mathematicians
Living people
20th-century Italian mathematicians
1956 births
Scientists from Naples
PDE theorists
Academic staff of the Australian National University
Academic staff of the University of Florence
Academic journal editors
Variational analysts
University of Naples Federico II alumni
Members of the Lincean Academy
European Research Council grantees |
https://en.wikipedia.org/wiki/Mahler%20polynomial | In mathematics, the Mahler polynomials gn(x) are polynomials introduced by in his work on the zeros of the incomplete gamma function.
Mahler polynomials are given by the generating function
Which is close to the generating function of the Touchard polynomials.
The first few examples are
References
Polynomials |
https://en.wikipedia.org/wiki/Bessel%E2%80%93Maitland%20function | In mathematics, the Bessel–Maitland function, or Wright generalized Bessel function, is a generalization of the Bessel function, introduced by . The word "Maitland" in the name of the function seems to be the result of confusing Edward Maitland Wright's middle and last names. It is given by
References
Special functions |
https://en.wikipedia.org/wiki/Str%C3%B6mgren%20integral | In mathematics and astrophysics, the Strömgren integral, introduced by while computing the Rosseland mean opacity, is the integral:
discussed applications of the Strömgren integral in astrophysics, and discussed how to compute it.
References
External links
Stromgren integral
Special functions
Astrophysics |
https://en.wikipedia.org/wiki/Norm%20group | In number theory, a norm group is a group of the form where is a finite abelian extension of nonarchimedean local fields. One of the main theorems in local class field theory states that the norm groups in are precisely the open subgroups of of finite index.
See also
Takagi existence theorem
References
J.S. Milne, Class field theory. Version 4.01.
Algebraic number theory |
https://en.wikipedia.org/wiki/Finite%20extensions%20of%20local%20fields | In algebraic number theory, through completion, the study of ramification of a prime ideal can often be reduced to the case of local fields where a more detailed analysis can be carried out with the aid of tools such as ramification groups.
In this article, a local field is non-archimedean and has finite residue field.
Unramified extension
Let be a finite Galois extension of nonarchimedean local fields with finite residue fields and Galois group . Then the following are equivalent.
(i) is unramified.
(ii) is a field, where is the maximal ideal of .
(iii)
(iv) The inertia subgroup of is trivial.
(v) If is a uniformizing element of , then is also a uniformizing element of .
When is unramified, by (iv) (or (iii)), G can be identified with , which is finite cyclic.
The above implies that there is an equivalence of categories between the finite unramified extensions of a local field K and finite separable extensions of the residue field of K.
Totally ramified extension
Again, let be a finite Galois extension of nonarchimedean local fields with finite residue fields and Galois group . The following are equivalent.
is totally ramified
coincides with its inertia subgroup.
where is a root of an Eisenstein polynomial.
The norm contains a uniformizer of .
See also
Abhyankar's lemma
Unramified morphism
References
Algebraic number theory |
https://en.wikipedia.org/wiki/Boas%E2%80%93Buck%20polynomials | In mathematics, Boas–Buck polynomials are sequences of polynomials defined from analytic functions and by generating functions of the form
.
The case , sometimes called generalized Appell polynomials, was studied by .
References
Polynomials |
https://en.wikipedia.org/wiki/B%C3%B6hmer%20integral | In mathematics, a Böhmer integral is an integral introduced by generalizing the Fresnel integrals.
There are two versions, given by
Consequently, Fresnel integrals can be expressed in terms of the Böhmer integrals as
The sine integral and cosine integral can also be expressed in terms of the Böhmer integrals
References
Special functions |
https://en.wikipedia.org/wiki/List%20of%20Olympic%20Games%20records%20in%20track%20cycling | This is a list of Olympic records in track cycling.
Men's records
♦ denotes a performance that is also a current world record. Statistics are correct as of 4 August 2021.
Women's records
♦ denotes a performance that is also a current world record. Statistics are correct as of 6 August 2021.
* In 2016, the 3000 m team pursuit with 3 riders will be replaced by a 4000 m team pursuit with 4 person riders.
References
Cycling
Track cycling at the Summer Olympics
Track cycling records
Olympic Games |
https://en.wikipedia.org/wiki/Meier%20function | In mathematics, Meier function might refer to:
Kaplan–Meier estimator
Meijer G-function |
https://en.wikipedia.org/wiki/Hahn%E2%80%93Exton%20q-Bessel%20function | In mathematics, the Hahn–Exton q-Bessel function or the third Jackson q-Bessel function is a q-analog of the Bessel function, and satisfies the Hahn-Exton q-difference equation (). This function was introduced by in a special case and by in general.
The Hahn–Exton q-Bessel function is given by
is the basic hypergeometric function.
Properties
Zeros
Koelink and Swarttouw proved that has infinite number of real zeros.
They also proved that for all non-zero roots of are real (). For more details, see . Zeros of the Hahn-Exton q-Bessel function appear in a discrete analog of Daniel Bernoulli's problem about free vibrations of a lump loaded chain (, )
Derivatives
For the (usual) derivative and q-derivative of , see . The symmetric q-derivative of is described on .
Recurrence Relation
The Hahn–Exton q-Bessel function has the following recurrence relation (see ):
Alternative Representations
Integral Representation
The Hahn–Exton q-Bessel function has the following integral representation (see ):
Hypergeometric Representation
The Hahn–Exton q-Bessel function has the following hypergeometric representation (see ):
This converges fast at . It is also an asymptotic expansion for .
References
Special functions
Q-analogs |
https://en.wikipedia.org/wiki/Backus%E2%80%93Gilbert%20method | In mathematics, the Backus–Gilbert method, also known as the optimally localized average (OLA) method is named for its discoverers, geophysicists George E. Backus and James Freeman Gilbert. It is a regularization method for obtaining meaningful solutions to ill-posed inverse problems. Where other regularization methods, such as the frequently used Tikhonov regularization method, seek to impose smoothness constraints on the solution, Backus–Gilbert instead seeks to impose stability constraints, so that the solution would vary as little as possible if the input data were resampled multiple times. In practice, and to the extent that is justified by the data, smoothness results from this.
Given a data array X, the basic Backus-Gilbert inverse is:
where C is the covariance matrix of the data, and Gθ is an a priori constraint representing the source θ for which a solution is sought. Regularization is implemented by "whitening" the covariance matrix:
with C′ replacing C in the equation for Hθ. Then,
is an estimate of the activity of the source θ.
References
Backus, G.E., and Gilbert, F. 1968, "The Resolving power of Gross Earth Data", Geophysical Journal of the Royal Astronomical Society, vol. 16, pp. 169–205.
Backus, G.E., and Gilbert, F. 1970, "Uniqueness in the Inversion of inaccurate Gross Earth Data", Philosophical Transactions of the Royal Society of London A, vol. 266, pp. 123–192.
Inverse problems
Linear algebra |
https://en.wikipedia.org/wiki/Bateman%20polynomials | In mathematics, the Bateman polynomials are a family Fn of orthogonal polynomials introduced by . The Bateman–Pasternack polynomials are a generalization introduced by .
Bateman polynomials can be defined by the relation
where Pn is a Legendre polynomial. In terms of generalized hypergeometric functions, they are given by
generalized the Bateman polynomials to polynomials F with
These generalized polynomials also have a representation in terms of generalized hypergeometric functions, namely
showed that the polynomials Qn studied by , see Touchard polynomials, are the same as Bateman polynomials up to a change of variable: more precisely
Bateman and Pasternack's polynomials are special cases of the symmetric continuous Hahn polynomials.
Examples
The polynomials of small n read
;
;
;
;
;
;
Properties
Orthogonality
The Bateman polynomials satisfy the orthogonality relation
The factor occurs on the right-hand side of this equation because the Bateman polynomials as defined here must be scaled by a factor to make them remain real-valued for imaginary argument. The orthogonality relation is simpler when expressed in terms of a modified set of polynomials defined by , for which it becomes
Recurrence relation
The sequence of Bateman polynomials satisfies the recurrence relation
Generating function
The Bateman polynomials also have the generating function
which is sometimes used to define them.
References
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Kamp%C3%A9%20de%20F%C3%A9riet%20function | In mathematics, the Kampé de Fériet function is a two-variable generalization of the generalized hypergeometric series, introduced by Joseph Kampé de Fériet.
The Kampé de Fériet function is given by
Applications
The general sextic equation can be solved in terms of Kampé de Fériet functions.
See also
Appell series
Humbert series
Lauricella series (three-variable)
References
External links
Hypergeometric functions |
https://en.wikipedia.org/wiki/Ex-tangential%20quadrilateral | In Euclidean geometry, an ex-tangential quadrilateral is a convex quadrilateral where the extensions of all four sides are tangent to a circle outside the quadrilateral. It has also been called an exscriptible quadrilateral. The circle is called its excircle, its radius the exradius and its center the excenter ( in the figure). The excenter lies at the intersection of six angle bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors (supplementary angle bisectors) at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect (see the figure to the right, where four of these six are dotted line segments). The ex-tangential quadrilateral is closely related to the tangential quadrilateral (where the four sides are tangent to a circle).
Another name for an excircle is an escribed circle, but that name has also been used for a circle tangent to one side of a convex quadrilateral and the extensions of the adjacent two sides. In that context all convex quadrilaterals have four escribed circles, but they can at most have one excircle.
Special cases
Kites are examples of ex-tangential quadrilaterals. Parallelograms (which include squares, rhombi, and rectangles) can be considered ex-tangential quadrilaterals with infinite exradius since they satisfy the characterizations in the next section, but the excircle cannot be tangent to both pairs of extensions of opposite sides (since they are parallel). Convex quadrilaterals whose side lengths form an arithmetic progression are always ex-tangential as they satisfy the characterization below for adjacent side lengths.
Characterizations
A convex quadrilateral is ex-tangential if and only if there are six concurrent angles bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors at the other two vertex
angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect.
For the purpose of calculation, a more useful characterization is that a convex quadrilateral with successive sides is ex-tangential if and only if the sum of two adjacent sides is equal to the sum of the other two sides. This is possible in two different ways:
or
This was proved by Jakob Steiner in 1846. In the first case, the excircle is outside the biggest of the vertices or , whereas in the second case it is outside the biggest of the vertices or , provided that the sides of the quadrilateral are
A way of combining these characterizations regarding the sides is that the absolute values of the differences between opposite sides are equal for the two pairs of opposite sides,
These equations are closely related to the Pitot theorem for tangential quadrilaterals, where the sums of opposite sides are equal for the two pairs of opposite sides.
Urquhart's theorem
If opposite sides in a convex quadrilateral intersect |
https://en.wikipedia.org/wiki/Burchnall%E2%80%93Chaundy%20theory | In mathematics, the Burchnall–Chaundy theory of commuting linear ordinary differential operators was introduced by .
One of the main results says that two commuting differential operators satisfy a non-trivial algebraic relation.
The polynomial relating the two commuting differential operators is called the Burchnall–Chaundy polynomial.
References
Ordinary differential equations |
https://en.wikipedia.org/wiki/John%20Greig%20%28mathematician%29 | John Greig (1759–1819) was an English mathematician. He died at Somers Town, London, on 19 January 1819, aged 60.
Works
He taught mathematics and wrote:
The Young Lady's Guide to Arithmetic, London, 1798; many editions, the last in 1864.
Introduction to the Use of the Globes, 1805 ; three editions.
A New Introduction to Arithmetic, London, 1805.
A System of Astronomy on the simple plan of Geography, London, 1810.
Astrography, or the Heavens displayed, London, 1810.
The World displayed, or the Characteristic Features of Nature and Art, London, 1810.
References
Attribution
1759 births
1819 deaths
18th-century English mathematicians
19th-century English mathematicians
Mathematicians from London
18th-century English educators
19th-century English educators |
https://en.wikipedia.org/wiki/Jackson%20q-Bessel%20function | In mathematics, a Jackson q-Bessel function (or basic Bessel function) is one of the three q-analogs of the Bessel function introduced by . The third Jackson q-Bessel function is the same as the Hahn–Exton q-Bessel function.
Definition
The three Jackson q-Bessel functions are given in terms of the q-Pochhammer symbol and the basic hypergeometric function by
They can be reduced to the Bessel function by the continuous limit:
There is a connection formula between the first and second Jackson q-Bessel function ():
For integer order, the q-Bessel functions satisfy
Properties
Negative Integer Order
By using the relations ():
we obtain
Zeros
Hahn mentioned that has infinitely many real zeros (). Ismail proved that for all non-zero roots of are real ().
Ratio of q-Bessel Functions
The function is a completely monotonic function ().
Recurrence Relations
The first and second Jackson q-Bessel function have the following recurrence relations (see and ):
Inequalities
When , the second Jackson q-Bessel function satisfies:
(see .)
For ,
(see .)
Generating Function
The following formulas are the q-analog of the generating function for the Bessel function (see ):
is the q-exponential function.
Alternative Representations
Integral Representations
The second Jackson q-Bessel function has the following integral representations (see and ):
where is the q-Pochhammer symbol. This representation reduces to the integral representation of the Bessel function in the limit .
Hypergeometric Representations
The second Jackson q-Bessel function has the following hypergeometric representations (see , ):
An asymptotic expansion can be obtained as an immediate consequence of the second formula.
For other hypergeometric representations, see .
Modified q-Bessel Functions
The q-analog of the modified Bessel functions are defined with the Jackson q-Bessel function ( and ):
There is a connection formula between the modified q-Bessel functions:
Recurrence Relations
By the recurrence relation of Jackson q-Bessel functions and the definition of modified q-Bessel functions, the following recurrence relation can be obtained ( also satisfies the same relation) ():
For other recurrence relations, see .
Continued Fraction Representation
The ratio of modified q-Bessel functions form a continued fraction ():
Alternative Representations
Hypergeometric Representations
The function has the following representation ():
Integral Representations
The modified q-Bessel functions have the following integral representations ():
See also
q-Bessel polynomials
References
Special functions
Q-analogs |
https://en.wikipedia.org/wiki/List%20of%20power%20stations%20in%20Northern%20Ireland | This is a list of electricity-generating power stations in Northern Ireland, sorted by type and name, with installed capacity (May 2011).
Note that the Digest of United Kingdom energy statistics (DUKES) maintains a comprehensive list of United Kingdom power stations, accessible through the Department of Energy and Climate Change here.
A red background denotes a power station that is no longer operational.
List
Joint venture with Scottish and Southern Energy
Tidal Power
Northern Ireland was home to the world's first commercially viable tidal stream generator. Trials were begun in Scotland then in England, before Marine Current Turbines installed the thousand-tonne SeaGen turbine at the mouth of Strangford Lough. The lough was chosen because it has one of the fastest tidal flows in the world. The installation went live and was connected to the grid in mid-December, 2008, injecting an extra 1.2 megawatts of electricity.
The turbine is scheduled to produce power for five years, though Marine Current Turbines were reported to have asked for an extension beyond their 2013 contract. By March 2010, the turbine had passed an operating time of over 1,000 hours - a first for any marine energy device.
Impact to the environment was closely scrutinised. The device, built in Belfast's famous Harland and Wolff shipyard, is rigged with a sonar device which stops the motion of the rotor blades when it detects marine lifeform near it. While there has been no negative affect to the environment - a special protected wildlife area - it has been noticed that porpoises stop communicating while passing the device.
See also
Northern Ireland Electricity
List of power stations in England
List of power stations in Scotland
List of power stations in Wales
List of power stations in the Republic of Ireland
References
External links
Department of Energy & Climate Change
Digest of United Kingdom energy statistics 2011
Power stations in the United Kingdom
Northern Ireland
North
Lists of buildings and structures in Northern Ireland |
https://en.wikipedia.org/wiki/Hopf%20theorem | The Hopf theorem (named after Heinz Hopf) is a statement in differential topology, saying that the topological degree is the only homotopy invariant of continuous maps to spheres.
Formal statement
Let M be an n-dimensional compact connected oriented manifold and the n-sphere and be continuous. Then if and only if f and g are homotopic.
References
Theorems in differential topology |
https://en.wikipedia.org/wiki/Erdelyi%E2%80%93Kober%20operator | In mathematics, an Erdélyi–Kober operator is a fractional integration operation introduced by and .
The Erdélyi–Kober fractional integral is given by
which generalizes the Riemann fractional integral and the Weyl integral.
References
Fractional calculus |
https://en.wikipedia.org/wiki/Mehler%E2%80%93Fock%20transform | In mathematics, the Mehler–Fock transform is an integral transform introduced by and rediscovered by .
It is given by
where P is a Legendre function of the first kind.
Under appropriate conditions, the following inversion formula holds:
References
Integral transforms |
https://en.wikipedia.org/wiki/Mehler%E2%80%93Heine%20formula | In mathematics, the Mehler–Heine formula introduced by Gustav Ferdinand Mehler and Eduard Heine describes the asymptotic behavior of the Legendre polynomials as the index tends to infinity, near the edges of the support of the weight. There are generalizations to other classical orthogonal polynomials, which are also called the Mehler–Heine formula. The formula complements the Darboux formulae which describe the asymptotics in the interior and outside the support.
Legendre polynomials
The simplest case of the Mehler–Heine formula states that
where is the Legendre polynomial of order , and the Bessel function of order 0. The limit is uniform over in an arbitrary bounded domain in the complex plane.
Jacobi polynomials
The generalization to Jacobi polynomials is given by Gábor Szegő as follows
where is the Bessel function of order .
Laguerre polynomials
Using generalized Laguerre polynomials and confluent hypergeometric functions, they can be written as
where is the Laguerre function.
Hermite polynomials
Using the expressions equivalating Hermite polynomials and Laguerre polynomials where two equations exist, they can be written as
where is the Hermite function.
References
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Weisner%27s%20method | In mathematics, Weisner's method is a method for finding generating functions for special functions using representation theory of Lie groups and Lie algebras, introduced by . It includes Truesdell's method as a special case, and is essentially the same as Rainville's method.
References
Generating functions |
https://en.wikipedia.org/wiki/Al-Salam%E2%80%93Ismail%20polynomials | In mathematics, the Al-Salam–Ismail polynomials are a family of orthogonal polynomials introduced by .
References
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Chihara%E2%80%93Ismail%20polynomials | In mathematics, the Chihara–Ismail polynomials are a family of orthogonal polynomials introduced by , generalizing the van Doorn polynomials introduced by and the Karlin–McGregor polynomials. They have a rather unusual measure, which is discrete except for a single limit point at 0 with jump 0, and is non-symmetric, but whose support has an infinite number of both positive and negative points.
References
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Ismail%20polynomials | In mathematics, Ismail polynomials may refer to one of the families of orthogonal polynomials studied by Mourad Ismail, such as:
Al-Salam–Ismail polynomials
Chihara-Ismail polynomials
Rogers–Askey–Ismail polynomials |
https://en.wikipedia.org/wiki/Chihara%20polynomials | In mathematics, Chihara polynomials may refer to one of the families of orthogonal polynomials studied by Theodore Seio Chihara, including
Al-Salam–Chihara polynomials
Brenke–Chihara polynomials
Chihara–Ismail polynomials |
https://en.wikipedia.org/wiki/List%20of%20FC%20Porto%20records%20and%20statistics | Futebol Clube do Porto is a Portuguese sports club based in Porto, which is best known for its professional association football team. They played their first match in 1893, but only won their first trophy in 1911. Two years later, Porto began competing in a regional championship, and in 1922 they won the inaugural edition of the Campeonato de Portugal, the first nationwide club competition, to become the first Portuguese champions. In 1934, an experimental two-tier league competition was introduced in Portuguese football; four years later, the first-level Primeira Liga was officially established as the top-tier league championship, from which Porto have never been relegated.
Involved in international competitions since 1956, the club beat Bayern Munich in the 1987 European Cup Final to win its first continental silverware.
Porto have won 30 league titles – including an unparalleled series of five consecutive top-place finishes from 1994 to 1999 – and lifted the Taça de Portugal on 19 occasions and 1 Taça da Liga. In addition, they have more Supertaça Cândido de Oliveira trophies (23) than every other winning club combined. Internationally, Porto is the most successful Portuguese club, with a total of seven titles. Former captain João Pinto and striker Fernando Gomes hold the club records for most appearances (587) and goals (352), respectively. In international competitions, these records belong respectively to Vítor Baía (99) and Radamel Falcao (22). Baía is also the club's most successful player, with a total of 25 titles. José Maria Pedroto is the club's longest-serving coach, overseeing 327 matches in nine seasons.
This list includes the honours won by Porto at all levels and all-time statistics and records set by the club, its players and its coaches. The players section includes the club's top goalscorers and those who have made most appearances in first-team competitive matches. It also displays international achievements by players representing Porto, and the highest transfer fees paid and received by the club. The club's attendance records since moving to the Estádio das Antas in 1952 and to the Estádio do Dragão in 2004 are also included.
All figures are updated as of match played on 23 May 2022.
Honours
Porto won the inaugural José Monteiro da Costa Cup tournament in 1911, securing its first-ever trophy. Three years later, the club clinched the first of a total of 30 regional championship titles. In 1922, their regional success expanded to a national level, after victory in the inaugural staging of the Campeonato de Portugal crowned Porto as the first Portuguese champions. The club then won its first Primeira Liga title in 1934–35, when it was still a provisional competition, and again in 1938–39, when it became the official domestic top-tier championship.
In 1955–56, Porto lifted the Taça de Portugal for the first time, and in doing so secured their first league and cup double. The following season saw the club's international |
https://en.wikipedia.org/wiki/Orthogonal%20polynomials | In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.
The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases.
The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by P. L. Chebyshev and was pursued by A. A. Markov and T. J. Stieltjes. They appear in a wide variety of fields: numerical analysis (quadrature rules), probability theory, representation theory (of Lie groups, quantum groups, and related objects), enumerative combinatorics, algebraic combinatorics, mathematical physics (the theory of random matrices, integrable systems, etc.), and number theory. Some of the mathematicians who have worked on orthogonal polynomials include Gábor Szegő, Sergei Bernstein, Naum Akhiezer, Arthur Erdélyi, Yakov Geronimus, Wolfgang Hahn, Theodore Seio Chihara, Mourad Ismail, Waleed Al-Salam, Richard Askey, and Rehuel Lobatto.
Definition for 1-variable case for a real measure
Given any non-decreasing function on the real numbers, we can define the Lebesgue–Stieltjes integral
of a function f. If this integral is finite for all polynomials f, we can define an inner product on pairs of polynomials f and g by
This operation is a positive semidefinite inner product on the vector space of all polynomials, and is positive definite if the function α has an infinite number of points of growth. It induces a notion of orthogonality in the usual way, namely that two polynomials are orthogonal if their inner product is zero.
Then the sequence of orthogonal polynomials is defined by the relations
In other words, the sequence is obtained from the sequence of monomials 1, x, x2, … by the Gram–Schmidt process with respect to this inner product.
Usually the sequence is required to be orthonormal, namely,
however, other normalisations are sometimes used.
Absolutely continuous case
Sometimes we have
where
is a non-negative function with support on some interval in the real line (where and are allowed). Such a is called a weight function. Then the inner product is given by
However, there are many examples of orthogonal polynomials where the measure has points with non-zero measure where the function is discontinuous, so cannot be given by a weight function as above.
Examples of orthogonal polynomials
The most commonly used orthogonal polynomials are orthogonal for a measure with support in a real interval. This includes:
The classical orthogonal polynomials (Jacobi polynomials, Laguerre polynomials, Hermite polynomials, and their special cases Gegenbauer polynomials, Chebyshev poly |
https://en.wikipedia.org/wiki/Alireza%20Ramezani%20%28footballer%2C%20born%201984%29 | Alireza Ramezani (born September 1, 1984) is an Iranian footballer who plays for Malavan in the Persian Gulf Pro League.
Club career statistics
References
External links
Alireza Ramezani player profile from iranproleague.net
1984 births
Living people
Persian Gulf Pro League players
Azadegan League players
Malavan F.C. players
F.C. Shahrdari Bandar Abbas players
Sanat Sari F.C. players
Iranian men's footballers
Men's association football wingers
Men's association football fullbacks
Footballers from Bandar-e Anzali |
https://en.wikipedia.org/wiki/Brenke%E2%80%93Chihara%20polynomials | In mathematics, Brenke polynomials are special cases of generalized Appell polynomials, and Brenke–Chihara polynomials are the Brenke polynomials that are also orthogonal polynomials.
introduced sequences of Brenke polynomials Pn, which are special cases of generalized Appell polynomials with generating function of the form
Brenke observed that Hermite polynomials and Laguerre polynomials are examples of Brenke polynomials, and asked if there are any other sequences of orthogonal polynomials of this form. found some further examples of orthogonal Brenke polynomials. completely classified all Brenke polynomials that form orthogonal sequences, which are now called Brenke–Chihara polynomials, and found their orthogonality relations.
References
Orthogonal polynomials |
https://en.wikipedia.org/wiki/1986%20S%C3%A3o%20Paulo%20FC%20season | The 1986 season was São Paulo's 57th season since club's existence.
Statistics
Scorers
Managers performance
Overall
{|class="wikitable"
|-
|Games played || 80 (38 Campeonato Paulista, 34 Campeonato Brasileiro, 8 Friendly match)
|-
|Games won || 30 (11 Campeonato Paulista, 17 Campeonato Brasileiro, 2 Friendly match)
|-
|Games drawn || 38 (20 Campeonato Paulista, 13 Campeonato Brasileiro, 5 Friendly match)
|-
|Games lost || 12 (7 Campeonato Paulista, 4 Campeonato Brasileiro, 1 Friendly match)
|-
|Goals scored || 120
|-
|Goals conceded || 64
|-
|Goal difference || +56
|-
|Best result || 6–1 (H) v Ponte Preta - Campeonato Brasileiro - 1986.12.10
|-
|Worst result || 1–4 (A) v Juventus - Campeonato Paulista - 1986.4.16
|-
|Top scorer || Careca (33)
|-
Friendlies
Taça dos Campeões Rio-São Paulo
Trofeo Teresa Herrera
Official competitions
Campeonato Paulista
First stage
Matches
Second stage
Matches
Record
Campeonato Brasileiro
First round
Matches
Second stage
Eightfinals
Quarterfinals
Semifinals
Final
Record
External links
official website
1986
1986 in Brazilian football |
https://en.wikipedia.org/wiki/William%20Charles%20Brenke | William Charles Brenke (April 12, 1874, Berlin – 1964) was an American mathematician who introduced Brenke polynomials and wrote several undergraduate textbooks.
He received his PhD in mathematics at Harvard under Maxime Bôcher. Brenke taught at the University of Nebraska-Lincoln mathematics department from 1908 to 1944 and was chair of the department from 1934 to 1944. He retired in 1943 but his successor, Ralph Hull, was put on official leave to do war work and returned from leave in 1945.
Publications
References
External links
NEGenWeb Project - Lancaster County Who's Who in Nebraska, 1940
1874 births
1964 deaths
20th-century American mathematicians
Harvard Graduate School of Arts and Sciences alumni
University of Nebraska–Lincoln faculty
Emigrants from the German Empire to the United States |
https://en.wikipedia.org/wiki/Jordan%27s%20theorem%20%28symmetric%20group%29 | In finite group theory, Jordan's theorem states that if a primitive permutation group G is a subgroup of the symmetric group Sn and contains a p-cycle for some prime number p < n − 2, then G is either the whole symmetric group Sn or the alternating group An. It was first proved by Camille Jordan.
The statement can be generalized to the case that p is a prime power.
References
External links
Jordan's Symmetric Group Theorem on Mathworld
Permutation groups
Theorems about finite groups |
https://en.wikipedia.org/wiki/Erwin%20Schr%C3%B6dinger%20Prize | The Erwin Schrödinger Prize (German: Erwin Schrödinger-Preis) is an annual award presented by the Austrian Academy of Sciences for lifetime achievement by Austrians in the fields of mathematics and natural sciences. The prize was established in 1958, and was first awarded to its namesake, Erwin Schrödinger.
Prize criteria and endowment
The prize is awarded at the discretion of the Austrian Academy of Sciences to scholars resident in Austria for excellence and achievements in the mathematical and scientific disciplines in the broadest sense. The prize is not awarded to full members of the Academy.
The award ceremony is held annually in October. The prize includes an annual stipend currently of € 15 000, paid monthly.
Prize winners
Source:
1956 Erwin Schrödinger
1958 Felix Machatschki
1960 Erich Schmid
1962 Marietta Blau
1963 Ludwig Flamm and Karl Przibram
1964 Otto Kratky
1965 Fritz Wessely
1966 Georg Stetter
1967 Berta Karlik and Gustav Ortner
1968 Hans Nowotny
1969 Walter Thirring
1970 Erika Cremer
1971 Richard Biebl
1972 Fritz Regler and Paul Urban
1973 Hans Tuppy
1974 Otto Hittmair and Peter Weinzierl
1975 Richard Kiefer and Erwin Plöckinger
1976 Herbert W. König and Ferdinand Steinhauser
1977 Viktor Gutmann and Helmut Rauch
1978 Edmund Hlawka and Günther Porod
1979 Heinz Parkus
1980 Peter Klaudy and Hans List
1981 Kurt Komarek
1982 Othmar Preining
1983 Josef Schurz and Peter Schuster
1984 Leopold Schmetterer and Josef Zemann
1985 Adolf Neckel and Karl Schlögl
1986 Walter Majerotto and Horst Wahl
1987 Edwin Franz Hengge and Franz Seitelberger
1988 Wolfgang Kummer and Fritz Paschke
1989 Johannes Pötzl
1990 Manfred W. Breiter and Karl Kordesch
1991 Siegfried J. Bauer and Willibald Riedler
1992 Josef F. K. Huber and Karlheinz Seeger
1993 Benno F. Lux and Oskar F. Olaj
1994 Tilmann Märk and Heide Narnhofer
1995 Heinz Gamsjäger and Jürgen Hafner
1996 Alfred Kluwick
1997 Werner Lindinger and Thomas Schönfeld
1998 Peter Zoller
1999 Johann Mulzer
2000 Erich Gornik and Hans Troger
2001 Bernhard Kräutler and Siegfried Selberherr
2002 Ekkehart Tillmanns
2003 Erwin S. Hochmair and Hildegunde Piza
2004 Anton Stütz and Jakob Yngvason
2005 Franz Dieter Fischer and Rainer Kotz
2006 Rainer Blatt
2007 Georg Brasseur and Thomas Jenuwein
2008 Georg Wick
2009 Bernd Mayer
2010 Walter Kutschera
2011 Gerhard A. Holzapfel
2012 Jürgen Knoblich
2013 Nick Barton
2014 Denise P. Barlow
2015 und Jiří Friml
2016 Ortrun Mittelsten Scheid and Jürgen Sandkühler
2017 Francesca Ferlaino
2018 Elly Tanaka and Peter Jonas
2019 Karlheinz Gröchenig and Helmut Ritsch
2020 László Erdős and Markus Arndt
2021 Christoph Bock
2022 Robert Seiringer
See also
Schrödinger Medal
References
External links
Austrian science and technology awards |
https://en.wikipedia.org/wiki/1975%20Cricket%20World%20Cup%20statistics | This is a list of statistics for the 1975 Cricket World Cup.
Team statistics
Highest team totals
The following table lists the ten highest team scores during this tournament.
Batting statistics
Most runs
The top ten highest run scorers (total runs) in the tournament are included in this table.
Highest scores
This table contains the top five highest scores of the tournament made by a batsman in a single innings.
Highest partnerships
The following tables are lists of the highest partnerships for the tournament.
Bowling statistics
Most wickets
The following table contains the five leading wicket-takers of the tournament.
Best bowling figures
This table lists the top five players with the best bowling figures in the tournament.
Fielding statistics
Most dismissals
This is a list of the wicketkeepers who have made the most dismissals in the tournament.
Most catches
This is a list of the outfielders who have taken the most catches in the tournament.
References
External links
Cricket World Cup 1975 Stats from Cricinfo
statistics
Cricket World Cup statistics |
https://en.wikipedia.org/wiki/Konhauser%20polynomials | In mathematics, the Konhauser polynomials, introduced by , are biorthogonal polynomials for the distribution function of the Laguerre polynomials.
References
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Sieved%20orthogonal%20polynomials | In mathematics, sieved orthogonal polynomials are orthogonal polynomials whose recurrence relations are formed by sieving the recurrence relations of another family; in other words, some of the recurrence relations are replaced by simpler ones. The first examples were the sieved ultraspherical polynomials introduced by . Mourad Ismail later studied sieved orthogonal polynomials in a long series of papers. Other families of sieved orthogonal polynomials that have been studied include sieved Pollaczek polynomials, and sieved Jacobi polynomials.
References
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Enrico%20Giusti | Enrico Giusti (born Priverno, 1940), is an Italian mathematician mainly known for his contributions to the fields of calculus of variations, regularity theory of partial differential equations, minimal surfaces and history of mathematics. He has been professor of mathematics at the Università di Firenze; he also taught and conducted research at the Australian National University at Canberra, at the Stanford University and at the University of California, Berkeley. After retirement, he devoted himself to the managing of the "Giardino di Archimede", a museum entirely dedicated to mathematics and its applications. Giusti is also the editor-in-chief of the international journal, dedicated to the history of mathematics "Bollettino di storia delle scienze matematiche".
One of the most famous results of Giusti, is the one obtained with Enrico Bombieri and Ennio De Giorgi, concerning the minimality of Simons' cones, and allowing to disprove the validity of Bernstein's theorem in dimension larger than 8. The work on minimal surfaces was mentioned in the citation of the Fields medal eventually awarded to Bombieri in 1974.
Giusti has a sustained interest in the history of mathematics, e.g. the mathematics of Pierre de Fermat (see Giusti 2009). He is currently the director of the Garden of Archimedes, a museum devoted to mathematics in Florence, Italy.
Awards
Giusti won the Caccioppoli Prize of the Italian Mathematical Union in 1978 and in 2003 was awarded with the national medal for mathematics by the Accademia Nazionale delle Scienze (dei XL).
Selected publications
"Minimal cones and the Bernstein problem" (with E. Bombieri e E. De Giorgi), Inventiones Mathematicae 7 (1969) 243–268
"Harnack's inequality for elliptic differential equations on minimal surfaces" (with E. Bombieri), Inventiones Mathematicae 15 (1972), 24–46
.
"On the equation of surfaces of prescribed mean curvature. Existence and uniqueness without boundary conditions", Inventiones Mathematicae 46 (1978), 111–137
"On the regularity of the minima of variational integrals" (with M. Giaquinta), Acta Mathematica 148 (1982), 31–46
"Differentiability of minima of nondifferentiable functionals" (with M. Giaquinta), Inventiones Mathematicae 72 (1983), 285–298
"The singular set of the minima of certain quadratic functionals" (with M. Giaquinta), Annali della Scuola Normale Superiore di Pisa Classe di Scienze (Serie 4) 11 (1984), 45–55.
, translated in English as .
Giusti, Enrico, Les méthodes des maxima et minima de Fermat. Ann. Fac. Sci. Toulouse Math. (6) 18 (2009), Fascicule Spécial, 59–85.
See also
Hilbert's nineteenth problem
Plateau's problem
References
External links
Site of Caccioppoli Prize
21st-century Italian mathematicians
Living people
1940 births
PDE theorists
Variational analysts
Academic staff of the University of Pisa |
https://en.wikipedia.org/wiki/Q-Konhauser%20polynomials | In mathematics, the q-Konhauser polynomials are a q-analog of the Konhauser polynomials, introduced by .
References
Orthogonal polynomials |
https://en.wikipedia.org/wiki/1979%20Cricket%20World%20Cup%20statistics | This is a list of statistics for the 1979 Cricket World Cup.
Team statistics
Highest team totals
The following table lists the ten highest team scores during this tournament.
Batting statistics
Most runs
The top five highest run scorers (total runs) in the tournament are included in this table.
Highest scores
This table contains the top five highest scores of the tournament made by a batsman in a single innings.
Highest partnerships
The following tables are lists of the highest partnerships for the tournament.
Bowling statistics
Most wickets
The following table contains the five leading wicket-takers of the tournament.
Best bowling figures
This table lists the top five players with the best bowling figures in the tournament.
Fielding statistics
Most dismissals
This is a list of the wicketkeepers who have made the most dismissals in the tournament.
Most catches
This is a list of the outfielders who have taken the most catches in the tournament.
References
External links
Cricket World Cup 1979 Stats from Cricinfo
1979 Cricket World Cup
Cricket World Cup statistics |
https://en.wikipedia.org/wiki/Argument%20of%20a%20function | In mathematics, an argument of a function is a value provided to obtain the function's result. It is also called an independent variable.
For example, the binary function has two arguments, and , in an ordered pair . The hypergeometric function is an example of a four-argument function. The number of arguments that a function takes is called the arity of the function. A function that takes a single argument as input, such as , is called a unary function. A function of two or more variables is considered to have a domain consisting of ordered pairs or tuples of argument values. The argument of a circular function is an angle. The argument of a hyperbolic function is a hyperbolic angle.
A mathematical function has one or more arguments in the form of independent variables designated in the definition, which can also contain parameters. The independent variables are mentioned in the list of arguments that the function takes, whereas the parameters are not. For example, in the logarithmic function the base is considered a parameter.
Sometimes, subscripts can be used to denote arguments. For example, we can use subscripts to denote the arguments with respect to which partial derivatives are taken.
The use of the term "argument" in this sense developed from astronomy, which historically used tables to determine the spatial positions of planets from their positions in the sky (ephemerides). These tables were organized according to measured angles called arguments, literally "that which elucidates something else."
See also
References
External links
Elementary mathematics |
https://en.wikipedia.org/wiki/Sieved%20ultraspherical%20polynomials | In mathematics, the two families c(x;k) and B(x;k) of sieved ultraspherical polynomials, introduced by Waleed Al-Salam, W.R. Allaway and Richard Askey in 1984, are the archetypal examples of sieved orthogonal polynomials. Their recurrence relations are a modified (or "sieved") version of the recurrence relations for ultraspherical polynomials.
Recurrence relations
For the sieved ultraspherical polynomials of the first kind the recurrence relations are
if n is not divisible by k
For the sieved ultraspherical polynomials of the second kind the recurrence relations are
if n is not divisible by k
References
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Little%20q-Jacobi%20polynomials | In mathematics, the little q-Jacobi polynomials pn(x;a,b;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of their properties.
Definition
The little q-Jacobi polynomials are given in terms of basic hypergeometric functions by
Gallery
The following are a set of animation plots for Little q-Jacobi polynomials, with varying q;
three density plots of imaginary, real and modulus in complex space; three set of complex 3D plots
of imaginary, real and modulus of the said polynomials.
References
Orthogonal polynomials
Q-analogs
Special hypergeometric functions |
https://en.wikipedia.org/wiki/Big%20q-Jacobi%20polynomials | In mathematics, the big q-Jacobi polynomials Pn(x;a,b,c;q), introduced by , are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.
Definition
The polynomials are given in terms of basic hypergeometric functions by
References
Orthogonal polynomials
Q-analogs
Special hypergeometric functions |
https://en.wikipedia.org/wiki/Continuous%20q-Jacobi%20polynomials | In mathematics, the continuous q-Jacobi polynomials P(x|q), introduced by , are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.
Definition
The polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by
References
Orthogonal polynomials
Q-analogs
Special hypergeometric functions |
https://en.wikipedia.org/wiki/Q-Jacobi%20polynomials | In mathematics, the q-Jacobi polynomials may be the
Big q-Jacobi polynomials
Continuous q-Jacobi polynomials
Little q-Jacobi polynomials |
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